Structural Engineering & Statics
where the load goes
Concept

Stiffness matrix — where it appears

The array relating the forces at a structure's freedoms to the displacements there. It is assembled by adding each member's own matrix into the rows and columns of the freedoms that member touches. Nothing in the assembly is an approximation; it is the same equilibrium and compatibility a hand method writes, in an order a machine can follow.

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

five floors · 300 t eachmode 10.6 s · 1.65 Hz88.0% of the massno nodemode 20.21 s · 4.83 Hz8.7% of the massone nodemode 30.13 s · 7.61 Hz2.4% of the masstwo nodes

A structure has more than one period

One mass on one spring has one period. A building with eight floors has eight, each with a shape of its own, and the second one bends the building into a curve nobody drew. They are not harmonics, they do not interfere, and each behaves as though the others were not there.

dynamics · mode shapes 5010015020000.20.40.60.811.2slenderness (effective length ÷ radius of gyration)they cross at λ = 85squashingEuler bucklingreal columns, which are neither

The stiffness the load takes away

Buckling is usually taught as an event — a critical load, a bifurcation, a mode. Written as a matrix it stops being an event at all. A compressive load subtracts a stiffness from the structure, the subtraction grows with the load, and the critical load is simply where what is left reaches zero.

stability · geometric stiffness 0123four nodes, three freedoms eachthe marked member touches six of the twelvethe global matrix, twelve by twelvethirty-six entries added in; the rest untouched

The answer that depends on how it was divided

Every computed answer in this collection came out of a structure chopped into pieces — elements, strips, stations, trial positions. The chopping is invisible in the result and it is not neutral: some divisions give the exact answer, some give one that is always too stiff, and one of them changes nothing but the cost of getting there.

deflection · discretisation

Named alongside it

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

Degrees of freedomEigenvalueCompatibilityMode shapeRayleigh quotientAmplificationBandwidthBucklingConvergenceCritical loadDiscretisationEffective length

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