Hutton - Fundamentals of Finite Element Analysis, страница 87

Inthe frontal solution technique, the entire system stiffness matrix is not assembledas such. Instead, the method utilizes the fact that a degree of freedom (an unknown) can be eliminated when the rows and columns of the stiffness matrix corresponding to that degree of freedom are complete. In this context, eliminating adegree of freedom means that we can write an equation for that degree of freedom in terms of other degrees of freedom and forcing functions. When such anequation is obtained, it is written to a file and removed from memory.

As isshown, the net result is triangularization of the system stiffness matrix and thesolutions are obtained by simple back substitution.U6, F6U5, F556U4, F445U3, F334U2, F223U1, F1121Figure C.1 A system of bar elements used to illustrate the frontalsolution method.xHutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix C: SolutionTechniques for LinearAlgebraic Equations© The McGraw−HillCompanies, 2004C.4 Frontal SolutionFor simplicity of illustration, let each element in Figure C.1 have characteristic stiffness k.

We begin by defining a 6 × 6 null matrix [K] and proceed withthe assembly step, taking the elements in numerical order. Adding the elementstiffness matrix for element 1 to the system matrix, we obtain   F1 k −k 0 0 0 0  U1   U2  −k k 0 0 0 0  F2  00 0 0 0 0 U3 = F3(C.34) 00 0 0 0 0 U4   F4  00 0 0 0 0U  F5   5  00 0 0 0 0U6F6Since U1 is associated only with element 1, displacement U1 appears in none ofthe other equations and can be eliminated now.

(To illustrate the effect on thematrix, we do not actually eliminate the degree of freedom from the equations.)The first row of Equation C.34 iskU 1 − kU 2 = F1(C.35)and can be solved for U1 once U2 is known. Mathematically eliminating U1 fromthe second row, we have  F1 U1 k −k 0 0 0 0  U2 0 0 0 0 0 0F1 + F2  0 0 0 0 0 0   U3   F3 (C.36) 0 0 0 0 0 0   U4  =  F4  0 0 0 0 0 0   U5   F5   0 0 0 0 0 0U6F6Next, we “process” element 2 and add the element stiffness matrix terms to theappropriate locations in the coefficient matrix to obtain  F1 k −k 0 0 0 0 U1  U2  0 k −k 0 0 0    F1 + F2  0 −k k 0 0 0   U3   F3 =(C.37)0 00 0 0 0U4 F4 0 00 0 0 0U  F5   5 0 00 0 0 0U6F6Displacement U 2 does not appear in any remaining equations and is now eliminated to obtain  U1 F1k −k 0 0 0 0  0 k −k 0 0 0   U2 F1 + F2 0 00 0 0 0 U3 = F1 + F2 + F3(C.38)0 00 0 0 0U4 F40 00 0 0 0U F5  5 0 00 0 0 0U6F6471Hutton: Fundamentals ofFinite Element Analysis472Back MatterAPPENDIX CAppendix C: SolutionTechniques for LinearAlgebraic Equations© The McGraw−HillCompanies, 2004Solution Techniques for Linear Algebraic EquationsIn sequence, processingprocedure results ink −k 00 0 k −k 00 0k −k0 00k0 0000 000the remaining elements and following the elimination000−kk−k  0 U1 F10 UF+F2120 UF+F+F3123=U  F1 + F2 + F3 + F4 0  4−k  UF51 + F2 + F3 + F4 + F5   U6F6k(C.39)Noting that the last equation in the system of Equation C.39 is a constraint equation (and could have been ignored at the beginning), we observe that the procedure has triangularized the system stiffness matrix without formally assemblingthat matrix.

If we take out the constraint equation, the remaining equations areeasily solved by back substitution. Also note that the forces are assumed to beknown.The frontal solution method has been described in terms of one-dimensionalelements for simplicity. In fact, the speed and efficiency of the procedure are ofmost advantage in large two- and three-dimensional models. The method is discussed briefly here so that the reader using a finite element software package thatuses a wave-type solution has some information about the procedure.Hutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix D: The FiniteElement PersonalComputer ProgramA P P E N D I X© The McGraw−HillCompanies, 2004DThe Finite ElementPersonal ComputerProgramWith permission of the estate of Dr.

