One-way slabs

Slabs make up most of the concrete in a building. A slab supported on two opposite sides, or on beams much farther apart in one direction than the other, bends mainly in one direction and is called a one-way slab. It is designed as a series of wide, shallow beams. This lesson covers the analysis of continuous one-way slabs by moment coefficients, the design of a one-metre or one-foot strip, and the detailing rules that make slabs durable, with the garage parking level as the example.

One-way and two-way action

A rectangular panel supported on all four edges bends in both directions, but the load goes mostly to the closer supports, because a shorter span is much stiffer. For a panel whose long side is more than about twice its short side, nearly all the load is carried in the short direction, and the panel can be designed as one-way. Panels closer to square are two-way slabs, covered in lesson 9.

The upper parking level of the garage is a 200 mm slab spanning between beams at 6.0 m centres. The beams span 9.0 m between the garage frames, so each panel is 9.0 m by 6.0 m, a ratio of 1.5. Strictly that is two-way, but because the frames themselves are stiffer than the slab and the beams carry the slab load to them, the team designs the slab one-way between the beams, the common practice for beam-and-slab parking floors, and adds the minimum steel in the long direction.

A one-way slab is designed by taking a strip one unit wide, spanning between the supports, and treating it as a rectangular beam of that width. Because the slab is continuous over many beams, the strip is a continuous beam with many spans.

Thickness

Slab thickness is usually set by deflection, by fire resistance, or by cover requirements in exposed slabs. The span-to-depth limits of lesson 5 give \(l_n/20\) for simple spans, \(l_n/24\) for one end continuous and \(l_n/28\) for both ends continuous. For the garage, with a clear span of 6.0 − 0.45 = 5.55 m (the team rounds to 5.6 m), \(l_n/28 = 200\) mm for interior spans and \(l_n/24 = 233\) mm for end spans. Owen uses 200 mm throughout and checks the end span deflection by calculation, because the larger 60 mm top cover of a parking slab reduces the effective depth.

Fire resistance ratings in the building code require minimum slab thicknesses (about 100 to 150 mm, or 4 to 6 in., for one to two hours depending on aggregate type) and minimum cover to the bottom bars. In practice most floor slabs in buildings are 150 to 250 mm (6 to 10 in.) thick.

Moments and shears by coefficients

For continuous beams and one-way slabs that meet certain conditions, A23.3 permits moments and shears from tabulated coefficients instead of a full analysis. The conditions are that there are two or more spans, adjacent spans differ by no more than 20 per cent, the loads are uniformly distributed, the live load is not large compared with the dead load, and the members are prismatic. The coefficients approximate the envelope of moments from patterned live loads.

The coefficients are a little conservative compared with an elastic analysis, because each one is the worst value at that location from different live load patterns. The first interior support is the most heavily loaded location, with a negative moment about 40 per cent greater than the positive moment in an interior span. When the conditions are not met, for example with a long cantilever or very unequal spans, the slab is analysed as a continuous beam with patterned live load, using the methods of CIV-321.

Reinforcement requirements

Slabs follow their own minimum steel rules. A23.3 requires at least \(0.002A_g\) of reinforcement in each direction, which serves as both the minimum flexural steel and the shrinkage and temperature steel. In the main direction the spacing may not exceed 3 times the slab thickness or 500 mm. The shrinkage and temperature steel, perpendicular to the span, may not be spaced more than 5 times the thickness or 500 mm.

Slabs are usually reinforced with the same bar size throughout, spaced in multiples of 25 mm or 1/2 in., and with spacing rounded down. Top bars over supports are extended into the spans beyond the points of inflection, commonly to about a third of the clear span past the face of each support, unless an analysis shows they can be shorter, and at least half of the bottom bars are continued into the supports.

Shear rarely governs slabs. A slab without stirrups resists \(V_c = \phi_c\lambda\beta\sqrt{f'_c}b_wd_v\) with \(\beta = 230/(1000 + d_v)\), and its thickness is normally set by deflection long before shear matters. Stirrups are impractical in slabs thinner than about 250 mm.

Crack control in an exposed slab

Parking slabs crack where the top surface is in tension over the beams, which is also where salt-laden water collects. Owen checks the crack control parameter for the top bars with \(f_s = 0.6f_y = 240\) MPa and \(d_c = 60 + 8 = 68\) mm. The effective tension area per bar is \(2d_c s\). At 250 mm spacing, \(z = 240\sqrt[3]{68 \times 2 \times 68 \times 250} = 31\,700\) N/mm, above the 30 000 N/mm interior limit, even though the covered deck is protected by a membrane. Closing the spacing to 200 mm gives \(z = 29\,500\) N/mm. The larger cover needed for durability makes cracks wider at the surface, so crack control, not strength, sets the top steel.

