Three Bars Where the Drawings Showed Four
An existing office building is being renovated. GPR on an interior simply supported beam (24 ft span) shows 3-#8 bottom bars continuous, not the 4-#8 on the 1974 drawings. The beam is 12 in. wide with d = 20 in. Cores confirm f′c = 4,000 psi; the bar grade is taken as Grade 60 pending a tensile test. The renovation adds a rooftop unit and a storage mezzanine, raising the factored moment at midspan to Mu = 205 kip-ft. Separately, a 4 in. diameter sleeve was found drilled through the web at mid-depth, at midspan, by a prior tenant.
1. Compute the beam's design flexural capacity and determine whether it is tension-controlled.
Read the worked answer
Steel area: As = 3 × 0.79 = 2.37 in²
Compression block depth: a = As fy / (0.85 f′c b) = (2.37 × 60,000) / (0.85 × 4,000 × 12) = 142,200 / 40,800 = 3.49 in.
Nominal moment: Mn = As fy (d − a/2) = 2.37 × 60 × (20 − 1.74) = 142.2 × 18.26 = 2,596 kip-in = 216 kip-ft
Confirm φ: c = a/β1 = 3.49/0.85 = 4.11 in. εt = 0.003 × (20 − 4.11)/4.11 = 0.0116, well above 0.005 → tension-controlled, φ = 0.90
φMn = 0.90 × 216 = 195 kip-ft
Also check the steel ratio: ρ = 2.37/(12 × 20) = 0.0099, comfortably under the tension-controlled limit, so the 0.90 factor is not the issue - the area is.
2. Is the beam adequate for Mu = 205 kip-ft? State the demand-capacity ratio, and rank the remedial options you would present to the owner.
Read the worked answer
No. D/C = 205/195 = 1.05. A five-percent overstress is still an overstress; you cannot round a strength check in the unsafe direction, and the capacity itself rests on assumed fy = 60 ksi and on the tested f′c being representative of the whole pour.
Ranked options:
- Reduce the demand. Re-detail the mezzanine framing or relocate the rooftop unit to a column line. Cheapest, fastest, and adds no new risk - always evaluated first.
- Add bottom flexural reinforcement by exposing the soffit and welding or epoxying plates, or by adding external post-tensioning. Field work in an occupied building, fireproofing reinstatement, and cover requirements all apply.
- FRP flexural strengthening on the soffit. Fast and low-profile, but it is a brittle system, requires a qualified installer, needs fire protection in an occupied building, and depends entirely on the concrete substrate strength and surface prep.
- Increase the section (concrete jacket). Only where headroom permits.
Before anything is designed: verify the bar grade by tensile test (if it is A615 Grade 40, capacity drops substantially and the deficiency is far larger than 5%), and confirm the beam is truly simply supported - partial restraint at the ends would change both the moment diagram and the crack pattern.
3. How does the 4 in. sleeve at mid-depth, midspan affect your evaluation, and what would change your answer?
Read the worked answer
Shear. At midspan of a simply supported beam, shear is near zero, so a 4 in. hole at mid-depth sits in the region least sensitive to shear disruption - this is the most favorable possible location for a web penetration.
Flexure. The hole is at mid-depth, between the compression block (top ~3.5 in.) and the tension steel (bottom 4 in. of the section). It does not intersect the tension chord, so the flexural lever arm is essentially unaffected.
Judgment: openings in a beam web still require an explicit check - not a pass by inspection. Confirm the hole is actually at mid-depth and not drifting into the tension zone, check the reduced web area for shear at the section (not just at midspan), check for any cut stirrups, and confirm the sleeve is not a source of water intrusion into the beam.
What would change the answer: a hole in the tension zone or cutting bottom bars (immediate capacity loss), a larger or multiple openings, an opening near a support where shear is high, or evidence of existing diagonal cracking near the penetration. Those cases require a designed reinforcement scheme around the opening.