Structural Engineering & Statics
where the load goes
Materials

The property that appears in none of the equations

Ductility is in no design formula on this site. Every method on this site depends on it — and a brittle structure does not merely fail early, it makes the analysis wrong.

Assumes The stress at which nothing in particular happens and After the first yield, which is not the end.

16 min read 6 figures Which failure arrives firstThe load must go somewhereOne support too many

Search this site for the word ductility in a formula and there is nothing to find. The plastic hinge is computed from the yield stress and the section’s geometry. The collapse load comes out of a work equation containing moments and rotations. The redundant beam’s moments come from stiffnesses. Every number quoted anywhere is produced without ever asking how far the material can stretch before it comes apart.

And yet almost every one of those methods is silently conditional on the answer. Ductility is not a term in the equations; it is a term in the derivations, and it drops out of them by being assumed sufficient. Which means the failure mode it guards against is not a larger number in a familiar calculation. It is the calculation ceasing to describe the structure.

Two materials pulled until they stopTwo stress-strain curves — mild steel, cast iron — plotted to a strain of 2.0%. One of them has a plateau, so the stress at which yielding starts is something the specimen does rather than something anyone chooses. No offset construction is drawn.00.5%1%2%2%050100150200250300350strainstress, N/mm²mild steelcast iron
Fig. 1 Two structural materials at the extremes. Mild steel yields at 275 N/mm² and goes on stretching for a hundred and fifty times that strain before it breaks. Cast iron in tension follows a nearly straight line to 150 N/mm² and then stops existing, at a strain of about 0.15% — barely more than the strain at which the steel has only just begun. Every method below works on the first and fails on the second, and no strength comparison between them says so.

What the methods are actually assuming

The plastic hinge assumes rotation. A propped cantilever collapses when two hinges have formed, and the second one forms only after the first has rotated far enough for the moment at the other critical section to catch up. If the first hinge fractures at the rotation the second needs, the mechanism never assembles and collapse arrives at the load that formed the first hinge — which for that beam is 0.686 of the load the analysis predicts.

The collapse mechanism of a propped cantileverA collapse mechanism, with the hinge position found by searching rather than quoted. Every position gives an upper bound on the collapse load; the lowest is 7.29, at a hinge 58.6 per cent along, which is a coefficient of 11.657 times Mp over the square of the span.sagging hinge at 4.69hinge at the fixed endlowest upper bound: 7.29every hinge position gives an upper bound on the collapse loadassumed position of the sagging hingecoefficient 11.66 Mp ÷ L²
Fig. 2 The mechanism the collapse calculation assumes. The kink at each hinge is a rotation, and the work equation multiplies the plastic moment by it. Nothing in that equation asks whether the material can supply the rotation — the hinge is drawn as a point and a point can rotate indefinitely. A real hinge is a region a few section depths long, and how much it can turn through before something tears is the quantity none of this contains.

Moment redistribution assumes rotation too. The whole benefit of continuity — the moment over the support falling as the mid-span moment rises — is a statement that once the support section has yielded, further load goes elsewhere. It goes elsewhere by the support section rotating at constant moment. A section that cannot rotate holds its elastic share of the load right up to the moment it breaks, and the redistribution never happens.

The lower-bound theorem assumes it everywhere at once. The theorem says that any set of internal forces in equilibrium with the load and nowhere exceeding the section capacity is a safe estimate of collapse. It is the most useful result in plastic theory and the licence for an enormous amount of practical design, including every masonry arch on this site. Its proof requires that the structure be able to reach the assumed state, which means every section that has to yield can yield and stay yielded while the others catch up. Without that, a lower-bound solution is not a bound at all.

A bolt group assumes it. A line of bolts in a lap joint does not share the load equally: the end bolts take much more, because the plates stretch between them. The design assumption that all the bolts carry the same force is only true once the end ones have yielded and shed their excess to their neighbours. A group of brittle high-strength bolts fails when the first one reaches capacity, and the joint carries a fraction of what the sum of its bolt capacities suggests.

Redundancy itself assumes it. The reason one support too many is worth having is that a structure with two routes to the ground survives losing one. It survives by the remaining route taking up the load, which requires the failed route to deform without shedding what it was already carrying, and requires the remaining one to accept a substantial overload while it does. Both are ductility demands, and a brittle redundant structure is a structure with two routes that both fail on the same afternoon.

