27 A Complete Plane Stress FEM Program IFEM Ch 27 Slide 1 - PDF document

Introduction to FEM 27 A Complete Plane Stress FEM Program IFEM Ch 27 – Slide 1

Introduction to FEM The 3 Basic Stages of a FEM-DSM Program Preprocessing : defining the FEM model Processing : setting up the stiffness equations and solving for displacements Postprocessing : recovery of derived quantities and presentation of results IFEM Ch 27 – Slide 2

Introduction to FEM Plane Stress Program Configuration User prepares Problem script for each Driver problem Utilities: (e.g. final exam Tabular Printing, problems) Analysis Graphics, etc Driver Built in Internal BC Assembler Equation Force Recovery Application Solver Presented in Element Element previous Chapters Stresses & Int Stiffnesses Forces Element Library IFEM Ch 27 – Slide 3

Introduction to FEM Problem Definition Data Structures Geometry Data Set: NodeCoordinates Element Data Set: ElemTypes, ElemNodes, ElemMaterials, ElemFabrications Degree of Freedom Activity Data Set: NodeDOFTags, NodeDOFValues Processing Data Set: ProcessOptions IFEM Ch 27 – Slide 4

Introduction to FEM Benchmark to Illustrate Problem Definition (one-element models) (c) y (b) (a) q = 10 ksi 75 kips 75 kips 25 kips 100 kips 25 kips � � A B C B C B C 3 1 1 7 4 � 1 � 1 2 8 5 � � 4 6 H D J 2 �� �� 9 J �� �� �� 3 12 in J D D � x � � E = 10000 ksi Model (I): Model (II): ν = 0.25 4 nodes, 8 DOFs, 9 nodes, 18 DOFs, h = 3 in 1 bilinear quad 1 biquadratic quad Global node numbers shown E G F q 10 in IFEM Ch 27 – Slide 5

Introduction to FEM Benchmark Problem: Plate with Central Circular Hole used in final exam and part of today's demo (a) Node 8 is exactly midway between 1 and 15 (c) (b) y 75 kips 37.5 kips 25 kips 37.5 kips 25 kips 100 kips q = 10 ksi � � B C B 8 8 C 15 A B C 15 1 1 Note: internal point of a 1 9-node quadrilateral is placed � � at intersection of the medians 7 2 2 9 9 � 2 � 1 16 3 3 J 10 16 8 10 H D K � � 12 in 22 4 17 17 x 11 4 11 R = 1 in 2 22 23 � � � 5 18 � � � 5 18 12 23 12 19 6 19 E = 10000 ksi 24 6 13 24 13 ν = 0.25 25 25 20 7 7 20 � � � � 26 26 h = 3 in J J 14 27 14 27 21 21 28 28 34 33 32 31 30 35 34 33 32 31 30 29 35 29 D K F E K G D � � � � � � � � � � � � � � � � � � � � � � � q 10 in Model (I): 35 nodes, 70 DOFs, Model (II): 35 nodes, 70 DOFs, 24 bilinear quads 6 biquadratic quads IFEM Ch 27 – Slide 6

Introduction to FEM Geometry Data: 4-Node Quad Model y (a) 75 kips 75 kips q = 10 ksi � B C A B C 3 1 1 � � � J 2 4 D H J D 12 in � x � � E = 10000 ksi ν = 0.25 h = 3 in F E G q 10 in IFEM Ch 27 – Slide 7

Introduction to FEM Geometry Data: 9-Node Quad Model y 25 kips 100 kips 25 kips q = 10 ksi � B C A B C 7 1 4 � 1 � 2 8 5 � � � 6 �� �� 9 3 J D H D J 12 in � x � � E = 10000 ksi ν = 0.25 h = 3 in E F G q 10 in IFEM Ch 27 – Slide 8

Introduction to FEM Element Data: 4-Node Quad Model 75 kips 75 kips � B C � 3 1 1 � � 2 4 J � D � � � IFEM Ch 27 – Slide 9

