1
Large Deformation Analysis of Inflated Membrane Boom Structures with Various Slenderness Ratios
Jin-Ho Roh*, Eun-Jung Yoo**, In Lee†and Jae-Hung Han‡
Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, Korea
A comparison of numerical schemes for the nonlinear behavior of inflated membrane boom structures due to the wrinkling is investigated. For a finite element analysis, two different numerical schemes are applied. In the first scheme, the structure is numerically modeled using membrane elements, and taut, wrinkled and slack membrane states are determined at each load step using the Tension Field theory. In the second scheme, the structure is modeled with shell elements, and the nonlinear deformation is analyzed using a post-buckling analysis. A comparative study of the wrinkling problems of the inflated membrane boom in bending is carried out.
Nomenclature
= stress
= strain
D = effective elasticity matrix
2 = minor principal stress
1 = major principal strain
= principal angle Fs = stretching force Fc = critical collapse load Mc = collapse bending moment
I. Introduction
The inflatable structure is a specific application of a membrane that usually implies that it has a thin and low- modulus. Inflatable structures are generally found either in tensioned-planner or inflated-curved configurations.
Compared with traditional mechanical structures, inflatable structures have the advantage of a much lower cost, weight, and packaging volume, as well as more a favorable thermal gradient and damping characteristics. These structures can be packaged into smaller volumes, and therefore can reduce the overall cost of a space mission by reducing the launch vehicle size requirements. However, surface distortion of the structure may be easily induced by the boundary conditions, thickness variations, wrinkling, thermal distortions, membrane inflation level, and surface roughness in the membrane material itself.
One of the most harmful situations for the membrane structure is the growth of wrinkling, as this phenomenon can rapidly induce the total collapse of the structure. For this reason, much research on regarding the wrinkling phenomenon has been performed over the years. Conventionally, the wrinkling phenomenon has been treated as a plane stress problem; wrinkled regions can be determined using the Tension Field (TF) theory [1]. By eliminating compressive stress through a modification of the constitutive relationships, TF theory enables the prediction of the wrinkled region and wrinkling orientations in membrane structures. Following a modified version of the TF theory, Miller and Hedgepeth [2] performed a finite element analysis using a recursive stiffness-modification procedure
*Research Engineer, [email protected]
**Graduate Student
†Professor, [email protected], Associate Fellow AIAA.
‡Associate Professor, [email protected], AIAA Member.
48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con
23-26 April 2007, Honolulu, Hawaii
AIAA 2007-1807
termed the Iterative Membrane Properties (IMP) method. Several computational efforts [3, 4] have used the IMP and penalty-based formulations of the TF theory by implementing appropriate routines into the nonlinear finite element program, ABAQUS. However, the major shortcoming of the TF scheme is its inability to predict the wavelengths and amplitudes of wrinkles. This shortcoming may be overcome by modeling thin-film structures using shell elements that include both membrane and bending deformations. Recent advances in nonlinear computational programs support viable analyses of highly nonlinear wrinkled deformations in thin membranes using shell elements. Using shell elements in the nonlinear finite element software, ABAQUS, Wong et al. [5] investigated the wrinkling phenomenon of a membrane structure. Their simulation involved the use of buckling eigenvectors to describe small out-of-plane geometric imperfections over the entire spatial domain of a membrane and could predict the wrinkled deformation through a post-buckling analysis. However, obtaining stable equilibrium states using shell elements was found to be a challenging computational problem [6]. These aforementioned numerical schemes on wrinkling analyses using either the Tension Field theory or a post-buckling analysis are well established. They have been applied mainly to the wrinkling problems of a flat stretched membrane subjected to a pure bending or twisting moment. In the analysis of wrinkling behaviors of a flat stretched membrane, two different approaches were compared each other and similar numerical results were observed [7]. However, for inflated boom structures, very few studies have sought to verify or compare two different numerical schemes for predicting the wrinkling behavior of inflated membrane boom structures.
In this research, the applicability of numerical schemes to the nonlinear analysis of inflated membrane boom structures due to wrinkling is investigated. For the finite element analysis, two different numerical schemes are applied. In the first scheme, the structure is numerically modeled using membrane elements, and the membrane states of taut, wrinkled and slack are calculated at each load step with a wrinkling algorithm based on the Stein- Hedgepeth membrane model. In the second scheme, the structure is modeled with shell elements, and the nonlinear deformation is investigated using wrinkling mode shapes and a post-buckling analysis. The first and second schemes are termed the Tension Field (TF) scheme and the post-buckling (PB) scheme, respectively, in this study. A comparative study on the wrinkling problems of an inflated membrane boom in bending is carried out in order to understand whether the two different numerical schemes are applicable to the description of wrinkling behaviors of the inflated membrane boom structures. The nonlinear behaviors of the structure are investigated for various slenderness ratios, and the results are compared with experimental results.
