基于初始缺陷敏感性的轴压薄壁圆柱壳屈曲分析研究进展

doi:10.3901/JME.2021.22.114

摘要/Abstract

摘要： 屈曲是轴压薄壁圆柱壳结构最主要的失效模式之一。围绕轴压下薄壁圆柱壳结构的承载能力对初始缺陷具有高度敏感性这一问题，系统回顾半个多世纪以来国内外学者在考虑初始缺陷对轴压薄壁圆柱壳屈曲临界载荷影响方面所做的理论、试验以及数值模拟等工作。研究表明，目前已形成了两类等效考虑初始缺陷影响的方法，即：基于数值模拟的确定性分析方法和基于概率统计的不确定性分析方法。其中，前者主要针对初始几何缺陷，而后者则还包括非传统初始缺陷，如材料、厚度及加载缺陷等。两类方法在开展轴压薄壁圆柱壳屈曲载荷预测和设计方面各具特点和优势，相较于试验结果，多点扰动载荷法和单边界扰动法的预测精度最高，两者均能达到90%以上的预测精度且同时具有一定的安全裕度。最后展望未来基于初始缺陷敏感性的轴压薄壁圆柱壳屈曲研究方向和亟待开展的研究工作，为相关研究的开展提供了一定的参考和指导。

关键词: 薄壁圆柱壳, 轴压屈曲, 初始缺陷敏感性, 缺陷表征分析

Abstract: Carbon fiber reinforced thermoplastic composites (CFRTP) have gradually become a new generation of lightweight material after aluminum alloys and high-strength steels due to their superior mechanical properties, low processing cost and recyclability. Domestic and foreign research institutions and materials suppliers have invested lots of money and manpower to conduct related research. Some universities and companies have begun to explore the application of CFRTP to the manufacture and assembly of airframes and vehicle bodies. Ultrasonic welding is one of the most suitable methods for joining thermoplastic materials, and it is also one of the key enabling technologies for the large-scale assembly of CFRTP components. It systematically describes the research progress in CFRTP ultrasonic welding from five aspects:basic process, joint type, numerical simulation, quality monitoring and dissimilar material joining. Future challenges for research on CFRTP ultrasonic welding are summarized, aiming to provide references for large-scale assembly applications of CFRTP components.

Key words: thin-walled cylindrical shells, axial load buckling, initial imperfections sensitivity, imperfection characterization analysis

中图分类号:

TU33
O317

陈志平, 焦鹏, 马赫, 顾亚楠, 葛鹏. 基于初始缺陷敏感性的轴压薄壁圆柱壳屈曲分析研究进展[J]. 机械工程学报, 2021, 57(22): 114-129.

CHEN Zhiping, JIAO Peng, MA He, GU Yanan, GE Peng. Advances in Buckling Analysis of Axial Compression Loaded Thin-walled Cylindrical Shells Based on Initial Imperfection Sensitivity[J]. Journal of Mechanical Engineering, 2021, 57(22): 114-129.

参考文献

[1] 乔丕忠, 王艳丽, 陆林军. 圆柱壳稳定性问题的研究进展[J]. 力学季刊, 2018, 39(2):223-236. QIAO Pizhong, WANG Yanli, LU Linjun. Advances in stability study of cylindrical shells[J]. Chinese Quarterly of Mechanics, 2018, 39(2):223-236.
[2] 王博, 郝鹏, 田阔. 加筋薄壳结构分析与优化设计研究进展[J]. 计算力学学报, 2019, 36(1):1-12. WANG Bo, HAO Peng, TIAN Kuo. Recent advances in structural analysis and optimization of stiffened shells[J]. Chinese Journal of Computational Mechanics, 2019, 36(1):1-12.
[3] 王博, 郝鹏, 杜凯繁, 等. 计及缺陷敏感性的网格加筋筒壳结构轻量化设计理论与方法[J]. 中国基础科学, 2018, 20(3):28-31. WANG Bo, HAO Peng, DU Kaifan, et al. The lightweight design theory and method of reinforced cylindrical shell structure considering defect sensitivity[J]. China Basic Science, 2018, 20(3):28-31.
