Wall subjected to an eccentric load

Middle-third rule

If the resultant load acts within the middle third of the thickness ($e \le t/6$), the whole section is in compression. If $e > t/6$, tension appears on the far face, and masonry (weak in tension) may crack.

Data

$P = 60$ kN/m, $t = 230$ mm, $e = 45$ mm, $b = 1000$ mm (per metre).

$$\frac{t}{6} = \frac{230}{6} = 38.3\ \text{mm} < e = 45\ \text{mm}$$

The load is outside the middle third, so some tension is expected.

Stresses

$$\begin{aligned} A &= 230 \times 1000 = 2.3\times10^5\ \text{mm}^2 \\ Z &= \frac{1000 \times 230^2}{6} = 8.817\times10^{6}\ \text{mm}^3 \\ M &= Pe = 60\,000 \times 45 = 2.7\times10^{6}\ \text{N·mm} \\ \frac{P}{A} &= \frac{60\,000}{2.3\times10^5} = 0.261\ \text{N/mm}^2 \\ \frac{M}{Z} &= \frac{2.7\times10^6}{8.817\times10^6} = 0.306\ \text{N/mm}^2 \end{aligned}$$

Or, using the formula $\sigma = \dfrac{P}{A}\left(1 \pm \dfrac{6e}{t}\right)$ with $6e/t = 1.174$:

$$\begin{aligned} \sigma_{max} &= 0.261\,(1 + 1.174) = 0.567\ \text{N/mm}^2 \ (\text{compression}) \\ \sigma_{min} &= 0.261\,(1 - 1.174) = -0.045\ \text{N/mm}^2 \ (\text{tension}) \end{aligned}$$

Checks

• Compression: $0.567 < 0.90$ N/mm². Safe.
• Tension: $0.045 < 0.07$ N/mm². Within the permissible flexural tension.

Both limits are satisfied. If the tension were more than allowed, the eccentricity would have to be reduced (for example, by centring the beam bearing with a bearing plate), or the wall thickness increased, or the wall reinforced.

Answer: $\sigma_{max} = 0.567$ N/mm² (compression) and $\sigma_{min} = -0.045$ N/mm² (tension); both within permissible values, so the wall is safe.