Structural Engineering & Statics
where the load goes
Concept

Released structure — where it appears

The determinate structure left when a redundant structure's extra restraints are removed. The force method computes movements on it to find the redundants, and a unit load may be placed on any such release to find a deflection of the real structure.

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

Two different structures released, and one bending moment diagram. The bending moment in a continuous beam of 8, 10, 8 m under 12 kN/m, solved twice by the force method with different redundants. The first release puts a hinge over each interior support, so the released structure is a row of simple spans and the redundants are moments. The second removes each interior support, so the released structure is one simple span of the whole length and the redundants are reactions. The two released structures have nothing in common — different shapes, different deflections, different everything — and the diagrams they produce lie on top of each other to 9e-15 of the peak moment. Which restraints are released is a choice about the arithmetic and not about the structure, which is a fact worth trusting: it means a hand calculation can pick whichever release makes the sums easiest and be sure of the answer.

Choose what to take away

The other machine for a redundant structure works by removing restraints until what is left can be solved by statics, then putting back exactly enough force to close the gaps that opened. Which restraints are removed does not change the answer at all, and changes the arithmetic completely — one choice gives a tridiagonal matrix a person can solve on paper, and another gives a full one.

deflection · Force method 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 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.

deflection · Virtual work

Named alongside it

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

CompatibilityVirtual workIndeterminacyUnit load methodEquilibriumForce methodProduct integralRedundantBending momentContinuous beamDriftFlexibility

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