Design of a square timber column

Method

The permissible stress depends on slenderness, which depends on the section, so we try sections and compare the capacity with 150 kN.

Effective length $l_e = L = 3600$ mm (hinged both ends). For a square section $d \times d$, $l_e/d = 3600/d$.

$$K_8 = 0.702\sqrt{\frac{E}{\sigma_{cc}}} = 0.702\sqrt{\frac{10500}{8}} = 0.702 \times 36.23 = 25.43$$

For $11 < l_e/d < K_8$ (intermediate):

$$\sigma_c' = 8\left[1 - \frac{1}{3}\left(\frac{l_e/d}{25.43}\right)^4\right]$$

Trial sections

Check of 160 mm × 160 mm

$$\begin{aligned} \frac{l_e}{d} &= \frac{3600}{160} = 22.5 < 25.43 \\ \sigma_c' &= 8\left[1 - \frac{1}{3}(0.8848)^4\right] = 8\,[1 - 0.2043] = 6.37\ \text{N/mm}^2 \\ \sigma_{c,\text{actual}} &= \frac{150\,000}{160 \times 160} = 5.86\ \text{N/mm}^2 < 6.37\ \text{N/mm}^2 \end{aligned}$$

The column is safe. Its self-weight (about $0.16^2 \times 3.6 \times 7 \approx 0.6$ kN) is less than 0.5 percent of the load and raises the stress only to about 5.88 N/mm², which is still below 6.37 N/mm².

Answer: Provide a 160 mm × 160 mm timber column (capacity 163 kN > 150 kN, $l_e/d = 22.5$).