Solid Sal column, 500 kN + 50 kNm, 2 m

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 $S = 2.00$ m = 2000 mm. Axial load $N = 500$ kN, bending moment $M = 50$ 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 not stated: inside. Bending about the major axis; buckling about the minor axis.

Trial section

Try 225 mm × 375 mm. Area $A = 84,375$ mm²; least dimension $d = 225$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{2000}{225} = 8.9 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{12.30}} = 18.75 \end{aligned}$$

$S/d \le 11$, so it is a short column and $f_c = f_{cp} = 12.30$ N/mm².

$S/d = 8.9 < 50$ (cl. 7.6.1.4). OK.

Combined axial compression and bending (cl. 7.7.1)

$$\begin{aligned} f_{ac} &= \frac{N}{A} = \frac{500000}{84,375} = 5.926\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2}\ \text{(bending about the strong axis)} = \frac{6\times50000000}{225\times375^2} = 9.481\ \text{N/mm}^2 \end{aligned}$$

Permissible bending stress $f_b = 19.60$ N/mm²; depth 375 mm $> 300$ mm so the form factor $K = 0.81\,(d^2 + 89400)/(d^2 + 55000) = 0.952$ is applied: $f_b = 18.67$ N/mm².

$$\frac{f_{ac}}{f_c} + \frac{f_{ab}}{f_b} = \frac{5.926}{12.30} + \frac{9.481}{18.67} = 0.990 \le 1.0$$

Safe.

Answer

Provide a solid Sal column of 225 mm × 375 mm cross-section (Select grade, inside location). Combined stress ratio $= 0.99 < 1$.