S18-1PAT301, Section 18, October 2003 SECTION 18 BRIEF INTRODUCTION TO THE FINITE ELEMENT METHOD

S18-2PAT301, Section 18, October 2003

S18-3PAT301, Section 18, October 2003 Engineering Analysis Classical Methods Numerical Methods Closed-form Approximate Finite Element Finite Difference Boundary Element METHODS FOR SOLVING ENGINEERING PROBLEMS As show below, the finite element method is one of several methods for solving engineering problems

S18-4PAT301, Section 18, October 2003 METHODS FOR SOLVING ENGINEERING PROBLEMS (Cont.) Classical Methods: Closed-form solutions are available for simple problems such as bending of beams and torsion of prismatic bars Approximate methods using series solutions to governing differential equations are used to analyze more complex structures such as plates and shells The classical methods can only be used for structural problems with relatively simple geometry, loading, and boundary conditions

S18-5PAT301, Section 18, October 2003 METHODS FOR SOLVING ENGINEERING PROBLEMS (Cont.) Numerical Methods: Boundary Element Method Solves the governing differential equation for the problem with integral equations over the boundary of the domain. Only the boundary surface is meshed with elements Finite Difference Method Replaces governing differential equations and boundary conditions with corresponding algebraic finite difference equations

S18-6PAT301, Section 18, October 2003 METHODS FOR SOLVING ENGINEERING PROBLEMS (Cont.) Numerical Methods (cont.) Finite Element Method (FEM) Capable of solving large, complex problems with general geometry, loading, and boundary conditions Increasingly becoming the primary analysis tool for designers and analysts The Finite Element Method is also known as the Matrix Method of Structural Analysis in the literature because it uses matrix algebra to solve the system of simultaneous equations.

S18-7PAT301, Section 18, October 2003 WHAT IS THE FINITE ELEMENT METHOD? The Finite Element Method (FEM) is a numerical approximation method. It is a method of investigating the behavior of complex structures by breaking them down into smaller, simpler pieces. These smaller pieces of structure are called (finite) elements. The elements are connected to each other at nodes. The assembly of elements and nodes is called a finite element model. The piston head shown on the next page is an example of a finite element model.

S18-8PAT301, Section 18, October 2003 SAMPLE FINITE ELEMENT MODEL Element Sample Finite Element Model Node

S18-9PAT301, Section 18, October 2003 FINITE ELEMENTS Finite elements have shapes which are relatively easy to formulate and analyze. The three basic types of finite elements are beams, plates, and solids. Beam (1D) Plate (2D) Solid (3D)

S18-10PAT301, Section 18, October 2003 ONE DIMENSIONAL ELEMENTS 1D beam elements are used to model long, slender structural members as demonstrated in this communications tower finite element model

S18-11PAT301, Section 18, October 2003 TWO DIMENSIONAL ELEMENTS 2D plate elements are used to model thin structural members such as aircraft fuselage skin or car body

S18-12PAT301, Section 18, October 2003 THREE DIMENSIONAL ELEMENTS 3D solid elements are used to model thick components such as the piston head show below:

S18-13PAT301, Section 18, October 2003 BUILDING A FINITE ELEMENT MODEL The Finite Element Method approximates the behavior of a continuous structure with a finite number of elements As one increases the number of elements (and hence, decrease the size of the elements), the results become increasingly accurate, but the computing time also increases. MSC.Patran provides numerous modeling tools to help the user build finite element models with the right balance between accuracy and computing time.

S18-14PAT301, Section 18, October 2003 HOW DOES FEM WORK ? Basic Approach A given problem is discretized by dividing the original domain into simply shaped elements. Elements are connected to each other by nodes. X Y Z

S18-15PAT301, Section 18, October 2003 HOW DOES FEM WORK ? (Cont.) Three translations (u x, u y, u z ) Three rotations ( x, y, z ) {u} = displacement vector = { u x u y u z x y z } Each node is capable of moving in six independent directions: three translations and three rotations. These are called the degrees of freedom (DOF) at a node.

S18-16PAT301, Section 18, October 2003 HOW DOES FEM WORK ? (Cont.) n The relationship between an element and its surrounding nodes can be described by the following equation: [ k ] e { u } e = { f } e n The elemental stiffness matrix [ k ] e is derived from geometry, material properties, and element properties. n The elemental load vector { f } e describes the forces acting on the element. n The displacement vector { u } e is the unknown in this equation. It describes how the nodes are moving as a result of the applied forces. [ k ] e { u } e = { f } e Elemental Equation

S18-17PAT301, Section 18, October 2003 HOW DOES FEM WORK ? (Cont.) n Next the elemental stiffness matrices are assembled into a global stiffness matrix. The loads are also assembled into a global load vector. This results in the following matrix equation for the overall structure: [ K ] { u } = { F } [ k ] e { u } e = { f } e Elemental Equation Global Equation

S18-18PAT301, Section 18, October 2003 HOW DOES FEM WORK ? (Cont.) n Next apply the boundary condition to the model (constrain the model). Mathematically this is achieved by removing rows and columns corresponding to the constrained degrees of freedom from the global matrix equation. Boundary Condition [ K ] { u } = { F } Global Matrix Equation with boundary condition applied

S18-19PAT301, Section 18, October 2003 HOW DOES FEM WORK ? (Cont.) n Finally the global matrix equation is solved to determine the unknown nodal displacements. n Element strains and stresses are then computed from the nodal displacements. Deformation Plot Stress Fringe Plot

S18-20PAT301, Section 18, October 2003 n Summary of the finite element method HOW DOES FEM WORK ? (Cont.) Assemble loads into a global load vector {F} Represent continuous structure as a collection of discrete elements connected by nodes Derive element stiffness matrices from material properties, element properties, and geometry Assemble all element stiffness matrices into a global stiffness matrix [K] Apply boundary conditions to constrain the model Solve the matrix equation [K] {u} = {F} for nodal displacements Compute strains and stresses from displacement results

S18-21PAT301, Section 18, October 2003 REFERENCES V. Adams Building Better Products with Finite Element Analysis OnWord Press, 1999 K. J. Bathe Finite Element Procedures in Engineering Analysis Prentice-Hall, 1982 R. D. Cook Concepts and Applications of Finite Element Analysis John Wiley & Sons, 1989 R. H. MacNeal Finite Elements: Their Design and Performance Marcel Dekker, 1994

S18-22PAT301, Section 18, October 2003 REFERENCES (Cont.) NAFEMS A Finite Element Primer Department of Trade and Industry, UK, 1986 J. S. Przemieniecki Theory of Matrix Structural Analysis McGraw-Hill, 1968 B. A. Szabo and I. Babuska Finite Element Analysis John Wiley & Sons, 1991 O. C. Zienkiewicz The Finite Element Method McGraw-Hill, 1994