CambridgeMolecularEngineering/StressEngineeringLiterature

text stringlengths 0 1.19M
n,
my,
z) within each one of more than 220 different loading scenarios, plus the equations defining the limits of that loading scenario, the presentation of such an interaction criterion is not practical.Therefore, a simplified analytical interaction criterion has been developed in order to make the evaluation of the cross-section plastic resistance more feasible. This simplified interaction criterion will be presented in the next section.However, in order to appraise the results given by this simplified interaction criterion, it would be useful to have access to an exact solution, which would be used as a reference. An alternative method for getting this exact solution lies on the use of the following semi-analytical procedure.The dimensions of each one of these three elementary rectangles are given in the expressions The bending axis and the neutral axis of the global I-section are the same as those of each elementary rectangular section. The cross-section area and the second moments of inertia of the global I-section, about its main axes, may be related with the equivalent characteristics of the elementary rectangle sections:This method is based on the section decomposition approach The exact plastic interaction criterion, used for each elementary rectangular cross-section (e, i and w), subjected to axial loading and biaxial bending, was developed by the author of this work, and it involves only three different loading scenarios This interaction criterion for rectangular cross-sections is based on the same basic hypothesis, indicated in , that were adopted in this work for the deduction of the I-shape analytical criteria defined by Eqs. . As mentioned before, the presentation of the analytical criterion obtained by the same method for the global I-section, when subjected to axial loading and biaxial bending, is not practical, since it involves the use of different analytical expressions for each one of more than two hundred different loading scenarios.In alternative, using the section decomposition approach, the reduced internal forces of the global I-section may, then, be evaluated by means of the following relations:npl,my,mz=∑jcn,jNpl,my,mz,jNel=npl,my,mz,e−αhnpl,my,mz,i+αbαhnpl,my,mz,wkAmpl,n,mz,y=∑jcmy,jMpl,n,mz,y,jMel,y=mpl,n,mz,y,e−αh2mpl,n,mz,y,i+αbαh2mpl,n,mz,y,wkI,ympl,n,my,z=∑jcmz,jMpl,n,my,z,jMel,z=mpl,n,my,z,e−αhmpl,n,my,z,i+αb2αhmpl,n,my,z,wkI,z shows a representation of the plastic limit state criterion for an IPE80A section obtained by the semi-analytical procedure described in The plastic limit state of an I-section subjected to biaxial bending with axial loading may be evaluated by means of a simplified criterion such as the one presented hereafter, which gives an alternative to the exact criteria mentioned before and to other simplified criteria usually adopted in practice for the design of these cross-sections.mpl,n,my,z=mpl,n,z1−c2−kI,y22kI,zmpl,n,y2c12−c2mz1=1kI,z31−1−kA1−npl,my,mz2−2kI,ympl,n,mz,ymz2=321−αhkI,z1−1−kA1−npl,my,mz1−αh2−kI,y2mpl,n,mz,y22kI,z1−αh3According to this criterion, the reduced value of the z component of the resisting plastic moment, mpl,
n,
my,
z, may be obtained after the given reduced values for the axial force npl,
my,
mz and for the y component of the resisting plastic moment, mpl,
n,
mz,
y. The values of npl,
my,
mz, mpl,
n,
mz,
y and mpl,
n,
my,
z are supposed to be positive; if not, their absolute values shall be used in the calculations and the result shall be affected by the correct sign.The value of mpl,
n,
y may be obtained by means of Eqs. , and the value of mpl,
n,
z may be obtained by means of Eqs. show a comparison between this simplified PLSapr criterion, defined by Eqs. , and the exact PLS criterion obtained by the semi-analytical method described in , for a HD 400 × 1086 section and a HE 1000 × 584 section, respectively. The values of the parameters αb and kA for the HD 400 × 1086 section are αb
= 0.1718 (Eq. ); therefore αb/kA
= 0.3207; in the case of the HE 1000 × 584 section, αb
= 0.1146 (Eq. These figures show an excellent agreement between the exact results and those given by this simplified PLSapr criterion, when αb/kA
≤
npl,
my,
mz
≤ 1.For smaller values of the axial force, when 0 ≤
npl,
my,
mz
<
αb/kA, the PLSapr criterion still gives a good estimation of the exact values of the cross-section resisting internal forces. The results obtained for these and other I-sections seem to indicate that the solutions given by Eq. , when 0 ≤
npl,
my,
mz
<
αb/kA, are usually conservative for the largest values of npl,
my,
mz ( when npl,
my,
mz
= 0.4), but they may also lead to a slight overestimation of the section resistance when npl,
my,
mz is close to 0. Yet, it should be reminded that Eqs. give the exact solution when npl,
my,
mz
= 0; this exact solution may be used as an upper bound for the results given by Eq. , in order to restrain the overestimation of the cross-section resistance. for the verification of cross-sections subjected to biaxial bending with axial loading. If the parameters α1
= 2 and α2
= 5
n (but α2
≥ 1), for I and H sections, are introduced in Eq. is employed, we obtain the following expressions for their PLS:When0≤npl,my,mz≤0.2:Mpl,n,mz,yMpl,n,y2+Mpl,n,my,zMpl,n,z=1When0.2≤npl,my,mz≤1:Mpl,n,mz,yMpl,n,y2+Mpl,n,my,zMpl,n,z5n=1where n
=
npl,
my,
mz and the values of Mpl,
n,
y are obtained by means of Eq. , and those of Mpl,
n,
z are evaluated by means of Eq. show a comparison between this Eurocode 3 criterion (EC3) and the exact PLS criterion, obtained by the semi-analytical method described in , for a HD 400 × 1086 section and a HE 1000 × 584 section, respectively.The results obtained for these and other cross-sections seem to indicate that the solutions given by Eqs. may be, in some situations, rather conservative, specially when the values of Mpl,
n,
mz,
y are large for a same value of n. If the HD 400 × 1086 section is subjected, for instance, to an axial force n
= 0.8 and a bending moment mpl,
n,
mz,
y
= 0.2848, the maximum value of the mpl,
n,
my,
z component given by the EC3 criterion is mpl,
n,
my,
z
= 0.1751 while the one given by the exact PLS criteria is mpl,
n,
my,
z
= 0.5566, which is 3.2 times larger than the previous one (On the other hand, when the values of Mpl,
n,
mz,
y are small for a same value of n, Eq. may overestimate the value of Mpl,
n,
my,
z up to 7% of the cross-section plastic moment Mpl,
z, as in the case of some HE sections for instance. shows that if a HE 1000 × 584 section is submitted, for instance, to an axial force n
= 0.7 and a bending moment mpl,
n,
my,
z
= 1.1780, the maximum value of the mpl,
n,
mz,
y component given by the EC3 criterion is mpl,
n,
mz,
y