The picture of what it costs

The measurable version of ductility for a section in bending is its rotation capacity: how far past the curvature at first yield it can be bent while still carrying its plastic moment. That is a quantity the moment-curvature curve answers directly, and it varies enormously between shapes carrying the same load.

What it costs to reach the plastic moment, for three shapesMoment against curvature for three cross-sections of identical area (3000 mm²) and identical depth (200 mm), in mild steel, each divided by its own first-yield moment and its own first-yield curvature. The rectangle has a shape factor of 1.50 and reaches 98% of its plastic moment at 4.3 times the curvature at first yield; The I-section has a shape factor of 1.09 and reaches 98% of its plastic moment at 1.2 times the curvature at first yield; The tee has a shape factor of 1.78 and reaches 98% of its plastic moment at 8.4 times the curvature at first yield. The dashed lines are the rigid-plastic moments, computed from the equal-area axis rather than read off the curves, and no curve reaches its own.02468101200.511.52curvature ÷ curvature at first yieldmoment ÷ moment at first yieldrectangle: 1.50× the yield moment, at 4.3× the yield curvatureI-section: 1.09× the yield moment, at 1.2× the yield curvaturetee: 1.78× the yield moment, at 8.4× the yield curvature
Fig. 3 Three sections of identical area and identical depth. The rectangle reaches 98% of its plastic moment at 4.1 times its first-yield curvature; the I-section at 1.2 times; the tee at 9.0. The tee is the section that has to bend furthest to deliver what the plastic analysis promises, and it is also the one whose plastic moment is 78% above its first-yield moment — the section with the most in reserve is the section that has to travel furthest to collect it.

That last sentence is the trade that a shape factor by itself conceals, and it has an essay of its own. The relevant point here is that a demand for rotation is generated by the analysis and supplied by the section, and nothing in the analysis checks that the second covers the first.

Which free body produced the number

The rotation figures above come from the section itself, taken as a free body under a curvature. Slice the cross-section into fibres, give each fibre the mild steel curve, impose a straight-line strain distribution of slope κ\kappa, and require the stresses to integrate to zero axial force. That fixes the position of the neutral axis; integrating the same stresses against distance gives the moment.

Sweeping κ\kappa upward gives the whole curve, and the numbers quoted are read off it: the curvature at which the first fibre reaches 0.131% strain, and the curvature at which the moment reaches 98% of the rigid-plastic value computed independently from the equal-area axis. For the rectangle those are in the ratio 4.10.

Two independent checks make that trustworthy. At small curvature the computed moment divided by EIκEI\kappa is 1.000000, which it has to be if the fibre integration reproduces elastic beam theory. And the rigid-plastic moment for the rectangle comes out as 275×100×2002/4275 \times 100 \times 200^2/4 to within one part in a million of the closed form, computed from areas with no stress-strain law involved at all.

The rotation capacity of a member is a harder quantity and the figure above does not claim it. A hinge occupies a length, and turning a curvature into a rotation requires knowing that length, which depends on the moment gradient and on how far the yielded zone spreads along the beam. What the section-level calculation gives is the ratio between shapes, and the ratio is what the argument needs.

What a brittle redundant structure does instead

The cleanest way to see what is being assumed is to take a structure whose whole design rationale is redundancy and remove the ductility from it.

A propped cantilever under a uniform load has one redundancy. Elastically, the fixed end carries wL2/8wL^2/8 and the span carries about 9wL2/1289wL^2/128 — the support moment is 1.78 times the span moment, so the support section reaches its capacity first, at a load well below what the beam as a whole could carry.

One support too manyThe same uniformly loaded beam with three sets of restraints, and the bending moment in each. Adding restraint moves moment from mid-span to the supports and lowers the peak — but only the first case can be solved by statics.simply supportedstatics alonesag 32.0propped at one endneeds stiffnesssag 18.0hog 32.0built in at both endsneeds stiffnesssag 10.7hog 21.3the load never changes; only what is holding the endsthe built-in case peaks at two-thirds of the simple span's moment
Fig. 4 The elastic moment distribution in a propped cantilever, with the fixed end carrying nearly twice what the span does. Everything the plastic analysis subsequently gains comes from that imbalance being allowed to even out, and evening out means the fixed end holding its moment while the beam bends further and the span moment climbs to meet it.

Grant ductility and the story is the familiar one: a hinge forms at the fixed end, the beam becomes simply supported for further load, a second hinge forms in the span, and the collapse load is 11.66 Mp/L211.66\,M_p/L^2 — 46% above the load that first yielded the support. Withhold ductility and the beam fails when the support section reaches its moment, at 8 Mp/L28\,M_p/L^2, and the entire 46% is not merely unavailable but was never there.

