Structural Engineering & Statics
where the load goes
Concept

Moment-curvature — where it appears

A section's own load-deformation curve, relating the moment it carries to the curvature it takes. The length of its plateau is the section's rotation capacity, and its shape decides whether the member can redistribute anything.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

What it costs to reach the plastic moment, for two shapes. Moment against curvature for two 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 4.1 times the curvature at first yield; The I-section has a shape factor of 1.09 and reaches 98% of its plastic moment at 1.1 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.

The section that yields from the outside in

A rectangle has half again as much moment in reserve past first yield as its elastic capacity suggests, and an I-section has a seventh. Read as a ranking that gets it backwards — the reserve is bought with curvature, and the rectangle pays four times as much of it.

materials · Moment-curvature A I-section at 80% of its plastic moment. The same I-section 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: 0% of the area has yielded, working inward from both faces, and the neutral axis sits at 70.9 mm against a centroid at 100.0 mm. The compression resultant is 322.5 kN and the tension resultant 322.5 kN, on a lever arm of 180.1 mm, which multiplies back to the 58.1 kNm the section is carrying. A rolled residual stress pattern of ±30% of yield is locked in before any load arrives.

The stress that was there before the load

A rolled steel section leaves the mill carrying eighty N/mm² of stress with nothing applied to it, in a pattern that sums to no force and no moment. It is invisible to every calculation and it is the knee in every column curve.

materials · Residual stress Where the class limits come from. The width-to-thickness ratio at which two kinds of plate reaches its own elastic critical stress at the yield stress, for three steel grades. A flange outstand (buckling coefficient 0.43) derives to 18.6, 15.2, 13.3 at 235, 355, 460 N/mm², against quoted limits of 14.0, 11.4, 10.0; A web, in bending (buckling coefficient 4) derives to 56.8, 46.2, 40.6 at 235, 355, 460 N/mm², against quoted limits of 42.0, 34.2, 30.0. The derived number is the larger every time, and by the same factor at every grade — flange outstand 1.33, web, in bending 1.35 — because both the derivation and the quoted limit go as one over the root of the yield stress. A constant ratio is what a fixed knockdown looks like: the derivation is for a perfect plate and the quoted limit is for a rolled one, carrying residual stress and not quite flat.

The section that cannot reach its own strength

A section classification looks like a table of arbitrary numbers. Set a plate's buckling stress equal to the yield stress and the numbers fall out of the plate buckling formula — larger than the quoted ones by a constant factor, at every grade.

materials · Section classification The same concrete, held sideways. Two stress-strain curves for one concrete. The lower is a cylinder test: it peaks at 30 N/mm² near a strain of 0.002 and has nothing left by 0.0035, because it fails by splitting apart sideways. The upper is the same material inside a 12 mm hoop at 100 mm centres, which cannot stop it expanding but can make the expansion stretch steel: the lateral pressure of 2.48 N/mm² — 8% of the strength it is multiplying — takes the peak to 44.4 and the ultimate strain to 0.028. The strength gain is 1.48 times and the strain gain 8.0; the area under the curve, which is the toughness, goes up by 11. It is the third number the confinement is provided for.

Squeezed sideways into a different material

Concrete in a cylinder test fails by splitting apart sideways under a load pushing it down. Put a hoop round it and the splitting has to stretch steel — and a lateral pressure of a twelfth of the strength raises the strength by half and the ultimate strain by eight.

materials · Confinement What is left after the first fibre yields, which is a property of shape. The shape factor — plastic modulus over elastic — for six sections, computed by finding each one's equal-area axis and summing ±f_y over it. The numbers contain no dimension, no stress and no material: a rectangle is exactly 3/2 whatever its size, a diamond exactly 2, a circle 16/3π. The spread is the argument. An I-section keeps only 13 per cent in reserve past first yield, because nearly all its material is already at the extreme fibre and there is nothing further in to recruit; a diamond keeps 100 per cent, because most of its material is near the middle and doing very little elastically. So the section shapes that are best at elastic bending are the ones with the least left afterwards, which is exactly backwards from the way the reserve is usually described.

What is left after the first fibre yields

The elastic section modulus stops at the moment the outermost fibre reaches yield. Nothing else in the section has, so it goes on taking load — and how much more it takes turns out to be a property of the shape alone, with no dimension, no stress and no material anywhere in the answer.

sections · Shape factor Wrong in shape, right in two integrals. The compression zone of a C30 section with its neutral axis 150 mm down, drawn twice. The curved outline is the real parabolic-rectangular stress distribution — the material's own law read off the linear strain profile plane sections supplies. The rectangle over it is what every design office uses instead: intensity η f_cd = 16.5 MPa over a depth λx = 125 mm. The two shapes are visibly different and give the same answer, because a bending calculation asks a stress distribution only two questions — how much compression there is, and where its resultant acts. Both are 619 kN at 62.4 mm from the face. The factors are α = 0.8095 and β = 0.4160, and λ = 2β follows from wanting the same centroid. A triangle and a full rectangle match neither integral and are nowhere near.

Deliberately the wrong shape

Concrete in compression follows a curve, and no design office has ever integrated it. Every code in the world replaces it with a rectangle of reduced depth and reduced intensity, and the answer is right to a fraction of a per cent — not because the shapes are similar, which they visibly are not, but because a bending calculation only ever asks a stress distribution two questions.

sections · Stress block Eight slices is enough, and nobody would have guessed it. The error in a cracked section's moment capacity against the number of strips it was integrated with, for a 300 × 450 mm section with 1200 mm² of steel, measured against the same computation at 2048 strips. The point of the fibre method is that it contains no formula: slice the section, give every strip the strain the assumed curvature puts it at, move the neutral axis until the axial force balances, and sum. It handles a cracked section, a confined one, a prestressed one and a composite one with the same twenty lines. The discretisation costs 1.4% at 2 strips and 0.088% at 8 — and the convergence is not smooth, because what the error actually depends on is where the neutral axis falls relative to a strip boundary rather than on the strip count as such.

The section calculation with no formula in it

Every ordinary section result is a closed form, and each was derived once for one arrangement of material. Slice the section instead, give each strip the strain a curvature puts it at, and move the neutral axis until the axial force balances — and the same twenty lines answer for a cracked section, a confined one, a prestressed one and a composite one, having been told nothing about any of them.

sections · Fibre model 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 axis moves when the section yields. Six sections, each drawn to its own scale, with their elastic neutral axis — the centroid, dashed — and their plastic neutral axis, the equal-area axis, solid. For the symmetric ones the two lines are the same line and the distinction never arises, which is why it is so easily missed. For the tee they are 23% of the depth apart, because the axis that makes the first moment of area vanish is not the axis that makes the two areas equal. The shape factors run from 1.144 to 1.800 across these six, and they are ratios of moduli taken about two DIFFERENT axes — which is also why an asymmetric section has two elastic section moduli, one to each extreme fibre, and only one plastic modulus. The tee's two elastic moduli differ by a factor of 2.78; a fully plastic section does not care which fibre reached yield first, so it has nothing to be two of.

The axis that moves when the section yields

An elastic section bends about its centroid. A fully plastic one bends about the axis that halves its area, and for anything symmetric those are the same line — which is why the distinction is almost never met. For a tee they are a fifth of the depth apart, and three things follow that the elastic calculation gives no warning of.

sections · Equal-area axis

Named alongside it

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

DuctilityNeutral axisPlane sectionsSection modulusStress blockLever armPlastic hingePlastic modulusRotation capacityShape factorCentroidComposite action

All concepts