By J Prescott

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Justification of the neglect of the shear strains in beams. points where to these practically small values of of the beam M Of course zero or very small but the strains are themselves negligible, and over beam M is of the order IF, and the this order. ^ -7- is of the order ideal thin rod is is due to bending moments of is So For the the length of the beam. is / M the whole of the /i stress ^ a force of the same. order as the loads on the beam, while of the order is there due is — , and for most beams this is a small fraction.

Something in an elastic body has certainly given way when permanent set is brought about, and it seems reasonable to regard the point at which permanent set begins as the beginning of elastic failure. The theory of elasticity does not help us to make calculations beyond the elastic limit because this theory is all based on the assumption that Hooke's law is true, and there is no adequate mathematical theory beyond that point. Experiment has not settled' at what particular state of stress or strain rupture or permanent set occurs.

Unfortunately, however, pure mathematics has not reached the stage of solving these equations in general terms, and therefore we have to be content very often with the reverse process of finding any solutions of the equations and then finding out the problems of which they are the solutions. That is, the real problem is; given the forces X, Y, Z, and the accelerations f^, f^, f^, calculate the displacements and consequently the stresses. The easier problem is assume some displacements and calculate the forces.

### Applied elasticity by J Prescott

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