Structural Engineering & Statics
where the load goes
Concept

Product integral — where it appears

The integral along a member of one moment diagram multiplied by another, divided by the bending stiffness. With a real diagram and a unit-load diagram it is a deflection, and because both are straight lines and parabolas it reduces to a table of standard shapes.

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

The deflection at x = 4, by virtual work. Three diagrams: the moment from the real load, the moment from a unit load placed where the answer is wanted, and their product. The area under the third, divided by EI, is the deflection — 213.33 here. No standard case was consulted, so the method works for any load pattern at all.

One deflection, without solving everything

To find how far one point of a structure moves, put an imaginary force of one unit there, multiply two moment diagrams together, and integrate. The answer arrives without ever solving for the deflected shape.

deflection · Virtual work A unit load carried by the prop taken away. A beam fixed at its left end and propped at its right, under 4 kN/m over 8 m, asked how far it moves at 4 m. The real moment is the propped cantilever's own, with -32.0 kNm at the wall. The unit load is carried by the prop taken away, whose moment diagram peaks at 4.00. Their product has 98.92 of area on one side and -13.33 on the other, and the net, divided by EI, is 85.33 — the propped cantilever's closed-form deflection, 85.33, which no part of this calculation was given.

Any structure will carry the unit load

Virtual work has two readings and each is free exactly where the other is bound. A unit load needs only something to stand on in equilibrium, so the deflection of a beam statics cannot solve comes out of a cantilever statics can. A virtual displacement needs only to fit together, so a reaction comes out of pushing a mechanism — and on a redundant beam the unknown cancels out of the equation and nothing is found at all.

deflection · Virtual work Five ways to share one deflection among four members. The sway at the top of the left column of a portal 12 m wide and 6 m high with fixed bases, an IPE 500 beam on HEB 300 columns, carrying 25 kN/m on the beam and 40 kN sideways at the top of the left column, 10.38 mm, found five times by the unit load: on the frame itself and on four releases of it. Each bar is split into what each member's integral contributes — left column, left half of the beam, right half, right column — and the red line is the total, which every bar reaches. In millimetres, member by member: the frame itself −3.72, 2.16, 0.66, 11.28; left column alone 10.38, 0.00, 0.00, 0.00; right column alone 0.00, 0.17, 0.17, 10.04; three hinges −15.78, 4.90, 1.41, 19.84; pin and roller −31.43, 27.10, 14.55, 0.15. A member's share can be negative, and on one structure it is the whole answer.

One column is enough to find the sway

The unit load that finds a deflection may be carried on any structure that can carry it, so in a fixed-base portal the sway can be found from one column's two end moments and nothing else. Carry it on a three-hinged frame instead and the same sway comes out of all four members, one of them contributing a negative amount. Every structure gives the same total and a different account of where it came from — which means "how much of the sway is axial shortening" has no answer until a structure is named, and only one structure's answer is worth having.

deflection · Virtual work

Named alongside it

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

CompatibilityUnit load methodVirtual workForce methodReleased structureStiffnessEquilibriumIndeterminacyMechanismMoment diagramPlastic hingePoint of contraflexure

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