________________________________________________________________________ PROBLEMS (10.48 through *10.51) A simple beam loaded as shown in Figs. P10.48 through P10.51. Determine: (a) The equation of the elastic curve. (b) The angle of rotation at the end A. (c) The deflection at the midspan. SOLUTION (10.48) (a) w y A L/2 wL/8 C L/2 B x 3wL/8 1 w L EIv '' = wLx − < x − > 2 8 2 2 1 w L EIv ' = wLx 2 − < x − >3 +C1 16 6 2 1 w L EIv = wLx3 − < x − > 4 +C1 x + C2 48 24 2 Using boundary conditions: v(0) = 0 : C2 = 0 (a) (b) 1 wL4 7 wLx 4 − + C1 L = 0, C1 = − wL3 48 384 384 Equations (a) and (b) become w L v' = [24 Lx 2 − 64 < x − >3 −7 L3 ] 384 EI 2 w L v= [8Lx 3 − 16 < x − > 4 −7 L3 x] 384 EI 2 (b) Make x=0 in Eq.(1): 7 wL3 θA = 384 EI v( L) = 0 : (1) (2) (c) Make x=L/2 in Eq.(2): 5 wL4 vC = ↓ 768 EI Continued on next slide Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. SOLUTION (10.49) (a) y M0 A a C b L M0/L x B M0/L M0 x − M 0 < x − a >0 L M EIv ' = 0 x 2 + M 0 < x − a > +C1 2L M0 3 M0 EIv = x + < x − a > 2 +C1 x + C2 6L 2 Boundary conditions: EIv '' = v(0) = 0 : C2 = 0 (a) (b) v( L) = 0 : C1 = − M0 (3b 2 − L2 ) 6L Equations (a) and (b) become M0 v' = [−3x 2 − 6 L < x − a > −(3b 2 − L2 )] 6 EIL M0 v= [− x3 + 3L < x − a > 2 −(3b 2 − L2 ) x] 6 EIL (b) Make x=0 in Eq.(1): M0 θA = ( L2 − 3b 2 ) 6 EIL (1) (2) (c) Make x=L/2 in Eq.(2): M 0 9 L3 3 vM = − 3aL2 + 3La 2 − b 2 L) ( 6 EIL 8 2 SOLUTION (10.50) (a) y P A a k P/2 C a B x P/2 Continued on next slide Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. P x−P < x−a > 2 P P EIv ' = x 2 − < x − a > 2 +C1 4 2 P 3 P EIv = x − < x − a >3 +C1 x + C2 12 6 EIv '' = (a) (b) Boundary conditions: P PEI v(0) = − : C2 = − 2k 2k 3 8Pa Pa 3 PEI v(2a) = 0 : − + 2C1a − , 12 6 2k C1 = − Pa 2 PEI + 4 4ak Equations (a) and (b) become P EI v' = [ x 2 − 2 < x − a >2 −a 2 + ] 4 EI ak P EI 6 EI v= [ x 3 − 2 < x − a >3 +3(− a 2 + ) x − ] 12 EI ak k (1) (2) (b) Make x=0 in Eq.(1): P EI θA = (−a 2 + ) 4 EI ak (c) Make x=a in Eq.(2): P 3EI vC = (2a 3 + )↓ 12 EI k SOLUTION (*10.51) (a) 2 x wo x w0 L 2w0 y B A L/2 C L/2 w0L/4 B w0L/4 (a) Actual load = A C x w0L/4 w0L/4 4 w0 ( x − L ) 2 L 2w0 (b) Equivalent load Continued on next slide Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Refer to Fig.(b): wL x 1 4w L L 1 2 x2 1 M = 0 x− ( w0 ) + [ 0 < x − > 2 ] < x − >3 4 2 L 3 2 L 2 3 2 Thus, wL w 2w L EIv '' = 0 x − 0 x 3 + 0 < x − >3 4 3L 3L 2 w0 L 2 w0 4 w0 L EIv ' = x − x + < x − > 4 +C1 8 12 L 6L 2 wL w w L EIv = 0 x 3 − 0 x 5 + 0 < x − >5 +C1 x + C2 24 60 L 30 L 2 (a) (b) Boundary conditions: v(0) = 0 : C2 = 0 v( L) = 0 : C1 = − 5 w0 L3 192 Equations (a) and (b) become w0 L v' = [24 L2 x 2 − 16 x 4 + 32 < x − > 4 −5 L2 ] 192 EIL 2 w0 L v= [8L2 x 3 − 3.2 x 5 + 6.4 < x − > 4 −5 L3 x] 192 EIL 2 (1) (2) (b) Substitute x=0 into Eq.(1): 5 w0 L2 θA = 192 EI (c) Substitute x=L/2 into Eq.(2): w0 L4 vC = vmax = ↓ 120 EI Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.