text: Strength of Materials Strength of Materials: A Unified Theory Surya N. Patnaik Dale A. Hopkins An Imprint of Elsevier Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo Butterworth±Heinemann is an imprint of Elsevier. Copyright # 2004, Elsevier (USA). All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (44) 1865 843830, fax: (44) 1865 853333, e-mail: email@example.com. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting `Customer Support' and then `Obtaining Permissions'. Recognizing the importance of preserving what has been written, Elsevier prints its books on acid-free paper whenever possible. Library of Congress Cataloging-in-Publication Data Pataik, Surya N. Strength of materials: a unified theory / Surya N. Pataik, Dale A. Hopkins. p. cm. Includes bibliographical references and index. ISBN 0-7506-7402-4 (alk. paper) 1. Strength of materials. I. Hopkins, Dale A. II. Title. TA405.P36 2003 620.1H 12Ðdc21 2003048191 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. The publisher offers special discounts on bulk orders of this book. For information, please contact: Manager of Special Sales Elsevier 200 Wheeler Road Burlington, MA 01803 Tel: 781-313-4700 Fax: 781-313-4882 For information on all Butterworth±Heinemann publications available, contact our World Wide Web home page at: http://www.bh.com 10 9 8 7 6 5 4 3 2 1 Printed in the United States of America Contents Preface ix Chapter 1 Introduction 1 1.1 Systems of Units 4 1.2 Response Variables 7 1.3 Sign Conventions 15 1.4 Load-Carrying Capacity of Members 16 1.5 Material Properties 28 1.6 Stress-Strain Law 30 1.7 Assumptions of Strength of Materials 37 1.8 Equilibrium Equations 42 Problems 50 Chapter 2 Determinate Truss 55 2.1 Bar Member 55 2.2 Stress in a Bar Member 68 2.3 Displacement in a Bar Member 72 2.4 Deformation in a Bar Member 74 2.5 Strain in a Bar Member 74 2.6 Definition of a Truss Problem 76 2.7 Nodal Displacement 85 2.8 Initial Deformation in a Determinate Truss 96 2.9 Thermal Effect in a Truss 99 2.10 Settling of Support 101 v 2.11 Theory of Determinate Analysis 104 2.12 Definition of Determinate Truss 113 Problems 122 Chapter 3 Simple Beam 129 3.1 Analysis for Internal Forces 131 3.2 Relationships between Bending Moment, Shear Force, and Load 149 3.3 Flexure Formula 153 3.4 Shear Stress Formula 159 3.5 Displacement in a Beam 164 3.6 Thermal Displacement in a Beam 179 3.7 Settling of Supports 183 3.8 Shear Center 184 3.9 Built-up Beam and Interface Shear Force 197 3.10 Composite Beams 202 Problems 209 Chapter 4 Determinate Shaft 217 4.1 Analysis of Internal Torque 218 4.2 Torsion Formula 222 4.3 Deformation Analysis 224 4.4 Power Transmission through a Circular Shaft 233 Problems 236 Chapter 5 Simple Frames 239 Problems 259 Chapter 6 Indeterminate Truss 263 6.1 Equilibrium Equations 266 6.2 Deformation Displacement Relations 268 6.3 Force Deformation Relations 269 6.4 Compatibility Conditions 269 6.5 Initial Deformations and Support Settling 270 6.6 Null Property of the Equilibrium Equation and Compatibility Condition Matrices 273 6.7 Response Variables of Analysis 273 6.8 Method of Forces or the Force Method 274 6.9 Method of Displacements or the Displacement Method 274 6.10 Integrated Force Method 275 Problems 305 vi Contents Chapter 7 Indeterminate Beam 311 7.1 Internal Forces in a Beam 315 7.2 IFM Analysis for Indeterminate Beam 317 7.3 Flexibility Matrix 329 7.4 Stiffness Method Analysis for Indeterminate Beam 337 7.5 Stiffness Method for Mechanical Load 339 7.6 Stiffness Solution for Thermal Load 341 7.7 Stiffness Solution for Support Settling 343 7.8 Stiffness Method Solution to the Propped Beam 350 7.9 IFM Solution to Example 7-5 355 7.10 Stiffness Method Solution to Example 7-5 360 Problems 366 Chapter 8 Indeterminate Shaft 371 8.1 Equilibrium Equations 372 8.2 Deformation Displacement Relations 373 8.3 Force Deformation Relations 373 8.4 Compatibility Conditions 375 8.5 Integrated Force Method for Shaft 376 8.6 Stiffness Method Analysis for Shaft 379 Problems 401 Chapter 9 Indeterminate Frame 405 9.1 Integrated Force Method for Frame Analysis 407 9.2 Stiffness Method Solution for the Frame 421 9.3 Portal FrameÐThermal Load 425 9.4 Thermal Analysis of the Frame by IFM 427 9.5 Thermal Analysis of a Frame by the Stiffness Method 429 9.6 Support Settling Analysis for Frame 431 Problems 436 Chapter 10 Two-Dimensional Structures 441 10.1 Stress State in a Plate 441 10.2 Plane Stress State 442 10.3 Stress Transformation Rule 445 10.4 Principal Stresses 448 10.5 Mohr's Circle for Plane Stress 453 10.6 Properties of Principal Stress 456 10.7 Stress in Pressure Vessels 463 10.8 Stress in a Spherical Pressure Vessel 463 10.9 Stress in a Cylindrical Pressure Vessel 466 Problems 470 Contents vii Chapter 11 Column Buckling 475 11.1 The Buckling Concept 475 11.2 State of Equilibrium 478 11.3 Perturbation Equation for Column Buckling 479 11.4 Solution of the Buckling Equation 481 11.5 Effective Length of a Column 487 11.6 Secant Formula 488 Problems 492 Chapter 12 Energy Theorems 497 12.1 Basic Energy Concepts 498 Problems 550 Chapter 13 Finite Element Method 555 13.1 Finite Element Model 557 13.2 Matrices of the Finite Element Methods 562 Problems 592 Chapter 14 Special Topics 595 14.1 Method of Redundant Force 595 14.2 Method of Redundant Force for a Beam 605 14.3 Method of Redundant Force for a Shaft 613 14.4 Analysis of a Beam Supported by a Tie Rod 615 14.5 IFM Solution to the Beam Supported by a Tie Rod Problem 618 14.6 Conjugate Beam Concept 622 14.7 Principle of Superposition 626 14.8 Navier's Table Problem 629 14.9 A Ring Problem 633 14.10 Variables and Analysis Methods 637 Problems 640 Appendix 1 Matrix Algebra 645 Appendix 2 Properties of a Plane Area 659 Appendix 3 Systems of Units 677 Appendix 4 Sign Conventions 681 Appendix 5 Mechanical Properties of Structural Materials 685 Appendix 6 Formulas of Strength of Materials 687 Appendix 7 Strength of Materials Computer Code 703 Appendix 8 Answers 717 Index 741 viii Contents Preface Strength of materials is a common core course requirement in U.S. universities (and those elsewhere) for students majoring in civil, mechanical, aeronautical, naval, architectural, and other engineering disciplines. The subject trains a student to calculate the response of simple structures. This elementary course exposes the student to the fundamental concepts of solid mechanics in a simplified form. Comprehension of the principles becomes essential because this course lays the foundation for other advanced solid mechanics analyses. The usefulness of this subject cannot be overemphasized because strength of materials principles are routinely used in various engineering applications. We can even speculate that some of the concepts have been used for millennia by master builders such as the Romans, Chinese, South Asian, and many others who built cathedrals, bridges, ships, and other structural forms. A good engineer will benefit from a clear comprehension of the fundamental principles of strength of materials. Teaching this subject should not to be diluted even though computer codes are now available to solve problems. The theory of solid mechanics is formulated through a set of formidable mathematical equations. An engineer may select an appropriate subset to solve a particular problem. Normally, an error in the solution, if any, is attributed either to equation complexity or to a deficiency of the analytical model. Rarely is the completeness of the basic theory ques- tioned because it was presumed complete, circa 1860, when Saint-Venant provided the strain formulation, also known as the compatibility condition. This conclusion may not be totally justified since incompleteness has been detected in the strain formulation. Research is in progress to alleviate the deficiency. Benefits from using the new compatibility condition have been discussed in elasticity, finite element analysis, and design optimization. In this textbook the compatibility condition has been simplified and applied to solve strength of materials problems. The theory of strength of materials appears to have begun with the cantilever experiment conducted by Galileo1 in 1632. His test setup is shown in Fig. P-1. He observed that the ix Section at x - x x x FIGURE P-1 Galileo's cantilever beam experiment conducted in 1632. strength of a beam is not linearly proportional to its cross-sectional area, which he knew to be the case for a strut in tension (shown in Fig. P-2). Coulomb subsequently completed the beam theory about a century later. Even though some of Galileo's calculations were not developed fully, his genius is well reflected, especially since Newton, born in the year of Galileo's death, had yet to formulate the laws of equilibrium and develop the calculus used in the analysis. Industrial revolutions, successive wars, and their machinery requirements assisted and accel- erated the growth of strength of materials because of its necessity and usefulness in design. Several textbooks have been written on the subject, beginning with a comprehensive treatment by Timoshenko,2 first published in 1930, and followed by others: Popov,3 Hibbeler,4 Gere and Timoshenko,5 Beer and Johnston,6 and Higdon et al.7 just to mention a few. Therefore, the logic for yet another textbook on this apparently matured subject should be addressed. A cursory discussion of some fundamental concepts is given before answering that question. Strength of materials applies the basic concepts of elasticity, and the mother discipline is analytically rigorous. We begin discussion on a few basic elasticity concepts. A student of strength of materials is not expected to comprehend the underlying equations. The stress (s)-strain (e) relation {s} [k] {e} is a basic elasticity concept. Hooke (a contemporary of Newton) is credited with this relation. The genus of analysis is contained FIGURE P-2 Member as a strut. x Preface in the material law. The constraint on stress is the stress formulation, or the equilibrium equation (EE). Likewise, the constraint on strain becomes the strain formulation, or the compatibility condition (CC). The material matrix [k] along with the stress and strain formulations are required to determine the response in an elastic continuum. Cauchy developed the stress formulation in 1822. This formulation contained two sets of equations: the field equation tij,j bi 0 and the boundary condition pi tij nj ; here 1 i, j 3; tij is the stress; bi , pi are the body force and traction, respectively; and tij,j is the differentiation of stress with respect to the coordinate xj . The strain formulation, developed in 1860, is credited to Saint-Venant. This formulation contained only the field equation. When expressed in terms of strain e it becomes q2 eij q2 ekl q2 eik q2 ejl À À 0 qxk qxl qxi qxj qxj qxl qxi qxk Saint-Venant did not formulate the boundary condition, and this formulation remained incomplete for over a century. The missing boundary compatibility condition (BCC) has been recently completed.*8 The stress and strain formulations required to solve a solid mechanics problem (including elasticity and strength of materials) are depicted in Fig. P-3. I III Stress formulation Strain formulation Field Field II IV Boundary Boundary Missed until recently FIGURE P-3 Stress and strain formulations. * Boundary compatibility conditions in stress (here n is Poisson's ratio): qÈ À Á É qÈ À Á É anz sy À nsz À nsx À any 1 ntyz any sz À nsx À nsy À anz 1 ntyz 0 qz qy qÈ À Á É qÈ À Á É anx sz À nsx À nsy À anz 1 ntzx anz sx À nsy À nsz À anx 1 ntzx 0 qx qz qÈ À Á É qÈ À Á É any sx À nsy À nsz À anx 1 ntxy anx sy À nsz À nsx À any 1 ntxy 0 qy qx Preface xi The discipline of solid mechanics was incomplete with respect to the compatibility condition. In strength of materials the compatibility concept that was developed through redundant force by using fictitious ``cuts'' and closed ``gaps'' is quite inconsistent with the strain formulation in elasticity. The solid mechanics discipline, in other words, has acknowledged the existence of the CC. The CC is often showcased, but sparingly used, and has been confused with continuity. It has never been adequately researched or under- stood. Patnaik et al.8 have researched and applied the CC in elasticity, discrete analysis, and design optimization. The importance of the compatibility concept cannot be overstated. Without the CC the solid mechanics discipline would degenerate into a few determinate analysis courses that could be covered in elementary mechanics and applied mathematics. The compatibility concept makes solid mechanics a research discipline that is practiced at doctoral and postdoctoral levels in academia and in large research centers throughout the world. The problem of solid mechanics was solved despite the immaturity with respect to the CC. An elasticity solution was obtained by using the information contained in the three- quarter portion of the pie diagram and skillfully improvising the fourth quarter. The fidelity of such a solution depended on the complexity of the problem.9 Likewise, the strength of materials problem was solved by manipulating the equilibrium equations while bypassing most of the compatibility condition. Airy's beam solution was erroneous because the CC was bypassed. Todhunter's remark is quoted in the footnote.y Strength of Materials The concepts used to solve strength of materials problems are reviewed next. A simple truss is employed to illustrate the principles. (As it turns out, analysis appears to have begun with a truss problem.)10 The truss (shown in Fig. P-4) is assembled out of two steel bars with areas p A1 and A2 and lengths l 2 and l. The bars are hinged at nodes 1, 2, and 3. The free node 1 is subjected to a load with components Px and Py. It is required to calculate the response consisting of two bar forces F1 and F2 and two nodal displacements u and v. The truss is called determinate because the number of force and displacement variables is the same, two. The concepts are described first for determinate analysis and then expanded to indeterminate analysis. y ``Important Addition and Correction. The solution of the problems suggested in the last two Articles were givenÐ as has already been statedÐon the authority of a paper by the late Astronomer Royal, published in a report of the British Association. I now observe, howeverÐwhen the printing of the articles and engraving of the Figures is already completedÐthat they cannot be accepted as true solutions, inasmuch as they do not satisfy the general equations (164) of § 303 [note that the equations in question are the compatibility conditions]. It is perhaps as well that they should be preserved as a warning to the students against the insidious and comparatively rare error of choosing a solution which satisfies completely all the boundary conditions, without satisfying the fundamental condition of strain [note that the condition in question is the compatibility condition], and which is therefore of course not a solution at all.'' Todhunter, I. A History of the Theory of Elasticity and the Strength of Materials. UK: Cambridge, Cambridge University Press, 1886, 1983. xii Preface y z 2 3 F1 F2 v u 1 Px Py 1′ FIGURE P-4 Determinate truss. Determinate Analysis Sign Convention Both the load {P} and the internal bar force {F} are force quantities, yet they follow different sign conventions. The sign convention in strength of materials is more than just the sense of a vector. The additional deformation sign convention5 has no parallel in elasticity and may not be essential. This textbook follows a unified sign convention, which reduces the burden especially in constructing the bending moment and shear force diagrams in a beam. Equilibrium Equation (EE) Force balance yields the equilibrium equation. The two EE of the truss (in matrix notation [B]{F} {P}) are solved to determine the two bar forces. The treatment of the EE is uniform across all strength of materials textbooks, including this one on unified theory. Force Deformation Relation (FDR) Hooke's law is adjusted to obtain the FDR that relates bar deformations b1 and b2 to bar forces F1 and F2 through the flexibility matrix [G] as {b} [G]{F}. The treatment of FDR is about the same in most textbooks. Deformation Displacement Relation (DDR) The DDR, which links bar deformations b1 and b2 to nodal displacements X1 u and X2 v, is easily obtained by transposing the equilibrium matrix ([B]T ) as {b} [B]T {X}). Solution of displacement from the DDR requires a trivial calculation because typically [B] is a sparse triangular matrix. The DDR is avoided, neither used nor emphasized, in most standard textbooks. Instead, displacement is calculated either by using a graphical procedure (see Fig. P-4) or by applying an energy theorem. The counterpart of the DDR is the strain displacement relation in elasticity. This key relation finds wide application in elasticity, yet it is not used in strength of materials. Its nonuse can make displacement computation circui- tous. The unified theory calculates displacement from the DDR. The treatment of problems Preface xiii with initial deformations becomes straightforward. Textbooks either avoid or dilute the treatment of initial deformation because of temperature and settling of supports, especially in determinate structures. Analysis of determinate structure becomes straightforward through a simultaneous appli- cation of the three sets of basic equations: equilibrium equation, force deformation relation, and deformation displacement relation. Indeterminate Analysis three-bar truss obtained by adding the third bar (shown in Fig. P-5) with area Consider next ap A3 and length l 2 to the two-bar truss as depicted in Fig. P-6. The number of displacement variables u and v remained the same (being m 2) between the two trusses, but the number of bar forces F1 , F2 , and F3 increased to three (n 3) from two. An indeterminate structure is obtained when n is bigger than m and their difference r n À m 1 represents the degree of indeterminacy, one. The principles discussed for determinate structures are expanded to obtain indeterminate analysis. Sign Convention The sign convention is not changed between determinate and indeterminate analysis. The equilibrium concept remains the same. However, the EE matrix [B] is expanded to accom- modate the third bar force F3 . It becomes a 2  3 rectangular matrix [B] with two rows and three columns, and it cannot be solved for the three bar forces. A third equation is required: the compatibility condition. The CC in this unified theory is written as [C][G]{F} {0}, where [C] is the r  n compatibility matrix. Simultaneous solution of the two EE q2 4 F3 1 q1 FIGURE P-5 Force F3 in the third bar. 2 3 4 1 2 3 v 1 u Px Py FIGURE P-6 Indeterminate truss. xiv Preface ([B]{F} {P}) and one CC ([C][G]{F} {0}) yields the force variable. Displacement is back-calculated from the force deformation and the deformation displacement relation. This direct force determination formulation is called the integrated force method (IFM). Such a method was also envisioned by Michell,z and Love's remark is quoted in the footnote. Compatibility Matrix [C] The matrix [C] is generated from the deformation displacement relation {b} [B]T {X}. The DDR relates three (n 3) bar deformations b1 , b2 , and b3 to two (m 2) displacements X1 u and X2 v. Eliminating the two displacements yields one (r n À m 1) compat- ibility condition [C]{b} {0} with the r  n (here 1  3) coefficient matrix [C]. The compatibility generation procedure is analogous to Saint-Venant's strain formulation in elasticity. Determinate or indeterminate problems of strength of material can be solved by applying four types of equation: 1. Equilibrium equation: [B]{F} {P} 2. Deformation displacement relation: {b} [B]T {X} 3. Compatibility condition: [C]{b} {0} 4. Force deformation relation: {b} [G]{F} For determinate problems the compatibility condition degenerates to an identity [q] À [q] [0], and it is not required. Using the remaining three sets of equations makes the solution process simple and straightforward. Traditional strength of materials methods employ the equilibrium equations and the force deformation relation but improvise the deformation displacement relation and the compatibility condition. This procedure makes analysis obscure and can degrade solution fidelity especially for complex solid mechanics problems.9 Strength of materials analysis, despite immaturity with respect to the compat- ibility condition, progressed but only through two indirect methods: the stiffness method and the redundant force method. In such methods force (the dominating variable in design) is not the primary unknown; it is back-calculated. Stiffness Method The stiffness method considers displacement as the primary unknown. For the three-bar truss example, there are two unknown displacements u and v. Two equilibrium equations are available. The two EE, when expressed in terms of the two displacements, yield a set of two stiffness equations [K]{X} {P} with a 2  2 symmetrical stiffness matrix [K]. Solution of the stiffness equations yields the two displacements, from which the bar forces {F} can be z ``It is possible by taking account of these relations [the compatibility conditions] to obtain a complete system of equations which must be satisfied by stress components, and thus the way is open for a direct determination of stress without the intermediate steps of forming and solving differential equations to determine the components of displacements.'' Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity. UK: Cambridge, Cambridge University Press, 1927. Preface xv R FIGURE P-7 Redundant force R. back-calculated. Generalization of this procedure, which is credited to Navier, became the popular displacement, or stiffness, method. Redundant Force Method The redundant force method considers redundant force as the primary unknown. For example, one bar of the three-bar truss is ``cut'' to obtain an auxiliary two-bar determinate truss as shown in Fig. P-7. Determinate analysis yields the displacement ÁP at the cut because of the applied load. The solution process is repeated for a fictitious load R, referred to as the redundant force, in place of the third bar; and the displacement ÁR at the cut is obtained in terms of the redundant force R. Because the physical truss has no real cut, the ``gap'' is closed (ÁP ÁR 0), and this yields the value of the redundant force. The solution for the indeterminate three-bar truss is obtained as the response of the determinate structure subjected to two loads: the given external load P and a known redundant force R. Generalization of the procedure became the redundant force method, which was popular at the dawn of computer-automated analysis. Currently, for all practical purposes the redundant force method has disappeared because it was cumbersome and had limited scope. This method can be derived from the integrated force method (IFM) with some assumptions. Other Methods Two other methods have also been formulated: the hybrid method and the total formulation. The hybrid method considers force and displacement as the simultaneous unknowns. The total formulation considers force, displacement, and deformation as the primary variables. For the sake of completeness the analysis methods are listed in Table P-1. A formulation can also be derived from a variational functional listed in the last column. An undergraduate student is not expected to comprehend all the information contained in Table P-1. For a strength of materials problem the calculation of the primary variable, such as force in IFM (or displacement in the stiffness method), may consume the bulk of the calculation. Back- calculation of other variables from the primary unknown requires a small fraction of the computational effort. Therefore, the force method (or IFM) and the displacement (or stiff- ness) method are the two popular methods of analysis. The hybrid method and the total formulation may not be efficient and are seldom used. The impact of a less mature state of development of the compatibility condition is sketched in Fig. P-8. The compatibility barrier blocked the natural growth of the force xvi Preface TABLE P-1 Methods of Solid Mechanics with Associated Variational Functionals Method Method Primary Variables Variational Functional Number Elasticity Strength of Elasticity Strength of Materials Materials 1 Completed Integrated force Stress Force IFM variational functional Beltrami-Michell method (IFM) formulation (CBMF) 2 Airy formulation Redundant force Stress function Redundant force Complementary energy method 3 Navier formulation Stiffness method Displacement Displacement or Potential energy deflection 4 Hybrid method Reissner method Stress and Force and Reissner functional displacement deflection 5 Total formulation Washizu method Stresses, strains, Force, deformation, Washizu functional and displacements and deflection Determinate Indeterminate structures structures Stiffness method d ho et m Compatibility barrier s NO YES es n iff St Method Integrated of force force method Re d NO YES un da Redundant nt fo force method rc (disappeared) e 2nd half of 20th century FIGURE P-8 Compatibility barrier prevented extension of force method for indeterminate structures. method for the indeterminate problem. The analysis course split into the stiffness method and the redundant force method. The IFM can be specialized to obtain the two indirect methods, but the reverse course cannot be followed. The augmentation of the compatibility condition facilitates easy movement between the strength of materials variables: from displacement to deformation, from force to deformation, and from force to displacement, as well as from IFM to other methods of analysis. Unified Theory of Strength of Materials We return to the logic for this textbook. The compatibility condition is required for analysis, but analysis could not benefit from the CC because it was not fully understood. Now we understand the compatibility condition and use the CC to solve strength of materials problems. The use of the CC has systematized analysis. This textbook, by combining the new compatibility concept with the existing theory, has unified the strength of materials theory. For determinate structures the calculation of displacement becomes straightforward. A new direction is given for the analysis of indeterminate structures. Treatment of initial deformation by the IFM is straightforward because it is a natural parameter of the compat- ibility condition: as load is to equilibrium, so initial deformation is to compatibility. Learning IFM will expose the student to almost all strength of materials concepts because the direct force determination formulation uses them all. The student must learn the stiffness method and should be able to solve simple problems by using the redundant force method. This textbook provides for all three methods and traditional techniques. This textbook does not duplicate any existing textbook. The first chapter introduces the subject. Chapters 2 through 5 treat determinate structures: truss, beam, shaft, and frame. Chapters 6 through 9 are devoted to indeterminate structures: truss, beam, shaft, and frame. Two-dimensional stress analysis is the subject matter of Chapter 10. Column buckling and energy theorems are given in Chapters 11 and 12, respectively. Chapter 13 introduces the xviii Preface finite element method. Special topics, including Navier's table problem, are discussed in the last chapter. Appendixes 1 through 6 discuss matrix algebra, properties of plane area, systems of units, sign convention, properties of materials, and strength of materials formulas, respec- tively. Appendix 7 introduces the reader to the computer code for solving strength of materials problems by using various methods of analysis. The FORTRAN code supplied with the solution manual of the textbook is also available at the website http://www.patnaik-ue.org/ifm/. The last appendix lists the answers to the problems. The textbook covers standard topics of strength of materials. Both SI (International System of Units) and USCS (U.S. Customary Systems) system of units are employed. The treatment of the material is suitable for sophomore and junior engineering students. The book includes more material than can be covered in a single course. The teacher may choose to select topics for a single course; for example emphasizing fundamental concepts while reducing rigor on theoretical aspects. Alternatively, determinate analysis (first five chapters) along with principal stress calculation in Chapter 10 and column buckling in Chapter 11 can be covered in a first course in the sophomore year. Indeterminate analysis can be covered as a sequel in the junior year. Freshmen graduate students can benefit from the advanced materials included in the book. It is written to serve as a standard textbook in the engineering curriculum as well as a permanent professional reference. It is impossible to acknowledge everyone who has contributed to this book. We express gratitude to them. A major debt is owed to our two NASA (Glenn Research Center) colleagues: James D. Guptill and Rula M. Coroneos. Historical Sketch Some of the scientists who contributed to the strength of materials are listed in Fig. P-9. The subject was developed over centuries, beginning with Leonardo da Vinci, circa 1400s, to the present time. Leonardo, born a century before Galileo, understood the behavior of machine components and the work principle. Galileo Galilei began mechanics and material science and describes them in his book Dialogues Concerning Two New Sciences.1 Robert Hooke, a contemporary of Newton, conducted tests on elastic bodies and is credited with the material law. Isaac Newton has given us the laws of motion. The Bernoulli brothers (Jacob and Johan) have contributed to virtual displacement and the elastic curve for a beam. Leonhard Euler is credited with the beam elastic curve and the nonlinear coordinate system. Charles-Augustin de Coulomb made contributions to torsion and beam problems. Joseph-Louis Lagrange formulated the principle of virtual work. Simeon-Denis Poisson has given us Poisson's ratio. Claude Louis Marie Henri Navier is credited with the displacement method. Jean Victor Poncelet is acknowl- edged for vibration of a bar due to impact load. Thomas Young introduced the notion of modulus of elasticity. Augustin Cauchy is credited with the stress formulation. È August Ferdinand Mobius contributed to the analysis of determinate truss. Gabrio Piola is acknowledged for analysis of the stress tensor. Felix Savart conducted acoustic experiments. Charles Greene contributed to the graphical analysis of bridge trusses. Gabriel Lame con- cluded the requirement of two elastic constants for an isotropic material. Jean-Marie   Duhamel contributed to vibration of strings. Adhemar Jean Claude Barre de Saint-Venant is credited with the strain formulation. Franz Neumann contributed to photoelastic stress Preface xix Gallagher (1927–1997) Levy (1886–1970) Timoshenko, S. (1878–1966) Ritz, W. (1878–1909) Michell (1863–1940) Love (1863–1940) Foppl (1854–1924) Muller-Breslau (1851–1925) Voigt (1850–1919) Engesser (1848–1931) Castigliano (1847–1884) Rayleigh (1842–1919) Boussinesq (1842–1929) Beltrami (1835–1899) Winkler (1835–1888) Mohr (1835–1918) Clebsch (1833–1872) Maxwell (1831–1879) Kirchhoff (1824–1887) Jourawski (1821–1891) Stokes (1819–1903) Whipple (1804–1888) Ostrogradsky (1801–1861) Airy (1801–1892) Scientists Clapeyron (1799–1864) Neumann, F. E. (1798–1895) Saint-Venant (1797–1886) Duhamel (1797–1872) Lamé (1795–1870) Green (1793–1841) Savart (1791–1841) Piola (1791–1850) Mobius (1790–1868) Cauchy (1789–1857) Young (1788–1829) Poncelet (1788–1867) Navier (1785–1836) Poisson (1781–1840) Lagrange (1736–1813) Coulomb (1736–1806) D'Alembert (1717–1783) Euler (1707–1783) Daniel Bernoulli (1700–1782) Jakob Bernoulli (1654–1705) Leibnitz (1646–1716) Newton (1642–1727) Hooke (1635–1703) Century of Mariotte (1620–1684) geniuses Galileo (1564–1642) Leonardo da Vinci (1452–1519) 1450 1475 1500 1525 1550 1575 1600 1625 1650 1675 1700 1725 1750 1775 1800 1825 1850 1875 1900 1925 1950 1975 2000 Life span FIGURE P-9 Scientists who contributed to strength of materials.  analysis. Emile Clapeyron is credited with a strain energy theorem. George Biddel Airy introduced the stress function in elasticity. Mikhail Vasilievich Ostrogradsky is known for his work in variational calculus. Squire Whipple was the first to publish a book on truss analysis.10 George Gabriel Stokes is recognized for his work in hydrodynamics. Dimitrii Ivanovich Zhuravskii, also known by D. J. Jourawski, developed an approximate theory for shear stress in beams. Gustav Robert Kirchhoff is credited with the theory of plates. James Clerk Maxwell completely developed photoelasticity. Alfred Clebsch calculated beam deflection by integrating across points of discontinuities. Otto Mohr is credited with the graphical representation of stress. Emile Winkler is recognized for the theory of bending of xx Preface curved bars. Eugenio Beltrami contributed to the Beltrami-Michell stress formulation. Joseph Boussinesq calculated the distribution of pressure in a medium located under a body resting on a plane surface. Lord Rayleigh formulated a reciprocal theorem for a vibrating system. Alberto Castigliano's contribution included the two widely used strain energy theorems. Friedrich Engesser made important contributions to buckling and the energy method. Woldemar Voigt settled the controversy over the raiconstant and multiconstant theories. H. Muller-Breslau developed a method for drawing influence lines by applying unit load. August Foopl contributed to space structures. The outstanding elastician Augustus Edward Hough Love is credited with discovering earthquake waves, referred to as Love-waves. John Henry Michell showed that the stress distribution is independent of the elastic constants of an isotropic plate if body forces are absent and the boundary is simply connected. Walter Ritz developed a powerful method for solving elasticity problems. Stephen Prokofyevich Timoshenko11 revolutionized the teaching of solid mechanics through his 12 textbooks. Maurice Levy calculated stress distribution in elasticity problems. Richard H. Gallagher made outstanding contributions to finite element analysis. References 1. Galilei, G. Dialogues Concerning Two New Sciences. Evanston, IL: Northern University Press, 1950. 2. Timoshenko, S. P. Strength of Materials. New York: D. Van Nostrand Company, 1930. 3. Popov, E. P. Engineering Mechanics of Solids. Englewood Cliffs, NJ: Prentice-Hall, 1999. 4. Hibbeler, R. C. Mechanics of Materials. Englewood Cliffs, NJ: Prentice-Hall, 2000. 5. Gere, J. M., and Timoshenko, S. P. Mechanics of Materials, MA: PWS Engineering, Massachusetts, 2000. 6. Beer, P. F., and Johnston, E. R. Mechanics of Materials. New York: McGraw-Hill, 1979. 7. Higdon, A., Ohlsen, E. H., Stiles, W. B., Weese, J. A., and Riley, W. F. Mechanics of Materials. New York: John Wiley & Sons, 1985. 8. Patnaik, S. N. ``The Variational Energy Formulation for the Integrated Force Method.'' AIAA Journal, 1986, Vol. 24, pp 129±137. 9. Hopkins, D., Halford, G., Coroneos R., and Patnaik, S. N. ``Fidelity of the Integrated Force Method,'' Letter to the Editor. Int. J. Numerical Methods in Eng., 2002. 10. Whipple, S. ``An Essay on Bridge Building,'' Utica, NY, 1847, 2nd ed. ``An elementary and practical treatise on bridge building'' New York, Van Nostrand, 1899. 11. Timoshenko, S. P. History of Strength of Materials, New York, McGraw-Hill Book Co., 1953. Preface xxi 1 Introduction Strength of materials is a branch of the major discipline of solid mechanics. This subject is concerned with the calculation of the response of a structure that is subjected to external load. A structure's response is the stress, strain, displacement, and related induced variables. External load encompasses the mechanical load, the thermal load, and the load that is induced because of the movement of the structure's foundation. The response parameters are utilized to design buildings, bridges, powerplants, automobiles, trains, ships, submarines, airplanes, helicopters, rockets, satellites, machinery, and other structures. A beam, for example, is designed to ensure that the induced stress is within the capacity of its material. Thereby, its breakage or failure is avoided. Likewise, the magnitude of compressive stress in a column must be controlled to prevent its buckling. Excessive displacement can crack a windowpane in a building or degrade the performance of a bridge. The magnitude and direction of displacement must be controlled in the design of an antenna to track a signal from a satellite. Displacement also plays an important role in calculating load to design airborne and spaceborne vehicles, like aircraft and rockets. Strain is important because the failure of a material is a function of this variable. Response calculation, a primary objective of the solid mechanics discipline, is addressed at three different levels: elasticity, theory of structures, and strength of materials. If the analysis levels are arranged along a spectrum, elasticity occupies the upper spectrum. Strength of materials, the subject matter of this book, belongs to the bottom strata. The theory of structures is positioned at the middle. The fidelity of response improves as we move from the lower to the upper spectrum methods, but at a cost of increased mathematical complexity. For example, calculating an elasticity solution to a simple beam problem can be quite difficult. In contrast, the strength of materials ballpark solution is easily obtained. This solution can be used in preliminary design calculations. Quite often the strength of materials result is considered as the benchmark solution against which answers obtained from advanced methods are compared. The upper spectrum methods have not reduced the 1 importance of strength of materials, the origin of which has been traced back to Galileo, who died in 1642 on the day Newton was born. An understanding of the strength of materials concepts will benefit an engineer irrespective of the field of specialization. It is, therefore, a common core course in almost all engineering disciplines. This course, in a simplified form, will expose students to the fundamental concepts of solid mechanics. Comprehension of other solid mechanics subjects can be challenging if the student is not well versed in this course because strength of materials principles lay the foundation for other advanced courses. This chapter introduces the vocabulary, parameters, units of measurement, sign conven- tion, and some concepts of strength of materials. If enburdened, a reader may skip sections of this chapter and proceed to the subsequent chapter. It is recommended, however, that the reader revisits this chapter until he or she has a complete comprehension of the material. The parameters and variables of strength of materials problems separate into two distinct categories. The first group pertains to the information required to formulate a problem, such as the configuration of the structure, the member properties, the material characteristics, and the applied loads. This group we will refer to as parameters, which become the input data when the problem is solved in a computer. These parameters must be specified prior to the commence- ment of calculations. The response of the structure, such as the stress, strain, and displacement determined from the strength of materials calculation, forms the second group. We will call these the response variables that constitute the output data when the problem is solved in a computer. The important variables, keywords, and concepts discussed in this chapter are listed in Table 1-1. A description of the keywords cannot be given in the sequence shown in Table 1-1 because they are interrelated. For example, we have to discuss stress and strain, which belong to the response category, prior to describing other material parameters. Structures and Members Structures are made of a few standard members, also referred to as elements. Our analysis is confined primarily to bar, beam, shaft, and frame members, and the structures that can be made of these elements. The structure types discussed in this book are 1. A truss structure or simply a truss: It is made of bar members. The truss shown in Fig. 1-1a is made of four bar members and four joints, also called nodes. It is subjected to load P1 at node 2 along the x-coordinate direction and to load P2 at node 3 along the negative y-direction. TABLE 1-1 Key Words in Strength of Materials Structure Parameters Response Concepts Others (Member) (Input Data) Variables Truss (bar) Member property Internal force Equilibrium Measurement unit Beam (beam) Load Stress Determinate Sign convention Shaft (shaft) Material characteristic Displacement Indeterminate Frame (frame) Deformation Shell Strain 2 STRENGTH OF MATERIALS y x 1 1 Bar member 2 P2 4 4 3 1 1 2 3 2 P Beam member 1 2 1 P1 1 2 3 1 2 (a) Four- bar truss. (b) Two- member beam. 2 3 2 P 3 T 3 3 4 1 4 Frame element (c) Shaft. 1 (d) Portal frame. FIGURE 1-1 Structures and members. 2. Beam: Simple and continuous beams are made of beam members. The beam shown in Fig. 1-1b has two members and three nodes. It supports load P at node 3 along the negative y-direction. 3. Shaft: Torsional shafts are made of shaft members. A shaft subjected to a torque T is shown in Fig. 1-1c. 4. Frame: Frame structures are made of frame members. A frame element combines the action of a bar and a beam member. The portal frame shown in Fig. 1-1d has three members and four nodes. It is subjected to load P along the x-coordinate direction at node 3. 5. Shell: An elementary treatment of simple shells is also included. Introduction 3 1.1 Systems of Units Measurement of the parameters and variables requires the adoption of a system of units. There are two popular systems: the International System of Units (SI) and the U.S. Customary System (USCS). This book uses both SI and USCS systems. Both systems have several base units from which all other units are derived. In the USCS, the base units of importance are pound-force (lbf), foot (ft), second (s), Fahrenheit ( F), and degree ( ). Pound-force is the unit of force; for example, a man can easily apply a 100-lbf while pushing a stalled car. A foot is the unit of length: the height of a person can be 6 ft. A second is the unit of time in both the USCS and SI systems: a tiger can leap 70 ft in 1 s. A degree Fahrenheit is the unit of temperature: the outside temperature is 70 F. A degree is the unit to measure angles: a right angle is equal to 90 . The base units in the SI system are kilogram (kg), meter (m), second (s), kelvin (K), and radian (rad). The kilogram is the unit of mass: the mass of a beam is 25 kg. The meter is the unit of length: the height of a person can be 1.83 m. The second is the unit of time: a tiger can leap 21.35 m in 1 s. Kelvin is the unit of temperature, an alternate unit being degree Celsius ( C). The two temperature scales differ by the absolute zero tempera- ture, which is set at À273:15 C; [t( C) t(K) À 273:15]. We will use the Celsius scale. The outside temperature is 21 C or 294.15 K. The radian is the unit to measure angles: a right angle is 1.571 rad. The dimensions of a quantity can be expressed in terms of the base units of mass M, length L, time T, temperature t, and angle y. Take, for example, force F. The dimension of force, which is the product of mass and acceleration, can be derived as F mass  acceleration M  acceleration time rate of change of velocity M  velocity=time MT À1  time rate of change of distance MT À1 distance=time MLT À1  LT À1 MLT À2 F MLT À2 The dimension of force is: mass  length Ä time squared, or MLT À2 . Table 1-2 provides the dimensions of a few quantities along with the factor to convert from USCS to SI units and vice versa. EXAMPLE 1-1 The cantilevered beam shown in Fig. 1-2 is 30 ft long, 2 ft deep, and 6 in. thick. It is made of steel with a mass density of 15.2 slug/ft3. It supports a 1000-lbf load at a 30 inclination to the vertical in the x±y plane. The temperature of the beam varies linearly 4 STRENGTH OF MATERIALS y x P = 1000 lbf z 30° 2 ft 6 in. 30 ft Support FIGURE 1-2 Cantilever beam. from 250 F at the support to 70 F at its free end. Calculate its mass and weight, and convert the parameters into SI units. Solution Length: ` 30 ft 30  12 360 in: ` 30  0:305 9:15 m 915 cm Depth: d 2 ft 24 in: d 2  0:305 0:61 m 61 cm Thickness: t 6 in: 0:5 ft t 0:5  0:305 0:152 m 15:24 cm Volume: V `dt 30  2  1=2 30 ft3 30 cft V 30  123 51:84  103 in:3 V 9:15  0:61  0:152 0:85 m3 850:0  103 cm3 Mass density in USCS units can be measured either in slug per cubic foot or pound mass (lbm) per cubic foot. 1 lbf 1 slug 1 ft=s2 1 lbf 1 lbm 1 lbf=g 32:17 ft=s2 Introduction 5 1 lbf 1 slug ft=s2 32:17 lbm ft=s2 1 slug 32:17 lbm Mass Density of Steel: rm 15:2 slug=ft3 489 lbm=ft3 15:2  14:59 rm 7831 kg=m3 2:832  10À2 Mass: m Vrm 30  15:2 456 slug 14;669:52 lbm m 0:85  7831 6656 kg Weight density is equal to the product of the mass density and the gravity accel- eration (rw rm G) rw 15:2  32:17 489 lbf=ft3 rw 7831  9:81 76;822 N=m3 76:8 kN=m3 Weight: W Vrw 30  489 14;670 lbf W 0:85  76:8 65:3 kN 1 N 14;670=65:3  103 0:2248 lbf Load: P 1000 lbf 1 kip 4:45 kN Angle: y 30 0:524 rad Axial Load: Px P sin 30 P=2 p 3P Shear Load: Py ÀP cos 30 À 2 Px 500 lbf 0:5 kip 2:22 kN Py À866 lbf À0:866 kip À3:85 kN Temperature: At support: Ts 250 F Ts 5=9 250 À 32 121:1 C Ts 121:1 273:15 394:25 K At free end: Ts 70 F 21:1 C 294:25 K The problem stated in SI units: The cantilevered beam shown in Fig. 1-2 is 9.15 m long, 0.61 m deep, and 0.152 m thick. It is made of steel with a mass density of 7831 kg/m3. It supports a 4.45-kN load at a 0.524 rad inclination to the vertical in the x±y plane. The temperature of the beam varies linearly from 121.1 C at the support to 21.1 C at its free end. 6 STRENGTH OF MATERIALS 1.2 Response Variables Internal force, stress, displacement, deformation, and strain are the response variables of a strength of materials problem. A brief description is given for each variable. The discussion is confined to the two-dimensional plane because this course addresses planar structures. The concepts, however, are easily extended into the third dimension. Force Force is a vector quantity and it has the dimension of F MLT À2 (see Table 1-2). It is defined by its magnitude, line of action, sense (positive or negative), and location. Its magnitude cannot be measured directly, but it can be experienced, for example, through the muscular effort required to push a stalled car. In USCS units, it is measured in units of pound-force (lbf). In engineering, force is often measured in units of 1000 lbf, which is 1 kilopound-force or kip (1 kip 1000 lbf). In SI units, it is measured in newton (N). It can also be measured in units of kilogram-force (kgf), which is equal to 9.807 newton (1 kgf 9:807 N). Conversions between the two systems of measurement for force, which is accurate for strength of material calculations, are given in Table 1-3. A newton, at about 1/10 kgf or less than 1/4 lbf, is a small force. The weight of a small apple is about one newton. Four apples weight about one pound force. A convenient SI unit to measure force is a metric ton (MT, which is 1000 kgf) or in units of kilonewton (1 kN 1000 N). A force P is applied to a beam as shown in Fig. 1-3a. Its point of application is the location B. The force is represented by a line with a single arrowhead. Its line of action is along the line b±b that contains the force. Its x and y components are Px and Py. These are obtained by projecting the force vector P into x- and y-coordinate axes. F Px P cos y 1-1a V ÀPy ÀP cos 90 À y ÀP sin y 1-1b The included angle y is between the line AC and the line of action b±b. The magnitude of force is calculated as the sum of the squares of the components. r P P2 P 2 x y 1-1c Strength of materials calculations emphasize the components. The component along the x-coordinate axis is called the axial force or normal force (F Px ). Likewise, the component along the y-coordinate axis is called the shear force or transverse force (V ÀPy ). The response of the beam subjected to force P is identical to the response obtained when P is replaced by the axial force and the shear force, as shown in Fig. 1-3b. Traditionally, the x-coordinate axis of the beamÐalso called its structural axisÐis chosen along its length as shown in Fig. 1-4a, and y and z are the orthogonal coordinate axes. The Introduction 7 TABLE 1-2 SI and USCS Units Quantity Dimension International System (SI) U.S. Customary System Conversion factor (USCS) SI to USCS USCS to SI a Mass M Kilogram (kg) Slug 6:852  10À2 slug/kg 14.59 kg/slug Length L Metera (m) 100 Foota (ft) 12 inches (in.) 3.281 ft/m 0.3048 m/ft centimeters (cm) Time T Seconda (s) Seconda (s) 1 1 Temperature t Celsiusa ( C) Fahrenheita ( F) t F (9/5)t C 32 t C (5/9)(t F À 32) Angle y Radiana (rad) Degreea ( ) 57.31 deg/rad 1:745  10À2 rad/deg Force F MLT À2 Newton (N) Pound-forcea (lbf) 0.2248 lbf/N 4.448 N/lbf Area A L2 Square meter (m2) ft2 10.76 ft2/m2 9:29  10À2 m2 /ft2 Volume (solid) V L3 Cubic meter (m3) ft3 35.31 ft3/m3 2:832  10À2 m3 /ft3 Gravity acceleration g LT À2 9.81 m/s2 32.17 ft/s2 (3.281 ft/s2)/(m/s2) (0.3048 m/s2)/(ft/s2) Pressure p MLÀ1 T À2 Pascal (Pa) N/m2 lbf per square inch (psi) 1:45  10À4 psi/Pa 6895 Pa/psi a Base unit. TABLE 1-3 Conversion between USCS and SI Units Variable USCS to SI SI to USCS Length 1 in: 2:54 cm 1 cm 0:3937 in: 1 ft 12 in: 30:48 cm 0:3048 m 1 m 100 cm 3:28 ft 39:37 in: 1 yd 3 ft 0:9144 m 1 m 1:094 yd Force 1 lbf 4:448 N 0:4535 kgf 1 N 0:225 lbf 0:1020 kgf 1 kip 1000 lbf 4448 N 1 kgf 9:807 N 2:205 lbf 1 kip 453:5 kgf 0:4535 MT 1 MT 1000 kgf 2:205 kip Moment, torque, energy, or work 1 ft-lbf 1:3558 N-m 13:83 kgf-cm 1 N-m 0:738 ft-lbf 1 in:-k 112:98 N-m 1 kgf-cm 0:0981 N-m 0:0723 ft-lbf Stress or pressure 1 psi 1 lbf/in:2 6895 Pa 1 Pa 1 N/m2 0:145  10À3 psi 145 mpsi (mpsi millipsi 10À3 psi) 1 ksi 1000 psi 6:895  106 Pa 6:895 MPa 1 Mpa 145:5 psi 0:145 ksi y b x V = –Py P Py F = Px A Px B C B b (a) Force P is applied. (b) Components Px and Py are applied. FIGURE 1-3 Axial force F and shear force V. y Support block t y a3 a4 a3 z z a4 x O a2 d Oc a1 a1 a2 z y (a) Cantilever beam. (b) Cross-section at a1– a2 – a3 – a4. FIGURE 1-4 Dimensions of a beam. origin of the coordinate system at point O lies in the support block. The dimensions of the beam are its length ` along the x-coordinate axis, depth d, and thickness (or width) t measured along the y- and z-coordinate axes, respectively. The beam has a uniform rectangular cross-section (a1 Àa2 Àa3 Àa4 ) that is parallel to the y±z plane, and it is symme- trical about both the y- and z-axes as shown in Fig. 1-4b. Symmetry about the y- and z-axes is y z marked as yÀ" and zÀ", respectively. The centroid of the cross-section (Oc) is the intersec- tion of the axes of symmetry. The origin of the coordinate system O is the projection of Oc onto the support block. Axial Force An axial force F acts along the x-coordinate axis, which is also the structural axis. For the purpose of analysis, it is assumed that the axial force is applied along the centerline passing through the centroid of the beam cross-section marked Oc in Fig. 1-5a. An axial force F e applied at an off-center eccentric location E, with eccentricities ey and ez, is not admissible. 10 STRENGTH OF MATERIALS y ez y E ey V (Admissible) V e (Inadmissible) x Oc z Eccentricity of Fe V Ve B ez (Inadmissible) Fe Oc B F x Eccentricity of Ve (Admissible) (a) Axial force. (b) Shear force V. pe in x– y plane p in x–y plane p pe ez (c) Admissible line (distributed) load p. (d) Inadmissible distributed load pe. FIGURE 1-5 Line of action of axial force and shear force. Shear Force A shear force V is the transverse force. The y-coordinate axis is its line of action. Shear force V passes through the centroid marked Oc as shown in Fig. 1-5b. An eccentric shear force V e applied at B with an eccentric ez as shown in Fig. 1-5b is not admissible. Distributed transverse load p along a line can be applied in the x±y plane as shown in Fig. 1-5c. The distributed load pe is not admissible because it is applied with an eccentricity ez (see Fig. 1-5d). Bending Moment Bending moment is the product of shear force V and a distance ` measured along the beam x-coordinate axis (M V`). It is applied in the x±y plane and directed along the z-coordinate axis, which represents its line of action. A positive moment can be shown by a semicircle with an arrowhead oriented along the counterclockwise direction, or alternately by a double- headed arrow directed along the z-coordinate axis. The bending moment M, shown in Fig. 1-6a, is an admissible moment. The moment Me shown in Fig. 1-6b is not admissible because it has an eccentricity ez, even though it lies in a parallel x±y plane. Introduction 11 y x x–y plane with V z x–y plane V eccentricity ez ez M=V Me M Me (b) Inadmissible moment Me in (a) Admissible moment M in x–y plane. eccentric x–y plane. FIGURE 1-6 Bending moment in xÀy plane. Torque A torque (T) is also a moment, but its line of action is along the beam length or the x-coordinate axis, and it lies in the y±z plane. A positive torque can be shown by a semicircle with an arrowhead oriented along the counterclockwise direction in the y±z plane. A double-headed arrow directed along the x-coordinate axis can also show its line of action. The torque T shown in Fig. 1-7 is admissible because it lies in the y±z plane and its line of action coincides with the x-coordinate axis. The torque T e is not admissible because its direction has an eccentricity e, even though it lies in the y±z plane. Both bending moment and torque have the dimension of force times distance, or T F` ML2 T À2 . Length in the USCS system is measured in feet. In SI units, it is measured in meters. Moment or torque in USCS units can be measured in foot pound-force (ft-lbf) or inch-kip (in.-k). In SI units, it can be measured in newton meters (N-m) or kilogram-force-centimeters (kgf-cm). Conversion between the two systems is given in Table 1-3. Force Eccentricity and Approximation A beam is a slender structural member. The slenderness ratio SR is defined as the ratio of beam length ` to its cross-sectional area A (SR `/A). The slenderness ratio of a typical beam exceeds 20 (SR `/A ! 20); that is, the length is at least 20 times greater than its cross-sectional area. Force eccentricity, which is associated with the dimensions of the area, induces a small error in the results because a beam has a high slenderness ratio. 12 STRENGTH OF MATERIALS y x z T e Te FIGURE 1-7 Torque T on a circular shaft. Eccentricity of force can be accommodated through an equivalence concept that replaces the original force by axial force, shear force, bending moment, and torque. This concept is illustrated for an eccentrically applied axial force F e shown in Fig. 1±8. The axial force F e with eccentricity ey and ez with respect to the y- and z-coordinate axes, respectively, can be replaced by an equivalent set of forces consisting of an axial force F a, and two bending moments (My and Mz). The force F a is equal to the applied force. The eccentricity ey ez a3 a4 y e ey M y = ez F e x Oc M z = –F ey a1 a2 z Cross- section at a1– a2 – a3 – a4 a e F =F a3 a4 Fe a2 a1 (b) Equivalent forces. e (a) Eccentrically applied axial force F . FIGURE 1-8 Equivalence concept for axial force. Introduction 13 produces the moment Mz, which is directed along the negative z-coordinate axis. Likewise, the eccentricity ex produces the moment My with the y-coordinate axis as its line of action. Fa Fe M y F e ez M z ÀF e ey 1-2 Strength of materials calculations can accommodate the axial force (F a F). The moment Mz can be added to the moment M because both are directed along the z-coordinate direction. However, the two-dimensional theory cannot accommodate the moment My, which is directed along the y-coordinate direction. Again, we emphasize that the moments My and Mz are small and their effects can be neglected without consequence. EXAMPLE 1-2 A square column with a cross-sectional area A 4 ft2 and length ` 16 ft is subjected to an axial load (P À10 kip). The load applied in the x±y plane has eccentricities (ey ez 3 in:) and an inclination (y 5 ) to the vertical, as shown in Fig. 1-9. Calculate the equivalent loads in USCS and SI units. 5° T ey P M x z P y ez V x 16 ft y 2 ft (a) Eccentric load P. (b) Two-dimensional model. FIGURE 1-9 Eccentrically loaded column. 