FEA by nitin gokhale

The main weaknesses of CFD are as follows :1. Acceptance of results : Wind tunnel simulation results are usually much accepted in theengineering industry due to a belief that they represent the real world correctly. Turbulencemodeling and the use of RANS, large eddy simulation have offered a lot of success and furtherresearch is improving the technology continuously. Validation studies have been iaken tominimize and eliminate uncertainties and errors which creep in due to a lack of knowledgeand otherwise. Many CFD simulation results are often criticized by academics as insufficientlyexact even though the solutions were found \" satisfactory\" for engineering purposes.2. Software skills & experienced people : CFD results can be erroneous as there aremany problem areas where CFD results do not coincide with the real world results in certaincircumstances.This can be corrected by using experienced people using the software and whocan understand the flow physics and interpret the results in much better way. Sometimes thesize ofthe project modelled is limited by the computing power and the software.Typicalcasesare simulations of flows in stadiums and flows around a number of tall building surroundedby small buildings and trees where atmosphericboundary layer is so important. Wind Tunnelsimulations can better represent such kind of effects as there i s no limitation on the size andcomplexity of the model.Lot of groups are taking care now to use \" best practice guidelines\" to enhance the quality andtrust in industrial CFD. See e.g. ERCOFTACspecial interest group guidelines. It requires a lotofjustification on which level of approximation should be used and a thorough check on theComputational model before one can ensure the validity of the results.Some guidelines for executing a practical CFD analysis :CFD is jokingly calledsometimesas \"Colourful Fluid Dynamics': We must take sufficient careto ensure that the numerical model must represent the physical problem as close as possibleand avoid the\" falling in love with the model\" as otherwise it can lead to a severe discrepancywith what exists in naturel practice and what was solved. lt is always good practice to treat theCFD solution with caution and find out meaningful tests to determine whether the solutionis valid or atleast serves the purpose of analysis.Any CFD project can be thought of consisting of the following steps and we suggest someguidelines based on Our experience. 1. Problem definition with accuracy level desired. 2. The FiniteVolume or FiniteElement mesh generation. 3. Solution step and postprocessing of results in software.1. Probiem definition: It should be remembered al1the time that the purposeof computingis insightand not just generating numbers (or beautiful plots!). Most important istocheck theapplication of resultsthat aregoing to be used.This will dictate the level of accuracydesired andreduce the inappropriate simulation on a too fine mesh or using a complex turbulence model.It is always desirableto have an estimate of the probable accuracy of CFD solutions which canbe obtained by comparison with known validation test case studies. One can also carry out

the sensitivity studies such as grid refinement, flow parameters and report the influences ofassumptions made, boundary conditions apriori so as to clearly mention a standard deviationlimit and confidence level.2. Grid generation considerations : A Very complex geometry can lead t o a distorted niesh thus offering problems with convergenceandquality of results.Aithoughit might be necessary to use asimplegeometry to improve the mesh Quality , it poses the risk of aitering the flow pattern. Better would be to run both cases of with and without simplification to ensure sufficient confidence level if it affects the flow. Whether alternative modeling strategy or type of mesh can be used is another question that need to be answered. The finite volume mesh should be such that it should be fine enough to resolve the flow features. An aspect ratio of <5.0 is recommended. (See the table fortypical mesh quality parameters ). However in boundary layers long thin elernents are acceptable. In turbulent flows, cells should have correct range of y + values if wall functions are being used . Ensure that there are a sufficient cells within a boundary layer thickness of the wall for an adequate representation.Use exponential biasing and beli curve biasing features to represent the boundary layer properly. Resolution of the vertical grid is much important than the horizontal one for boundary layers. Always run initial simulationson a coarser mesh ratherthan too fine. Advantage of this is that al1initial problems with model set up are dealt on a model which converges quickiy thus saving a lot of computational time. It i s always better to use topoiogicaily fitting (body fitted meshes) rather than using Cartesian grids. For flows with heat transfer, use large number of points for representing the thermal boundary layer. Grid sensitivity studies should be carried out to show that solution is grid independent or how much the solution changed due to grid refinement. Expert users can also use advanced tools such as dynamic mesh adaptivity rather than using conventional practice of doubling the entire mesh.3. Solution and post processing checks : Convergence of the solution to steady state andchecking the physics as solution develops are two most important things for the CFD anaiyst.The residual history which is a global measure of the error plotted against iteration numberin a suitable norm (L'or maximum norm) gives a very convenient indication of the numericalconvergence of the model. A necessary condition is that residuals should be sufficiently low(min. of le-06 rather than 1e~'O)and that it should remain constant from iteration to iteration.There should not be sudden spikes or oscillations in the residue plot as they indicate that themesh is too coarse or inappropriate mesh type was used.Sensitivity studies should be carried out to understand whether the correct level of physicalapproximation was used or not . There can be a number of possibiiities such as 2-D vs. 3-D,

Steady vs. Unsteady, Viscid vs. Inviscid, choice of a numerical sclieme (second order vs. firstorder), effect of boundary conditions of pressure and velocity on solution. Here the expertisehelp of CFD specialist play a very important role. A good understanding of physics of flowis one of the few ways of protecting the analyst from presenting non physical or incorrectsolution. The checks that can beuseful during or attheendofthe run togive a basic confidenceare proper flow entrance at inlet and outlets, mas8 conservation, sensible pressure drops andtemperature, velocity profiles.The tendency of just giving colourful contour plots should be avoided. It is well accepted thatCFD solutions generate a huge amount of data but the analyst must present the x-y plotssuch as those representing the fluid and thermal boundary layer, pressure vs. length of theairfoil which is much more physically meaningful. Particle tracing animations can also be ofimmense help.We now present here a detailed CFD ProjectTrackingSheet along with a solution sheet details. CFD PROJECT TRACKINGSHEET



Co~nputationaFlluid Dynamics SOLUTION VALIDATION: 1. Hand Calculations 2 . ExperimentalData 3. Pr€--validated Simulations SOLUTION ISSUESIF ANY : 16.16Typical Applications of Computational Fluid Dynamics in Various Industries: There are lot of applications of CFD and some typical applicationsare as follows : AerospaceEngineering: 1. To determine flow structures over wings, rudders. 2. Toestimate the drag and the lift coefficients. 3. Propulsion fluid dynamics where we study the interna1 flow and interest is in computing the distortion at compressor face for various points in flight

envelope, computing reacting flows througii combustors.Automobile Engineering : 1. Estimation ofdrag and streamlining of the vehicfe. 2. Rear flow, study of spoilers, impact on vehicle noise. 3. Engine CFD involving reaction chemistry. 4. Underhood flow and thermal management of vehicle. 5. Disc Brake cooling. 6. Radiators. 7. Computation of side mirrornoise wherea pressuredistribution on the structure is first necessary.Civil Engineering :Computing the flow patterns over a tall building and simulating the effect of the atniosphericboundary layer.Flow simulation of metal castings and plastic components:Solidification of castings is a nonilnear transient phenomena inciuding a change of phase withliberation of latent heat from a moving liquid -solid boundary .The influence of the locationof the ingate and the poring rate as weil as the varying rates of heat transfer in different partsof the mould owing to cores, feeders are some of the factors that must be taken care of.CAE simulation of castings models the physics so that important process variables can beidentified and controlled resulting in significant benefits.Tiiis leads to prediction of shrinkageporosity defects, effects of metal fluid flow and solidification. Progressive and directionalsolidification contours can be used to predict the hot spots or the last freezing regions.Alternate designs of castings and the system,thickness map analysis, placement of chilis andgating system can be quickly carried out on available software easily.Casting simulation softwares - Magmasoft, CastCAE, SOLIDCast, NovaFlow/NovaCast, ProCast,PAMCast / SIMULOR, SIMTEC etc.The same technology has also been used to analyse the filling phase of injection mouidingprocess of plastic components. Typical software capabilities of such a software include sizingof the runners to balance the flow in multi-cavity and family mould layouts, determine thebest gate location for a given part design ,prediction of flow front temperatures, pressures, airtrap locations. lnterested readers can refer to [IO]for more details.Plastic simulation softwares: Moldflow, Moldex etc.

