Finite Element Simulations with ANSYS Workbench 12 Theory Applications Case Studies

Huei-Huang Lee PUBLICATIONS SDCSchroff Development Corporationwww.schroff.com Better Textbooks. Lower Prices.Preface4Chapter 1Introduction91.1 Case Study:Pneumatically Actuated PDMS Fingers101.2 Structural Mechanics:A Quick Review231.3 Finite Element Methods:A Conceptual Introduction311.4 Failure Criteria of Materials361.5 Problems42Chapter 2Sketching462.1 Step-by-Step:W16x50 Beam472.2 Step-by-Step:Triangular Plate582.3 More Details692.4 Exercise:M20x2.5 Threaded Bolt762.5 Exercise:Spur Gears802.6 Exercise:Microgripper862.7 Problems89Chapter 32D Simulations913.1 Step-by-Step:Triangular Plate923.2 Step-by-Step:Threaded Bolt-and-Nut1023.3 More Details1153.4 Exercise:Spur Gears1253.5 Exercise:Filleted Bar1303.6 Problems141

Chapter 43D Solid Modeling1434.1 Step-by-Step:Beam Bracket1444.2 Step-by-Step:Cover of Pressure Cylinder1504.3 Step-by-Step:Lifting Fork1624.4 More Details1704.5 Exercise:LCD Display Support1754.6 Problems180Chapter 53D Simulations1825.1 Step-by-Step:Beam Bracket1835.2 Step-by-Step:Cover of Pressure Cylinder1935.3 More Details2005.4 Exercise:LCD Display Support2045.5 Problems209Contents 1ContentsChapter 6Surface Models2116.1 Step-by-Step:Bellows Joints2126.2 Step-by-Step:Beam Bracket2226.3 Exercise:Gearbox2326.4 Problems243Chapter 7Line Models2457.1 Step-by-Step:Flexible Gripper2467.2 Step-by-Step:3D Truss2587.3 Exercise:Two-Story Building2687.4 Problems280Chapter 8Optimization2828.1 Step-by-Step:Flexible Gripper2838.2 Exercise:Triangular Plate2968.3 Problems304Chapter 9Meshing3069.1 Step-by-Step:Pneumatic Fingers3079.2 Step-by-Step:Cover of Pressure Cylinder3269.3 Exercise:3D Solid Elements Convergence Study3389.4 Problems350Chapter 10Buckling and Stress Stiffening35210.1 Step-by-Step:Stress Stiffening35310.2 Step-by-Step:3D Truss36410.3 Exercise:Beam Bracket36810.4 Problems372Chapter 11Modal Analyses37411.1Step-by-Step:Gearbox37511.2 Step-by-Step:Two-Story Building38011.3Exercise:Compact Disk38711.4Exercise:Guitar String39511.5Problems402Chapter 12Structural Dynamics 40412.1Basics of Structural Dynamics 40512.2 Step-by-Step:Lifting Fork41412.3 Step-by-Step:Two-Story Building42612.4Exercise:Ball and Rod43312.5Exercise:Guitar String44112.6Problems4522 ContentsChapter 13Nonlinear Simulations45413.1Basics of Nonlinear Simulations45513.2Step-by-Step:Translational Joint46613.3Step-by-Step:Microgripper47913.4Exercise:Snap Lock49413.5Problems508Chapter 14Nonlinear Materials51014.1Basics of Nonlinear Materials51114.2Step-by-Step:Belleville Washer52014.3Step-by-Step:Planar Seal53714.4Problems550Chapter 15Explicit Dynamics55215.1Basics of Explicit Dynamics55315.2Step-by-Step:High-Speed Impact55915.3Step-by-Step:Drop Test56715.4Problems578Index580Contents 3Usage of the BookLearningnite elementsimulations needs muchbackgroundknowledge, notjustatextbooklike this.The bookis a guidanceinlearningniteelementsimulations. This textbookisdesignedmainlyforgraduate studentsandsenior undergraduatestudents.Itisdesignedforuseinthreekindsofcourses:(a)asarstcourseofniteelement simulationbefore youtake anytheory-intensivecourses, suchas Finite Element Methods, (b)asanauxiliaryparallel tutorial in acourse such as Finite Element Methods, or(c) as an advanced(in an application-orientedsense)course after you took a theoretical course such as Finite Element Methods.Why ANSYS?ANSYS hasbeenasynonym of niteelementsimulations.I'vebeenusing ANSYSbothasalearningplatform in a course of niteelementsimulationsandas aresearchtoolintheuniversityforover20years. ThereasonsI love ANSYS are due to its multiple physics capabilities, completeness of on-line documentations, and popularity among both academiaandindustry. Equippingengineeringstudents withinterdisciplinarycapabilities is becominganecessity. A complete documentation allows the students nding solutions themselves independently, especiallyfor those problems nottaughtinthe classroom.Popularity, implyingahighpercentageofmarketshare, means thatafterthe students graduate and work as CAE engineers, they will be able to work with the software without any further training. Recent years, I have another reason to advocate this software, the user-friendliness.ANSYS WorkbenchThe Workbenchhas evolvedfor yearsbutmatured morein recentyears, andtheversion12 has been an important bench mark, worth a "wow" or 4.5 stars. Before the Workbench gets mature enough, I have been using the Classic(nowitis dubbed ANSYS APDL).The Classicis essentially driven bytext commands(itsGUI providesno essentialadvantagesovertextcommands). The user-unfriendlylanguageimposesunnecessaryconstraintsthatmakethe useofthesoftwareextremelydifcultand painful. Thedifcultycomesfrommanyaspects, forexamples, modelinggeometries, settingupcontactsorjoints, settingupnonlinear materialproperties, transferring databetweentwo analysis systems. As a result, thestudents or engineers often restrict themselves within limited types of problems, forexample, working on mechanical component simulations rather than mechanical system simulations. ComparingwiththeClassic,therealpoweroftheWorkbenchisitsuser-friendliness.Itreleasesmany unnecessary constraints.In a cliche, the only limitation is engineers' imagination.Why a New Tutorial?Preparing a tutorial for the Workbench needs much more effort than thatfor the Classic, due to the graphicnature of the interface.I think that is whythe number of books for the Workbench is still so limited.So far, the most complete tutorial, tomyknowledge,isthetrainingtutorialspreparedbyANSYS Inc.However, theymaynotbesuitablefor directuseasauniversitytextbook forthefollowingreasons.First, thecasesusedinthesetutorialsareeither too trivial or too complicated.Some cases are too complicated for students to create from scratch.The students need to rely on the geometry les accompanied with the tutorials.Students usually obtain a better comprehension by working fromscratch.Second, thetutorialcoverstoolittleontheoryaspectwhiletoomuchonthesoftwareoperations aspect.Many of nonessential software operations should not be included for a semester course.On the other hand, it contains limitedtheoreticalbackground aboutsolidmechanics andthe nite elementmethods. Besides, the tutorials are not available in any bookstores.To access the tutorials, the students need to attend the trainingcourses offered by ANSYS, Inc. or authorized rms.Other reasons include that they are in a form ofPowerPoint presentation les; much of effort is needed to furnish it to a university textbook, for example, adding homework problems.4 Preface PrefaceStructure of the BookThe structure of the book will be detailed in Section 1.1.Here is an overall picture. With the help of a case study, Section 1.1 overviews the Workbench simulation procedure.During the overview, asmoreconceptsortoolsare needed, specicchaptersor sections willbepointedoutto the students.In-depth discussionwillbeprovidedinthesechaptersor sections. The restof Chapter1 providesnecessarybackgroundof structuralmechanics, which willbe used in thelaterchapters. These backgrounds includeequations thatgovernthe behavior of amechanicalor structuralsystem, the nite elementmethods thatsolve thesegoverning equations, and thefailurecriteriaofmaterials.Chapter1isthe onlychapterthatdoesn'thaveanyhands-onexercises.Itisso designed because, in the very beginning of a semester, students may not be able to access the software facilities yet. Chapters2and3introduce2D geometricmodelingandsimulations.Chapters4-7introduce3Dgeometric modeling and simulations. Up to Chapter 7, we almostrestrictourdiscussion on linear staticstructuralsimulations. Chapter8is dedicatedto optimizationandChapter 9to Meshing.Chapter10dealswithbucklinganditsrelated topic: stress stiffening. Chapters 11 and 12 discuss dynamicsimulations.Chapters 13 and 14dedicate to amore in-depth discussion of nonlinear simulations, although several nonlinear simulations have been performedin the previous chapters.Chapter15devotesto an excitingtopic: explicitdynamics, whichis becominganecessarydisciplinefor a simulation engineer.Features of the BookComprehensiveness andcomprehensibility are the ultimate goals of everytextbook.There is no exception for this book.To achieve these goals, following features are incorporated into the design of the book. Real-World Cases.There are 45 step-by-step hands-on exercises in this book; each exercise is conducted in a single section.These exercises center on 27 cases.These cases are neither too trivial nor too complicated.Many of them are industrial or research projects; pictures of prototypes are presented in many cases.The size ofthe problems are not too large so that they can be simulated in an academic version ofANSYS Workbench 12, which has a limitation on the number of nodes or elements.Theyare nottoo complicatedso that the students can build eachprojectstep by step by themselves.Throughout the book, the students don't need any supplement les to work on these exercises. The les in the DVD thatcomes with the book are provided forthe students onlyin cases theyneed(see Usage of the Accompanying DVD). BackgroundKnowledge.Relevantbackgroundknowledgeisprovidedwhenevernecessary, suchassolid mechanics,niteelementmethods,structuraldynamics,nonlinearsolutionmethods(Newton-Raphsonmethods), nonlinear materials, explicit integration methods, etc.To be efcient, the teaching methods are conceptual rather than mathematical, short, yet comprehensive.The last four chapters cover more advanced topics, and each chapter begins a section that gives basics of that topic in an efcient way to facilitate the subsequent learning. LearningbyHands-onExperiencing.Alearningapproachemphasizinghands-onexperience spreads through the entire book.Inmyownexperience, this is the bestway to learn acomplicatedsoftware such as ANSYS Workbench.A typical chapter, such as Chapter 3, consists of 6 sections.The rsttwo sections provide two step-by-stepexamples. Thethirdsectiontriestocomplementtheexercisesbyprovidingamoresystematicviewofthe chapter subject.The following two sections provide more exercises. Mostoftheseadditional exercisesin the book are also presented in a step-by-step fashion.The nal section provides review problems. Learning by Building Motivation and Curiosity.After complete anexercise in a section, the students oftenraisemorequestionsthanwhattheyhavelearned.Forexample,wewillintroduceproblemsinvolving nonlinearities as earlyas in Chapter 3, without further in-depth discussion.Nonlinearities will be formally discussed in Chapters 13 and 14.Learning is more efcient after building enough motivation and curiosity. Key Concepts.Keyconceptsare insertedin placeswheneverappropriate.Must-knowconcepts, suchas structuralerror, nite elementconvergence, stress singularity, are taughtby usingdesigned hands-onexercises, rather thanbyabstractlecturing.Forexample, howniteelementsolutions convergeto theiranalyticalsolutions, asthe meshesgetner and ner, isillustratedbyguidingthestudentsto plotconvergencecurves. Thatway, the students should have strong knowledge ofthe nite elements convergence behaviors (and, after hours of working, they will not forget it for the rest of their life).Step-by-step guidingthe students to poltcurves to illustrate important concepts is one of the featuring teaching methods in this book. InsideBlackbox.Howthe Workbenchinternallysolves amodelis conceptuallyillustrated throughoutthe book.Understanding these procedures, at least conceptually, is crucial for a simulation engineer.Preface 5 On-lineReference.Oneoftheobjectivesofthisbookistoserveasaguidingbooktowardthehuge repositoryof ANSYS on-line documentation.As mentioned, the ANSYS on-line documentation is so complete that it even includes a theorymanual; itshouldbe awell ofknowledge formanystudents and engineers.The discussionsin the textbook often point to the on-line documentation as a further study aid whenever helpful. HomeworkExercises.Additionalexercisesorextensionresearchproblemsareprovidedashomework exercises at the ending section of each chapter. Summary of Key Concepts.Key concepts are summarized at the ending section of each chapter.One goal ofthis textbook is to train the engineering studentto comprehend the