import math
# Garage one-way slab, CSA A23.3: moment coefficients and a 1 m design strip
fc, fy, phic, phis = 35.0, 400.0, 0.65, 0.85
h, ln = 200.0, 5.6                       # mm, clear span m (beams at 6.0 m, webs 400 mm)
wf = 1.25 * (0.2 * 23.5 + 0.5) + 1.5 * 2.4
a1 = max(0.85 - 0.0015 * fc, 0.67)
d_top, d_bot = h - 60 - 8, h - 30 - 8    # 60 mm top cover (parking exposure), 15M bars
print(f"wf = {wf:.2f} kPa, thickness check ln/28 = {ln*1000/28:.0f} mm (h = {h:.0f})")

def As_req(Mf, d, b=1000.0):
    k = (phis * fy) ** 2 / (2 * a1 * phic * fc * b)
    return (phis * fy * d - math.sqrt((phis * fy * d) ** 2 - 4 * k * Mf * 1e6)) / (2 * k)

As_min = 0.002 * 1000 * h                # also the shrinkage and temperature steel
s_lim = min(3 * h, 500)
cases = [("exterior support, interior face (top)", 1 / 24, d_top),
         ("end span midspan (bottom)",           1 / 14, d_bot),
         ("first interior support (top)",       1 / 10, d_top),
         ("interior span midspan (bottom)",      1 / 16, d_bot),
         ("other interior supports (top)",       1 / 11, d_top)]
for name, c, d in cases:
    Mf = c * wf * ln**2
    As = max(As_req(Mf, d), As_min)
    s = min(200 * 1000 / As, s_lim)
    s = 25 * math.floor(s / 25)
    print(f"{name:38s} Mf = {Mf:5.1f} kN.m/m  As = {As:4.0f} mm2/m  15M @ {s:.0f}")
print(f"shrinkage and temperature (transverse): {As_min:.0f} mm2/m, 15M @ {min(200e3/As_min, 5*h, 500):.0f} max")

# One-way shear at the first interior support, no stirrups
Vf = 1.15 * wf * ln / 2
dv = max(0.9 * d_top, 0.72 * h)
beta = 230 / (1000 + dv)
Vc = phic * beta * math.sqrt(fc) * 1000 * dv / 1000
print(f"shear: Vf = {Vf:.1f} kN/m, dv = {dv:.0f} mm, beta = {beta:.3f}, Vc = {Vc:.0f} kN/m")

# Crack control for the top bars over the supports (fs = 0.6 fy, interior limit, membrane-protected deck)
fs = 0.6 * fy
dc = 60 + 16.0 / 2
for s in (275, 250, 200):
    A = 2 * dc * s                        # effective tension area per bar in a slab
    z = fs * (dc * A) ** (1 / 3)
    print(f"top 15M @ {s}: z = {z:.0f} N/mm -> {'OK' if z <= 30000 else 'too wide'}")

wf = 10.10 kPa, thickness check ln/28 = 200 mm (h = 200)
exterior support, interior face (top)  Mf =  13.2 kN.m/m  As =  400 mm2/m  15M @ 500
end span midspan (bottom)              Mf =  22.6 kN.m/m  As =  421 mm2/m  15M @ 475
first interior support (top)           Mf =  31.7 kN.m/m  As =  745 mm2/m  15M @ 250
interior span midspan (bottom)         Mf =  19.8 kN.m/m  As =  400 mm2/m  15M @ 500
other interior supports (top)          Mf =  28.8 kN.m/m  As =  674 mm2/m  15M @ 275
shrinkage and temperature (transverse): 400 mm2/m, 15M @ 500 max
shear: Vf = 32.5 kN/m, dv = 144 mm, beta = 0.201, Vc = 111 kN/m
top 15M @ 275: z = 32760 N/mm -> too wide
top 15M @ 250: z = 31735 N/mm -> too wide
top 15M @ 200: z = 29460 N/mm -> OK

Owen adds three details that the analysis does not show. Top bars over the beams extend 1.9 m (about a third of the clear span) past each face so that they cover the negative moment region with margin. Transverse top bars are added over the slab, not just at the supports, to control shrinkage cracking at the exposed surface. And a drainage slope of at least 1.5 per cent is formed into the top of the slab, because ponded water and salt are the main enemy of parking structures.

Other one-way systems

• One-way joist (ribbed) slabs use closely spaced ribs with a thin top slab, formed with reusable pans. They save concrete for longer spans and light loads; the ribs are designed as T-beams and the top slab as a short one-way slab between ribs.
• Slabs on grade, such as the garage ground floor, are supported directly on soil and are designed for wheel and rack loads, joint layout and shrinkage rather than by these methods.
• Precast hollow-core planks, with a cast-in-place topping, span one way and are common for parking levels when speed of erection matters.
• Composite steel deck slabs act one-way and are used on steel-framed floors.