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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. Three theories have been advanced but no decisive experiments seem to have deterjnined which is right. The first theory states that failure occurs when the greatest principal stress reaches a certain definite limit for a given material; the second that the greatest principal strain is the deciding factor; and the third, difference between the that failure point reaches a fixed amount.

3-55) forces are zero. — = (m — n) A + 2n du CJX ^ |(i — 20) {m — n)-\-2n^aZ' = \20n-\- 2n}az , p ^ ^ ^ = 2naz{i -{-o) = Eaz = {(i — 2o){m — n) — 2on]az = F. = Ui - 2o)(m — n)— 2on}az = (3-56) p,, ' . 60) us suppose that our equations apply to a rod of uniform and we section, the let P^, and this is = Ea^ Pi Now — x)=^^o not zero is <»" 1 before the strain, the jr-axis was of centres normal sections of the rod. any cross section where the inertia of the The line in P^ stress is zero is called Thus axis of the section the the neutral axis of the section in dA Let neutral the ,^-axis is 15.

In finding the tensile strength of any given material, small bars^ whose longitudinal sections are as usually of circular cross-section, shown and 19(b) are gripped in figs, igi-a) at the ends, tension they break, till between till the fracture tension occurs. and extension being observed The graph showing and strain for such a test-piece of stress in Fig. 19b Fig. iga continuously and put the relation steel is shown in fig. 20. OA is a straight line, but AB is slightly curved, the The portion lower side being the concave side.

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