Structural Engineering & Statics
where the load goes
Concept

Brittle failure — where it appears

A failure that arrives with no plastic deformation to give warning, at a load that the elastic calculation may have been perfectly right about. What distinguishes it is not the load but the absence of a second mechanism after the first one goes, which is why the checks that guard against it compare two states of a member rather than a demand with a capacity.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

A strength that is a property of the specimen. Nominal strength against size for geometrically similar specimens of one material. On the left the specimen is too small for a crack to run and the strength is a plateau — a plastic limit, and the regime laboratory specimens sit in. On the right a crack releases more energy than it consumes as soon as it starts and the strength falls as the inverse square root of size, which is the regime real structures sit in. The turn happens at D₀ = 120 mm. A 100 mm specimen reads 3.10 N/mm² and a 1500 mm member of the same material carries 1.14: the test overestimates the structure by a factor of 2.71.

The bigger one is the weaker one

Two geometrically similar beams of the same concrete should fail at the same nominal stress, because a strength is supposed to be a material property. They do not. The large one fails at less, and the reason is that a crack releases energy in proportion to a volume and consumes it in proportion to an area.

materials · Size effect The steel that is sized by the concrete. Ultimate moment of a 1000 × 400 mm section against the area of tension steel in it, with the moment that cracks the section drawn across. The cracking moment is 77.2 kNm and contains no steel at all — it is f_ctm times the gross section modulus, 2.90 N/mm² times bh²/6 — so it is a horizontal line, and every section to the left of where the two meet is one whose first crack is its failure. The crossing is at 507 mm², and rearranging the two expressions gives 0.245·(f_ctm/f_yk)·bd against the 0.26 the codes print — the constant is a section modulus divided by a lever arm and not a fitted number. The rule as printed asks for 538 mm² here, which is 6% more than the derivation needs, and that margin is the whole of the safety in a check whose failure mode is sudden.

The steel the concrete asks for

Every other bar in a concrete member is there because of an action. This one is there because of the member itself — enough steel that the cracked section can carry more than the moment that cracked it, so that the first crack is not also the failure. The requirement contains no load, and both of its consequences run the wrong way round.

sections · Minimum reinforcement The strength that does not keep pace. Concrete's mean tensile strength against its characteristic compressive strength, with the ratio of the two on the same axis, scaled. The tensile strength goes as f_ck^⅔, so it rises from 1.57 to 5.04 N/mm² over a sixfold rise in the compressive strength, and the ratio between them falls from 11.1% at C20 to 6.0% at C80 — a factor of 1.83. Nothing in a bending or a column calculation ever uses the lower curve, and everything that decides a transition does: when the section cracks, how much minimum steel it needs, how far a bar has to be lapped, what a member without links carries. So a stronger concrete needs more minimum reinforcement, longer laps and a bigger crack-control check, in a member whose ultimate capacity has barely moved. Its real scale is a length: E·G_F/f_ct² is 299 mm here, which is the size at which a member stops behaving plastically and starts behaving like a fracture problem.

The strength that is never used

Concrete's tensile strength appears in no bending calculation, no column calculation and no shear calculation with links in it. The whole design philosophy is that it cracks and the steel takes over. And it decides where nearly every transition in the subject sits — when a section cracks, how much minimum steel it needs, how far a bar has to be lapped, what a member without links can carry, and how wide a crack opens.

materials · Tensile strength The same steel, brittle in January. Fracture toughness against temperature for a 25 mm ferritic plate, from Wallin's master curve — 30 + 70·exp(0.019(T − T₀)) in MPa√m, whose shape is the same for every ferritic steel and whose only free parameter is the reference temperature T₀ = -60 °C. Beside it, and on its own scale, is the crack length that toughness will tolerate at 200 N/mm², which goes as the square of it. At +20 °C this plate carries a 764 mm flaw and at -20 °C it carries 199 mm — a factor of 3.8 for a forty-degree change in the weather, in a steel that met its specification on both days. Thickness moves the curve as well, and the wrong way: constraint at a crack tip suppresses the yielding that would have blunted it, so a 100 mm plate of this steel tolerates 27% of the flaw a 10 mm plate does. Loading it in a millisecond shifts the whole curve another forty degrees.

The same steel, brittle in January

Every other material property in this collection is a number. Toughness is a curve, and the axis it runs along is temperature. Between twenty degrees and minus twenty a structural steel does not get gradually weaker — it changes the mechanism by which it fails, and the flaw it will tolerate falls by a factor of four.

materials · Transition temperature 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 perimeterEffective depthFlat slabEccentricitySize effectShear reinforcementCharacteristic lengthCharacteristic strengthCrack widthCracking momentDuctility

All concepts