Finite Element Methods (in Solid and Structural Mechanics)

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1 CEE570 / CSE 551 Class #1 Finite Element Methods (in Solid and Structural Mechanics) Spring 2014 Prof. Glaucio H. Paulino Donald Biggar Willett Professor of Engineering Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign Acknowledgements: J. Kim, Z. Zhang, S. Song, C. Le and K. Park 1

2 Acknowledgements The slide series for this class has been developed with the help from the following people. J. Kim S. Song Z. Zhang Chau Le K. Park 2

3 Content Teaching Team Prerequisites Course Logistics Course Outline (See Handout) Course Information (See Handout) Course Schedule (See Handout) Course Objectives Text books FEM: The big picture 3

4 Handouts Paper Handouts Course outline Course information Course schedule Note: These handouts are clipped together. Electronic Handouts MATLAB Review of elasticity (CEE471) This ppt presentation 4

5 CEE570/CSE551 Teaching Team Instructor TA TA TA TA TA Prof. Paulino Professor Willett Prof. of Eng. Faculty: CEE, CSE Sofie Leon Ph.D. Student CEE Tomas Zegard Ph.D. Student CEE Junho Chun Ph.D. Student CEE Heng Chi Ph.D. Student CEE Xiaojia (Shelly) Zhang Ph.D. Student CEE 5

6 CEE 471 (Structural Mechanics) Hjelmstad, Keith D. Fundamentals of Structural Mechanics 2nd ed., 2005, XIV, 480 p., Hardcover ISBN: X This course is an introduction to basic continuum mechanics that emphasizes variational formulations and some basic theoretical/numerical aspects. It provides the prerequisite information for finite elements for graduate and advanced undergraduate students in engineering. 6

7 Pre-requisites for CEE570/CSE551: FEM CEE 471 (Structural Mechanics) Tensors; geometry of deformation; constitutive relations; energy principles; boundary value problem; beam theory; plate theory; static stability theory; computational methods. OR TAM 551 (Solid Mechanics I) Mechanics of elastic deformable bodies, based on the fundamental concepts of modern continuum mechanics: kinematics, balance laws, constitutive equations; classical small-deformation theory; formulation of initial boundary-value problems of linear elastodynamics and boundary-value problems of linear elastostatics; variational formulations, minimum principles; applications of theory to engineering problems 7

8 Course Logistics URL: Computer Labs.: EWS: Software Required (available at EWS) MATLAB PATRAN ABAQUS 8

9 CEE570/CSE551: FEM Class Description Theory and applications of the finite element method; General procedure for solving a structural problem using FEM; Variational Methods; Weighted Residual (WR) Methods Review of one-dimensional elements; Techniques for constructing element stiffness matrices; Various two- and three-dimensional solid elements; Isoparametric formulation; Plates and shells elements; Stress recovery and post processing; Other topics: error, convergence, solving linear system of equations Textbook Concepts and Applications of Finite Element Analysis, R.D. Cook, D.S. Malkus, M.E. Plesha, R.J. Witt., 4th edition, John Wiley & Sons, Inc

10 Concepts and Applications of Finite Element Analysis; Cook, Malkus, Plesha, Witt., 4th Ed, John Wiley, Our book has something for everyone 1. Introduction 2. One-Dimensional Elements and Computational Procedures 3. Basic Elements 4. Formulation Techniques: Variational Methods 5. Formulation Techniques: Galerkin and Other Weighted Residual Methods 6. Isoparametric Elements 7. Isoparametric Triangles and Tetrahedra 8. Coordinate Transformation and Selected Analysis Options 9. Error, Error Estimation, and Convergence 10. Modeling Consideration and Software Use 11. Finite Elements in Structural Dynamics and Vibrations 12. Heat Transfer and Selected Fluid Problems 13. Constraints: Penalty Forms, Locking, and Constraint Counting 14. Solids of Revolution 15. Plate Bending 16. Shells 17. Nonlinearity: An Introduction 18. Stress Stiffness and Buckling Practical Application Very Important Theoretical 10

11 Recommended Textbooks K.-J. Bathe (1995) Finite Element Procedures Prentice Hall R. D. Cook (1995) Finite Element Modeling for Stress Analysis John Wiley and Sons 11

12 Recommended Textbooks T. J. Hughes (2000) The Finite Element Method: Linear Static and Dynamic Finite Element Analysis Dover Publications J. N. Reddy (2005) An Introduction to the Finite Element Method 3nd Edition, McGraw Hill J. N. Reddy (2004) An Introduction to Nonlinear Finite Element Analysis Oxford University Publication 12

13 Recommended Textbooks M. Asghar Bhatti (2005) Fundamental Finite Element Analysis and Applications: with Mathematica and Matlab Computations John Wiley and Sons 13

14 Recommended Textbooks O. C. Zienkiewicz and R. L. Taylor (2000) The Finite Element Method 4th Edition, John Wiley and Sons Vol 1: The Basis Vol 2: Solid Mechanics Vol 3: Fluid Mechanics 14

15 Recommended Textbooks O. C. Zienkiewicz, et. al (2005) The Finite Element Method 6th Edition, Butterworth-Heinemann Vol 1: The Basis Vol 2: Solid Mechanics Vol 3: Fluid Mechanics 15

16 FEM: The big picture General Structural Analysis Problem Structure Geometry Material properties Boundary conditions: Essential BCs (prescribed displacement) Natural BCs (tractions) We want to know the states (displacement, strains, stresses ) throughout the structure. 16

17 Governing relations to be satisfied: (1) Constitutive relations: relation between stresses and strains (determined by material properties) (2) Kinematic relations: relation between displacement and strains (usually in the form of differential equations) (3) Equilibrium of forces (usually in the weak form of the Principle of Virtual Work, the Principle of Stationary Potential Energy, or the vanishing of weighted residuals) Closed form solution is not possible for a general case. Approximate numerical solution is employed: the FEA. 17

18 Simulation process (broad view) Beam theory Plate theory Finite element methods IDEALIZATION DISCRETIZATION SOLUTION Physical system Mathematical system Discrete Model Discrete Solution Modeling + discretization + solution error Discretization + solution error Solution error change models (e.g. beam solid) e.g. adaptive remeshing e.g. iterative refinement The above flowchart is based on the one given in Felippa s book 18

19 Finite Element Analysis Procedure Discretization (divide the structure into small, simple elements) z Localization (obtain the behavior of each element) x y K e u e e f Globalization (Assembly) (relate all elements based on the connectivity) Ku f Solution and post processing (solve for state variables and recover quantities of interest, such as stress) 19

20 Next Class Introduction to FEM History of FEM Advantages and shortcomings of FEM Misuses of FEA programs 20

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