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Introduction to Finite Element Method (FEM) - Intranet home

UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Introduction to Finite Element Method (FEM). Dr. J. Yang, School of Civil Engineering 2011. What is FEM. FEM, also called as Finite Element analysis (FEA), is a Method for numerical solution of field problems. 1. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Element Discretisation Node Mesh Alternative numerical methods Other numerical solution methods : Finite differences Approximates the derivatives in the differential equation using difference equations.

UNIVERSITY OF BIRMINGHAM ABAQUS Training Introduction to Finite Element Method (FEM) 5 Types of finite elements 1-D (line ) element 2-D (plane) element 3-D (solid) element How does the FEA work Consider the bar of different cross-sections subjected to

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Transcription of Introduction to Finite Element Method (FEM) - Intranet home

1 UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Introduction to Finite Element Method (FEM). Dr. J. Yang, School of Civil Engineering 2011. What is FEM. FEM, also called as Finite Element analysis (FEA), is a Method for numerical solution of field problems. 1. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Element Discretisation Node Mesh Alternative numerical methods Other numerical solution methods : Finite differences Approximates the derivatives in the differential equation using difference equations.

2 Useful for solving heat transfer and fluid mechanics problems. Works well for two-dimensional regions with boundaries parallel to the coordinate axes. Cumbersome when regions have curved boundaries. Weighted residual methods (not confined to a small subdomain): Collocation Subdomain Least squares*. Galerkin's Method *. Variational methods * (not confined to a small subdomain). 2. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Advantages of FEA. Versatile No geometric restriction Boundary conditions and loading are not restricted Material M t i l properties ti are nott restricted ti t d Components that have different behaviours An FE structures can closely resembles the actual body The approximation is easily improved by grading the mesh or increasing the order of elements Can deal with multi-physics problems.

3 Terminologies Interpolation elements Nodes DOF. History of FEA. 1943 Courant (Variational Method ). 1956 Turner, Clough, Martin and Topp (Stiffness). 1960 Clough ( Finite Element Element , plane problems). 1970s Applications on mainframe computers 1980s Microcomputers, pre- and post- processors 1990s Analysis of large structural systems 3. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Available commercial FEM software packages ANSYS. SDRC/I-DEAS (complete CAD/CAM/CAE. package). Nastran ABAQUS. COSMOS. ALGOR. PATRAN. HyperMesh LS-Dyna Solving a problem by FEA.

4 Classification of problems Model Idealization Preliminary analysis Finite Element analysis Preprocessing Numerical analysis Postprocessing Check and interpret results Reiteration Structure Model Discretized models s sAt=P. P P. P A t A t 1. 1 A 1. 2. 2 A 2. 3. 3 A 3. A b Rigid support A b Ground 4. Physical Finite Element representation representation 4. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Types of Finite elements 1-D (line ) Element 2-D (plane) Element 3-D (solid) Element How does the FEA work Consider the bar of different 1.

5 Cross-sections subjected to F1. three point loads as shown EA1,l1 1. 2. in the figure. Determine the F2 EA2,ll2 2. displacements at the loading 3. points. F3. EA3,l3 3. 4. F4. nodes elements Element stiffness matrices Equilibrium equation k1 k1 u1 P11 k i EA i k . k1 u 2 P12 . for Element 1 li 1. Equilibrium equation k2 k 2 u 2 P21 . k . k 2 u 3 . for Element 2. 2 P22 . Equilibrium equation k3 k 3 u 3 P31 . k . k 3 u 4 P32 . for Element 3. 3. 5. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Equilibrium of nodes 1 F1. 2 EA2,l2.

6 P11 P11. P22 P22. 1 EA1,l1 3. F3. P12 P12 P31 P31. 2 F 3. 2 EA3,l3. P21 P21 P32 P32. 2 EA2,l2 4. F4. P11 F1 P22 P31 F3. P12 P21 F2 P32 F4. Assembly of stiffness matrices k1 k1 0 0 u 1 P11 . k 1 k1 0 0 u 2 P12 . For Element 1. 0 0 0 0 u 3 0 .. 0 0 0 0 u 4 0 . 0 0 0 0 u1 0 . 0. k2 k2 0 u 2 P . 21 . For Element 2. 0 k2 k2 0 u 3 P22 .. 0 0 0 0 u 4 0 . Element equilibrium equations in the global coordinate system Assembly of stiffness matrices 0 0 0 0 u1 0 . 0 u 0 . 0 0 0 2 . 0 0 k3 k3 u3 P31 For Element 3.. 0 0 k3 k3 u 4 P32 . k1 k1 0 0 u1 P11 F1 . k k1 k 2 k2 0 u2 P12 P21 F2.

7 1 . 0 k2 k 2 k3 k3 u3 P22 P31 F3 .. 0 0 k3 k3 u4 P32 F4 . 6. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Boundary conditions and nodal loads Applying boundary conditions Stiffness matrix u1 0, F1 R. Global equilibrium equations k1 k1 0 0 0 R . k k1 k 2 k2 0 u2 F2 . 1 . 0 k2 k 2 k3 k3 u3 F3 .. 0 0 k3 k3 u4 F4 . Solve for nodal displacements from the global equilibrium equations Solve for reaction forces from the global equilibrium equations Calculate the strain for each Element from obtained displacements Calculate the stress for each Element from obtained displacements Properties of stiffness matrices Nonnegative Kii Symmetry Sparsity p y Singularity (no support or mechanism).

8 7. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Checking results Deformed shape of the structure Balance of the external forces Order of magnitudes of results A generalised approach to formulate stiffness matrix xi xj Let =2x/l and move x the origin into the = -1 =1 middle of the bar =0. Ni 1 1 . Nj Ni Nj . 2 2. 1 1. = -1 = 0 = 1 = -1 = 0 = 1. x N i ( ) xi N j ( ) x j Element displacement field u ( x) N i ( )u1 N j ( )u 2. u u = u2. u = u1. = -1 =0 =1. u Ni N j uu .. i j . 8. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM).

9 Element strain field du dN i dN j u i . ( x) . dx dx dx u j . dx dN i dN j x x x xi l xi xj i j j . d d d 2 2 2 2. dN i dN i d 1 2 .. dx d dx 2 l . dN j dN j d 1 2 .. dx d dx 2 l . 1 1 ui .. l l u j . Element stress field dN i dN j ui . ( x) E ( x) E . dx dx u j . E E ui .. l l u j . Note: Element strain and Element stress are constant Element internal force dN dN j ui . N ( x) ( x) A EA i . dx dx u j . EA EA ui . N . l l u j . 9. UNIVERSITY OF BIRMINGHAM. ABAQUS Training Introduction to Finite Element Method (FEM). Element strain energy E . 1 1 ui .. V. 1 Al 2 0.

10 U dV ui u j l . 2 . E l l u j . l . EA EA .. 1.. uj l l ui . EA u j . ui 2 EA.. l l . EA EA .. [K ] l l . EA EA Element stiffness matrix . l l . Work in the Element Pi q(x) Pj W qudx Pi u i P j u j . l Ni . q ui . u j dx Pi u i P j u j . l N j . ql . Pi . ui . u j 2 ui u j . ql . Pj . 2 . Principle of minimum potential energy for a single Element EA EA ql .. 1. U W ui l l ui u .. u j 2 ui Pi . EA EA u j . uj i uj . 2 l Pj . q . l l 2 . EA EA ql .. l l ui 2 Pi 0.. uT EA EA u j ql Pj .. l l 2 . EA EA ql . Pi . l l ui 2. EA EA u j l Element equilibrium equation q Pj.


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