14 STRENGTH OF MATERIALS Solution Load Components: The orientation of the load components is shown in Fig. 1-9a. Px P cos 5 À9:962 kip À9:962  4448 À44:31 kN Py P sin 5 0:872 kip 3:88 kN Pz 0 Mx ÀPy ez À0:872  3 À2:62 in:-k À0:296 kN-m My ÀPx ez À9:962  3 À29:89 in:-k À3:38 kN-m Mz Px ey 9:962  3 29:89 in:-k 3:38 kN-m The force components for two-dimensional analysis in the x±y plane marked in Fig. 1-9a are as follows: Axial Load: P0 Px À9:96 kip À44:31 kN Shear Load: V Py 0:87 kip 3:88 kN Bending Moment: M Mz 29:89 in:-k 3:38 kN-m Torque: T Mx À2:62 in:-k À0:296 kN-m The two-dimensional analysis may neglect the bending moment My. 1.3 Sign Conventions A consistent sign convention must be followed to solve strength of materials problem. Two sets of sign conventions have to be used. The first, is referred to as the normal (n) sign convention. It applies to vector quantities like external load, reaction, and displacement. The second, which is called the tensor (t) sign convention, applies to stress and some other response variables. n-Sign Convention Consider a bar subjected to an axial load P1 as shown in Fig. 1-10. Axial load is considered positive when it is directed along the positive x-coordinate axis. The axial load P1 is positive, whereas P2 is negative. Transverse load is considered positive when it is directed along the y-coordinate axis. The transverse load Q1 is positive, whereas Q2 is negative. The sign convention for applied or external bending moment is illustrated in Fig. 1-11a. Bending moment is positive when its line of action is directed along the z-coordinate axis. In vector notation, it is shown by a double-headed arrow in Fig. 1-11b. In the two-dimensional x±y plane, a positive moment can also be shown by a circle with a dot, indicating the arrowhead pointing out of the x±y plane or along the z-coordinate direction, as shown in Fig. 1-11c. In this book, we will follow engineering notation for bending moment, Introduction 15 y x Q1 P1 Positive axial load P1 Positive transverse load Q1 Q2 P2 Negative axial load P2 Negative transverse load Q2 FIGURE 1-10 Sign conventions for load. represented by a semicircle with an arrowhead in the counterclockwise direction, as shown in Fig. 1-11d. All three representations refer to the same positive applied moment M. A negative moment M À for the three representations is shown in Figs. 1-11e, f, and g. The moment M À in Fig. 1-11e is negative because the double-headed arrow points toward the negative z-coordinate axis. The circle with a cross representing the tail of an arrow indicates the negative moment M À in Fig. 1-11f. The clockwise direction in Fig. 1-11g is the engineering convention for a negative moment. The sign convention for an external torque is shown in Fig. 1-12. Positive torque T can be shown by a counterclockwise curved arrow in the y±z plane, as in Fig. 1-12a. Alternatively, it can be shown by a double-headed arrow directed along the positive x-coordinate axis, as shown in Fig. 1-12b. The torque T À in Fig. 1-12c is negative because it is shown by a clockwise curved arrow or it is directed along the negative x-coordinate axis (see Fig. 1-12d). Right-Hand Rule The sign convention for torque and moment follows the right-hand rule, which is defined with an arrowhead (thumb) and a curl (fingers). The thumb of the right hand aligned along the double arrowhead represents the direction of moment or torque. The curled fingers indicate the rotational tendency. The right-hand rule is illustrated in Fig. 1-13a for a positive torque T. It is illustrated for positive torque T and a positive moment M in Fig. 1-13b. It is illustrated for negative torque T À and moment M À in Fig. 1-13c. 1.4 Load-Carrying Capacity of Members Load-carrying capacity is the basis to classify structural members into bar, beam, shaft, and frame elements. A bar member resists only an external axial load P. The bar resists the load by inducing an internal force F that is uniform across its length. Internal force can be 16 STRENGTH OF MATERIALS y M x M z (b) Vectorial notation for positive (a) Positive bending moment M. moment. y y M M x x (d) Counterclockwise moment is (c) Two- dimensional representation. positive. y M– y M– x x z (e) Negative moment. (f) Arrow is into the x–y plane. y M– x (g) M oment is clockwise. FIGURE 1-11 Sign conventions for external moment. considered as a stress resultant (F sdA), here s is the axial stress and A is the bar area. A stress resultant is obtained by integrating stress over the cross-sectional area. Both parameters (stress-s and area-A) are used to define the sign convention of the stress resultant (F). t-Sign Convention This convention is based on the product of two factors (f and n). The factor f refers to the direction of the variable, and it is assigned unity (f 1) when directed along the positive Introduction 17 y x y z T x T (a) Counterclockwise torque T (b) Vectorial notation for a positive is positive. torque T. y T– y x T– x z (c) Clockwise orientation is negative. (d) Negative torque T. FIGURE 1-12 Sign conventions for torque. coordinate axis. It is negative (f À1) when directed along the negative axis. The second factor is the orientation of the normal to the cross-sectional area. It is assigned unity (n 1) when the normal is directed along the positive coordinate direction. It is negative (n À1) when pointing along the negative coordinate direction. The t-sign convention is illustrated for a bar member that induces an axial force (F), as shown in Fig. 1-14a. The positive load P elongates the bar, inducing a tensile internal force F. At the cross section a±a, the direction of the two quantities is negative (fx À1 and nx À1 but fx nx 1). Force (F) is directed along the negative x-coordinate axis, or (fx À1). Likewise the normal to the bar area points along the negative x-axis, or (nx À1). The product is positive (fx nx 1), or F is a positive internal force. Forces acting in the block a±a±b±b are marked in Fig. 1-14b. In the right side of the block fx 1, nx 1, and fx nx 1 or F is positive. It is likewise positive in the left side because fx À1, nx À1, and fx nx 1. A positive bar force is shown by arrowheads that point at each other (see Fig.1-14c). It stretches the bar and induces tension. A negative bar force is shown by arrowheads that point away from each other as shown in Fig. 1-14d. It induces compression and contracts the bar. A beam resists a transverse external load Q that is applied along the y-coordinate direction, as shown in Fig. 1-15a. It resists the load by inducing a shear force V and a bending moment M. The internal forces (V andM) can vary along the length of the beam. The bending moment is a stress resultant (M szdA), here s is the axial stress, z is the distance from the neutral axis, and A is the cross-sectional area of the beam. The shear force 18 STRENGTH OF MATERIALS y Curl (fingers) T y indicate direction x of rotation M T x Thumb is z direction Right hand of moment z (b) Positive torque T and positive (a) Right- hand rule. moment M. y T– M– x O z (c) Negative torque T– and negative moment M –. FIGURE 1-13 Right-hand rule for torque T and moment M. is also a stress resultant (V tdA); here t is the shear stress, and A is the cross-sectional area of the beam. Both (M and V) follow the t-sign convention. For moment M acting in the cross-section marked a±a in Fig. 1-15b, both factors are negative (fm À1 and nx À1). The factor fm À1 because the moment M is clockwise. The factor fx À1 because the normal is along the negative x-coordinate axis. Because fm mx 1, M is positive. In the right side in Fig. 1-15c, fm 1, nx 1, and fm nx 1 or M is positive. Likewise, M is positive in the left side because fm À1, nx À1, and fm nx 1. A positive moment stretches the bottom fiber of the block a±a±b±b, inducing tension; and contracting the top fiber, creating compression as shown in Fig. 1-15c. In engineering, it is referred to as sagging moment, as shown in Fig. 1-15d. A hogging moment, so-called because it hogs the section of a beam, is negative with tension in the top fiber and compression in the bottom fiber, as shown in Fig.1-15e. Negative moment is marked on the block in Fig. 1-15f. The induced shear force V follows the t-sign convention. At the cross-section marked a±a in Fig. 1-15b, for the shear force V, the two factors are negative (fv À1 and nx À1) or V is positive. In the right side of Fig. 1-15c, fv 1, nx 1, and fv nx 1 or V is positive. Introduction 19 y x b a P b a F F a F P fx = –1 fx = 1 a fx nx = –1 nx = 1 nx (a) Bar in tension. (b) Block a–a–b–b shown enlarged. P F P P F P (c) Positive axial force F. (d) Negative axial force F. FIGURE 1-14 t-sign convention for axial force in a bar. Likewise, V is positive in the left side because fv À1, nx À1, and fv nx 1. Shear force and bending moment in Fig. 1-15f are negative. A shaft resists an external torque load T 0 that is applied along the x-coordinate direction. External torque follows the n-sign convention. It resists the load by inducing an internal torque T that is uniform across its length, as shown in Fig. 1-16. The induced torque T, which is also a stress resultant (T trdA), follows the t-sign convention. For the purpose of sign conven- tion, a shaft member is similar to a bar member. A positive torque is shown by arrowheads that point at each other. A negative torque is shown by arrowheads that point away from each other. A frame member resists axial load P, transverse load Q, and an external bending moment M 0 by inducing axial force F, shear force V, and bending moment M. A frame member combines the action of a beam and a bar. A portal frame subjected to load P, Q, and M 0 is shown in Fig. 1-17a. Consider the member BC. Its internal force state is not known because the analysis has not yet been completed. Analysis is initiated with a positive internal force state as shown in Fig. 1-17b. The orientation of the internal forces can be adjusted from the analysis results. The assumed internal force F is positive because arrows point at each other. The positive sense of bending moment M and shear force V is easily ascertained by comparing it to Fig. 1-15c. 20 STRENGTH OF MATERIALS y x Q Q b a M a fm = –1 nx = –1 b a a ∆ V (a) Beam under transverse load. (b) Section at a-a. M M M M V V (c) Positive M in block a-a–b-b. (d) Sagging moment is positive. V M M M M V (e) Hogging moment is negative. (f) Negative M. FIGURE 1-15 t-sign convention for bending moment and shear force in a beam. y x T0 T0 T0 T Positive torque T T0 T0 T Negative torque T FIGURE 1-16 t-sign convention for internal torque T. Introduction 21 y Q M0 x E C B P M M V F B C D (b) Internal forces in member BC. A (a) Portal frame. FIGURE 1-17 Internal forces in a frame member. The four members of strength of materials can be considered as special cases of a general beam member of the solid mechanics discipline. This beam can resist three forces and three moments, as shown in Fig. 1-18. Axial force (F Fx ) and torque (T Mx ) act along the x-coordinate axis. Along the y-coordinate axis, the beam resists shear force (V Fy ) and a bending moment My. The shear force Fz and the bending moment (M Mz ) are along the z-coordinate axis. A bar member is obtained from the general beam by retaining the axial force along the x-coordinate axis (F Fx ) and setting the other five force components to zero (Fy Fz Mx My Mz 0). A beam is obtained by retaining the shear force and bending moment (V Fy , M Mz ) but setting the other four force components to zero (Fx Fz Mx My 0). A shaft element is obtained by retaining the torsional moment (T Mx ) and setting the other five force components to zero (Fx Fy Fz My Mz 0). A frame y My x Fy z Fx Mx Fz Mz FIGURE 1-18 A general beam member resists three forces and three moments. 22 STRENGTH OF MATERIALS y y d x x (a) Bar member. (b) Beam, shaft, or frame member. FIGURE 1-19 Sketches for structural members. member is obtained by retaining three force components (F Fx , V Fy , M Mz ) and setting the other three force components to zero (Fz Mx My 0). The sketches we will use in this book to illustrate the structural members are depicted in Fig. 1-19. A bar member is sketched by a line representing its length and the x-coordinate axis. Beam, frame, and shaft members are sketched by a rectangle. The length ` is along the beam centerline, which coincides with the x-coordinate axis. The y-coordinate axis, which is perpendicular to the beam centerline, coincides with the member depth d for a rectangular cross-section or with a diameter when the cross-section is circular. The bar area and the width of a rectangular beam are not shown in the sketch. EXAMPLE 1-3 The loads on a general beam of length ` 5 m in the two-dimensional space shown in Fig. 1-20a, are as follows: P 1 kip Q À1 kN M 0 20 ft-k T 0 1 kN-m Calculate the forces in SI and USCS units if the general beam is replaced by a bar, a shaft, a beam, and a frame member. Solution The axial load, bending moment, and torque are positive, whereas the shear load is negative. A bar member carries an internal axial force (F P) as shown in Fig. 1-20b. Loads at the nodes (PA and PB) are obtained from the equilibrium along the x-coordinate (PA PB P 1 kip 4:448 kN). Introduction 23 M0 T0 PA = P F=P PB = P P A B Q (a) Loads in a beam. (b) Bar member. MB = M0 VA = Q TA = T0 T= T0 TB = T0 MA = VB – MB VB = Q (c) Shaft member. (d) Beam member. MB VA F MA VB (e) Frame member. FIGURE 1-20 Forces on structural members. A shaft member carries an internal torque (T T 0 ), which follows the t-sign con- vention, as shown in Fig. 1-20c. Torques at the nodes (TA and TB) are obtained from the rotational equilibrium about the x-axis (TA TB T 0 1 kN-m 738 ft-lbf). A beam member carries internal shear force and bending moment, which follows the t-sign convention as shown in Fig. 1-20d. At node B, VB Q, and MB M 0 . The shear force at node A is obtained from the EE written along the y-coordinate direction (VA Q). The shear forces (VA and VB À1 kN À225 lbf) as marked in Fig. 1-20d are negative. The bending moment at node A is obtained from the EE written about the z-coordinate direction (MA VB ` À MB À16:3 ft-k À22:1 kN-m). The moments are different at nodes A and B because shear force has to be accounted in the calculation of moment. A frame member is obtained by combining the bar member and the beam member. It carries three internal forces at each node (FA, VA, MA and FB, VB, MB) as shown in Fig. 1-20e. Traditionally, the torque load is not included in a two-dimensional frame model. 24 STRENGTH OF MATERIALS Displacement Displacement is a vector quantity that can be observed and measured. The attributes of displacement are quite similar to that of external load. Most rules developed for load are applicable to displacement, which follows the n-sign convention. Load and displacement separate into four distinct pairs, as follows: 1. Axial displacement u and axial load P 2. Transverse displacement v and transverse load Q 3. Angle of rotation y and external bending moment M 0 4. Angle of twist j and applied torque T 0 Axial displacement u is directed along the x-coordinate axis as shown in Fig. 1-21. The transverse displacement v is directed along the y-coordinate axis. Rotation y lies in the x±y plane. A counterclockwise curved arrow represents its positive direction. When a rotation is shown by a double arrowhead, it is positive when directed along the positive z-coordinate axis. Angle of twist j lies in the y±z plane. A counterclockwise curved arrow represents its positive direction. When the angle of twist is shown by a double arrowhead, it is positive when directed along the positive x-coordinate axis. Displacements u and v have the dimen- sion of length. In USCS units, displacement is measured in inches. In SI units, it is measured in millimeters or centimeters. The rotation and angle of twist are dimensionless quantities. These two variables are measured either in degrees in USCS units or in radians in SI units. Deformation A structural member deforms under the action of internal force. Deformation is a measure of the relative displacements along two locations in the member length. For a bar, the deform- ation (bbar u2 À u1 ) is equal to the relative axial displacements (u2 and u1) of its end points (2 and 1), respectively. A beam deformation (bbeam y2 À y1 ) is equal to the relative angular rotations (y2 and y1 ) at two locations (2 and 1) in the beam length, respectively. Shear force induces shear deformation in a beam. Shear deformation is small and it is neglected. A torque deforms a shaft member. Its deformation (bshaft j2 À j1 ) is equal to the relative angles of twist (j2 and j1 ) at two locations (2 and 1) along its length, respectively. Deformation of a frame member is the sum of the bar deformation and the y x v z u FIGURE 1-21 Sign conventions for displacement. Introduction 25 beam deformation. A rigorous treatment of deformation makes the theory of strength of materials straightforward. Stress Stress can be defined as the intensity of force per unit area. There are two types of stress: normal stress and shear stress. Normal stress (s) is the intensity of normal or axial force per unit area. Shear stress (t) is the intensity of shear force per unit area. Its dimension is force divided by area (s F/A FLÀ2 MLÀ1 T À2 ). Stress in USCS units is measured in pound- force per square inch (psi) or in units of ksi, which is equal to 1000 psi (1 ksi 1000 psi). Stress in SI units is measured in pascals (Pa), which is 1 newton force per one square meter. In engineering, stress is measured in megapascals (MPa), which is 1 million pascals (1 MPa 106 Pa) because one Pa is a very small stress. Conversion between the two systems to measure stress is given in Table 1-3. In the two dimensions, there are three stress components: two normal stresses (sx and sy ) and a shear stress (t txy ), as depicted on an elemental area in Fig. 1-22a. Stress follows the t-sign convention. As it turns out, elasticians adopted the t-sign convention for the stress tensor, and we use this in strength of materials. Normal stress (sx ) is positive if the product (nf) of the normal (n) and direction (f) is positive. It is easy to verify that stresses marked in Fig. 1-22a are positive, whereas those marked in Fig. 1-22b are negative. Consider the shear (n = 1, f = –1) y y x d c C n y om io x ns x pr Te es si on yx a b (n = f = –1) (n = f = 1) y x x xy c n=1 y f =–1 b (a) Positive stress. (b) Negative stress. FIGURE 1-22 Stress in two dimensions. 26 STRENGTH OF MATERIALS stress (t) in the face b±c in Fig. 1-22b. This stress component is negative because the normal is along the positive x-coordinate axis, or n 1, but shear stress points along the negative y-direction, or ft À1, and the product (nt ft À1) is negative. The positive normal stress sx in Fig. 1-22a stretches the elemental area along the x-coordinate direction and sx is tensile. The negative normal stress sx in Fig. 1-22b contracts the elemental area along the x-coordinate direction and sx is compressive. The shear stress can be transformed to obtain tension along the leading diagonal and compression along the other diagonal, as shown in Fig. 1-22a. Strain Strain is defined as the intensity of deformation and it is a dimensionless quantity. There are two types of strains: normal strain e and shear strain g. Normal strain is the intensity of axial elongation or contraction in a member. Shear strain is the intensity of angular deformation of an initial right angle. Like stress, there are three strain components in two dimensions; two normal strains (ex and ey ) and one shear strain (g gxy ). Strain, like the stress, follows the t-sign convention. Positive normal strain (ex ) induces a sympathetic negative normal strain (ey Ànex ). This phenomenon is called Poisson's effect and is illustrated in Fig. 1-23a. Positive or tensile strain (ex ) elongates the member along the x-coordinate direction, while simultaneously contracting it along the y-coordinate direction that induces compressive strain of magnitude ey Ànex . The constant n is referred to as Poisson's ratio. It is a positive fraction and for steel and aluminum it is about n 0:3. Shear strain g deforms an original right angle p/2 to p/2 À g along the leading diagonal, and it increases the right angle in the other diagonal by the same amount (p/2 g), as shown in Fig. 1-23b. y x y = –υ x /2 + x x xy = π/2 y /2 – (a) Normal strain. (b) Shear strain. FIGURE 1-23 Strain in two dimensions. Introduction 27 EXAMPLE 1-4 The displacements and rotations measured at the two locations in the circular beam shown in Fig. 1-24 are as follows: at location x1 50 cm, the displacements are u1 1 cm, v1 1 cm, y1 5 , and f1 2 ; and at x2 100 cm, u2 2 cm, v2 3 cm, y2 7 , and f2 5 . Calculate deformations in the segment of the beam between the two locations. Solution Axial deformation bbar u2 À u1 2 À 1 1 cm 0:394 in: The elongation along the x-coordinate axis at location 2 relative to location 1 is 1:0 cm 0:394 in: The angular deformation bshaft f2 À f1 5 À 2 3 0:0524 rad: The shaft is twisted at location 2 relative to location 1 by 3 or 0:0524 rad: The flexural deformation bbeam y2 À y1 7 À 5 2 0:035 rad: The beam rotates at location 2 relative to location 1 (by 2 0:035 rad:): The beam has a relative displacement (dbeam v2 À v1 3 À 1 2 cm 0:788 in:): The beam deflects at location 2 relative to location 1 by 2 cm or 0:788 in: y x z v1 v2 1 u1 2 u2 1 2 x1 x2 FIGURE 1-24 Deformation in a beam. 1.5 Material Properties Structures are made of engineering materials that include steel, aluminum, composites, and wood. Our treatment is confined to material that is homogeneous, isotropic, and elastic. A homogeneous material has the same properties at every location throughout its volume. An isotropic material has the same properties in any direction. An elastic material returns to its original size and shape without any permanent deformation upon the removal of the force that 28 STRENGTH OF MATERIALS deforms it. A material has several characteristic properties. We discuss density r, coefficient of linear expansion a, Young's modulus E, shear modulus G, Poisson's ratio n, and ductility. Density Density r is the mass contained in a unit volume of a material. It is defined as the mass per unit volume or as a ratio of mass Ám contained in an elemental volume ÁV. Ám r 1-3a ÁV Weight density rw is sometimes preferred, especially in calculating the weight of a structure. It is defined as the ratio of weight Áw contained in an elemental volume ÁV. It is also equal to the density multiplied by the terrestrial gravitational acceleration g. Áw rw gr 1-3b ÁV Density in USCS units is measured in slug (or pound-mass) per cubic inch. In SI units, it is measured in kilogram mass per cubic centimeter. Density is used to calculate weight and the inertia force in dynamic analysis. EXAMPLE 1-5 A 10 in:3 piece of aluminum weighs 1.0 lbf. Calculate its weight and mass densities in USCS and SI Units. Solution Weight density: rw weight=volume 1=10 0:1 lbf=in:3 173 lbf=ft3 27:15 kN=m3 Mass density: rm rw =g 27; 150=9:81 2767 kg=m3 2767= 35:31  0:4535 172:7 lbm=ft3 172:7=32:17 5:34 slug=ft3 Coefficient of Linear Expansion An engineering material can alter its dimensions when its temperature is changed. A material expands when the temperature increases, and it contracts with a decrease in the temperature. The deformation caused by a change in temperature is measured through a coefficient of linear expansion a. It is defined as the increment of length in a unit-length for a rise in temperature of 1 . Introduction 29 The dimension of the coefficient of expansion is per degree of temperature. In USCS units, it is measured in units of per degree Fahrenheit (1/ F or FÀ1 ). The unit of a in SI units is per degree Celsius (1/ C or CÀ1 ). The formulas to convert temperature (T ) from one scale to another are 9 T F T C 32 1-4a 5 9 T F T K À 459:67 1-4b 5 5 T C T F À 32 1-4c 9 5 T K T F À 32 273:15 1-4d 9 T C T K À 273:15 1-4e Change of temperature ÁT induces normal thermal strain (eT aÁT), but no thermal shear strain (gT 0) is induced. EXAMPLE 1-6 Calculate the thermal strain in a stainless steel material for a change in temperature of ÁT 300 K. The coefficient of thermal expansion of the material is a 9:60  10À6 / F. Solution The temperature given in kelvin (K) can be changed into F using Eq. (1-4b). ÁT 0 F 9=5300 À 459:67 80:33 F This temperature (ÁT) produces a thermal strain eT : eT aÁT 9:60  10À6 = F  80:33 F 771:2  10À6 Strain, being a dimensionless quantity, remains the same in SI and USCS units. 1.6 Stress-Strain Law Stress in a material induces strain and vice versa. Robert Hooke (1635±1703), a contempor- ary of Newton (1642±1727) postulated the concept through his statement: ``the extension is proportional to force.'' The concept was further developed into Hooke's law. According to 30 STRENGTH OF MATERIALS Cross-sectional area at grip T L B A B W C Test section R Grip G FIGURE 1-25 Standard tension coupon. this law, stress at any point in an elastic solid is a function of the strain at that point. The material relationship is determined by experiment through standardized laboratory tests. The tension test is a popular method for the purpose. This test is carried out on a standard specimen, also called a test coupon. In the United States, the American Society for Testing and Materials (ASTM) publishes guidelines for such tests. An ASTM test coupon is shown in Fig. 1-25. Its dimensions are designed to produce uniform normal stress and normal strain states over a section of the coupon, referred to as the gauge length G. An acceptable set of dimensions in inches for the specimen follows: Gauge length, G 8 in: Width inside gauge section, W 1:5 in: Total coupon length, L 18 in: Width at grip, C 2 in: Thickness of coupon, T 0:25 in: Length of grip section, B 3 in: Length of reduced section, A 9 in: Radius of filler, R 1 in: The cross-sectional area AG of the specimen in the gauge section is obtained as the product of the width and thickness (AG WT 0:375 in:2 ). The coupon is tested in a uniaxial testing machine sketched in Fig. 1-26. The bottom grip section of the flat coupon is held firm by a wedge grip. The top grip section of the specimen similarly held is mounted on a movable crosshead. The crosshead is moved very slowly and uniformly at a constant rate, providing a uniaxial load P to the specimen. The load P is read directly from a measuring device called the load cell. The normal stress, which is uniform across the gauge section, is calculated as the ratio of load P to area AG . P P s 1-5a AG 0:375 Stress cannot be measured directly from an experimental setup. It is determined from the measured load P and calculated area AG . Strain is amenable to direct measurement through strain-measuring devices called strain gauges. Two strain gauges are bonded to Introduction 31 Load P Movable Grip crosshead Thickness Gauge length Wedge grip P FIGURE 1-26 Uniaxial testing. the specimen inside the gauge section, as shown in Fig. 1-27. The gauge n mounted along the line of action of the load is read to obtain the normal strain en . Likewise, normal strain in the lateral direction e` is read from the gauge ` mounted perpendicular to the gauge n. The experiment is repeated for a different load setting P, and a set of values for stress and strain is obtained. A stress-strain diagram is constructed by plotting strain e along the x-axis versus the stress s along the y-axis. The shape of the stress-strain diagram depends on the material being tested. Figure 1-28 shows such a diagram (which is not drawn to scale) for a ductile steel material. The stress-strain diagram is credited to Bernoulli (1654±1705). The diagram exhibits four distinct zones: a linear region, a yielding or perfectly plastic zone, and a strain-hardening section followed by the necking region. Elastic Region The segment at the beginning of the diagram marked ``OAea '' is the elastic region. In this region, stress and strain are linearly proportional to each other. This region is elastic because the original shape and size of the coupon are restored upon removal of the load. The proportional limit sp corresponding to the location marked A in Fig. 1-28 is defined as the Strain gauges P P n FIGURE 1-27 An enlarged view of test section. 32 STRENGTH OF MATERIALS Ultimate stress u U Yield stress F y= e B C Fracture ( f) Proportional p A limit E 1 a c u f O Linear Perfect Strain Necking region plasticity hardening or yielding Plastic region FIGURE 1-28 Stress-strain diagram of a ductile steel material (not to scale). upper limit of stress inside the linear elastic region. The strength of materials calculation is confined to the elastic region. Young's Modulus The proportionality of stress and strain or the slope of the stress-strain diagram is the Young's modulus E of the material. Credited to Young (1773±1829), it is defined as the ratio of stress and strain: normal stress s E 1-5b normal strain e The definition of Young's modulus is valid provided stress is within the proportional limit of the material (s sp ). Young's modulus, which is also referred to as the modulus of elasticity, is used frequently in strength of materials calculations. It has the unit of stress because strain is a dimensionless quantity. Its unit of measure in USCS units is either pound force per square inch or kilopound force per square inch. In SI units, it can be measured in pascal or megapascal. The stress-strain relation (s Ee) derived from the definition of Young's modulus is referred to as Hooke's law. Poisson's Ratio Elongation of a member under a tensile load is accompanied by a sympathetic lateral contraction. The deformation for a small portion of the tension coupon in an exaggerated Introduction 33 Axial expansion ts trac Con P Expand P s Lateral contraction FIGURE 1-29 Deformation of a section of text component in an exaggerated scale. scale is sketched in Fig. 1-29. The section elongates, inducing tensile strain en along the direction of load. It also contracts along the transverse direction, inducing a compressive strain e` . Poisson's ratio n is defined as the negative ratio of the lateral, or transverse, strain e` to the longitudinal strain en . tranverse strain e` nÀ À 1-5c longitudinal strain en Poisson's ratio can be determined directly as the ratio of the strains measured from gauges mounted along perpendicular directions, as shown in Fig. 1-27. Poisson's ratio is a positive fraction because the two strain components have opposite signs. It is a dimensionless quantity and has an approximate value of n 0:3 for steel and aluminum. Shear Modulus Within the proportional limit of a material, shear modulus G is defined as the ratio of the shear stress t to the shear strain g: (G t/g). The shear modulus can be measured from a torsion test, or alternatively, it can be calculated from the Young's modulus and Poisson's ratio of a material using a closed-form elasticity relationship. E G 1-5d 2 1 n The calculated shear modulus from the formulas is accurate. The dimension of the shear modulus G is identical to that of the Young's modulus. Yielding Zone A stress value slightly more than the proportional limit sp is called the elastic limit se . The material is elastic inside the small region bounded by sp and se , marked A and B in Fig. 1-28, but the stress-strain relation becomes nonlinear. Beyond the elastic limit, the material deforms permanently. The minimum value of stress that initiates the permanent deformation is called the yield stress (sy ) and the yield point B. Upon the initiation of yielding, the strain increases 34 STRENGTH OF MATERIALS up to a point C without any increase in the stress. This region between B and C is referred to as the yielding or the perfectly plastic region. In this plateau, the strain increases under constant load P or stress P/AG . Strain-Hardening Section The material begins to harden beyond the perfectly plastic region marked by C and U in Fig. 1-28 and this phenomenon is referred to as ``strain hardening'' because stress increases with strain. The stress-and-strain relationship is nonlinear in the strain- hardening region, which extends up to the peak stress point U referred to as the ultimate stress su . Necking Zone At about the ultimate stress point, the cross-sectional area of the coupon AG , hitherto considered to be a constant, begins to decrease in a localized region. This phenomenon is called necking, and it continues until the specimen breaks at a stress value referred to as the fracture stress sf . Fracture stress is lower than the ultimate stress (sf < sU ). In this zone (U±F in Fig. 1-28), the strain increases but load is relieved or stress decreases. The combined region, from yielding to the fracture point (which contains the yielding, strain hardening, and necking regions) is referred to as the plastic region. Ductility The ability of a material to accumulate inelastic deformation without breaking is called ductility. It is measured as the percent elongation at fracture or 100 times the strain at fracture ef . Lf À LG % elongation 100 ef 100 1-5e LG where Lf is the length of the test section at fracture and LG is the gauge length. The stress-strain diagram of a ductile material is shown in Fig. 1-30a. This material exhibits considerable elongation prior to fracture. The shaded area under the stress-strain diagram is called the density of the strain energy that has been absorbed by the material prior to fracture. A ductile material absorbs a considerable amount of strain energy prior to fracture. The toughness of a material is proportional to the strain energy density. A ductile material is also tough. The stress-strain diagram of a brittle material is shown in Fig. 1-30b. This material breaks with little accumulation of plastic deformation. It absorbs a very small amount of strain energy, which corresponds to low toughness. Failure in a ductile material is accompanied with warning during the process of elongation. Failure in a brittle material can be sudden because it has a small percent of elongation at failure. Ductile material is preferred in engineering construction. Ductile material includes mild steel, aluminum and its alloys, copper, and nickel. Glass is a good example of a brittle material. Introduction 35 Fracture Fracture stress stress u f Stress Stress y Strain Strain energy energy density density Strain f f Strain (a) Ductile material. (b) Brittle material. FIGURE 1-30 Stress-strain diagram of a ductile and a brittle material. EXAMPLE 1-7 During a uniaxial tensile test conducted on the coupon shown in Fig. 1-25, the microstrains (m 10À6 ) recorded at a load step (P 750 lbf) are 67:50 m along the load direction and À20:25 m along the transverse direction. Calculate the properties of the coupon material. Solution Cross-sectional area of coupon AG 0:375 in:2 242 mm2 Load P 750 lbf 3336 N Stress sp P=AG 2000 psi 13:8 N-mmÀ2 13:8 MPa Strain ep 67:50 m 67:50  10À6 Strain et À20:25  10À6 Stress 2000 Young's Modulus: E 29:63  106 psi Strain 67:50  10À6 13:8  106 E 204:44 GPa 67:50  10À6 Here, GPa stands for gigapascal (or 1 GPa 1000 MPa 109 Pa) 36 STRENGTH OF MATERIALS et 20:25 Poisson's Ratio: n À 0:3 ep 67:50 E E 29:63 Shear Modulus: G  106 11:4  106 psi 2 1 n 2:6 2:6 204:44 E GPa 78:63 GPa 2:6 The material of the coupon appears to be steel with a Young's modulus of E 30  106 psi 207 GPa, a Poisson's ratio of n 0:3, and a shear modulus of G 11:4  106 psi 79 GPa. EXAMPLE 1-8 A square aluminum plate of area 1 in.2 is subjected to a tensile stress of 30 MPa along the x-coordinate direction. The Young's modulus of the material is E 10,000 ksi and its Poisson's ratio is n 0:28. Calculate the shear modulus and the induced strains. Solution Young's modulus given in ksi is changed to MPa to obtain E 10,000  6895  103 68,947:6 MPa 68:95 GPa. Stress sx 30 MPa and strain along the x-coordinate direction: ex sx /E 30  106 /68:95  109 0:435  10À3 0:0435 percent. Strain is induced along the y-coordinate direction because of the Poisson's effect: ey Àn  ex À0:28  0:435  10À3 À0:122  10À3 À0:0122 percent: A compressive strain is induced in the y-coordinate direction because of a tensile strain along the x-coordinate direction. Poisson's effect applies to strain but not to stress. No stress is induced along the y-coordinate direction because of sx . E Shear modulus G 2(1n) 10,000/(2  1:28) 3906 ksi 26:93 GPa. 1.7 Assumptions of Strength of Materials The theory of strength of materials is based on two primary assumptions. The first pertains to the material of the structure, whereas the second is related to the magnitude of the displacement. The assumptions make analysis linear, leading to the linear elastic Introduction 37 theory of strength of materials. The assumptions do not unduly restrict the scope of the subject because the behavior of a vast majority of structures falls in the linear domain. Material Linearity Structures are made of material of one kind or other. The stress-strain diagram shown in Fig. 1-28 contains almost all the material information required for analyzing and design- ing a structure. Strength of materials analysis is confined to the elastic region marked ``Oaea '' in Fig. 1-28, wherein the stress is linearly proportional to strain and Hooke's law is linear. For a structure made of mild steel, the material linearity is satisfied provided the induced stress s is below the proportional limit (s sp ; sp 35 ksi or 241 MPa) and the induced strain e is less than the proportional strain (e ep ; ep 0:0012 in:/in: 0:12 percent). The Young's modulus (E 30,000 ksi), shear modu- lus (G 11,538 ksi), and Poisson's ratio (n 0:3) remain constant throughout the elastic region. The magnitude of strain at e 0:0012 in:/in: is very small compared with the magnitude of stress at 35 ksi or the modulus of elasticity at 30,000 ksi. The magnitude of this nondimensional quantity is expressed as a percent strain (e 0:0012 is equal to e 0:12 percent strain). Geometrical Linearity and Small-Displacement Theory In a structure, the displacement induced due to the application of load is much smaller than the geometrical dimensions of the structure. Take, for example, a long-span bridge that is susceptible to displacement. We do not experience any movement of the bridge floor while driving on it because the displacement is really small. In analysis, the displacement is assumed to be small, and it is a valid assumption for the vast majority of real-life structures. Because the displacement is small, equations of strength of materials written for undeformed configurations remain valid even for the deformed structure. The attributes of geometrical linearity are illustrated through the example of the cantilever beam shown in Fig. 1-31. It is a rectangular beam of length `, depth d, thickness t, and Young's modulus E. It is clamped at location A, and it is free at B, as shown in Fig. 1-31a. The free end B is subjected to a load P applied along the negative y-coordinate direction. The coordinate system (x, y), prior to the application of load, is referred to as the initial, or undeformed, coordinate system. The load deforms the cantilever, inducing a displacement v, which can be small (vs), as shown in Fig. 1-31b, or large (v` ), as depicted Fig. 1-31c. For a small displacement, the deformed coordinate system (xs, ys) is not too far away from the undeformed coordinates (x, y). In other words, equations written with respect to the undeformed coordinates (x, y) can be considered to be valid even for the deformed system (xs, ys). The intensity of deformation in the beam can be expressed through a curvature parameter , which is defined as the reciprocal of the radius of curvature R. In the small displacement theory, the curvature is equal to the second derivative of the displacement, whereas rotation or slope y is equal to the first derivative of the displacement. 1 d 2 vs k 2 1-6a R dx 38 STRENGTH OF MATERIALS y x P ys P vs A B d x xs (b) Response under small displacement (a) Cantilever beam under load P. theory. y S Load P y x P v O Displacement v x (c) Response with geometrical (d) Linear load displacement nonlinearity. relation. Load P Displacement v (e) Nonlinear load displacement relation. FIGURE 1-31 Geometrical linearity and small displacement theory. dvs y 1-6b dx Under these assumptions, the load displacement relation is linear, as shown in Fig. 1-31d. The displacements of the cantilever beam can be calculated from the formulas. 4Px3 vs x 1-6c Etd 3 4P`3 vsmax 1-6d Etd Introduction 39 Here vs(x) is the displacement at location x and the maximum displacement vsmax occurs at the free end B. The load displacement relationship is linear, as shown in Fig. 1-31d. If the magnitude of the load P is increased, the displacement can grow into the geometrically nonlinear domain, as shown in Fig. 1-31c. The deformed coordinate system (x` , y` ) has moved substantially from the undeformed system (x, y) and the load direction might also have changed. A curvilinear coordinate system may have to be employed for its analysis. Equations have to be written with respect to the deformed coordinates (x` , y` ). The load displacement relation becomes nonlinear (see Fig. 1-31e), and the curvature term has to be redefined to include the square of rotation because it is no longer small compared to unity. v` x vs x À f n P; E; d; t 1-6e 2 ` d v 1 dx`2 k 4 ` 2 53=2 1-6f R dv 1 dx` The curvature definition given by Eq. (1-6f) is reduced to Eq. (1-6a) when the square of the rotation is much smaller than unity f(dv` /dx` )2 ( 1g and coordinate x` xs x. The function f n (P, E, d, t) is nonlinear in load P, modulus E, and sizes d and t. The nonlinear strain displacement relation becomes linear when the displacement is small. In summary, the assumptions of the small displacement theory are: 1. Displacement is small in comparison to the smallest dimension of the structure. 2. The rotation is small and the square of the rotation is much smaller than unity. 3. The strain is small and its square can be neglected. 4. The equations written in the undeformed configuration of the structure are valid even for the deformed configuration. 5. The strain displacement relationship is linear. The geometrical linearity can be independent of the material linearity and vice versa. In strength of materials, we will assume both material linearity and geometrical linearity. Because of this assumption, all the equations of strength of materials are linear. The nonlinear analysis, which is much more complex than the linear theory, is covered in advanced solid mechanics courses. Nonlinear theory has special applications, but the linear theory is routinely used in almost all design calculations. EXAMPLE 1-9 The deformed shape of an initially square aluminum plate with 2-in.-long sides is shown in Fig. 1-32a. Calculate the average strains in the plate. 40 STRENGTH OF MATERIALS FIGURE 1-32 Strains in a plate. Solution The average normal strain is obtained as the ratio of the average elongation to the initial length. The shear strain is obtained as the change of the original right angle. Average Displacements along the x-Coordinate Direction: Displacements (uA and uB ) along with the x-coordinate direction as marked in Fig. 1-32b are as follows: 200 uA 100  10À6 in: 2 250 uB 150 275  10À6 in: 2 Deformation (dx ) is dx uB À uA 175  10À6 in: Along the y-Direction: vC 0 50 75 vD 62:5  10À6 in: 2 dy vD À vC 62:5  10À6 in: Rotations in the xÀy Plane: 200  10À6 y1 100  10À6 rad 2 Introduction 41 250  10À6 y2 125  10À6 rad 2 Normal Strain Components: dx 175  10À6 ex 87:5  10À6 0:009 percent a 2 dy 62:5  10À6 ey 31:25  10À6 0:003 percent a 2 Shear Strain: gxy y1 y2 225  10À6 0:0225 percent 1.8 Equilibrium Equations The equilibrium equations (EE) are the most important equations of analysis. Fortunately, these equations are simple and straightforward. Force balance is the central concept behind the equilibrium equations. The equilibrium concept matured over a long period of time as eminent scientists and engineers worked in different applications. It is credited to Newton (1642±1727). However, it was used earlier by Archimedes (287±212 B.C.) in levers and pulleys, by Leonardo da Vinci (1452±1519) in virtual work principles, by Tyco Brahe (1564±1601) and Kepler (1571±1630) in planetary motion, just to mention a few different applications. We will introduce the equilibrium equations in the three-dimensional space. These will be specialized to strength of materials applications in the subsequent chapters of this book. Consider a Cartesian coordinate system and the displacements u, v, and w along the x, y, and z directions, as shown in Fig. 1-33. Also consider the three rotations (yx , yy , yz ) about the same three axes. In general, a point in a structure can displace by an amount u along the x-coordinate direction and by amounts v and w along the y- and z-coordinate directions, respectively. The same point can also rotate by yx , yy , and yz amounts about the coordinate directions, respectively. The six variables (u, v, w, yx , yy , yz ) are referred to as the six displacement degrees of freedom (dof). Displacements encompass transla- tions, or translatory displacements (u, v, w), and rotation or rotatory displacements (yx , yy , yz ). One equilibrium equation can be written along each of the six displacement direc- tions, yielding six equilibrium equations. The equations along the displacement directions (u, v, w) represent the three force balance conditions. The equations along the rotation directions (yx , yy , yz ) represent the three moment balance conditions. The six EE along the six directions follow. 42 STRENGTH OF MATERIALS y y v x u x O w z z FIGURE 1-33 Reference frame to formulation of equilibrium equations. The sum of forces along the x-axis or u-direction is zero: F0 1-7a x-axis along the y-axis or n-direction, it is zero: F0 1-7b y-axis along the z-axis or w-direction, it is zero: F0 1-7c z-axis The sum of moments about the x-axis or yx -direction is zero: M0 1-7d about x-axis about the y-axis or yy -direction, it is zero: M0 1-7e about y-axis about the z-axis or yz -direction, it is zero: M0 1-7f about z-axis Sign convention for the EE is discussed in Appendix 4. The nature of the equilibrium equation is explained by considering two examples: a three-legged table and a four-legged Introduction 43 table. These problems lead to the definition of the determinate problem and indeterminate problem in strength of materials. Three-Legged Table Problem A three-legged circular wooden table is shown in Fig. 1-34a. The three legs of the table form an equilateral triangle with sides a and centroid at G. The height of the table is h. The tabletop is circular with radius r. The table is resting on a level floor made of a rigid material, such as stone, and the tabletop is also considered to be rigid. (A rigid material cannot deform under any circumstance.) The centroid G is considered as the origin of the x, y, z coordinate system. A load of magnitude P is applied along the negative y-coordinate direction with eccentricities ex and ey. The problem is to determine the three reactions (R1, R2, R3) along the three legs of the table. Step 1ÐStability of the Table The table without any restraint is free to displace as a rigid body by u, v, and w inches along the x-, y-, and z-coordinate directions. Likewise, it can rotate as a rigid body by yx , yy , and yz radians along the three axes (see Fig. 1-33a). The analysis model must be stable; that is, it must not move as a rigid body. A stable model requires the restraining of at least six displacement degrees of freedom, as shown in Fig. 1-34b. 1. Three degrees of freedom are restrained by attaching leg 1 to the foundation through a hinge or u1 v1 w1 0. z, w y, v z y x, u ex P P ey x G F3 h F1 F2 a a R3 a R1 R2 (a) Circular table under load P. (b) Table with boundary restraints. FIGURE 1-34 Three-legged determinate table problem. 44 STRENGTH OF MATERIALS 2. Two degrees of freedom are restrained by preventing the movement of legs 2 and 3 along the z-coordinate direction (w2 w3 0). 3. Despite the five restraints, the table can rotate about the z-coordinate, with leg 1 as the axis. A restraint at the second leg along the x-coordinate direction (u2 0) prevents this motion. The six restraints (u1 v1 w1 w2 w3 u2 0) have eliminated the rigid body motion of the table and it is a stable model for analysis. Step 2ÐSimplification of Analysis Each of the six restraints, in principle, can induce six reactions. Because the rigid tabletop cannot stretch in the x±y plane, we assume that there is no movement of the legs in a parallel (x, y) plane at the support level. Since no load is applied in either the x- or y-coordinate directions, it is assumed that there are no reactions along the x- or y-axis. In consequence, the problem has three unknown reactions (R1, R2, R3) along the three legs, as shown in Fig. 1-34a. Step 3ÐCalculation of Reactions The equilibrium equations must be employed to calculate the three reactions. The six equilibrium equations, Eqs. (1-7a) to (1-7f ), for the table problem degenerate into three independent equations. Equilibrium Eqs. (1-7a) and (1-7b) are trivially satisfied because there are no forces along the x or y directions. The EE [Eq. (1-7c)] is obtained as the algebraic sum of the three reactions and the load along the z-coordinate. ÀR1 À R2 À R3 ÀP 1-8a The EE [Eq. (1-7d)] is obtained as the algebraic sum of the moments about the x-axis. a p R1 R2 À 2R3 Àey P 1-8b 2 3 Likewise, rotational equilibrium about the y-coordinate axis yields a ÀR1 R2 ex P 1-8c 2 The rotational equilibrium equation along the z-direction [Eq. (1-7c)] is satisfied trivially because there are no loads in the x±y-plane. Three Eqs. (1-8) containing three unknowns are solved to obtain the reactions. P p R1 (a À 3ey À 3ex ) 1-9a 3a P p R2 (a À 3ey 3ex ) 1-9b 3a P p R3 (a 2 3ey ) 1-9c 3a Introduction 45 When the load is applied at the centroid G with no eccentricities (ex ey 0), then each leg carries one-third of the load (R1 R2 R3 P/3). Determinate Problem The internal forces along the legs of the table are under compression and in magnitude are equal to the reactions (F1 R1 , F2 R2 , and F3 R3 ). If a problem can be solved for the internal forces including the reactions through an application of the equilibrium equations alone, then it is called a determinate problem. Such problems are also referred to as statically determinate problems because they can be solved by an application of the equilibrium equations of statics. The three-legged table is a determinate problem because the application of the equi- librium equations produced the solution for the forces and the reactions. The solution of a determinate problem is straightforward, and in this book it is treated first. If the internal forces and reactions cannot be determined from the equilibrium equations alone, then it is called an indeterminate problem. Navier's table, discussed in this chapter and solved in Chapter 14, is an example of an indeterminate problem. EXAMPLE 1-10 The uniform beam of length ` 10 ft shown in Fig. 1-35a weighs 1000 N and it is subjected to P 1 kip gravity load at the one-quarter span location. Calculate the reactions at the supports. For this problem, the weight is specified in SI units whereas the length and load are given in USCS units. All parameters must be converted either to USCS units or SI units prior to the solution of the problem. Here, we solve the problem in USCS units. Weight 1000 N 1000=4:45 224:7 lbf For calculating the reactions, the beam weight can be lumped at the midspan. The analysis model is shown in Fig. 1-35b. The reaction RA at node A is obtained by taking the moment about node B and setting it to zero. 1000 224.7 P A B 5 ft 10 ft RA RB 2.5 (a) Beam. (b) Analysis model. FIGURE 1-35 Reaction in a beam. 46 STRENGTH OF MATERIALS Moment about Node B: RA  10 À 1000  7:5 À 224:7  5 0 or RA 862:35 lbf Likewise, the reaction (RB) at node B is obtained by taking the moment about node A and setting it to zero. Moment about Node A: ÀRB  10 224:7  5 1000  2:5 0 or RB 362:35 lbf The sum of the reactions must be equal to the applied load. This condition is satisfied because (RA RB 862:35 362:35 1224:7) (weight P 1000 224:7 1224:7 lb). The reactions in SI units are RA 862:35  4:45 3:84 kN and RB 362:35  4:45 1:61 kN: Navier's Table Problem Navier attempted to determine the four reactions along the legs of a table, which is referred to as Navier's table problem. The symmetrical table is made of wood, rests on a level stone floor, and is subjected to a load P with eccentricities ex and ey, as shown in Fig. 1-36. The table height is h and the distances between the legs along the x- and y-directions are 2a and 2b, respectively. The tabletop is considered to be rigid. The problem is to calculate the four reactions (R1, R2, R3, R4) along the four legs of the table. The solution process is similar to that for the earlier table problem except for the number of reactionsÐfour instead of three for the circular table. The table has only three independent equilibrium equations consisting of one transverse equilibrium equation along the z-coordinate direction and two moment equilibrium equations along the x- and y-coordinate directions. The three equations are as follows: ÀR1 À R2 À R3 À R4 À P 1-10a P z y A ey x ex h R4 2a R3 2b R1 R2 FIGURE 1-36 Navier's indeterminate table problem. Introduction 47 ey R1 R2 À R3 À R4 À P 1-10b a ex ÀR1 R2 R3 À R4 P 1-10c b These three equilibrium equations contain four unknown reactions. The four reactions cannot be determined from the three equations alone. The equilibrium equations are inde- terminate because the number of unknown reactions exceeds the number of equilibrium equations. The equilibrium equations must be augmented with an additional equation to solve for the four reactions. The additional equation is called the compatibility condition. Navier's indeterminate table problem requires one compatibility condition that must be added to the three equilibrium equations, yielding four equations that can be solved for the four reactions. The vast majority of engineering problems are indeterminate in nature. The solution of an indeterminate problem requires the simultaneous application of the equilibrium equations =2M =3M td2 td2 M M Galileo Galilei Solution by Jacob Bernoulli Solution by (1564–1642) Galileo (1654–1705) Bernoulli Historical prediction of stress in a beam. M =6 td2 t d M Charles A. Coulomb Correct solution (1736–1806) by Coulomb FIGURE 1-37 Galileo's cantilever experiment conducted in 1632. 48 STRENGTH OF MATERIALS and the compatibility conditions. The formulation of the compatibility conditions will be discussed in Chapter 6 of this book. The analysis of an indeterminate problem becomes straightforward when the compatibility conditions are fully understood and used. As it turned out, the compatibility conditions were not fully understood until recently, even though the theory of strength of materials dates back to the cantilever experiment of Galileo conducted in 1638 (the setup is shown in Fig. 1-37). Even though some of his calculations were underdeveloped, Galileo's genius is well reflected in the solution of the problem. Historical predictions of stress (sA ) at the root of the rectangular cantilever beam depicted in the experimental setup in Fig. 1-37 are as follows: 2MA Galileo's Solution: sA 1-11a td 2 3MA Bernoulli's Formula: sA 2 1-11b td 6MA Correct Coulomb's Solution: sA 2 1-11c td where t, d, and MA are the beam thickness, depth, and moment at A, respectively. The prediction by Galileo is one-third of the correct solution given by Coulomb. Bernoulli's solution is 50 percent accurate. EXAMPLE 1-11 A wooden cantilevered beam (similar to the one shown in Fig. 1-37) has the following dimensions: length ` 10 ft, depth d 6 in:, and thickness t 12 in: Calculate the stress at its base for a 1000-lbf gravity load applied at its free end using the formulas given by Galileo, Bernoulli, and Coulomb. Neglect the weight of the beam. Moment MA at the base of the cantilever (A in Fig. 1-37) is MA À1000  10  12 À120;000 in:-lbf: M All three scientists recognized the stress and moment relationship to be s k td2 . The value of the coefficient k differed between the scientists. For Galileo it was k 2 and for Bernoulli it was k 3. Finally, Coulomb correctly predicted it to be k 6. MA 120;000 C À À277:78 td2 12  62 2MA Galileo's Stress: sG À555:26 psi À3:83 MPa td 2 3MA Bernoulli's Stress: sB 2 À833:33 psi À5:75 MPa td 6MA Coulomb's Stress: sC 2 À1666:67 psi À11:50 MPa td Introduction 49 EXAMPLE 1-12 Calculate the breaking load of the cantilevered beam in Example 1-11. Assume wood to break at a stress of sB 0:3 ksi. According to the three scientists, the load-carrying capacities of the beam (Pmax) are as follows: M sB td 2 Pmax ` k` Galileo's stress formula expressed in terms of load becomes PGmax sB t d2 =2` 0:3  12  62 = 2  120 0:54 kip Bernoulli's predictionX PBmax 0:36 kip Coulomb's prediction PCmax 0:18 kip The beam in Example 1-11 is likely to break when designed using the formula given by Galileo or Bernoulli because their predictions are nonconservative. They predict lower than actual stress. Problems Use material properties given in Appendix 5 to solve the problems. 1-1 Derive the dimensional formula of the quantities given in Table P1-1. Use SI base units of mass (M, kg), length (L, m), time (T, s), temperature (t, C), and angle (y, rad); and USCS TABLE P1-1 Galileo's cantilever beam experiment conducted in 1632 Quantity Value (USCS Units) Quantity Value (USCS Units) Angle of twist, j, deg 1.5 Poisson's ratio 0.3 Angular deformation, deg 1.0 Rotation, y, deg 3 Area, A, in.2 10 Shear modulus, ksi 3700 Axial deformation, in. 0.25 Strain, percent 0.06 Coefficient of thermal 6.6 Stress, psi 6000 expansion, (10À6 )/F Curvature, in:À1 10À4 Temperature, F 100 Displacement, u, in. 1.0 Torque, T, ft-lbf 10,000 Force, F, lbf 1000 Volume, V, in.3 100 Length, in. 150 Weight, lbf 1000 Mass, slug 1000 Weight density, lbf/ft3 490 Mass density, slug/ft3 15.2 Yield strength, ksi 20 Moment, M, ft-lbf 50,000 Young's modulus, ksi 10,000 50 STRENGTH OF MATERIALS base units of force (F, lbf), length (L, ft), time (T, s), temperature (t, F), and angle (y, y ). Convert the numerical values given in USCS units into SI units. 1-2 The two-bar aluminum truss shown in Fig. P1-2 has a span of 10 m and a height of 6 m. The cross-sectional areas of bars 1 and 2 are 800 and 1000 mm2, respectively. The truss supports a 5-kN load at an inclination of 0.78 rad to the x-coordinate axis. The bar temperatures are 100 and 125 C, respectively. Convert the parameters of the problem into SI units and calculate its mass and weight. y P = 5 kN x 0.78 rad 3 1 2 6m 1 2 10 m FIGURE P1-2 1-3 The uniform symmetrical steel cantilever beam shown in Fig. P1-3 supports two loads (Px 1 kN and Py 10 kN). The dimensions are marked in the figure. Calculate the mass, weight, and equivalent forces in the beam in USCS and SI units. Px = 1 kN 0.25 m 0.15 m 3m Py = 10 kN FIGURE P1-3 1-4 The displacements and rotations are measured at the top of a uniform concrete column at location A, as shown in Fig. P1-4. The column is 16 ft high and has a square cross- sectional area of 1 ft2. Along the z-axis, the displacement is wA 0:5 cm; along the x-axis, it is uA 1 cm. Rotation about the y-axis is yA 5 ; about the z-axis, it is jA 2 . Calculate the deformations in the column. Introduction 51 z y x A 16 ft 1 ft FIGURE P1-4 1-5 A one-cubic-meter piece of steel weighs 77 kN. Calculate its mass and weight densities in USCS and SI units. 