EN175-24

EN175-24 - EN0175 11 / 30/ 06 Finite element method...

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EN0175 11 / 30/ 06 Finite element method (continued) Summary of essential concepts: FEM analysis begins with calculations on the element level. Element 1 #2 # 3 # Element nodal displacement: = 3 # 2 3 # 1 2 # 2 2 # 1 1 # 2 1 # 1 u u u u u u u el Element interpolation function: () 2 1 1 , x x N , ( ) 2 1 2 , x x N , ( ) 2 1 3 , x x N Element displacement field: el u N N N N N N u u = 3 2 1 3 2 1 2 1 0 0 0 0 0 0 Element strain field: 1
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EN0175 11 / 30/ 06 el el u B u x N x N x N x N x N x N x N x N x N x N x N x N = = = 1 3 2 3 1 2 2 2 1 1 2 1 2 3 2 2 2 1 1 3 1 2 1 1 12 22 11 0 0 0 0 0 0 2 ε Element stress field: el u B D D E = = = = ν σ 12 22 11 2 12 22 11 2 2 1 0 0 0 1 0 1 1 Element strain energy: el el T el V ij ij el u K u V U el 2 1 d 2 1 = = εσ B D B A K T el el = Element nodal force: el el A i i V i i u F S u t V u f el el = + d d After the element quantities are calculated, the next step is to assemble the global stiffness matrix. Global strain energy: u K u u K u U U T elements el el T el elements el 2 1 2 1 = = = 1 6 3 # 2 3 # 1 2 # 2 2 # 1 1 # 2 1 # 1 × = u u u u u u u el Æ ( ) () 1 2 3 2 3 1 2 2 2 1 1 2 1 1 × = n u u u u u u u M ( ) 6 6 × el K Æ n n K 2 2 × Element connectivity ( ) c b a , , 3 # , 2 # , 1 # ( ) ( ) c c b b a a z 2 , 1 2 , 2 , 1
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This note was uploaded on 01/10/2010 for the course EN 0175 taught by Professor Huajiangao during the Spring '06 term at Brown.

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EN175-24 - EN0175 11 / 30/ 06 Finite element method...

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