9.3Combined Axial Force and Bending Moment

9.3.1.1 Plastic and compact sections

In the design of members subjected to combined axial force (tension or compression) and bending moment, the following should be satisfied:

$\left(\frac{M_{\mathrm{y}}}{M_{\mathrm{ndy}}}\right)^{\alpha_{1}}+\left(\frac{M_{\mathrm{z}}}{M_{\mathrm{ndz}}}\right)^{\alpha_{2}} \leq 1.0$

Conservatively, the following equation may also be used under combined axial force and bending moment:

$\frac{N}{N_{\mathrm{d}}}+\frac{M_{\mathrm{y}}}{M_{\mathrm{dy}}}+\frac{M_{\mathrm{z}}}{M_{\mathrm{dz}}} \leq 1.0$

where

• $\begin{aligned} M_{y}, M_{z} & =\text { factored applied moments about the } \\ & \text { minor and major axis of the cross-section } \\ & \text { respectively; }\end{aligned}$
• $M_{\mathrm{ndy}}, M_{\mathrm{ndz}}=$ design reduced flexural strength under combined axial force and the respective uniaxial moment acting alone (see 9.3.1.2);
• $\begin{aligned} N & =\text { factored applied axial force (Tension, } T \\ & \text { or Compression, } P) ;\end{aligned}$
• $\begin{aligned} N_{\mathrm{d}} & =\text { design strength in tension, } T_{\mathrm{d}} \text { as obtained } \\ & \text { from 6 or in compression due to yielding } \\ & \text { given by } N_{\mathrm{d}}=A_{\mathrm{g}} f_{\mathrm{y}} / \gamma_{\mathrm{m} 0} ;\end{aligned}$
• $M_{\mathrm{dy}}, M_{\mathrm{dz}}=$ design strength under corresponding moment acting alone (see 8.2);
• $A_{\mathrm{g}}=$ gross area of the cross-sectio

$\alpha_{1}, \alpha_{2}=$ constants as given in Table 17; and

$\gamma_{\mathrm{m} 0}$ = partial factor of safety in yielding

9.3.1.2

For plastic and compact sections without bolts holes, the following approximations may be used for evaluating $M_{\mathrm{ndy}}$ and $M_{\mathrm{ndz}}$ :

• a) Plates

$M_{\mathrm{nd}}=M_{\mathrm{d}}\left(1-n^{2}\right)$

• b) Welded I or H section

$M_{\mathrm{ndy}}=M_{\mathrm{dy}}\left[1-\left(\frac{n-a}{1-a}\right)^{2}\right] \leq M_{\mathrm{dy}}$ where $n \geq a$

$M_{\mathrm{ndz}}=M_{\mathrm{dz}}(1-n) /(1-0.5 a) \leq M_{\mathrm{dz}}$

where

$n=N / N_{\mathrm{d}}$ and $a=\left(A-2 b t_{\mathrm{f}}\right) / A \leq 0.5$

• c) For standard I or H sections for $n \leq 0.2 \quad M_{\mathrm{ndy}}=M_{\mathrm{dy}}$ for $n>0.2 \quad M_{\mathrm{ndy}}=1.56 M_{\mathrm{dy}}(1-n)(n+0.6)$ $M_{\mathrm{ndz}}=1.11 M_{\mathrm{dz}}(1-n) \leq M_{\mathrm{dz}}$
• d) For rectangular hollow sections and welded box sections

When the section is symmetric about both axes and without bolt holes

$\begin{aligned} M_{\mathrm{ndy}} & =M_{\mathrm{dy}}(1-n) /(1-0.5 a_{\mathrm{f}}) \leq M_{\mathrm{dy}} \\ M_{\mathrm{ndz}} & =M_{\mathrm{dz}}(1-n) /(1-0.5 a_{\mathrm{w}}) \leq M_{\mathrm{dz}}\end{aligned}$ where

$a_{\mathrm{w}}=\left(A-2 b t_{\mathrm{f}}\right) / A \leq 0.5$

$a_{\mathrm{f}}=\left(A-2 h t_{\mathrm{w}}\right) / A \leq 0.5$

• e) Circular hollow tubes without bolt holes

Mnd 1.04 M (1-n¹7) ≤ Ma

9.3.1.3 Semi-compact section

In the absence of high shear force (see 9.2.1), semi-compact section design is satisfactory under combined axial force and bending, if the maximum longitudinal stress under combined axial force and bending, $f_{x}$ satisfies the following criteria:

$f_{x} \leq f_{y} \gamma_{\mathrm{m} 0}$

For cross-section without holes, the above criteria reduces to,

$\frac{N}{N_{\mathrm{d}}}+\frac{M_{\mathrm{y}}}{M_{\mathrm{dy}}}+\frac{M_{\mathrm{z}}}{M_{\mathrm{dz}}} \leq 1.0$

where

$N_{\mathrm{d}}, M_{\mathrm{dy}}, M_{\mathrm{dz}}$ are as defined in 9.3.1.1.

