Problem 7.159 w = w0 cos px 2L y For the beam and loading shown, (a) write the equations of the shear and bending-moment curves, (b) determine the magnitude and location of the maximum bending moment. A x B L

Solving Problems on Your Own w = w0 cos px 2L y For the beam and loading shown, (a) write the equations of the shear and bending-moment curves, (b) determine the magnitude and location of the maximum bending moment. A B x L For beams supporting a distributed load expressed as a function w(x), the shear V can be obtained by integrating the function -w(x) , and the moment M can be obtained by integrating V (x).

( ) ( ) ( ) ò ò dV dx = -w = -w0 cos V = - wdx = - w0 sin + C1 dM y w = w0 cos px 2L Problem 7.159 Solution The shear V can be obtained by integrating the function -w(x) and the moment M can be obtained by integrating V (x). A x B x L dV dx px 2L = -w = -w0 cos ( ) 2L p px 2L V = - wdx = - w0 sin + C1 ò dM dx ( ) 2L p px 2L = V = - w0 sin + C1 ( ) 2L p 2 px 2L M = Vdx = w0 cos + C1x + C2 ò

( ) ( ) V = - w0 sin + C1 2 M = w0 cos + C1x + C2 w = w0 cos y w = w0 cos px 2L Problem 7.159 Solution The shear V can be obtained by integrating the function -w(x) and the moment M can be obtained by integrating V (x). A x B x L V = - w0 sin + C1 ( ) 2L p px M = w0 cos + C1x + C2 ( ) 2L p 2 px Boundary conditions At x = 0 : V = C1 = 0 C1 = 0 At x = 0 : M = w0 (2L/p)2 cos (0) + C2 = 0 C2 = -w0 (2L/p)2

( ) ( ) ( ) ( ) ( ) V = - w0 sin 2 2 M = w0 cos - w0 2 y w = w0 cos px 2L Problem 7.159 Solution The shear V can be obtained by integrating the function -w(x) and the moment M can be obtained by integrating V (x). A B x x L V = - w0 sin ( ) 2L p px ( ) 2L p 2 px 2L ( ) 2L p 2 M = w0 cos - w0 ( ) 2L p 2 px 2L M = w0 ( -1 + cos ) ( ) 2L p 2 Mmax = w0 [-1 + 0] Mmax at x = L: 4 p2 Mmax = w0 L2