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

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

Sep 18, 2022 374 likes •582 views

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

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SLIDE 1

Introduction to FEM

27

A Complete Plane Stress FEM Program

IFEM Ch 27 – Slide 1

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SLIDE 2

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

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SLIDE 3

Introduction to FEM

Plane Stress Program Configuration

Analysis Driver Assembler Element Stiffnesses Internal Force Recovery Element Stresses & Int Forces Built in Equation Solver Utilities: Tabular Printing, Graphics, etc BC Application Presented in previous Chapters Problem Driver User prepares script for each problem (e.g. final exam problems)

Element Library

IFEM Ch 27 – Slide 3

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SLIDE 4

Problem Definition Data Structures

Introduction to FEM

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

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SLIDE 5

Benchmark to Illustrate Problem Definition (one-element models)

Introduction to FEM

10 in

y x

q = 10 ksi q

B B B C C C D D D E F G A H J J J

75 kips 25 kips 100 kips 25 kips 75 kips

1 1 3 7 4 9 2 6 8 2 3 5 4

Global node numbers shown (a) (b) (c) 12 in

E = 10000 ksi ν = 0.25 h = 3 in

  • Model (I):

4 nodes, 8 DOFs, 1 bilinear quad Model (II): 9 nodes, 18 DOFs, 1 biquadratic quad 1 1

IFEM Ch 27 – Slide 5

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SLIDE 6

Benchmark Problem: Plate with Central Circular Hole used in final exam and part of today's demo

Introduction to FEM

1 3 4 5 6 7 2 9 16 8 15 11 10 12 13 14 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 34 33 35 1 3 4 5 6 7 2 9 16 8 15 11 10 12 13 14 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 34 33 35

Model (I): 35 nodes, 70 DOFs, 24 bilinear quads Model (II): 35 nodes, 70 DOFs, 6 biquadratic quads

B C D J K B C D J K

37.5 kips 37.5 kips 25 kips 100 kips 25 kips 75 kips

Node 8 is exactly midway between 1 and 15

  • 1

1 2 2 7 8

10 in

y x

q = 10 ksi q

B C D E F G A H

(a) (b) (c)

12 in

E = 10000 ksi ν = 0.25 h = 3 in

K

J R = 1 in

Note: internal point of a 9-node quadrilateral is placed at intersection of the medians

IFEM Ch 27 – Slide 6

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SLIDE 7

Geometry Data: 4-Node Quad Model

Introduction to FEM

10 in

y x

q = 10 ksi q

B B C C D D E F G A H J J

75 kips 75 kips

1 3 4 2

(a) 12 in

E = 10000 ksi ν = 0.25 h = 3 in

  • 1

IFEM Ch 27 – Slide 7

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SLIDE 8

Geometry Data: 9-Node Quad Model

Introduction to FEM

10 in

y x

q = 10 ksi q

B C D E F G A H J

12 in

E = 10000 ksi ν = 0.25 h = 3 in B C D J

25 kips 100 kips 25 kips

1 7 9 2 6 8 3 5 4

  • 1

IFEM Ch 27 – Slide 8

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SLIDE 9

Element Data: 4-Node Quad Model

Introduction to FEM

B C D J

75 kips 75 kips

1 3 4 2

  • 1

IFEM Ch 27 – Slide 9

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SLIDE 10

Element Data: 9-Node Quad Model

Introduction to FEM

B C D J

25 kips 100 kips 25 kips

1 7 9 2 6 8 3 5 4

  • 1

IFEM Ch 27 – Slide 10

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SLIDE 11

Freedom Activity Data: 4-Node Quad Model

Introduction to FEM

B C D J

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1 3 4 2

  • 1

IFEM Ch 27 – Slide 11

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SLIDE 12

Freedom Activity Data: 9-Node Quad Model

Introduction to FEM

B C D J

25 kips 100 kips 25 kips

1 7 9 2 6 8 3 5 4

  • 1

IFEM Ch 27 – Slide 12

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SLIDE 13

A Complete Problem Script Cell Part 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]; B C D J

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1 3 4 2

  • 1

IFEM Ch 27 – Slide 13

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SLIDE 14

Introduction to FEM

1 1 2 3 4

One element mesh - 4 node quad

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One element mesh - 9 node quad

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

One element mesh - 4 node quad

Mesh Plot Showing Element & Node Numbers

Produced by previous script

IFEM Ch 27 – Slide 14

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SLIDE 15

Introduction to FEM

A Complete Problem Script Cell Part 2: Processing

(* Solve problem and print results *) {NodeDisplacements,NodeForces,NodePlateCounts,NodePlateStresses, ElemBarNumbers,ElemBarForces}= PlaneStressSolution[ NodeCoordinates,ElemTypes,ElemNodes, ElemMaterials,ElemFabrications, NodeDOFTags,NodeDOFValues,ProcessOptions]; B C D J

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1 3 4 2

  • 1

IFEM Ch 27 – Slide 15

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SLIDE 16

A Complete Problem Script Cell Part 3: PostProcessing

Introduction to FEM B C D J

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1 3 4 2

  • 1

PrintPlaneStressSolution[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 16

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SLIDE 17

Solution Printout (Required in Exam Problems)

Introduction to FEM B C D J

75 kips 75 kips

1 3 4 2

  • 1

Computed Solution: node x-displ y-displ x-force y-force sigma-xx sigma-yy sigma-xy 1 0.0000 0.0060 0.0000 75.0000 0.0000 10.0000 0.0000 2 0.0000 0.0000 0.0000 −75.0000 0.0000 10.0000 0.0000 3 −0.0013 0.0060 0.0000 75.0000 0.0000 10.0000 0.0000 4 −0.0013 0.0000 0.0000 −75.0000 0.0000 10.0000 0.0000

IFEM Ch 27 – Slide 17

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SLIDE 18

Stress Contour Plots (Not Required in Exam Problems)

Introduction to FEM

Nodal stress sigxx Nodal stress sigyy Nodal stress sigxy

IFEM Ch 27 – Slide 18

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SLIDE 19

Stress Contour Plots (cont'd)

Introduction to FEM

sigma-yy stress contour plot reconstructed

  • ver complete plate

IFEM Ch 27 – Slide 19