n,

my,

z) within each one of more than 220 different loading scenarios, plus the equations defining the limits of that loading scenario, the presentation of such an interaction criterion is not practical.Therefore, a simplified analytical interaction criterion has been developed in order to make the evaluation of the cross-section plastic resistance more feasible. This simplified interaction criterion will be presented in the next section.However, in order to appraise the results given by this simplified interaction criterion, it would be useful to have access to an exact solution, which would be used as a reference. An alternative method for getting this exact solution lies on the use of the following semi-analytical procedure.The dimensions of each one of these three elementary rectangles are given in the expressions The bending axis and the neutral axis of the global I-section are the same as those of each elementary rectangular section. The cross-section area and the second moments of inertia of the global I-section, about its main axes, may be related with the equivalent characteristics of the elementary rectangle sections:This method is based on the section decomposition approach The exact plastic interaction criterion, used for each elementary rectangular cross-section (e, i and w), subjected to axial loading and biaxial bending, was developed by the author of this work, and it involves only three different loading scenarios This interaction criterion for rectangular cross-sections is based on the same basic hypothesis, indicated in , that were adopted in this work for the deduction of the I-shape analytical criteria defined by Eqs. . As mentioned before, the presentation of the analytical criterion obtained by the same method for the global I-section, when subjected to axial loading and biaxial bending, is not practical, since it involves the use of different analytical expressions for each one of more than two hundred different loading scenarios.In alternative, using the section decomposition approach, the reduced internal forces of the global I-section may, then, be evaluated by means of the following relations:npl,my,mz=∑jcn,jNpl,my,mz,jNel=npl,my,mz,e−αhnpl,my,mz,i+αbαhnpl,my,mz,wkAmpl,n,mz,y=∑jcmy,jMpl,n,mz,y,jMel,y=mpl,n,mz,y,e−αh2mpl,n,mz,y,i+αbαh2mpl,n,mz,y,wkI,ympl,n,my,z=∑jcmz,jMpl,n,my,z,jMel,z=mpl,n,my,z,e−αhmpl,n,my,z,i+αb2αhmpl,n,my,z,wkI,z shows a representation of the plastic limit state criterion for an IPE80A section obtained by the semi-analytical procedure described in The plastic limit state of an I-section subjected to biaxial bending with axial loading may be evaluated by means of a simplified criterion such as the one presented hereafter, which gives an alternative to the exact criteria mentioned before and to other simplified criteria usually adopted in practice for the design of these cross-sections.mpl,n,my,z=mpl,n,z1−c2−kI,y22kI,zmpl,n,y2c12−c2mz1=1kI,z31−1−kA1−npl,my,mz2−2kI,ympl,n,mz,ymz2=321−αhkI,z1−1−kA1−npl,my,mz1−αh2−kI,y2mpl,n,mz,y22kI,z1−αh3According to this criterion, the reduced value of the z component of the resisting plastic moment, mpl,

