Circular Sal column, open shed, 200 kN

Approach. IS 883:1994 cl. 7.6.1.6: the permissible load on a circular column must not exceed that of a square column of equal cross-sectional area. So the circular column is designed through its equivalent square side $a$, with $a^2 = \pi D^2/4$, i.e. $a = 0.886D$.

Data and permissible stresses

Height (unsupported length) $S = 3.5$ m, axial load $N = 200$ kN, pin ends.

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

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

An open shed is an outside location; the grade is not stated, so Grade I is used.

Trial section

Try diameter $D = 180$ mm. Area $A = \pi D^2/4 = 25,447$ mm². Equivalent square side:

$$a = 0.886D = 159.5\ \text{mm}$$

Slenderness and permissible stress

$$\begin{aligned} S/a &= \frac{3500}{159.5} = 21.94 \\ K_8 &= 0.584\sqrt{\frac{E}{f_{cp}}} = 0.584\sqrt{\frac{12670}{9.40}} = 21.44 \end{aligned}$$

$S/a = 21.94 > K_8 = 21.44$, so it is a long column ($S/a < 50$, OK):

$$f_c = \frac{0.329E}{(S/a)^2} = \frac{0.329\times12670}{21.94^2} = 8.66\ \text{N/mm}^2$$

Check

$$P = f_cA = 8.66\times25,447/10^3 = 220\ \text{kN} > 200\ \text{kN}$$

Safe.

Answer: provide a circular Sal column of diameter 180 mm. (Its permissible load equals that of a 160 mm square column of the same area.)