1. What is analysis?Imagination384,400 kilometers 200 meters"In 1 million to 10 million years theymight be able to make a plane thatwould fly." -The New York Times, 1903Application ModelAbhishek & Jitendra2

2. Methods of analysisMode ofanalysis Analytical &numerical Empirical methodsMethods Upper & Slip Line FiniteFiniteLowerVisioElementary fieldSimilarity plastictheoryElement DifferenceBoundtheorytheoryMethod method (FDM)method method (FEM)Image courtesy: Lecture notes, Fundamental of solving methods,Prof Dr. Ing. G. Hirt Abhishek & Jitendra 3 3. Elementary plasticity theoryApproach:Establishment of kinetics relativeto the process.Establishment of differentialequations suiting the process &the simplification. Abhishek & Jitendra 4 4. Upper & Lower boundary method Method approximates the values of deformingforces to be higher or lower than actual forces.Any estimate of the collapse load of a structure made by equating the rate of the energy dissipation internally to the rate at which external forces do work, in some assumed pattern of the deformation will be greater than or equal to the correct load-W.F. Hosford & R.M CaddellAssumptions of the method: Material being deformed is isotropic &homogenous There is no effect of work hardening William Hosford No friction exists between work piece & toolinterface. Plane strain conditions assumed.Abhishek & Jitendra5 5. Slip-line theoryHere flow pattern from point to point whiledeformation is considered & analyzed. due to frictionlessSlip line refers to the planes of maximumupsetting shear stress which are inclined at 45o to the principle planes. Slip line on due toAssumptions of the method: the Symmetryedge Material being deformed is isotropic &homogenous Symmetry There is no effect of work hardening & strainplanerate. No friction exists between work piece & toolinterface. Plane strain conditions assumed.Image courtesy: Lecture notes, Fundamental of solving methods, Effect of temperature , strain rate, & time Prof Dr. Ing. G. Hirtneglected.Abhishek & Jitendra6 6. ComparisonFEM Material flow analysis &local states of stress & straindescribed. Various boundary conditionscan be applied. Multi-axial stress inconsiderationAnalytical methods Only Global analysis is done. Material homogeneity is assumed. 2-deminsional conditions. Temperature effects neglected.Abhishek & Jitendra 7 7. Finite Element AnalysisA brief history Concept was developed by the works of RichardCourant & Alexander Hrennikoff (early 40s). Idea was originated to solve complex problemsof civil engineering & structural analysis.Richard Courant Idea was promoted by Boeing to compute sweepof airplane wings (mid 50s). M.J Turner & Ray W. Clough articles establishedthe applications of FEA (mid 50s). Idea was also used to compute roof of MunichOlympic stadium (late 60s) Abhishek & JitendraR.W.Clough8 8. Areas of applications Engineering Fluid mechanics Thermodynamics Metal Forming etc Biological Sciences Botany Zoology Archeological Anthropology Paleontology General application Geology Astrophysics Abhishek & Jitendra9 9. Engineering applicationsNumerical assessment of static & seismic behavior ofDepartment of Atomic & Solid state Hochschule Regensburg,the Basilica of Santa Maria allImpruneta (Italy)physics University of CornellBiomechanik Department of electronics & C-blade Forging & Lehrstuhl Numerische Mathematik, Telecommunications, University of Manufacturing Ruprecht-Karls-Universitt, Heidelberg Naples,Italy10 Abhishek & Jitendra 10. Hierarchy of FEMPhysical ProblemEstablish Finite element model ofthe physical problem Solve the problemInterpret the resultAbhishek & Jitendra11 11. Space IncrementationFinite Elements: Every model is sub-divided into finite elements. Their junction points are called as nodes. Model assumes that forces act at nodes & stresses & strain exist at the finite element. Reliability of FEA depends on number of finite elements. Abhishek & Jitendra 12 12. Stiffness MatrixExample: Beams protruding from fixed surfaceuE A 1 1u1F1L1 1 u2 F2 K u F Stiffness Coeffecient Displacement ForceF1F2u1u2 E A 1 1 0 u1F1 F1(u 2u1 )E AL 12 1u2 F2L E A01 1u3 F3 F2(u1 u2 )L Stiffness Matrix 13 13. Space IncrementationElement types: Image courtesy: Lecture notes, Fundamental of solving methods, Prof Dr. Ing. G. Hirt Abhishek & Jitendra14 14. Space IncrementationMeshingNetwork of nodes is called a mesh.There are 2 broad mesh-generation methods. Unstructured( Formedautomatically) A Structured (Formed by grid basedsub-dividing of geometry) BAbhishek & Jitendra 15 15. Space IncrementationMeshing: Accuracy of results always depends on the assumptions. Fine mesh is considered where there are stress & strain gradients. A coarse mesh is used in the areas of reasonably constant stress or areas of interest.Abhishek & Jitendra 16 16. Protocols Gaps