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hw4-2008sol

# hw4-2008sol - 1 Set the center of Z-shaped beam cross...

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1. Set the center of Z-shaped beam cross section as origin of coordinate. The moment of inertia I yy = integraldisplay A y 2 dA = 2 3 ( h - 2 t ) ( t 2 ) 3 + 2 3 t parenleftbigg ( t 2 ) 3 + ( b - t 2 ) 3 parenrightbigg I zz = integraldisplay A z 2 dA = 2 3 t ( h 2 - t ) 3 + 2 3 b parenleftbigg ( h 2 ) 3 - ( h 2 - t ) 3 parenrightbigg I yz = integraldisplay A yzdA = 1 2 ( bt - b 2 )( t 2 - ht ) plug in numbers, t = 3 4 in, b = 8 in, h = 20 in, we have I yy = 222 . 3379 in 4 I zz = 1508 in 4 I yz = 418 . 6875 in 4 Young’s Modulus is E = 200 GPa, L = 12 in. (a) Generalized flexure method, coordinates of point A is ( y, z ) = ( - 10 , - 7 . 625) , M z = - 15000 in-lbs, M y = 0 in-lbs, use transformation formulas for the moments of inertia θ = 1 2 tan 1 parenleftbigg - 2 I yz I yy - I zz parenrightbigg = 16 . 54 o Iy 1 y 1 = I yy + I zz 2 + I yy - I zz 2 cos2 θ - I yz sin2 θ = 98 in 4 Iz 1 z 1 = I yy + I zz 2 - I yy - I zz 2 cos2 θ + I yz sin2 θ = 1632 . 31 in 4 Iy 1 z 1 = I yy - I zz 2 sin2 θ + I yz cos2 θ = 0 in 4 σ = ( M y I zz + M z I yz ) z - ( M z I yy + M y I yz ) y I yy I zz - I 2 yz = 90 . 86 lb/in 2 κ y = 2 ν y ∂x 2 = M z I yy + M y I yz E ( I yy I zz - I 2 yz ) = - 1 . 04 × 10 10 κ z = 2 ν

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hw4-2008sol - 1 Set the center of Z-shaped beam cross...

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