The failure that punches a cone
A flat slab is the minimalist’s floor: no beams, no drops, just a plate of concrete resting directly on columns. Its price is a concentrated argument at every column, where the slab’s whole tributary load funnels into a small contact area. If the concrete there loses that argument, failure comes as two-way shear — punching shear — with the column driving a truncated cone or pyramid up through the slab thickness. It is a brittle mechanism: little warning, no ductile redistribution, and once one column punches, its load shifts to neighbors that are next in line.
Punching differs from the one-way shear checked in beams, which acts across a single plane. Two-way shear wraps around the column on all available sides, so the resisting section isn’t a beam width — it’s a perimeter times the slab’s effective depth. ACI 318 formalizes the check in three moves: define the critical perimeter, cap the concrete’s shear stress by the least of three expressions, and compare the factored stress demand to the reduced capacity. This guide runs one column through all three: an interior 400 × 400 mm column (about 16 in square) in a slab with effective depth d = 200 mm (8 in), concrete f′c = 30 MPa (≈4,350 psi), carrying a factored shear Vu = 400 kN (about 90 kips).
The control perimeter lives at d/2
The punching cone spreads outward as it rises through the slab, so the code checks stress not at the column face but on a control perimeter b0 drawn at d/2 from each face — halfway through the slab’s effective depth. For an interior column the perimeter closes on all four sides: b0 = 2(c1 + d) + 2(c2 + d). The example: b0 = 2(400 + 200) + 2(400 + 200) = 2,400 mm. Multiply by the depth and the resisting area is b0 × d = 480,000 mm² — the concrete actually on trial.
Geometry is destiny here, because edge and corner columns lose sides of the perimeter to the slab boundary. The same 400 mm column at a slab edge keeps only three sides, b0 = 2(c1 + d/2) + (c2 + d) = 1,600 mm; at a corner, two sides, b0 = (c1 + d/2) + (c2 + d/2) = 1,000 mm — 42% of the interior perimeter. Their tributary loads are smaller too, but rarely in proportion, and the code compounds the penalty through the position constant αs (40 interior, 30 edge, 20 corner) in the third stress limit below. Corner columns are where flat-slab designs go to be humbled.
Three ceilings on the concrete stress
ACI 318-19 Table 22.6.5.2 caps the concrete’s two-way shear stress vc at the least of three expressions (SI form): a baseline 0.33·λ·√f′c; an aspect-ratio limit 0.17·(1 + 2/β)·λ·√f′c, where β is the column’s long side over its short side; and a perimeter-size limit 0.083·(2 + αs·d/b0)·λ·√f′c. The λ is the lightweight-concrete factor, 1.0 for normalweight. All three scale with √f′c — concrete’s shear resistance grows with the square root of its strength, not linearly, so doubling f′c buys only 41% more stress capacity.
The second and third limits are corrections for cases where the baseline is unconservative. An elongated column loads the ends of its perimeter harder than its long sides; run the algebra and the β limit only drops below the baseline once β exceeds about 2.1 — a column more than twice as long as it is wide. The perimeter limit likewise punishes very large perimeters relative to depth, where stress distributes unevenly; for an interior column it governs only when b0 exceeds roughly 20d. For the compact square example — β = 1, b0/d = 12 — the three limits compute to 1.81, 2.79, and 2.42 MPa: the baseline governs, vc = 0.33·√30 = 1.81 MPa.
Demand against capacity: the worked verdict
The demand side divides the factored load over the trial area: vu = Vu/(b0·d) = 400,000/(2,400 × 200) = 0.833 MPa. The capacity side discounts the nominal stress by the shear strength-reduction factor φ = 0.75: φvc = 0.75 × 1.81 = 1.356 MPa. The check is one inequality — 0.833 ≤ 1.356 — and the utilisation ratio makes it quotable: 0.833/1.356 = 0.61. The slab is working at 61% of its punching allowance, with a force-unit capacity of φvc·b0·d = 1.356 × 2,400 × 200 = 651 kN against the 400 kN demand.
The punching shear calculator runs exactly this chain — perimeter by column position, all three vc limits with the governing one flagged, φ = 0.75, capacity in kN and utilisation — from six inputs, which makes it easy to interrogate the design the way the next section does. Two honest caveats travel with the concentric check, and the tool states them: unbalanced moment transferred between slab and column raises the peak shear stress (through a γv factor), and shear reinforcement — studs or stirrups — raises capacity; both need the longer analysis, so a utilisation near 1.0 on the concentric check is a design smell, not a pass.
What actually moves the number
Play the levers on the example. Concrete strength works through √f′c: jumping from 30 to 40 MPa lifts capacity about 15% — real but expensive. Depth is the star, because it enters twice: the resisting area is b0·d, and b0 itself contains d. Thickening the slab so d goes from 200 to 250 mm stretches the interior perimeter from 2,400 to 2,600 mm and the capacity from 651 to 881 kN — 35% more punching strength for 25% more depth, before the extra self-weight is counted. This double-dip is why drop panels and column capitals, which deepen the slab only where the perimeter lives, are the classic flat-slab fixes.
Column size works the same way through c1 and c2 — a bigger column is a longer perimeter — and position is the lever you usually can’t pull: the corner version of the example column, even at the same governing stress, offers only 1,000 mm of perimeter and 271 kN of capacity, well under half the interior figure, with αs = 20 further trimming the third limit for large corner perimeters. When geometry and concrete run out, shear studs around the column add capacity directly — the modern replacement for the bent-bar details of older slabs, and the point where this concentric check hands off to detailed design.
The check in one paragraph
The interior column, end to end: at d/2 from the 400 × 400 face, the control perimeter is b0 = 2,400 mm over d = 200 mm of depth. Of the three ACI stress ceilings — 1.81 MPa baseline, 2.79 for aspect (β = 1), 2.42 for perimeter size (b0 = 12d) — the baseline 0.33√30 governs. Applying φ = 0.75 gives 1.356 MPa of design capacity against a demand of vu = 400,000/480,000 = 0.833 MPa: utilisation 0.61, capacity 651 kN versus 400 kN of load, adequate with margin. The same slab at an edge keeps 1,600 mm of perimeter, at a corner 1,000 — which is why the punching check is run per column position, and why the corner column, not the middle of the floor plate, usually writes the slab’s thickness.