[4] 陈志平, 唐小雨, 苏文强, 等. 薄壁圆柱壳轴压屈曲研究技术进展[C]//第九届全国压力容器学术会议, 中国安徽合肥, 2017. CHEN Zhiping, TANG Xiaoyu, SU Wenqiang, et al. Research progress in buckling of thin-walled cylindrical shell under axial compression[C]//Proceedings of the 9th National Conference on Pressure Vessels, Anhui, Hefei, China, 2017.
[5] JIAO P, CHEN Z, TANG X, et al. Design of axially loaded isotropic cylindrical shells using multiple perturbation load approach-Simulation and validation[J]. Thin-Walled Structures, 2018(133):1-16.
[6] WAGNER H N R, HÜHNE C. Robust knockdown factors for the design of cylindrical shells under axial compression:Potentials, practical application and reliability analysis[J]. International Journal of Mechanical Sciences, 2018(135):410-430.
[7] 陈铁云, 沈惠申. 结构的屈曲[M]. 上海:上海科学技术文献出版社, 1993. CHEN Tieyun, SHEN Huishen. Buckling of structures[M]. Shanghai:Shanghai Science and Technology Literature Publishing House, 1993.
[8] 吴连元. 板壳理论[M]. 上海:上海交通大学出版社, 1989. WU Lianyuan. Theory of plate and shell[M]. Shanghai:Shanghai Jiao Tong University Press, 1989.
[9] VON KARMAN T, TSIEN H S. The buckling of thin cylindrical shells under axial compression[J]. Journal of the Aeronautical Sciences, 1941, 8(8):303-312.
[10] KOITER W T. On the stability of elastic equilibrium[D]. Delft:Delft University, 1945.
[11] STEIN M. The effect on the buckling of perfect cylinders of postbuckling deformations stresses, induced by edge support[R]. NASA TND-1510, 1962.
[12] HUTCHINSON J W, KOITER W T. Postbuckling theory[J]. Applied Mechanics Reviews, 1970(23):1353-1356.
[13] 钱基宏. 薄壳失稳机理浅析[J]. 计算力学学报, 2003, 20(3):366-371. QIAN Jihong. A buckling research of thin shells[J]. Chinese Journal of Computational Mechanics, 2003, 20(3):366-371.
[14] ARBOCZ J, STARNES J, NEMETH M. Towards a probabilistic criterion for preliminary shell design[C]//39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, 1998:2051.
[15] WAGNER H N R, SOSA E M, LUDWIG T, et al. Robust design of imperfection sensitive thin-walled shells under axial compression, bending or external pressure[J]. International Journal of Mechanical Sciences, 2019(156):205-220.
[16] 田阔. 基于多保真度建模的多层级筒壳屈曲分析及优化方法研究[D]. 大连:大连理工大学, 2018. TIAN Kuo. Research on buckling analysis and optimization methods of hierarchical cylindrical shells based on multi-fidelity modeling[D]. Dalian:Dalian University of Technology, 2018.
[17] HAO P, WANG B, LI G, et al. Worst multiple perturbation load approach of stiffened shells with and without cutouts for improved knockdown factors[J]. Thin-Walled Structures, 2014(82):321-330.
[18] 刘正兴, 李德建. 几何缺陷对圆柱壳安全性能影响的数值分析[J]. 上海交通大学学报, 1990, 24(5):148-155. LIU Zhengxing, LI Dejian. A numerical analysis of geometric imperfection influence on the safety of cylindrical shells[J]. Journal of Shanghai Jiao Tong University, 1990, 24(5):148-155.
[19] 郝鹏. 面向新一代运载火箭的网格加筋柱壳结构优化研究[D]. 大连:大连理工大学, 2013. HAO Peng. Optimum design of stiffened shell structures for new generation launch vehicle[D]. Dalian:Dalian University of Technology, 2013.
[20] DEGENHARDT R. New design scenario for future composite launcher structures[C]//Proceedings of the30th Congress of the International council of the Aeronautical Sciences (ICAS), 2016.