Coiiipiitntionol Fluid Dynornics References : Dale A Anderson, John CTannehill and Richard H Pietcher (1984 ): Computational Fluid Mechanicr and Heat Transfer, Hemisphere Publishing Corporation. (Clarsical Referencefor CFD with fantastic description of model convection and diffusion equations dong with Grid Generation and physicr of fluid flow) Pietier Wesseling (2004): Principlesof Computational Fluid Dynamics, SpringerVerlag Joiin D Anderson (1995):Computationai Fluid Dynarnics -The basicswith applications, McGrawHill Joel L Ferrigerand Milovan H Peric (1999): Coniputatianal Methods for Fluid Dynaniics, SpringerVeriag T. J. Chung (2002): Computational Fluid Dynamics, Cambridge University Press .O . C Zienklewicz and RL Taylor (2000). Finite Element Method Vo1.3, Butterworth Heinemann ERCOFTACSIG Best Practice Guidelines for industrial CFD Veirteeg H K and Maialasekara W (1995): An introduction to Cornputational Fluid Dynarnics.The FiniteVolume Method, Longman Group Ltd. Wilcox:Turbulence Modeiing for CFD, DCW Industries www. MoIdflow.com, www.imtechdesign.com

Fatigue Analysis> Fatigue accounts for 90 % o f service failures.> Manufacturer8 give warranty on the components (in terms of kms or years).Fatigue analysis helps in predicting life of the component in design phase itself. Static or dynamic analysis can tell us about stress, displacernent, acceleration etc. but not how long the cornponent will survive.> Many a times static or dynamic analysis predicts location of failure not matching with lab test or field failure and then analyst keep on thinking whether some thing is wrong with boundary conditions or material properties or geometry of the component. But when fatigue analysis is carried out using same static or dynamic results, it reveals correct location of failure.> Failures or crack usually initiates at surface. Life of the component depends on surface condition (likegrinding, induction hardening, shot preening etc.). Static or dynamic analysis can not take in to account these details while fatigue can.17.\"Life of structure when it is subjected to repetitive load\"Failures were observed even after designing the componeiits with maximum stress value well 295

t otigiie A i i d y i i i below yieldlultimate stress. Laterit was concluded that thefailure is because of variation of load with respect to\"time\"not being taken in to account.Tests were then carried out for time varying loads. Results proved that the component fails at values below yield stress when subjected to tinie varying load. lt was also observed that below a specific stress value components were not failing a t all.This stress value was termed as endurance limit. For exampleyield stress for general steel is around 250 Nlmm2 and endurance limit 160 Nlmm2. Objective of fatigue analysis is tocalculate Life of structure when it is subjectedto repetitive load. Doesit meanthat for random load (Le. n o t repetitive) fatigue calculations are not possible. No, No ! Fatigue calculations are possible for any kind of loading. Basic Fatigue test data and formulae are based on constant amplitude loading. Any random load history can be converted to series of constant amplitude sine waves using various techniques like rain flow counting. Life is calculated for each individual constant amplitude load and then superposition of results using minor's rule would lead to fatigue life for random load. Fatigue accounts for 90% of failures in mechanical engineering application. Typically these failures are observed at stress concentration points or welded, bolted joints etc. FEA based fatigue analysis of metals (iron, steel, aluminium etc.) has been well established. But same is not true with non nietals (like polymers etc.). The nonlinear fatigue behaviour is not yet completely understood and is still in research phase. The term\"Fatigue\" was suggested by Frenchengineer Monsieur Poncelet. German language word \"Betriebsfestigkeit\"(operational strenqth) is a better descriptor of Fatigue phenomenon. 1820's - It was recognized that the metal subjected to repetitive or time varying load will fail at a stress much lower tlian that required to cause fracture on a single application of load. -1870's Wohler conducted experiments and proved a safe alternating stress below which failure would not occur. 1880's - Bauschinger developed mirror extensometer capable of measuring strain upto one micro strain. He suggesteda natural elastic limit (measuredin cyclic tests) below which fatigue would not occur. Elastic limit measured from uniaxial test was not found equal to the one predicted by cyclic testThis set basis for monotonic & cyclic yield strength of the material. 1900'5 - Ewing & Humphrey used optical microscopy to study the same region of the specimen a t various stages of the fatigue life. They suggested'To and Fro slip theory'for fatigue failures. -1920's Jekin, suggested \"spring slider modei\"for simuiating the stress strain behaviour of metals & to study cyclic deformation. Griffith published paper on fracture & proved that fatigue failure is because of brittle fracture caused by cyclic growth of a fatigue crack to an unstable length. Moore and Kommerspublished book \"The fatigue of Metals\"based on

- Practicai Finite Elenient Anaiysistheir practical applications for real life problems.This book provided guidelinesfor designapplications.-1950's Coffin and Mansonproved effect of plastic strain on fatigue life.1960's- lrwin and others work helpedin the development of fracture mechanics.-1970'5 Fatigue analysis theory was quite established and commercial softwares eitherbased on FEA or practicallymeasured strains were launchedin the market.17.5 DefinitionsWhat is Durability ,Reliabllity and Fatigue ?Durabiiity, Reliability and Fatigue are often looselyused for describing Fatigue related analysis.There is slight differencein these three terms.Durability describes overall life requirement, like to last for 2 years (or warranty period).Reliability includes a probabilityof failure, such as to have a 95 %chance of survival (if we test100 specimen, 95 will pass and 5 might fail).Fatigueis thefailure caused by applicationof repetitiveload by the processof initiation of cracksand growth.What is Life 7Total -- Crack + CrackLife Initia ' '.ife Growth Life Crack growthlife - Ductile material /' Brittle material $What is the criteria for transition from crackinitiation t o crackgrowth life?Life till crack of the size 2 mm detected is crack initiation iife and remaininglife after detection ofcrack is crack propagation or crack growthlife.

