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CCHHAAPPTTEERR 99 PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 1373 B A y L P x PROBLEM 9.1 For the loading shown, determine (a) the equation of the elastic curve for the cantilever beam AB, (b) the deflection at the free end, (c) the slope at the free end. SOLUTION 0: ( ) 0JM M P L x\uf053 \uf03d \uf02d \uf02d \uf02d \uf03d ( )M P L x\uf03d \uf02d \uf02d 2 2 ( ) d yEI P L x PL Px dx \uf03d \uf02d \uf02d \uf03d \uf02d \uf02b 2 1 1 2 dyEI PLx Px C dx \uf03d \uf02d \uf02b \uf02b 1 10, 0 : 0 0 0 0 \uf0e9 \uf0f9\uf03d \uf03d \uf03d \uf02d \uf02b \uf02b \uf03d\uf0ea \uf0fa\uf0eb \uf0fb dyx C C dx 2 3 1 2 1 1 2 6 EIy PLx Px C x C\uf03d \uf02d \uf02b \uf02b \uf02b 2 2[ 0, 0] : 0 0 0 0 0\uf03d \uf03d \uf03d \uf02d \uf02b \uf02b \uf02b \uf03dx y C C (a) Elastic curve. 2 (3 ) 6 Pxy L x EI \uf03d \uf02d \uf02d \uf074 (2 ) 2 dy Px L x dx EI \uf03d \uf02d \uf02d (b) at .y x L\uf03d 2 3 (3 ) 6 3B PL PLy L L EI EI \uf03d \uf02d \uf02d \uf03d \uf02d 3 3B PLy EI \uf03d \uf0af \uf074 (c) at .dy x L dx \uf03d 2 (2 ) 2 2B dy PL PLL L dx EI EI \uf03d \uf02d \uf02d \uf03d \uf02d 2 2B PL EI \uf071 \uf03d \uf074\uf020 PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 1374 BA y L x M0 PROBLEM 9.2 For the loading shown, determine (a) the equation of the elastic curve for the cantilever beam AB, (b) the deflection at the free end, (c) the slope at the free end. SOLUTION 0 0 0 : 0KM M M M M \uf03d \uf02d \uf02b \uf03d\uf0e5 \uf03d 2 0 0 1 d yEI M M dx dyEI M x C dx \uf03d \uf03d \uf03d \uf02b 0 1 1 0, 0 : 0 \uf0e9 \uf0f9\uf03d \uf03d \uf03d \uf02b \uf03d \uf02d\uf0ea \uf0fa\uf0eb \uf0fb dyx L M L C C M L dx 20 1 2 1 2 EIy M x C x C\uf03d \uf02b \uf02b 2 2 20 0 2 2 0 1 1[ , 0] 0 2 2 x L y M L M L C C M L\uf03d \uf03d \uf03d \uf02d \uf02b \uf03d (a) Elastic curve: 2 20 ( 2 ) 2 My x Lx L EI \uf03d \uf02d \uf02b \uf074\uf020 20 ( ) 2 My L x EI \uf03d \uf02d \uf074\uf020 (b) at 0:y x \uf03d 20 ( 0) 2A My L EI \uf03d \uf02d 2 0 2A M Ly EI \uf03d \uf0ad \uf074\uf020 (c) at 0:dy x dx \uf03d 0 0 0( ) ( 0)dy M M M LL x L dx EI EI EI \uf03d \uf02d \uf02d \uf03d \uf02d \uf02d \uf03d \uf02d 0\uf071 \uf03dA M LEI \uf074\uf020 PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 1375 B A y w L x PROBLEM 9.3 For the loading shown, determine (a) the equation of the elastic curve for the cantilever beam AB, (b) the deflection at the free