Charles E. Knight, the Finite ElementPersonal Computer (FEPC) program is available to users of this textvia the website www.mhhe.com/hutton<www.mhhe.com/hutton>.FEPC is a finite element software package supporting bar, beam, plane solid, andaxisymmetric solid elements and hence is limited to two-dimensional structural applications. Dr. Knight’s A Finite Element Method Primer for MechanicalDesign is available via the website and includes basic concepts as well as anappendix delineating FEPC capabilities and limitations.

The following materialpresents a general description of the programs’ capabilities and limitations. Acomplete users guide is available on the website.FEPC is actually a set of three programs that perform the operations of preprocessing (model development), model solution, and postprocessing (resultsanalysis).

FEPCIP is the input processor used to input and check a model and prepare data files for the solution program FEPC. The output processor is FEPCOPand this program reads solution output files and produces graphic displays.All the programs are menu driven, with automatic branching to submenuswhen appropriate. The Files menu of the input processor is used to recall a previously stored model or store a new model. Models are stored as filename.MODwhere filename is user specified and can contain a maximum of 20 characters.The analysis file, which becomes the actual input to the FEPC solution phase, isfilename.ANA.D.1 PREPROCESSINGModel definition is activated by the Model Data menu.

Selection of Model Dataleads to a submenu used to define element type, material properties, nodes,473Hutton: Fundamentals ofFinite Element Analysis474Back MatterAPPENDIX DAppendix D: The FiniteElement PersonalComputer Program© The McGraw−HillCompanies, 2004The Finite Element Personal Computer Programelements, restraints (displacement constraints), and loads. Element type is limited to bar (truss), beam, plane stress, plane strain, or axisymmetric.

Only asingle element type can be used in a model. Up to 10 material property sets canbe used in a model and should be defined in numerical order. Nodes can bedefined by direct input of node number and the X, Y coordinates of the node.Nodes for truss and beam elements are always defined in this manner. Nodes forplane and axisymmetric elements can also be defined by direct input but anautomeshing capability for two-dimensional (2-D) areas is included and discussed subsequently. Similarly, elements are defined by specification of thenodes and material property number.

For truss and beam elements, the order ofnode specification is of no consequence. However, for plane or axisymmetricelements, the nodes must be specified in a counterclockwise order around theelement area. Displacement constraints are applied by setting the values andselecting the node to which the values apply. Loads are applied as nodal forces orelement edge pressures for 2-D solid elements.The 2-D Automesh Generation section of the program is used for area meshgeneration of two-dimensional plane stress, plane strain, or axisymmetric models.The approach used is a coordinate transformation mapping of a grid of squareelements in an integer area into a grid of elements in the geometric area. The geometric area is defined using points that are subsequently used to define lines andarcs that enclose the area.

The integer grid is bounded by lines that correspond tothe lines and arcs of the geometric area. Grid points bound the square elements inthe integer area map to element nodes in the geometric area. The mapping processis iterative and consists of distorting the integer area grid to fit into the geometricarea. Thus the user has some control over element size and shape via the size ofthe integer grid.D.2 SOLUTIONWhen a model has been defined and saved as filename.ANA, the solution is generated by the FEPC.EXE program. During execution, several other files are generated and stored. Also, screen messages are issued to report on progress of thesolution phase. A listing of all printed output is stored in filename.LST.

This filecontains the input data, all numerical output, and any error messages issued during execution. Two additional files are created for use by the output processor.The node and element data is placed in filename.MSH and nodal displacementand element stress data is stored in filename.NVL.D.3 POSTPROCESSINGThe output processor FEPCOP.EXE is used to display the solution results ingraphic form. (The printed form of the numerical output data is in the .LST file.)Displacement results can be displayed as a plot of the deformed element meshHutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix D: The FiniteElement PersonalComputer Program© The McGraw−HillCompanies, 2004D.3 Postprocessingsuperimposed over a plot of the undeformed model. Displacements are scaledsuch that the deformed shape is exaggerated for clarity.For two-dimensional solid models, stress components can be displayed ascontour plots.

The stress components available from the solution are the normal,shear, and von Mises stress components for plane stress and plane strain, and theradial, axial, shear, and hoop stress components for axisymmetric models. Formodels using truss or beam elements, stress components are plotted as bar charts.475Hutton: Fundamentals ofFinite Element AnalysisEBack MatterAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004A P P E N D I XProblems for ComputerSolutionThe following problems are intended for solution using finite elementanalysis software. In general, the problems associated with Chapters 3, 4,and 9 can be solved using the FEPC software (Appendix D) if anothersoftware package is not available.