singularity functions

singularity functions - Shear and bending moment diagrams...

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Shear and bending moment diagrams can be rather complex when there are a number of different loads acting on a beam. To write explicit equations we must consider separate free body diagrams for each section of the beam where the loads change. It would be nice if we could express the shear force and bending moment in terms of functions that are valid for the entire beam. Such functions are called singularity functions. Example: consider a concentrated load, P, on a beam at x = a x=a P for x < a V =0 M=0 for x > a P V=P x-a M=P(x-a)
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Here, we can express the shear force and bending moment in terms of ordinary functions in a piece-wise manner or we can write these expressions in terms of singularity functions where are called singularity functions ( 29 ( 29 ( 29 0 0 x a V x P x a x a M x P x a x a < = < = - ( 29 ( 29 0 1 V x P x a M x P x a = - = - ( 29 0 n n x a x a x a x a < - = -
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As a second example consider a uniform load that starts at x=a (and continues forever): w lb/ length x=a for x < a V =0 M=0 for x >a w(x-a) x-a (x-a)/2 V = w(x-a) M=w(x-a)2/2
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singularity functions - Shear and bending moment diagrams...

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