Out-of-plane bending of a masonry wall

(a) Pressure and moment

Self-weight of wall per unit area $= 19 \times 0.23 = 4.37$ kN/m².

$$\begin{aligned} q &= 0.2 \times 4.37 = 0.874\ \text{kN/m}^2 \\ M_{max} &= \frac{qH^2}{8} = \frac{0.874 \times 3.0^2}{8} = 0.983\ \text{kN·m per m} \end{aligned}$$

 roof load 25 kN/m
      v
 -----+----- roof (hinge)
      |  ->  q = 0.874 kN/m2
      |  ->
 3.0 m|  ->  (UDL on wall face)
      |  ->
 -----+----- floor (hinge)

(b) Stresses at mid-height

$$Z = \frac{1000 \times 230^2}{6} = 8.817\times10^{6}\ \text{mm}^3, \qquad \sigma_b = \frac{0.983\times10^6}{8.817\times10^6} = 0.1115\ \text{N/mm}^2$$

Vertical load at mid-height $= 25 + 19\times0.23\times1.5 = 31.56$ kN/m:

$$\sigma_a = \frac{31\,560}{230 \times 1000} = 0.137\ \text{N/mm}^2$$ $$\sigma_{max} = 0.137 + 0.1115 = 0.249\ \text{N/mm}^2 \ (\text{compression}), \qquad \sigma_{min} = 0.137 - 0.1115 = +0.026\ \text{N/mm}^2$$

$\sigma_{min}$ is positive (compression), so there is no tension. The wall is safe.

(c) Load needed for no tension

$\sigma_a \ge \sigma_b$: $N \ge 0.1115 \times 230 \times 1000 = 25\,650$ N/m $\approx 25.7$ kN/m at mid-height.

(d) No roof load

Only the self-weight above mid-height acts: $19\times0.23\times1.5 = 6.56$ kN/m, so $\sigma_a = 0.0285$ N/mm².

$$\sigma_{min} = 0.0285 - 0.1115 = -0.083\ \text{N/mm}^2 \ (\text{tension}) > 0.07\ \text{N/mm}^2$$

The tension exceeds the permissible value, so the wall would crack. A wall without a roof load must have its height limited or its thickness increased, or it must be reinforced horizontally or stiffened with cross-walls, pilasters or bands.

Answer: $q = 0.874$ kN/m², $M = 0.983$ kN·m/m; $\sigma$ at mid-height varies from +0.026 to +0.249 N/mm² (no tension, safe); minimum vertical load ≈ 25.7 kN/m; without roof load tension of 0.083 N/mm² > 0.07, unsafe.