1-6 Calculate the thermal deformation in the column of Problem 1-4 for a uniform variation in temperature of 75 K. Use both SI and USCS units. (Thermal deformation thermal strain  length.) 1-7 The two-bar truss shown in Fig. P1-7 has a height H and a bar inclination y. Calculate the reactions Rx and Ry. Graph the reaction versus the angle y in the range 90 to 0 . y Ry Ry x Rx Rx 2 3 1 2 H 1 P FIGURE P1-7 52 STRENGTH OF MATERIALS 1-8 A square aluminum plate of area 625 mm2 is subjected to a 1-ksi compressive stress along the y-direction (see Fig. P1-8). Calculate the induced strains. y 1 ksi 5m 25 m x z 1 ksi FIGURE P1-8 1-9 The deformed shape of an initially rectangular steel plate with 25- and 15-mm sides is shown in Fig. P1-9. Calculate the average strains in the plate. y 100a x 125a 50a 25a 15 mm 25 mm a = 10–6 mm FIGURE P1-9 1-10 For the beam in Example 1-11, recalculate the stress when its orientation is changed by switching the depth and thickness and vice versa. Compare your answer to the solution in Example 1-11. 1-11 Calculate the breaking load of the cantilevered beam in Problem 1-10. Assume wood to break at a stress of sb 0:3 ksi. 1-12 During a uniaxial tensile test conducted on the coupon shown in Fig. 1-25, the gauge length at a load step (P 2 kN) expanded by 100 m-in. along the load direction, while the lateral contraction was 5 m-in. Calculate the properties of the coupon material. Introduction 53 2 Determinate Truss A truss structure, or simply a truss made out of bar members, is the simplest of the five structure types listed in the first column of Table 1-1 of Chapter 1. A truss can be used in bridges, building roofs, transmission towers, helicopter bodies, and other applications. In practice, a truss is fabricated by connecting bars with rivets, bolts, or welds. For the purpose of analysis, the real connections are replaced by frictionless hinges or pin connec- tions. This idealization makes the analysis simple, yet the error introduced is small. If required, the error can be quantified through an auxiliary calculation, referred to as a ``secondary stress'' analysis. The truss shown in Fig. 2-1 is made of nine bars (which are circled) and six joints, also referred to as nodes (shown in boxes). The connections at nodes l to 3, which are shown by solid lines, represent the real joints. Idealized pin connections are shown for nodes 4 to 6. Truss analysis treats all joints (1 to 6) as pin connections. A truss that can be analyzed for the bar forces only through an application of the equilibrium equation is called a determinate truss, and this chapter is devoted to its analysis. The technique devel- oped for the analysis of the determinate truss, with some modifications, can be used to analyze other structure types. 2.1 Bar Member The bar shown in Fig. 2-2a is a straight structural member with a length ` and a uniform cross-sectional area A. The shape of the cross section is not relevant. It can be circular, annular, rectangular, angular, or of any other shape, as shown in Fig. 2-2b. It is a slender member because its cross-sectional area is small compared with its length. The slenderness ratio (SR `/A) is defined as the ratio of its length ` to its cross-sectional area A. A slenderness ratio greater than 20 (SR ! 20) is typical. For analysis, a bar is represented by its centerline, which is also considered as the x-coordinate axis, and it passes through the 55 y x 3 6 5 h 1 3 5 7 9 1 6 2 4 8 2 4 1 2 3 FIGURE 2-1 Six-node, nine-bar truss. Real joints, nodes 1, 2, and 3; idealized pin connection, nodes 4, 5, and 6. centroid of its uniform area. A bar can be made of steel, aluminum, wood, or any other type of structural material. Its analysis requires the Young's modulus E and the coefficient of thermal expansion a of the material. A bar member is sketched by a line segment joining its two nodes, 1 and 2, as shown in Fig. 2-2c. Node 1 is considered as the origin of the x-coordinate axis. The quantities required for its analysis are length `, area A, modulus E, density r, and coefficient of thermal expansion a. Centroid G Area A Cross-section at a–a A A x G G x Circular Rectangular 2 a a A A G G Annular Angular 1 (a) Physical bar. (b) Typical cross-section for a bar. (c) Sketch of bar. FIGURE 2-2 Bar member. 56 STRENGTH OF MATERIALS Force Analysis in a Bar A bar can resist only an external axial load P that must be applied along the x-coordinate axis. It resists the load by inducing an internal force F that also acts along the x-coordinate axis. Consider a bar of length `, area A, modulus E, density r, and coefficient of thermal expansion a that is hinged to a ceiling at node 1 and is free at node 2, as shown in Fig. 2-3a. The hinged node represents the foundation or support. The free node 2 is subjected to an external load P. The problem is to determine the induced internal force F in the bar. Consider 1 c1 c1 y y F1 a Cross- section ci ci at y–y 2 F2 P c2 c2 x P (a) Physical bar. (b) Analysis model. R F1 F2 F F P F2 F1 Node 2 Location Node 1 Tensile Compressive at a (c) Free-body diagram. (d) Internal force. FIGURE 2-3 Force analysis of a bar. Determinate Truss 57 node 1 as the origin and the x-coordinate direction from node 1 to node 2. The line diagram, or the analysis model, of the bar is depicted in Fig. 2-3b. The solution is obtained by employing the concept of a free-body diagram. Such a diagram can be constructed in three steps: Step 1ÐFree Body A free body is a sketch of a small portion that is isolated from the rest of the bar member. A small portion can be isolated at any location in the bar. It can be at the origin, at node 2, or at an intermediate location at a distance a as shown in Fig. 2-3b. Step 2ÐMark Forces Consider the free body at node 2, which is obtained by cutting the bar at c2±c2 just above node 2, as shown in Fig. 2-3c. It is subject to the given load P and an unknown internal force F2. The x-coordinate axis is the line of action of both forces P and F2. Step 3ÐEstablish Equilibrium A free body must be in equilibrium. Equilibrium is satisfied when the algebraic sum of all forces along the x-coordinate direction is zero. F0 x P À F2 0; or F2 P 2-1a The internal force F2 at node 2 is equal to the load P. Repeat the three steps for a free body at an intermediate location. Step 1ÐFree Body A free body is isolated for an intermediate location at an arbitrary distance a from node 1 by cutting the bar at ci±ci as shown in Fig. 2-3c. Step 2ÐMark Forces The free body has two forces, F1 and F2. From Eq. (2-1a), F2 P. Step 3ÐEstablish Equilibrium Equilibrium along the x-coordinate direction yields F 2 À F1 0 or F1 F2 P 2-1b The internal forces are equal (F1 F2 ). Repeat the three steps for a free-body diagram at the support node 1. Step 1ÐFree Body A free body at node 1 is isolated by cutting the bar at ci±ci as shown in Fig. 2-3c. 58 STRENGTH OF MATERIALS Step 2ÐMark Forces The two forces are F1 and an unknown reactive force R. Step 3ÐEstablish Equilibrium The EE along the x-direction yields F1 À R 0 or F1 R P 2-1c The force solution for the bar is obtained as F1 F 2 F R P 2-1d Forces F1 and F2, are called the internal forces because these forces are internal to the body of the bar member. These forces are designated by the letter F. The applied load is also a force, but it is external to the body of the bar and is designated by the letter P. The reaction is the force induced at the support; it is designated by the letter R. For the bar member, the internal force is equal to the external load, which is also equal to the reaction. Positive Direction for Forces The external load and the reaction follow the n-sign convention. These are positive when directed along the positive x-coordinate axis. The external load P is positive, whereas the reaction is negative (see Fig. 2-3c). The induced reaction is opposite to the direction of load. The internal force F follows the t-sign convention. It is positive when it is tensile, which stretches or elongates the bar. It is shown by a pair of arrowheads that point at each other as in Fig. 2-3d. It is negative when it is a compressive force, which shortens or contracts the bar. A compressive force is shown by a pair of arrowheads that point in opposite directions as in Fig. 2-3d. The force analysis of the bar is independent of its length, area, density, modulus, or coefficient of expansion. Interface Forces A free-body diagram created by cutting a member exposes an interface. For example, the structure shown in Fig. 2-4a is cut at a±a to create an interface. The forces at the interface must be in equilibrium. This mandatory condition is satisfied by changing the direction of the forces acting at the left and right sections of the interface. If the left section is subjected to a force F directed along the positive x-coordinate axis, then the right section must be subjected to an equal magnitude force but applied along the negative x-coordinate axis, as shown in Fig. 2-4b. The force in the left section can be considered as the action, whereas the force in the right section is the reaction; and we know that action reaction. A reverse orientation for the interface forces, as shown in Fig. 2-4c, is valid. The interface forces marked in Fig. 2-4d are not admissible because equilibrium is violated. The shear force at the interface in Fig. 2-4e is admissible, but the moment as marked in Fig. 2-4f is not admissible. Moment and torque at interfaces in Figs. 2-4g and 2-4h are admissible. The axial force F, the shear force V, and the moment M at the interface, shown in Fig. 2-4i, are admissible. Determinate Truss 59 y a Py x Left Interface Right section section a Px F (a) Structure. (b) Admissible axial force. F F V (c) Admissible (d) Inadmissible (e) Admissible shear force. axial force. force. M M (f) Inadmissible bending moment. (g) Admissible bending moment. T M F V (h) Admissible torque. (i) Admissible forces. FIGURE 2-4 Forces at an interface. Force Analysis of a Composite Bar A composite bar made of two members is suspended from a ceiling as shown in Fig. 2-5a. Bar 1, made of steel with Young's modulus Es 30,000 ksi, is `1 in: long and has a solid cross-sectional area A1. Bar 2, made of aluminum with modulus Ea 10,000 ksi, is `2 in: long and has an annular cross-sectional area A2. It is subjected to a P1 kip load at the interface and a P2 kip load at its end. Determine the internal force in each bar and the reaction at the support. 60 STRENGTH OF MATERIALS y Support r r O x a a 1 Bar 1 Section at a–a, area A1 F1 F2 P1 b b 2 P1 Bar 2 Section at b–b, P2 area A2 P2 P2 x (a) Composite bar. (b) Section at b–b. (c) Section at a–a. R R F1 P1 P1 F2 P2 P2 (d) Section at r–r. (e) All forces. FIGURE 2-5 Analysis of a composite bar. A concept called the method of section is introduced through the composite bar. The three steps of this method are explained by considering bar 2 as an example. Step 1ÐTake a Section Take a section at any location along the length of bar 2, such as at b±b. Separate the free body as shown in Fig. 2-5b. Step 2ÐMark Forces The forces that are marked on the free body are the external load (P2) and the bar force (F2). Determinate Truss 61 Step 3ÐEstablish Equilibrium Equilibrium is established by writing the EE along the x-coordinate direction. F0 x P2 F2 0; or F2 ÀP2 2-2a In magnitude, the internal force (F2) is equal to the load (P2), but its direction is opposite to the load, or along the negative x-coordinate axis. To calculate the force in bar 1, we take a section at a±a and repeat the three steps. Step 1ÐTake a Section Take a section at a±a, and separate the portion shown in Fig. 2-5c. Step 2ÐMark Forces Mark the three forces: loads P1 and P2 and the bar force F1. Step 3ÐEstablish Equilibrium The EE along the x-coordinate direction yield P 2 P1 F1 0 or F1 À P1 P2 2-2b In magnitude, the internal force in bar 1 is equal to the sum of the two loads, but it acts along the negative x-coordinate direction. To calculate the reaction, we take a section at r±r and repeat the three steps. Step 1ÐTake a Section Take a section at r±r and separate the portion shown in Fig. 2-5d. Step 2ÐMark Forces There are three forces: loads P1 and P2 and the reaction R. Step 3ÐEstablish Equilibrium The EE along the x-coordinate direction yield R P1 P 2 0 or R À P1 P 2 2-2c In magnitude, the reaction is equal to the sum of the two loads, but it acts along the negative x-coordinate direction. The bar forces, loads, and reaction are shown with the correct directions in Fig. 2-5e. The reaction (R) is in equilibrium with the loads (P1 and P2). Both bar forces (F1 and F2) are tensile and are independent of materials and cross- sectional areas. 62 STRENGTH OF MATERIALS Force Analysis of a Tapered Bar The frustum of a cone or a tapered bar made of steel with Young's modulus Es 30,000 ksi, weight density rw 0:289 lbf/in:3 , and length ` 60 in: is suspended from a ceiling as shown in Fig. 2-6a. It has a solid cross-sectional area A. The minimum (d1) and maximum (d2) diameters of the frustum are 2 in. and 4 in., respectively. The bar is subjected to an external load P of 5 kip at its free end. Determine the internal force in the bar and the reaction at the support without neglecting the self-weight of the structure. The x-coordinate axis is selected along the bar centerline with its origin at the support. The three steps of the method of section yield the force analysis of the bar. Step 1ÐTake a Section The bar is cut at a±a at a distance x from the support. The lower portion of the bar is shown in Fig. 2-6b. The bar diameter d(x) at the location x can be calculated as 4À2 x d x 2 x2 2-3a 60 30 F 2 + x/30 2 C –x x Wf a a hc P (b) Section at a–a. 4 in. R Cross-sectional P = 5 kip area at a–a x P + Wb (a) Tapered bar. (c) Section at support. FIGURE 2-6 Analysis of a tapered bar. Determinate Truss 63 The volume (V) of the frustum of a cone with diameters d1 and d2 and height h is given by the formulas ph À 2 2 Á V d1 d 2 d1 d 2 2-3b 12 The volume of the frustum (Vf) of the cone shown in Fig. 2-6b is obtained by substituting for d1 2 x/30, d2 4, and h 6` À x. & ' p x 2 2 x Vf ` À x 2 4 4 2 12 30 30 p ` À x x2 4x or Vf 28 2-3c 12 900 15 The weight (Wf) of the frustum is obtained as the product of volume and density (r) prw ` À x x2 4x Wf rVf 28 2-3d 12 900 15 The weight acts at the centroid, which is located on the x-coordinate axis at a distance hc from the bigger base. ` À x 1 2k 3k2 hc 2-3e 4 1 k k2 d2 4 k x d1 2 30 The centroidal height (hc) given by Eq. (2-3e) is given for information but not used in our calculations. Step 2ÐMark Forces It is subjected to three forces: the external force P, weight Wf, and internal force F, as marked in Fig. 2-6b. Step 3ÐEstablish Equilibrium The equilibrium of the forces along the x-direction yields P Wf F 0 & ' À Á pr ` À x x2 4x or F À P Wf À P w 28 2-3f 12 900 15 64 STRENGTH OF MATERIALS The direction of the internal force F is opposite to the direction of load P and weight Wf. If the weight is neglected, then the internal force F and the load P become equal in magnitude but opposite in sign (F ÀP). Calculation of Reaction The three steps are repeated to calculate the reaction for a section taken at the support. Step 1ÐTake a Section Cut the bar at the support as shown in Fig. 2-6c. Step 2ÐMark Forces The section is subjected to three forces: the reaction (R), the weight of bar (Wb), and the external load P. Wb rw Vb 2-3g The volume of bar (Vb) is 60p À 2 Á Vb 2 42 2  4 439:82 in:3 2-3h 12 Wb rb Vb 439:82  0:289 127:11 lbf 2-3i Step 3ÐEstablish Equilibrium The equilibrium equation along the x-coordinate direction yields R P Wb 0 2-3j or R À P Wb À 5000 127:11 À5127:11 lbf 2-3k The direction of the reaction is along the negative x-coordinate axis. If the weight of the bar is neglected, then the reaction is R À5000 lb, and it has a 2.54 percent error. EXAMPLE 2-1: Analysis of an Octahedral Bar A regular octahedral bar made of steel is suspended from a ceiling as shown in Fig. 2-7a and b. It is 2 m long and its perimeter is 80 cm. Calculate the internal force and reaction for the following cases: 1. Load (P 10 kN) applied at the center of gravity; neglect bar weight 2. Weight only 3. Combined action of load and weight Determinate Truss 65 R 1 1 c c x F1 R 2m 3 3 P x/2 10 kN x Fsw 2 10 cm 2 x F(x) (d) Free- body (c) Free- body (a) Bar. (b) Cross- section. diagram with diagram. weight. FIGURE 2-7 Analysis of an octahedral bar. Solution The external load is applied along the bar axis, which is also considered as the x-coordinate axis. The bar is modeled by three nodes: node 1 is the support, node 2 is the free node, and node 3 is the centroid. The free-body diagram for the first load case is shown in Fig. 2-7c. Load Case 1 The reaction (R) is equal to the applied load (P). R P 10 kN a Internal Force Along the segment 0 to 1 m, the internal force (F) is equal to the applied load (P). For; 0 x 1 m; F P 10 kN b The span segment 1 to 2 m is free of internal force. For; 1m x 2 m; F0 c Load Case 2 Consider a section x from node 1. The forces are marked in Fig. 2-7d. The self-weight (Fsw) is equal to the product of the weight density of steel (rs ) and volume (V). 66 STRENGTH OF MATERIALS Fsw rs V V xA d The area of a regular polygon with n sides with length of side s is given by the formula. 1 2 180 Ap ns cot e 4 n perimeter p here; s number of side n 80 For an octagon, s 10 cm 8 A Ap8 4:828 s2 483 cm2 4:83  10À2 m2 V 4:83  10À2 x Weight density of steel: rs 77 kN/m3 Fsw rs V 77  4:83  10À2 x 3:72x kN f Total Weight (W) of the bar (x ` 2 m) W 3:72  2 7:44 kN R W 7:44 kN g The EE for the free-body diagram shown in Fig. 2-7d yields F x Fsw À R 0 F x R À Fsw F x 7:44 À 3:72x h At node 1 F x 0 7:44 kN node 2 F x 2 m 0 node 3 F x 1 m 3:72 kN Determinate Truss 67 Load Case 3 Solution to load case 3 is obtained by adding the responses of the two cases. Reaction X R P W 17:44 kN Internal Force Along the segment 0 to 1 m, the internal force (F) is For; 0 < x < 1 m; F 17:44 À 3:72x i In the span segment from 1 to 2 m, For; 1 m < x < 2 m; F 7:44 À 3:72x At node1 F x 0 17:44 equal to the reaction R node 3 F x 1 m 13:72 kN node 2 F x 2 m 0 free end j The internal force peaks at the support at F 17:44 kN. The free end has no internal force. 2.2 Stress in a Bar Member Stress is defined as the intensity of force per unit area. There are two types of stress: normal stress and shear stress. Shear stress is neglected in a bar member. Only one normal stress is induced in a bar member, and it is designated by the Greek letter sigma (s). Consider a symmetrical elastic solid subject to an external load (P) that passes along the line of symmetry, which is also considered as the x-coordinate axis, as shown in Fig. 2-8a. Take a section (a1±a2±a3±a4) in the y±z plane of symmetry. A separated portion of the solid is shown in Fig. 2-8b. The external load (P) must be balanced by an internal force (F) that is normal to the cut section. It is further assumed that F is the only internal force and that it is directed along the x-coordinate axis. The force (F) represents the sum (or resultant) of all forces in the cut section. It need not be distributed uniformly across the section, but it can vary as shown in Fig. 2-8c. Divide the section by a grid to create n number of elemental areas (ÁA1 , ÁA2 , F F F , ÁAn ). Let ÁFi represent the force on area ÁAi , as shown in Fig. 2-8d. The sum of all the distributed forces (ÁFi ) is equal to the internal force (F). n F ÁFi 2-4a i1 68 STRENGTH OF MATERIALS y a3 a4 P P x P F z a2 a1 (a) Solid subjected to a load. (b) Internal force F. ∆Ai ∆Fn σi = t ∆Ai 0 ( ∆F ) ∆A i i ∆Fi ∆Fi ∆F4 ∆F1 (c) Distribution of internal force. (d) Force on an elemental area. FIGURE 2-8 Normal stress. The direction of the force (ÁFi ) is normal to the elemental area (ÁAi ). In other words, the direction of the normal to the elemental area is also the direction of the force. Stress (si ) at location i is defined as the ratio of the force (ÁFi ) to the area (ÁAi ) in the limit as ÁAi approaches zero. ÁFi si Lt ÁAi 3 0 2-4b ÁAi In a bar member, it is assumed that the normal stress is uniform across its cross-sectional area. This assumption simplifies Eq. (2-4b) to F P s saverage 2-4c A A Determinate Truss 69 This simple formula (s F/A) is adequate for the analysis of bar and truss structures. The attributes of stress (s) follow: 1. The stress is normal to the section a1±a2±a3±a4, which lies in the y±z plane of symmetry. The stress is directed along the x-coordinate axis. In advanced solid mechanics, it is designated with the subscript x as sx that represents the x-coordinate axis. Because we deal with a single stress component, the subscript x is dropped. 2. A general solid mechanics problem has six stress components. In truss analysis, five of the six stress components are set to zero. 3. Normal stress is the intensity of internal force. It is directed along the x-coordinate axis. It is perpendicular to the cross-sectional area that lies in the y±z plane. 4. The dimension of stress is force divided by area (s F/A F/LÀ2 MLÀ1 T À2 ). Stress in USCS units is measured in pound-force per square inch (psi) or in units of ksi, which are 1000 psi (1 ksi 1000 psi). Stress in SI units is measured in pascal (Pa), which is one newton force per one square meter area. In engineering, stress is measured in units of megapascal (MPa), which is one million pascals (1 MPa 106 Pa), because one Pa is a very small stress. Conversion between the two systems to measure stress is given in Table 1-3. 5. The magnitude of stress is a big number like 20,000 psi, 138.8 million Pa, or 138.8 MPa. EXAMPLE 2-2: Stress in an Octahedral Bar Calculate stress in the octahedral bar of Example 2-1 for each of the three load cases. Solution Stress is obtained as the ratio of the internal force to the cross-sectional area (s F/A). The bar has a uniform area (A 4:83  10À2 m2 ). Load Case 1ÐStress in the Segment 0 to 1 m 10 kN s 207 kPa 4:83  10À2 m2 For, 1 m x 2m s0 Load Case 2 For, 0 x 2 m, 7:44 À 3:72x s x 154 À 77x kPa 4:83  10À2 70 STRENGTH OF MATERIALS Load Case 3 For, 0 x 1 m, s x 361 À 77x kPa For, 1 m x 2 m, s x 154 À 77x kPa Sign Convention for Stress Stress follows the t-sign convention. Consider the stress on a rectangular bar member that is subjected to a tensile load (P) as shown in Fig. 2-9a. Select an elemental section of length Á`, shown separately in Fig. 2-9b. In the right side of the block, the direction of the stress sr is along the positive x-coordinate axis, whereas in the left side, stress s` is along the negative x-coordinate axis. Both s` and sr are positive stresses because tension is induced in the block. The sign of stress is determined as the product of two factors: the first pertains to the normal (n) to the area, and the second is the direction ( f ) of stress. y x ∆ z σr σ P P ∆ (a) Stress at section a1– a2 – a3 – a4. (b) Stress: left (σ ) and right (σr ). ∆x a b Area P– P– O n=1 a b (c) Outward normal n. (d) Bar under compressive load. σ σr P– P– P– F– F– P– (e) Compressive stress. (f) Compressive internal force. FIGURE 2-9 Normal stress in a bar. Determinate Truss 71 The orientation of an area is positive (n 1) if an outward normal drawn to the area is directed along the positive coordinate axis, as shown in Fig. 2-9c. The direction of stress is positive ( f 1) when directed along the positive coordinate axis. Stress is positive when the product (nf ) is positive. Consider the stress (s` ). The orientation of the area is negative (n À1), and the stress (s` ) is directed along the negative x-axis ( f À1). The product (nf À1  À1 1) is positive, or the stress (s` ) is positive. For the stress (sr ), n 1 and f 1. The product (nf 1) is positive, or the stress (sr ) is positive. A positive stress is also called a tensile stress. Negative stress induces compression. Consider next a bar that is subjected to a load (PÀ ) applied along the negative x-coordinate axis, as shown in Fig. 2-9d. The free-body diagrams depicting stress and internal force are shown in Figs. 2-9e and 2-9f, respectively. The elemental section is subjected to a stress (sr PÀ /A). The stress is compressive because for stress sr , the orientation of the area is negative (n À1), but the stress is directed along the positive x-coordinate axis ( f 1), and their product (nf À1) is negative. The same section is subjected to negative internal force (F À PÀ ) as shown in Fig. 2-9f. Internal force also follows the t-sign conven- tion. It is easy to observe that the elemental section has a tendency to contract, inducing compressive stress. The force (F À ) is also compressive. A negative internal force is called a compressive force. 2.3 Displacement in a Bar Member A bar member in the x±y plane, with an orientation y, and nodes 1 and 2, is shown in Fig. 2-10a. Each node can displace along the x- and y-coordinate directions. At node 1, the displacement along the x-coordinate axis is u1, and it is v1 along the y-coordinate axis. Likewise, its displacements at node 2 are u2 and v2. Consider a new set of orthogonal coordinate axes x` and y` , as marked in Fig. 2-10b. In this system, the x` -coordinate axis is along the length of the bar, and the y` -axis is perpendicular to it. Resolve the displacement components (u1, v1 and u2, v2) to obtain equivalent displace- ments (ua , vt1 , ua , vt2 ) in the (x` , y` ) system. 1 2 ua u1 cos y v1 sin y 1 ua u2 cos y v2 sin y 2 2-5 vt1 Àu1 sin y v1 cos y vt2 Àu2 sin y v2 cos y The displacement components (ua and ua ) are along the bar axis (x` ), whereas the 1 2 components (vt1 and ut2 ) are along the perpendicular y` -axis. It is important to observe that the displacement components along the bar axis (ua and ua ) induce stress in the bar. No stress 1 2 72 STRENGTH OF MATERIALS y y x v2 v 2t θ u2a 2 u2 2 y v1 u 1a v 1t θ 1 u1 1 x x (b) Orientation with x-axis along the (a) Orientation of a bar member. bar length. u2a ua(x) u1a u2a u1a x x x (c) Axial displacement. (d) Linear displacement function. FIGURE 2-10 Displacement in a bar member. is induced by the two transverse components (vt1 and vt2 ). In other words, a bar has four displacement degrees of freedom, but stress is induced only by the two axial displacement components. The nodal displacements (ua and ua ) at nodes 1 and 2 are assumed to be distributed 1 2 linearly along the bar length, as shown in Fig. 2-10d. The displacement function is ` ` x` x ua x u1a 1 À u2a 2-6a ` ` The displacements at the nodes of the bar are obtained as u x` 0 u1a 2-6b ` u x ` u2a 2-6c The displacement has the dimension of length. In USCS units, it is measured in inches, and in SI units, it can be measured in centimeters or millimeters. Displacements are small quantities, in magnitude ranging from a fraction of an inch to a few inches. Determinate Truss 73 2.4 Deformation in a Bar Member Deformation (b) is a measure of relative displacement. In a bar it is a measure of the relative axial displacements at two neighboring locations in the member. Consider the locations: (1) at x and (2) at x Áx, as shown in Fig. 2-11. The deformation b is the relative displacement across the differential length (Áx). b x ua x Áx À ua x 2-7a The deformation is expressed in nodal displacements using the function given by Eq. (2-6a). & ' n x Áx x Áx x xo b x u1a 1 À u2a À u1a 1 À u2a ` ` ` ` Áx or b x u2a À u1a 2-7b ` The deformation (b) across the entire length (Áx `) of the bar is equal to the relative displacement of its nodes: b u2a À u1a 2-7c Deformation is the expansion of the bar length when it is stretched by a tensile force. It is the contraction when the bar length is shortened by a compressive force. Deformation is positive when a bar expands, and it is negative when it contracts. Bar deformation has the dimension of displacement. It has the unit of length and can be measured in inches in USCS units or in centimeters or millimeters in SI units. Appropriate use of deformation, which is sometimes overlooked in strength of materials calculations, can systematize the subject. u 1a u a(x) ∆x u 2a u a (x + ∆x) 1 2 x x (b) Displacement across elemental (a) Displacement in a bar. length ∆x. FIGURE 2-11 Deformation in a bar. 2.5 Strain in a Bar Member Strain (e) is the intensity of deformation. A bar member has one normal strain component along its axis. This normal strain designated by the Greek letter epsilon (e) is defined as b x u2a À u1a Áx u2a À u1a e x e 2-8a Áx ` Áx ` 74 STRENGTH OF MATERIALS Strain in a bar is the ratio of the relative nodal displacement to its length. For a bar member, the strain is uniform across its length. In general, strain is defined as the ratio of relative displacement over a differential length. In the limit, strain becomes the derivative of the displacement with respect to the x-coordinate: ua x Áx À ua x Áua dua e x Lt Áx 3 0 Lt Áx 3 0 2-8b Áx Áx dx The differentiation of the displacement function given by Eq. (2-6a) yields the definition of strain that is identical to Eq. (2-8a). dua b u2a À u1a e 2-8c dx ` ` Strain is a tensor quantity, and it follows the t-sign convention. For a bar, a positive strain is also called a tensile strain. It is the ratio of the elongation to the bar length. A negative strain, also called a compressive strain, corresponds to the ratio of the contraction to the length. Strain is a dimensionless quantity. Its magnitude is small and ranges between 0 and 0.2 percent (0 and 0.002). Stress and strain are related through Hooke's law, as discussed in Chapter 1. If a stress (s) is known, then the strain (e) can be calculated as the ratio of the stress (s) to Young's modulus (E) as e s/E and vice versa (s Ee). Hooke's law is a cause-and-effect class of relation. Stress can be the cause and strain the effect, and vice versa. EXAMPLE 2-3: Strain in an Octahedral Bar Calculate strain and deformation in the octahedral bar of Example 2-1 for each of the three load cases. Use Young's modulus for steel (E 200 GPa). Load Case 1ÐStrain in the Segment 0 to 1 m s 207 kPa e 1:04  10À6 E 200 GPa For, 1 m x 2m e0 Load Case 2 For, 0 x 2 m, s x 154 À 77x kPa e x E 200 GPa 0:77 À 0:39x  10À6 Determinate Truss 75 Load Case 3 For, 0 x 1 m, 361 À 77x e x  10À6 200 1:8 À 0:39x  10À6 For, 1 x 2 m, 153:94 À 76:97x e x  10À6 200 0:77 À 0:39x  10À6 2.6 Definition of a Truss Problem The information developed in the earlier sections is used to define and solve a truss problem. The problem is defined by the truss configuration (which is specified by the coordinates of its nodes given in Table 2-1), support conditions, member properties, and external loads. This information, also referred to as input data, must be available prior to the beginning of analysis. The input data are illustrated by considering the example of a four-node truss with five bars as shown in Fig. 2-12. EXAMPLE 2-4: Five-Bar Truss Coordinates of the Nodes The geometry of the truss is specified by the x- and y-coordinates of its nodes. The coordinates of the nodes with the origin chosen at node 1 for the five-bar truss are marked in Fig. 2-12a. The coordinates can be used to plot the geometry of the truss, shown in Fig. 2-12a. It is a plane truss because all nodes lie in the x±y plane. The vertical distance (along the y-axis) between the support nodes (1 and 4) is referred to as the height (h) of the truss. Its span (s) is the horizontal distance measured along the x-coordinate axis between nodes 1 and 2. The span and height provide an overall size of the truss. It is a square truss because its height and span are equal (h s 100 in:). The inclination of the diagonal bar is 45 . Support Conditions A truss must be supported to prevent its motion as a rigid body, such as an automobile. Loads from the truss are transferred to the foundation through the supports. The truss in Fig. 2-12a is supported at nodes 1 and 4. It is hinged to a foundation at node 4. A hinge support restrains movement along both the x- and y-coordinate directions. Node 1 is on a roller support, which allows movement along the y-coordinate direction, but displacement is restrained along the x-coordinate direction. A reaction is induced at the support node along the direction of restraint. The truss has two reactions (R4x and R4y) at node 4 and one reaction (R1x) at node 1, as shown in Fig. 2-12b. 76 STRENGTH OF MATERIALS y x P3y = –1 R4y v3 4 (0, 100) 3 F4 u3 4 (100, 100) 45° R4x 5 2 3 P2y = –5 F5 F2 F3 v1 v2 u2 P2x = 2 1 R1x F1 1 (0, 0) 2 (100, 0) (a) Truss parameters. (b) Response variables. v1 F5 F1 R 1x 1 (c) Free- body diagram at node 1. FIGURE 2-12 Five-bar truss. TABLE 2-1 Coordinates and Loads for Truss in Fig. 2-12 Node x-Coordinate, y-Coordinate, Load along Load along Number In. In. x-Coordinate, kip y-Coordinate, kip 1 0 0 ± ± 2 100 0 2 À5 3 100 100 ± À1 4 0 100 ± ± Member Properties A truss bar connects two nodes. Bar 1 connects node 1 and 2, bar 2 connects nodes 4 and 2, bar 3 connects nodes 3 and 2; bar 4 connects nodes 4 and 3, and bar 5 connects nodes 4 and 1. The properties of a bar required for analysis are its length, size, and material. The length is calculated from the nodal coordinates, and this need not be given again. The size of the bar is specified by its cross-sectional area (A). For the truss, the areas of the horizontal bars (1 and 4) are considered to be equal to 2 in.2 (A1 A4 2 in:2 ). The areas of the other three bars are Determinate Truss 77 considered to be equal to 1 in.2 (A2 A3 A5 1 in:2 ). A bar can be made of steel, aluminum, wood, or any other type of structural material. Truss analysis requires only the Young's modulus (E) of the material. The entire truss can be made of a single material, steel for example, or different bars can be made of different materials. The nature of the material does not increase the complexity of analyses. The bars of the truss in Fig. 2-12 are made of steel with Young's modulus E 30,000 ksi. External Loads A truss can support loads at its nodes. Load at a node can be applied along the x- and y-coordinate directions as listed in Table 2-1. Load is also referred to as applied force, or external force. It is designated by the letter P. Node 2 of the truss in Fig. 2-12a is subjected to a load (P2x 2 kip) applied along the x-coordinate direction and a load (P2y À5 kip) directed along the negative y-coordinate direction. Node 3 is subjected to a load (P3y À1 kip) applied along the negative y-coordinate direction. A load at a support node should not be applied along the direction of restraint because such a load is directly transmitted to the foundation. Truss analysis requires four pieces of information: 1. Coordinates of the nodes 2. Support conditions 3. Member properties 4. External loads The objective of the analysis is to determine the response of the truss consisting of the following six variables: (GS±1) Bar force (GS±2) Bar stress (GS±3) Reaction (GN±1) Bar strain (GN±2) Bar deformation (GN-3) Nodal displacement The variables are separated into a stress group (GS), which includes the force, stress, and reaction; and a strain group (GN), which contains the strain, deformation, and displacement. For a determinate truss, it is straightforward to determine the variables of the stress group first and then back-calculate the variables of the strain group. For the five-bar truss, the (GS±1 and GN±3) response variables are marked in Fig. 2-12b. 1. Bar forces: The truss has five bar forces (F1, F2, F3, F4, F5). To begin analysis, one should consider the bar forces, which are not yet known, to be positive and marked by arrowheads that point at each other. 2. Reactions: The truss has three restraints, one at node 1 and two at node 4. It develops three reactions along the direction of restraints, consisting of R1x at node 1 and R4x and R4y at node 4. Mark all the reactions to be positive and directed along the coordinate axis because these are also unknowns. 78 STRENGTH OF MATERIALS 3. Loads: Mark the specified external loads. It is preferable to mark the load directions along the positive coordinate axis. A negative load is specified with a negative value, as shown in Fig. 2-12a. The bar stress, strain, deformation, and nodal displacements are back-calculated from the forces. The equilibrium equations yield the bar forces and reactions of a determinate truss. We will formally define a determinate truss later in this chapter. For the time being, assume that the five-bar truss shown in Fig. 2-12 is a determinate truss. Equilibrium Equations (EE) Consider a general determinate truss with m nodes and n bars. Two equilibrium equations can be written at each node as the summation of the forces along the x- and y-coordinate directions. xF 0 and yF 0 i 1; 2; F F F ; m nodes 2-9 i i There are 2m EE for a truss with m modes. In truss analysis, the 2m equations are written in two stages. In the first stage, the EE are written at the nodes that can displace but not along the restrained directions. Solution of these EE yields the bar forces. In the second stage, the EE are written at the support nodes along the direction of restraints. Solution of these equations yields the reactions. EE for the Five-Bar Truss The five-bar truss has five displacement components (v1, u2, v2, u3, v3), as marked in Fig. 2-12b. Node 1 can displace along the y-coordinate direction by an amount v1, as marked at node 1. Likewise, mark the displacements u2 and v2 at node 2 and u3 and v3 at node 3. The fully restrained node 4 has no displacement. Five EE can be written along the five displacement directions. EE at Node 1 One EE can be written along the displacement direction v1. To write this EE, separate a free body for node 1 as shown in Fig. 2-12c. The summation of forces along the displacement direction v1 yields the EE: EE along v1 X F5 0 2-10a The bar force is zero (F5 0) because this is the only force in the v1 direction. In truss analysis, it is not necessary to separate a free-body diagram to write the EE. The same equation (2-10a) is generated by writing the EE at node 1 along the v1 direction shown in Fig. 2-12b. Determinate Truss 79 EE at Node 2 This node has two displacements (u2 and v2), and one EE can be written along each direction. EE along u2 X P2x À F1 À F2 cos 45 0 F2 F1 p 2 2-10b 2 EE along v2 X P2y F2 sin 45 F3 0 ÀF2 p À F3 À5 2-10c 2 The two EE containing three unknown forces cannot be solved for F1, F2, and F3. EE at Node 3 This node has two displacements (u3 and v3), and the EE are EE along u3 X F4 0 2-10d EE along v3 X P3y À F3 0 or F3 À1 2-10e Solution to the five EE Eqs. (2-10a) to (2-10e) yields the five forces: F1 À4 kip p F2 6 2 kip F3 À1 kip F4 0 F5 0 2-10f Equilibrium Equations in Matrix Notation It is advantageous to write the EE using matrix notation. The reader should become familiar with matrix analysis because the structural analysis computer codes used by engineers in industry adopt this notation. The five equilibrium equations, Eqs. (2-10a) to (2-10e), written in matrix notation (following the rules of algebra given in Appendix 1) have the following form: P QV W V W u2 1 À p 1 À0 À0 À0 b F1 b b 2 b 2 b b b b b b b b v2 T 0 T À p 1 2 À1 À0 À0 Ub F2 b b À5 b U` a ` a v3 T 0 T À0 À1 À0 À0 Ub F3 b b À1 b U 2-11a u3 R 0 À0 À0 À1 À0 Sb F4 b b 0 b b b b b b b X Y X b b Y v1 0 À0 À0 À0 À1 F5 0 80 STRENGTH OF MATERIALS In the matrix equation (2-11a), the EE have been rearranged to obtain an upper triangular diagonal matrix. Solution of the matrix equation (2-11a) yields the bar forces. V W V W b F1 b b b b b À4 b p b b b b b b b b ` F2 a ` 6 2b a F3 À1 2-11b b b b F4 b b b b b b b b b b 0b b b X Y X Y F5 kip À0 Equilibrium equations given by Eq. (2-11a) in matrix notation can be written as BfF g fPg 2-12a The coefficient matrix [B] is called the equilibrium matrix. It is a square matrix of dimension n, here n 5. For a square matrix, the number of columns is equal to the number of rows. The unknown {F} is a (n 5) component force vector. The known load {P} is also a (n 5) component vector. The matrix Eq. (2-12a) is inverted to obtain the force vector. fF g BÀ1 fPg 2-12b The inversion of a matrix is discussed in Appendix 1. Remarks on the Matrix Equation A row of the equilibrium matrix [B] represents an equation written along a displacement direction. For example, the first row in Eq. (2-11a) is the EE written along the displacement direction u2. Likewise, the third row is the EE written along the displacement direction v3 and so forth. We will follow a sign convention for EE. In the EE ([B]{F} {P}), the load vector {P} must be aligned along the positive coordinate direction. For example, the first load component must be directed along u2 that is, along the positive x-coordinate direction at node 2. Likewise, the third load component must be directed along v3 that is, along the positive y-coordinate direction at node 3, and so forth. The rows of the EE that included the load can be shuffled without any consequence. For example, the equation along the displacement direction (u2) can be written first, and the EE along the direction v1 can be written last, as given in Eq. (2-11a). The reshuffled pattern shown in Eq. (2-11a), with zeros below the diagonal is called an upper triangular matrix. An upper triangular matrix equation is easily solved. Solve the last equation containing one unknown to calculate the value of the last variable; here, Eq. (2-11a) (the fifth equation) is solved to obtain F5 0. Next proceed to the last but one equation; here, Eq. (2-11a) (the fourth equation). This equation has two unknowns, but one force (F5) is known. Calculate the value of the remaining unknown (F4). Repeat the process to solve all the equations. The solution to a triangular system of equations can be obtained by calculating the value of one unknown at a time. This process is called back-substitution. Equilibrium equations of many determinate trusses can be cast into a triangular form by shuffling the nodal EE, and such equations are trivially solved. Determinate Truss 81 Calculation of Reactions The reactions are also determined from an application of the equilibrium equations, which are written at the support nodes along the direction of the restraint. Take, for example, the five-bar truss. Its three reactions can be obtained from three EE written along R1x at node 1, and R4x and R4y at node 4. The EE at support node 1 along the direction of restraint, which is the positive x-coordinate direction, yields EE along R1x X R1x F1 0 or R1x ÀF1 4 2-13a Consider next the support node 4, which is restrained in both the x- and y-coordinate directions and has two reactions (R4x and R4y). The EE along the two directions yields two equations. 1 EE along R4x X R4x F4 p F2 0 2 or R4x À6 2-13b 1 EE along R4y X R4y À F5 À p F2 0 2 or R4y 6 2-13c The reactions are: R1x 4 R4x À6 2-13d R4y 6 The three equations, (2-13a) to (2-13c), can be written in matrix notation as V W P Qb F1 b V b b W À1 ÀÀ 0 À0 À0 À0 b F2 b ` R1x a b b ` a T À0 À p 1 À0 À1 À0 U F3 R4x R 2 S 2-14a b b X Y À0 À p 1 À0 À0 À1 b F4 b b b b b R4y 2 X Y F5 This equation can be abridged to obtain BR fF g fRg 2-14b 82 STRENGTH OF MATERIALS The equilibrium matrix [BR] for the reactions is a rectangular matrix. It has as many equations, or rows, as the number of reactions, say nR. It can have a maximum of n 5 columns. The dimension of the matrix [BR] is nR  n. The reaction matrix of the five-bar truss has three rows (nR 3) and five columns (n 5). The equilibrium matrix [BR] will be utilized in the analysis of the settling of support. Equilibrium of Reaction and Load The overall equilibrium, that is the equilibrium of an entire structure, requires the balance of all three types of forces: the internal force (F), the reactions (R), and the load (P). However, the internal force is already in equilibrium by itself. In a truss, for example, the internal force in a bar is in self-equilibrium because it is represented by a pair of collinear arrowheads with opposite directions. It is easily observed in Fig. 2-12b that the forces in all five bars are in equilibrium. Therefore, the reactions must be in equilibrium only with the external loads, and such EE we will refer to as the external equilibrium equations. The external equilibrium in two dimensions is satisfied by three equations, which consist of two force balance EE and one moment balance EE. Force equilibrium along the x-direction R P 0 2-15a x Along the y-direction R P 0 2-15b y Moment equilibrium in the x±y plane or along the z-direction M R M P 0 2-15c z Here, R and P represent reactions and loads, respectively. MR and MP indicate moments due to reaction and load, respectively. The reactions are correct provided the three EE [Eqs. (2-15a) to (2-15c)] are satisfied. For a determinate structure, the reactions can be calculated from Eq. (2-15) without determining the internal forces. For the five-bar truss, the equilibrium of the reactions and loads is easily verified. R P 0X R4x R1x P2x 0 3 À6 4 2 0 x R P 0X R4y P2y P3y 0 3 6 À 5 À 1 0 y MR MP 0X hR1x hP2x hP2y hP3y 0 3 h 4 2 À 5 À 1 0 node 4 Determinate Truss 83 The external EE can also be written as R4x R1x ÀP2x À2 R4y À P2y P3y 6 100R1x À100 P2x P2y P3y 400 The solution of the external EE yields the reaction R1x 4 R4x À6 R4y 6 The reactions calculated from the external EE are in agreement with Eq. (2-13d). Bar Stress The stress in a bar is obtained as the ratio of the force to the area (s F/A). The five bar stresses are s1 F1 =A1 À4=2 À2 ksi compression p p s2 F2 =A2 6 2=1 6 2 ksi tension s3 F3 =A3 À1=1 À1:0 ksi compression s4 F4 =A4 0 s5 F5 =A4 0 2-16 Bars 1 and 3 are in compression. Bar 2 is in tension. Bar 4 and bar 5 are stress-free. Bar Strain The strain in a bar is obtained from Hooke's law as the ratio of the stress to Young's modulus (e s/E). The five bar strains are s1 2 e1 À À6:67  10À5 À6:67  10À3 percent E 30;000 p s2 6 2 e2 À 28:28  10À5 28:28  10À3 percent E 30;000 s3 1 e3 À À3:34  10À5 À3:34  10À3 percent E 30;000 s4 e4 0 E s5 e5 0 2-17 E 84 STRENGTH OF MATERIALS Bars 1 and 3 are in compression. Bar 2 is in tension. Bar 4 and bar 5 are not strained. Bar Deformation The deformation in a bar is obtained as the product of the strain and its length (b e`). The length of a bar is calculated from its nodal coordinates. Consider a bar connecting two nodes (m1 and m2). Let the x- and y-coordinates of the nodes be xm1 and ym1, and xm2 and ym2, respectively. The length of the bar is calculated from the formula q ` xm2 À xm1 2 ym2 À ym1 2 2-18a The lengths of the five bars are calculated as q `1 100 À 02 0 À 02 100 in: q p `2 100 À 02 0 À 1002 100 2 in: q `3 100 À 1002 0 À 1002 100 in: q `4 100 À 02 100 À 1002 100 in: q `5 0 À 02 100 À 02 100 in: 2-18b The deformation of the bars are as follows: b1 e1 `1 À6:67  10À5  100 À6:67  10À3 in: p b2 e2 `2 28:3  10À5  100 2 40  10À3 in: b3 e3 `3 À3:34  100  10À5 À3:34  10À3 in: b4 e4 `4 0 in: b5 e5 `5 0 in: 2-18c Bars 1 and 3 contract while bar 2 expands. Bars 4 and 5 are not deformed. 2.7 Nodal Displacement The determination of nodal displacement is more involved than the back-calculation of stress, strain, and deformation. We discuss three methods to calculate the displacement. Method 1ÐDeformation Displacement Relation (DDR) In this method, the nodal displacements are calculated from member deformations. Consider the five-bar truss as an example. Mark the deformations {b} and displacements {X} as shown in Fig. 2-13. The deformations should be marked with arrows pointing at each other Determinate Truss 85 y v3 4 u3 2 5 3 v1 v2 u2 1 x FIGURE 2-13 Deformations and displacements. because b follows the sign convention for internal force. Deformation, by definition, is the relative axial displacement (b u2a À u1a ). For the truss, the five deformations are expressed in terms of the five nodal displacements by observation of Fig. 2-13: b1 u2 1 b2 u2 cos 45 À v2 cos 45 p u2 À v2 2 b3 v3 À v2 b4 u3 b5 Àv1 2-19 Equation (2-19) in matrix notation can be written as V W P QV W b b1 b b b 1 0 0 0 0 b u2 b b b b b2 b T p À p b b 0 0 Ub v2 b b b ` a T 12 1 2 0 U` a T0 b3 T À1 1 0 0U U v3 2-20a b b R bb b b b b 4b b b 0 0 0 1 0 Sb u3 b b b b b X Y X Y b5 0 0 0 0 À1 v1 or fbg DfX g 2-20b Here, V W b u2 b b b b b b b ` v2 a f X g v3 b b b u3 b b b b b X Y v1 and the coefficient matrix in Eq. (2-20a) is [D]. 86 STRENGTH OF MATERIALS Equation (2-20), which relates deformation to displacement, is called the deformation displacement relation (DDR). Compare the matrix [D] in the DDR to the equilibrium matrix [B] in Eq. (2-11a). Matrices [B] and [D] are the transpose of each other ([D] [B]T ). In EE [Eq. (2-11a)], [B] is an upper triangular matrix, whereas it is a lower triangular matrix in the deformation displacement relations, given by Eq. (2-20a). The deformation displacement relation can be written using the transpose of the equilibrium matrix [B] as fbg BT fX g 2-21a À ÁÀ1 fXg BT BÀT fbg 2-21b The triangular matrix Eq. (2-21) is easily solved by back-substitution to obtain the displacements from known deformations. The nodal displacements for the five-bar truss are obtained as the solution to Eq. (2-19) [or Eq. (2-20a)] using deformations from Eq. (2-18c). b1 u2 À6:67  10À3 u2 v2 b2 p À p 40:0  10À3 2 2 b3 Àv2 v3 À3:34  10À3 b4 u3 0 b5 v 1 0 The nodal displacements are obtained as a solution to the DDR. u1 0 boundary condition v1 0 u2 À6:7  10À3 in: v2 À63:24  10À3 in: u3 0 v3 À66:57  10À3 in: u4 0 boundary condition v4 0 boundary condition 2-22 The maximum displacement occurs at node 3 along the negative y-coordinate direction (v3 À66:57  10À3 in:). Calculation of displacement from the DDR ({b} [B]T {X}) is straightforward. This should be the method of choice. Method 2ÐGraphical Determination of Displacement Nodal displacements of a truss can be determined graphically. This technique is credited to the French engineer J. V. Williot (1843±1907). It is based on the motion or displacements of Determinate Truss 87 y a4 4 4' 3 4 3' a3 r a3 a4 5 2 3 2' a1 a2 ∆ ∆n 1 ∆ ∆a 1 2 1 1' 2 x a2 r 2' a1 (a) Bar. (b) Williot's diagram. FIGURE 2-14 Graphical determination of displacement. the nodes of a bar, as depicted in Fig. 2-14. It is assumed that the position of node 1 in Fig. 2-14a is known. It has either zero displacement (u1 v1 0) or known displacement (u1 and v1). With respect to node 1, node 2 can move along the bar axis by Áa and perpendicular to it by Án. In other words, the bar rotates about node 1 by Án with the radius ` Áa, and the process displaces node 2 to the new location 2H . The displaced bar occupies the position 1±2H , and the vector (Á) between positions 2 and 2H is the displacement vector of node 2. The magnitude of axial elongation Áa is known because it equals the deformation of the bar (Áa b). The magnitude of the transverse displacement Án is not known, but it lies along the arc (rÀr) drawn with its center at node 1 and the radius R ` Áa. The magnitude of Án will be determined from subsequent construction. Repetition of this simple concept provides the displacement diagram of a truss. This technique is accurate for the small displacement theory. Williot's technique for the five-bar truss is depicted in Fig. 2-14b. The diagram is constructed in the following steps. Step 1ÐSelect a Node (i) That Has a Zero or Known Displacement Node 4 is selected because it is a boundary node with zero displacement (u4 v4 0). Step 2ÐSelect a Bar That Connects Node (i = 4) Bar 5 with nodes 4 and 1 is selected. Bar 5 is selected instead of bar 2 or bar 4 because the other node of the bar (node 1) has a simple motion. It is constrained to move only in the y- coordinate direction. Node 1 has no displacement because the deformation b5 0. In other words, nodes 1 and 4 retain their original position even after the application of load. 88 STRENGTH OF MATERIALS Step 3ÐLocate the Displacement of Another Node (i + 1 = 2). Bar 1, connecting nodes 1 and 2, contracts by the deformation amount (b1 À6:67  10À3 in:). Draw an arc (a1 Àa1 ) with radius r `1 À b1 and center at node 1. The displacement of node 2 must lie in the arc a1 Àa1 . The displaced position of node 2 can be located by considering the deformation of bar 2H . Bar 2 expands by the amount b2 40  10À3 in. Draw an arc (a2 Àa2 ) with radius r `2 b2 and center at node 4. The displacement of node 2 must lie in the arc a2 Àa2 . The intersection of the arcs (a1 Àa1 and a2 Àa2 ) locates the position of node 2H after deformation. Step 4ÐLocate the Displacement of the Next Node (i + 2 = 3) Bar 4 has no axial deformation (b4 0). Draw an arc (a4 Àa4 ) with radius r `4 and center at node 4. The displacement of node 4 must lie in this arc. Bar 3 contracts by the deformation amount (b3 À3:34  10À3 in:). Draw an arc (a3 Àa3 ) with radius r `3 À b3 and center at the deformed node 2H . The displacement of node 3 must lie in this arc. The intersection of the arcs (a4 Àa4 and a3 Àa3 ) locates the position of node 3H after deformation. Step 5ÐConnect the Deformed Nodes Connect the deformed nodes (1H À2H ), (2H À3H ), (3H À4H ), (4H À2H ), and (4H À1H ) to obtain the deformed configuration of the truss. The nodal displacements can be measured from the Williot's diagram. Displacements at Node 3 Displacement components (u3 and v3) at node 3 are obtained by projecting Á 3À3H onto the x- and y-axes. u3 Áx3 projection of vector 3À3H along the x-axis 0 in: v3 Áy3 projection of vector 3À3H along the y-axis À66:57  10À3 in: 2-23a Displacements at Node 2 u2 Áx2 projection of vector 2À2H along the x-axis À6:7  10À3 in: v2 Áy2 projection of vector 2À2H along the y-axis À63:24  10À3 in: 2-23b Nodes 1 and 4 have no displacements. The graphical method can degrade the accuracy of the displacements. Method 3ÐDisplacement Calculation from Energy Principle Calculating displacements from the energy principle is a popular approach. This approach is based on energy and work concepts. Since both concepts are discussed in Chapter 12, here we provide a cursory account for trusses. The reader should refer to Chapter 12 for more information. Strain Energy Energy is stored in a bar when it deforms. The stored energy is called strain or deformation energy. An undeformed bar has no strain energy stored in it. Strain energy (U) is defined in Determinate Truss 89 terms of its density (ud), which is the strain energy per unit volume (V). Strain energy and its density are defined as Ubar ud dv 2-24a e ud sde 2-24b 0 For a bar member, the strain energy expression simplifies because the stress (s F/A), strain (e s/E F/AE), and internal force (F) are uniform across the volume (V A`) of the bar with length ` and area A. Strain energy expressed in stress, strain, and internal force has the following form: e 1 1 s2 ud eEde Ee2 2-24c 0 2 2E 2 1 1F ` Ubar Ee2 A` 2-24d 2 2 AE The strain energy can be written in terms of force and deformation (b e`) as 1 1 Ubar EeA e` Fb 2-24e 2 2 Strain energy stored in a truss with n bars is obtained by adding the individual contribu- tions from the n bars. 2 1 n F i `i 1 n `F Utruss Fi  2-24f 2 i1 Ai Ei 2 i1 AE i The strain energy equation expressed in terms of the force and deformation has the following form: 1 n 1 Utruss Fi bi fFgT fbg 2-24g 2 i1 2 The matrix term ({F}T {b}) is the dot product of the force and deformation vectors. Work Work is done when a load P moves through a displacement X. If both P and X are assumed to be collinear acting in the same direction, and displacement is proportional to load, then work (W) is defined as X 1 W PdX PX 2-25a 0 2 90 STRENGTH OF MATERIALS For a truss the definition of work is generalized for m loads (P1 , P2 , F F F , Pm ) that move through m displacements (X1 , X2 , F F F , Xm ), respectively. It is obtained as the sum of individual contributions for the m-component load and displacement: 1 m 1 Wtruss Pi Xi fPgT fXg 2-25b 2 i1 2 Work-Energy Conservation Theorem According to this theorem, the strain energy stored in the structure is equal to the work done by the external load. UW 2-26a 1 1 fF gT fbg fPgT fX g 2-26b 2 2 or fF gT fbg fPgT f X g 2-26c In other words, the work done by the load is stored as the internal energy in the bars of the truss. EXAMPLE 2-5 For the five-bar truss shown in Fig. 2-12a, calculate the strain energy and the work, and illustrate the conservation theorem. Solution Strain energy Utruss is calculated from Eq. (2-24g) as 1 Utruss F1 b1 F2 b2 F3 b3 F4 b4 F5 b5 2 1n À Á p À Áo À4  À6:67  10À3 6 2  40  10À3 À 1  À34  10À3 2 1 f26:68 339:41 3:4g  10À3 185  10À3 in:-k 2-27a 2 Work done by loads is calculated from Eq. (2-25b) Determinate Truss 91 1È É Wtruss P2x u2 P2y v2 P3y v3 2 1È À Á À Á À ÁÉ 2  À6:67  10À3 À 5  À63:24  10À3 À 1  À66:57  10À3 2 1 À13:34 316:20 66:57  10À3 185  10À3 in:-k 2-27b 2 The conservation theorem is satisfied because the strain energy is equal to the work (Utruss Wtruss 185 in:-k). Deformation Displacement Relation The conservation theorem is used to derive the deformation displacement relation ({b} [B]T {X}) that was obtained earlier from the geometrical consideration for the five- bar truss; see Eq. (2-21a). In this derivation, we utilize matrix notation for the equilibrium equation ([B]{F} {P}), which is Eq. (2-12a) and the transpose of the EE ({F}T [B]T {P}T ). The conservation theorem (U W) can be written as 1 1 fFgT fbg fPgT fXg 2-28a 2 2 Eliminate load {P} in favor of the internal force {F} by using the EE ([B]{F} {P}) to obtain 1 1 fFgT fbg fFgT BT fXg 2 2 or 1 À Á fFgT fbg À BT fXg 0 2 Because the force vector is not a null vector ({F} T 0) fbg BT fXg 2-28b For a determinate structure, the EE matrix [B] is a square matrix, and Eq. (2-28b) can be inverted to obtain the displacement. As we have seen for the five-bar truss, the determination of displacement does not require a formal inversion of the EE matrix because most often it is a triangular matrix. Force Deformation Relations (FDR) The force (F) in a bar is related to its deformation (b) through the force deformation relations (FDR). Deformation (bi ) for the ith member of a truss can be written as s` F` ` bi e`i Fi gi Fi 2-29a E i AE i AE i 92 STRENGTH OF MATERIALS À `Á Here, gi AE i is called the flexibility coefficient for the ith bar. The bar deformation is equal to its flexibility coefficient (bi gi ) for a unit value of the bar force (Fi 1). For an n bar truss, the deformation force relationship can be written as b1 g1 F1 b2 g2 F2 F F F bn gn Fn 2-29b In matrix notation, Eq. (2-29b) can be written as fbg GfFg 2-29c The coefficient matrix [G] is called the flexibility matrix. It is a diagonal matrix. The diagonal coefficient (gii) is defined as ` gii i À 1; 2; F F F ; n 2-29d AE i The flexibility matrix of a truss with n bars can be written as P Q g11 T g22 0 U T U T gii U G T U 2-29e T FF U R 0 F S gnn Elimination of deformations between the deformation displacement relation ({b} [G]{F}) and the deformation displacement relation {b} [B]T {X} yields the force displacement relations. fbg GfFg BT fXg fXg BÀT GfFg 2-30 Displacement can be back-calculated from the force using the force displacement rela- tionship given by Eq. (2-30). The notation [B]ÀT represents the inverse of the transpose of the matrix [B], [B]ÀT [[B]T ]À1 . Castigliano's Second Theorem Displacement can also be calculated using Castigliano's second theorem, discussed in Chapter 12. A cursory account of the method is given here for completeness. According Determinate Truss 93 to the theorem, displacement is equal to the derivative of strain energy with respect to load as qU Xj j 1; 2; F F F ; m 2-31a qPj The displacement (Xj) and load (Pj) act at the same location or node of a truss, and both Xj and Pj are collinear, acting in the same direction. The derivative (qU/qPj ) can be expanded by using the chain rule of differentiation. qU qU qFi n 2-31b qPj i1 qFi qPj Here, Fi is the force in the ith bar, and the strain energy stored in it is Ui. Strain energy in terms of internal force can be written as n ` n U Ui F2 2-31c i1 i1 2AE i i qU ` Fi bi i 1; 2; F F F ; n 2-31d qFi AE i The partial derivative of the strain energy with respect to bar force is equal to the member deformation. The derivative (qFi /qPj ) is calculated from the equilibrium equation BfFg fPg 2-32 The EE matrix [B] is independent of load (P), and the EE can be differentiated with respect to Pj to obtain & ' & ' q qF qP È É BfFg fPg B Nj 2-33a qPj qPj qPj {Nj} is a unit vector with unity at the jth location and zero at every other location. For example, {N1} and {N4} for a five-component (m 5) load vector are V W V W b1b b b b0b b b b b b b b b b b b0b b b b0b b b ` a ` a fN1 g 0 and fN4 g 0 2-33b b b b b b b b b b b b0b b b b1b b b b b b b b b X Y X Y 0 0 94 STRENGTH OF MATERIALS For a determinate truss, the square equilibrium matrix [B] can be inverted to obtain (qF/qPj ) as & ' qF È É È É BÀ1 Nj jth column of BÀ1 f j 2-33c qPj The vector {f j} represents the n-component internal force vector of a truss that is subjected to a unit load {Nj} defined in Eq. (2-33b). A component of the vector {f j} is designated by ( fi j ). The vector {f j} is also called the influence coefficient vector. Substitution of Eqs. (2-31d) and (2-33c) into Eq. (2-31b) yields the displacement formula. ` n Xj Fi f j j 1; 2; F F F m 2-34 i1 AE i i For a truss with m displacement unknowns, Eq. (2-34) has to be repeated m times. Mathematically, it can be proven that the formula for displacement given by Eq. (2-34) is identical to the force displacement relations given by Eq. (2-30). Calculation of displacement using Eq. (2-34) is illustrated considering the five-bar truss shown in Fig. 2-12a as an example. Let us calculate the displacement u2. The first EE given by Eq. (2-11a) corresponds to the displacement (u2 X1 ). The influence coefficient vector {f 1} is obtained as the solution to the following equation: È É B f 1 fN1 g 2-35a or P QV 1 W V W 1 p 1 0 0 0 b f1 b b 1 b b b b b T 2 Ub 1 b b b b b b b T0 À p 1 À1 0 Ub f b b 0 b 0 Ub 2 b b b b b b b T b b b b T 2 U` 1 a ` a T0 0 1 0 U f 0 Ub 3 b b 0 b 2-35b T T Ub b b b b 1b b b T0 R 0 0 1 0 Ub f4 b b 0 b Sb b b b b b b b b b b b b b b b X 1Y X Y 0 0 0 0 1 f5 0 The solution to Eq. (2-35b) yields V 1W V W b f1 b b 1 b b b b b b b b b bf1 b b0b b b b b b 2b b b b b b b ` a ` a f1 0 2-35c b 3b b b b 1b b b bf b b0b b 4b b b b b b b b b b b b b b b X 1Y X Y f5 0 Determinate Truss 95 ` 5 ` 1 u2 F1 f1 Fi f 1 2-35d AE 1 i2 AE i i 100 u2  À4 1 À6:67  10À3 in: 2-35e 2  30,000 Displacement v2 corresponds to the second EE in Eq. (2-11a). The vector {f2} is obtained as the solution to Eq. (2-35b) for the right side vector {N2}. V 2W V W b f1 b b 1b b b b p b b b b b 2b b b b bf b bÀ 2b b b b b 2b b b b b b b ` a ` b a 2 0 f3 b b b b b b b b bf2 b b b b b b b 4b b b b b 0bb b b b b b b b b b X 2Y X Y f5 0 p ` 2 ` 2 100 100 2 À pÁ v2 F1 f1 F2 f2  À4 1  8:485 À 2 AE 1 AE 2 2  30;000 1  30;000 v2 À63:24  10À3 in: 2-35f Likewise, displacement v3 is obtained: V 3W V W b f1 b b 1 b b b p b b b b b b bf3 b bÀ 2b b b b b b 2b b b b b b b ` a ` a 3 1 f3 b b b b 3b b b b bf b b b 4b b 0bb b b b b b b b b b b b X 3Y X b Y f5 0 ` 3 ` 3 ` v3 F1 f 1 F2 f2 3 F3 f3 À66:57  10À3 in: 2-35g AE 1 AE 2 AE 3 The displacements u3 0 and v1 0 because the product ( fk Fk ) is zero. The displace- ments given by Eqs. (2-34e), (2-34f ), and (2-34g) are in agreement with the same given by Eq. (2-22). 