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Deflection of Loaded Beam, Simply Supported

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I'm trying to determine the position and magnitude of the maximum bending moment. As well as plotting a graph of the deflection along the length of the beam at 1 m intervals. I'm trying to use Macaulay's method and have got equations for slope and delection and the constants of integration but believe these are wrong. Attached is my BM and SF diagram and the beam. Also attached is my attempt thus far.

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I think one mistake occured when applied the condition y(0) = 0, since by replacing x = 0 in your expression of y(x), (x-a)^3 ...

Solution Summary

The deflection of a loaded beam in an arbitrary location can be computed by determining the equation of deflected mean line of the beam.