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441_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

441_Mechanics SolutionInstructors_Sol.Manual-Mechanics_Materials_7e.book_Gere_light.1

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SECTION 5.7 Nonprismatic Beams 435 Solution 5.7-3 (a) F IND MAX . BENDING STRESS FOR TAPERED CANTILEVER FIG . ( A ) numerical data s ( x ) ± M ( x ) S ( x ) M ( x ) ± Px + M 0 S ( x ) ± b c h A a 1 + x 2 L bd 2 6 S ( x ) ± bh ( x ) 2 6 I ( x ) ± bh ( x ) 3 12 S ( x ) ± I ( x ) h ( x ) 2 M 0 ± 4 5 PL M 0 ± 800 in.-lb h A ± 2 in. h B ± 3 in. b ± 1 in. P ± 50 lb L ± 20 in. h ( x ) ± h A a 1 + x 2 L b agrees with plot at left Evaluate max. stress & stress at B using numerical data (b) F IND MAX . BENDING STRESS FOR TAPERED CANTILEVER , FIG . ( B ) I ( x ) ± bh ( x ) 3 12 S ( x ) ± I ( x ) h ( x ) 2 S ( x ) ± bh ( x ) 2 6 q 0 ± 3 P L M 0 ± 4 5 PL M 0 ± 800 in.-lb h ( x ) ± h A a 1 + x 2 L b ; s max s B ± 1.042 s B ± s (20) s B ± 1200 psi ; s max ± s (8) s max ± 1250 psi ; x max ± 8 in. so x ± 2 1 PL ² M 0 2 P OR simplifying c ² 24 L 2 ² 2 PL + Px + 2 M 0 bh A 2 (2 L + x ) 3 d ± 0 ² 2 1 24 Px ³ 24 M 0 2 L 2 bh A 2 (2
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Unformatted text preview: L ³ x ) 3 ± 24 P L 2 bh A 2 (2 L + x ) 2 d dx c 24 1 Px + M 2 L 2 bh A 2 (2 L + x ) 2 d ± d dx s ( x ) ± then solve for x max s ( x ) ± 24 1 Px + M 2 L 2 bh A 2 (2 L + x ) 2 s ( x ) ± Px + M b c h A a 1 + x 2 L bd 2 6 10 20 500 1000 1500 2000 M(x) (in.-lb) x (in.) 10 20 1200 1220 1240 1260 σ (x) (psi) x (in.) 05Ch05.qxd 9/24/08 4:59 AM Page 435...
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