. Differential and integral calculus. Fig. 72. In this case we have _ ? x wl w, . _ N J/ = #/# • ,3; = — (jt — /.*:). 22 2 v 7 Hence ^^ 7£/ = Tml* --U)\ Mechanical Applications 391 hence dy dx 2 EI (Ft)- / dyWhen x = - i -j- = o 2 tf# C = .1 7£/324^7 Hence 7i If Ix2 P 2 12 dx 2 EI k w { x4 lxs Px ?• y = —wy \ r1 2 -ZSY / 12 6 12 Since # = o, y = o; .-. C = o. / ., 5 wl< = 5 ET/3 384^/ 384^/ If # = - , then 2 Cor. Comparing the value of S of § 259 with S of thisarticle, we find 8 = f8, , the deflection produced by a load concentrated at the cen-ter of a beam is f of that produced by the
. Differential and integral calculus. Fig. 72. In this case we have _ ? x wl w, . _ N J/ = #/# • ,3; = — (jt — /.*:). 22 2 v 7 Hence ^^ 7£/ = Tml* --U)\ Mechanical Applications 391 hence dy dx 2 EI (Ft)- / dyWhen x = - i -j- = o 2 tf# C = .1 7£/324^7 Hence 7i If Ix2 P 2 12 dx 2 EI k w { x4 lxs Px ?• y = —wy \ r1 2 -ZSY / 12 6 12 Since # = o, y = o; .-. C = o. / ., 5 wl< = 5 ET/3 384^/ 384^/ If # = - , then 2 Cor. Comparing the value of S of § 259 with S of thisarticle, we find 8 = f8, , the deflection produced by a load concentrated at the cen-ter of a beam is f of that produced by the same load whenuniformly distributed. 261. Shape and deflection of a beam fixed at both ends anduniformly Fig. 73- This case is similar to that of the preceding except that anunknown moment m acts on the portion of the beam OB; hence, M=-(x*-/x) + m. (0 392 Integral Calculus d2y i ( wx2 wlx Hence ^=ei\~ - — + *}• W Integrating and noting that when x = o,. — = o, and there-lore C = o, we have, dy i ( wxs wlx2 chc £I{ 6 4 +mX\ W dyAt the point C where x = I, — = o. If we substitute these values in the last expression, we find after reduction wl2m = j (c) 12 W for the moment of the unknown couple acting at the points ofsupport. Substituting this value of m in (p) and integrating,we find, since x = o gives y = o, and therefore C = # = - in (d) we find o = 384^/ 384^/ Comparing this value of 8 with that of S in § 260, we find * = s s. that is, by fastening the ends of a beam at its points of support,the deflection caused by a uniform load is only one-fifth ofwhat it would be if the beam merely rested on its supports. Again, making m = in (1), we have wx2 wlx wl2M — 1 2 ,2
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