What makes this worth stating carefully is that the elastic analysis of the brittle beam is perfectly correct. Nothing about the moment distribution above is wrong. What has gone is the step that follows it — the claim that reaching capacity somewhere is not reaching capacity everywhere — and that step is not in any equation. It is in the sentence between two equations.

Where the model stops

Rotation capacity is not a material property, and it is not a section property either. It is a property of the section, the material, the moment gradient, the axial load and the local buckling behaviour of the plates the section is made of, and the last of those usually decides. A section whose flanges buckle locally at the strain the analysis wants sheds its moment immediately, and its rotation capacity is nearly zero however ductile the steel is. That is the subject of section classification, and it means a ductile material can be assembled into a brittle member.

Ductility measured on a coupon is not ductility available in a joint. A welded connection is a region of altered metallurgy, residual stress and geometric discontinuity, and the strain it can accept is far below the 20% a tensile test reports. Nearly every structural collapse attributed to brittleness has occurred at a joint.

Nor is a coupon’s ductility available where the material is restrained. A tensile specimen thins as it stretches, and the thinning is what lets it stretch. Material in the middle of a thick plate, or at the root of a notch, cannot thin — the surrounding material holds it — and it is therefore pulled in three directions at once. Triaxial tension of that kind suppresses yielding entirely while leaving the stress that drives fracture untouched, so the same steel that reports 20% elongation on a coupon can break at a fraction of a per cent in a restrained detail. This is why plate thickness appears in toughness requirements at all, and why it is the geometry of a joint rather than the specification of its steel that usually decides whether it behaves.

Temperature moves it discontinuously. Mild steel is ductile at 20°C and can be brittle at −20°C — not a little less ductile, but capable of failing at a nominal stress it passes every test at, with no plastic deformation whatever. The transition is sharp and the material’s tensile test at room temperature contains no hint of it.

Three materials pulled until they stopThree stress-strain curves — mild steel, timber, along the grain, concrete — plotted to a strain of 0.6%. One of them has a plateau, so the stress at which yielding starts is something the specimen does rather than something anyone chooses. No offset construction is drawn.00.1%0.2%0.3%0.4%0.5%0.6%050100150200250300350strainstress, N/mm²mild steeltimberconcrete
Fig. 5 Three materials chosen for how differently they run out. Steel yields and travels. Timber is brittle in tension and ductile in compression, which is the opposite arrangement to steel and decides which face of a timber beam fails first. Concrete has almost no tensile capacity at all, which is why it is reinforced — and why the reinforcement is required to yield before the concrete crushes, since that is the only way a concrete member gets any ductility.

The generalisation

The pattern is that an assumption that appears in a derivation and not in its result is invisible to everyone who uses the result.

This site has met the same shape before. Plane sections stay plane is nowhere in σ=My/I\sigma = My/I, and it is the reason that expression fails for a deep beam. The pin-jointed idealisation is nowhere in a truss’s member forces, and it is the reason those forces are missing a bending stress that can reach a quarter of them. Ductility is the same class of thing, and it is the most consequential member of the class because its absence does not degrade an answer gracefully — it removes the guarantee entirely.

The practical response the profession has settled on is not to compute ductility but to legislate the conditions under which it can be assumed: limits on plate slenderness so that local buckling cannot pre-empt yielding, limits on the amount of moment that may be redistributed, minimum reinforcement so that a concrete section cannot crack and break in the same instant, toughness requirements tied to thickness and service temperature, and rules on where welds may be placed. Every one of those is a rule about geometry or specification standing in for a calculation nobody performs.

How much of a section has to have given up

There is one more thing worth measuring, because it explains why the ductility demand is so much larger than intuition suggests. Reaching the plastic moment does not mean the extreme fibre has just yielded. It means nearly the whole section has.

