A rod AB. whoso unstretched length is L, is originally oriented .it angle theta

Question

A rod AB. whoso unstretched length is L, is originally oriented .it angle theta counterclockwise from the +x axis. If the rod is free to rotate about a fixed pin at A. what is the extensional strain in the rod when end B moves a distance u in the + x direction to point B*? Express your answer in terms of the displacement u. the original length L, and the original angle theta. Simplify the answer you obtained for Part , obtaining a small-displacement approximation that is valid if u L. (Note: This result will be used in Section 3.10.)

Explanation / Answer

It is easy to see that angle ABB* = 180o-. From law of cosines, we have

L* = [L2+u2-2Lucos(180o-)] = (L2+u2+ 2Lu cos)

(a) = (L*-L)/L = [(L2+u2+ 2Lu cos)-L]/L = [1+(u/L)2+2 (u/L) cos]-1

(b) For small deformation u<<L, (u/L)2 is negligible, we have

     = [1+2 (u/L) cos]-1

Use the Taylor series of at 0, we have

     = [1+(u/L)cos - (1/2)(u/L)2 + ...] - 1

Ignore the higher order term, we have

     1+(u/L)cos - 1 = (u/L) cos

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