Structural Engineering & Statics
where the load goes
Concept

Tensile strength — where it appears

The stress at which a material parts in tension, which for concrete is a tenth of its compressive strength and the smallest number in any reinforced calculation. It is deliberately ignored in strength design and it decides cracking, splitting and durability, which are the limits that arrive first.

Named by 15 essays across 5 fields — each of them below, with the objects they name alongside it.

The beam is stiffer than its cracked section and softer than its gross one. Moment against mid-span deflection for a 300 × 550 mm beam spanning 8.0 m, with the two bounds it lies between. The uncracked line is what the gross transformed section gives; the cracked line is what the section at a crack gives; and the curve between them is the member, because between the cracks the concrete is still carrying tension and the average curvature is not either section's. At the service load the deflection is 23.2 mm — span over 345 — against 8.3 uncracked and 25.2 fully cracked, a factor of 3.03 between the bounds. The interpolation ζ = 1 − β(M_cr/M)² sits it 88 per cent of the way across, and β falls from one to a half under sustained or repeated load because the bond that does the dragging deteriorates.

Stiffer than its cracked section says

At a crack the concrete below the neutral axis has gone and the steel carries the tension alone. Between the cracks it has not gone — bond drags it back into tension, the steel strain drops, and the curvature averaged over a length of beam is neither section's.

deflection · Tension stiffening Cover enters twice, and the strength of the concrete enters once. How long a 20 mm bar has before the cover over it splits, against the cover, split into the two halves it is always split into. Initiation is the time for the chloride front to reach the bar, which goes as the square of the cover — Fick's law and nothing else — and it is 9.1 years at 35 mm and 36.2 at 70. Propagation is the time from there to a split cover, which is short: 0.9 years, because the cover cracks at a section loss of 0.21% and no strength check in this collection would notice a loss that small. The pressure the cover can take grows with the cover too, so cover appears in both terms and the concrete's own tensile strength appears in one of them, linearly. That asymmetry is why every durability clause in every code is about cover and crack width, and hardly at all about strength.

The load that comes from inside

Every action in this collection has been applied from outside — a weight, a pressure, a movement, a temperature. Corrosion is not applied at all, and the reason it belongs to statics rather than to durability is that what does the damage is a load: rust occupies three times the volume of the steel it came from, and the only place to make room is by pushing the cover apart.

materials · Corrosion The middle third, computed. The kern of a 300 × 600 mm rectangle, computed by asking, for every direction, how far the resultant can move before the far face would be pulled. It reaches ±100.0 mm vertically and ±50.0 mm horizontally, which are h/6 and b/6 exactly, and the region between is a rhombus rather than the ellipse the two numbers suggest — a resultant on a diagonal has less room than either axis allows.

The strength thrown away on purpose

Masonry, concrete and soil are all analysed as though they had no tensile strength whatever. Each of them has some. The decision to set it to zero is the single most consequential modelling assumption in the subject, it is safe for one kind of check and unsafe for another, and almost nothing that uses it says which.

materials · No tension A base plate, and when the bolts start working. A 500 × 500 mm plate carrying 600 kN and 180 kN·m, so the resultant sits 300 mm from the centre against a kern of 83.33 mm. The plate is in bolts engaged: bearing over 150.88 mm at a peak of 20 N/mm², with the holding-down bolts carrying 154.42 kN. The plate lifts at 50 kN·m and crushes at 126 kN·m, and the bolts are not needed until 150 kN·m.

The failure that is in the concrete

An anchor bolt is a steel component and its capacity is usually decided by something else entirely — a cone of concrete pulled out around it, failing in tension, in a material every other calculation on the project has assumed cannot take tension at all. The exponent in the capacity says so: it is not the square the geometry implies.

connections · Anchor breakout 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 same steel, and a crack three times as wide. Calculated crack width against bar diameter, with the area of steel held at 1340 mm² per metre throughout — so the spacing changes with the square of the diameter and the amount of reinforcement does not change at all. The width runs from 0.173 mm at 8 mm bars to 0.372 at 25, a factor of 2.15 for identical steel. The reason is in the crack spacing: after a crack forms the bar has to re-anchor the concrete's tensile force before the next one can, and the length that takes is proportional to the bar's diameter. Of the 337 mm spacing drawn, 35% is the cover term and 65% is the bar term — and the cover term is the one that puts crack control and durability in opposition, because cover protects the bar and widens the crack that reaches it.

