Why do the closed-form and unit-load answers match?

Both come from the same Euler-Bernoulli relation EI v″ = M. The unit-load method uses virtual work to extract the displacement at one point, while the closed form directly integrates the beam equation.

When is the unit-load method useful?

It is especially useful for frames, trusses, and beams with complicated loading, where one displacement is needed but writing the full deflection curve is awkward.

Beam Deflection: Unit Load Method

Simply supported Euler-Bernoulli beam with point load P and UDL w: closed-form deflection vs virtual-work unit-load integral.

The unit-load method is virtual work: multiply the real bending moment by the bending moment from a unit load at the point where deflection is requested. Shear deformation, large deflection, yielding, and support flexibility are not included.

Live graphs

About this model

This simulator compares two equivalent ways to compute small deflection of a simply supported Euler-Bernoulli beam. Closed-form formulas are used for a point load P at position a and a uniform load w over the full span. The same deflection at a selected probe point x0 is also computed by the virtual-work unit-load method, δ(x0) = ∫ M(x)m(x;x0)/(EI) dx, where M is the real bending moment and m is the moment diagram from a unit load at x0. The comparison makes the energy method concrete and shows why unit-load integration is useful when closed forms are not convenient. The model omits shear deformation, large deflection, yielding, support settlement, variable EI, and dynamic effects.

Who it's for: Structural analysis, mechanics of materials, civil engineering, mechanical design, and finite-element method introductions.

Key terms

Beam deflection
Unit load method
Virtual work
Euler-Bernoulli beam
Bending moment

How it works

Simply supported Euler-Bernoulli beam: compare closed-form deflection from point and uniform loads with the virtual-work unit-load integral ∫Mm/EI dx.

Key equations

δ(x0) = ∫₀ᴸ M(x) m(x; x0) / (EI) dx

UDL closed form: v(x)= w x(L³−2Lx²+x³)/(24EI)

Frequently asked questions

Why do the closed-form and unit-load answers match?: Both come from the same Euler-Bernoulli relation EI v″ = M. The unit-load method uses virtual work to extract the displacement at one point, while the closed form directly integrates the beam equation.
When is the unit-load method useful?: It is especially useful for frames, trusses, and beams with complicated loading, where one displacement is needed but writing the full deflection curve is awkward.

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Beam Deflection: Unit Load Method

Live graphs

About this model

How it works

Key equations

Frequently asked questions

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Beam Deflection: Unit Load Method

Live graphs

About this model

How it works

Key equations

Frequently asked questions