{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

09-05 Chap Gere

# 09-05 Chap Gere - SECTION 9.9 Castigliano's Theorem 601...

This preview shows pages 1–4. Sign up to view the full content.

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 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 A B L M 0

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 14

09-05 Chap Gere - SECTION 9.9 Castigliano's Theorem 601...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online