Essential Considerations for Buckling Analysis

Transcription

1 Worlwie Aerospace Conference an Technology Showcase, Toulouse, France, Sept , 2001 Essential Consierations for Buckling Analysis Sang H. Lee MSC.Software Corporation, 2 MacArthur Place, Santa Ana, CA Phone: (714) sang.lee@mscsoftware.com Abstract There are three main characteristics to examine in orer to etermine the integrity of a structure in a esign process: stresses, ynamics, an stability. Buckling analysis assesses the stability characteristics ictating the integrity limit of a structure. A linear buckling analysis use to suffice for esign requirements, but rigorous nonlinear buckling analysis has become more popular as the tools have become available an the esigns more sophisticate. In this paper, the methos for linear an nonlinear buckling analysis are reviewe with a few illuminating examples to elicit the best approach for the buckling analysis proceure. 1. Introuction An accurate solution to a buckling problem requires more meticulous efforts than just following a numerical proceure. There are a number of factors to consier before a buckling solution can be accepte with confience. Such points along with a few examples are iscusse by comparing solutions obtaine from ifferent methos. A starting step shoul be a linear buckling analysis. The linear buckling analysis capability in SOL 105 uses two Subcases up to Version 70.5: one Subcase for a linear static analysis with a buckling loa, followe by the secon Subcase for an eigenvalue analysis. Frequently, there may be other static loas sustaine by the structure, in aition to the buckling loa in question. In orer to provie more versatile capability for general buckling cases, a new capability was introuce in MSC.Nastran Version 70.7, allowing constant preloas in the linear buckling analysis. Nonlinear buckling analysis capability has been available in SOL 106 by restarts [1-2]. While serving the purpose of a nonlinear buckling analysis following a static nonlinear analysis, this buckling analysis proceure is cumbersome to the user because it requires a restart. To provie a more convenient an versatile capability for nonlinear buckling analysis, an option for a buckling analysis in a col-start run with PARAM, BUCKLE, 2 was introuce in MSC.Nastran Version This option allows a buckling analysis in any Subcase that has the METHOD comman specifie for an eigenvalue analysis. There are two more ways in nonlinear analysis to estimate the critical buckling loa. One metho is the arc-length metho that can provie solutions past the critical buckling loa into the post-buckling state [3]. Using the arc-length metho, the noal isplacement of a point with the most noticeable motion shoul be trace to illustrate the peak point. The applie loa at this peak resembles the critical buckling loa. Another metho is to use Newton s metho until the solution cannot be obtaine ue to ivergence, in which case the aaptive bisection metho is activate in the vicinity of the critical buckling loa an stops at the limit loa, close to the critical buckling point. A similar metho was applie to the nonlinear 1

2 ynamic stability analysis using aaptive bisection algorithm in time omain for buckling or bifurcation preiction in the implicit irect-time integration [4], but the ynamic stability is not contemplate in this paper. 2. Theoretical Backgroun Let us start with a linear equation of motion for a preloae structure, i.e., where P(t) M C, K, an Mu& + Cu& + Ku + K u = P(t) K are mass, viscous amping, material an ifferential stiffness matrices, an is a forcing function in time omain. The ifferential stiffness is create by the initial stress ue to preloas, an it may inclue the follower stiffness if applicable. Ignoring the amping term to avoi complex arithmetic, an eigenvalue problem may be formulate as 2 [( K + K ) ω M ]{ φ} = {0} which is a governing equation for normal moe analysis with a preloa, but it coul be recast for a ynamic buckling analysis at a constant frequency. For a static buckling, the inertia term rops out because the frequency of vibration is zero, i.e., [ K + K ]{ φ} = {0} in which {φ } represents virtual isplacements. A non-trivial solution exists for an eigenvalue that makes the eterminant of [ K ] K + vanish, which leas to an eigenvalue problem [ K + λ K ]{ φ} = {0} where λ is an eigenvalue which is a multiplier to the applie loa to attain a critical buckling loa. If there exists static preloas other than the buckling loa in question, the above equation shoul inclue aitional ifferential stiffness, i.e., [ + λ K ]{ φ} = {0} K + K preloa buckle in which ifferential stiffnesses are istinguishe for constant preloa an variable buckling loa. Notice that no eigenvalue solutions are meaningful if the preloa makes the structure buckle, i.e., [ K + ] K preloa is not positive efinite. The buckling analysis with an excessive preloa can rener wrong solutions. The correct solution can be ascertaine by a buckling analysis with the preloa as a buckling conition, unless a safeguar is built into the analysis coe. The nonlinear buckling analysis is base on an extrapolation using two incremental solutions. The governing equation for an eigenvalue analysis is with where [ + λ K]{ φ} = {0} K n K K n an n 1 = K n K n 1 K are the stiffness matrices evaluate at the known solution points near instability. The critical isplacements on instability may be estimate as with { U } = { U } + λ { U} cr { U } = { U n } { Un 1} Base on the virtual work principle, where n T T { U} { Pcr } = { U} { Fcr } F cr Ucr 1 = Fn + K u Fn + [ Kn + λ K] U U n 2 λ. 2