The same steel, and a wider crack

A crack's width is the distance between cracks times the strain the steel carries over that distance. Neither of those is decided by how much reinforcement there is. The spacing is a bond length, so it goes as the bar diameter; the strain is set by the stress in the steel. Two arrangements of identical steel can differ by a factor of two in crack width, and the one that wins is the one with more, smaller bars.

sections · Crack width 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 Prestress buys shear as a square root, not as a sum. The shear stress an uncracked web can take before the principal tension reaches the concrete's tensile strength, against the axial compression the prestress put there. With no prestress it is 1.35 N/mm², the tensile strength itself, because pure shear has a principal tension of exactly its own magnitude at forty-five degrees. Adding compression gives √(f_ct² + σ_cp·f_ct), which is a square root and therefore flattens: the first newton of prestress is worth far more than the last. At the 7.14 N/mm² drawn the limit is 3.39 N/mm², a gain of 2.51, and doubling the prestress from there takes it only to 4.59. The straight line is what a rule that simply added the two strengths would have promised.

The crack that never reached forty-five degrees

Every shear expression for reinforced concrete is a curve fitted to tests, because a cracked section has no free body worth drawing. An uncracked prestressed web has one — a single point, a Mohr's circle and a principal tension — and it is the only shear check in the subject that is derived rather than measured.

internal-forces · Prestressed shear Steel and concrete happen to match, and nothing else on the list does. The mismatch strain a 40 degree change produces in seven pairs of materials that engineering bonds together, which is the difference of their coefficients of expansion times the temperature. Steel against concrete is 80 microstrain — 17 per cent of the larger coefficient, and by far the smallest on the list. It puts 0.223 N/mm² of tension into the concrete, 7.7 per cent of its tensile strength and 1.9 per cent of the 12 N/mm² a fully restrained member would have carried. Reinforced concrete works because of a coincidence in the third significant figure of two numbers nobody chose, and the same bar in aluminium would put in two and a third times as much.

The coincidence reinforced concrete stands on

Steel expands at twelve microstrain per degree and concrete at ten. Nobody chose either number, they are not equal, and the seventeen per cent between them is the smallest mismatch of any pair of materials engineering bonds together — which is the reason the most-used structural material on earth does not tear itself apart every summer.

materials · Thermal compatibility The ductility number depends on the ruler. Elongation after fracture against the gauge length it was measured over, in units of √S₀. A tensile specimen extends uniformly until the ultimate load and then localises into a neck, so the total is a strain (16 per cent here) plus a length (7.2 mm), and dividing a length by the gauge length is what makes the curve fall. At the two standard gauges the same steel reports 27.5 per cent over 5.65√S₀ and 21.8 over 11.3√S₀ — a ratio of 1.264, and 42 per cent of the first number is a property of the specimen rather than of the material.

The ductility that depends on the ruler

Percentage elongation after fracture is the most quoted ductility measure in the subject and one of the least well defined. A specimen stretches uniformly until the ultimate load and then localises, so the number is a strain plus a length — and dividing a length by the gauge length makes the answer a property of the specimen.

materials · Gauge length The curve the machine drew and the curve the material was on. The same tensile test twice. Force over the ORIGINAL area against extension over the original length is the engineering curve, which peaks at 431 MPa and 24.6 per cent and then falls. Force over the ACTUAL area against the natural logarithm of the length ratio is the true curve, which passes 538 MPa at the same instant and keeps rising. Nothing softens anywhere on this figure: the descending branch is the original area still being divided by, and the material is hardening the whole way. The two separate at the very first plastic strain and the gap between them is exactly e^ε, which is 1.25 at the ultimate load. Past that instant the deformation stops being uniform, so the engineering curve is drawn dashed: the real one falls faster than this, because the extension is now happening in one short length of the bar and the strain axis is still dividing it by the whole gauge.