Table 17 Constants $\alpha_{1}$ and $\alpha_{2}$

(Clause 9.3.1.1)

9.3.2.1 Bending and axial tension

The reduced effective moment, $M_{\mathrm{eff}}$, under tension and bending calculated as given below, should not exceed the bending strength due to lateral torsional buckling, $M_{\mathrm{d}}$ (see 8.2.2).

$M_{\mathrm{eff}}=\left[M-\psi T Z_{\mathrm{ec}} / A\right] \leq M_{\mathrm{e}}$

where

• $M, T=$ factored applied moment and tension respectively;
• $A=$ area of cross-section
• $Z_{\mathrm{ec}}=$ elastic section modulus of the section

with respect to extreme compression fibre; and

$\begin{aligned} \psi & =0.8 \text {, if } T \text { and } M \text { can vary independently } \\ & \text { or otherwise }\end{aligned}$

= 1.0.

9.3.2.2 Bending and axial compression

Members subjected to combined axial compression and biaxial bending shall satisfy the following interaction relationships:

$\begin{aligned} \frac{P}{P_{\mathrm{dy}}} & +K_{y} \frac{C_{\mathrm{my}} M_{y}}{M_{\mathrm{dy}}}+K_{\mathrm{LT}} \frac{M_{z}}{M_{\mathrm{dz}}} \leq 1.0 \\ \frac{P}{P_{\mathrm{dz}}} & +0.6 K_{y} \frac{C_{\mathrm{my}} M_{y}}{M_{\mathrm{dy}}}+K_{z} \frac{C_{\mathrm{mz}} M_{z}}{M_{\mathrm{dz}}} \leq 1.0\end{aligned}$

where

$C_{\mathrm{my}}, C_{\mathrm{mz}}=$ equivalent uniform moment factor as per Table 18;

• $\begin{aligned} P & =\text { applied axial compression under factoree } \\ & \text { load; }\end{aligned}$
• $\begin{aligned} M_{y}, M_{z} & =\text { maximum factored applied bending } \\ & \text { moments about } y \text { and } z \text {-axis of the } \\ & \text { member, respectively; }\end{aligned}$
• $P_{\mathrm{dy}}, P_{\mathrm{dz}}=$ design strength under axial compression as governed by buckling about minor (y) and major (z) axis respectively;
• $M_{\mathrm{dy}}, M_{\mathrm{dz}}=$ design bending strength about y (minor) or z (major) axis considering laterally unsupported length of the cross-section (see Section 8);

$\begin{aligned} K_{y} & =1+\left(\lambda_{y}-0.2\right) n_{y} \leq 1+0.8 n_{y} \\ K_{z} & =1+\left(\lambda_{z}-0.2\right) n_{z} \leq 1+0.8 n_{z} ; \text { and }\end{aligned}$

$K_{\mathrm{LT}}=1-\frac{0.1 \lambda_{\mathrm{LT}} n_{y}}{\left(C_{\mathrm{mLT}}-0.25\right)} \geq 1-\frac{0.1 n_{y}}{\left(C_{\mathrm{mLT}}-0.25\right)}$ where

• $n_{y}, n_{z}=$ ratio of actual applied axial force to the design axial strength for buckling about the $y$ and $z$ axis, respectively, and
• $C_{\mathrm{mLT}}=$ equivalent uniform moment factor for lateral torsional buckling as per Table 18 corresponding to the actual moment gradient between lateral supports against torsional deformation in the critical region under consideration.

Table 18 Equivalent Uniform Moment Factor

(Clause 9.3.2.2)

For members with sway buckling mode, the equivalent uniform moment factor $C_{n y}=C_{m z}=0.9$.

$M_{y}$ for $C_{my}$ $M_{z}$ for $C_{mz}$ for $C_{mLT}$