. The strength of materials; a text-book for engineers and architects. untilthe plate is flat; we then have 8 = 8,„ and we get fromequation (1) the following value for the test or proof load W^ Stress in Plates.—The stress in the plates will be con-stant along their length because their depth as well as theirmoment of inertia is constant. •••/ = M d I ^ 2 Wl d . nb d^^ 123W? 2 ~ 2nbd ^^^ Derivation of Deflection from Resilience.—Theformula for deflection may be derived from the resilience asfollows— 42 Resilience (see p. 273) = J^ .*. Total work done in stressing = y!^ x vol. - ^-- Id — 6E •^^
. The strength of materials; a text-book for engineers and architects. untilthe plate is flat; we then have 8 = 8,„ and we get fromequation (1) the following value for the test or proof load W^ Stress in Plates.—The stress in the plates will be con-stant along their length because their depth as well as theirmoment of inertia is constant. •••/ = M d I ^ 2 Wl d . nb d^^ 123W? 2 ~ 2nbd ^^^ Derivation of Deflection from Resilience.—Theformula for deflection may be derived from the resilience asfollows— 42 Resilience (see p. 273) = J^ .*. Total work done in stressing = y!^ x vol. - ^-- Id — 6E •^^- 2 JWS _ f^lndb^ •*• 2 ~ 12 E dWUHn^db ~ In^b^ d^ xl2E ^ 3W^ P ~ I6nbd^ . E .*. d = c,T->- 1. Ti as beiore. SPRINGS 355 Numerical Example.—A lamiyiated plate spring of 40inches span has 12 plates, each -375 inch thick and 3*40 incheswide. Calculate the deflection when carrying a central load of4 tons, taking E = 11,600 tons per sq. in. By formula (2) we have 3WP 8 = S^nbd^ 3 X 4 X 40 X 40 X 40 8 X 11,600 X 12 X 3-40 x -3753= 3-84 Fig. 154.—Piston Rings. Piston Rings.—Springs in the form of split-rings areplaced around pistons in oil, gas and steam engines to preventescape of the working fluid past the piston, and such ringsshould be designed so as to give as constant a pressure aspossible all round the cylinder. The necessary variation inthickness has been investigated by Professor Robinson inthe following manner. Let T>, Fig. 154, be the point of maximum thickness toat the centre of the ring which was initially circular on theoutside and is sprung into position so that it is still circularon the outside. Consider a length a B of the ring, the thickness of thering at the point b being te and the breadth throughout 356 THE STPxEXGTH OF MATERIALS being b. ^ is the angle which the arc b D subtends at thecentre. Let R be the radius of the spring when bent and let R„be the radius at b when in the unstrained condition indicatedin dotted l
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