2.094 – Finite Element Analysis

of Solids and Fluids

Fall 1998 Course Outline

The objective is to teach in a unified manner the fundamentals of finite element analysis of solids, structures and fluids. This includes the theoretical foundations and appropriate use of finite element methods.

The topics of the course are:

• The formulation of finite element methods for linear static analysis of solids and structures :
  • two- and three-dimensional solids;

  • beam, plate and shell structures.

• The displacement-based finite element procedures, when they are effective, and mixed finite element methods for almost incompressible media and beams, plates and shells.

• The formulation of finite element methods for nonlinear static analysis, including

  • geometric nonlinearities (large displacements and large strains);

  • material nonlinearities (nonlinear elasticity and elasto-plasticity).

• The formulation of finite element methods for the analysis of heat transfer in solids: conduction, convection and radiation conditions.
• The formulation of finite element methods for fluid flows:
• incompressible flows;

• Navier-Stokes equations including heat transfer;

• interactions with structures.

• The appropriate use of finite element procedures:

  • setting up an appropriate model;

  • interpreting the results and assessing the solution error.

The methods studied in this course are practical procedures that are employed extensively in the mechanical, civil, ocean and aeronautical industries.

Increasingly, the methods are used in computer-aided design.

Undergraduate statics, undergraduate mathematics.

K.J. Bathe. Finite Element Procedures. Prentice Hall, Inc., Englewood Cliffs, NJ, 1996.

You will find many references in this book that you may refer to when studying special topics.

The student's course grade is based on the performance in:

• the weekly homeworks, given out each Thursday and to be handed in the following Thursday;
• the course project, due November 24, 1998;
• two 1½ hour exams: on October 22, 1998, and on December 3,1998.

The reading assignments will be given in the lectures and will refer to the textbook Finite Element Procedures. We will discuss specific material in Chapters 1, 3, 4, 5, 6, 7, and Section 8.4 .

Every student is required to complete a term project. The objective of this task is that each student obtains hands-on experience in solving analysis problems using a typical finite element code. The term project can address a problem solution in solids and structures or fluid flows using ADINA.

The following projects are suggested:

1. Large deformation analysis of a rubber sheet with holes.
2. Large deformation analysis of a thick-walled rubber cylinder subjected to internal pressure.
3. Large displacement collapse analysis of an elastic thin structure (beam, plate or shell structure).
4. Elasto-plastic collapse analysis of a structure, for example a truss bridge.
5. Thermal stress analysis of a structure.
6. Analysis of fluid flow in a chamber or around an obstruction.
7. Analysis of a forced or natural convection fluid flow problem.
8. Analysis of a problem related to your research.

Note: Please choose a (tractable) problem that you can analyze in depth in the very limited time available.

The project work is typically started at the beginning of October and typically involves the following steps:

• Choose a problem and simplify it in its data (geometry, material response, boundary conditions) such that in the first instance you can compare your analysis results with some analytical results.
• Solve this "simple" problem using ADINA. Obtain an accurate solution.
• Now increase the complexity of the problem (for example, assume that the material response is nonlinear) and re-solve the problem. Obtain an accurate solution using different finite elements, different meshes. Ask "what if" questions and experiment with the finite element method.
• In each case, interpret the calculated response.