Structural Engineering & Statics
where the load goes
Concept

Discretisation — where it appears

The division of a continuous structure into a finite set of elements, strips, stations or trial positions so that it can be computed. Where the true field lies in the family the division can represent, the answer is exact; where it does not, restricting a displacement field can only stiffen, so the error has a known sign.

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

Eight slices is enough, and nobody would have guessed it. The error in a cracked section's moment capacity against the number of strips it was integrated with, for a 300 × 450 mm section with 1200 mm² of steel, measured against the same computation at 2048 strips. The point of the fibre method is that it contains no formula: slice the section, give every strip the strain the assumed curvature puts it at, move the neutral axis until the axial force balances, and sum. It handles a cracked section, a confined one, a prestressed one and a composite one with the same twenty lines. The discretisation costs 1.4% at 2 strips and 0.088% at 8 — and the convergence is not smooth, because what the error actually depends on is where the neutral axis falls relative to a strip boundary rather than on the strip count as such.

The section calculation with no formula in it

Every ordinary section result is a closed form, and each was derived once for one arrangement of material. Slice the section instead, give each strip the strain a curvature puts it at, and move the neutral axis until the axial force balances — and the same twenty lines answer for a cracked section, a confined one, a prestressed one and a composite one, having been told nothing about any of them.

sections · Fibre model One member's stiffness, scattered into the freedoms it touches. A member's own six-by-six stiffness matrix relates the forces at its two ends to the displacements there, and it is written in the member's own axes. Assembly is two operations and no physics: rotate it into the structure's axes, then add each of its thirty-six entries into the row and column of the global freedom that entry belongs to. Every member does the same, and the sum is the structure. The shaded rows and columns are the six freedoms this one member reaches; every other entry it contributes is exactly zero, and that is the whole reason a global stiffness matrix is sparse. Nothing here is an approximation — the result is the same equilibrium and the same compatibility a hand method writes, in an order a machine can follow.

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 Two load sets with the same resultant and different work. The two ways of putting a uniform load of 10 kN/m onto a beam element 6.00 m long. Both put 30.0 kN at each node, so both have the same resultant and the same moment about any point — they are equivalent for a rigid body. The consistent set adds a couple of 30.0 kN·m at each end, in opposite senses, which is what makes it do the same virtual work over the element's shape functions as the real load does. The couples cancel in the resultant, which is exactly why the resultant cannot see them, and they are the whole difference between an exact answer and one that is a third out.

Equivalent in work, not in resultant

Two force systems with the same resultant and the same moment about every point are interchangeable — for a rigid body. A finite element is not a rigid body, and substituting one for the other on a beam element leaves the tip of a cantilever a third too low with no warning of any kind.

equilibrium · Force couple

Named alongside it

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

ConvergenceShape functionVirtual workBandwidthCompatibilityComposite actionConfinementCracked sectionCurvatureDeflectionDegrees of freedomEigenvalue

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