GT ME 6443 - Introduction to Finite Element Methods (13 pages)

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Introduction to Finite Element Methods



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Introduction to Finite Element Methods

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Pages:
13
School:
Georgia Institute of Technology
Course:
Me 6443 - Variational Methods
Variational Methods Documents

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Introduction to Finite Element Methods ME6443 Dr A A Ferri x u1 u2 Within each element we have a particular E A L u1 u2 x Assume a linear displacement field exact for statics Need to relate the 2 generalized coordinates for each element to the displacement field within each element Assume a linear displacement field u x u1N1 x u2 N 2 x 1 x N1 x 1 L x N 2 x L 1 The functions N x are termed shape functions and they are just Ritz functions that are local to each element Element potential energy 2 1 L V EA u dx 2 0 We can also consider a distributed force per unit length on each element u1 u2 f x P1 P2 x Elemental virtual work can be expressed as L dW 0 f x du dx P1 du 0 P2 du L P1 du1 P2 du2 Let s take P1 P2 zero for now it will be easy to put them back later dV dW 0 Variational Principle L du EA u 0 f x du dx 0 L 2 2 2 EA u N d u N f x d u N j j dx 0 i i i i 0 i 1 i 1 j 1 L 2 d u u EA Ni N i j j dx 0 i 1 j 1 2 kij f x N dx i 0 0 L fi EA 1 1 ke 1 1 L Assuming that the ui s are independent and arbitrary we get 2 u j kij j 1 fi 0 i 1 2 ke ue f e What if there are 2 elements element 1 element 2 E1 A1 L1 u1 u11 u12 combine E2 A2 L2 u11 u12 u12 k1 u1 f1 k1 0 u22 k2 u2 f 2 0 u1 k2 u2 f2 uG FG KG f1 Thus the global or unassembled static equations are KG uG FG But the global coordinate vector uG u11 u12 u12 u22 T has 1 too many degrees of freedom if the elements are connected U1 U2 U3 The coordinates of uG and U U1 U2 U3 T are related as follows u11 1 1 u2 0 2 0 u1 2 0 u2 0 1 1 0 0 U1 0 U2 0 U3 1 uG S U 4x3 compatibility matrix Substitue S into global relationship KG S U FG Now premultiply by S T to eliminate equations and preserve symmetry S T K G S U S T FG K U F Any nodes that are fixed have zero displacement Therefore we can just delete the rows and columns of K and the rows of F corresponding to any fixed nodes Nodal forces If a force P is applied to node 1 of element n then the additional contribution to the elemental force vector is P 0 T likewise if the force P is



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