________________________________________________________________________ PROBLEM (*10.15) A cantilever beam is loaded as shown in Fig. P10.15. Using the multiple-integration method, determine: (a) The equation of the elastic curve. (b) The deflection at the free end. (c) The reactions at the fixed support. *SOLUTION (a) EIv '''' = − w0 cos EIv ''' = −( 2 w0 L πx 2L πx + C1 2L 4 w L2 πx + C1 x + C2 EIv '' = ( 02 ) cos π 2L 8w L3 πx 1 + C1 x 2 + C2 x + C3 EIv ' = ( 03 ) sin π 2L 2 4 16w0 L πx 1 1 + C1 x 3 + C2 x 2 + C3 x + C4 ) cos EIv = −( 4 π 2L 6 2 π ) sin Boundary conditions: 16w0 L4 v(0) = 0 : C4 = 4 π v '(0) = 0 : C3 = 0 2w L v(0) ''' = 0 : C1 = 0 π v(0) '' = 0 : C2 = − 2 w0 L2 π Therefore, w πx − π 3 Lx3 + 3π 3 L2 x 2 − 48 L4 ] v = − 40 [48L4 cos 3π EI 2L (b) Let x=L in the above expression: w w L4 vB = − 20 (−π 3 L4 + 3π 3 L4 − 48L4 ) = 0.04795 0 ↓ 3π EI EI (c) Expressions for shear & moment are 2w L πx V = 0 (1 − sin ) π 2L 2w0 L πx M = 2 (cos + π x − π L) π 2L 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. These result in, for x=0: 2w L RA = 0 ↑ π 2(π − 2) w0 L2 MA = π2 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.