abaqus經(jīng)典聲學(xué)分析例題l3-static

1、Buckling, Postbuckling, and Collapse Analysis with ABAQUSLecture 3Static Postbuckling and Snap-Through AnalysisCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.2OverviewIntroductionABAQUS Implementation ABAQUS UsageSnap-Through Problems Postbuckling ProblemsIntr

2、oducing Imperfections for Postbuckling Simulations Postbuckling ExamplesUsage Hints LimitationsSummaryCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSIntroductionCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.4Introduction E

3、igenvalue buckling analysis is useful for the analysis of “stiff” structures. The method is not suitable if large geometry changes occur prior to buckling and can provide very misleading results if the structure is imperfection sensitive. In cases where the eigenvalue buckling procedure is not appli

4、cable or its results are questionable, a fully nonlinear transient analysis is required.A transient analysis can be done dynamically or by addition of viscous forces to the static problem.The disadvantage of such analyses is that it is hard to understand thecharacteristics of the structure after the

5、 load maximum is reached.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.5Introduction To avoid any of the effects of the stabilizing forces, we would like to obtain a solution to the static equilibrium equation without adding such forces. An algorithm is neede

6、d in which the applied loads are adapted automatically. The solution algorithm must solve simultaneously for loads and displacements. As a consequence another quantity must be selectedto measure the progress of the solution. For this we choose the “arc length,” l, which is the length along the stati

7、c equilibrium path in load-displacement space. A form of this method is activated in ABAQUS by adding the RIKS parameter on the *STATIC option.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.6Introduction Static analysis of snap-through and postbuckling problem

8、s with the arc- length method provides valuable information about the characteristics of structures in the unstable regime. The method works well if the equilibrium path in the load-displacement space is smooth and does not branch.Otherwise, convergence and incrementation problems may occur.Generall

9、y, this means that the method should always be applied to imperfect geometriesthe “perfect” structures initial coordinates should be perturbed to create a suitable imperfection.This converts a pure bifurcation behavior into a snap-through problem.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, an

10、d Collapse Analysis with ABAQUSABAQUS ImplementationCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.8ABAQUS Implementation In the Riks method we always deal with proportional loading within an analysis step. The load is assumed to consist of: An initial load ve

11、ctor, P0, which has already been applied at the start of the step and remains constant throughout the step. A load, lP, where P is a nominal load vector and l is the “l(fā)oad proportionality factor” that ABAQUS will find as part of the solution. In the simplest case P0 will be zero and P will be the re

12、sult of distributing, for example, a uniform pressure of unit magnitude onto the structure. In general, P is obtained as the difference between the reference load Prefspecified in the Riks step and the dead load, P0 :P = Pref - P0 .Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Anal

13、ysis with ABAQUSL3.9ABAQUS Implementation The Riks procedure will then provide a series of values of the pressure magnitude l following a static equilibrium path. The Riks method works in a scaled solution space, consisting of the displacement degrees of freedom in the model (rotations are not inclu

14、ded) and the forces (not moments), each scaled to make the dimensions on the displacement and load axes about the same magnitude.Pbe the load P scaled in this For the purpose of this explanation, letumanner and letbe the scaled displacement. The “arc length” is then the length along the equilibrium

15、path in this scaledload-displacement space.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.10ABAQUS Implementation ABAQUS begins an increment with a “target” arc length, Dl. This value is equal to the initial “time increment” specified for the step and is adjus

16、ted if the increment fails to converge. Assume you start an increment from point A0 (see the figure on next page). The value of Dl to advance the solution to point A1 is obtained by solvingK0 v0 = Pand then choosing the value of Dl for the increment so as to satisfy()lv22D v+1= Dl .0000 The value of

17、 Dl is initially suggested by the user and is adjusted automatically based on the convergence rate.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.11ABAQUS Implementation Choosing Dl as described earlier effectively applies Dl along the tangentto the equilibriu

18、m curve at the equilibrium point at the start of the increment. This determines point A1 in the figure below:Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.12ABAQUS Implementation The solution is then corrected onto the equilibrium path in the plane passing th

19、rough A1 and orthogonal to (n 0;1). For each iteration i:(ni ;1).Consider the setWe try to find a correction vector such thatthe new solution is located on this plane:ci= ci + gni ,where ci is the solution to the equilibrium residual equation,K ci= (l0 + Dli ) P- I ,vi is the solution vector to the

