Masonry structures under lateral loads

Masonry walls resist lateral load (earthquake, wind) in two ways, depending on the direction of the force relative to the wall plane.

In-plane behaviour

Force acts parallel to the plane of the wall; the wall acts as a shear wall and is strong and stiff in this direction. Failure modes:

1. Sliding shear (horizontal cracks): sliding along a bed joint, usually near the base or at a damp-proof course when vertical load is small.
2. Diagonal shear (X-cracks): principal tension exceeds the tensile strength of masonry; stair-stepped cracks through joints or cracks through bricks; the most common failure of squat walls.
3. Rocking (flexural, toe crushing): tall slender wall or pier rotates about its toe; horizontal cracks at the heel and crushing at the toe.
4. Bed-joint sliding combined with diagonal cracking in piers between openings (short piers fail in shear, spandrels in cracking).

 Diagonal shear     Sliding shear      Rocking / toe crush
 +--------+         +--------+         +--------+
 |\      /|         |        |         |        |
 | \    / |         |        |         |  ----  |  horiz. crack
 |  \  /  |   F ->  |========| F ->    |        |
 |  /  \  |         |  slide |         |        |
 | /    \ |         +--------+         +--/---\-+
 +--------+                              toe crush

Out-of-plane behaviour

Force acts perpendicular to the wall plane; the wall acts as a vertical (and horizontal) slab in bending, but has very low tensile strength and no ductility, so it is the weakest direction.

1. Overturning / rocking as a cantilever: gable or free-standing wall, parapet, tilts about its base (horizontal crack at base).
2. Vertical bending between floor and roof: horizontal crack at mid-height; wall spans one way like a beam.
3. Horizontal bending between cross walls: vertical crack at the corners where walls meet, then wall bulges and collapses outward.
4. Two-way bending (arching action): when supported on all sides, cracks run diagonally from the wall corners to the centre (like a yield-line pattern).
5. Delamination of leaves in thick or multi-leaf walls and separation of walls at corners because of poor connection.

 Corner separation        Horizontal bending crack
   |   |                   +------------+
 --+   +--               ->|            |   <- force
   | crack |               |~~~~~~~~~~~~|  mid-height crack

Remedies

Provide RCC bands and tie beams at plinth, lintel and roof levels; vertical bars at corners and jambs; adequate cross walls and rigid diaphragm roofs; limit wall height and length (H/t and L/t); keep openings small and away from corners; good mortar and bond.

In a masonry building, lateral forces from wind or earthquake are picked up by floors and roof and sent to the walls that run parallel to the force; these then carry the force to the foundation. The main resisting elements are:

1. Shear walls (load-bearing masonry walls) in the direction of the force: resist the lateral force by in-plane shear and flexure. The shear capacity of the wall depends on length, thickness, vertical load and mortar bond. They are the principal elements.
2. Floor and roof diaphragms: RCC slabs or rigid roofs act as horizontal deep beams that transmit the inertia load to shear walls; flexible diaphragms (timber, GI sheet roof) distribute poorly and need bracing.
3. Cross walls and piers or buttresses (transverse walls): stiffen the walls against out-of-plane forces by acting as vertical supports; connection at corners must be strong.
4. Bands (ring beams) and tie columns: RCC or timber horizontal bands at plinth, lintel, floor and roof tie the walls together and give box action; vertical reinforcement at corners and jambs.
5. Foundation: transmits the load into the ground; a rigid continuous plinth band is preferred.

 Load path
 Earthquake force -> Roof diaphragm -> Shear walls (in-plane)
                          |                |
                          v                v
                   Cross walls       Foundation -> ground
                   (out-of-plane support)

For good behaviour all these elements must form a box with continuous load path; roofs must be tied to walls, walls tied to each other at corners, and openings small.

