The American Society of Mechanical Engineers

Reprinted From PVP - Vol. 176. Computational Experiments

Editors: W. K. Liu, P. Smolinski, R. Ohayon, J. Navickas, and J. Gvildys

Book No. H00491 - 1989

AXIAL BUCKLING OF A THIN CYLINDRICAL SHELL: EXPERIMENTS AND CALCULATIONS

S. W. Kirkpatrick and B. S. Holmes SRI International

Menlo Park. California

ABSTRACT

Thin cylindrical shells were tested under axial compression beyond the critical huckling load. Both pretest and posttest finite element calculations were performed to calculate the huckling loads and post- huckling deformations. Finite element simulations of the shell included pretest measured imperfections in shell geometry and asymmetry in the axial load. Results show that the axial collapse load is sensitive to imperfections in both the shell geometry and the load distribution. Careful modeling of the imparfec- Zions resulted in accurate predictions of the buckling load and postbuckling deformations for the shells.

INTRODUCTION AND BACKGROUND

One of the major design criteria of thin shell structures that experience compressive loads is the huckling load limit. Bucking of a thin shell struc- ture can often lead to a catastrophic failure; there- fore, it is important that the huckling loads he known. Typically, thin shell structures are designed with large safety factors to preclude the possibility of a buckling failure. However, large safety factors result in thicker and heavier structures. For many applications, such as in the aerospace industry, the excess weight and material required for a large safety factor can greatly increase the cost of the structure. In these applications, it is important to be able to accurately predict the huckling load. This paper discusses a numerical modeling technique using finite element methods that can accurately predict the static buckling load for an axially compressed cylindrical shell.

Buckling of thin shell structures can occur under static or dynamic loading conditions. When the load duration is long compared to the response time of the structure, the structure is said to buckle statically, as in the case of an aluminum can crushed in a trash compactor. However, even though the process is called static huckling, the transition from the prehuckling to the postbuckling state is definitely a dynamic process for thin shells. Dynamic huckling occurs when

the load duration is shorter than the structural response time. In this dynamic "pulse" buckling, the amplitude of the load is of necessity higher than the static buckling load and, in general, the buckling mode is different. Examples of dynamic pulse buckling occur frequently in impact problems such as the impact of an arrow on a solid target or the response of a structure to blast loading. In both the quasistatic and dynamic huckling cases, the huckling process is initiated by imperfections in the materials and geometry of the structure and nonuniformities in the load application. In general, the critical buckling load is governed by the amplitude of the imperfections, so that structures with large imperfections have lower critical loads. This is the reason for the large scatter in data of many experimental buckling studies.

Because of the importance of imperfections on the buckling load, a third type of buckling problem is possible. In this problem, the structure is subjected to a static load below the critical static load level and also a dynamic load. The dynamic load introduces transient shape imperfections that can trigger buck- ling under the action of the static load. An example of this problem occurs when a rocket in flight is exposed to blast loading. In this study, we solved the problem of a thin cylindrical shell under a static axial load; however, the methods used were chosen so that the analysis could easily be extended to include the effects of combined static preload and dynamic external radial loads.

The stability limit for an axially compressed elastic cylindrical shell was determined more than 75 years ago by Lorenz. Southwell, and Timoshenko (1961) from linear equilibrium equations. These equations were derived for a long cylindrical shell with simply supported boundaries under the action of a uniform axial load. However, comparison of the theoretical huckling load with experimental data for thin, axially compressed shells showed that buckling occurs at load levels significantly below those predicted by this solution. This discrepancy in the huckling loads has been the subject of considerable study. Possible explanations for the difference

between the experimental buckling load and the theoretical load limit included the effects of end conditions, imperfections in shell geometry, imperfec- tions in shell thickness, nonuniformities in material properties, and nonuniformities in the load. In general, geometric imperfections are the primary reason for the discrepancy between experiment and classical theory for thin shells.