Fatigue Analysis .High cyclefatigue S - NCurve: lEndurance strength 1 -LoWcyclefatigue- Life (abscissa)is alwaysplotted on log scale whitealternatingstress on either linear or log.Low Cycle Fatigue (LCF) : Life of component is less than 100000 cycles, applicable for heavyduty application loading,.High CycleFatlgue (HCFI: Component subjectto less sever loads and life > I O 5cycles.InfiniteLife: Stress levelbelow which material neverfailsis known as endurancelimit or fatiguelimit. Never fails or infinite life is a relative term. For steel, test is stopped after 2 * 10QycIes (incase if till then failure is not detected) and said to have infinite life.This is the point where S-Ncurve slop changes and it becornesparallelto x-axis.Unlike steel, non ferrousalloyshaveno specific endurance limit (S-Ncurve neverbecomeparallelto x-axis). Pseudo-endurancelimit for thesematerials isstressvalue correspondinqt olife= 5x1@cycles (something similai. to proportionality limit for brittle materials).S-N curve shown aboveis basedon constant amplitude rotating bendingtest (Shaft subjectedtopure alternatingbending stress). Similar test couldbeconductedfor tension, compression, shearand torsional stress. Bending fatigue strength is higher than tension / compressionand torsionalfatigue sfrength is the lowest.DamageandEnduranceFactor of Safety% Damage is calculated for Low Cycle Fatigue (LCF)applications.Damage= n/N = no. of cycles applied/Total lifeDamage < 1 -=,safeDamage > 1 fail

9 EnduranceFactor of safety is calculated for High Cycle Fatigue (HCF) applications.Endurancefactor of safety =Endurance strengthlFE stressEndurancefactor of safety < 1 3 fail=Endurance factor of safety > 1 safe17.6Various Approaches in Fatigue Analysis hriî#uahwlyrlr lExperlmental data based Finlte Eiement analyslsbasedn-code, FEMFAT StrainPhysical prototype is necessaiy, FEMFAT, MSC FATIGUE, LMS,Expenswetest set up FE SAFE etc. - No physicalprototype required Many iterationscouldbe carriedout at less cort as well lesstimeS@ss L i k Straln Llfe rpriiru ReyonançeeffeaMCiS=h Approach haclureMwhantcsHcghcycle ihtigue lnput data for Low cyctefatigue LEFM, EPFM fatigue calculationsTotal Life is dynamic analysis Crackinitiationlife Rate of crack growth results based onFi~tfatigueanaiysis Transient, Oevelopedin 1960% Life leitmethod to be Frequencydomaindeveloped Eiasticandplastic Could be used in or Power Spectral strains combinationwith Density inputStress and strain elastic strain ilfeapproachto StrainVs. reversais predlct total lifeUses stressVr No.ofcyclesplot (5-N curvel Data (E-N cuwel Static Fatigue - DynamicFatigue Exc~tation frequency > 113 rd. Whenthe excitationfrequencylsvery fundamental frequency less than fundamentaifrequency. . Vibrationfatigue.. No resonanceeffect. - input data is dynamlc stress results Inputdata for fatigueanalysisis static Acnirateloaddata(variationwrt time or frequency) should be specified- stress results Time is not important for input load &excitations ln theform offulior half cycle 1s sufficlent

My Company does n o t have any Fatigue analysis software b u t 1 wan't t o have approximateidea about the life of component. Can 1achieve i t using linear static results?Yes, for basic calculations one can use S-N curve. But the resuits wouid be approximate as it cannot take in to account weiding, bolted joints, localizedeffects etc.For exaniple, Say static stress resuits for a steel component = 260 N Imm2andapplication of ioadis alternating in nature. Amplitude stress for this case = 1260-(-260))/ 2 = 260. Plot the stress onS-N curve for steel as shown below.Alteinating Life for stress 260 N/rnm2= 3V05cyclesStress 260 3*105 No of cyclesStress life or S-N method was the first method used for fatigue calcuiation. it was the standardfatigue design method for 100 years (before developments of other methods like Strain life andLEFM).Advantages of stress life approach: Easy to use and simple approach based on 5-N Curve (aisoknown as Wohier diagram) i.e. alternating stress's 'versus No. of cycles'N:The curve is generated by conducting rotating bending test (constant amplitude, uniaxial ioading). Work very weli for high cycle fatigue (stress within elastic iimit)Limitations: It works with engineering stress and ignores true stress - strain behaviour (treats ail strains as elastic). lt has been proved now that failure is caused due to iocalizedplastic strain and failure mechanisini s to-fro slip.

stress range : Au= orna-%a , \" astress amplitude: = (uM. - on) 12,+meanstress: O, = (_ orn,,,)/2 stress ratio: R = umf/Onrnuamplitude ratio: A = %/arnMean stress considerationThe standard S-N curveused for fatigue calculations i s based on pure alternatingload (constantamplitude). Meanstressfor this test is zero.Presence of meanstressaffectsthe resultsand shouldbe considered in the design. Mean stress would be present for al1the loading conditions otherthanpure alternating. ltisalso generateddue to processeslike rollingor heat treatment, bolt pre-stresses or constant (dead weight) loading applications.Tensile mean stress decreases life whilecompressive meanstress increase the life.Various curves (theories) are in used for considerationof tensile mean stress effect as shown betow. Goodman curve is m a t commonly used due tomathematical simplicltyand slightlyconservative results.Goodman diagram 1AlternatingStress o,t.Compression Tension+Soderberg: 4 / O_ urn/ oy= 1en+Goodman: ua1 O,,, / ou= 1Gerber : oa1 4+ (um10J2= 1

Haigh siagram : (aaand am)vs. LifeBy plotting amplitude and meanstress on above graph one can easily know the range of fatiguelife. Same graph could be dispiayed in sliyhtly different format as below (this diagram is alsoknown as master diayram). In the past material fatigue properties used to be represented bymaster diayram.

1 rractical FiniteElement AnalysisMiner's Rule:Formulas discussed so far were basedon the assumption of constant amplitude loading. Miner'sruie i s used for calculating the damage for variable amplitude loading. +Total damage= nl/ NI nJN, + n& n = applied cycles N =cycles t o faiiureDifferent types o f loads:1. Constant: Bolt torque, Pressure, Shrinkfit etc.

2. Pulsating: Name pulsating is derived from pulse (theone which doctors check when we areill).3. Alternating:lmaginea rod being puiled as well as pushedat equal tinie interval. Loadvaryingas: Zero - Maximum-Zero - Minimum - Zero.4. RandomFactors affecting fatigue analysis:i Foliowing factors will reduce value of endurance limit: - Tensile mean stress - Large section size - Large test specimen size shows less endurance limit - Rough surface finish - Macliined surfaces have better iife than the casting parts. - Chrome and nickel plating - Decarburization(due to forging and hot rolling): Decarburizationis 108s of carbon atoms from the surface, causes low strength & tensile residual stresses - Stress concentration, geometrical discontinuity - Corrosive Environment

P Following factors will increasevalueof endurance limit: - Compressiveresidualstress - Nitriding - Flame and induction hardening - Carburisation - Shot peening - Cold rolling - Rolled threads have higher strength than the cut or machined threadsFatigue results are very sensitive to FE-mesh and staticldynamicstressvalues:Followingfigure shows effect of rnesh densityand element type on fatigue analysis. Convergedstatic or dynamic stresses should be usedas input for fatiq-ueanalysis. Çub-modelinq- could alsohelp in achieving better results.l;a,X & M = 0.w 2. Fine tefromode1 3. Corne hexsubmodel4. Finehexrubmodel ix =bosedonrerulfsof 1 5Ft 5.Fine hexmadd bosed foccuratererultr) Moterioi typ:Nodulorcostiron, UT54W MPo Laading:PulsatingbendinaFuil - EnduronreFactorofSnfetyCourtery: FEMFATSofhwre(Engineering CenferSteyrGmbH&Co KG,AustriaJ> Whenloadis doubled, what i s the eïfect on fatigue life ?Consider plate with holefixed atone end and subjected to load on other. Plate dimensions: 100x 100 x 1 mm, Holediameter:12 mm. Material- Steel, out= 450 N/mm2,oy=240 Nlmm2.