Data and assumptions

• Exterior wall of a single-storey warehouse, height 3.5 m, tied by metal anchors at floor and roof levels, so it spans vertically as simply supported. Vertical roof load $P=30$ kN/m (assumed axial); wind pressure 750 N/m$^2$.
• Anchors give lateral restraint only (no rotational restraint), so the effective height is $h_{ef}=1.0H$ (IS 1905 Table 4 Sl. 3).
• Design by IS 1905:1987 cl. 5.4.1, 5.4.2, 5.5.2, 5.5.3 (NBC 109 follows the same method). Unit weight of masonry 19 kN/m$^3$. Per metre run of wall.

     roof anchor (support)
        |<-- wind w
  ------+-----------   load P (roof)
        |   |
        | wall, H=3.5 m
        |   |  M = wH^2/8 at mid-height
  ------+-----------
        floor anchor (support)

Step 1: Wind moment

$$\begin{aligned} w &= 0.750\ \text{kN/m (per m of wall)}\\ M &= 0.750\times3.5^2/8 = 1.148\ \text{kNm/m} \end{aligned}$$

Step 2: Trial thickness 230 mm, load and slenderness

$$\begin{aligned} \text{Self weight of wall} &= 0.230\times3.50\times19 = 15.30\ \text{kN/m}\\ N\ (\text{at mid-height}) &= 37.65\ \text{kN/m}\\ h_{ef} &= 1.0\times3.5=3.5\ \text{m} \\ SR &= \frac{3.500}{0.230} = 15.22\ (<27) \end{aligned}$$

Step 3: Stresses at the mid-height section

$$\begin{aligned} \sigma_a &= \frac{N}{A}=\frac{37.65\times10^3}{230\times10^3} = 0.1637\ \text{N/mm}^2\\ \sigma_b &= \frac{6M}{t^2} = \frac{6\times1.148\times10^6}{1000\times230^2} = 0.1303\ \text{N/mm}^2\\ \sigma_{max} &= \sigma_a+\sigma_b = 0.2939,\qquad \sigma_{min} = \sigma_a-\sigma_b = 0.0334\ \text{N/mm}^2 \end{aligned}$$

$\sigma_{min}>0$: the whole section is in compression, no tension (the resultant eccentricity $e=M/N=0.0305$ m, $e/t=0.133<1/6$).

Step 4: Required basic compressive stress

Resultant eccentricity ratio $e/t = M/(Nt) = 0.133$. From Table 9: $k_s=0.676$ ($SR=15.22$). $k_a=1$, $k_p=1$. Since $1/24<e/t\le1/6$, a 25 percent increase is allowed (cl. 5.4.1.4 a).

$$f_b \ge \frac{\sigma_a}{k_sk_ak_p\times1.25} = \frac{0.1637}{0.676\times1.25} = 0.194\ \text{N/mm}^2$$

Step 5: Brick and mortar (IS 1905 Table 8)

Answer: 230 mm thick wall (one brick). Bricks of at least 3.5 N/mm$^2$ crushing strength in cement mortar M1 (1:5) or richer; practically use 5 N/mm$^2$ bricks with 1:5 mortar. Maximum stress 0.294 N/mm$^2$ (compression) and minimum 0.033 N/mm$^2$; slenderness ratio 15.2 < 27. Provide metal anchors at roof and floor at close spacing to pass the wind load into the roof diaphragm and floor.

Data and assumptions

• Exterior wall of a single-storey warehouse, 3.5 m high, fixed at base (continuous with the foundation) and tied with a metal anchor at roof level: a propped cantilever. $P=25$ kN/m (axial); wind 860 N/m$^2$.
• Fixed at the base and laterally supported (not rotationally restrained) at the top: $h_{ef}=0.85H$ (Table 4 Sl. 2). Maximum moment is at the fixed base, $M=wH^2/8$ (hogging); sagging moment $9wH^2/128$ is smaller.
• Design by IS 1905:1987 cl. 5.4.1, 5.4.2, 5.5.2, 5.5.3 (NBC 109 follows the same method). Unit weight of masonry 19 kN/m$^3$. Per metre run of wall.

     roof anchor (support)
        |<-- wind w=860 N/m2
  ------+-----------   load P=25 kN/m
        |   |
        | wall, H=3.5 m
        |   |  M = wH^2/8 at the fixed base
  ======+===========   foundation (fixed)

