Course Description:

The finite element method is a widely used numerical procedure that can be applied to obtain solutions to a large class of engineering problems. A robust understanding of the finite element method assures that analyses are performed correctly, confidently and efficiently. This course presents the fundamentals of the finite element method and modeling techniques for structural analysis applications.

Theory of the finite element method, using several approaches is covered throughout the course. The direct method is mostly covered, but the Principle of Minimum Potential Energy is also mentioned. The direct method, the Principle of Minimum Potential Energy, various methods of Weighted Residual and a Variational approach are introduced. The formulations of several element types are discussed. The isoparametric formulation of the elements stiffness matrices and numerical integration are covered.

This demonstrates how software solves large scale problems and provide an understanding which aids with setting up analyses and post- processing results, and sets the basis for more advanced uses. An introduction to non-linear finite element analyses and modal analyses is provided at the end of the course. The concepts covered in this course are enforced with practical problems and examples, and discussion on common errors and typical use of the finite element method in the aerospace industry. Lastly, the concepts of verification and validation, their requirements and methods are presented throughout the course.

Course Includes:

• A course notebook prepared by your instructors
• Supplemental material to enhance the subject matter

Course Textbook:

Logan, Daryl L., A First Course in the Finite Element Method, Cengage Learning- Global Engineering Publisher, Sixth Edition, 2016

Course Outline:

### Day 1 (7 Hours)

• Theoretical component
• Introduction – 1.5 Hrs.
• Review of Mathematical Fundamentals – Linear Algebra – 1 Hr.
• Introduction to Finite Element Method – 1.5 Hrs.
• Finite Element Analysis Using Direct Method – 1.5 Hrs.
• Rod and springs Element – 1.5 Hrs.
• Application component
• Use of FEM to solve statically indeterminate problems and examples – 1 Hr.
• Assignment
• Selected truss problems to be solved by hand

### Day 2 (7 Hours)

• Theoretical component
• Finite Element Analysis Using Minimum Potential Energy – 1.5 Hrs.
• Introduction of the concept
• Derivation of the stiffness matrix for a bar/rod element
• Example
• Introduction to Beam Element  – 3 Hrs.
• Bernoulli and Timoshenko formulations
• Extending the concept of minimum potential energy to a beam element
• Examples with FEMAP, show orientations
• Application component
• Design and analysis of bolted joints – 2.5 Hrs.
• Assignment 2
• Load transfer of a bolted joint using FEM (by hand or with software)

### Day 3 (7 Hours)

• Theoretical component
• Introduction of 2-D Solid Elements – 3 Hrs.
• Triangle element
• Constant strain elements
• Element limitations
• Examples and in-class demo
• Isoperimetric Formulation – 2 Hrs.
• Mapping of coordinate systems from physical to natural and vice versa
• Numerical integration (integration points) – Midpoint Rule, Simpson Rules, Newton Cotes, Gaussian Quadrature, and Gaussian Points
• Examples
• Applications component
• Stress concentrations, intro to fatique, and use of the FEM in conjunction with S-N Curves – 2 Hrs.
• Assignment 3
• Calculation of stress concentrations using FEM (requires use of software)

### Day 4 (8 Hours)

• Theoretical component
• 3-D Solid Elements – 1 Hr.
• Show element types and formulations and shape functions
• Solve example problems
• Class exercise
• Introduction to Shell Elements – 1 Hr.
• Introduce and discuss element types and formulations and shape functions
• Kirchhoff and Mindlin formulations
• Examples and in-class demo
• Higher Order Elements – 1 Hr.
• Introduction to Non-linear Analyses – 1.5 Hrs.
• Sources of non-linearity
• Quick mathematical background
• FEMAP/ANSYS/ABAQUS demonstration
• Introduction to dynamic FEM – 1 Hr.
• Applications component
• How to go from CAD to FEM – 0.5 Hr.
• Demo of non-linear analyses with FEM – 1 Hr.
• Demo of implicit / explicit fem – 1 Hr.
• Assignment 4

### Day 5 (8 Hours)

• Theoretical component
• Modal Analysis – 1.5 Hrs.
• Introduction to Natural Frequency
• Computation techniques using FEM
• Introduction to buckling
• Computation techniques using FEM
• Examples and in-class demo
• Applications component
• Introduction to FEM as a tool in fracture mechanics – 2 Hrs.
• Linear Elastic Fracture Mechanics
• Elasto-Plastic Fracture Mechanics
• Examples and in-class demo
• Closing – 0.5 Hr

Course Project:

Beam bending modeled with beam elements, rods and plates, shells, solid elements. Comparison with close form solution and computational cost.