Understanding punching shear in slabs and footings
A concentrated load or reaction — a column landing on a slab, or a column pushing down into a footing — can fail the concrete around it in a completely different way than a beam fails in ordinary shear.
Ordinary (one-way) shear is what a beam resists along its length — a fairly straightforward "slice across the section" failure. Punching shear is a different failure mode entirely, specific to situations where a concentrated force acts over a relatively small area on a slab or footing that isn't supported by beams underneath it: a column landing on a flat slab, or a column's load bearing down into an isolated footing. Instead of failing across a straight cut, the concrete can fail by a roughly cone- or pyramid-shaped surface punching straight through the slab or footing around the loaded area — which is exactly where the name comes from.
Why the check happens away from the column face
Because the failure surface is a cone radiating outward and downward (or upward) from the loaded area rather than a vertical cut right at its edge, codes don't check punching shear stress at the face of the column itself — they check it around a critical perimeter set back some distance from the face (commonly related to the section's effective depth, d). That critical perimeter is where the sloped failure cone is expected to actually reach the level of the reinforcement, which is a genuinely different geometric idea from a one-way shear check and is why punching shear needs its own dedicated calculation rather than being folded into ordinary shear design.
This is also why punching shear specifically matters most where there's no beam to help: a flat slab supported directly on columns, or an isolated footing under a single column, has no beam action to spread the load out gradually — the connection between the concentrated load and the flat element around it has to survive on its own. RC Slab Design and Foundation Design both check punching shear explicitly for exactly this reason; RC Column Design feeds the column load that drives this check in the first place (see the related tools below).