HOL - The Use of High-fidelity Analysis for Reliable Buckling Load Calculations

7/27/2019 HOL - The Use of High-fidelity Analysis for Reliable Buckling Load Calculations

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THE USE OF HIGH-FIDELITY ANALYSIS FOR RELIABLE BUCKLING LOADCALCULATIONS

THE USE OF HIGH-FIDELITY ANALYSIS FOR RELIABLEBUCKLING LOAD CALCULATIONS

Jan Hol1

and Johann Arbocz2

Delft University of Technology, The Netherlands

SUMMARY

It is shown that using the measured midsurface and boundary imperfections, the so-calledmanufacturing signature of a fabrication process, it is possible to derive a safe, verifiedknockdown factorby applying a hierarchical high fidelity analysis approach (Ref [22]).Thusthe analysis and design phase will be completed faster and only the reliability of the finalconfiguration needs to be verified by structural testing.

1: Introduction

With the arrival of the era of supercomputing there is a tendency to replace the relativelyexpensive experimental investigations by numerical simulation. The use of large general

purpose computer codes for the analysis of different types of aerospace, marine, and civilengineering structures is by now well accepted. These programs have been used successfully

to calculate the stress and deformation patterns of very complicated structural configurationswith the accuracy demanded in engineering analysis.

However, there exist numerous complex physical phenomena where only a combinedexperimental, analytical, and numerical procedure can lead to an acceptable solution. Onesuch problem is the prediction of the behaviour of buckling sensitive structures under thedifferent loading conditions that can occur in everyday usage.

The axially compressed cylindrical shell represents one of the best known examples of the

very complicated stability behaviour that can occur with thin-walled structures. For thin shellsthat buckle elastically initial geometric imperfections [1,2] and the effect of the differentboundary conditions [3-5] have been identified as the main cause for the wide scatter of theexperimental results. However, this knowledge had not been, as yet, incorporated into thecurrent shell design manuals.

1 Assistant Professor, Aerospace Structures, Faculty of Aerospace Engineering

2 Professor Emeritus, Aerospace Structures, Faculty of Aerospace Engineering, Fellow AIAA


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These design recommendations [6-8] all adhere to the so-called lower bound designphilosophy and as such recommend the use of the following buckling formula:

aP PF.S.

crit (1)

where aP = allowable applied load; critP = lowest buckling load of the perfect structure; = knock-down factor; and F.S. = factor of safety.

The empirical knockdown factor is so chosen that when it is multiplied with critP , thelowest buckling load of the perfect structure, a lower bound to all available experimental datais obtained. Using this approach for isotropic shells under axial compression, in 1965

Weingarten et al [9] have published the following expression for a lower bound curve

1 R

16 h

c

P1 0.901 (1 e )

P

= =

l

(2)

where

cP =l classical critical load = c2 RN l

(3)

22

cEh

N ; c 3(1cR

= = l )

and R = shell radius, h = shell wall-thickness, E = Youngs modulus, = Poissons ratio.

The central goal of the research reported in this paper is the development ofimproved shelldesign criteria. The improvements with respect to the presently recommended shell design

procedures are primarily sought in a more selective approach by the definition of theknockdown factor . The proposed new improved shell design procedure can be represented

by the following formula

aaP

F.S.

cP (4)


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where aP = allowable applied load; cP = lowest buckling load of theperfect structurecomputed via one of the shell codes; a = verified high fidelity (higher) knockdown factor;and F.S. = factor of safety.

The steps involved in the derivation of such a verified high-fidelity (higher) knockdownfactor a are the subject of this paper.

2: Mesh Convergence Study

At the beginning of any stability investigation, the accuracy of the discrete model usedshould be checked against available analytical or semi-analytical results or finite elementmodels with appropriate mesh convergence studies to assure that the analysis using the

discrete model converges to a meaningful result. This step is part of a mandatory studyneeded in order to establish the dependence of the buckling load predictions on the meshdistribution used and to assure that the analytical solution is consistent with the physics of the

problem being solved. Furthermore, as has been pointed out in the past by Byskov [10], if onecarries out imperfection sensitivity investigations, which involve an extension of the solutioninto the postbuckling response region, further mesh refinement may be needed since thewavelength of the dominant large deformation pattern may often decrease significantly. Eachlikely response mode must be properly represented in the discrete model. Unfortunately, someof the critical shorter wave-length modes associated with the nonlinear response of thestructure may not be activated until well into the load-response history for the structure.

A test series of seven isotropic shells carried out by Arbocz and Babcock [11] at Caltech isused to illustrate how a hierarchical high fidelity analysis can be carried out. The platform forthe multi-level computations, used for an accurate prediction of the critical buckling loads anda reliable estimation of their imperfection sensitivity, is provided by DISDECO, the DelftInteractive Shell DEsign COde [12]. With this open ended, hierarchical interactive computercode the user can access from his workstation a succession of programs of increasingcomplexity that are suitable for various nonlinear aspects of the response.

