Structural Engineering & Statics
where the load goes
Concept

Point of contraflexure — where it appears

A station where the bending moment passes through zero, so the member changes from sagging to hogging and its curvature reverses.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

00.050.10.150.20.250.30.3501020304050overhang, as a fraction of the spanbest at 21% — peak 8.8sagging, mid-spanhogging, over the supportmoving the supports in by a third of the way halves the worst moment

Where to put the supports, which is not at the ends

Moving the supports of a uniformly loaded beam inward by about a fifth of its length halves the worst bending moment. The load has not changed and nor has the beam.

internal-forces · support layout moment19.6 sagging24.5 hogging30.6 if the spans were simplereactions 14.0 38.5 38.5 14.0 — the inner supports carry far more than a sharethe continuous case needed stiffness; the comparison did not

The moment over the support, and what it buys

Run a beam over its supports instead of stopping at each one, and the mid-span moment falls by a third while a new moment appears where there was none. Nothing was added but continuity.

internal-forces · continuity 10centroid at 2.00 mM/EIarea = 180.00 · first moment = 720.00by double integration: 720.00

The area of a diagram is a rotation

A deflection is the double integral of a bending moment, and the two constants of integration are the whole difficulty. Mohr's theorems replace them with two pictures — an area, and where that area's centre of gravity sits.

deflection · moment area contraflexure at the middle of the holefrom the moment: 70.3 N/mm²from the shear: 167.6 N/mm²total 237.9 of 355second moment lost: 30%deflection up by 0.7%the shear the tees must carry: 63 kN, half each, over 400 mm

The hole that costs nothing, and everything

A service opening removes 30% of a beam's second moment and 0.7% of its deflection. What it costs is not that. Across the opening the shear has nowhere to go but through the two tees, and a tee carrying shear over a length bends.

sections · web opening 100 kNchord moment 25.0 kNm from the panel shear · axial 200 kN from the global moment6.20 mm against 3.18 mm triangulated — 68% of it is chord bending

The truss with no diagonals

A rectangle without a diagonal is a mechanism, so a Vierendeel girder makes its corners rigid instead. The shear a diagonal would have carried as axial force now travels as bending in the chords, and bending is a far more expensive way to move a force.

structures · vierendeel

Named alongside it

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

StiffnessContinuityDeflectionHogging and saggingLoad pathMoment envelopeSecondary bendingVirtual workCentroidChord forceDisturbed regionFirst moment of area

All concepts