The pioneering work on the effect of geometrical imperfections on shell stability was performed by Donnell (1934) and Donnell and Wan (1950). A number of publications have used the nonlinear Donnell shell equations to examine the effects of initial imperfec- tion distributions (Arbocz, 1974) on the axial huck- ling load. By using imperfections in a single made or a combination of a few modes, the nonlinear shell equations can be solved to obtain the buckling load. With this approach, it has been shown that an imper- fection in the shape of the preferred buckling mode at a realistic amplitude can account for the discrepancy between the experimental and theoretical buckling loads. However, use of this method to predict the buckling load requires knowledge of both the preferred buckling mode and the "effective amplitude" of the shape imperfection in that mode. Solution of the Donnell equations for a more general description of the initial imperfections is considerably more complex. In addition, this method does not allow calculation of the postbuckling strength of the shell.

In the theoretical work described above, the amplitudes of the geometric imperfections were unknown. Typically, the imperfection amplitude was chosen such that the analytical prediction of the buckling load matched experimental data. More recently, surveys have been performed to measure the imperfections that naturally exist in cylindrical shell structures. These data have been collected into "data banks," and Arbocz (1982) suggests that they be used to improve design criteria for buckling load limits of thin shells. The imperfection data banks can be used in calculations to more accurately model the characteristics of the imperfections present in tbin cylindrical shell structures.

Recent computational analyses of dynamic buckling problems (Kirkpatrick and Holmes, 1988a.b) have applied this approach, incorporating in the numerical model the actual imperfections measured in the shell or a statistically accurate description of the imper- fections. In these studies, the dynamic buckling of a thin cylindrical shell subjected to an external radial impulse was modeled using finite element methods. The imperfections measured on the shells or a statistically accurate description of the imperfections was included in the finite element mesh for the shells. This tech- nique was very successful for predicting the dynamic buckling response for thin shells and was applied to the static axial buckling problem in our study.

Many static buckling problems have been solved numerically. Many authors prefer using static analysis for the buckling process, while others prefer dynamic analysis. Each of these techniques bas advantages depending on the information required by the analyst (Kroplin and Dinkler, 1986). Static analysis can rely on static equilibrium states and simplifies the numerical analvsis. However. static solutions mav not simulate the transition from the prebuckling to the postbuckling state and often must account for the existence of a large number of bifurcation paths. Dynamic analysis leads to a physically unique post-

buckling pattern. For this study, we were interested in the postbuckling defolmations and strength in addition to the buckling load limit. We also wanted to be able to extend the analysis to the case of buckling under a static axial prestress in combination with an external impulsive pressure. For these reasons, we chose to use a dynamic analysis that could cope with the difficult dynamic phenomena between the pre- and postbuckling states following a unique solution path.

The following sections describe the geometry of the problem and the measurement of the imperfections in the shell, the experiments and the experimental results, and the numerical methods used in the analysis. We then compare the predicted buckling response from the pretest calculation with the experimental results. Finally, we perform posttest calculations that incorporate the load asymmetry measured in the experiments and again compare the calculated and experimental results.

PROBLEM GEOMETRY AND IMPERFECTION MEASUREMENT

In our experiments, two tbin 6061-T6 aluminum shells with radius-to-thickness ratios ( a h ) of 150 were tested under an axial load (Figure 1). The shells used in the experiments were manufactured by spinning 30.5-cm-diameter, 22.9-cm-long cylinders on a mandrel. A 2.5-cm length at either end of the shell was clamped between a shrink-fit, 5.1-cm-thick end plate and a clamping ring to produce a 17.8-cm-long test section in the shell. When mounted in the testing machine, the fixtures produced clamped end conditions.