3 The critical buckling loa may be estimate as with where { P } = { P } + α { P} cr n { P } = { P n } { Pn 1} T 1 λ{ U} [ Kn + λ K]{ U } 2 α = T { U} { P} The unerlying assumption for this formulation is that the tangent stiffness matrix is proportional to the isplacement increments, an not to the external loas. 3. Valiation of a Panel Buckling An example of input ata for a linear buckling analysis with a preloa is shown below: ID plate105, v2001 $ SHL 2/23/01 DIAG 8, 15 $ SOL 105 TIME 60 CEND TITLE = MSC.Nastran job create on 20-Feb-01 SPC = 2 $ clamp the bottom ege DISP=ALL SUBCASE 10 SUBTITLE= preloa tensile ege force 300 lb/in LOAD = 2 SUBCASE 20 Subtitle= top pressure 100 lb/in. LOAD = 7 SUBCASE 30 SUBTITLE=buckling STATSUB(preloa)=10 STATSUB(buckle)=20 METHOD=1 BEGIN BULK PARAM POST 0 EIGRL ENDDATA This is an example of a plate moel (5 in. by 10 in., 0.1 in thick, steel, clampe on the bottom ege) with lateral tension by an ege loa (in x) of 300 lb/in for a preloa an vertical ege loa (in y) of 100 lb/in for a buckling loa. Alternatively, the preloa an the buckling loa may be switche to perform a similar analysis. The first buckling moe is a bening moe at vertical loa of lb/in, as shown in Figure 1. The secon buckling moe in bening at 593 lb/in is also shown in Figure 1, but it is of no physical significance an shown here for verification purposes only. If the lateral loa is in compression instea of tension, the plate coul buckle in a twist moe at lb/in in the absence of a vertical loa, as shown in Figure 2. A parametric stuy, conucte with various combinations of lateral an vertical loas, resulte in a buckling envelope (or locus) on a loaing plane as shown in Figure 3. The buckling shape is a bening moe above the curve an a twist moe in the left sie of the curve. Bifurcation behavior can be observe in the transitional region near the corner point, where either the twist or bening moe coul appear as the first an the secon moe. Lower-right sie of the curve is the safety zone. 3

4 The nonlinear buckling analysis proceure in MSC.Nastran allows buckling analysis with preloas. However, the nonlinear proceure follows a ifferent numerical proceure. Therefore, it coul provie a means for valiation of linear solutions. A parametric stuy using nonlinear buckling analysis renere ata points labele Nonlinear 1 an 2 in Figure 3, using the lateral force an vertical force as preloas, respectively. Although two ifferent methos have been use in the nonlinear buckling preiction: namely, (1) eigenvalue estimation by extrapolation an (2) etection of the critical loa by consecutive bisections with no recourse ue to ivergence, all the nonlinear solution points fell, by an large, on the same curve. This figure also shows that the problem is basically linear, because the Large Displacement option introuce no visible effects. 4. Nonlinear buckling analysis of a spherical cap An example of an input ata for a col-start nonlinear buckling analysis is shown below: ID NLSOLIDB,V2001 $ SHL 4/27/00 SOL 106 $ DIAG 8,50 $ TIME 60 $ FOR VAX CEND TITLE=ELASTIC-PLASTIC BUCKLING OF IMPERFECT SPHERICAL CAP SUBTITLE=HYDROSTATIC PRESSURE APPLIED,PERIPHERY CLAMPED SET 10 =100,200,109,209,119,219,131,231 SET 20 =10,11,15,20,25 SET 30 = 1000 ECHO=UNSORT DISP=10 OLOAD=30 STRESS(PLOT)=20 SPC=10 $ buckling analysis in col-start PARAM BUCKLE 2 $ SUBCASE 1 LOAD=10 NLPARM=10 SUBCASE 2 LOAD=20 NLPARM=20 METHOD=30 $ for buckling analysis SUBCASE 3 LOAD=30 NLPARM=30 METHOD=30 $ for buckling analysis BEGIN BULK $ FOR BUCKLING ANALYSIS EIGB 30 SINV $ PARAMETERS PARAM POST 0 PARAM LGDISP 1 NLPARM 10 2 AUTO 7 30 yes NLPARM 20 5 AUTO 1 30 yes NLPARM 30 2 AUTO 1 yes $ LOADING LOAD

