Structural Engineering & Statics
where the load goes
Concept

Nonlinearity — where it appears

Any behaviour in which the response is not proportional to the load, whether because the material yields, the geometry changes, or a contact opens. It removes superposition, so load cases can no longer be added and each combination has to be analysed as it stands.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

The windward guy tightens, the leeward one gives way. A 120 m mast on three guy levels, at a wind of 3 N/mm, with the deflection drawn 0.54 times its true size. The guys start at 160 kN each and end at 259 against 95, 299 against 83, 221 against 112 kN. The leeward guys still carry a real force — the lowest keeps 28 per cent of its partner's tension — and supply almost none of the restraint, because their tangent modulus has fallen to 54 per cent of the steel's. The mast top moves 100 mm, its worst bending moment is 554 kNm at 40 m, and it is carrying 801 kN of axial load that nothing but the guys put there.

Held by something that goes soft

A guy is a cable, so it has no stiffness of its own — what resists a mast's movement is the guy's geometry changing, and how much of that there is depends on the tension already in it. Wind pushes the mast towards the leeward guy, which is the one losing tension.

structures · Guyed mast One pulse, two blocks of the same shape. Rotation as a fraction of the toppling angle, under a single 0.8 s sine pulse of 1.00 g, for two blocks of identical proportion whose sizes differ by a factor of 3. Both lift off at the same instant, because uplift depends on the shape alone. The small one reaches the toppling angle and goes over; the large one reaches 46%. The kinks are impacts: there is no dashpot anywhere in this model, and the only energy the block loses is lost when it lands on its other corner, at a velocity ratio of 0.926 per landing — 14% of the energy each time, decided by the block's shape and by nothing else.

The only damping is the landing

A rocking block has no dashpot in it. The only energy it ever loses is lost at the instant it lands on its other corner, and how much is a property of the block's proportions — 14 per cent for a slender one and 38 for a stocky one. That single number decides whether it settles or goes over, and a real base does not deliver the value the theory computes.

dynamics · Rocking The linear answer has the ground holding the ends down. The contact pressure along a strip 16 m long, EI 5.40 × 10⁵ kN·m², on a bed of 50 × 10³ kN/m² per metre of its length, under a column of 1,000 kN. With springs that pull as readily as they push (dashed) the bed presses 197 kN/m under the column and pulls down on the ends at up to 34 kN/m — tension the ground cannot supply. With the pulling springs released (solid), the strip touches the ground only from 3.97 m to 12.03 m, 8.05 m in all, and presses 213 kN/m under the column, 8 per cent more; beyond that it carries nothing.

The ends the ground was holding down

Winkler's springs pull as readily as they push, and on a long footing strip the linear answer quietly uses that: it has the ground holding the strip's ends down. Take the pull away and the correction under the column is modest — eight or nine per cent on the pressure and the moment — but the shape of the answer changes completely. A weightless strip longer than about eight metres keeps exactly eight metres of itself on the ground, whatever length was poured, and the rest rises off as straight cantilevers carrying nothing. The middle-third rule turns out to be a property of rigid footings: an eight-metre strip lifts at a fiftieth of the eccentricity the kern allows.

internal-forces · Elastic foundation A hanger that would push is a hanger that is not there. The force in each hanger of a tied arch of 100 m span and 17 m rise on 32 crossed hangers, carrying 60 kN/m of dead load over the span and 120 kN/m of live load over its left half, against the hanger's position along the tie. Solved as though every hanger could push (pale bars), 4 come out in compression, the worst at 185 kN. Solved with those hangers taken out until no remaining one is compressed (dark bars), 8 are slack — more than the linear solution has in compression, because each hanger that goes passes its share to its neighbours and some of them follow.

The deck that keeps the hangers in the truss

Crossing the hangers of a tied arch turns it from a frame that bends into a truss that does not, as long as every hanger is pulling. A hanger cannot push, and under a load on half the span some of them would have to. Solved as it really is — with those hangers taken out, and the ones that follow them — the arch keeps the linear answer exactly until the live load on half its span is about equal to its own dead weight, and past that the tie's moment doubles and then quadruples. The arch's own weight is the pretension nobody had to apply, and a heavier deck is a stiffer network.

structures · Network arch

Named alongside it

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

CableAnchorBeam-columnBucklingCharacteristic lengthContact pressureDampingDead loadEccentricityElastic foundationElastic supportEnergy dissipation

All concepts