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1 S2-1 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation SECTION 2 INTRODUCTION TO THE FINITE ELEMENT METHOD

2 S2-2 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation

3 S2-3 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation Engineering Analysis Classical Methods Numerical Methods Closed-form Approximate Finite Element Finite Difference Boundary Element METHODS FOR SOLVING ENGINEERING PROBLEMS n As show below, the finite element method is one of several methods for solving engineering problems

4 S2-4 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation METHODS FOR SOLVING ENGINEERING PROBLEMS (CONT.) n Classical Methods: u Closed-form solutions are available for simple problems such as bending of beams and torsion of prismatic bars u Approximate methods using series solutions to governing differential equations are used to analyze more complex structures such as plates and shells u The classical methods can only be used for structural problems with relatively simple geometry, loading, and boundary conditions

5 S2-5 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation METHODS FOR SOLVING ENGINEERING PROBLEMS (Cont.) n Numerical Methods: u Boundary Element Method l 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 u Finite Difference Method l Replaces governing differential equations and boundary conditions with corresponding algebraic finite difference equations

6 S2-6 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation METHODS FOR SOLVING ENGINEERING PROBLEMS (Cont.) n Numerical Methods (cont.) u Finite Element Method (FEM) l Capable of solving large, complex problems with general geometry, loading, and boundary conditions l Increasingly becoming the primary analysis tool for designers and analysts l 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.

7 S2-7 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation WHAT IS THE FINITE ELEMENT METHOD? n 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. n These smaller pieces of structure are called elements. The elements are connected to each other at nodes. n 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.

8 S2-8 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation SAMPLE FINITE ELEMENT MODEL Element Sample Finite Element Model Node

9 S2-9 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation FINITE ELEMENTS n Finite elements have shapes which are relatively easy to formulate and analyze. The three basic types of finite elements are beams, shells, and solids. Beam (1D) Shell (2D) Solid (3D)

10 S2-10 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation ONE DIMENSIONAL ELEMENTS n 1D beam elements are used to model long, slender structural members as demonstrated in this communications tower finite element model

11 S2-11 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation TWO DIMENSIONAL ELEMENTS n 2D shell elements are used to model thin structural members such as aircraft fuselage skin or car body

12 S2-12 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation THREE DIMENSIONAL ELEMENTS n 3D solid elements are used to model thick components such as the piston head show below:

13 S2-13 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation BUILDING A FINITE ELEMENT MODEL n The Finite Element Method approximates the behavior of a continuous structure with a finite number of elements n 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. n MSC.Patran provides numerous modeling tools to help the user build finite element models with the right balance between accuracy and model size.

14 S2-14 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation HOW DOES FEM WORK ? n Basic Approach u A given problem is discretized by dividing the original domain into simply shaped elements. u Elements are connected to each other by nodes. X Y Z

15 S2-15 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation 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 } n 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.

16 S2-16 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation 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

17 S2-17 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation 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

18 S2-18 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation 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

19 S2-19 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation 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

20 S2-20 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation 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

21 S2-21 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation TYPES OF FINITE ELEMENT METHODS n There are two different types of finite element methods - the displacement method and the force method. In both methods, equilibrium, compatibility, and stress-strain relations are used to generate a system of equations that represent the behavior of the structure. n In the displacement method, the grid point displacements are the basic unknowns in the system of equations. n In the force method, the member forces are the basic unknowns in the system of equations. n Both methods can be used to solve structural problems. The displacement method is used by most modern finite element codes, including MSC.Nastran.