2.8 Initial Deformation in a Determinate Truss A determinate truss can experience deformation because of the change in the temperature and the settling of its support. The temperature is assumed to be ambient at the time the truss was manufactured and at that time it was free from initial deformation. Let the temperature 96 STRENGTH OF MATERIALS increase over the ambient be (ÁT). The problem is to calculate the response of the truss due to the change in the temperature. Likewise, assume that the original foundation of the structure was firm and free from initial deformation. After assembly, one or more of the truss supports may settle by small amounts ÁX relative to the firm foundation. The problem is to determine the response of the truss due to settling of its supports. Problems with temperature changes and support settling are referred to as initial deform- ation problems. We begin their solution by adjusting Hooke's law to include initial deform- ation in the strain. Hooke's law relates stress and the elastic strain. Actual strain in the structure is referred to as the total strain (et ), and it is the sum of two components: the elastic strain (ee ) and initial strain (e0 ). et ee e0 2-36a Only the elastic strain (ee ) induces stress. The stress-strain Hooke's law can be written as s Eee 2-36b or À Á s E et À e0 2-36c Initial strain due to change in temperature (ÁT) is the product of the coefficient of thermal expansion (a) and ÁT. e0 aÁT 2-36d Hooke's law for a problem with thermal strain can be written as s E et À aÁT 2-36e Stress induced because of a change in the temperature is illustrated in Example 2-6. EXAMPLE 2-6 Consider a uniform steel bar of cross-sectional area A 1 in:2 and length ` 100 in: supported between two rigid walls, as shown in Fig. 2-15. Its temperature is changed by ÁT 50 C. Calculate the response of the bar. For steel, the Young's modulus is E 30,000 ksi and the coefficient of thermal expansion is a 12  10À6 / C. The bar has no displacement (u 0) because it is constrained at both nodes (1 and 2). The total strain, which is equal to the derivative of the displacement with respect to the x-coordinate, is also zero. du et 0 2-37a dx Determinate Truss 97 y ∆T °C 1 2 x = 100 in. FIGURE 2-15 Temperature stress in a bar. The thermal strain in the bar is e0 aÁT 2-37b Elastic strain can be back-calculated from the total strain as et ee e0 0 ee et À aÁT ÀaÁT 2-37c The stress from Eq. (2-36a) becomes s Eee ÀEaÁT 2-37d The total deformation (bH ) is zero because the total strain is zero. The initial deformation and elastic deformation are bt et ` 0 2-37e b0 e0 ` 2-37f be Àe0 ` 2-37g The numerical values of the response variables for the bar are as follows: ee Àe0 À12  10À6  50 À0:6  10À3 et 0 bt 0 u1a u2a 0 e b Àb0 À0:6  10À1 s À30;000  0:6  10À3 À18 ksi F sA À18 kip 2-37h 98 STRENGTH OF MATERIALS A change of temperature of ÁT 50 C induced a 18-ksi compressive stress and a compressive strain of magnitude 0:6  10À3 . The reader should remember that stress is induced because of elastic strain, whereas displacement is induced because of total strain. 2.9 Thermal Effect in a Truss The response calculation because of thermal effect is illustrated by considering the five-bar truss shown in Fig. 2-12a as an example. Calculate the response of the structure for a change in temperature (ÁT 50 C) for each of its five bars. The thermal response of the truss is determined in the following steps. Step 1ÐInternal Force For a determinate truss, the solution of the equilibrium equation ([B]{F} {P}) yields the forces {F}. The change in the temperature has no effect in the equilibrium matrix [B]. No mechanical load is induced in a determinate system because of temperature, or ({P} {0}). Therefore, the internal force is a null vector ({F} {0}), and the truss is stress-free. No reaction ({R} {0}) is induced because there is no load. Step 2ÐInitial Deformation {b0 } The initial deformation in the ith bar because of the change in the temperature is b0 e0 `i aÁT`i i i 2-38a The initial strain vectors of the five-bar truss are V 0W V W b e1 b b b b1b b b b 0b b b b b b b be b b 2b b1b b b ` a ` a 0 0 feg e3 aÁT 1 2-38b b b b 0b b b b b be b b b b1b b b b 4b b b b b b b X 0Y X Y e5 1 The initial deformation vector ({b}0 {e`}0 ) is V 0W V W b b1 b b b b 1b b p b b 0b b b b b b bb b b 2b b 2b b b b ` a ` a 0 0 fbg b3 aÁT ` 100 1 2-38c b 0b b b b b b bb b b 4b b 1b b b b b b b b b b b b X 0Y X Y b5 1 Total deformation {b}t in the absence of elastic deformation is equal to the initial deformation. Determinate Truss 99 V W b 1b b p b b b b ` 2a b t e 0 0 fbg fbg fbg fbg aÁT` 1 2-38d b 1b b b b b b X b Y 1 Displacement {X}0 is induced because of deformation. It is calculated as the solution to the DDR {b}t [B]T {X}0 . V W P QV W 1 b p b 1 0 0 0 À0 b u0 b b 2b b b b b b b b b b 2 b T p À p 0 0 À0 Ub v0 b b b T 1 1 b ` b a T Ub 2 b b b 2 2 U` 0 a 100 aÁT 1 T 0 À1 1 0 À0 U U v 2-38e b b b T b T 3 Ub 0 b b b b 1 b R0 b b 0 0 1 À0 Sb u3 b b b b b b b b b b b X Y b 0b X Y 1 0 0 0 0 À1 v1 Solution to Eq. (2-38e) yields the displacements: u0 60  10À3 in: 2 v0 À60  10À3 in: 2 v0 0 3 u0 60  10À3 in: 3 v0 À60  10À3 in: 1 2-38f Initial configuration Deformed configuration 4 3 1 2 FIGURE 2-16 Displacement caused by temperature. 100 STRENGTH OF MATERIALS The displacement pattern of the truss is shown in Fig. 2-16. It expands in the positive x- and negative y-directions without violating the boundary conditions. In a determinate truss, the change in temperature has no effect on stress parameters like internal force, reaction, and stress, but displacement, deformation, and strain are induced. 2.10 Settling of Support The response caused by support settling is obtained by calculating the initial deformation and then the displacement. Consider a truss with m modes. Let its jth support node settle by ÁXj amount along the direction of the reaction Rj. The deformation caused by this settling is formulated using the deformation displacement relation (DDR). The DDR formulation requires the EE along the direction of the reaction Rj n  à Rj BRi Fi or Rj BRj fFg 2-39a i1 Here [BRj ] is a row matrix, and it is written along the direction of the ÁXj support displacement (corresponding to reaction Rj). The DDR becomes  ÃT À Á fbg0 BRj ÀÁXj ÀfBRJ gÁXj 2-39b The transpose of the row [BRj] is replaced by a column {BRJ } since [BRj ]T {BRJ }. The direction of reaction Rj is opposite to the settling ÁXj . For example, in the truss shown in Fig. 2-17, the settling of support Áv4 at node 4 along the negative y-coordinate direction induces a reaction R4y along the positive y-direction. The negative sign in the DDR given by Eq. (2-39b) accounts for the opposite direction of Rj and ÁXj . Equation (2-39b) is generalized for simultaneous settling of p number of supports by adding the individual contributions: fbg0 À fBR1 gÁX1 fBR2 gÁX2 ; F F F ; fBRP gp ÁXp 2-39c fbg0 ÀBR T fÁX g 2-39d nÂ1 nÂp pÂ1 Here, the p columns of the matrix [BR ]T corresponds to the p rows of the EE matrix [BR] defined in Eq. (2-14b). After the determination of the initial deformation {b}0 , the method used for thermal load is adopted to calculate the response of the structure. Settling of the Support in a Truss The response calculation because of support settling is illustrated by considering the five-bar truss shown in Fig. 2-17 as an example. A settling of 1/2 in. is assumed for support node 4 in the negative y-coordinate direction (Áv4 Áy À0:5 in:). The response from the settling of the support of the truss is determined in the following steps. Determinate Truss 101 Initial configuration Deformed configuration y 4 x 3 y= v4 R 4y y 1 2 FIGURE 2-17 Deformation due to settling of support. Step 1ÐInternal Force For a determinate truss, the solution of the equilibrium equation ([B]{F} {P}) yields the forces {F}. The settling of the support has no effect on [B] and {P}. The internal force {F} is a null vector, and no reaction is induced. 0 Step 2ÐInitial Deformation {b } The settling corresponds to the reaction R4y, which is the third reaction in the matrix equa- " tion Eq. (2-14a). The initial deformation for ÁX3 Áv4 À0:5 in. is obtained 0 ({b} {BR3 }ÁX " 3 ) as V W V W b 0b b 1 b b 0 b b b b p b b b b 0:354 b b b ` 2a ` a 0 fbg À 0 À0:5 0 2-40a b b b 0b b 0 b b b b b b b b b b b X Y X Y 1 3rd row of BR in Eq: 2-14a 0:5 The total deformation is equal to the initial deformation because the elastic component is zero ({b} {b}t {b}0 ). 102 STRENGTH OF MATERIALS Displacements are obtained from the solution of the DDR ({b} [B]T {X}). V W V WV W b 0 b b1 b b b 1 0 0 0 0 bb u 2 b bb b b 0:354 b b p À p b b b 1 0 0 0 bb v2 b bb b ` a ` 2 2 a` a 0 0 À1 1 0 0 v 2-40b b b 0 b b0b b bb 3 b b b b b b b 0 0 1 0 bb u 3 b bb b bb b X Y X YX Y 0:5 0 0 0 0 À1 v1 Solution to Eq. (2-40b) yields V W V W b u2 b b b b b 0bb b v2 b b À0:5 b b b b b ` a ` a fXg v3 À0:5 2-40c b b b b u3 b b b b b b b b b 0bb b X Y X Y v1 À0:5 in: The displacements for all nodes of the truss are obtained by augmenting equations (2-40c) with the prescribed support settling {ÁX}0 and other boundary displacements: fXgtruss fXg fÁXg0 V W V W b u1 b b 0:0 b b b b b b b b b b v b b À0:5 b b b b b 1b b b b b b b b b b b b b b b b b u b b 0:0 b b b b b 2b b b b b b b b b b b b b b b b b b b b ` v2 a ` À0:5 b b b b a truss fXg 2-40d b u3 b b 0:0 b b b b b b b b b b b b b b b b b b v3 b b À0:5 b b b b b b b b b b b b b b b b b b b b b b u b b 0:0 b b b b b 4b b b b b b b b b b b b b b b X Y X b Y v4 À0:5 The total deformation {b}t and strain {e}0 are back-calculated from the DDR as {b}t {`e}t [B]T {X}. V WV W V W b1 b 1 0 0 0 0 bb bb 0b b 0 b b b b b b p À p 1 0 0 0 bb À0:5 b b 0:354 b bb b b b ` 2 2 a` a ` a t t fbg fe`g 0 À1 1 0 0 À0:5 0 2-40e b b0 bb b b b b b 0 0 1 0 bb bb bb 0b b 0 b b b b b b b X YX Y X Y 0 0 0 0 À1 À0:5 0:5 Determinate Truss 103 The deformed configuration of the truss resulting from the settling of support node 4 by À1/2 in. in the y-coordinate direction is shown in Fig. 2-17. The entire structure undergoes uniform displacement in the negative y-coordinate direction. This motion is called rigid- body displacement of the truss. A rigid body displacement does not induce any elastic strain in the bars of the member because the total deformation is equal to the initial deformation given by Eq. (2-40e). Response for Load, Temperature, and Support Settling The response of a truss that is subjected to load, temperature, and support settling is obtained as the superposition of the three individual responses: Bar forceX fFg fFgload fFgthermal fFgsettling 2-41a ReactionX fRg fRgload fRgthermal fRgsettling 2-41b StressX fsg fsgload fsgthermal fsgsettling 2-41c DisplacementX fXg fXgload fXgthermal fXgsettling 2-41d StrainX feg fegload fegthermal fegsettling 2-41e Deformation fbg fbgload fbgthermal fbgsettling 2-41f 2.11 Theory of Determinate Analysis The theory of a determinate analysis is contained in the analysis of the five-bar truss. In subsequent chapters, this formulation is modified to solve other structure types like beams, shafts, and frames. The modifications pertain to the details of member characteristics, but the underlying theory is changed little. The analysis steps are formalized considering the example of the nine-bar determinate truss shown in Fig. 2-18. Step 0ÐSketch The analysis of a determinate structure is centered around the equilibrium equations. The formulation of the EE requires a figure of the truss. Such a sketch depicting the geometry of the truss with the support conditions should be prepared from the coordinates of the nodes or the dimensions of the bars. In the sketch, mark the n number of internal forces {F}, m nodal displacements {X}, and mR reactions {R}. Other response variables back-calculated include the n bar stress {s}, the n bar deformations {b}, and the n bar strains {e}. The solution of a determinate problem has the following four basic steps: Step 1ÐFormulate the Equilibrium Equations The EE are formulated using the sketch. Solution of the EE yields the internal forces and the reactions. 104 STRENGTH OF MATERIALS v3 (P3y = –1.0) v5 (–1.0) u3 (0.5) u5 (0.75) 3 F6 5 F1 F3 F5 F7 F9 y v2 (–1.5) v4 (–1.5) u6 45° u2 u4 6 1 F2 F4 F8 x 2 4 R1x R1y R6y (a) Forces. 3 5 1 2 4 6 (b) Thermal displacement (exaggerated). 3 5 2 4 6 1 (c) Displacement caused by support settling (exaggerated). FIGURE 2-18 Nine-bar, six-node truss. Step 2ÐFormulate the Force Deformation Relations Deformations are calculated from the FDR. Step 3ÐFormulate the Deformation Displacement Relations Calculate displacements from the DDR. Step 4ÐBack-Calculate Stress and Strain Determinate Truss 105 EXAMPLE 2-7: Analysis of a Six-Node, Nine-Bar Truss The truss shown in Fig. 2-18 is made of steel with a Young's modulus E 30,000 ksi and a coefficient of thermal expansion a 12  10À6 / C. The coordinates, boundary conditions, loads, and member properties are given in Tables 2-2 and 2-3. The truss is also subjected to a change of temperature (ÁT 50 C). The support at node 1 settles by 1/2 in. along the negative y-coordinate direction. Calculate the response of the truss for the mechanical load, thermal load, support settling, and their combined effect. Solution Step 0ÐSketch The geometry of the truss with nodes, support conditions, external load (shown in parenthesis), internal force, and nodal displacement are depicted in Fig. 2-18. The truss has nine bar forces (n 9; F1 , F2 , F F F , F9 ), nine displacements (m 9; u2 , v2 ; u3 , v3 ; u4 , v4 ; u5 , v5 ; u6 ), and three reactions (mR 3; R1x , R1y , R6y ). It is subjected to a load at node 2 in the negative y-direction (P2y À1:5 kip), at node 3 (P3x 0:5 kip, P3y À1:0 kip, at node 4 (P4y À1:5 kip), and at node 5 (P5x 0:75 kip, P5y À1:0 kip). The included angle between the bars is 45 . TABLE 2-2 Coordinates, Load, and Boundary Restraints for the Truss in Fig. 2-18 Node Coordinates, In. Load, kip Boundary Restraints x y x y 1 0 0 ± ± Both directions 2 100 0 ± À1:5 ± 3 100 100 0.50 À1:0 ± 4 200 0 ± À1:5 ± 5 200 100 0.75 À1:0 ± 6 300 0 ± ± y-direction TABLE 2-3 Member Properties for the Truss in Fig. 2±18 Member Connecting Nodes Area, In.2 1 1±3 1.0 2 1±2 1.0 3 2±3 0.75 4 2±4 1.0 5 3±4 0.75 6 3±5 1.0 7 4±5 0.75 8 4±6 1.0 9 5±6 1.0 106 STRENGTH OF MATERIALS Step 1ÐEquilibrium Equations ([B] {F} = {P}) The truss has m 9 EE. The EE along the displacement directions are written as 1 along u2 X À F2 F4 0 2 along v2 X F3 À 1:5 0 F1 F5 3 along u3 X À p p F6 0:5 0 2 2 F1 F5 4 along v3 X À p À F3 À p À 1:0 0 2 2 F5 5 along u4 X À F4 À p F8 0 2 F5 6 along v4 X p F7 À 1:5 0 2 F9 7 along u5 X À F6 p 0:75 0 2 F9 8 along v5 X À F7 À p À 1:0 0 2 F9 9 along u6 X À F8 À p 0 2 The EE ([B]{F} {P}) in matrix notation becomes P QV W V W u2 X 0 1 0 À1 0 0 0 0 0 b F1 b b 0b b b b b b b b b T Ub b b b v2 X T 0 T 0 À1 0 0 0 0 0 0 Ub F2 b b À1:5 b b b b b b b Ub b b b b b T p 1 À p 1 Ub b b b F b b 0:5 b b b b b b u3 X T 2 T 0 0 0 2 À1 0 0 0 Ub 3 b b Ub b b b b b b b b T p b b b Ub b bb b b b 1 p 1 v3 X T 2 0 1 0 2 0 0 0 Ub F4 b b À1:0 b b 0 Ub b b b b T b b b ` a ` b a T p 1 U u4 X T 0 T 0 0 1 2 0 0 À1 0 U F5 Ub b b 0 T Ub b b b b p 1 b b b b v4 X T 0 T 0 0 0 À 2 0 À1 0 0 Ub F6 b b À1:5 b b b b b b b Ub b b b b b b b b T 1 Ub u5 X T 0 0 0 0 0 1 0 0 À p Ub F7 b b 0:75 b b b b b b b b b T 2 Ub b b bb b b b T Ub b b 1 b b b b b v5 X T 0 R 0 0 0 0 0 1 0 p Ub b F8 b b À1:0 b 2 Sb b b b b b b b b b b b b b b p 1 X Y X Y u6 X 0 0 0 0 0 0 0 1 2 F9 0 2-42a Determinate Truss 107 Solution of the equation to the EE yields the forces: V W V W b F1 b b À2:95 b b F b b 3:33 b b b b b 2b b b b b b b b b F b b 1:50 b b b b b 3b b b b b b b b b b b b b F4 b b 3:33 b b b b ` a ` b a F5 À0:59 2-42b b F6 b b À2:17 b b b b b b b b b b b b b b F7 b b 1:92 b b b b b b b b b b b b b b b b b F8 b b 2:92 b b b b b b b b b X Y X b Y F9 À4:12 kip Reactions are calculated from the EE that are written along the directions of restraint: F1 EE along R1x or u1X R1x F2 p 0 2 F1 Along R2y or v1X R1y p 0 2 F9 Along R6y or v6X R6y p 0 2-43a 2 The EE to calculate reactions can be written in matrix notation as V W P À p À1 1 0 QV W V W ` R1x a 2 ` F1 a ` À1:25 a T 1 0 U F2 R1y R À p 0 S 2:08 2-43b X Y 2 X Y X Y R6y 0 0 À p 1 F9 2:92 kip 2 The accuracy of the reactions is ascertained from the overall EE. P R 0 R1x P3x P5x 0 x À1:25 0:5 0:75 0 P R 0 R1y R6y P2y P3y P4y P5y 0 y 2:08 2:92 À 1:5 À 1:0 À 1:5 À 1:0 0 MP MR 0 R6y  300 P2y  100 P3y  100 P4y  200 node 1 P5y  200 À P3x  100 À P5x  100 0 100 3  2:92 À 1:5 À 1:0 À 2  1:5 À 2  1 À 0:5 À 0:75 0 Step 2ÐForce Deformation Relations ({b} = [G]{F}) The deformations for mechanical load are obtained from the FDR: 108 STRENGTH OF MATERIALS p F` 2:95  100 2 b1 À À1:39  10À2 in: AE 1 1  30;000 F` 3:33  100 b2 1:11  10À2 in: AE 2 1  30;000 F` 1:5  100 b3 6:67  10À3 in: AE 3 0:75  30;000 F` 3:33  100 b4 1:11  10À2 in: AE 4 1  30;000 p F` 0:59  100 2 b5 À À3:70  10À3 in: AE 5 0:75  30;000 F` 2:17  100 b6 À À7:23  10À3 in: AE 6 1  30;000 F` 1:92  100 b7 8:53  10À3 in: AE 7 0:75  30;000 F` 2:92  100 b8 9:73  10À3 in: AE 8 1  30;000 p F` 4:12  100 2 b9 À À1:94  10À2 in: 2-44a AE 9 1  30;000 Step 3ÐDeformation Displacement Relations ([B] T{X} = {b}) Nodal displacements are calculated from the DDR. V W P p 1 p 1 QV W V W b b1 b 0 0 0 0 0 0 0 b u2 b b À1:39 b b b T bb b b 2b T 1 0 0 2 0 2 0 0 0 0 Ub v b b 1:11 b b b b 0 Ub 2 b b b b b b b b b b b b T b b b b3 b T 0 À1 0 1 0 0 0 0 Ub b b 6:67 b bu b b b 3b b 0 Ub b b b b b b b b T b b b b b b b b b4 b T ` a À1 0 0 0 1 0 0 0 0 Ub v3 b b 1:11 b b b b U` a ` b a T U b5 T 0 0 À p 1 p 1 p À p 1 1 0 0 0 U u4 À3:70  10À2 b b T b b6 b T 0 2 2 2 2 Ub b b b b b b b 0 À1 0 0 0 1 0 0 Ub v4 b b À7:23 b b b b b b b T bb b T 0 Ub b b b b b b u5 b b 8:53 b b b b 7b b b T 0 0 0 0 À1 0 1 0 Ub b b b b b b b b b bb b R 0 Ub b b b b 8b b b 0 0 0 À1 0 0 0 1 Sb v5 b b 9:73 b b b b b b b b b X Y 1 X Y X Y b9 0 0 0 0 0 0 À p 1 2 p 1 2 p 2 u6 À1:94 2-44b Determinate Truss 109 Solution of the DDR yields the displacements: V W V W b X1 b u2 b b 11:11 b b b b b b b b bX b 2 b b v2 b b À53:98 b b b b b b b b b b b b b b 27:67 b b b b b X3 b b b u3 b b b b b b b b b b b b b b X4 b b b À47:31 b b b v3 b b b b ` a ` a fX i g X5 u4 22:22 Â10À3 2-44c b b b b b b b b b X6 b b v4 b b À47:52 b b b b b b b b b b b b b X7 b b b 20:45 b b b u5 b b b b b b b b b b b b b b b b b b b b b X8 b v5 b b À39:00 b b b b b X b b Y X b Y X9 u6 31:94 in: Step 4ÐBack-Calculate Stress and Strain Stress is calculated from the bar force and area (s F/A). Strain is obtained from Hooke's law (e s/E). V W b s1 F2 b V W V W b b A2 b b b À2:95 b b b s2 F1 b b b b b À0:098 b b b b A1 b b b b b b 0:111 b b b b b b b 3:33 b b b b b b b b b b b b b b b b b s3 b b A3 b b 2:0 b F3 b b b b b b b 0:066 b b b b b b b b b b b b b b b b b b b F4 b b b b b s4 b b A4 b b 3:33 b b b b b b b b 0:111 b b b b b ` a ` a 1 ` a F5 fsg s5 A5 À0:79 feg fsg À0:026  10À3 b b b b b b b b E b b b b b b s6 b b F6 b b À2:17 b b b b b A6 b b b b b b b À0:072 b b b b b b b b b b 2:56 b b b b b 0:085 b b b bs b 7 F7 b b A7 b b b b b b b b b b b b b b b b b b b b 2:92 bb b 0:097 b b b b bs b b F8 b b b b b b b b b 8 b b A8 b b b X b b Y b X b Y b X b Y À4:12 ksi À0:137 F9 s9 A9 2-44d Thermal Load The change in temperature has no effect on the internal force, reaction, and stress. It induces total strain, deformation, and displacement. Total strain {e}t is equal to thermal strain {e}t ({e}T ) because there is no elastic strain ({e}e {0}). Initial deformation is calculated as b0 (`eT )i (`aÁT)i . Displacements are determined from the DDR i ({b}0 [B]T {X}). 110 STRENGTH OF MATERIALS V W V p W b 0:6 b b 2b b 0:6 b b b b b b b 1 b b b b b b b b b b 0:6 b b b b b b 1 b b b b b b b b b b b b b 0:6 b b b b 1 b b ` a ` p b À a Á fegt fegT faÁtg 0:6  10À3 fbg0 f`et g 2  6  10 À2 2-45a b 0:6 b b b b b b b b 1 b b b b b b b b b 0:6 b b b b b b b 1 b b b b b b b b b 0:6 b b b b b 1 b b b p b b b X b Y b X b Y 0:6 2 Substitution of the initial deformation {b} in the DDR of Step 3 given by Eq. (2-44b) and solution yield the displacements. V W V W b u1 b b b b 0:00 b b b b b b b b b b v1 b b b b 0:00 b b b b b b b b b b b b b b b b u2 b b b b 60:00 b b b b b b b b b b b b b b b b v2 b b b b 0:00 b b b b b b b b b b b b b b b bu b b 3b b 60:00 b b b b b b b b b b b b b b b `v a ` 60:00 b b a all nodes 3 fXg  10À3 in: 2-45b b b b b b u4 b b 120:00 b b b b b b v b b 0:00 b b b b b b 4b b b b b b b b b b b b u b b 120:00 b b b b b 5b b b b b b b b b b b b b b b b b b b b b b b b v5 b b 60:00 bb b b b b b b b b b u6 b b 180:00 b b b b b b b b b b b b b b b b b X Y X Y v6 0:00 The nodal displacements are graphed in Fig. 2-18b. It consists of a uniform expansion in the y-direction. In the x-direction, the displacement is proportional to the bar length along the bottom chord. The displacement along the x-direction peaks at node 6 at u6 0:18 in. The thermal displacements confirm to the boundary conditions. Because the truss is restrained along the y coordinate direction at both the boundary nodes 1 and 6, it moves at nodes (3 and 5) along the y-direction. It moves along the x-direction at node 6 because it is a roller support. Support Settling The settling of a support has no effect on the internal force, reaction, and stress. It induces total strain, deformation, and displacement in the structure. Total strain ({e}t ) and initial strain ({e}0 ) are equal ({e}t {e}0 ) because there is no elastic strain ({e}e {0}). Initial deformation caused by the settling of node along the y-direction is calculated as Determinate Truss 111 " fbg0 ÁX fBR2 g 2-46a " Here, ÁX À0:5 is the settling of node 1 along the y-direction, or along the reaction R1y . {BR2 } is the second row in Eq. (2-43b). It is the EE along the reaction R1y . V W0 V 1 W ` b1 a ` À p a 2 b À0:5 0 2-46b X 2Y X Y b3 0 in: The full deformation vector {b}0 is obtained by adding zero for the other components. V W0 V W V W b b1 b b 0:35 b b b b b b 2:48  10À3 b b b bb b b 0 b b b b b b b 2b b b b b b b b b b b 0 b b b bb b b 0 b b b b b b b 3b b b b b b b b b b b 0 b b b b b b b b4 b b 0 b b & 0' `b b b b ` a ` a b 0 a 0 0 fbg b5 0 fe g 0 2-47a b b b b b6 b b 0 b b ` b b b b b b b b b b b b b b 0 b b b b b bb b b 0 b b b b b b b 7b b b b b b b b b 0 b b b b b bb b b 0 b b b b b b b 8b b b b b b b b b 0 b b X Y X Y X Y b9 0 in: 0 Substituting for {b}0 in the DDR ({b}0 [B]T {X}0 ) of Step 3 and solving yields the displacements. The displacements at the support nodes are added to obtain the displacements for all nodes, as follows: V W V W b u1 b b 0:00 b b v b b À0:50 b b b b b b 1b b b b b b b b b b b u b b 0:00 b b b 2b b b b b b b b v b b À0:33 b b b b b 2b b b b b b b b b b b b b u3 b b À0:16 b b b b b b b b b b b b ` a ` b a all nodes v3 À0:33 {Xg 2-47b b u4 b b 0:00 b b b b b b v4 b b À0:16 b b b b b b b b b b b b b b u5 b b À0:16 b b b b b b b b b b b b b b v5 b b À0:16 b b b b b b b b b b b b b b u b b 0:00 b b b b b 6b b b b b b b X Y X b Y v6 0:00 in: The displaced structure caused by the settling of node 1 is shown in Fig. 2-18c. The structure rotates with node 6 as the center. Displacements along the negative y-direction at nodes 4 and 2 are proportional to their distance from node 6. Nodes 3 and 5 tilt by the same amount along the negative x-direction. 112 STRENGTH OF MATERIALS 2.12 Definition of Determinate Truss Consider a two-dimensional truss with n bars and m nodes. The truss must be stable for analysis. It is stable when it is supported at least at two nodes and one node has two restraints. One restraint must be along the x-direction to prevent its motion as a rigid body along that direction. Likewise, another restraint along the y-direction prevents such motion along the y-direction. The third restraint prevents the rigid body rotation in the x±y plane. It has a total of three nodal restraints (mr 3) and three reactions. Its displacement degree-of- freedom is dof 2m À mr 2m À 3. Its number of bar forces is n, and its number of reactions is mr 3. A truss is determinate when its n bar forces and three reactions can be determined only from an application of the equations of equilibrium. Determinacy of a stable truss is separated into external determinacy and internal determinacy. 1. External determinacy: A truss is externally determinate provided the total number of nodal restraints is equal to three (mr 3) and there are three reactions. This is referred to as the rule of reaction. 2. Internal determinacy: A truss is internally determinate provided the number of bar forces is equal to the displacement degree-of±freedom (n 2m À mr dof). This we will refer to as the member rule. 3. A determinate truss must be stable and satisfy both the reaction and member rules. Consider the N-truss shown in Fig. 2-19a. It has 17 bars (n 17), 10 nodes (m 10), 3 support restraints (mr 3), and 3 reactions (R1x, R1y, R9y). Its dof is 17 (dof 2  10 À 3 17). The truss satisfies the reaction rule (mr 3) and the member rule (dof n 17). It is a stable, determinate truss. The N-truss shown in Fig. 2-19b has one restraint along the y-coordinate direction at each of its three nodes (1, 5, 9). Both the reaction rule (mr 3) and the member rule (dof 17 n) are satisfied. However, it is an unstable truss because it can roll along the x-coordinate direction. The analysis of unstable trusses is not included in strength of materials analysis. Consider the two-bar truss. It has three nodes (m 3), two members (n 2), two restraints each at node (2 and 3) (mr 4), and four reactions (R2x, R2y, R3x, R3y). It appears to fail the reaction rule but passes the member rule (dof 2 n). The truss, by definition, may not be considered externally determinate. The two-bar truss is determinate. This is because of additional available information. The direction of the reaction must be along the bar. At node 2, the reaction is along bar 1, which connects nodes 1 and 2; and at node 3, the reaction is along the bar 3, which connects nodes 1 and 3. The two bar forces can be calculated from two EE written at node 1. The EE at the support yield the reactions. Consider next the case when node 3 is on a roller support as shown in Fig. 2-19c. This case satisfies the reaction rule because mr 3, but the truss is unstable because the support condition cannot resist a horizontal reaction component resulting from the force in bar 2. Consider the cantilevered N-truss shown in Fig. 2-19d. It satisfies both the reaction rule (mr 3) and the member rule (dof n 17). The truss is unstable because it has an unstable bay, as marked in Fig. 2-18d. Determinate Truss 113 y x 2 4 6 8 10 Move 1 9 3 5 7 R1x R 1y R5y R9y R1y R9y (a) N- truss. (b) Unstable N- truss. 2 4 6 8 10 R2y R3y R1x R3x Unstable bay 1 9 2 3 3 5 7 1 2 3 R 1x R1y R7y 1 (c) Two- bar truss. (d) Cantilevered N- truss. FIGURE 2-19 Determinate truss. The member and reaction rules may have some limitations, but they perform satisfactorily for regular types of trusses. For a determinate truss, the equilibrium matrix [B] is a square matrix with a nonzero determinant, or it is a nonsingular square matrix. Calculation of Force in an Individual Bar The force in an individual bar in a determinate truss can be calculated without analyzing the entire structure. This is possible because the bar force depends on load and the truss geometry but it is independent of the bar area and material. The method of section can be followed for the calculation. A section is taken through the bar in question. Next a free body is separated. The equilibrium of forces acting on the free body yield the bar force. Reactions, if required, can be determined from the overall EE. Force calculation in an individual bar can be used to verify an available analysis result or in the redesign of a critical member. The method cannot be extended to calculate the displacement at a node of a truss because displacement is dependent on bar area and material and it is a global variable. The method is illustrated by considering an example. 114 STRENGTH OF MATERIALS EXAMPLE 2-8: Bar Force in a Truss A 17-bar, 10-node truss with a span of 20 m and a height of 2 m is shown in Fig. 2-20a. It is subjected to a load (P 10 kN) at node 9 along the negative y-coordinate direction. Calculate the forces in bars 10, 11, and 12, and in bars 5 and 7. Forces in Bars (10, 11, and 12) Take a section (C1±C1) that cuts the group of bars as shown in Fig. 2-20a. A free body can be taken either to the left or right of C1±C1. The right free body shown in Fig. 2-20b is preferred because this portion is not associated with any reaction. The bar forces and loads marked in this free body are in equilibrium. Bar forces are calculated from the EE. Force F10: The moment at node 8 yields F10. M8 0X À 2F10 À 5  10 0 F10 À25 kN 2-48a y C2 C1 5m x 2 4 8 6 8 10 4 12 16 2m 1 3 5 7 9 11 13 15 17 1 1 2 6 10 14 3 5 7 9 C2 C1 10 kN R1x 20 m R1y (a) 17-bar truss. F12 4 F8 F11 F7 F5 p F10 F2 3 10 kN 10 (b) Free body at C1–C1. (c) Free body at C2–C2. FIGURE 2-20 Method of section for a truss. Determinate Truss 115 Force F12: The moment at node 5 yields F12. M5 0X 2F12 À 10  10 0 F12 50 kN 2-48b Force F11: The EE along the y-coordinate direction yields F11. F 0X À F11 cos y À 10 0 Y F11 À10= cos y 2 cos y p 0:37 2 2 52 sin y 0:93 F11 À27 kN 2-48c The accuracy of the calculation is ascertained from the three EE written for the entire free body shown in Fig. 2-20b. F 0X À F10 F12 F11 sin y 0 x À À25 50 À 25 0 F 0X À F11 cos y 10 0 Y À À10 10 0 M5 0X À 2F12 10  10 0 À 100 100 0 2-48d Forces in Bars (5 and 7) Take a section (C2±C2) that cuts the bars as shown in Fig. 2-20a. The left free body shown in Fig. 2-20c is considered. It has two reactions (R1x and R1y). The reactions are calculated from the overall EE for the truss shown in Fig. 2-20a. 116 STRENGTH OF MATERIALS F 0X R1x 0 2-49a x À Á M5 0X À R1y  10 10  10 0 R1y À10 kN 2-49b The bar forces and reactions are marked in the free-body diagram shown in Fig. 2-20c. Force F2: The moment at node 4 yields F2. M4 0X 2F2 10  5 0 F2 À25 kN 2-49c Calculation of F7 requires the values for F8 and F5. In other words, all three forces (F4, F8, and F7) have to be calculated simultaneously as the solution to the three equations consisting of M3 0 and the EE along the x- and y-coordinate directions. M3 0X À 2F8 5  10 À pF7 0 x F 0X F2 F8 F7 sin y 0 y F 0 À 10 F5 F7 cos y 0 The three EE and F2 À25, yield the bar forces. F8 50 F7 0 F5 À27 kN 2-49d Standard Shape of a Bar Member To construct a truss, designers can select bar members from the standard rolled shapes adopted by the American Institute of Steel Construction. Of course, other materials and shapes desired by the designer can be used. A design course addresses the properties of many different types of available rolled shapes. We introduce two properties relevant to analysis: the net cross-sectional area and the equivalent cross-section of a bar member. Determinate Truss 117 Net Cross-Sectional Area A truss is manufactured by fastening bar members by bolting, riveting, or welding. The fastening process reduces the cross-sectional area of a bar. Consider, for example, the angle section of lengths L1 and L2 and thickness t, shown in Fig. 2-21. It is traditionally specified as L1  L2  t. Its gross area (Ag) and net area (An) are calculated as follows: Ag L1 L2 À tt 2-50a An 0:85Ag 2-50b The reduction (t2) in Eq. (2-50a) accounts for the rounding of the corners. Analysis, preferably, should use net area, which is less than the gross area (An < Ag ). Net area (An) can be calculated by reducing the gross area to account for the effect of fastening. For example, if the truss is fastened by rivets the areas of such holes must be accounted. The gross area is reduced to obtain the net area (An < Ag ) using an approximate reduction factor of 85 percent as (An 0:85Ag ) for angle sections. Centerline of angle Rounded t corners L2 t/2 Centerline of angle t/2 L1 FIGURE 2-21 Dimensions of an angle section. EXAMPLE 2-9: Net Area of Angle Section For the angle iron (6  4  0:25) shown in Fig. 2-21, calculate the gross area and the net cross-sectional areas. L1 6 in: L2 4 in t 0:5 in: Ag 6 4 À 0:5  0:5 4:75 in2 An 0:85  4:75 4:04 in2 2-51 118 STRENGTH OF MATERIALS In the analysis of a truss made of angle iron, the net area (An 4:04 in:2 ) is used. Net area depends on the shape of the cross-section, such as angle section, T section, channel section, I-section, round or circular bar, square pipe, and others, as well as on the nature of the fastening. In case details are not available, then analysis should proceed either assuming the net area from the formula given by Eq. (2-50b), or even the gross area can be used. Equivalent Cross-Sectional Area A truss member can be made of two or more different materials, such as steel and aluminum. What is the area of the bar for the purpose of analysis? In the analysis of a composite bar any one material can be selected as the base material. The bar cross-section is then transformed in terms of the base material. Consider a bimetallic bar, shown in Fig. 2-22a as an example. It is made of an inner aluminum core of diameter da and an outer steel shell of diameter ds, as depicted in Fig.2-22b. It is assumed that the bar is adequately designed to act as a single integral unit throughout its use without any gap or debonding at the aluminum-steel interface, from the support point at A to the load application point at B. An equal strain concept is used to calculate the equivalent cross-sectional area. According to this concept, the induced strain at any location in the interface is the same in steel and in aluminum. At the interface the stress (which is a product of strain and modulus) is not the same for the two materials. Consider a small segment of the bar of length a, as shown in Fig. 2-22c. The strain (e) in steel (es e) is identical to that in aluminum (ea e). The segment has equal deformation A P P x x δ/2 δ2 δ1 a L ds δ/2 δ4 δ3 da B P P Admissible Inadmissible P (a) Bimetallic bar. (b) Section at x–x. (c) Deformation. FIGURE 2-22 Deformation in a composite bar. Determinate Truss 119 (d ae aes aea ). This admissible state of expansion (d/2) at each end is marked in Fig. 2-22c. The inadmissible deformation, which is caused by uneven expansions in steel (d2 and d3 ) and aluminum (d1 and d4 ), is marked in Fig. 2-22c. The differential deformation induces different strains at the interface of aluminum and steel materials (es ea ), and this is not T admissible. The stresses in steel and aluminum are related to the strain through Hooke's law. sa eEa ss eEs 2-52a Force in aluminum (Fa Aa sa ) and in steel (Fs As ss ) are in equilibrium with the load (Fa Fs P). The EE in terms of the stress can be written as Aa s a As s s P Aa Ea e As eEs P e Aa Ea As Es P eEa Aa As Es =Ea P eEs As Aa Ea =Es P sa Aa As Es =Ea ss As Aa Ea =Es P 2-52b The load is equated with the aluminum equivalent area (Aae) and steel (Aas) as sa Aae P ss Ase P 2-52c The equivalent areas for aluminum (Aae) and steel (Aas) are defined as Es Aae Aa As 2-52d Ea Ea Aas As Aa 2-52e Es The aluminum equivalent area is defined as the sum of the aluminum area added to the steel area that is prorated by the ratio of the elastic modulus of steel to aluminum. The steel equivalent area is likewise defined. When a bar is made of different materials, the base material must be selected: for example, consider this to be steel. The analysis should use the base material (steel) and its equivalent area (steel equivalent area). 120 STRENGTH OF MATERIALS EXAMPLE 2-10: A Composite Bar The composite shaft shown in Fig. 2-22a carries a load (P 10 kip). Calculate the equivalent area and the internal forces in steel and aluminum for a shaft with a length of 100 in., diameters of ds 4 in: and da 3 in:, and moduli of Es 30 million psi and Ea 10 million psi. The ratios (R) of the modulus, areas for aluminum (Aa), steel (As), gross area, and equivalent area for aluminum (Aae) and steel (Ase) are calculated as Es R 3 Ea d2 Aa p a 7:07 in:2 4 pÀ 2 Á 2 As ds À da 5:5 in:2 4 Ag Aa As 12:57 in:2 Aae 7:07 R  5:5 23:57 in:2 Ase 5:5 7:07=R 7:86 in:2 2-53 The geometrical area of the cross-section is Ag 12:57 in:2 . The aluminum equiva- lent area at Aae 23:57 in:2 is more than the geometrical area (Ag 12:57 in:2 ). The steel equivalent area at Ase 7:86 in:2 ) is less than the geometrical area (Ag 12:57 in:2 ). The strain, which is the same for both materials, is obtained as s P 1 10  103 ea 42:43  10À6 E Aae Ea 23:57  10  106 P 1 10  103 es 42:43  10À6 Ase Es 7:86  30  106 e ea es 2-54a The stress and internal force in steel are obtained as ss Es e 30  106  42:43  10À6 1273 psi 2-54b Fs As ss 5:5  1273 7:0 kip 2-54c The stress and internal force in aluminum are obtained as sa Ea e 10  106  42:43  10À6 424:3 psi 2-54d Fa Aa sa 7:7  424:3 3:0 kip 2-54e Determinate Truss 121 The internal forces in the steel and aluminum sections add up to the applied load (Fs Fa P). Load sharing in a composite bar is proportional to the elastic modulus times the area of the material (AE). The steel part, with a higher coefficient (As Es 165 106 ), carries more load. The load fraction is obtained as the ratio (As Es /(As Es Aa Ea ) 165/235:7 0:7), or it carries 70 percent of the load. The aluminum material, with a lower coefficient (AEa 70:7  106 ) carries (70:7/235:7 0:3), or 30 percent of the load. The axial deformation (b `e) is obtained as the product of strain and bar length. d e` 42:43  10À6  100 4:24  10À3 in: 2-54f The axial displacement is equal to the deformation because one boundary is restrained. The axial displacement is 4:24  10À3 in. Problems Use the material properties given in Tables A5-1 and A5-2 in Appendix 5 to solve the problems. 2-1 A 30-ft-long aluminum bar is suspended from the ceiling as shown in Fig. P2-1. The bar has a uniform cross-sectional area of 1 in.2 It is subjected to an external load (P 1 kip) and a change of temperature (ÁT 100 F). Calculate the internal force, the stress, the displacement, the deformation, the strain at the midheight of the strut, and the support reaction. Compare the response when aluminum is replaced by steel. c c Section at c–c P FIGURE P2-1 122 STRENGTH OF MATERIALS 2-2 The composite column shown in Fig. P2-2 has three sections. The base section is made of concrete, the middle section of steel, and the top section of aluminum. The dimensions and cross-sectional properties are marked in the figure. The column supports a compressive load of 5 kN, and the steel section is subjected to a temperature differential (ÁT 250 F). Calculate the internal force, stress, displacement, deformation, and strain at the middle of each section. Do not neglect the weight of the material. P Aluminum R = 6 in. a a 8 ft Section t = 0.25 in. at a–a b Steel b Section 10 ft at b–b c Concrete c 4 ft 8 ft Octagonal section at c–c. FIGURE P2-2 2-3 A composite column is made of aluminum and steel with the dimensions marked in Fig. P2-3. The upper aluminum segment is subjected to a change of temperature (ÁT À250 F), and the steel column carries a 100-kN load. Calculate the displacement at the free end of the column. Determinate Truss 123 a a 0.125 m 2m Section at a–a b b 0.15 m 2.5 m Section P 1m at b–b FIGURE P2-3 2-4 A wooden strut is supported by a steel and an aluminum post through a shaft as shown in Fig. P2-4. The wooden strut carries two loads (P1 À2 kip, and P2 1 kN). The posts are subjected to a temperature differential (ÁT 250 F). The dimensions are marked in the figure. Calculate the reactions in the posts, the displacement at A, and the rotation at the center of the shaft CD. Neglect the weight. 3 ft P1 B 1 ft C D Steel a a s s 2 ft Wood 4 ft Aluminum w w A P2 R = 4 in. Section at a–a, w–w, s–s FIGURE P2-4 124 STRENGTH OF MATERIALS 2-5 A two-bar, three-node steel truss is subjected to a load (P 1 kip) at angle y as shown in Fig. P2-5. Calculate the maximum values of the support reaction at node 1 as the load rotates through a full revolution (0 y 360) . The problem parameters are marked in the figure. 3 0.2 m t = 10 mm Section at 2 a–a and b–b 4m b P b a 1 1 a 2 4m FIGURE P2-5 2-6 For the truss in Problem 2±5, calculate the response (the bar force, stress, strain, deformation, and displacement) when it is subjected to a load (P 1 kip) applied at the angle y 1:5 p rad. 2-7 Deformations have been calculated for the six-bar truss shown in Fig. P2-7 (b1 À20 mm, b2 À56:6 mm, b3 16:66 mm, b4 À47:13 mm, b5 16:66 mm, b6 0:0). Draw the displacement diagram using Williot's method. Can you draw the diagram if one of the six deformations, for example b5 , is not available to you? 4 1 1 4 2 100 in. 6 3 5 3 2 100 in. FIGURE P2-7 Determinate Truss 125 2-8 Draw the displacement diagram using Williot's method for the truss shown in Fig. P2-8 for the following cases. 1. For an increase in the temperature from its ambient manufacturing environment at 25 C to 100 C. 2. For settling of support node 1 by 0.5 in. in the x-coordinate direction and by 1 in. the negative y-coordinate axis at node 3. 3. For simultaneous occurrence of cases 1 and 2. 5 6 /2 /2 4 3m /2 /2 3 1 2 4m 5m FIGURE P2-8 2-9 Calculate the response (internal force, reaction, stress, strain, deformation, and displacement) of the steel truss shown in Fig. P2-9 for the following load conditions. Assume that the areas of the inclined bars are 880 mm2 and that those of the other bar are 500 mm2. P7y = 4 kN 8 7 6 5 P5x = 1 kN 7 6 5 8 9 10 11 12 13 4 3m 1 2 3 1 2 3 4 x y a a=3m a FIGURE P2-9 126 STRENGTH OF MATERIALS 1. For a mechanical load as shown in the figure. 2. For a temperature increase of ÁT 100 C for the bottom chord members (1, 2, 3). 3. For settling of support node 1 by 12 mm in the x-coordinate direction and for node 4 by 25 mm in the negative y-coordinate direction. 4. For simultaneous action of the mechanical load, thermal load, and support setting. 2-10 The steel truss shown in Fig. P2-10 is made of bars with areas of 1 in.2 Other dimensions are marked in the figure with a 3 m. Calculate the reactions and the stresses in the bars joining nodes 8±9, 9±7, 6±10, and 10±5. 9 10 0.5 kN a/2 8 6 5 7 a 1 2 3 4 1 kN 1 kN a a a FIGURE P2-10 2-11 The roof truss shown in Fig. P2-11 is made of equal-leg steel angle sections of the dimensions 203  203  25:4 mm3 . Calculate the response in the midspan diagonal bar only (do not analyze the entire truss). 3m 50 kN 2m 2m 2m FIGURE P2-11 2-12 A composite angle section (102  51  12:67 mm3 ) made of aluminum and steel has the dimensions shown in Fig. P2-12. For each material, calculate the stress, strain, and displacement at its midlength for the following load cases. 1. A 10-kN mechanical load. 2. A uniform temperature change (ÁT 50 C) throughout its volume. 3. An axial displacement of 1 in. at its free node. Determinate Truss 127 c c Steel Aluminum 3m Cross-section at c–c P = 10 kN FIGURE P2-12 128 STRENGTH OF MATERIALS 3 Simple Beam A beam is a horizontal structural member that supports a vertical load. Its length is much larger than its cross-sectional area. For a typical beam, the slenderness ratio SR, defined as the ratio of length ` to the cross-sectional area A (SR `/A), can be 20 or more (SR `/A ! 20). A beam cross-section can be rectangular, circular, or annular, or it can be a rolled I-section or a built-up section. Beams are fabricated of steel, aluminum, concrete, wood, and composite materials. They are used in buildings, bridges, aircraft, machinery, and other types of structures. A simple beam rests on two or fewer supports. A lintel placed across the space between two columns is an example of a simple beam. A cantilever projecting out of a single supporting wall is also a simple beam. Simple beams are determinate structures. In contrast, a continuous beam on more than two supports is an indeterminate structure. This chapter is devoted to the analysis of simple determinate beams. A cantilevered beam is depicted in Fig. 3-1a. It has a length ` with a uniform rectangular cross-sectional area (A dt), depth d, and thickness t. It is subjected to a transverse load (P). Galileo began the solid mechanics discipline around 1638 with the analysis of a cantilevered beam (see Fig. 1-37). A uniform rectangular beam resting on two supports, referred to as a simply supported beam, is depicted in Fig. 3-1b. This beam is subjected to a distributed transverse load (p) that is uniform along a portion of its span (a). Analysis of most other simple beams can be obtained as an extension of the solution of the two examples. For analysis, a beam is sketched as a rectangle with length ` and depth d, as shown in Fig. 3-2a. The length ` is along the beam centerline, which coincides with the x-coordinate axis. The y-coordinate axis coincides with the beam depth d. The thickness, or width, of the beam, which is along the z-coordinate axis, plays a passive role in the analysis and is not shown in the sketch. The beam cross-section must be symmetrical about the y-axis. The cross-section can be rectangular, circular, annular, or a rolled I-section, as shown in Fig. 3-2b. In general, a beam section need not be geometrically symmetrical about the z-coordinate 129 p per unit length P A a D C B a a p d d t t Section a–a (a) Cantilever beam. (b) Simply supported beam. FIGURE 3-1 Example of simple beams. y y d/2 d/2 z d/2 y P Rectangular Circular p a y y x z a d/2 d/2 (a) Beam axes. z z Annular I–Section (b) Symmetrical beam cross-sections. FIGURE 3-2 Beam coordinate axes. axis, but in this chapter it will be thus assumed. In other words, the beam section is assumed to be symmetrical about both the y- and z-coordinate axes. The x-z plane, which cuts the beam into two equal halves, is referred to as the neutral plane, as shown in Fig. 3-3. Since the beam thickness is not shown in a typical sketch, engineers popularly refer to the x-axis as the neutral axis, but they actually imply the neutral plane. 130 STRENGTH OF MATERIALS y y Neutral plane x d Neutral z axis FIGURE 3-3 Neutral plane of a beam. Analysis of beam is more involved than truss analysis because its response variables are functions of the x-coordinate. Beam analysis is introduced in stages. A general formulation is given at the conclusion of all the stages. 3.1 Analysis for Internal Forces A beam resists an external load by inducing internal forces. The external load can be a point load (P), a distributed load (p) (see Fig. 3-1), or an applied bending moment, as shown in Fig. 3-4. The loads (P and p) must be applied along the y-coordinate direction and are positive when directed along the positive y-axis. The moment (M 0 ) is positive when its line of action is along the z-axis, and it is shown by a counterclockwise arrow. The external moment (m0 ) can also be distributed over a portion of the beam span. In USCS units, load can be specified in pound-force (lbf) and beam dimensions in inches. Concentrated load (P) is specified in pound-force, distributed load ( p) in pound-force per inch; moment (M 0 ) in inch pound-force, and distributed moment (m0 ) in inch pound-force per inch. In SI units, load P can be specified in kilonewton, p in kilonewton per meter, moment M 0 in kilonewton- meter, and m0 in kilonewton-meter per meter. The beam resists the applied load by inducing an internal bending moment (M) and a shear force (V ) at a location (x), as shown in Fig. 3-5. The internal bending moment and shear force follow the t-sign convention as discussed in Chapter 1. Accordingly, both moments M` (with n` 1 and f ` 1) and Mr (with nr À1 and f r À1) are positive. Likewise, both shear forces V` (with n` 1 and f ` 1) and Vr (with nr À1 and f r À1) are positive. Equality of internal force is established through the transverse or shear EE and the moment EE written at location x. y M0 m0 x z FIGURE 3-4 Applied bending moments. Simple Beam 131 P y x x M Mr z V Vr y f r = –1 n r = –1 n =1 f =1 FIGURE 3-5 Sign convention for bending moment and shear force. F 0X V` À Vr 0 or V ` V r V 3-1a y M 0X M` À Mr 0 or M ` M r M 3-1b z M and V designate the bending moment and shear force at a section, respectively. The equilibrium formulation must follow the n-sign convention. Boundary Conditions The free end B of the cantilevered beam shown in Fig. 3-6a has no restraints, and it is free to T displace and rotate. In other words, the displacement [v(x `) 0], and the slope T [y(x `) 0] are nonzero quantities. Displacement in a beam is specified along its neutral axis. Displacement along the beam length, also referred to as the elastic curve, is depicted in Fig. 3-6a. There is no reaction at the free end. The moment and shear force are zero (M V 0). This information constitutes the boundary condition at the free end and is stated as M x ` 0 3-2a V x ` 0 3-2b v x ` T 0 3-2c y x ` T 0 3-2d The built-in end B of the beam is fully restrained. This end can neither displace nor rotate [v(x 0) 0], and the slope [y(x 0) 0] is zero. There are two reactions at the fixed end. 132 STRENGTH OF MATERIALS y P P R C D A B x v r MR Rr R (a) Cantilevered beam. (b) Simply supported beam. FIGURE 3-6 Boundary conditions of beams. The moment and shear force are nonzero [M(x 0) T 0 and V(x 0) T 0]. This informa- tion constitutes the boundary condition at the fixed end A. M x 0 T 0 3-3a V x 0 T 0 3-3b v x 0 0 3-3c y x 0 0 3-3d Consider next the simply supported beam shown in Fig. 3-6b. It is hinged to the founda- tion at C. This support condition suppresses displacements but allows rotation. The displace- ments (u, v) along the x- and y-coordinate directions are zero [u(x 0) 0 and v(x 0) 0], but the slope exists [y(x 0) T 0]. There are reactions along the x- and y-coordinate directions, but the moment is zero (M 0). This information constitutes the boundary condition at the simply supported end C. M x 0 0 3-4a V x 0 T 0 3-4b u x 0 0 3-4c v x 0 0 3-4d y x 0 T 0 3-4e The end at D is supported on rollers. The hinged conditions given by Eq. (3-4) are modified to obtain the boundary conditions at the roller support. Such a support allows displacement along the x-coordinate direction (u(x `) T 0). M x ` 0 3-5a V x ` T 0 3-5b u x ` T 0 3-5c v x ` 0 3-5d y x ` T 0 3-5e Simple Beam 133 For the force analysis of the determinate beam, we will use the boundary conditions imposed on the shear force and bending moment. The conditions on displacement will be utilized during the calculation of displacement. The hinge boundary condition (u v 0) ensures the stability of a beam. A boundary condition on axial displacement u(x) is usually neglected in simple beam analysis because it is small. Bending Moment and Shear Force Diagrams Consider a beam with specified dimensions, boundary conditions, and external loads. The problem is to determine the bending moment M(x) and shear force V(x) at any location x along the beam span. The graphsÐmoment versus the x-coordinate and shear force versus the x-coordinateÐare called the bending moment and shear force diagrams, respectively. The problem is to determine both diagrams. The determination of the diagrams is illustrated through several beam examples. EXAMPLE 3-1 Determine the bending moment and shear force diagrams for the cantilever beam shown in Fig. 3-7a. It is ` units long and subjected to a load P at its free end along the negative y-coordinate direction. Solution The solution is obtained in two steps. First the reactions are calculated. The reactions are used next to determine the bending moment (BM) and shear force (SF) diagrams. Step 1ÐCalculation of Reactions As mentioned earlier, the three equilibrium equations yield the three reactions of a plane structure. A simple beam, however, has no load along the x-coordinate direction. The EE along this direction is satisfied trivially and not explicitly written. The two EE required to calculate the reaction are given by Eq. (3-1). The cantilevered beam has two reactions at its fixed end AÐBM (MR) and transverse force (R)Ðbut it has no reaction at its free end. The forces acting on the beam are shown in the free-body diagram in Fig. 3-7b. Since the reactions are not yet known, we assume them to be positive. The moment is along the counterclockwise direction, and the transverse force is along the y-coordinate direction. Application of Eq. (3-1) yields the values of the reactions. F 0X R À P 0 or RP 3-6a y M 0X MR À P` 0 or MR P` 3-6b A The assumed directions for the reactions MR and R turn out to be correct. 