Nonlinear Postbuckling Analyses of Cylindrical Structures using Hybrid-Grid Systems for Launch Vehicles Chang Hoon Sim 1, Han-Il Kim2, Ye-Lin Lee3, Min-Jeong Hwang4, Joo-Won Bang5, and Jae-Sang Park6 Chungnam National University, Daejeon, 34134, Republic of Korea and Keejoo Lee7 Korea Aerospace Research Institute, Daejeon, 34133, Republic of Korea
This work conducts the nonlinear postbuckling analyses of the stiffened cylinders for the structure of launch vehicles. Two grid systems using stiffeners -conventional orthogrid and hybrid-grid systems- are considered. The hybrid-grid system uses two different configurations of stiffeners such as major and minor stiffeners. In addition, the baseline and minimum weight cylinders are considered for each stiffener grid system. The single perturbation load approach is used in order to represent the initial imperfection of cylinders. A commercial finite element analysis code, ABAQUS, is used for the present nonlinear analyses. The Newton-Raphson method using displacement control scheme is applied to obtain the relations of the load and axial displacement of the postbuckled stiffened cylinders under compressive loads. Through the present numerical simulations, the global and local instabilities are investigated for both stiffened cylinders using orthogrid and hybrid-grid systems. The global buckling load for the hybrid-grid cylinder is higher than that for the conventional orthogrid cylinder for both the baseline and minimum weight models. In addition, the shell Knockdown factors are derived for using obtained numerical analysis results for the stiffened cylinders. Since the shell Knockdown factor of the hybrid-grid cylinder is higher than that of the conventional orthogrid cylinders for both the baseline and minimum weight models, the stiffened cylinder using a hybrid-grid system is more efficient in buckling design as compared to the conventional orthogrid cylinder.
Nomenclature (Ncr)imperfect = global buckling load for a cylinder with imperfection (Ncr)perfect = global buckling load for a perfect cylinder without imperfection
I. Introduction
T
HIN cylindrical structures are used as the main and important element for the structure of space launch vehicles. However, they are seriously weak in buckling under axial compressive loads. Therefore, the buckling load is often considered as the critical criteria in the structural design of launch vehicles. Generally, the measured buckling load in tests are much smaller than the theoretical value from a linear bifurcation buckling analysis for a geometrically perfect cylinder. This large discrepancy between tests and analyses mainly results from the initial imperfection of a cylinder. To account for this, the empirical Knockdown factor has been used in the buckling design shell structures, so that the theoretical buckling load is reduced (or knocked down) appropriately. NASA
1
Graduate Student, Department of Aerospace Engineering,
[email protected]. Undergraduate Student, Department of Aerospace Engineering,
[email protected]. 3 Undergraduate Student, Department of Aerospace Engineering,
[email protected]. 4 Undergraduate Student, Department of Aerospace Engineering,
[email protected]. 5 Undergraduate Student, Department of Aerospace Engineering,
[email protected]. 6 Assistant Professor, Department of Aerospace Engineering,
[email protected]. 7 Senior Researcher, Launcher Performance Team,
[email protected]. 1 American Institute of Aeronautics and Astronautics 2
space vehicle design criteria1 provides the lower bound of Knockdown factor with respect to various slenderness ratios (ratios of shell radius to equivalent thickness), as given in Figure 1.
Figure 1. Lower bound of Knockdown factor1 This Knockdown factor is based on the collected test data from the 1930s to the 1960s; therefore, it cannot consider modern and advanced technologies for the development of launch vehicles. As a result, the design using Knockdown factors provided by NASA SP-80071 is likely to be very conservative, thus the structure may be designed to be over-weighted2. Recently NASA’s SBKF (Shell Buckling Knockdown Factor) project3 and EU’s DESICOS (New Robust DESign Guideline for Imperfection Sensitive COmposite Launcher Structures) project4 were conducted in order to establish newly the shell Knockdown factor based on analyses for the design of the lightweight structure of future launch vehicles. In the two projects, the SPLA (single perturbation load approach5, Figure 2) was used to model the worst initial imperfection of cylinders. In the SPLA, a concentrated single perturbation load is applied in radial direction at the middle of the cylindrical structure. For both SBKF and DESICOS projects, extensive postbuckling analyses6 - 8 as well as tests were conducted. In the numerical analyses using the finite element method, the single perturbation load (Q) is increased and a nonlinear postbuckling analysis is conducted for each lateral perturbation load. The two-step approaches are used for each lateral perturbation load as follows. In the first step, the lateral perturbation load is applied at the middle of a cylinder, and then in the second step, an axial displacement at the cylinder’s end is applied while the lateral perturbation load is applied as a fixed value. This approach is repeated as the lateral perturbation load is increased until a global buckling is observed.
