A variational differential quadrature formulation for buckling analysis of anisogrid composite lattice conical shells

doi:10.1007/s10483-025-3310-9

摘要/Abstract

Abstract:

Anisogrid composite lattice conical shells, which exhibit varying stiffness along their cone generators, are widely used as interstage structures in aerospace applications. Buckling under axial compression represents one of the most hazardous failure modes for such structures. In this paper, the smeared stiffness method, which incorporates the effect of component torsion, is used to obtain the equivalent stiffness coefficients for composite lattice conical shells with triangular and hexagonal patterns. A unified framework based on the variational differential quadrature (VDQ) method is established, leveraging its suitability for asymptotic expansion to determine the critical buckling loads and the $b$ -imperfection sensitivity parameter of lattice conical shells with axially varying stiffness due to rib layout. The influence of pre-buckling deformation is taken into account to enhance the accuracy of predictions on the linear buckling loads. The feasibility of the present equivalent continuum model is verified, and the differences in buckling behaviors for composite lattice conical shells with both triangular and hexagonal unit cells are numerically evaluated through the finite element (FE) simulations and the VDQ method.

Key words: lattice conical shell, smeared stiffness method, variational differential quadrature (VDQ) method, parametric finite element (FE) model, buckling

中图分类号:

O317

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(11): 2155-2176.

Yongqi LIU, Jianwei WANG, Dong DU, Guohua NIE. A variational differential quadrature formulation for buckling analysis of anisogrid composite lattice conical shells[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(11): 2155-2176.

图/表 15

$ϕ 1 / (∘)$	$H = 2$ mm, $b h = b c = 8$ mm	$H = 4$ mm, $b h = b c = 4$ mm	$H = 8$ mm, $b h = b c = 2$ mm
16.47	87.47	189.03	420.20
24.40	126.83	271.11	595.33
32.01	162.45	344.70	746.34
37.83	186.74	393.57	844.49
44.73	212.29	443.88	942.19

$ϕ 1 / (∘)$	$H = 2$ mm, $b h = b c = 8$ mm	$H = 4$ mm, $b h = b c = 4$ mm	$H = 8$ mm, $b h = b c = 2$ mm
16.47	78.12	177.32	182.68*
24.40	118.88	261.65	334.54*
32.01	155.93	338.09	502.24*
37.83	185.05	392.08	640.50*
44.73	213.07	448.34	796.11*
* local in-plane buckling

$ϕ 1 / (∘)$	$H = 2$ mm, $b h = b c = 8$ mm	$H = 4$ mm, $b h = b c = 4$ mm	$H = 8$ mm, $b h = b c = 2$ mm
16.47	76.84	163.20	339.64
24.40	116.71	241.91	496.42
32.01	154.33	315.71	645.72
37.83	181.23	369.71	755.76
44.73	210.70	426.78	870.87

$ϕ 1 / (∘)$	$H = 2$ mm, $b h = b c = 8$ mm	$H = 4$ mm, $b h = b c = 4$ mm	$H = 8$ mm, $b h = b c = 2$ mm
16.47	$P V > P T > P H$	$P V > P T > P H$	$P V > P H > P T$
24.40	$P V > P T > P H$	$P V > P T > P H$	$P V > P H > P T$
32.01	$P V > P T > P H$	$P V > P T > P H$	$P V > P H > P T$
37.83	$P V > P T > P H$	$P V > P T > P H$	$P V > P H > P T$
44.73	$P T > P V > P H$	$P T > P V > P H$	$P V > P H > P T$
* $P V$ denotes the critical buckling load calculated by the VDQ method based on the equivalent continuum model; $P T$ and $P H$ refer to the results by the FE model of composite lattice conical shells with triangular unit cell and hexagonal unit cell, respectively