[21] 杜凯繁. 航天薄壁筒壳结构高精度稳定性实验系统设计与应用研究[D]. 大连:大连理工大学, 2019. DU Kaifan. On the design and application of high-precision stability experimental system for thin-walled aerospace shell structures[D]. Dalian:Dalian University of Technology, 2019.
[22] ARBOCZ J, SECHLER E E. On the buckling of axially compressed imperfect cylindrical shells[J]. Journal of Applied Mechanics, 1974, 41(3):737-743.
[23] CALLADINE C R. Understanding imperfectionsensitivity in the buckling of thin-walled shells[J]. Thin-walled Structures, 1995, 23(1):215-235.
[24] 周承倜. 复合材料圆柱壳屈曲时缺陷敏感度的Koiter理论[J]. 大连大学学报, 1992, 2(1):43-54. ZHOU Chengti. Sensitivity of initial imperfections for the buckling of composite cylindrical shells with Koiter's theory[J]. Journal of Dalian University, 1992, 2(1):43-54.
[25] THOMPSON J M T. Advances in shell buckling:Theory and experiments[J]. International Journal of Bifurcation and Chaos, 2015, 25(1):1530001.
[26] PETERSON J P, SEIDE P, WEINGARTEN V I. Buckling of thin-walled circular cylinders[S]. NASA Space Vehicle Design Criteria, NASA SP-8007, 1965.
[27] HUTCHINSON J W. Knockdown factors for buckling of cylindrical and spherical shells subject to reduced biaxial membrane stress[J]. International Journal of Solids and Structures, 2010, 47(10):1443-1448.
[28] ARBOCZ J, STARNES JR J H. Future directions and challenges in shell stability analysis[J]. Thin-Walled Structures, 2002, 40(9):729-754.
[29] HÜHNE C, ROLFES R TESSMER J. A new approach for robust design of composite cylindrical shells under axial compression[C]//European Conference on Spacecraft Structures, Materials and Mechanical Testing. Nordwijk, 2005.
[30] HILBURGER M W, NEMETH M P, STARNES Jr J H. Shell buckling design criteria based on manufacturing imperfection signatures[J]. AIAA Journal, 2006, 44(3):654-663.
[31] DEGENHARDT R, BRTHGE A, KLING A, et al. Probabilistic approach for improved buckling knock-down factors of CFRP cylindrical shells[C]//18th Engineering Mechanics Division Conference (EMD2007), 2007.
[32] 秦旭东, 龙乐豪, 容易. 我国航天运输系统成就与展望[J]. 深空探测学报, 2016, 3(4):315-322. QIN Xudong, LONG Lehao, RONG Yi. The achievement and future of China space transportation system[J]. Journal of Deep Space Exploration, 2016, 3(4):315-322.
[33] HILBURGER M W, LOVEJOY A E, THORNBURGH RP, et al. Design and analysis of subscale and full-scale buckling-critical cylinders for launch vehicle technology development[C]//53rd AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics and Materials Conference, Honolulu:AIAA, 2012:1865.
[34] HILBURGER M W, HAYNIE W T, LOVEJOY A E, et al. Sub-scale and full-scale testing of buckling-critical launch vehicle shell structures[C]//53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu:AIAA, 2012:1688.
[35] HILBURGER M W, STARNES J H. Effects of imperfections of the buckling response of composite shells[J]. Thin-Walled Structures, 2004, 42(3):369-397.
[36] HILBURGER M W. Developing the next generation shell buckling design factors and technologies[C]//53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu:AIAA, 2012:1686.
[37] WINTERSTETTER T A, SCHMIDT H. Stability of circular cylindrical steel shells under combined loading[J]. Thin-Walled Structures, 2002, 40(10):893-910.
[38] ARBOCZ J, ABRAMOVICH H. The initial imperfection data bank at the Delft University of Technology[R]. Delft:Delft University of Technology, 1979.
[39] ARBOCZ J. The imperfection data bank, a mean to obtain realistic buckling loads[C]//Proceedings Buckling of Shells-A-state-of-the-Art Colloquium, Berlin:Springer Verlag, 1982:535-567.
[40] WANG B, DU K, HAO P, et al. Numerically and experimentally predicted knockdown factors for stiffened shells under axial compression[J]. Thin-Walled Structures, 2016(109):13-24.