134 STRENGTH OF MATERIALS P y P A B x R MR (a) Cantilever beam. (b) Free-body diagram. M(x) R C x V(x) MR (c) Bending moment and shear force at x. FIGURE 3-7 Free-body diagrams to calculate bending moment and shear force. Step 2ÐBending Moment and Shear Force Diagrams Consider a location C at a distance x from the origin as shown in Fig. 3-7c. Mark the internal moment M(x) and shear force V(x) along the positive y-coordinate direction. The transverse equilibrium equation and the rotational EE at C yield: F 0X R V x 0 3-7a y M 0X MR À Rx M x 0 3-7b C V x ÀP 3-7c M x Px À P` 3-7d The shear force V(x) is the negative of the applied load P, and it is constant throughout the span of the cantilever. The bending moment M(x) is a linear function in the x-coordinate. It peaks at the origin with M(x 0) ÀP`, and it has a zero value at the free end. The graph of the shear force V(x) versus the x-coordinate, or the shear force diagram, is depicted in Fig. 3-8a. Likewise, the graph of the bending moment M(x) versus the x-coordinate, or the bending moment diagram, is shown in Fig. 3-8b. Both the shear force and bending moment are negative throughout the span of the beam. Simple Beam 135 y A x y y M(x) V(x) x= x x –P –P (a) SF diagram. (b) BM diagram. FIGURE 3-8 SF and BM diagrams for Example 3-1. EXAMPLE 3-2 Determine the bending moment and shear force diagrams for the simply supported beam shown in Fig. 3-9a. The beam is ` units long and is subjected to a load P along the negative y-coordinate direction at its quarter span. Solution Step 1ÐCalculation of Reactions The simply supported beam has two reactions (RA and RB), as shown in Fig. 3-9b. The reactions are obtained from the transverse EE and rotational EE written at A. F 0 X R A RB À P 0 3-8a y ` M 0 X RB ` À P 0 3-8b A 4 3P RA 3-8c 4 P RB 3-8d 4 The support A, which is nearer to the load, carries three-quarters of the load, while the far support at B has to share one-quarter of the load. Both reactions (RA and RB) are along the positive y-coordinate direction. 136 STRENGTH OF MATERIALS y A x P P A C B D E /4 RA RB (a) Simply supported beam. (b) Free-body diagram. M(x) P M(x) V(x) V(x) A D E x x RA R = 3P/4 (c) BM and SF in span AC. (d) BM and SF in span CB. FIGURE 3-9 SF and BM in a simply supported beam. Step 2ÐBending Moment and Shear Force Diagrams Consider a location D at a distance x from the origin inside the beam span AC (0 x `/4) as shown in Fig. 3-9c. Mark the internal moment M(x) and shear force V(x), considering them positive. The transverse equilibrium equation and the rota- tional EE at D yield V x RA 0 M x À RA x 0 or W V x À 3P a 4 ` for 0 x 3-8e Y 4 M x 3Px 4 Simple Beam 137 Likewise, consider a location E at a distance x from the origin inside the beam span CB (`/4 x `), as shown in Fig. 3-9d. Mark the internal moment M(x) and shear force V(x), considering them positive. The transverse equilibrium equation and the rotational EE at E yield 3P V x À P 0 4 ` 3 M x P x À À Px 0 4 4 or A V x P 4 ` for x ` 3-8f M x P` À Px 4 4 4 The shear force V(x) is graphed in the span AC (0 x /4) using Eq. (3-8e) and in the span CB (`/4 x `) from Eq. (3-8f ), as shown in Fig. 3-10a. The bending moment M(x) is graphed in the span AC (0 x `/4) using Eq. (3-8e) and in the span CB (`/4 x `) from Eq. (3-8f), as shown in Fig. 3-10b. The SF diagram exhibits a discontinuity at C that is the point of application of the load P. The shear force at the left (V ` À3P/4) and right (V r P/4) of location C differ by the load value P. The application of load P created a discontinuity in the shear force diagram. The shear force V(x) is negative in the span AC {0 x `/4}, but it is positive in the span CB (`/4 x `). The bending moment at C is continuous with a positive slope in the left and a negative slope in the right of the load application point C. The bending moment is positive in the entire length of the beam. y A x y M(x) P/4 V(x) A C B x –3P/4 P 3P /16 A C B x (a) SF diagram. (b) BM diagram. FIGURE 3-10 SF and BM diagrams for Example 3-2. 138 STRENGTH OF MATERIALS EXAMPLE 3-3 Determine the bending moment and shear force diagrams for the simply supported beam shown in Fig. 3-11. It is ` units long and is subjected to a moment M 0 at its midspan. Solution Step 1ÐCalculation of Reactions The two reactions (RA and RB) of the beam are obtained from the transverse EE and rotational EE written at A. R A RB 0 3-9a 0 RB ` M 0 3-9b 0 M RA 3-9c ` M0 RB À 3-9d ` y A x M0 M(x) V(x) A B C D x RA /2 RA RB (a) Beam subjected to moment. (b) Span AC. M0 M(x) A E C V(x) x RA (c) Span CB. FIGURE 3-11 Bending moment and shear force for concentrated moment. Simple Beam 139 The reaction RA is positive at support A. It is negative (RB) at support B. Step 2ÐBending Moment and Shear Force Diagrams Consider a location D at a distance x from the origin inside the portion of the beam span AC (0 x `/2), as shown in Fig. 3-11b. Mark the internal moment M(x) and shear force V(x), considering them positive. The transverse equilibrium equation and the rotational EE at D yield M0 V x ÀRA À 3-9e ` 0x M x RA x M 3-9f ` Likewise, the BM and SF are obtained for the span CB {`/2 x `} using Fig. 3-11c. M0 V x ÀRA À 3-9g ` M x M 0 À RA x 0 M0 M x x À ` 3-9h ` The shear force diagram is constructed for the entire span, because there is no transverse load, and it is depicted in Fig. 3-12a. The shear force V is uniform across the span and satisfies Eqs. (3-9e) and (3-9g). The bending moment diagram is constructed for span AC (0 x `/2) using Eq. (3-9f ) and for span CB (`/2 x `) from Eq. (3-9h), as shown in Fig. 3-12b. The BM diagram, as expected, shows an abrupt discontinuity of magnitude M0 at C, which is the point of application of the moment. V(x) Mo/2 M(x) –Mo A C B x A C B x –Mo/2 (a) SF diagram. (b) BM diagram. FIGURE 3-12 SF and BM diagrams for Example 3-3. 140 STRENGTH OF MATERIALS EXAMPLE 3-4 Determine the bending moment and shear force diagrams for the simply supported beam shown in Fig. 3-13a. It is ` units long and is subjected to a distributed load p per unit length. Solution Step 1ÐCalculation of Reactions For the calculation of the reactions, the distributed load in the entire span can be lumped by an equivalent load (P p`) acting at the center of the span, as shown in Fig. 3-13b. The two reactions (RA and RB ) of the beam are obtained from two rotational EE written at A and B, respectively. This procedure is equivalent to the earlier method of writing one transverse EE and one rotational EE. ` RB ` À p` 0 2 ` À RA ` p` 0 2 p` RA RB 3-10a 2 Step 2ÐBending Moment and Shear Force Diagrams Consider a location D at a distance x from the origin as shown in Fig. 3-13c. Mark the internal moment M(x) and shear force V(x), considering them positive. Also lump the distributed load px px at the location x/2 from the origin. The transverse equilibrium equation and the rotational EE at D yield p` V x À px 0 2 ` V x p x À 3-10b 2 x p` M x px À x0 2 2 px M x ` À x 3-10c 2 The shear force diagram is constructed from Eq. (3-10b). It is linear with a negative value at A(V(x 0) Àp`/2), and it is positive at B(V(x `) p`/2), as shown in Fig. 3-14a. The bending moment diagram is constructed from Eq. (3-10c). It is positive throughout the span, as shown in Fig. 3-14b. It has a parabolic shape with a maximum value {Mmax M(x `/2) p`2 /8} at the midspan (x `/2). Simple Beam 141 y A x p per unit length P=p B A /2 RA RB (a) Beam subjected to distributed load. (b) Support reactions. M(x) x/2 px V(x) D x RA (c) BM and SF. FIGURE 3-13 BM and SF diagrams for Example 3-4. V(x) p /2 A p 2/8 M(x) B x –p /2 A B x (a) SF diagram. (b) BM diagram. FIGURE 3-14 SF and BM diagrams for Example 3-4. EXAMPLE 3-5 Determine the bending moment and shear force diagrams for a simply supported beam with an overhang as shown in Fig. 3-15a. The beam is 180 in. long with a 60-in. overhang. It is subjected to a distributed load of 100 lbf/in. along the center 100 in. of its span. The overhang carries a load of 1 kip long the negative y-coordinate direction. 142 STRENGTH OF MATERIALS y 1000 lbf 10,000 1000 lbf A x 100 lbf/in. B B A C C 100 F E G 40 50 180 in. 60 RA RB (a) Overhung beam. (b) Support reactions. 100 (x – 40) M(x) M(x) V(x) V(x) F x x RA RA (c1) Segment AF. (c2) Segment FG. 10,000 1000 M(x) M(x) V(x) C G x V(x) 240 – x RA x (c3) Segment GB. (c4) Segment BC. FIGURE 3-15 BM and SF diagrams for Example 3-5. Solution Step 1ÐCalculation of Reactions The two reactions (RA and RB ) of the beam are obtained from a transverse EE and a rotational EE written at E, which is the midpoint of the distributed load p. FX RA RB À 11;000 0 y MX À90RA 90RB À 1000  150 0 E Simple Beam 143 RA 4667 lbf 4:667 kip 3-11a RB 6333 lbf 6:333 kip 3-11b The accuracy of the reaction can be checked by writing the moment EE at any convenient location. Select locations at A and C because the moment is zero at these points (MA MC 0). MA À1  240 6:333  180 À 10  90 0 3-11c MC À4:667  240 10  150 À 6:333  60 0 3-11d Step 2ÐBending Moment and Shear Force Diagrams Because of the nature of load distribution, the BM and SF diagrams have to be constructed separately for four segments: AF, FG, GB, and BC, as shown in Fig. 3-15(c1) to (c4). The BM and SF for each segment follow. Segment AF (0 x 40): V x ÀRA À4667 3-11e M x 4667x 3-11f Segment FG: V x À4667 100 x À 40 100x À 8667 3-11g 100 M x 4667x À x À 402 4667x À 50 x À 402 3-11h 2 Segment GB: V x À4667 10;000 5333 3-11i M x 4667x À 10;000 x À 90 900;000 À 5333x 3-11j Segment BC: For simplicity, the calculation for this section uses the loads in the overhang. M(x) and V(x) are positive as marked in Fig. 3-15(c4). V x À1000 3-11k M x À1000 240 À x 3-11l 144 STRENGTH OF MATERIALS The SF and BM are verified for few locations in the beam: Location F: The SF and BM calculated at F (x 40) from segments AF and FG must be in agreement. Segment AF: VF À4667 MF 4667  40 186;680 Segment FG: VF V x 40 100  40 À 8667 À4667 3-12a MF M x 40 4667 x 186;680 3-12b Location G: The SF and BM calculated from segments FG and GB are in agreement. VG V x 140 5333 3-12c MG M x 140 15;338 3-12d Location B: The SF and BM calculated from segments GB and BC are in agreement. VB V x 180 À1000 3-12e MB M x 180 À60;000 3-12f The SF and BM for the overhang BC are calculated from support point C. Such a calculation is legitimate as long as the moment M(x) and shear force V(x) are considered to be positive and are appropriately marked. The BM at B is marked with a clockwise arrow because the normal to the section is negative (n À1). For the same reason, the shear force is directed along the negative y-coordinate direction. The x-coordinate is measured from the origin at A. (186.7) (153.4) 5.3 V(x) RB x M(x) x A F G B C A F G B C –4.6 1 (–60) (a) SF diagram in kip. (b) BM diagram in in.- k. FIGURE 3-16 SF and BM diagrams for Example 3-5. Simple Beam 145 The SF diagram is constructed for the four segments using the relevant equations: for segment AF, Eq. (3-11e); for segment FG, Eq. (3-11g); for segment GB, Eq. (3-11i); and for segment BC, Eq. (3-11k). In segment AF, the shear force has a constant value of À4:667 kip because there is no distributed load. In the segment FG, the SF variation is linear because of the distributed load. The SF is constant in segments GB and BG. An abrupt variation occurs at B because of reaction RB . The BM diagram is also constructed for the four segments using the relevant equations: for segment AF, Eq. (3-11f); for segment FG, Eq. (3-11h); for segment GB, Eq. (3-11j); and for segment BC, Eq. (3-11l). The BM diagram has a linear variation in segment AF because there is no distributed load. In the segment FG, the BM variation is quadratic because of distributed load. The BM is linear in segments GB and BG. It attains a maximum value of Mmax (x 40 in:) 186:68 in.-k at F. EXAMPLE 3-6 Determine the bending moment and shear force diagrams for a simply supported beam with two overhangs as shown in Fig. 3-17a. It is 100 in. long with 50-in. overhangs. It is subjected to symmetrical load as shown in the figure. Solution Step 1ÐCalculation of Reactions The two reactions (RA and RB ) are calculated from the moment EE written at A and B, respectively. MX RB  100 À 1  150 À 5  75 À 5  25 1  50 0 A RB 6 kip 3-13a MX À RA  100 1  150 5  75 5  25 À 1  50 0 B RA 6 kip 3-13b It is easily verified that the reactions satisfy the transverse equilibrium equation. FX RA RB À 1 5 5 1 0 3-13c Y The reactions are equal (RA RB ) because of symmetry. Each reaction is equal to half the applied load. 146 STRENGTH OF MATERIALS y C x P2 = 5 P2 = 5 P1 = 1 M(x) M(x) P1 = 1 kip 1 1 V(x) V(x) A E F B D A C x x RA = 6 25 25 50 100 in. 50 RA RB (a) Beam with two overhangs. (b) Segment CA. (c) Segment AE. M(x) 1 5 V(x) M(x) 1 E V(x) C D F B RA = 6 x 50 in. x 150 – x (d) Segment EF. (e) Segment FB. M(x) 1 V(x) B D x 200 – x (f) Segment BD. FIGURE 3-17 Analysis of beam with overhangs. Step 2ÐBending Moment and Shear Force Diagrams Because of the nature of load distribution, the BM and SF diagrams have to be constructed separately for five segments: CA, AE, EF, FB, and BD. The BM and SF for the segments are illustrated in Figs. 3-17b to 3-17f. Segment CA (0 x 50): V x 1 3-13d M x Àx 3-13e Simple Beam 147 Segment AE (50 x 75): V x À5 3-13f M x 5x À 300 3-13g Segment EF (75 x 125): V x 0 3-13h M x 75 3-13i Segment FB (125 x 150): V x 5 3-13j M x À5x 700 3-13k Segment BD (150 x 200): V x 1 3-13l M x 200 À x 3-13m We construct the SF diagram for the five segments by using the relevant equations: for segment CA, Eq. (3-13d); for segment AE, Eq. (3-13f); for segment EF, Eq. (3-13h); for segment FB, Eq. (3-13j); and for segment BD, Eq. (3-13l). The shear force diagram is asymmetrical with the beam center G. The shear force is constant (V 1) in the overhangs, and it is zero in the central segment EF. In segments AE and FB, shear force is uniform but negative (V À5 kip) and positive (V 5 kip), respectively. The BM diagram is also constructed for the five segments using the relevant equations: for segment CA, Eq. (3-13e); for segment AE, Eq. (3-13g); for segment EF, Eq. (3-13i); for segment FB, Eq. (3-13k); and for segment BD, Eq. (3-13m). The bending moment diagram is symmetrical about the beam center G. The BM has a linear variation in the overhangs, and it is negative. It is constant at M 75 in:-k in the center segment EF. In segments AE and FB, the bending moment is linear, but it changes sign, as shown in Fig. 3-18b. V(x) (75) (75) M(x) 6 5 60 1 C A E F B D C A E G F B D –5 (–50) (–50) (a) SF diagram in kip. (b) BM diagram in in.- k. FIGURE 3-18 SF and BM diagrams for Example 3-6. 148 STRENGTH OF MATERIALS 3.2 Relationships between Bending Moment, Shear Force, and Load The bending moment, shear force, and transverse load in a beam are related. These relationships can be used to verify the accuracy of the bending moment and shear force diagrams. The relationships between the three variables are obtained from the transverse EE and rotational EE of an elemental block of length Áx, as shown in Fig. 3-19. The forces acting on the block are marked as follows. 1. Distributed transverse load p is considered positive when directed along the positive y-coordinate axis. 2. Internal bending moment is M in the left face and it is (M ÁM) on the right face. Both moments are positive according to the t-sign convention. The moment increases by ÁM between the two faces. 3. Internal shear force is V in the left face and it is (V ÁV) on the right face. Both forces are positive according to the t-sign convention. The shear force increases by ÁV between the two faces. The block is in equilibrium under the action of the forces. The equilibrium equation along the y-coordinate direction yields F 0X V ÁV pÁx À V 0 y Dividing by Áx and taking the limit as Áx tends to zero & ' ÁV Lt Àp Áx30 Áx y A x p M M+ M Beam depth V z V+ V x FIGURE 3-19 Forces in an elemental beam block. Simple Beam 149 dV Àp 3-14a dx The rate of change of shear force with respect to distance x along the beam axis is equal to the negative of the distributed transverse load p. The moment equilibrium equation at a point z, as marked in Fig. 3-19, yields Áx M V ÁV Áx M ÁM À M pÁx Á 0 z 2 & ' ÁM Áx Lt V p 0 Áx30 Áx 2 dM ÀV 3-14b dx V x À pdx C1 3-14c M x À Vdx C2 3-14d Second-order terms are neglected. The term (ÁV) is neglected because it is small compared to V. In the limit (pÁx) becomes small and is neglected. The rate of change of the bending moment with respect to distance x along the beam axis is equal to the negative of the shear force V. The relationships can be integrated to obtain two more formulas given by Eqs. (3-14c) and (3-14d). The constants of integration (C1 and C2) have to be determined from the boundary conditions of the problem. For the calculation of constant C1 in Eq. (3-14c), a known value of shear force V V a at a location (x a) has to be used. Likewise, for the calculation of constant C2 in Eq. (3-14d), a known value of bending moment M M a at location (x a) can be used. EXAMPLE 3-7 Verify the SF diagram from the BM diagram in Example 3-6. Solution As observed earlier, the SF and BM diagrams have five segments. Verification has to be performed individually. Segment CA (0 x 50): M x Àx 3-13e dM x V x À 1 dx 150 STRENGTH OF MATERIALS Segment AE (50 x 75): M x 5x À 300 3-13g dM x V x À À5 dx Segment EF (75 x 125): M x 75 3-13i dM x V x À 0 dx Segment FB (125 x 150): M x À5x 700 3-13k dM x V x À 5 dx Segment BD: M x 200 À x 3-13m dM x V x À 1 dx The shear force that is back-calculated from the moment is verified successfully for each segment. EXAMPLE 3-8 Verify the bending moment diagram from the shear force diagram in Example 3-6. Solution The bending moment diagram has to be verified separately for each segment. Segment CA (0 x 50): V x 1 M x À V xdx C1 Àx C1 To calculate the constant of integration C1, one must know and use the value of M(x) at any location in the segment CA (0 x 50). BM is zero at C; that is, at x 0, M(0) 0, or C1 0. Simple Beam 151 M x Àx Segment AE (50 x 75): V x À5 M x 5dx C1 M x 5x C1 The constant C1 is determined from the condition at x 50; M(50) À50 as calculated from segment CA because this segment also includes the point x 50. À50 5  50 C1 C1 À300 M x 5x À 300 Segment EF (75 x 125): V x 0 M x C1 At x 75, M(x) 75 C1 M x 75 Segment FB (125 x 150): V x 5 M x À5x C1 At x 125, M(x) 75 À5  125 C1 C1 700 M x À5x 700 Segment BD (150 x 200): V x 1 M x Àx C1 At x 200, M(x) 0 À200 C1 C1 200 M x 200 À x 152 STRENGTH OF MATERIALS 3.3 Flexure Formula Stress is induced in a beam because of the internal bending moment and the shear force. Bending moment induces normal stress, whereas shear force induces shear stress. The relationship between the normal stress s and the bending moment M is referred to as the flexure formula. This important formula is developed in this section. The shear stress formula is developed in Section 3.4. The flexure formula to be derived is illustrated in Fig. 3-20, and it has the following form. Às M 3-15 y I y z x M y t (y) M d/2 d y d x Neutral axis dx (a) Three-dimensional model. (b) Two-dimensional model. y td3 z d Iz = I = 12 t (c) Moment of inertia. FIGURE 3-20 Distribution of bending stress in a beam. Simple Beam 153 Here s is the stress at a distance y from the neutral axis. M is the bending moment, and I is the moment of inertia of the beam cross-section. For a beam with a rectangular cross-section, the stress s is uniform across its thickness as shown in Fig. 3-20a. The uniform distribution across the thickness allows the use of a two-dimensional illustration without any con- sequence, as shown in Fig. 3-20b. The top fiber of the beam at depth y d/2 is in compres- sion, whereas the bottom fiber at depth y Àd/2 is in tension. The stress in a beam changes sign along its depth, and it is zero at y 0. The fibers with no stress lie in the x±z plane at y 0, and this is referred to as the neutral plane. In the two-dimensional representation, this plane degenerates to the neutral axis. The parameter I in the flexure formula is the moment of inertia of the beam cross-section about the neutral axis. For a symmetrical, rectangular cross- section, the moment of inertia about the z-coordinate axis, as shown in Fig. 3-20c, is 1 3 I Iz td 3-16 12 The moment of inertia property of a beam cross-section is further discussed in Appendix 2. The development of the flexure formula took nearly two centuries. It is credited to Coulomb (1736±1806) and to Navier (1785±1836). This formula correctly predicts the absolute maximum stress at jsmax j 6jM/td 2 j at the beam top or bottom fiber at (y Æd/2). An earlier version of the formula by Galileo (1564±1642) predicted stress at jsmax jGalileo 2jM/td 2 j, which is one-third the correct value. Another formula by Bernoulli (1654±1705) produced half the correct value for stress at jsmax jBernoulli j3M/td 2 j. The error is attributed to the positioning of the neutral axis. The derivation of the flexure formula is based on assumptions that pertain to the material property, the pure flexure condition, and the kinematics of deformation. Each assumption is discussed. Material Property The material of the beam is assumed to be linearly elastic. The stress and strain are assumed to lie inside the linear elastic domain (see the diagram in Fig. 1-28). Poisson's effect of sympathetic deformation is neglected because the Poisson's ratio (n 0) is set to zero, and the elastic modulus E is considered to be a constant. In beam bending, the Poisson's ratio produces an anticlastic curvature, as sketched in Fig. 3-21a. The beam curvature (which is in the x±y plane because the moment M is applied in that plane) is referred to as the normal or longitudinal curvature k. Its reciprocal is the radius of curvature (r 1/k). Poisson's ratio induces the transverse anticlastic curvature kn in the y±z plane. It is the product of the Poisson's ratio and the curvature (kn nk). The associated radius of curvature (rn 1/kn r/n) is larger than the radius of curvature because Poisson's ratio is less than unity. The reader can bend a rectangular rubber eraser to observe both curvatures. In our analysis, Poisson's effect is neglected to obtain the deformation pattern in the x±y plane, as shown in Fig. 3-21b. The beam deforms into a cylindrical shape with radius r at the neutral axis. Pure Flexure Condition Under this assumption, the beam is in a state of pure bending without the presence of the shear force or axial force. A pure flexure test case is easily setup as shown in Fig. 3-22a. The center half span of the beam of length ` is subjected to a constant bending moment 154 STRENGTH OF MATERIALS y z x M 0' x Neutral M axis M M Neutral surface Beam ρ Axis >ρ υ (b) Curvature neglecting Poisson's (a) Antielastic curvature. effect. FIGURE 3-21 Curvature in a beam. (M P`/4) when two equal loads (P) are applied at the `/4 and 3`/4 span locations. The beam has no axial force. However, in a typical beam location both bending moment and shear force occur together. For example, at the one-eighth span location the moment M P`/8 and the shear force V ÀP (see Fig. 3-22c). The beam theory assumes a pure flexure condition. Kinematics Assumption It is assumed that a beam segment is initially straight, as shown in Fig. 3-23a. For the purpose of simplification, let us assume a rectangular cross-section for the beam with the neutral axis at mid depth. This assumption is not mandatory but imposes little limitation to the theory. The deformed pattern of the beam under pure moment is sketched in Fig. 3-23b. It is assumed that an initially straight fiber (ad) across the beam depth d remains straight (aH d H ) even after deformation. The fiber (ad) rotates to the deformed position (aH dH ). The perpendi- cularity between ad and ef is maintained after deformation (aH dH and eH f H ). The fiber (ef) does Simple Beam 155 y A x P P (P /4) (P /4) M(x) A B /4 /2 /4 (a) Loads. (b) Bending moment diagram. V(x) P –P (c) Shear force diagram. FIGURE 3-22 Pure flexure in a beam. not deform, and it is identical to the fiber (eH f H ). This fiber (ef or eH f H ) is referred to as the neutral axis. A parallel fiber at a distance y d/4 above the neutral axis contracts, say by amount ÀÁ`. A similarly positioned fiber below the neutral axis at a distance y Àd/4 expands by the same amount Á`. The deformed shape, including the depth dimension of the beam, is shown in Fig. 3-23c. The undeformed cross-sectional plane (s1 ±s2 ±s3 ±s4 ) with the normal n; deforms into another plane (sH1 ±sH2 ±sH3 ±sH4 ) with the normal nH . Both (s1 Às2 Às3 À s4 ) and (sH1 ±sH2 ±sH3 ±sH4 ) are planes. The rotation of the normals is equal to the rotation of the planes. The line (g1 ±g2 ) and the associated plane (g1 ±g2 ±g3 ±g4 ) do not deform. This plane, which is free from deformation, is called the neutral plane. A block a±b±c±d, shown separately in an enlarged scale in Fig. 3-23d, has the following characteristics: ab ef cd undeformed length; aaH bbH ddH ccH deformation. Maximum strain emax occurs at the fiber dc located at y d/2 from the neutral axis. 2cH c emax 3-17a ef Application of the rule of proportionality to the triangles fccH and fttH yields the value of strain e(y) at a distance y from the neutral axis. 156 STRENGTH OF MATERIALS y Neutral axis (na) y q d c s M q' s' M d' c' O O d O' na O' e f x G z P e' f' p p' r' a b r t p' Section a' b' (a) Straight beam segment (enlarged). (b) Deformed beam (enlarged). d d' c' c s2 t' t' s1 s'2 t t g3 s'1 d/2 g4 y g1 g2 e f na s'3 n s'4 s3 s4 n' a' b' a b (c) Deformation including depth dimension. (d) Deformation in an enlarged block. t max dy y (f) Linear stress variation. FIGURE 3-23 Flexure formula. ttH y ccH d=2 ccH y ttH d=2 2ttH 2ccH y e y À À ef ef d=2 Àemax y e y 3-17b d=2 The strain e(y) is compressive when y is positive. In other words a positive moment induces compressive stress and strain in the upper fibers with positive y coordinate. Simple Beam 157 Hooke's law yields Àsmax y s y Ee y 3-17c d=2 where, smax Eemax . Consider an elemental area (dA tdy) at a location y from the neutral axis, as shown in Fig. 3-23f. The elemental force (dF), which is the product of the stress s times the elemental area (dA tdy), can be written as: dF s ydA The internal force F is obtained by integration as d=2 smax d=2 Àsmax t y2 F dF s ydA À ytdy 0 3-17d d=2 Àd=2 d=2 2 Àd=2 No internal force (F 0) is induced in a beam subjected to pure bending. The moment of inertia (I) about the neutral axis is defined as d=2 td 3 I y2 tdy 3-17e Àd=2 12 Moment (M) is the product of force (F) times distance (y). It can be written as an integral: d=2 smax M ydF À ty2 dy 3-17f d=2 Àd=2 If one uses the definition of moment of inertia given by Eq. (3-17e), the moment can be rewritten as I MÀ smax 3-17g d=2 The flexure formula is obtained by eliminating smax between Eqs. (3-17c) and (1-17g). Às M 3-15 y I The stress variation along the beam depth is shown in Fig. 3-23f. The stress peaks at outermost fibers. The magnitude of the stress is reduced as the neutral axis is approached. A positive moment induces compression in the upper fibers where y is positive and tension in 158 STRENGTH OF MATERIALS the lower fiber corresponds to a negative value for y. The flexure formula is routinely used in the design of structural members. Laboratory tests support the assumptions behind this important formula. Its formulation, as mentioned earlier, required the time span of the two centuries between Galileo, Navier, and Coulomb. 3.4 Shear Stress Formula Consider two wooden planks, one placed on top of the other but not bonded to each other, as shown in Fig. 3-24a. The structure is subjected to a load P. The deformed structure when the planks are unbonded is depicted in Fig. 3-24b. The unbonded planks slide at the interface. The slide is prevented when the planks are bonded as shown in Fig. 3-24c. The bonding prevents motion (slide) but induces shear strain and shear stress at the interface. A beam is an integral block, and it acts like the bonded planks, inducing shear stress. The tendency to slippage is because of differential shear force along the beam depth. A small elemental beam length (Áx) subjected to moments M and M ÁM is shown in Fig. 3-24d. The flexure formula is used to replace the moments with stresses (s ÀMy/I) and s Ás À(M ÁM)y/I) in Fig. 3-24e. The stresses on a small block (k) at height y from the neutral axis are shown in Fig. 3-24f. The unequal normal stresses in the left and right side of the block k are balanced by the induction of shear stress (t) at the bottom of the block. The forces on the left side (F ` ), right side (Fr), and bottom (ÁF b ) acting on the block k of thickness t and length dx, as marked in Fig. 3-24g, are calculated as F` s` dA 3-18a A Fr sr dA 3-18b A ÁF b ttdx 3-18c The force balance equilibrium condition yields F ` À F r ÁF b 0 3-19a s` dA À sr dA ttdx 0 3-19b Stress is eliminated in favor of moment using the flexure formula ÀMy M dM s` Y sr À y I I 1 fÀM M dM gydA ttdx 0 3-19c I Simple Beam 159 y x P (a) Unformed planks. (b) Unbonded planks. M M + dM P x (c) Bonded planks. (d) Moments. + k r y y F Fb Fr z dy t dx (e) Stress. (f) Shear stress. (g) Forces in block k. FIGURE 3-24 Shear stress in a beam. 1 dMydA ttdx 0 3-19d I Bending moment, M, is taken outside the integration because it has no variation along the beam depth. dM À ydA À ttdx 0 3-19e I define Q ydA 3-19f A dM À Q À tt dx 0 3-19g I 160 STRENGTH OF MATERIALS dM Q tÀ 3-19h dx It But from Eq. (3-14b) dM V À dx The shear formula is obtained by eliminating moment in favor of shear force (V ). VQ t 3-20 It The parameters in the shear stress formula given by Eq. (3-20) are: the thickness of the beam t, and its moment of inertia I. The parameter Q is referred to as the first area of the cross-section defined in Eq. (3-19f). The shear force is V, and the shear stress is t. The flexure and shear formulas are illustrated next. EXAMPLE 3-9 A simply supported beam of length ` 240 in. is subjected to a concentrated load (P 10 kip) at the center span, as shown in Fig. 3-25a. Determine the BM and SF diagrams. Calculate the normal stress and shear stress assuming a uniform rectangular cross-section with depth d 12 in. and thickness t 2 in. Solution The reactions are equal (RA RB P/2) because of symmetry. The bending moment and shear force diagrams are shown in Figs. 3-25b and 3-25c, respectively. The shear force and bending moment peak at the midspan, Vmax P 10 kip and Mmax P`/4 600 kip-in. The calculation of stress requires the moment of inertia I and the moment of area Q. For a rectangular cross-section with depth d and thickness t, these two parameters are 1 3 1 I td 2 123 288 in:4 3-21a 12 12 d=2 t d2 À Á Q y ydA ytdy 2 À y 36 À y2 in:3 3-21b y 2 4 The moment of inertia is a constant at I 288 in:4 The moment of area (Q) for a rectangular cross-section is a quadratic function of the distance (y) from the neutral axis. The parabolic variation of moment of area is shown in Fig. 3-26a. It is symme- trical about the neutral axis, and it peaks (Q td 2 /8 36 in:3 ) at the neutral axis. It reduces to zero at the top (y d/2) and bottom (y Àd/2) fibers. Simple Beam 161 P A B V(x) P/2 x –P/2 /2 /2 R A = P/2 R B = P/2 (a) Beam under concentrated load. (b) SF diagram. M(x) P /4 x (c) BM diagram. FIGURE 3-25 SF and BM diagrams for Example 3-9. The normal stress and shear stress at the midbeam span that correspond to M Mmax 600 in.-k and Vmax 10 kip are as follows: Mmax y 600 smax À À y À2:083y 3-22a I 288 Vmax Q 10 Q tmax 17:36  10À3 36 À y2 3-22b It 288  2 Vmax 10 tav 0:4166 ksi 3-22c area 12  2 tmax y0 0:625 ksi 3-22d smax at y d=2 À12:5 ksi 3-22e The distribution of the normal and shear stress are shown in Figs. 3-26c and 3-26d, respectively. From these two diagrams we observe 1. The normal stress due to bending moment M has a linear distribution. It attains a peak value at the top and bottom fibers of the beam, which are equal in magnitude but opposite in sign. The bottom fiber of the beam is in tension, whereas its top fiber is in compression. The bending stress is zero at the neutral axis. 162 STRENGTH OF MATERIALS Q(y) (y = d/2) (y = 0) Qmax = 36 av (y = –d/2) av = 0.42 ksi (a) Moment of area function Q(y). (b) Average shear stress. –12.5 ksi y=0 0.625 max = 1.5 av 12.5 ksi (c) Normal stress distribution. (d) Shear stress distribution. FIGURE 3-26 Distributions of normal and shear stresses. 2. The shear stress due to shear force V has a parabolic distribution. It attains a peak value at the neutral axis. It is zero at both the top and bottom fibers of the beam. 3. The normal stress and the shear stress peak at different locations of the beam cross-section. Bending stress peaks at the extreme fiber where shear stress attains zero value. The shear stress peaks at the neutral axis, where the bending stress is zero. 4. The shear stress at tmax 0:625 ksi is small in comparison to the normal stress (smax À12:5 ksi). The average value of shear stress (tav V/A 10/24 0:42) can be obtained as the ratio of the shear force V to the cross-sectional area A. For a rectangular cross- section, the maximum shear stress calculated from the formula given by Eq. (3-20) is equal to 1.5 times the average stress. tmax 1:5tav 3-23 Simple Beam 163 EXAMPLE 3-10 Calculate the maximum normal strain and shear strain in the beam in Example 3-9. Consider the beam material to be steel with Young's modulus E 30,000 ksi and Poisson's ratio n 0:3. Solution smax 12:5 emax 0:417  10À3 0:042 percent E 30;000 tmax gmax G E 30;000 G 11:54  103 ksi 2 1 n 2:6 0:625 gmax 0:054  10À3 0:005 percent 11:54  103 The material property (consisting of the Young's modulus and Poisson's ratio) is required to calculate strain. For a determinate beam, the material property is not required to calculate moment, shear force, normal stress, and shear stress. 3.5 Displacement in a Beam The determination of beam displacement, also referred to as ``deflection,'' is credited to Leonhard Euler (1707±1783). Earlier, Jacob Bernoulli (1759±1789) had shown the propor- tionality of bending moment and curvature with an incorrect proportionality constant. The beam deflection pattern is also called the elastic curve, and it is obtained as the solution to a differential equation. Euler calculated the elastic curve from moment curvature relations, as well as by minimizing the strain energy. We will derive the governing equation from Hooke's law and also from the moment curvature relation. The underlying principle is the same in both methods with some variation in the details. The basic assumption is illustrated through the example of a simply supported beam, shown in Fig. 3-27a. The deflection is due to bending moment only. Shear strain and the associated shear deflection are neglected. The deformation along the beam depth is neglected. For example, the beam deflection (v(x)) at span location x is the same along the depth at A, B, and N, (v(x) vA (x) vB (x) vN (x)); see Fig. 3-27b. In other words, deflection pertains to the displacement of the neutral axis, as shown in Fig. 3-27c, and it is referred to as the elastic curve. 164 STRENGTH OF MATERIALS Neutral Top fiber y axis A Deformed position A x N v B v(x) v d Undeformed position d(x) d A x B x t (a) Beam displacement. (b) No variation of v along depth. v Elastic v(x) curve Neutral axis x x (c) Elastic curve. FIGURE 3-27 Deflection in a beam. Strain in a Beam Hooke's law, which relates the stress and strain variables, is used to back-calculate strain from stress that is already known. Normal strain (e) due to bending is obtained from the stress-strain formula (s Ee) as À s Ee M y I Àe M 3-24a y EI My Àe 3-24b EI The strain, like the stress, has a linear variation over the depth of the beam. The quantity EI, which is the product of the Young's modulus E and the moment of inertia I in the strain formula is called the bending rigidity of a beam. Strain is inversely proportional to the beam rigidity EI, as shown in Eq. (3-24b). For a beam made of a rigid material like stone, the rigidity (EI) approaches infinity while the strain is reduced to zero. Shear strain g in a beam is obtained using the shear stress and shear strain formula (t Gg). The shear modulus (G) and Young's modulus (E) are related by the formula Simple Beam 165 G 0:5E/(1 n). For metals like steel and aluminum, Poisson's ratio can be approximated at n 0:3. For such materials, the shear modulus is 38.5 percent of the Young's modulus (G 0:3846E). Shear strain in a beam is VQ t Gg 3-25 It VQ VQ g 2 1 v 3-26 GIt EIt The shear strain follows the parabolic variation of shear stress over the depth of the beam. In strength of material analysis, the shear strain is neglected (g 0). Method 1ÐHooke's Law In this method, the differential equation for the elastic curve is obtained by using Hooke's law that relates stress and strain. Consider an elemental beam block as shown in Fig. 3-28a. According to beam theory, a plane section before deformation remains plane even after deformation, and the unde- formed right angle (boa p/2) is maintained after deformation (bH oaH p/2). The axial displacement u measured from o±b (which is parallel to the y-coordinate axis) is shown in Fig. 3-28b. It is zero at the neutral axis and has a linear variation along the beam depth (u yy). Normal strain (e), which is compressive, is obtained as the derivative of displace- ment (u), which is negative. Àdu Àe 3-27a dx Slope is equal to the derivative of the transverse displacement (v) because both angles < boa and < bH oaH are right angles. dv x Slope y 3-27b dx d d Àu yy À y 3-27c 2 2 dv Àu y 3-27d dx Hooke's law is used to relate strain to stress, which is obtained from the beam formula. Àdu d2 v s My Àe y 2À 3-27e dx dx E EI M d2 v 3-27f EI dx2 166 STRENGTH OF MATERIALS y x b' b b' b a' u y o a d o 1 b b' dx (a) Enlarged block. (b) Displacement u = y. FIGURE 3-28 Strain displacement relationship for a beam. The axial displacement u is small, but it plays a critical role in the derivation of the moment curvature relation given by Eq. (3-27f ). The steps used to derive the moment curvature relationship can be summarized as: 1. The slope is defined as the derivative of the transverse displacement v with respect to the x-coordinate, as in Eq. (3-27b). 2. The axial displacement at a location y given by Eq. (3-27c) is obtained from the geometry in Fig. 28(b), which defines the undeformed orientation (bb) and the deformed orientation (bH bH ). 3. Strain e is related to the derivative of the transverse displacement (v), the stress (s/E), and the moment (My/EI) in Eq. (3-27e). 4. Simplification of Eq. (3-27e) yields the moment curvature relationship for a beam. d2 v M k 3-28 dx2 EI In the linear theory, the term (d2 v/dx2 k) represents the beam curvature. Method 2ÐGeometrical Derivation The moment curvature relation can be also be derived via the radius of curvature r. The elastic curve of the beam is shown in Fig. 3-29a. An elemental beam length in an enlarged scale is depicted in Fig. 3-29b. The change in slope in the elemental length is Dy. The rate of change of the slope with respect to the beam length is the inverse of the radius of curvature r. Áy dy dy 1 LtÁs 3 0 9 3-29a Ás ds dx r Simple Beam 167 x y =d ds v x x dx (a) Elastic curve. (b) Radius of curvature. S ds dv O T dx (c) Relation between ds and dx. FIGURE 3-29 Radius of curvature of the elastic curve. Angle y is eliminated in favor of displacement using Eq. (3-27b) to obtain 1 d2 v M k 3-29b r dx2 EI Equation (3-29) adds the reciprocal of the radius of curvature into the moment of curvature relation given by Eq. (3-28). Remarks on the Moment Curvature Relationship (MCR) The MCR given by Eq. (3-29) assumes that ds dx, y tan y, and cos y 1 because of the linear small displacement theory. MCR can be derived without the assumptions. The arc length (ds), axial distance (dx), and included angle (y) are marked in Fig. 3-29c. Because dx is small and in the limit, the length (ds) can be assumed to be the diagonal of the right triangle OST. The following relationships can be written for the triangle OST. dx cos y ds dv tan y dx 168 STRENGTH OF MATERIALS ds2 dx2 dv2 3-30a dy d À1 dv d À1 dv dx tan tan 3-30b ds ds dx dx dx ds Differential calculus yields d2 v d À1 dv dx2 tan À Á2 3-30c dx dx 1 dv dx Differentiate ds2 dx2 dv2 with respect to x to obtain ds dv 2ds 2dx 2dv 3-30d dx dx Divide both sides of Eq. (3-30d) by dx and simplify to obtain 2 2 ds dv 1 dx dx 4 2 51=2 ds dv 1 3-30e dx dx Substitute Eqs. (3-30c) and (3-30e) into Eq. (3-30b) to obtain d2 v dy dx2 h ÀdvÁ2 i3=2 3-30f ds 1 dx Curvature, by definition, is k dy/ds 1/r. The curvature can be written as 2 d v 1 dx2 k h ÀdvÁ2 i3=2 3-30g r 1 dx The curvature when (dv/dx)2 is small compared with unity (1 ( (dv/dx)2 ) becomes the linear theory formula (k d2 v/dx2 ). Our treatment uses the simple formula. In the moment curvature relation, the bending moment is expressed in terms of the shear force (V) and the uniformly distributed load (p) using the appropriate definitions: Simple Beam 169 dV Àp dx dM ÀV dx d2 M dV À p dx2 dx d2 v M dx2 EI d3 v 1 dM V 3 À dx EI dx EI d4 v d2 M 1 d2 M 4 2 dx dx EI EI dx2 d4 v p 4 dx EI The three important formulas to calculate displacement are as follows: 1. For distributed load (p) d4 v p 4 3-31a dx EI 2. For shear force (V) d3 v V À 3-31b dx3 EI 3. For moment (M) d2 v M 3-31c dx2 EI The relationship between the fourth derivative of the displacement (v) with respect to the x-coordinate, which is equal to the ratio of the uniformly distributed load p to the beam rigidity EI is given by Eq. (3-31a). Equation (3-31a) is a fourth-order ordinary differential equation. Solution of this equation requires four boundary conditions (BC). Displacement follows the load, or (p and v) have the same direction. The moment curvature relation given by Eq. (3-31c), is integrated to obtain the elastic curve of a beam. The process is illustrated for different types of beams subjected to various types of loads. 170 STRENGTH OF MATERIALS EXAMPLE 3-11: Simply Supported Beam Subjected to a Uniformly Distributed Load p A simply supported beam of length ` and rigidity EI is subjected to a uniformly distributed load of intensity p, as shown in Fig. 3-30a. Calculate its elastic curve. Note: The gravity load is specified as Àp, or load p is along the y-coordinate direction. Solution The four boundary conditions (BC) of the beam shown in Fig. 3-30a are as follows: BC1X at x 0; vA 0 BC2X at x `; vB 0 d2 v BC3X at x 0; M 0 dx2 d2 v BC4X at x `; M 0 dx2 The first boundary condition (BC1) constrains the displacement to zero at x 0, which corresponds to A in Fig. 3-30a. Likewise, the second condition (BC2) constrains the displacement to zero at x `, which corresponds to B in Fig. 3-30a. For a simply supported beam, the moment is zero (M 0) at A and B. When the moment is zero, the curvature can be set to zero because M EI d2 v/dx2 and the stiffness is not zero (EI T 0). We replace moment to obtain the conditions imposed on the curvatures. The third condition (BC3) constrains the curvature to zero at x 0. Likewise, the fourth condition (BC4) constrains the curvature to zero at x `. The fourth-order differential equation, Eq. (3±31a), and the four boundary condi- tions define the displacement determination problem of the beam. The fourth-order differential equation is integrated in four steps to obtain the following four equations: y –p per unit length B V(x) x A max max Vmax (a) Simply supported beam under (b) Displacement function. uniform load. FIGURE 3-30 Displacement function for Example 3-11. Simple Beam 171 d3 v EI px C1 3-32a dx3 d2 v x2 EI p C1 x C2 3-32b dx 2 2 dv x3 x2 EI p C1 C2 x C3 3-32c dx 6 2 x4 x3 x2 EI v x p C1 C2 C3 x C4 3-32d 24 6 2 The final displacement v(x), which is given by Eq. (3-32d), is a fourth-order polynomial. It is expressed in terms of four constants (C1 to C4), which are also called the constants of integration. These four constants are obtained from the four boundary conditions as follows: BC1X v x 0 0 yieldsX C4 0 `4 `3 `2 BC2X v x ` 0X p C1 C2 C3 ` 0 24 6 2 d2 v BC3X 0X C2 0 dx2 d2 v `2 ` BC4X 0X p C1 ` 0 or C1 Àp dx2 2 2 The value of the constant (C3 p`3 /24) is obtained from BC2 by substituting C2 C4 0 and C1 Àp`/2. The values for the four constants are C1 Àp`=2 C2 0 C3 p`3 =24 C4 0 Substitution of the value for the four constants into Eq. (3-32d) yields the displace- ment function v(x) as px À 3 Á v x x À 2`x2 `3 3-33 24EI The displacement function is a cubic polynomial in the x-coordinate. The location of the maximum value of the displacement is obtained by setting its first derivative to zero: 172 STRENGTH OF MATERIALS dv p À 3 Á 4x À 6`x2 `3 0 3-34a dx 24EI x À 0:5` x À 1:37` x 0:37` 0 3-34b x 0:5` 3-34c Only the solution for x 0:5` is valid. The other two solutions (x 1:37` and x À0:37`) are impractical because one is more than the span and the other is negative. Displacement attains the maximum value (vmax 5p`4 /384EI) at the mid- span, where the slope is zero. ` 5p`4 vmax v x 3-34d 2 384EI ` y x 0 3-34e 2 dv p`3 ymax y x 0 3-34f dxx0 24EI dv p`3 y x ` À 3-34g dx x`=2 24EI The slope peaks at x 0 and x `, with magnitude ymax p`3 /24EI. The elastic curve is symmetrical about the midspan as depicted in Fig. 3-30b. In Eq. (3-33), both displacement and load are positive. Displacement has a tendency to follow the direction of load. Displacement as shown in Fig. 3-30b is along the negative y-coordinate direction because load is negative (Àp). EXAMPLE 3-12 A cantilever beam is subjected to a uniformly distributed load of intensity Àp. It has a length of ` and a rigidity of EI, as shown in Fig. 3-31a. Calculate its elastic curve. Solution The displacement solution (expressed in terms of the four constants) as given by Eq. (3-32d) is still valid. The four constants have to be adjusted for the four boundary conditions of the cantilever beam. The displacement is zero at the origin. For a cantilever, the slope at the origin is zero (y 0 at x 0). The free end of a cantilever beam can neither resist a moment nor a shear force (M V 0 at x `). These four boundary conditions in terms of the displacement function are as follows: Simple Beam 173 y –p per unit length x x vmax max (a) Cantilever beam under uniform load. (b) Displacement function. FIGURE 3-31 Displacement function for Example 3-12. BC1X at x 0; v0 3-35a dv BC2X at x 0; y 0 3-35b dx 2 M d v BC3X at x `; 0 3-35c EI dx2 V d3 v BC4X at x `; À 30 3-35d EI dx Substitution of the displacement function into the boundary conditions yields the following equations: BC1 yields C4 0 3-35e BC2 yields C3 0 3-35f 2 ` BC3 yields p C1 ` C2 0 3-35g 2 BC4 yields p` C1 0 3-35h Solution of the four BCs provides values of the integration constants: C1 Àp` 3-35i 2 p` C2 3-35j 2 C3 C4 0 3-35k The displacement function for the cantilever is obtained as px2 À 2 Á v x x À 4x` 6`2 3-36a 24EI 174 STRENGTH OF MATERIALS The magnitude of the displacement and slope are maximum at the free end of the cantilever, as shown in Fig. 3-31b. p`4 vmax v x ` 3-36b 8EI dv p`3 ymax x ` 3-36c dx 6EI The displacement function shown in Fig. 3-12a is negative because the distributed load is a gravity load (Àp). EXAMPLE 3-13 A simply supported beam is subjected to a concentrated gravity load P at the two- thirds span location, as shown in Fig. 3-32a. It has a length of ` and a rigidity of EI. Calculate its elastic curve. Solution The moment curvature relationship given by Eq. (3-31c) is used to solve the problem. The bending moment diagram of the beam is shown in Fig. 3-32b. The moment function is defined separately for span segments AC and CB. Span AC (0 £ x £ 2a): P M x x 3-37a 3 Span CB (2a £ x £ 3a): P 2P M x x À P x À 2a 3a À x 3-37b 3 3 The displacement function has to be defined separately in each span segment because of the discrete nature of the moment functions. The elastic curve is obtained by integrating the moment curvature relationship given by Eq. (3-31c). Span AC (0 £ x £ 2a): d2 v M Px 3-37c dx 2 EI 3EI dv Px2 C1 3-37d dx 6EI Simple Beam 175 P C B A M(x) 2Pa/3 2a a A C Bx R A = P/3 R B = 2P/3 (a) Simply supported beam under (b) BM diagram. concentrated load. V(x) A B C x (c) Elastic curve. FIGURE 3-32 Elastic curve diagrams for Example 3-13. Px3 v C1 x C2 3-37e 18EI SpanCB (2a £ x £ (3a = ø)): d2 v 2P 3a À x 3-37f dx2 3EI dv 2P x2 3ax À C3 3-37g dx 3EI 2 2P 3ax2 x3 v À C3 x C4 3-37h 3EI 2 6 The elastic curve is defined in terms of four constants. These are C1 and C2 for segment AC, and C3 and C4 for segment CB. The moment and shear force boundary conditions cannot be used when the displacement calculations are initiated from the moment function because these conditions were already used during the calculation of the moment functions. Only the displacement and slope conditions can be used. The slope conditions cannot be used because these are not known at either x 0 or x `. The two displacement boundary conditions that can be used are 176 STRENGTH OF MATERIALS BC1X at x 0; v0 3-38a BC2X at x `; v0 3-38b Two additional boundary conditions are required to calculate the four constants of integration. These conditions come from displacement continuity conditions. For an elastic structure, displacement is always continuous. The continuity condition is satisfied by specifying the displacement and slope to be finite at every location (x) in the beam span. v x finite 3-39a dv x finite 3-39b dx The conditions given by Eqs. (3-39a) and (3-39b) are applied to the beam at location C, at x 2a, which is common to the two beam segments. Let the displace- ment calculated at C in the span segment AC be vleft (x 2a). Designate the same displacement, but calculate in it segment CB as vright (x 2a). The displacement continuity condition at C is specified as BC3X vleft jx2a vright jx2a 3-38c Likewise, the continuity of the slope at C yields dvleft dvright BC4X jx2a jx2a 3-38d dx dx The four constants of integration (C1 to C4) are obtained from the four boundary conditions given by Eqs. (3-38a) to (3-38d). These four equations are simplified to obtain BC1 yieldsX C2 0 3-40a 2P 27a3 27a3 BC2 yieldsX À 3C3 a C4 0 3EI 2 6 6Pa3 or, 3C3 a C4 0 3-40b EI BC3 is simplified to obtain 8Pa3 2C1 a À 2C3 a À C4 À 3-40c 3EI Simple Beam 177 Likewise, BC4 yields 2Pa2 C1 À C3 À 3-40d EI The solution to Eqs. (3-40a) to (3-40d) yields the values of the four constants. 4 C1 Pa2 3-41a 9EI C2 0 3-41b 2 22Pa C3 3-41c 9EI 4Pa3 C4 À 3-41d 3EI The displacement functions of the beam are obtained by substituting the integration constants in Eqs. (3-37e) and (3-37h). For span segment AC (0 x 2a), P À 3 Á v(x) x À 8a2 x 3-42a 18EI For span segment CB (2a x 3a), P À 3 Á v(x) Àx 9ax2 À 22a2 x 12a3 3-42b 9EI The elastic curve is shown in Fig. 3-32c. In Eq. (3-42), the load direction has been incorporated, or ÀP is along the positive y-coordinate axis. It can be assumed from observation that the displacement peaks in the longer segment AC. The location of maximum displacement xm is obtained by setting the slope to zero. dv 0 0 x 2a dx P À 2 Á 3x À 8a2 0 3-42c 18EI r 8 xm Æ a 3 This location (xm ) has two values, but the negative value is not admissible. The value of xm 1:63a, which is inside segment AC, is admissible. xm 1:63a 3-42d 178 STRENGTH OF MATERIALS The maximum displacement for gravity load (P) is Pa3 vmax v xm À0:484 3-42e EI 3.6 Thermal Displacement in a Beam A beam can deform because of a change of temperature. Consider the simply supported beam shown in Fig. 3-33a, which is subjected to a uniform temperature variation of T u at the top fiber and T ` at the bottom fiber and has a linear variation across the depth d, as shown in Fig. 3-33b. The Young's modulus is E, and the coefficient of thermal expansion is a. The deformation due to the temperature variation is sketched in Fig. 3-33c. The beam deflects along the positive y-direction because the upper fiber with the higher temperature (T u > T ` ) expands more than the lower fiber. The problem is to calculate the displacement function. Temperature variation induces a normal thermal strain (et aÁT), but no shear strain is induced (gt 0). In a determinate structure, the solution of the equilibrium equations yields the internal force variables. Because the temperature variation does not affect the equili- brium equations, the internal forces are not altered. For a determinate structure, the internal forces are zero when it is only subjected to temperature variation. No thermal stresses are induced because a determinate beam does not offer any resistance to the thermal deforma- tion. It undergoes free thermal deformation, which is expansion at the top fiber and contraction in the bottom fiber. This process bows the beam upwards as shown in Fig. 3-33c. The linear temperature variation with a mean temperature T0 can be written as Tu T` T0 3-43a 2 À u Á T À T` ÁT y T0 y 3-43b d The mean temperature expands the beam length, but it makes no contribution to flexural stress. There is no flexural deformation when the temperatures are equal at the upper and lower fibers of the beam cross-section. Consider the deformation of an elemental length dx as shown in Fig. 3-33d. The temperature elongates the top fiber to ÁSu and contracts the bottom fiber to ÁS` from its initial dimension of ÁS Áx at the neutral axis. The thermal deformations sketched in Fig. 3-33d are as follows u d ÁS ÁT y aÁx aT u Áx 3-43c 2 Simple Beam 179 y y a Tu d To a x T (a) Simply supported beam. (b) Temperature variation along depth. su d/2 x s= d vt s t t t x (c) Thermal deformation. (d) Enlarged view of deformation. FIGURE 3-33 Thermal deformation in a simply supported beam. Likewise, ÁS` aT ` Áx 3-43d À u Á t dSu À dS` T À T` Áy À Àa Áx 3-43e d d À u Á Áyt dyt T À T` L t Áx 3 0 Àa Áx dx d The slope y has a maximum value at x 0 in Fig. 3-33c, and it decreases with an increase in the x-coordinate. This fact, as can be observed, is accounted for by the negative sign in Eq. (3-43e). 180 STRENGTH OF MATERIALS The thermal curvature (rt ) can be expressed in terms of temperature as 1 d 2 vt Áyt dyt kt 2 L tÁx30 3-43f rt dx Áx dx À u Á d 2 vt T À T` Àa 3-43g dx 2 d The curvature temperature relation given by Eq. (3-43f) can be integrated to obtain the thermal displacement (vt ) of the beam. The thermal displacement vt is independent of the Young's modulus but depends on the coefficient of thermal expansion. The integration of the curvature temperature relationship yields the slope and the displacement as À u Á t dvt T À T` y Àa x C1 3-44a dx d À Á t a Tu À T` 2 v À x C1 x C2 3-44b 2d The thermal slope (yt ) is expressed in terms of a single integration constant C1. The thermal displacement is expressed in terms of two constants (C1 and C2). The treatment here assumed a uniform temperature distribution along the top and bottom fibers of the beam. This temperature variation induces linear variation in slope and a quadratic variation in the displacement. A nonuniform temperature variation along the beam length (T u (x) and T ` (x)) can be accommodated during the integration of the curvature displacement relationship. The constants (C1 and C2) are adjusted for a simply supported beam as BC1X vt x 0 0 C2 0 3-45a À Á t a Tu À T` BC2X v x ` 0 C1 ` 3-45b 2d a À u ÁÀ Á vt T À T ` x` À x2 3-46 2d The simply supported beam bows upward when the temperature of the upper fiber is greater than the temperature of the lower fiber (T u > T ` ). Its displacement at midspan is t ` a À u Á v x T À T ` `2 3-47 2 8d A simply supported beam bows downward when the upper-fiber temperature is smaller than the lower fiber (T u < T ` ). The thermal deflection of a cantilever beam is obtained by adjusting the boundary conditions as follows: BC1X vt x 0 0 C2 0 3-48a Simple Beam 181 y Tu (Tu>T ) x T v t( ) FIGURE 3-34 Thermal deformation in a cantilever beam. BC2X yt x ` 0 C1 0 3-48b a À u Á vt À T À T ` x2 3-48c 2d a À u Á vt x ` À T À T ` `2 3-48d 2d The thermal displacement for a higher upper-fiber temperature (T u > T ` ) is sketched in Fig. 3-34. The beam deflects in the negative y-coordinate direction because the fibers above the neutral axis are stretched. The deflection will be in the positive y-coordinate direction when the lower-fiber temperature is higher (T u > T ` ). EXAMPLE 3-14 Calculate the thermal displacement in the simply supported beam shown in Fig. 3-33a for the following parameters. Length (` 200 in:), depth (d 7 in:), upper and lower fiber temperatures are (T u 200 F and T ` 100 F) and coefficient of thermal expansion (a 6:6  10À6 F). Calculate the response when the simply supported beam is replaced by a cantilever beam shown in Fig. 3-34. Solution The displacement function for the simply supported beam is obtained by substituting numerical values for the coefficient of thermal expansion, depth, temperatures, and length in Eq. (3-46). À Á vt 0:47  10À4 200x À x2 3-49a Differentiation of the displacement function yields the slope. yt 0:94  10À4 100 À x 3-49b Maximum displacement occurs at mid span (vt max 0:47 in:). The slope at (x 0) is (y0 9:43  10À3 rad), and at (x 200 in:) it is (y200 À9:43  10À3 rad). The beam 182 STRENGTH OF MATERIALS bows outwards because the upper fiber with higher temperature (T u 200 F) stretches more than the lower fiber with temperature (T ` 100 F), as shown in Fig.3-33c. Likewise the displacement function is obtained for the cantilever beam using Eq. (3-48c). vt À0:47  10À4 x2 3-49c Differentiation of the displacement function yields the slope. yt À0:94  10À4 x Maximum displacement occurs at the free end (vt max À1:88 in:). The slope at (x 0) is (y0 0), and at the free end (x 200 in:) it is (y200 À18:86  10À3 rad). The cantilever bends downwards as shown in Fig. 3-34. The response of the cantilever is opposite to that of the simply supported beam, but for both beams the top fibers elongate while the bottom fibers contract. 