Knockdown factor of buckling load for axially compressed cylindrical shells: state of the art and new perspectives
doi: 10.1007/s10409-021-09035-x
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摘要: 薄壁筒壳结构作为航空航天的主承力构件, 在轴压载荷下易发生屈曲失稳, 并且对几何缺陷表现为强敏感性. 尽管经过数十年的研究, 精准地预测几何缺陷影响下轴压筒壳的折减因子仍是一个非常具有挑战的科学难题. 直至现在, NASA 20世纪提出航天薄壁轴压筒壳折减因子下限设计准则SP-8007仍被广泛使用, 随着制造工艺的进步和发展, SP-8007已经被证明过于保守. 近年来, 学者们基于数值方法和实验技术发展了一系列折减因子的确定方法, 本文对这些方法进行了全面综述, 其中作者提出的多点最不利扰动载荷法(WMPLA)是最具代表性的方法之一, 其以合理的方式有效地降低了传统折减因子预测方法的保守程度. 另外, 本文还基于搜集的1990~2020年轴压筒壳屈曲实验数据, 对原始的折减因子下限准则曲线进行了改进. 可以发现, 相比于原始的SP-8007准则, 新折减因子准则曲线整体上提升了0.1~0.3, 并且使用WMPLA预测的折减因子非常接近于新准则曲线的边界. 本文的研究工作有助于新一代轴压筒壳设计准则的建立, 并可服务于大直径航天运载器主承力薄壁结构的轻量化设计.Abstract: Thin-walled structures are commonly utilized in aerospace and aircraft structures, which are prone to buckling under axial compression and extremely sensitive to geometric imperfections. After decades of efforts, it still remains a challenging issue to accurately predict the lower-bound buckling load due to the impact of geometric imperfections. Up to now, the lower-bound curve in NASA SP-8007 is still widely used as the design criterion of aerospace thin-walled structures, and this series of knockdown factors (KDF) has been proven to be overly conservative with the significant promotion of the manufacturing process. In recent years, several new numerical and experimental methods for determining KDF have been established, which are systematically reviewed in this paper. The Worst Multiple Perturbation Load Approach (WMPLA) is one of the most representative methods to reduce the conservatism of traditional methods in a rational manner. Based on an extensive collection of test data from 1990 to 2020, a new lower-bound curve is approximated to produce a series of improved KDFs. It is evident that these new KDFs have an overall improvement of 0.1-0.3 compared with NASA SP-8007, and the KDF predicted by the WMPLA is very close to the front of the new curve. This may provide some insight into future design guidelines of axially compressed cylindrical shells, which is promising for the lightweight design of large-diameter aerospace structures.
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1. Schematic diagrams of buckling shells. a The CZ-5 launch vehicle; b buckling of stringer stiffened cylindrical shell; c buckling of grid stiffened cylindrical shell; d buckling of unstiffened cylindrical shell.
3. Schematic diagrams of the energy barrier and non-destructive experimental technique for cylindrical shells proposed in Ref. [52].
8. Loading system for axial compression. a Schematic diagram of 3D model; b photograph of the real device [61].
9. Loading system for axial compression combined with external pressure. a Schematic diagram of 3D model; b photograph of the real device.
11. KDFs obtained by the experimental test of axially compressed cylindrical shells from 1957 to 2020 compared with NASA SP-8007 and DLR threshold. a KDFs of cylindrical shell tests from 1990 to 2020; b KDFs of cylindrical shell tests from 1968 to 2020 combined with data in NASA SP-8007.