[41] HILBURGER M W. On the development of shell buckling knockdown factors for stiffened metallic launch vehicle cylinders[C]//2018 AIAA/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference, Kissimmee, 2018:1990.
[42] PRIYADARSINI R S, KALYANARAMAN V, SRINIVASAN S M. Numerical and experimental study of buckling of advanced fiber composite cylinders under axial compression[J]. International Journal of Structural Stability and Dynamics, 2012, 12(4):1250028.
[43] WANG B, ZHU S, HAO P, et al. Buckling of quasi-perfect cylindrical shell under axial compression:Acombined experimental and numerical investigation[J]. International Journal of Solids and Structures, 2018(130-131):232-247.
[44] WANG B, DU K, HAO P, et al. Experimental validation of cylindrical shells under axial compression for improved knockdown factors[J]. International Journal of Solids and Structures, 2019(164):37-51.
[45] HÜHNE C, ROLFES R, BREITBACH E, et al. Robust design of composite cylindrical shells under axial compression-Simulation and validation[J]. Thin-Walled Structures, 2008, 46(7-9):947-962.
[46] LINDGAARD E, LUND E. Nonlinear buckling optimization of composite structures[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37-40):2319-2330.
[47] TENG J G, SONG C Y. Numerical models for nonlinear analysis of elastic shells with eigenmode-affine imperfections[J]. International Journal of Solids and Structures, 2001, 38(18):3263-3280.
[48] 张建, 周通, 王纬波, 等. 模态缺陷条件下复合材料柱形壳屈曲特性[J]. 复合材料学报, 2017, 34(3):588-596. ZHANG Jian, ZHOU Tong, WANG Weibo, et al. Buckling of a composite cylindrical shell considering mode imperfections[J]. Acta Material Composite Sinica, 2017, 34(3):588-596.
[49] HAYNIE W T, HILBURGER M W. Comparison of methods to predict lower bound buckling loads of cylinders under axial compression[C]//51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Orlando:AIAA, 2010:2532.
[50] EN 1993-1-6. Eurocode 3:Design of steel structures[S]. Brussels:European Committee for Standardizations, 1999.
[51] WAGNER H N R, HÜHNE C, NIEMANN S, et al. Robust design criterion for axially loaded cylindrical shells-Simulation and validation[J]. Thin-Walled Structures, 2017, 115:154-162.
[52] ISMAIL M S, PURBOLAKSONO J, ANDRIYANA A, et al. The use of initial imperfection approach in design process and buckling failure evaluation of axially compressed composite cylindrical shells[J]. Engineering Failure Analysis, 2015(51):20-28.
[53] ESSLINGER M. High-speed recordings of the bulging process of thin-walled, axially loaded cylinders[J]. Steel Structures, 1970(39):73-76.
[54] CASTRO S G P, ZIMMERMANN R, ARBELO M A, et al. Geometric imperfections and lower-bound methods used to calculate knock-down factors for axially compressed composite cylindrical shells[J]. Thin-Walled Structures, 2014(74):118-132.
[55] HAYNIE W T, HILBURGER M W. Validation of lower-bound estimates for compression-loaded cylindrical shells[C]//53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu:AIAA, 2012:1689.
[56] DEGENHARDT R, CASTRO S G P, ARBELO M A, et al. Future structural stability design for composite space and airframe structures[J]. Thin-Walled Structures, 2014(81):29-38.
[57] KHAKIMOVA R, WARREN C J, ZIMMERMANN R, et al. The single perturbation load approach applied to imperfection sensitive conical composite structures[J]. Thin-Walled Structures, 2014(84):369-377.
[58] SIM C, PARK J, KIM H, et al. Postbuckling analyses and derivations of knockdown factors for hybrid-grid stiffened cylinders[J]. Aerospace Science and Technology, 2018(82-83):20-31.
[59] CASTRO S G P, ZIMMERMANN R, ARBELO M A, et al. 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[J]. Thin-Walled Structures, 2013(72):76-87.
[60] ORIFICI A C, BISAGNI C. Perturbation-based imperfection analysis for composite cylindrical shells buckling in compression[J]. Composite Structures, 2013(106):520-528.
