09-03 Chap Gere - SECTION 9.5 Method of Superposition 571 q0 Problem 9.5-11 Determine the angle of rotation B and deflection B at the free end of a

09-03 Chap Gere - SECTION 9.5 Method of Superposition 571...

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SECTION 9.5 Method of Superposition 571 Problem 9.5-11 Determine the angle of rotation B and deflection B at the free end of a cantilever beam AB supporting a parabolic load defined by the equation q 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 element of load q q 0 x 2 L 2 T ABLE G-1, C ASE 5 (Set a equal to x ) q 0 6 EIL 2 L 0 ( x 4 )(3 L x ) dx 13 q 0 L 4 180 EI 1 6 EI L 0 ¢ q 0 x 2 L 2 ( x 2 )(3 L x ) dx B L 0 ( qdx )( x 2 ) 6 EI (3 L x ) q 0 2 EIL 2 L 0 x 4 dx q 0 L 3 10 EI u B L 0 ( qdx )( x 2 ) 2 EI 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 A at the left-hand support and the deflection max at the midpoint. Solution 9.5-12 Simple beam (partial uniform load) A B L a a q L OAD : qdx 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 max Pa 24 EI (3 L 2 4 a 2 ) q 24 EI ( L 3 6 a 2 L 4 a 3 ) u A L 2 a qdx 2 EI ( x )( L x ) q 2 EI L 2 a ( xL x 2 ) dx u A Pa ( L a ) 2 EI A B L / 2 L / 2 a a x x qdx qdx (Continued)
572 CHAPTER 9 Deflections of Beams A LTERNATE SOLUTION (not recommended; algebra is extremely lengthy) Table G-2, Case 3 q 24 EI ( L 3 6 La 2 4 a 3 ) u A q ( L a ) 2 24 LEI [2 L ( L a ) ] 2 qa 2 24 LEI (2 L a ) 2 q 384 EI (5 L 4 24 a 2 L 2 16 a 4 ) q 24 EI L 2 a (3 L 2 x 4 x 3 ) dx max L 2 a qdx 24 EI ( x )(3 L 2 4 x 2 ) max q 384 EI (5 L 4 24 L 2 a 2 16 a 4 ) 6 L ¢ L 2 2 2 ¢ L 2 3 R qa 2 24 LEI B La 2 4 L 2 ¢ L 2 a 2 ¢ L 2 4 L ( L a ) ¢ L 2 2 L ¢ L 2 3 R 4 L 2 ( L a ) 2 2( L a ) 2 ¢ L 2 2 max q ( L 2) 24 LEI B ( L a ) 4 4 L ( L a ) 3 A B q q q = a a a L – 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 3 a L B PL 3 48 EI Qa ¢ L 2 16 EI 0 (b) D EFLECTION AT POINT D Table G-2, Case 4; Table G-1, Case 4; Table G-2, Case 7 P Q 16 a ( L a ) 3 L 2 D PL 2 16 EI ( a ) Qa 3 3 EI Qa ¢ L 3 EI ( a ) 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 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 L 3 L 3 L 6 L 6 W total weight EI flexural rigidity F REE BODY DIAGRAM (the part of the strip above the table) q W L T ABLE G-2, C ASES 1 AND 10 But : 19 WL 3 31,104 EI q W L 19 qL 4 31,104 EI 5 qL 4 31,104 EI qL 4 1296 EI 5 q 384 EI ¢ L 3 4 M 0 8 EI ¢ L 3 2 L / 6 L / 6 q M 0 M 0 qL 2 18 = Problem 9.5-15 An overhanging beam ABC with flexural rigidity EI 15 k-in. 2 is supported by a pin support at A and by a spring of stiffness k at point B (see figure). Span AB has length L 30 in.

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