SHELL CHARACTERISTICS

- 6061-T6 Aluminum - 30.5 cm Diameter -a/h=150 - U2a = 0.58 - Displacement Ecundary Conditions

Fig. 1 Problem definition

Geometrical imperfections of both the shells were measured before testing. Each shell was mounted in a modified lathe bed so that it could he rotated freely. A potentiometer was used to measure the angular posi- tion of the shell, and a linear voltage displacement transducer (LVDT) was used to measure the radial position of the shell surface. Outputs from the LVDT

and potentiometer were recorded on a digital oscil- loscope while the shell was rotated. Approximately 1800 data points were obtained around the shell, and this measurement was repeated at 19 evenly spaced axial locations.

Projected three-dimensional images of the two imperfection sets are shown in Figure 2. The spatial resolution of the points plotted in the figure corres- ponds to the refinement of the mesh used later in the finite element calculations. The amplitude of the imperfections was multiplied by a factor of 500 in the figure to make them visible. The imperfections seen on the first shell (Figure 2a) are similar in character to the imperfections measured on shells made by other manufacturing techniques (Arbocz, 1982, and Kirkpatrick and Holmes, 1988). In particular, the dominant imper- fections appear as ridges along the axial direction.

A Fourier analysis of the radial imperfections was performed at the axial location at midheight on the first shell. Figure 3 shows the modal amplitude of the radial imperfections around the circumference expressed as a percentage of the wall thickness. Over the range of modes from 1 to 160, the magnitude of the modal imperfections measured on this shell is 3 to 10 times less than that of the modal imperfections measured on rolled and welded shells used in previous studies (Kirkpatrick and Holmes, 1988). The imper- fections on the shells in this study are smaller because of the spinning process used to manufacture the shells. The spinning process forms a more perfect cylindrical shell than rolling and welding, and because these shells are imperfection sensitive, the spun shells will have correspondingly higher buckling loads.

(a) Shell 1 Imperfections MODE NUMBER

'0 &J

Fig. 3 Modal composition of measured imperfections

(b) Shell 2 Imperfections

Fig. 2 Measured imperfection distributions

In contrast to the first shell, the second shell has imperfections of a different character (Figure 2b). The imperfections on this shell have some dominant features that run a significant distance around the circumference. The character of these imperfections may be a result of the spinning process, which could produce imperfections with different character from shell to shell. Another possible explanation for the differences between the first and second shells is that strain gages were applied to the second shell. Strain gage application requires polishing the aluminum surface under the gage for a good adhesive bond and warming the shell to cure the adhesive. It is possible that this process altered the character of the imperfections.

EXPERIMENTS

The cylinders were tasted in e 450-kN MTS servo- hydraulic testing machine operated in the displacement crmtrol mode. The top end plate was rigidly mounted to the upper support of the MTS machine to produce a rigid clamped end condition at the upper edge of the cylindrical shell. A load cell was mounted between

the top end plate and the support to measure the load time history of the shell. The lower end plate of the shell was loaded by a parallel plate mounted rigidly on the stroke arm of the MTS machine. The load was

by the loading plate moving upward at a uniform velocity during the experiments.

The first experiment was performed at a loading rate of 1.26 cm/s. Shell 1 buckled at a critical load of 205 kN, which corresponds to an axial buckling stress of 210 MPa. The shell buckled into a mode number of 9, with a buckling pattern that was not uniform around the circumference. The postbuckling deformations for the first shell experiment are shown in Figu~e 4. The asymmetry of the buckling pattern around the circumference suggested that the load may not have been perfectly uniform around the circum- ference of the shell. Misalignment of the lower end plate and the loading plate on the MTS arm would produce the asymmetry in the load.

To address the question of load asymmetry, we placed axial strain gages at five locations around the circumference of the second shell to measure the

uniformity of the strain distribution produced by the load. When the shell was first mounted in the MTS machine and an initial preload was applied, the uniformity of the strains around the circumference varied by approximately + 5 % . Shims were used between the loading plate and the end plate of the shell to reduce the load asymmetry to approximately +4%. When this level of uniformity was achieved, the second experiment war performed.