The curve was rising the whole time

Every tensile curve ever printed turns over and falls, and nothing about the material does. The fall is the original area still being divided by after the specimen has stopped having it — and correcting the two denominators turns a strength, a ductility and a failure into one exponent.

materials · Gauge length What a crack makes of a notch, against how sharp the notch is. The fatigue notch factor against the root radius, at a fixed elastic factor of 3 in a 430 MPa steel. K_t is a property of the shape and does not move along this axis at all — the dashed line — while the factor fatigue actually feels climbs toward it from below. At a 1 mm root the answer is 2.40, which is 30 per cent of the notch relieved; at 0.1 mm it is 1.38, and a notch that concentrates by 3 elastically is barely felt. Nothing has changed about the stress field: what has changed is that the peak is confined to a smaller volume than the material's own process size, so the crack starts against an average rather than against a maximum.

The notch a crack does not feel in full

The elastic concentration factor is a property of shape and knows nothing about size, which is what makes it so useful and so misleading. A fatigue crack starts against an average over a volume the material owns, so two notches with the same factor and different radii have different fatigue strengths — and the stronger the steel, the less of that relief it gets.

materials · Stress concentration Two restraints, and only one of them cares how much was imposed. Crack width against the restrained strain, for the same 20 m wall restrained two ways. Edge restraint — a wall cast on a base — gives a width proportional to the strain, because the concrete has to accommodate the movement and the cracks are where it does. End restraint — a bay cast between two that have hardened — gives 0.32 mm at every strain on the axis, because the crack opens only until the steel can push the cracking force back into the concrete, and that force is a property of the section. At the 304 microstrain this wall is asked for, the two are 0.17 and 0.32 mm, and the movement is divided into 35 cracks and 19.

The bay that is cast last

A wall cast on a base cracks in a way the steel controls: more movement gives wider cracks, and reinforcement decides how many. A bay cast between two walls that have already hardened cracks in a way that has nothing to do with how much movement there was — and the threshold that separates a controlled crack from a single wide one is a quantity of steel rather than a limit on anything.

internal-forces · Restraint cracking The deflection a building can take, and which way it is bending. The limiting deflection ratio — the sag or hog of a building divided by its length — at a critical tensile strain of 0.075 per cent, against how long the building is compared with its height. Two curves, and the gap between them is the whole finding: hogging is worse than sagging by exactly 2.0 times, because the neutral axis of a hogging building sits near the bottom and the tension face is the full height of the wall. The minimum is at L/H = 1.6, where the two mechanisms cross: a squat building cracks diagonally in shear and a long low one cracks in bending at the extreme fibre.

The crack is a strain, not a slope

Angular distortion is what every table of settlement limits is written in, and it is a proxy. What cracks a building is a tensile strain in it, and reading the building as a deep beam says which of two mechanisms supplies it, why hogging is exactly twice as damaging as sagging, and why a horizontal stretch of a few millimetres takes half the capacity away before the settlement starts.

deflection · Differential settlement Heave puts the pile in tension where settlement put it in compression. For a 24 m pile 600 mm across carrying 200 kN, in clay whose top 18 m moves, by 100 mm at the surface: the axial force down the pile, compression positive, when the ground heaves (solid) and when it settles as far (dashed). Settling, the pile's compression peaks at 910 kN at 17.2 m and its toe carries 324 kN. Heaving, it is in tension below 8.9 m, peaking at 520 kN at 17.3 m, and its toe carries nothing: lifted, it parts from the soil beneath, so everything holding the pile down comes from the shaft below the plane.

The pile the ground pulls apart

Clay that swells after its trees are felled rises past the piles in it, and the same compatibility that put a pile into compression in settling ground puts it into tension in heaving ground, at nearly the same depth. Nearly everything else changes. A lifted toe parts from the soil and holds nothing, so the shaft alone anchors the pile; the head load that added to the downdrag now subtracts from the tension; and the tension peaks 17 metres down, far below the short cage a lightly loaded pile is given. A house's piles in swelling clay are tension members nobody reinforced.

internal-forces · Downdrag

Named alongside it

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

BondCrack widthServiceabilitySize effectCharacteristic strengthDuctilityLoad pathMinimum reinforcementShrinkageAnchorageCoverCracking moment

All concepts