3.7 Settling of Supports A simply supported beam is obtained by restraining its movement along the y-coordinate direction at the boundary points x 0 and x `. No restraint is imposed on its boundary slope y. The beam is free to rotate, or y is finite at x 0 and x `. A simply supported beam can settle along the y-coordinate direction by v(x 0) ÀÁA and v(x `) ÀÁB as shown in Fig. 3-35a. A cantilever beam is obtained by restraining its movements, consisting of displacement (v(x 0) 0) and slope (y(x 0) 0) at the origin. No restraint is specified at its free boundary. A cantilever beam can settle by v(x 0) ÀÁA and rotate by y(x 0) ÀÁyA at its fixed support, as shown in Fig. 3-35b. Because the amount of settling is small, the equilibrium equations, written in the initial undeformed configuration, remain valid even for the settled structure. The internal moment and shear forces that are determined from an application of the EE are not changed by support settling. For a determinate structure, the internal forces are zero when the beam is subjected only to support settling. No stress is induced because a determinate beam does not offer any resistance to support settling. Support settling moves the beam as a rigid body. For a simply supported beam, the displacement function is a line joining the two settling supports at v(x 0) ÀÁA and v(x `) ÀÁB , as shown in Fig. 3-35a. & ' ÁB À ÁA v x À ÁA x 3-50a ` For a cantilever beam, the settling of its fixed support by v(x 0) ÀÁA and its rotation by y(x 0) ÀÁyA is depicted in Fig. 3-35b. The beam displacement has two components: a uniform displacement (ÀÁA ) and a linear rigid-body rotation (ÀÁyA x). Simple Beam 183 y y A B A B x x A A B A (a) Simply supported beam. (b) Cantilever beam. FIGURE 3-35 Settling of support in a beam. v x ÀfÁA ÁyA xg 3-50b The displacement functions given by Eqs. (3-50a) and (3-50b) are referred to as rigid- body displacements. Their derivative with respect to the x-coordinate is a constant (dv/dx constant). No bending moment (M EId 2 v/dx2 0) or shear force (V ÀEId 3 v/dx3 0) is induced in the beam. 3.8 Shear Center Consider a uniform cantilever beam of length `, subjected to a uniformly distributed load of magnitude p per unit length and a concentrated load P, as shown in Fig. 3-36a. In Chapter 1, it was mentioned that the beam will flex when the line of action of the load (P or p) passes through the centroid of the cross-section. An eccentrically applied load (P or p) can induce secondary torque in a beam. There are situations associated with specific shapes of beam cross-sections that can induce torque even when the applied load has no eccentricity. The issue is illustrated by considering three different cross-section types for the beam: a rectan- gular section, an I-section, and a channel section, as shown in Fig. 3-36b. The rectangular beam under the action of a load applied along the y-coordinate axis bends or flexes along its neutral axis, which is the x-coordinate axis. The beam will experience only flexural deformation along its length (or x-coordinate axis) when the load (P or p) passes through the centroid (G) with the y-axis as its line of action. Likewise, a beam with an I-section will experience only flexural deformation along its length when the load passes through the centroid G with the y-axis as its line of action. A uniform beam with a cross-section that is symmetrical about both the y- and z-coordinate axes will experience flexure along the x-axis when the load is applied along the y-coordinate direction passing through the centroid. Consider next a uniform beam made of a channel section with equal flange width b. Let G be the centroid of the cross-sectional area as shown in Fig. 3-36b. Consider the y- and 184 STRENGTH OF MATERIALS Line of y (P, p) action of load (P, p) y (P, p) y p per unit length P c z z O G z d G, O d G, O x x c b Rectangular I-section Channel section section (a) Uniform beam. (b) Cross-section. P P Shear center O Line of e symmetry G Shear axis (c) Twisting accompanies flexing. (d) Pure flexure. p P (e) Load through shear center. FIGURE 3-36 Shear center for different beam shapes. z-coordinate axes with origin G, as marked in the figure. Apply a load (P or p) along the y-coordinate axis that passes through the centroid. This load will cause the beam to bend along the x-axis as well as twist in the y±z plane, as shown in Fig. 3-36c. Twisting occurred without the application of a torque load. Let us shift the line of application of the same load (P or p) to a parallel axis that passes through the shear axis at O (using an angle attachment) as shown in Fig. 3-36d. The beam bends, but it does not twist for this line of action of the Simple Beam 185 load. Point O is referred to as the ``shear center,'' and its distance is e from the wall of the web. Point O lies in the line of symmetry of the channel section. The shear axis passes through point O, and line of action is parallel to the y-coordinate axis. The shear center of a cross-section composed of thin parts like the web and flange, such as in the channel section, is important because such a cross-section provides a large resistance to bending. But it offers a small resistance to twisting, and is, thereby, prone to shear buckling. The twisting can be avoided when the line of action of load passes through the shear center. In aircraft construction, a heavy load is applied to a channel beam at the shear center through an angle attachment, as shown in Fig. 3-36e. Determination of the shear center is based on the concept of shear flow: its direction and magnitude. Direction of Shear Flow For the discussion of shear flow, the example of a uniform simply supported beam made of a thin-walled I-section shown in Fig. 3-37a is considered. An elemental section in an enlarged scale is shown in Fig. 3-37b. The bending moments (M and M ÁM) induce compression in the top flange AB and tension in the bottom flange CD. Forces in an elemental length at B are marked in Fig. 3-37c. The moments (M and M ÁM) induce internal forces (F and F ÁF), and the balancing force is ÁF. The elemental force (ÁF qÁx) is equal to the product of the shear flow (q) and the elemental length (Áx). In other words, the shear flow is the rate of change of ÁF as q ÁF/Áx and, in the limiting case, q dF/dx. The shear flow follows the shear stress notation, and it is marked in Fig. 3-37c. Shear flow along the x-coordinate is balanced among adjacent faces of the element. At location B, the shear flow along the z-direction is from B toward A. A similar analysis is performed using the forces marked in Fig. 3-37d to obtain the direction of shear flow at location A along the z-coordinate direction, which is from A to B. Likewise at the bottom flange (which is in tension), the shear flow directions are marked at locations C and D, in Fig. 3-37b. The shear flow in the web follows the direction of the shear force (V), which is along the negative y-coordinate direction, as shown in Fig. 3-37e. Magnitude of Shear Flow The magnitude of shear flow is calculated by considering the example of a symmetrical I-section with depth (d), flange width (b), and thickness (t), as shown in Fig. 3-38a. The shear flow (q) (which is the rate of change of force) is also equal to the product of shear stress (t VQ/It) and thickness (t), as (q tt VQ/I). Its calculation requires two geometrical properties of the section about the neutral axis: the moment of inertia (I) and the moment of area (Q). The moment of inertia of the I-section about the neutral axis (nÀa) shown in Fig. 3-38a is calculated by adding the flange and web contributions as bt3 btd 2 td 3 I 3-51a 6 2 12 The higher power of the thickness (t3) is neglected because t is assumed to be small compared with the depth (d) and flange width (b). The moment of inertia formula simplifies to 186 STRENGTH OF MATERIALS Adjacent elements y P y M B t x z A M + dM x D x C (a) I-beam. (b) Elements of I-beam (enlarged). y z F F x dF x q dF Shear flow q at B F + dF F + dF (c) Internal forces at B. (d) Force and shear flow at A. M R = P/2 V (e) Force in the web. FIGURE 3-37 Shear flow in an I-beam. td 2 d I9 b 3-51b 2 6 The moment of the area about the neutral axis depends on the location of the area in the cross-section. The moment of area Qf(z) for an elemental area (dA t dz) in the flange located along the positive z-axis is defined as b=2 b=2 d dt dt Qf z dA dz b=2 À z 3-52 z 2 2 z 2 Simple Beam 187 y dz Ff Ff t (1) 2 z y z d n a n a Fw d/2 Ff Ff b b (a) I-section. (b) Force because of (c) Qw in web. shear flow. FIGURE 3-38 Magnitude of shear flow. The moment of area at z b/2, or location 2 in Fig. 3-38b, is zero (Qf (z b/2) 0). At location 1, z 0 or it is maximum (Qf (z 0) bdt/4). It has a linear variation between locations 2 and 1. The shear flow at locations 1 and 2 are as follows: VQ Vdt b=2 À z V b=2 À z q 2 I td b d=6 d b d=6 q2 0 at z b=2 3-53a Vb q1 at z0 2d b d=6 The average shear flow is obtained as q1 q2 Vb qa 3-53b 2 4d b d=6 The force in the top flange Ff, along the positive z-coordinate axis is obtained as the product (Ff qa b/2). b Vb2 Vtdb2 Ff q a 3-53c 2 8d b d=6 16I The forces are marked in other flange segments in Fig. 3-38b. The magnitude of the force Ff, is given by Eq. (3-53c) and the directions are shown in Fig. 3-38b. 188 STRENGTH OF MATERIALS For the web, the moment of area Qw(y) at the location y shown in Fig. 3-38c is calculated as the sum of contributions from the flange (btd/2) and the part of the web of height (d/2 À y). Thickness, being small, is neglected to obtain d/2 À t d/2. d=2 btd Qw y t ydy 2 y d=2 btd t À 2 Á Qw y y 2 2 y 2 ! w btd t d 2 Q y Ày 2 2 4 btd td 2 ty2 Qw y À 3-54a 2 8 2 The moment of the area Qw(y), being a quadratic function in y, is not changed for positive y or negative y coordinate. The shear flow in the web (qw) is obtained as VQw Vt d2 2 qw bd À y 3-54b I 2I 4 The shear flow in the web qw has a parabolic variation along the y-coordinate. The force in the web (Fw) is obtained by integrating the shear flow (qw) over the depth as Vt d=2 d2 2 Fw qw dy bd À y dy 2I Àd=2 4 d=2 Vt d 2 y y3 bdy À 2I 4 3 Àd=2 Vtd 2 2b d=3 3-54c 4I 2 Because I td (2b d), the force (Fw) is equal to the shear force (V). 4 3 Fw V 3-54d The web force (Fw) and shear force (V) relation (Fw V) could have been obtained from the equilibrium along the y-coordinate direction. This relation assumes that the thickness of flange is thin and that the web carries the shear force. EXAMPLE 3-15 A simply supported I-beam is subjected to a P 100 kN load at the center span as shown in Fig. 3-39a. The beam is 10 m long. The dimensions of the I-section are marked in Fig. 3-39b. Calculate the shear stress distribution, and locate the shear center at one-quarter span of the beam. Simple Beam 189 /4 y 1 =0 c P = 100 kN 5 mm d x 2 3 c 1m = max R = 50 kN = 10 m R 0.25 m (a) I-beam. (b) Section at c–c. z –6.1 –0.6 M C L z –1.21 –1.0 G, O R V=R C L 6.1 kN/cm2 I-beam Bending Shear Average (c) Free-body diagram. (d) Stress distribution in kN/cm2. FIGURE 3-39 Stress distribution in an I-beam. Solution The shear force at the one-quarter span is obtained as V À0:5P À50 kN from the free-body diagram shown in Fig. 3-39c. The shear force (V) is negative, whereas the moment (M P`/8) is positive (the t-sign convention is followed). The shear stress has a symmetrical distribution about the neutral axis, and it peaks at the neutral axis (tmax ), with its zero value at the top surface of the flange (t 0). The shear stress is calculated at three locations: (1) the top of the flange, (2) the bottom of the flange, and (3) the neutral axis. At Location 1 (top of flange) (y = d/2 = 50 cm) The moment of area Q1 A1 d/2 0 because A1 0. The shear stress is (t1 VQ1 /It 0). The top flange surface is shear stress-free. At Location 2 (bottom of flange) (y = ( d/2) À ( t/2) = 50 À 0.25 = 49.75 cm) The moment of the top flange area Q2 A2 y 25  0:5  49:75 621:88 cm3 . Engineers may neglect the flange thickness in calculating distance y and set (y 50) to obtain Q2 A2 d=2 25  0:5  50 625:0 cm3 190 STRENGTH OF MATERIALS The moment of inertia of the I-section is obtained as the difference between the two rectangular cross-sections. I 1=12 25  1003 À 24:5  993 102:3  103 cm4 The shear stress is t2 VQ2 =It À50  621:88= 102:3  103  0:5 À0:608 kN=cm2 At Location 3 (neutral axis) (y = 0) The moment of area (Q3) above the neutral axis is obtained by adding the flange and the web contributions as Q3 Aflange  d À t=2 Aweb  d=2 À t=2 25  0:5  50:0 À 0:25 0:5  0:5  50 À 0:52 1234:4 cm3 The shear stress is t2 VQ3 =It À50  1234:4= 102:3  103  0:5 À1:21 kN=cm2 The average shear stress in the web (ta V/web area) is ta À50=0:5  100 À 1 À1:0 kN=cm2 The compressive bending stress at the top flange surface (y d/2) at location x `/4 is s`=4 My=I Pd`=16I À6:1 kN=cm2 The shear stress and the bending stress at the one-quarter span location are graphed in Fig. 3-39d. The shear stress peaks at the neutral axis at tmax À1:21 kN/cm2 . It has a parabolic variation along the depth, and it attains t2 0:61 kN/cm2 at the bottom flange surface and reduces to zero at the top flange surface, as shown in Fig. 3-39d. The average stress calculated from the simple formula ta V/A À1:0 kN/cm2 is close to its peak value. The bending stress at the one-quarter span location is about five times the shear stress, and it occurs at the top flange location, where the shear stress is zero. The centroid (G) and shear center (O) coincide because the beam is symmetrical about the y- and z-axes as marked in Fig. 3-39d. Simple Beam 191 EXAMPLE 3-16 Develop an analytical expression for the shear center of a symmetrical channel section with dimensions as marked in Fig. 3-40a. The moment of inertia (I) of the channel section about the neutral axis is btd 2 1 I td 3 3-55 2 12 The moment of the flange area (Qf) as a function of z about the neutral axis is dt Qf z b À z 3-56 2 The moment of the area of the channel web at a location y, about the neutral axis, is (see Eq. (3-54a)) t d2 Qw y bd À y2 3-57 2 4 qf qf max b Ff z V C y d qw max d n a O O e G Fw t qw t qf max Ff (a) Channel section. (b) Shear flow. (c) Shear center. t = 1/4 in. V e V C G 12 O e z 6 (d) Load through an attachment. (e) Shear center and centroid. FIGURE 3-40 Shear center for a channel section. 192 STRENGTH OF MATERIALS The shear flow (qf) and force (Ff) in the flange are VQf V dt qf z b À z I I 2 V Vb qfmax bdt 3-58 2I d b d=6 b b Vdt b Vtd z2 Ff qf dz b À zdz bz À o 2I o 2I 2 o Vtd 2 Vtb2 Ff b 4I 2 btd td 2 =6 Vb2 3Vb2 Ff 3-59 2 db d2 =6 d 6b d The variation of shear flow in the flange is shown in Fig. 3-40b. It has a linear variation and attains the maximum value [qf max Vb/fd(b d/6)g] at z 0. The shear flow (qw) in the web is Vt d2 2 qw bd À y 2I 4 Vtd qwmax b d=4 2I d=2 Fw qw dy V 3-60 Àd=2 The shear flow distribution in the web is shown in Fig. 3-40b. It has a parabolic distribution along the web depth about the neutral axis. It attains the maximum value (qf max [(Vtd/2I)(b d/4)] at the neutral axis. The forces in the flange and web are marked in Fig. 3-40c. The flange and web forces (Ff and Fw) do not equilibrate when the load (V) is applied at the centroid G. The nonequilibrating forces are equivalent to a torque load that twists the section. The twisting can be alleviated provided the load (V) is applied along the neutral axis at O with eccentricity e. The location 0 is referred to as the shear center. The shear center is determined from an equilibrium of the forces acting in the channel section, as shown in Fig. 3-40c. M 0 F f d À Fw e 0 Ff d e Fw db2 t I d  Àd3 bd2 Á 4I t 12 2 Simple Beam 193 3b2 e 3-61 d 6b The derivation of Eq. (3-61) uses both flange and web forces (Ff and Fw). The same expression can be derived without using the web force when equilibrium is taken about the web center C, as shown in Fig. 3-40c. Mc 0 Ff d À Ve Ff d 3b2 e 3-62 V d 6b The distance of the shear center (e) depends on the depth (d) and width (b), but it is independent of the thickness (t), which is considered to be uniform for the entire section. If, however, the flange and web have different thicknesses (tf and tw), respectively, the formula will include the thickness ratio k tw /tf as 3b2 3b2 e À Á 3-63 6b d tf =tw 6b kd The load at the shear center can be applied through an angle attachment as shown in Fig. 3-40d. For a channel with depth d 12 in:, width b 6 in:, and thickness t 0:25 in:, as shown in Fig. 3-40e, the shear center is calculated as 3  62 9 e 2:25 in: 3-64 12 6  6 4 The centroid of the channel section can be obtained as the ratio of the moment of area ÆAz to area A as ÆAz z " ÆA The cross-sectional area of the channel section is A ÆA 2bt dt The moment of area about the web center is b ÆAz 2tb tb2 2 The centroidal distance from the web center is b2 z " 3-65 2b d 194 STRENGTH OF MATERIALS The distance to the centroid (G) from the web center is 62 z " 1:5 in 3-66 12 12 The distance to the shear center from the web center is 2.25 in. The shear center and the centroid are on opposite sides of the web, and the distance z between them is e " 3:75 in: The channel section will experience a torque (T 3:75 V) when the load is applied at the centroid. The section will be free from torque when the load is applied at the shear center through an attachment, as shown in Fig. 3-40d. We provide a definition for the shear center for a beam cross-sectional area with symmetry about the neutral axis. The shear center is the point on the cross-sectional area of a beam through which the resultant of the transverse load must be applied, so that the beam undergoes bending and stress can be determined from the flexure formula. EXAMPLE 3-17 Determine the shear center for the angle section shown in Fig. 3-41. The angle section is symmetrical about the z-coordinate axis, which is also the neutral axis. It has equal legs of length (b) and thickness (t). It is subjected to a shear load (V). Solution The moment of inertia (I) about the neutral axis is given by the formula b3 t I 3-67 3 The moment of area of a segment r, as marked in Fig. 3-41, is p Q r rt b À r=2= 2 3-68 The shear stress in segment r is VQ 3rV r t r p bÀ It 2b3 t 2 Shear stress peaks at r b, which is the intersection of the angle legs. 3V V tmax p 2:12 3-69 2 2bt 2bt Simple Beam 195 y b F r q /2 O z z b F t max q Shear stress FIGURE 3-41 Shear center for an angle section. V V tav A 2bt The shear stress has a quadratic variation. It is zero at the boundary (r 0) and p attains the maximum value at r b (tmax 1:5 V/bt 2), as depicted in Fig. 3-41. It is about twice the average shear stress (ta V/2bt). The shear flow is the product of the stress and the thickness (q tt). The internal force (F1) in the top leg is obtained as b b b 3V r2 F1 qdr ttdr p br À dr 0 0 2b3 o 2 V F1 p 2 F2 F 1 3-70 The force (F2) in the bottom leg is equal to that in the top leg because of symmetry. The two forces in the legs intersect at O, which is the shear center. The shear force (V) is equal to the resultant of the forces in the legs. For an angle section, the intersection of the two legs locates the shear center O. 196 STRENGTH OF MATERIALS R O O d O z b z b z=O z= 2 + d/3b z= 4 R (a) Channel section. (b) Angle section. (c) Semicircular section. R w t O O h 2 z O z t z=O 2 R(sin – cos ) b1 b z= – sin cos 2 2 3(b – b1 ) z= for b1 < b (w/t)h + 6(b + b1) (d) T-section. (e) Sector section. (f) Unequal I-section. FIGURE 3-42 Shear centers for different cross-sections. The position of the shear centers for six typical cross-sectionsÐa channel section with uniform thickness, an equal leg angle section with uniform thickness, a semi- circular section with uniform thickness, a T-section with uniform thickness, a sector, and an unequal I-sectionÐare given in Fig. 3-42. The distance (z) to the shear center O, is measured from a reference location. For the channel section (and also for an I-section) it is measured from the web center. For semicircular and sector sections it is measured from the center of the circle. 3.9 Built-up Beam and Interface Shear Force A built-up beam, which is assembled from separate components, is quite often used in engineering practice. In such a beam, an interface is created when two component parts are joined together by nailing, bonding, welding, or other means. The beam acts as a single unit Simple Beam 197 provided the interface does not fail. Such a failure can be prevented by designing the fastening to withstand the induced shear force at the interface. Excessive shear force can promote failure at an interface. The shear stress and shear force formula are used to calculate interface shear force; this is illustrated through examples. EXAMPLE 3-18 The box beam shown in Fig. 3-43a is assembled from four wooden planks, consisting of two flange planks and two web planks. The planks are fastened together by screws. The dimensions of the beam and the spacing of the screws are marked in the figure. Calculate the capacity and spacing of the screws (s), shown in Fig. 3-43b, for the beam to carry a shear load (V). V Screw t s h y d Neutral z axis b (a) Box beam. (b) Elevation. b – 2t y y d–h 2 (c) Shear flow at y–y. FIGURE 3-43 Analysis of a boxed beam. 198 STRENGTH OF MATERIALS Solution The moment of inertia of the box beam about the neutral axis is obtained as the difference between the outer and inner rectangular areas as 1 h 3 i I bd À b À 2t d À 2h3 3-71 12 The shear flow is transmitted from the upper (and lower) flange planks to the two web planks. The screws must resist this shear flow. The shear flow at the location of the screws marked ``y±y'' in Fig. 3-43c is obtained from the formula qy VQy /I. The moment of area (Qy ) is calculated for the plank cross-section shown in Fig. 3-43c as dÀh Qy "A y b À 2th 3-72 2 The shear flow at the screw is VQy 6V d À h b À 2th qy 3-73 I bd 3 À b À 2t d À 2h3 The shear flow (qy ) must be resisted by the two rows of screws. Let us assume the capacity or the resistance offered by one screw in shear to be Rs . We assume the spacing of the screws to be uniform at the s unit apart. The capacity per unit length for a screw pair (since the flange is fastened to the web by two rows of screws) is qs 2Rs /s. The fastener will not fail provided the resistance (qs ) exceeds or equals the shear flow (qy ). Rs 2 qy s 2Rs s 3-74 qy The numerical value for spacing is calculated for the following parameters: d 12 in:, b 10 in:, t 0:5 in:, h 1:5 in:, V 2:5 kip, and Rs 250 lbf. 1 h i I 10  123 À 10 À 1 12 À 33 893:25 in:4 12 12 À 1:5 Qy 10 À 11:5 70:88 in:3 2 VQy 70:88 qy 2500  198:38 lbf=in: I 893:25 2  250 s 2:52 in: 3-75 198:38 Simple Beam 199 The spacing of the screws should be at least s 2:54 in: It can be rounded to a lower spacing at 2.5 in. EXAMPLE 3-19 The T-beam shown in Fig. 3-44a is assembled of two wooden planks. The planks are fastened together by screws. The dimensions of the beam are marked in the figure. Calculate the spacing of the screws for the following parameters: d 500 mm, b 250 mm, t 50 mm,V 10 kN, and the resistance of the screw at Rs 1 kN. Solution The shear flow formula requires the moment of inertia about the neutral axis, which is obtained by equating the moment of area taken about the top fiber of the flange. btt t d 2 À t2 yc fbt t d À tg 2 2 bt2 t d2 À t2 yc 2 bt td À t2 250  502 50 5002 À 502 yc 2 250  50 50  500 À 502 13:0  106 yc 185:7 mm 3-76 70  103 The neutral axis is biased toward the top fiber (yc 185:7 mm) because of the flange area. The dimensions of the T-section in millimeter are marked in Fig. 3-44b. The moment of inertia (I) and the moment of flange area (Qf) about the neutral axis are as follows: V b 250 t 50 yc 185.7 n a Neutral axis d 450 500 t (a) T-beam. (b) Dimensions in millimeters. FIGURE 3-44 Built-up T-beam. 200 STRENGTH OF MATERIALS 1 I 250  503 250  50  185:7 À 252 12 1  50 500 À 503 50  450  275 À 185:72 12 I 885:33  106 mm4 Qf 50  250  185:7 À 25 Qf 2:01  106 mm3 3-77 The shear flow in the flange is obtained as VQf 2:01 q 104  22:70 N=mm 3-78 I 885:33 The spacing of the screws is obtained as the ratio of the resistance (Rs) to the shear flow as Rs 1000 S 44:05 mm 3-79 q 22:70 The spacing of the screws must not exceed 44.05 mm. It may be rounded to s 40 mm. Remarks on Shear Flow We have provided an elementary treatment for shear stress and shear flow in thin-walled cross-sections. The subject is further discussed in advanced strength of materials and elasticity. Our calculation is based on the shear stress formula (t VQ/It) and the shear flow formula (q VQ/I). Internal force analysis provides the shear force (V). The para- meters t, Q, and I are geometrical properties of the cross-section, and only the section with one axis of symmetry is considered. The moment of inertia (I) is calculated about the neutral axis. The moment of area (Q) is calculated about the neutral axis, but its value depends on the location of the area. The shear flows in four typical cross-sections are depicted in Fig. 3-45. The shear center is designated by the letter (O) and centroid by (G). Shear flow is along the thin walls of a cross- section. In the web the shear flow is vertical (along the negative y-axis), which is the direction of the shear force (V). In the flange, the shear flow is horizontal (or along the x- axis). The shear flow has a quadratic variation in the web, and it attains the maximum value at the neutral axis where the normal stress due to flexure is zero. In the flange, the shear flow is linear and it is zero at a free surface. For an angle section, the shear flow has a quadratic distribution along the legs and peaks at the neutral axis. For a section with two axes of symmetry, the torque induced from the shear flow equilibrates. A beam made of such a cross-section will flex without twisting when the line Simple Beam 201 y V qmf z qmf 2qmf V qmw qmw O G O, G (a) Symmetrical I-section. (b) Symmetrical channel section. V V qm qm qm O G G O qm qm (c) Angle section with equal legs. (d) Box section. FIGURE 3-45 Shear flow in a typical cross-section. of action of the load passes through the centroid of the cross-section. For such sections, the shear center (O) coincides with the centroid (G). When a transverse shear load (V) is applied to a cross-section with one axis of symmetry, it will bend as well as twist. The twist can be avoided when the load is applied at the shear center, which lies along the axis of symmetry. Its location is calculated by setting the torque due to shear flow and shear force to zero. For a symmetrical angle and T-section, the shear center lies at the intersection of the legs, which need not be at the centroid of the section. The shear center is a property of the cross- section, and it is independent of the applied loads. The shear flow formula is used to estimate the interface shear force in segments of a built- up cross-section. The spacing of a fastener with known resistance can be calculated by equating the shear force to the strength of the fastener. 3.10 Composite Beams The normal stress (s) in a beam due to a moment (M) has a linear variation with zero stress (sna 0) at the neutral axis and maximum stress at the top and bottom fibers (smax ), as shown in Fig. 3-46a. For a rectangular cross-section, shown in Fig. 3-46b, the material at the outer fiber is subjected to a peak stress and the cross-section must be designed for this 202 STRENGTH OF MATERIALS y max na =0 Full Poor z utilization utilization (a) Stress variation (b) Rectangular (c) I-beam. ( = My/I). cross-section. Steel Steel rod Wood Concrete Interface (d) Wood-steel beam. (e) Reinforced concrete. FIGURE 3-46 Composite beams. condition to avoid failure. The material can be utilized to its capacity at the top and bottom fiber locations. The material cannot be used to its full strength capacity around the neutral axis because of the low stress level. This design can become inefficient and heavy. Engineers can improve the design by using different cross-sections, as depicted in Figs. 3-46c to 3-46e. A single material I-section shown in Fig. 3-46c is quite efficient because its flange area is subjected to high stress. Its web is proportioned thin, thereby only a small amount of web material is not efficiently utilized. The same purpose can be achieved by building a beam of two (or more) materials. A strong material is used at the outer fiber, and a weak material is placed around the neutral axis. A wooden beam with steel plates, as shown in Fig. 3-46d, is such an example. Concrete can be used in a beam design by reinforcing it with steel rods as shown in Fig. 3-46e. Consider a beam made of a strong material (m1 with Young's modulus E1) at the outer fiber and a weak material (m2 and E2) elsewhere, as shown in Fig. 3-47a. Let the beam deform, inducing strain due to moment M. It is assumed that the beam is adequately designed 1 m1 t m1 n a m2 m2 b 2 (a) Composite beam. (b) Strain variation. (c) Stress variation. FIGURE 3-47 Strain and stress variations in a composite beam. Simple Beam 203 to act as a single integral unit throughout its use without any gap or debonding at the two material interfaces. The concept of equality of strain developed in Chapter 2 for the analysis of the composite bar is extended to analyze the composite beam. Interface failure (or delamination) is avoided when the induced strain at any location at the interface is the same in both materials. The variation of the flexural strain across the depth is shown in Fig. 3-47b. It has a linear variation with no discontinuity. At the top interface, strain (et ) is the same for both m1 and m2 materials, as shown in Fig. 3-47b. Likewise, the strain is eb in both materials at the bottom interface. The strain variation is linear across the beam depth for both the composite beam and the single material elastic beam. The stress (s), which is a product of the strain (e) and the modulus (E), changes depending on its value for m1 and m2 materials. The stress variation is not linear across the beam depth. Its variation is linear within a material, but it jumps at the interface, as shown in Fig. 3-47c. The two stresses at the top interface are sm1 et Em1 and sm2 et Em2 . Consider, for example, that material 1 is steel with modulus Em1 200 GPa and material 2 is aluminum with modulus Em2 70 GPa. Assume that there is an interface strain of one-tenth of a percent (eb 0:001). At the interface, the stresses are sm1steel 0:001  200 200 Mpa, and sm2aluminum 0:001  70 70 MPa. There is a stress jump at the interface from 200 MPa (for steel) to 70 MPa (for aluminum), but the strain is the same (ebm1steel ebm2aluminum 0:001). The strain equality principle is used to calculate an equivalent section, and the flexure formula can be used to calculate the stress in a composite beam. The procedure is illustrated through two examples. EXAMPLE 3-20 A composite beam with the dimensions shown in Fig. 3-48a is made of wood with steel plates strapped to the top and bottom faces. It is subjected to a moment of 100 in.-k. Determine the stress variation over the cross-section. For Young's modulus, use Es 30,000 ksi and Ew 2000 ksi for steel and wood, respectively. Solution The solution of the problem is illustrated through a steel model and a wood model. Steel Model: The ratio (n) of the two moduli is Ew 2000 1 n 3-80 Es 30;000 15 The wood section is converted into an equivalent steel section. Such a section has the same depth (11.5 in.), but its original width at 8 in. is reduced by the ratio n (8/n 0:53 in:). Strain equality is the basis of the transformation (e sw /Ew ss /Es ). The steel model is shown in Fig. 3-48b. It is like an I-beam that can utilize the material to its full capacity. The flexural formula is used to calculate stress (s My/I); here the sign is not important and neglected. 204 STRENGTH OF MATERIALS y = 2917 Steel z 2796 1/4 in. y 12 in. 8/15 in. n a z 6 in. Wood 1/4 in. 186 8 in. (a) Composite beam. (b) Steel model. (c) Stress variation in psi. 120 in. z w s (d) Strain variation. (e) Wood model. FIGURE 3-48 Analysis of a composite beam. 1 3 1 8 1 3 Is 8  12 À 8À 12 À 2  205:68 in:4 12 12 15 4 ÀMy À100  103 s y À468:2y psi Is 205:68 At y 6 in:; s À2917 psi 1 y 6 À in:; s À2796 psi 4 Stress in wood at y 5:75 in., ss ss sw Ew 186:4 psi 3-81 Es n Simple Beam 205 The stress variation across the cross-section is shown in Fig. 3-48c. In the steel material, the stress is s 2917 psi at the top fiber and s 2796 psi at the interface. In the wood material, the maximum stress is s 186 psi. A stress jump of Ás 2796 À 186 2610 psi occurs at the interface. The strain at the interface is continuous [e es ( 2796/Es 93  10À6 ) ew ( 186/Ew 93  10À6 )] as shown in Fig. 3-48d. The strain continuity avoids delaminating at the interface. Wood Model: The equivalent width is obtained using the inverse ratio (1/n 15). The steel section is converted into an equivalent wood section. Such a section has the same depth, but its width is increased (8n 120 in:). The wood model is shown in Fig. 3-48e. It is also like an I-beam that can utilize the material to its full capacity. The stress is calculated from the flexural formula for the wood model (s ÀMy/I). 1 1 Iw  120  123 À  120 À 8  11:53 12 12 112 17,280 À  1520:875 12 17:28 À 14:195  103 3:085  103 in:3 My 100 y sw 32:41y I 3:085 sw at y 6 in: 32:41  6 194:48 psi 194:48 ss  Es 194:48  15 2917 psi Ew ss at y 5:75 in: 32:41  5:75  15 2795:6 psi sw at y 5:75 in: 32:41  5:75 186:4 3-82 The stress in the wooden model is changed to obtain the stress in steel. The actual stress variation is identical to that in Fig. 3-48c. Either the steel or wood model can be used, and the choice is not important. EXAMPLE 3-21 A reinforced concrete beam made of concrete with steel reinforcement is shown in Fig. 3-49a. It is 50 cm deep and 25 cm wide. Two 25-mm-diameter steel bars that are located 50 mm from the bottom surface reinforce it. Determine the stress variation over the cross-section for a moment (M 80 kN-m). Use Es 200 GPa and Ec 25 GPa as Young's modulus for steel and concrete, respectively. 206 STRENGTH OF MATERIALS 25 cm Concrete dn n 25-mm-diameter a 50 cm 45 cm steel bar 50 mm Steel 25 cm 31.4 cm (a) Reinforced beam. (b) Equivalent concrete model. 11.4 25.2 kPa 202 (c) Stress variation. FIGURE 3-49 Analysis of a reinforced concrete beam. Solution Concrete is a good material in compression, but it is weak in tension. It is customary to neglect the tensile strength of concrete in the analysis of a reinforced beam. The calculation uses meter as the length unit and the dimensions are d 0:5 m b 0:25 m Es 200 n 8 Ec 25 The area of steel reinforcement is pdbar p 25 2 2 As 2 982  10À6 m2 3-83 4 2 1000 The steel area can be considered as an equivalent rectangle with a depth equal to the bar diameter and a width calculated as the ratio of area to diameter as Simple Beam 207 As 982  10À6 bs 39:3  10À3 m 3-84a dbar 25  10À3 The equivalent concrete dimension (bcc ) is obtained as the product of bs and n. bcc bs  n 39:3  10À3  8 314  10À3 m 3-84b For the calculation, the concrete model shown in Fig. 3-49b is used. The depth of the neutral axis is dn and is not yet known. It is assumed that the concrete is in compression above the neutral axis, but this is neglected below the neutral axis because it might have cracked due to tension. The tensile strength is provided by steel, which is shown as a rectangle with a depth of 25 mm and a width of 31.4 cm located 45 cm from the top fiber. To use the beam formula (s My/I), we have to calculate the location of the neutral axis and the moment of inertia about this axis. The depth of the neutral axis is calculated by equating the moment of area of concrete and steel to be zero about the neutral axis. In such a calculation, the contribu- tion that the steel depth makes to the moment of area is neglected because it is a small quantity. The neutral axis calculation requires the solution of a quadratic equation that is obtained by taking the moment of concrete and steel about the neutral axis. dn 0:25dn À 8  982  10À6  0:45 À dn 0 2 2 1000 dn 62:84dn À 28:32 0 p À62:85 Æ 62:852 4  1000  28:32 dn 2  1000 dn 0:14; À0:20m 3-85 The quadratic equation has a positive root at dn 0:14 m. Its negative root (dn À0:20 m) is discarded. For the concrete model, the moment of inertia about the neutral axis is calculated as 1 0:14 2 I  0:25  0:143 0:25  0:14  12 2 8  982  10À6  0:45 À 0:142 57:17 171:5 755  10À6 984  10À6 m4 My 80 sÀ y 81:3y  103 kPa I 984  10À6 Stress (sc ) at the top concrete fiber, 208 STRENGTH OF MATERIALS sc À81:3  103  0:14 11:38 MPa compression Stress (ss ) in steel, ss 8  81:3  0:45 À 0:14  103 202 MPa tension 3-86 The stress distribution is shown in Fig. 3-49c. The stress at the top concrete fiber is sc 11:4 kPa, and it is compressive. In the steel reinforcement, the stress is tensile (ss 202 kPa). If we assume that the concrete has not cracked, then near the steel reinforcement, the stress would be ssc 202/8 25:2 kPa. The stress jump at the inter- face would be 176.8 kPa. Concrete cracks because it cannot sustain 176.8 kPa tensile stress. Problems Use the material properties given in Appendix 5 to solve the problems. 3-1 Draw the shear force and bending moment diagrams for the following beam examples. (a) A cantilever beam of length ` 3a is subjected to a uniform load (p 1 lbf/unit length) along the mid third of its span, as shown in Fig. P3-1a. (b) A cantilever beam of length ` 4 m is subjected to a uniform load ( p 1 kN/m) for the quarter span and a concentrated load of magnitude P 3 kN at the midspan, as shown in Fig. P3-1b. (c) A 30-ft-long simply supported beam is subjected to a uniformly increasing distributed load ( p 20 (` À x) lbf per ft), as shown in Fig. P3-1c. (d) A 5-m-long cantilever beam is subjected to a midspan moment (M0 50 kN-m), as shown in Fig. P3-1d. (e) A simply supported beam of span ` is subjected to two concentrated loads of magnitude P, as shown in Fig. P3-1e. (f) A simply supported beam of span 15-ft is subjected to two concentrated loads of magnitude P at one-third and two-thirds span locations, as shown in Fig. P3-1f. (g) A simply supported beam with a 10-m span is subjected to a uniformly distributed load ( p 4 kN/m) along 2 m, as shown in Fig. P3-1g. (h) A simply supported beam with a 20-ft span is subjected to uniformly increasing distributed loads of magnitude p 1 to 5 kip/ft, as shown in Fig. P3-1h. (i) A 40-ft-long simply supported beam is subjected to a uniformly distributed load of magnitude p 40 lbf/ft along the central half span, as shown in Fig. P3-1i. (j) The 10-m-long simply supported beam has a 4-m overhang, as shown in Fig. P3-1j. It is subjected to a moment (M0 ) at its free end. (k) The 10-m-long simply supported beam has 2-m overhangs, as shown in Fig. P3-1k. It is subjected to a moment (M0 40 kN-m) at one end and a load (P) at the other end. Simple Beam 209 y P p lbf/length p per meter x a a a /2 /4 (a) (b) y p = po( – x) Mo x 5m 30 ft (c) (d) /4 /4 1 kip 1 kip P P 5 ft 5 ft 5 ft (e) (f) 4 kN/m 5 kip/ft 1kip/ft 10 m 2m 20 ft (g) (h) 40lbf/ft 10 ft Mo 40 ft 6m 4m (l) (j) P Mo 2m 6m 2m 5 kN/m 2m 6m 2m M = 40 kN-m (k) (l) FIGURE P3-1 210 STRENGTH OF MATERIALS (l) The 10-m-long beam has 2-m overhangs on either side, as shown in Fig. P3-1l. It is subjected to a moment (M0 ) at the midspan and to a uniform load (p 5 kN/m) along the overhang. 3-2 Verify the shear force diagram using the differential relationship between bending moment and shear force for the beams in Problems 3-1c, 3-1f, and 3-1`. 3-3 Verify the bending moment diagram using the integral relationship between the bending moment and shear force for the beams in Problems 3-1a and 3-1k. 3-4 Determine the distribution of normal stress and strain and shear stress and strain, along the depth of the beam in Problem 1-f for the following two cases: (a) The beam is made of steel with a rectangular cross-section that is 20 cm deep and 10 cm thick. (b) The beam is an aluminum I-beam with the dimensions shown in Fig. P3-4b t t = 6 mm t 30 cm 15 cm FIGURE P3-4b 3-5 Generate the elastic curves for the beam problems (Problems 3-1a, 3-1c, and 3-1g). Assume a uniform Young's modulus (E) and moment of inertia (I). 3-6 The aluminum beam shown in Fig. P3-6 has a uniform temperature variation over the ambient along its length, as shown in the figure. Calculate its displacement function due to the temperature variation. tu t u = 100°F y 12 in. t = 25°F x a a 4 in. t Section Temperature variation 20 ft 10 ft at a–a along depth FIGURE P3-6 Simple Beam 211 3-7 The temperature variation in Problem 3-6 is confined to the overhang portion of the beam. Generate its elastic curve. 3-8 The roller support of the aluminum beam in Problem 3-6 has a settling of 2 in. along the negative y-coordinate direction. Calculate the displaced shape of the beam. 3-9 For the four beam cross-sections shown in Fig. P3-9, (a) Calculate the shear stress and shear flow for a unit value of shear force applied along the negative y-coordinate axis. (b) Locate the shear center for each cross-section. y z 8 1 12 0.2 20 0.4 0.4 12 in. 20 cm (a) I-Section. (b) Angle section. 10 t 8.5 in. t 0.715 20 t=1 8 0.375 (c) T-Section. (d) Channel section. FIGURE P3-9 3-10 A 25-ft-long simply supported beam subjected to a load (P 10 kip) at the center span is shown in Fig. P3-10. Calculate the resistance of the screws for a 3 in. spacing for the two built-up sections shown in the figure. 212 STRENGTH OF MATERIALS 10 kip 25 ft 9 in. 9 in. t t t 18 in. t 18 in. 2 2 t = 1in. t FIGURE P3-10 3-11 A 4-m-long cantilever beam is subjected to a load (P 10 kN), as shown in Fig. P3-11. Calculate, at the mid-span of the beam, the stress and strain along the beam depth and at P 10kN 4m 25 cm 25 cm 5 mm Concrete 50 cm 40 cm Wood Aluminum 25-mm- diameter 5 mm 5 cm Steel bar Aluminum-wood construction Reinforced concrete FIGURE P3-11 Simple Beam 213 the interfaces for the composite cross-sections shown in figure. The first cross-section is made of wood with aluminum plates at the top and bottom faces as shown in the figure. The concrete beam is reinforced with two 25-mm bars in the second section. 3-12 Verify the dimensional stability of the following eight beam formulas in SI base units by calculating the dimensions of the left and right variables. The definitions of the variables are identical to those discussed in this chapter. (a) Flexure formula: s À MyI (b) Shear stress formula: t VQ It (c) Shear flow formula: q VQ I (d) Bending moment and shear force relationship: V À dM dx (e) Displacement formulas: d4 v p 1 dx4 EI d3 v V 2 3 À dx EI d2 v M 3 2 dx EI (f) Hooke's law: s Ee E (g) Material property: G 2(1) (h) Relations: MR P`, and R Pa` 3-13 For a determinate structure, answer ``true or false'' with illustration if required. (a) The number of boundary restraints cannot exceed three. (b) Four restraints can make a cantilever beam unstable. (c) Equilibrium equations are sufficient to calculate internal moment and shear force. (d) The number of equilibrium equations sometimes may be insufficient for the calculation of all the reactions. (e) Reactions can be a function of the beam material. (f) Reaction, load, and applied moment follow the t-sign convention. (g) Internal moment and shear force follow the n-sign convention. (h) Stress and strain follow the n-sign convention. (i) Displacement follows the n-sign convention. (j) The shear force and bending moment are independent of each other. (k) Shear force and bending moment depend on one or both of the material properties: Young's modulus and Poisson's ratio. (l) The normal and shear stresses are independent of the beam material. (m) The normal and shear strains are independent of the beam material. (n) No stress is induced in the beam because of the change in temperature. (o) No strain is induced in the beam because of the change in temperature. (p) No strain is induced in the beam because of support settling. (q) Temperature variation can induce a reaction. 214 STRENGTH OF MATERIALS (r) There is no deflection in a beam because of support settling. (s) Temperature variation cannot induce displacement. (t) In displacement calculation, the shear strain is not accounted. (u) Beam deflection is due only to flexural strain. (v) A plane section before deformation remains a plane even after deformation. (w) There is no rotation of a plane section during deformation. (x) Stress can be discontinuous at the interface of a composite beam. (y) Strain can be discontinuous at the interface of a composite beam. (z) The shear center and the centroid coincide for a cross-section that has two axes of symmetry. (aa) The location of the shear center depends on the magnitude of the shear force. (bb) The shear center is a property of the shape of the cross-section. (cc) Even when a shear load is applied along the shear center, the cross-section can twist in addition to flexing. (dd) Stress is always continuous. (ee) Strain is sometimes continuous. (ff) Displacement can be discontinuous at times. (gg) The screws fastening a two-material interface resist the normal stress at the location. (hh) The screws fastening a two-material interface resist the shear stress at the location. (ii) Concrete is reinforced because it is weak in compression. Simple Beam 215 4 Determinate Shaft A shaft is a structural member that is used to transmit power as well as rotational motion. A ship's propeller shaft and an automotive drive shaft transmit power. A shaft provides the axis of rotation for gear trains and flywheels. This structural member has applications in automotive and aircraft engines, pulleys and sprockets, clock mechanisms, and other machinery. A typical shaft can be a straight cylinder, a stepped cylinder, and the frustum of a cone with a solid or hollow core. The cross-section of a shaft can also take a noncircular shape. A shaft can resist torsional load (or torque), bending moment, and axial load. This chapter is confined to the analysis of a determinate shaft of simple geometry that is subjected only to torque. A circular shaft cantilevered out of a wall is depicted in Fig. 4-1a. It has a length ` with a uniform circular cross-section and radius R. It is subjected to a torque load (T ` ) that acts along the axis of the shaft, which is considered as the x-coordinate axis. The shaft can be made of an isotropic elastic material with shear modulus G. Its response variables consist of an internal torque (T), shear stress (t), and angle of twist (j). The goal of analysis is to calculate the three variables (T, t, and j). The applied torque, or the torque load (T ` ) follows the n-sign convention, and it is positive when it is directed along the x-coordinate axis. The induced or internal torque shown in Fig. 4-1b is positive, and it follows the t-sign convention. The shear stress (t) follows the t-sign convention and it has a linear variation along the radius as shown in Fig. 4-1c. The angle of twist as shown in Fig. 4-1d is positive and follows the n-sign convention. The stress and deformation in a shaft are primarily due to the applied torque because temperature and support settling have little effect. Temperature does not change the response because no shear strain is induced in a shaft made of an isotropic material. Settling of the support of a shaft, which represents a slippage at the boundary, induces a uniform rigid body rotation (j0 ) throughout its length. Because of the circular symmetry, this rotation is inconsequential, except that it changes the reference to measure the angle of twist (j) without any change to the angle (j) itself. The analysis, in other words, is for the applied 217 T T T T T x= dx dx x (a) Shaft under torque. (b) Internal torque T. max R (r) r (c) Variation of shear stress in (d) Deformation, or angle of twist, . cross-section. FIGURE 4-1 Torsion of a shaft. torque load, and it is introduced in stages that included the analysis of internal torque, torsion formula, and deformation analysis. 4.1 Analysis of Internal Torque A shaft resists an external torque load by inducing internal torque. The external load can be applied at a point (T ` ), or it can be distributed over a portion of the shaft length (t` ), as shown in Fig. 4-2a. The line of action of the load must be along the shaft axis, or the x-coordinate axis. In USCS units torque is specified in units of inch-pound force (in.-lbf) and distributed torque (t` ) in units of inch-pound force per inch (in.-lbf/in.). In SI units, torque can be specified in kN-m and t` in units of kN-m/m. The internal torque (T) is determined by applying the rotational equilibrium along the yx direction, which is also the direction of torque and rotation j. The single equilibrium equation represents the summation of torques or moments (Mx ) along the rotational x-coordinate direction. ÆMx ÆT 0 4-1 The equilibrium equation is illustrated through the solution of examples. In these exam- ples, two-dimensional sketches, which are sufficient to illustrate the analysis, are employed. 218 STRENGTH OF MATERIALS A CT D t B TR TR TAC TCD TDB x A C D B /2 /4 (a) Shaft in Example 4-1. (b) Reaction and internal torques. C TCD TDB t B TR A T x –x (c) Torque in span CD ( /2 ≤ x ≤ 3 /4). (d) Torque in span DB (3/4 ≤ x ≤ ). TR A C T D t TDB x (e) Torque in span DB (3/4 ≤ x ≤ ). FIGURE 4-2 Analysis of a cantilevered shaft. EXAMPLE 4-1 Determine the internal torque in a circular cantilever shaft of diameter d and length ` that is subjected to a torque load (T ` ) at the midspan and a distributed torque (t` ) in the fourth quarter of the span, as shown in Fig. 4-2a. Solution The three internal torques: (T AC ) in span segment AC (0 x `/2); (T CD ) in CD (`/2 x 3`/4); and (T DB ) in DB (3`/4 x `) are marked in Fig. 4-2b. Reaction (T R ) marked in Fig. 4-2a is calculated to equilibrate the applied loads. `t` TR T` À 0 4 `t` T R ÀT ` 4-2a 4 Torque (T AC ) is calculated from EE at A using the free-body diagram shown in Fig. 4-2b. T AC T R 0 Determinate Shaft 219 `t` T AC T ` À 4-2b 4 The internal torque (T CD ) is obtained from the equilibrium of torques in the free- body diagram shown in Fig. 4-2c. T CD T ` T R 0 `t` T CD ÀT R À T ` À 4-2c 4 The internal torque (T DB ) is obtained from the equilibrium of torques in the free- body diagram for the short span BD shown in Fig. 4-2d. ÀT DB À t` ` À x 0 T DB Àt` ` À x 4-2d An identical answer is obtained when torque (T DB ) is recalculated from the longer segment as shown in Fig. 4-2e. 3` T DB À t` x À T` TR 0 4 3` T DB t` x À À T` À TR 4 T DB t` x À ` 4-2e Calculation of internal torque is quite simple because it requires only one equilib- rium equation. The torque analysis may use the length of the shaft, but neither the material property nor the diameter is required. EXAMPLE 4-2 A 120-in.-long shaft rests on two bearings at A and D. Two gears are mounted at the one-third and two-third span locations, as shown in Fig. 4-3a. The small gear at B rotates in the clockwise direction and exerts a torque of T B À2 in:-k. The big gear rotating in the counterclockwise direction exerts a torque T C 10 in:-k. The bearing at A is locked, whereas the shaft is free to rotate at D. Determine the internal torque in the shaft, neglecting the weight of the shaft and the gears. 220 STRENGTH OF MATERIALS C B D A TB TC TR A B C D 40 in. 40 40 (a) Gear-mounted shaft. (b) Cantilever shaft model. TR TAB TBC TCD TR TB TBC A B C D A B C (c) Reaction and internal torque. (d) Torque in span segment BC. TBC TC B C D (e) Torque in BC. FIGURE 4-3 Analysis of a gear-mounted shaft. Solution For torque calculation, the cantilever model depicting the load (T B , T C ) and reaction (T R ) is shown in Fig. 4-3b. It is fixed at A because the rotor is locked, and it is free at D. The counterclockwise motion of the big gear induces a positive torque (T C 10 in:-k) at C and a negative torque (T B À2 in:-k) at B because the small gear is rotating in the clockwise direction. The internal torque in span segment AB (0 x 40) is (T AB ), in BC (40 x 80) it is (T BC ), while the segment CD (with 80 x 120) is torque free (T CD 0); see Fig. 4-3c. The EE of the applied torques and the reaction (T R ) in Fig. 4-3b yields TR TB TC 0 T R ÀT B À T C À 8 in:-k 4-3a The EE at A in the free-body diagram in Fig. 4-3c yields the internal torque (T AB ) in span segment AB as T R T AB 0 T AB 8 in:-k 4-3b Determinate Shaft 221 Torque in the segment BC (40 x 80) can be calculated using the neighboring segments either to the left or to the right of BC. For the left neighboring segment shown in Fig. 4-3d, the EE yield T R T B T BC 0 T BC 8 2 10 in:-k 4-3c For the right neighboring segment in Fig. 4-3e, the EE yield T C À T BC 0 T BC T C 10 in:-k 4-3d An identical value is obtained for torque (T BC ) from both Eqs. (4-3c) and (4-3d), as expected. The torque is constant (T BC 10 in:-k) in span segment BC (40 x 80). The torque in span segment AB (T AB 8 in:-k); in BC (T BC 10 in:-k); CD CD (T 0) and the reactive torque is (T R À8 in:-k). The material and diameter of the shaft are not used in the torque calculation. 4.2 Torsion Formula The internal torque induces shear stress in the shaft. The relationship between the stress (t) and the torque (T) is referred to as the torsion formula. It is developed in this section and would have the following form. Tr t 4-4 J Here, t t (r) shear stress at r r distance from center T torque J polar moment of inertia, which is a property of the cross-section The formula is illustrated for a shaft with a solid circular cross-section in Fig. 4-4a and a tube with an annular cross-section in Fig. 4-4b. The shear stress attains its maximum value (tmax ) at the outer fiber (r R) for both the solid and annular shafts. The shear stress attains a zero value (t 0) at the center of the shaft cross-section. The shear stress has a linear variation in the fiber at a distance r from the center as shown in Fig. 4-4a. For an annular cross-section, the shear stress attains minimum value (tmin ) at the inner wall (r R0 ), as shown in Fig. 4-4b. The polar moment of inertia (J) of a cross-section in Eq. (4-4) is a parameter analogous to the 222 STRENGTH OF MATERIALS max max (r) min R R r Ro (a) Solid cross-section. (b) Annular cross-section. Undeformed (x) plane ( ) a b b' T T T O1 T O2 Deformed x x x plane (c) Pure torsion. (d) Pure shear stress. FIGURE 4-4 Pure torsion of a shaft. moment of inertia (I) of a beam in the flexure formula. The torsion formula is credited to Coulomb (1736±1806) for a circular cross-section and to Saint-Venant (1797±1886) for a general cross-section. The derivation is based on assumptions pertaining to material prop- erty, a pure torsion condition, and the kinematics of deformation. The shaft material is assumed to be linearly elastic. The shear stress (t) and shear strain (g) are related through the shear modulus (G) as (t Gg). A shaft is twisted by torque (T) acting at its ends, as shown in Fig. 4-4c. An arbitrary cross-section at a distance x from the origin is subjected to the same torque (T). Such a uniform torque state is referred to as ``pure torsion.'' Pure torsion induces pure shear stress (t), as shown in the elemental block (Áx) in Fig. 4-4d. Because of the circular symmetry of the shaft cross-section, the shear stress varies linearly from the center to the outer radius as shown in Fig. 4-4a. tmax t r r 4-5a R Here, R radius of the shaft tmax maximum value of shear stress Shear stress (t) is replaced by shear strain (g) using Hooke's law (t Gg), and a linear variation is obtained for the strain. Determinate Shaft 223 t tmax gmax r r 4-5b G RG Twisting of a shaft under uniform torque is shown in Fig. 4-4c. If the left end of the shaft is restrained, the right end will rotate by angle j. The angle j is referred to as the angle of twist. During the process of twisting, an undeformed plane marked ``o1 abo2 '' will deform into the position ``o1 abH o2 ,'' which is also a plane. In other words, a plane section before deformation remains a plane after deformation. 4.3 Deformation Analysis Deformation analysis establishes the relationship between the angle of twist and the shear strain. It is illustrated by considering an elemental block of length (Áx), as shown in Fig. 4-5a. The right-hand section twists with respect to the left by a small angle. Before deformation, an elemental area on its outer wall is marked ``z1 Àz2 Àz3 Àz4 .'' Let this elemental block twist by the angle Áj. The location of the deformed area is shown as z1 ÀzH2 ÀzH3 Àz4 . The definition of shear strain (g) is also illustrated in Fig. 4-5a. The angle z2 Àz1 Àz4 is a right angle (p/2) in the undeformed sector ``z1 Àz2 Àz3 Àz4 ,'' and it is located at the outer surface of the shaft. The deformed sector occupies the position marked ``z1 ÀzH2 ÀzH3 Àz4 .'' The original right angle z2 Àz1 Àz4 has changed to ``zH2 Àz1 Àz4 ,'' which is (p/2 À g). The change in the right angle is defined as the shear strain (g) and it is the angle z2 Àz1 ÀzH2 . z2 p2 z'2 z1 max z'2 p1 z2 z4 z3 R R z'3 x x (a) Strain ( ) and twist d . (b) Maximum shear strain and angle of twist. F A O r (c) Torque in a shaft cross-section. FIGURE 4-5 Relationship between strain and angle of twist. 224 STRENGTH OF MATERIALS Assuming small deformation; the length of the elemental arc z2 ÀzH2 shown in Fig. 4-5a can be calculated from two different directions. 1. Along the longitudinal direction, z2 ÀzH2 Áxgmax 4-6a The shear strain attains the maximum value (gmax ) at the outer surface of the shaft, as shown in Fig. 4-5b. The plane (p1) before deformation remains a plane (p2) even after deformation. 2. Along the radial direction, z2 ÀzH2 RÁj 4-6b The relationship between the shear strain and angle of twist is obtained by equating the arc length defined in Eqs. (4-6a) and (4-6b) as Áxgmax RÁj Áj dj gmax R LtÁx 3 0 R 4-6c Áx dx Let the angle of twist per unit length, or the rate of change of angle of twist with respect to length, be defined as y. dj y 4-7a dx The angle y is a constant under pure torsion because every section along the length of the shaft is subjected to the same torque. If a shaft of length ` produces an angle of twist j, then the angle y can be expressed as dj j y 4-7b dx ` The relationship between shear strain, the angle of twist (j), and angle y becomes Rj gmax Ry 4-7c ` The torsion formula is obtained by linking the shear stress to torque. The force in an elemental cross-sectional area (ÁA) of the shaft shown in Fig. 4-5c is obtained as the product of stress (t) and the elemental area as ÁF t rÁA 4-8a In the limit (ÁA 3 0), (dF t(r)dA). The corresponding torque about the center 0 is obtained as the product of ÁF and the radial distance r as Determinate Shaft 225 dT rdF rt rdA Eliminate t(r) in favor of tmax using Eq. (4-5a) to obtain tmax 2 dT r dA 4-8b R Integration over the cross-sectional area yields the torque T as tmax T r 2 dA 4-8c R A The polar moment of inertia (J) is defined as J r 2 dA 4-8d tmax T J 4-8e R TR tmax 4-8f J r Since (t(r) tmax R) the torsion formula is obtained as Tr t r 4-9a J Strain can be calculated from Hooke's law as Tr g r 4-9b JG Substitute strain for the angle of twist from Eq. (4-7c) into the stress-strain relationship to obtain GRj tmax Ggmax 4-10 ` Elimination of stress between Eqs. (4-8f) and (4-10) yields the angle of twist and torque relationship. TR GRj J ` T` j 4-11 JG 226 STRENGTH OF MATERIALS The factor JG is called the shear rigidity of the shaft. The angle of twist has a linear variation along the length of the shaft. The angle j(x) at an intermediate location x in the shaft axis is obtained by replacing x in place of ` in Eq. (4-11). Tx j x 4-12 JG In a shaft, stress, strain, and angle of twist are calculated from Eqs. (4-9a), (4-9b), and (4-12), respectively. EXAMPLE 4-3 A solid shaft and a hollow shaft, each of length 100 in. are subjected to a 10-in.