Structural Engineering & Statics
where the load goes
Concept

Effective depth — where it appears

The distance from a concrete section's compression face to the centroid of its tension reinforcement, which is the lever arm every check divides by. Nearly every quantity in a concrete section is proportional to it or to its square, which is why it is the dimension worth arguing over and the cover is the dimension that erodes it.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

internal-forces · Punching shear A basement is a boat. A 20 by 30 m substructure dug 6 m into ground whose water table stands 2 m down. The head on the underside of the base slab is 4.0 m, so the pressure there is 39.2 kN/m² over the whole plan — 23.5 MN of it, pushing upward. Nothing about the structure changes that number. What resists it is weight: 18.7 MN of concrete and whatever is built above, giving a factor of 0.80. The structure floats if the water reaches 2.82 m below the ground, and a base slab alone would have to be 1.64 m thick to hold it down.

A basement is a boat

Every load in this collection presses down and is resisted by strength. Hydrostatic uplift presses up, is resisted by weight, and does not care what is built on it — so the check contains no material property at all. It is a ratio of two weights, and one of them is water.

equilibrium · Uplift The neutral axis is wherever the first moment vanishes. A 300 by 600 section with 1800 mm² of steel at a depth of 540, carrying 250 kNm after the tension side has cracked. The neutral axis is no longer at mid-depth: it has risen to 234.5 mm from the top, which is where the first moment of the compression zone plus the transformed steel vanishes. The compression is 15.4 N/mm² at the top fibre and the steel carries 301 N/mm²; the resulting couple is 541 kN on a lever arm of 462 mm, which multiplies back to the 250 kNm applied. The uncracked section would have had 6673×10⁶ mm⁴ against the cracked 3809×10⁶ — a loss of 43% of the stiffness.

Where the steel is, not how much of it

A section in bending resists a moment with a couple, and a couple is a force times a distance. The force is bought — it is an area of steel at a stress. The distance is free, decided by where the bars were put, and it is the variable almost nobody optimises because it does not appear on an order.

sections · Lever arm The dimension the capacity rides on is not a drawn dimension. Moment capacity of a 225 mm slab against the cover to the reinforcement. The line is very nearly straight, because capacity is A_s·f_yd·z and z is about 0.9d — so the capacity is proportional to a dimension that is not on the drawing. What is on the drawing is the overall depth, and the effective depth is what the cover, the link and half a bar diameter leave of it: 225 − 30 − 0 − 8 = 187 mm here. Every one of those three is a site tolerance rather than a design decision. Ten millimetres of bar position is 5.7% of this slab's capacity and 1.8% of a 600 mm beam's — the same workmanship costs 3.0 times as much in the shallow member, and the shallow member is the one whose steel is walked on before the pour.

The dimension nobody can measure

Every flexural capacity in reinforced concrete is proportional to the effective depth, and the effective depth is not on the drawing. It is what is left of the thickness after a cover, a link and half a bar diameter have been taken off it — and each of those is a site tolerance. In a slab, ten millimetres of workmanship is six per cent of the strength.

sections · Effective depth Three specimens cannot see the tail. The factor k applied to the sample's own scatter when a characteristic value is estimated from n specimens. With the scatter known in advance it is z·sqrt(1 + 1/n) and barely moves; with the scatter estimated from the same n results it is the Student t quantile instead, and it runs from 7.73 at two specimens to 1.73 at 30. At n = 4 the characteristic strength comes out at 23.8 N/mm² against 27.9 for a population known exactly — 15% lower, for a material that is identical. A small test programme does not report a worse estimate of the strength; it reports a worse strength.

The strength that belongs to the test programme

A characteristic strength is a fractile of a distribution, and a distribution estimated from four specimens is not the same object as one known exactly. The same material tested four times reports a strength 15 per cent below what it reports when its scatter is known — and nothing about the material differs.

materials · Characteristic strength 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.

internal-forces · Punching shear 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.

internal-forces · Punching shear 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.

internal-forces · Punching shear The shadow an opening casts on every perimeter. One column of a 260 mm flat slab (d = 225 mm) carrying 12 kN/m² on 400 × 400 mm columns, the shear 300 mm off centre, on a 10.0 m bay, to scale, with a 300 × 300 mm opening 225 mm (1.0d) from the column. The two tangents from the column's centre to the opening's corners (dashed) enclose 38.9 degrees, and every perimeter loses the part between them: 458 mm of the control perimeter 2d out (solid), 1,093 mm of the outer perimeter (dashed), the lost parts drawn faint. With the opening the outer perimeter stands 6.13d from the face instead of 5.36d, and the studs need 7 perimeters instead of 6; one of the 12 rails runs into the hole.

A hole takes a tenth of every perimeter

A 300 mm riser cut a slab's depth from a column throws a shadow from the column's centre, and EN 1992-1-1 deletes whatever part of a punching perimeter falls inside it. The outer perimeter, being longest, loses the most length — but it loses the same share as the control perimeter, a tenth of each, because the shadow is an angle. The cost arrives somewhere less obvious: a seventh ring of studs, 24 per cent more steel, and studs needed in a bay a third of a metre shorter than before.

internal-forces · Punching shear A core drilled into a slab the studs were laid out for without it. One column of a 260 mm flat slab (d = 225 mm) carrying 12 kN/m² on 400 × 400 mm columns, the shear 300 mm off centre, on a 10.0 m bay, to scale, its studs laid out for the slab without a hole — six rings of 14 mm studs on twelve rails, the last ring 4.25d from the face — and a 150 mm core 225 mm (1.00d) from the face. Its shadow from the column's centre takes 20.3 degrees of every perimeter. The studs' outer perimeter stands where the last ring leaves it, 5.75d out (solid); with the core the slab needs it at 5.74d (dashed), so the outer check is at 1.00 of its capacity, and the control check's studs at 0.79. No rail runs into it.

The core drilled after the slab was built

A building in use acquires new services, and a core drilled through a flat slab beside a column is a routine instruction. The studs round the column were laid out for a slab without the hole, and the drill finds whatever is there. Whether it cuts a stud matters less than it looks. What decides is the outer perimeter, which the hole pushes outward while the studs stay where they are, and whether the slab survives is set by a margin nobody chose: the room left when the studs were rounded up to a stock size and the rings to a whole number.

internal-forces · Punching shear

Named alongside it

The objects these essays reach for when they reach for this one.

Punching shearControl perimeterFlat slabBrittle failureEccentricityShear reinforcementSize effectToleranceBondCharacteristic strengthCoverCracked section

All concepts