[61] WAGNER H N R, HÜHNE C, NIEMANN S. Robust knockdown factors for the design of axially loaded cylindrical and conical composite shells-Development and validation[J]. Composite Structures, 2017(173):281-303.
[62] WAGNER H N R, HÜHNE C, NIEMANN S. Constant single-buckle imperfection principle to determine a lower bound for the buckling load of unstiffened composite cylinders under axial compression[J]. Composite Structures, 2016(139):120-129.
[63] KHAKIMOVA R, CASTRO S G P, WILCKENS D, et al. Buckling of axially compressed CFRP cylinders with and without additional lateral load:Experimental and numerical investigation[J]. Thin-Walled Structures, 2017(119):178-189.
[64] KRIEGESMANN B, ROLFES R, HÜHNE C, et al. Probabilistic design of axially compressed composite cylinders with geometric and loading imperfections[J]. International Journal of Structural Stability and Dynamics, 2012, 10(4):623-644.
[65] FRIEDRICH L, SCHMID-FUERTES T, SCHRÖDER K. Comparison of theoretical approaches to account for geometrical imperfections of unstiffened isotropic thin walled cylindrical shell structures under axial compression[J]. Thin-Walled Structures, 2015(92):1-9.
[66] KRIEGESMANN B, JANSEN E L, ROLFES R. Design of cylindrical shells using the single perturbation load approach-Potentials and application limits[J]. Thin-Walled Structures, 2016(108):369-380.
[67] ARBELO M A, DEGENHARDT R, CASTRO S G P, et al. Numerical characterization of imperfection sensitive composite structures[J]. Composite Structures, 2014(108):295-303.
[68] ASMOLOVSKIY N, TKACHUK A, BISCHOFF M, Numerical approaches to stability analysis of cylindrical composite shells based on load imperfections[J]. Engineering Computations, 2015, 32(2):498-518.
[69] ARBELO M A, CASTRO S G P, KALNINS K, et al. Experimental characterization of buckling load on imperfect cylindrical shells using the multiple perturbation load approach[C]//Proceedings of European Conference on Spacecraft Structures, Materials and Environmental Testing, 2014.
[70] SIM C, KIM H, LEE Y, et al. Derivations of knockdown factors for cylindrical structures considering different initial imperfection models and thickness tatios[J]. International Journal of Aeronautical and Space Sciences, 2018, 19(3):626-635.
[71] WAGNER H N R, HÜHNE C, ROHWER K, et al. Stimulating the realistic worst case buckling scenario of axially compressed unstiffened cylindrical composite shells[J]. Composite Structures, 2017(160):1095-1104.
[72] TIAN K, WANG B, HAO P, et al. A high-fidelity approximate model for determining lower-bound buckling loads for stiffened shells[J]. International Journal of Solids and Structures, 2018(148-149):14-23.
[73] WAGNER H N R, PETERSEN E, KHAKIMOVA R, et al. Buckling analysis of an imperfection-insensitive hybrid composite cylinder under axial compression-Numerical simulation, destructive and non-destructive experimental testing[J]. Composite Structures, 2019(225):111152.
[74] WAGNER H N R, HÜHNE C, NIEMANN S, et al. Robust knockdown factors for the design of cylindrical shells under axial compression:Analysis and modeling of stiffened and unstiffened cylinders[J]. Thin-Walled Structures, 2018(127):629-645.
[75] WAGNER H N R, HÜHNE C, NIEMANN S. Buckling of launch-vehicle cylinders under axial compression:Acomparison of experimental and numerical knockdown factors[J]. Thin-Walled Structures, 2020(155):106931.
[76] WAGNER H N R, HÜHNE C, ELISHAKOFF I. Probabilistic and deterministic lower-bound design benchmarks for cylindrical shells under axial compression[J]. Thin-Walled Structures, 2020(146):106451.
[77] WAGNER H N R, HÜHNE C, JANSSEN M. Buckling of cylindrical shells under axial compression with loading imperfections:An experimental and numerical campaign on low knockdown factors[J]. Thin-Walled Structures, 2020(151):106764.