Step 1: Wind moment

$$\begin{aligned} w &= 0.860\ \text{kN/m (per m of wall)}\\ M &= 0.860\times3.5^2/8 = 1.317\ \text{kNm/m} \end{aligned}$$

Step 2: Trial thickness 230 mm, load and slenderness

$$\begin{aligned} \text{Self weight of wall} &= 0.230\times3.50\times19 = 15.30\ \text{kN/m}\\ N\ (\text{at base}) &= 40.30\ \text{kN/m}\\ h_{ef} &= 0.85\times3.5=2.975\ \text{m} \\ SR &= \frac{2.975}{0.230} = 12.93\ (<27) \end{aligned}$$

Step 3: Stresses at the base section

$$\begin{aligned} \sigma_a &= \frac{N}{A}=\frac{40.30\times10^3}{230\times10^3} = 0.1752\ \text{N/mm}^2\\ \sigma_b &= \frac{6M}{t^2} = \frac{6\times1.317\times10^6}{1000\times230^2} = 0.1494\ \text{N/mm}^2\\ \sigma_{max} &= \sigma_a+\sigma_b = 0.3246,\qquad \sigma_{min} = \sigma_a-\sigma_b = 0.0258\ \text{N/mm}^2 \end{aligned}$$

$\sigma_{min}>0$: the whole section is in compression, no tension (the resultant eccentricity $e=M/N=0.0327$ m, $e/t=0.142<1/6$).

Step 4: Required basic compressive stress

Resultant eccentricity ratio $e/t = M/(Nt) = 0.142$. From Table 9: $k_s=0.753$ ($SR=12.93$). $k_a=1$, $k_p=1$. Since $1/24<e/t\le1/6$, a 25 percent increase is allowed (cl. 5.4.1.4 a).

$$f_b \ge \frac{\sigma_a}{k_sk_ak_p\times1.25} = \frac{0.1752}{0.753\times1.25} = 0.186\ \text{N/mm}^2$$

Step 5: Brick and mortar (IS 1905 Table 8)

Answer: 230 mm thick wall; bricks not less than 3.5 N/mm$^2$ with cement mortar M1 (1:5) or richer (practically 5 N/mm$^2$ bricks, 1:5 mortar). Base section: $\sigma_{max}=0.325$, $\sigma_{min}=0.026$ N/mm$^2$ (all compression). Anchor the roof to the wall and tie the wall to the footing with dowels to ensure fixity.

Data and assumptions

• Exterior wall of a single-storey warehouse, 3.5 m high, tied by metal anchors at floor and roof levels (simply supported vertically). $P=25$ kN/m (axial); wind pressure 860 N/m$^2$.
• Anchors give lateral restraint only: $h_{ef}=1.0H$ (Table 4 Sl. 3).
• Design by IS 1905:1987 cl. 5.4.1, 5.4.2, 5.5.2, 5.5.3 (NBC 109 follows the same method). Unit weight of masonry 19 kN/m$^3$. Per metre run of wall.

     roof anchor (support)
        |<-- wind w
  ------+-----------   load P=25
        |   |
        | wall, H=3.5 m
        |   |  M = wH^2/8 at mid-height
  ------+-----------
        floor anchor (support)

Step 1: Wind moment

$$\begin{aligned} w &= 0.860\ \text{kN/m (per m of wall)}\\ M &= 0.860\times3.5^2/8 = 1.317\ \text{kNm/m} \end{aligned}$$

Step 2: Trial thickness 230 mm, load and slenderness

$$\begin{aligned} \text{Self weight of wall} &= 0.230\times3.50\times19 = 15.30\ \text{kN/m}\\ N\ (\text{at mid-height}) &= 32.65\ \text{kN/m}\\ h_{ef} &= 1.0\times3.5=3.5\ \text{m} \\ SR &= \frac{3.500}{0.230} = 15.22\ (<27) \end{aligned}$$

