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AppendixB - APPENDIX B FIXED-END FORCES AND DEFLECTIONS...

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Unformatted text preview: APPENDIX B FIXED-END FORCES AND DEFLECTIONS Beam Loading Moments, Shears and Deflections 2 2 (a) Ma y L w A C B Mb M a = - Mb = wL wL ; Mc = 12 24 x Va 0.211L 0.211L Vb wL2/8 V a = V b = wL 2 2 wx 2 y = (2Lx - L - x 2 ); 24EI yc = - wL 384 EI 4 (b) y Ma A L/2 C P L/2 Mb B Va L/4 PL/4 Vb x M a = - M b = PL ; M c = PL 8 8 P V a= Vb = 2 P (A to C ) y = - (3Lx 2 - 4x 3 ) 48EI 3 PL yc =- 192EI 2 2 M a = Pab ; M b = - Pa 2b ; M c = 2Pa 3b 2 L L L 2 2 Pb Pa Va= (3a + b ); V b = (3b + a) 3 3 L L 2 Pb x 2 A to C :y = 3 (3ax + bx - 3aL ) 6EIL 2 2 (c) y Ma A a P C b Mb B x Vb Va Pab/L C to B :y = (d) y Ma A C Pa2 (L - x ) 6EIL 3 2 [(3b + a)(L - x ) - 3bL )] L w B Mb x Vb Va 0.237 L 0.192L wL wL Ma= ; Mb = - 30 20 2 M c = 0.0214wL at x = 0.548 L 3wL 7wL Va= ; Vb = 20 20 wL x5 y = (3x 3 - 2Lx 2 - 2 ) 120EI L 2 2 Appendix B - Fixed-end forces and deflection B1 (e) y Ma Va A Ma= Mo a C B Mo L 2 (3a 2 + L - 4La );M b = 6M o 3 2 Mo L 2 (3a 2 - 2La ) Mb b x Vb V a= -Vb = (a 2 - La ) Mo L 1 A to C :y = 6EI (3M a x 2 + V a x 3 ) 2 2 1 (M a + M o )(3x - 6Lx + 3L ) C to B :y = - 6EI 2 3 + V a (3xL - x 3 - 2L ) (f) Ma y h A L Tt Tb B Mb M a = - Mb = V a = V b = 0; EI (T b - T t ) h y =0 Fa Va Ma Fb Vb Mb = Coefficient of thermal expansion T = Temperature change Axial forces due to uniform change T : F a = F b = - AE T 2 2 (g) A C L w B Mb x Vb Va L/4 M b = - wL ; M c = 9wL at x = 3 L 8 8 128 3wL 5wL Va= ; Vb = 8 8 w 3 - 2x 4 - L 3 x ) y = (3Lx 48EI wL2/8 P (h) A L/2 L/2 Mb C B x Va PL/4 V 3L/11 b M b = - 3PL ; M c = 5PL 16 32 5P 11P Va= ; Vb = 16 16 P 2 A to C :y = 96EI (5x 3 - 3L x ) 3 2 C to B :y = P [5x 3 - 16(x - L ) - 3L x )] 96EI 2 Appendix B - Fixed-end forces and deflection B2 (i) y A a P C b Mb B x Vb Va Pab/L Pab Pab 2 Mb = - 2 (L + a); Mc = (2L + a) 2L 2L3 Pb 2 Pa Va = 3 (3L - b); Vb = 3 (3L2 - a 2 ) 2L 2L 1 V (x 3 - 3xL2 ) + 3Pxb2 A to C : y = 3 a 6EIL 3 2 1 Va (x - 3xL ) C to B : y = 6EIL3 + P 3xb2 - (x - a)3 [ [ L (j) A C B w Mb x Va 0.225L V b (k) y A C Mo B Mb x Vb Va a b Mo M b = - wL 15 2 M c = 0.0298wL at x = 0.4474 L V a = wL ; V b = 2wL 10 5 w 3 x5 y = (2Lx 3 - L x - ) 120EI L Mo a2 Mb = - (1 - 3 2 ) 2 L 3M o 2 V a= -Vb = - (L - a 2 ) 3 2L M o (L - a) 3x x3 A to C :y = - 3 ) - 4x (L + a) ( 4EI L L 3 x 1 2 2 3x M o 4 (L - a )( L - 3 ) - Lx L C to B :y = EI 1 2 + (x + a 2 ) 2 3EI (T b - T t ) 2 (l) Fa y L Tt Mb B h A Va Tb Fb Vb Mb 2h 3 EI (T b - T t ) V a= -Vb =- 2hL (T b - T t ) x 3 y =- ( + Lx - 2x 2 ) 4h L = Coefficient of thermal expansion T = Temperature change Axial forces due to uniform change T : F a = F b = - AE T Mb = - Appendix B - Fixed-end forces and deflection B3 ...
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