FatigueAflolysis Total Force: LoodCare 1-333333 N Lo#dCase2 - 666.66NLife calculations for Load =6666.66 N: Applied cycles = 1E5 cycles,damage = n 1N => Life N= 1EOS 1damageLife = le05 10.0102 = 9.80E06 cyclesStatic stress doubled when load was doubled but fatigue life reduced by a factor of 30,510 !> Alternating load is more severthan pulsating load:Above problem was solved for load = 6666.667 N for alternating and pulsating load cycles.Results show alternatingloading is more sever than pulsating

> Better surface finish meansbetter fatigue life:Crack initiates at outer surface, and better surface finish results in better fatigue life. Please notejust machininga cast surface shows around 1.37 times betterfatigue life. Similarlyshot peeninginducescompressive residual stresses and is recommended.9 What type of stress (max. principal or vonMisesor max. shear stress) is used for fatigue calculations.Commercial softwares provide option for max. principal, vonMises as well as max. shear stress.Fatigue calculations are based on absolute max. principal stress or signed vonMises or signedmax. shear stress.Signed principal stress or absolute principal stress:This term is commoniy used in fatigueanalysis. Fatigue calculationsare basedon amplitude and meanstress. Ithas been observed thatif the calculations are just based on only maximum principal stress or only minimum principalstress then stress range (_ - om,Jis less and leads to higher fatigue life. Remedy is to findmax. value out of the two at a point over given period of time and then find the stress range oramplitude and mean stress based on this data(saymax. principal stress a t a nodeat time 1 sec.+is 250 and min. principal is - 400 then absolute principal stress = -400, collect the data over aperiodof say 10 sec and then find stress range based on max. and min absolutestress values outof the 10)Signed vonMises stress: vonMises and max. shear stress values are always positive. If thesevalues are used for fatigue calculations then stress range wouid be reduced to half resulting inhigher fatigue life. Remedy is tofind sign of absoiute principal stressat the point a t a given timeinstance and assignitto correspondingvalue of vonMisesor max. shear stress. Say absolute max.principal stress is -300attime 2 sec andcorresponding vonMisesstress at time= 2 sec is 315 thensigned vonMises stress would be-315.In general max. principal stress is recommended over vonMises and max. shear. vonMises andmax. shear stress are not directional i.e. directionof crack propagationcould be better answeredby usingmax. principalstress.> For fatigue calculations what kind of stress i s recommended: average stress or unaverage, element (centroidal)or nodal stressesNodal stresses are recommended over elemental similarly unaverage over average. In somesituations this might lead to lesser fatigue life than actual but from design point of view it isalways safer and recommended.

Fatigue Aiialysis train Life Approach Strain life approach is also known as Crack Initiation approach or Local Stress Strain approachor CriticalLocation approach Stress life approach is based on stress, Strain iife approach is based on strain. lt is now well accepted that fatigue is strain & not stress controlled. This method calculates crack initiation life. It is based on strain history at a point where failure is likely to occur. = Consideration for plastic strain = Recommended for Low Cycle Fatigue In the High Cycle Fatigue region, stress and strain levels are low and they are linearly related. However, when stress is higher than yield (lowcycle fatigue),strain based approach. which takes in to account plasticity and non linear relation between stress - strain, gives better result. Research has shown that crack always initiates in the plastic region & damage is dependent on plastic deformation or strain. In strain life approach the plastic strain or deformation is directly measured and quantified. Strain StrainAway from crack location At cracktip (Elastic(Linear Elartic) - Plastic1Away from the crack tip structure remains elastic and stress value below yield. At crack tip dueto geometrical discontinuity, stress may go in elastic - plastic region giving rise to plastic strainsand reversals as shown in the figure.S-N (alternating stressvs. life) curve is generated via rotating bending test is the basis for stresslife approach. while E - N (Alternating strain Vs. Life) is the base for strain life caiculation. Smoothspecimen i s loaded in fix grip and subjected to tension-compression loading.Test is conductedin strain control environment (component's deformation and original length kept constant, sothat the strain i s controlled)

E - N Curve:Total strain ( E,) = Elastic strain ( E ) + Plastic strain ( EJ El = ((of(Nl)b 1E l + kt (2 NI 1')2 NI= Number of reversals to faiiurec= Regression slop calledfatigue ductility exponentDifferencein reversaiandcycle?1 cycle=? reversals,reversalis morecommonlyused withstrainhistory (hysteresisloop,counting)whiie cycles are more common with stress histories and stress life approach.Steps for Strain - Life calculations1) Material properties obtained from smooth specimen strain controlled test (cyclic stress strain data & strain life data)2) Stress Strain history at the critical location3) Cycle counting4) Meanstress correction5 ) DamagecalculationsElastic and plastic strainsTotal strain (E,) = Eiastic strain (EJ +Plastic strain ( EJE~=OIEPlastic strain as per Ramberg -0sgood empirical formulaEp= (O1 K)\"\"K = Strength coefficient, n=Strainhardening exponent

Stress-strain (Hysteresis) Loop AB -Strerr ronge dç-Totolrtrainronge E A -~Elorricrlroin ronge ~ A p P I o r t i c r t r o i nrange nrea ioiide the bop-dii~ipoted energyper unit volume (plastic work donelHow t o determine plastic stress 1strain at critical area !Strain life method requires local stresses and strains (plasticity consideration) for fatiguecalculations.This couid be achieved by Conduct Test : Strain gauge measurement at the critical location. Non Linear Analysis : Elastic - Plastic finite element analysis (non linear solver) Linear Analysis and Neuber's rule: Assumptionto linearstatic anaiysis is stress-strain curve is straight line even after crossing the yield point, usuallythis leadstovery high and unrealistic stresses, Neuber's rule helps in determining equivalent non linear stress - strain from linear FEA results.

i Practical Finite Eiement Anaiysis-Elastic Plastic correction : Neuber's equationK,'=KeXKSK. = Theoretical (basedon geometryshape, linear elastic behaviour) stress concentration.K: = True strain concentrationfactor = Notch (plastic) strain 1Nominal strainKo = True stress concentration factor =Notch (plastic)stress I Nominal stress17.9 FractureMechanics ApproachFracture mechanics is used for calculating \"remaining life\" after crack initiation i.e. crackpropagation life. There are two methods namely LEFM (Linear Elastic Fracture Mechanics)and EPFM (Elastic Plastic Fracture Mechanics). For Automobile industries usually detection ofcrack itself is considered as failure and the subject is more important for Aeroplanes designsand maintenance. We will just discuss the basics now and reader is requested to refer booksmentionedin the referencesforfurther details.In strength of materials we assume material is free from al1the defects (and cracks), in fracturemechanics the starting point itself is to assume presence of a finite length crack.Strength of materials deals with stresses (normal& shear) developed due to various forces andmoments while fracture mechanics is al1 about calculating stress intensity factor and crackgrowthrate for basic three crack openingmodes as shown below. Modesof CmckOpening'Mode I Mode2 Mode 3 SlidingMode TeoringModeOpening Made

