Serviceability Worked Examples

These examples keep service-level stiffness, long-term deflection, and crack-control quantities distinct. SI empirical equations are used only with their stated SI inputs.

Example 1 — A rectangular beam has b = 300 mm, h = 500 mm, and f'c = 28 MPa. Calculate the cracking moment for normal-weight concrete.

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Example 2 — For the beam in Example 1, provide three 25 mm tension bars at d = 450 mm. Take Es = 200,000 MPa and service moment Ma = 60 kN·m. Calculate c, Icr, and Branson Ie.

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Example 3 — Treat the beam from Example 2 as simply supported over L = 6.0 m with uniform service load producing Ma = 60 kN·m. Using Ec = 24,870 MPa and Ie = 1.953×10⁹ mm⁴, calculate immediate midspan deflection.

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Example 4 — A beam has immediate sustained-load deflection Δi,sust = 15 mm and compression reinforcement ratio ρ' = 0.005. Calculate the additional long-term and final sustained-load deflections after at least 5 years using ξ = 2.0.

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Example 5 — Reinforcement nearest a beam tension face has calculated service stress fs = 250 MPa and clear cover cc = 40 mm. Determine the ACI 318-14 maximum spacing from the direct crack-control equations.

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Example 6 — For historical comparison, take fs = 250 MPa, dc = 50 mm, effective tension area per bar A = 10,000 mm², and β = 1.2. Calculate the Gergely-Lutz z parameter and SI empirical surface crack-width estimate.

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Example 7 — A one-way slab is 200 mm thick. For a 1000 mm design strip and Grade 420 deformed reinforcement, calculate the minimum shrinkage-and-temperature reinforcement area and the general maximum spacing stated in the lesson.

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Example 8 — A reinforced concrete balcony is exposed to frequent moisture and airborne/seawater chlorides. A design note labels the condition 'P—permeability' and addresses durability only through a numerical crack-width limit. Diagnose the ACI 318-14 errors.

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Example 9 — Use the 300 × 500 mm beam from Example 1 with f'c = 28 MPa, Ec = 24,870 MPa, Ig = 3.125×10⁹ mm⁴, L = 6.0 m, and a service moment Ma = 30 kN·m. Determine the stiffness state and immediate deflection for the simple-span uniform-load teaching model.

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Example 10 — A 6.0 m RC floor beam has φMn = 180 kN·m and Mu = 150 kN·m, so its stated flexural-strength check passes. Brittle partitions are then attached. The later immediate live-load deflection is 5.0 mm and the additional long-term deflection occurring after attachment is 9.6 mm. Check the L/480 after-attachment serviceability limit.

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