Essentials in the theory of framed structures . utations are as follows:Area-moment about A X 54 X 72 = 3,724, X 54 X 144 = 7,448, X 54 X 180 = 8,138,000 9,310,000 ? . 19,310,000 , . A„ax. = -^^—^ = m. 29,000,000 MThus it is clear that, if the ordinates in the y -diagram were relatively close together, say at every foot or closer; or even ifI were expressed as an exact function of x in Eq. (9) and theintegration performed, the results in either case would notdiffer materially from those obtained above. 152. Beams with Cover Plates.—The plate girder in consis
Essentials in the theory of framed structures . utations are as follows:Area-moment about A X 54 X 72 = 3,724, X 54 X 144 = 7,448, X 54 X 180 = 8,138,000 9,310,000 ? . 19,310,000 , . A„ax. = -^^—^ = m. 29,000,000 MThus it is clear that, if the ordinates in the y -diagram were relatively close together, say at every foot or closer; or even ifI were expressed as an exact function of x in Eq. (9) and theintegration performed, the results in either case would notdiffer materially from those obtained above. 152. Beams with Cover Plates.—The plate girder in consists of a 24 by % web plate, four angles 5 by 3^^ by% and two cover plates 12 by % by 24 ft. symmetricalabout the center line. The M-diagram is shown in Fig. i486, Mthe I-diagram in Fig. 148c, and the ^-diagram in Fig. i^Sd. 238 THEORY OF FRAMED STRUCTURES Chap. V Area moment about A X 36 X 48 = X 18 X 84 = X 18 X 96 = X 54 X 144 = X 54 X 180 = 12,278,300 24,o7790o 2000O 1,840,500 95 ,637,1007,367,000. The deflection at the center is 24,077,900 o . ^ _ ^ — = m. 29,000,000 The solution by integration may be obtained as follows: LetM\ represent the bending moment for values of x between oand 9; and Mi the bending moment for values of x between 9and 18. Then M\ = 3o,oooit; Mi = 10,000a; + 180,000Then for values of x between o and 6, EIi = 408,417,000 ft^.- Sec. V DEFLECTION OF BEAMS 239 lb. and for values of x between 6 and i8, E/2 = 688,750,000ft*.-lb. The deflection at the center expressed in feet is I /** I T I r^* L = t = -=rr I Mxocdx + -=rr I Mio^x + ^tt I Mixdx The solution by integration is much more simple in thisproblem than in the preceding one, for in this problem / isconstant between certain limits of x and is therefore not a func-tion of x. The geometric treatment by area-moments is by far thesimplest and best method now known for obtaining a solutionof any practical problem involving the deflection of beams. CHAPTER VI
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