# Compression members with moment to BS 5950-1:2000

According to clause 4.8.3.2 the cross-section capacity for non-slender section is: $$\frac{F_c}{A_g p_y}+\frac{M_x}{M_{cx}}+\frac{M_y}{M_{cy}}\leq 1 \qquad (1)$$

According to clause 4.8.3.3.1 simplified method of member buckling resistance, the following relationship should be satisfied: $$\frac{F_c}{P_c}+\frac{m_x M_x}{p_y Z_x}+\frac{m_y M_y}{p_y Z_y}\leq 1\quad (2)$$

Let's consider a beam-column with class 1 section (doubly-symmetric) under uniform moment (so that the equivalent moment factor $$m_x = m_y =1$$ according to Table 26) and axial compression, and ignore lateral-torsional buckling. In this case, equation (2) is always critical than equation (1), because $$P_c \leq A_g p_y$$ and $$p_y Z_x \leq M_{cx}$$, $$p_y Z_y \leq M_{cy}$$.

My questions are:

1. What is the use of equation (1), or clause 4.8.3.2, if buckling resistance is always more critical? Is there any case that equation (1) is more critical?
2. If the member is under biaxial bending but without axial force, i.e. $$F_c=0, M_x>0, M_y>0$$, is checking equation (2), or clause 4.8.3.3 still required?

I understand that the simplified method is at the cost of conservatism, but making one criterion obsolete seems too much to me.

Edit 1:

1. I am aware of that the equivalent moment factor can be taken as minimum as $$m_x = m_y =0.4$$ for double curvature member, which should make equation (2) less critical if the axial compression is not large. However, I would like to confine the discussion in the case of single curvature and uniform moment.
2. I welcome any example that a non-doubly-symmetric class 1 section has $$p_y Z_x \geq M_{cx}$$, $$p_y Z_y \geq M_{cy}$$.
• Your proof that Eq.2 is more critical than Eq.1 is only for doubly-symmetric sections. How can you then say that "buckling resistance is always more critical"? What about non-doubly-symmetric sections? – Wasabi Oct 29 '18 at 18:53
• And what about loading conditions other than uniform moment? – alephzero Oct 29 '18 at 19:22
• @Wasabi I welcome any example / counterexample for non-doubly-symmetric section that has $p_y Z_x \geq M_{cx}$, $p_y Z_y \geq M_{cy}$ – York Tsang Oct 30 '18 at 1:10
• @alephzero thank you for your comment, however I would like to confine the discussion to uniform moment. It is true that the equivalent moment factor can be taken as minimum as $m_x = m_y =0.4$ for double curvature member, which should make equation (2) less critical if the axial compression is not large. – York Tsang Oct 30 '18 at 1:11

$$P_{cr} = \dfrac{\pi^2 EI}{L_{cr}^2}$$
If $$L_{cr} \rightarrow 0$$ then $$P_{cr} \rightarrow \infty$$.
Hence, equation one can be more critical and $$P_{cr}$$ can be greater than $$Af_y$$.