Masonry walls in flexure

Most masonry walls in low-rise buildings span vertically between a floor slab or footing and a roof diaphragm, and their main job, apart from carrying the roof, is to resist wind pushing on their face. A wall loaded this way bends like a one-way slab. This lesson designs the pump house walls for out-of-plane wind by the strength design method of TMS 402, first as unreinforced masonry, which turns out to be too weak, and then as reinforced masonry with bars in grouted cores. Lesson 9 adds the roof load and the second-order effects that come with slender walls.

The pump house walls

The pump house is 32 ft by 24 ft in plan with 8 in. CMU walls 16 ft high from the slab to the roof bearing. The roof is a steel deck on open-web steel joists spanning the 24 ft direction, and the deck acts as a diaphragm that supports the top of each wall. The bottom of the wall is doweled into the slab. For out-of-plane loading, each vertical strip of wall is a simply supported beam 16 ft long. The strength-level wind on the wall, combining internal and external pressure for this exposure, is 30 psf, so

Unreinforced masonry

An unreinforced wall resists bending through the modulus of rupture of the masonry normal to the bed joints, \(f_r\), which TMS 402 gives in a table by mortar type and grouting. For hollow units in Type S portland cement-lime mortar it is of the order of 84 psi, a small and variable number. The design strength is \(\phi M_n = \phi f_r S_n\) with \(\phi = 0.60\), plus a contribution from axial load, which is neglected here.

# Lesson 8: pump house wall, out-of-plane wind, 8 in. CMU, span 16 ft (TMS 402 strength design)
t, fs_t, h = 7.625, 1.25, 16.0
fm, fy = 2000.0, 60000.0              # psi
wu = 1.0 * 30.0                        # psf, strength-level wind on the wall
Mu = wu * h**2 / 8                     # lb-ft/ft
print(f"wu = {wu:.0f} psf, Mu = {Mu:.0f} lb-ft/ft")

# Unreinforced, ungrouted: face shells only, modulus of rupture normal to bed joints
y = t / 2 - fs_t / 2
I = 2 * (12 * fs_t**3 / 12 + 12 * fs_t * y**2)
S = I / (t / 2)
fr = 84.0                              # psi, hollow units, type S PCL mortar (approx.)
phiMn_urm = 0.60 * fr * S / 12
print(f"Unreinforced: S = {S:.1f} in3/ft, phi Mn = {phiMn_urm:.0f} lb-ft/ft -> {'OK' if phiMn_urm >= Mu else 'not enough'}")

d = t / 2
for bar, Ab, s in [("#4", 0.20, 32), ("#4", 0.20, 24), ("#5", 0.31, 48), ("#5", 0.31, 32)]:
    As = Ab * 12 / s
    b_eff = min(s, 6 * t)                         # compression width per bar
    a = Ab * fy / (0.80 * fm * b_eff)
    phiMn = 0.9 * As * fy * (d - a / 2) / 12
    flag = "within face shell" if a <= fs_t else "exceeds face shell"
    print(f"{bar} @ {s} in.: As = {As:.3f} in2/ft, a = {a:.2f} in. ({flag}), phi Mn = {phiMn:.0f} lb-ft/ft, Mu/phiMn = {Mu/phiMn:.2f}")

fr_g = 106.0                           # psi, partially grouted, approx. (between hollow and fully grouted)
print(f"Cracking moment with grouted cells (fr about {fr_g:.0f} psi) = {fr_g*S/12:.0f} lb-ft/ft")

wu = 30 psf, Mu = 960 lb-ft/ft
Unreinforced: S = 81.0 in3/ft, phi Mn = 340 lb-ft/ft -> not enough
#4 @ 32 in.: As = 0.075 in2/ft, a = 0.23 in. (within face shell), phi Mn = 1247 lb-ft/ft, Mu/phiMn = 0.77
#4 @ 24 in.: As = 0.100 in2/ft, a = 0.31 in. (within face shell), phi Mn = 1645 lb-ft/ft, Mu/phiMn = 0.58
#5 @ 48 in.: As = 0.077 in2/ft, a = 0.25 in. (within face shell), phi Mn = 1285 lb-ft/ft, Mu/phiMn = 0.75
#5 @ 32 in.: As = 0.116 in2/ft, a = 0.36 in. (within face shell), phi Mn = 1899 lb-ft/ft, Mu/phiMn = 0.51
Cracking moment with grouted cells (fr about 106 psi) = 715 lb-ft/ft

The face shells give a section modulus of 81.0 in³ per foot, and the unreinforced wall resists only 340 lb·ft/ft, about a third of the 960 lb·ft/ft required. Unreinforced block walls 16 ft high are therefore limited to sheltered interior walls or lower heights; the pump house walls must be reinforced.