end, (c) the slope at the free end. SOLUTION 2 0: ( ) 0 2 1 2 J xM wx M M wx \uf053 \uf03d \uf02b \uf03d \uf03d \uf02d 2 2 2 3 1 1 2 1 6 d yEI M wx dx dyEI wx C dx \uf03d \uf03d \uf02d \uf03d \uf02d \uf02b 3 31 1 1 1, 0 : 0 6 6 \uf0e9 \uf0f9\uf03d \uf03d \uf03d \uf02d \uf02b \uf03d\uf0ea \uf0fa\uf0eb \uf0fb dyx L wL C C wL dx 3 31 1 6 6 dyEI wx wL dx \uf03d \uf02d \uf02b 4 3 2 1 1 24 6 EIy wx wL x C\uf03d \uf02d \uf02b \uf02b 4 4 2 1 1[ , 0] 0 0 24 6 x L y wL wL C\uf03d \uf03d \uf03d \uf02d \uf02b \uf02b \uf03d 4 42 1 1 3 24 6 24 C wL wL\uf0e6 \uf0f6\uf03d \uf02d \uf03d \uf02d\uf0e7 \uf0f7\uf0e8 \uf0f8 (a) Elastic curve. 4 3 4( 4 3 ) 24 wy x L x L EI \uf03d \uf02d \uf02d \uf02b \uf074 (b) at 0.y x \uf03d 4 43 24 8A wL wLy EI EI \uf03d \uf02d \uf03d \uf02d 4 8A wLy EI \uf03d \uf0af \uf074\uf020 (c) at 0.dy x dx \uf03d 3 6A dy wL dx EI \uf03d 3 6A wL EI \uf071 \uf03d \uf074 PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 1376 w0 x B A y L PROBLEM 9.4 For the loading shown, determine (a) the equation of the elastic curve for the cantilever beam AB, (b) the deflection at the free end, (c) the slope at the free end. SOLUTION Use Free body AJ. 0:\uf053 \uf03dJM 2 0 0 2 3 \uf02b \uf0d7 \uf03dw x xM L [ , 0] , 0 \uf03d \uf03d \uf0e9 \uf0f9\uf03d \uf03d\uf0ea \uf0fa\uf0eb \uf0fb x L y dyx L dx 3 01 6 \uf03d \uf02d w xM L 2 3 01 6 \uf03d \uf02dd y w x dx L 4 0 1 1 24 \uf03d \uf02d \uf02bdy w xEI C dx L 5 0 1 2 1 120 \uf03d \uf02d \uf02b \uf02bw xEIy C x C L , 0 :dyx L dx \uf0e9 \uf0f9\uf03d \uf03d\uf0ea \uf0fa\uf0eb \uf0fb 3 0 1 1 0 24 w L C\uf02d \uf02b \uf03d 31 0 1 24 \uf03dC w L [ , 0]\uf03d \uf03dx L y 4 40 0 21 1 0120 24\uf03d \uf02d \uf02b \uf02b \uf03dEIy w L w L C 42 0 1 30 C w L\uf03d (a) Elastic curve. 5 4 50 ( 5 4 ) 120 \uf03d \uf02d \uf02d \uf02bwy x L x L EIL \uf074 4 40 ( ) 24 \uf03d \uf02d \uf02bdy w x L dx EIL (b) @ 0:\uf03dy x 4 0 30 \uf03d \uf02dA w Ly EI 4 0 30 \uf03d \uf0afA w Ly EI \u25c4 (c) @ 0:\uf03ddy x dx 3 0 24 \uf03d A dy w L dx EI 3 0 24 \uf071 \uf03dA w LEI \uf074 PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 1377 y A w B C x P 5 23 wa a2a PROBLEM 9.5 For the cantilever beam and loading shown, determine (a) the equation of the elastic curve for portion AB of the beam, (b) the deflection at B, (c) the slope at B. SOLUTION Using ABC as a free body, 20: 2 0 3 4 4 3 3 y A A F R wa wa R wa wa \uf053 \uf03d \uf02b \uf02d \uf03d \uf03d \uf02d \uf03d \uf0af 20: (2 )( ) (3 ) 0 3A A M M wa a wa a\uf0e6 \uf0f6\uf053 \uf03d \uf02d \uf02b \uf02d \uf03d\uf0e7 \uf0f7\uf0e8 \uf0f8 0AM \uf03d Using AJ as a free body, 2 40: ( ) ( ) 0 3 2 1 4 2 3 J xM M wa x wx M wx wax \uf0e6 \uf0f6 \uf0e6 \uf0f6\uf053 \uf03d \uf02b \uf02d \uf03d\uf0e7 \uf0f7 \uf0e7 \uf0f7\uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf03d \uf02d 2 2 2 3 2 1 1 4 2 3 1 2 6 3 d yEI wx wax dx dyEI wx wax C dx \uf03d \uf02d \uf03d \uf02d \uf02b 1 10, 0 : 0 0 0 0 dyx C C dx \uf0e9 \uf0f9\uf03d \uf03d \uf03d \uf02d \uf02b \uf05c \uf03d\uf0ea \uf0fa\uf0eb \uf0fb 4 3 2 1 2 24 9 EIy wx wax C\uf03d \uf02d \uf02b 2 2[ 0, 0]: 0 0 0 0x y C C\uf03d \uf03d \uf03d \uf02d \uf02b \uf05c \uf03d (a) Elastic curve over AB. 4 3(3 16 ) 72 wy x ax EI \uf03d \uf02d \uf074\uf020 \uf020 \uf020 3 2( 4 ) 6 dy w x ax dx EI \uf03d \uf02d (b) at 2 .y x a\uf03d 410 9B way EI \uf03d \uf02d 410 9B way EI \uf03d \uf0af \uf074 (c) at 2 .dy x a dx \uf03d 34 3B dy wa dx EI \uf0e6 \uf0f6 \uf03d \uf02d\uf0e7 \uf0f7\uf0e8 \uf0f8 34 3B wa EI \uf071 \uf03d \uf074\uf020 FBD ABC: FBD AJ: PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part. 1378 y A w B L a C x MC 5 wL2 6 PROBLEM 9.6 For the cantilever beam and loading shown, determine (a) the equation of the elastic curve for portion AB of the beam, (b) the deflection at B, (c) the slope at B. SOLUTION Using ABC as a free body, 0:\uf053 \uf03dyF 0\uf02d \uf03d \uf03dA AR wL R wL [ 0, 0] 0, 0 x y dyx dx \uf03d \uf03d \uf0e9 \uf0f9\uf03d \uf03d\uf0ea \uf0fa\uf0eb \uf0fb 0:\uf053 \uf03dAM 2 2 ( ) 0 2 6 1 3 \uf0e6 \uf0f6\uf02d \uf02d \uf02b \uf03d\uf0e7 \uf0f7\uf0e8 \uf0f8 \uf03d \uf02d A A L wLM wL M wL Using AJ as a free body (portion AB only), 0:\uf03dJM ( ) 02 \uf0e6 \uf0f6\uf02b \uf02d \uf02d \uf03d\uf0e7 \uf0f7\uf0e8 \uf0f8 A A xM wx R x M 21 2 \uf03d \uf02d \uf02b \uf02bA AM wx R x M 2 21 1 2 3 \uf03d \uf02d \uf02b \uf02dwx wLx wL 2 2 2 2 1 1 2 3 \uf03d \uf02d \uf02b \uf02dd yEI wx wLx wL dx 3 2 1 1 1 1 6 2 3 \uf03d \uf02d \uf02b \uf02d \uf02bdyEI wx wLx wLx C dx 1 10, 0 : 0 0 0 0 0 dyx C C dx \uf0e9 \uf0f9\uf03d \uf03d \uf02d \uf02b \uf02d \uf02b \uf03d \uf03d\uf0ea \uf0fa\uf0eb \uf0fb 4 3 2 2 1 1 1 24 6 6 \uf03d \uf02d \uf02b \uf02d \uf02byEI wx wLx wLx C 2 2[ 0, 0]: 0 0 0 0 0\uf03d \uf03d \uf02d \uf02b \uf02d \uf02b \uf03d \uf03dx y C C (a) Elastic curve over AB. 4 3 2 2( 4 4 ) 24 \uf03d \uf02d \uf02b