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Home/Engineering/Beam Deflection: Unit Load Method

Beam Deflection: Unit Load Method

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

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

Beam and stiffness

6 m
200 GPa
18000000 mm4

Loads and probe

22 kN
3 m
4 kN/m
3 m

Measured values

Closed-form deflection46.250mm
Unit-load integral46.250mm
Numerical difference0.0006%
Maximum deflection46.250mm
Flexural rigidity EI3600kN·m²

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.