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problem7_sol

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

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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 ν =0.25 Boundary conditions: At z=0: At z=L: u=v=w=0 u=v=0 M 0 =0 M L =0 F z =0 (a) According to the beam theory: σ z ε z = + ε zo ; where ε zo = 0.01 ε gz E furthermore , ε z = ε za k y x ; where ε za = dw and k y = d θ y = d 2 u 2 dz dz dz therefore , dw d 2 u σ z = E ( 2 x 0.01 ε gz ) dz dz At any cross section, M y = σ z xdA = M 0 = 0 A In the meanwhile, M y = σ z xdA = E ( ε za k y x 0.01 ε gz ) xdA A A where u,v,w, ε za and k y are only functions of z ,and ε 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

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Therefore, M y = − E ε za ( z ) dA + Ek y ( z ) x 2 dA + 0.01 EC 1 N z ( z ) 15 1 + 0 0 . . 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 x 2 dA = 2 0 2 x 2 wdx + T 2 x 2 Tdx = 4.6722 × 10 6 A 2 N ( x ) xdA = 15 xdA + 7.5 x 2 dA = 3.504 × 10 5 x A A A N x ( x ) 2 xdA = 225
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problem7_sol - 22.314/1.56/2.084/13.14 Fall 2006 Problem...

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