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1.19M
Mv and α
=
π/2 (The distribution of the cross-section longitudinal strains (associated to its normal stresses) is based on the classical Bernoulli hypothesis that, after deformation, the cross-sections remain plane and normal to the structural element longitudinal axis. Therefore, the strain field may be defined by the following expression:where εN, χy e χz represent the cross-section global deformations (axial deformation εN, and bending curvatures χy and χz, about the cross-section main axes); y and z are the coordinates of a cross-section point regarding the same cross-section main axes.The cross-section neutral axis is parallel to the bending axis v, and it is defined by:The change between the coordinate systems associated to the (v, w) axes or to the (y, z) axes may be carried out by means of the following equations:If the following relations, defined on Eqs. , are considered, and the expressions from Eq. , a new definition of the cross-section strain field may be obtained, which is expressed by Eq.  may be considered as the sum of a uniform strain field, associated to the axial deformation εN, with a linear strain field associated to bending about the v axis, which contains the cross-section centre of gravity (The resistant properties of an I-section may be characterised by a set of geometrical and mechanical characteristics, as follows where σel is the cross-section material yield strength; A and Nel are the cross-section area and the yielding axial force, respectively.where Iy, Mel,
y and Mpl,
y are the second moment of area, the yielding and the plastic moments about the y axis, respectively (where Iz, Mel,
z and Mpl,
z are the second moment of area, the yielding and the plastic moments about the z axis, respectively (where Mel,
v and Mel,
w are the yielding moments about the v and the w axes.The graphical representation of the variation of the parameters kel,
v and kel,
w is shown in . The parameter kel,
v is always positive; it has a minimum value and two maximum values, kel,
v
= 1 in the case of simple bending about the y axis (when α
= 0˚ and β
= 0) and kel,
v
=
αrkI,
z/kI,
y in the case of simple bending about the z axis (when α
= 90˚ and β
=
∞). The parameter kel,
w has a minimum value too, and it is always negative except in the case of bending about the cross-section main axes (when β
= 0 or β
=
∞), where kel,
w
= 0.The analytical formulation of the cross-section limit states criteria is carried out in a system of reduced (normalised) variables These equations lead to the following relations between the values of these reduced bending moments, in the elastic domain of the cross-section behaviour:The main interest in using reduced variables lies on the separation of the physical phenomena from the numerical quantification of the variables, associated to a specified system of units The elastic limit state of the cross-section is achieved when the most stressed point reaches the material yield strength σel, under the combined effects of the cross-section axial load and biaxial bending moment; in this analysis, the effects of warping normal stresses due to bimoments, as well as those of shear stresses due to bending, uniform torsion and warping are supposed to be very small and they are not taken in account.The limit state criterion for an I-section at its elastic limit state, when subjected to an axial force and a biaxial bending moment (), may be expressed in several different ways, depending on the variables considered on its definition:The values of all the variables in Eqs.  are supposed to be positive. If any of these internal forces is negative, its absolute value shall be used in the calculations and the result shall be affected by the correct sign. defining the direction of the cross-section bending axis may be evaluated by one the following expressions:) defining the position of the cross-section neutral axis may be evaluated by one the following expressions:wo,el=1kθnel,my,mzmel,n,mz,yh2=βkθnel,my,mzmel,n,my,zh2=1+βkθnel,my,mzmel,n,mw,vh2The parameter wo,
el may also be expressed in terms of the cross-section global deformations, from Eqs. , as indicated on the following expressions:In alternative, the position of the cross-section neutral axis may also be defined by the reduced (normalised) value τo,
el of the zo,
el coordinate of the intersection between the cross-section z axis and its neutral axis (The plastic limit state (PLS) criterion presented in this paper for I-shaped cross-sections is based on the hypothesis of a full cross-section yielding at the plastic limit state, the material being elastic–plastic without strain-hardening. The cross-section strength depends only on normal strains and stresses due to axial loading and biaxial bending; the possible effects of warping normal stresses due to bimoments are neglected, as well as those of shear stresses due to bending, uniform torsion, and warping.The deduction of a global exact criterion for the interaction between the I-section internal forces at its plastic limit state is a complex problem, since this plastic stress distribution (), which depends on the axial force level.The influence of the first two factors may be taken into account by means of the parameter β