Step 3: Stresses at the mid-height section

$$\begin{aligned} \sigma_a &= \frac{N}{A}=\frac{32.65\times10^3}{230\times10^3} = 0.1419\ \text{N/mm}^2\\ \sigma_b &= \frac{6M}{t^2} = \frac{6\times1.317\times10^6}{1000\times230^2} = 0.1494\ \text{N/mm}^2\\ \sigma_{max} &= \sigma_a+\sigma_b = 0.2913,\qquad \sigma_{min} = \sigma_a-\sigma_b = -0.0074\ \text{N/mm}^2 \end{aligned}$$

$\sigma_{min}$ is a small tension of 0.0074 N/mm$^2$ (resultant $e/t=0.175>1/6$). This is below the permissible flexural tension 0.07 N/mm$^2$ for mortar M1 or better in vertical bending (IS 1905 cl. 5.4.2), so the wall is safe against cracking provided the mortar is M1 or richer.

Step 4: Required basic compressive stress

Resultant eccentricity ratio $e/t = M/(Nt) = 0.175$. From Table 9: $k_s=0.653$ ($SR=15.22$). $k_a=1$, $k_p=1$.

$$f_b \ge \frac{\sigma_a}{k_sk_ak_p\times1.00} = \frac{0.1419}{0.653\times1.00} = 0.218\ \text{N/mm}^2$$

Step 5: Brick and mortar (IS 1905 Table 8)

For comparison, IS 1905 Table 11 (free-standing walls, 860 N/m$^2$ gives $H/t=5$) is for walls without anchors: it would need $t=3.5/5=0.70$ m, which shows the importance of the anchors.

Answer: 230 mm thick wall; bricks not less than 3.5 N/mm$^2$ in cement mortar of grade M1 (1:5) or richer (needed for the 0.07 N/mm$^2$ flexural tension allowance); practical choice 5 N/mm$^2$ bricks with 1:5 mortar. Stresses at mid-height: $\sigma_{max}=0.291$ and $\sigma_{min}=-0.007$ N/mm$^2$; slenderness ratio 15.2.

Under earthquake or wind, unreinforced masonry buildings show characteristic damage because the material is heavy, brittle and weak in tension.

1. Diagonal (X-shaped) shear cracks in walls and piers parallel to the force; stair-stepped through joints or through bricks.
2. Horizontal sliding cracks along bed joints near the base, plinth or above the lintel.
3. Vertical cracks at corners and wall junctions where walls separate because of poor connection and out-of-plane bending.
4. Out-of-plane collapse of walls, gables and parapets (overturning, bulging, mid-height horizontal cracking).
5. Cracks around openings: diagonal cracks from corners of doors and windows; piers between openings fail in shear.
6. Delamination of wall leaves in thick two-leaf or rubble walls.
7. Roof/floor damage and pounding: slab pulling out from supporting walls; roof falls when walls collapse.

 Diagonal cracks      Corner crack         Opening cracks
 +--------+           |   |                +----------+
 |\      /|         --+   +--              | /|  []  |\ |
 | \    / |           crack                |/ |      | \|
 +--------+                                +----------+

These damages can be reduced by bands, corner reinforcement, proper roof-wall anchorage and good mortar.

Data and assumptions

• The wall is fixed at top and bottom, so $h_{ef}=0.75H=2.25$ m (full restraint, Table 4 Sl. 1). No vertical load other than self weight is given. The pressure is taken as stated, 16 N/m$^2$ (0.016 kN/m$^2$) [reading: a very small wind pressure, so stresses are tiny and slenderness controls the thickness].
• End moment of a wall fixed at both ends under uniform load: $M=wH^2/12$ at each end (sagging $wH^2/24$ at mid-height). Section checked at the base, where self weight is the largest.
• Design by IS 1905:1987 cl. 5.4.1, 5.4.2, 5.5.2, 5.5.3 (NBC 109 follows the same method). Unit weight of masonry 19 kN/m$^3$. Per metre run of wall.