Introduction to FEM Element Data: 9-Node Quad Model 25 kips 100 kips 25 kips � B C 1 7 4 � 1 � 2 8 5 � 6 � 9 � 3 J D � � � � IFEM Ch 27 – Slide 10

Introduction to FEM Freedom Activity Data: 4-Node Quad Model 75 kips 75 kips � B C � 3 1 1 � � 2 4 J � D � � � IFEM Ch 27 – Slide 11

Introduction to FEM Freedom Activity Data: 9-Node Quad Model 25 kips 100 kips 25 kips � B C 1 7 4 � 1 � 2 8 5 � 6 � 9 � 3 J D � � � � IFEM Ch 27 – Slide 12

Introduction to FEM A Complete Problem Script Cell Part 1: Preprocessing 75 kips 75 kips ClearAll[Em, ν ,th]; � Em=10000; ν =.25; th=3; aspect=6/5; Nsub=4; B C Emat=Em/(1- ν ^2)*{{1, ν ,0},{ ν ,1,0},{0,0,(1- ν )/2}}; � 3 1 (* Define FEM model *) 1 NodeCoordinates=N[{{0,6},{0,0},{5,6},{5,0}}]; � PrintPlaneStressNodeCoordinates[NodeCoordinates,"",{6,4}]; ElemNodes= {{1,2,4,3}}; � 2 4 � J D 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]; IFEM Ch 27 – Slide 13

Introduction to FEM Mesh Plot Showing Element & Node Numbers Produced by previous script One element mesh - 9 node quad One element mesh - 4 node quad One element mesh - 4 node quad 1 4 7 1 3 1 8 15 1 7 2 9 16 2 8 3 13 2 5 8 10 17 1 3 1 9 4 14 11 18 4 22 10 5 15 12 19 23 5 6 11 13 16 24 6 20 7 14 12 17 25 19 21 3 6 9 2 4 26 18 20 27 21 28 22 23 24 35 34 33 32 31 30 29 IFEM Ch 27 – Slide 14

Introduction to FEM A Complete Problem Script Cell Part 2: Processing 75 kips 75 kips � B C 3 1 � 1 � 4 J 2 � � D (* Solve problem and print results *) � � � {NodeDisplacements,NodeForces,NodePlateCounts,NodePlateStresses, ElemBarNumbers,ElemBarForces}= PlaneStressSolution[ NodeCoordinates,ElemTypes,ElemNodes, ElemMaterials,ElemFabrications, NodeDOFTags,NodeDOFValues,ProcessOptions]; IFEM Ch 27 – Slide 15

Introduction to FEM A Complete Problem Script Cell Part 3: PostProcessing PrintPlaneStressSolution[NodeDisplacements,NodeForces,NodePlateCounts, NodePlateStresses,"Computed Solution:",{}]; (* Plot Displacement Components Distribution - skipped *) 75 kips 75 kips �� B C (* Plot Averaged Nodal Stresses Distribution *) 3 1 sxx=Table[NodePlateStresses[[n,1]],{n,numnod}]; syy=Table[NodePlateStresses[[n,2]],{n,numnod}]; 1 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"]; 4 J 2 � D 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 16

Introduction to FEM Solution Printout (Required in Exam Problems) 75 kips 75 kips � B C 3 1 � 1 � � 4 J 2 � D Computed Solution: � � node x-displ y-displ x-force y-force sigma-xx sigma-yy sigma-xy 1 0.0000 0.0000 0.0000 0.0000 0.0060 75.0000 10.0000 0.0000 0.0000 2 0.0000 0.0000 −75.0000 0.0000 10.0000 3 −0.0013 0.0000 0.0060 0.0000 75.0000 0.0000 10.0000 4 −0.0013 0.0000 −75.0000 0.0000 0.0000 0.0000 10.0000 IFEM Ch 27 – Slide 17

Introduction to FEM Stress Contour Plots (Not Required in Exam Problems) Nodal stress sig � xx Nodal stress sig � yy Nodal stress sig � xy IFEM Ch 27 – Slide 18

Introduction to FEM Stress Contour Plots (cont'd) sigma-yy stress contour plot reconstructed over complete plate IFEM Ch 27 – Slide 19