---
title: "Out-of-plane bending of a masonry wall"
subject: "Design of Timber and Masonry Structures"
chapter: 6
key: "semester_5/design_of_timber_and_masonry_structures:out-of-plane-wall-flexure-check"
source: https://www.bctengineer.com/q/semester_5/design_of_timber_and_masonry_structures/out-of-plane-wall-flexure-check
site: BCT Engineering
---

# Out-of-plane bending of a masonry wall

Design of Timber and Masonry Structures · Chapter 6: Masonry Structures Under Lateral Loads

- Asked in: year not given, 6 marks
- Video: [Out-of-Plane Design of Reinforced Masonry Bearing Walls](https://www.youtube.com/watch?v=0Z8tMRPcXu4) — SKGAssociates
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/design_of_timber_and_masonry_structures/out-of-plane-wall-flexure-check
- Whole chapter: https://www.bctengineer.com/md/semester_5/design_of_timber_and_masonry_structures/chapter-6

## Question

A 230 mm thick unreinforced brick wall, 3.0 m high, spans vertically between the floor and the roof and may be treated as simply supported. It carries a vertical roof load of 25 kN/m at the top. For out-of-plane seismic action take an equivalent lateral coefficient of 0.2 applied to the self-weight of the wall (unit weight 19 kN/m³). The permissible flexural tensile stress is 0.07 N/mm². (a) Find the out-of-plane pressure and the bending moment per metre. (b) Check the mid-height section. (c) What minimum vertical load at mid-height avoids tension? (d) What happens if the roof load is absent?

## Short answer

q = 0.874 kN/m², M = 0.983 kN·m/m. Mid-height stresses: 0.026 to 0.249 N/mm² (safe, no tension). Minimum vertical load to avoid tension: 25.7 kN/m. Without roof load, tension 0.083 N/mm² > 0.07, unsafe.

## Answer

#### (a) Pressure and moment
Self-weight of wall per unit area $= 19 \times 0.23 = 4.37$ kN/m².

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

```text
 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.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How are the out-of-plane pressure and moment found?
   A. q = 0.2 x 19 x 0.23 = 0.874 kN/m^2 and M = qH^2/8 = 0.983 kN·m/m
   B. q = 25 x 0.2 and M = qH^2/2
   C. q = 0.2 x 25 kN/m and M = qH/4
   D. q = 19 x 0.23 and M = qH^2/12
   Answer: A. The coefficient is applied to self-weight per area, and a simply supported span under UDL gives qH^2/8. (Key point: Pressure and moment)

2. How do the stresses at mid-height combine?
   A. Bending stress is ignored
   B. Axial stress 0.137 plus and minus bending stress 0.1115 N/mm^2, giving +0.249 and +0.026
   C. Axial 0.137 minus bending only
   D. Only the maximum is checked, against tension
   Answer: B. The axial stress from roof load and half-wall is added to and subtracted from the bending stress; minimum remains positive. (Key point: Mid-height check)

3. What condition avoids tension at mid-height?
   A. Axial stress equal to zero
   B. Axial stress must be less than the bending stress
   C. Axial stress >= bending stress, i.e. N >= about 25.7 kN/m
   D. Bending stress must exceed 0.07
   Answer: C. Setting sigma_a >= sigma_b gives N >= 0.1115 x 230 x 1000 = 25.65 kN/m. (Key point: No-tension load)

4. What happens without the roof load?
   A. The wall becomes safer
   B. Nothing changes
   C. Compression increases
   D. Axial stress falls to 0.0285 so tension of 0.083 > 0.07 occurs and the wall cracks
   Answer: D. Only 6.56 kN/m remains above mid-height, so bending produces tension beyond the permissible value. (Key point: No roof load)

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