  ======+===========  fixed (top)
        |<-- wind w
        |  wall H=3 m
        |  M = wH^2/12 at ends
  ======+===========  fixed (bottom)

Step 1: Wind moment

$$\begin{aligned} w &= 0.016\ \text{kN/m (per m of wall)}\\ M &= 0.016\times3.0^2/12 = 0.012\ \text{kNm/m} \end{aligned}$$

Step 2: Trial thickness 115 mm, load and slenderness

$$\begin{aligned} \text{Self weight of wall} &= 0.115\times3.00\times19 = 6.56\ \text{kN/m}\\ N\ (\text{at base}) &= 6.56\ \text{kN/m}\\ h_{ef} &= 0.75\times3.0=2.25\ \text{m} \\ SR &= \frac{2.250}{0.115} = 19.57\ (<27) \end{aligned}$$

Step 3: Stresses at the base section

$$\begin{aligned} \sigma_a &= \frac{N}{A}=\frac{6.56\times10^3}{115\times10^3} = 0.0570\ \text{N/mm}^2\\ \sigma_b &= \frac{6M}{t^2} = \frac{6\times0.012\times10^6}{1000\times115^2} = 0.0054\ \text{N/mm}^2\\ \sigma_{max} &= \sigma_a+\sigma_b = 0.0624,\qquad \sigma_{min} = \sigma_a-\sigma_b = 0.0516\ \text{N/mm}^2 \end{aligned}$$

$\sigma_{min}>0$: the whole section is in compression, no tension (the resultant eccentricity $e=M/N=0.0018$ m, $e/t=0.016<1/6$).

Step 4: Required basic compressive stress

Resultant eccentricity ratio $e/t = M/(Nt) = 0.016$. From Table 9: $k_s=0.619$ ($SR=19.57$). $k_a=1$, $k_p=1$.

$$f_b \ge \frac{\sigma_a}{k_sk_ak_p\times1.00} = \frac{0.0570}{0.619\times1.00} = 0.092\ \text{N/mm}^2$$

Step 5: Brick and mortar (IS 1905 Table 8)

Answer: a half-brick wall of 115 mm is sufficient for stress (SR = 19.6 < 27), but for stability and practical robustness adopt a 230 mm (one-brick) wall; bricks of 3.5 N/mm$^2$ with cement mortar M1 (1:5). Stresses are very small: $\sigma_{max}=0.0624$, $\sigma_{min}=0.0516$ N/mm$^2$, far below the permissible values.

Data and assumptions

• Wall 3.0 m high, free at the top and fixed at the base: a cantilever (free-standing) wall. Uniform wind pressure taken as stated, 12 N/m$^2$ (0.012 kN/m$^2$) [reading: given value is very small, so stability is easily satisfied; if the wind pressure is larger, repeat with the same steps].
• Design by IS 1905:1987 cl. 5.5.2.1 and Table 11, Table 4 Sl. 4. Per metre run; masonry 19 kN/m$^3$; trial thickness 230 mm.

        ^ free top
        |<-- w = 12 N/m2
        |  wall H = 3.0 m
        |  M = wH^2/2 at base
  ======+======  fixed base

Step 1: Slenderness (cl. 4.3, Table 4 Sl. 4)

For a free top and a fixed base $h_{ef}=1.5H=4.5$ m, so $SR = 4.5/0.23 = 19.6<27$ (cement mortar), but the slenderness check alone would give $t\ge4.5/27=0.167$ m.

Step 2: Bending and stresses

$$\begin{aligned} M &= \frac{wH^2}{2} = \frac{0.012\times3^2}{2} = 0.054\ \text{kNm/m}\\ \text{Self weight} &= 0.23\times3\times19 = 13.11\ \text{kN/m}\\ \sigma_a &= \frac{13.11\times10^3}{230\times10^3} = 0.0570\ \text{N/mm}^2\\ \sigma_b &= \frac{6M}{t^2} = \frac{6\times0.054\times10^6}{1000\times230^2} = 0.0061\ \text{N/mm}^2\\ \sigma_{max} &= 0.0631,\ \ \sigma_{min} = 0.0509\ \text{N/mm}^2 \end{aligned}$$

No tension at the base ($\sigma_{min}>0$), and the compressive stress is far below the allowable value for the lowest brick grade.