Laser Sensor
Compressor
Pressure Indicator
Pulley
Weight
(a) Test configuration (b) Inflated membrane boom models
II. Inflated Membrane Boom in Bending
The bending behavior of inflated membrane boom structures is experimentally and numerically investigated in this study. The experimental model was fabricated of a thin ( 30 mµ ) Kapton film, a thin film product of the Dupont company. In the numerical analysis, two different numerical schemes based on the Tension Filed (TF) theory using membrane elements and the post-buckling (PB) analysis using shell elements were applied.
The bending test was carried out on inflated membrane booms with various slenderness ratios (Fig. 1) in order to prove that the measured bending stiffness and the maximum load limit are consistent with the numerical predictions.
The test frame was designed to stand the inflated membrane boom up vertically and measure the deflection of the end tip when it was loaded in a lateral direction. Load was applied via weighting at the tip point of the boom, which was covered with an acryl hoop. Tip deflection was measured by a laser sensor and recorded using a data acquisition system. Nitrogen gas was used as the inflation gas, and the internal pressure was measured by a pressure sensor attached to the bottom of the frame.
Figure 2 shows a numerical model of the inflated membrane boom. A rigid body constraint was applied to the upper edge. When the boom structure was pressurized, the membrane was tensioned longitudinally as well as laterally. Considering the pressure load to the upper end cap surface, the stretching force (F ) was applied. Thes thickness, Young’s modulus, and Poisson’s ratio of the Kapton film were t=30µm, E=3 PaG , and =0.34, respectively. In the finite element analysis, a Tension Field (TF) scheme using 660 membrane elements (M3D4) and a post-buckling analysis using 5,000 shell elements (S4R5) were applied. In the post-buckling (PB) scheme, the mesh density could significantly influence the results. Considering the convergence test and computational time, a mesh with 100 elements in the longitudinal direction and 50 elements around the circumference was used. However, in the TF scheme, the membrane states of taut, slack and wrinkled were calculated at each load step. Therefore, the mesh density of the model using the TF theory is not as important as that if using the PB scheme. For a reduction of the computational time, a mesh 33 elements in length and 20 elements around the circumference was used in the TF scheme.
Y X Z
Rigid body D Fs
F
L
Fig. 2 Numerical model of inflated membrane boom.
A. Tension Field (TF) scheme using membrane elements
In order to investigate the wrinkled region in the membrane, the wrinkling model of Stein and Hedgepeth [1] is applied. This wrinkle theory applies to membranes that are elastic, isotropic, have no bending stiffness and cannot carry compressive stress. If both principal stresses are tensile, the membrane is taut. If both principal stresses are zero, the membrane is slack. However, if one principal stress (minor) is zero and the other (major) is tensile, the membrane is wrinkled. A numerical algorithm of the Stein-Hedgepeth wrinkling model may be expressed as:
=D (1)
where, =
{
xx, yy, xy}
T, ={
xx, yy, xy}
T, and D is the effective elasticity matrix that relates stresses and elastic strain within a membrane element. A useful algorithm for choosing the D matrix, the Stress-Strain Criteria is used [8]:Stress-Strain Criteria
2 >0: taut (2a)
1 0 : slack (2b)
1>0 & 2 0: wrinkled (2c)
where 2 and 1 are a minor principal stress and a major principal strain, respectively.
The effective elasticity matrix at each membrane state can be expressed as follows:
2
1 0
1 0
1 1
0 0 2
Taut
= E
D (3a)
0 0 0
0 0 0
0 0 0
Slack =
D (3b)
( )
( )
2 1 0
0 2 1
4 1
Wrinkled
P Q
E P Q
Q Q
+
=
D (3c)
where, P=cos 2
( )
,Q=sin 2( )
, and is the principal angle. The effective elasticity matrix when taut follows the linear elastic isotropic material constitutive formulation. For slackness, the effective elasticity matrix is formulated based upon the assumption that no stress is formed. The wrinkled elasticity matrix is formulated from the constitutive relationships using the variable Poisson’s ratio [1].The numerical analysis involves nonlinear stress-strain behavior; consequently, an iterative solution is required.
