Department of Mechanical Engineering and Materials Sciences Durham, NC 27708
Michael L. Tinker*
Structural Dynamics and Loads Group/ED21 Structures, Mechanics, and Thermal Department
NASA Marshall Space Flight Center Huntsville, AL 35812
Dr. Lawrence N. Virgin‡Duke University
Department of Mechanical Engineering and Materials Sciences Durham, NC 27708
Kara N. Slade**
Structural Dynamics Branch NASA Langley Research Center
Hampton, VA 23681
GEOMETRIC SCALING PROPERTIES OF INFLATABLE STRUCTURES FOR USE IN SPACE SOLAR POWER GENERATION
Abstract
Investigation of the geometric scaling properties of polyimide film inflatable booms is described. These structures have considerable potential for use in space with solar concentrators, solar sails, and space solar power systems including solar arrays. Multiple cylindrical test articles were fabricated, utilizing two different thicknesses of Kapton polyimide film and seven aspect (slenderness) ratios. Numerous static bending and axial buckling experiments were conducted and compared to computer simulations using the MSC/NASTRAN program. Both beam element models and shell element models were developed for several inflatable struts and compared to experimental test results. Several problems encountered during the construction, experimentation, and finite element analyses are described. These included creating the proper experimental setup for static testing and establishing the optimum finite element analysis process for the analytical models. Many of these problems were overcome in the course of the research. Using the results from both experimental and analytical aspects of the research effort, guidelines for appropriate analysis techniques and experimental test article design were determined. These can be used to approximately determine properties of large-scale structures, which can not be tested in laboratory experiments. ______________________ †Graduate Research Assistant ‡Associate Professor, Department of Mechanical Engineering and Materials Science *Structural Dynamics Lead Engineer; Associate Fellow AIAA **Research Engineer, Member AIAA Copyright cO 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Introduction and Background
Inflatable structures have been the subject of renewed interest in recent years for space applications such as communications antennas, solar thermal propulsion, and space solar power. A major advantage of using inflatable structures in space is their extremely low mass. The ability to deploy in orbit and space savings during launch and pre-orbital flight is a second advantage. A recent technology demonstrator flight for inflatable structures was the Inflatable Antenna Experiment (IAE) that was deployed on orbit from the Shuttle Orbiter. Although difficulty was encountered in the inflation/deployment phase, the flight was successful overall and provided valuable experience in the use of such structures (Ref. 1).
The particular emphasis of this paper is space solar power applications of thin film inflatable structures. Power generation in space for consumption on earth requires extremely large structures in orbit for collecting solar energy (Figs. 1-3). Sizes on the order of kilometers may be necessary to generate the levels of power needed for practical use. Obviously for very large structures to be feasible for construction in space, they must be easily deployable and as lightweight as possible. Otherwise, the number of launches required to place the structures in orbit becomes cost prohibitive. For these reasons, inflatable structures technology is attractive for consideration in space solar power systems.
Many investigators have considered the use of inflatable structures for space applications. Perhaps the earliest was Frei Otto (Ref. 2), who in 1962 published ideas for inflated tubular frames for use in structures such as orbiting platforms. A more recent proposed application involves the use of inflatable beam segments to replace solid segments of the Space Shuttle remote
43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con22-25 April 2002, Denver, Colorado
manipulator system and thus reduce storage space and inertia of the arm (Ref. 3).
Several papers on static structural analysis of inflated cylinders have been written, describing different techniques such as linear shell theory, and nonlinear and variational methods (Refs. 4-12), but very little work had been done in dynamics of inflatable structures until recent years. In 1988 Leonard (Ref. 13) indicated that elastic beam bending modes could be utilized in approximating lower-order frequencies of inflatable beams. Main, et al. wrote a significant 1995 paper describing results of modal tests of inflated cantilever beams and the determination of effective material properties (Ref. 14). Changes in material properties for different pressures were also discussed, and the beam model was used in a more complex structure. The paper demonstrated that conventional finite element analysis packages could be very useful in the analysis of complex inflatable structures. Reference 15 describes an investigation of the dynamics of polyimide thin-film inflated cylinders, and Refs. 16-24 discuss recent dynamic tests/analyses and potential applications of inflatable solar concentrators.