Step 3: Stability (cl. 4.2.2.4 a)

Free-standing wall: stability moment must be at least 1.5 times the overturning moment:

$$M_{stab}=W\frac{t}{2} = 13.11\times0.115 = 1.51\ \text{kNm/m}\ \gg\ 1.5\times0.054 = 0.081\ \text{kNm/m}$$

Step 4: Table 11 check

IS 1905 Table 11 allows a free-standing wall without calculation (seismic zones I and II, mortar not leaner than M1) when $H/t \le 10$ for pressures up to 285 N/m$^2$: $t\ge3.0/10=0.30$ m. Our wall satisfies stress and stability by calculation; if the simplified table is preferred, adopt 345 mm (one and a half brick).

Answer: the wall is safe with 230 mm thickness by calculation ($\sigma_{max}=0.063$ N/mm$^2$, stability factor 28); bricks of at least 3.5 N/mm$^2$ in cement mortar M1 (1:5), practical 5 N/mm$^2$. If the wall is built without calculation use $t=345$ mm (Table 11, $H/t\le10$). Provide a damp-proof course and a coping to protect the top.

In-plane behaviour (force parallel to the wall)

The wall acts as a shear wall and is stiff and strong. Failure modes:

• Sliding shear: horizontal crack along a bed joint when vertical load is small.
• Diagonal shear (X-crack): diagonal tension cracking, through joints (stair-step) or units.
• Rocking / toe crushing: slender wall or pier rotates about its toe, horizontal cracks at heel and crushing at the toe.

Out-of-plane behaviour (force perpendicular to the wall)

The wall acts as a thin plate or cantilever in bending; it has little tensile capacity and fails suddenly:

• Overturning / rocking of parapets, gables and free-standing walls about the base.
• Vertical bending: horizontal crack at mid-height between floor and roof.
• Horizontal bending: vertical cracks and collapse at wall junctions or between cross walls.
• Two-way bending: diagonal cracks from corners when supported on all edges.

 In-plane              Out-of-plane
 +--------+            ---|||---> force
 |\  X  /|             |wall|  crack at mid-height
 | \  / |  F ->        |~~~~|
 +--------+            ====== base

Remedial and preventive measures

1. RCC or timber bands (plinth, lintel, roof) to tie walls and give box action.
2. Vertical reinforcement at corners, junctions and jambs, anchored in foundation and roof band.
3. Limit height-to-thickness and length-to-thickness ratios; provide cross walls, buttresses or pilasters.
4. Small openings (not more than one-third of wall length), located away from corners, with lintel band.
5. Rigid diaphragm (RCC slab) or properly braced roof anchored to walls.
6. Good quality brick and mortar, proper bond and curing.
7. Repair and retrofit of existing buildings: grouting of cracks, steel mesh with plaster, ferrocement jacketing, installation of bands.

In-plane and out-of-plane behaviour

• In-plane: force acts parallel to the wall; the wall is a shear wall that develops its shear strength and fails by sliding, diagonal tension (X-cracks) or rocking and toe crushing. The behaviour is relatively stiff and strong.
• Out-of-plane: force acts perpendicular to the wall; the wall bends as a vertical or two-way plate. It has little tensile strength, so it fails by overturning, horizontal mid-height cracking, or vertical corner cracks. This is the weaker and more dangerous direction.

Ductile behaviour

Ductility is the ability to deform plastically without losing strength, giving warning and absorbing energy.

• Unreinforced masonry (URM): brittle. After the first cracks the strength falls sharply; very small ductility (displacement ductility about 1 to 2) and low energy dissipation, so collapse is sudden. Some limited ductility is shown by rocking walls.
• Reinforced masonry (RM) / confined masonry: steel bars in cores or RCC bands and tie columns carry tension and shear after the masonry cracks. The load-displacement curve has a long plateau with large deformation (ductility about 3 to 5), cracks are spread, and energy is dissipated.

