problem7_sol - 22.314/1.56/2.084/13.14 Fall 2006 Problem...

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Unformatted text preview: 22.314/1.56/2.084/13.14 Fall 2006 Problem Set VII Solution Solution: x z y Beam model Cross section Geometry and material properties: L=2.4m; W=200mm; T=7mm and E=75 GPa and =0.25 Boundary conditions: At z=0: At z=L: u=v=w=0 u=v=0 M =0 M L =0 F z =0 (a) According to the beam theory: z z = + zo ; where zo = 0.01 gz E furthermore , z = za k y x ; where za = dw and k y = d y = d 2 u 2 dz dz dz therefore , dw d 2 u z = E ( 2 x 0.01 gz ) dz dz At any cross section, M y = z xdA = M 0 = 0 A In the meanwhile, M y = z xdA = E ( za k y x 0.01 gz ) xdA A A where u,v,w, za and k y are only functions of z ,and gz = C 1 N + C 2 N 2 = C 1 N x ( x ) N z ( z ) + C 2 ( N x ( x ) N z ( z )) 2 1 Therefore, M y = E za ( z ) dA + Ek y ( z ) x 2 dA + 0.01 EC 1 N z ( z ) 15 1 + . . 1 2 x xdA + 0.01 EC 2 N z ( z ) 2 N x ( x ) 2 xdA A A A A where dA = 0 for symmetry A T w...
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problem7_sol - 22.314/1.56/2.084/13.14 Fall 2006 Problem...

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