Timber column under axial load and bending

Interaction rule

For members under combined direct compression and bending (IS 883), the condition is

$$\frac{\sigma_{c,cal}}{\sigma_c'} + \frac{\sigma_{bc,cal}}{\sigma_{bc}} \le 1$$

where $\sigma_c'$ is the permissible axial compressive stress allowing for slenderness. (Magnification of moment due to deflection is ignored here for simplicity.)

Step 1: Actual stresses

$$\begin{aligned} A &= 150 \times 200 = 30\,000\ \text{mm}^2 \\ \sigma_{c,cal} &= \frac{80\,000}{30\,000} = 2.67\ \text{N/mm}^2 \\ Z &= \frac{150 \times 200^2}{6} = 1.0 \times 10^{6}\ \text{mm}^3 \\ \sigma_{bc,cal} &= \frac{5.0 \times 10^6}{1.0 \times 10^6} = 5.0\ \text{N/mm}^2 \end{aligned}$$

Step 2: Permissible axial stress

Least dimension $d = 150$ mm, so $l_e/d = 3000/150 = 20$.

$$K_8 = 0.702\sqrt{\frac{9000}{8}} = 0.702 \times 33.54 = 23.55$$

Since $11 < 20 < 23.55$, the column is intermediate:

$$\sigma_c' = 8\left[1 - \frac{1}{3}\left(\frac{20}{23.55}\right)^4\right] = 8\,[1 - 0.1735] = 6.61\ \text{N/mm}^2$$

Step 3: Interaction check

$$\frac{2.67}{6.61} + \frac{5.0}{10.0} = 0.403 + 0.500 = 0.903 < 1$$

The column is safe; about 10 percent reserve remains.

Answer: Interaction value = 0.90 < 1.0, so the 150 mm × 200 mm column is safe.