The resulting model was analyzed with the same loads, and the process was repeated until no compressive stresses remained in the membrane. To simulate the wrinkling phenomenon, a numerical algorithm of wrinkling based on the Stein-Hedgepeth membrane model was developed and implemented using the user-defined material (UMAT) subroutine supported by the ABAQUS finite element program (Fig. 3).
START
INITIAL CONDITIONS
STRESSES AND STRAIN
PRINCIPAL STRESSES AND STRAINS PRINCIPAL STRESS
ANGLE
CHECK TAUT CRITERIA
CHECK WRINKLED CRITERIA
CREATE TAUT ELEMENT PROPERTY
CREATE WRINKLED ELEMENT PROPERTY
CREATE SLACK ELEMENT PROPERTY
RETURN
RETURN
RETURN YES
NO
YES
NO
UPDATE STATE VARIABLES
UPDATE STATE VARIABLES
UPDATE STATE VARIABLES
Fig. 3 Numerical algorithm of wrinkling.
B. Post-Buckling (PB) scheme using shell elements
An eigenvalue buckling analysis was carried out to obtain the wrinkling mode shapes of the inflated membrane boom by applying internal pressure and bending load to the boom structure. The wrinkling mode shapes were applied as an imperfection with a normalized magnitude of 10% of the membrane thickness in the post-buckling analysis. If the wrinkling final shape is already known, it is possible to determine the wrinkling mode shape that corresponds to the final pattern of the wrinkled structure. The analysis will converge faster if an eigenmode that resembles the final wrinkling pattern is used. Wrinkling mode shapes and eigenvalues were used to perform the post-buckling analysis. The eigenvalues and wrinkling mode shapes correspond to the load magnitudes and shapes of the possible wrinkling modes of the membranes.
C. Bending behaviors of the inflated membrane booms
The bending behaviors of inflated membrane booms with various slenderness ratios (2 /L D) were numerically and experimentally investigated. Figure 4 shows the tip load and deflection curves with an internal pressure of 5,710Pa (0.7psi). For the slenderness ratio, 2L/D=5 (length, L=0.48m and diameter, D=0.19m), the TF scheme predicts the nonlinear behavior of the boom closer to the experimental results compared to the post-buckling (PB) scheme. In particular, when the PB scheme is applied, the value of the critical collapse load (F ) defined as in Fig. 5c is lower than the value in the experimental result. In addition, there is a slight deviation in the linear bending stiffness between the numerical and experimental results. For the slenderness ratio, 2 /L D=11 (L=1m and D=0.19m), both numerical schemes can appropriately predict the nonlinear behavior of an inflated membrane boom.
Despite the fact that the model using the post-buckling analysis shows a small amount of deviation in the linear bending stiffness, the critical collapse load in good agreement with the experimental results. However, for the slenderness ratio, 2L/D=20 (L=1.2m and D=0.12m), both the numerical results of the TF and the PB schemes show several errors both in the linear bending stiffness and in the critical collapse load.
0.000 0.002 0.004 0.006 0.008 0.010 0
5 10 15 20 25
Tipload(N)
Tip deflection (m)
Internal pressure, P=5170Pa (0.7psi) TF scheme
PB scheme Experiment
(a) Slenderness ratio, 2L/D=5
0.000 0.005 0.010 0.015 0.020
0 2 4 6 8 10
Tipload(N)
Tip deflection (m)
Internal pressure, P=5170Pa (0.7psi) TF scheme
PB scheme Experiment
(b) Slenderness ratio, 2L/D=11
0.00 0.01 0.02 0.03
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Tipload(N)
Tip deflection (m)
Internal pressure, P=5170Pa (0.7psi) TF scheme
PB scheme Experiment
(c) Slenderness ratio, 2L/D=20
Fig. 4 Load-deflection curves with various slenderness ratios.
Veldman et al. [10] summarized various theoretical bending models of inflated membrane booms and found that the collapse bending moment changes with the length of the structure. As indicated in Fig. 6, the collapse bending moment decreases as the length increases in the intermediate length range [10]. For a very short inflated membrane boom, the collapse bending moment approaches Stein’s collapse moment. However, for a long inflated membrane boom, the bending moment is approximately equal to Brazier’s collapse moment. The variation in the collapse bending moment (Mc =Fc×L) with respect to the slenderness ratios is shown in Fig. 7 when the internal pressure P is 3,000Pa and the diameter D is 0.19m. When the slenderness ratio is approximately 10, the values of the collapse bending moment calculated using both schemes are identical. In the TF scheme, the collapse bending moment is nearly constant as the length increases. However, for the PB scheme, the collapse bending moment increases linearly. Below a slenderness ratio of 2 /L D=10, the collapse bending moment using the PB scheme is lower than that of the TF scheme. However, the value of the PB scheme is much higher than that of the TF scheme when the slenderness ratio is increased. Therefore, the numerical schemes should be carefully applied with respect to the slenderness ratio of inflated membrane booms.
