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Boundary conditions real to conjugate beam

Short answer

In the conjugate beam, slope equals shear and deflection equals moment. Thus a real fixed end (zero slope and deflection) becomes a conjugate free end (zero shear and moment). A real free end becomes fixed. A real simple support remains simple. An interior support becomes an internal hinge, and an internal hinge becomes an interior roller.

Explain how the boundary conditions are changed while converting a real beam to a conjugate beam with reference to the conjugate beam theorems.

In the conjugate beam method, a real beam is replaced by a conjugate beam so that slope = shear and deflection = bending moment in the conjugate beam. For this to hold, the support conditions of the conjugate beam must reproduce the boundary conditions of the real beam.

Condition in real beamSlope θ\thetaDeflection yyNeeded in conjugateConjugate support
Fixed end00V′=0V' = 0, M′=0M' = 0Free end
Free end≠0\ne 0≠0\ne 0V′≠0V' \ne 0, M′≠0M' \ne 0Fixed end
Simple (pin/roller) end≠0\ne 00V′≠0V' \ne 0, M′=0M' = 0Simple (pin/roller) end
Interior supportcontinuous, ≠0\ne 00V′V' continuous, M′=0M' = 0Internal hinge
Internal hingediscontinuous≠0\ne 0V′V' jumps, M′≠0M' \ne 0Interior roller support
 Real:      fixed -----------  free       simple --- interior --- simple
 Conjugate: free  -----------  fixed      simple --- hinge ------ simple

The conjugate beam must be statically stable and determinate, which these rules guarantee when the real beam is determinate.

✓ Answer checked by Claude Opus 5.5 (high effort)

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