=
αrαθ parameters; the different positions of the neutral axis are characterised by the parameter τo,
plIn order to simplify, in practice, the evaluation of the resisting cross-section internal forces at its plastic limit state, the plastic interaction criteria will be presented, in a first step, for some particular combinations of these internal forces.After that, the global solution for the simultaneous actuation of an axial force and bending moments about both the main axes of inertia will be presented by means of two different approaches, concerning an exact semi-analytical procedure and a simplified analytical interaction criterion.In the particular case of axial loading together with bending about the y main axis, when mz
= 0 (β
= 0), the plastic interaction criterion for an I-section is given by Eq. In the case of axial loading together with bending about the z main axis, when my
= 0 (β
=
∞), the plastic interaction criterion for an I-section is given by Eq. αbkA≤n≤1⇒mpl,n,z=mpl,z1−αhkM,z1−1−kA1−n1−αh2These plastic criteria have been obtained by means of an exact integration (within the frame of the hypotheses adopted in this study) of the cross-section axial stress field The European Standard EN 1993 (Eurocode 3) shows a representation of the plastic criterion adopted by the Eurocode 3 (EC3) , and of the plastic criterion given by Eqs. , for the particular cases of two European I-sections, HD400
1086 and IPE80A, subjected to an axial force and bending about their strong axis. The results given by the EC3 are usually conservative in the case of the HD400
1086 section and, in the case of the IPE80A, they match quite well the results obtained by means of Eqs. , except for small values of the axial force where the EC3 criterion overestimates the cross-section plastic resistance, due to the linearization of this simplified criteria. shows a representation of the plastic criterion adopted by the Eurocode 3 (EC3) , and of the plastic criterion given by Eqs. , for the particular cases of two European I-sections, HE1000
1086, subjected to an axial force and bending about their weak axis. In these cases, the results given by the EC3 always overestimate the cross-section plastic resistance, except when n
≈ 0 or n
≈ 1.If the geometrical proportions of an I-section are within the domain of the current I-shapes for steel construction (0.0250 ≤
αb
≤ 0.0589 and 0.5606 ≤
αh
≤ 0.8879), the plastic interaction criterion for this I-section, when subjected to biaxial bending without axial force (n
= 0), is given by Eqs. Mpl,mz,y=3kM,yβ−1+αb3β3+3αh2β−2αh32kI,yβMel,yMpl,mz,y=−αb3β3+3αbαh2β−2αh3+22kI,yβMel,yMpl,my,z=2αb3β3+31−αhβ2+αh3−12kI,zβ2Mel,z for the verification of I and H sections subjected to biaxial bending, with the parameters α1
= 2 and α2
= 5
n, but α2
≥ 1.where n
=
NEd/Npl.
Rd and the values of MN,
y,
Rd are obtained by means of Eq. , and those of MN,
z,
Rd are evaluated by means of Eq. If the notation used before is adopted, the plastic limit state criterion adopted by the Eurocode 3 takes the form of Eq. In order to compare this criterion, Eq. , with the one proposed in this paper (PLS), Eqs. , a graphical representation of these expressions is presented in for two different European I-shapes: HE1000
This figure shows the reduced values of the maximum plastic bending moments of these I-sections, mpl,
mz,
y and mpl,
my,
z, in the absence of an axial force (when n
= 0). These reduced values may be obtained by introducing the relations given by Eqs. , may also be written according to the condensed forms given by Eqs. ; in the other cases, the direct use of Eqs. When0≤β≤αh:2kI,ympl,mz,y+kI,z2αb3mpl,my,z2=3kM,yWhenαh/αb≤β<∞:kI,y2mpl,mz,y2+2kI,zmpl,my,z=3kM,z may also be used, as an alternative to Eqs. When0≤β≤αh:mpl,mz,y+kI,z22kI,yαb3mpl,my,z2=mpl,yWhenαh/αb≤β:kI,y22kI,zmpl,mz,y2+mpl,my,z=mpl,z shows that the PLS and the EC3 criteria are in perfect agreement in the case of simple bending (n
= 0) about one of the cross-section main axes (when mpl,
my,
z
= 0 or mpl,
mz,
y
= 0). In the case of biaxial bending (when mpl,
my,
z
≠ 0 and mpl,
mz,
y
≠ 0), the results obtained for these and other cross-sections seem to indicate that the EC3 always gives conservative results, specially for intermediate values of the mpl,
my,
z/mpl,
mz,
y ratio, such as 0, 5 <
mpl,
my,
z/mpl,
mz,
y
< 5 for instance.As mentioned before, there is a very large number of possible scenarios resulting from the possible combinations of the parameters β, αb, αh and τo,
pl, which affect the plastic resistance of an I-section when subjected to an axial force and biaxial bending, regarding its main axes of inertia. Since an exact criterion, resulting from the analytical integration of the plastic stress distribution, would include several hundreds of analytical expressions, one for each internal force (npl,
my,
mz, mpl,
n,
mz,
y and mpl,