The second experiment was performed at a loading rate of 0.20 m/s. Shell 2 buckled at a critical load of 228 kN, which corresponds to an axial buckling stress of 234 MPa. This shell buckled in a mode number of 9, wich a uniform buckling pattern. The postbuckling deformations of the second shell are shown in Figure 5. The two uniform rows of buckles are slightly offset toward the upper end of the shell. Uniformity of the axial strains measured in the ex- periment was within +3.6%. The measured postbuckling collapse load was approximately 76 kN, which corres- ponds to a postbuckling average axial stress of approximately 80 MPa.

Fig. 4 Buckled shape of Shell 1

Fig. 5 Buckled shape of Shell 2

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PRETEST CALCULATION

The DYNA3D computer code used in the analysis of shell response was developed at Lawrence Livermore National Laboratory (Hallquist and Benson, 1986). DYNA3D is an explicit finite element code for the nonlinear dynamic analysis of structures in three dimensions. The equations-of-motion are integrated in time using the central difference method. Spatial discretization was achieved with the Belytschko-Tsay shell element (Belytschko and Tsay, 1981). This is a four node element with single point integration and a stabilization procedure for the kinematic modes (Belytschko and Tsay, 1983). The Belytschko-Tsay element offers the advantage of allowing a time step size that is insensitive to shell thickness and larger than that for corresponding brick elements. This feature permits more economical solutions and is very important in this application, where many elements are required for a good simulation.

The material model used in the calculation was an elastic-plastic model with combined isotropic/kinematic hardening (Krieg and Key, 1976), a Mises yield func- tion, and an associated flow rule. A combination of 20% isotropic and 80% kinematic hardening was used. A yield stress of 276 MPa was used in the material model, with an elastic modulus of 69 GPa and a hardening modulus of 586 MPa. These values were baaed on quasi- static tensile tests of 6061-T6 aluminum sheet stock.

The mesh for the pretest calculation had 100 elements uniformly distributed around the circumference and 20 elements along the length. This distribution produced elements with an aspect ratio of approximately 1 in the circumferential and axial directions. Thi mesh used in the calculations is shown in Figure 6. Because of the measured imperfections that are super- imposed on the mesh generated for a perfect cylindrical shell, no planes of symetry exist in the mesh. The use of symmetry planes also places constraints on both the locations in which buckles can form and the modes into which the shell can buckle. The mesh used in these calculations modeled the entire shell to avoid these difficulties

Both rotational and translational degrees of freedom for the nodes at the upper end of the cylindri- cal mesh were fixed. Similarly, at the lower end of the mesh, all degrees of freedom except the translation along the axis of the shell were fixed. Nodal velocities along the shell axis were specified for these nodes at a uniform rate of 10.2 cm/s. To facilitate a smooth load application at this rate, we applied a linear initial velocity field to all the nodes in the mesh, with initial velocities varying from 0.0 em/= at the fixed end to 10.2 cm/s at the loaded end. This velocity field eliminates any stress wave reverberations along the shell that would occur if the velocity were specified only at the end of the shall. Additional calculations that demonstrate the effect of loading rate are discussed below.

The postbuckling deformations predicted by the pretest calculation are shown in Figure 7. The calculation predicts that two uniform rows of buckles around the circumference, slightly offset toward the loaded end, will be produced by the axial load. The predicted buckling made number is 10 at a critical buckling load of approximately 235 kN, which produces an axial buckling stress of 240 MPa.

Comparison of the pretest calculation with the results of the first experiment shows that the calcu- lation overpredicted the critical buckling load by approximately 15%. In addition, the calculation predicted a uniform buckling pattern with a mode number of 10, but a nonuniform buckling pattern with a mode number of 9 was observed in the experiment. Although the agreement between the analysis and experiment is reasonable, we investigated further to explain the 15% difference observed. In the next section, we describe how more accurate solution requires introducing the effect of load asymmetry.

