Masonry Structures Under Lateral Loads

Lateral loads (wind, earthquake) act on a masonry building in any direction. A wall resists lateral load in two ways depending on its orientation to the load.

• In-plane: the load acts parallel to the length of the wall; the wall acts as a shear wall.
• Out-of-plane: the load acts perpendicular to the wall face; the wall acts as a thin slab or cantilever in bending.

   In-plane (shear wall)         Out-of-plane (bending)
      ---> F                         F ->  |
    +--------+                            | wall bends
    |        |                         ___|___  floor
    +--------+                            |

In-plane failure modes

1. Diagonal tension (shear) cracking: stepped cracks through the mortar or through the units forming an X under cyclic load; common in squat walls with low axial load.
2. Sliding shear: horizontal sliding along a bed joint when friction is low (low precompression).
3. Rocking / flexural failure: slender piers rock about the toe and crush at the compressed toe, with horizontal cracks at the base.

Out-of-plane failure modes

1. Vertical bending between floor and roof: horizontal crack near mid-height and overturning.
2. Horizontal bending between cross-walls when the wall is long.
3. Overturning of gable and parapet walls which have no top support.
4. Separation of walls at corners and delamination of leaves in multi-leaf walls.

Improving the behaviour

Out-of-plane performance improves by reducing $h/t$ and wall length, tying walls to floors and cross-walls, providing RC bands and corner stitching, and anchoring roofs. In-plane performance improves with adequate wall density in each direction and sufficient vertical load and reinforcement.

Masonry is brittle, weak in tension and heavy, so damage is typically cracking, separation and collapse in recognisable patterns.

1. Diagonal X-cracks in piers and walls (in-plane shear)

Caused by in-plane shear when the principal tensile stress exceeds the masonry tensile strength. Cracks reverse with cyclic motion and form an X.

 +-------+
 |\     /|
 | \   / |      X-cracks in
 |  \ /  |      pier or wall
 |  / \  |
 | /   \ |
 |/     \|
 +-------+

2. Cracks at openings

Stress concentration at corners of doors and windows leads to diagonal cracks starting at the opening corners, and horizontal cracks at sill and lintel levels; narrow piers between openings fail first.

3. Vertical cracks at wall corners and junctions

Orthogonal walls vibrate out of phase, so the corner or T-junction separates (vertical crack). It is worse if there is no bonding or bands.

4. Out-of-plane bulging, overturning and collapse

Long, tall or thin walls bend outward; horizontal cracks at mid-height or at floor level result, and the wall falls out. Gable walls and parapets overturn because they are not restrained at the top.

5. Delamination of wall leaves and collapse of stone walls

In rubble masonry the outer leaves separate from the loose core, and the wall sheds as a pile of stones.

6. Roof and floor damage

Heavy roofs on weak walls cause collapse; timber joists pull out of walls because of poor anchorage.

7. Damage due to pounding, soft storey, torsion

Irregular plans (L, T shapes), unsymmetrical openings or stiff walls on one side lead to torsion and concentrated damage at re-entrant corners.

Ductility is the ability of a structure to undergo large inelastic deformation without significant loss of strength before collapse. It is measured by the displacement ductility $\mu = \Delta_u/\Delta_y$, the ratio of ultimate to yield displacement.

Unreinforced masonry (URM)

• Masonry units and mortar are brittle with low tensile strength. After first diagonal cracking or toe crushing, strength drops quickly.
• Typical failure modes (diagonal shear, sliding, out-of-plane collapse) are brittle with small energy dissipation and ductility close to 1–1.5.
• URM can show limited ductility by rocking of squat walls under low axial load, but this is unreliable.

Reinforced masonry (RM)

• Vertical reinforcement in cells or cavities (grouted) and horizontal reinforcement in bond beams or bed joints carry tension after masonry cracks.
• A flexural failure with yielding of steel is preferred to shear failure. Walls then show a wide hysteresis loop, absorb energy and can sustain large displacement.
• Shear failure remains brittle unless enough horizontal steel is given; the design should force flexure to govern ("strong shear, weak flexure").

 Load                         URM: peak, then drops suddenly
  |    /\___ RM: ductile      
  |   /     \___________      
  |  /  /\                    URM
  | /  /  \__ (brittle)
  |/__/_________________ Displacement

Confined masonry

Masonry walls enclosed by RC tie-columns and tie-beams cast after the wall are not reinforced masonry but show better ductility: the frame confines the wall after cracking and prevents collapse.

Improving ductility

1. Provide reinforced concrete or timber bands (plinth, lintel, roof) to tie the walls and control cracks.
2. Use vertical bars at corners, junctions and sides of openings, anchored in the foundation and the roof band.
3. Use bed-joint reinforcement or steel mesh; grout hollow blocks.
4. Use good bond between units and mortar, and avoid weak mortar or dry stone.
5. Keep walls with low axial stress, limit slenderness and opening size.
6. Ensure box action by rigid floor diaphragms and tied roofs.

The code reduction (response reduction) factor $R$ is higher for reinforced masonry than for URM because of this ductility.

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

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

The wall would crack; a free (unloaded) wall must be limited in height or thickened, or horizontally reinforced or stiffened with cross-walls/pilasters/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.

Principle

A rigid diaphragm gives the same horizontal displacement $\Delta$ to all walls, so each wall takes force in proportion to its stiffness $k = V/\Delta$.

For a wall fixed at both ends (top rotation prevented):

$$\Delta = \frac{Vh^3}{12EI} + \frac{1.2\,Vh}{GA} \quad\Rightarrow\quad k = \frac{1}{\dfrac{h^3}{12EI} + \dfrac{1.2h}{GA}}$$

where $I = tL^3/12$ and $A = tL$ (the factor 1.2 is the shape factor for rectangular shear).

Data: $h = 3000$ mm, $t = 230$ mm, $E = 2500$ N/mm², $G = 1000$ N/mm².

Stiffness of each wall

Sum $\Sigma k = 273\,810$ N/mm.

Shares of 240 kN

$$V_i = V \frac{k_i}{\Sigma k}$$

Displacement of the slab: $\Delta = 240\,000/273\,810 = 0.88$ mm (same for all walls).

Remarks

• The longest wall (3.0 m) takes 52.5 percent of the force although it is only 3 times the length of the shortest, because stiffness grows faster than length.
• Slender walls are weaker in resisting lateral force; a pier with $L/h<1$ is dominated by flexure. If the walls are treated as cantilevers (free at top), shares become 8.8, 27.1, 57.2 and 147.0 kN, so the long wall attracts even more.
• If only shear deformation were considered, the shares would be in proportion to the wall lengths.

Answer: Shares are 14 kN, 36 kN, 64 kN and 126 kN for the 1.0, 1.5, 2.0 and 3.0 m walls respectively.