Keywords: non-linear shell finite element, random imperfections, autoregressive moving average model, evolutionary power spectrum.

The buckling behaviour of shell structures is generally influenced by their initial imperfections, which occur during the manufacturing and construction stages. Thus, the analysis of imperfection sensitive shells has attracted the attention of many researchers in the past. Although these research efforts resulted in achieving predictions close to the experimental results, it was soon realized that the wide scatter in measured buckling loads of shell structures could only be approximated through modeling taking into account the randomness of the imperfect geometries [1]. In addition to initial geometric imperfections, other sources of imperfections such as the variability of thickness, material properties, boundary conditions and misalignment of loading are also responsible for the reduction as well as the scatter of the buckling load of shell structures [2]. In the majority of studies these influencing parameters have not been treated as stochastic variables in a rational manner. An accurate prediction of the buckling behaviour of shells would therefore require a realistic description of all uncertainties involved in conjunction with a robust finite element formulation that can efficiently and accurately handle the geometric as well as physical non-linearities of shell type structures.

In the present paper the effect of material and thickness imperfections on the buckling load of a thin-walled isotropic axially compressed cylinder is investigated. This type of shell structure is selected as an example of an imperfection-sensitive shell in the sense that small deviations from its perfect geometry may result in a dramatic reduction in its buckling strength. The concept of an initial "imperfect" structure is introduced involving not only geometric deviations of the shell structure from its perfect geometry but also a spatial variability of the modulus of elasticity as well as of the thickness of the shell. These combined "imperfections" are incorporated in an efficient and cost effective non-linear stochastic finite element formulation of the TRIC shell element [3,4] using the local average method for the derivation of the stochastic stiffness matrix, while the variability of the limit loads is obtained by means of the Monte Carlo Simulation (MCS) procedure.

Initial geometric imperfections are described as a two-dimensional uni-variate stochastic field with statistical properties derived from a data bank of measured initial imperfections [5]. The resulting stochastic field is clearly non-homogeneous with substantially varying first and second order properties. The non-homogeneous characteristics of the stochastic field of initial geometric imperfections are modelled in the context of the spectral representation method using an autoregressive moving average model with evolutionary power spectra [6,7]. The elastic modulus and the shell thickness are described as two-dimensional uni-variate non-correlated homogeneous stochastic fields with assumed statistical properties.

The numerical tests performed demonstrate the decisive role that the material and thickness variability play in the buckling behaviour of imperfection-sensitive shell structures. Furthermore, it is found that predictions of the scatter of the buckling load reasonably close to the experimental results can be obtained provided that the material and thickness imperfections are incorporated in a rational manner to the model of initial geometric imperfections.

1

C.A. Schenk and G.I. Schueller, "Buckling analysis of cylindrical shells with random geometric imperfections", Int Journal of Non-Linear Mechanics, 38,1119-1132, 2003. doi:10.1016/S0020-7462(02)00057-4

2

Y.W. Li, I. Elishakoff, J.H. Starnes Jr, and D. Bushnell, "Effect of the thickness variation and initial imperfection on buckling of composite shells: asymptotic analysis and numerical results by BOSOR4 and PANDA2", Int. J, Solids Structures, 34, 3755-3767, 1997. doi:10.1016/S0020-7683(96)00230-2

3

J.H. Argyris, L. Tenek and L. Olofsson, "TRIC, A simple but sophisticated 3node triangular element based on 6 rigid-body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells", Comp. Meth. Appl. Mech. & Engrg., 145, 11-85, 1997. doi:10.1016/S0045-7825(96)01233-9

4

J.H. Argyris, L. Tenek. M. Papadrakakis and C. Apostolopoulou, "Postbuckling performance of the TRIC natural mode triangular element for isotropic and laminated composite shells", Comp. Meth. Appl. Mech. & Engrg., 166, 211-231, 1998. doi:10.1016/S0045-7825(98)00071-1

5

J. Arbocz and H. Abramovich, "The initial imperfection data bank at the Delft University of Technology Part 1", Technical Report LR-290, Delft University of Technology, Department of Aerospace Engineering, 1979.

6

M. Shinozuka and Y. Sato, "Simulation of nonstationary random processes", J. of Eng. Mech., ASCE., (EM1)", 11-40, 1967.

7

G. Deodatis and M. Shinozuka, "Auto-Regressive model for nonstationary stochastic processes", J. of Eng. Mech., ASCE., 114(11), 1995-2012, 1988. doi:10.1061/(ASCE)0733-9399(1988)114:11(1995)

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