5 LOAD LOAD $ unit pressure is 1000 psi PLOAD PLOAD $ a ummy force applie to a fixe point to output critical buckling loa FORCE GRID ENDDATA The moel consists of 67 noal points an 16 soli elements (15 HEXAs an 1 PENTA) with geometric an material nonlinearity. The buckling analysis is requeste at the en of Subcases 2 an 3 in this example. Notice that the KSTEP fiels of NLPARM entries for both Subcases 2 an 3 are set to 1 to upate the stiffness matrix at every solution step. The PARAM, BUCKLE, 2 is place above Subcases, but it coul be place in the Bulk Data section or uner the esire Subcases. The user oes not know where the buckling point is until the analysis is performe. That is the reason the user may want to ivie the loa into a number of Subcases an perform buckling analysis in multiple steps. In this example, pressure loas of 2000, 3000 an 3500 psi have been applie in three Subcases, respectively. The secon Subcase has five increments with an incremental pressure of 200 psi. At the en of the secon Subcase, an eigenvalue analysis is performe ue to PARAM, BUCKLE, 2, which resulte in α= Then the critical buckling loa is calculate as P cr = P + α P = α * 200 = n The thir Subcase applies a pressure of 3500 psi in two increments, followe by a buckling analysis. The buckling analysis resulte in α=0.1135, an the buckling loa from this eigenvalue analysis reners P cr = P + α P = α * 250 = n Returning to the example input ata, notice that there is an OLOAD request on GRID 1000 that is fixe in space. However, a FORCE of 1000 is applie on this ummy noe with the same scaling in the LOAD Bulk Data as the pressure loa. This OLOAD request makes the applie loa printe for GRID 1000 at the eigenvalue solution step uner the heaing LOAD VECTOR. This applie loa at buckling is the critical buckling loa ientical to those manually calculate above. Therefore, the user oes not have to exercise such a manual calculation. This example is a spherical cap with a ouble curvature subjecte to an external pressure; moele by a 10- egree segment with the periphery clampe, simulating an axisymmetric moel. The problem involves large isplacements an plastic eformation. When this problem was analyze using an arc-length metho (CRIS), the solution went through the peak loa of psi. When the same problem was analyze using Newton s metho with the bisection proceure, the solution trace up to the maximum loa of psi before ivergence ue to instability. When the moel was re-meshe with about 100 times more elements, the buckling analysis proceure preicte a critical buckling loa at psi. In this refine moel, the arc-length metho peake at psi, an Newton s metho with bisection converge up to psi (consiere the most accurate solution). For a reference, the nonlinear buckling solution without the material nonlinearity (elasto-plastic) reners psi when estimate at 3500 psi (or psi when estimate at 3000 psi). When it is further reuce to a linear problem (without geometric nonlinearity), the buckling solution reners psi. This emonstrates that the problem is severely nonlinear both in material an geometric characteristics. The same example is further stuie by remoeling with 2667 noal points an noe TETRA elements, which is shown in Figure 4 (left sie). The pressure loa was applie in three Subcases: 2000 psi in the first Subcase, 3000 psi in the secon Subcase, an 4000 psi in the thir Subcase that aopte an arc-length metho. The first Subcase has two increments an the eformation is within the elastic limit. The secon Subcase has five increments unergoing plastic eformation, which was complete without any bisection. The thir Subcase proceee with an aaptive arc-length metho using an initial increment of 5

