1. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-1 MAR120, Section 3, December 2001 SECTION 3 ANALYSIS PROCEDURES

2. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-2 MAR120, Section 3, December 2001 TABLE OF CONTENTS SectionPage 3.0 Analysis Procedures Overview ………………………………………………………………………… ………………………………....3-3 Distinction Between Perturbation (Linear) And General (Nonlinear) Procedures …………………………..3-4 Structural Procedures Supported By MSC.Patran Marc Preference………………………………………….3-7 MSC.Marc Analysis Procedures…………………………………………………………………………………..3-8 Structural Analysis Procedures: Linear Static………………………………………………………………….3-9 Structural Analysis Procedures: Nonlinear Static……………………………………………………………..3-10 Structural Analysis Procedures Normal Modes……………………………………………………………….3-13 Structural Analysis Procedures: Frequency Response……………………………………………………….3-16 Structural Analysis Procedures: Buckling………………………………………………………………………3-17 Buckling: Eigenvalue Problem Formulation…………………………………………….……………………….3-18 Structural Analysis Procedures: Direct Linear Transient……………………………………………………..3-19 Structural Analysis Procedures: Modal Linear Transient……………………………………………………..3-20 Structural Analysis Procedures: Nonlinear Transient Dynamics…………………………………………….3-21 Structural Analysis Procedures Frequency Response………………………………………………………….3-23 Structural Analysis Procedures: Response Spectrum………………………………………………………..3-24 Structural Analysis Procedures: Creep (Time Dependent Plasticity)………………………………………..3-25 Thermal Analysis Procedures: Steady State Heat Transfer………………………………………………….3-26 Thermal Analysis Procedures: Transient Heat Transfer……………………………………………………...3-27 More On General And Perturbation Procedures………………………………………………………………...3-28

3. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-3 MAR120, Section 3, December 2001 OVERVIEW Discussion of Available Procedures in MSC.Patran  Structural Procedures (MSC.Patran 2001)  Thermal Procedures (MSC.Patran 2001)  Coupled Thermal-Structural Procedures (MSC.Patran 2002) Highlights of Theoretical Features Graphical example of analysis for each procedure

4. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-4 MAR120, Section 3, December 2001 If the structure (1) is deformed after applying a general (nonlinear) procedure it is said that it has arrived to a new base state (2). A perturbation (linear) procedure leaves the base state unchanged. Thus a modal analysis (for example) of the deformed structure (2) will return different natural frequencies that the same procedure applied to the original structure (1) but leave the base state in (2) unchanged. A later general procedure will use this base state (2). Preloaded (new base state) Undeformed (original base state) The Base State Preloading DISTINCTION BETWEEN PERTURBATION (LINEAR) AND GENERAL (NONLINEAR) PROCEDURES

5. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-5 MAR120, Section 3, December 2001 If a point of the structure reaches the inelastic region, the curve that represents the stress- strain relation will have in general a slope different to the elastic Young modulus. A new base state will in general have different points of the structure at different points in the curve. Typically most areas will remain in the elastic region. A perturbation procedure will use this local tangent to the nonlinear curve as the Young modulus for that part of the structure. As long as the tangent does not separates appreciably from the actual curve, the linear analysis will be correct. Linear Stress-Strain relation Nonlinear Materials in Linear Analysis DISTINCTION BETWEEN PERTURBATION (LINEAR) AND GENERAL (NONLINEAR) PROCEDURES (CONT.) Linear Perturbation about new base state E Nonlinear Stress-Strain relation new base state

6. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-6 MAR120, Section 3, December 2001 If a nonlinear analysis results in contact or a change in contact conditions between two parts and the condition remains in the new base state, a subsequent perturbation procedure will use the contact as established. Nonlinear Contact Conditions in Linear Analysis DISTINCTION BETWEEN PERTURBATION (LINEAR) AND GENERAL (NONLINEAR) PROCEDURES (CONT.) The contact will not change during or as a result of the linear analysis (the base state is not modified by the perturbation) yet the linear procedure accounts for the contact. The natural frequencies will often change dramatically due to contact. Therefore the response to a oscillatory excitation may also change dramatically if the new natural frequencies are present in the excitation.

7. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-7 MAR120, Section 3, December 2001  Linear Static  Nonlinear Static  Normal Modes  Euler Buckling  Direct Linear Transient  Modal Linear Transient  Nonlinear Transient  Frequency Response  Spectrum Response  Creep STRUCTURAL PROCEDURES SUPPORTED BY MSC.PATRAN MARC PREFERENCE

8. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-8 MAR120, Section 3, December 2001 MSC.MARC ANALYSIS PROCEDURES Thermal procedures supported by MSC.Patran Marc Preference Steady State Heat Transfer Thermal Heat Transfer

9. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-9 MAR120, Section 3, December 2001 STRUCTURAL ANALYSIS PROCEDURES: LINEAR STATIC Inertia effects are neglected Model response defined by linear elastic stiffness at the base state (the state of deformation and stress at the beginning of the step) For Hyperelastic and Hyperfoam materials:  Base step Contact Conditions cannot change during step

10. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-10 MAR120, Section 3, December 2001 1 2 3 4 STRUCTURAL ANALYSIS PROCEDURES: NONLINEAR STATIC Solves problems where there are one or more of up to the three forms of nonlinearity:  Material Nonlinearity  Geometrical Nonlinearity  Boundary Nonlinearity (Contact) and for which time is not a variable

11. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-11 MAR120, Section 3, December 2001 Incremental-iterative solution with automatic “Adaptive” control or “Fixed” (constant fractional incrementation) control. The increments represent load incrementation rather than time incrementation. Advanced optional user control include various criteria to drive the Adaptive procedure. STRUCTURAL ANALYSIS PROCEDURES: NONLINEAR STATIC (CONT)

12. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-12 MAR120, Section 3, December 2001 Basic method based on Newton- Raphson and related techniques (discussed later) Additional methods based in the Arclenth method (discussed later) Generally, coupled nonlinear equations for each degree of freedom Basic statement of equilibrium: Balance between internal forces {I} and external forces {P}: {K}{u} - {P} = 0 {I} = [K]{u} Generally, coupled nonlinear equations for each degree of freedom STRUCTURAL ANALYSIS PROCEDURES: NONLINEAR STATIC (CONT)

13. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-13 MAR120, Section 3, December 2001 n Uses eigenvalue techniques to extract the frequencies of vibration of the structure n Generally, (-  2 [M] +  [C] + [K]){  } = 0 where:  = circular frequency  = mode of vibration associated to  [M],[C],[K] = Mass, Damping, and Stiffness Matrices Basic method based on Newton-Raphson STRUCTURAL ANALYSIS PROCEDURES: NORMAL MODES

14. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-14 MAR120, Section 3, December 2001 In this metal forming example a tube is (step 1) deformed plastically, then (step 2) unloaded. Subsequently one may (step 3) extract the natural frequencies and associated modes. In the same job we may (step 4) apply other loads on a nonlinear static procedure thus deforming the tube past its previous base state and possibly without producing additional plastic strain yet reaching another base state. Without unloading we would (step5) extract the natural frequencies corresponding to the base state reached at the end of the previous nonlinear step 4. STRUCTURAL ANALYSIS PROCEDURES: NORMAL MODES (CONT) Uses the stiffness of the base step, so that small vibrations of a preload condition (previous step) can be modeled.

15. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-15 MAR120, Section 3, December 2001 STRUCTURAL ANALYSIS PROCEDURES: NORMAL MODES (CONT) A number of “Linear Perturbation” procedures require having run a previous Natural Frequency step. These are:  Modal Linear Transient  Frequency Response  Spectrum Response  Viscoelastic (Frequency Domain) In MSC.MARC, [C]=0 for the purpose of computing   Methods available: Inverse Power Sweep and Lanczos algorithms  Example: One- and Two-DOF Spring-Mass-Dashpot Systems

16. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-16 MAR120, Section 3, December 2001 For the cantilever beam shown here (figure at top), and a cosine (Harmonic) forcing function presented as a tip load, the Frequency Response procedure finds a solution that matches the theory. The first natural frequency is 325 Hz Plotting the tip displacement magnitude as a function of the frequency of the harmonic excitation (figure at bottom) one can clearly see the static solution and the resonance when the first natural frequency is reached. Resonance Static solution  = 0 STRUCTURAL ANALYSIS PROCEDURES: FREQUENCY RESPONSE

17. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-17 MAR120, Section 3, December 2001 buckling STRUCTURAL ANALYSIS PROCEDURES: BUCKLING Classical “Euler” buckling. Eigenvalue and critical load estimates. “Stiff” Structures. (they carry loads primarily by axial or membrane actions) “Snap-through” problems. (they have large displacements/rotations and small strains)  These should not be solved with this procedure. Instead, use the Nonlinear Static procedure with the RIKS method option.

18. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-18 MAR120, Section 3, December 2001 Bifurcation buckling is useful for “Stiff” structures. The method is not suitable if large geometry changes occur prior to buckling. Material is assumed to be linear elastic before buckling. The method can provide misleading results if the structure is imperfection sensitive. If results are questionable, run a Nonlinear Transient Dynamics analysis or preload the structure with a Nonlinear Static procedure. BUCKLING: EIGENVALUE PROBLEM FORMULATION Structure under “dead load” P 0, stiffness [K 0 ] A “live” load is added, equal to lDP. As long as the response is stiff and linear elastic, the stiffness changes to [K 0 ] + l[DK] This poses the eigenproblem: ([K 0 ] + l[DK]){v} = {0} Nontrivial solutions l cr define the Critical Buckling Load P 0 + l cr DP

19. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-19 MAR120, Section 3, December 2001 STRUCTURAL ANALYSIS PROCEDURES: DIRECT LINEAR TRANSIENT Integrates equations of motion through time Various integration methods available: Newmark, Houbolt, Central Difference, Fast Explicit, Single Step Houbolt

20. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-20 MAR120, Section 3, December 2001 Computationally inexpensive where = Eigenmode = amplitude of mode STRUCTURAL ANALYSIS PROCEDURES: MODAL LINEAR TRANSIENT Provides the model response as a function of time based on a given time dependent loading The number of modes used is a matter of user judgment More Modes = More Accurate Less Modes = Less Expensive

21. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-21 MAR120, Section 3, December 2001 STRUCTURAL ANALYSIS PROCEDURES: NONLINEAR TRANSIENT DYNAMICS Same definitions as for Nonlinear Static, except: Internal forces here include inertial and damping forces, not just stiffness forces

22. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-22 MAR120, Section 3, December 2001 STRUCTURAL ANALYSIS PROCEDURES: NONLINEAR TRANSIENT DYNAMICS (CONT) Same time operators used for Direct Linear Transient, Fixed Increments. In addition, Adaptive Incrementation is available for Newmark, Fast Explicit, and Single Step Houbolt time integration methods.

23. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-23 MAR120, Section 3, December 2001 Excitation must be Stationary, thus and ergodic (different samples of the excitation yield the same time average) STRUCTURAL ANALYSIS PROCEDURES: FREQUENCY RESPONSE Steady State Response to a continuous excitation containing a specified set of frequencies. Requires previous Normal Modes analysis.

24. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-24 MAR120, Section 3, December 2001 STRUCTURAL ANALYSIS PROCEDURES: RESPONSE SPECTRUM Inexpensive approach to estimating the peak response of a model subjected to “base motion”. Behavior is assumed to be linear. total response = relative response excitation + base motion Useful for seismic analyses of buildings Setting up in MSC.Patran requires the input of a non-spatial Field (frequency dependent) representing the Spectrum.

25. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-25 MAR120, Section 3, December 2001 STRUCTURAL ANALYSIS PROCEDURES: CREEP (TIME DEPENDENT PLASTICITY) Analysis of materials described in the CREEP Material form. Explicit and Implicit procedures. Adaptive or Fixed incrementation. Relative and Absolute accuracy control.

26. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-26 MAR120, Section 3, December 2001 A non-physical “time” incrementation is used to allow management of prescribed temperatures and fluxes through the analysis and control the output accordingly THERMAL ANALYSIS PROCEDURES: STEADY STATE HEAT TRANSFER Independent from stress and deformation state May include conduction, boundary convection and radiation May include gap radiation, conductance, and heat generation between contact surfaces May be linear or nonlinear “Steady State” means that the rate of change of temperature is null all over the domain.

27. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-27 MAR120, Section 3, December 2001 THERMAL ANALYSIS PROCEDURES: TRANSIENT HEAT TRANSFER Temperature rate is significant Time incrementation corresponds to physical time. Automatic time incrementation optional user control Transient results may be used for a sequentially coupled thermo-structural transient analysis (the stress and deformation depend on the transient temperature field but the opposite is not necessarily true).

28. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-28 MAR120, Section 3, December 2001 MORE ON GENERAL AND PERTURBATION PROCEDURES A General analysis procedure is one in which nonlinear effects are included. The response is generally nonlinear, but one may obtain a linear response using a nonlinear procedure.  The starting condition of a general step is the ending condition from the last general step  Total time increases throughout the general, nonlinear analysis  Each step also has its own time, which begins at zero in each step  The “step time” may have actual physical meaning or not  Nonlinearity requires us imagining what will happen to properly planned meshing, loading, and sequencing.

29. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-29 MAR120, Section 3, December 2001 Example: Step 1: Nonlinear Static: Apply P = 500 Lb load (with contact condition) Step 2: Natural Frequency extraction Step 3: Nonlinear Static: Heat the beam (Thermal Expansion, changing area of contact) Step 4: Natural Frequency extraction Where does the contact happen? To the right of the circle’s top. A Perturbation analysis procedure is one in which a linear response is computed about a “base state”. The response is always linear, but the base state may be the result of a previous nonlinear step. 1.To the left of the circle’s top ? 2.On the circle’s top ? 3.To the right of the circle’s top ? MORE ON GENERAL AND PERTURBATION PROCEDURES(CONT.)

30. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-30 MAR120, Section 3, December 2001 A Nonlinear Transient Dynamics step may not be interrupted to perform a perturbation analysis. Before performing a perturbation analysis, the structure must be brought into static equilibrium. Nonlinear effects may only be included in the base state for a linear perturbation step MORE ON GENERAL AND PERTURBATION PROCEDURES(CONT.) The starting condition of a perturbation step is the ending condition from the last general step - if there was any - or the undeformed structure. This is the “base state”. The ending condition of a perturbation step will be ignored by subsequent perturbation steps. That is, the structure reverts to the “base state” at the end of the perturbation step. The step time of linear perturbations is never accumulated into total time Procedures that are General:  Nonlinear Static  Nonlinear Transient Dynamic

31. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-31 MAR120, Section 3, December 2001 MORE ON GENERAL AND PERTURBATION PROCEDURES(CONT.) Plasticity and other inelastic effects are ignored during the perturbation Hyperelastic properties will be used by their value at the base state Contact conditions cannot change during the perturbation Frictional slipping is not allowed during the perturbation Purely perturbation analysis cannot be simulated with alternative general procedures but may be preloaded to a modified base state. The following procedures are considered “purely” linear perturbation analyses:  Bifurcation Buckling  Natural Frequency  Modal Dynamics  Response Spectrum  Direct Steady State Dynamics  Modal Steady State Dynamics

32. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S3-32 MAR120, Section 3, December 2001 MORE ON GENERAL AND PERTURBATION PROCEDURES(CONT.) Other analyses may be performed in principle by either general or perturbation procedures, although using alternative procedures  Creep (Requires preloading with Nonlinear Static)  Buckling Collapse using Arc Length method (Nonlinear Static procedure)