Course Description:
The finite element method is a widely used numerical technique that is commonly applied to solve a large class of engineering problems. The two part course is specially designed to best equip engineers to maximize their understanding and application of this great tool in various areas of engineering. In addition, this set of courses presents the fundamentals of the finite element method and modeling techniques for structural analysis applications.
Part 1 of this course provides a robust understanding of the finite element method and assures that analyses are performed correctly, confidently and efficiently. Theory of the finite element method and applications, using several approaches is covered in Part 1. 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 postprocessing results, and sets the basis for more advanced uses, which are covered in Part 2. An introduction to nonlinear finite element analyses and modal analyses is provided at the end of Part 1. 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.
Part 2 of the course builds upon the fundamentals covered in Part 1 and presents a wellbalanced discussion on the theory and application of finite element modeling techniques for an array of disciplines. In Part 2 nonlinear analyses, structural dynamics, stability, contacts, analysis of composite materials, and use of the finite element modeling in damage tolerance and fracture mechanics are discussed in greater detail.
Course Includes:
 A course notebook prepared by your instructors
 Supplemental material to enhance the subject matter
 Refreshments served each morning and afternoon
Course Textbook:
Logan, Daryl L., A First Course in the Finite Element Method, Cengage Learning Global Engineering Publisher, Fifth Edition, 2012
Course Outline:
Day 1
 Introduction (1 Hour)
 Why FEA?
 No fear FEA – When done right!
 History
 Applications – various examples including mechanical, thermal, fluid, electromagnetic, and bio engineering
 Need for verification and validation of the results
 Introduction to types of errors with examples
 Review of Mathematical Fundamentals – Linear Algebra (1 Hour)
 Vectors and matrices
 Matrix equations and operations
 Eigenvalues and eigenvectors
 Introduction to Finite Element Method (1.5 Hours)
 Illustrations of nodes, degrees of freedom (DOF) and element (type, order)
 Numbering convention and notations
 Element to global stiffness matrix assembly
 Boundary Conditions enforcement and model solution
 Interpretation of results
 Finite Element Analysis Using Direct Method (1.5 Hours)
 Applying direct method to derive the stiffness matrix for a bar/rod element
 Shape functions
 Solving a multi uniaxialbar (1D truss) problem
 Applying direct method to derive the stiffness matrix for a bar/rod element
 Structural Analysis (2 Hours)
 Design requirements
 Application of principles of mechanics
 Structural behavior subject to forces
 Equilibrium of forces
 Stress and strain analysis
Day 2
 Examples (2 Hours)
 Solving a case with prescribed forces
 Solving a case with prescribed displacements
 Assignment
 Element rotations – Add DOF
 2D Truss problem – Show the importance of putting matrices together
 Examples – Classical and FE solutions of a bridge
 Finite Element Analysis Using Minimum Potential Energy (2 Hours)
 Introduction of the concept
 Derivation of the stiffness matrix for a bar/rod element
 Example – show similarity with the direct method
 Introduction to Beam Element (3 Hours)
 Bernoulli and Timoshenko beams
 Formulations
 Shape functions
 Bernoulli and Timoshenko beams

 Extending the concept of minimum potential energy to a beam element
 Example – Assembly of beam element stiffness matrix
Day 3
 Finite Element Analysis Using Variational Methods of Weighted Residuals and Galerkin’s Method (3.5 Hours)
 Introduction of the concept
 Applying the concepts to derive the stiffness matrix for a bar/rod element
 Example – Comparison to direct and minimum potential energy methods
 Applying the concepts to derive the stiffness matrix for a beam element
 Example – Comparison to direct and minimum potential energy methods
 Class Exercise – Solve a spring problem in tension with MPE using classical analysis and FEA
 Introduction of 2D Solid Elements (3.5 hours)
 Triangle element
 Constant strain elements
 Element limitations
 Triangle element

 Quadrilateral element through various examples
 Applications: Use FEMAP to solve a problem using 3 element types: constant strain, linear strain, and equilateral
 Problems with trielement
Day 4
 Isoperimetric Formulation (2.5 Hours)
 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
 Example 1 – 4 element model with loads applied, solve natural vs. physical coordinates
 Example 2 – Calculate shape functions for a distorted quadrilateral element with straight sides
 Example 3 – Calculate shape functions for a distorted quadrilateral element with curved sides
 Introduction to 3D Solid Elements (1.5 Hours)
 Show element types and formulations and shape functions
 Solve example problems
 How to go from CAD to FEM
 Class Exercise
 Introduction to Shell Elements (2 Hours)
 Introduce and discuss element types and formulations and shape functions
 Kirchhoff and Mindlin elements
 Introduce and discuss element types and formulations and shape functions

 Class Exercise
 Introduction to Higher Order Elements (1 Hour)
Day 5
 Practical Considerations Using MultiElement Static Problem (1.5 Hours)
 Applications – combination of road/beam and plates/shells
 Use of symmetry to reduce computational cost
 Reflective vs period symmetry
 Examples
 Floor grid models
 Full fuselage model
 Application for loads analysis
 Introduction to submodeling – further discussion in Part 2’s course

 Example – stress and failure analyses
 Introduction to Nonlinear Analyses (1 Hour)
 Sources of nonlinearity
 Quick mathematical background;
 FEMAP demonstration
 Modal Analysis (1.5 Hours)
 Introduction to Natural Frequency
 Importance of natural frequency
 Computation techniques using FEM
 Instances of real life events
 Example – Vibration of beams
 Introduction to buckling
 Introduction to Natural Frequency
 Verification and validation of results (3 Hours)
 Requirement
 Means
 Definitions from NAFEMS
 Closing (0.5 Hour)