# Stress and Dynamic Analysis (326MAE)

Stress and Dynamic Analysis (326MAE)
Stress Assignment – For
hand-in on or before Friday 22nd November 2013

Question
1
a) A
hollow shaft is subjected to a bending moment of 10 kNm and a torque of 5 kNm. The internal diameter is 0.75 times the outer
diameter. If the yield stress is 300 MPa
and a safety factor of 1.5 is required, determine the minimum inner and outer
diameters required according to the Maximum Shear Stress theory (Tresca).
(10
marks)

b)
Figure Q1a on the next page shows a jib crane. When a load is lifted by the crane a force is
reacted through the support frame on the left of Figure Q1a.

The support frame basically consists of
two 50 mm x 50 mm square columns made from plates 5 mm thick, joined by a 6 mm thin
web as shown in the X – X cross-section in Figure Q1b. A force of 100 kN is applied to the top of
the support frame at an angle that creates a 3-4-5 triangle with the vertical
and horizontal directions.

There is a point A identified on the
support frame. It is 350 mm from the
top, on the web adjacent to the right hand column.

(i)
Calculate the axial stress at A.
(4 marks)

(ii)
Calculate the bending stress at A.
(4 marks)

(iii) Calculate the shear stress at A.
(4 marks)

(iv)
Show the state of stress on an infinitesimal aligned to the vertical and
horizontal directions.
(3
marks)

.jpg”>

Question
2
The stress tensor for a
three-dimensional system for some x, y z axis system is

.gif”> MPa

a) Find
the normal and shear stress on a plane whose normal makes an angle of 50° with x-axis and 70° with y-axis. Also give the
direction cosines for the shear stress on the plane.
(8
marks)

b) Find
the principal stresses for the stress tensor.
(5 marks)

c) What angles does the y-axis make with the second
principal direction? (3
marks)

d)
Carry out an analysis to determine if this stress tensor fails the von
Mises criterion when the yield stress of the material is 250 MPa? If not, what
is the factor of safety?

(4 marks)

(You
may use MATLAB for this problem if you wish.)

Question
3
A
steel cylinder with closed ends has an internal diameter of 160 mm and external
diameter of 320 mm OD. It is subject to
an internal pressure of 150 MPa.

a) Determine
the radial and tangential stress distributions and plot the results in
EXCEL.
(10
marks)

b) Also determine the maximum shear stress in
the cylinder.
(5 marks)

(You
may use Lame’s equations without proof.)

Question 4

A cantilever beam with length L and flexural rigidity EI has a
rectangular cross-section. The height in
the y-direction is 2c. It has an end
load P as shown in Figure Q4.
The deflection, including shear effects, can be written as:
.gif”>
.gif”>
a)
Determine σx, σy, and τxythroughout
the beam from the displacement field and verify that these results agree with
the basic strength of materials theory for beams.
(This means that you have to work out
normal and shear stresses from.gif”> and .gif”> and show the
answers are the same as from the displacement functions.)
(12 marks)
b) Find
the vertical displacement of the centre line.gif”> and the slope
of the beam along the centreline Ф(x,0).
Macaulay’s method and discuss any differences.
(8
marks)

.jpg” alt=”stress2.bmp”>
Figure Q4

Question
5
a) Show
that.gif”> is a permissible Airy stress function. (4 marks)
b) Derive expressions for the stresses
from this Airy function. (5 marks)
c) These general stresses may be used to
solve the problem of a taped cantilever beam of unit thickness carrying a
uniformly distributed load q per unit length as shown in Figure Q5. Show that the derived stresses in b) satisfy
all the boundary conditions along the edges θ = 0◦ and θ = α. (6
marks)
d) Hence obtain a value for the constant Cin terms of q and α and therefore that
.gif”> ,when θ = 0◦.

(2 marks)
e) Compare the value obtained from this
formula with the bending stress obtained from simple bending theory.gif”>when α = 30◦. Give the percentage difference between the
stress given by the formula and that from simple bending. Identify clearly which is the bigger.
(3 marks)

q per unit length

.gif”>

Figure Q5

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