Essentials in the theory of framed structures . (b)Fig. 158. Elh = (72 X 6 X 8) + (72 X 9 X 18) + M{i5 X 20)Elh = M{i5 X 20) ii = —iiwhence M — — as the reactions have been determined, the bendingmoments at C and D, and the deflection at any point may becomputed. The elastic curve when drawn will have the generalconfiguration shown in Fig. 159. 2s6 THEORY OF FRAMED STRUCTURES Chap. VI i6i. In Fig. 1580, PQS is the M-diagram when AC h con-sidered as a simple beam, and SOTV is the M-diagram whenCB is considered as a simple beam; then the area PUV \s addedto provide the bending m
Essentials in the theory of framed structures . (b)Fig. 158. Elh = (72 X 6 X 8) + (72 X 9 X 18) + M{i5 X 20)Elh = M{i5 X 20) ii = —iiwhence M — — as the reactions have been determined, the bendingmoments at C and D, and the deflection at any point may becomputed. The elastic curve when drawn will have the generalconfiguration shown in Fig. 159. 2s6 THEORY OF FRAMED STRUCTURES Chap. VI i6i. In Fig. 1580, PQS is the M-diagram when AC h con-sidered as a simple beam, and SOTV is the M-diagram whenCB is considered as a simple beam; then the area PUV \s addedto provide the bending moment on account of the the tangent to the elastic curve be drawn through C andlet t\ and h represent the tangential deviations at A and B |<. Pig. 159. respectively, then Elh = (9,000 X 6 X 6) + M(6 X 8) Elh = (6,000 X 3 X 4) + (6,000 X 6 X 9) + (6,000 X 3 X 14) + M{g X 12)h- —ti::i2:i8whenceM = —6,300 The M-diagram may now be drawn to scale as shown in statics —6,300 = 12R1 — 18,000 ^1 = 975— 6,300 = i8i?3 — 6,000 — 12,000R3 — 650Ri = s,ooo - 975 - 650 = 3,375 Sec. II RESTRAINED AND CONTINUOUS BEAMS 257 162. The general expressions for Ri, R2 and R3 will now bedeveloped in connection with Fig. 159. Let the tangent to theelastic curve be drawn through C and let h and h represent thetangential deviations at A and B respectively; then Elh = Pk{i - k)h /i/iVfyfe/i +^(i - k)h] + Mik.)[\ = m(^.)(^ ti. — = —till Since k is less than unity, ib — ^^ is positive; hence M is anegative bending moment, and Rz is a negative reaction actingdownward. When the two spans are of equal length I, the reactions are ^1 = -{k -5^ + 4) 4 (9) i?2 = -{-2k^ + 6k)4 (10) Rz = -{k - k)4 (11) 163. In Fig.
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