Stochastic stability analysis of steel tubes with random initial imperfections Isaak Vryzidis a , George Stefanou a,b,n , Vissarion Papadopoulos a a Institute of Structural Analysis & Antiseismic Research, School of Civil Engineering, National Technical University of Athens, 9 Iroon Polytechniou Street, Zografou Campus, 15780 Athens, Greece b Institute of Structural Analysis & Dynamics of Structures, Department of Civil Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece article info Article history: Received 19 December 2012 Received in revised form 6 August 2013 Accepted 4 September 2013 Available online 11 October 2013 Keywords: Shell finite element Stochastic fields Evolutionary spectrum Random imperfections Buckling load variability abstract In this paper, the effect of initial geometric imperfections on the buckling load of steel tubes (relatively thick cylindrical shells) under axial load and lateral pressure is investigated. The geometric imperfections are modeled as a 2D-1V non-homogeneous Gaussian stochastic field simulated using the spectral representation method. The evolutionary power spectrum of the non-homogeneous field is derived from available experimental measurements using the recently proposed method of separation. For the determination of the limit load variability of the tubes, a stochastic formulation based on Monte Carlo simulation is implemented. It is shown that the imperfections can lead to a substantial reduction of the buckling load and thus should be taken into account via a realistic description through stochastic field modeling. & 2013 Elsevier B.V. All rights reserved. 1. Introduction The failure of shell-type structures is often due to buckling phenomena mainly triggered by the initial geometric imperfec- tions which occur during the manufacturing process. Therefore, the study of imperfect shell structures raised the interest of many researchers in the recent years. The main issues when dealing with this problem are the big discrepancy between theory and experi- ment as well as the large scatter in the measured buckling loads. Both deterministic and probabilistic approaches have been used to address the aforementioned issues. It was soon realized that a realistic approach to the problem could only be achieved by taking into account the inherent randomness of the imperfect geometries [10,15,18]. Buckling analysis based on such approach allows for a robust modeling of the buckling load scatter produced by different manufacturing processes as well as of the observed dispersion of experimental results [8,9,12]. Various methods have been developed to take into account the uncertainty in the geometry of the shell. Some methods use the Fourier series analysis of measured initial imperfections consider- ing the series coefficients as random variables [5]. The idea of using two-dimensional Fourier series with random coefficients resulted from the analytical solution of the problem of stability of cylindrical shells which leads to the representation of buckling modes by series of this kind. These methods have in common that the limit load is computed analytically or semi-analytically. More recent research has proposed the use of stochastic fields to simulate the imperfections in conjunction with the finite element (FE) method to solve the stability problem [1–4,6,11,13,16,17, 19–28,30–34]. The quality of the results obtained from the stochastic approach depends largely on the existence of data (experimental measure- ments) validating the assumptions made for the probabilistic char- acteristics (probability distribution, correlation structure) of the initial geometric imperfections. Many researchers have dealt with this issue in recent years. The result of their investigations has led to the conclusion that the variance does not remain constant in space and therefore the stochastic field describing the geometric imperfec- tions cannot be considered homogeneous. In addition, the histo- grams of the computed buckling loads show that the probability density function is highly skewed and hence buckling loads follow a non-Gaussian distribution [9,23]. It is thus evident that the existence of databanks is important in order to avoid false assumptions and to achieve a realistic simulation of initial imperfections. In this frame- work, a novel approach has been recently proposed for the estima- tion of the evolutionary power spectra of non-homogeneous stochastic fields [26]. This approach, called “the method of separa- tion”, combines computational efficiency and accuracy as it achieves optimum simultaneous resolution in space and frequency domains. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/finel Finite Elements in Analysis and Design 0168-874X/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.finel.2013.09.002 n Correspondence to: School of Civil Engineering, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece. Tel.: þ30 2107722997. E-mail address: stegesa@mail.ntua.gr (G. Stefanou). Finite Elements in Analysis and Design 77 (2013) 31–39