 Load
  |      ____ RM (ductile plateau)
  |    /
  |   /  \
  |  / URM \_____ sudden drop
  | /
  +------------------- Displacement

Hence reinforcing or confining the masonry converts the sudden brittle failure into a ductile one and greatly improves earthquake performance.

Reading of the question: the end of the question is cut off in the scan, so it is read as: "List the elements of masonry structure resisting lateral loads. Describe the stepwise load path (design procedure) for lateral loads."

Elements resisting lateral loads

1. Shear walls: load-bearing masonry walls parallel to the lateral force (in-plane shear and rocking resistance).
2. Roof and floor diaphragms: rigid horizontal slabs that distribute the force to shear walls.
3. Cross (transverse) walls, piers and buttresses: support walls against out-of-plane bending.
4. RCC/timber bands and tie columns: give box action and tie walls together.
5. Foundation and plinth band.

Stepwise procedure

1. Find the lateral force: seismic base shear $V_B = A_h W$ (IS 1893 / NBC 105) or wind load (IS 875 Part 3) on the walls and roof.
2. Distribute the force to the floors and roof in proportion to mass and height.
3. Transfer the force from each diaphragm to the shear walls in proportion to their stiffness (rigid diaphragm) or tributary area (flexible).
4. Check each shear wall for in-plane shear stress $\tau = V/(L\,t)$ against the permissible shear stress $f_s = 0.1 + f_d/6 \le 0.5$ N/mm$^2$ (IS 1905 cl. 5.4.3).
5. Check overturning and the extreme-fibre stresses (axial plus bending), with no tension in masonry.
6. Check each wall for out-of-plane bending using vertical or horizontal spans, flexural tension 0.07 N/mm$^2$ for M1 mortar.
7. Provide bands, corner steel and connections to complete the load path to the foundation.

 Force -> Diaphragm -> Shear walls -> Foundation -> Ground
             (cross walls support walls out-of-plane)

The seismic performance of a masonry building is improved by:

1. Box action: tie walls together with horizontal RCC bands (plinth, lintel, roof) and vertical bars at corners and junctions (IS 4326, NBC 109).
2. Rigid roof/floor diaphragm of RCC slab, anchored to the walls.
3. Limit openings (small, symmetrically placed, not near corners) and limit wall height and length.
4. Regular, symmetric plan and good cross walls.
5. Better materials and workmanship: strong brick, richer mortar (1:4 or 1:6), good bond and curing.
6. Confined or reinforced masonry to give ductility.
7. For existing buildings, retrofit by ferrocement or wire-mesh plaster jacketing and added bands.

Under gravity (vertical) load

• Compression failure: crushing of units or splitting; vertical cracks (tensile splitting of bricks) when load is high or units/mortar are weak.
• Buckling of slender walls and columns: high slenderness ratio, large eccentricity.
• Local bearing failure under concentrated loads from beams and lintels; no bed block.
• Differential settlement causes diagonal or vertical cracks.
• Deficiencies: poor workmanship, voids in joints, raked joints, weak mortar, unfilled perpends, thick joints, no bond.

Under lateral load

• In-plane: diagonal shear, sliding shear and rocking failures.
• Out-of-plane: overturning, mid-height bending cracks, corner separation.
• Deficiencies: lack of bands and ties, heavy flexible roofs not tied, large openings near corners, weak connection between walls, no cross walls, slender walls, soft mortar and poor-quality bricks.

Result

Brittle, sudden failure with little warning. Measures: bands, corner bars, controlled slenderness and openings, good materials and workmanship.

Wall without opening

A solid wall acts as a monolithic shear wall (in-plane) or plate (out-of-plane). The lateral force is carried by one wide section and the stiffness and strength are high. Typical failure: single large diagonal crack or sliding or rocking at the base. Capacity is calculated on the full length $L$: $V = f_s\,L\,t$.

Wall with openings

Doors and windows break the wall into piers (vertical strips between openings) and spandrels (horizontal strips above and below).