terminologies and use them properly.Thatis notsoeasyforsomestudents.Forexample, wheneverasked"Whatareshapefunctions?"mostofthestudents cannotsatisfyinglydenetheterminology.Yes,manytextbooksspendpagesteachingstudentswhattheshape functions are, but the challenge is how to dene or describe a term in less than two lines of words.This part of the textbookdemonstrates howto dene or describeaterm inanefcientway, for example, "Shapefunctions serveas interpolating functions, to calculate continuous displacement elds from discrete nodal displacements." OrderedSpeechBubbles.Screenshotswithorderedspeechbubblesareusedthroughoutthebook. Although not an orthodox way for a university textbook, ithas been proven to be very efcientin myclassroom.My students love it.I personally feel proud of creating this way of presentation for a textbook. Classroom Tryout.The entire book has been tried out on myclassroom for a semester.The purpose is to minimize mistakes.How the tryout proceeds is described as follows.To Instructor:How I Use the TextbookI use this textbook in a course offered each fall semester.There are 3 classroom hours a week; and the semester lasts 18 weeks.The progress is one chapter per week, except Chapter I, which takes 2 weeks to complete. Thetextbookis designedmuchlikeaworkbook. Thestudentsmustcompleteallthe hands-onexercisesand read the textof a chapter before theygo to my classroom.Every student has to prepare an one-page report and turns it in attheendofthe class.The one-page report should includequestionsand comments.The students must propose their questions in theclassroom.In myclassroom, there areonlydiscussionsof students'questions: NO traditional lecturing.The instructor's main responsibility in the classroom is to answer the students' questions.Imark andgradethe one-page reportsas partof performanceevaluations. The main purposeof the one-page report is to ensurethatthestudents compete the exercises and thoroughlyreadthetext of thechaptereachweek.The ideais that a student who completes the exercises and reads the textmust be full ofquestions in his/her mind, and a teacher should be able to grade the students'comprehensionfrom the levelofthe questions.The emphasis here is thatwe grade students' performance according to their questions, not their answers. The course load is not light as all; some chapters are as lengthyas 50 pages.Nevertheless, most of students were willingtospendhoursworkingonthesestep-by-stepexercises, becausetheseexercisesaretangible, ratherthan abstract.Students of this generationare usuallybetter in pickingup knowledgethrough tangiblesoftwareexercises rather than abstract lecturing. At the end of the semester, eachstudenthas to turn ina project. Students are free to choose topics fortheir projects as long as theyuse ANSYS Workbench to complete the project.Students who are working as engineers may choosetopicsrelatedtotheirjob.Other students who areworkingontheirthesesmaychoosetopicsrelatedto theirstudies.Theyare also allowed to repeataprojectfrom journal papers, as longas theygo through all details by themselves. Thepurposeofthenalprojectistoensurethatthestudentsarecapableofcarryingoutaproject independently, which is an ultimate goal of the course, not just following the step-by-step procedure in the textbook.To Students:How My Students Use the BookManystudentsinmyclassroomreportedtomethat, whenfollowingthestepsinthetextbook, theyoftenmade mistakes and endedup with completely differentresults from thatin the textbook. In manycases theycannotgure out which steps the mistakeswere made. In these case, they have to redo the exercise from the beginning.Itis not uncommon that they redid the exercise twice and nally saw the beautiful results. What I want to say is thatyou may come across the same situation, but you are not wasting your time when you redo the exercises.You are learning from the mistakes.Each time you x a mistake, you gain more insight.After you obtain the same results as the textbook, redo it and try to gure outif there are other ways to accomplish the same results.That's how I learn nite element simulations when I was a young engineer.6 Preface Finiteelementmethodsand solidmechanics are the foundationof mechanicalsimulations. If youhaven'ttaken these courses, plan to take them after youcomplete thiscourse of simulation. If you've alreadytaken them andfeel not "solid" enough, review them.Why Different Numerical Results?Manystudentsoftenpuzzledbecausetheyobtainedslightlydifferencenumericalresults,buttheyinsistthatthey followed exactly the same steps in the textbook.One of the reasons is thatdifferentway of creating ageometrymay end up with slightly different mesh, and this in turn ends up with slightly different numerical results.For example, when you drawa straightline, the order of the end points mayaffectmesh slightly.Limited differences in numerical values arenormal,particularlywhenthemesharecoarse.Asthemeshbecomesner,thesolutionwillconvergetoa theoretical value, which will be independent of mesh variations, and this kind of puzzle should be resolved.Usage of the Accompanying DVDThe les in the DVD that accompanies with the book is organized according to the chapters and sections of the book. Each folder of a section stored nished project les for that section.If everythingworks smoothly, you may not need the DVD at all.Every project can be built from scratch according to the steps described in the book.We provide this DVD just in some cases you need it.For examples, when you want to skip the creation of geometry, or when you run into troubles followingthestepsandyoudon'twanttoredofrom the beginning, youmayndthattheselesare useful.Another situation may happen when you have troubles following the geometry details in the textbook, you may need to look up the geometry details in the DVD les. However, Itis suggested that, in the beginningofastep-by-step exercise when previously savedprojectles are needed, youuse theprojectlesstoredin the DVD rather thanyourown les, inorder to obtainresults thathave exact the same numerical values as shown in the textbook.Numbering and Self-Reference SystemTo efciently present the material, the writing of this textbook is not always done in a traditional format.Chapters and sectionsarenumberedinatraditionalway.Eachsectionisfurtherdividedinto subsections,forexample, the8th subsectionof the 3rdsectionof Chapter 4 isdenoted as "4.3-8." Eachspeech bubble inasubsectionis assigned a number. The numberis enclosed bya pair of square brackets (e.g., [9]).Whenneeded, we may refer to thatspeech bubblesuchas"4.3-8[9]." Whenreferringtoaspeechbubbleinthesamesubsection,wedropthesubsection identier, for the foregoingexample, we simplywrite "[9]." Equations are numbered in asimilar way, except thatthe equationnumberis enclosedbyapairofroundbrackets(parentheses)ratherthansquare brackets.Forexample, "1.2-3(2)" refers to the 2nd equation in the Subsection 1.2-3.Numbering notations are summarized as follows: 1.2-3 The number after a hyphen is a subsection number. [1], [2], ... Square brackets are used to number speech bubbles. (1), (2), ... These notations are used to number equations (a), (b), ... These notations are used to number items in the text. Reference1, 2Superscripts are used to number references. Angle brackets are used to highlight Workbench keywords.Workbench KeywordsThere areliterallythousandsof keywordsused in the Workbench.For example: DesignModeler, ProjectSchematic, etc.To maintain readabilityandefciencyof the text, Workbench keywords are normally enclosed byapairof angle brackets,forexamples,,.Sometimes,however,theanglebracketsmaybe dropped, whenever it doesn't cause any readability or efciency problems.Preface 7AcknowledgementI feelthankfulto the studentswho hadeversatinmyclassroom,listeningto mylectures. Theyarespreadingout acrosstheworld, workingasengineersordedicatedresearchers.Someofthemstilldiscussproblemswithme through e-mail.I hope that, as theybecome aware ofthis textbook bytheirold-time professor, theywill go get one andrefresh their knowledgerightaway. Itismystudents, pastandpresent, thatmotivated meto give birthto this textbook.Thanks. Many ofthe cases discussed in this textbook are selected from turned-in nal projects of my students.Some are industrycaseswhileothers arethesis-relatedresearch topics. Withoutthese real-worldcases, thetextbookwould never be useful.The following is a list of the names who contributed to the cases in this book. "Pneumatic Finger" (Sections 1.1 and 9.1) is contributed by Che-Min Lin and Chen-Hsien Fan, ME, NCKU. "Microgripper" (Sections 2.6 and 13.3) is contributed by C. I. Cheng, ES, NCKU and P. W. Shih, ME, NCKU. "Cover of Pressure Cylinder" (Sections 4.2 and 9.2) is contributed by M. H. Tsai, ME, NCKU. "Lifting Fork" (Sections 4.3 and 12.2) is contributed by K. Y. Lee, ES, NCKU. "LCD Display Support" (Sections 4.5 and 5.4) is contributed by Y. W. Lee, ES, NCKU. "Bellows Tube" (Section 6.1) is contributed by W. Z. Liu, ME, NCKU. "Flexible Gripper" (Sections 7.1 and 8.1) is contributed by Shang-Yun Hsu, ME, NCKU. "3D Truss" (Section 7.2) is contributed by T. C. Hung, ME, NCKU. "Snap Lock" (Section 13.4) is contributed by C. N. Chen, ME, NCKU.Many ofthe original ideas of these projects came from the academic advisors of the above students.I also owe them a debt of thanks. Specically, the project"Pneumatic Finger" is an unpublishedwork led byProf. Chao-Chieh Lan of the DepartmentofME, NCKU. The project"Microgripper"originatesfrom aworkledbyProf. Ren-JungChangofthe DepartmentofME,NCKU. ThankstoProf.LanandProf. Changforlettingmeusetheiroriginalideas, including detailed geometries and some of the pictures. Thetextbookhadbeentriedoutinmyclassroom.Manystudentsvolunteeredtoproofreadthetextand pointed out manyerrors.Theywrote down those errors intheir one-page reports that I collected atthe end of the class.Thanks to these students. Much ofinformation about the ANSYS Workbench are obtained from training tutorials prepared by ANSYS Inc.Ididn'tspecicallycitetheminthetext, butIappreciatethesetrainingtutorialsverymuch. As Imentioned, these training tutorials are one of the most comprehensive tutorials about the ANSYS Workbench. I'mthankfulfortheenvironmentprovidedbyNationalChengKungUniversityandtheDepartmentofEngineeringScience. Thecampusiscozy, the libraryfacilityisexcellent, andthe workingatmosphere is sofreeof pressure that I was able to accomplish this textbook within a short time. I want to thank Mrs. Lilly Lin, the CEO, and Mr. Nerow Yang, the general manager, of Taiwan Auto Design, Co., the partner ofANSYS, Inc. in Taiwan.Thecouple, mylong-term friends, provided much ofsubstantial supportduringthe writing of this book. Special gratitude is due to Professor Sheng-Jye Hwang, of the ME Department, NCKU, and Professor Durn-Yuan Huang, ofChung Hwa University ofMedical Technology.They are my long-term research partners.Together, we have accomplished many projects, and, in carrying out these projects, I've learned much from them. Lastly, thanks to myfamily, includingmy wife, myson, and the dogs (Penny, Beagle, andShiba), for their patience and sharing the excitement with me.Huei-Huang LeeAssociate ProfessorDepartment of Engineering ScienceNational Cheng Kung UniversityTainan, Taiwanhhlee@mail.ncku.edu.tw8 Preface 10 Chapter 1IntroductionSection 1.1Case Study:Pneumatically Actuated PDMS Fingers1About the Pneumatic FingersThepneumaticngers[1]aredesignedaspartofasurgicalparallelrobot system which is remotely controlled by a surgeon through the Internet2. TherobotngersaremadeofaPDMS-based(polydimethylsiloxane) elastomer material.The geometry ofa nger is shown in the gure [2].Note that 14 air chambers are built in the nger.The purposes of this section are to (a)overviewthe functionalityof the ANSYS Workbench through acase study, (b) presentan overallstructureof the textbook bybringingup topics ofthe chapters through a case study, and (c) build motivationfor learningthe topics in Sections 2, 3, 4 of this chapter: structural mechanics, nite elementmethods, and the failure criteria. Although this case studyis presented in a step-by-step fashion, it does notintend to guide the students working in frontof acomputer.In fact, only the relevant stepsare presented, and some steps are purposely omittedto make the presentation more instructional.There will be many hands-on exercises in the later chapters.So, be patient.1.1-1Problem Description The chambers are located closer to the upper face than the bottom face so that when the air pressure applies, thengerbends downward[3].Notethatonlyhalf of the modelisrendered, so youcan seethe chambers. The undeformed model is also shown in the gure [4].Note:In this book, each speech bubble has a unique number in a subsection.The number is enclosed with a pair of square brackets.When you read gures, please follow the order of numbers; the order is important. These numbers also serve as reference numbers when referred.