22 S2-22 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation KEY CONCEPTS IN FEM n The Displacement Method n Formulation of the Element Stiffness Matrix n Matrix Assembly and Decomposition

23 S2-23 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation THE DISPLACEMENT METHOD n All structural engineering analyses must satisfy the following three general conditions: 1. Equilibrium of forces and moments: F = 0, M = 0 2. Strain-Displacement relations (also called compatibility of deformations): Ensures that the displacement field in a deformed continuous structure is free of voids or discontinuities

24 S2-24 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation THE DISPLACEMENT METHOD (CONT.) 3. Stress-Strain relations (also called constitutive relations): u For a linear material, the generalized Hookes law states { } = [E] { } where { } = { x y z xy yz zx } { } = { x y z xy yz zx } [E] = 6 x 6 matrix of elastic constants

25 S2-25 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation THE DISPLACEMENT METHOD (CONT.) n These three conditions can be used to generate a system of equations in which the displacements are unknown. n The stiffness matrix [K] is used to relate the forces acting on the structure and the displacements resulting from these forces in the following manner: {F} = [K] {u} Where {F} = forces acting on the structure [K] =stiffness matrix [k ij ] where each k ij term is the force of a constraint at coordinate i due to a unit displacement at j with all other displacements set equal to zero {u} =displacements resulting from {F} n Boundary conditions are applied to prevent rigid body motions, and the system of linear equations is solved for the unknown {u}.

26 S2-26 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation FORMULATION OF THE ELEMENT STIFFNESS MATRIX n A key step in the displacement method is the formulation of the element stiffness matrix n Each element in a finite element model is represented by an element stiffness matrix [K] e n A single-rod case study is used to demonstrate the element stiffness matrix formulation for a rod element

27 S2-27 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: ROD ELEMENT STIFFNESS MATRIX n Consider an elastic rod of uniform cross section A and length L under axial load. n Axial translations u 1 and u 2 are the only displacements at grid points 1 and 2. Thus, this element has two degrees of freedom. F1F1 F2F2 X 1 2 u1u1 u2u2 L X = 0 A

28 S2-28 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation n Step 2: Relate strain to displacements Assume that the rod changes length by an amount L due to the axial load. The strain in the rod is n Step 1: Satisfy static equilibrium CASE STUDY: ROD ELEMENT STIFFNESS MATRIX (CONT.) (1) (2)

29 S2-29 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation n Step 3: Relate stress to strain n Step 4: Relate force to stress CASE STUDY: ROD ELEMENT STIFFNESS MATRIX (CONT.) (3) (4) and

30 S2-30 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation n Step 5: Relate force to displacement u Substitution of Equations 2 and 3 into Equation 4 yields CASE STUDY: ROD ELEMENT STIFFNESS MATRIX (CONT.) or similarly EA (5) (6)

31 S2-31 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation {F} = [K] e {u} n Equations 5 and 6 represent two linear equations with two unknowns. Rewrite them in matrix form: CASE STUDY: ROD ELEMENT STIFFNESS MATRIX (CONT.) (6) or [K] e Where [K] e = [k ij ], the known 2x2 rod element stiffness matrix {F} = vector of known applied forces {u} = vector of unknown displacements

32 S2-32 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation n The method used in the previous case study to derive the rod element stiffness matrix is called the direct method or the stiffness method. This method works well for simple elements such as rods and beams. n For more complex 2D and 3D elements, the variational method is used. u The variational method is also known as the Rayleigh-Ritz method u Assumed element shape functions and energy principles are used to derive the element stiffness matrices u The variational method is covered in detail in text books on the finite element method. A list of reference books on the finite element method is included at the end of this section FORMULATION OF THE ELEMENT STIFFNESS MATRIX

33 S2-33 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation n The stiffness matrix for a rod element under torsion is shown below: ADDITIONAL EXAMPLES OF ELEMENT STIFFNESS MATRIX e T1T1 T2T2 X 1 2 x1 L X = 0 J x2

34 S2-34 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation n The stiffness matrix for a beam element under in-plane shear and bending is shown below: ADDITIONAL EXAMPLES OF ELEMENT STIFFNESS MATRIX e F

35 S2-35 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: TWO-ROD ASSEMBLY n The following case study demonstrates the assembly of the the individual element stiffness matrices and the solution to the entire problem.