Table 1. Comparison of different KDF determination methods and test results [81]
Method NASA SP-8007 Eigenmode imperfection method SPLA WMPLA Test KDF 0.386 0.505 0.675 0.566 0.569-0.897 
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Table 2. Summary of experimental data of axially compressed cylindrical shells
No. Ref. Year Material Structural type R (mm) R/t L (mm) Number of specimens Buckling load (kN) KDF Note 1 Harris [107] 1957 2S-H18 aluminum Cylindrical shell 222 1750 546 2 – 0.28-0.29 – Hard stainless steel 1006 4 – 0.76-0.89 – Hard stainless steel 2734 8 – 0.14-0.23 – 2 Ricardo [108] 1967 Mylar Cylindrical shell 102 400 279.4 9 0.75-0.82 0.60-0.66 – 3 Muggeridge [69] 1969 Photo-elastic plastic Cylindrical shell 67/100 110-280 188-282 11 – 0.57-0.92 – 4 Caswell [109] 1971 Epoxy Cylindrical shell 100 100-198 259-279 14 – 0.53-0.76 – 5 Hutchinson [110] 1971 Epoxy Cylindrical shell 100 175-219 279 7 – 0.35-0.92 – 6 Carri [111] 1973 Boron-Epoxy Cylindrical shell 46.35 19.90 464.3 2 713.82-753.92 0.75-0.79 [45/−45/05/90/45/−45/90/05/45/−45] Boron-Epoxy 46.6 20.00 426.7 1 512.44 0.83 [45/−45/04/90/90/04/45/–45] Boron-Epoxy 46.6 20.00 426.7 1 592.24 0.97 [45/−45/04/90/90/04/45/−45] Ti-6AL-4V/Boron-Epoxy 48.9 18.81 381 1 1116.15 0.73 [45/−45/013/0.026Ti] Ti-6AL-4V/Boron-Epoxy 48.5 26.08 381 1 679.86 0.79 [45/−45/07/0.026Ti] Ti-6AL-4V/Boron-Epoxy 48.25 33.51 381 2 442.66-472.39 0.76-0.81 [45/−45/04/0.026Ti] 7 Wilkins [112] 1975 Graphite-Epoxy Narmco5505/T300 Cylindrical shell 190.5 354 381 1 56.81 0.68 [45/−45]s 8 Herakovich [113] 1978 Boron-Epoxy Cylindrical shell 75.36 81 355.6 1 55.6 – [0/0/0/0/0/0/0/0] Boron-Epoxy 76.25 139 508.8 1 25.8 – [82.5/30/20/−82.5] Boron-Epoxy 76.58 69 507.2 1 61.6 – [−45/−45/45/45]s Graphite-Epoxy 76.38 132 508.8 1 20.1 – [−82.5/30/20/−82.5] Graphite-Epoxy 75.87 115 333.4 1 18.24 – [−45/45]s Boron-Epoxy 75.59 74 508 1 136.78 – [0/45/−45/90]s Graphite-Epoxy 75.59 60 508 1 54.27 – [−45/−45/45/45]s Graphite-Epoxy 75.54 74 428.6 1 100.08 – [0/45/−45/90]s Graphite-Epoxy 75.54 73 509.6 1 85.63 – [0/45/−45/90]s 9 Arbocz [65] 1979 Copper Cylindrical shell 101.6 844-915 196.85-203.20 7 3.04-3.85 – – Nickel Cylindrical shell 101.6 1030-1042 196.85 3 2.68-3.91 – – Brass Cylindrical shell 101.6 386-681 140.97-196.85 4 7.18-16.66 – – Welded stainless steel Cylindrical shell 117.86-118.49 258-261 148.59 6 41.70-56.05 – – Aluminum Stringer stiffened cylindrical shell 101.6 362-517 139.7 3 14.29-22.36 – – Aluminum Ring stiffened cylindrical shell 101.6 384-492 133.35-146.30 11 9.74-24.95 – – 10 Uemura [114] 1982 T300A/Epoxy Cylindrical shell 100 100 300 2 82.51-84.36 – [0/90/0/90]s 100 300 2 64.65-65.89 – [0/0/0/0/90/90/90/90] 200 300 1 12.16 – [−20/−20/20/20] 100 300 2 55.29-56.83 – [−20/−20/−20/−20]as 100 300 1 82.51 – [−20/20/−20/20]as 200 300 1 13.24 – [−45/45/45/−45] 200 300 1 13.24 – [−45/−45/45/45] 100 300 1 49.01 – [−45/−45/−45/−45]as 100 300 2 56.83-56.83 – [−45/45/−45/45]as 100 300 4 74.51-79.43 – [−70/70/−70/70]s 200 300 1 18.63 – [−70/70/−70/70] 100 300 2 58.80-58.80 – [−70/−70/−70/−70]as 100 300 2 78.2-80.66 – [−70/70/−70/70]as 11 Hirano [115] 1983 T300/S305 Cylindrical shell 100 122.85 300 3 61.8-63.7 0.75-0.77 [20/−20/0/0/40/−40] 3 43.9-51.3 0.69-0.81 [20/−20/40/−40/0/0] 3 32.7-34.8 0.78-0.83 [40/−40/20/−20/0/0] 12 Tennyson [116] 1983 3M SP 288 T300 Cylindrical shell 83.9 74.91 282.7 1 129.83 – [0/45/90/−45/0/45/90/−45] 83.82 84.67 287.8 1 124.04 0.92 [0/45/−45/90/90/−45/45/0] 83.85 83.85 284.7 1 89.35 1.03 [0/0/45/45/−45/−45/90/90] 83.79 90.10 274.1 1 97.09 – [0/45/−45/−45/45/0/0/0] 83.82 89.17 269.2 1 96.04 0.90 [0/0/45/45/−45/−45/0/0] 83.79 90.10 282.2 1 60.71 – [45/−45/−45/45/0/0/0/0] 83.57 181.67 282.7 1 15.22 0.90 [0/90/90/0] 83.57 194.35 267.7 1 17.18 1.17 [90/0/0/90] 13 Kobayashi [117] 1984 CFRP Cylindrical shell 100 238.1 200 1 13.27 0.956 [20/−20/90] 173.01 1 27.34 0.97 [0/45/−45/90] 111.23 1 88.39 0.756 [30/−30/−30/30/90/90] 98.33 1 98.19 0.904 [0/60/−60/−60/60/0] 14 Foster [118] 1987 Epoxy Cylindrical shell 50 255-398 49-58 4 0.15-0.38 0.84-0.99 No imposed imperfections 288-334 62-73 4 0.12-0.24 0.52-0.83 With imposed imperfections 15 Sun [119] 1987 AS/3501-6 Cylindrical shell 83.31 166.62 152.4 1 30.10 0.86 [26/−42/76/−3] 166.62 1 18.64 0.86 [0/90/90/0] 167.34 1 21.84 0.83 [−59/5/51/−59] 169.07 1 18.37 0.92 [90/0/0/90] 166.62 1 18.50 0.97 [−83/37/15/−86] 16 Dancy [66] 1988 Low carbon steel Cylindrical shell 33 330 100 33 3.05-4.68 – – 17 Geier [120] 1991 CFRP Cylindrical shell 250 200 510 2 221.48-227.76 0.80-0.82 [60/−60/0/0/68/−68/52/−52/37/−37] 200 2 90.12-93.46 0.92-0.95 [51/−51/45/−45/37/−37/19/−19/0/0] 200 2 227.76-287.23 0.83-1.02 [30/−30/90/90/22/−22/38/−38/53/−53] 400 1 68.92 1.03 [51/−51/90/−90/40/−40] 18 Giavotto [121] 1991 Kevlar fabric Cylindrical shell 350 336.54 550 2 32.48 0.88 [0/90/90/0] 31.79 0.84 [45/−45/−45/45] 19 Krishnakumar [122] 1991 Araldite LC 261/hardener LC 