Structural Engineering & Statics
where the load goes
Concept

Force method — where it appears

A way of analysing an indeterminate structure by releasing its redundants and restoring compatibility with the forces that requires. Each redundant comes back as a ratio of two flexibility integrals, which makes visible which flexibility in the structure is deciding the answer.

Named by 9 essays across 4 fields — 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 The tie is a redundancy, so its stiffness decides the thrust. Thrust and rib bending for a 60 m tied arch of 0.15 rise ratio, against the stiffness of its tie. Cut the tie and the structure is a curved simply supported beam, so the tie force is the one redundant and the force method gives it: with a rigid tie the answer is 2227 kN, within 1.0 per cent of the funicular wL²/8f, and the shortfall is the arch's own axial shortening. A real tie stretches 68 mm and returns 2166 kN — 2.7 per cent of the flexibility is the tie — and whatever thrust the arch does not get, it carries as bending: 760 kNm at 30 m. A tenth of the tie stiffness is not a tenth of the problem; it is a different structure.

The thrust that never reaches the ground

Every arch on this site has ended at the same sentence — the foundation is where an arch is really decided. A tie changes the sentence without changing the arithmetic: the horizontal force is still there, still the same size, and it now closes on itself through a bar at deck level.

structures · Tied arch 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 Five millimetres short, and a hundred kilonewtons in every member. An X-braced bay 6 m by 4 m in which one diagonal was fabricated 5 mm short, with the force that leaves in every member. Nothing is applied to this frame. The forces are the self-stress state the frame's one redundancy supports, scaled so that the diagonal is pulled back to the length it should have been: tension in both diagonals at 100 kN, compression in the four members round the outside, and the whole set in equilibrium with nothing. That is 25% of the force the diagonal was sized to carry, and it is there for the life of the structure. Take one diagonal out and the frame becomes determinate: the short member then simply puts the joint somewhere else, and the structure is in the wrong place instead of under stress. Redundancy is bought, and this is the price.

Built to the wrong length

A redundant structure's members do not have independent lengths. Choose all but one and geometry decides the last, so a member made a different length has to be pulled or pushed into place — and the force required stays in the structure for as long as the structure does. Nothing has been applied to it, there is no load case and no factor, and the members are carrying real force.

connections · Fit-up The answer arrives in instalments. The hogging moment at support 1 of a three-span beam, cycle by cycle. It starts at the fixed-end moment of 98.0 kNm — the value with every joint clamped — and settles at 156.9 kNm against an exact 156.9. The error falls by about a factor of four per cycle: 21.03, 5.92, 1.54, 0.60 kNm after one, two, three and four. Two cycles is an engineering answer and nobody had to invert anything.

Why it converges, and how fast

Moment distribution is an iteration, and iterations do not always converge. This one always does, at a rate the beam's own proportions fix — about a factor of four per cycle on a regular beam and considerably worse on an irregular one, which is where the method's reputation for two cycles being enough comes from and where it stops being true.

deflection · Moment distribution Three moment diagrams for one pair of beams. Two 15 m spans carrying 12.0 kN/m, drawn sagging downward. As built they are simple spans: 337.5 kN·m at each midspan and nothing over the middle support. Built monolithic they would carry 337.5 kN·m of hogging over the support and 168.8 at midspan. Loaded at 28 days and made continuous at 60, creep takes them 46 per cent of the way from the first to the second: 155.2 kN·m over the support and 259.9 at midspan, after twenty years in which nothing about the loading changed. The support had no moment on the day it was cast and no drawing of the finished structure shows why it has one now.

The support that had no moment when it was cast

Two beams are set on their bearings, carry their own weight for a month, and are then stitched together over the middle support. Nothing about the loading changes afterwards. Twenty years later the stitch is carrying 155 kilonewton-metres, because the concrete went on creeping and the joint would not let it — and how much arrives is decided by a crane schedule.

materials · Creep 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 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

Named alongside it

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

CompatibilityRedundantVirtual workFlexibilityBending momentCastiglianoComplementary energyEquilibriumImposed deformationIndeterminacyMoment distributionProduct integral

All concepts