Fatigue Atiolysis Stress intensity factor 'K' = Y o ( r ~a)\"' (Knownas Paris Erdogan Equation) Y - compliance (geometry) function, a- crack length, o = nominal stress Rate of crack propagation: da/dN = C (AK)\" da/dN= current rate of crack propagation, C & m materialconst. 17.10 Cycle Countin In real life components are very rarely subjected to pure constant amplitude loading. But problem is most of the available data & empirical formulas are based on constant amplitude loading. So the question is how to use this data for solving real life problem subjected random or variable amplitude loading. This i s achieved by cycle counting. Converting random load history to a number of constant amplitude events is termed as cycle counting. Different methods for cycle counting 1) Level crossing counting 2) Peak counting 3) Simple- range counting 4) Rain flow counting Level crossing counting Real Life variableanipiirudedara LeveiCrorring Method Divide Strain /Stress axis in to number of level (increments)312

. A count for each time a positively sloped portion of strain history crosses a level A count for each time a negatively sloped portion of strain history crosses a level Countsarecombined toformcompletedcyclesasfollows(frommost damaging combination of counts) : First form a largest possible cycle followed by next largest possible cycle & soPeak counting Real Lifevariable amplitude data LevelCrorrRig Merhad Based on peak counts Le. max. & min. stress / strain values Axis divided in to number of levels(increments) Positions of al1max. (peak) strain & ali min. (valley)strain values are tabulated. Countsarecombinedtoform completed cyciesasfollows(frommostdamagingcombination of counts):First form a largest possible cycle by combining largest peak and smallest valiey followed by next largest possible cycle & so on.Limitation of early methods of counting No consideration of order in which cycles are applied. Due to nonlinear relation between stress & strain (plastic behaviour) e.g.

FatigueAnalysis CycleA Cycle B StqrtingPoint TensileLondlng Starting Point CompressiveLoadhg CycfeC cycle O Smnllnmplitudeat the beginning Lorgeamplitude at thebeginningCounting as per above methods will show exactly samecyclesand same damage(fatigue life) forA, B, C & D. But in practice different lifesare observed for above cases.Rain flow countingJapanese engineers Matsuishi& Endowere carrying out research on cycle counting and fatigue.They got idea of this method while watching flow of rain flowing through Pagodaroof houseandhence its named as rain flow counting. Cycles i4 7i 2-3 5-6 E, Counrlng ofHystersisLoop E~Draw Stress 1Strain time history with t h e axis oriented vertically and greatest magnitude ofstrain value at the start and end of strain history curve (this step eliminates counting of halfcycles)

- Practicai Finite Element Analysis1) A flow of rain begins at each valley and peak (strainreversal) and allowed to flow unless a) The rain began at local peak falls opposite a local max. point greater than that from whichit originated. b) The rain began at valley and falls opposite a local min. point greater than that from which i t originated. C) It comeacrossprevious rain flow. i. Rain flows from pt. 1 over points 2 & 4 and continues to the end of history since noneof the conditionsfor stopping rain flow are satisfied ii.Rain flows from pt. 2 over 3 and stops oppositept. 4 since both 2 and 4 are local max. and the magnitude of 4 > 2 iii. Rain flows from pt. 3 and must stop upon meeting the rain flow from pt. 1 (and continuing through 2).There are 3 cycles of constant amplitude 1-4,2-3,s-6 (closed hysteresis loops) each having itsown meanstress & range value. Cumulative damagecould be determinedby using Miner's ruleD=lJN,+l/N,+I/N,At present manyrain flow techniques are in use like- - Original rain flow method - Range-pair counting - Three point cycle counting - Four point cyclecounting - Hysteresis loopcounting - Race track method - Ordered overallrange counting -- Rangepair range counting Hayes method etc.17.11 Multi-AxialFatigue

FatigueAnalysis Stress lifeapproach,strain lifeapproachandothertopicscoveredsofar werebasedonassumption of uniaxialloadins with constant amplitude cycles. In real life seldom componentsare subjected to thiskind of loading. ~ o sotf the times it's a multiaxial loadingwith variableamplitude. hpular fatigue analysis softwares providea separatemodule to take careof multiaxial applications. Stress Me, strain life or LEFM concentrates only on a singfestress usuallynormal stress. Stress is a tensor & has 9 components. For real life applications more than one of 9 stresses are non zero. e otlnput stress I I I Uni-Axial .Multi Axial Proportional .Multi AxialNon ProporlionalPrincipalplane hx Stresses vary in simple Direction of Principal proportion so that plane wry during theexistingtheories ihke directions of the principal cycle &fundonof tirne.stressor strain lifeapproach plane rernains constant Criticalplane approach with tirne. . Equivalentstress approach like theoriesof faiiuresin staticanaiysisLimitations ofpro~ortionalloadinotheorv:Proportionalloading was the only method used for multi axial fatiguecalculationsbefore 70's.Ithas following limitations- There is no consideration fora'fatiguebeing a directional process: Damage (crack) takes place on particularplane.- No considerationfor phaseangle.- Whether or not the loading is proportional and how much it deviates from proportional can be determined by observing load variation with time, stress ratios & principal stress direction.Non-proportional loading:Though there are several theories for non proportional loading but most commonly used infatigue analysis softwares is\"Criticalplane approach Itrecognize fatigue as directional processandcalculates damage for al1possibleplanes Lat Say 100intervals) &the worst or critical plane isreported.17.12 Welding AnalysisMany a times static or dynamic analysis predicts a location of failure not matching with actualfield or test failure. In particular this probiem is quite common when failure is a t weld locations.For exampleconsiderwheel disc and rim analysis, static analysis will always show failure a t eitherbolt locations or vent holes (ellipticalholes) but never at the weld a t disc and rim junction. But

=Practical Finite Eiement Analysisin real life failures are usually observed at welding location and we keep on thinking what iswrong with analysis. Whether constraints are wrong or loading. But when fatigue analysis usingweld modulei s carriedout based on same statiddynamicanalysis as input data, it clearly showsfailurea t welding.SPOT weld fatigue analysisCommercial softwares providesspecial provision for spot and arc weld fatigue analysis.We will be carrying out a simple exercise to show \"Spot weld is stronger in shear and weak innormal loading? Emudelength =MOmm No. ofspots =5 Spor piich =40 mm Startingspot =N)mm from edge TotalFarce= TOWNSpot; Normalstress ' Shearrtreress? o r h d a nspot Normollwdonspot(Courtesy: FEMFATSohvare, EngineernigCenterSteyr GmbH &Co KG, Austria)In the original mesh spots are represented by beam elements. Prior to linear static analysis spotweld elementswere speciallypre-processedin FEMFAT.Thepre-processor creates specialpatternof elements with appropriate materialproperties around the spots.

Fatigue Anolysis Spot welds are stronger in shear and weak in normal (tension, compression, bending) loading. Many a times just rearranging spots (say orientation of spot changed by 90 degree) works well and solve the problem. Arc Welding: Static or dynamic analysis cannot take into account various aspects of welding such as type of welding, heat zone, surfacefinishetc. Forexampleabuttjoint betweentwo small thickness parts (via shell mesh at the mid plane) will be representedas below It does not differentiatebetween various joint types like square or single V, double V, Single U, double U etc. and hence analysis result will be the same for all. But this is not the case in real life.

