abaqus經(jīng)典聲學(xué)分析例題l2-eigenvalue

1、Buckling, Postbuckling, and Collapse Analysis with ABAQUSLecture 2Eigenvalue Buckling AnalysisCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.2OverviewIntroductionEigenvalue Problem Formulation ABAQUS UsageClosely Spaced EigenvaluesConcluding RemarksCopyright 2

2、004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSIntroductionCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.4Introduction The stability of structures is a problem that analysts face frequently. These problems predominantly occur in beam

3、 and shell structures. Such stability studies usually require two types of analyses: Eigenvalue buckling analysis Postbuckling or collapse analysis We focus on eigenvalue buckling analysis, which very often is a required step for the more general collapse or load-displacement response analysis. Eige

4、nvalue buckling analysis is used to obtain estimates of the critical load at which the response of a structure will bifurcate, assuming that the response prior to bifurcation is essentially linear. The simplest example is the Euler column, which responds very stiffly to a compressive axial load unti

5、l a critical load is reached, at which point it bends suddenly and exhibits much lower stiffness.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.5Introduction Load-displacement response of an Euler columnCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Co

6、llapse Analysis with ABAQUSL2.6Introduction Deformed configurations of an Euler columnCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.7Introduction The purpose of eigenvalue buckling analysis is to investigate singularities in a linear perturbation of the struc

7、tures stiffness matrix. The resulting estimates will be of value in design only if the linear perturbation is a realistic reflection of the structures response before it buckles. Therefore, eigenvalue buckling is useful for “stiff” structures (structures that exhibit only small, elastic deformations

8、 prior to buckling). In most cases of stiff structures, even when inelastic response may occur before collapse, eigenvalue buckling analysis provides a useful estimate of the collapse mode shape. Only in quite restricted cases (linear elastic, stiff response; no imperfection sensitivity) is it the o

9、nly analysis needed to understand the structures collapse limit.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.8Introduction In many cases the post-buckled response is unstable; the collapse load will then depend strongly on imperfections in the original geome

10、try. This is known as “imperfection sensitivity.” In this case the actual collapse load may be significantly lower than the bifurcation load predicted by eigenvalue buckling analysis. Thus, eigenvalue buckling analysis provides a nonconservative estimate of the structures load carrying capacity. Eve

11、n if the pre-buckling response is stiff and linear elastic, nonlinear load- displacement response analysis (of the imperfect structure) is generally recommended to augment the eigenvalue buckling analysis. This is essential if the structure is imperfection sensitive.Copyright 2004 ABAQUS, Inc.Buckli

12、ng, Postbuckling, and Collapse Analysis with ABAQUSEigenvalue Problem FormulationCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.10Eigenvalue Problem Formulation The objective of an eigenvalue buckling analysis is to find the load level at which the equilibrium

13、 becomes unstableor estimate the maximum load level which the structure can sustain. The load level depends on the stiffness following from the virtual work equation, as described in Lecture 1. The stiffness is dependent on the internal stress and, if the load follows the structure, on the applied l

14、oad.It will be assumed that the loading is conservative, so the stiffness matrix is symmetric.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.11Eigenvalue Problem Formulation First apply a “dead” load, P0. This defines the stiffness of the base state K0effects

15、(even if NLGEOM is not used).which includes preload Now add a “l(fā)ive” load, lDP, where l is the magnitude of the live load being added and DP is the pattern of the live load. As noted earlier, the method works best if the response is linear prior to bifurcation. Thus, as long as the response is stiff

16、 and linear elastic, the stress and, hence, the structural stiffness will change proportionally with l,asK0 + lDK,Stiffness change is proportional to l.andDK = KDs + KDP .Due to incremental loading pattern; made up of two parts: the internal stress and the applied load.Copyright 2004 ABAQUS, Inc.Buc

17、kling, Postbuckling, and Collapse Analysis with ABAQUSL2.12Eigenvalue Problem Formulation We need to find values of l, which provide singularities in this tangent stiffness; this poses the eigenproblem.In other words, a loss of stability occurs when the total stiffness matrix is singular:( K0+ lDK )

18、V = 0.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.13Eigenvalue Problem Formulation Values lcr, which provide nontrivial solutions to this problem, define the critical buckling load as P0 + lcrDP with corresponding buckling mode shapes V . Buckling mode shap

19、es V are normalized vectors, just as modes of free vibration are, and do not represent actual magnitudes of deformation at the critical load. They are often the most useful outcome of the eigenvalue analysis since they predict the likely failure mode of the structure. The mode shapes are also often

20、used to generate perturbations ingeometry for collapse analysis. They are typically scaled to a fraction of a relevant structural dimension (such as a beam cross-section or a shell thickness) before they are used to perturb the geometry.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse

21、 Analysis with ABAQUSL2.14Eigenvalue Problem Formulation Estimating the maximum load levelNonlinear pre-buckling makes the method approximate.The estimate is more accurate if the structure is preloaded to a level close to the pre-buckling load capacity.Generally alternative solution techniques (e.g.

