ch9-5 - SECTION 9.9 Castigliano's Theorem 601 Castigliano's...

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Castigliano’s Theorem The beams described in the problems for Section 9.9 have constant flexural rigidity EI. Problem 9.9-1 A simple beam AB of length L is loaded at the left-hand end by a couple of moment M 0 (see figure). Determine the angle of rotation u A at support A . (Obtain the solution by determining the strain energy of the beam and then using Castigliano’s theorem.) Solution 9.9-1 Simple beam with couple M 0 A B L M 0 SECTION 9.9 Castigliano’s Theorem 601 (downward) 5 M 0 ¢ 1 2 x L M 5 M 0 2 R A x 5 M 0 2 M 0 x L R A 5 M 0 L S TRAIN ENERGY C ASTIGLIANO S THEOREM (clockwise) (This result agree with Case 7, Table G-2) u A 5 dU dM 0 5 M 0 L 3 EI U 5 # M 2 dx 2 EI 5 M 2 0 2 EI # L 0 ¢ 1 2 x L 2 dx 5 M 2 0 L 6 EI A B L M 0 x Problem 9.9-2 The simple beam shown in the figure supports a concentrated load P acting at distance a from the left-hand support and distance b from the right-hand support. Determine the deflection d D at point D where the load is applied. (Obtain the solution by determining the strain energy of the beam and then using Castigliano’s theorem.) Solution 9.9-2 Simple beam with load P AB D L ab P M DB 5 R B x 5 Pax L M AD 5 R A x 5 Pbx L R B 5 Pa L R A 5 Pb L S TRAIN ENERGY C ASTIGLIANO S THEOREM (downward) d D 5 dU dP 5 Pa 2 b 2 3 LEI U 5 U AD 1 U DB 5 P 2 a 2 b 2 6 LEI U DB 5 1 2 EI # b 0 ¢ Pax L 2 dx 5 P 2 a 2 b 3 6 EIL 2 U AD 5 1 2 EI # a 0 ¢ Pbx L 2 dx 5 P 2 a 3 b 2 6 EIL 2 U 5 # M 2 dx 2 EI D L P xx
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Problem 9.9-3 An overhanging beam ABC supports a concentrated load P at the end of the overhang (see figure). Span AB has length L and the overhang has length a . Determine the deflection d C at the end of the overhang. (Obtain the solution by determining the strain energy of the beam and then using Castigliano’s theorem.) Solution 9.9-3 Overhanging beam 602 CHAPTER 9 Deflections of Beams AB C La P (downward) M CB 52 Px M AB R A x Pax L R A 5 Pa L S TRAIN ENERGY C ASTIGLIANO S THEOREM (downward) d C 5 dU dP 5 Pa 2 3 EI ( L 1 a ) U 5 U AB 1 U CB 5 P 2 a 2 6 EI ( L 1 a ) U CB 5 1 2 EI # a 0 ( 2 Px ) 2 dx 5 P 2 a 3 6 EI U AB 5 1 2 EI # L 0 ¢ 2 Pax L 2 dx 5 P 2 a 2 L 6 EI U 5 # M 2 dx 2 EI C P x x Problem 9.9-4 The cantilever beam shown in the figure supports a triangularly distributed load of maximum intensity q 0 . Determine the deflection d B at the free end B . (Obtain the solution by determining the strain energy of the beam and then using Castigliano’s theorem.) Solution 9.9-4 Cantilever beam with triangular load A L B q 0 P 5 fictitious load corresponding to deflection d B M Px 2 q 0 x 3 6 L S TRAIN ENERGY C ASTIGLIANO S THEOREM (downward) (This result agrees with Cases 1 and 8 of Table G-1.) S ET P 5 0: d B 5 q 0 L 4 30 EI d B 5 0 U 0 P 5 PL 3 3 EI 1 q 0 L 4 30 EI 5 P 2 L 3 6 EI 1 Pq 0 L 4 30 EI 1 q 2 0 L 5 42 EI U 5 # M 2 dx 2 EI 5 1 2 EI # L 0 ¢ 2 Px 2 q 0 x 3 6 L 2 dx A L B q 0 P x
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Problem 9.9-5 A simple beam ACB supports a uniform load of intensity q on the left-hand half of the span (see figure).
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ch9-5 - SECTION 9.9 Castigliano's Theorem 601 Castigliano's...

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