n,

my,

z, may be obtained after the given reduced values for the axial force npl,

my,

mz and for the y component of the resisting plastic moment, mpl,

n,

mz,

y. The values of npl,

my,

mz, mpl,

n,

mz,

y and mpl,

n,

my,

z are supposed to be positive; if not, their absolute values shall be used in the calculations and the result shall be affected by the correct sign.The value of mpl,

n,

y may be obtained by means of Eqs. , and the value of mpl,

n,

z may be obtained by means of Eqs. show a comparison between this simplified PLSapr criterion, defined by Eqs. , and the exact PLS criterion obtained by the semi-analytical method described in , for a HD 400 × 1086 section and a HE 1000 × 584 section, respectively. The values of the parameters αb and kA for the HD 400 × 1086 section are αb

= 0.1718 (Eq. ); therefore αb/kA

= 0.3207; in the case of the HE 1000 × 584 section, αb

= 0.1146 (Eq. These figures show an excellent agreement between the exact results and those given by this simplified PLSapr criterion, when αb/kA

≤

npl,

my,

mz

≤ 1.For smaller values of the axial force, when 0 ≤

npl,

my,

mz

<

αb/kA, the PLSapr criterion still gives a good estimation of the exact values of the cross-section resisting internal forces. The results obtained for these and other I-sections seem to indicate that the solutions given by Eq. , when 0 ≤

npl,

my,

mz

<

αb/kA, are usually conservative for the largest values of npl,

my,

mz ( when npl,

my,

mz

= 0.4), but they may also lead to a slight overestimation of the section resistance when npl,

my,

mz is close to 0. Yet, it should be reminded that Eqs. give the exact solution when npl,

my,

mz

= 0; this exact solution may be used as an upper bound for the results given by Eq. , in order to restrain the overestimation of the cross-section resistance. for the verification of cross-sections subjected to biaxial bending with axial loading. If the parameters α1

= 2 and α2

= 5

n (but α2

≥ 1), for I and H sections, are introduced in Eq. is employed, we obtain the following expressions for their PLS:When0≤npl,my,mz≤0.2:Mpl,n,mz,yMpl,n,y2+Mpl,n,my,zMpl,n,z=1When0.2≤npl,my,mz≤1:Mpl,n,mz,yMpl,n,y2+Mpl,n,my,zMpl,n,z5n=1where n

=

npl,

my,

mz and the values of Mpl,

n,

y are obtained by means of Eq. , and those of Mpl,

n,

z are evaluated by means of Eq. show a comparison between this Eurocode 3 criterion (EC3) and the exact PLS criterion, obtained by the semi-analytical method described in , for a HD 400 × 1086 section and a HE 1000 × 584 section, respectively.The results obtained for these and other cross-sections seem to indicate that the solutions given by Eqs. may be, in some situations, rather conservative, specially when the values of Mpl,

n,

mz,

y are large for a same value of n. If the HD 400 × 1086 section is subjected, for instance, to an axial force n

= 0.8 and a bending moment mpl,

n,

mz,

y

= 0.2848, the maximum value of the mpl,

n,

my,

z component given by the EC3 criterion is mpl,

n,

my,

z

= 0.1751 while the one given by the exact PLS criteria is mpl,

n,

my,

z

= 0.5566, which is 3.2 times larger than the previous one (On the other hand, when the values of Mpl,

n,

mz,

y are small for a same value of n, Eq. may overestimate the value of Mpl,

n,

my,

z up to 7% of the cross-section plastic moment Mpl,

z, as in the case of some HE sections for instance. shows that if a HE 1000 × 584 section is submitted, for instance, to an axial force n

= 0.7 and a bending moment mpl,

n,

my,

z

= 1.1780, the maximum value of the mpl,

n,

mz,

y component given by the EC3 criterion is mpl,

n,

mz,

y