6 five. The arc-length process unerwent a snap-through at psi before terminating at 30 specifie increments near the loa of 4000 psi. The eforme shapes (isplacement in true scale) before an after the snap-through are shown in Figures 4 (right sie) an 5. The nonlinear buckling analysis preicte a critical buckling loa at psi. These results inee show a goo agreement with the preceing solutions base on HEXA element moels. 5. Demonstration of a Cyliner Buckling A hollow cyliner is use to emonstrate an interesting behavior in buckling. Its geometric attributes inclue: 50 in. in iameter, 100 in. in height an in. wall thickness with close ens of 0.5 in. in thickness. The cyliner, mae of steel, is subjecte to a combine loa of external pressure on the cylinrical surface an vertical ege force on the top, while the bottom is fixe in z. The cyliner coul buckle in two ifferent moes epening on the combination of loas as shown in Figures 6 an 7, where the first moe with vertical creases appear when the external pressure is preominant (showing a star-shape cross-section at the mi-span) an the secon moe with horizontal creases appear when the vertical ege force is more ominant. A failure curve (or buckling envelope) can be constructe by a parametric stuy using various magnitues of the preloas with external pressure or vertical ege loa, as shown in Figure 8. The first buckling shape appears above the curve an the secon buckling in the right sie of the curve. A bifurcation occurs at the corner, where either buckling moe may occur. The cyliner is free from buckling in the lower-left region of the curve. It is note that the buckling resistance against the vertical loa oes not increase even if the pressure irection is reverse to an internal pressure (creating a tensile hoop stress), although the buckling loa increases against the external pressure as the vertical compression ecreases. The pressure loa creates follower stiffness, which is inclue in this analysis. The critical buckling loa is normally over-estimate without the follower stiffness, but the effect of the follower stiffness seems negligible in this problem. Nonlinear buckling solution points are ae to the curve in Figure 9, where two categories of nonlinear solutions are presente. Solution points in square symbol are from the eigenvalue analysis using an extrapolation scheme, an those in iamon symbol are from the limit process using consecutive bisections upon ivergence uring the incremental/iterative processes. Although the secon buckling moes in the right sie of the curve preicte by nonlinear analyses are almost at ientical loas to the linear solutions, nonlinear solutions for the first buckling shape above the curve are wiely isperse as shown in Figure 9. This shows that the geometric nonlinearity has significant effects on the first kin with vertical creases, while the geometric nonlinear effect is negligible in the secon kin. It also signifies that the nonlinear eigenvalue analysis base on extrapolation is an approximation an it can vary appreciably epening on where the extrapolation is applie. In aition, the limiting process by aaptive bisection shows some scatter epening on the solution strategy as well as the loaing path. One particular concern lies in the iscrepancy exhibite by the nonlinear solution points without any vertical ege loa, where the scatter is most pronounce. It was conjecture that nonlinear buckling solution proceure coul overestimate the critical loa if there is no isturbance or imperfection introuce to the moel in orer to inuce the buckling. This hypothesis was verifie by introucing a vertical loa initially an removing it before buckling stage is reache. Inee, it verifie the tren of the inaccuracy, but it i not fully remove the iscrepancy. 6. Discussion The linear buckling analysis with preloas encompasses all the static buckling problems an provies a general capability for linear buckling analysis. Most real-life structures belong to this category in terms of buckling. This new capability, using STATSUB Case Control comman for preloas, eliminates limitations in the linear buckling analysis as well as the ynamic response analyses in MSC.Nastran. 6