[1] Five ngers compose a robot hand, which is remotely controlled by a surgeon.[2] The ngers size is 80x5x10.2 (mm).There are 14 air chambers built in the PDMS nger, each is 3.2x2x8 (mm).[4] Undeformed shape.[3] As the air pressure applies, the nger bends downward.46 Chapter 2Sketching Chapter 2SketchingA simulation projectstarts with the creation of a geometricmodel.To be pro0cient atsimulations, an engineer has to be pro0cient atgeometric modeling 0rst.In a simulation project, itis not uncommon to take the majority of human-hours to create a geometric model, that is particularly true in a 3D simulation. Acomplex3Dgeometrycanbeviewedasacollectionofsimpler3D solidbodies.Eachsolidbodyisoften createdby0rstdrawingasketch onaplane, andthenthe sketchisusedtogenerate the 3D solidbodyusing tools such as extrude, revolve, sweep, etc.In turn, to be pro0cient at 3D bodies creation, an engineer has to be pro0cient at sketching 0rst.Purpose of the ChapterThe purpose of this chapteris to provide exercisesfor the students so thattheycanbe pro0cientatsketchingusing DesignModeler.Five mechanical parts are sketchedin this chapters.Although eachsketch isusedto generate a 3D models, the generation of3D models isso trivial thatwe should be able to focus on the 2D sketcheswithoutbeing distracted.More exercises of sketching will be provided in later chapters.About Each SectionEach sketchofamechanicalpartwillbe completedinasection.Sketchesinthe0rsttwo sectionsare guidedina step-by-step fashion.Section 1 sketches across section of W16x50; the cross section is then extruded to generate a solid model in 3D space.Section 2 sketches atriangular plate; the sketch is then extrudedto generate asolidmodel in 3D space. Section3doesnotmeantoprovide ahands-oncase.Itoverviewsthesketchingtoolsinasystematicway, attempting to complement what were missed in the 0rst two sections. Sections 4, 5, and6 provide three cases formore exercises.Sketches in these sectionsare in anot-so-step-by-step fashion; we purposely leave some room for the students to 0gure out the details.Section 2.1Step-by-Step:W16x50 Beam Section 47Section 2.1Step-by-Step:W16x50 BeamConsider astructuralsteelbeam witha W16x50cross-section [1-4] and alength of 10ft. Inthis section, we willcreatea 3D solid body for the steel beam.2.1-1About the W16x50 BeamW16x502.1-2Start Up 16.25" .628" .380" 7.07" R.375"[1] Wide-;ange I-shape section. [2] Nominal depth 16".[3] Weight 50 lb/ft.[4] Detail dimensions[2] After a while, the shows up.[3] Click the plus sign (+) to expand the .Note that the plus sign become minus sign.[4] Double-click to place a system in the .[6] Double-click to start up DesignModeler.[5] If anything goes wrong, click here to show message.[1] From Start menu, click to launch the Workbench.48 Chapter 2SketchingNotes:Inastep-by-step exercise, wheneveracircle isused withaspeech bubble, it is to indicatethatmouse or keynoard ACTIONS mustbetaken in thatstep (e.g., [1, 3, 4, 6, 8, 9]). The circle maybe smallor large, ;lledwith white coloror un;lled, dependingonwhichever gives more information.A speechbubble withouta circle (e.g., [2, 7]) or with a rectangle (e.g., [5]) is used for commentary only, no mouse or keyboard actions are needed.2.1-3Draw a Rectangle on [9] Click .Note that, after clicking , the length unit connot be changed anymore.[8] Select as the length unit.[7] After a while, the DesignModeler shows up.[1] is already the current sketching plane.[2] Click to enter the sketching mode.[4] Click tool.[3] Click to rotate the coordinate axes, so that you face the .[5] Draw a rectangle (using click-and-drag) roughly like this.Section 2.1Step-by-Step:W16x50 Beam Section 49Impose symmetry constraints...Specify dimensions...[6] Click toolbox.[8] Click tool.[9] Click the vertical axis and then two vertical lines on both sides to make them symmetric about the vertical axis.[10] Right-click anywhere on the graphic area to open the context menu, and choose .[11] Click the horizontal axis and then two horizontal lines on both sides to make them symmetric about the horizontal axis.[7] If you don't see tool, click here to scroll down to reveal the tool.[12] Click toolbox.[13] Leave as the default tool.[17] In the , type 7.07 (in) for H1 and 16.25 (in) for V2.[14] Click this line, move the mouse upward, and click again to create H1.[15] Click this line, move the mouse rightward, and click again to create V2.[17] Click .[16] The segments turn to blue color.Colors are used to indicate the constraint status.The blue color means that the geometric entities are well constrained.50 Chapter 2Sketching2.1-4Clean up the Graphic AreaThe ruler occupies space and is sometimes annoying; let's turn it off...Let's display dimension values (in stead of names) on the graphic area...[2] The ruler disappears.It creates more space for the graphic area.For the rest of the book, we always turn off the ruler to make more space in the graphic area.[1] Pull-down-select to turn the ruler off.[3] If you don't see tool, click here to scroll all the way down to the bottom.[4] Click tool.[5] Click to turn it off.The automatically turns on.[6] The dimension names are replaced by the values.For the rest of the book, we always display values instead of names, so that the sketching will be more ef8cient.Section 2.1Step-by-Step:W16x50 Beam Section 512.1-5Draw a PolylineDraw a polyline; the dimensions are not important for now...Copy the newly created polyline to the right side, ;ip horizontally...2.1-6Copy the Polyline[1] Select toolbox.[2] Select tool.[3] Click roughly here to start the polyline.Make sure a (coincident) appears before clicking.[4] Click the second point roughly here.Make sure an (horizontal) appears before clicking. [5] Click the third point roughly here.Make sure a (vertical) appears before clicking.[6] Click the last point roughly here.Make sure an and a appear before clicking.[7] Right-click anywhere on the graphic area to open the context menu, and select to end the tool.[4] Right-click anywhere on the graphic area to open the context menu, and select .[1] Select toolbox.[2] Select tool.[3] Control-click (see [11, 12]) the three newly created segments one by one.52 Chapter 2SketchingContext menu is used heavily...Basic Mouse OperationsAt this point, let's look into some basic mouse operations[10-16].Skill of these operations is one ofthe keys to be prorst two clicks de>ne the two centers, and the third click de>nes the radius.Circle by 3 TangentsSelect three edges, then a circle tangent to these three edges will be created.Remember that an edge can be a line or a curve.Arc by TangentClick a point on an edge, an arc starting from that point and tangent to that edge will be created; click a second point to de>ne the other end point of the arc.SplineA spline is either rigid or ?exible.The difference is that a ?exible spline can be edited or changed by imposing constraints, while a rigid spline cannot.After de>ning the last point, you must right-click to open the context menu, and select an option [2]: either open end or closed end; either with >t points or without >t points.[1] By default, DesignModeler is in mode, both globally and locally.You can turn them off whenever cause troubles.[1] toolbox.Section 2.3More Details 73Construction Point at IntersectionSelect two edges, a construction point will be created at the intersection.Delete EntitiesThere are no tools in the to delete entities.To delete entities, select them and right-click-select .Multiple selection methods (e.g., control-selection and sweep-selection, see Section 2.1-6 and 2.2-12[2]), can be used to select entities.Abort a ToolTo cancel a tool in any of toolbox, simply press .2.3-7 Tools4CornerClick two entities, which can be lines or curves, the entities will be trimmed or extended up to the intersection point and form a sharp corner.The clicking points decide which sides to be trimmed.SplitThis tool split an edge into several segments depending on the options [2].: you click an edge, the edge will be split at the clicking point.: you click a point, all the edges passing through that point will be split at that point.: you select an edge, the edge will be split at all points on the edge.: You specify the value n, and select an edge, the edge will be split equally into n segments.DragDrag a point or an edge to a new position.All the constraints and dimensions are preserved.CutIt is the same as , except the originals are deleted.MoveIt is equivalent to a followed by a .ReplicateIt is equivalent to a followed a .DuplicateIt is equivalent to , except the entities are pasted on the same place as the originals and become part of the current sketch.It is often used to duplicate plane boundaries.Spline EditIt is used to modify 7exible splines.You can insert, delete, drag the 6t points, etc.For details, see the reference4.[2] Right-click and select one of the options to complete the tool.[1] toolbox.[2] Context menu for tool.[3] Context menu for .74 Chapter 2Sketching2.3-8 Tools5Semi-AutomaticThis tool will display a series of dimensions automatically to help you fully dimension the sketch.EditClick a dimension name or value, it allows you to change its name or value.2.3-9 Tools6FixedIt applies on any entity to make it fully constrained.HorizontalIt applies on a line to make it horizontal.VerticalIt applies on a line to make it vertical.PerpendicularIt applies on two edges to make them perpendicular to each other.TangentIt applies on two edges, one of which must be a curve, to make them tangent to each other.CoincidentSelect two points to make them coincident.Select a point and an edge, the edge or its extension will pass through the point.There are other possibilities, depending on how you select the entities. MidpointSelect a line and then a point, the midpoint of the line will coincide with the point.SymmetrySelect a line or an axis, as the line of symmetry, and either select 2 points or 2 lines.If select 2 points, the points will be symmetric about the line of symmetry.If select 2 lines, the lines will form the same angle with the line of symmetry.ParallelIt applies on two lines to make them parallel to each other.[1] toolbox.[1] toolbox.Section 2.3More Details 75ConcentricIt applies on two curves, whichmaybecircle, arc, or ellipse, to make their centers coincident.Equal RadiusItappliesontwocurves,whichmaybecircleorarc,tomaketheir radii equal.Equal LengthIt applies on two lines to make their lengths equal.Equal DistanceItappliesontwodistancestomakethemequal. A distancecanbe de6nedbyselectingtwopoints, two parallellines,oronepointand one line.2.3-10 Tools7[2] You can turn on the grid display.References1. ANSYS Help System>DesignModeler>2D Sketching>Auto Constraints2. ANSYS Help System>DesignModeler>2D Sketching>Constraints Toolbox>Auto Constraints3. ANSYS Help System>DesignModeler>2D Sketching>Draw Toolbox4. ANSYS Help System>DesignModeler>2D Sketching>Modify Toolbox5. ANSYS Help System>DesignModeler>2D Sketching>Dimensions Toolbox6. ANSYS Help System>DesignModeler>2D Sketching>Constraints Toolbox7. ANSYS Help System>DesignModeler>2D Sketching>Settings Toolbox[1] toolbox.[3] You can turn on the snap capability.[4] If you turn on the grid display, you can specify the grid spacing.[5] If you turn on the snap capability, you can specify the snap spacing.76 Chapter 2SketchingSection 2.4Exercise:M20x2.5 Threaded BoltConsider apairofthreadedboltandnut. Thebolthas externalthreadswhilethenuthas internalthreads. This exerciseisto createdasketchandrevolvethesketch 360