Reinforced masonry in flexure

Reinforced masonry is designed on the same assumptions as reinforced concrete: plane sections remain plane, the masonry carries no tension, the steel yields, and the masonry compression is represented by an equivalent rectangular stress block. The differences are in the numbers: the masonry crushing strain, the stress block intensity and the effective width of the compression zone.

In TMS 402, the stress block has an intensity of \(0.80f'_m\) over a depth \(a = 0.80c\), and the maximum usable strain is 0.0025 for concrete masonry and 0.0035 for clay masonry. With one layer of steel at depth \(d\), equilibrium and the moment about the compression resultant give

For a bar centered in an 8 in. wall, \(d = 3.81\) in. In a partially grouted wall the compression zone is the face shell plus the grouted core, and its width per bar is limited to the bar spacing, six times the wall thickness and 72 in.; if the stress block depth \(a\) falls within the 1-1/4 in. face shell, the section acts as a rectangle of that width. If \(a\) extends past the face shell, the ungrouted cores must be allowed for, as with a T-beam.

Limits on reinforcement

Like concrete, reinforced masonry should fail by yielding of the steel, not crushing of the masonry. TMS 402 limits the reinforcement by requiring the steel to reach a multiple of its yield strain when the masonry reaches its maximum usable strain, with the multiple depending on the seismic design category and the type of member, and it sets minimum reinforcement through prescriptive seismic requirements. The design strength must also exceed the cracking moment, so that the wall does not fail suddenly the first time it cracks; the pump house wall has \(\phi M_n = 1899\) lb·ft/ft against an estimated cracking moment of about 715 lb·ft/ft.

Bars in masonry are also limited in size relative to the cell. A bar should not exceed about one quarter of the least clear dimension of the cell, and the total steel area in a cell is limited, so there is room for the grout to flow around the bar and bond to it. Lap splices are long, often 50 to 60 bar diameters or more, because the bond strength of grout in a block cell is lower than that of concrete.

Walls spanning horizontally and in two directions

A wall supported by columns or cross walls at its ends can span horizontally, relying on tension parallel to the bed joints in unreinforced work or on horizontal joint reinforcement and bond beams in reinforced work. Walls supported on three or four edges span in two directions, and design methods based on yield line analysis allow the load to be shared between the two spans, which can make unreinforced walls workable at heights that would fail as one-way vertical spans. The pump house corners act this way near the ends of each wall, but the design conservatively treats each wall as spanning vertically.

Deflection and the control of cracking

A reinforced wall that cracks under wind becomes much more flexible, because only the steel and the compression face shell resist bending. For walls that support brittle finishes or glazing, the deflection under service wind is checked. Lesson 9 computes it with the cracked moment of inertia and shows that the pump house walls deflect only a few millimeters under service load. Horizontal cracks in the bed joints open and close with each storm; they are harmless in a reinforced wall but would admit water in an unreinforced one.

Supporting the top of the wall

The simply supported strip assumed in design needs real supports. At the bottom, dowels from the slab lap with the vertical bars and the slab resists the horizontal reaction. At the top, the wall reaction, half the wind on the wall, must pass into the roof diaphragm. For the pump house, a grouted and reinforced bond beam runs around the top of the walls, steel angles bolted to the bond beam carry the deck edge, and anchor bolts at close spacing transfer the reaction in both directions, so that suction on the wall cannot pull it away from the roof. The bond beam also acts as the diaphragm chord and ties the corners together. Many failures of masonry walls in storms start at this connection rather than in the wall itself.

Basement and retaining walls

Masonry walls below grade, such as the walls of a small pump chamber or a basement, resist soil pressure as well as the load from above. Soil pressure increases with depth, so the maximum moment is lower in the wall than for wind and the reinforcement is placed toward the tension face, the inside, for larger \(d\). Below-grade walls are usually fully grouted for water tightness and durability, and they need drainage behind them so that water pressure does not build up. The lift station below the pump house was designed in reinforced concrete in CIV-341, so the masonry starts at the slab.