Greschik et al. (Ref. 25) recently presented work discussing the scale model testing of inflatable structures using a constant thickness scaling rule. The current paper attempts to build upon that work and explore computational and experimental issues arising in the modeling and testing of scale models of inflatable structures. These issues include finite element model complexity and accuracy and differences between analytical and experimental results.
Fabrication Methodology
The Kapton samples for testing were constructed by wrapping film around a length of PVC tubing with a one-inch overlap. A wire or string was placed between the film and the tubing in order to facilitate removing the bonded tube from the PVC. The film was bonded together using silicon gel (Fig. 4). End caps were constructed from closed-form polystyrene (Fig. 5). Silicon gel was used to bond the end caps to the film and to fill in any micro-holes and tears in the film that caused significant air leakage. An eighth-inch hole was drilled into one of the end plugs to facilitate air inflation and pressurization. Samples were constructed from both 1-mil (.001 inch) and 2-mil (.002 inch) thick Kapton-HN film. For static bending testing, three aspect ratios of samples were constructed, which were 12:1, 6:1, and 4:1 (Fig. 6). For axial buckling testing, four test sample sizes were constructed, which were 10:1, 13:1, 15:1, and 20:1.
Static Bending Experiments and Results
Much of the impetus for this study comes from the results seen in load-deflection testing to large amplitudes of an inflated strut fabricated of 2-mil Kapton, 8 feet long and 6 inches in diameter (fully described in Ref. 15). The strut was suspended in the cantilevered configuration as seen in Fig. 7 and tested at two pressures, 0.5 and 1.0 psig. A typical load-deflection curve measured near the free end of the strut, Fig. 8, shows a distinct deviation from linear behavior at a deflection of approximately 1.5 inches. At this point in the test, a distinct ‘popping’ sound could be heard and was noted to correspond to the deviation in linearity and to the onset of wrinkling near the cantilevered end of the strut.
Rotation of the test article by 90 degrees places the 1-inch longitudinal seam of the strut in a different orientation with respect to the applied load, and a smaller degree of softening may be noted in Fig 9. The fitted slopes for the linear portions of the load-deflection curves (Table 1) indicate that there is indeed a significant difference in stiffness between the two orientations, i.e. the rotated case is stiffer. (It is to be noted that the units of the slope are the inverse of stiffness, α=1/K.)
Static bending testing was performed on six smaller constructed samples to measure the effects of material thickness, aspect ratio, and seam location on static inflatable strut behavior. A vertical test stand was constructed to mount all test samples and provide constant 1 psig air pressure during testing (Fig. 10). All test samples were incrementally loaded and end deflection measurements were taken. Each test sample was loaded in three different orientations, with the seam located at 0°, 90°, or 180° relative to the direction of the loading. Multiple tests were run on each sample in each orientation. In all cases, deflection was measured until the test sample failed due to localized material wrinkling at the cantilevered base, after which no further load could be sustained.
The results of the static bending testing are presented in various manners, including load-deflection plots for different seam orientations. Fig. 11 shows a sample plot of the load deflection curves for a 2-mil thick strut with a 6:1 aspect ratio. The experimental results point to the influence that seam location has on the static bending stiffness and the onset of behavior deviatoric to linear beam theory. The behavior of the structure is relatively linear for low level loading, but as loading increases nonlinear material effects begin to arise. The onset of these effects is clearly related to the beam orientation.