to generate asolidbodyforaportionofthebolt[1] threadedwithM20x2.5 [2-6]. InSection 3.2, we willusethissketchagain to generate a2D solidmodel. The 2D model is then used for a static structural simulation.2.4-1About the M20x2.5 Threaded BoltM20x2.5 H= ( 3 2)p= 2.165 mmd1= d (5 8)H 2=17.294 mmExternalthreads(bolt)Internalthreads(nut) H H4 H8 32 11

p=27.5 p p d1 d Minor diameter of internal thread d1 Nominal diameter d 60o[2] Metric system. [3] Nominal diameterd = 20 mm.[4] Pitchp = 2.5 mm.[1] The threaded bolt created in this exercise.[5] Thread standards.[6] Calculation of detail sizes.Section 2.4Exercise: M20x2.5 Threads 772.4-2Draw a Horizontal Line2.4-3Draw a PolylineDrawapolyline (totally3segments)and specifydimensions(30o, 60o, 60o, 0.541, and2.165)asshownbelow. Note that, to avoid confusion, we explicitly specifyall the dimensions.You may applyconstraints instead.For example, using constraint in stead of specifying an angle dimension [1].Launch .Create a System.Save the project as "Threads."Start up .Select as length unit. Draw a horizontal line on the .Specify the dimensions as shown [1].[1] Draw a horizontal line with dimensions as shown.[1] You may impose a constraint on this line instead of specifying the angle.78 Chapter 2Sketching2.4-4Draw FilletsDraw two vertical lines and specify their positions (0.271 and 0.541).Draw an arc using .If the arc is not in blue color, impose a constraint on the arc and one of its tangent line [1].2.4-5Trim Unwanted Segments2.4-6Replicate 10 TimesSelect all segments except the horizontal one (totally 4 segments), and replicate 10 times.You may need to manually set the paste handle [1].You may also need to use the tool [2].[1] Tangent point.[1] Set Paste Handle at this point.[2] .[1] The sketch after trimming. Section 2.4Exercise: M20x2.5 Threads 792.4-7Complete the SketchFollow the steps [1-5] to complete the sketch.Note that, in step [4], you don't need to worry about the length.After step [5], you can trim the vertical segment created in step [4].2.4-8Revolve to Create 3D SolidReferences1. Zahavi, E., The Finite Element Method in Machine Design, Prentice-Hall, 1992; Chapter 7.Threaded Fasteners.2. Deutschman, A. D., Michels, W. J., and Wilson, C. E., Machine Design: Theory and Practice, Macmillan Publishing Co., Inc., 1975; Section 16-6.Standard Screw Threads.Revolve the sketch to generate a solid of revolution.Select the Y-axis as the axis of revolution. Save the project and exit from the Workbench.We will resume this project again in Section 3.2.[1] Create this segment by using .[3] Specify this dimension.[2] Draw this segment, which passes through the origin.[4] Draw this vertical segment.You can trim it after next step. [5] Draw this horizontal segment.80 Chapter 2Sketching The 8gure below shows apairof identical spur gears in mesh [1-12]. Spur gears have their teeth cut parallel to the axis of the shaft on which the gears are mounted.Spur gears are used to transmit powerbetween parallelshafts.In orderthattwomeshinggearsmaintainaconstantangularvelocityratio,theymustsatisfythefundamentallawof gearing: the shape of the teeth must be such that the common normal at the point of contact between two teeth must always pass through a 8xed point on the line of centers1 [5].This 8xed point is called the pitch point [6].Theanglebetweenthelineof actionandthecommontangent[7]isknownas thepressureangle[8]. The parameters de8ning a spur gear are its pitch radius (rp = 2.5 in) [3], pressure angle ( = 20o) [8], and number of teeth (N = 20).In addition, the teeth are cut with a radius ofaddendum ra = 2.75 in [9] and a radius of dedendum rd = 2.2 in [10].The shaft has a radius of 1.25 in [11].The 8llet has a radius of 0.1 in [12].The thickness of the gear is 1.0 in.2.5-1About the Spur GearsSection 2.5Exercise:Spur GearsGeometricdetails ofspur gearsare importantforamechanicalengineer.However, if you arenotconcernedabout these geometric details for now, you may skip the 8rst two subsections and jump directly to Subsection 2.5-3.[7] Common tangent of the pitch circles.[6] Contact point (pitch point).[8] Line of action (common normal of contacting gears).The pressure angle is 20o.[3] Pitch circlerp = 2.5 in.[9] Addendumra = 2.75 in.[10] Dedendumrd = 2.2 in.[1] The driving gear rotates clockwise.[2] The driven gear rotates counter-clockwise.[4] Pitch circle of the driving gear.[5] Line of centers.[12] The 8llet has a radius of 0.1 in.[11] The shaft has a radius of 1.25 in.Section 2.5Exercise:Spur Gears 81To satisfy the fundamental law of gearing, most ofgear pro8les are cut to an involute curve [1].The involute curve may be constructed bywrapping astring around a cylinder, called the basecircle[2], and then tracing the path ofa point on the string. Given the gear'spitch radius rpand pressureangle, we cancalculated the coordinates of each pointon the involutecurve. For example, consideranarbitrarypointA [3]ontheinvolute curve; we wantto calculate its polar coordinates (r,) , as shown in the 8gure.Note that BA and CP are tangentlines ofthe base circle, andF is afootof perpendicular.2.5-2About Involute Curves

A C O P B rb rp r

D rb rb E F

Since APF is an involute curve and BCDEF

is the base circle, by the de8nition of involute curve, BA= BC

+ CP= BCDEF

(1) CP= CDEF

(2)From OCP , rb= rpcos(3)From OBA , r=rbcos (4)Or equivalently, = cos1rbr (5)To calculate , we notice that DE

= BCDEF

BCD

EF

Dividing the equation with rb and using Eq. (1), DE

rb=BArb

BCD

rb

EF

rbIf radian is used, then the above equation can be written as = (tan) 1 (6)The last term

1 is the angle EOF , which can be calculated by dividing Eq. (2) with rb, CPrb=CDEF

rb, or tan= +1, or

1= (tan) (7)Eqs. (3-7)areallwe needto calculate polarcoordinates (r,) . The polarcoordinatescanbeeasilytransformedto rectangular coordinates, using O as origin and OP as y-axis, x=r sin, y= r cos(8)

1[4] Contact point (pitch point).[2] Base circle.[5] Line of action.[6] Common tangent of pitch circles.[7] Line of centers; this length is the pitch radius rp.[1] Involute curve.[3] An arbitrary point on the involute curve.82 Chapter 2Sketching Numerical CalculationsIn our case, the pitch radius rp= 2.5in, and pressure angle = 20o; from Eqs. (2) and (7), rb= 2.5cos 20o= 2.349232 in

1= tan20o

20o180o= 0.01490438The calculated coordinates are listed in the table below.Notice that, in using Eqs. (6) and (7), radian is used as the unit of angles; in the table below, however, we translated the unit to degrees.rin.

