10-01 Chap Gere

# 10-01 Chap Gere - 10 Statically Indeterminate Beams...

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Differential Equations of the Deflection Curve The problems for Section 10.3 are to be solved by integrating the differential equations of the deflection curve. All beams have constant flexural rigidity EI. When drawing shear-force and bending-moment diagrams, be sure to label all critical ordinates, including maximum and minimum values. Problem 10.3-1 A propped cantilever beam AB of length L is loaded by a counterclockwise moment M 0 acting at support B (see figure). Beginning with the second-order differential equation of the deflection curve (the bending-moment equation), obtain the reactions, shear forces, bending moments, slopes, and deflections of the beam. Construct the shear-force and bending-moment diagrams, labeling all critical ordinates. Solution 10.3-1 Propped cantilever beam 10 Statically Indeterminate Beams A L y x B M 0 M A R A R B M 0 5 applied load Select M A as the redundant reaction. R EACTIONS ( FROM EQUILIBRIUM ) (1) R B 52 R A (2) B ENDING MOMENT ( FROM EQUILIBRIUM ) (3) D IFFERENTIAL EQUATIONS (4) EIv ¿ 5 M A L ¢ x 2 2 2 L x 1 M 0 x 2 2 L 1 C 1 EIv 5 M 5 M A L ( x 2 L ) 1 M 0 x L M 5 R A x 2 M A 5 M A L ( x 2 L ) 1 M 0 x L R A 5 M A L 1 M 0 L B . C . 1 [ C 1 5 0 (5) B . C . 2 v (0) 5 0 [ C 2 5 0 B . C . 3 v ( L ) 5 0 [ M A R EACTIONS ( SEE E QS . 1 AND 2) S HEAR FORCE ( FROM EQUILIBRIUM ) B ENDING MOMENT ( FROM E Q . 3) M 5 M 0 2 L (3 x 2 L ) V 5 R A 5 3 M 0 2 L R B 3 M 0 2 L R A 5 3 M 0 2 L M A 5 M 0 2 5 M 0 2 EIv 5 M A L ¢ x 3 6 2 L x 2 2 1 M 0 x 3 6 L 1 C 2 v ¿ (0) 5 0 (Continued) 633

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Problem 10.3-2 A fixed-end beam AB of length L supports a uniform load of intensity q (see figure). Beginning with the second-order differential equation of the deflection curve (the bending-moment equation), obtain the reactions, shear forces, bending moments, slopes, and deflections of the beam. Construct the shear-force and bending-moment diagrams, labeling all critical ordinates. Solution 10.3-2 Fixed-end beam (uniform load) 634 CHAPTER 10 Statically Indeterminate Beams S LOPE ( FROM E Q . 4) D EFLECTION ( FROM E Q . 5) v 52 M 0 x 2 4 LEI ( L 2 x ) v ¿ M 0 x 4 LEI (2 L 2 3 x ) S HEAR - FORCE AND BENDING - MOMENT DIAGRAMS 3 M 0 2 L V O M o 2 M o M O 2 L 3 A L y x B M B M A R A R B q Select M A as the redundant reaction. R EACTIONS ( FROM SYMMETRY AND EQUILIBRIUM ) M B 5 M A B ENDING MOMENT ( FROM EQUILIBRIUM ) (1) D IFFERENTIAL EQUATIONS (2) B . C . 1 [ C 1 5 0 (3) B . C . 2 v (0) 5 0 [ C 2 5 0 B . C . 3 v ( L ) 5 0 [ M A 5 qL 2 12 EIv M A x 2 2 1 q 2 ¢ L x 3 6 2 x 4 12 1 C 2 v ¿ (0) 5 0 EIv ¿ M A x 1 q 2 ¢ L x 2 2 2 x 3 3 1 C 1 EIv 5 M M A 1 q 2 ( L x 2 x 2 ) M 5 R A x 2 M A 2 qx 2 2 M A 1 q 2 ( L x 2 x 2 ) R A 5 R B 5 qL 2 R EACTIONS S HEAR FORCE ( FROM EQUILIBRIUM ) B ENDING MOMENT ( FROM E Q . 1) S LOPE ( FROM E Q . 2) D EFLECTION ( FROM E Q . 3) d max v ¢ L 2 5 qL 4 384 EI v qx 2 24 EI ( L 2 x ) 2 v ¿ qx 12 EI ( L 2 2 3 L x 1 2 x 2 ) M q 12 ( L 2 2 6 Lx 1 6 x 2 ) V 5 R A 2 qx 5 q 2 ( L 2 2 x ) M A 5 M B 5 qL 2 12 R A 5 R B 5 qL 2
Problem 10.3-3 A cantilever beam AB of length L has a fixed support at A and a roller support at B (see figure). The support at B is moved downward through a distance d B .

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## This note was uploaded on 03/19/2008 for the course E M 316 taught by Professor Korkolis during the Spring '08 term at University of Texas.

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10-01 Chap Gere - 10 Statically Indeterminate Beams...

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