Structural Engineering & Statics
where the load goes
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Virtual work — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · deflection
  2. 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.

    part 2 · deflection
  3. 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.

    part 3 · deflection
  4. The column line's sum agrees with every analysis, because it cannot see the beams. The roof sway of a 4-bay, 10-storey concrete frame, 8 m bays and 3.5 m storeys, 600 mm columns and 400 × 700 mm beams, 40 kN/m on every beam and 40 kN of wind at every floor, against the beams' stiffness as a share of their gross value, from the whole frame (line) and from the windward column line's integral of the moments that analysis gives it (dots). At the gross stiffness, 15.7 mm; at 0.5, 25.4; at 0.35, the usual allowance for cracked beams, 33.3; at 0.2, 52.0. The windward column's base moment moves from 126 to 235 kN·m, and the line's integral follows the sway exactly at every one.

    The check that cannot see the beams

    The unit load that finds a frame's sway may be carried on any structure that can carry it, and in a frame of four bays and ten storeys it can be carried down one line of columns alone. Then the roof sway is that line's real moments against the unit load's, a few dozen products by hand — exact for the frame as drawn, for its beams cracked to a third of their stiffness, for its joints made springs and for its bases made springs, the last only once the bases' rotation is given its own term. It is exact because the unit load never enters a beam. Which makes it a check on the analysis's arithmetic that cannot say anything about the analysis's beams.

    part 4 · deflection

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