22、, Riks) are required for accurate prediction.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSABAQUS UsageCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.16ABAQUS Usage ABAQUS will calculate the initial stress and load stiffne

23、ss matrix corresponding to the live load directly. The *BUCKLE step is a linear perturbation step, and the magnitude of the live load is not important. The live load is specified in the*BUCKLE step. The *BUCKLE step may be preceded by a*STATIC step in which the dead load is applied.Copyright 2004 AB

24、AQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.17ABAQUS Usage The buckling mode shapes of symmetric structures are either symmetric or antisymmetric. For such structures it is more efficient to model only part of the structure and to perform the buckling analysis twice: once w

25、ith symmetric boundary conditions and once with antisymmetric boundary conditions. The live load pattern is usually symmetric, so symmetric boundary conditions are needed for the calculation of the perturbation stresses used in the formation of the initial stress stiffness matrix.Copyright 2004 ABAQ

26、US, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.18ABAQUS Usage The boundary conditions must be switched to antisymmetric in the*BUCKLE step to obtain theantisymmetric modes.This is done by giving the antisymmetric boundary conditions with LOAD CASE=2 on the *BOUNDARY option in th

27、e*BUCKLE step.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.19ABAQUS UsageExample 1:Antisymmetric buckling of a symmetric structureFinite element modelBoundary conditions for load case 1Boundary conditions for load case 2 B21 elements Rectangular cross-sectio

28、n (1 in 1in) Linear elastic material:E = 30E6 psin = 0Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.20ABAQUS UsagePartial input for antisymmetric buckling of a symmetric structure*headingantisymmetric ring buckling eigenvalue estimate:*nset, nset=left 4*trans

29、form, nset=left1., 1., 0., -1., 1.,*boundary right, 2right, 6left, 2left, 6*step, name=Step-1*buckle 3,*boundary, load case right, 2right, 6left, 1*dsload ring, p, 1.*end step0.= 2,op=newCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.21ABAQUS Usage The dead an

30、d live loads can be point loads, distributed loads, or thermal loads; dead loads can also include nonzero prescribed boundary conditions. If the live loads include uniform motion of a boundary, use multi-point constraints to constrain these nodes to a single point and load that point. The dead load,

31、 P0, and the live load, DP, can be entirely different in magnitude and in nature. Multiple buckling modes and associated critical load values can beobtained in a single eigenvalue buckling step. Obtaining multiple buckling modes is often useful since many common systems (such as short cylindrical sh

32、ells) have several closely spaced critical modes.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.22ABAQUS Usage Negative eigenvalues are sometimes obtained.These values can indicate that there is a buckling mode corresponding to the load applied in the opposite

33、 direction. For example, a pressure vessel under internal pressure might buckle under external pressure.However, they can also point to spurious modes if nonlinearities occur before buckling.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.23ABAQUS Usage The loa

34、d stiffness can have a significant effect on the critical buckling load. ABAQUS will use the symmetric form of load stiffness in eigenvalue calculations. Follower force effects are associated with pressure, hydrostatic pressure, buoyancy effects, centrifugal loading, and Coriolis loading and also wi

35、th concentrated loads with the FOLLOWER option. Eigenvalue buckling analysis with concentrated FOLLOWER loads will not yield the correct results since the follower force effects are not taken into account (the load stiffness is generally not symmetric).Copyright 2004 ABAQUS, Inc.Buckling, Postbuckli

36、ng, and Collapse Analysis with ABAQUSL2.24ABAQUS Usage If temperature-dependent elastic properties are used, the elastic stiffness will be based on the temperatures prior to the eigenvalue buckling step. Modifying temperatures in an eigenvalue buckling step will not change the elastic stiffness matr

37、ix. When nonlinear material properties such as hyperelasticity are present in a model, ABAQUS/Standard ignores the nonlinear effects during the eigenvalue buckling analysis. The material response during the buckling analysis is based on the linear elastic stiffness in the (potentially nonlinear) bas

