. The strength of materials; a text-book for engineers and architects. be p and q en s, = E ~ E s,= E E .-. {s. -^J- iv- - 9) (1 + -n)E • •• {Sx- 5jG = Av- -g)(l+r7)G - ^^ ^^ because E - 2 G (1 -f q) (p. 11). .*. Equivalent maximum shear stress =-- It will be noted that the equivalent and actual maximumshear stresses come the same, whereas the equivalent andactual principal stresses are different. STRAIN, STRESS, AND ELASTICITY 33 Resilience.—The work done per unit volume of a materialin producing strain is called resilience. Consider the case of abody subjected to a simple tensile strain. In
. The strength of materials; a text-book for engineers and architects. be p and q en s, = E ~ E s,= E E .-. {s. -^J- iv- - 9) (1 + -n)E • •• {Sx- 5jG = Av- -g)(l+r7)G - ^^ ^^ because E - 2 G (1 -f q) (p. 11). .*. Equivalent maximum shear stress =-- It will be noted that the equivalent and actual maximumshear stresses come the same, whereas the equivalent andactual principal stresses are different. STRAIN, STRESS, AND ELASTICITY 33 Resilience.—The work done per unit volume of a materialin producing strain is called resilience. Consider the case of abody subjected to a simple tensile strain. In going from thepoint A to the point b. Fig. 12, very near to it, the averagestress acting is /. Therefore, if a b = x, the work done bythe force / in straining the material from the point a to thepoint B will be equal to / x x. Now, if x is the increase inunital strain and / is the intensity of stress, the volume ofmaterial acted upon is unity. Now, a B is assumed to bevery small, and f x x is equal to the area of the shadedportion of the stress-strain Fig 12.—Resilience. Therefore, the resilience is equal to the area of the stress-strain curve up to the point m, resilience = area of A p m x _ 1. — 2 ^ ^ Now, — = Youngs modulus = E resilience in tension ^ 2E similarly in shear the resilience = ^^ where s is the shear and Strains due to Sudden or DynamicLoading.—If a load is applied suddenly to a structure, D 34 THE STRENGTH OF MATERIALS vibration will ensue, and the strain—and thus the stress—will reach twice the value which would occur if the loadwere gradually applied. This will be made clear from considering a diagram, Fig. 13(1), where the force is plotted against the strain. We haveseen that, with gradual loading of an elastic body, the curverepresenting the relation between the strain and the load indirect stress is represented by a straight line a d, the areabelow the line giving the work done up to a given point.
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