[78] CROLL J G A. Towards a rationally based elastic-plastic shell buckling design methodology[J]. Thin-Walled Structures, 1995, 23(1):67-84.
[79] SOSA E M, GODOY L A, CROLL J G A. Computation of lower-bound elastic buckling loads using general-purpose finite element codes[J]. Computers&Structures, 2006, 84(29-30):1934-1945.
[80] 孙海虹, 陈铁云. 轴压随机几何缺陷圆柱壳可靠性分析[J]. 计算结构力学及其应用, 1996, 13(4):85-92. SUN Haihong, CHEN Tieyun. Reliability analysis of cylindrical shells with random geometric defects under axial compression[J]. Computational Structural Mechanics and its Applications, 1996, 13(4):85-92.
[81] ELISHAKOFF I. Probabilistic resolution of the twentieth century conundrum in elastic stability[J]. Thin-Walled Structures, 2012(59):35-57.
[82] KRIEGESMANN B, ROLFES R, HÜHNE C, et al. Fast probabilistic design procedure for axially compressed composite cylinders[J]. Composite Structures, 2011(93):3140-3149.
[83] KRIEGESMANN B, ROLFES R, HUHNE C, et al. Probabilistic second-order third-moment approach for design of axially compressed composite shells[C]//51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Orlando:AIAA, 2010:2534.
[84] ARBOCZ J, HILBURGER M W. Toward a probabilistic preliminary design criterion for buckling critical composite shells[J]. AIAA Journal, 2005, 43(8):1823-1827.
[85] BIAGI M, DEL MEDICO F. Reliability-based knockdown factors for composite cylindrical shells under axial compression[J]. Thin-Walled Structures, 2008, 46(12):1351-1358.
[86] BUDIANSKY B, HUTCHINSON J W. A survey of some buckling problems[J]. AIAA Journal, 1966, 4(9):1505-1510.
[87] ELISHAKOFF I, MANEN S V, VERMEULEN P G, et al. First-order second-moment analysis of the buckling of shells with random imperfections[J]. AIAA Journal, 1987, 25(8):1113-1117.
[88] ELISHAKOFF I, ARBOCZ J. Reliability of axially compressed cylindrical shells with random axisymmetric imperfections[J]. International Journal of Solids and Structures, 1982, 18(7):563-585.
[89] ELISHAKOFF I. Simulation of space-random fields for solution of stochastic boundary-value problems[J]. The Journal of the Acoustical Society of America, 1979, 65(2):399-403.
[90] BROGGI M, SCHUËLLER G I. Efficient modeling of imperfections for buckling analysis of composite cylindrical shells[J]. Engineering Structures, 2011, 33(5):1796-1806.
[91] BROGGI M, CALVI A, SCHUELLER G I. Reliability assessment of axially compressed composite cylindrical shells with random imperfections[J]. International Journal of Structural Stability and Dynamics, 2011, 11(2):215-236.
[92] KEPPLE J, HERATH M T, PEARCE G, et al. Stochastic analysis of imperfection sensitive unstiffened composite cylinders using realistic imperfection models[J]. Composite Structures, 2015(126):159-173.
[93] VRYZIDIS I, STEFANOU G, PAPADOPOULOS V. Stochastic stability analysis of steel tubes with random initial imperfections[J]. Finite Elements in Analysis and Design, 2013(77):31-39.
[94] SCHENK C A, SCHUELLER G I. Buckling analysis of cylindrical shells with random geometric imperfections[J]. International Journal of Non-Linear Mechanics, 2003, 38(7):1119-1132.
[95] HILBURGER M W, STARNES Jr J H. High-fidelity analysis of compression-loaded composite shells[C]//42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA, 2001:1394.
[96] 王俊奎, 仝立勇. 关于圆筒壳稳定性中的初始缺陷[J]. 力学进展, 1988, 18(2):215-221. WANG Junkui, TONG Liyong. On initial imperfections in stability of circular cylindrical shells[J]. Advances in Mechanics, 1988, 18(2):215-221.
[97] ELISHAKOFF I. Probabilistic methods in the theory of structures, strength of materials, random vibrations and random buckling[M]. Singapore:World Scientific, 2014.