7 The linear buckling solution provies a goo initial solution as a cross-reference to the nonlinear buckling solution. Since the critical buckling point is not known a priori, an eigenvalue analysis with extrapolation for a nonlinear buckling solution can be performe in any Subcase of SOL 106. The extrapolation is more accurate as the two solution points for extrapolation approach the critical loa, unless the two points are too closely space. The limit process by an aaptive bisection upon ivergence, in general, preicts the critical buckling loa more accurately for nonlinear problems. The nonlinear buckling point can also be preicte by the arc-length metho, but it preicts a slightly higher value for a buckling loa than other methos. Therefore, the arc-length metho is not conservative an inappropriate for a buckling solution even if it may be more accurate. Different methos of nonlinear buckling analysis rener somewhat ifferent solutions epening on the nonlinear solution strategies, such as stiffness matrix upate scheme. The nonlinear buckling solution also requires a provision to inuce buckling for an accurate solution, unless the buckling shape is ientical to the static eformation shape. This can be achieve by introucing a geometric imperfection, or applying ummy loas (to trigger a buckling shape) to be remove before reaching the buckling stage. Because of such characteristics, the nonlinear buckling solution is eeme path-epenent (in terms of loa or eformation) to some egree, even if there is no other path-epenency (i.e., plasticity or friction) in the moel. 7. Concluing Remarks As the mechanical esign becomes more critical for weight reuction or other constraints, nonlinear buckling analysis may be require for more accurate assessment of the structural stability. The linear buckling analysis with a preloa is a goo starting point to preict a critical loa even for a nonlinear problem. The buckling solution may nee to be more accurate but still conservative. The linear solution enhances the confience level for a nonlinear solution, since the nonlinear solutions vary epening on the methoology an the parameter setting use in the moel. A number of nonlinear methos, as well as linear buckling analysis, provie alternative means to cross-check the result. Conventional nonlinear buckling analysis is performe by extrapolation near the buckling point. The accuracy of the solution from this extrapolation proceure epens heavily on the ata points use for extrapolation. This lack of certainty ictates that the nonlinear buckling analysis be performe repeately uring the incremental process until an instability conition is encountere. MSC.Nastran provies a user-frienly proceure to allow repetitive buckling analysis. The eigenvalue analysis for the buckling preicts a more accurate critical loa as the two solution points for extrapolation approach the actual buckling point. Comparison of the results confirms that the limit loa obtaine from Newton s metho with aaptive bisection process closely preicts the critical buckling loa. The arc-length metho tens to preict slightly higher values for the buckling loa, an therefore a non-conservative solution. References: [1]. Lee, S. H., Hanbook for Nonlinear Analysis (for MSC/NASTRAN), The MacNeal- Schwenler Corp., March [2]. Lee, S. H., an Herting, D. N., A note on A Simple Approach to Bifurcation an Limit Point Calculation, Int. J. Numerical Methos in Engineering, John Wiley & Sons, Vol. 21, pp , [3]. Lee, S. H., Hsieh, S. S., an Bock, T. L., Aaptive Arc-Length Methos in MSC/NASTRAN, Proceeings of 1990 MSC Worl Users Conference, 5/1-22, Los Angeles, March 28, [4]. Lee, S. H., Nonlinear Dynamic Stability Analysis using Aaptive Bisection Algorithm in Implicit Integration, Proceeings, 1992 ASME CIE Conference, Vol. 2, pp , San Francisco, August 2-6,

8 Figure 1. Panel buckling shapes for the first two bening moes Figure 2. Panel buckling in twist moe uner lateral compression Cantilevere Panel Buckling vertical compression in lb/in Thousans lateral tension in lb/in linear buckling Nonlinear 1 with lateral force as preloa Nonlinear 2 with vertical force as preloa Figure 3. Cantilevere panel buckling envelope 8

9 Figure 4. Uneforme shape an eforme at buckling ( psi) Figure 5. Post snap-through at psi, uneforme shape superimpose 9

10 Figure 6. Cyliner buckling shape 1 with vertical creases. Figure 7. Cyliner buckling shape 2 with horizontal creases. 10

11 external pressure in psi 30 Cantilevere Cyliner Buckling (linear solution) Thousans linear buckling with constant pressure linear buckling with constant axial force axial force in lb/in Figure 8. Cyliner buckling envelope (linear analysis) Cantilevere Cyliner Buckling external pressure in psi Thousans axial force in lb/in linear buckling with constant pressure linear buckling with constant axial force Nonlinear eigenvalue analysis Nonlinear by limiting process Figure 9. Cyliner buckling envelope (with nonlinear solutions) 11