Structural Engineering & Statics
where the load goes
Concept

Equilibrium path — where it appears

The whole curve of load against displacement a structure follows, whose maxima are limit points and whose branches are its alternatives. Following it past a limit point needs displacement control, which is why a testing machine that pushes gives a different answer from one that pulls to a stroke.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

A load with a maximum in it, and nothing bifurcates. Load against apex movement for a two-bar frame of half-span 1000 mm and rise 150 mm. The load rises to 133.4 kN at a movement of 64 mm — well short of the 150 mm that would bring the apex level — and then falls. Past that point the frame can only be held by taking load away, so under a dead weight it goes: 260 mm of movement at constant load, arriving inverted and in tension. The minimum on the path is -133.4 kN, the exact negative of the maximum, because the geometry is symmetric about the flat position and the arithmetic knows it.

The roof that jumps

Every stability failure in this collection so far has been a bifurcation — a straight thing discovering it can be bent. A shallow frame does something else entirely. It stays perfectly symmetric, deforms steadily, and at some point the load it can carry starts to fall while it is still moving in the direction it was pushed.

stability · Snap-through Three paths out of the same critical load. Load against sideways movement past the critical load, for three systems whose critical loads are identical. The stable one climbs, so a real structure with a small crookedness reaches nearly the full load and keeps going. The unstable one falls symmetrically, so the imperfect structure has a maximum below the critical load and it matters not at all which way it leans. The asymmetric one falls one way and climbs the other, so the direction of the imperfection decides everything. All three are drawn at an imperfection of 0.02 radians.

A third of what the theory promised

A column with a small crookedness reaches almost its full Euler load. A cylinder with the same relative crookedness reaches a third of its classical one, and the theory is not wrong — what separates them is the slope of the path just past the critical load, which no calculation of the critical load itself can see.

stability · Imperfection sensitivity Two frequencies that meet, and a determinant that never moves. The two natural frequencies of Ziegler's two-bar column against the follower load Pℓ/k, with the determinant of its stiffness matrix drawn along the top. The determinant is k² at every load — it varies over this whole axis by 1.1e-16 of itself, which is round-off — so a static buckling analysis of this structure finds no critical load whatever and reports it as stable everywhere. The frequencies say otherwise: they approach, meet at Pℓ/k = 2.0858, the closed form (7 − 2√2)/2, and become a complex pair, which is oscillation that grows. With no damping that merge is also where the column goes unstable. Adding any internal damping at all drops the load at which that happens to 1.4643, which is 41/28 and 30% below the undamped value; the limit of the damped system is not the undamped system, which is the paradox Ziegler found in 1952 and which was taken for an arithmetic error for a decade.

The load it cannot buckle under

Every stability calculation on this site rests on an assumption nobody states: that the load has a potential, so a critical load is where a total potential energy stops being a minimum. A load that turns with the structure it is pushing has no potential, and the static analysis of such a column returns no critical load at all — a determinant that never vanishes, for a column that fails at a perfectly finite one.

stability · Follower force Three grounds, three tipping pushes. The push against the lean it produces, in thousandths of a radian, for the 6 m square, 3,000 kN block on ground that bears 300 kPa at most, half of it mobilised at 7.5 mm of settlement, pushed 8 m up, its weight 5.0 m up, on the three grounds matched at their limit and at half of it; dashed, the rigid-plastic limit, 813 kN. Linear, capped: 776 kN at a lean of 13.2 thousandths; hyperbolic: 701 kN at a lean of 26.2 thousandths; S-shaped: 770 kN at a lean of 15.2 thousandths. The S-shaped ground behaves almost exactly like the capped one; the hyperbola, which never quite reaches its limit, tips 10 per cent sooner at twice the lean.

The ground that never quite gives way

On a bed of springs capped at the ground's bearing limit, a block pushed sideways tips at a definite push, below the limit the rigid-plastic parabola gives. Real ground is curved: it softens all the way to its limit and never quite reaches it. On such ground the block has no tipping push of its own. Its resistance creeps toward the limit, and the peak it does have is made by its own weight leaning with it — 14 per cent below the limit with the weight 5 m up, nearly at it with the weight at the base, where the lean simply runs away.

equilibrium · Overturning

Named alongside it

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

BucklingBifurcationImperfection sensitivityStabilityAeroelastic instabilityBearing capacityContact pressureCritical loadDampingDivergenceEigenvalueFactor of safety

All concepts