参考文献 58

[1]	VASILIEV, V. V., BARYNIN, V. A., and RASIN, A. F. Anisogrid lattice structures — survey of development and application. Composite Structures, 54(2-3), 361–370 (2001)
[2]	VASILIEV, V. V., BARYNIN, V. A., and RAZIN, A. F. Anisogrid composite lattice structures — development and aerospace applications. Composite Structures, 94(3), 1117–1127 (2012)
[3]	GIUSTO, G., TOTARO, G., SPENA, P., DE NICOLA, F., DI CAPRIO, F., ZALLO, A., GRILLI, A., MANCINI, V., KIRYENKO, S., DAS, S., and MESPOULET, S. Composite grid structure technology for space applications. Materials Today: Proceedings, 34, 332–340 (2021)
[4]	TOTARO, G. Multilevel Optimization of Anisogrid Structures for Aerospace Applications, Ph. D. dissertation, Delft University of Technology (2011)
[5]	FAN, H. L., FANG, D. N., and JIN, F. N. Mechanical properties of lattice grid composites. Acta Mechanica Sinica, 24(4), 409–418 (2008)
[6]	FAN, H. L. and YANG, W. An equivalent continuum method of lattice structures. Acta Mechanica Solida Sinica, 19(2), 103–113 (2006)
[7]	KIDANE, S., LI, G. Q., HELMS, J., PANG, S. S., and WOLDESENBET, E. Buckling load analysis of grid stiffened composite cylinders. Composites Part B: Engineering, 34(1), 1–9 (2003)
[8]	BURAGOHAIN, M. and VELMURUGAN, R. Buckling analysis of composite hexagonal lattice cylindrical shell using smeared stiffener model. Defence Science Journal, 59(3), 230–238 (2009)
[9]	ZHENG, Q., JIANG, D. Z., HUANG, C. F., SHANG, X. L., and JU, S. Analysis of failure loads and optimal design of composite lattice cylinder under axial compression. Composite Structures, 131, 885–894 (2015)
[10]	LI, M., LAI, C. L., ZHENG, Q., and FAN, H. L. Multi-failure analyses of carbon fiber reinforced anisogrid lattice cylinders. Aerospace Science and Technology, 100, 105777 (2020)
[11]	LOPATIN, A. V., MOROZOV, E. V., and SHATOV, A. V. Axial deformability of the composite lattice cylindrical shell under compressive loading: application to a load-carrying spacecraft tubular body. Composite Structures, 146, 201–206 (2016)
[12]	LOPATIN, A. V., MOROZOV, E. V., and SHATOV, A. V. An analytical expression for fundamental frequency of the composite lattice cylindrical shell with clamped edges. Composite Structures, 141, 232–239 (2016)
[13]	LOPATIN, A. V., MOROZOV, E. V., and SHATOV, A. V. Buckling of uniaxially compressed composite anisogrid lattice plate with clamped edges. Composite Structures, 157, 187–196 (2016)
[14]	TOTARO, G. Flexural, torsional, and axial global stiffness properties of anisogrid lattice conical shells in composite material. Composite Structures, 153, 738–445 (2016)
[15]	SHI, H. G., FAN, H. L., and SHAO, G. J. Equivalent continuum method for anisogrid composite lattice conical shells with equiangular, equidistant and geodesic spiral ribs. Composite Structures, 275, 114472 (2021)
[16]	FORMAN, S. E. and HUTCHINSON, J. W. Buckling of reticulated shell structures. International Journal of Solids and Structures, 6(7), 909–932 (1970)
[17]	TOTARO, G. Local buckling modelling of isogrid and anisogrid lattice cylindrical shells with triangular cells. Composite Structures, 94, 446–452 (2012)
[18]	TOTARO, G. Local buckling modelling of isogrid and anisogrid lattice cylindrical shells with hexagonal cells. Composite Structures, 95, 403–410 (2013)
[19]	MOROZOV, E. V., LOPATIN, A. V., and NESTEROV, V. A. Finite-element modelling and buckling analysis of anisogrid composite lattice cylindrical shells. Composite Structures, 93, 308–323 (2011)
