. The strength of materials; a text-book for engineers and architects. Fig. 202.—^Fixed Beam with Isolated Load. ditions to be satisfied are that the free and end bendingmoment diagrams a b d and a G^ H b respectively must beequal and op^iosite in area, and must have their centroids uponthe same vertical Fig. 203.—^Fixed Beam with Isolated Load. The first condition gives us —^^^——- = area of A a d b = i a b . r ])I .Wab Wab 1 2 • (1) FIXED AND CONTINUOUS BEAMS 427 The end bending moment diagram a Gj. h b may be con-sidered as divided up into two triangles a g^ B, b g^ h, whosecentroids
. The strength of materials; a text-book for engineers and architects. Fig. 202.—^Fixed Beam with Isolated Load. ditions to be satisfied are that the free and end bendingmoment diagrams a b d and a G^ H b respectively must beequal and op^iosite in area, and must have their centroids uponthe same vertical Fig. 203.—^Fixed Beam with Isolated Load. The first condition gives us —^^^——- = area of A a d b = i a b . r ])I .Wab Wab 1 2 • (1) FIXED AND CONTINUOUS BEAMS 427 The end bending moment diagram a Gj. h b may be con-sidered as divided up into two triangles a g^ B, b g^ h, whosecentroids act in the third Unes x x and y y respectively. We have next to calculate the position of the centroid g ofthe free bending moment diagram. According to the ordinaryrule, the centroid g will be one-third of the way up the medianline c D. 1 1 fl, •*• ^ 6 ^^ ~6 3V2 y ~3 Regarding the areas of the triangles ag^b, bg^h, adbas concentrated in the lines x x, y y, and g g respectively,we have by taking moments about the line x x Area ofAADB xa; = area of A b g^ h x ^ o i. e. from (1), —^— . .t = | M,, ? x ^ M,J2 ^Wab _Wab a? • 6 ~ 2 ^ ~ 2 3 _ w an ~ 6• M = ^^^ • • -^-Ljj 72 Similarly, M^ = „— As a check (M, + M„) = ^f ^ + ^ |^ Wa6. ,, Wab , Wab= —^ (a 4- 6) = ^2 ^= I (M, -f
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