MIT2_094S11_hw_8_sol

# MIT2_094S11_hw_8_sol - 2.094 FINITE ELEMENT ANALYSIS OF...

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2.094 F INITE E LEMENT A NALYSIS OF S OLIDS AND F LUIDS S PRING 2008 Homework 8 - Solution Instructor: Prof. K. J. Bathe Assigned: 04/10/2008 Due: 04/17/2008 Problem 1 (20 points): Since , we only consider the displacement in the , Hh b ± u 1 x -direction with the plane stress assumption. Total Lagrangian formulation Applied force: 2 1 Bt tt f x ρ ω = . Thickness: 1 0 0 1 0 x bx a tH h ab ba ⎛⎞ −− =+ ⎜⎟ ⎝⎠ , Page 1 of 4 11 x a h t The displacement field with a single two node element is 1 12 1 2 2 1 1 ( ) (1 ) ) ) ) 22 2 2 u uh r u hr u r u r u r r u ⎡⎤ = + + = + ⎢⎥ ⎣⎦ The Jacobian matrices are 0 0 1 1 2 r J x == ∂− and 1 1 2 t t r J x The linear strain components can be written as () 0 00 0 0 0 0 11 1 1 1 1 2 1 1, 2 , 2 r t r uu u u r u r u r e xx x r x r x r x u JJ rr r u hJJ h u ∂∂ = + ⎧⎫ + ⎨⎬ ⎩⎭

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() ( ) ( ) 1 12 03 3 00 20 2 11 02 0 1 1 tt t uu uh u hh u e xx x x u ⎡⎤ ⎢⎥ =+ = + + ⎣⎦ where 1, 2 2, 1 t rr u hu hu r , and 1 1 2 2 t u hu 1 0 22 ba ba xr + Therefore, 2, 0 2 1 2 2 t t t L JJ hJJ h B −− ⎧⎫ ∂∂ ++ ⎨⎬ ⎩⎭ = The nonlinear strain components are 1 1 0 2 11 1 1 2 u u u x r r δδ ηη δ ⎛⎞ =→ == ⎜⎟ ⎝⎠ 2 3 3 1 2 0 1 u x =
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MIT2_094S11_hw_8_sol - 2.094 FINITE ELEMENT ANALYSIS OF...

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