# ch9-3 - SECTION 9.5 Method of Superposition 571 q0 Problem...

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SECTION 9.5 Method of Superposition 571 Problem 9.5-11 Determine the angle of rotation u B and deflection d B at the free end of a cantilever beam AB supporting a parabolic load defined by the equation q 5 q 0 x 2 / L 2 (see figure). Solution 9.5-11 Cantilever beam (parabolic load) A B q 0 x y L L OAD : qdx 5 element of load q 5 q 0 x 2 L 2 T ABLE G-1, C ASE 5 (Set a equal to x ) 5 q 0 6 EIL 2 # L 0 ( x 4 )(3 L 2 x ) dx 5 13 q 0 L 4 180 EI 5 1 6 EI # L 0 ¢ q 0 x 2 L 2 ( x 2 L 2 x ) dx d B 5 # L 0 ( qdx )( x 2 ) 6 EI (3 L 2 x ) 5 q 0 2 EIL 2 # L 0 x 4 dx 5 q 0 L 3 10 EI u B 5 # L 0 ( qdx x 2 ) 2 EI 5 1 2 EI # L 0 ¢ q 0 x 2 L 2 x 2 dx A B qdx a L Problem 9.5-12 A simple beam AB supports a uniform load of intensity q acting over the middle region of the span (see figure). Determine the angle of rotation u A at the left-hand support and the deflection d max at the midpoint. Solution 9.5-12 Simple beam (partial uniform load) A B L a a q L OAD : qdx 5 element of load T ABLE G-2, C ASE 6 Replace P by qdx Replace a by x Integrate x from a to L /2 T ABLE G-2, C ASE 6 Replace P by qdx Replace a by x Integrate x from a to L /2 d max 5 Pa 24 EI L 2 2 4 a 2 ) 5 q 24 EI ( L 3 2 6 a 2 L 1 4 a 3 ) u A 5 # L / 2 a qdx 2 EI ( x L 2 x ) 5 q 2 EI # L / 2 a ( xL 2 x 2 ) dx u A 5 Pa ( L 2 a ) 2 EI A B L / 2 L / 2 a a x x qdx qdx

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572 CHAPTER 9 Deflections of Beams A LTERNATE SOLUTION (not recommended; algebra is extremely lengthy) Table G-2, Case 3 5 q 24 EI ( L 3 2 6 La 2 1 4 a 3 ) u A 5 q ( L 2 a ) 2 24 LEI [2 L 2 ( L 2 a )] 2 2 qa 2 24 LEI (2 L 2 a ) 2 5 q 384 EI (5 L 4 2 24 a 2 L 2 1 16 a 4 ) 5 q 24 EI # L / 2 a (3 L 2 x 2 4 x 3 ) dx d max 5 # L / 2 a qdx 24 EI ( x )(3 L 2 2 4 x 2 ) d max 5 q 384 EI L 4 2 24 L 3 a 2 1 16 a 4 ) 2 6 L ¢ L 2 2 1 2 ¢ L 2 3 R 5 qa 2 24 LEI B 2 La 2 1 4 L 2 ¢ L 2 1 a 2 ¢ L 2 2 4 L ( L 2 a ) ¢ L 2 2 1 L ¢ L 2 3 R 1 4 L 2 ( L 2 a ) 2 1 2( L 2 a ) 2 ¢ L 2 2 d max 5 q ( L / 2) 24 LEI B ( L 2 a ) 4 2 4 L ( L 2 a ) 3 A B aa q q L-a q A a = Problem 9.5-13 The overhanging beam ABCD supports two concentrated loads P and Q (see figure). (a) For what ratio P / Q will the deflection at point B be zero? (b) For what ratio will the deflection at point D be zero? Solution 9.5-13 Overhanging beam (a) D EFLECTION AT POINT B Table G-2, Cases 4 and 7 P Q 5 3 a L d B 5 PL 3 48 EI 2 Qa ¢ L 2 16 EI 5 0 (b) D EFLECTION AT POINT D Table G-2, Case 4; Table G-1, Case 4; Table G-2, Case 7 P Q 5 16 a ( L 1 a ) 3 L 2 d D 52 PL 2 16 EI ( a ) 1 Qa 3 3 EI 1 Qa ¢ L 3 EI ( a ) 5 0 A D C B a P Q L 2 L 2
Problem 9.5-14 A thin metal strip of total weight W and length L is placed across the top of a flat table of width L /3 as shown in the figure. What is the clearance d between the strip and the middle of the table? (The strip of metal has flexural rigidity EI .) Solution 9.5-14 Thin metal strip SECTION 9.5 Method of Superposition 573 d L 3 L 3 L 6 L 6 W 5 total weight EI 5 flexural rigidity F REE BODY DIAGRAM (the part of the strip above the table) q 5 W L T ABLE G-2, C ASES 1 AND 10 But : d 5 19 WL 3 31,104 EI q 5 W L 5 19 qL 4 31,104 EI 52 5 qL 4 31,104 EI 1 qL 4 1296 EI d 5 q 384 EI ¢ L 3 4 1 M 0 8 EI ¢ L 3 2 L / 6 L / 6 q M o M o q-L 2 18 = Problem 9.5-15 An overhanging beam ABC with flexural rigidity

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ch9-3 - SECTION 9.5 Method of Superposition 571 q0 Problem...

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