Math 2.0

Mechanics of deformable solids Definition

Conjugate beam method

Builds on Euler–Bernoulli beam equationBending momentShear and moment diagram

Definition

Let a real beam occupy 0≤x≤L0 \le x \le L and follow the Euler–Bernoulli beam equation with bending stiffness EI(x)EI(x). Let it have bending moment M(x)M(x) and deflection w(x)w(x) in the direction of the load, so that w′′=−M/(EI)w'' = -M/(EI). The conjugate beam is a beam of the same length with these properties.

  1. It carries the distributed load qˉ(x)=M(x)/(EI(x))\bar q(x) = M(x)/(EI(x)).
  2. Its supports are changed from those of the real beam. A clamped end becomes a free end, and a free end becomes a clamped end. A simply supported end stays simply supported. An interior support becomes an internal hinge, and an internal hinge becomes an interior support.

The conjugate beam method computes the shear force Vˉ\bar V and bending moment Mˉ\bar M of the conjugate beam by statics, and uses w′(x)=Vˉ(x),w(x)=Mˉ(x).w'(x) = \bar V(x), \qquad w(x) = \bar M(x).

Remarks

The rule works because Mˉ′′=−qˉ=−M/(EI)=w′′\bar M'' = -\bar q = -M/(EI) = w'', and the support changes of clause 2 turn the conditions on ww and w′w' into the same conditions on Mˉ\bar M and Vˉ\bar V. For example a clamped end has w=w′=0w = w' = 0, which for the conjugate beam reads Mˉ=Vˉ=0\bar M = \bar V = 0, the conditions of a free end.

Example: a cantilever of length LL, clamped at x=0x = 0, with a force PP at x=Lx = L has M(x)=−P(L−x)M(x) = -P(L - x). The conjugate beam is free at 00 and clamped at LL, and carries a triangular load of total size PL2/(2EI)PL^2/(2EI) whose centroid is at distance 2L/32L/3 from x=Lx = L. Its moment at the clamped end has size PL2/(2EI)⋅2L/3=PL3/(3EI)PL^2/(2EI) \cdot 2L/3 = PL^3/(3EI), which is the tip deflection of the real beam.

Non-example: a conjugate beam with the same supports as a real cantilever would be clamped at 00 and would give w(0)≠0w(0) \neq 0 in general. This contradicts the clamped end, so clause 2 is needed.

The conjugate beam of a statically determinate real beam is also determinate. That of an indeterminate beam can be unstable as a structure but still in equilibrium under its load.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Bending moment, concept, no evidence levelBending momentconceptEuler–Bernoulli beam equation, concept, no evidence levelEuler–Bernoulli beam …conceptShear and moment diagram, concept, no evidence levelShear and moment diag…conceptConjugate beam method, definition, no evidence levelConjugate beam methoddefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Bending moment concept
  2. Euler–Bernoulli beam equation concept
  3. Shear and moment diagram concept
  4. Conjugate beam method definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Conjugate_beam_method
About this page
TagW78WHCMS03
URIm2://W78WHCMS03
Kinddefinition
WritingW0 stub
MSC74K10, 74G05
DefinesConjugate beam method
UsesEuler–Bernoulli beam equation, Bending moment, Shear and moment diagram
Changed3 November 2026

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