Unformatted text preview: APPENDIX B
FIXEDEND 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  Fixedend 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  Fixedend 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  Fixedend forces and deflection B3 ...
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 Winter '08
 Dr.HA

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