. Differential and integral calculus. i nbas _ Aa24 4 DEFLECTION AND SLOPE OF Formula. From mechanics we have^ EI 00 for the relation between the moment of the extraneous forces Mechanical Applications 387 (AT) and the moment of the internal resistance I— J about the neutral axis of any section. In this formula E = coeffi-cient of elasticity of the material of which the beam is made;/= moment of inertia of section, and p = radius of curvatureof the curve of mean fiber, at the point in which it pierces thesection. But § 134, P + n d*ydx2 Hence, M= EI d*y ~dx? 1 + /dy\2Since f -=-) =
. Differential and integral calculus. i nbas _ Aa24 4 DEFLECTION AND SLOPE OF Formula. From mechanics we have^ EI 00 for the relation between the moment of the extraneous forces Mechanical Applications 387 (AT) and the moment of the internal resistance I— J about the neutral axis of any section. In this formula E = coeffi-cient of elasticity of the material of which the beam is made;/= moment of inertia of section, and p = radius of curvatureof the curve of mean fiber, at the point in which it pierces thesection. But § 134, P + n d*ydx2 Hence, M= EI d*y ~dx? 1 + /dy\2Since f -=-) = tan2 a, , the square of the slope of the beam, and since in practice the value is small, we may omit it andwrite d*y M=EI dx ? (0 Formula (2) is sufficiently accurate for all practical purposes,and is in general use. 257. Slope and deflection of a beam loaded at one end and fixedat the other. Let / = length of beam, andW = weight applied at its endA. Let OA be the mean fiber,and S a plane _L to OA at adistance x from O; then. Fig. 69. M= W(l- x). 388 Integral Calculus d2y WHence, § 256 (2), — = ^(/- *) When x = o, — = o, since the tangent at O is coincident with X\ hence C — o. dx Integrating, W1EI W (2 Ix — x2). ^ = 6J/^-4 («) since when :r = o, y — o and therefore C = o. Equa. («;) isthe equation of the curve OA, , the equation of the curvewhich the mean fiber takes under the action of the load W. If in (a) we make x = I and let 8 = value of y when x = /,we have Wlz for the maximum deflection of the beam. 8 = 258. Shape and deflection of a beam fixed at one end anduniformly loaded.
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