Figure 2. SPLA (Single Perturbation Load Approach) for the initial imperfection modelling In order to increase the stiffness and strength of cylinders to the buckling under axial compressive loads, a large number of stiffeners in the circumferential and axial directions of a cylinder are used. These stiffeners are used in the forms of grids such as orthogrid and isogrid, as given in Figure 3. Generally, a single configuration (shape, thickness, height, and so on) for the stiffener is used in both the conventional orthogrid system (Figure 4(a)) and 2 American Institute of Aeronautics and Astronautics
isogrid system. However, the hybrid-grid system (Figure 4(b)) uses two different configurations of stiffeners such as major and minor stiffeners. The major and minor stiffeners have the same cross-sectional shape (rectangle) but their cross-sectional areas (width and height) are different. This hybrid-grid system can provide the reduction of the shell structural weight while buckling load is maintained or improved when the design optimization is appropriately applied9. The relations between the axial load and axial shortening have not been studied in depth for the stiffened cylinder using the hybrid-grid system except that the previous work10 conducted the postbuckling analyses partly although the design optimization for hybrid-grid cylinders was conducted.
(a) Orthogrid
(b) Isogrid
Figure 3. Stiffened shells: orthogrid and isogrid systems
(a) Conventional orthogrid system
(b) Hybrid-grid system
Figure 4. Stiffened cylinders: conventional grid system (orthogrid) and hybrid-grid system Therefore, the present work conducts the postbuckling analysis using the SPLA to investigate the relation of the axial load and displacement of the stiffened cylindrical structures. Two different grid systems using stiffeners are considered such as the conventional orthogrid and hybrid-grid systems. In addition, the baseline cylinder and minimum weight cylinder models obtained from the previous work11 are used for each grid system. For this numerical analysis, a commercial finite element analysis code, ABAQUS (version 6.15), is used. Four node shell elements with reduced integration (S4R element) are used for the finite element modelling for both skin and stiffeners. The displacement control method is adopted to examine the behavior of the cylinder in the postbuckled state. The artificial damping is added in order to study the local instabilities of a stiffened cylinder. Therefore, the present postbuckling analysis investigates both the local and global instabilities of the stiffened cylinders. Through the present study, the similarity and difference of postbuckling behaviors of the cylinder with a conventional orthogrid system and the cylinder using a hybrid-grid system are investigated. In addition, the Knockdown factors are derived using the obtained results of numerical analyses.
II. Analysis Methods A. Analysis model The geometric parameters, boundary conditions, and material properties of the present stiffened cylinder models (orthogrid and hybrid-grid systems) are based on the stiffened cylinders used in the reference11. The geometry properties of the stiffened cylinders are described in Figure 5 and Tables 1 and 2. All the stiffened cylinders have radius (R) of 1.5 m and total length (L) of 2 m. In this study, two different models of stiffened cylinders are 3 American Institute of Aeronautics and Astronautics
considered for each stiffener grid system. The first models in Table 1 are for the baseline cylinders with the same weight of 354 kg for both the orthogrid and hybrid-grid systems. The second models in Table 2 are for the minimum weight cylinders designed by the optimization11 for each grid system. The optimization was conducted to minimize the structural weight of stiffened cylinders while improving global buckling loads11. The equally spaced stiffeners in both axial and circumferential directions are used for both grid systems. The material properties for the present stiffened cylinders are the elastic modulus of 70 GPa, Poisson’s ratio of 0.33, and density of 2700 kg/m3.
