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445_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

445_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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Solution 5.7-5 Tapered cantilever beam F ROM E Q . (5-32), E XAMPLE 5-9 Eq. (1) F IND THE VALUE OF THAT MAKES A M AXIMUM N p c d A + ( d B d A ) a x L b d 3 [32 P ] Let s 1 u v d s 1 dx v a du dx b u a dv dx b v 2 N D s 1 x s 1 32 Px p c d A + ( d B d A ) a x L b d 3 After simplification: d s 1 dx N D 32 P c d A 2( d B d A ) x L d p c d A + ( d B d A ) a x L b d 4 D p 2 c d A + ( d B d A ) x L d 6 N 32 p P c d A + ( d B d A ) a x L b d 2 c d A 2( d B d A ) x L d [32 Px ][ p ] [3] c d A + ( d B d A ) a x L b d 2 c 1 L ( d B d A ) d SECTION 5.7 Nonprismatic Beams 439 OR simplifying (288 L 2 ) c 9 PL 2 36 PxL + 5 Px 2 9 M 0 L d c p b A h A 2 (3 L + x ) 4 d 0 * L 2 p b A h A 2 (3 L + x ) 4 d 0 3 1 864 P x L 864 M 0 L 1440 Px 2 2 d dx s ( x ) c (864 PL 2880 P x ) L 2 p b A h A 2 (3 L x ) 3 * L 2 p b A h A 2 (3 L + x ) 3 d 0 d dx c 288 1 3 Px L 3 M 0 L + 5 P x 2 2 OR Solving for x max : solution agrees with plot above, evaluate using numerical data ; s B 0 MPa s B s ( L ) ; s A 210 MPa s A s (0) ; s max s ( x max ) s max 214 MPa x max 0.105m 9 PL 2 36 PxL + 5 Px 2 9 M 0 L 0 Problem 5.7-5 Refer to the tapered cantilever beam of solid circular cross section shown in Fig. 5-24 of Example 5-9.
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