Eq. (4), degrees

Eq. (5), degreesx y2.349232 0.000000 -0.853958 -0.03501 2.348972.449424 16.444249 -0.387049 -0.01655 2.449372.500000 20.000000 0.000000 0.00000 2.500002.549616 22.867481 0.442933 0.01971 2.549542.649808 27.555054 1.487291 0.06878 2.648922.750000 31.321258 2.690287 0.12908 2.746972.5-3Draw an Involute CurveLaunch .Create a system.Save the project as "Gear."Start up .Select as length unit.Start to draw sketch on the XYPlane. Draw six and specify dimensions as shown (the vertical dimensions are measured down to the X-axis).Note that the dimension values display three digits after decimal point, but we actually typed with @ve digits (refer to the above table).Impose a constraint on the Y-axis for the point which has a Y-coordinate of 2.500. Connect these six points using tool, keeping option on, and close the spline with .Note that you could draw directly without creating @rst, but that would be not so easy.[1] Y-axis.Section 2.5Exercise:Spur Gears 832.5-4Draw CirclesDraw three circles [1-3].Let the addendum circle "snap" to the outermost construction point [3].Specify radii for the circle of shaft (1.25 in) and the dedendum circle (2.2 in). 2.5-5Complete the Pro4leDraw a line starting from the lowest construction point, and make it perpendicular to the dedendum circle [1-2].Note that, when drawing the line, avoid a auto-constraint. Draw a 4llet [3] of radius 0.1 in to complete the pro4le of a tooth.[3] Let addendum circle "snap" to the outermost construction point.[1] The circle ofshaft.[2] Dedendum circle.[2] This segment is a straight line and perpendicular to the dedendum circle.[3] This 4llet has a radius of 0.1 in.[1] Dedendum circle.84 Chapter 2Sketching 2.5-6Replicate the Pro:leActivate tool, type 9 (degrees) for .Select the pro:le (totally 3 segments), , , , and .End the tool. Note that the gear has 20 teeth, each spans by 18 degrees.The angle between the pitch points on the left and the right pro:les is 9 degrees.2.5-7Replicate Pro:les 19 TimesActivate tool again, type 18 (degrees) for .Select both left and right pro:les (totally 6 segments), , , and .Repeat the last two steps (rotating and pasting) until :ll-in a full circle (totally 20 teeth). As the geometric entities is getting more and complicated, the computer's processing time may be getting slower, depending on your hardware con:guration. Save your project once a while by clicking the tool in the toolbar.[1] Replicated pro:le.[1] Section 2.5Exercise:Spur Gears 85References1. Deutschman, A. D., Michels, W. J., and Wilson, C. E., Machine Design: Theory and Practice, Macmillan Publishing Co., Inc., 1975; Chapter 10.Spur Gears.2. Zahavi, E., The Finite Element Method in Machine Design, Prentice-Hall, 1992; Chapter 9.Spur Gears.2.5-8Trim Away Unwanted Segments2.5-9Extrude to Create 3D SolidExtrude the sketch 1.0 inch to create a 3D solid as shown.Save the project and exit from .We will resume this project again in Section 3.4.Trim away unwanted portion on the addendum circle and the dedendum circle.86 Chapter 2Sketching Section 2.6Exercise:MicrogripperManymanipulatorsaredesigned as mechanisms, thatis, theyconsistof bodiesconnectedbyjoints, suchas revolute joints, sliding joints, etc., and the motions are mostly governed by the laws of rigid body kinematics. The microgripper discussed here [1-2] is a structure rather than amechanism; the mobility are providedby the 4exibility of the materials, rather than the joints. The microgripper is made of PDMS (polydimethylsiloxane, see Section 1.1-1).The device is actuated bya shape memoryalloy(SMA) actuator [3], ofwhichthe motioniscausedbytemperaturechange, andthe temperatureisin turn controlled by electric current.2.6-1About the Microgripper In the lab, the microgripperistestedbygrippingaglass bead of a diameter of 30 micrometer [4]. Inthissection,wewillcreateasolidmodelforthe microgripper.The model will be used for simulation in Section 13.3 to assess the gripping forces on the glass beadunder the actuation of SMA actuator. 480 144 176 280 400140 212 32 92 77 47 87 20 R25R45 D30 Unit: mThickness: 300 m[2] Actuation direction.[1] Gripping direction.[3] SMA actuator.[4] Glass bead.Section 2.6Exercise:Microgripper 872.6-2Create Half of the ModelLaunch .Create a system.Save the project as "Microgripper."Start up .Select as length unit.Start to draw sketch on the XYPlane. Draw the sketch as shown on the right side [1].Note that two of the three circles have equal radii.Trim away unwanted segments as shown below [2].Note that we drew half of the model, due to the symmetry.Extrude the sketch 150 microns both sides of the plane symmetrically (total depth is 300 microns) [3].Now we have half of the gripper [4].[1] Before trimming.[2] After trimming.[3] Extrude both sides symmetrically.[4] Half of the gripper.88 Chapter 2Sketching 2.6-2Mirror Copy the Solid Body2.6-3Create the BeadCreate a new sketch on XYPlane and draw a semicircle as shown [1-4].Revolve the sketch 360 degrees to create the glass bead.Note that the two bodies are treated as two parts.Rename two bodies [5].Wrap UpClose , save the project and exit .We will resume this project in Section 13.3.[3] Select the solid body and click .[2] The default type is (mirror copy).[6] Click .[3] Remember to impose a constraint here.[2] Remember to close the sketch by draw the vertical line.[5] Right-click to rename two bodies.[4] Select the in the model tree and click .If doesn't appear, see next step. [1] The semicircle can be created by creating a full circle and then trim it using the axis.[4] Remember to specify the dimension.[5] If doesn't appear, clicking the yellow area will make it appear.[1] Pull-down-select . 2.7-1Key ConceptsSketching ModeAn environment under DesignModeler, con8gured for drawing sketches on planes.Modeling ModeAn environment under DesignModeler, con8gured for creating 3D or 2D bodies.Sketching PlaneThe plane on whichasketch iscreated. Eachsketch must be associatedwith aplane; each plane mayhave multiple sketches on it.Usage of planes is not limited for storing sketches.EdgeIn , an edges may be a (straight) line or a curve.A curve may be a circle, ellipse, arc, or spline.SketchA sketch consists of points and edges. Dimensions and constraints may be imposed on these entities.Model TreeA model tree is the structured representationof a geometryand displayed on the in DesignModeler. A model tree consists of planes, features, and a part branch, in whichtheir order is important.The parts are the only objects exported to .BranchA branch is an object of a model tree and consists one or more objects under itself.ObjectA leaf or branch of a model tree is called an object.Context MenuThe menu that pops up when you right-click your mouse.The contents of the menu depend on what you click.Auto ConstraintsWhile drawingin, bydefault, DesignModelerattemptstodetectthe user'sintentionsandtryto automatically impose constraints on points or edges.Detection is performed over entities on the active plane, not just active sketch. can be switched on/off in the toolbox.Section 2.7Problems 89Section 2.7ProblemsSelection FilterA selection 5lter 5lters one type of geometric entities.When a selection 5lter is turned on/off, the corresponding type ofentities become selectable/unselectable.In Mode, there are two selection 5lters which corresponding to points and edges respectively.Along with these two 5lters, face and body selection 5lters are available in .Paste HandleA reference point used in a copy/paste operation.The point is de5ned during copying and will be aligned at a speci5ed location when pasting.Constraint StatusIn mode, entities are color coded to indicate their constrain status: greenish-blue forunder-constrained; blue and black for well constrained (i.e., 5xed in the space); red for over-constrained; gray for inconsistent.2.7-2Workbench ExercisesCreate the Triangular Plate with Your Own WayAfterso manyexercises, youshouldbe able to 5gure outanalternative wayofcreatingthe geometricmodel forthe triangular plate (Section 2.2) on your own.Can you 5gure out a more ef5cient way?90 Chapter 2Sketching102 Chapter 32D Simulations Section 3.2Step-by-Step:Threaded Bolt-and-Nut3.2-1About the Threaded Bolt-and-Nut The plane of symmetry The axis of symmetry 17 mmThe threaded boltwe created in Section 2.4 is partof abolt-nut-plate assembly[1-4].The bolt is preloaded with a tension. Thepretensionisappliedbytighteningthenutwithtorque. The pretension can be calculated bymultiplying the maximum torque with a coefcient, which is empirically determined.The pretension in our case is 10 kN.We want to know the stress at the threads under such a pretension condition. Pretensionisaready-to-useenvironmentconditionin 3D simulations, in which apretension can apply on abody or cylindricalsurface.Itis,however,notapplicablefor2D simulations. In this 2D simulation, we willmakesome simplication.Assumingasymmetrybetweenupperandlowerpart,we modelonlyupperpartoftheassembly[5].Theplateis removed, to reduce the problem size and alleviate the contact nonlinearity, and its boundary surface with the nut is replaced by a frictionless support [6]. The pretension is replaced by a uniform force applied on thelowerfaceofthebolt.Themodelsomewhatdeviates fromthereality,whichwewilldiscussattheendof this section, but for accessing the stress, it should be acceptable. The coefcient of friction betweenthe boltand the nut is estimated to be 0.3.[1] Bolt. [2] Nut. [3] Plates.[4] Section view.[5] The 2D simulation model.[6] Frictionless support.150 Chapter 43D Solid Modeling Section 4.2Step-by-Step:Cover of Pressure Cylinder4.2-1About the Cylinder CoverThe pressure cylinder [1] contains gas of 0.5 MPa.The cylinder cover [2-4] is madeofcarbon-berreinforcedplastic.Wewanttoinvestigatethe deformationofthecylindercoverundersuchworkingpressure. Wewill create a3D solidmodelinthis section; the modelwillbe usedforastatic structural simulation in Section 5.2. Unit: mm. 30.3 25.3 21.0 1.3 31.0 3.010.0 R8.5 R7.5 R19.0 62.0 2.3 1.6 7.4 7.4 62.0 R4.9R3.2 R9.0R14.5 R18.1 R25.4 R27.8 R3.4[1] Pressure cylinder.[2] Cylinder Cover.[3] A close-up view of the cylinder cover.[4] Back view of the cover.Section 4.5Exercise:LCD Display Support 175Section 4.5Exercise:LCD Display SupportTheLCDDisplaysupportismadeof an ABS(acrylonitrile-butadiene-styrene)plastic. Thethicknessoftheplasticis3 mm [1].Details of the hinge design is not shown in the gure but will be shown in 4.5-4 [2].Thesolidmodelwillbe usedin Section5.4 forastatic structuralsimulationtoassessthedeformationandstress under a design load.4.5-1About the LCD Display Support 200 90 60 4410 50 42 2017 Unit: mm[1] The thickness of the plastic is 3 mm.[2] Details of the hinge design will be shown in 4.5-4.212 Chapter 6Surface Models Section 6.1Step-by-Step:Bellows JointsThebellowsjoints[1-2]areusedasexpansionjoints, whichabsorbthermalorvibrationalmovementinapiping system thattransports high pressuregases. As partofthe pipingsystem, the bellowsjointsare designedto sustain internal pressure as wellas external pressure. The external pressure mustbe considered whenthe piping system is used across theocean.Under the internal pressure, the engineers mostly concern about its radialdeformation(due toanengineeringtoleranceconsideration)andhoopstress(due to the safetyconsideration).Undertheexternal pressure, buckling is the main concern (see an exercise in Section 10.4-2).6.1-1About Bellows Joints Inthissection,wewillcreateafull3Dsurface modelforthebellowsjointandperformastatic structuralsimulationundertheinternalpressureof0.5 MPa. Abucklingsimulationunder the externalpressure will be left as an exercise in Section 10.4-2. Notethattheproblemisaxisymmetricbothin geometryandloading. We could take advantageofthis property and model the problem as 2D solid body or 2D linebody(bothas axisymmetric models).The latter, 2D linemodel,isnotsupportedinthecurrentversionof (itissupportedthroughAPDL).The former, 2D solid model, usually resultsa poorer solution thansurfacebody,forthisparticularcase,becausethe bending dominates its structural behavior. R315 28 R315 28 20 Unit: mm.[1] The bellows joints are made of SU316 steel, which has a Young's modulus of 180 GPa and Poisson's ratio of 0.28.[2] All arcs have radii of 7 mm.The thickness is 0.8 mm.Section 11.4Exercise:Guitar String 395Section 11.4Exercise:Guitar StringThe guitar string inour case is made of steel, which has amass densityof 7850 kg/m3, a Young's modulus of 200 GPa, anda Poisson's ratio of0.3.Ithas acircular cross section of diameter 0.28 mm anda length of 1.0 m. The stringis stretched with a tension T, and is in tune with astandard A note (la), which has been dened to be exactly440 Hzin the modern music.In the next subsection (11.4-2), we will perform a modal analysis to nd the required tension T. Before the simulation, let's make some simple calculation.According to the basicphysics, the wave traveling on a string has a speed of v =TWhere is the linear density(kg/m) of the string.The standingwave correspondingto the lowestfrequencyis called the rst harmonic mode, which has awavelength of twice the stringlength (2L).According to the relation between the velocity, the frequency, and the wavelength f =v