(b) Orthogrid system
(a) Stiffened cylinder
(c) Hybrid-grid system
Figure 5. Schematic diagrams of a stiffened cylinder and stiffeners Table 1. Geometric properties of stiffeners for baseline cylinders11 (a) Orthogrid system (baseline model) Skin thickness, ts 0.0040 m Stiffener thickness, tr 0.0090 m Stiffener length, h 0.0150 m Number of circumferential stiffeners 25 Number of axial stiffeners 90 Effective thickness, teff 0.0078 m (b) Hybrid-grid system (baseline model) Skin thickness, ts 0.0040 m Minor stiffener thickness, tn 0.0090 m Major stiffener thickness, tj 0.0090 m Minor stiffener length, hn 0.0115 m Major stiffener length, hj 0.0230 m Number of circumferential minor stiffeners 18 Number of axial minor stiffeners 60 Number of circumferential major stiffeners 7 Number of axial major stiffeners 30 Effective thickness, teff 0.0093 m Table 2. Geometric properties of stiffeners for minimum weight cylinders11 (a) Orthogrid system (minimum weight model) Skin thickness, ts 0.0050 m Stiffener thickness, tr 0.0031 m Stiffener length, h 0.0192 m Number of circumferential stiffeners 17 4 American Institute of Aeronautics and Astronautics
Number of axial stiffeners 129 Effective thickness, teff 0.0083 m (b) Hybrid-grid system (minimum weight model) Skin thickness, ts 0.0028 m Minor stiffener thickness, tn 0.0031 m Major stiffener thickness, tj 0.0063 m Minor stiffener length, hn 0.0115 m Major stiffener length, hj 0.0230 m Number of circumferential minor stiffeners 7 Number of axial minor stiffeners 6 Number of circumferential major stiffeners 47 Number of axial major stiffeners 94 Effective thickness, teff 0.0151
B. Single perturbation load approach The SPLA is known as the best methods to represent the initial imperfection of shell structures, since it can provide the realistic and worst geometric imperfection of cylindrical structures12-15. In the present analysis using ABAQUS, the SPLA is modelled through three-step process that is slightly different from the original modelling16 as follows. First, a single perturbation load in radial direction is applied at the middle of the stiffened cylinder using an orthogrid or hybrid-grid system in order to represent the initial imperfection of a cylinder. The application point of this lateral perturbation load is located at the intersection of a circumferential (major) stiffener and an axial (major) stiffener (Figure 6). In this step, the nonlinear static analysis is conducted to obtain the deformed configuration of a stiffened cylinder under a lateral perturbation load. Second, the deformed shape is implemented into a perfect cylinder. This second process is for modelling of a cylinder considering the initial imperfection with stress-free, which is more realistic modelling for the initial imperfection, as compared to the original SPLA modelling16 which is briefly summarized in Section 1. Third, the axial compressive forces are applied to the deformed stiffened cylinder but in stress-free state. In this step, the nonlinear postbuckling analysis with artificial damping is used to obtain the relation between the axial load and displacement of a stiffened cylinder (Figure 7). This process is repeated as the lateral perturbation load is increased until the global buckling load is converged
Figure 6. Application point of single perturbation load
Figure 7. Load – displacement curve
In the SPLA, as a single perturbation load increases, the buckling load decreases as compared to the buckling load of a perfect cylinder without an initial imperfection (Figure 8). When the lateral perturbation load exceeds a certain perturbation load (Q1), the buckling load is not reduced and nearly constant although the perturbation load is increased further. The corresponding buckling load is defined as the design load (N1) of the shell structure. When the lateral perturbation load exceeds the minimum perturbation load (Q1) that provides constant global buckling load, a small drop behavior may be shown in the curve of a load-axial displacement (Figure 9). This small drop behavior 5 American Institute of Aeronautics and Astronautics
is for the local buckling or the snap-through with a sudden radial deflection at the application point of perturbation load.
Figure 8. Global buckling load in terms of perturbation load
Figure 9. Global and local instabilities
C. Techniques for finite element modelling and analyses Nonlinear postbuckling analyses are performed to investigate the behavior between the load and axial displacement. The present nonlinear analyses using ABAQUS are conducted using Newton-Raphson method with displacement control method. In addition, the artificial damping is applied to stabilize the nonlinear analysis for local instabilities such as snap-through, local buckling, wrinkling, and so on, since the artificial damping dissipates the released strain energy when a local instability occurs. S4R shell elements are used for the finite element (FE) models (Figure 10) of the present stiffened cylinders. At each end of the stiffened cylinders, nodes are coupled with a control node using rigid link (Figure 11) to apply a uniform displacement condition. The enforced displacement in the axial direction is applied to a control node to use the displacement control method. The postbuckling analyses using displacement control method examine the local snap-through behavior as well as the global instability.