POSTTEST CALCmATIONS

The first posttest calculation was performed using the same mesh as the pretest calculation that incor- porated the imperfections measured on Shell 1. The objective of the calculation was to investigate the effects of the load asymmetry on the critical buckling load and the postbuckling deformations. On the basis of initial measurements of the strain in the second experiment, before adjustments were made to reduce the load asymmetry, a side to side load variation of 15% was used in the calculation. The imperfection of the load was input as a fractional change in the end velocity proportional to 15% of the uniform velocity (10.2 cm/s) times the cosine of the angular position. The comparison of the postbuckling deformations predicted by this calculation with the experimental results are shown in Figure 8a. The shell buckled in a mode number of 9 at an axial stress of approximately 195 MPa, which corresponds to a buckling load of 185 . This buckling load is approximately 8% lower than the value measured in the first experiment. In addition, the calculation properly predicts the mode number and the nonuniform character of the buckling pattern observed in the experiment. These results suggest that the load asymmetry was responsible far the initial disagreement between the pretest calculation and the experimental results for Shell 1.

Fig. 6 Mesh used for finite element calculations

Fig. 7 Buckle pattern from pretest calculation of Shell 1

(a) Shell 1

(b) Shell 2

Fig. 8 Comparison of posttest caledlations with experiments

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The second posttest calculation was performed to duplicate the the second experiment performed on Shell 2. The mesh used the imperfection set measured on Shell 2 (Figure 2b) and an imperfection in the load equal to 3.6% of the uniform end velocity (10.2 cm/s) times the cosine of the angular position plus a phase shift chosen to approximate the measured distribution. A comparison of the buckled shape predicted by this calculation with the experimental result for Shell 2 is shown in Figure 8b. The calculation predicted that the shell would buckle at an axial load of 232 kN, which corresponds to an axial buckling stress of 238 MPa. This buckling load is within 2% of the experimentally observed buckling load for Shell 2. The calculation predicts a buckling mode number of 9, with a nonuniform buckling pattern around the circumference. The experi- mentally observed mode number was also 9; however two uniform rows of buckles were produced, with a regular buckling pattern around the circumference.

The final two posttest calculations were performed to demonstrate the effects of the loading rate on the calculated buckling response and postbuckling strength. The mesh from Shell 2 was used with loading rates of 20.3 and 5.1 cm/s, which are, respectively, twice as fast and twice as slow as those used in the previous calculations. The buckling patterns predicted by the three calculations were very similar. A comparison of the average axial stresses for the three calculations is shown in Figure 9. This figure shows that there is a slight increase in the peak compressive load each time the loading rate is doubled. An explanation for these differences is that buckling is initiated at a critical average axial stress of 232 MPa but the buckling process requires approximately 150 ps before the axial stress begins to drop off. For the slowest loading rate, the end displacements that occur during this time increase the peak axial stress over the buckling stress by only 3 MPa. However, at loading rates of 10.2 and 20.3 cm/s, the peak axial stresses are 6 and 12 MPa greater than the buckling stress, respectively. These increases correspond to the magnitude of the differences in the peak axial stresses seen in Figure 9.

An additional observation made from Figure 9 is that all three calculations predict the postbuckling average axial stress to he approximately 115 MPa, which corresponds to a postbuckling collapse load of approxi- mately 112 kN. This value for the calculated posthuck- ling strength overpredicts the measured postbuckling strength by nearly 50%. The difference may partly be due to fundamental differences between the experiments and the numerical simulations. These differences can be seen in Figure 10, which plots the average axial stress measured in the experiment (measured load divided by shell cross-sectional area) against the testing machine stroke arm displacement. A comparison of Figures 9 and 10 shows that the compliance of the cylinder and testing machine together is approximately 2.5 times the compliance of the shell alone from the calculated response. Therefore, when the shell buckles in the experiments, larger dirplacsments are produced in the shell because of the compliance of the testing machine and the stored energy that is released (increased shell end displacements) as the load drops. In the calculations, the end displacements are specified and there is no jump in the end displacements when the shell buckles. Therefore, the buckles in the calculation immediately after the buckling occurs are not as large as those in the experiment. The more developed buckling pattern in the experiments may well produce a lower postbuckling strength.