are not permitted during meshing. Nodes are numbered sequentially. Abhishek & Jitendra 17 17. Space IncrementationApproaches:LAGRANGES approach Mesh is bound to the material Mesh will be distorted with increasing deformation.Courtesy: FHWA. USAAbhishek & Jitendra 18 18. Space IncrementationEULERS Approach Mesh is fixed & not bound to the material. Material flows through fixed mesh. Abhishek & Jitendra 19 19. Space IncrementationRemeshingWhy is it necessary? Formation of unacceptable shapes due to large local deformations. High relative motion between die surface & deforming material. Large displacement causes computational problems. Difficulties encountered in incorporating die boundary shapes withincrease in relative displacement.To overcome above difficulties, periodic redefining of mesh is necessary Abhishek & Jitendra 21 20. Space IncrementationRemeshing comprises offollowing steps:1. Assignment of new meshsystem to work piece2. Transfer of information(strain, strain rate, &temperature) from the old tothe new mesh throughinterpolation. Image courtesy: emerald.com Abhishek & Jitendra 22 21. SolversFor simulation of metal forming, following 2 solutionsare used: Implicit method ( Stable, iterative, highcomputational effort) Explicit method (conditionally stable, no iteration,less computational effort) Abhishek & Jitendra 23 22. Implicit solvers Studies reveal that this solver isuseful in smaller & 2Dproblems. Each time step or incrementhas to be treated asunconditionally stable process. Large time steps lead to largeriterations & process do notconverge. (Newton Raphson method )Non- linear analysis of reinforced Abhishek & Jitendra concrete beam 24 23. Implicit SolversIn the implicit approach a solution to the set of finiteelement equations involves iteration until a convergencecriterion is satisfied for each increment. Here computation is divided into several calculation timesteps. At the end of each time step(increment) the equilibriumbetween internal & external load must be reached. Else iteration continues.Abhishek & Jitendra 25 24. Explicit solvers The finite element equations in the explicit approach arereformulated as being dynamic. In this form they can be solved directly to determine thesolution at the end of the increment, without iteration.Two methods are followed for time step calculations.Abhishek & Jitendra 26 25. Explicit solversHere largest allowable time step for a stable solution depends on: Highest Eigen frequency occurring (max )in the system Corresponding damping () tm (2/max)* ((1+2)0.5-)Sonic frequency & smallest element Le are estimated as follows: t Le /C with C=(E/)0.5To compensate the disadvantage of extremely small time step, will bereduced through increasing the density or shortening the process time. Abhishek & Jitendra 27 26. Computational time required for explicit/implicit methods(Calculation time)ComplexityEfficiencyImplicitImplicit Explicit ExplicitModel-size Statics StructuralHighly dynamicsdynamicimplicit: Complexity ~ number of freedom degrees x wave frontexplicit: Complexity ~ number of freedom degrees Image courtesy: Lecture notes, Fundamental of solving methods, Prof Dr. Ing. G. Hirt 27. Non-linearities in FEM Following Non-linearities are encountered during the simulations. Geometrical Non-linearity Material Non-linearity Contact variance (Change of boundary conditions) FrictionAbhishek & Jitendra 29 28. Geometrical Non-linearity In practical cases it is not uncommon to encounter strain of magnitude 2 or more due to : Large elongation Large rotationCourtesy: MRF tyres, India Portions of rigid body movementsAbhishek & Jitendra30 29. Geometrical Non linearityHydroforming Operation Tools Upper part TubeLower part In consideration ofGeometrical nonlinearity geometrical nonlinearity neglectedImage courtesy: Lecture notes, Fundamental of solving methods,Prof Dr. Ing. G. Hirt 30. Material Non-linearityOccurs when: Transition of elastic to plasticphase Depends on ,, , CPNote: This non-linearity is important when considering thermal effects z.B hot forming or for calculation of temperature Courtesy: COMSOL, USA increase during forming process.Abhishek & Jitendra 32 31. Material Non-linearity duringtensile testconsideredMaterial-nonlinearity (flow curve) Not consideredkf = kf ( v, v)Initial meshkf = 100 N/mm2 = const.(strain hardened!)Image courtesy: Lecture notes, Fundamental of solving methods,Prof Dr. Ing. G. Hirt 32. Contact Non-linearityChanging contact changes:a.) Mechanical Boundariesb.) Thermal Boundaries.Types of contacts in metal forming1.) Contacts with rigid tools2.) Contacts with deforming tools Courtesy: ICS, Switzerland3.) Self contact Abhishek & Jitendra34 33. FrictionFriction is non-linear. Friction leads to asymmetricalequation system. This increases the calculationcomplexity.Categorization:1.