Introduction to Finite Elements (Matrix Methods)

1 Introduction to Finite Elements (Matrix Methods) forME 345 Modeling and SimulationSpring 2005 Professor Frank FisherNotes: These notes are borrowed extensively from a Finite Elements primer written byProfessors Belytschko and Brinson (January 1995) and revised for Engineering Analysis2 by Professors Moran and Krishnaswamy (February 1997) at Northwestern OF CONTENTST able of Contents ..iiIntroduction to Static, Linear Stress Analysis by Finite ..1 Fundamental assumptions..1 Basic equations of linear stress analysis..4 Element Stiffness matrix for 1D Spring..6 Assembly of global equations in one dimension..8 Stiffness matrix of a rod element..17 Stiffness Method for 2D Trusses..20 Assembly in 2D and 3D..25 Matlab for Finite Element Matlab for Finite Element Analysis ..28 Graphics Window ..32 Appendix: Mechanics of Deformable Bodies ..341 Introduction TO STATIC, LINEAR STRESS ANALYSISBY Finite of the most commonly used methods of stress analysis is the Finite elementmethod a matrix based method of solving problems which was developed forstructural analysis of aircraft and later recognized as a versatile tool for which a rigorousmathematical foundation could be laid.

2 It is currently one of the most-widely usedanalysis tools in all aspects of engineering and sicence. Almost any manufactured part orsystem, including components such as gears, cams, and suspensions, to completesystems, such as automobiles and aircraft, are analyzed by Finite Elements to ensure thatthey have the durability and reliability required and that they meet performancerequirements. Finite element methods are also used in biomedical application such asthe design of prostheses, modeling the nonlinear dyanamics of the human heart oranalysing gait, for example. Finite element programs are also used for many other typesof analysis and design: fluid dynamics, heat transfer, electromagnetic fields are major reason for the popularity of the Finite element method is that a singleprogram can perform the analysis of almost any component or structure: the geometryof the object and its loads are defined by mesh data and the program then sets up thegoverning equations in a straightforward manner.

3 Examples of Finite element meshesfor some industrial problems are shown in Figures 1 to 4. Finite element analyses todayare usually performed by general purpose programs which can do a large variety ofanalyses: stress analysis, vibration analyses, optimum design are a sample of some ofthe functions of general purpose programs. Furthermore, many of these programs canperform both linear and nonlinear type of stress analysis which will be taught in this course is linear, static stressanalysis. A large part of stress analysis in industry is linear, static analysis. Most stressanalysis taught at the undergraduate level is linear and static. Nonlinear analysis isusually used only for evaluating the performance under extreme assumptions which are made in linear static stress analysis are:1. The displacements of the structure are small compared to its The material is linear and The response of the structure is static (or steady-state).

4 234 The first assumption implies the geometry of the structure does not changeappreciably due to the application of the loads. Almost all stress analyses considered inundergraduate strength of materials courses are small displacement assumption of a linear, elastic material implies two characteristics:i. the relation between the stress and strain, and hence the relation between theforces and displacements, is the strains are reversible, that is, when the loads are removed, the strains anddisplacements return to their original values, which are usually zero (this iscalled elastic behavior). Some elastic materials are not linear; an example is rubber, for which the strainsare completely reversible (a rubber band will return to its original length afterstretching), but the stress is not a linear function of the metals are linear, elastic when the stresses remain below the yield-point, orelastic limit.

5 Above the yield point, the metal sustains plastic strains which are notreversible, which means they will be deformed after release of the load. As an example,consider the bending of a paper clip. For small deformations, the clip will revert to itsoriginal shape after it is deformed, but if the deformations are large enough, the clipwill be permanently deformed, which is evidence of irreversible, plastic assumption of static behavior implies that dynamic or inertial effects arenegligible because of the slow rate at which the loads are the displacements of a structure are large or the material behavior isnonlinear, nonlinear analysis must be used. Nonlinear analysis will not be studied in equations of linear stress analysis of stresses in a static body must satisfy the following:1. equilibrium2. stress-strain law3. compatibility and strain-displacement equationsEquilibrium requires that the sum of the forces and the moments vanish at allpoints of the structure.