=v2Lwe can estimate the required tension T = 2fL( )2= 7850 (0.00028)242440 1.0( )2= 374.32 N11.4-1About the Guitar StringMeaningsofsoundqualitymaybedifferentfromthepoints of viewbetweenengineersandmusicians. Thissection tries to build a bridge for the engineers to the territory of music.When designing or improvinga musicalinstrument, anengineer must know the physicsof music. On theotherhand, to fullyappreciate the theoryof music, a musician needs to know the physics behind the music. We will useaguitar string to demonstrate someof the physics of musicin this section and Section12.5. For those students who are notinterested inmusictheoryat all, youcanread only the rst two subsections(11.4-1and 11.4-2)and skip the restof this section.On the other hand, if you want to introduce this article to a friend who does not have enough background in modal analyses, he can skip the rst two subsections and jump to 11.4-3 directly.11.4-2Perform Modal AnalysisLaunch the Workbench.Create a System.Save the project as "String."Drag-and-drop a analysissystem to the cellofthe system. Inthe , makesurethe material properties for the are consistent with those of the guitar string. Enter the DesignModeler(using as lengthunit), create asketch anduse the sketch to create a line body of 1.0 m.Create a circular cross section of radius 0.14 mm, and associate the line body with the cross section. Before starting up , don'tforgetto turnoninthe (7.1-7[2]). In the ,specifyenvironmentconditionsunderthe[1]:a[2],a [3], a [4], and a [5].Note that we suppressed all rigid body modes.Section 11.4Exercise:Guitar String 39911.4-4Just Tuning System1Whydo somenotessoundpleasingtoourearswhenplayedtogether,whileothersdonot? We knowfromthe experience that whentwo notes have asimple frequencyratio, theysoundharmonious with each other.The simpler the ratio, the more harmonious it sounds.The details will be explained in Section 11.4-6.For now, we simply believe it. In Western music, an 8-tone musical scale has traditionally been used.When learning to sing, we identify the eight tones in the scale by the syllables do, re, mi, fa, sol, la, ti, do.For a C-major scale in a piano, there are 8 white keys from a C to the higher pitch of C [1].The two C's has a frequency ratio of 2:1, and are said to be an octaveapart.Ifwe play two notes an octave apart, theysound very similar.Infact, we often have difcultytellingthe difference between two notes having an octave apart.This is because that, except the fundamental harmonicof the lower note, two notes have most of the same higher harmonics. For the following discussion, let's arbitrarilyassume the frequencyof the lower pitch C as 1. (Ina modern piano, the middle C has a frequencyof 261.63 Hz; see the table in Section 11.4-5.)Then the frequency of the higher pitch C is 2.Beforebeingreplacedbythe"equaltemperament"(Section11.4-5)intheearly20thcentury, the"justtuning" systems prevailin the musicworld.In ajust tuning musicsystem, the frequencies ofthe notes betweenthe 2 C's are chosen according to the "simple ratio" rule, in order to be harmonious to each other.The result is as shown [2].Note that we didn't show the frequency ratios for the black keys (the semitones) to simplify our discussion. Now, you can appreciate that if we play the notes do and sol together, the sound is pleasing to our ears, since they have the simplest frequency ratio between 1 and 2.You also can appreciate that the major cord C consists of the notes do, me, sol, do, the simplest frequency ratios (but not too "close," to avoid beats; see Section 11.4-6) between 1 and 2. The problem of the justtuning system is that itis almostimpossible to play in another key.For example, when we play in D key, then the frequency ratio between D and its fth (A) is no longer 3/2.Instead, the frequencyratio is an awkward 40/27; the two notes are not harmonious enough any more. C D E F G A B C do re mi fa sol la ti do C# D# F# G# A# C# Db Eb Gb Ab Bb Db 1 98 54 43 32 53 158 211.4-5Twelve-Tone Equally Tempered Tuning System2Modern Western music is dominated by a 12-tone equally tempered tuning system, or simply equal temperament.The idea istocompromise onthefrequencyratiosbetweenthenotes, so thattheycanbe playedindifferentkeys.Inthis system, anoctaveisequallydividedinto12tones(includingsemitones)inlogarithmicscale.Inotherwords,the adjacent tones has a frequency ratio of 2112, or 1.05946.For example, the frequencyratio between the C# and the C is 2112; the frequencyratio between the A and the lower C is 25 12.According to this idea, frequencies ofthe notes can be calculated and listed in the table below.For comparison, we also list the frequencies of the notes in the just tuning system.The data in the table are plotted into a chart as shown [1, 2].The compromised frequencies are close enough to the just tuning system that most of musicians are satised with this system for centreis.[1] There are 8 notes across an octave.[2] The frequency ratios in a "just tuning" system.410 Chapter 12Structural Dynamics Recognizing that the damping is small for astructure and the globalbehavior is similar regardless of the sources ofdamping, the Workbenchassumesthatthe hysterisdampingforce isproportionalto the velocityof thestructural displacement, the same as the viscous damping, FD = c x(5)However, we cannot characterize a material using a dampingcoefcient c.As mentionedin the end of Section 12.1-2, thedampingcoefcientc is not an intrinsicproperty of amaterial.To lteroutother factors, such as geometry, we need more elaboration.Eq. (2)shows how the coefcient c relates to the mass m and stiffness k for the case of single degree of freedom; for cases of multiple degrees of freedom, the relation is notso simple.In engineeringpractice, an efcientwaytocharacterizeamaterial is proposingamathematicsform withparametersandthendeterminingthe parameters using datatting.We assume thatthe coefcientc is alinear combination of the mass m and the stiffness k, that is, c = m+ k(6)Now, theparameters and are usedto characterizethedampingpropertyofamaterial. Thestudentsmay wonderwhydon'twejustassume c =mk ,whichwouldbe closerto theformof Eq. (2),andcharacterizethe materialbyasingleparameter. Thereasonisthat, ingeneral, we are dealingwithmultipledegrees offreedom system, where m and k are matrices, and the simple relation of Eq. (2) doesn't exist. Using c = 2m inEq. (2)and k = m2inEq. 12.1-2(5), Eq. (6)canberewrittenintermsof frequencyand damping ratio, 2 =+ 2 (7)Ifwe can make asingle materialspecimen andmeasurethe damping ratios

i underdifferent excitationfrequencies

i, ormakeseveralmaterial specimens of different sizes, andmeasurethe damping ratios