36 S2-36 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: TWO-ROD ASSEMBLY (CONT.) n Write the following element stiffness equations based on the previous derivation of stiffness matrix for a rod element [K] 1 [K] 2

37 S2-37 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: TWO-ROD ASSEMBLY (CONT.) n Rewrite the stiffness matrices in simpler terms: where and

38 S2-38 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: TWO-ROD ASSEMBLY (CONT.) n Assemble the two stiffness matrices by superposition. The resulting matrix is called the global stiffness matrix. ( ) Global Stiffness Matrix [K]

39 S2-39 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: TWO-ROD ASSEMBLY (CONT.) n Apply external loads to the structure F 1 = -PF 2 = 0F 3 = 0

40 S2-40 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: TWO-ROD ASSEMBLY (CONT.) n Next impose the boundary condition u The right end is fixed so u 3 = 0. This is achieved by discarding row 3 and column 3 from the global stiffness matrix.

41 S2-41 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: TWO-ROD ASSEMBLY (CONT.) n Now solve the matrix equation n One way to solve this equation is to multiply both sides by the inverse of [K] or {F} = [K] {u} [K] -1 {F} = {u} n In actual practice, the inversion of the stiffness matrix to solve the system of equations is highly inefficient. MSC.Nastran uses a more efficient matrix decomposition procedure rather than the matrix inversion method.

42 S2-42 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: TWO-ROD ASSEMBLY (CONT.) n Inversion of the [K] matrix requires that [K] be square and that det[K] 0 (i.e., nonsingular) n If rigid body motion or mechanisms are not prevented (constrained), the structure is unstable and the stiffness matrix will be singular. n Always remember that MSC.Nastran is working in a 3-D space when considering rigid body motion. Therefore the set of constraints you apply must be able to prevent any possible rigid body motion in 3-D space.

43 S2-43 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation CASE STUDY: TWO-ROD ASSEMBLY (CONT.) Example of Inadequate Constraints Example of Adequate Constraints

44 S2-44 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation n The same procedure used for the two-rod model can be extended to a general structure such as the aircraft structure shown below: n The two highlighted stringer elements are represented by the two element stiffness matrices developed in the previous case study. Element 100 Element 200 PROCEDURE FOR GENERAL STRUCTURES

45 S2-45 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation n The stiffness characteristics of the rest of the aircraft are obtained by assembling the individual element stiffness matrices to the global stiffness matrix using the same procedure as used in the two-rod model. k 1 -k 1 0 -k 1 (k 1 + k 2 ) -k 2 0 -k 2 k 2 Stiffness contributions from the rest of the aircraft N x N PROCEDURE FOR GENERAL STRUCTURES (CONT.)

46 S2-46 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation n Rule of thumb for computer resources (cpu time) used by MSC.Nastran for a problem with N DOF u Overhead (~ constant) u Stiffness matrix assembly (~ N) u Solution cost ( ~ N 2 ) u Data recovery ( ~ N) PROCEDURE FOR GENERAL STRUCTURES (CONT.)

47 S2-47 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation OTHER APPLICATIONS OF FINITE ELEMENT METHOD n In general, the finite element method can be applied to any continuum described by partial differential equations. u Example: Steady-state heat conduction l Replace the structural stiffness matrix with the matrix of thermal conductivities l Single DOF at each node (temperature) u Other fields l Fluid flow/wave propagation l Electromagnetics l Dynamics

48 S2-48 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation 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, 2002 R. H. MacNeal Finite Elements: Their Design and Performance Marcel Dekker, 1994

49 S2-49 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation REFERENCES (CONT.) NAFEMS A Finite Element Primer Department of Trade and Industry, UK, 1992 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

50 S2-50 NAS120, Section 2, May 2006 Copyright 2006 MSC.Software Corporation EXERCISE Perform Workshop 2 Simply Supported Beam in your exercise workbook.

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