Structural Engineering & Statics
where the load goes
Series

Punching shear — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.606 N/mm² against a resistance of 0.658.

    A check made on a perimeter, not on a section

    Every shear check in this collection is made on a plane cut through a member. A slab sitting on a column has no such plane, because the shear leaves in every direction at once — so the check is made on a closed line, and a line grows with the column while the load grows with the square of the bay.

    part 1 · internal-forces
  2. A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.697 N/mm² against a resistance of 0.658.

    Turn the column, and the slab passes

    A flat slab that is comfortable under gravity fails its punching check the moment a moment arrives at the column, and nothing about the load has changed. The fix is not more concrete. It is the column's plan shape and, at equal area, which way round it is turned — worth more than adding half again as much column.

    part 2 · internal-forces
  3. The perimeter the edge leaves, and where its middle is. A 400 × 400 mm column at the edge of a 260 mm slab (d = 225 mm) carrying 12 kN/m², on a 7.2 m end span. Heavy line: the control perimeter 2d out, cut by the slab's edge — 2,614 mm, against 4,427 for the same column inside the slab. Wide band beneath it: the reduced perimeter u1 of EN 1992-1-1, 2,214 mm, its legs cut back from the edge. Open circle: the perimeter's centroid, 363 mm in from the column's centre. Filled: the load's resultant, 631 mm in, set by the slab's end moment of 170 kN·m on a shear of 269 kN — 269 mm beyond it, into the slab. The general multiplier on the shear is 1.78; u1/u1 gives 1.18 and the shortcut 1.4.

    The perimeter whose middle is not the column's

    At an edge column the slab's own edge cuts the control perimeter, and what is left has its centroid 363 mm inside the column's centre. The slab's end moment moves the shear's resultant inward too, by a distance that grows with the end span. What twists the perimeter is the gap between the two: nothing at a 5 m end span, enough by 7.2 m to put the check at 1.23 of its resistance where the 1.4 shortcut reads 0.97. And a bigger column makes it worse.

    part 3 · internal-forces
  4. The studs reach out to where the concrete is enough. One column of a 260 mm flat slab (d = 225 mm) carrying 12 kN/m² on 400 × 400 mm internal columns, the moment it hands the column putting the shear 300 mm off centre, on a 10.0 m bay, to scale. The inner check, on the control perimeter 2d out (solid), needs reinforcement: 12 rails of studs, 6 perimeters of them from 0.5d at 0.75d spacing, 1,415 mm² on each perimeter. The outer perimeter (dashed), where the concrete alone carries the shear, stands 5.4d from the face — 9,173 mm long against the control perimeter's 4,427 — and the last studs must be within 1.5d inside it, 3.9d from the face.

    The studs that send the check outward

    Shear studs round a column fix a failing punching check, and they turn one check into three. The studs carry the control perimeter; past them the concrete alone must carry a perimeter long enough to need no help, and that outer perimeter stands further from the column the larger the load, so the studs follow it out. At a 10 m bay they reach four effective depths from the face, and the steel they need grows as the shear to the power 3.5. The third check, at the column face, is the one no stud reaches, and it ends the series at 11.8 m.

    part 4 · internal-forces

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