[20]	MOROZOV, E. V., LOPATIN, A. V., and NESTEROV, V. A. Buckling analysis and design of anisogrid composite lattice conical shells. Composite Structures, 93, 3150–3162 (2011)
[21]	FRULLONI, E., KENNY, J. M., CONTI, P., and TORRE, L. Experimental study and finite element analysis of the elastic instability of composite lattice structures for aeronautic applications. Composite Structures, 78, 519–528 (2007)
[22]	BELARDI, V. G., FANELLI, P., and VIVIO, F. Structural analysis and optimization of anisogrid composite lattice cylindrical shells. Composites Part B: Engineering, 139, 203–215 (2018)
[23]	BELARDI, V. G., FANELLI, P., and VIVIO, F. Design, analysis and optimization of anisogrid composite lattice conical shells. Composites Part B: Engineering, 150, 184–195 (2018)
[24]	TONG, L. Y., TABARROK, B., and WANG, T. K. Simple solutions for buckling of orthotropic conical shells. International Journal of Solids and Structures, 29(8), 933–946 (1992)
[25]	TONG, L. Y. and WANG, T. K. Simple solutions for buckling of laminated conical shells. International Journal of Mechanical Sciences, 34(2), 93–111 (1992)
[26]	TONG, L. Y. Free vibration of composite laminated conical shells. International Journal of Mechanical Sciences, 35(1), 47–61 (1993)
[27]	SOFIYEV, A. H. The buckling of an orthotropic composite truncated conical shell with continuously varying thickness subject to a time dependent external pressure. Composites Part B: Engineering, 34(3), 227–233 (2003)
[28]	SOFIYEV, A. H. Non-linear buckling behavior of FGM truncated conical shells subjected to axial load. International Journal of Non-Linear Mechanics, 46(5), 711–719 (2011)
[29]	XIE, X., JIN, G. Y., YE, T. G., and LIU, Z. G. Free vibration analysis of functionally graded conical shells and annular plates using the Haar wavelet method. Applied Acoustics, 85, 130–142 (2014)
[30]	MEHRI, M., ASADI, H., and WANG, Q. Buckling and vibration analysis of a pressurized CNT reinforced functionally graded truncated conical shell under an axial compression using HDQ method. Computer Methods in Applied Mechanics and Engineering, 303, 75–100 (2016)
[31]	FAGHIH SHOJAEI, M. and ANSARI, R. Variational differential quadrature: a technique to simplify numerical analysis of structures. Applied Mathematical Modelling, 49, 705–738 (2017)
[32]	ANSARI, R. and TORABI, J. Numerical study on the buckling and vibration of functionally graded carbon nanotube-reinforced composite conical shells under axial loading. Composites Part B: Engineering, 95, 196–208 (2016)
[33]	ANSARI, R., FAGHIH SHOJAEI, M., ROUHI, H., and HOSSEINZADEH, M. A novel variational numerical method for analyzing the free vibration of composite conical shells. Applied Mathematical Modelling, 39, 2849–2860 (2015)
[34]	ANSARI, R., TORABI, J., and HASRATI, E. Postbuckling analysis of axially-loaded functionally graded GPL-reinforced composite conical shells. Thin-Walled Structures, 148, 106594 (2020)
[35]	KOITER, W. T. On the Stability of Elastic Equilibrium, Ph. D. dissertation, Delft University of Technology (1945)
[36]	BUDIANSKY, B. Theory of buckling and post-buckling behavior of elastic structures. Advances in Applied Mechanics 14, 1–65 (1974)
[37]	BYSKOV, E. Elementary Continuum Mechanics for Everyone, Springer, Netherlands (2013)
[38]	HENRICHSEN, S. R., WEAVER, P. M., LINDGAARD, E., and LUND, E. Post-buckling optimization of composite structures using Koiter’s method. International Journal for Numerical Methods in Engineering, 108, 902–940 (2016)