-k torque as shown in Fig. 4-6a. The solid shaft, which is made of steel with Young's modulus E 30,000 ksi, Poisson's ratio n 0:3, and weight density r 0:284 lbf/in:3 , has a diameter of 12 in. The hollow shaft, which is made of the same material, has an inside diameter that is 75 percent of its outside diameter, but its weight is equal to that of the solid shaft, as shown in Fig. 4-6b. Calculate the maximum stress and the angle of twist for both the shafts. Solution The equilibrium of load and reaction T R shown in Fig. 4-6a yields (T R ÀT ` À10 in.-k). The internal torque (T) is uniform across the shaft length, and its value is obtained from the equilibrium of the free-body diagram shown in Fig. 4-6c. T T ` 10 in.-k 4-13a The polar moment of inertia (J S ) of the solid shaft is p 4 p JS d  124 2036 in:4 4-13b 32 32 The weight of the solid shaft (W S ) with diameter (d 12 in:) is pd2 p W s rV s rA` r ` 0:284   122  100 3:21 kip 4-13c 4 4 The weight of the hollow shaft with outer diameter d0 and inner diameter di 0:75d0 is obtained as a function of the outer diameter as pÀ 2 Á ph 2 i W H rV H rAH ` r d0 À di2 ` r d0 À 0:75d0 2 ` 9:759d0 lbf 4-13d 2 4 4 Determinate Shaft 227 ro ri a T = 10 TR a 12 in. 100 in. Solid shaft Hollow shaft (a) Shaft under torsion. (b) Cross section. T T (c) Free-body diagram at b–b. FIGURE 4-6 Analysis of the shaft in Example 4-3. The outer diameter of the hollow shaft is obtained by equating the two weights (W S W H ) as r H s 2 3 3:21  103 W W X 9:759 d0 3:21  10 d0 18:14 in: 4-13e 9:759 di 0:75  18:14 13:60 in: 4-13f The polar moment of inertia for the hollow shaft is pÀ 4 Á pÀ Á JH d0 À di4 18:144 À 13:604 7266 in:4 4-13g 32 32 The maximum shear stress at the outer fiber of the two shafts is TR 10  103  6 tS max 29:5 psi 4-13h Js 2036 TR 10  103  9:07 tH max 12:5 psi 4-13i JH 7266 The shear modulus is calculated from the Young's modulus and the Poisson's ratio, and it is E 30,000 G 11,538 ksi 4-13j 2 1 n 2 1 0:3 228 STRENGTH OF MATERIALS The angles of twist at the free end of the solid and hollow shafts are T` 10  103  100 js 4:26  10À5 rad 2:44  10À3 deg 4-13k J S G 2036  11;538  103 T` 10  103  100 jH 1:19  10À5 rad 0:68  10À3 deg 4-13l J HG 7266  11;538  103 tH max 12:5 Stress ratio 0:42 4-13m tS max 29:5 jH 0:68  10À3 Angle of twist ratio 0:28 4-13n jS 2:44  10À3 The maximum shear stress is less in the hollow shaft. It is only 42 percent that of the solid shaft. The hollow shaft has less angle of twist, which is 28 percent that of the solid shaft. A hollow shaft with weight equal to that of the solid shaft induces lower stress as well as a smaller angle of twist. For the problem, the hollow shaft should be preferred because it is more efficient than the solid shaft. EXAMPLE 4-4 A stepped cantilevered shaft of length 120 in. is made of two members, as shown in Fig. 4-7a. The first steel shaft is a 80-in.-long tube with an outer radius r0 of 6 in. and an inner radius ri of 5 in. The second solid aluminum shaft has a radius rs of 5 in. The first shaft carries a torque of T `1 6 in:-k at its center span, whereas the second shaft is subjected to a torque of T `2 3 in:-k, also at its center. The Young's modulus of steel is E1 30,000 ksi and of aluminum is E2 10,000 ksi. The Poisson's ratio for both materials is ns na n 0:3. Calculate the shear stress and angle of twist for the shaft. Solution The polar moments of inertia (J1 and J2) of the hollow and solid shafts, respectively, are pÀ 4 Á pÀ Á J1 r0 À ri4 64 À 54 1054 in:4 4-14a 2 2 p 4 J2 rs 981:7 in:4 4-14b 2 Determinate Shaft 229 a B b D A C E TR T 1 T 2 E J1 a b J2 T 1 T 2 (b) Reaction at A. 1 = 80 2 = 40 in. x AB 6 in. A B T 9 5 5 (c) Torque in span AB. a–a b–b A B TBC C Cross-sections 9 6 (a) Stepped shaft. (d) Torque in span BC. A B C TCD D A B C D TDE E 9 6 9 6 3 (e) Torque in span CD. (f) Torque in span DE. FIGURE 4-7 Analysis of the stepped shaft in Example 4-4. The shear moduli and rigidities of the shafts are E1 G1 11;538 ksi 2 1 v E2 G2 3846 ksi 4-14c 2 1 v J1 G1 12:16  106 in:2 -k J2 G2 3:78  106 in:2 -k 4-14d The reaction (T R) is obtained from the equilibrium of reaction and loads as shown in Fig. 4-7b. T R T `1 T `2 0 4-14e T R À9 in:-k Likewise, the EE for the free-body diagrams in Figs. 4-7c to 4-7f yield the internal torque for the segments AB (T AB), BC (T BC), CD (T CD), and DE (T DE), respectively. 230 STRENGTH OF MATERIALS T AB 9 in:-k T BC 9 À 6 3 in:-k T CD 9 À 6 3 in:-k T DE 0 in:-k 4-14f The shear stresses at location A, and in the segments AB, BC, CD, and DE obtained from the torsion formula, are TR 9000  6 tA r0 51:23 psi J1 1054 T AB 9000  6 tAB r0 51:23 psi J1 1054 T BC 3000  6 tBC r0 17:1 psi J1 1054 T CD 3000  5 tCD rs 15:3 psi J2 981:7 T DE tDE rs 0 4-14g J2 The angles of twist for segments AB, BC, CD, and DE are T AB  `AB 9000  40 ÁjAB 29:6  10À3 rad 1:70 deg J1 G1 12:16  106 T BC  `BC 3000  40 ÁjBC 9:87  10À3 rad 0:56 deg J1 G 1 12:16  106 T CD  `CD 3000  20 ÁjCD 15:87  10À3 rad 0:91 deg J2 G 2 3:78  106 T DE  `DE ÁjDE 0 4-14h J2 G 2 Determinate Shaft 231 Angle of Twist in a Composite Shaft Calculating the angle of twist in a composite shaft with a variation of parameters like internal torque and shear rigidity requires an extension of the formula (j T`/JG) given by Eq. (4-11). The formula for a very small length (Á`) can be written as TÁ` Áj JG In the limit Á` 3 0, Td` dj JG ` Td` j 4-15 0 JG Consider a composite shaft of total length ` made of p number of segments with the following lengths and shear rigidities: {`1 ,(JG)1 , `2 ,(JG)2 , F F F , `p ,(JG)p }. For such a shaft, Eq. (4-15) is integrated to obtain `1 `2 FFF`p `1 `2 `p Td` Td` Td` Td` j ; Á Á Á ; JG JG JG JG 0 0 `1 `pÀ1 T 1 `1 T2 `2 Tp `p j ; Á Á Á ; 4-16a JG1 JG2 JGp The total angle of twist is the sum of the individual contributions p j Áji i1 T` Áji 4-16b JG i Consider a shaft with three segments with varying properties (T, J, G, and `), as shown in Fig. 4-8. The individual contributions to the angle of twist (Áj) are T1 `1 T` T` T` Áj1 Áj2 Áj3 J1 G1 JG 1 JG 2 JG 3 In a typical term, for example in term 2, the torque (T2 ) and the rigidity (JG)2 are assumed to be constant over the span segment `2 . The term Áj2 (T`/JG)2 is the contribution to the angle of twist from segment 2. The angles of twist at locations D, C, B, and A are obtained as 232 STRENGTH OF MATERIALS A B C D 1 2 3 (T1, 1, J1, G1) (T, , J, G)2 (T, , J, G)3 FIGURE 4-8 Three-segment shaft. jD Áj1 Áj2 Áj3 jC Áj1 Áj2 jB Áj1 jA 0 4-16c Returning to Example 4-4, we can calculate the angles of twist at different shaft locations as follows. jE ÁjAB ÁjBC ÁjCD ÁjDE 55:4  10À3 rad 3:17 deg jD ÁjAB ÁjBC ÁjCD 55:34  10À3 rad 3:17 deg jC ÁjAB ÁjBC 39:47  10À3 rad 2:26 deg jB ÁjAB 29:6  10À3 rad 1:70 deg jA 0 4-17 The angle of twist (jE ) at the free end E includes the contributions from segments AB, BC, CD and null contribution from the torque-free segment (DE). 4.4 Power Transmission through a Circular Shaft Power from a motor is transmitted through a circular shaft. The stress and deformation induced in the shaft can be calculated by relating power to torque. Power produced by a motor is rated in terms of shaft horsepower (shp) at a specified rotational speed (n). Power is defined as work done per unit time (1 hp 550 ft-lbf/sec). The speed n can be measured in units of revolutions per minute (rpm) or revolutions per second (frequency, f), which is also called hertz (Hz). Speed (n) in rpm and frequency ( f ) in Hz are related as n f 4-18a 60 The motor torque (T m) is related to horsepower and speed in rpm (and Hz) in the following formulas: Determinate Shaft 233 550 hp Tm 4-18b 2pf 33;000 hp Tm 4-18c 2pn 1 hp 550 ft-lbf/sec 746 watts 4-18d The derivation of the horsepower-torque formula can be found in standard physics text- books. It relates horsepower and torque at a specified rotational speed (n in rpm or f in Hz). Equation (4-18) is sufficient to solve the torsion problem associated with the transmission of power through shaft. EXAMPLE 4-5 Calculate the torque produced by a 40-hp motor at a frequency of f 20 Hz. Solution The torque from Eq. (4-18b) is 550  40 Tm 175 ft-lbf 2:1 in.-k 4-19 2p  20 A shaft that is subjected to torque T m can be analyzed using regular torsion theory. EXAMPLE 4-6 A 50-hp motor at a speed of 20 Hz is driving two gears that are mounted on a steel shaft of radius r 3 in: and length ` 10 ft, as shown in Fig. 4-9a. The power consumed by gear A is twice that of gear B. Calculate the stress and deformation in the shaft neglecting the weight of the shaft and gears and the friction at bearing support C. Solution The analysis model of the shaft is shown in Fig. 4-9b. The frictionless bearing support at C offers no resistance to its twisting motion, and it need not be considered in the analysis model. The shaft horsepower (T m) is obtained as 550  50 Tm 219 ft-lbf 2:63 in.-k 4-20a 2p  20 234 STRENGTH OF MATERIALS A C Tm TA 0.5 TA B D 50 hp A B D 20 Hz 4 ft 3 ft D TDA A TAB B 10 ft (a) Gear locations. (b) Analysis model. Tm TDA D A TAB B Tm T A TM D A (c) Torque in segment DA. (d) Torque in segment AB. FIGURE 4-9 Analysis of the gear-mounted shaft in Example 4-6. The motor torque shown in Fig. 4-9b is assumed positive and along the counter- clockwise direction. Let us assume torque from gear A to be (ÀT A ) and from gear B as (À0:5 T A ). The EE of the model in Fig. 4-9b yields T m À T A À 0:5T A 0 1:5T A 2:63 T A 1:75 in.-k 4-20b The internal torques (T DA and T AC) are marked in Fig. 4-9b. The EE of the segment DA in Fig. 4-9c yields T m T DA 0 4-20c DA m T ÀT À2:63 in.-k Likewise, the EE of the segment BD in Fig. 4-9d yield T AB À T A T m 0 T AB 1:75 À 2:63 À0:88 in.-k The polar moment of inertia and shear rigidity are p J 34 127:23 in:4 2 Determinate Shaft 235 G 11;538 ksi steel JG 1:468  106 in:2 -k 4-20d The shear stresses in segments AB (tAB ) and BC (tBC ), are calculated from the torsion formula. T DA r 2630  3 tDA À À62:0 psi 4-20e J 127:23 T BC r 880  3 tAB À À20:75 psi 4-20f J 127:23 The angles of twist in segments AB (ÁjAB ) and BC (ÁjBC ) are T DA `DA 2:630 ÁjDA À  4  12 À0:086  10À3 rad JG 1:468  106 À4:93  10À3 deg 4-20g T AB `AB 0:88  6  12 ÁjAB À À0:043  103 rad À2:47  10À3 deg JG 1:468  106 4-20h The angles of twist jB at B and jA at A are as follows: jB ÁjDA ÁjAB À7:4  10À3 deg 4-20i jA ÁjDA À4:93  10À3 deg 4-20j jD 0 4-20k The angle of twist is zero near the motor, and it is maximum at the free end B. Problems Use the material properties given in Appendix 5 to solve the problems. 4-1 Verify the dimensional consistency of the torsion formulas in USCS base units by calculating the dimensions of the left and right variables. The definition of the variable is identical to that described in this chapter. Tr Stress t J 236 STRENGTH OF MATERIALS Tr Strain g JG T` Angle of twist j JG 4-2 A tubular aluminum shaft is subjected to a 1-kN force through a wrench as shown in Fig. P4-2. Calculate the torque, the variation of shear stress and shear strain along the radius, and the maximum angle of twist. 1 kN c 0.3 3m A m c 0.5 m B 1 kN t = 15 mm 15 cm Section at c–c FIGURE P4-2 4-3 The cantilevered shaft shown in Fig. P4-3 has three segments, and it is subjected to two torque loads. Calculate the internal torque, maximum shear stress, and maximum angle of twist for the following two design cases: 4m 4m 4m a b c Ta Tc c 2 a b 2 T a = 2T c = 10 kN-m 20 cm 16 8 Section at a–a b–b c–c FIGURE P4-3 Determinate Shaft 237 (a) The entire shaft is made of aluminum with a solid circular cross-section with diameter 14 cm. (b) The shaft is made of three steel tubes with outer diameters as shown in the figure. Each tube is 12 mm thick. 4-4 A 40-hp motor at 20 Hz drives the two gears that are mounted at locations B and C along a solid steel shaft of diameter 4 in. as shown in Fig. P4-4. The support at D offers no resistance to twisting of the shaft. The gear at C requires three times more torque than the gear at B. Gear B has a counterclockwise motion while gear C moves in the opposite direction. Calculate the stress and angle of twist in each segment. B C D A 40 hp 20 Hz a a/2 a a = 60 in. FIGURE P4-4 4-5 Two gears and shafts have the dimensions shown in Fig. P4-5. The solid shafts and the gears are made of steel. (a) Calculate the torque in each gear in the unmated condition for circumferential displacements of s1 and s2, as marked in the figure. (b) The gears are mated next. In this condition, the angle of twist at the circumference of the smaller gear was measured at j1 0:2 . Calculate the torque and angle of twist in each shaft. = 60 in. s1 b 1 R b s2 R = 8 in. s1 = 0.15 in. 2 R a a R = 16 in. s2 = 0.3 in. 1 = 0.2 8 in. Section at a–a and b–b FIGURE P4-5 238 STRENGTH OF MATERIALS 5 Simple Frames Simple frames can be built as an assemblage of truss, beam, and shaft members. The frames can be analyzed for internal force, stress, and strain using the theory that has already been developed for the three types of members. Displacement analysis, when attempted through an extension of the formulas of the three member types, may become cumbersome, at least for some kind of frames. It can, however, be handled elegantly through energy theorems based on the internal energy and external work concepts, as discussed in Chapter 12. For simplicity, this chapter is confined to the calculation of force parameters. The displacement calculation will be examined in a subsequent chapter. The frame analysis requires no new theory, and it is illustrated through the solution of a set of typical examples. The analysis has four basic steps: 1. The frame is separated into two (or more) substructures. A substructure can be a truss, a beam, or a shaft. In other words, the frame is made of truss, beam, and shaft members. 2. The first substructure is analyzed for internal forces and reactions (also stress and strain) from the data given for the problem. 3. The analysis for the next substructure is set up using the reactions and forces determined for the first substructure, along with the data of the problem. The second substructure is solved, and the process is repeated for the remaining substructures. 4. The substructure results are combined to obtain the solution to the given problem. EXAMPLE 5-1: A Support Frame The aluminum frame shown in Fig. 5-1 supports a load (P 500 lbf) at its free end A. It is fixed to the foundation at D and hinged at locations A, B, and C. The frame is built of three aluminum tubes whose dimensions are marked in Fig. 5-1. Analyze the frame. 239 y P = 500 lbf x 40 in. c A B c b 23.1 in. b 30 ° t = 0.5 in. C 4 in. 120 in. Sections b–b a a and c–c 1 in. D 9 in. Section a–a FIGURE 5-1 Analysis of a support frame for Example 5-1. Solution Step 1ÐSeparate the Frame into Two Substructures, a Truss and a Beam The truss, shown in Fig. 5-2a, is made of two bars (AB and AC), hinged at A, B, and C. Locations B and C are considered as supports. The beam DCB is shown in Fig. 5-2b. Step 2ÐAnalysis of the Truss (or the First Substructure) The truss is analyzed for bar forces and the reactions at B and C, along with stress and strain. Two EE are written at node A, and their solution yields the two internal forces (Fba and Fca ). ÀFba À Fca cos 30 0 ÀFca sin 30 À 500 0 5-1 Fba 866 lbf Fca À1000 lbf 5-2 The truss bar BA is in tension while bar CA is in compression. The four reactions (Rbx , Rby , Rcx , and Rcy ), shown in Fig. 5-2a are back-calculated from EE written at B and C as 240 STRENGTH OF MATERIALS 500 lb B A Fba Rbx B Fba A D 30° Rby D C B y int Fca Fcay Fb Fba 30° Fca Fba D C x Fcax Rcx Rcy (a) Truss. (b) Member DCB. (c) Interface force. D C B D Fdc C Fcb B Fba Fcax Fcay = 500 lbf (d) Beam. (e) Truss bar. D C B Md M Rd C B 886 866 D Mdc 20 in.-k y (g) Bending moment diagram. 20 Vdc Mcb C B –66 lbf 120 – y Vcb 866 (h) Shear force diagram. (f) Moment and shear force calculation. FIGURE 5-2 Analysis of internal forces for Example 5-1. Rbx Fba 0 Rby 0 Rcx Fca cos 30 0 Rcy Fca sin 30 0 5-3 Rbx À866 lbf Rby 0 Rcx 866 lbf Rcy 500 lbf 5-4 Simple Frames 241 The vertical reaction (Rby ) is zero because the connecting bar BA is horizontal. Step 3ÐAnalysis of Member DCB (or the Second Substructure) The member DCB is shown in Fig. 5-2b with the y-coordinate axis positioned horizontally and the x-coordinate axis shown vertically down. This position is obtained by a 90 clockwise rigid body rotation that has no effect on the analysis. Member DBC interacts with the truss at locations B and C. The load is transferred to member DCB from the truss at these locations. The value of the load at B is calculated from the force interface diagram shown in Fig. 5-2c. The interface force (Fb int ) is obtained from the EE written at a cut in the bar BA close to B as Fb int À Fba 0 Fb int Fba 866 lb 5-5 The interface force (Fb int ) is identical in magnitude to the bar force (Fba ), and its direction is that of the bar force at B for bar BA. It is also equal to the reaction in magnitude but acts in opposite direction. Likewise, the interface force is applied at location C, as shown in Fig. 5-2b. At C, this force is equal to the bar force Fca both in magnitude and direction. The interface force is resolved to obtain the components: Fcax Fca cos 30 À866 lbf Fcay Fca sin 30 À500 lbf Member DBC is decomposed into a beam subjected to two transverse loads (Fba and Fcax ), as shown in Fig. 5-2d, and a truss bar subjected to an axial load (Fcay ), as shown in Fig. 5-2e. The force response of member DBC is obtained by superposing the beam and truss solutions. Step 3aÐAnalysis of the Beam The beam DCB with loads is shown in Fig. 5-2f. The reactions Rd and Md are obtained from the EE of the beam. Moment EE at D: Md DC 866 À DB 866 0 Md À96:9  866 120 866 20 in:-k Transverse EE: Rd 866 À 866 0 Rd 0 5-6a Forces in segment DC: The moment Mdc and shear force Vdc are obtained from EE of the free body of the segment DC as shown in Fig. 5-2f. 242 STRENGTH OF MATERIALS Vdc 0 Mdc Md 0 Mdc À20 5-6b Likewise, the moment Mcb and shear force Vcb are calculated from the EE of segment CB, as shown in Fig. 5-2f. ÀMcb À 866 120 À y 0 Mcb 866 y À 120 ÀVcb À 866 0 Vcb À866 Mcbmax À20 kip at y 96:9 in: 5-6c The axial forces for segments DC and CB are obtained from equilibrium of forces in Fig. 5-2e. Axial force in Segment DC: Fdc Fcay À500 lb 5-6d Segment CB: There is no axial force Fcb 0 5-6e The bending moment and shear force diagrams constructed from Eq. (5-6b and 5-6c) are depicted in Figs. 5-2g and 5-2h, respectively. Moment is uniform in the segment DC and it has a linear variation in CB. The cross-sectional properties are not required to calculate forces and reactions. Step 4ÐStress and Strain Analysis The calculation of stress and strain require the area, moment of area and moment of inertia of the members. These parameters follow. Bar member AB and AC: Area of bar AB and AC (Aab and Aac ), respectively, are calculated as À 2 Á Aab p r0 À ri2 r0 4=2 2 ri 2 À 1=2 1:5 Aab p 4 À 2:25 5:5 in:2 Aac Aab 5:5 in:2 5-6f Simple Frames 243 Beam Member BCD: The area (Abcd ), moment of area (Qbcd) and moment of inertia (Ibcd) are r0 4:5 and ri 3:5 À Á Abcd p 4:52 À 3:52 25:13 in:2 pÀ 4 Á Ibcd r0 À ri4 204:20 in:4 4 2À 3 Á Qbcd r0 À ri3 32:17 in:3 5-6g 3 In truss bar AB, the axial stress is obtained as the ratio of axial force to the bar area. Fba 866 sab 157:5 psi 5-6h Aab 5:5 Strain in bar AB if required can be calculated as (eab sE 0:016  10À3 ) ab In truss bar AC, Fac À1000 sac À181:8 psi 5-6i Aac 5:5 In member BCD, there is an axial force along span segment DC, but the segment CB has no axial force. The axial stress in segment DC is obtained as the ratio of the axial force to area as sa 0 cb Fdc À500 sa dc À19:9 psi 5-6j Abcd 25:13 Member BCD is also subjected to bending moment and shear force. Both the bending stress (which is an axial stress) and shear stress are induced in this member. Bending stress changes along the beam depth. The flexural formula is applied to obtain the bending stress. In span segment DC, the bending stress at the outer fiber is My 20  4:5 sb dc Æ Æ Æ440:7 psi 5-6k I dc 204:2 The shear stress is zero in span DC because of the absence of shear force. The shear stress at any point in the span segment CB, calculated from the shear stress formula, is VQ À866  32:17 tcb À68:2 psi 5-6l Ib CB 204:2  f2  r0 À ri g 244 STRENGTH OF MATERIALS Concept of a Beam Column The internal force in a truss bar can be either tensile or compressive. When a bar carries a predominantly tensile force, it is called a tensile strut or a tie whereas it is called a column when compression is the dominating force. A simple beam carries only the bending moment and the associated shear force. When a member carries both bending moment and axial force, along with the associated shear force, it is called a beam column or a frame number. We can calculate the internal forces (bending moment, axial force, and shear force) in a member disregarding their interaction. Finally, the stresses calculated from the internal forces are combined, which is the last step in frame analysis. Step 5ÐCombined Stress Consider the midspan of segment DC. The internal forces and stresses at this location are a F Fdc À500 sa À19:9 psi c M À20 in:-k sb Æ440:7 psi V0 t0 F My s À19:9 Æ 440:7 psi A I t0 Strain is obtained as the ratio of stress to the modulus. s e À0:002 Æ 0:044  10À3 5-6m E The annular cross-section and the axial stress distribution are shown in Fig. 5-3a and Fig. 5-3b, respectively. The axial stress is compressive (s F/A À19:9 psi). The linearly varying flexural stress (s My/I) due to the bending moment is shown in Fig. 5-3c. The flexural stress is zero at the neutral axis, and it peaks at outer fibers (r Ær0 ) at (s Æ440:7 psi). Combined stress is obtained by adding the axial stress and the flexural stress (s F/A My/I), as shown in Fig. 5-3d. The stress is relieved at (r Àr0 ) to 421 psi in tension, but it is increased at r r0 to À460:7 psi in compres- sion. This segment has no shear stress because of the absence of shear force. –20 –441 –461 n a r0 Mr 421 F/A I 0 F My (a) Cross-section. (b) Axial stress in psi. (c) Bending stress. (d) Combined stress = A + I . FIGURE 5-3 Combined stress in psi for Example 5-1. Simple Frames 245 EXAMPLE 5-2: L-Frame The steel frame shown in Fig. 5-4, supports a load (P 1 kip) at its free end A. It is fixed to the foundation at C. The dimensions are marked in the figure. Analyze the frame. Solution Step 1ÐThe Frame Is Separated into Two Substructures Beam BA is the first substructure and the beam column BC is the second, as shown in Fig. 5-4b. Step 2ÐAnalysis of the Beam BA It is analyzed as a cantilever beam subjected to a concentrated load, as shown in Fig. 5-5a. The reactions at the support (MR and R) are obtained as MR À 1  60 0 or MR 60 in:-k RÀ10 or R 1 kip The EE at location x yields moment M(x) and shear force V(x) M x M R À xR 0 or M x x À 60 V x R 0 or V x À1 5-7a y x P = 1 kip 60 in. b B b A 6 in. 120 in. 60 in. 12 in. B A a a Sections a–a and b–b P 120 in. C C (a) L-frame. (b) Analysis model. FIGURE 5-4 Analysis of an L-frame for Example 5-2. 246 STRENGTH OF MATERIALS R 1 kip B A MR V(x) M(x) R V(x) x –1000 lbf x MR (a) Cantilever BA. (b) Shear force diagram. P VB M MB B F(y) M(y) V y y M(x) C –60 in.-k MR MR Ry Ry (c) Bending moment diagram. (d) Reaction. (e) Free-body diagram. FIGURE 5-5 Analysis of internal force for Example 5-2. The shear force (V À1000 lbf) is uniform across the span, as shown in Fig. 5-5b. The bending moment has a linear variation with a peak value of M À60 in:-k at B, as shown in Fig. 5-5c. Step 3ÐAnalysis of the Beam-Column BC The load on the beam-column is obtained from the reaction of the cantilever beam BA. The compressive axial load (P) and a bending moment (M) shown in Fig. 5-5d are as follows. M MB 60 in:-k P VB 1 kip The beam reactions are Ry V B 1 Rx 0 MR M 60 The internal forces are obtained using the free body diagram shown in Fig. 5-5e. Simple Frames 247 Axial force: F y ÀRy À1 kip Shear force: V y 0 Bending moment: M y ÀMR À60 in:-k 5-7b The bending moment and the axial force are uniform across span BC. The beam has no shear force. For the cantilever beam BA, the depth is (d 6 in:) and the thickness is (b 12 in.). Its area (Aba), moment of inertia (Iba), and first moment of area (Qba) are as follows: Aba 12  6 72 in:2 1 3 1 Iba bd  12  63 216 in:4 12 12 bd 2 12  62 Qba Qneutral axis 54 in:3 5-7c 8 8 For the beam column BC, the depth is (d 12 in:) and the thickness is (b 6 in:). Its area (Abc), moment of inertia (Ibc), and first moment of area (Qbc) are as follows: Abc 72 in:2 1 3 1 Ibc bd 6  123 864 in:4 12 12 6  122 Qbc 108 in:3 5-7d 8 Stress and strain are as follows: Beam column BA: Axial force F 0; sa 0 VQ À1000  54 Shear force (V À1000), t À20:83 psi Ib 216  12 t 20:83 g À1:8  10À6 G 11:54  106 My 60;000 Bending moment (M À60,000), sb Æ Æ Â 3 Æ833:3 psi I 216 833:3 eb Æ Æ27:7  10À6 30  106 248 STRENGTH OF MATERIALS Beam column BCX F À1000 Axial force F À1000; sa À13:9 psi A 72 À13:9 ea À4:6  10À7 30  106 Shear force V 0; t0 60;000 Bending moment M À60;000; sb Æ Â 6 Æ416:3 psi 864 eb Æ13:8  10À6 5-7e Step 4ÐCombined Stress The stresses for the beam and the beam column are shown in Fig. 5-6. The bending moment (M À60 in:-k) is the same for both the beam and the beam column. Both members have the same cross-sectional area (A 72 in:2 ), but the normal stress due to bending in the beam at sb 833:3 psi, as shown in Fig. 5-6c, is twice that of the beam-column as sb 416:3 psi, marked in Fig. 5-6f. This is because of the orienta- tion of the cross-sectional area. The beam cross-section experiences the moment along its shorter (or weaker) side with a depth of 6 in. and a moment of inertia I of 216 in:4 . In contrast, the cross-section of the beam column experiences the moment along its longer (or stronger) side with a depth of 12 in. and a moment of inertia I of 864 in.4. y σb 833 y 12 in. σ –21 psi 6 in. n a (a) Cross-section in beam BA. (b) Shear stress. (c) Bending stress. Beam BA y –416 –430 –14 psi 6 in. n a 12 in. 416 402 x x (d) Cross-section in (e) Axial stress. (f) Beam stress. (g) Combined beam column BC. stress. Beam column BC FIGURE 5-6 Stress distribution for Example 5-2. Simple Frames 249 In a beam, it is advantageous to orient the longer side of a rectangular cross-section along the bending moment. The shear stress peaks at the neutral axis of the beam cross-section, as shown in Fig. 5-6b, where the bending stress is zero. The bending stress peaks at the outer fiber where the shear stress is zero. Combined stress for the beam column is obtained by adding the axial stress to the bending stress. The compressive value of the normal stress is increased to (430.2 psi) while the tensile component is reduced to (402.4 psi), as shown in Fig. 5-6g. EXAMPLE 5-3: L-Joint This structure is made of a steel pipe with a stem and a rigid arm joined at 90 , as shown in Fig. 5-7a. The stem is built into a wall, and the arm supports a load. The steel stem of the L-joint is 60 in. long and has an annular cross-section with outer and inner radii at 3 and 2.5 in., respectively, as shown in Fig. 5-7b. The 36-in.-long rigid arm has a rectangular cross-section with a depth of 6 in. and a width of 2 in., as shown in Fig. 5-7b. The arm supports a gravity load (P 2 kip). Analyze the L-joint. y x z 60 in. A a B y 6 in. a /2 b 2.5 in. 6 in. z P = 2 kip b 3 in. 36 in. C 2 in. a-a b-b (a) L-joint. (b) Sections. y P=2 y B C A B T TA b z x 36 in. MB MA VA Pb VB M(z) M(x) V(z) TA VA V(x) VB z x T(x) MB MA (c) Model for a rigid arm. (d) Model for stem. FIGURE 5-7 Analysis of an L-joint for Example 5-3. 250 STRENGTH OF MATERIALS Solution Step 1ÐIdealizing the L-Joint into Two Substructures The first substructure is the rigid arm BC in the y±z±coordinate plane, as shown in Fig. 5-7c. The beam axis is oriented along the positive z-coordinate direction, and the load P 2 kip is applied along the negative y-coordinate direction at C. The second substructure is the shaft AB in the x±y±coordinate plane, as shown in Fig. 5-7d. The transverse reaction from the arm is transferred as a load inducing bending in the member AB. The reactive bending moment from the arm is transferred as torque. The shaft, in other words, is subjected to combined bending and torque loads. Step 2ÐAnalysis of the Rigid Arm The arm is analyzed as a cantilever beam subjected to a concentrated load as shown in Fig. 5-7c. The reactive moments (MB) and shear force (VB), marked in the figure are determined from the moment and transverse EE. MB À 36P 0 MB 72 in.-k VB À P 0 VB P 2 kip V z ÀVB À2 M z À72 2z 5-8a The shear force (V(x) À2 kip) is uniform across the span, and the bending moment has a linear variation with a peak of MB À72 in.-k at B, as shown in Fig. 5-8a. The arm experiences bending moment and shear force even though it is V(x) A B x –2 kip SF diagram M(x) V(z) B C –120 x z in.-k –2 kip BM diagram SF diagram T(x) M(z) B C –72 in.-k z 72 in.-k BM diagram Torque diagram x (a) SF and BF in a rigid arm. (b) SF, BM, and torque diagrams for a stem. FIGURE 5-8 Analysis of internal forces for Example 5-3. Simple Frames 251 rigid because the internal forces in a determinate structure are independent of the material. The bending moment and shear force cannot deform the rigid member; thus, no strain or stress is induced. Step 3ÐAnalysis of the Steel Stem The load in the stem is calculated from the reaction at B. The stem is subjected to a torque (Tb) and load (Pb), as shown in Fig. 5-7d. Pb VB 2 kip Tb Mb 72 in.-k 5-8b The load is equal to the applied load (Pb 2 kip) and it acts along the negative y direction. The moment from the arm is transferred to the stem as a torque (Tb 72 in.-k) along the positive direction. The member is analyzed as a cantilever beam and as a shaft. The stress and strain in the beam and the shaft can be calculated independent of each other because they do not interact. The reactions are calculated by observation of the forces marked in Fig. 5-7d. The reactive torque is (TA ÀTb À72 in.-k:); moment is (MA 60 Pb 120 in.-k) and the transverse reaction is (VA Pb 2 kip). VA À Pb 0 VA Pb 2 kip MA À 60 Pb 0 MA 120 in.-k TA T b 0 TA ÀTb À72 in.-k 5-8c The shear force, bending moment, and torque at a location x in the stem, shown in Fig. 5-7d are as follows: V x ÀVA À2 kip M x MA À xVA 0 M x À120 2x T x ÀTA 72 in.-k 5-8d The shear force and torque are uniform across the span, whereas the bending moment has a linear variation, as shown in Fig. 5-8b. Step 4ÐStress and Strain At location A on the shaft, the bending stress (sx ) and associated shear stress (txy ), and the stress associated with the torsion (txz ) are calculated from the flexure and torsion formulas. 252 STRENGTH OF MATERIALS The bending moment induces normal stress (sx ), and there is no other normal stress induced in the x-coordinate direction. The shear stress (t) due to the shear force (V) more appropriately should be written with two subscripts (x and y) as t txy . The first subscript (x) indicates that the cross-section is normal to the x-coordinate axis, whereas the second subscript indicates that its direction is along the y-coordinate axis. The shear stress (t) due to the torque (T) should also be written with two subscripts (x and z) as t txz . The first subscript (x) indicates that the cross-section is normal to the x-coordinate axis, whereas the second subscript indicates that its direction is along the z-coordinate axis, because the torque is in the y±z plane. The two shear stresses (txy and txz ), which have the same normal but different directions, cannot be added in a straightforward manner because stress is a tensor. Stress, however, can be combined through formula discussed in Chapter 10. At location A on the shaft, the bending stress (sx ) associated and shear stress (txy ), and the stress associated with the torsion (t xz ) are calculated from the flexure and torsion formulas. pÀ 4 Á Iab r0 À ri4 32:94 in:4 4 2À 3 Á Qab r0 À ri3 7:58 in:3 3 pÀ 4 Á Jab r0 À ri4 65:88 in:4 2 M A r0 120  3 sx Æ Æ Æ10:93 ksi Iab 32:94 sx 10:93 ex 0:36  10À3 E 30  103 VA Qab À2  7:58 txy À0:46 ksi Iab b 32:94f2 r0 À ri 1g txy À0:46 exy À4:0  10À5 G 11:54  103 Ta r0 72  3 txz 3:28 ksi Jab 65:88 txz 3:28 gxz 0:28  10À3 5-9 G 11:54  103 There is no interaction between the shear stress components because txy (directed along y) and txz (directed along z) have different orientations. Simple Frames 253 EXAMPLE 5-4: Leaning Column A tower made of reinforced concrete, for a preliminary estimation, is modeled as a stepped column, as shown in Fig. 5-9a. After construction, it developed a 5 tilt to the vertical, as shown in Fig. 5-9b. Calculate the stress at the foundation level for the two positions. Solution The top structure is modeled as a cylindrical column with a diameter of 50 ft and a height of 250 ft. The bottom structure, also modeled as a cylindrical column, has a diameter of 100 ft and a height of 200 ft. In the leaning position, the column is off-centered from the vertical (y-coordinate axis) by 5 . The crushing around the toe at c1 Àc1 , as shown in Fig. 9-9b, is neglected. The weights of the structures are lumped at their centers of gravity. The weight density of reinforced concrete is assumed to be rc 145 lbf/ft3 with Young's modulus Ec 3500 ksi. y 50 ft a a θ = 5° a' a' 250 ft Wt Wth Wtv b b 100 ft b' b' 200 ft Wbv Wbh Wb c c c' x c' Foundation (a) Original structure. (b) Inclined structure. FIGURE 5-9 Leaning column for Example 5-4. 254 STRENGTH OF MATERIALS The weight of the columns and their cross-sectional properties are as follows: Weight of top column (Wt )X Wt (pr 2 )hrc p  252  250  145 71:2  106 lbf 71:2  103 kip Weight of bottom column (Wb ) X Wb p  502  200  145 228  106 lbf 228  103 kip The tilted top-column weight component along original y-axisX Wtv Wt cos 5 71  103 kip The weight component along the original x-axisX Wth Wt sin 5 6:2  103 kip The tilted bottom-column weight component along the original y-axis X Wbv Wb cos 5 227:1  103 kip The weight components along the original x-axis X Wbh Wb sin 5 19:9  103 kip Area of bottom column: Ab p  502 7854 ft2 Moment of inertia for the bottom column: p Ib  504 4:91  106 ft3 4 First moment of area about diameter for bottom columnX 2 Qb  503 83:3  103 ft3 5-10 3 Compressive stress: In the original configuration the compressive stress at cÀc, which is uniform across the base area, is obtained as the ratio of weight to area. À Wt Wb À 71:2 22:8  103 sao À38:1 kip=ft2 À264:6 psi Ab 7854 In the inclined condition, the stress has two components. The axial component is obtained as the ratio of the axial load to the base area. The second component is due to the bending moment. Simple Frames 255 The axial component is: À Wtv Wbv À 70:1 227:1  103 sai À37:8 kip=ft2 À262:8 psi Ab 7854 Moment (M i ) at the base for the inclined position: Mi Wth 200 125 Wbh 100 Mi 100 3:25  6:2 19:9  103 4006  103 ft-k Bendingstress (sbi )X Mi y 4:006  106  50 sbi Æ Æ Æ40:8 kip=ft2 Æ283:3 psi I 4:91  106 Shear stress at the center of the base: Shear force V Wth Wbh 26:1  103 kip Shear stress VQb 26:1  103  83:3  103 ti 4:43 kip=ft2 30:1 psi 5-11a Ib b 4:91  106  100 Combined stress in the inclined position at c1 Àc1 is obtained from the formula: F My s Ç A I s max sai sbi À262:8 À 283:3 À546:1 psi smin À262:8 283:3 20:5 psi 5-11b At the foundation level, the tensile stress is 20.5 psi. The stability of the structure cannot be assured because the stress is tensile. EXAMPLE 5-5: Galileo's Problem The beginning of strength of materials is credited to Galileo (1564±1642). One of his problems relating to strength and resistance is discussed in this example (see his cantilever experimental setup in Fig. 1-37). His puzzle is explained in the following quotation: SAGREDO: I am more puzzled . . . the strength of resistance against breaking increases in a larger ratio than the amount of material. Thus, for instance, if two nails be driven into a wall, the one which is twice as big as the other will support not only twice as much weight as the other, but three or four times as much. 256 STRENGTH OF MATERIALS d B a P T a V (a) Cantilever beam. (b) Section at a–a. FIGURE 5-10 Galileo's problem for Example 5-5. . . . this problem of resistance opens up a field of beautiful and useful ideas . . . (circa 1638; Galileo, G., Dialogue Concerning Two New Sciences. Northern University Press, Illinois, 1950) The problem is illustrated through a steel cantilever beam that resembles Galileo's nail (see Fig. 1-37) of length ` and a uniform circular cross-section with diameter d, as shown in Fig. 5-10. The resistance is obtained in closed form separately for a normal load (P), a transverse load (V), and a torque (T). It is assumed that the yield strength of steel is sy 36 ksi. The resistance under normal stress is considered to be equal to the yield strength (s0 sy 36 ksi), and the shear strength is assumed to be half the yield strength (t0 sy /2 18 ksi). The resistance is defined as the load when the maximum normal stress is equal to the strength (smax s0 36 ksi) or as the load under shear when (tmax t0 s0 /2 18 ksi). Solution The parameters of the beam are as follows: pd2 Area of beam: A 4 Volume: V A` d3 Moment of area about neutral axis: Q 12 pd 4 Moment of inertia: I 64 pd4 Polar moment of inertia: J 32 The axial stress, normal stress in bending, shear stress in bending, and shear stress in torsion are as follows: Simple Frames 257 P 4P Axial stress: sb max A pd2 My 32V` Normal bending stress for M V`X sb max I pd3 VQ 16 V Shear stress in bending: tb max Ib 3 d2 Tr 16T Shear stress due to torque TX tT max 3 5-13 J pd The resistance to the different types of load are calculated by equating stress to strength. Resistance to axial load: 4P sa s0 max pd2 ps0 2 P d s0 A 5-14a 4 Resistance to flexural load: 32 V` sa s0 max pd3 p ps0 3 s0 A A V d p 5-14b 32` 4` p Shear resistance to flexural load: s0 16 V tb t0 max 2 3d2 3 3 V d2 s0 s0 A 5-14c 32 8p Resistance to torque load: s0 16 T tT t0 max 2 pd3 p s0 p T s0 d3 p A A 5-14d 32 4 p For an axial load, the resistance is proportional to the area, see Eq. (5-14a). The load capacity of a bar with a 1-in.2 cross-sectional area is 36 kip, and it increases to 72 kip when the area doubles to 2 in.2. Galileo presumed to have known the bar capacity for axial load. 258 STRENGTH OF MATERIALS For transverse load, the resistance in flexure is proportional to the 3/2 power of the area, see Eq. (5-14b). The load-carrying capacity for a 1-ft-long bar with a 1-in.2 cross-section (and 12 in.3 volume) is V 0:423 kip, but it increases to V 1:20 kip when the area is doubled to 2 in.2 (with volume at 24 in.3), representing a 2.83-fold increase in the load-carrying capacity. Galileo's observationÐ3 or 4 times, but not 8 timesÐis in the correct range because a cantilever typically fails in flexure. For a transverse load, the resistance in shear is proportional to the area, see Eq. (5-14c). The load-carrying capacity for a bar with a 1-in.2 cross-section is V 4:3 kip, and it increases to V 8:6 kip when the area is doubled at 2 in.2. For a torsion load, the resistance in shear is proportional to the 3/2 power of area (see Eq. (5-14d)). The torque-carrying capacity for a bar with a 1-in.2 cross-section is T 5:08 in.-k, but it increases to T 14:36 in.-k when the area is doubled to 2 in.2, representing a 2.83-fold increase in the load-carrying capacity. The load-carrying capacity of the bar with a 1-in.2 cross-sectional area is the least of the four values: (36 kip, 0.423 kip, 4.3 kip, or 5.08 in.-k in torsion). The failure in flexure dominated the other three types of failure modes. Problems Use the material properties given in Appendix 5 to solve the problems. 5-1 A square balcony support frame made of steel is shown in Fig. P5-1. Assume that the beam CB is simply supported on the cantilever beams AB and DC. Calculate the internal force, stress and strain in each of the three members of the frame. Use the following z y x a a B A c d c P d C D t d Sections at a–a, c–c, and d–d FIGURE P5-1 Simple Frames 259 values for: length ` 2 m, depth d 30 cm, thickness t 15 cm. A ten kN load (P 10 kN) is applied at the beam center. 5-2 The post (ABE) supports a truss (ABDC), which carries a load (P) as shown in Fig. P5-2. Consider hinge connections at A, B, C, and D and a fixed condition at E. The truss bars are made of aluminum, and the post is made of steel. Analyze the structure for the following values of the parameters: length a 4 ft and load P 1 kip. The truss bars are tubular with outer and inner radii of r0 4 in: and ri 3:5 in:, respectively. The post has an annular section with outer and inner depths of d0 8 in: and di 7 in:, and inner and outer thickness of t0 4 in: and ti 3:5 in. 2a A C a B D c c P t0 3a d0 Section at c–c E FIGURE P5-2 5-3 The steel u-frame shown in Fig. P5-3 can be used to construct a shed. Considering a triangular reactive pressure distribution at the foundation, calculate the response of the a p per unit length h b0 d0 1.5a pmax FIGURE P5-3 260 STRENGTH OF MATERIALS frame for the following parameters: length a 2 m, height h 3 m, and load p 1 kN/m. The frame is made of annular section with outer and inner dimensions of d0 20 cm, di 18 cm:, b0 10 cm, and bi 9 cm. 5-4 The steel L-frame shown in Fig. P5-4 supports a mass (m0). The mass swings in the x±z plane. For a swing angle (y 150 ), calculate the response of the frame for the following parameters: lengths, a 8 ft, b 4 ft, and c 6 ft, and mass m0 2 slug. The frame is made of a solid circular cross-section with a 4-in. radius. z C c x y b B a A m0 FIGURE P5-4 Simple Frames 261 6 Indeterminate Truss An indeterminate truss is obtained by adding extra bars to a determinate truss, by increasing the number of support restraints, or both. The extra bars and restraints are referred to as redundant members and redundant support conditions, respectively. An indeterminate truss remains stable even when some or all of the redundant bars or restraints are removed. In contrast, a determinate truss will collapse when a single bar or a support restraint is eliminated. An indeterminate truss is preferred because of its stability and the increased strength that emerge from the redundant members. Temperature and support settling induce stress in an indeterminate truss but not in a determinate truss. The analysis of an indetermi- nate truss requires all three sets of equations of determinate analysis, along with an additional set of constraints called the compatibility conditions: 1. Equilibrium equations (EE) 2. Deformation displacement relations (DDR) 3. Force deformation relations (FDR) 4. Compatibility conditions (CC) Any indeterminate truss (or a structure like a beam, a frame, or a shaft) can be analyzed through an application of the four types of equations: EE, DDR, CC, and FDR. The types of response variables are not increased between determinate and indeterminate analysis. They remain the same: namely, bar forces {F}, reactions {R}, nodal displacements {X}, bar deformations {b}, stress (s), and strain (e). Their individual numbers can increase; for example, the number of bar forces increases between determinate and indeterminate trusses, and this is specified through the term referred to as ``the degree of indeterminacy.'' The degree of indeterminacy is illustrated by considering a general two-dimensional truss with the following parameters: 263 md total number of nodes, including the support nodes nF number of bar forces, or bars (n) in the truss (n nF ) nR number of support restraints, which is also equal to the number of reactions m (2md À nR ); number of displacement components The degree of indeterminacy (r) of the truss is equal to the sum of the bar forces and the reactions (nF nR ) less twice the number of nodes (2md ). r nF nR À 2md 6-1a Equation (6-1a) is rearranged to obtain a simpler form: r n À m nF n À f 2md À nR mg 6-1b The indeterminacy is equal to the difference in the number of internal forces (n) and the number of displacement components. If r 0, then the truss is determinate. If r < 0, then it is a mechanism. Our treatment is confined to the analysis of determinate and indeterminate trusses. The truss shown in Fig. 6-1a has four nodes (md 4), five bars (nF 5), and three reactions (nR 3). Its degree of indeterminacy is zero (r 5 3 À 8 0), or it is a determinate truss. The truss shown in Fig. 6-1b has four nodes (md 4), six bars (nF 6), and four reactions (nR 4). It is a two-degree indeterminate truss (r 6 4 À 8 2). The truss shown in Fig. 6-2 has 12 nodes (md 12), 26 bars (nF 26), and 5 reactions (nR 5). Its degree of indeterminacy is seven (r 26 5 À 24 7). The degree of indeterminacy increases to nine (r 26 7 À 24 9) when node 10 is also restrained in both the x- and y- coordinate directions. Engineers often separate indeterminacy (r) into external indeterminacy (re ) and internal indeterminacy (ri ): r re ri . External indeterminacy pertains to the support restraints, y Rv3 x Rv3 v1 v1 3 1 1 1 Ru3 u1 Ru3 u1 2 5 2 3 5 3 Rv4 6 v2 v2 4 4 Ru4 4 2 u2 Ru4 u2 v4 (a) Determinate truss. (b) Indeterminate truss. FIGURE 6-1 Examples of determinate and indeterminate trusses. 264 STRENGTH OF MATERIALS Rv10 Rv10 2 4 6 8 10 12 1 11 3 5 7 9 Ru1 Rv1 Ru11 Rv7 Rv11 FIGURE 6-2 Twenty-six±bar truss. whereas internal indeterminacy is associated with the number of bars in the truss. The amount of external indeterminacy is equal to the number of support restraints in excess of three. re nR À 3 r À ri 6-1c The truss shown in Fig. 6-1a has three support restraints. Therefore, its degree of external indeterminacy is zero (re nR À 3 0), or it is externally determinate. The three reactions of the truss can be determined from the EE of the truss. The truss shown in Fig. 6-1b has four restraints and the same number of reactions. Its degree of external indeterminacy is one (re 4 À 3 1). The truss shown in Fig. 6-2 has five restraints, and its degree of external indeterminacy is two (re 5 À 3 2). The degree of external indeterminacy is increased to four when node 10 is fully restrained (re 7 À 3 4). The internal indeterminacy can be calculated from the following formula: ri nF 3 À 2md r À re 6-1d The addition of Eqs. (6-1c) and (6-1d) yields Eq. (6-1a): (r re ri ). The truss shown in Fig. 6-1a has no internal indeterminacy because ri 5 3 À 8 0. The truss shown in Fig. 6-1b is one-degree indeterminate (ri 6 3 À 8 1). The truss shown in Fig. 6-2 has five degrees of internal indeterminacy (ri 26 3 À 24 5). The degree of internal indeterminacy does not increase when node 10 is restrained. A determinate truss is zero-degree indeterminate internally and zero-degree indeterminate externally. An indeterminate truss can be indeterminate internally, externally, or both. The degree of indeterminacy is independent of the external loads and the material properties of the truss. The analysis equations are different for the determinate and indeterminate trusses. The differences pertain to the number of equations, rather than to the underlying concepts. The differences are discussed for the equilibrium equations, deformation displacement relations, and the force deformation relations. The additional equations, referred to as the compatibility conditions, are then discussed. Indeterminate Truss 265 EXAMPLE 6-1 The nature of the equations of indeterminate analysis is discussed by considering the six-bar truss shown in Fig. 6-3 as an example. It is made of aluminum with a Young's modulus E of 10  103 ksi and a coefficient of thermal expansion (a) of 6:0  10À6 per F. It has six bars (nF n 6), four nodes (md 4), and four reac- tions (nR 4). It has four displacements (m 4). The cross-sectional areas of mem- p bers 1, 3, 5, and 6 are 1 in.2 and those of members 2 and 4 are 2/2. The truss has to be analyzed for the following three load cases: y P = 1 kip = 0.1 x2 3 1 F1 x1 F2 20 in. F6 F3 F4 x4 x3 4 F5 2 x 20 in. FIGURE 6-3 Six-bar truss. Load Case 1: Mechanical load P 1 kip at node 1 along the y-coordinate direction. Load Case 2: The temperature increases (DT 100 F) for member 3 only. Load Case 3: Support 3 settles along the y-coordinate direction by 0.1 in. For simplicity, we will begin the discussion with the mechanical load and then proceed to the other two load cases. 6.1 Equilibrium Equations The equilibrium equations are written at each truss node along the direction of free dis- placement. We avoid writing the EE along the restrained displacement directions. Standard sign convention is followed. A tensile bar force is positive, and it is shown with arrowheads pointing at each other (t-sign convention). Displacement and loads are positive when directed along the positive coordinate directions (n-sign convention). Loads in the EE must be directed along positive directions. 266 STRENGTH OF MATERIALS The indeterminate truss shown in Fig. 6-3 has a total of four equilibrium equations that are written in terms of six bar forces (F1 , F2 , F F F , F6 ). There are two EE at node 1 (along displacement u1 x1 and v1 x2 ) and two at node 2 (along u2 x3 and v3 x4 ). EE are not written at the boundary nodes because these are restrained. p EE along u1 x1 X F1 F2 = 2 0 v1 x2 X F2 =v2 F3 1000 p u2 x3 X F4 = 2 F5 0 p v2 x4 X ÀF3 À F4 = 2 0 The EE in matrix notation can be written as ([B]{F} {P}) V W P p Qb F1 b V b b b b W 1 1/p2 0 0 0 0 b F2 b b 0 b b b b b b ` b T0 ` a a T 1/ 2 1 0p 0 0 U F3 U 1000 R0 0 0 2 1/p 1 Sb F4 b b 0 b 0 b b b 6-2 b b X b Y 0 0 À1 À1/ 2 0 0 b F5 b b b b b 0 X Y F6 Observe the following characteristics of the equilibrium equation (Eq. 6-2), [B] {F} {P}: 1. The coefficient equilibrium matrix [B] of an indeterminate truss is a rectangular matrix, with more columns (n 6) than rows (m 4); n > m and r n À m 2. For a determinate truss, the equilibrium matrix [B] is a square matrix with m n and r 0. 2. The EE are written along the displacement {X} directions, but {X} does not explicitly appear in the EE. The number of EE is exactly equal to the number of displacements (m 4). 3. A column of the EE matrix [B] can be a null column. The force associated with the null column is zero. Force F6 will eventually turn out to be zero because the sixth column is a null column, and the sixth bar cannot carry any load because it is restrained at both ends. This column is retained for the purpose of illustration. A row of an equilibrium matrix must not be a null row, because such a condition will represent instability of the structure. 4. The EE of an indeterminate truss (m 4) cannot be solved to obtain the (n 6) bar forces and it is called an indeterminate problem. The problem remains unresolved for a truss (or a structure) that is made of a rigid material. The problem, however, can be solved for elastic structures, which is our interest. 5. The equilibrium equations are independent of the material properties (such as the Young's modulus and the coefficient of thermal expansion), bar temperatures, and support settling. Indeterminate Truss 267 6.2 Deformation Displacement Relations The deformation displacement relations (b [B]T {X}) derived earlier for a determinate truss still remain valid for an indeterminate truss. The number of DDR differs for determi- nate and indeterminate analysis. The number of deformations (n) increases, but the number of displacements (m) can remain the same (n > m). Because the DDR are defined through the equilibrium matrix [B], their generation does not require additional effort, once the equilibrium matrix is known. The DDR for the six-bar truss example can be written as V W P Q b b1 b b b 1 p 0 p 0 0 V W b b2 b T 1/ 2 1/ 2 b b b b T 0 0 U b x1 b b b ` a T Ub b ` a b3 0 1 0 p À1p U x2 T U 6-3a b b4 b T 0 b b 0 1/ 2 U b x3 b À1/ 2 Ub b b b T bb b R 0 X Y b 5b b b 0 1 0 S x4 X Y b6 0 0 0 0 b1 x1 x1 x2 b2 p 2 b3 x2 À x4 6-3b x3 À x4 b4 p 2 b5 x3 b6 0 Six deformations (b1 , b2 , F F F , b6 ) are expressed in terms of four displacements (x1 , x2 , x3 , and x4 ). For an indeterminate truss, the deformation components (n) outnumber the displace- ment components (m). Their difference is the degree of indeterminacy (r n À m > 0). The six deformations are not independent of each other, but are related through the nodal displacements. For a determinate truss, the number of deformation components is equal to the number of displacement components (r n À m 0). The DDR are geometrical rela- tions and are independent of the material properties (such as the Young's modulus and the coefficient of thermal expansion), the bar temperatures, and the external loads. The interpretation of the DDR is straightforward. Deformation in bar 1 is equal to the displacement along the x-coordinate direction at node 1 (b1 x1 ) because node 3 is restrained (see Fig. 6-3). Likewise, the deformation of bar 5 is b5 x3 because node 3 is restrained. Deformation of bar 3 is the relative displacement of its two nodes (1 and 2)p along the y-coordinate directions (b3 x2 À x4 ). The deformation of bar 2 (b2 (x1p x2 )/ 2) is its elongation along the diagonal. The deformation of bar 4 (b4 (x3 À x4 )/ 2) is also a diagonal elongation, but the displacement component (x3 ) expands the bar, whereas x4 contracts it. Bar 6 has no deformation (b6 0) because both of its nodes are restrained. 268 STRENGTH OF MATERIALS 6.3 Force Deformation Relations Forces {F} and deformations {b} are related through the force deformation relations. The FDR does not change between determinate and indeterminate analysis. In the FDR, ({b}e {b} À {b}0 [G]{F}),{b}e is the elastic deformation, {b} is the total deformation, {b}0 is the initial deformation, and the flexibility matrix [G] is a diagonal matrix with n entries that correspond to the n bars of the truss. The flexibility coefficient is gii (`/AE)i , where `i is the length, Ai is the area, and Ei is the modulus of elasticity of the bar i; see Eq. (6-4a). The FDR for the six-bar truss with a mechanical load (and with no initial deforma- tions, b0 0 and b be ) has the following form: `1 F1 be b1 À b0 1 1 A1 E1 20F1 20 be 1 b1 F1 E E 40 b2 F2 E 20 b3 F3 E 40 b4 F4 E 20 b5 F5 E 20 b6 F6 6-4a E The FDR ({b} [G]{F}) in matrix notation can be written as V W P QV W b b1 b b b 1 b F1 b b b b b b b2 b U b F2 b b b b b T b b T 2 Ub b b b ` a 20 T T U ` F3 a b3 1 U 6-4b b b4 b b b E T T 2 U b F4 b Ub b b b bb b R S b F5 b b b b 5b b b 1 b b b b X Y X Y b6 1 F6 The diagonal flexibility matrix [G] is easily generated. It contains the material property and parameters of the truss bar. 6.4 Compatibility Conditions The n-deformations (b1 , b2 , F F F , bn ) of an indeterminate truss are not independent of one another. They are controlled by linear equality constraints, which are the compatibility conditions. An r-degree indeterminate truss has r compatibility conditions, like f1 (b1 , b2 , F F F , bn ) 0, f2 (b1 , b2 , F F F , bn ) 0, and fr (b1 , b2 , F F F , bn ) 0. The two-degree indeterminate six-bar truss shown in Fig. 6-3 has two compatibility conditions. Indeterminate Truss 269 The deformation displacement relations are the raw material from which the compatibility conditions are derived. In the DDR ({b} [B]T {X}), n-deformations {b} are expressed in terms of m-displacements {X}, and n is bigger than m, (r n À m > 0). Elimination of the m displacements from the n DDR yields (r n À m) constraints on deformations, which constitute the compatibility conditions. C fbg f0g 6-5a Equation (6-5a) represents the r compatibility conditions of the indeterminate structure with n force and m displacement unknowns. In CC {b} represents the total deformation. The compatibility matrix [C] with r rows and n columns has full row rank r. The compatibility matrix [C] is independent of the material of the structure, and it is also a geometrical relation. The total deformation {b} is decomposed into the elastic component {be } and initial component {b0 } as fbg fbge fbg0 6-5b n o Cfbg C fbge fbg0 0 fdRg ÀC fbg0 6-5c The r component vector, {dR} in the CC, is the called the effective initial deformation vector. The CC in terms of elastic deformation can be written as C fbge fdRg 6-5d The compatibility condition, when expressed in terms of total deformation, is a homogeneous equation, such as Eq. (6-5a). The CC becomes a nonhomogeneous equation when it is written in terms of elastic deformations, such as in Eq. (6-5d), with {dR} as the right side. 6.5 Initial Deformations and Support Settling The initial deformation discussed for the determinate truss remains valid for the indeterminate truss. Such a deformation, when due to thermal effects, is equal to the product of temperature strain (et ) and the bar length (`) as (b0 `et `aDT). Here, the coefficient of thermal expansion is a, the temperature change is DT, and the bar length is `. The deformation vector is fbg0 f`et g f`aDtg 6-6 The initial deformations {b}0 due to support settling, which was explained in Chapter 2, have the following form: 270 STRENGTH OF MATERIALS " fbg0 ÀBR T fX g 6-7 " Here, {X} is a p component vector that corresponds to a p number of simultaneous support settlings by amounts ("1 , "2 , F F F , "p ). The matrix [BR ] is associated with the EE x x x written along the p component of the displacements {X}. " The CC is discussed next, considering the example of the six-bar truss. It has two compatibility conditions. The CC is obtained by eliminating the four displacements from the six deformation displacement relations given by Eq. (6-3b). b6 0 6-8a p p b1 À 2b2 b3 À 2b4 b5 0 6-8b The two CC in matrix notation can be written as V W b b1 b b b b b b b !b b2 b & ' b b ` a 0 0 p 0 0 p 0 1 b3 0 6-8c 1 À 2 1 À 2 1 0 b b4 b b b 0 b b bb b b 5b b b X Y b6 The first CC (b6 0) has a single entity because this solitary member 6 is connected to fully restrained supports at nodes 3 and 4. Such a situation arises when the EE matrix [B] contains a null column (here, the 6th column is a null column). The second CC controls the deformations of the other five bars. The CC given by Eq. (6-8c) are specialized for the three load cases. Load Case 1ÐMechanical Load This load case has no initial deformation. The total deformation is equal to the elastic deformation ({b} {b}e ). The CC is obtained from Eq. (6-8c) by replacing {b} in favor of {b}e as V eW b b1 b b eb b b b b !b b2 b & ' b eb ` a 0 0 p 0 p 0 0 1 b3 0 6-8d 1 À 2 1 À 2 1 0 b be b b 4b 0 b eb bb b b 5b b eb X Y b6 Load Case 2ÐThermal Load The initial deformation due to temperature variation {b}0 (`aDT) for the truss is obtained as follows: Indeterminate Truss 271 b0 `aDT1 0 1 b0 0 2 b0 20  6:0  10À6  100 12  10À3 in: 3 6-8e b0 0 4 b0 0 5 b0 0 6 V W b b 0 b b b b b b b b 0 b b ` À3 a 0 12  10 fbg 6-8f b b 0 b b b b b b b b 0 b b X Y 0 The CC for thermal load is obtained by adding {dR} terms in Eq. (6-5d). It is discussed later in this chapter. Load Case 3ÐSupport Settling The initial deformation due to settling of the foundation, given by Eq. (6-7), requires the definition of the matrix [BR ]. This matrix is obtained by writing the EE along the direction of support settling. For this example, the row matrix [BR ] is obtained by writing the EE at support node 3 along the y-coordinate direction, being the direction of settling. F4 Rv3 p F6 6-8g 2 h i BR 0 0 0 p 0 11 2 6-8h The initial deformation due to the settling of the support by 0.1 in. along the y-coordinate direction is Q P V W 0 b 0 b b b T0U b 0 b b b b b T U b ` b a T0U 0p fbg ÀBR fXg ÀT p Uf0:1g À 0 T " T 1 U 6-8i T 2U b 0:1/ 2 b b b b R0S b 0 b b b b b X b Y 1 0:1 The compatibility conditions accommodate the initial deformations that may be due to temperature variation or support settling. 