Critical Collapse Load, F Tipload(N) c
Tip deflection (m)
Stein
Brazier
Intermediate Long Short
Very short
Collapsebendingmoment
Length
Fig. 5 Critical collapse load. Fig. 6 Change of collapse bending moment with length.
10 20 30 40
3 6 9 12 15
Collapsebendingmoment(Mc)
Slenderness ratio (2L/D) TF scheme
PB scheme
Fig. 7 The variation of collapse bending moment.
The deformed shapes of an inflated membrane boom ( 2 /L D=11) with a tip deflection of 0.015m are illustrated in Fig. 8. A detailed wrinkling deformed shape and wrinkling amplitude can be described by the PB scheme using shell elements. However, the TF scheme using membrane elements can identify only the wrinkled region.
Y X
Z
Minor Principal Stress (MPa) 13
11 10 8 6 5 3 1 -0 -2
Wrinkled deformation
X Y Z
(a) Post-buckling scheme using shell elements (b) Tension Field scheme using membrane elements Fig. 8 Deformation of the inflated membrane boom ( 2 /L D=11).
0.00 0.02 0.04 0.06 0.08 0.10
0 2 4 6 8 10 12
Internal pressure 2590Pa (0.38psi) 4310Pa (0.63psi) 5170Pa (0.75psi)
Tipload(N)
Tip deflection (m) TF scheme Experiment
Fig. 9 Load-deflection curves with various internal pressures ( 2 /L D=11).
The load and deflection curves with several internal pressures are shown in Fig. 9. There are very good correlations between the experiment and the numerical results using the TF scheme. By increasing the internal pressure, the critical load that initiates a collapse is also increased. The expansion of the wrinkling area and
Y X Z
Y X
Z
Y X
Z
(a) Deflection at 0.005m (b) Deflection at 0.020m (c) Deflection = 0.078m Fig. 10 Deformed shapes with wrinkled area (internal pressure, P=2,590Pa).
Fig. 11 Wrinkling extension along the circumferential direction.
The wrinkled area is developed at the bottom region along the circumferential direction according to increase in the deflection. At a tip deflection of 0.078m, the inflated membrane boom is totally collapsed (Fig. 10(c)). The wrinkled state can be seen not only at the bottom side area but also at the topside opposite to the tip load. It is considered that the wrinkling is generated at the topside opposite to tip load via a compressive load that results from the rotation of the top rigid line. The membrane elements at the bottom region are totally collapsed due to the wrinkling extension. The right side remains nearly straight and the deflection is increased as the wrinkled elements can no longer sustain the applied load and only serve as hinge. The left sideline is curved and the mid-section of the inflated membrane boom cannot maintain a perfect cylindrical shape. Experimentally, the wrinkling extension in the circumferential direction can be certified as shown in Fig. 11.
III. Conclusion
The applicability of numerical schemes for the nonlinear behavior of inflated membrane boom structures resulting from wrinkling is investigated. For a finite element analysis, two different numerical schemes, one based on the Tension Field theory and the other on a post-buckling analysis, are applied. A comparative study on the wrinkling problems of an inflated membrane boom in bending was carried out. For the inflated membrane boom in bending, the numerical models show different nonlinear behaviors depending on the slenderness ratio ( 2 /L D ).
Therefore, the numerical scheme for an inflated membrane boom should be carefully applied according to the slenderness ratio of the structure. In case of the short ( 2L D=5) inflated membrane boom, the Tension Field theory is more applicable than the post-buckling analysis for describing a large deformation due to wrinkling. For an intermediate ( 2L D=10) inflated membrane boom, both numerical schemes using the Tension Field theory and the post-buckling analysis are applicable. However, the results of both numerical schemes show some deviation from experiments in the description of the nonlinear behavior of a long ( 2L D=20) inflated boom structure.
Acknowledgments
This work was supported by the Brain Korea 21 Project in 2006. The authors also acknowledge the support by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2005-041-D00168).
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