Figure 10. ABAQUS FE model of the stiffened cylinders.
Figure 11. Rigid links and a control node
Equation (1) defines the shell knockdown factor, γ, that is a ratio between two global buckling loads of a stiffened cylinder with and without initial imperfection, (Ncr)perfect and (Ncr)imperfect, respectively. Therefore, the shell Knockdown factor in this paper can be defined using numerical analysis results without test data.
6 American Institute of Aeronautics and Astronautics
γ=
( N cr )imperfect ( N cr )perfect
(1)
III. Results of Postbuckling Analyses A. Baseline model: orthogrid cylinder Figure 12 shows the result of the nonlinear postbuckling analyses for the baseline orthogrid cylinder defined in Table 1(a). The predicted buckling load for the baseline orthogrid cylinder without initial imperfection is 11,100 kN. The behavior between the load and axial displacement is investigated with increasing the perturbation load (Q). As seen in the figure, as the lateral perturbation load increases, the curves of load and axial displacement are converged therefore, the global buckling load (②) is converged to the value of 6,500 kN. Figure 13 shows the deformed shapes as the axial load increases while the perturbation load is constant as the value of 15 kN. The snap-through occurs due to the local buckling caused by perturbation load (① ). Buckling waves spread in circumferential direction, resulting in the global buckling (②). It is observed that buckling waves spread to the edge of the cylinder as the axial load increases further in the postbuckled state (③).
Figure 12. Load - displacement curve of the orthogrid cylinder (baseline model) ①
Local buckling
②
Global buckling
③
Postbuckling
Figure 13. Deformed shapes of the orthogrid cylinder (baseline model) The Knockdown factor in terms of the perturbation load is calculated using the results of the present nonlinear postbuckling analyses of the baseline orthogrid cylinder (Figure 14). As shown in Figure 14, when the lateral perturbation load exceeds 15 kN, the Knockdown factor is converged to 0.58. The derived Knockdown factor is higher than the value of 0.48 given in NASA SP-80071 for the effective thickness of 0.0078 m.
7 American Institute of Aeronautics and Astronautics
Figure 14. Knockdown factor in terms of perturbation load for the orthogrid cylinder (baseline model)
B. Baseline model: hybrid-grid cylinder The nonlinear postbuckling analyses for the baseline hybrid-grid cylinder defined in Table 1(b) are conducted in Figure 15. The calculated buckling load for the baseline hybrid-grid cylinder without initial imperfection is 10,900 kN. The global buckling load (②) of the baseline hybrid-grid cylinder with initial imperfection is converged to 7,300 kN as the lateral perturbation load increases. The deformed shapes of the baseline hybrid-grid cylinder are shown in Figure 16. As seen in the figure, like to the previous result in Figure 13, buckling waves spread in circumferential direction in the state of the global buckling (②) and then buckling waves spread to the edge of the cylinder in the postbuckled state (③) as the axial load increases further. Figure 17 shows the Knockdown factor of the baseline hybrid-grid cylinder in terms of the perturbation load. The Knockdown factor is derived using the results of the present nonlinear postbuckling analyses. As shown in Figure 17, the value of Knockdown factor is converged to 0.67, when the lateral perturbation load exceeds 45 kN. The derived Knockdown factor (0.67) is higher than both the corresponding value (0.51) given in NASA SP-80071 and the Knockdown factor of the baseline orthogrid cylinder (0.58).
Figure 15. Load – displacement curve of the hybrid-grid cylinder (baseline model)
8 American Institute of Aeronautics and Astronautics
①
Local buckling
②
Global buckling
③
Postbuckling
Figure 16. Deformed shapes of the hybrid-grid cylinder (baseline model)
Figure 17. Knockdown factor in terms of perturbation load for the hybrid-grid cylinder (baseline model) The structural weights of two baseline cylinders are identical as the value of 354kg. As seen in Figures 12 and 15, the baseline hybrid-grid cylinder is more effective to resist the buckling than the baseline orthogrid cylinder. In addition, since the shell Knockdown factor for the baseline hybrid-grid cylinder is higher than that for the orthogrid cylinder, the baseline hybrid-grid cylinder is more appropriate for the lightweight design of future space launch vehicles. However, Knockdown factors of the two baseline stiffened cylinders both are higher than the corresponding values given in NASA SP-80071.
