Brick column 35 cm x 50 cm allowable load

Data and assumptions

• Column 350 mm $\times$ 500 mm, height 4 m, fully restrained at top and bottom. The strengths of brick and mortar are not given, so brick 7.5 N/mm$^2$ and cement mortar 1:5 (grade M1) are assumed (IS 1905 Table 8: $f_b=0.74$ N/mm$^2$).
• Brick $200\times100\times150$ mm: height/width $=150/100=1.5$, so the shape factor is $k_p=1.30$ (Table 10, 7.5 N/mm$^2$).

Step 1: Slenderness ratio (IS 1905 cl. 4.3, 4.6.2)

With full restraint at both ends, $h_{ef}=0.75H = 0.75\times4 = 3.0$ m. For a column the greater slenderness ratio governs, so the least dimension (350 mm) is critical:

$$SR=\frac{3.0}{0.35}=8.57\ (<12,\ \text{OK})$$

Step 2: Axially loaded column

$$\begin{aligned} A &= 0.35\times0.5 = 0.175\ \text{m}^2 \ (<0.2)\ \Rightarrow\ k_a = 0.7+1.5A = 0.9625\\ k_s &= 0.933\ \text{(Table 9, } e=0,\ SR=8.57)\\ f_c &= f_b k_s k_a k_p = 0.74\times0.933\times0.9625\times1.30 = 0.864\ \text{N/mm}^2\\ P &= f_c A = 0.864\times0.175\times10^3 = 151.2\ \text{kN} \end{aligned}$$

Step 3: Eccentricity 110 mm about the major axis

Bending about the major axis means the eccentricity is measured along the 500 mm side: $e/D = 110/500 = 0.22$. This exceeds $1/6$ (middle-third limit $=83$ mm), so part of the section would be in tension; the tension area is neglected (cl. 5.4.1.4 b, 5.4.4). The stress on the compressed part is triangular, over a depth of

$$3\left(\frac{D}{2}-e\right)=3\times(250-110)=420\ \text{mm},\qquad A_{eff}=350\times420=0.147\ \text{m}^2$$

$k_s=0.899$ (Table 9, $SR=8.57$, $e/D=0.22$, interpolated). Since $e/D > 1/6$, a 25 percent increase in permissible stress is allowed (cl. 5.4.1.4 b):

$$\begin{aligned} f_c &= 0.74\times0.899\times0.9625\times1.30 = 0.832\ \text{N/mm}^2\\ f_{c,\text{allow}} &= 1.25\times0.832 = 1.040\ \text{N/mm}^2\ \text{(maximum stress at the compressed edge)}\\ P_{ecc} &= \tfrac{1}{2}\,f_{c,\text{allow}}\,A_{eff} = \tfrac{1}{2}\times1.040\times0.147\times10^3 = 76.4\ \text{kN} \end{aligned}$$

Answer: axially loaded: allowable stress 0.864 N/mm$^2$, $P = 151.2$ kN. With 110 mm eccentricity: allowable (maximum) compressive stress 1.040 N/mm$^2$, allowable load $\approx 76.4$ kN (for brick 7.5 N/mm$^2$, mortar 1:5). Without the 25 percent increase these become 0.832 N/mm$^2$ and 61.2 kN. All values change in proportion to $f_b$ if other brick and mortar are used.