Interestingly, the influence of the seam orientation increases as the aspect ratio of the test samples rises. Fig. 12 shows a plot of the flexural rigidities calculated for the various static bending tests performed. These
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values are normalized compared to the theoretical flexural rigidities of a cantilever beam and plotted versus the aspect ratio of the sample. As Fig. 12 shows, the spread between the experimental flexural rigidities is low when the aspect ratio is low. This effect is amplified as the aspect ratio increases. In all the samples tested, the seam was kept at a constant size in order to facilitate construction. It was not scaled with the aspect ratio. However, the differences in the moment of inertia caused by the seam are taken into account in the computation of the flexural rigidities. This leads to the conclusion that for the high aspect ratio samples, there is a significant contribution to the flexural rigidity from the silicon bonding gel used, resulting in the spread of the data. In the case of the 12:1 aspect ratio struts, the seam area accounts for nearly 17% of the surface area of the strut. By comparison, the 6:1 aspect ratio strut, which did not exhibit large differences in flexural rigidity with respect to seam orientation, has a seam area which is less than 10% of the overall surface area.
The general trend of the experimental data is upwards, approaching the theoretical flexural rigidity of a cantilever beam as the aspect ratio increases.
Axial Loading Experiments and
Results
Axial loading experiments were also conducted in an attempt to determine the critical aspect ratio for transition between localized material buckling and Euler column buckling. A vertical test stand was constructed to mount each sample and provide constant pressure air supply. Each test sample was mounted to simulate a vertical cantilever beam. Fig. 13 shows the test apparatus used. Axial load was supplied by placing incremental mass on the top of the samples. The occurrence of buckling was noted through visual inspection and noting the decrease in the bending natural frequency as the load increased. A constant air pressure of 1.0 psi was maintained during testing. All samples used were constructed from 2 mil (.002 inch) thick Kapton and were 3.50 inches in diameter.
From the experimental aspect ratio testing, the transition ratio from localized to Euler buckling was estimated. Table 2 contains the results of the different aspect ratio testing. It was found that for an aspect ratio of less than 15:1, the type of buckling that occurred in the inflatable samples was local surface buckling at the base of the sample. For booms with an aspect ratio of 15:1 or higher, Euler column buckling occurred instead of crush buckling. Fig. 14 shows an example of the Euler buckling that was observed in a 20:1 aspect ratio sample. The sample was statically stable as long as the internal pressure was maintained constant. The struts that exhibited crush buckling completely failed once this phenomenon was initiated. There was no observable post-buckled stiffness. However, in the case
of the Euler buckled columns, the struts exhibited a small amount of post-buckled stiffness and would straighten again if the load was reduced.
The buckling load of the experimental samples was normalized with respect to the Euler buckling load of a built-in column and the resulting values are plotted in Fig. 15. In all cases, the critical buckling load is below the normalized Euler buckling load. However, because buckling is an imperfection sensitive phenomenon, this is to be expected. It is again noted that as the aspect ratio of the test samples increases, the load rises to approach the theoretical buckling load.
Finite Element Analysis General Parameters
In addition to experimental results, finite element models were also used in an attempt to predict the behavior of the various test samples. Certain common parameters were used in the modeling and analysis of the inflatable structures during the course of this research effort. They include the material properties and types of finite elements used. MSC/NASTRAN was utilized in all cases as the analysis program.
The materials common to all the analyses include Kapton-HN polyimide film and polystyrene. The manufacturers provided the Kapton material modulus, density, and Poisson’s ratio. The values used were
E=360 ksi, ν=0.34, and ρ=0.0513 lbm/in3
. The closed-form polystyrene material was modeled with properties
of E= 464 ksi, ν=0.25, and ρ=0.00108 lbm/in3
.All beam element models were created using 1-
dimensional first order beam elements. All shell models were created using first order isoparametric quadrilateral plate elements. Both .001 inch and .002 inch thick plate elements were used for the Kapton film in analysis. Isoparametric quadrilateral elements were also used for the polystyrene end plugs, with a thickness of 0.75 inches.
