Structural Engineering & Statics
where the load goes
Concept

Virtual work — where it appears

3 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.
real Mpeak 32.0a unit load, here and nowhere elseunit mM × marea ÷ EI = 213.33the unit load is the only place the question 'deflection where?' is asked

One deflection, without solving everything

To find how far one point of a structure moves, put an imaginary force of one unit there, multiply two moment diagrams together, and integrate. The answer arrives without ever solving for the deflected shape.

deflection · virtual work 100 kNthe gap is the shear: 16.3% of the totalbending alone, and what the beam really doesthe shear part alone, magnified 3 times furthertwo straight lines meeting under the loadshearγ = V/GAs is a slope the section is racked through, not a curvature —so this diagram is integrated once, where the moment diagram above it is integrated twice

The deflection that is not bending

Every deflection on this site so far has been the second integral of a moment, and that calculation silently drops a term. The beam also shears, and the shear deflection is not a correction to the curve — it is a different shape, and for a deep member it is most of the answer.

deflection · shear deflection δ = 631.06 read here10 kN at every top node — each member drawn at the width of its own sharetop chord (four members)47.2%bottom chord (six members)28.5%diagonal (six members)22.3%vertical (five members)2.0%the members that moved the roof, ranked — a symmetric pair is two members and appears twicetop chord, at mid-span14.80%F -52.9 × f -1.765top chord, at mid-span14.80%F -52.9 × f -1.765bottom chord, at mid-span8.77%F 47.1 × f 1.176top chord, near the left support8.77%F -47.1 × f -1.176bottom chord, at mid-span8.77%F 47.1 × f 1.176top chord, near the right support8.77%F -47.1 × f -1.176diagonal, near the left support6.20%F -38.6 × f -0.772diagonal, near the right support6.20%F -38.6 × f -0.772virtual work and a stiffness solution agree to 3.6e-15 — two methods sharing no arithmetic

Which member moved the roof

A beam sags because it curves. A truss has no curvature anywhere — it comes down because every one of its members changes length, and the sum of those changes, weighted member by member, is a ranking that names which ones are worth stiffening. Usually not the ones a designer worries about.

deflection · truss deflection

Named alongside it

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

StiffnessDeflectionUnit load methodChord forceCompatibilityForce methodLoad pathMethod of jointsMoment diagramPlane sectionsProduct integralShear deflection

All concepts