• Stiffness and strength fall sharply; the lateral force is shared among piers in proportion to their stiffness (stiffness $\propto$ $t$ and falls with $(h/L)^3$ for slender piers).
• Stress concentrations at corners of openings cause diagonal cracks radiating from the corners.
• Short, stocky piers fail in shear; slender piers fail in rocking (flexure); spandrels crack in shear and flexure.
• Out-of-plane: weaker section at the opening, cracks run from the corners of the opening to the edge; the wall can fail at lintel or sill levels.

 No opening          With opening
 +---------+         +----+-----+----+
 |  \   /  |         | pier|open| pier|
 |   \ /   |   ->    | /   |    |  \  |
 +---------+         +----+-----+----+
  single X-crack      cracks from the corners

Hence IS 4326 limits total opening length (not more than 50 percent of wall length for one storey, 42 percent for two storeys, 33 percent for three storeys), the distance of an opening from the inside corner (not less than 600 mm) and the width of the pier between openings (not less than 340 mm), and require lintel bands and sill bands, or reinforcement around openings.

Causes of out-of-plane failure

Out-of-plane failure occurs when the lateral (inertia) force acts perpendicular to the wall plane and the wall bends. The main causes are:

1. Low tensile and flexural strength of masonry and weak mortar bond.
2. High slenderness: large height-to-thickness or length-to-thickness ratio.
3. No lateral support: missing or weak connection to cross walls, floors and roof; flexible roof not acting as diaphragm.
4. No bands or ties at floor and roof; weak connection at wall corners.
5. Large openings and long unsupported spans, parapets and gables unbraced.
6. Heavy roof or floor that increases inertia force; poor construction (poor bond, thin walls).
7. Cantilever or free-standing parts (parapets, boundary walls).

Two typical crack patterns

1. Horizontal crack at mid-height (or base) of the wall: the wall bends vertically between floor and roof supports and breaks like a beam; parapets overturn with a horizontal crack at the base.
2. Vertical crack at wall corners or junctions (corner separation): the wall bends horizontally between cross walls, and the two walls separate and fall outwards. In walls supported on four sides diagonal cracks from corners to the centre appear.

 Horizontal crack           Corner crack
  |    |                     |   |
  |~~~~|  <- mid-height     -+   +-
  |    |                    crack

A masonry wall under lateral load (wind, earthquake or earth pressure) is designed by working-stress method as per IS 1905 cl. 5.5.2 and 5.5.3.

Design steps

1. Establish the lateral load: wind pressure from IS 875 (Part 3) (as design wind pressure $p_z$) or seismic force; load per metre run $w$.
2. Support conditions: find the structural system: cantilever (free-standing), simply supported between floor and roof (anchored), propped cantilever (fixed base, tied top), or panel supported on 3 or 4 edges (App. D).
3. Bending moment: cantilever $M=wH^2/2$; simply supported $M=wH^2/8$; fixed both ends $M=wH^2/12$; propped cantilever $M=wH^2/8$ at the fixed end.
4. Vertical load: add the vertical loads and self weight at the critical section.
5. Effective height, slenderness ratio (Tables 4, 5, 7) and eccentricity $e=M/N$ (or load eccentricity).
6. Stresses: $\sigma = N/A \pm 6M/(t^2)$ per metre; check no tension, or tension limited to 0.07 N/mm$^2$ (vertical bending, M1 mortar, cl. 5.4.2) and 0.14 N/mm$^2$ along bed joints.
7. Compression check: $\sigma \le f_b k_s k_a k_p$, with 25 percent increase if $1/24<e/t\le1/6$ (cl. 5.4.1.4).
8. Free-standing wall: stability moment $\ge 1.5\times$ overturning moment, or $H/t$ from Table 11.
9. Shear check for in-plane forces: $f_s=0.1+f_d/6 \le 0.5$ N/mm$^2$ (cl. 5.4.3).
10. Detail: anchors, damp-proof course, bands, reinforcement if the masonry cannot take tension.

 Cantilever      Simply supported     Fixed-propped
   ^ free           anchor                anchor
   | w           ->|w               ->|w
   |              |                    |
   =====          anchor              =====
 M=wH^2/2        M=wH^2/8           M=wH^2/8 (base)