Question $1$Question $1$

The doubly symmetric I$-$section beam shown in Fig. $1$ is loaded by a uniformly distributed load $(q)$ acting

through the bottom flange of the cross$-$section. The flexural$-$torsional buckling load of the beam is to be

determined using an energy analysis, as follows:

a$)$ Beginning from Eq$.(271)$ in the course notes, explain why the energy equation for this problem can be written

as$,$

$V_T=0$ where $V_T=\frac{1}{2}{\displaystyle \underset{L}{\overset{\phantom{}}{}}}[E{I}_y{u}_b^{\text{'}2}+E{I}_w{}_b^{\text{'}2}+GJ{}_b^{\text{'}2}+M_x(2{}_b{u}_b^{**})+q(y_q-y_0){}_b^2]dz$

where $M_x$ is the major axis bending moment in the beam at incipient buckling.

b$)$ Assume the beam is simply supported at the ends and the flexural$-$torsional buckling load $(q)$ is based on the

displacement field,

$\frac{u_b}{A_{iu}}=\frac{{}_b}{A_{}}=\frac{z}{L}-\frac{z^2}{L^2}$

Determine the flexural$-$torsional buckling load $(q)$ based on the above displacement field.

c$)$ In solving the energy equation in Question $1$b to determine the flexural$-$torsional buckling load, you would

have obtained two potential solutions via the quadratic equation. Explain the physical and numerical differences

between the two solutions.

d$)$ For the case where the load is acting at the shear center, the accurate value of buckling moment for this

problem is given by$,$

$M_c={}_m\sqrt[\phantom{2}]{\frac{{}^{2}E{I}_y}{L^2}[GJ+\frac{{}^{2}E{I}_w}{L^2}]}$

where $M_c$ is the maximum moment at the mid$-$span and ${}_m=1\mathrm{.}13.$ For both buckling loads obtained earlier,

determine the error on your buckling solution for this case where the uniformly distributed load is acting at the

shear center. It is assumed that the torsion rigidity $($GJ$)$ is negligible.

Figure $1.$

Hints:

The bending moment distribution is given by$,$

$M_x=\frac{1}{2}q(zL-z^2)$

The doubly symmetric I$-$section beam shown in Fig. $1$ is loaded by a uniformly distributed load $(q)$ acting

through the bottom flange of the cross$-$section. The flexural$-$torsional buckling load of the beam is to be

determined using an energy analysis, as follows:

a$)$ Beginning from Eq$.(271)$ in the course notes, explain why the energy equation for this problem can be written

as$,$

$V_T=0,$ where $,V_T=\frac{1}{2}{\displaystyle \underset{L}{\overset{\phantom{}}{}}}[E{I}_y{u}_b^{\text{'}\text{'}2}+E{I}_w{}_b^{\text{'}\text{'}2}+GJ{}_b^{\text{'}2}+M_x(2{}_b{u}_b^{\text{'}\text{'}})+q(y_q-y_0){}_b^2]dz$

where $M_x$ is the major axis bending moment in the beam at incipient buckling.

b$)$ Assume the beam is simply supported at the ends and the flexural$-$torsional buckling load $(q)$ is based on the

displacement field,

$\frac{u_b}{A_u}=\frac{{}_b}{A_{}}=\frac{z}{L}-\frac{z^2}{L^2}$

Determine the flexural$-$torsional buckling load $(q)$ based on the above displacement field.

c$)$ In solving the energy equation in Question $1b$ to determine the flexural$-$torsional buckling load, you would

have obtained two potential solutions via the quadratic equation. Explain the physical and numerical differences

between the two solutions.

d$)$ For the case where the load is acting at the shear center, the accurate value of buckling moment for this

problem is given by$,$

$M_c={}_m\sqrt[\phantom{2}]{\frac{{}^{2}E{I}_y}{L^2}[GJ+\frac{{}^{2}E{I}_w}{L^2}]}$

where $M_c$ is the maximum moment at the mid$-$span and ${}_m=1\mathrm{.}13.$ For both buckling loads obtained earlier,

determine the error on your buckling solution for this case where the uniformly distributed load is acting at the

shear center. It is assumed that the torsion rigidity $($GJ$)$ is negligible.