Under-Reinforced Roof Beam Found During Audit
During a structural audit of a two-storey school building, you inspect a simply supported roof beam of 230 mm width and 450 mm overall depth, with an effective cover of 40 mm. The span is 6.0 m. The factored uniformly distributed load, including self-weight, is 20 kN/m. The concrete is M20 and the steel is Fe415. The contractor has provided three 16 mm diameter bars at the bottom. The drawing notes say the beam was designed for the same load using standard flexural theory. You are asked to check whether the provided steel is adequate, using an effective depth of 410 mm, and to state the immediate action. Assume a clear cover that gives the stated effective depth and ignore compression steel. No revision of the loading is expected.
1. Compute the flexural capacity of the provided three 16 mm bars and compare it with the factored moment.
Read the worked answer
Area of steel provided: Ast = 3 x (pi/4) x 16^2 = 603 mm2.
Factored moment demand: Mu = w L^2 / 8 = 20 x 6^2 / 8 = 90 kNm.
For M20 and Fe415, using b = 230 mm and d = 410 mm: Ast fy / (b d fck) = 603 x 415 / (230 x 410 x 20) = 0.1327. Lever-arm factor = 1 - 0.1327 = 0.8673.
Mu capacity = 0.87 fy Ast d x lever-arm factor = 0.87 x 415 x 603 x 410 x 0.8673 = 77.4 x 10^6 Nmm = 77.4 kNm.
77.4 kNm is less than the demand of 90 kNm. The provided steel is short by about 14% in moment capacity.
2. What area of steel is required, and is four 16 mm bars adequate?
Read the worked answer
Solve for the required Ast using Mu = 0.87 fy Ast d [1 - Ast fy / (b d fck)]: 90 x 10^6 = 0.87 x 415 x Ast x 410 x [1 - Ast x 415 / (230 x 410 x 20)]. This gives Ast approximately 723 mm2.
Four 16 mm bars give Ast = 4 x 201 = 804 mm2, which exceeds 723 mm2. So four 16 mm bars are adequate in flexure, subject to checking bar spacing and clear cover. The minimum steel requirement is also satisfied by the original three bars, but minimum steel does not make an under-strength beam safe.
3. State the immediate action, including what to do if the contractor argues that the concrete is only a few months old and the drawing is old.
Read the worked answer
Treat the beam as deficient until proved otherwise. Stop loading or propping the affected area if there is any sign of distress, mark the beam for investigation, and inform the client in writing. Require a redesign check by a competent structural engineer and a strengthening scheme such as additional reinforcement, jacketing, or external strengthening. Check other similar beams from the same drawing. If the contractor argues that the concrete is young or the drawing is old, do not accept verbal assurance: the audited section and the actual factored load govern. If concrete strength is also doubtful, use cores or non-destructive tests, because lower concrete strength would reduce capacity further.