[39]	GOLDFELD, Y., SHEINMAN, I., and BARUCH, M. Imperfection sensitivity of conical shells. AIAA Journal, 41(3), 517–524 (2003)
[40]	GOLDFELD, Y. Imperfection sensitivity of laminated conical shells. International Journal of Solids and Structures, 44(3-4), 1221–1241 (2007)
[41]	GOLDFELD, Y. Elastic buckling and imperfection sensitivity of generally stiffened conical shells. AIAA Journal, 45(3), 721–729 (2007)
[42]	WHITE, S. C., RAJU, G., and WEAVER, P. M. Initial post-buckling of variable-stiffness curved panels. Journal of the Mechanics and Physics of Solids, 71, 132–155 (2014)
[43]	RAHMAN, T. and JANSEN, E. L. Finite element based coupled mode initial post-buckling analysis of a composite cylindrical shell. Thin-Walled Structures, 48, 25–32 (2010)
[44]	ANSARI, R. and TORABI, J. Semi-analytical postbuckling analysis of polymer nanocomposite cylindrical shells reinforced with functionally graded graphene platelets. Thin-Walled Structures, 144, 106248 (2019)
[45]	SHI, H. G., FAN, H. L., and SHAO, G. J. Dynamic theory of composite anisogrid lattice conical shells with nonconstant stiffness and density. Applied Mathematical Modelling, 115, 661–690 (2023)
[46]	SHARGHI, H., SHAKOURI, M., and KOUCHAKZADEH, M. A. An analytical approach for buckling analysis of generally laminated conical shells under axial compression. Acta Mechanica, 227(4), 1181–1198 (2016)
[47]	ABBASI, M. and GHANBARI, J. A comprehensive analytical model for global buckling analysis of general grid cylindrical structures with various cell geometries. International Journal for Computational Methods in Engineering Science and Mechanics, 22(6), 477–499 (2021)
[48]	ZHANG, F., PAN, F., and CHEN, Y. L. Revisiting the stiffness of lattice plates with micromechanics modeling. Composite Structures, 286, 115276 (2022)
[49]	VASILIEV, V. V., RAZIN, A., TOTARO, G., and DE NICOLA, F. Anisogrid conical adapters for commercial space application. Proceedings of the AIAA/CIRA 13th International Space Planes and Hypersonic Systems, American Institute of Aeronautics and Astronautics, Capua (2005)
[50]	SHU, C. Differential Quadrature and Its Application in Engineering, Springer, London (2000)
[51]	SHARGHI, H., SHAKOURI, M., and KOUCHAKZADEH, M. A. An analytical approach for buckling analysis of generally laminated conical shells under axial compression. Acta Mechanica, 227(4), 1181–1198 (2016)
[52]	SPAGNOLI, A. Koiter circles in the buckling of axially compressed conical shells. International Journal of Solids and Structures, 40(22), 6095–6109 (2003)
[53]	CIVALEK, Ö. Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Engineering Structures, 26(2), 171–186 (2004)
[54]	FUNG, T. C. Imposition of boundary conditions by modifying the weighting coefficient matrices in the differential quadrature method. International Journal for Numerical Methods in Engineering, 56(3), 405–432 (2003)
[55]	ABAQUS. ABAQUS/Standard User’s Manual, Hibbitt, Karlsson, & Sorensen, Inc. (2005)
[56]	MARTINEZ, O. Micromechanical Analysis and Design of an Integrated Thermal Protection System for Future Space Vehicles, Ph. D. dissertation, University of Florida (2007)
[57]	RIKS, E. An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures, 15(7), 529–551 (1979)
[58]	CRISFIELD, M. A. A fast incremental/iterative solution procedure that handles “snap-through”. Computers & Structures, 13, 55–62 (1981)