6 This is a consequence of Newton s second law of motion whichstates that the resultant force acting on a body is equal to the rate of change of linearmomentum (recall that linear momentum is equal to mass times velocity) and theresultant moment is equal to the rate of change of angular momentum (equal tomoment of inertia times angular velocity). See Bedford and Fowler Dynamics Chapters 25and 3 and Bedford and Fowler Statics Chapters 2 and 3 for a more detailed discussion ofNewton s =ddt(mv)M =ddt(I )( )Thus, for a body which is at rest (or moving at a constant velocity) the sum of theforces and the sum of the moments of the forces acting on it must vanish. In this casethe equilibrium equations are written as:EquilibriumF=0 M=0 ( )Stress Strain LawThe stress-strain law depends on the material. For axial stretching of a rod, we saw that: = E ( )where is the stress, E is Young s modulus and is the strain.

7 This is called Hooke slaw. Note that Young s modulus has dimensions of stress (N/m2).For a spring, the force-elongation relation is analogous to the stress strain law for amaterial, ,f=k ( )where fis the force in the spring,k is the spring constant (which depends on the springgeometry and coil arrangement and is the elongation of the spring. Note that thespring constant has dimensions of force per unit length (N/m).CompatibilityCompatibility requires that the displacements be continuous everywhere in the body. Inlater courses, you will see how to represent displacements as a function of position andwill then be in a position to describe compatibility in a more general sense involvingrestrictions on strain and on the continuity of displacements. For now, it suffices for usto think of compatibility as the restriction that the displacements be continuous. Inparticular, if two bodies (or parts of a single body) are connected at a point, each part of6the body experiences the same displacement at that point.)

8 We will also require that thebodies or parts of the body not break apart. These conditions imply that there are nogaps or overlaps in the Stiffness matrix for1D element stiffness relates the element nodal internal forces fe (the superscript "int"has been omitted) to the element nodal displacements de byfe=Kede( )The subscripts e indicate that the matrices pertain to an element. When it is clearthat the matrices are related to Elements , the subscripts e are often dropped. Note thatnodal displacements are denoted by deThe element stiffness matrix is derived by requiring equilibrium, the spring law(which is the counterpart of a stress-strain law), and compatibility to be satisfied on anelement element nomenclature is defined belowfe=fIfJ de=dIdJ The spring law states that the tension t in the spring is given byt = k ( )where k is the spring constant and is the elongation of the +1fI(ext)fJ(ext)fIefJedIdJx7fJtJeFrom a free-body diagram, we can see that equilibrium of the part of element eshown gives (the superscript denoting the element number is dropped for convenience)fJ = t( )and equilibrium of the whole element givesfI = -fJ( )By the definition of elongation = dJ - dI( )Combining ( ), ( ) and ( ) givesfJ = t = k = k(dJ-dI)( )and ( ) and ( ) givefI = -fJ = -k(dJ-dI) = k(dI-dJ)( )Writing ( ) and ( )

9 In matrix formfIfJ = k +1 -1-1 +1 dIdJ Ke( )The above equation gives the element stiffness matrix Ke for the 1D spring. Note thatKe= KeT , Ke is 1. The nodal forces and nodal displacements are always defined in finiteelement methods so that they are positive in the positive coordinate direction. This iscrucial for easy assemby of the global equations, as will be seen 2. The work done by the forces is proportional to force times displacement. This work is equal tothe energy stored in the stretched body. The nodal forces and displacements are arranged so that dTf isproportional to the work done; otherwise Ke is not of global equations in one of the most attractive features of the Finite element method is its ability to treat alarge variety of geometries, loadings and boundary conditions in a single program. Thisversatility arises from the fact that a Finite element program does not incorporate aspecific geometry or loading in its coding, but generates the global equations from theelement equations by assembling the element equations according to the input data.

10 Inthis section, the assembly operation is described and illustrated for one assembly operation is designed to meet the basic requirements of a solution: i. compatibility ii. equilibriumiii. stress-strain lawThe assembly operation will now be described for a 3 element mesh for the problemshown in Table 1A. In general purpose programs, the code is written so that theprocedure is independent of the node numbering. When elaborate meshes aregenerated for complicated industrial components (Figures 1-4) the resulting nodenumbering is usually haphazard and a code which can account for this is required. Forthe one-dimensional spring and rod codes used here, the node numbering is taken to besequential, as shown in Table 1A. This makes for a simpler code, and also for an easierillustration of the method. The element numbers are enclosed by circles. The elementstiffness matrices are shown below the mesh.