Fatatigu analysis softwares providesprovision far mostofthe variablesand henm@es fealisticanswes.17.1 3 CAE (Fatigue) and Test Data Correlation Courte3y: FEMFATSafhvare (Engineering CentererSfeyrGmbH&CoKG, AusfriolWhat do you feel, isthis correlation acceptable ?In static analysis, correlation is said to established if difference is < 15 %. For Fatigue analysisacceptance criteria is factor of 3 (i.e. if test result i s 3*105cycles and fatigue analysis softwareis predicting life in the range of IO5to 9*105 cycles). Acceptable difference may go as high asfactor of 10 when loading, material properties and locaiized effects such as welding not takenin to consideration properly. Fatigue is quite a complicated phenomenon and even two testresults (if we conduct test on two identical specimen) won't match. If the test is conducted onseveralidentical components then a factor of 3 is observed (standarddeviation, scattereddata).Accuracy of fatigue analysis dependson1. Meshing ( coarse mesh would lead to lower life, convergence of stress 1 strain result i s recommended before exportingit for fatigue anaiysis)2. Load data and boundaryconditions3. Materialproperties4. Process effects and residual stresses, localized effects likeweld, boit, riveted joints etc.lmporting load history data in fatigue analysis sotwarerFollowingare the common formats supported by mostof the commercial softwares RPC ASCII File ADAMS Request file Tec Math ASCII file DlAdemdata file ADAMS spread sheet FIC file RPC binary file n CodeDAC file Rainflow load spectrum (TechMath-ASCII-RFM)

Fotigiie Annlysis References J.A. Bannantine, J.J. Corner, J. L. Handrock, Fundamentais of Metal Fatigue anaiysii, Prentice Hall Dr.N.W.M.Bishop, Dr.F.Sherrat:FiniteElementBasedFatigueCalculations,NAFEMS-TheinternationalAssociation for the Engineering Analysis community Yung-LILee, Jwo Pan, Richard Hathaway, Mark Barkey: FatigueTesthg and analysis -Theory and Practice, Elsevier Publication Tore Dahlberg, Anders Ekberg : FailureFracture Fatigue, An Introduction, Oveoeas Press (India) Pvt. Ltd. G. E. Dieter: Mechanical Metailurgy, McGraw Hill D. Broek: Elementary Engineering Fracture Mechanics, Martinus Nijhoff,The Hague FEMFAT4.6 Help. Atlas of Stress -5train Curves - ASM International,The Materials Information Society Atlas of Fatigue Curves - ed. by H. E. Bayer, 1986, ISBN081702142

lmoge5ource:Alloir Colendor2005 Courrery :Muhindro & MohindroLtd. At M&M HypetWorkr oridrS Dynoorrurrdfur v rruolvolidorian for Impact Anoiyrir Thlr imogereprerentr the FF modelof the SCORPIOvehicle modeliedin tiyperhlerh forconourtmg O uorh/impacl onolyrir with 15 DYNA HyperWorksPmcerrMonoger ls uredas thefrome work foroutomofing key CAEprocesserPast few decades have seen an increasing application of CAE for simulation of crashphenomenon particularly due to the development of high computing machines and parallelcomputing techniques. The increasein safety standards canbe attributed to the improvementof structural crashworthinessperformancethrough FiniteElement Analysis.The effect of crash and impacts on structures is one problem and the second one which is ofprime importance is the safety of occupants. We find that occupant safety simulation offerstoday reasonably accurate results which can Save a lot of testing time and overall designcycle t h e . Today's dummies have closer dynamic properties and also include the correctload carrying capacity to allow interaction with the structure as compared to earlier dummymodels which didn't take into account loads entering into the body. The CAE developmentfor these application was delayed due to unavailability of high end computing power andit can be said that such simulations are barely 20 years old. Although there has been atremendous increase in application of softwares to solve problems related to automotiveindustry, aerospace industry and drop test of components, still the best is yet to come andresearch has been very active in al1aspects of this application ranging from basic physicalphenomena understanding to development of efficient numerical algorithms and finally thedevelopment of a general purposesoftware which can solvealmost al1 kinds of problems. Wenow find the simulation of tests prescribedby the ECE and FMVSS standards and regulationsquitecommon application.Other area widely used is the bird strike impact analysis and jetengine blade containment analysis pertaining to FAR in aerospaceindustry.

CrashAnalysis Themain aim of this chapteris to introduce basics of structural crashworthiness simulations in terms of the solution methodology and discuss some of the practical simulations performed in the industry. 183 What do We Solve iir StructuralCrashworthiness ? Theseare basically the equilibriumequationsof transient dynamicsasdescribedin the chapter 3 and these are written below as follows: Where M is the mass matrix, C is the damping matrix and K is the stiffness matrix. All that we do is just determine the evolution of the basic quantities such as displacement, velocityand acceleration given initial conditions on displacement and velocity with respect to time. Ali other quantities can be derived from these and most important are the element stresses, plastic strains, contact forces and the energies such as kinetic, potential energy and overall energy absorption characteristics popularlycalled as energy management. Most softwares would commonly solve the dynamic equilibrium equation in an implicit way but the most popular way that shouldbe used for highly nonlinear problems is to use explicit time integrationscheme such as a central difference scheme. There are several advantages of such a procedure and the most important is that it leads to an algorithm which can be easily programmed, does not require any matrix inversion procedure and further is extremely suitablefor a fast parallelcomputing Methodology. The explicit philosophy is used in softwares such as LSDYNA , MSC - DYTRAN , MADYMO, ABAQUS Explicit,PAMCRASH, RADIOSS. The implicit methodsare Beta Newmarkintegrationschemeand theHilbert-HughesTaylor and common softwares using these are LSDYNA-Implicit, ABAQUS -Standard, Radioss - lmplicit 18.3 Transient Dynamics Solution Methodology When solving dynamic problems with FiniteElement Method, it must be remembered that we use FEM only for the spatial discretization and the temporal (Time discretization is always by using the FiniteDifference Method.This approach is called as the Semi Discrete Galerkin as the space t h e finite element concept was a total failure. We divide the total response t h e into much smaller time intervals called t h e steps or increments . The equilibrium equations are +solved and the valueof unknowns are determined a t (t A t) based on the knowledge of their values at t h e t.