Thus the shell designer can study the buckling behavior of a specified shell, calculate itscritical buckling load quite accurately, and make a reliable prediction of the expected degreeof imperfection sensitivity of the critical buckling load. The proposed procedure consists of ahierarchical approach, where the analyst proceeds step-by-step from the simpler (Level-1)methods used by the early investigators to the more sophisticated analytical and numerical(Level-2 and Level-3) methods used presently. This approach allows different aspects of thenonlinear response to be represented properly as these aspects begin to affect the response.

In an earlier paper [13] the buckling predictions obtained using Level-1 methods withmembrane prebuckling analysis [3-5] and Level-2 methods with rigorous nonlinear

prebuckling analysis [14,15] have been reported.

The Level-3 solutions used in this paper are based either on a two-dimensional finitedifference [16] or finite element [17,18] formulation. In both cases, if one uses the appropriate


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meshes, one can obtain rigorous solutions where all nonlinear effects are properly accountedfor. Thus, initially a convergence study must be carried out in order to establish the mesh sizeneeded to model accurately the response and buckling behavior of the shell in question. Forthis purpose, the asymmetric bifurcation responses from a nonlinear prebuckling path solutionoption was used, whereby the earlier results obtained with the Level-2 DISDECO moduleANILISA [19] listed in Reference 13 serve as a reference.

Figure 1: Figure 1STAGS-A [10] convergence study using Caltech isotropic shell A-8 [11].In the convergence study, at first, for a fixed number of mesh points in the axial direction

(NR = 161 for this example) the number of mesh points in the circumferential direction (NC)was increased until the bifurcation load approached a horizontal tangent. As can be seen fromFig. 1, the results converge to a limiting value from below at about NC = 261. Next, for afixed number of mesh points in the circumferential direction (NC = 161), the number of rows(NR) was varied. This time the convergence is from above, as can be seen from Figure 1, andthe horizontal tangent is reached at about NR = 261.

Figure 2: Maximum normal displacement versus axial compression curves using refined meshes


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To illustrate the difference between using coarser meshes to speed up the computations andfiner meshes which produce (nearly) converged solution, the axial compression versusmaximum normal displacement curves for the isotropic shell A-8 with a short-waveasymmetric imperfection [13] are displayed in Figure 2. Using a mesh of 161 rows and161 columns (a model with 79708 degrees of freedom and a maximum semi-bandwidthof 635) and SS-3 boundary conditions (NX = -N0, v = w = Mx = 0) the results do not deviatesignificantly from the results of a mesh with 261 rows and 261 columns (a model with207508 degrees of freedom and a maximum semi-bandwidth of 1037) and the same SS-3

boundary conditions. Thus to speed up the computations in the following the coarser 161x161mesh will be used.

3: Midsurface Initial Imperfections

In order to apply the theory of imperfection sensitivity with confidence, one must know thetype of imperfections that occur in practice. In 1969 Arbocz and Babcock [11] published theresults of buckling experiments where, for the first time, the actual initial imperfections andthe prebuckling growth of the midsurface of electroplated isotropic shells were carefullymeasured and recorded by means of an automated scanning mechanism.

Figure 3: Measured initial imperfections of Caltech isotropic shell A-8 [11].As can be seen from Figure 3, the measured initial midsurface imperfections of shell A-8

[11] show a rather general distribution dominated by an n 2= mode. One can use thefollowing half-wave cosine double Fourier series

7 18 19 99

io o o k i 1 1 1 k,

116

kk,

W x y y x yW cos i W cos W sin W cos k cos

h L R R L R

x yW cos k sin

L R

= = =

= + + +

+

l l ll l l

ll

l l l

l(5)


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to represent the measured initial imperfections accurately, where Fourier coefficients withabsolute values less than 0.001 are neglected.

For a more accurate estimate of the buckling load, one must use the measured initialimperfections in codes like STAGS [16,17] to carry out two-dimensional nonlinear collapseanalysis. Employing a user-written subroutine WIMP to input the double Fourier series ofequation (5), the 161x161 STAGS model with SS-3 boundary condition (NX = -N0,v = w = Mx = 0) yielded a collapse load of sP 900.1= lbs, whereas the same model with C-4

boundary condition (u = u0, v = w = w,x = 0) yielded a collapse load of sP lbs. Bothof these values are significantly higher than the experimental buckling load of

976.5=

expP 825.9= lbs.

It is shown in References [20,21] that boundary imperfections can have a significantdegrading effect on the buckling loads of axially compressed cylindrical shells. The flatnessof the end ring attached to the load-cell of the Caltech test set-up used to test shell A-8 wasmeasured. As can be seen in Fig. 4 there is a very large amplitude wavy pattern in the plane ofthe end support, with maximum deviations of about 3 wall thicknesses.

