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by: Jackeline Wuckert


Jackeline Wuckert
U of L
GPA 3.51

Roger Bradshaw

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Roger Bradshaw
Class Notes
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This 14 page Class Notes was uploaded by Jackeline Wuckert on Friday October 23, 2015. The Class Notes belongs to ME 512 at University of Louisville taught by Roger Bradshaw in Fall. Since its upload, it has received 45 views. For similar materials see /class/228366/me-512-university-of-louisville in Mechanical Engineering at University of Louisville.

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Date Created: 10/23/15
Gaussian Quadrature gt Isoparametric element formulation 0 Integral equations for stiffness matrix K 0 Integral equations for nodal forces r KllBT EBJdV 0 Once obtained standard fundamental FEA equation K d r gt Integrals are dif cultimpossible to evaluate in closed form 0 Use numerical integration to determine approximate values gt Gaussian quadrature 7 typical FEA method 0 Consider the integral I of a I function fx over X E 71 l I f id 4 0 Gaussian Auadrature a A roximates this using 1 oint evaluations 7 Poinw are called Gauss points or Gauss quadrature points I u Ifrmax E M W aquot X i I 0 x1 7 location of Gauss points 0 Wi 7 Gauss point weight parameters 411 Gaussian Quadrature gt Why does Gaussian quadrature GQ work 0 Suppose you use n poinm for GQ Integral is exact for fx that is a I f Co 5 I polynomial of orderN 2n71 or less GQ Poms 79 MW l n x w l o 8 i39 2 5 5M4 in qx ooze Cx gr 4395le 139 C x5 Nsl MA dag cx 4 owl chquot 439 cwd Jr chs r us gt Finite element analysis 0 Typically use n l or n 2 Gauss Quadrature 1 Point gt Consider the integral for a line 0 Exact for fx given by a line N 2n 71 l 0 Equation of line is fx C0 C1 X Shae Cl 584d gt Estimate integral by 1 point Gauss quadrature n l 0 GQ point x1 0 gt evaluate f0 C0 0 GQ weight W1 2 gt integral estimate is 2C0 2 X C0 rectangle 0 Area error for x gt 0 balances area error for x lt 0 0 As a result the Gauss quadrature value is exactly correct I fisc 1 200 l Gauss Quadrature Plane Problems gt GQ on previous slides good for 1D problems I A I a frank a g m Ha gt Plane problems involve 2 integrals 0 Simply repeat procedure in two dimensions I I U A ffawwmg 2 anny quot LJ 0 1D 1 point GQ gt 2D 1 pointl X l 0 1D 2 point GQ gt 2D 4 poinm 2 X 2 0 1D 3 point GQ gt 2D 9 poinm 3 X 3 gt Example 2 X 2 GQ 7 xy pairs i r xiVJ 5m 3972 A 5quot Mia l mamm Examples Using Gauss Quadrature Consider several numerical examples Demonstratlon of Gauss Quadrature GO Roger Bradshaw Fall 2007 Exam ev1D Case 1 Suppose we want a know 1 it v d for x r 1 Exam answer is 1 39 2 1 1 pclnl GO W1 1 pom we1gh1 x1 1 pmnt localion L111 WllFl l or Llpl 7 11101 111111 l 171p I 1511711111 mo Enilpr 4m 2 point GO 1w z perm weights x2 2 2 polnl locations 1 1 11 2I 1ix21 l 111 of 1911 11 fl 1 1137 1 1 LL 1 pr 1 gm1122 1113px I 1110 111172111 711 111 FEA Application of Gauss Quadrature V How is Gauss quadrature used in FEA Recall isoparametric element stiffness matrix K grgf Lg We Evaluate this and similar quantities by Gauss quadrature IJ M if g W1W58TE 8 J39 1 l 1 39 5r 7 tJ ANSYS evaluates onlJ a few 1 oints to vet Kel gt assemble for K What order of Gauss quadrature is best to use Finite element models are usually stiffer than reality Higher order GQ gt prevents certain deformation gt stiffer model Lower order GQ gt prevents certain deformation gt instabilities Typical FEA 7 use lowest order GQ that is acceptable ANSYS uses 2 x 2 GQ for both Q4 and Q8 elements Stiffness Matrix Via Gauss Quadrature gt Flowchart to evaluate K61 using 2 X 2 Gauss quadrature m ANSYS Use of Gauss Quadrature gt ANSYS uses Gauss quadrature to nd K matrix Handout from AN SYS documentation m yuulmlLu ml Him I I 1 l mu 39m 1 n mtwmhnu uhnucxuhw um I Vruh n rhruir quotmin 1 ittwiudl i t mu inmn in mi innit m nh mum JIlill 39ili2iul39J 39 mu M 