B-3Frame instability analysis

B-3.1 Analysis

Frame instability, as treated here, is related to the design of multi-storey rigid-jointed frames subject to side sway. The elastic critical load factor, $\lambda_{\mathrm{cr}}$ may be determined using the deflection method as given in B-3.2 or any other recognized method. This is used to calculate the amplified sway moments for elastic designs and to check frame stability in plastic designs. The elastic critical load factor, $\lambda_{\mathrm{cr}}$ of a frame is the ratio by which each of the factored loads would have to be increased to cause elastic instability.

B-3.2 Deflection Method

An accurate method of analysis (ordinary linear elastic analysis) should be used to determine the horizontal deflections of the frame due to horizontal forces applied at each floor level, which is equal to the notional horizontal load in 4.3.6. Allowance should be made

for the degree of rigidity of the base as given in B-3.2 in this deflection calculation.

The base stiffness should be determined by reference to 4.3.4.

The elastic critical load factor, $\lambda_{\mathrm{cr}}$ is calculated as:

$\lambda_{\mathrm{cr}}=\frac{1}{200 \phi_{\mathrm{s}, \mathrm{Max}}}$

where

$\begin{aligned} \phi_{\mathrm{s}, \mathrm{Max}} & =\text { largest value of the sway index, } \phi_{\mathrm{s}} \text { given } \\ & \text { by: }\end{aligned}$

$\phi_{\mathrm{s}}=\frac{\delta_{\mathrm{u}}-\delta_{\mathrm{L}}}{h}$

where

• $h=$ storey height
• $\begin{aligned} \delta_{\mathrm{u}} & =\text { horizontal deflection of the top of the storey } \\ & \text { due to the combined gravity and notional } \\ & \text { loads, and }\end{aligned}$
• $\begin{aligned} \delta_{\mathrm{L}} & =\text { horizontal deflection of the bottom of the } \\ & \text { storey due to gravity and notional load. }\end{aligned}$

B-3.3 Partial Sway Bracing

In any storey the stiffening effect of infill wall panels may be allowed for by introducing a diagonal strut in

that storey of area A, given by:

$A=\frac{k_{3} \sum K_{\mathrm{c}}}{h(h / b)}\left[1+\left(h / b\right)^{2}\right]^{3 / 2}$

where

$h=$ storey height;

$b \quad=$ width of the braced bay

$\sum K_{\mathrm{c}}=$ sum of the stiffness $I / L$, of the column in that storey;

$k_{3}=\frac{h^{2} \sum S_{\mathrm{p}}}{80 E \sum K_{\mathrm{c}}} \leq 2 ;$ and

$\begin{aligned} \sum S_{\mathrm{p}} & =\text { sum of spring stiffness horizontal force } \\ & \text { per unit horizontal deflection of all the } \\ & \text { panels in that storey determined from: }\end{aligned}$

$S_{\mathrm{p}}=\frac{0.6 h / b}{\left[1+(h / b)^{2}\right]^{2}} t$

where

• $t_{\mathrm{p}} \quad=$ thickness of the wall panel, and

$E_{\mathrm{p}}=$ modulus of elasticity of the panel material.