The greatest difficulty encountered in modeling and analysis was obtaining convergence of the nonlinear static pressurization solution in MSC/NASTRAN. This difficulty is due to the extremely thin polyimide material of which the inflatable struts are constructed. The ratios of film thickness to overall geometric dimensions such as strut cross-section appear to be the critical parameters in the nonlinear pressurization analysis. For example, it was found that increasing the film thickness to 10 mils (.01 inches) resulted in solutions that converged easily. The reason for this phenomenon appears to be that extremely thin films tend to have large displacements and large stiffness changes when pressure loading is applied. Obviously, increased thickness reduces the magnitude of this sensitivity.
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The method for analyzing the finite element models had to be carefully considered. All nonlinear finite element solvers, including the algorithms used in MSC/NASTRAN, rely on an incremental loading approach in order to reach a convergent solution. Due to rapid stiffness changes, displacements and instability of the static solutions during the early stages of analysis, a true Newton-Raphson tangent stiffness update method was employed in the analysis of the shell models. Although computationally demanding, this method helped to keep the analysis stable in the early stages until the model stiffened due to the internal pressure. Another reason that a Newton-Raphson method was employed was in order to guard against numerical errors late in the solving process that could arise in a quasi-Newton solution scheme. It is recognized that quadrilateral plate elements do not capture the true nature of the Kapton material. This could not be avoided in order to obtain a converged solution. Numerous attempts were made to model the membrane behavior of the material, none of which were successful. In all cases, singularities in the stiffness matrix quickly arose or the stiffness matrix become non-positive definite and could not be inverted. The Newton-Raphson solver was chosen to expose the load values at which a solution would become numerically unsolvable immediately.
Finite Element Modeling Approach and Results for Static Bending Analysis
The finite element modeling technique for the static bending analysis utilized two different modeling approaches. The first method was to create beam element models using 1-dimensional elements with the appropriate material properties for Kapton, the bending load at the correct location, and the appropriate cross sectional dimensions for each test sample. The second method was to create a more complex shell model of the beams and apply the appropriate internal pressure as well as the bending loading.
These models were analyzed through a series of load cases similar to the load increments of the static experiments. The beam models were analyzed using a linear static analysis using MSC/NASTRAN. The shell models were analyzed using a Newton-Raphson nonlinear static analysis in MSC/NASTRAN. This was done instead of a linear analysis to account for large displacements and pressure stiffening effects of the internal pressure on the Kapton material. In both cases the maximum tip displacement was noted for comparison with the experimental results.
The solutions for the beam element models and the shell element models were only slightly different for all the test cases analyzed, regardless of material thickness or aspect ratio. The resulting normalized flexural
rigidity of the finite element models is plotted along with the experimental results on Fig. 12. The results for both types of models are so similar that they are indistinguishable when plotted separately. Hence one curve is employed. Very little impact in the static bending results appeared to occur as a result of including internal pressure in the shell models. However, in almost all cases, the finite element models exhibited much higher flexural rigidity than their experimental counterparts. This exposes the limitations of conventional finite element analysis to capture the membrane material behavior. The finite element models as a result are only slightly less stiff than theory.
Because of the complexity of the models and the additional computation time for the nonlinear static analysis, the shell models were more computationally complex. However, little was gained as a result using the more complex geometry and including internal pressure in the analysis.
We conclude from this set of results and the earlier experiments that the finite element model results were better for the larger aspect ratio struts.
Finite Element Modeling Approach and Results for Axial Loading Analysis
The finite element modeling technique for the axial loading analysis was similar to that of the static bending analysis. Two types of models were created for each geometry of beam tested: a beam element model and a shell element model. In all cases, the appropriate material properties for Kapton-HN film and cross sectional geometry were used. Gravity loading was also included in the analysis of both the beam and shell element models.
These models were analyzed using MSC/NASTRAN to determine the critical buckling load and mode shape. In the case of the beam models, a reference load of 1 lbf was applied in the axial direction of the beam. A linear static analysis was performed to deform the geometry and achieve the necessary geometric and bending stiffness matrices. This was followed by a buckling eigenvalue analysis. The resulting lowest eigenvalue, when multiplied by the reference load, corresponded to the critical (analytical) buckling load of the beam. The normalized eigenmode corresponding to the lowest eigenvalue provided the resulting buckled shape.