272 STRENGTH OF MATERIALS 6.6 Null Property of the Equilibrium Equation and Compatibility Condition Matrices The product of the equilibrium matrix [B] and the compatibility matrix [C] is a null matrix ([B][C]T [0] or [C][B]T [0]). The null matrix can be verified from the definition of the DDR and the CC as follows: W DDR 3fbg BT fXg b a CC 3 Cfbg 0 6-9 b Y CBT fXg 0 Because the displacement {X} is arbitrary and it is not null a vector, its coefficient matrix must vanish, or CBT BCT 0 6-10 The equilibrium matrix [B] and the compatibility matrix [C] are related. The null property should be verified after the generation of the two matrices: [B] and [C]. 6.7 Response Variables of Analysis The response variables of a truss include bar forces, reactions, deformations, displacements, and stress and strain. The six-bar truss, for example, has a total of 32 response variables, consisting of 1. Six bar forces and four reactions 2. Six deformations 3. Four displacements 4. Six bar stresses 5. Six bar strains The 32 response variables are seldom calculated simultaneously. First, we calculate a single set of variables, called the primary variables. The remaining response variables are back-calculated from the primary variables. For a typical problem, the determination of the primary variables requires the bulk of the effort, which may exceed 80 percent of the total calculations required to solve the problem. Back-calculation requires a small fraction of the total effort. Two solution methods have been developed on the basis of the selection of the primary variables. 1. The method of forces, with forces as the primary unknowns. 2. The method of displacements, with displacements as the primary unknowns. Both methods satisfy all analysis equations (EE, CC, DDR, and FDR) and yield all response variables. Indeterminate Truss 273 6.8 Method of Forces or the Force Method The method of forces considers all the n internal forces as the primary unknowns of the problem. Reactions are not included in this set. This method requires a set of n equations to calculate the n forces. The n equations are obtained by coupling the m equilibrium equations to the r compatibility conditions (n m r). Solution of the n equations yields the primary force unknowns. All other response variables are back-calculated from the n forces. Conceptually, such an analysis can be represented by the following symbolic expression: ! & ' Equilibrium equation Mechanical load fForceg 6-11 Compatibility condition Initial deformation A balance of the internal force {F} and external mechanical load {P} is achieved through the equilibrium equation, which forms the upper part of Eq. (6-11). The compliance of force and initial deformation is achieved through the compatibility condition, representing the lower portion of Eq. (6-11). This symbolic expression, which bestows appropriate emphasis on equilibrium and compatibility, provides both necessary and sufficient conditions for determining forces in an elastic indeterminate structure. In advanced finite element structural analysis, this method is referred to as the Integrated Force Method (IFM), and the same name will be maintained for strength of materials analysis. Equation (6-11) in matrix notation is written as SfFg fPgà 6-12 Here, the coefficient matrix [S] has the dimensions of n  n. The right side vector {P}à includes both the mechanical load {P} and the effective initial deformation vector {dR}. 6.9 Method of Displacements or the Displacement Method The method of displacements considers all the m-nodal displacements as the primary unknowns of the problem. This method requires a set of m equations to calculate the m displacements. Navier (1785±1836) in 1822 transmuted the m equilibrium equations to generate m equations, which are expressed in terms of the m displacements. KfXg fPg 6-13 Here, the coefficient matrix [K] with dimensions of m  m is called the stiffness matrix. The right side vector {P} is the load vector. Other response variables are back-calculated from the displacements. This method is also called the stiffness method. The equation set (6-13) can also be obtained by manipulating the IFM equations. This solution strategy is referred to as the Dual Integrated Force Method (IFMD). The governing equations of IFMD, which resemble the stiffness equations, will be written as [D]{X} {P}. Both the primal IFM and the IFMD yield identical solutions. The dual method and the stiffness method also yield identical solutions to simple strength of materials problems. 274 STRENGTH OF MATERIALS The performance of the dual method and the stiffness method can differ for complex solid mechanics problems. This textbook emphasizes IFM and the stiffness method, but it also illustrates IFMD for a truss problem. In summary, there are two major methods. The method of forces, or IFM, which calculates n forces, {F}, by solving the n equations ([S]{F} {P}) and then back- calculating other response variables. On the other hand, the method of displacements, also called the stiffness method, calculates m displacements {X} by solving the m equations ([K]{X} {P}) and then back-calculates the other variables. The IFMD is a variation of the Integrated Force Method. There is also the traditional redundant force method, which is suitable for small problems and is discussed in Chapter 14. We recommend that readers learn all the methods and compare their relative performances. 6.10 Integrated Force Method The Integrated Force Method (IFM) requires both the equilibrium equations and the compat- ibility conditions to calculate the internal forces. The m EE are expressed in terms of the forces ([B]{F} {P}), but the r CC are expressed in terms of the deformations ([C]{b}e {dR}). The CC has to be expressed in terms of the forces before these can be coupled to the EE. The CC is expressed in forces by eliminating deformations in favor of forces by using the force deformation relations. Cfbge fdRg 6-14a fbge GfFg 6-14b CGfFg fdRg 6-14c The EE and CC are coupled to obtain the governing IFM equation. BfFg fPg 6-15a CGfFg fdRg 6-15b ! & ' B P fFg 6-16a CG dR or SfFg fPgà 6-16b ! B S 6-16c CG & ' P fPgà 6-16d dR Indeterminate Truss 275 Forces are obtained as a solution to Eq. (6-16a). Other response variables can be back- calculated from the n forces. The basic steps required to solve an indeterminate problem using IFM are as follows: Procedures for Analysis Step 0ÐSolution Strategy. At the initial problem-formulation stage, the internal forces and nodal displacements are identified. The number of equilibrium equations, compatibility conditions, and degree of indeterminacy are determined. Step 1ÐFormulate the Equilibrium Equations. Step 2ÐDerive the Deformation Displacement Relations. Step 3ÐGenerate the Compatibility Conditions. Step 4ÐFormulate the Force Deformation Relations. Step 5ÐExpress the Compatibility Conditions in Terms of Forces. Step 6ÐCouple the Equilibrium Equations and Compatibility Conditions to Obtain the IFM Equations, and Solve for the Forces. Step 7ÐBack-Calculate the Displacements and Other Response Variables, as Required. The IFM solution strategy is illustrated by considering the six-bar truss (Example 6±1) and a three-bar truss. Each truss is analyzed for mechanical load, thermal load, and support settling. Load Case 1ÐSolution for Mechanical Loads Step 1ÐFormulate the Equilibrium Equations The four EE along the displacement directions (x1 , x2 , x3 , x4 , also derived in Eq. 6-2) are V W P p Qb F1 b V b b b b W 1 1/p2 0 0 0 0 b F2 b b 0 b b b b b b ` b T 0 1/ 2 1 U` a a T p 0 0 U F3 1000 0 6-17a R0 0 0 1/p2 1 0 b b b Sb F4 b b 0 b b b X b Y 0 0 À1 À1/ 2 0 0 b F5 b b b b b 0 X Y F6 Step 2ÐDerive the Deformation Displacement Relations The six DDR ({b} [b]T {X}), also derived in Eq. (6-3) have the following form: b1 x x1 x2 b2 p 2 b3 x2 À x4 x3 À x4 b4 p 2 b5 x3 b6 0 6-17b 276 STRENGTH OF MATERIALS Step 3ÐGenerate the Compatibility Conditions The two CC, which are obtained by eliminating the four displacements from the six DDR, can be written in matrix notation as V W b b1 b b b b b b b !b b2 b & ' b b ` a 0 p0 0 p0 0 1 b3 0 6-17c 1 À 2 1 À 2 1 0 b b4 b b b 0 b b bb b b 5b b b X Y b6 The null property ([B][C]T [0]) can be verified from the one EE and the CC matrices as P Q P p Q 0 1 p P Q 1 1=p2 0 0 0 0 T0 T À 2U U 0 0 T0 1= 2 1 0p 0 0 UT 0 1U T0 0U T UT p U T U 6-17d R0 0 2 0 1=p 1 0 ST 0 À 2 U R0 0S T U 0 0 À1 À1= 2 0 0 R0 1S 0 0 1 0 Step 4ÐFormulate the Force Deformation Relations The FDR (b F`/AE) for the six truss bars are as follows: `1 F1 20F1 40F4 b1 b4 A1 E1 E E 40F2 20F5 b2 b5 E E 20F3 20F6 b3 b6 6-17e E E Step 5ÐExpress the Compatibility Conditions in Terms of Forces The CC is expressed in forces by eliminating deformations between the CC and FDR to obtain V W b F1 b b b b b b b ! b F2 b & ' b b ` a 20 0 0 0 p 0 0 p 1 F3 0 6-17f E 0 À2 2 1 À2 2 1 0 b F4 b b b 0 b b b F5 b b b b b X Y F6 Indeterminate Truss 277 Step 6ÐCouple the Equilibrium Equations and Compatibility Conditions to Obtain the IFM Equations, and Solve for Forces P p QV W V W 1 1=p2 0 0 0 0 b F1 b b 0 b b b b b T 0 1= 2 1 b b b Ub b b b b T p0 0 0 Ub F2 b b 1000 b b b b ` a ` b a T0 0 0 1= p2 1 0 U U F3 0 T 6-17g T0 0 À1 À1= 2 0 0 Ub F4 b b 0 b T Ub b b b b b b b R0 Sb b b b 0 0 p p0 0 1 b F5 b b 0 b b b b X Y X b Y 1 À2 2 1 À2 2 1 0 F6 0 The CC is scaled, by setting (20/E) to unity because of the homogeneous nature of the equation. Solution of the IFM equation yields the forces: V W V W b F1 b b À545:5 b b b b b b F2 b b 771:4 b b b b b b b b b b b b ` a ` b a F3 454:5 6-17h b F4 b b À642:8 b b b b b b F5 b b 454:5 b b b b b b b b b b b b b X Y X Y F6 0:0 lbf Step 7ÐBack-Calculate the Displacement, if Required, from the Deformation Displacement Relations Displacements are back-calculated from the DDR. Displacement calculation requires only (m 4) out of the (n 6) DDR. Any m DDR can be chosen. Other (n À m) DDR are satisfied automatically. 20F1 x1 b1 À1:090  10À3 in: E p 20 p x2 Àb1 2b2 À F1 À 2 2F2 5:454  10À3 in: E 20F5 x3 b5 0:909  10À3 in: E p 20 p x4 À b1 À 2b2 b3 À F1 À 2 2F2 F3 4:545  10À3 in: 6-17i E Calculation of Reactions The reactions are back-calculated from the bar forces. The reaction (R3x ) at node 3 in the x-coordinates direction is obtained as p 642:8 R3x À F1 F4 = 2 545:5 p 1000 lb 2 Likewise, other reactions are calculated as 278 STRENGTH OF MATERIALS V W V W b R3x b b 1000:00 b b b b b ` a ` a R3y À454:6 6-17j b R4x b b À1000:00 b b b b b X Y X Y R4y À545:4 lbf The reactions satisfy the overall EE (SFx SFY SM 0). Bar stress (s F/A), strain (e s/E), and deformation (b [B]T {X}) are as follows: V W V W V W V W b s1 b b À545:5 b b b b b b À54:6 b b b b À1:09 b b b b b b b b b b b b b b b s2 b b 1091:1 b b b b b b b b 109:1 b b b b b b 3:09 b b b b ` a ` a ` a ` a s3 454:5 45:5 0:91 fsg feg  10À6 fbg Â10À3 b s4 b b À909:2 b b b b b b À90:9 b b b b 2:57 b b b b s5 b b 454:5 b b b b b b 45:5 b b b b 0:91 b b b b b b b b b b b b b b b b b b b X Y X Y X Y X Y s6 0 psi 0 0 in: 6-17k Load Case 2ÐSolution for Thermal Loads Only the right side of the CC, or {dR}, has to be modified for thermal analysis: fR g ÀCfbg0 V W b 0 b b b b b b 0 b b b b b b b b b b b b b ` a`DT b b a 0 fbg b 0 b b b b b b b b b b 0 b b b b b b b b b b X b Y 0 V W b 0 b b b b b b b 0 b b b b !b b ` b b 0 0 0 0 0 1 a`DT a fdRg ÀCfbg0 À p p 1 À 2 1 À 2 1 0 b b b 0 b b b b b b b b b 0 b b b X b Y 0 or & ' & ' 0 0 fdRg 6-17l Àa`DT À1:20  10À2 Indeterminate Truss 279 For thermal loads only, the IFM equations can be written as P p QV W V W 1 1=p2 0 0 0 0 b F1 b b b b b 0 b b b b b b T 0 1= 2 1 T 0 p 0 0 Ub F2 b b Ub b b b b b 0 b b b T0 ` a ` U F3 a T 0 0 2 1= p 1 0U 0 T0 6-17m T 0 À1 À1= 2 0 0 Ub F4 b b Ub b b 0 b b R0 0p 0 0p 0 Sb F5 b b b b b b b b 1 b b b 0 b b b X Y X b 3Y 1 À2= 2 1 À2= 2 1 0 F6 À6:0  10 The {dR} in Eq. (6-17m) is normalized with respect to 20/E. Forces due to DT 100 F in member 3 are obtained by solving the IFM equation as V W V W b F1 b b À545:45 b b b b b b F2 b b 771:39 b b b b b b b b b b b b ` a ` b a F3 À545:45 6-17n b F4 b b 771:39 b b b b b b F5 b b À545:45 b b b b b b b b b b b b X Y X b Y F6 0 lb Calculation of Displacements Displacements at the nodes are calculated from the DDR as follows: x1 b1 be bt1 1 `F be 1 À1:090  10À3 AE 1 bt1 0 x1 À1:090  10À3 in: 6-17o Likewise, other displacements can be calculated: V W V W V W b x1 b b b b b b1p b b À1:091 b b b b ` a ` a ` a x2 Àb1 2b2 5:454  10À3 in: 6-17p b x3 b b À b b b b5 p Á b b À1:091 b b b b X Y X Y X Y x4 À b1 À 2b2 b3 À5:454 The reactions back-calculated from forces are as follows: V W V W b R3x b b b b b 0:0 b b ` a ` a R3y 545:4 6-17q b R4x b b b X b b Y X 0:0 b b Y R4y À545:4 lbf The reactions self-equilibrate, (R3y R4y 0), because there is no external load. 280 STRENGTH OF MATERIALS The calculation of stress and strain is not repeated because these are straightforward. The total deformations are calculated as {b} [B]T {X} from the known EE matrix [B] and displacements {X}. b1 x1 À1:09  10À3 in: x1 x2 b2 p 3:09  10À3 in: 2 b3 x2 À x4 0:91  10À3 in: x3 x4 b4 p À3:64  10À3 in: 2 b5 x3 0:909  10À3 in: b6 0 6-17r Load Case 3ÐSolution for Support Settling The expression {dR} has to be modified for the settling of support node 3 by 0:1 in: along the y-coordinate direction. The initial deformation {b}0 calculated in Eq. (6-8i) is V W b b 0 b b b b b b b b 0 b b ` a " 0 fbg0 BR T fXg b b À0:0707 b b b b b b b b 0 b b X Y À0:10 V W b b 0 b b b b b b !b b 0 b b ` a 0 p0 0 p0 0 1 0 fdRg ÀCfbg0 À 1À 2 1À 2 1 0 b b À0:0707 b b b b b b b b 0 b b X Y À0:1 or & ' À0:1 fdRg 6-17s 0:1 The IFM equations for support settling can be written as P p QV W V W 1 1=p2 0 0 0 0 b F1 b b b b b 0bb b b b b T 0 1= 2 1 0p 0 0 Ub F2 b b Ub b b 0bb T T0 ` a b b b ` U F3 b a T 0 2 0 1=p 1 0U 0 T0 6-17t T 0 À1 À1= 2 0 0 Ub F4 b b Ub b b 0bb b b b b R0 0p 0 0 p 0 1 Sb F5 b b À50;000 b b b b b b b b b X Y X Y 1 À2= 2 1 À2= 2 1 0 F6 50;000 Indeterminate Truss 281 The {dR} in Eq. (6-17t) is normalized with respect to 20/E. Forces due to the settling of support 3 by 0.1 in. in the y-coordinate direction are obtained by solving the IFM equation as V W V W b F1 b b À4545:0 b b b b b b F2 b b 6428:0 b b b b b b b b b b b b ` a ` b a F3 À4545:0 6-17u b F4 b b 6428:0 b b b b b b F5 b b À4545:0 b b b b b b b b b b b b X Y X b Y F6 50;000 lbf Calculation of Displacements Displacements at the nodes are calculated from the DDR as follows: x1 b1 be bt1 1 `F be 1 À9:1  10À3 AE 1 b0 0 1 x1 À9:1  10À3 in: 6-17v Likewise, other displacements can be calculated: V W V W V W b b x1 b b b b bep 1 b b À9:09 b b b b b b b b b b 45:46 b b b ` x2 b b a ` Àbe 2be 1 2 a ` b a e x3 b p À9:09  10À3 in: 6-17w b x4 b b ÀÀbe À 2be be Á b b 5 b b b b b b b b b b b b b b 3 b À54:54 b b X Y X 1 2 Y bX b Y x5 " x prescribed 100:00 Reactions are back-calculated from forces V W V W b R3x b b b b b 0bb ` a ` a R3y 54:54 6-17x b R4x b b b X b b Y X 0bb Y R4y À54:54 kip The formula ({b} [B]T {X}) cannot be used to calculate the deformations because the EE matrix [B] does not include the reactions as primary forces, hence the prescribed displacement is not included in the DDR. The total deformation is back-calculated from the elastic deformation and the initial deformation. 282 STRENGTH OF MATERIALS À4545  20 F` b1 be 1 9:09  10À3 in: 1 1  10  106 AE p 6428  20 2 b2 be 2 p 25:71  10À3 in: 1= 2  107 4545  20 b3 be À 3 À9:09  10À3 in: 1  107 p e 0 6428  20 2 b4 b4 b4 p 0:0707 À45  10À3 in: 1= 2  107 À4545  20 b5 be 5 À9:09  10À3 in: 1  107 50;000  20 b6 be b0 6 6 À 0:1 0 in: 6-17y 1  107 EXAMPLE 6-2 A three-bar truss, shown in Fig. 6-4a, is made of steel with a Young's modulus E of 30,000 ksi and a coefficient of thermal expansion a of 6:6  10À6 per F. The areas of its three bars (A1, A2, A3) are (1.0, 1.0, and 2.0) in.2, respectively. Analyze the truss for the following three load conditions: 1. Load Case 1: Mechanical loads (Px 50 kips and Py 100 kips) as shown in Fig. 6-4a. 2. Load Case 2: Two cases of temperature variations. V W V W F V W V W F ` DT1 a ` 100:0 a ` DT1 a ` 100:0 a DT 200:0 and DT À200:0 X 2Y X Y X 2Y X Y DT3 case1 300:0 DT3 case2 À300:0 Here DTi is the temperature variation in bar i. 1. Load Case 3: Settling of support node 4 by 1 in. in the y-coordinate direction. Load Case 1ÐSolution for Mechanical Loads Step 0ÐSolution Strategy A coordinate system (x, y) with origin at node 1 is shown in Fig. 6-4b. The three bar forces (F1, F2, F3) are the three (n 3) force unknowns. It has two (m 2) displace- ments at node 1: (x1 and x2). The truss is one degree indeterminate (r n À m 1). It has two equilibrium equations and one compatibility condition. Indeterminate Truss 283 100 in. 100 in. 2 3 4 F2 F3 F1 2 (Py, x2) 100 in. 5 3 (Px, x1) 1 (a) Truss. R2y = R2 R3y = F2 R4y = R4 2 2 R2 = F1 R4 = F3 R3x = 0 R2x = –R2 R4x = R4 2 2 y x (b) Reactions. FIGURE 6-4 Three-bar truss. Step 1ÐFormulate the Equilibrium Equations The two EE ([B]{[F]} {P}) of the problem are obtained from the force balance condition at the free node 1 along displacements x1 and x2. V W 4 p p 5b F1 b & ' & ' 1= 2 0 À1= 2 ` a Px 50 p p F 6-18 À1= 2 À1 À1= 2 X Yb 2b Py 100 F3 One CC is required for the analysis of the three-bar truss. 284 STRENGTH OF MATERIALS Step 2ÐDerive the Deformation Displacement Relations The DDR ({b} [B]T {X}) of the truss has the following form: x1 x2 b1 p À p 2 2 b2 Àx2 x1 x2 b3 À p À p 6-19a 2 2 Here, b1 , b2 , and b3 are the bar deformations corresponding to forces F1, F2, and F3, respectively. Step 3ÐGenerate the Compatibility Condition The single CC for the problem is obtained by eliminating two displacements from the three DDR in Eq. (6-19b). p b1 À 2b2 b3 0 6-19b The CC can be written in matrix notation as V W  p Ã` b1 a 1 À 2 1 b 0 6-19c X 2Y b2 The null property ([B][C]T [0]) of the EE and CC matrices can be verified as P Q p p ! 1 ! 2 1=p 0 À1=p2 R p S 0 À 2 6-19d À1= 2 À1 À1= 2 0 1 Step 4ÐFormulate the Force Deformation Relations The FDR for the barsof the truss can be obtained as (b F` /AE). The lengths of the p p three bars are: (100 2, 100, and 100 2) in., and their areas are (1.0, 1.0, and 2.0) in.2, respectively. Indeterminate Truss 285 p ` 1 F1 100 2 b1 F1 A1 E1 E 100 b2 F2 E p 50 2 b3 F3 6-19e E Step 5ÐExpress the Compatibility Conditions in Terms of Forces The CC is obtained in terms of forces by eliminating deformations between the CC and FDR. V W p ` F1 a 100 2 1 À1 1=2 F2 f0g 6-19f E X Y F3 Step 6ÐCouple the Equilibrium Equations and Compatibility Conditions to Obtain the IFM Equations, and Solve for Forces P p p QV W V W 1=p 2 0 À1=p2 ` F1 a ` 50 a R À1= 2 À1 À1= 2 S F2 100 6-19g X Y X Y 1 À1 1=2 F3 0 Solution of the IFM equation yields the forces as V W V W ` F1 a ` 5:025 a F À 42:893 6-19h X 2Y X Y F3 75:736 kips Step 7ÐBack-Calculate Displacement, If Required, from the Deformation Displacement Relations 100F2 x2 Àb2 À 0:143 in: E p 100 x 1 2 b 1 À b2 2F1 À F2 0:110 in: 6-19i E Load Case 2ÐSolution for Thermal Loads Thermal analysis requires the inclusion of the nontrivial {dR} in the right side of the CC: 286 STRENGTH OF MATERIALS fdRg ÀC fbg0 6-19j where V 0W V W b b1 b ` a b DT1 `1 b ` a fbg0 b0 a DT2 `2 2b b Y X 0 b X b Y b3 DT3 `3 V W b DT1 `1 b ` a  p à fdRg À 1 À 2 1 a DT2 `2 b X b Y DT3 `3 or p fdRg À100 2a DT1 À DT2 DT3 fdRgcase1 À0:187 6-19k Likewise, the {dR} calculated for the temperature increase for case 2 becomes fdRgcase2 0 6-19l The nontrivial thermal distribution for case 2 represents a compatible temperature distribution that does not induce any stress. For thermal distribution case 1, the IFM equation can be rewritten to include the {dR} term as follows: P p p QV W V W 1= p2 0 À1=p2 ` F1 a ` 0 a R À1= 2 À1 À1= 2 S F2 0 6-19m X Y X Y 1 À1 1=2 F3 À39:6 The compatibility condition in Eq. (6-19m) has been scaled with respect to 100/E. Solution of the IFM equation yields forces as V W V W ` F1 a ` À13:59 a F À 19:22 6-19n X 2Y X Y F3 À13:59 kips Calculation of Displacement Displacement can be calculated from the DDR as Indeterminate Truss 287 p x1 2 b 1 À b2 b1 be bt1 1 and b2 be bt2 2 upon substitution, x1 À0:155 in: À Á x2 Àb2 À be bt2 À0:196 in: 2 6-19o The compatible temperature, thermal load case 2, produces trivial forces (F1 F2 F3 0) and reactions, but nonzero displacements, which are also calcu- lated from the DDR: V W b b0:0933 b b ` a e t t fbg fbg fbg fbg À0:132 b b b b X Y À0:2800 p x1 2 bt1 À bt2 0:264 in: x2 bt2 0:132 in: 6-19p Load Case 3ÐSolution for Support Settling The {dR} is modified to account for settling of support node 4 by D 1 in: along the y-coordinate direction: V W V W b 0 b b b b b 0 b b b ` b a b ` b a 0 T fbg ÀBR f"g À x 0 f 1g 0 b 1 b b 1 b b p b b X b Y b À p b b X b Y 2 2 V W b 0 b b b b & ' b 0  p Ã` 0 a 1 fdRg À 1 À 2 1 p 6-19q b 1 bb 2 b À p b b X Y 2 P p p QV W V W 1= 2 0 À1= 2 b F1 b b b b b 0 b b T p p U` a ` a T À1= 2 À1 À1= 2 U F2 0 6-19r R Sb b b b b b b X Y X b Y 1 À1 1=2 F3 150;000 The compatibility condition in Eq. (6-19r) has been scaled with respect to p 100 2/E, see Eq. (6-19f). Solution of the IFM equation yields forces as 288 STRENGTH OF MATERIALS V W V W ` F1 a ` 51; 472 a F À À72; 792 6-19s X 2Y X Y F3 51; 472 Calculation of Displacement Displacement can be calculated from the DDR as p x1 2b À b 2 À Á 1 À Á b1 be bt1 0 1 and b2 be bt2 0 2 upon substitution x1 0:586 in: À Á x2 Àb2 À be 0:243 in: 2 6-19t Reactions are back-calculated from forces: V W V W b ÀF1 b V p W b R2x b b 2 b b À36:4 b b b b F1 b b b b b b b R2y b b p b b 36:4 b b b b b b 2b b b b b b b b b ` b b b ` a ` a a R3x 0 0 6-19u b R3y b b F2 b b À72:8 b b b b F b b b b R4x b b p3 b b 36:4 b b b b b b b b b 2b b b b b X Y b F b b b b b X b Y R4y b p3 b X Y 36:4 2 The reactions self-equilibrate in x- and y-coordinate directions because there are no external loads. Theory of Dual Integrated Force Method The dual Integrated Force Method uses the same set of equations as IFM, such as the equilibrium equations, the deformation displacement relations, force displacement relations, as well as compatibility conditions. In IFMD the nodal displacements are calculated first as a solution to its m-governing equations ([D]{X} {P}). The square matrix [D] of dimension (m  m) is obtained from two basic IFM matrices, (EE matrix [B], and flexibility matrix [G]) as ([D] [B][G]À1 [B]T ). The dual matrix [D] is a symmetrical matrix. The forces are back-calculated from the displacement also using a set of equations ({F} [G]À1 [B]T {X} À [G]À1 {b}0 ). The compatibility matrix is not used explicitly but it is satis- fied. The equations of IFMD are formulated in longhand and in matrix notation using the three-bar truss Example 6±2. The IFM equations are listed in Table 6-1. Indeterminate Truss 289 TABLE 6-1 Basic Equations for Three-Bar Truss Example 6-2 Type Longhand Notation Matrix Notation F F Equilibrium p1 2 À p3 Px 50 2 BfF g fPg of forces V W (EE) 4 5 & ' & ' p 1 0 À p 1 ` F1 a p Px 50 ÀF1 À F2 À p3 Py 100 F B 2 2 1 Y fPg Y fF g F2 2 2 À p 1 À1 À p Py 100 X Y 2 2 F3 T Deformation b1 be b0 x1p 2 1 1 Àx 2 fbg B fX g fbge fbg0 displacement V W P 1 Q relation (DDR) ` b1 a p À p & ' 1 & ' T 02 2 À1 U 1 Y x x1 b2 be b0 Àx2 b R S fX g 2 2 X 2Y 1 1 x2 x2 b3 Àp Àp 2 2 x b3 be b0 À x1p 2 3 3 2 p Force deformation be F11`1 100 E 2F1 1 A E fbge GfF g relation (FDR) be F22`2 100F2 2 P Q A E E `1 P p Q p A 1 E1 100 2 T `2 U 1R S be F33`3 3 A E 50 2F3 E G R A 2 E2 SE 100 p `3 50 2 A 3 E3 In dual method IFMD, force is expressed in displacement using the force deformation relations and deformation displacement relations, as given in Table 6-2. The EE are expressed in displacement by eliminating forces between force displacement relation and the EE. For the three-bar truss problem, the equations of IFMD are also given in closed form in Table 6-2. The dual method is numerically illustrated next. IFMD Solution to Example 6-2 In the dual method IFMD, the displacement is calculated first from the equation ([D]{X} {P} {P}0 ), being the governing equation of IFMD. The two-bar truss has two displacements and [D] is a (2  2) symmetrical matrix. The dual matrix [D] is calculated from the equilibrium matrix [B] and the inverse of the flexibility matrix [G] as follows QP P A1 E p p 1 Q 4 5 1 À p p 01 p À1 T ` 2 UT 2 2 D BGÀ1 BT 2 2 A2 E ` T R UR 0 S À1 U S p À1 À1 p À1 2 2 p A3 E À p À p 1 2 1 2 ` 2 P Q P Q 4 1 p 5 1 À1 p 1 3 ! E 2 0 À p T 2 1 2 2 U E R2 2 2 2 S p 3:18 1:06 R 0 À1 S p  105 6-20a ` À p À1 À p 1 2 1 2 ` p 32 2 1 p 1:06 6:18 À1 À1 2 2 2 2 The dual matrix [D] is a (2  2) symmetrical matrix. It remains the same for mechanical, thermal, and support settling load cases. The right side of the governing equation ([D]{X} {P} {P}0 ) changes depending on the nature of the load. Load Case 1ÐSolution for Mechanical Load The right side of the dual equation is the mechanical load {P} and the contribution from initial deformation ({P}0 {0}) is set to zero. & ' 50;000 fP g 100;000 lbf The IFMD equation becomes !& ' & ' 3:18 1:06 x1 50;000 105 6-20b 1:06 6:18 x2 100;000 & ' & ' x1 0:110 6-20c x2 0:143 in: Forces, in the dual method, are obtained from the equation ({F} [G]À1 [B]T {X}) as V W P QP p Q V W ` F1 a 1 p 1 À p & 1 ' ` 5:025 a E 2 2 ST 0 À1 U 0:110 À 42:893 F p R 2 R S 6-20d X 2Y ` 2 0:143 X Y F3 2 À p À p 1 2 1 2 75:736 kip Indeterminate Truss 291 TABLE 6-2 IFMD Equations for Three Bar Truss Example 6-2 Type Matrix Notation Longhand Notation p e T 0 Eliminate deformation fbg GfF g B fX g À fbg be F11`E 100 E 2F1 x1p 2 À b0 1 A 1 Àx 2 1 between DDR and FDR to obtain force displacement relation n o fF g GÀ1 BT fX g À GÀ1 fbg0 be F22`E 100F2 Àx2 À b0 2 A 2 E 2 p x be F33`E 50 E2F3 À x1p 2 À b0 3 A 3 2 3 F1 200 x1 À x2 À 100p b0 E E 2 1 F2 À 100 x2 À 100 b0 E E 2 F3 À 100 x1 x2 À 50E b0 E p 2 3 À Á Governing equation BGÀ1 BT fX g À BGÀ1 fbg0 fPg E p 3x1 200 2 x2 50 200 b0 À 2b0 E 1 3 of IFMD is obtained by writing EE B{F} displacement DfX g fPg fPg0  À pÁà À Á D BGÀ1 BT Y fPg0 BGÀ1 fbg0 E p 200 2 x1 3 2 2 100 À 200 b0 2b0 2b0 E 1 2 3 n o Recover forces fF g GÀ1 BT fX g À GÀ1 fbg0 from force displacement relations Solution yields x1 and x2. Forces are back-calculated. The displacements and forces obtained by IFMD are in agreement with the IFM solution. Calculation of reactions and other variables follow the IFM procedure. Load Case 2ÐSolution for Thermal Load An equivalent load (Pt ) is calculated for the temperature change as È É fPt g BGÀ1 b0 where V W È 0 É ` aDT1 `1 a b aDT2 `2 X Y aDT3 `3 PA E Q 1 1 V W P Q aDT1 `1 b Ub `1 p 1 0 À p T 1 b b 2 2 T U` a fP g R t ST A2 E2 U aDT2 `2 1 T Ub `2 b À p 1 2 À1 Àp R 2 Sb X b Y A3 E3 aDT3 `3 `3 V W P Q AEaDT 1 b V W p 1 0 p b 1 À 2 ` b b a ` p DT1 À 2DT3 1 a 2 2 R S AEaDT Ea p À p À1 1 À p b 1 b 2 b b X À p ÀDT 2DT 2DT Á Y 1 2 2 X Y 2 1 2 3 AEaDT 3 p Since, Ea/ 2 140, the thermal load for the problem becomes & ' t DTpÀ 2DT3 1 À Á fP g 140 6-20e À DT1 2DT2 2DT3 The temperature load for the two thermal loads becomes @ A @ A 100 À 600 À7:0 t fP gcase1 140 À p Á Â104 6-20f À 100 200 2 600 À13:76 lbf @ A @ A 100 À 600 9:8 fPt gcase2 140 À p Á  104 6-20g À 100 À 200 2 À 600 10:96 Displacements for both thermal load cases are obtained as a solution to the IFMD equation. !& ' & 'case1 & 'case2 3:18 1:06 x1 À7:0 9:8 105  Â104 Y Â104 6-20h 1:06 6:18 x2 À13:76 10:96 Indeterminate Truss 293 The displacements are & ' & ' & ' & ' x1 À0:155 x1 0:264 Y 6-20i x2 case1 À0:196 in: x2 case2 0:132 in: The forces, in the dual method, are calculated from the formula È È ÉÉ fF g GÀ1 BT fX g À b0 V W V W V W ` p x1 Àx2 2 a ` 9:33 a ` 9:33 a T case1 case2 fbg B fXg Àx2 Yfb0 g 13:20 Â10À2 Yfba g À13:20 Â10À2 XÀ Y x1 x2 p X Y X Y 2 28:00 À28:00 V W V W ` 2:93 a ` 9:33 a fbgcase1 19:6  10À2 and fbgcase2 À13:2  10À2 X Y X Y 24:82 À28:00 V W V W ` 2:93 À 9:33 a ` À6:4 a fbgeÀcase1 19:6 À 13:2  10À2 6:40  10À2 X Y X Y 24:82 À 28:00 À3:20 V W 6:40  10À2 b b b b À p b b b V b W 100 2 b b b b ` À13:59 a À2 ` a 6:40  10 fFg GÀ1 fbge 30  106 19:22 6-20j b b 100 b X b Y b b 3:20  2  10 bÀ2 b À13:59 kip b bÀ b b X p Y 100 2 V W V W ` 9:33 À 9:33 a ` 0:0 a fbge-case2 À13:2 13:20  10À2 0:0 6-20k X Y X Y À28:0 28:0 0:0 No force is induced in the bars ({F} {0}) because the elastic deformation is zero ({b}e {0}). Load Case 3ÐSettling of Support The initial deformation due to the settling of support calculated earlier for the IFM solution is valid even for the dual method. 294 STRENGTH OF MATERIALS W V 0 a ` " fbg0 ÀBR T fX g 0p X Y À1= 2 The load induced by the support settling is fPgs BGÀ1 fbg0 P QV W A1 E P Q b 0b p 1 0 À p T 1 `1 Ub b b b 2 2 T U` a fPgs R ST T A2 E `2 U Ub 0 À p 1 À1 À p R 1 Sb b b 2 2 b X 1 b p Y A3 E À 2 `3 V W P Qb 0 b b b p 1 0 À p b 1 ` b a @ 1 A @ 212:13 A 2 2 A3 E R S 0 À 6-20l À p 1 À1 À p b A3 E b 1 2 X b p b b b 2`3 1 212:13 kip 2 À` 2 Y 3 The equation of the dual method becomes !& ' & ' 5 3:18 1:06 x1 212;130 10 6-20m 1:06 6:18 x2 212;130 Solution to Eq. (6-20m) yields the displacements. & ' & ' x1 0:585 6-20n x2 0:242 in: Total deformations {b} are obtained as V W V x1 Àx2 W V W ` b1 a ` p a ` 0:242 a 2 b Àx2 À0:242 6-20o X 2 Y X À x1p 2 Y X x Y b3 2 À0:585 The elastic deformations ({b}e {b} À {b}0 ) are V W V e W ` b1 a ` 0:242 a b À0:242 X 2Y X Y b3 0:121 Indeterminate Truss 295 The member force are V W ` 51:47 a fFg GÀ1 fbge À72:80 6-20p X Y 51:47 kip Theory of Stiffness Method The stiffness method is quite similar to the dual Integrated Force Method. Its governing equation, also called the stiffness equation ([K]{X} {P}) resembles IFMD equation ([D]{X} {P}). For strength of materials problems the two matrices are equal identically, ([K] [D]). In the stiffness method a bar force (F) is expressed in terms of the displace- ments of its nodes, or in terms of four displacement components, consisting of (u1, v1) at node 1 and (u2, v2) at node 2. The force equilibrium equation ([B]{F} {P}) is obtained in terms of displacement, and this yields the stiffness equation. The stiffness method is illustrated considering the three-bar truss shown in Fig. 6-4 as an example. The force (F) and nodal displacements (u1, v1, u2, v2) for a truss bar shown in Fig. 6-5 are related in the following steps. Axial displacements u1a and u2a are obtained by projecting the nodal displacements (u1, v1 and u2, v2) along the bar axis as shown in Fig. 6-5. u1a u1 cos y v1 sin y 6-21a u2a u2 cos y v2 sin y 6-21b The deformation (b) in the bar is obtained as the difference between the axial displace- ments (u1a and u2a). b u2a À u1a u2 À u1 cos y v2 À v1 sin y 6-21c v2 u2a y u2 2 F v1 u1a u1 x FIGURE 6-5 Truss bar. 296 STRENGTH OF MATERIALS The strain (e) in the bar of length (`) is obtained as e b=` 6-21d The stress (s) is obtained from Hooke's law s Ee 6-21e The bar force (F) is calculated as F AEe 6-21f The bar force is expressed in displacements AE AE F u2a À u1a f u2 À u1 cos y v2 À v1 sin yg 6-21g ` ` The bar force (F) is resolved along the displacement directions to obtain four components (Fx1, Fy1, Fx2, Fx2) along the displacement directions (u1, u2, u3, u4), respectively. AE È É Along u1 X Fx1 F cos y u2 À u1 cos2 y v2 À v1 sin y cos y 6-21h ` AE È É Along v1 X Fy1 F sin y u2 À u1 sin y cos y v2 À v1 sin2 y 6-21i ` ÀAE È É Along u2 X Fx2 ÀF cos y u2 À u1 cos2 y v2 À v1 sin y cos y 6-21j ` ÀAE È É Along v2 X Fy2 ÀF sin y u2 À u1 sin y cos y v2 À v1 sin2 y 6-21k ` The EE at a node of a truss is obtained as the summation of all bar forces and load at that node. The m-EE of the truss is expressed in displacements because bar force is written in that variable. These m-EE in displacement become the stiffness equations. Consider next the example of the three-bar truss (Example 6-1) shown in Fig. 6-4. As before, the nodal EE should be written along the displacement degrees of freedom, or along x1 and x2 displace- ments as depicted in Fig. 6-6. The EE along the x1 direction obtained from the contribution of the three-bars after suppressing the boundary displacements as shown in Fig. 6-6 is p Bar 1X u1 x1 ; v1 x2 ; u2 0; v2 0; y 135 ; A1 1; `1 100 2; and E 30  106 1 Fx1 0:21  106 À0:5x1 0:5x2 6-21l Bar 2X This bar has no contribution to the EE along the x1 direction because y 90 : p Bar 3X u1 x1 ; v1 x2 ; u2 v2 0; y 45 ; A3 2; `2 100 2; and E 30  106 Indeterminate Truss 297 0 0 0 0 1 2 = 135° x2 x2 x1 x1 Bar 1 Bar 2 0 0 3 x2 x1 Bar 3 FIGURE 6-6 Three-bar truss model for stiffness method. 3 Fx1 0:42  106 À0:5x1 À 0:5x2 6-21m The EE at node 1 along x1 is 1 3 Fx1 Fx1 Px 0 6-21n or 0:21  106 À1:5x1 À 0:5x2 50;000 0 Likewise, the EE along the x2 direction is obtained as contributions from the three bars. 298 STRENGTH OF MATERIALS 1 Fx2 0:21  106 0:5x1 À 0:5x2 6-21o 2 Fx2 0:3  106 Àx2 6-21p 3 Fx2 0:42  106 À0:5x1 À 0:5x2 6-21q The EE at node 1 along x2 is 1 2 3 Fx2 Fx2 Fx3 Py 0 6-21r or 0:21  106 À0:5x1 À 2:93x2 100;000 0 The stiffness equation is obtained by coupling the EE along the x1 and x2 directions. !& ' & ' 5 3:18 1:06 x1 50;000 10  6-22a 1:06 6:18 x2 100;000 The equations of the stiffness method and IFMD are identical. Solution of the stiffness equation yields the displacements: & ' & ' x1 0:110 x2 0:143 in: Bar forces are calculated from Eq. (6-21g) as AE F1 fÀx1 cos 135 À x2 sin 135g ` 1 0:110 0:143 6 0:21  10 p À p À5:025 kip 6-22b 2 2 AE F2 Àx2 À42:89 kip 6-22c ` 2 AE x1 x2 F3 À p À p À75:73 kip 6-22d ` 3 2 2 Indeterminate Truss 299 Stiffness Method for Thermal Load Adding the temperature strain in Hooke's law accommodates the thermal load. This is accomplished by rewriting Hooke's law to account for the temperature variation. s E e À et E e À aDT 6-23a AEb F sA AEe À AEaDT À AEaDT 6-23b ` The bar force in displacements is obtained by adding the thermal contribution into Eq. (6-21g). AE F f u2 À u1 cos y v2 À v1 sin yg À AEaDT 6-23c ` Likewise, the bar force is resolved along the displacement directions adding thermal terms in Eq. (6-21) to obtain AE È É Along u1 X Fx1 u2 À u1 cos2 y v2 À v1 sin y cos y À AEaDT cos y 6-23d ` AE È É Along v1 X Fy1 u2 À u1 sin y cos y v2 À v1 sin2 À AEaDT sin y 6-23e ` AE È É Along u2 X Fx2 À u2 À u1 cos2 y v2 À v1 sin y cos y AEaDT cos y 6-23f ` AE È É Along v2 X Fx2 À u2 À u1 sin y cos y v2 À v1 sin2 y AEaDT sin y 6-23g ` Compare Eqs. (6-21h) to (6-21k) with Eqs. (6-23d) to (6-23g), for mechanical and thermal loads, respectively. The thermal load case retains all terms for mechanical load while adding the contribution due to temperature change. The added term is called equivalent thermal load {P}th due to temperature variation. The thermal load for a bar is obtained as V W bÀ cos y b b b ` a À sin y fPgth AEaDT 6-23h b cos y b b b X Y sin y 300 STRENGTH OF MATERIALS First Thermal Load Calculation of the thermal load at a node is illustrated considering the three-bar truss as an example. The EE for thermal load at node 1 along the x1-direction is obtained by adding the contributions from the bars along x1 X Pth1 Pth2 Pth3 . The contribution from bar 1 is obtained by setting y y1 135 , DT1 100 F, and A1 1:0 in Eq. (6-23h). p Along x1 X Pth1 AEa1 100 À cos 135 50 2 Ea p Along x2 X Pth2 AEa1 100 À sin 135 À50 2 Ea The contributions from bar 2 are Along x1 X Pth2 Ea 200 0 0 Along x2 X Pth2 Ea 200 À1 À200 Ea The contributions from bar 3 are p p Along x1 X Pth3 2 Ea 300  À1= 2 À300 2 Ea p p Along x2 X Pth2 2Ea 300  À1= 2 À300 2 Ea The thermal load for the truss is obtained by adding the contributions from the three bars: pt`Àx1 pth1 pth2 pth3 p p p Ea 50 2 À 300 2 À250 2Ea pt`Àx2 pth1 pth2 pth3 p p p Ea À50 2 À 200 À 300 2 À 200 350 2 Ea & ' È t` É À7:0  104 P À13:76  104 lbf The stiffness equation for thermal load becomes !& ' & ' 3:18 1:06 x1 À7:0  104 105 À 6-23i 1:06 6:18 x2 À13:76  104 The displacement solution for thermal load is & ' & ' x1 0:155 À 6-23j x2 0:196 in: Indeterminate Truss 301 The bar forces are back-calculated from Eq. (6-23c) as follows: For bar 1: u1 x1 , v1 x2 , and u2 v2 0. AE F1 f Àx1 cos 135 Àx2 sin 135g À AEaDT 1 ` 1 0:2121  106 À0:1096 0:1386 À 1:98  104 or F1 À13:59 kip Likewise, forces F2 and F3 are calculated. AE F2 Àx1 À AEaDT 2 19:22 kip ` 2 & ' AE x1 x2 F3 p À AEaDT 2 À13:59 kip ` 3 2 The bar forces for the change in temperature, case 1, are V Wcase 1 V W ` F1 a ` À13:59 a F 19:22 6-23k X 2Y X Y F3 À13:59 kip Second Thermal Load The response for the second thermal load is obtained following the procedure given for the first thermal load. The thermal load obtained is & ' case 2 98:00 fP g 109:60 kip The stiffness equations are !& ' & ' 5 3:18 1:06 x1 98:00 10  103 6-24a 1:06 6:18 x2 109:60 The displacement solution is & ' & ' x1 0:264 6-24b x2 0:132 in: The forces back-calculated from the displacements are zero. 302 STRENGTH OF MATERIALS V W V W ` F1 a ` 0:0 a F 0:0 6-24c X 2Y X Y F3 0:0 The thermal load induced displacements but not force in the truss bar. Stiffness Method for Support Settling x Support settling is accommodated by adding the specified initial displacement " in the force displacement relation. Let us assume that the specified displacement for a bar are ut1 u1 "1 u t " u2 u2 u2 vt1 v1 "1 v vt2 v2 "2 v 6-25a Here, the total displacement (ut) is composed of the unknown displacement (u) and the u prescribed displacement ". The bar force in displacements becomes AE À t Á À Á F u2 À ut1 cos y vt2 À vt1 sin y 6-25b ` The components of the bar force along the displacement components become AE ÈÀ t Á À Á É Along u1 X Fx1 u2 À ut1 cos2 y vt2 À vt1 sin y cos y ` AE ÈÀ t Á À Á É Along v1 X Fy1 u2 À ut1 sin y cos y vt2 À vt1 sin2 y ` AE ÈÀ t Á À Á É Along u2 X Fx2 u2 À ut1 cos2 y vt2 À vt1 sin y cos y ` AE ÈÀ t Á À Á É Along v2 X Fy2 u2 À ut1 sin y cos y vt2 À vt1 sin2 y 6-25c ` The subsequent analysis method proceeds inline with the mechanical load. The method is illustrated for the three-bar truss with the settling of support node 4 by 1 in. along y-coordinate direction: "4 1:0 in. v The force components for bar 1 along the displacement directions x1 and x2 are obtained as before because initial displacement at node 4 has no effect for this bar. 1 Fx1 0:21  106 À0:5x1 0:5x2 1 Fx2 0:21  106 0:5x1 À 0:5x2 Indeterminate Truss 303 Likewise for bar 2, they are 2 Fx1 0 2 Fx2 0:3  106 Àx2 The settling of support node 4 affects the force displacement relationship for bar 3. Its parameters are ut1 x1 vt1 x2 ut4 0 vt4 1:0 in: is the preassigned displacement 3 Fx1 0:42  106 À0:5x1 À 0:5x2 0:5 3 Fx2 0:42  106 À0:5x1 À 0:5x2 0:5 The EE along the displacement directions x1 and x2 are obtained as 1 2 3 Fx1 Fx1 Fx1 Px1 0 0 1 2 3 Fx2 Fx2 Fx2 Px2 0 0 6-25d The EE in displacement variables become 0:21  106 À0:5x1 0:5x2 À x1 À x2 1 0 0:21  106 0:5x1 À 0:5x2 À 1:44x2 À x1 À x2 1 0 The EE has the following form in the matrix notation: !& ' & ' 3:18 1:06 x1 0:21 105  106 6-25e 1:06 6:18 x2 0:21 The prescribed displacement is included in the stiffness equation as an equivalent load. The solution of the stiffness equation yields the two displacement components (x1 and x2). The displacements including the prescribed displacement become V W V W ` x1 a ` 0:585 a x2 0:242 6-25f X bY X Y x3 1:0 in: Bar forces are back-calculated from displacements 304 STRENGTH OF MATERIALS 6 0:585 À 0:242 F1 0:21  10 p 51:47 kip 2 F2 0:3  106 À0:242 À72:80 kip & ' 6 À0:585 1:0 À 0:242 F3 0:43  10 p 51:47 kip 6-25g 2 The forces and displacements calculated via IFM, IFMD, and the stiffness method are in agreement. Both IFMD and the stiffness method have identical governing equations. These are of dimension (m  m) and are symmetrical. The compatibility matrix [C] is not used by either method. This is because the compatibility conditions are automatically satisfied by displacement. For example the CC ([C]{b} [C][BT ]{X} {0} since [C][BT ] [0]). IFM calculates forces first and then back-calculates the displacement. If the desire is to calculate forces alone, then there is no need to formulate the dual or stiffness equations in displacements. IFM may be preferred. If, however, it is required to calculate both forces and displacements, then any one of the three methods can be used. Problems Use the properties of material given in Tables A5-1 and A5-2 to solve the problems. 6-1 For the six trusses shown in Fig. P6-1, identify the number of: 1. Force, displacement, deformation, and reaction variables. 2. The number of equilibrium equations and compatibility conditions. 3. The degree of internal and external indeterminacies. P1 P2 1 P1 P3 P2 (a) Truss a. (b) Truss b. Indeterminate Truss 305 P3 P4 P1 P2 P1 P2 P3 P4 (c) Truss c. (d) Truss d. P P (e) Truss e. (f) Truss f. FIGURE P6-1 6-2 Answer either true or false to the following statements. Statement True False (1) The deformation {b} in CC ([C]{b} {0}) represents: (a) Total deformation (b) Initial deformation (c) Elastic deformation (d) None of the above (2) The deformation {b} in FDR ({b} [G]{F}) represents: (a) Total deformation (b) Initial deformation (c) Elastic deformation (d) None of the above (3) The deformation {b} in DDR ({b} [B]T {X} represents: (a) Total deformation (b) Initial deformation 306 STRENGTH OF MATERIALS (c) Elastic deformation (d) None of the above (4) The deformation {b} in a truss bar can be measured: (a) In unit of length, like inch, or centimeter (b) In unit of radian or degree (c) It is a dimensionless quantity (5) Stress is induced in a truss bar because of: (a) Total deformation (b) Initial deformation (c) Elastic deformation (6) CC is homogenous equation in total deformation. (7) CC is nonhomogenous equation in elastic deformation. (8) A truss has a (6  8) EE matrix [B]. It has: (a) Eight bars (b) Eight EE and six CC (c) Six EE and 2 CC (9) Degree of indeterminacy is dependent on (a) Applied load (b) Settling of support (c) Temperature variation (d) Young's modulus 6-3 Analyze the square diamond truss of size three meters, as shown in Fig. P6-3 by all three methods: Integrated Force Method. Dual Integrated Force Method. Stiffness Method. The outer four bars of the truss are made of steel with one square inch cross-sectional area. The diagonal bars are made of aluminum with nine square centimeters cross- sectional area. The truss is subjected to mechanical load as well as a change of temperature and settling of support as follows: 6 kN 2 y 1 x 3 a = 3m 4 5 kip FIGURE P6-3 Indeterminate Truss 307 (a) Loads are applied at nodes 2 and 4: a 6 kN along the x-coordinate direction at node 2 and 5 kip along the negative y-coordinate direction at node 4. (b) Temperatures of the diagonal bars are increased by 75 C. (c) The support at node 3 is moved along the negative x-coordinate direction by 0.5 in. 6-4 Model the column shown in Fig. P6-4(a) as three bar members and analyze by IFM and stiffness method. The column is made of steel. It has a total length of 9 meters. Its cross- sectional area is 2 in.2 for the central one-third span, while it is 1 in.2 for the remainder of its length. It is subjected to three load cases: Load case 1: Mechanical load (P1 10 kip and P2 20 kip) applied at the one-third and two-thirds span locations as shown in Fig. P6-4(b). Load case 2: A uniform temperature variation (DT 200 F) along the central one-third span, as shown in Fig. P6-4(c). x a a 3m Area = A Cross-section a-a b b T = 200° F T = 200° F 3m P1 Area = 2A Cross-section b-b c c P2 3m Area = A Cross-section c-c y (a) Geometry. (b) Case 1— (c) Case 2— (d) Case 3— Mechanical Central span Uniform load. temperature. temperature. FIGURE P6-4 308 STRENGTH OF MATERIALS Load case 3: A uniform temperature variation (DT 200 F) along the entire column length, as shown in Fig. P6-4(d). 6-5 Find bar forces and nodal displacements for the truss shown in Fig. P6-5 by IFM and stiffness method. The truss is made of steel with one square in. bar areas and the length (` 48 in:). It is subjected to three load cases: Case 1: Mechanical load (P 10 kN) at node 4 along the x-coordinate axis. Case 2: A uniform temperature variation (DT 100 F) for the truss. Case 3: A settling of support 3 by 1 cm along the y-coordinate direction. y x 3 4 P 45° 45° 1 2 FIGURE P6-5 6-6 A four-feet diameter symmetrical hexagonal wagon wheel with 6 aluminum spokes and a steel rim is modeled as a truss, as shown in Fig. P6-6. The bar area is 1 in.2 for the 4 P3 5 3 y 7 x 6 P7 2 1 P2 FIGURE P6-6 Indeterminate Truss 309 spokes, while it is 2 in.2 for the rim. Analyze the truss by force and displacement method for the following load cases. Case 1: Gravity load (P7 5 kip) applied at node 7 along the negative y-coordinate direction. Case 2: Bar connecting nodes 2 and 3 is stretched by equal load but with opposite directions (P3 ÀP2 1 kip) at nodes 2 and 3. Case 3: The temperature in the steel rim is increased by (DT 200 F). Case 4: The temperature is decreased by (DT 100 F) in the spokes. Note: Formulate the equations. Verify by substituting the given answers. 310 STRENGTH OF MATERIALS 7 Indeterminate Beam An indeterminate beam is obtained by adding extra restraints at the supports of a determinate beam, by increasing the number of spans, or both. A beam with additional restraints and a multispan continuous beam can more efficiently transfer external load to the foundation than a determinate beam. Furthermore, it remains stable even when some or all of the extra spans and the redundant restraints are removed. Indeterminate beams are used in bridges, buildings, and machinery. Temperature and support settling induce stress in an indeterminate beam, but not in a determinate beam. To analyze an indeterminate beam, we add compatibility conditions to the determinate beam formulation. The four sets of equations required for its analysis are the 1. Equilibrium equations (EE) 2. Deformation displacement relations (DDR) 3. Force deformation relation (FDR) 4. Compatibility conditions (CC) The types of response variables remain the same for determinate and indeterminate beams. These are a bending moment (M), shear force (V), reaction (R), displacement (v), rotation (y), curvature (k) that also is the beam deformation (b k), stress (s), and strain (e). Their individual numbers can be greater for indeterminate beams. The amount of increase in the number of force variables over determinate beam becomes the degree of indeterminacy. It is defined in terms of four parameters. md total number of nodes, which includes the support nodes nF number of internal forces (each span has two forcesÐa moment and a shear force) nR number of support restraints, which is also equal to the number of reactions m (2md À nR ) number of displacement components 311 Each node has two displacementsÐa transverse displacement (or deflection) and a rotation. The degree of indeterminacy (r) of a beam is equal to the sum of the beam forces and the reactions (nF nR ), less twice the number of nodes (2md) because a node has two EE. r nF nR À 2md 7-1a Equation (7-1a) is rearranged using the formula m 2md À nR to obtain a simpler form. r nF À m 7-1b The degree of indeterminacy (r) is equal to the difference between the number of internal forces (nF) and the number of displacement components (m). If r 0, then the beam is determinate. If r > 0, then it is an indeterminate beam. If r < 0, then it is a mechanism, and this book does not address the analysis of mechanism. To illustrate the degree of indeterminacy, we consider the beams shown in Figs. 7-1a to 7-1c. The propped beam shown in Fig. 7-1a has two nodes (md 2), two internal forces y, v x, u 1 2 1 2 3 (a) Propped beam. (b) Continuous beam. V Rm1 M θ 1 2 3 Rv1 (c) Clamped beam. Rv2 (d) Free-body diagram of beam (a). V1 V2 V1 V2 2 Rm3 M1 M2 Rm1 M1 M2 Rm3 1 Rv3 Rv1 Rv2 Rv3 Rv1 Rv2 (e) Free-body diagram of beam (b). (f) Free-body diagram of beam (c). FIGURE 7-1 Indeterminate beams. 312 STRENGTH OF MATERIALS (nF 2), three support restraints (nR 3), and a single displacement component (m 1), as marked in Fig. 7-1d. Bending moment M and shear force V are the internal forces. It has two reactions (Rm1 and Rv1) at support node 1 and one reaction (Rv2) at node 2. Its single displacement is the rotation (y) at node 2. The degree of indeterminacy (r) is r 23À2Â21 see Eq: 7-1a r 2 À 1 1 see Eq: 7-1b The degree of indeterminacy (r), calculated from either formula, is one. The parameters for the continuous beam shown in Fig. 7-1b are the number of nodes (md 3), forces (nF 4), restraints (nR 4), and displacements (m 2), as marked in Fig. 7-1e. The internal forces are V1, M1, V2, and M2; the reactions are Rv1, Rv2, Rv3, and Rm3; and the displacements are y1 and y2 . The degree of indeterminacy (r) is r 44À2Â32 see Eq: 7-1a r 4 À 2 2 see Eq: 7-1b For the two-span clamped beam shown in Fig. 7-1c, md 3, nF 4, nR 5, and m 1, as marked in Fig. 7-1f. The internal forces are V1, M1, V2, and M2; the reactions are Rv1, Rm1, Rv2, Rv3, and Rm3; and the displacement is y. The degree of indeterminacy (r) is r 4 5 À 2  3 3 see Eq: 7-1a r 4 À 1 3 see Eq: 7-1b The degree of indeterminacy is a property of the structural configuration, and it depends on the number of spans and support restraints. It is independent of member dimensions, material properties, and loads. EXAMPLE 7-1 Calculate the degree of indeterminacy of the beams shown in Figs. 7-2a to 7-2c. The clamped beam shown in Fig. 7-2a has two nodes (md 2); two forces: V and M (nF 2); four restraints (nR 4) and reactions: Rv1, Rm1, Rv2, and Rm2; but it has no explicit displacement at either beam nodes (m 0), as marked in Fig. 7-2d. Its degree of indeterminacy (r) is r 24À2Â22 see Eq: 7-1a r 2 À 0 2 see Eq: 7-1b Indeterminate Beam 313 The beam with an overhang shown in Fig. 7-2b has three nodes (md 3); four forces (nF 4): V1, M1, V2, and M2; two restraints (nR 2) and reactions: Rv1, Rv2; and four displacements (m 4)X y1 , y2 , v3 , and y3 , as marked in Fig. 7-2e. Its degree of indeterminacy (r) r 4 2 À 2  3 0 see Eq: 7-1a r 4 À 4 0 see Eq: 7-1b The overhang beam is a determinate structure. The three-span beam in Fig. 7-2c has four nodes (md 4); six forces (nF 6): V1, M1, V2, M2, V3, and M3; five restraints (nR 5), and reactions: Rv1, Rv2, Rv3, and Rm1; and three (m 3) displacements: y2 , y3 , and y4 , as marked in Fig. 7-2f. Its degree of indeterminacy (r) is r 6 5 À 2  4 3 see Eq: 7-1a r 6 À 3 3 see Eq: 7-1b The continuous beam is three degrees indeterminate. 1 2 1 2 3 (a) Clamped beam. (b) Beam with overhang. V M Rm1 1 2 3 4 Rm1 Rv1 Rv2 (c) Three-span beam. (d) Free-body diagram of beam (a). V1 V2 V1 V2 V3 M1 M2 M1 M2 M3 2 1 V3 Rm1 3 Rv1 Rv1 Rv2 Rv2 Rv3 Rv4 (e) Free-body diagram of beam (b). (f) Free-body diagram of beam (c). FIGURE 7-2 Indeterminate beams of Example 7-1. 314 STRENGTH OF MATERIALS 7.1 Internal Forces in a Beam A beam member has two nodes, a span ` and two internal forces. There are two traditional systems to select the internal forces. The first system uses a shear force (V) and a bending moment (M) at node 2, as shown in Fig. 7-3a. The second one uses two bending moments (M1 and M2) at nodes 1 and 2, as shown in Fig. 7-3b. The two systems are equivalent and follow the t-sign convention. Either system can be selected to solve a problem. Some engineers prefer the system with two moments because both variables have the same dimension. The choice of a force system neither increases nor decreases the analysis complexity. We can back-calculate the forces at the nodes in terms of the internal forces by using the transverse (or shear) equilibrium equations and rotational (or moment) EE. The equilibrium equations follow the n-sign convention. For convenience the n-sign convention is also used for the four nodal forces (Vv1 , My1 , Vv2 , My2 ) as shown in Fig. 7-3c. The nodal forces can be expressed in terms of an internal force system 1, consisting of (M) and (V) as: V W V W P Q b Vv1 b b b b b ÀV b b À1 0 b b b b b b b b T U@ A ` My1 b b a ` À`V À M b b a T À` À1 U V T U T U 7-2a b Vv2 b b b b b Vb T 1 b R 0U M b b b b b b b b S b X b Y b X b Y My2 M 0 1 The four nodal forces (Vv1 , My1 , Vv2 , My2 ) are in equilibrium with the independent internal shear force and bending moment (V, M). Likewise, the nodal forces are obtained when M1 and M2 are considered as the internal forces. y y V M 1 2 (a) System 1: M and V. Vv1 Vv2 M1 M2 M 2 1 2 M 1 (b) System 2: M1 and M2. (c) Nodal forces. FIGURE 7-3 Two internal force systems for a beam. Indeterminate Beam 315 V W V W P Q bV b b v1 b b M2 À M1 b b b 1 1 b b b b b b ` b b TÀ b b b b b b b b T ` ` U& ` My1 b b a b ` b U ' M1 a T À1 0 U M1 T T U 7-2b b Vv2 b b M1 À M2 b T 1 b b b b 1 U M2 b b b b b b b R b À U b b b b b b b b b ` b b b ` `S X Y X Y 0 1 My2 M2 The four nodal forces (Vv1 , My1 , Vv2 , My2 ) are in equilibrium with the independent internal bending moments (M1 and M2). The internal forces follow the t-sign convention, while the nodal forces follow n-sign convention. EXAMPLE 7-2 Two moments (M1 À6 kN-m and M2 13 kN-m) are considered as the internal forces of the beam of span ` 8 m, as shown in Fig. 7-4a. Calculate the alternate internal force systems and the nodal forces. The nodal forces for the two-moment internal force system are calculated first. For the moments M1 À6 kN-m and M2 13 kN-m, the shear force (V Vv1 ) at node 1 is obtained from the rotational EE written at node 2. V Vv1 M2 À M1 =8 2:375 kN The transverse EE yields the shear force at node 2 as (V Vv2 À2:375 kN). The nodal forces are: (Vv1 , My1 , Vv2 , My2 ) (2:375 kN, 6 kN-m, À2:375 kN, 13 kN-m): The nodal forces for the shear force and moment internal force system are obtained by inspection. V = –2.375 M1 = –6 kN-m M2 = 13 kN-m M = 13 1 2 8m (a) Moments as internal forces. (b) Moment and shear as internal forces. Vv1 = 2.375 Vv2 = –2.375 M 2 = 13 M 1=6 (c) Nodal forces. FIGURE 7-4 Internal force for the beam in Example 7-2. 316 STRENGTH OF MATERIALS The internal forces are (V Vv2 À2:375 kN and M My2 13 kN-m). The nodal forces remain unchanged even for this internal system. The internal forces are marked in Fig.7-4b. Nodal forces are marked in Fig.7-4c. The nodal forces satisfy the equilibrium equations. Transverse EEX Vv2 Vv1 À2:375 2:375 0 Rotational EEX My2 My1 `Vv2 13 6 À 2:375  8 0 The shear forces (Vv1 , Vv2 ) are equal in magnitude but opposite in direction. The moments (My1 , My2 ) need not be equal either in magnitude or in direction. 7.2 IFM Analysis for Indeterminate Beam The IFM to analyze indeterminate beam follows the steps of indeterminate truss analysis discussed in Chapter 6. The following steps are modified for flexure to obtain the beam analysis. Step 0ÐSolution Strategy. Step 1ÐFormulate the Equilibrium Equations. Step 2ÐDerive the Deformation Displacement Relations. Step 3ÐGenerate the Compatibility Conditions. Step 4ÐFormulate the Force Deformation Relations. Step 5ÐExpress the Compatibility Conditions in Terms of Forces. Step 6ÐCouple the Equilibrium Equations and Compatibility Conditions to Obtain the IFM Equations, and Solve for the Forces. Step 7ÐBack-Calculate the Displacements and Other Response Variables, as Required. The analysis steps are developed considering a clamped beam, Example 7-3, shown in Fig. 7-5a. EXAMPLE 7-3: Clamped Beam with a Mechanical Load, a Thermal Load, and Settling of Support A clamped beam has a uniform depth d, thickness b, and moment of inertia I. It is made of steel with a Young's modulus of E and a coefficient of expansion of a per F. It is clamped at both ends (A and B) as shown in Fig. 7-5a. Analyze the beam for the following load cases: Load Case 1: Transverse load P at the center of the span. Load Case 2: Uniform temperature along the length of the beam. Along the depth, the temperature variation is linear with values DT and ÀDT at the upper and lower surfaces, respectively, as shown in Fig. 7-5b. Indeterminate Beam 317 P y A C B x ∆A ∆B a = 2a (a) Clamped beam under a concentrated load. y ∆T ( °F) d z b ∆T ( °F) (b) Temperature distribution shown on an enlarged cross-section. v P RmB M1 M2 θ M3 M4 A B 1 C 2 RmA a a M2 – M1 M1 – M2 M4 – M3 M3 – M4 a a a a RvA RvB (c) Free-body diagram. θ (d) Displacement in the beam. FIGURE 7-5 Clamped beam under a concentrated load and a settling support. Load Case 3: Settling of supports A and B by ÁA and ÁB inches along the negative y-coordinate direction, respectively. Load Case 1ÐSolution for a Mechanical Load Step 0ÐSolution Strategy The coordinate system (x, y) with its origin at A is shown in Fig. 7-5a. The beam is divided into two members (1, 2) and three nodes (A, C, B). For the beam member, two 318 STRENGTH OF MATERIALS moments (M1, M2), as depicted in Fig. 7-3c, are considered as the internal force unknowns. Moments M1, M2, M3, and M4 are the force unknowns of the problem, and n 4. The beam has two restraints at A and at B. At A(x 0) and at B(x `), 1. Transverse displacement v(x) 0 2. Slope y