Select Sal column, 200 kN + 30 kNm

Approach. Working stress design to IS 883:1994 (cl. 7.6 for columns, cl. 7.7 for combined axial compression and bending). Ends are taken as pin-ended, so the effective length equals the unsupported length $S$ (cl. 7.6.1.5).

Data and permissible stresses

Length $S = 3.20$ m = 3200 mm. Axial load $N = 200$ kN, bending moment $M = 30$ kN·m.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Select grade (Grade I values multiplied by 1.16, IS 883:1994 cl. 6.3).

Permissible stresses (IS 883:1994 Table 1, inside location):

Location is not stated, so inside is taken. The bending moment acts about the major axis (depth $d$); buckling is checked about the minor axis (least dimension $b$).

Trial section

Try 200 mm × 275 mm. Area $A = 55,000$ mm²; least dimension $d = 200$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{3200}{200} = 16.0 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{12.30}} = 18.75 \end{aligned}$$

$11 < S/d = 16.0 < K_8 = 18.7$, so it is an intermediate column:

$$f_c = f_{cp}\left[1 - \frac{1}{3}\left(\frac{S/d}{K_8}\right)^4\right] = 12.30\left[1 - \frac{1}{3}\left(\frac{16.0}{18.7}\right)^4\right] = 10.12\ \text{N/mm}^2$$

$S/d = 16.0 < 50$ (cl. 7.6.1.4). OK.

Combined axial compression and bending (cl. 7.7.1)

$$\begin{aligned} f_{ac} &= \frac{N}{A} = \frac{200000}{55,000} = 3.636\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2}\ \text{(bending about the strong axis)} = \frac{6\times30000000}{200\times275^2} = 11.901\ \text{N/mm}^2 \end{aligned}$$

Permissible bending stress $f_b = 19.60$ N/mm²; the form factor is not applied for depths up to 300 mm (cl. 7.5.4).

$$\frac{f_{ac}}{f_c} + \frac{f_{ab}}{f_b} = \frac{3.636}{10.12} + \frac{11.901}{19.60} = 0.966 \le 1.0$$

Safe.

Answer

Provide a solid Sal column of 200 mm × 275 mm cross-section (Select grade, inside location). Combined stress ratio $= 0.97 < 1$.