The bottom two stories of a multistory frame are shown in Figure 12.6, where the second-story columns, at 3 and 4, are free to sway but are fixed against rotation. Beam 25 has a second moment of area of I in^4 and a plastic moment of resistance of 180 kip-in. All columns have a second moment of

Question

The bottom two stories of a multistory frame are shown in Figure 12.6, where the second-story columns, at 3 and 4, are free to sway but are fixed against rotation. Beam 25 has a second moment of area of I [latex] in^{4}[/latex] and a plastic moment of resistance of 180 kip-in. All columns have a second moment of area of 0.6 I and a plastic moment of resistance of 100 kip-in. Determine the load factor against elastic-plastic failure if the axial load in the columns is 0.28 of the Euler load.

Accepted Answer

The stiffness of the members of the frame shown at (i) are:

[latex]\begin{aligned}s_{21} &=s_{23} \&=q E(0.6 I) / 10 \&=3.62 imes 0.6 E I / 10 \&=0.217 E I \s_{25} &=2 E I / 30 \&=0.067 E I\end{aligned}[/latex]

The distribution factors at joint 2 are:

[latex]\begin{aligned}d_{21} &=d_{23} \&=0.217 / 0.501 \&=0.434 \d_{25} &=0.067 / 0.501 \&=0.132\end{aligned}[/latex]

The initial fixed-end moments in the frame are:

[latex]\begin{aligned}M_{25}^{F} &=-M_{52}^{F} \&=7.5 imes 360 / 12 \&=225 ext { kip-in }\end{aligned}[/latex]

The final moments in the frame are:

[latex]\begin{aligned}M_{21} &=M_{23} \&=-225 imes 0.434 \&=-98 ext { kip-in } \M_{25} &=196 ext {kip-in}\end{aligned}[/latex]

Thus, as loading is increased, plastic hinges are produced simultaneously at both ends of the beam at a load factor of:

N = 180/196

= 0.92

The deteriorated structure, at this load factor, is shown at (ii), and the deteriorated critical load is obtained by equating the stiffness of joint 2 to zero. Thus:

[latex]\begin{aligned}2 imes 0.6 n_{21} / 10+0 &=0 \n_{21} &=0\end{aligned}[/latex]

and:

[latex]\begin{aligned}W_{d 1} / P_{E} &=0.25 \N_{d 1} &=0.25 / 0.28 \&=0.89\end{aligned}[/latex]

This is less than the load factor required to produce the plastic hinges, and thus elastic-plastic failure occurs at a load factor of:

[latex]N_{f}=0.92[/latex]

Question 12.2.2: The bottom two stories of a multistory frame are shown in Fi...

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