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Department of Biomedical Engineering

**Finite Element Analysis**

**BMEN/ENGR 636 INTRODUCTION TO THE FINITE ELEMENT METHOD**

Spring Semester 2006

Last Updated 1/12/2006 by RTH

- Instructor: Dr. R.T. Hart
- Office hours: Generally by appointment, after 11 AM
- Boggs Center 534, 865-5889
- e-mail: rthart@tulane.edu
**Class Meetings:**TTh 8:00-9:15**Required Textbooks:**- Robert D. Cook, D.S. Malkus, M.E. Plesha, R.J. Witt,
*CONCEPTS**AND**APPLICATIONS**OF**FINITE**ELEMENT**ANALYSIS*,**Fourth****Edition**, John Wiley and Sons, 2002.

Getting started with ABAQUS, Version 6.5, from Hibbitt, Karlsson & Sorensen, Inc., 2005.

**Recommended Supplementary Text:**- S.J. Chapman,
*MATLAB Programming for Engineer*s, The Wadsworth Group, Brooks/Cole, Third Edition, 2004.

**Goals:**

The objective of the course is introduce finite element methods for approximate numerical solutions to engineering problems. The course concentrates on solution of structural problems, but also provides the basis for expanding that focus to other engineering field problems (

e.g., thermal, electromagnetic). The displacement method of finite element analysis is developed with emphasis on the isoparametric formulation. A balanced presentation is sought that gives a firm grounding in the theoretical fundamentals coupled some programming considerations and use of finite element computer programs to solve structural problems. Analyses will be conducted using ABAQUS (ABAQUS, Inc.), including the CAE pre- and post-processor for most assignments. A student project involves writing a MATLAB program for analysis of plane frame structures. The course will use the TIS RS6000 UNIX cluster (on-line here: TIS UNIX documentation).

**Course Specific Aims:**

1. Students will be able to derive element stiffness matrices for plane frame elements using the direct stiffness method.

- Relevant ABET Criteria:

- a: an ability to apply knowledge of mathematics, science, and engineering
- e: an ability to identify, formulate, and solve engineering problems

2. Students will write a MATLAB computer program to solve plane frame analysis problems, plot undefomed and deformed shapes, and describe the use and operations of the program with a complete user's manual.

- Relevant ABET Criteria:

- a: an ability to apply knowledge of mathematics, science, and engineering
- c: an ability to design a system, component, or process to meet desired needs
- e: an ability to identify, formulate, and solve engineering problems
- g: an ability to communicate effectively
- k: an ability to use the techniques, skills, and modern engineering tools necessary for engineering practice

3. Students will be able to use the principle of stationary potential energy and the Rayleigh-Ritz method to develop approximate solutions for structural problems and will be able to incorporate these concepts into developing the isoparametric family of finite elements.

- Relevant ABET Criteria:

- a: an ability to apply knowledge of mathematics, science, and engineering
- e: an ability to identify, formulate, and solve engineering problems

4. Students will be able to solve and analyze a set of linear elastic structures (plane frames, plane strain, plane stress, 3-D) using the finite element method with ABAQUS software.

- Relevant ABET Criteria:

- a: an ability to apply knowledge of mathematics, science, and engineering
- c: an ability to design a system, component, or process to meet desired needs
- e: an ability to identify, formulate, and solve engineering problems
- j: a knowledge of contemporary issues
- k: an ability to use the techniques, skills, and modern engineering tools necessary for engineering practice

5. Students will read and discuss finite element literature including the history of its development as well as recent and current applications. Results – including failures -- are discussed. Students read about applications in the current biomechanics literature.

- Relevant ABET Criteria:

- i: a recognition of the need for, and an ability to engage in life-long learning
- j: a knowledge of contemporary issues

**Prerequisites:**

- Statics (E241), Mechanics of Materials (E243), Biomechanics (BMEN 330, or consent of instructor)
- Programming experience (and ability to learn MATLAB independently)
- Senior or Graduate standing (or for exceptional circumstances, seek consent of instructor).

**Topics:**

- Introduction, Review of Matrix Algebra
- Direct Method: Springs
- ABAQUS analysis procedures
- Direct Method: Plane truss and beam elements
- Stiffness matrix transformations; Assembly of Global Stiffness Equations
- MATLAB Programming considerations
- Equation solving techniques: direct and interative
- Stationary Potential Energy Principle
- Introduction to Rayleigh-Ritz Method
- Energy formulations of plane truss and frame elements
- Review of Elasticity: Equilibrium equations, Strain-Displacement relations, Stress-Strain relations, Compatibility relations.
- Interpolation functions
- Isoparametric element formulation, Jacobian matrix
- Numerical integration techniques
- Plane 2-D elements, consistent nodal loads, convergence requirements
- Introduction to Dynamic and Non-Linear problems and/or Introduction to Biomechanics applications
- Introduction of finite element theory for non-structural applications

**FINAL EXAM:** The "final" exam is generally a take-home exam due near
the end of the semester.

**POLICIES:**

In order for the course to be effective, students must assume responsibility for being prepared to learn, and in particular to keep up to date with the reading and assignments -- students must be "active learners." This is a demanding course, especially for undergraduates, but can provide methods needed in a variety of biomechanics research projects.

Although you can discuss your homework with others, including me, (

e.g., if you get stuck), assignments are not intended to be done by a "team," and what you turn inmust be your own work.For the program project, you may ask others for MATLAB or computer help, but the project must represent your own work. The take home exam must be entirely your own work.

Compliance with the School of Engineering Honor Code is, of course, expected and assumed.

**Approximate Grading Criteria:**- Weekly homework 30%
- Frame project 30%
- Take home exam 40%

**Note: in accordance with BMEN Departmental policy, 2% of the
final grade will be determined based on whether the end-of-semester course
evaluation
is completed by the student. **

Some Useful Links:

*UNIX, Emacs:***MATLAB**

*Fortran**FEM**Pi*- Pi Through the Ages
- Archimede's method for approximating Pi
(from Mathematica Demo)

*Equation Solving*- R.J. Shewchuk's report: An Instroduction to the Conjugate Gradient Method, Without the Agonizing Pain (PDF)

*Course Handouts**References*- On-line list of FEM references
- Some General Mechanics References:
- Beer, F. P. and E. R. Johnston, 1981,
*Mechanics of materials*, New York, McGraw-Hill. - Gere, J. M. and S. Timoshenko, 1990,
*Mechanics of materials*, Boston, PWS-KENT Pub. Co. - Hughes, T.J.R., 1987,
*The Finite Element Method*, Prentice-Hall. - Jennings, A., 1977
*, Matrix Computation for Engineers and Scientists,*John Wiley and Sons. - Timoshenko, S. and J. N. Goodier, 1969,
*Theory of elasticity*, New York,, McGraw-Hill. - Timoshenko, S. and D. H. Young, 1968,
*Elements of strength of materials*, Princeton, N.J.,, Van Nostrand.