C. Minimum weight model: orthogrid cylinder Figure 18 shows the nonlinear postbuckling analyses for the minimum weight model of an orthogrid cylinder (weight of 320 kg) defined in Table 2(a). As previously described, this optimized model is obtained from the previous work11. The calculated buckling load for the minimum weight model of an orthogrid cylinder without initial imperfection is 12,900 kN. With increasing the lateral perturbation load (Q), the behavior between the load and axial displacement is examined. As seen in the figure, as the lateral perturbation load increases, the curves of axial load and shortening are converged. Thus, the global buckling load (②) is converged to the value of 6,900 kN. The deformed shapes are given in Figure 19 as the axial load increases while perturbation load is constant as the value of 45 kN. It is observed that the local buckling due to the perturbation load causes the snap-through (①). In the state of global buckling (②), it is seen that buckling waves spread definitely in circumferential direction. As the axial load is increased further, the buckling waves spread to the edge of the cylinder in the postbuckled state (③). As in the previous cases, the Knockdown factor in terms of the perturbation load is derived using the obtained analysis results for the minimum weight model of an orthogrid cylinder (Figure 20). As shown in the figure, when the lateral 9 American Institute of Aeronautics and Astronautics
perturbation load exceeds 45 kN, the Knockdown factor is converged to the value of 0.54. The derived Knockdown factor is higher than the value of 0.49 given in NASA SP-80071 for the effective thickness of 0.0083 m.
Figure 18. Load - displacement curve of the orthogrid cylinder (minimum weight model) ①
Local buckling
②
Global buckling
③
Postbuckling
Figure 19. Deformed shapes of the orthogrid cylinder (minimum weight model)
Figure 20. Knockdown factor in terms of perturbation load for the orthogrid cylinder (minimum weight model) D. Minimum weight model: hybrid-grid cylinder 10 American Institute of Aeronautics and Astronautics
Figure 21 shows the predicted behaviors between the axial load and displacement for the minimum weight model of a hybrid-grid cylinder (weight of 252 kg) defined in Table 2(b). This optimized model for the hybrid-grid cylinder is also obtained from the previous design11. The buckling load for the minimum weight model of a hybridgrid cylinder without initial imperfection is calculated as 11,800 kN. As the lateral perturbation load increases, the curves of axial load and displacement for the minimum weight model of a hybrid-grid cylinder are converged; however the behavior is more complex than the previous examples. The global buckling load (②) is converged to 7,500 kN when the lateral perturbation load exceeds 40 kN. The deformed shapes from the local buckling (①) to the post buckling (③) states are shown in Figure 22. Figure 23 derives the Knockdown factor for the minimum weight model of a hybrid-grid cylinder, using the obtained postbucking analyses. The Knockdown factor is derived as 0.64, which is higher than the Knockdown factor (0.54) for the minimum weight model of an orthogrid cylinder shown in the previous Figure 20. In addition, this Knockdown factor is higher than the corresponding value (0.583) given in NASA SP-80071. For the minimum weight models, the global buckling load and Knockdown factor of a hybrid-grid cylinder both are higher than those of an orthogrid cylinder.
Figure 21. Load - displacement curve of the hybrid-grid cylinder (Minimum weight model) ①
Local buckling
②
Global buckling
③
Postbuckling
Figure 22. Deformed shapes of the hybrid-grid cylinder (minimum weight model)
IV. Conclusion This study conducted the nonlinear postbuckling analyses for the stiffened cylinders using a conventional orthogrid system and a hybrid-grid system for both the baseline and minimum weight cylinder models. The single perturbation load approach was used in order to represent the initial imperfection of stiffened cylinders. ABAQUS based on the finite element method was used for the nonlinear numerical simulation. The nonlinear loaddisplacement relation of the stiffened cylinders under compressive loads was obtained by Newton-Rapson method with the artificial damping. In the present analyses, the local and global instabilities were observed for the stiffened cylinders. Although the structural weights of the baseline cylinder models were same, the global buckling load and shell Knockdown factor of a baseline hybrid-grid cylinder both were higher than those of a baseline orthogrid 11 American Institute of Aeronautics and Astronautics
cylinder. Although the weight of the minimum weight model for a hybrid-grid cylinder was much lighter than that of the minimum weight model for an orthogrid cylinder, the global buckling load and shell Knockdown factor of a hybrid-grid cylinder both were higher than those of an orthogrid cylinder for the minimum weight models. Therefore, it can be concluded that the hybrid-grid cylinder is more efficient in buckling design from the point of view of lightweight design for space launch vehicles.