249 Cylindrical shell 77 197.44-596.9 69.3-192.5 36 – 0.66-0.98 – 20 Waters [123] 1996 AS4/3502 Cylindrical shell 203.2 201.19 355.6 1 133.59 0.74 [45/0/90]s 203.4 100.69 355.6 1 328.89 0.74 [45/−45/−45/45]2s 203.3 101.14 355.6 1 656.26 0.86 [45/0/90]2s 203 104.10 355.6 1 557.63 0.91 [45/904/−45/45]s 203.4 100.69 355.6 1 408.66 0.59 [45/04/−45/45]s 21 Schneider [124] 1996 Polycarbonate plastic Cylindrical shell 19.1 15.04 38.1 3 13.41-13.61 – Perfect 2 5.26-7.37 – Axisymmetric imperfectionwo/t = 0.3 22 Bisagni [125] 1999 CFRP fabricCFRP unidirectional Cylindrical shell 350 265.15 540 4 151.62-172.88 0.63-0.72 [0/45/−45/0] 265.15 4 151.62-172.88 0.85-0.99 [45/−45]s 291.67 2 92.86-96.27 0.53-0.55 [45/−45]2s 291.67 2 92.05-99.54 0.54-0.59 [90/0]2s 233.33 2 185.94-196.23 0.64-0.68 [90/30/−30/90] 233.33 2 155.35-159.06 0.98-0.99 [45/−45] 23 Kim [126] 1999 IM7/997-2 Isogrid stiffened cylindrical shell 624.8 – 368.3 1 117.88 – – 24 Meyer-Piening [127] 2001 CFRP Cylindrical shell 250 200 510 2 208-212.6 – [60/−60/0/0/68/−68/52/−52/37/−37] 3 213-222 – [37/−37/52/−52/68/−68/0/0/60/−60] 8 206.6-228.2 – [53/−53/8/−8/90/−90/68/−68/38/−38] 12 186-249.7 – [53/−53/38/−38/22/−22/90/−90/30/−30] 4 88-92.4 – [0/0/19/−19/37/−37/45/−45/51/−51] 6 156-172.8 – [51/−51/45/−45/37/−37/19/−19/0/0] 25 Bisagni [128] 2003 CFRP fabricCFRP unidirectional Cylindrical shell 350 292 540 1 74.93 – [45/−45]2s 292 1 83.66 – [0/45/−45/0]2s 265 1 97.95 – [45/-45]s 265 1 140.2 – [0/45/−45/0] 26 Hilburger [129] 2006 AS4/3502 Cylindrical shell 203.2 200 406.4 3 123.6 0.929 [−45/45/0/0]s 142 0.879 [−45/45/90/90]s 151.6 0.821 [−45/45/0/90]s 27 Bisagni [91] 2006 CFRP Stringer stiffened cylindrical shell 350 – 540 2 360.2-380.3 2.78-3.68 [45/−45] 28 Hilburger [130] 2008 Al-Li Orthogrid stiffened cylindrical shell 1219.2 – 1981.2 1 3065.71 – – 29 Fan [94] 2009 T700/bisphenol epoxy CFRC sandwich cylindrical shell with kagome cores 312.5 – 375 1 524.6 – – 30 Degenhardt [131] 2010 IM7/8552 Cylindrical shell 250 500 540 10 21.32-25.69 0.68-0.82 [24/−24/41/−41] 31 Haynie [132] 2012 Aluminum alloy 2024 Cylindrical shell 228.6 225 787.4 3 168-169 – – 32 Priyadarsini [133] 2012 CFRP Cylindrical shell 150.5 150.5 400 4 77.1 – [0/45/−45/0] 98.6 – [0/45/−45/0] 99.8 – [0/45/−45/0] 98.2 – [0/45/−45/0] 33 Chen [95] 2013 T700/Epoxy Carbon fiber reinforced lattice-core sandwich cylindrical shell 600 30 1600 1 1200 – – 34 Bisagni [134] 2015 IM7/8552 Cylindrical shell 250 500 520 2 