A rectangle at 98% of its plastic momentThe same rectangle drawn three ways: the shape, the strain across its depth, and the stress that strain produces in mild steel. The strain diagram is a straight line, because plane sections stay plane whatever the material is doing. The stress diagram is not: 76% of the area has yielded, working inward from both faces, and the neutral axis sits at 100.0 mm against a centroid at 100.0 mm. The compression resultant is 362.0 kN and the tension resultant 362.0 kN, on a lever arm of 111.7 mm, which multiplies back to the 40.4 kNm the section is carrying.neutral axisrectanglestrainalways a straight linestressthe material's own curve, sidewaysC = 362.0 kN · T = 362.0 kN · lever arm 112 mm · M = 40.4 kNm76% of the area has yielded — 75 mm from the top, 75 mm from the bottom · Mp = 41.2 kNm · shape factor 1.50
Fig. 6 A rectangle at 98% of its plastic moment. The shaded material is what has passed the yield strain, and it is almost all of it — only a thin core either side of the neutral axis is still elastic. Getting there means straining the outermost fibre to many times the yield strain, and the strain diagram is what says how many: it is a straight line through the section, so the outer fibre’s strain is the curvature times half the depth, and the curvature is four times what it was at first yield.

That is the geometric reason the demand is large. The plastic moment is collected by recruiting the material nearest the neutral axis, and material near the neutral axis is at low strain by construction — so bringing it to yield requires the material at the surface to go far past it. A section’s shape factor and its ductility demand are two readings of the same fact, and the second is the one that has to be paid.

A surprising place this turns up

The most valuable thing ductility buys is not capacity. It is information.

A ductile structure approaching its limit deflects visibly, cracks audibly and holds. That interval — between the first sign and the collapse — is what makes it possible to evacuate a building, close a bridge, or notice a badly built structure before it is loaded. It is not in any calculation and it is worth more than most of the margins that are.

The mild steel plateau is the same observation at the scale of a specimen: what the plateau buys is a permanent set that somebody notices. Ductility is the property that converts a structural margin into a warning, and a structure that is exactly as safe but brittle offers the same margin with none of the notice.

Where the ladder goes next

Later rungs on this anchor: rotation capacity computed properly, with hinge length and moment gradient, against the demands that redistribution generates. The upper- and lower-bound theorems proved, and exactly which step of each proof consumes ductility. Section classification as a rotation-capacity requirement rather than a table. Robustness and tying — the design of structures to survive the removal of a member, which is a ductility argument applied at the scale of a building and arrived in codes after Ronan Point in 1968. Ductility in reinforced concrete, where the whole design philosophy is arranged so that the steel yields before the concrete crushes, and where over-reinforcing a section makes it stronger and brittle at once. And capacity design, the seismic idea of choosing where the ductility will be demanded and making everything else stronger than the chosen place.

Historically, the recognition that ductility was doing hidden work came out of failure rather than theory. Plastic design was developed at Cambridge from the mid-1930s in part from the observation that steel-framed buildings damaged by bombing had survived deformations no elastic analysis could account for — the frames had used ductility nobody had asked them for. The Liberty ships going the other way, splitting in two at ordinary stresses in cold water, made the same point in reverse, and the investigation that followed established that a steel meeting every strength requirement could be structurally useless if it could not deform.

Where this goes next

sagging hinge at 4.69hinge at the fixed endlowest upper bound: 7.29every hinge position gives an upper bound on the collapse loadassumed position of the sagging hingecoefficient 11.66 Mp ÷ L² After the first yield, which is not the end A steel beam whose extreme fibre has reached yield has not failed. It has started forming a hinge, and collapse waits until there are enough hinges to make a mechanism. 00.5%1%2%2%0100200300400500600strainstress, N/mm²the 0.2% proof stress: 460 N/mm²mild steelhigh-strength steelaluminium The stress at which nothing in particular happens One material in six has a yield point that a specimen actually does something at. For all the others the yield stress is a construction — a line drawn at an arbitrary offset — and every calculation on this site depends on it. 501001502000100200300400500600crack length, mmstress at failure, N/mm²275 N/mm² crosses at 33.5 mm460 N/mm² crosses at 12.0 mmfracture: the crack decides The flaw that sets the strength A member with a crack twenty millimetres long fails at its yield stress. Make the steel stronger and the crack that does it gets shorter, so the same flaw that was harmless in the weaker grade decides the stronger one. simply supportedstatics alonesag 32.0propped at one endneeds stiffnesssag 18.0hog 32.0built in at both endsneeds stiffnesssag 10.7hog 21.3the load never changes; only what is holding the endsthe built-in case peaks at two-thirds of the simple span's moment One support too many, and what it costs to know Add a redundant restraint and the load has two routes to the ground. Equilibrium cannot say how it splits, and the answer turns out to depend on stiffness — which is a different kind of question.

What this makes readable

Essays that name this one as a prerequisite.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

Brittle fractureDuctilityLoad pathLower bound theoremMoment redistributionPlastic hingeRobustnessRotation capacityStress strain