20、applied load,Kni= P ,and g is a scaling factor determined automatically by ABAQUS and represents the correction to the load proportionality factor increment.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.13ABAQUS Implementation With a suitable choice for g , w

21、e find a point Ai+1 on the plane through Ai, which is closer to the equilibrium surface. Continuation of this process provides a solution point. During this process the value of l (the load proportionality factor) is computed automatically, so the user has no control over the load magnitude.Hence, “

22、time” can no longer be used and is replaced by arc length.As a result, any effects involving time or strain rate (such as viscous damping, creep, and viscoelasticity) are no longer treated correctly and should not be used.Even the use of history-dependent material models such as rate- independent pl

23、asticity requires careful interpretation, since the unstable static equilibrium path followed by the Riks method is not likely to be the same as the path that the structure would follow in an actual loading.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.14ABAQ

24、US Implementation The Riks procedure involves only a few extra calculations: one additional back substitution in the tangent stiffness matrix and some vector manipulations. Besides being useful for unstable cases, the Riks method can also speed up convergence in ill-conditioned problems that do not

25、exhibit instability.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSABAQUS UsageCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.16ABAQUS Usage The Riks procedure is invoked by including the Riks parameter on the*STATIC option

26、.Since the method is generally used with geometrically nonlinear cases, the NLGEOM parameter is usually included on the *STEP option.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.17ABAQUS Usage A typical input sequence for a postbuckling analysis is*STEP, NLG

27、EOM (apply optional dead load)*STATIC(define the dead load and specify output requests)*END STEP*STEP, NLGEOM, INC= (postbuckling Riks*STATIC, RIKSstep)lend,Dlinit, lperiod,D lmin, D lmax,node, dof,umax(define the reference load and specify output requests)*ENDSTEPCopyright 2004 ABAQUS, Inc.Buckling

28、, Postbuckling, and Collapse Analysis with ABAQUSL3.18ABAQUS Usage The first two entries on the *STATIC, RIKS data line define the initial and total arc lengths associated with the load in this step.*STATIC, RIKSlend,Dlinit, lperiod,D lmin,D lmax,node,dof,umaxDlinitlperiodCopyright 2004 ABAQUS, Inc.

29、Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.19ABAQUS Usage The second two entries are optional and form bounds for the arc length increment, Dl.*STATIC, RIKSlend,Dlinit, lperiod,D lmin,D lmax,node,dof,umaxCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with A

30、BAQUSL3.20ABAQUS Usage The last four optional entries serve as alternate termination criteria.*STATIC, RIKSlend,Dlinit, lperiod,D lmin, D lmax,node, dof,umax lendis provided to terminate thestep when the load exceeds a certain magnitude. node, dof, umax are provided to terminate the step when a part

31、icular displacement component exceeds a given value.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.21ABAQUS Usage ABAQUS will not stop exactly at these values but will stop when the values are exceeded. If none of the above termination criteria is included, AB

32、AQUS will stop when the maximum number of increments is reached or when the solution fails (for example, because of excessive distortion).Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.22ABAQUS Usage Any amplitude references are ignored in the Riks procedure.A

33、ll loads are ramped from the initial (dead load) value to the reference value specified.If the reference load is equal to the dead load, the Riks procedure will fail.The load magnitudes are available as output. The status (.sta) file shows clearly that a step uses the Riks procedure.SUMMARY OF JOBMO

34、NITOR NODE:INFORMATION:1DOF:2STEPINCATTSEVERE DISCON ITERS000000EQUIL ITERSTOTAL ITERSTOTAL TIME/ FREQSTEP TIME/LPFINC OF TIME/LPFDOF MONITORIF RIKS1111111234561111113233333233330.04710.08820.1380.1840.2000.1770.047130.041100.049580.046380.01590-0.02278-0.195-0.397-0.714-1.22-2.00-2.93R R R R R Rfla

35、g indicating that this is a Riks analysislDlCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSSnap-Through ProblemsCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.24Snap-Through Problems The Riks method works very well for snap

36、-through problems.Classical snap-through problems are characterized by a smooth load- displacement curve and do not exhibit branching (bifurcation).As a result, the Riks procedure can solve this kind of problem with ease.Generally, you do not need to take any special precautions to ensure a successf

37、ul analysis. An example of a problem with a smooth load-displacement curve (taken from the ABAQUS Example Problems Manual) is the shallow-arch problem shown on the following pages.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.25Snap-Through ProblemsExample 1:

38、 Shallow circular arch under pressureCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.26Snap-Through ProblemsHistory input for shallow circular arch*step, nlgeom loading*static, riks0.05, 1.0,*dsload, 0.2,0.4arch, p, 5000.*endstepCopyright 2004 ABAQUS, Inc.Buckl

39、ing, Postbuckling, and Collapse Analysis with ABAQUSL3.27Snap-Through ProblemsLoad-displacement curve of shallow archCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSPostbuckling ProblemsCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with

40、ABAQUSL3.29Postbuckling Problems The Riks method can also be used to solve postbuckling problems, both with stable and unstable postbuckling behavior. The exact postbuckling problem cannot be analyzed directly because of the discontinuous response at the point of (bifurcation) buckling. To analyze t

41、he problem, you must turn it into a problem with continuous response instead of bifurcation. The problem can be converted by introducing an initial imperfection in the model so that there is some response in the buckling mode before the critical load is reached. If the imperfection is small, the def

42、ormation will be quite small (relative to the imperfection) below the critical load.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.30Postbuckling Problems The response will grow quickly near the critical load, introducing a rapid change in behavior. Such a rap

43、id transition is difficult to analyze. If the imperfection is large, the postbuckling response will grow steadily before the critical load is reached. The transition into postbuckled behavior will be smooth and relatively easy to analyze. Imperfections are usually introduced as perturbations in the

44、initialgeometry of the model. Imperfections can also be introduced by perturbations in the loads or the boundary conditions. Moreover, imperfections based on linear buckling modes can be useful for analyzing structures that behave inelastically prior to reaching peak load.Copyright 2004 ABAQUS, Inc.

45、Buckling, Postbuckling, and Collapse Analysis with ABAQUSIntroducing Imperfections for Postbuckling SimulationsCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.32Introducing Imperfections for Postbuckling SimulationsImperfections in otherwise “perfect” models ar

46、e necessary to develop models that are suitable for use in postbuckling analyses.Two methods are used to introduce imperfections into a model:1.Geometric imperfections are the most commonly used method.Perturbations in the models initial geometry cause the structure to buckle in the appropriate mann

47、er.Loading imperfections may also be used to ensure that the structure buckles in the appropriate manner.Small fictitious “trigger loads” are used to deform the model so that it buckles correctly.2.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.33Introducing I

48、mperfections for Postbuckling SimulationsGeometric imperfections: They are typically based on previous eigenvalue buckling analyses. A few of the buckling modes are used typically to perturb the geometry.However, the lowest buckling modes are assumed to provide the most critical imperfections, so us

49、ually the lower modes are scaled and added to the perfect geometry to create the perturbed mesh.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.34Introducing Imperfections for Postbuckling Simulations The magnitude of the imperfection should be chosen realistic

50、ally. For example, the size of the imperfections may be determined by manufacturing tolerances. Often the magnitude is chosen as a few percent of a relevant structural dimension such as a beam cross-section or a shell thickness. In shells the imperfection magnitude is typically chosen to be 1%100% o

51、f the thickness. Only the coordinates of the nodes are affected; the nodal normals are notmodified.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.35Introducing Imperfections for Postbuckling SimulationsLoading imperfections: The “trigger loads” should perturb

52、the structure in the expected buckling modes. Typically, these loads are applied as “dead” loads prior to the Riks step so that they have a fixed magnitude. The magnitude must be sufficiently small so that the trigger loads do not affect the overall postbuckling solution.Copyright 2004 ABAQUS, Inc.B

53、uckling, Postbuckling, and Collapse Analysis with ABAQUSL3.36Introducing Imperfections for Postbuckling SimulationsIntroducing geometric imperfections Use the *IMPERFECTION option to introduce geometric imperfections for postbuckling and collapse simulations. The geometric imperfection can be based

54、on nodal displacements written to the results (.fil) file during a previous simulation.The FILE parameter is used to identify the name of the results file from the previous simulation.The STEP parameter must be used to identify the step from the previous analysis containing the results that will def

55、ine the geometric imperfection.An imperfection that uses only a subset of the models nodes can be created by using the optional NSET parameter to identify the subset.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL3.37Introducing Imperfections for Postbuckling Simulations Imperfections can be formed by the superposition of weighted eigenmodes from a previous eigenvalue buckling or frequency extraction analysis. Any number of eigenmodes can be specified a