Introduction to FEM27A CompletePlane StressFEM ProgramIFEM Ch 27 – Slide 1Introduction to FEMThe 3 Basic Stages of a FEM-DSM ProgramPreprocessing : defining the FEM modelProcessing : setting up the stiffness equations and solving for displacementsPostprocessing : recovery of derived quantities and presentation of resultsIFEM Ch 27 – Slide 2Introduction to FEMPlane Stress Program Configuration Analysis DriverAssemblerElement StiffnessesInternal Force RecoveryElement Stresses & IntForcesBuilt in Equation SolverUtilities:Tabular Printing,Graphics, etcBCApplication Presented inprevious Chapters Problem DriverUser preparesscript for eachproblemElement LibraryIFEM Ch 27 – Slide 3Problem Definition Data Structures Introduction to FEMGeometry Data Set: NodeCoordinatesElement Data Set: ElemTypes, ElemNodes, ElemMaterials, ElemFabricationsDegree of Freedom Activity Data Set: NodeDOFTags, NodeDOFValuesProcessing Data Set: ProcessOptionsIFEM Ch 27 – Slide 4Benchmark to Illustrate Problem Definition(one-element models)Introduction to FEM10 inyxq = 10 ksiqBBBCCCDDDEFGAHJJJ75 kips 25 kips100 kips 25 kips75 kips1137492682354Global node numbers shown(a)(b)(c)12 inE = 10000 ksiν = 0.25h = 3 in;;;;;;;;;;;;;;;;;;;Model (I): 4 nodes, 8 DOFs,1 bilinear quadModel (II): 9 nodes, 18 DOFs,1 biquadratic quad11IFEM Ch 27 – Slide 5Benchmark Problem: Plate with Central Circular Holeused in final exam and part of today's demoIntroduction to FEM13456729168151110121314171819202122232425262728293031323433351345672916815111012131417181920212223242526272829303132343335Model (I): 35 nodes, 70 DOFs, 24 bilinear quads Model (II): 35 nodes, 70 DOFs, 6 biquadratic quadsBCDJKBCDJK37.5 kips37.5 kips25 kips100 kips25 kips75 kipsNode 8 is exactly midway between 1 and 15;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;11227810 inyxq = 10 ksiqBCDEFGAH(a)(b)(c)12 inE = 10000 ksiν = 0.25h = 3 inKJR = 1 inNote: internal point of a9-node quadrilateral is placed at intersection of the mediansIFEM Ch 27 – Slide 6Geometry Data: 4-Node Quad ModelIntroduction to FEM10 inyxq = 10 ksiqBBCCDDEFGAHJJ75 kips75 kips1342(a)12 inE = 10000 ksiν = 0.25h = 3 in;;;;;;;1IFEM Ch 27 – Slide 7Geometry Data: 9-Node Quad ModelIntroduction to FEM10 inyxq = 10 ksiqBCDEFGAHJ12 inE = 10000 ksiν = 0.25h = 3 inBCDJ 25 kips100 kips 25 kips179268354;;;;;;;;;;;;;1IFEM Ch 27 – Slide 8Element Data: 4-Node Quad ModelIntroduction to FEMBCDJ75 kips75 kips1342;;;;;;;;1IFEM Ch 27 – Slide 9Element Data: 9-Node Quad ModelIntroduction to FEMBCDJ 25 kips100 kips 25 kips179268354;;;;;;;;;;1IFEM Ch 27 – Slide 10Freedom Activity Data: 4-Node Quad ModelIntroduction to FEMBCDJ75 kips75 kips1342;;;;;;;;1IFEM Ch 27 – Slide 11Freedom Activity Data: 9-Node Quad ModelIntroduction to FEMBCDJ 25 kips100 kips 25 kips179268354;;;;;;;;;;1IFEM Ch 27 – Slide 12A Complete Problem Script CellPart 1: Preprocessing Introduction to FEM ClearAll[Em,ν,th]; Em=10000; ν=.25; th=3; aspect=6/5; Nsub=4;Emat=Em/(1-ν^2)*{{1,ν,0},{ν,1,0},{0,0,(1-ν)/2}};(* Define FEM model *)NodeCoordinates=N[{{0,6},{0,0},{5,6},{5,0}}];PrintPlaneStressNodeCoordinates[NodeCoordinates,"",{6,4}];ElemNodes= {{1,2,4,3}};numnod=Length[NodeCoordinates]; numele=Length[ElemNodes]; ElemTypes= Table["Quad4",{numele}]; PrintPlaneStressElementTypeNodes[ElemTypes,ElemNodes,"",{}];ElemMaterials= Table[Emat, {numele}]; ElemFabrications=Table[th, {numele}];PrintPlaneStressElementMatFab[ElemMaterials,ElemFabrications,"",{}];NodeDOFValues=NodeDOFTags=Table[{0,0},{numnod}];NodeDOFValues[[1]]=NodeDOFValues[[3]]={0,75}; (* nodal loads *)NodeDOFTags[[1]]={1,0}; (* vroller @ node 1 *)NodeDOFTags[[2]]={1,1}; (* fixed node 2 *)NodeDOFTags[[4]]={0,1}; (* hroller @ node 4 *)PrintPlaneStressFreedomActivity[NodeDOFTags,NodeDOFValues,"",{}];ProcessOptions={True};Plot2DElementsAndNodes[NodeCoordinates,ElemNodes,aspect, "One element mesh - 4-node quad",True,True];BCDJ75 kips75 kips1342;;;;;;;;1IFEM Ch 27 – Slide 13Introduction to FEM11234One element mesh - 4 node quad1123456789One element mesh - 9 node quad1234567891011121314151617181920212223241234567891011121314151617181920212223242526272829303132333435One element mesh - 4 node quadMesh Plot Showing Element & Node NumbersProduced byprevious scriptIFEM Ch 27 – Slide 14Introduction to FEMA Complete Problem Script CellPart 2: Processing (* Solve problem and print results *){NodeDisplacements,NodeForces,NodePlateCounts,NodePlateStresses, ElemBarNumbers,ElemBarForces}= PlaneStressSolution[ NodeCoordinates,ElemTypes,ElemNodes, ElemMaterials,ElemFabrications, NodeDOFTags,NodeDOFValues,ProcessOptions];BCDJ75 kips75 kips1342;;;;;;;;1IFEM Ch 27 – Slide 15A Complete Problem Script Cell Part 3: PostProcessing Introduction to FEMBCDJ75 kips75 kips1342;;;;;;;1PrintPlaneStressSolution[NodeDisplacements,NodeForces,NodePlateCounts, NodePlateStresses,"Computed Solution:",{}]; (* Plot Displacement Components Distribution - skipped *)(* Plot Averaged Nodal Stresses Distribution *)sxx=Table[NodePlateStresses[[n,1]],{n,numnod}];syy=Table[NodePlateStresses[[n,2]],{n,numnod}];sxy=Table[NodePlateStresses[[n,3]],{n,numnod}];{sxxmax,syymax,sxymax}=Abs[{Max[sxx],Max[syy],Max[sxy]}];ContourPlotNodeFuncOver2DMesh[NodeCoordinates,ElemNodes, sxx,sxxmax,Nsub,aspect,"Nodal stress sig-xx"];ContourPlotNodeFuncOver2DMesh[NodeCoordinates,ElemNodes, syy,syymax,Nsub,aspect,"Nodal stress sig-yy"];ContourPlotNodeFuncOver2DMesh[NodeCoordinates,ElemNodes, sxy,sxymax,Nsub,aspect,"Nodal stress sig-xy"];IFEM Ch 27 – Slide 16Solution Printout (Required in Exam Problems) Introduction to FEMBCDJ75 kips75 kips1342;;;;;;;1Computed Solution:node x-displ y-displ x-force y-force sigma-xx sigma-yy sigma-xy10.00000.00600.000075.00000.000010.00000.000020.00000.00000.0000−75.00000.000010.00000.00003−0.00130.00600.000075.00000.000010.00000.00004−0.00130.00000.0000−75.00000.000010.00000.0000IFEM Ch 27 – Slide 17Stress Contour Plots Introduction to FEMNodal stress sigxxNodal stress sigyyNodal stress sigxyIFEM Ch 27 – Slide 18Stress Contour Plots (cont'd)Introduction to FEMsigma-yy stress contour plot reconstructedover complete plateIFEM Ch 27 – Slide