5L ml l4 nurturqu m vn Innuulm ind m lhu m Ill Murmur Mun ur n Hu lurm39 z tzx1umnlt 3 I I L 11 hm nu Iumlmnmimuuuyt I mm x int m muummium luucwur nuc cmrldumlnl Instability Using Gauss Quadrature gt Recall that GQ only evaluates values at certain key points 0 Instability 7 deformation modes with no deformation at key points 0 Energy in the structure represented by no energy in FE model gt Instability using l X l Gauss quadrature 0 Each deformation below leads to no stressstrain at Gauss points gt Instability using 2 X 2 Gauss quadrature Hourglass instability 0 Can only occur in single element model 0 Do not worry about this in practice Stress Calculations in FEA gt Solution of FE model leads to nodal displacements 0 Evaluate stiffness matrix of model K 0 Evaluate nodal loads r and apply BCs Solve K d r to 11nd unknown nodal drsplacemenm d gt Stressstrain calculations 0 Straindisplacement matrix B leads to strains g 172d 4 N 0 Constitutive matrix E leads to stresses 039 g 39 Slveun s Hurl Jo vuzF N A A cauu o 3M5 wmlec 0 These field quantities can be evaluated anywhere in the model gt ANSYS stress evaluation 0 Perform evaluation at limited number of points 0 Use those values to predict stresses everywhere Stress Calculation in ANSYS gt Recall that certain matrices are already known by FEA Evaluations performed at the Gauss quadrature points Straindisplacement matrix B known Constitutive matrix E Known can vary 1n element 1n certaln cases 15quot Mir 9 Mi 5 7 t 39 gt AN SYS stress calculation also applies to strain J Evaluate at the Gauss points most accurate results at those locations Extrapolate obtained values over entire element 7 Fit using N constants if N Gauss quadrature points Example 7 2D element with 2X2 Gauss quadrature 7 Fit predicts stress everywhere in element D e t W Z Z 21gt ecmf My 2x2 639 SA Ma acoeds 39 572w 4 5 evzls 57 L739 ffgr gg 39 00115 4 21 Stress Calculations in ANSYS gt Element solution stresses previous slide Stresses calculated at Gauss points Extrapolate to predict stresses everywhere element call this 039 Element stress results are discontinuous from element to element ANSYS Contour Plots gt Element Solution gt Nodal solution stresses Use element solution stress to predict stress at nodes for each element Calculate the average stress at each node 7 Combine nodal stresses for all elements sharing a given node Evaluate stress eld similar to displacement 7 Use shape functions nodal stress value to predict everywhere Call this quantity the nodal stress eld element call this 6 Nodal stress results are continuous from element to element ANSYS Contour Plots gt Nodal Solution gt Element stress results most accurate use for NE 512 gt Nodal stress results look best smooth continuous ANSYS Stress Demonstration gt Square 2 X 2 X 010 thick aluminum plate Comer load 1000 lbs in 7X direction Calculate stresses 2 ways element solution nodal solution Model with 4 elements Q4 PLANE42 Q8 PLANESZ 1w moans Q4 Element Solution ox gt Q4 PLANE42 7 element solution Clearly discontinuous Minimum Value 26429 psi 7 Q4 Nodal Solution O39X gt Q4 PL Con ANE42 inodal solution inuous 7 100 s am m 395 ma QB Element Solution ox gt Q8 PL ANE82 7 element solution Clearly discontinuous Mini mum Value 42083 psi 4m 53gt 4 quot A 1 quotm QB Nodal Solution oX gt Q8 PLANESZ 7 nodal solution I Continuous 7 looks better I Minimum Value 42083 psi 7 Same location no averaging 7 same value ANSYS Documentation Gauss Point Stresses gt ANSYS rst evaluates stresses at the Gauss points I Same de nition as previously presented I These are then modi ed for output to the user 23L Integration Paint Strains and Stresses Ll and Fanm u 44 m geL EE klEllulelE h 359 U 01ng 27607 trams that cause stresses output as EPEL 7 r p ant mam e unrated at Integration pmnt u nodal dtsplncemeut actor am thermal strum enter 5 e cluihcxh mntnx ANSYS Documentation Element Stresses gt ANSYS extrapolates Gauss point stresses over element I These are referred to