10-02 Chap Gere

# 10-02 Chap Gere - SECTION 10.4 Method of Superposition 643...

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Problem 10.4-2 The propped cantilever beam shown in the figure supports a uniform load of intensity q on the left-hand half of the beam. Find the reactions R A , R B , and M A , and then draw the shear-force and bending-moment diagrams, labeling all critical ordinates. Solution 10.4-2 Propped cantilever beam SECTION 10.4 Method of Superposition 643 Select R B as redundant. E QUILIBRIUM R ELEASED STRUCTURE AND FORCE - DISPLACEMENT RELATIONS C OMPATIBILITY d B 5 ( d B ) 1 2 ( d B ) 2 5 0 Substitute for ( d B ) 1 and ( d B ) 2 and solve for R B : O THER REACTIONS ( FROM EQUILIBRIUM ) M A 5 9 qL 2 128 R A 5 57 qL 128 R B 5 7 qL 128 M A 5 qL 2 8 2 R B L R A 5 qL 2 2 R B S HEAR - FORCE AND BENDING - MOMENT DIAGRAMS M max 5 945 qL 2 32,768 x 1 5 57 L 128 A B M A R A R B q L 2 L 2 A B ( d B ) 1 5 7 qL 4 384 EI ( d B ) 2 5 R B L 3 3 EI AB L R B q L 2 L 2 x 1 R A R B V O 2 M O M max M A 2 Problem 10.4-3 The figure shows a propped cantilever beam ABC having span length L and an overhang of length a . A concentrated load P acts at the end of the overhang. Determine the reactions R A , R B , and M A for this beam. Also, draw the shear-force and bending-moment diagrams, labeling all critical ordinates. A L a B C M A R A R B P

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Solution 10.4-3 Beam with an overhang 644 CHAPTER 10 Statically Indeterminate Beams Select M A as redundant. E QUILIBRIUM R ELEASED STRUCTURE AND FORCE - DISPL . EQS . C OMPATIBILITY u A 5 ( u A ) 1 2 ( u A ) 2 5 0 Substitute for ( u A ) 1 and ( u A ) 2 and solve for M A : M A 5 Pa 2 R B 5 1 L ( M A 1 PL 1 Pa ) R A 5 1 L ( M A 1 Pa ) O THER REACTIONS ( FROM EQUILIBRIUM ) S HEAR - FORCE AND BENDING - MOMENT DIAGRAMS R B 5 P 2 L (2 L 1 3 a ) R A 5 3 Pa 2 L A B ( u A ) 1 ( u A ) 2 M A L ( u A ) 2 5 M A L 3 EI ( u A ) 1 5 PaL 6 EI C P a P 3 Pa 2 L V O 2 Pa 2 Pa M O 2 L 3 Problem 10.4-4 Two flat beams AB and CD , lying in horizontal planes, cross at right angles and jointly support a vertical load P at their midpoints (see figure). Before the load P is applied, the beams just touch each other. Both beams are made of the same material and have the same widths. Also, the ends of both beams are simply supported. The lengths of beams AB and CD are L AB and L CD , respectively. What should be the ratio t AB / t CD of the thicknesses of the beams if all four reactions are to be the same? Solution 10.4-4 Two beams supporting a load P P B D C A t AB t CD For all four reactions to be the same, each beam must support one-half of the load P . D EFLECTIONS d CD 5 ( P / 2) L 3 CD 48 EI CD d AB 5 ( P / L 3 AB 48 EI AB C OMPATIBILITY d AB 5 d CD or M OMENT OF INERTIA t AB t CD 5 L AB L CD L 3 AB t 3 AB 5 L 3 CD t 3 CD I CD 5 1 12 bt 3 CD I AB 5 1 12 bt 3 AB L 3 AB I AB 5 L 3 CD I CD
Problem 10.4-5 Determine the fixed-end moments ( M A and M B ) and fixed-end forces ( R A and R B ) for a beam of length L supporting a triangular load of maximum intensity q 0 (see figure). Then draw the shear-force and bending-moment diagrams, labeling all critical ordinates.

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10-02 Chap Gere - SECTION 10.4 Method of Superposition 643...

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