Structural Engineering & Statics
where the load goes
Concept

Complementary energy — where it appears

The area between a member's stress–strain curve and its stress axis, integrated over its volume: the integral of strain with respect to stress. Its derivative with respect to a load is the deflection at that load for any elastic material, where the strain energy's is right only for a linear one.

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

A derivative taken with a ruler, and the step that makes it worst. Castigliano's theorem says the deflection is ∂U/∂P, and the derivative here is taken numerically — two solves at ±dQ and a central difference. Against the unit-load answer of 1.720635e-2 it agrees to 1.6e-13, which for a linear structure it must: ∂N/∂P is exactly the force a unit load produces, so the two expressions are the same sum written twice. The error against step size is the classic pair of straight lines — truncation falling as the step shrinks, round-off rising as the difference of two nearly equal energies loses its digits — meeting near dQ = 1.2e+1. For a linear structure the truncation term is exactly zero, so what is drawn here is round-off alone.

The deflection that is a derivative

A structure's strain energy is one number. Differentiate it with respect to a load and out comes the displacement under that load — and the trick that makes it a method rather than an identity is that the load does not have to be there.

deflection · Strain energy Three routes to one deflection, and the one that is wrong. Two 2500 mm bars of 250 MPa proof stress meeting at a loaded apex, the drop of the apex against the load. The line is geometry: each bar's extension from its own stress, divided by the sine of its slope. The dots are the derivative of the total complementary energy with respect to the load, and lie on it. The dashed line is the derivative of the strain energy — Castigliano's theorem applied to a material that is not linear — which leaves the truth by ten per cent at 99 kN and at 170 kN gives 308 mm for a deflection of 46.0.

The other area under the curve

Castigliano's theorem says a deflection is the derivative of the strain energy with respect to the load, and it is true only while the material is linear. Past that, the right energy is the area on the other side of the stress–strain curve. On two aluminium bars at their proof stress the strain energy gives a deflection four times too large, and on a redundant truss minimising it picks a set of forces in perfect equilibrium that no deformed shape can produce.

deflection · Strain energy The strain-energy route halves the sag. The midpoint deflection of a cable 10.0 m long, pretensioned to 60 kN, with an axial stiffness EA of 64 MN, loaded across its span at midpoint, against the load, found three ways. Solid: the deflection itself, from the cable's geometry; the dots, the derivative of the complementary energy with respect to the load, lie on it. Dashed: the derivative of the strain energy. Dotted: the pretension alone, as if the cable's tension did not grow. At 10.0 kN the cable sags 212 mm; the strain-energy route gives 107 mm, 0.51 of it, and the pretension alone 417 mm. At 20.0 kN: 294, 128 and 833 mm. The strain-energy route understates the deflection of a member that stiffens by the same mechanism that made it overstate one that softens.

The sag the strain energy halves

The derivative of the strain energy overstates the deflection of a member that softens. Turn the curve the other way up — a cable that grows stiffer as it sags, a hanger that takes up its play — and the same derivative understates it: by half for a pretensioned cable carrying a modest load, by two thirds for one with no pretension, and by the whole of the slack for a member that has any. The error is one ratio in both directions, and the least-work shortcut built on it hands the load to the member that stiffens.

deflection · Strain energy The wrong energy moves load towards the cable furthest along its curve. Each cable's share of 50 kN, by compatibility and by minimising the strain energy. With lengths of 30, 22, 16 and 12 m and one pretension: 0.08, 0.14, 0.27, 0.51 against 0.04, 0.07, 0.22, 0.67. With one length of 20 m and pretensions of 20, 60, 120 and 240 kN: 0.16, 0.20, 0.26, 0.38 against 0.33, 0.28, 0.21, 0.18.

The load goes to the cable furthest along

Applied to one stiffening member, Castigliano's theorem halves or thirds its deflection and leaves its force alone, since there is only one member to carry it. A net has many, each at its own place on its own curve, and when the wrong energy has to divide a load among them it hands it to whichever is furthest along. The peak cable force comes out a fifth too high on one net and a quarter too low on another, by the same mistake. Only a net with no pretension at all gets the force right, and even that one gets the deflection wrong by a factor of three.

deflection · Strain energy

Named alongside it

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

Strain energyCastiglianoCompatibilityRedundantForce methodVirtual workAluminiumCable stiffnessDeflectionDummy loadFlexibilityGeometric stiffness

All concepts