RADiOSS is o finite-elemenf solver technology for explicrf orrmpircif onolysis. Leveroging o wide ronge of formulatianssuch os Logrongian, Eulerion, Arbrtrary Euler-Logtonge(ALE)and Smooth Hydrndynomic Particles (SPHl RADlOSS 1s orobust solver soiufron for mechonicoi, structuml, flud andfluid-structure rnteroction problems for rtatfc, dynamic orvonrientloading conditions.Explicit methodsare those in which the informationattime step n+l can be obtained in termsof previous time steps and there is no dependence on the current time step.lmplicit methods are those where the information at time step n+l is dependent on previoustime levels ( n ,n-1 ...)and the current i.e. n+l.See the application for a scalar waveequationin one dimension as given in the chapter on CFDand you will understandthis in a better way. Most common explicit methods are the centraldifferencescheme and implicit codes use betaNewmark integration scheme. Ampleliteratureon these methods for scalar equations, their stability as understood through von-Neumannand characteristic roots and hence we don'twish to present the same. Theinterestedreader isreferred to Belytschko et.al (1). We will discuss about what kind of applications these methodsare suited to and what are the reasonsfor using them.Explicit time integration schemes :The equationsof motion can be written down as follows:where'n'represents a time level index.Physically this means + +Inertialforce Damping force Stiffness force = External force ...................(.3.)Using second order accurateExplicitCentralDifference Operator,

Substituting the above equations into equation (1 ), we get 1 1 + -[Ml [ 2 i U ) n - ( U l o . , l + - [Cl {U] \"., ................................ (6) At' 2ntIftheproblem is linearthen {KI is evaluated ateachandeverytimestep.There isnocomputationaladvantage as still we shall have to invert the matrix present on LHS of equation ( 5 ) .Mass matrixcan be madediagonal/lumped rather than using a consistentone by using standardrow/ column sum lumping techniquesas described in most of the text books of FEA such as RDCook et.al. ( 8 1 whereas thedamping matrix has to be madediagonal by suitableapproximationsas otherwise the classical damping matrix i s usually non- symmetric. If the problem doesn'tinvolve damping then the solution technique as given by Eq. (51 is straightforward without anymatrix inversion but does require data on initial conditions which can be obtained as follows.WhereMost of the numerical algorithms use a two step formulation rather than the above procedureand this doesn't require any initial condition on the diagonality of IC 1 the damping matrix. {G}n+%- {G} n - % ...............................(.10) At n

The equations of motion is recasted as follows:Using the above equations,The method can be started by using the initial conditions and theapproximation I U ),and orby usingThe von-Neumann stability condition for the scheme yields a conditional stability governed bythe following condition : At< -2( m -5)................................(16) wn>axWhere Z is the fraction of critical damping a t the highest undamped frequency of the wholemodel.We can also write thefrequency in terms of the material property and the characteristic length

of the element and we get : length (Ax)(At)mh Elastic wave speed (C) ................................(17)Another convenient forni to remember is to recast the above equation as CAt - < 1 or Courant Number < 1 ............................. (18) AxAnd thiscorrespondstothestability condition ofthescalarwaveequationasgivenanddiscussedin details in the chapter of CFD.This is also called ascourant Stability orCFL (Courant -Fredrichsand Levy) conditions. The physical interpretation of this is that't'niust be small enough suchthat the information doesn't propagate across more than one element per time step.The elastic speed does depends on the wave propagation characteristics and different elementscan have different wave speeds.The reader is referred to the concerned software user's manualsfor the details as this has concern with the wave propagation in elastic medium. Most softwareuse a time step scaling factor to the maximum time step that can be allowed and this i s usuallyof the order of 0.9 with a provision to change this factor for highly nonlinear problems.If we use reduced integration elements, then effectively we are adding the hourglass force onthe RHS of the equilibrium equation as illustrated by thefollowing equation. lnertia Force + Damping force + Interna1force = External force + Hourglass forceCare must be taken to solve the system in such a way that it ensures a sufficient accuracy andthe fact that it doesn't mask any prominent physical behaviour of the system e.g. a fractureor breakage of the physical system completely masked by the hourglass effects.A quickclieck on the minimum mesh length that corresponds to a failure time step of 1 microsec for steel (Young'sModulus = 210 Gpa, Density= 7800 kg/m3) would tell you that it i s ofthe order of 5 m m . Thus you should set the global length in a pre processor software to 5 mmsize for crash simulation. Using a larger element length is always a better practice.Explicit schemes are economical and efficient for problems involving high frequency loads asthese problems require very small time steps to capture the associated phenomena and theschemes are stable for small time steps only.Other Popular schemes are Runge Kutta family methods and trapezoidal rule.The lmplicit integration schemes :In lmplicit methods, equilibrium is achieved at each time using an iterative procedure.Thus theaccuracy of the method depends much on the solution procedure and convergence tolerancesspecihed.The common algorithm used is the p - newmark time integration which uses a Taylor- Series

discretization as followsWhere B and y are parameters of the system.The scheme has been shown to be unconditionally stable for 2 p r y r 0.5 ........................................(21)For practical problems; we use the following iterative procedure + F,,,,,,,, NUIn+,)= (FeXJn+..,..............................(22) n+lBy substituting we get[Ml + F,,,,,,,, ((UIn+,)=(Fe,)\"+, [Ml ................................(25)- (U),,+,The same can rewritten asThis is solved by an iterative procedure as

Crash Ana/ysisYieldingA tolerance is specified for 'U' in various noms so as to ensure a correct convergence. Eachincrement in implicit consists of a minimum one iteration, in practise always greater than one.The computationalcost is thusproportional to the model size. In explicit method, there are noiterationsand the time marching cost is proportionalto the model size.lmplicit schemesshowa fastspeedonly uptosmall to moderatelargeproblemswhefeasexplicitschemes become muchfaster towardsthe high end application.Thecomparisonis shown in thefigure. Rqian I Camputational rmt in terms ofCPUHme, rpeedlmplicit schemesareefficientforstructural dynamics problemswith lowto moderatefrequency.content and since these are of longer duration, one can use comparatively large t h e stepsTypically al1static solution methodsuse an implicit procedure.Some other implicit methods used for computationare Houbolt,WilsonTheta, ParkStifflystablemethod and Hilbert-Hughes-Taylor scheme .18.4 lncreasingthe Speedof Explicit Methodsfor QuasistaticSimulationin statics, we totallyignore the dynamiceffects due to inertialforce.The majordifficulty in usinga dynamic code for getting a static simulation result are : 1. Presence of acceleration / inertial force term. Displacements are derived from solution of acceieration. Whereas in Statics, displacementsare primaryvariable and are directly obtained.

Practical FiniteElementAnalysis B2. Static stress analysis gives you stresses but dynamic code gives you rather stress waves. The effect of stress waves reflection a t boundaries are always present in a dynamic solution.A suddenapplicationof loads will result in stress waves of much higher magnitude than staticstresses, and hence ramping is necessary.From the Courant Stability Condition, 6length length 4Ë At s s Elasticwave speedIn order to increase the time step, we have following options: 1. lncreasethe elernentlength. ( Always preferredatleast for initial rum) 2. lncrease the density of the component thereby increasing the mass of component. 3. Decrease elasticity moduli of the material.Option 2 is called as the \" Mass Scaling \". Although this increases the computational speedconsiderably,caremust be taken to see that a bound is put on the percentageof mass increase.Secondly it should not change the physics of the problem i.e. m a s increase is allowed fordeformablebodies but if appliedto rigid bodies then results will be different than the originalmodel.Option 3 is calledas the\"Time scaling\"but this is not used muchinpractise.Conventionalway i sto use lmplicit codes for quasi Static simulationbut the major problem is issues of convergenceand efficiencyof the code when friction Isliding contact is present. Usually it has been foundthat the contacts work better in an explicit code rather than an implicit code.Mass scaling is much accepted technique among the CAE Community to increase the speed oftheexplicit software.18.5 Comparlsonof Explicit vs. lmplicitMethods