13.01-15.34 – [45/−45/−45/45] 2 12.75-14.33 – [45/−45/−45/46] 2 14.41-15.79 – [45/−45/−45/47] 35 Schillo [135] 2015 AS7/8552 Cylindrical shell 115 147 215 2 55.4-62.1 – [90/30/−30]s 36 Kalnins [136] 2015 IM7/8552 Cylindrical shell 251.13 479.99 500 1 25.38 0.65 [24/−24/41/−41] 251.8 481.27 500 1 25.64 0.66 [24/−24/41/−41] 150.4 574.92 300 1 1.6 0.37 [0/45] 150.4 574.92 300 1 2.44 0.56 [0/45] 150.52 383.59 300 1 6.22 0.46 [0/60/−60] 150.61 383.82 300 1 6.34 0.46 [0/60/−60] 150.22 382.82 300 1 7.28 0.53 [0/60/−60] 150.66 383.94 300 1 8.71 0.50 [0/45/−45] 150.76 384.2 300 1 8.5 0.49 [0/45/−45] 150.73 384.12 300 1 9.63 0.56 [0/45/−45] 151.32 289.22 300 1 28.96 0.73 [24/−24/41/−41] 150.76 288.15 300 1 26.85 0.68 [24/−24/41/−41] 151.16 288.91 300 1 21.1 0.53 [24/−24/41/−41] 151.01 288.63 300 1 25.47 0.65 [24/−24/41/−41] 37 Takano [137] 2016 TR/HSX Cylindrical shell 68.00 139.34 136 1 13.197 0.6 [−70/70/0/0/70/−70] 71.75 147.03 287 1 11.87 0.54 [−70/70/0/0/70/−70] 72.67 148.91 436 1 11.537 0.525 [−70/70/0/0/70/−70] 68.00 194.84 136 1 12.997 0.615 [−70/70/0/0/70/−70] 73.83 211.56 443 1 12.647 0.578 [−70/70/0/0/70/−70] 74.00 422.86 148 1 1.806 0.877 [−70/0/70] 73.83 421.9 443 1 1.437 0.697 [−70/0/70] 68.00 139.34 136 1 12.665 0.576 [−70/70/0/0/70/−70] 72.67 148.91 436 1 8.738 0.397 [−70/70/0/0/70/−70] 68.00 194.84 136 1 10.366 0.491 [−70/70/0/0/70/−70] 72.67 208.21 436 1 9.875 0.451 [−70/70/0/0/70/−70] 68.00 586.21 136 1 0.583 0.605 [−50/0/50] 72.67 626.44 436 1 0.464 0.492 [−50/0/50] 38 Wang [61] 2016 Aluminum alloy Stiffened cylindrical shell 2250 – 2200 1 3151 0.62 – 39 Li [96] 2016 T700/Epoxy Carbon fiber reinforced lattice truss sandwich cylindrical shell 312.5 – 375 1 328.03 – – 40 Wang [62,81] 2017 Aluminum alloy Cylindrical shell 500 333 600 5 335.13-519.39 0.57-0.89 – 41 Khakimova [92] 2017 CFRP Cylindrical shell 400 533 800 2 58.3-63.3 0.65-0.71 [34/−34/0/0/53/−53] 42 Rudd [103] 2018 Al 2219 Seamless orthogrid stiffened cylindrical shell 1225.6 – 2286 1 3302.79 – – 43 Hilburger [102] 2018 Al-Li Stiffened cylindrical shell 1219.2 – 1981.2 1 2878 – – 44 Wu [138] 2018 T700/Epoxy Hierarchical orthogrid stiffened cylindrical shell 312.5-315 – 590-596 2 132.74-359.24 – – 45 Li [97] 2018 T700/Epoxy Hierarchical isogrid stiffened cylindrical shell 312.5 – 431 1 741.4 – [0/60/−60]s 46 Li [98] 2020 T700/Epoxy Folded lattice-core sandwich cylindrical shell 312.5 – 375 1 293.4 – –
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