Support settlement in a two-span continuous beam
A two-span continuous reinforced-concrete beam supports an office floor. Each span is 7.2 m, the section is 300 mm wide by 650 mm deep, and the concrete is C30/37. Columns A and C are founded on stiff clay and have settled about 2 mm; column B, founded on a 1.8 m square pad within 2.5 m of compacted fill, has settled 14 mm over eight months. The original design analysis assumed rigid supports. The predicted long-term midspan deflection under service load was 11 mm. A floor-level survey now shows the first-span midspan 21 mm below the straight chord joining the tops of columns A and B. There is no visible cracking in the beam, but a 0.25 mm crack runs along the slab-beam interface near column B. The contractor has asked whether the beam can be accepted, and the checking engineer is concerned that the settled support changes the moment distribution and may overload the sagging reinforcement.
1. Estimate the change in support moment at B caused by the 14 mm settlement, and judge whether it can explain the measured 21 mm midspan deflection. State your assumptions.
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Use a two-equal-span prismatic beam with pinned outer supports and a settlement δ at B. From compatibility, the settlement-induced moment at B is M_δ = 3EIδ/L²; it acts in the direction that reduces the hogging moment from the rigid-support analysis and increases sagging in the spans. For the gross section, E ≈ 5000√30 ≈ 27,400 MPa and I_g = 300×650³/12 = 6.87×10⁹ mm⁴, so EI ≈ 1.88×10⁵ kNm². With δ = 0.014 m and L = 7.2 m, M_δ = 3×1.88×10⁵×0.014/7.2² ≈ 152 kNm. If the section is cracked and EI_eff ≈ 0.4EI, the change is about 61 kNm. This is a significant redistribution: the hogging moment at B is lower than the rigid-support value by 60–150 kNm, and the span sagging moment rises by roughly half that amount. However, using δ ≈ ΔM L²/(16EI_eff), the corresponding extra deflection is only about 2–6 mm, not 10 mm. Therefore the settlement is relevant to strength and redistribution, but it does not by itself explain the measured 21 mm. The original 11 mm prediction most likely used gross-section elastic stiffness and omitted long-term creep, shrinkage and load-induced cracking. The survey readings should be checked for datum and construction tolerance, and the deflection calculation repeated with cracked, long-term stiffness.
2. Recommend whether to accept the floor, and state the checks and monitoring you would require before sign-off.
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Acceptance depends on the governing serviceability limit, not on the average deflection alone. For a 7.2 m span, L/250 = 28.8 mm and L/500 = 14.4 mm; 21 mm passes the former but fails the latter. If the floor supports brittle partitions or sensitive finishes, 14.4 mm is the relevant limit and the floor is non-compliant. The 0.25 mm crack is within a usual 0.3 mm limit for moderate exposure, but it should be monitored. More importantly, the settlement has reduced the hogging moment at B and increased sagging; reanalyse the beam with a spring support at B and cracked stiffness, and check the sagging steel at midspan against the settled condition. If the original design used a rigid-support envelope, the sagging reinforcement may be unconservative. Survey the settlement trend: if B is still moving more than about 2 mm in three months, underpinning or compensation grouting should be considered before finishes. Require crack gauges, floor-level readings and a check of the beam-column joint at B. If the reanalysis shows adequate capacity and movement has stabilised, accept with monitoring; otherwise strengthen the spans or relieve the settlement. Do not sign off on the basis of the 21 mm deflection alone.