Crash Anaiysis SeatingSystemCrash Analysis (Covrtesy:TataJohnronControlsAutwnoriveLtd.,lndia) Contact treatment and algorithms: Contact between two components occurs when they try to come towards each other during the deformation process. When the bodies touch, a force is transmitted across the common interface. Usually there is friction present between the surfaces and hence forces normal and tangential to the surface will be created. This gives rise to a contact pressure and a shear stress. A high end computational process is required due to severe discontinuity with respect to boundary conditions. This discontinuity in boundary conditions arises as the boundary condition of no interpenetrationis applied only when the two parts are in contact.This should be removed when the components try to separate. Thus the boundary conditions resembte an ON /OFF switch and the software must detect contact and clearance continuously as the simulation proceeds. If there were no contacts defined then the components would simpiy penetrate into each other and this i s unphysical.The software by default doesn't have any



d.Contact pressure: Thisis thenormalcomponentofforceper unitarea at thecontact interface.There is absolutely no limit in the contact formulation on magnitude of contact pressure. Thesurfaces separate from each other when contact pressure becomes zero or negative and thekinematic constraint is removed . It is the dramatic change in contact pressure when a contactcondition changes from \" Open\"toWClosed\" making it difficult to analyse..e Friction : Stick and Slip condition : The most commonly used model is Coulomb frictionand this characterises the frictional behaviour between the surfacesusing coefficient of friction.The contacting interface will not slip ( slide relatively wlth respect to each other ) until shearstresses across the interface equals the limiting frictional shear stress. (This is similar to motionimpending in applied mechanics i.e. unless and until frictional force becomes equal to appliedforce, you can not have any motion)The frictional characteristics of contact are shown in thefigure below .The solid line indicates the behaviour of Coulomb friction model. A zero relativemotion /slip i s calledas \"Stick\"or \" Weld\" condition in some softwares.Sticking or slipping can result in convergence problems during contact simulation. So frictionshould be included in the contact analysis only when it has a significant influence on response.Also softwares allow an elastic slip which i s indicated by the dotted line. This is a small amount ofrelative motion between the surfaces that occurs when the surfaces would be sticking .There are two methods to analyse contact: a.The Penalty StiffnessFormulation in which stiffSprings are used between the contacting bodies. b.The Lagrange Multiplier which handles thekinematic constraint in a strict way but on the other hand i s computationally very costly.

Contact types :Theseareclassifiedwith respectto the elasticityand the typeofcontactingbodies. Thefollowingare possible ways : a. Rigid to DeformablelFlexible b. Deformableto DeformableRigid to Deformable : When one of the contacting bodies is highly stiffer than the other,then it can be considered as a rigid one and other deformable.The rigid body undergoes nodeformation and stresses. This is well suited to a softer material coming in contact with a hardmaterial.c. Deformable t o Deformable : In this the two contacting bodies have the same order ofelasticity and a typical example is a boited Range, the analysis of which has been given a t theend of Nonlinear AnalysisChapter.Classificationbased on the type of the contacting bodies:a. Point topointcontact :Iftheinteractiontakesplaceat a pointfor both thecontactingbodiesthenthe problem is classifiedas a point to point contact. Most commonlyused softwares haveGAP elements to define this and usually these types of contact problems involvesmall relativesticking .Typical example is a pipe whip model.Point to Point contact can also be used to solve a surface to surface interaction if the nodes ofthe two line up, relative sliding deformationi s negligibleand the deflection of the two surfacesis small e.g. interferenceor shrink fit.b. Point t o Surface contact: ln this type. the interaction takes place a t a point on one bodyand on a surface in another contacting body. This is very rare case of contact simulation and.an example can be a spinning top impacting a rigid floor or two beams contactingeach otherat a beam tipc. Surface to Surface contact : This is the most common way to handle al1 the engineeringproblemsproviding better results on contact pressureand frictional stress. it also supports largedeformation with a significant amount of slidingand friction efficiently. Examplesof this can bemany typically al1metalforming simulation and forging or deep drawing.Typical contact algorithm :The flow chart in the following figure shows a typical algorithm used for contact analysis. Itexamines the state of al1 contact pairs at the start of each increment to establish whetherslave nodesare open or ciosed. If a node is closed,it determines whetherit is slidingor sticking.A constraint i s applied for each closed node and removes the constraints from any where,contact state changes from closed to open. The procedure is repeated until the iteration iscompleted with nochange in contact states and this iterationbecomes thefirst equilibriumiteration. The algorithm then checks for normal equilibrium convergence checks. If theconvergence check fails,another iteration i s performed.The entire process is repeated for al1subsequentload steps.

Crash Anolysis18.6 Typlcal Issues in Contact Analysisa. lmpropersurfacenormals: This is themostcommon error and usuallythesoftwareshavean Automatic option to correct the orientation. Master surface normals should point towardthe slave surface.b. Compatible elements: Ail elements underlying a surface must be compatible. They mustbe either shellsor solids or of the same order (eitherfirst or second), al1deformable or al1rigid.That is you cannot have slaves where one is shell of first order and adjoining element i s asecond order. (a) Incorrect (b)correct

- PracticalFiniteElement Analysisc. No. of increments :Contact analysis requirea careful, logical approach. Dividethe analysisinto severalsteps if necessaryand apply loading slowlyto makesurethat the contactconditionsare established.d. Boundary conditions : There should not be any constraints applied for nodes on thecontacting surfaces in the direction of contact. If friction is included in the model, there shouldnot be any boundarycondition in any direction for the nodeson contact surfaces.(a) incorrect (bl Correcte. Extending the rigid surface : For Rigid to deformable contact simulation, rigid surfaceshould be large enough to ensure that slave nodes don't slide off and fall behind a rigidsurface. This will lead to a convergence failure. Extending the rigid surface as shown in thefigure can improve the convergence. Extended Master SurfacemSlave Surfae FR3 (a) Incorrect (b) CorrectExtending theMosterSuriacef. Mesh density of slave surface:Slave mesh must be refined enough to interact with al1important features on the rigid or master surface. Otherwise the master surface will just

penetrate into the slave as shown in the figure.(a) Incorrect (b) C o r r e c t Meih Denrity ofSlave Surfaceg. Proper selectlon of Master & Slave :The slave and master surface definitions should beselectedcarefully. The slave surface should be morefinely meshedthan the master.lfthe meshdensities are of the same order, slave surface is the one with lesser ElasticModuli.h. Noinitial penetration :There shouldnot beany initial penetration as shown in thefigure.Thisgives rise t o a simulation with sliding energy present at the initial conditions which i s physicallywrong and hence it should bestrictly seen that the penetration checkis applied through thepre processor as well as by carrying out the zero second termination run to cross check.

Practical FiniteEiement Analysis Marter Line dMasterLineM \ l\I[IIi Slave Line I 1 gave Line(a) incorrect (blCorrect No ininalpenerrariDni.Avoid sharpcorners :Avoid sharpcornersinrigidsurfacesastheycan posesevereconvergenceproblems. Better to have smooth rigid surfaces by providing fillets in the sharp corners. (a) Incorrect (bf Correct18.7 Some Aspects of Shell Element TechnologyThis topic has been discussed till depth in the books written by Belytschko et.al (1) and alsodescribed in theoretical manualsof the softwaressuchas the LS DYNA (2 & 3). We will make noattempt to dlscuss the advantagesof each overthe other. Althoughthe default shellformulationmay be fast due to several simplificationsrequiredfor the speed, there canbe severedrawbackssuchas ignoring change in warpageand one can easilyobserve the failure for standard twistedbeam with end load.