as Element Solution stresses I Element stress listing 7 evaluate at the element centroid 136r Nodal and Centroidal Data Evaluation G M V A t mrl u Twhl t t r r u Table 116 Assumed Data Variariun ofStresses a b r cLe r e hme icients s r r element natural coordinates 4729 gt ANSYS averages stress at common nodes I These are referred to as the Nodal Solution I Same de nition as previously described 1911 Derived Nodal Data Computation 5mm mLh Geomem 39mmneanuesquot through Chagrer s Acoustics Demedxmdrd dorms malloblefor send and shell Elements exceprslEEL 39 1an H39F39T so W39mw Mamd mm m rhmvwl u nr e w 9mm lm ld lm vvdnltmlt Human w ee w m we a u presiure accumulatedeq39un alentphsnr strain plastic state unable mdplnshc Mark POSTI 21 Uuk 1971 Um N k here 511 merage demed data eempenenu moods k em ed dam componinn ofelemenr J at node k number of elements eemeemg to node k 430 Error Estimation V Consider that you have solved a model 0 Certain regions may require re nement additional elements 0 First approach 7 identify these regions visually is there an automated procedure that can pertorm th1s task Recall the two types of stress previously presented 9 elwmi39 JRanaj cD 5545 V g v Jal Cqmvmagdu SMH De ne the difference in these as the stress error GE I a F 3 5 398 E V V Now consider the structural strain energy in each element elIo dVlIoD lodV ANSYS7SENE 2V 2V V Can similarly calculate using stress error em 50 dVIoED l6E dV ANSYS7SERR V V 431 Error Estimation V Strain energy error 7 provides method to assess model error 0 Value needs to be considered relative to strain energy 0 Can use in model re nement 7 more elements in high error regions Can view as contour plots or element result listings 0 Listed under Element Solution in postprocessor SENE 7 strain energy 8 in each element V 0 SERR 7 strain energy error 8mm in each element 0 Can also obtain values using GET command for a given element GET Postprccess39mg Items Entity 2 1511531 Luti 5y IT2 I ANSYS Documentation Error Estimation gt ANSYS estimates strain energy error I Can be used in subsequent mesh re nement 1971 Error Approximation Technique for DisplacememrBased Problems An nkin iein n 197103 where H stress ennr enter at nnde n nfelememi NZ UL m N reg averaged siressveciurai nude quote Nquot number er elements cnnnecnng in nude n H Stress ecmr nfnude n nfelementi rnen fer each element 1 er e Efw iAniiim iiAnidwui 197109 nnere e energy mar er eternenrr accessed nth TABLE SERR item command nlnlume nfthe element accessedmtn 14er ortnernj cncnrnand m skesirstxmn mm e aluated at reference temp erdrure A5 mess manecmr pmnts as needed emlunred from all Aon nfthis element Q4 Element Strain Energy SENE gt Q4 PLANE42 7 strain energy per element I Calculate per element solid plot I Maximum Value 1184 inlb energy measure QB Element Strain Energy SENE gt Q8 PLANE82 7 strain energy per element Calculate per element solid plot Maximum Value 2336 inlb energy measure Q4 Element Strain Energy Error SERR gt Q4 PLANE42 7 strain energy per element Maximum Value 0241 inlb energy measure Compare to strain energy Value 1184 inlb energy measure Q8 Element Strain Energy Error SERR gt Q8 PLANESZ 7 strain energy per element Maximum value 0977 inlb energy measure Compare to strain energy value 2336 in lb energy measure gt Adaptive Meshing gt Adaptive meshing Allow ANSYS to change mesh based on strain energy error gt Steps for adaptive meshing Build and load solid model 7 cannot use nodalelement loads Specify number of solutions and target error 7 Smaller error means better result but longer to solve Specify other adaptive meshing parameters if desired Solution gt Solve gt Adaptive Meshing gt Can use adaptive meshing selectively Unselect certain areas onl elements in active areas are re ned 8e CarCAJ a xxzu graces 475qu 9 7 wquot Skaf0 Cofyrrf ac 3796 t 3 7 9 AJ ioht Wilma 104 742 5391 hi of pcfvwemmj bu dues 53 mean mul ave anecyL 4 38


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