Solid Sal column, 75 kN + 12 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 $N = 75$ kN, bending moment $M = 12$ 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):

The loads are used as given (IS 883 is a working-stress code, so they are treated as the design loads); location and grade are not stated: inside, Grade I. Bending about the major axis.

Trial section

Try 125 mm × 225 mm. Area $A = 28,125$ mm²; least dimension $d = 125$ mm.

Slenderness and permissible axial stress (cl. 7.6.1)

$$\begin{aligned} S/d &= \frac{2000}{125} = 16.0 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{10.60}} = 20.19 \end{aligned}$$

$11 < S/d = 16.0 < K_8 = 20.2$, so it is an intermediate column:

$$f_c = f_{cp}\left[1 - \frac{1}{3}\left(\frac{S/d}{K_8}\right)^4\right] = 10.60\left[1 - \frac{1}{3}\left(\frac{16.0}{20.2}\right)^4\right] = 9.21\ \text{N/mm}^2$$

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

Combined axial compression and bending (cl. 7.7.1)

$$\begin{aligned} f_{ac} &= \frac{N}{A} = \frac{75000}{28,125} = 2.667\ \text{N/mm}^2 \\ f_{ab} &= \frac{6M}{bd^2}\ \text{(bending about the strong axis)} = \frac{6\times12000000}{125\times225^2} = 11.378\ \text{N/mm}^2 \end{aligned}$$

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).

$$\frac{f_{ac}}{f_c} + \frac{f_{ab}}{f_b} = \frac{2.667}{9.21} + \frac{11.378}{16.90} = 0.963 \le 1.0$$

Safe.

Answer

Provide a solid Sal column of 125 mm × 225 mm cross-section (Grade I (standard), inside location). Combined stress ratio $= 0.96 < 1$.