iunder theirrespective fundamental frequencies

i, or a combination of the above ideas, then we can evaluate the material parametersand by a standard data tting procedure. In the core of ANSYS, it does implementthe idea of Eq. (6).Using the APDL commands2, you can input aglobal value (using ALPHAD command) and aglobal value (usingBETAD command). Youalso can input value for each material as its property (using MP, DAMP). In the current version ofWorkbench, you cannotinputa global value (itassumes = 0 ), but it allows you to input a globalvalue [7].It also allows you to input avalue for each material as a material property [8, 9].[7] A global beta value can be input here.[8] Beta value as a material property can be included.[9] When included, the beta value of the material can be input here.414 Chapter 12Structural Dynamics Section 12.2Step-by-Step:Lifting ForkIn Section 4.3, we builta model for alifting fork andglass.The lifting fork [1]is usedin an LCD factory to handle the glass panel[2].The fork is modeled as solid body and the glass as surface body.The glass panel is so unprecedentedly large that the engineers concern about its vertical deections during the dynamic handling. The fork is made ofsteelwith adensityof 7850 kg/m3, Young's modulus of 200GPa, andPoisson's ratio of 0.3.The glass has a density of 2370 kg/m3, Young's modulus of 70 GPa, and Poisson's ratio of 0.22. In thissection we will perform astaticstructural simulationrst, to evaluate the vertical deection of the glass panelunder the gravitational force.This is acritical when determiningtheclearance of the processingmachine [3]. Duringthedynamichandling, theforkacceleratesupwardto6m/sin0.3secondandthendeceleratestostopin another0.3second, causingthe glass panel vibrate [4]. Wewantto knowthe time duration whenthe vibrationis settledto a certain amount so that the glass can be pushedinto the processingmachine [3].We also want to know themaximumstressduringthehandling.Beforethesimulationofthe dynamichandling,wewillperformamodal analysisto obtainthevibrationfrequencyofthesystem. Thisfrequencywillhelpusestimatetheinitialintegration time step.12.2-1About the Lifting Fork12.2-2Resume the Project "Fork"Launch .Open the project "Fork," saved in Section 4.3.[1] The lifting fork.[2] The glass panel.[3] Schematic of the processing machine. [1] Double-click to start up .[4] During the handling, the fork accelerates upward to a velocity of 6 m/s in 0.3 second, and then decelerates to a stop in another 0.3 second, causing the glass panel vibrate.426 Chapter 12Structural Dynamics Section 12.3Step-by-Step:Two-Story BuildingIn this section, we willdemonstrate the procedure of a harmonic response analysis.The two-storybuilding (Sections 7.3, 11.2) will be used to demonstrate the procedures.Harmonic Response AnalysisAttheendofSection11.2-3,wementionedthattherhythmicloadingontheoormaycauseasafetyissue.Is "dancingon the oor" reallyan issue? Since the building is designedto withstand alive load of 50 psf (lb/ft2), we will assume thatagroupofyoungpeopleof50psfisdancingon aside-spanoordeck[1]to simulate an asymmetric loadingthatwillcausethebuildingsidesway.Thedancingissohardthattheyoungpeoplegenerateavertical harmonic force of 10 psf, that is, the loading uctuates from 40 psf to 60 psf. Engineersusuallydon'tconsider"dancing"asaseriousissue.Let'slookatamorerealisticengineering consideration.Imaginethatanelectricmotororanyrotatorymachineisinstalledontheoordeck[1]. The operational speed of the machine is 3000 rpm.When started up, the machine's speed increases from zero up to 3000 rpm. Arethevibrationscausedbytherotatorymachineanissue?Inthissection,wewillperformaharmonic response analysis to answer the above questions.12.3-1About the Two-Story Building12.3-2Perform an Unprestressed Modal AnalysisLaunch .Open the project "Building," saved in Section 11.2.[1] Harmonic loading will apply on this oor deck. [1] In this section, we want to reuse this system.Remember the model in this system has no diagonal members.Section 13.1Basics of Nonlinear Simulations 45913.1-5Force Convergence and Displacement ConvergenceIn the last subsection, we stated that when the residual force FRis smaller than a criterion, then the substep is said to be converged.This statement is not strictly correct.There are at mostfour convergence criteria that can be activated underyourcontrol,namely,forceconvergence[1],displacementconvergence[2],momentconvergence[3],and rotationconvergence[4]. Themomentconvergenceandrotationconvergencecanbeactivatedonlywhenshell elementsorbeamelementsareusedinthemodel. Theseconvergencemonitoringmethodsarealldefaultto , thatis, the Workbenchautomaticallyturns onanyofthem whenitisappropriate. Youmay manually turn off or turn on any of them. Whenyou turnon anyof them, youmayspecifya, a, anda. The criterion is then Criterion = maximum(ToleranceValue,Minimum Reference)The force (or moment) convergence satises when FR< Criterion (1)The displacement (or rotation) convergence satises when D< Criterion (2)Where denotesthe normoftheunderlyingvector,andiscalledaconvergencevalue. The defaultsto ,whichusuallymeansthecurrentmaximumvalue.Forexample,intheexampleofSection 13.1-4,thecurrentmaximumforcevalueis F0,andthecurrentmaximumdisplacementvalueis D0.The defaults to 0.5%.Note that setting up a is to avoid anever-convergent situation when is near zero.[1] You can turn on and set the criterion.[2] You can turn on and set the criterion.[3] When shell elements or beam elements are used, can be activated.[4] When shell elements or beam elements are used, can be activated.Section 13.1Basics of Nonlinear Simulations 46513.1-11Other Advanced Contact SettingsPinball RegionThe pinball is a sphere region, its radius can be dened in .Consider again that a contacting point approaches a target face.Using the contacting point as the center of a pinball, for the nodes on the target face that are within the pinball region, they are considered to be in "near" contact and will be monitored.Nodes outside of the pinball region will not be monitored. If the type is specied, surfaces that have gap smaller than the pinball radius is treated as bonded.Interface TreatmentFor contact type, a large enough pinball radius may allow any gap between contacting faces to be ignore.For or contact types, an initial gap is not automatically ignored, no matter how large the pinball is used, since the gap may represents the geometry. If an initial gap is present [1] and a force is applied, one part may "y away" relative to another part [2] if the initial contact is not established right at the end of the time step. To alleviate situations where a gap (clearance) is modeled but needs to be ignored to establish initial contact for or contact types, the can internally offset the contact surfaces by a specied amount.Note that this treatment is intended for small gaps.Don't apply it in a large gap.Time Step Controls tries to enhance convergence by allowing adjustments of time step size based on changes in contact behavior. By default, contact behavior does not affect auto time stepping, since adjustment of time step based on contact behavior may increase computing time too much. When turning on , Workbench will adjust the time step size based on contact behavior. Update StiffnessBy default, structural stiffness is not updated upon the change of contact behavior, to save the computing time.Turning on enhance convergence, with the cost of computing time. Force[1] An initial gap is present.[2] This part may "y away" relative to another part.466 Chapter 13Nonlinear Simulations 6020 20 40Section 13.2Step-by-Step:Translational Joint1A translationaljointis used to connecttwo machine components, so thattherelativemotion of two components is restrictedtotranslateinaspecicdirection.Conventionally,translationaljointsaredesignedasmechanisms, composedbyparts,betweenwhichtheclearance orinterferenceareinevitable; theyeitherdecreaseprecisionor increase friction.The translational joint discussed in this section is not a mechanism; it is a unitary exible structure, in which no clearance or interference exist. The translational joint [1-4] is made of POM (polyoxymethylene, a plastic polymer), which has a Young's modulus of2 GPaandaPoisson'sratio of0.35. The mostimportantdesignconsideration is thattherigidityoftranslational direction should be much less than all other directions, so that the motion can be restricted in that direction only. Here,wewanttoexplorethegeometricnonlinearityofthestructure:howtheappliedforceincreases nonlinearlywiththetranslationaldisplacement.Forthispurpose,wewillmodelthestructureusinglinebodies entirely.The goal of the simulation is to plot a force-versus-displacement chart.The unit system used in the simulation is mm-s-N.13.2-1About the Translational Joint[3] All connectors have a cross section of 10x10 mm.[1] The translational joint is used to connect two machine components, so that the relative motion of the components is restricted in this direction.[4] A prototype of translational joint.Note that this prototype has no horizontal "wings" on it.[2] All leaf springs have a cross section of 1x10 mm.494 Chapter 13Nonlinear Simulations Section 13.4Exercise:Snap LockThe snap lock consists of two parts: the insert [1] and the prongs [2]; it is fastened when pushed into position [3].The snap lock has a thickness of 5 mm and is made of a plastic material with a Young's modulus of 2.8 GPaand a Poisson's ratio of 0.35.The coefcientof friction between the materials is 0.2.The purpose of the simulation is to nd outthe force required to push the insert into the position and the force required to pull it out. We will modelthe problem as a plane stress problem.Due to the symmetry, only one half of the model is used in the simulation.13.4-1About the Snap Lock 7 20 20 7 10 30 17 7 5 10 5 8 Unit: mm.All fillets has radius of 2 mm.[3] After snapping in.[1] The insert.[2] The prongs.Section 14.1Basics of Nonlinear Materials 513PART B.PLASTICITYStress (Force/Area)Strain (Dimensionless)Plasticitybehavior typicallyoccursin ductile metalssubjectto large deformation.Plastic strain results from slip between planes of atoms duetoshearstresses.Thisdislocationdeformationisa rearrangement of atoms in the crystal structure. IntheWorkbench,atypicalstress-strainrelation,suchas 14.1-2[2], isidealized to the one as shown [1-4]. Thestress-strain curveiscomposedofseveralstraightsegments. Theslopeofthe rstsegmentistheYoung'smodulus[3].Whenthestressis released,thestraindecreaseswithaslopeequaltotheYoung's modulus [4].This implies that ifthe stress/strain state is on the rst segment,thebehavioriselastic:noplasticstrainremainsafter releasingthestress. Thepointattheendoftherstsegmentis called elastic limit, or initial yield point.All points higher than the initial yield pointare calledsubsequent yieldpoints, sincetheyallrepresent yielding state. Astress-strainrelationsuchas[1-4]isnotsufcienttofully dene aplasticitybehavior.There are other questions thatmust be answered: (a) Whatis the yieldcriterion?(b) What is the hardening rule?14.1-3Idealized Stress-Strain Curve for Plasticity14.1-4Yield CriteriaAstress-straincurvesuchas14.1-3[1-4]isusuallyobtainedbyauniaxialtensiletest.Itprovidesaninitialyield strength

yofuniaxialtensiletest.Inthree-dimensionalcases,thestressstateismultiaxial. Accordingtowhat criteria can we say that a stress state reaches a yield state?The Workbench uses von Mises criterion (Section 1.4-5) as theyieldcriterion,thatis, astressstatereachesyieldstatewhenthevonMisesstress

eisequaltothecurrent uniaxial yield strength

y

, or 12o1 o2( )2+ o2 o3( )2+ o3 o1( )2

1]1 = oy(1)The yielding initiallyoccurs when

y

= y, andthe "current" uniaxial yield strength

y

maychange subsequently.As mentionedatthe endof Section 1.4-5, when plottedin the

1

2

3 space, Eq. (1)is a cylindricalsurface aligned with the axis

1 = 2 = 3 andwith aradiusof 2y

. Itis calleda von Mises yield surface[1]. If the stress state is inside the cylinder, no yieldingoccurs. If the stress state is on the surface, yielding occurs.No stress state canexist outside the yield surface.If the stress state is on the surface and the stress state continue to "push" the yield surface outward, the size (radius) or the location of the yield surface will change.The rule that describes how the yield surface changes its size or location is called a hardening rule. The conceptof yieldsurface is worthemphasisagain. Inauniaxialtest, we are talkingabout"yieldpoints" in stress axis.Inabiaxialcase, theyieldingstate forma"yieldline," while in a3D cases, the yieldingstateisa"yield surface."[1] Idealized stress-strain curve.[2] Initial yield point.[3] The stress-strain relation is assumed linear before Yield point, and the slope is the Young's modulus.[4] When the stress is released, the strain decreases with a slope equal to the Young's modulus. 514 Chapter 14Nonlinear Materials

1

2

3

1 = 2 = 314.1-5Hardening RulesTwo hardening rules are implemented in the Workbench: (a) kinematic hardening, and (b) isotropic hardening.It should be noted that, in metal plasticity, hardening behavior is often a mix-up of kinematicand isotropic.Since the Workbench implements only two extremities, you have to choose either one that is suitable to describe your application.Kinematic HardeningThekinematichardeningassumesthat,whenastressstatecontinuesto"push"ayieldsurfaceoutward,theyield surfacewillchangeitslocation, accordingto the "pushdirection," butpreservethesizeoftheyieldsurface.Ina uniaxialtest, Itisequivalenttosaythatthedifference betweenthetensileyieldstrengthandthe compressive yield strength remains a constant of 2y [1]. Kinematic hardening is generally used for small strain, cyclic loading applications.StressStrain

y 2y[1] This is a von Mises yield surface, which is a cylindrical surface aligned with the axis

