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-meter 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 an 8 in. slab spanning between beams at 20 ft centers. The beams span 30 ft between the garage frames, so each panel is 30 ft by 20 ft, 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 \(l/20\) for simple spans, \(l/24\) for one end continuous and \(l/28\) for both ends continuous. For the garage, with a clear span of 20 − 1.33 = 18.67 ft, \(l/28 = 8.0\) in. for interior spans and \(l/24 = 9.3\) in. for end spans. Owen uses 8 in. throughout and checks the end span deflection by calculation, because the 2 in. 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, ACI 318 permits moments and shears from tabulated coefficients instead of a full analysis. The conditions are that the members are prismatic, there are two or more spans, the longer of two adjacent spans is no more than 1.2 times the shorter, the loads are uniformly distributed, and the unfactored live load is no more than three times the unfactored dead load. 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 percent 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 analyzed as a continuous beam with patterned live load, using the methods of CIV-321.

Reinforcement requirements

Slabs follow their own minimum steel rules. ACI 318 requires at least \(0.0018A_g\) of Grade 60 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 18 in. The shrinkage and temperature steel, perpendicular to the span, may not be spaced more than 5 times the thickness or 18 in.

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. ACI 318-19 computes the strength of a slab without stirrups from the size-effect form, \(\phi V_c = 0.75 \times 8\lambda_s\lambda(\rho_w)^{1/3}\sqrt{f'_c}\,b_wd\); for slabs thinner than about 10 in. \(\lambda_s = 1\), and with typical steel ratios the result is 25 to 35 percent below the older \(2\sqrt{f'_c}\) value. Even so, its thickness is normally set by deflection long before shear matters. Stirrups are impractical in thin slabs.

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 bar spacing for the top bars with \(f_s = \tfrac{2}{3}f_y = 40{,}000\) psi and clear cover \(c_c = 2.0\) in.: \(s \le 15(1.0) - 2.5(2.0) = 10\) in. The 13.5 in. spacing from the strength design is too wide, so the top bars over all interior supports become #5 at 10 in. 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, ACI 318: moment coefficients and a 12 in. design strip
fc, fy, phi = 5000.0, 60000.0, 0.90
h, ln = 8.0, 20 - 16 / 12                 # in., clear span ft (beams at 20 ft, 16 in. webs)
wu = 1.2 * (100 + 10) + 1.6 * 40          # psf
d_top, d_bot = h - 2.0 - 0.3125, h - 0.75 - 0.3125   # 2 in. top cover (parking exposure), #5 bars
print(f"wu = {wu:.0f} psf, thickness check ln/28 = {ln*12/28:.1f} in. (h = {h:.0f})")

def As_req(Mu, d, b=12.0):                 # Mu in lb-in. per 12 in. strip
    Rn = Mu / (phi * b * d * d)
    return 0.85 * fc / fy * (1 - math.sqrt(1 - 2 * Rn / (0.85 * fc))) * b * d

As_min = 0.0018 * 12 * h                  # flexural minimum and shrinkage steel, Grade 60
s_lim = min(3 * h, 18)
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:
    Mu = c * wu * ln**2 * 12              # lb-in. per ft
    As = max(As_req(Mu, d), As_min)
    s = min(0.31 * 12 / As, s_lim)
    s = math.floor(s * 2) / 2
    print(f"{name:38s} Mu = {Mu/12000:4.2f} kip-ft/ft  As = {As:.3f} in2/ft  #5 @ {s:.1f} in.")
print(f"shrinkage and temperature (transverse): {As_min:.3f} in2/ft, #4 @ {min(0.20*12/As_min, 5*h, 18):.1f} in. max")

# One-way shear at the first interior support, no stirrups (size-effect form, #5 @ 10 in. top)
Vu = 1.15 * wu * ln / 2
rho_w = 0.31 * 12 / 10 / (12 * d_top)
lam_s = min(1.0, math.sqrt(2 / (1 + d_top / 10)))
phiVc = 0.75 * 8 * lam_s * rho_w ** (1 / 3) * math.sqrt(fc) * 12 * d_top
print(f"shear: Vu = {Vu/1000:.2f} kips/ft, rho_w = {rho_w:.4f}, lambda_s = {lam_s:.2f}, phiVc = {phiVc/1000:.2f} kips/ft")

# Crack control for the top bars over the supports (fs = 2/3 fy)
fs = 2 / 3 * fy
cc = 2.0
smax = min(15 * (40000 / fs) - 2.5 * cc, 12 * (40000 / fs))
print(f"crack control: top bar spacing must not exceed {smax:.1f} in. -> use #5 @ 10 in. at supports")

wu = 196 psf, thickness check ln/28 = 8.0 in. (h = 8)
exterior support, interior face (top)  Mu = 2.85 kip-ft/ft  As = 0.173 in2/ft  #5 @ 18.0 in.
end span midspan (bottom)              Mu = 4.88 kip-ft/ft  As = 0.173 in2/ft  #5 @ 18.0 in.
first interior support (top)           Mu = 6.83 kip-ft/ft  As = 0.275 in2/ft  #5 @ 13.5 in.
interior span midspan (bottom)         Mu = 4.27 kip-ft/ft  As = 0.173 in2/ft  #5 @ 18.0 in.
other interior supports (top)          Mu = 6.21 kip-ft/ft  As = 0.249 in2/ft  #5 @ 14.5 in.
shrinkage and temperature (transverse): 0.173 in2/ft, #4 @ 13.9 in. max
shear: Vu = 2.10 kips/ft, rho_w = 0.0055, lambda_s = 1.00, phiVc = 5.10 kips/ft
crack control: top bar spacing must not exceed 10.0 in. -> use #5 @ 10 in. at supports

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 percent 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.