Figure 4: Measured flatness of the Caltech load-cell end-ring [11].The measured boundary imperfections are decomposed into a one-dimensional Fourier

series

=

= = + +o b b o n nn 1

1 yu u (y) h{ a (a cosn b sinn )}

2 R

y

R (6)


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This series is then used in STAGS [16] to model the effect of boundary initialimperfections by using a modified C-4 boundary condition

u = ub(y), v = w = w,x = 0 (7)

and by taking advantage of the dual loading systems provided by the STAGS program.

Figure 5: End rings used in the Caltech test set-up for buckling tests [11].Because the grooves of the end rings were filled with liquid Cerrolow at the time shell A-8

was installed in the test apparatus, see Fig. 5, it is to be expected that after cooling thehardened liquid metal has filled-in all of the gaps. The effect of this unknown end support wasmodeled by varying the amplitude of the boundary imperfection b between 0 and 0.1 as can

be seen in Fig. 6.

4: Numerical Results

It is shown in Ref. [22] that including both the measured initial midsurface imperfectionsof Equation (5) and the measured boundary imperfections of Equation (6) with an amplitudeof b 0.04 = , the calculated collapse load of sP 748.6= lbs agrees closely with the

experimental buckling load of shell A-8 of expP 825.9= lbs.

Repeating the buckling load calculations for all 7 A-shells of Ref. [11] using theappropriate measured initial midsurface imperfections and the same measured boundaryimperfections of Equation (6) with an amplitude of B 0.04 = , one obtains the simulated

buckling loads s BP tabulated in Table 1. The experimental buckling loads( 0.04) = expP

exceed the simulated buckling loads sP in all cases, except for the 2 shells that buckle initiallyin a stable local buckle. However, looking at the results presented in Figure 6, one sees that if


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one uses a boundary imperfection amplitude of B 0.058 = , then the simulated buckling loadsyield a lower bound to all seven shells tested.

SS 3P C 4P s BP ( 0.04) = localP expP

A-7 -880.0 -910.2 -689.9 -633.5 -682.9

A-8 -900.1 -976.5 -748.6 -825.9

A-9 -911.7 -926.9 -694.9 -837.4

A-10 -1014.2 -1037.0 -805.6 -685.9 -718.7

A-12 -1009.4 -1044.0 -823.1 -866.2

A-13 -850.2 -876.6 -673.5 -698.9

A-14 -892.4 -905.5 -667.3 -774.0

Table 1 Summary of Level-3 Buckling Load Calculations (161x161 mesh; buckling loads in lbs.)

Figure 6: Combined effect of measured midsurface and boundary imperfections.


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To obtain a , the verified high fidelity knockdown factorof an imperfect shell, thesimulated buckling loads using both midsurface and boundary imperfections with B 0.058 = are normalzed by the lowest buckling load of the corresponding perfect shell with C-4 (u = u0,

v = w = w,x = 0) boundary conditions. From the results listed in Table 2 it is clear that usinga 0.6 = one obtains indeed a safe allowable axial load for all seven shells tested.

s BP ( 0.058) = 0.058C 4

PP

a C 4P C 4P expP

A-7 -545.2 0.599 -546.1 -216.6 -6821.6

A-8 -595.8 0.610 -585.9 -237.3 -825.9

A-9 -553.0 0.597 -556.1 -222.5 -837.4

A-10 -659.7 0.636 -622.2 -255.1 -718.7

A-12 -677.0 0.648 -626.4 -256.8 -866.2

A-13 -533.4 0.608 -526.0 -207.8 -698.9

A-14 -516.3 0.570 -543.3 -212.8 -774.0

Table 2 Comparison of simulated high-fidelity versus experimental buckling loads (buckling loads in lbs.)

It is interesting that the proposed high fidelity simulated allowable buckling loads areabout twice as big as the allowable buckling loads of the traditional lower bound designcriteria of NASA SP-8007.

5: Conclusions

It is shown that using the measured midsurface and boundary imperfections, the so-calledmanufacturing signature of a fabrication process, it is possible to derive a safe, verified highfidelity knockdown factor a by the numerical simulation approach described in Ref. [22] andin this paper. It is believed that in the end the use of high fidelity numerical simulation willalso lead to overall cost reduction, since the analysis and design phase will be completedfaster and only the reliability of the final configuration needs to be verified by structuraltesting.


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6: Acknowledgement

Part of the research reported in this paper has been carried out during the second authorstenure as an NRC Research Associate at the NASA Langley Research Center. This support isgratefully acknowledgement.

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HOL - The Use of High-fidelity Analysis for Reliable Buckling Load Calculations