User:Floquenbeam/SE Workbench/Conjugate beam method

The conjugate beam method is a structural analysis method for determining deformations in statically determinate and indeterminate beams developed by Otto Mohr (or possibly Harold M. Westergaard?). It was published in XXXX in XXXX? The method only accounts for flexural effects and ignores axial and shear effects. While other classical structural analysis methods are usually more convenient when determining reactions, this method is useful in hand calculation of deflections.

Introduction
The basis for the conjugate beam method is the analogous mapping between the basic differential equations governing shears and moments, and the basic differential equations governing slopes and deflections. A conjugate beam can be created with shear and moment boundary conditions corresponding to the actual beam's slope and deflection boundary conditions. When loads analogous to the original beam's shears and moments are applied to the conjugate beam, the resulting shears and moments in the conjugate beam correspond to the slopes and deflections in the actual beam. Since most people are more familiar with analysis of shears and moments, this can make the determination of beam deflections much more intuitive, even though the underlying mathematics is existentially the same.



add equations described above here

Implementation
To apply the conjugate beam method, the following steps are taken.

Determine shears and moments
If statically indeterminate, it's OK to write shears and moments in terms of unknowns

Conjugate beam
A conjugate beam is created who's shear and moment boundary conditions correspond to the actual beam's slope and deflection boundary conditions.

discuss boundary condition mapping

Apply analogous loads to conjugate beam
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Determine shears and moments in conjugate beam
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Correlate to slopes and deflections in actual beam
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Example
The statically indeterminate beam shown in the figure is to be analyzed.