The shell models were analyzed in a slightly different manner. A load equal to the theoretical Euler buckling load was applied to the shell models. Then a Newton-Raphson nonlinear static solution was performed. The output of this analysis was monitored to determine the load level at which the geometric stiffness matrix caused the global stiffness matrix to lose positive definiteness. The loading at which this occurred was noted. The loading on the model was then
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modified to slightly lower than this value and the nonlinear static analysis was repeated. If the solution converged without loss of stability, a buckling eigenvalue analysis was performed. The resulting eigenvalue, when added to the reference loading, gave the model’s buckling load.
The normalized results for the buckling analysis of the shell and beam element models are so similar to the theoretical critical loading that they are indistinguishable from each other. Due to the complexity and the extra analysis required to perform the nonlinear buckling analysis, the computational time required for the shell models is much greater than that of the beam models, with negligible improvements.
Formulation of Scaling Guidlines for Experimentation and Modeling Basedon Finite Element and Experimental
Results
Based on the comparison of the experimental results and the analytical models, general guidelines can be drawn for performing scale model testing of inflatable structures in laboratory testing and in modeling the static and buckling behavior of these structures on a large scale.
In order to better capture the true static behavior of a large-scale structure and minimize the nonlinear effects of the membrane material on flexural rigidity, scale models should use at least a 12:1 aspect ratio on structural members.
The static bending analysis pointed out the significance of the seam orientation and size with respect to the aspect ratio. Based on the spread of the data for the static bending testing, the width of the seam should be kept to a value of less than 10% of the circumference of the structural member and seams should be located in the same orientation on scale models as on full-scale structures.
Although significant work has been done to get reasonable dynamic response predictions based on finite element models, basic finite element models are insufficient to predict static and buckling behavior due to the membrane nature of the material. However, as the aspect ratio of the structural members rises, it appears that the experimental results approach the finite element results. Beam element models can be used to predict static for structures with aspect ratios above 12:1 for static bending, and 15:1 for axial loading.
Summary
Experimental and analytical results for static bending and buckling of inflatable struts were studied in this work. Varying aspect ratio, seam orientation, and
material thickness provided insight into proper modeling practices and scale model construction. The limitations of conventional finite element analysis tools to capture the behavior of such structures was explored and determined to be very significant for low aspect ratio structures. However, there is little significance between beam element and pressurized shell element model results. Clearly, a more sophisticated analysis is required to fully capture the behavior of these structures.
Acknowledgments
NASA Marshall Space Flight Center provided access to MSC/NASTRAN and support for testing. Sophia Santillon and Ingrid Abendroth provided support for test sample construction and static bending testing. Andrew Schnell provided support for test sample construction and axial deflection and buckling testing.
References
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11. Webber, J.P.H., “Deflections of Inflated Cylindrical Cantilever Beams Subjected to Bending and Torsion”, Aeronautical Journal 86(858), pp. 306-312, 1982.
12. Main, J.A., Peterson, S.W., and Strauss, A.M., “Load-Deflection Behavior of Space-Based Inflatable Fabric Beams”, Journal of Aerospace Engineering, 7(4), 1994.
13. Leonard, J.W., Tension Structures, McGraw-Hill, New York, 1988. 14. Main, J.A., Carlin, R.A., Garcia, E., Peterson, S.W., and Strauss, A.M., “Dynamic Analysis of Space-Based Inflated Beam Structures”, Journal of the Acoustical Society of America, 97(2), pp. 1035-1045, Feb. 1995.
