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MIT2_094S11_lec21

# MIT2_094S11_lec21 - 2.094 Finite Element Analysis of Solids...

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�� 2.094 Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 21 - Plates and shells Prof. K.J. Bathe MIT OpenCourseWare Timoshenko beam theory, and Reissner-Mindlin plate theory For plates, and shells, w , β x , and β y as independent variables. w = displacement of mid-surface, w ( x, y ) A = area of mid-surface p = load per unit area on mid-surface w = w ( x, y ) (21.1) w ( x, y, z ) = w ( x, y ) The material particles at “any z move in the z -direction as the mid-surface. (21.2) u ( x, y, z ) = β x z = β x ( x, y ) z v ( x, y, z ) = β y z = β y ( x, y ) z (21.3) (21.4) xx = ∂u ∂x = z ∂β x ∂x yy = ∂v ∂y = z ∂β y ∂y (21.5) (21.6) γ xy = ∂u + ∂v = z ∂β x + ∂β y (21.7) ∂y ∂x ∂y ∂x ∂β x ∂x xx = z (21.8) ∂β y yy ∂y γ xy ∂β x + ∂β y ∂y ∂x κ 90

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MIT 2.094 21. Plates and shells ∂w ∂u ∂w γ xz = ∂x + ∂z = ∂x β x (21.9) ∂w ∂v ∂w γ yz = ∂y + ∂z = ∂y β y (21.10) ⎤ ⎛ τ xx E 1 ν 0 xx τ yy = ν 1 0 ⎦ ⎝ yy
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MIT2_094S11_lec21 - 2.094 Finite Element Analysis of Solids...

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