Rectangular Sal column, 150 kN + 15 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 $S = 4.00$ m = 4000 mm. Axial load $N = 150$ kN, bending moment $M = 15$ kN·m.

Species: Sal (Shorea robusta), Group A. Location: inside. Grade: Grade I (standard) (Table 1 values are for Grade I).

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

Location and grade not stated: inside, Grade I (standard) assumed. Bending about the major axis.

Trial section

Try 175 mm × 250 mm. Area $A = 43,750$ mm²; least dimension $d = 175$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{4000}{175} = 22.9 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{10.60}} = 20.19 \end{aligned}$$

$S/d = 22.9 > K_8 = 20.2$, so it is a long column:

$$f_c = \frac{0.329E}{(S/d)^2} = \frac{0.329\times12670}{22.9^2} = 7.98\ \text{N/mm}^2$$

$S/d = 22.9 < 50$ (cl. 7.6.1.4). OK.

Combined axial compression and bending (cl. 7.7.1)

$$\begin{aligned} f_{ac} &= \frac{N}{A} = \frac{150000}{43,750} = 3.429\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2}\ \text{(bending about the strong axis)} = \frac{6\times15000000}{175\times250^2} = 8.229\ \text{N/mm}^2 \end{aligned}$$

Permissible bending stress $f_b = 16.90$ N/mm²; the form factor is not applied for depths up to 300 mm (cl. 7.5.4).

$$\frac{f_{ac}}{f_c} + \frac{f_{ab}}{f_b} = \frac{3.429}{7.98} + \frac{8.229}{16.90} = 0.917 \le 1.0$$

Safe.

Answer

Provide a solid Sal column of 175 mm × 250 mm cross-section (Grade I (standard), inside location). Combined stress ratio $= 0.92 < 1$.