38、e state at the end of the previous step. Inelastic behavior such as plasticity can be used if the eigenvalue buckling step is applied before the stress at any point has reached yield. ABAQUS will use the elastic stiffness, defined with linear elasticity, in the eigenvalue buckling procedure to calcu

39、late DK, not the tangent stiffness from the hardening curve.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.25ABAQUS UsageChoice of eigenvalue extraction method In ABAQUS/Standard you have the choice of using either the subspace iteration or the Lanczos method

40、to extract the eigenvalues. The Lanczos method is generally faster when a large number of eigenmodes are required for a system with many degrees of freedom.The subspace iteration method may be faster when only a few (less than 20) eigenmodes are needed.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckli

41、ng, and Collapse Analysis with ABAQUSL2.26ABAQUS Usage By default, the subspace iteration method is used when extracting the buckling modes.To use the Lanczos solver, include the EIGENSOLVER=LANCZOS parameter on the *BUCKLE option.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analy

42、sis with ABAQUSL2.27ABAQUS Usage When the Lanczos eigensolver is requested, you can also specify the minimum and/or maximum eigenvalues of interest on the data line. ABAQUS extracts eigenvalues until either the requested number of eigenvalues has been extracted in the given range or all the eigenval

43、ues in the given range have been extracted. The Lanczos eigensolver has the following restrictions for buckling simulations; it cannot be used with:A model containing hybrid elementsA model containing distributing coupling elementsA model containing contact pairs or contact elementsA model that has

44、been preloaded above the bifurcation (buckling) loadCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSClosely Spaced EigenvaluesCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.29Closely Spaced Eigenvalues Certain buckling probl

45、ems may have many buckling modes with very closely spaced critical loads. The eigenvalue algorithm may converge slowly in such a case. Cylindrical shells with axial compressive loads are a good example of this case.Radius = 100Length = 800Thickness = 0.25Youngs modulus = 30 106 Poissons ratio = 0.3C

46、opyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.30Closely Spaced EigenvaluesExample 2: Buckling of a thin cylinder S9R5 elements Linear elastic material Lanczos eigensolver Symmetry (only half the cylinder length is modeled)*headingcylinder buckling:*stepbottom*

47、buckle, 5,*cloadeigensolver=lanczosMidside nodes Corner nodesbot1, bot2,3, 65450.03, 32725.0top*boundarybottom, 1, 2 top, zsymm*node file u*end stepCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.31Closely Spaced Eigenvalues Critical stress as a function of mod

48、e shape (analytical results)Axial modesCircum. modesCritical stress1265849103737104459810711116107912840718.0943291.8644283.6544292.4844662.1344711.8744751.2944867.9444874.5644909.4945608.1845718.5445792.7745827.96Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2

49、.32Closely Spaced EigenvaluesCritical buckling mode of axially compressed cylinderCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.33Closely Spaced Eigenvalues Closely spaced eigenvalues are usually an indication that a structure is imperfection sensitive. The i

50、mperfections cause the buckling modes to interact and will trigger collapse at a much lower level than predicted by the eigenvalue buckling analysis. Hence, the actual buckling mode shape is usually not the same as the lowest buckling mode in the eigenvalue analysis. Convergence can be improved subs

51、tantially by applying part of the critical live load as a dead load, loading the structure to just below the buckling load. This provides a larger separation of the eigenvalues and much improved convergence. The process is equivalent to a dynamic eigenfrequency extraction with shift.Copyright 2004 A

52、BAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.34Closely Spaced Eigenvalues If you exceed the critical (lowest) buckling load of the structure when applying the dead load, the analysis may terminate prematurely because ABAQUS will not be able to find the lowest buckling loads

53、.In addition, the Lanczos method will fail in that case.Copyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.35Closely Spaced Eigenvalues Buckling of axially loaded cylinder without imperfectionsCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysi

54、s with ABAQUSL2.36Closely Spaced Eigenvalues Buckling of axially loaded cylinder with imperfectionsCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.37Closely Spaced EigenvaluesExample 3: Buckling of a thin cylinder with preload*headingcylinder buckling:*step, nl

55、geom*staticwithpreload*cloadbot2, 3, 130900.0bot1, 3, 261800.0preload*endstep*step*buckle, eigensolver=lanczos 5,*cloadbot2, 3, 32725.0bot1, 3, 65450.0*node file u*end stepCopyright 2004 ABAQUS, Inc.Buckling, Postbuckling, and Collapse Analysis with ABAQUSL2.38Closely Spaced Eigenvalues The analysis with the preload converges much faster. The analysis without