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FEnotes_part4

# FEnotes_part4 - Finite Element Method 2D Beam Element Now...

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1 Finite Element Method – 2D Beam Element Now we consider the 2D beam element , 6 DOF in local coordinates: u 1 – x displacement of node 1 u 2 - z displacement of node 1 u 3 - rotation of node 1 u 4 - x displacement of node 2 u 5 - z displacement of node 2 u 6 - rotation of node 2 How does this element deform? Axial and bending strain energy So we need an approximation for both u 0 (x) and w 0 (x). Properties: A, E, I yy , L displacements reaction forces

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Write u 0 (x), w 0 (x) in terms of u 1 , u 2 , u 3 , u 4 , u 5 , u 6 : Axial deformation: Only u 1 and u 4 are affected by an axial deformation. So we have the same linear assumption as for the truss element: Bending deformation: Four boundary conditions left, so we have: 2 Finite Element Method – 2D Beam Element   x a a x u 2 1 0     4 2 1 0 1 1 0 0 u L a a L u u a u   11 41 2 au uu a L   3 6 2 5 4 3 0 x a x a x a a x w         6 3 6 5 4 0 5 3 6 2 5 4 3 0 3 4 0 2 3 0 3 2 0 0 u L a L a a L w u L a L a L a a L w u a w u a w 3 4 2 3 u a u a boundary conditions boundary conditions 2 6 5 3 2 6 6 5 3 2 5 / 2 2 / 3 2 3 L u u L u u L a L u u L u u L a
3 Finite Element Method – 2D Beam Element Now, evaluate strain energy Substitute previous expressions for u 0 (x) and w 0 (x), take partial derivatives to find components of stiffness matrix: x d x d w d EI x d x d du EA U L y y L 0 2 2 0 2 0 2 0 2 1 2 1 j i ij u u U k 2     22 2 5 6 00 2 2 2 2 3 2 5 5 6 6 11 26 4 12 12 LL yy yy U EA a dx EI a a x dx EALa EI a L a a L a L      41 2 uu a L 2 6 5 3 2 6 6 5 3 2 5 / 2 2 / 3 2 3 L u u L u u L a L u u L u u L a   2 5 3 5 5 6 2 3 2 5 2 6 1 1 4 4 3 2 2 2 2 2 3 3 6 6 12 12 12 12 12 24 12 1 2 4 4 4 yy u u Lu u Lu u Lu u u u Lu u EI EA U u u u u L u L u u L u    

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Finite Element Method – 2D Beam Element 4   L EI L EI L EI L EI L EI L EI L EI L EI L EA L EA L EI L EI L EI L EI L EI L EI L EI L EI L EA L EA k 4 6 0 2 6 0 6 12 0 6 12 0 0 0 0 0 2 6 0 4 6 0 6 12 0 6 12 0 0 0 0 0 2 2 2 3 2 3 2 2 2 3 2 3 beam element stiffness matrix in local coordinate system
5 Finite Element Method – 2D Beam Element Distributed forces: Calculate external potential energy: Set this equal to potential due to “equivalent nodal forces”:    

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FEnotes_part4 - Finite Element Method 2D Beam Element Now...

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