Structural Engineering & Statics
where the load goes
Concept

Self excitation — where it appears

A load manufactured by the structure's own movement, which has a threshold rather than a magnification and grows without bound past it. Its energy comes from a steady flow rather than from an oscillating load, which is why a bridge can be destroyed by a wind that never changed speed.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The damping a crowd leaves behind, and the number of people that uses it up. Total damping ratio against pedestrians, for a structure with 0.60% of its own and a modal mass of 120 tonnes at 0.5 Hz. Each person walking laterally puts a force in phase with the deck's velocity into it — about 300 N·s/m per person — so the crowd subtracts from the damping, and at 30.16 people what is left is nothing. Below the threshold a disturbance dies away. Above it, the structure feeds itself and the motion grows out of nothing at all — which is what makes this a different mechanism from resonance rather than a severe case of it.

The bridge that was pushed by its own sway

A crowd walking on a bridge that moves sideways adjusts its footing to stay balanced, and the adjustment pushes the bridge the way it is already going. The crowd is a damper with the sign reversed, and past a certain number of people the total damping is negative.

dynamics · crowd synchronisation The damping a wind leaves behind, and the speed that uses it up. Total damping ratio against wind speed, for a structure with 0.60% of its own and a modal mass of 25 kg per metre at 1.2 Hz. The aerodynamic damping of a section whose lift falls with angle of attack is negative and grows with speed, so at 4.14 m/s what is left is nothing. Below the threshold a disturbance dies away. Above it, the structure feeds itself and the motion grows out of nothing at all — which is what makes this a different mechanism from resonance rather than a severe case of it.

The motion that feeds itself

A steady wind contains no frequency at all, and it can destroy a bridge. The force that does it is manufactured by the structure's own movement, so there is no excitation to resonate with — there is a wind speed above which the equilibrium is unstable, and below which nothing happens.

dynamics · aeroelastic instability 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 is (7 − 2√2)/2 = 2.0858 — and become a complex pair, which is oscillation that grows. 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

Named alongside it

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

DampingFlutterNegative dampingStability thresholdAeroelastic instabilityAeroelasticityBucklingCritical loadCritical velocityCrowd loadingDivergenceEigenvalue

All concepts