15. Slade, K.N., and Tinker, M.L., “Analytical and Experimental Investigation of the Dynamics of Polyimide Inflatable Cylinders”, AIAA-99-1518, Proceedings of the 40th Structures, Structural Dynamics, and Materials Conference, April 12-15, 1999, St. Louis, MO. 16. Tinker, M.L., “Passively Adaptive Inflatable Structure for the Shooting Star Experiment”, AIAA-98-1986, Proceedings of the 39th Structures, Structural Dynamics, and Materials Conference, April 20-23, 1998, Long Beach, CA. 17. Lassiter, J.O., “Shooting Star Experiment Prototype Inflatable Strut/Torus Assembly Modal Survey”, ED73(97-69), NASA Marshall Space Flight Center, Huntsville, AL, June 1997. 18. Engberg, R., and Lassiter, J.O., “Shooting Star Experiment, Pathfinder 2, Inflatable Concentrator Modal Survey in Vacuum Conditions”, Dynamics Test Branch, Marshall Space Flight Center Test Report SSE-DEV-ED97-120, March, 1998. 19. Engberg, R., and Lassiter, J.O., “Shooting Star Experiment, Pathfinder 3, Inflatable Concentrator Modal Survey in Thermal-Vacuum Conditions”, Dynamics Test Branch, Marshall Space Flight Center Test Report SSE-DEV-ED97-115, February, 1998. 20. Engberg, R., and Lassiter, J.O., “Shooting Star Experiment, Pathfinder 3, Inflatable Concentrator Terminator Test, Dynamics Test Branch, Marshall Space Flight Center Test Report SSE-DEV-ED98-046, July, 1998. 21. Lassiter, J.O., and Engberg, R., “Dynamic Testing of an Inflatable Structure Under Thermal-Vacuum Conditions”, AIAA-99-1519, Proceedings of the 40th Structures, Structural Dynamics, and Materials Conference, April 12-15, 1999, St. Louis, MO. 22. Slade, K.N., Tinker, M.L., Lassiter, J.O., and Engberg, Robert, “Comparison of Dynamic Characteristics for an Inflatable Solar Concentrator in Atmospheric and Thermal Vacuum Conditions”, AIAA-2000-1641, Proceedings of the 41st Structures, Structural Dynamics, and Materials Conference, April 3-7, 2000, Atlanta, GA.
23. Slade, K.N., "Dynamic Characterization of Thin Film Inflatable Structures", Ph.D. Dissertation, Duke University , Dept. of Mechanical Engineering and Materials Science, Durham, NC, 2000.
24. Smalley, K.B.., Tinker, M.L., and Fischer, R.T., “Investigation of Nonlinear Pressurization and Modal Restart in MSC/NASTRAN for Modeling Thin Film Inflatable Structures”, AIAA-2001-1409, Proceedings of the 42nd Structures, Structural Dynamics, and Materials Conference, April 16-19, 2001, Seattle, WA.
25. Greschik, G., Mikulas, M.M., and Freeland, R.E., “The Nodal Concept of Deployment and the Scale Model Testing of its Application to a Membrane Antenna”, AIAA-99-1523, Proceedings of the 40th Structures, Structural Dynamics, and Materials Conference, April 12-15, 1999, St. Louis, MO.
Figure 1. Space Solar Power ("SunTower") Concept Based on Inflatable Structures (Picture courtesy of
NASA)
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Primary Mirrors (36 per Clamshell)
Solar Array
Docking Ports
Transmitter Array
Transmitter Radiator
1.2 gW Delivered
Figure 2. Integrated Symmetrical Concentrator ("Clamshell") Concept Based on Inflatable Structures
(Picture courtesy of NASA)
Figure 3. "Abacus" Space Solar Power Concept Based on Lightweight Deployable Structures (Picture
courtesy of NASA
Figure 4. Seam Bonding of 6:1 Aspect Ratio Test Sample
Figure 5. Bottom View of 6:1 Test Article
Figure 6. Static Bending Test Stand and Test Articles
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Figure 7. The large-scale strut in its experimental configuration.
Figure 8. Load-deflection curve for the 2 mil strut, 0.5 psig, measurement point 2.
Figure 9. Load-deflection curve for the 2 mil strut, 0.5 psig, measurement point 2, rotated 90 degrees.
