Deflections of beams by conjugate beam method

A conjugate beam is a fictitious beam that corresponds to the real beam and loaded with M/EI diagram of the real beam so that the shear and moment at any point of the conjugate beam are equal, respectively, to the slope and deflection at the corresponding point on the real beam.

The conjugate beam method was developed by Christian Otto Mohr (1835 – 1918),

a German civil engineer and one of the most celebrated civil engineer in 19th century.

The method is based on the analogy between the relationships among load, shear, and bending moment and the relationships among M/EI, slope, and deflection.

Slope on real beam = Shear on conjugate beam Deflection on real beam = Moment on conjugate beam

Properties of Conjugate Beam

The length of a conjugate beam is always equal to the length of the actual beam.
The load on the conjugate beam is the M/EI diagram of the loads on the actual beam.
A simple support for the real beam remains simple support for the conjugate beam.
A fixed end for the real beam becomes free end for the conjugate beam.
The point of zero shear for the conjugate beam corresponds to a point of zero slope for the real beam.
The point of maximum moment for the conjugate beam corresponds to a point of maximum deflection for the real beam.

Supports of Conjugate Beam

Knowing that the slope on the real beam is equal to the shear on conjugate beam and the deflection on real beam is equal to the moment on conjugate beam, the shear and bending moment at any point on the conjugate beam must be consistent with the slope and deflection at that point of the real beam. Take for example a real beam with fixed support; at the point of fixed support there is neither slope nor deflection, thus, the shear and moment of the corresponding conjugate beam at that point must be zero. Therefore, the conjugate of fixed support is free end.

Real beam support and its corresponding conjugate beam support

Examples of Conjugate Beams for its Real (original) beams.

Real beam and its corresponding conjugate beam

Comments

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