1 = 2 = 3 and with a radius of 2y

, where

y

is the current yield strength.[1] The kinematic hardening assumes that the difference between tensile yield strength and the compressive yield strength remains a constant of 2y.516 Chapter 14Nonlinear MaterialsPART C.HYPERELASTICITY14.1-7Test Data Needed for HyperelasticityAs mentioned in Section 14.1-2, challenge of implementing nonlinear elastic models comes from that the strain may be as large as 100% or even 200%, such as rubber under stretching. In plasticityor linear elasticity, we use astress-strain curve to describe its behavior, and the stress-strain curve is usuallyobtained bya tensile test.Since onlytensionbehavioris investigated, other behaviors (compression, shearing) mustbe drawn from the tensile test data.In plasticity or linear elasticity, we implicitly made some assumptions: (a) The compressive behavior is symmetric to the tension behavior in the sense that they have the same Young's modulus, and thesamePoisson'sratio. Thesymmetrymaynotbetruewhenthestrainislarge. Wemayneedtoconducta compressive test to assessthe Young'smodulusandthe Poisson's ratio forthe compressive behavior. (b) The shear modulus G is related to the Young's modulus and the Poisson's ratio byEq. 1.2-8(2).Again, this assumption may not be truewhen the strainislarge. We mayneedto conduct a sheartestto assess theshear modulus for describingthe shearingbehavior. (c) We also assume thatthebulk modulus Bisrelatedto the Young's modulusandthe Poisson's ratio by B =E3(1 2)(1)Again, this assumption may not be true when the strain is large.We may need to conduct avolumetric test to assess thebulkmodulusfordescribingthevolumetricbehavior.Notethat,inmanycases,thebulkmodulusisalmost innitelylarge(i.e.,thematerialisincompressible).Forthesecases,weusuallyassumeincompressibilitywithout conducting a volumetric test. Further, when the strain is large, all the moduli (tensile, compressive, shear, and bulk) are no longer constant; they change along stress-strain curves.Nonlinear elasticity with large strain is also called hyperelasticity. As asummary, to describehyperelasticitybehavior, we need following testdata: (a)asetof uniaxial tensile test data, (b) a set of uniaxial compressive testdata, (c) a setof shear test data, and (d) a setof volumetric testdata if the material is compressible. Notethatitispossiblethatasetof testdataisobtainedbysuperposingtwo setsofothertestdata.For example, the set of uniaxial compressive test data can be obtained by adding a set of hydrostatic compressive test data to a set of equibiaxialtensile test data [1-3].Reasons of doingthis are as follows.(a)Biaxial tesile test may be easier to conductthan compressive testinsome testingdevices; (b) For incompressible materials, hydrostaticcompressive test data are trivial: all strains have zero values. An example of test data for hyperelasticity is shown below [4-6], which will be used in of Section 14.3.=+[1] Uniaxial compressive test.[2] Equibiaxial tensile test.[3] Hydrostatic compressive test.Section 14.1Basics of Nonlinear Materials 5170601201802403000 0.2 0.5 0.7Stress (psi)Strain (Dimensionless)14.1-8Strain Energy FunctionRaw test data such as 14.1-7[4-6] are not convenient for internal calculations in the Workbench.Itis usually preferable to use mathematical forms to describe materialbehavior (such asEq. 1.2-8(1)). The ideais to propose mathematical forms with parameters, and determine the parameters that best-t the test data. Since there are three sets oftest data, a rudimentary idea is to propose a mathematical form for each set of data.Thisideaisnotworkablesince,asdiscussedinSection1.2, componentsofastress stateorastrainstatearenot arbitrary values; they must satisfy some relations, such as Eq. 1.2-6(2).For example, given a strain state, it is possible to lookup the mathematicalforms andobtaincomponentsof a"stressstate," which is, however, not necessarilysatisfy the equilibrium relation (Eqs. 1.2-6(2) or 1.2-6(3)); the components may not be a "legal" stress state. We need better ideas. AsmentionedinSection14.1-2, hyperelasticityischaracterizedbythefactthatthestressingcurveandthe unstressingcurvearecoincident(14.1-2[1]):duringthestressingandunstressing,theenergyisconserved,or, equivalently, the stressing and unstressing are path independent.The stress state depends only on the strain state, and vice versa.Theyare independent ofthe stressing/unstressing history.This implies that there exists apotentialenergy function thatdependsonthe state ofthe stress or strain.It reminds us the strain energydensityfunction W, which does depend only on the state of stress or strain.We may propose a mathematical form for the strain energy W = W(ij) (1)And the stress can be calculated from the strain energy using

ij = W

ij (2)The strainstate

ijconsists of 6straincomponents(Eq. 1.2-4(4)). To furthersimplythestrainenergyfunctionand developacoordinate-independentexpression,wemayreplacethe6straincomponents(whicharecoordinate-dependent)with3straininvariants(whichare coordinate-independent). To gofurther, we needmorebackground. Let's refresh some terms in solid mechanics.[4] Uniaxial test data.[6] Shear test data.[5] Equibiaxial test data.Section 15.1Basics of Explicit Dynamics 553Section 15.1Basics of Explicit Dynamics15.1-1Implicit Integration MethodsConsider solving Eq. 12.1-4(1) again, M

D{ }+ C

D{ }+ K

D{ } = F{}Copy of 12.1-4(1)Consider atypicaltime step t = tn+1tn.Let Dn,

Dn, and

Dn be the displacement, velocity, and accelerationat tn, and Dn+1,

Dn+1, and

Dn+1 be those at tn+1.Consider a special case that the acceleration is linear over the time step (i.e.,

Dn =

Dn+1 = 0 ), then, by Taylor series expansions at tn,

Dn+1 =

Dn + t

Dn + t22

Dn(1) Dn+1 = Dn + t Dn + t22

Dn + t36

Dn (2)The quantity

Dn can be approached by

Dn =

Dn+1 Dnt (3)Substitution of Eq. (3) into Eqs. (1) and (2) respectively yields

Dn+1 =

Dn + t2

Dn+1 + Dn( ) (4) Dn+1 = Dn + t Dn + t216 Dn+1 +13 Dn

(5)Eqs. (4) and (5) can be regarded as a special case of the Newmark method,

Dn+1 =

Dn + t Dn+1 + (1)

Dn

(6) Dn+1 = Dn + t Dn +12 t22 Dn+1 + (1 2)

Dn

(7)If you substitute =1 2 and =1 6into Eqs. (6) and (7) respectively, you will obtain Eqs. (4) and (5). Eqs. (6)and(7)areused in analysissystem. Theparametersandare chosen to controlcharacteristics ofthe algorithm such as accuracy, numerical stability, etc.Itis called an implicit method because the calculation of

Dn+1 and Dn+1 requires knowledge of

Dn+1.Thatis, the response at the current time step depends on not only the historical information but also the current information; iterations are needed to solve Eqs. (6) and (7). Calculationof the response at time tn+1 is conceptually depicted in[1-6].Inthe beginning [1], the displacement Dn, velocity

Dn, andacceleration

Dnof the laststepare alreadyknown (Forn = 0, we mayassume

D0 = 0 ). Since

Dn+1 is needed in Eqs. (6) and (7), we use

Dn as an initial gauss of

Dn+1.Knowing Dn,

Dn, and

Dn+1, the quantities

Dn+1 and Dn+1can be calculated according to Eqs. (6) and (7)[2].The next step [3] is to substitute

Dn+1,

Dn+1, and Dn+1into Eq. 12.1-4(1). IfEq. 12.1-4(1) is satised, then the calculationof the response attime tn+1 iscomplete [5], otherwise,

Dn+1 isupdatedandanother iteration isinitiated[6]. Update of

Dn+1 [6] is similarto theNewton-Raphson method described in Section 13.1-4.554 Chapter 15Explicit Dynamics Yes No[1] Given the response of the last step, Dn,

Dn, and

Dn.Use

Dn as an initial gauss of

Dn+1.[2] Calculate

Dn+1 and Dn+1, according to Eqs. (6) and (7).[3] Substitute

Dn+1,

Dn+1, and Dn+1into Eq. 12.1-4(1).[4] Eq. 12.1-4(1) satised?[5] Response of the "current" step becomes that of the "last step."[6] Update

Dn+1.15.1-2Explicit Integration MethodsThe explicit method used in analysis system is based on half-step central differences

Dn =

Dn+12

Dn12t, or

Dn+12 =

Dn12 + Dnt(1)

Dn+12 =Dn+1Dnt, or Dn+1 = Dn +

Dn+12t(2)Eqs. (1)and (2)are called explicit methods because thecalculation of

Dn+12 and Dn+1requires knowledge of historical information only.That is, the response at the current time can be calculated explicitly; no iterations within a time step is needed.Therefore, itis very efcient to complete a time step, also called a cycle.One of the distinct characteristics of the explicit method is that its integration time step needs to be very small in order to achieve an accurate solution. With implicit methods, a typical integration time step is about 0.0001 to 0.01 seconds; a typical simulation time is about 0.1 to 10 seconds, which involves hundreds or thousands of integration time steps. Implicitmethodscanbeusedformostoftransientstructuralsimulations.However,forhighlynonlinear problems, it often fails due to convergence issues; for high-speed impactproblems, the integration time is so small that the computing time becomes intolerable.In such cases, explicit methods are more applicable.Section 15.1Basics of Explicit Dynamics 555[1] Given the initial conditions, D0and

D0.Set n = 0.[3] Calculate element strains and strain rates.[5] Calculate element stresses.[6] Calculate nodal forces.[7] Calculate nodal accelerations

Dn.[8] Calculate nodal velocity

Dn+12.[9] Calculate nodal displacements Dn+1.This completes a cycle. Set n = n + 1.[2] Given Dnand

Dn. [4] Calculate element volume changes and update their mass density. The procedure used in the analysis system is illustrated in [1-9].In the beginning of a cycle [2], the displacement Dn and velocity

Dn of the last cycle are already known.With these information, we can calculate thestrainandstrainrate foreachelement[3],usingtherelationssuchasEqs. 1.3-2(2)and1.2-7(1). Thevolume change foreachelementis then calculated, according to the equations ofstate, and the mass densityis updated[4]. The volumetric information is needed for the calculation of stresses.With these information, the element stresses can becalculated[5]accordingtoaconstitutivemodel,relationbetweenstressesandstrains/strainrates,suchasEq. 1.2-8(1).The stresses are integrated over the elements, and the external loads are added to form the nodal forces Fn [6].The nodal accelerations are then calculated [7] according to

Dn =Fnm +b

(3)where b is thebodyforce(see Eq. 1.2-6(2)), m is thenodal mass, and isthe massdensity. Thenodal velocities at tn+12 are calculated [8] according to Eq. (1) and the nodal displacements at tn+1 are calculated [9] according to Eq. (2). Withexplicitmethods,atypicalintegrationtimestepisabout1nanosecondto1microsecond;atypical simulation time is about 1 millisecond to 1 second, which will need many thousands or millions of cycles. Explicitmethodsisusefulforhigh-speedimpactproblemsandhighlynonlinearproblems.Forlow-speed problems, using explicit methods becomes impractical due to an enormous computing time, since it requires very small integration time steps.Section 15.2Step-by-Step:High-Speed Impact 559Section 15.2Step-by-Step:High-Speed ImpactImagine, duringanexplosion, analuminum pipethatwas blastedawayunder the explosivepressure. Thepipe hita steelsolidcolumn,deformed,andnallytornto fragmentsduetoexcessivestrain[1-6].Inthissection,wewill demonstrate the simulation of this scenario.We will use the default settings as much as possible to demonstrate that a complicated simulation like this can be done in analysis system with just a few input data. Both the aluminum pipe and the steel solid column have a diameter of 50 mm and a length of 200 mm.The steel columnismodeledasarigid bodyandxedinthe space. The aluminum pipehasathickness of 1mm and, when hittingthe pipe, has aspeedof 300 m/s (aboutthe speedof sound).The aluminum is modeled as a bilinear isotropic plasticity material(Section 14.1)usingthe material parametersstored in the with amodication that the tangent modulus is set to zero, i.e., the al