Column 500 x 900: stress intensities, critical corner

Distances are measured from the faces. The 170 mm distance is across the 500 mm side (from the 900 mm face) and the 360 mm distance is along the 900 mm side (from the 500 mm face). Hence $e_1 = 250-170 = 80$ mm and $e_2 = 450-360 = 90$ mm.

Data

Section 500 mm $\times$ 900 mm, $P=300$ kN. The load acts 170 mm from the 900 mm face and 360 mm from the 500 mm face.

Eccentricities

$$\begin{aligned} e_{1} &= 80\ \text{mm (across the 500 mm side)}\\ e_{2} &= 90\ \text{mm (along the 900 mm side)} \end{aligned}$$

Section properties

$$\begin{aligned} A &= 500\times900 = 450000\ \text{mm}^2\\ Z_1 &= \frac{900\times500^2}{6} = 37500000\ \text{mm}^3\\ Z_2 &= \frac{500\times900^2}{6} = 67500000\ \text{mm}^3 \end{aligned}$$

Stresses

$$\begin{aligned} \sigma_0 &= \frac{P}{A} = \frac{300\times10^3}{450000} = 0.6667\ \text{N/mm}^2\\ \sigma_{b1} &= \frac{Pe_1}{Z_1} = \frac{300\times10^3\times80}{37500000} = 0.6400\ \text{N/mm}^2\\ \sigma_{b2} &= \frac{Pe_2}{Z_2} = \frac{300\times10^3\times90}{67500000} = 0.4000\ \text{N/mm}^2 \end{aligned}$$

Corner stresses $\sigma=\sigma_0\pm\sigma_{b1}\pm\sigma_{b2}$ (positive = compression, N/mm²):

Check: $\sigma_0$ = average of the four corners = 0.6667. Corner D is in tension (-0.3733 N/mm²) because $6e_1/500+6e_2/900 = 1.560 > 1$, i.e. the load lies outside the kernel (middle-third) of the section.

Answer: A = 1.707, B = 0.907, C = 0.427, D = -0.373 N/mm² (compression positive); maximum 1.707 N/mm² at A.

Critical stress

• The maximum compressive stress is at corner A = 1.707 N/mm$^2$, the corner nearest the load. This must be compared with the permissible compressive stress of the masonry ($f_b k_s k_a k_p$).
• Corner D has a tension of 0.373 N/mm$^2$ because the load is outside the kernel of the section. Masonry and plain concrete have little tensile strength (IS 1905 cl. 5.4.2 permits none for vertical loads), so the corner would crack and the compression at A would be even higher than 1.707 N/mm$^2$.

Therefore corner A (compression) and corner D (tension) both govern; the column must be enlarged or the eccentricity reduced.