## Course

Course Number: | ENGRG 7540 |

Course Name: | Advanced Finite Element Method (Online) |

Course Description: | Introduces the finite element method. Emphasizes beam and frame analysis, plane strain, axisymmetric, and three-dimensional stress analysis. Includes dynamic analysis and field problems, such as heat transfer. Utilizes readily available finite element computer programs to solve stress analysis, heat transfer, thermal stresses, etc. P: BS in Engineering or related field. |

Prerequisites: | |

Level: | Graduate |

Credits: | 3 |

Format: | Online |

Program: | Masters of Science in Engineering |

NOTE: The information below is representative of the course and is subject to change. The specific details of the course will be available in the Desire2Learn course instance for the course in which a student registers.

## Additional Information

**Learning Outcomes**

The overall goals of ENGR 7540: Finite Element Method are that you be able to do the following:

- Demonstrate a working knowledge of computational methods in the finite element method that allow for the solution of stress analysis and heat-transfer problems.
- Select which element(s) to use and what model to construct for a specific problem in stress analysis and heat transfer.
- Assess practical situations where the finite element method is applicable.
- Develop skills that enable you to define and solve open-ended engineering design problems with the aid of a finite element computer program.

**Unit Descriptions**

*Unit 1: Preliminary and Basic Concepts*

The finite element method involves the effective use of matrix algebra and the setup and solution of simultaneous linear algebraic equations. To better understand the discussions, derivations, notations, and formulations of equations to follow in later units, it is imperative that you understand matrix algebra and solution procedures for simultaneous linear algebraic equations. Along with background material, Unit 1 presents a brief history of the finite element method, as well as some significant references for those wanting to examine the development of this method. Unit 1 outlines the general steps in a finite element formulation. It shows numerous applications of the finite element method. Finally, it introduces basic concepts on which the primary method, called the direct stiffness method, is based.

*Unit 2: Line Elements - Bar (Truss), Beam (Frame) - and Flowchart of Computer Program for Truss Analysis*

Line elements form the basis for the stress analysis of real structures. Such structures as two- and three-dimensional trusses, statically indeterminate beams, and two- and three-dimensional frames can be analyzed quite easily with the line elements introduced in this unit. Unit 2 then develops the stiffness matrices for bar, truss, beam, and frame elements. This unit also introduces concepts of transformation matrices, inclined or skewed supports, use of symmetry, distributed loading, and beam element with nodal hinge. Finally, Unit 2 describes a flowchart that is a basis for a computer program for the analysis of trusses.

*Unit 3: Two-Dimensional Elements for Plane Stress/Strain and Axisymmetric Problems and Computer Program Analysis*

Two-dimensional or plane elements are used for plane stress/strain analysis where the loading is applied in the plane of each element. These elements are used to analyze plates with holes, fillets, or other changes in geometry for stress concentration problems. An example would be a hydraulic cylinder rod end used in a piece of heavy equipment such as a backhoe tractor. Axisymmetric elements allow a simple method to model three-dimensional solids that are actually axisymmetric in their geometry. An example is a steel die used in the plastic film industry. Unit 3 then develops the stiffness matrices for the plane stress/strain and axisymmetric elements. This unit also introduces various concepts that should be considered when modeling any problem for solution using these elements. Further extension of the computer program to solve problems with these elements is also described. Numerous examples are included.

*Unit 4:*

Three-dimensional elements are used for three-dimensional (3-D) stress analysis where the loading is applied anywhere in space. These elements are either tetrahedral (4-sided) or hexahedral (6-sided or brick-type) elements and are used to model such bodies as engine blocks, thick-walled dams, foot pedals, and other truly 3-D structures. Unit 4 first develops the stiffness matrix for the tetrahedral element and describes how the hexahedral element stiffness matrix is developed. It goes on to develop the stiffness matrix for 1-D and 2-D heat transfer due to both conduction and convection, allowing the determination of temperature variation and heat transfer throughout a body. Then, for bodies subjected to variations in temperature, the unit examines how thermal stresses occur. Finally, Unit 4 formulates the equations necessary for a time-dependent loading problem, such as when a blast load is applied to a building frame.

**Grading Criteria**

This course is structured as a graduate-level course, with primary delivery via the World Wide Web, e-mail, and fax. Your success depends on a number of written assignments consisting primarily of longhand solving of problems using numerical analysis, more difficult solutions using the computer program, written engineering design projects (two), and written exams after each unit (four). All written assignments should draw on your knowledge gained from the assigned readings, which are listed both in the reading schedule and in the individual units. Also included are additional suggested readings to gain further insight into the material.

As a member of this class, you are expected to meet basic requirements as follows:

- Solve numerous longhand exercises (20 percent of your final grade)
- Solve numerous problems using the computer program (15 percent)
- Complete two open-ended design projects (25 percent)
- Complete exams at the end of each of four units (40 percent)