Figure 23. Knockdown factor in terms of perturbation load for the hybrid-grid cylinder (minimum weight model)
Acknowledgments This work was supported by research on the preceding technologies for geostationary satellite launch vehicle of the Korea Aerospace Research Institute (KARI)
References 1Weingarten,
V. I., Seide, P., and Peterson, J. P. 1968. Buckling of thin-walled circular cylinders. NASA SP 8007. 2Arbocz, J., and Starnes Jr, J. H., “Future directions and challenges in shell stability analysis. Thin-Walled Structures,” Vol. 40, No. 9, 2012, pp.729-754. 3Hilburger, M., “Developing the next generation shell buckling design factors and technologies,” Proceedings of the 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2012. 4DESICOS Project. URL: www.desicos.eu [cited 12 June 2017]. 5Haynie, W., and Hilburger, M., “Comparison of methods to predict lower bound buckling loads of cylinders under axial compression,” Proceedings of the 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2010. 6Casado, V. M., Hinsch, S., Gómez, J., and Castro, G. P., “Effect of initial geometrical imperfections on the buckling load of cylindrical sandwich shells under axial compression,” Proceedings of the 13th European conference on spacecraft structures, materials and environmental testing, 2014. 7Arbelo, M. A., Degenhardt, R., Castro, S. G., and Zimmermann, R., “Numerical characterization of imperfection sensitive composite structures,” Composite Structures, Vol. 108, 2014, pp. 295-303. 8Kriegesmann, B., Jansen, E. L., and Rolfes, R., “Design of cylindrical shells using the single perturbation load approach– Potentials and application limits,” Thin-Walled Structures, Vol. 108, 2016, pp. 369-380. 9Hao, P., Wang, B., Li, G., Meng, Z., Tian, K., and Tang, X., ”Hybrid optimization of hierarchical stiffened shells based on smeared stiffener method and finite element method,” Thin-Walled Structures, Vol. 82, 2014, pp. 46-54. 10Wang, B., Hao, P., Li, G., Zhang, J. X., Du, K. F., Tian, K. and Tang, X. H., “Optimum design of hierarchical stiffened shells for low imperfection sensitivity,” Acta Mechanica Sinica, Vol. 30, No. 3, 2014, pp. 391-402. 11Hao, P., Wang, B., Li, G., Meng, Z., Tian, K., and Tang, X., “Hybrid optimization of hierarchical stiffened shells based on smeared stiffener method and finite element method,” Thin-Walled Structures, Vol. 82, 2014, pp. 46-54.
12 American Institute of Aeronautics and Astronautics
12Castro, S. G., Zimmermann, R., Arbelo, M. A., and Degenhardt, R., “Exploring the constancy of the global buckling load after a critical geometric imperfection level in thin-walled cylindrical shells for less conservative knock-down factors,” Thinwalled structures, Vol. 72, 2013, pp. 76-87. 13Deml, M., and Wunderlich, W., “Direct evaluation of the worst imperfection shape in shell buckling,” Computer methods in applied mechanics and engineering, Vol. 149, No. 1-4, 1997, pp. 201-222. 14Haynie, W., Hilburger, M., Bogge, M., Maspoli, M., and Kriegesmann, B., “Validation of lower-bound estimates for compression-loaded cylindrical shells,” Proceedings of the 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2012. 15Castro, S. G., Zimmermann, R., Arbelo, M. A., Khakimova, R., Hilburger, M. W., and Degenhardt, R., “Geometric imperfections and lower-bound methods used to calculate knock-down factors for axially compressed composite cylindrical shells,” Thin-walled structures, Vol. 74, 2014, pp. 118-132. 16Hühne, C., Rolfes, R., Breitbach, E., and Temer, J., “Robust design of composite cylindrical shells under axial compression—simulation and validation,” Thin-walled structures, Vol. 46, No. 7, 2008, pp. 947-962.
13 American Institute of Aeronautics and Astronautics