0.02 0.04 0.06 0.08 0.10 0.12 END DISPLACEMENT (cm)

Fig. 9 Comparison of average stress histories for different loading rates

0 0.05 0.10 0.15 0.20 0.25 STROKE ARM DISPLACEMENT (cm)

Fig. 10 Measured average axial stress against stroke a m displacement for Shell 2

SUMMARY AND CONCLUSIONS

Careful imperfection surveys were performed on thin aluminum cylindrical shells tested under axial compression beyond the critical buckling load. The first shell buckled at an average axial stress of approximately 210 MPa. The buckling mode number was 9, and the buckling pattern was not uniform around the circumference or along the length. The nonunifomity of the buckling pattern suggested that the load was asymmetric. Strain gages were placed on the second

shell to obtain a measure of the uniformity of the load around the circumference. The initial side-to-side asymmetry wan approximately f15% when the shell was first placed in the test fixture. With careful adjust- ment, the load symmetry was reduced to approximately t3.6%, and the shell was then tested. The shell buckled at an average axial stress of approximately 238 MPa. The shell buckled into a mode number of 9, with two uniform rows of buckles around the circum- ference. The postbuckling strength produced an average axial postbuckling stress in the shell of approximately 80 MPa.

Two methods of analysis are available for perform- ing numerical simulations of the buckling process. The first approach is to perform a static analysis that treats the axial buckling as a bifurcation process. The static analysis approach allows simplifications in the numerical methods used hut has the difficulty that a number of different bifurcation paths are possible. The second approach, which was used in this study, is to perform a dynamic analysis of the buckling process. When the dynamics are included in the numerical analysis, a unique buckling solution is obtained.

Finite element simulations of the axial buckling process were performed using the explicit three dimen- sional code DYNA3D. Imperfections were included in the calculations by superimposing the measured imperfec- tions on the mesh for a perfect cylinder and including the load asymmetry in the displacement boundary conditions. The finite element calculation predicted the buckling mode and overpredicted the buckling load by approximately 2% when both the geometric and load imperfections were modeled. The calculations over- predicted the postbuckling strength by approximately 50%; however, this discrepancy may have resulted from the compliance of the testing machine, which was not accounted for in the calculations.

Both this study and previous work on dynamic buckling problems (Kirkpatrick and Holmes, 1988) have shown that, when the imperfections in the geometry of the structure and the imperfections in the applied load are accurately modeled, the buckling process can be accurately modeled by dynamic finite element analysis. However, in both of these buckling processes, small changes in the imperfections can result in comparative- ly large changes in the mode number or buckling load. The consequence of this imperfection sensitivity is that, although the solution obtained from a dynamic analysis is deterministic, it can also be considered chaotic. If the imperfections of a given structure are known, the buckling load for that structure can be accurately predicted, potentially within a few percent. However, there will be a statistical variation between the buckling load from the first structure and that from a second structure made by the same manufacturing process and loaded in similar fashion. Consequently, the numerical modeling approach used in this study is extremely useful when the buckling load for a single structure needs to be accurately predicted. However, this approach can also be useful for analyzing large groups of structures by investigating statistical variations in imperfections and the imperfection sensitivity of the structures.

We feel that these modeling techniques (accurately modeling the imperfections and using dynamic analysis) are very useful for analysis of buckling processes and potentially useful for many different problems beyond buckling that incorporate imperfection sensitivity, bifurcation, localization, or softening. Some examples include fracture of brittle materials under static loading, ductile or brittle fracture under dynamic loading, and development of turbulence in fluid flow.

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