[98] SCHULTZ M R, SLEIGHT D W, GARGNER N W, et al. Test and analysis of a buckling-critical large-scale sandwich composite cylinder[C]//2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Kissimmee, 2018:1693.
[99] SCHENK C A, SCHUËLLER G I. Buckling analysis of cylindrical shells with cutouts including random boundary and geometric imperfections[J]. Computer Methods in Applied Mechanics and Engineering, 2007, 196(35-36):3424-3434.
[100] SCHILLO C, RÖSTERMUNDT D, KRAUSE D. Experimental and numerical study on the influence of imperfections on the buckling load of unstiffened CFRPshells[J]. Composite Structures, 2015(131):128-138.
[101] FRIEDRICH L, SCHRÖDER K. Discrepancy between boundary conditions and load introduction of full-scale built-in and sub-scale experimental shell structures of space launcher vehicles[J]. Thin-Walled Structures, 2016(98):403-415.
[102] DEGENHARDT R, KLING A, BETHGE A, et al. Investigations on imperfection sensitivity and deduction of improved knock-down factors for unstiffened CFRPcylindrical shells[J]. Composite Structures, 2010, 92(8):1939-1946.
[103] 张德林. 采用概率随机扰动载荷法的薄壁圆柱壳下限轴压屈曲载荷预测[D]. 杭州:浙江大学, 2020. ZHANG Delin. Lower-bound axial buckling load prediction for thin-walled cylindrical shells using probabilistic random perturbation load approach[D]. Hangzhou:Zhejiang University, 2020.
[104] ZHANG D, CHEN Z, LI Y, et al. Lower-bound axial buckling load prediction for isotropic cylindrical shells using probabilistic random perturbation load approach[J]. Thin-Walled Structures, 2020(155):106925.
[105] ESSLINGER M, GEIER B. Buckling and postbuckling behavior of conical shells subjected to axisymmetric loading and of cylinders subjected to bending[J]. Theory of Shells, 1980:263-288.
[106] KRIEGESMANN B, HILBURGER M W, ROLFES R. The effects of geometric and loading imperfections on the response and lower-bound buckling load of a compression-loaded cylindrical shell[C]//53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Honolulu:AIAA, 2012:1864.
[107] PAPADOPOULOS V, IGLESIS P. The effect of non-uniformity of axial loading on the buckling behaviour of shells with random imperfections[J]. International Journal of Solids and Structures, 2007, 44(18-19):6299-6317.
[108] STEFANOU G. Response variability of cylindrical shells with stochastic non-Gaussian material and geometric properties[J]. Engineering Structures, 2011, 33(9):2621-2627.
[109] KEPPLE J, HERATH M, PEARCE G, et al. Improved stochastic methods for modelling imperfections for buckling analysis of composite cylindrical shells[J]. Engineering Structures, 2015(100):385-398.
[110] KRIEGESMANN B, MÖHLE M, ROLFES R. Sample size dependent probabilistic design of axially compressed cylindrical shells[J]. Thin-Walled Structures, 2014(74):222-231.
[111] PAPADOPOULOS V, CHARMPIS D C, PAPADRAKAKIS M. A computationally efficient method for the buckling analysis of shells with stochastic imperfections[J]. Computational Mechanics, 2009, 43(5):687-700.
[112] SCHILLO C, KRIEGESMANN B, KRAUSE D. Reliability based calibration of safety factors for unstiffened cylindrical composite shells[J]. Composite Structures, 2017(168):798-812.
[113] FINA M, WEBER P, WAGNER W. Polymorphic uncertainty modeling for the simulation of geometric imperfections in probabilistic design of cylindrical shells[J]. Structural Safety, 2020(82):101894.
[114] LIANG K, ZHANG Y, SUN Q, et al. A new robust design for imperfection sensitive stiffened cylinders used in aerospace engineering[J]. Science China Technological Sciences, 2015, 58(5):796-802.
[115] MEURER A, KRIEGESMANN B, DANNERT M, et al. Probabilistic perturbation load approach for designing axially compressed cylindrical shells[J]. Thin-Walled Structures, 2016(107):648-656.