Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 758959, 18 pages

http://dx.doi.org/10.1155/2015/758959

## Influence of Physical and Geometrical Uncertainties in the Parametric Instability Load of an Axially Excited Cylindrical Shell

^{1}School of Civil Engineering, Federal University of Goiás, Avenida Universitária, 1488, Setor Universitário, 74605-220 Goiás, GO, Brazil^{2}Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rua Marquês de São Vicente 225, 22451-900 Gávea, RJ, Brazil

Received 10 October 2014; Accepted 24 February 2015

Academic Editor: Huaguang Zhang

Copyright © 2015 Frederico Martins Alves da Silva et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work investigates the influence of Young’s modulus, shells thickness, and geometrical imperfection uncertainties on the parametric instability loads of simply supported axially excited cylindrical shells. The Donnell nonlinear shallow shell theory is used for the displacement field of the cylindrical shell and the parameters under investigation are considered as uncertain parameters with a known probability density function in the equilibrium equation. The uncertainties are discretized as Hermite-Chaos polynomials together with the Galerkin stochastic procedure that discretizes the stochastic equation in a set of deterministic equations of motion. Then, a general expression for the transversal displacement is obtained by a perturbation procedure which identifies all nonlinear modes that couple with the linear modes. So, a particular solution is selected which ensures the convergence of the response up to very large deflections. Applying the standard Galerkin method, a discrete system in time domain that considers the uncertainties is obtained and solved by fourth-order Runge-Kutta method. Several numerical strategies are used to study the nonlinear behavior of the shell considering the uncertainties in the parameters. Special attention is given to the influence of the uncertainties on the parametric instability and time response, showing that the Hermite-Chaos polynomial is a good numerical tool.

#### 1. Introduction

Theoretical and experimental results found in the literature show that cylindrical shells subjected to static loads are susceptible to buckling, and they may have a load capacity much lower than the theoretical critical load. This difference may be due to variations in physical and geometric properties [1–3], including geometric imperfections [4–6] or load noise [7, 8]. It may occur in the manufacturing process or during the service life of such structures.

The pioneering technique for quantifying the effect of randomness in problems involving structural systems was the method of Monte Carlo, which relies on repeated random sampling to obtain numerical results, that is, by running simulations many times over in order to calculate probabilities heuristically just like recording a large number of experiments [3–6, 9]. However, to access the influence of uncertainties on the nonlinear dynamics of structural systems, a large enough number of samples are necessary to obtain reliable results and, for each simulation, the equations of motion must be integrated numerically during a sufficiently long time to capture the behavior of the structure. Therefore, the Monte Carlo method requires a costly processing time. So, an alternative method not based on sampling processes is ideal for nonlinear dynamic systems.

Stochastic Galerkin method has been proposed as an alternative for solving stochastic problems. This method adapts the standard numerical methods in order to consider the uncertainties in the input parameters and to quantify their influence on the system solution. In the stochastic Galerkin method the statistical characteristics of random response, such as mean value and variance, are determined without the need of carrying out a usually high set of samples, as shown in [10].

Recently, Sepahvand et al. [11] and Ernst et al. [12] investigated the use of polynomial chaos in various problems and studied its convergence. The method of generalized polynomial chaos with the stochastic Galerkin method has been used to analyze several stochastic problems in applied mechanics [13–18].

The nonlinear stochastic static analysis of cylindrical shells has been carried out by [19, 20] and Silva et al. [8] studied the influence of uncertainties on the nonlinear dynamic response, parametric instability boundary, bifurcation diagrams, and basins of attraction of cylindrical shells. However, none of these works uses the stochastic Galerkin method together with the expansion of the generalized polynomial chaos.

Cylindrical shells are one of the most common structural elements with applications in nearly all engineering fields. They are particularly suited to withstand axial loads and lateral pressure. Under these loading conditions thin-walled cylindrical shells usually display a complex nonlinear response due to modal coupling and interaction and high imperfection sensitivity. The study of the nonlinear vibrations of cylindrical shells goes back to the middle of the last century [21–25]. In these works either the Ritz or the standard Galerkin method is used to discretize the shell. For this, a modal expansion for the displacement field is necessary. The development of consistent modal solutions capable of describing the main modal interactions observed in cylindrical shells has received much attention in the literature. A detailed review of this subject was published by Amabili and his coauthors [26, 27].

So far, we cannot find in the literature any work that uses the stochastic Galerkin method together with the expansion of the generalized polynomial chaos to study the influence of uncertainties of physical or geometrical parameters on the nonlinear dynamics of cylindrical shells. Usually, previous works were conducted studying numerically a very large set of samples so that the results considering uncertainties in a certain parameter can be statistically reliable. But this leads to a time consuming numerical procedure [1–8]. The Galerkin method seems to be a reliable alternative, especially when only the lower bound of buckling loads is desired. Therefore, the aim of this work is to study the effects of randomness in Young’s modulus, in the shells thickness, and in the amplitude of an initial geometrical imperfection on the dynamic buckling of axially excited cylindrical shell using the stochastic Galerkin method, associated with the Hermite-Chaos polynomials, and a consistent procedure to describe the nonlinear displacement field of a cylindrical shell. In this case, the proposed methodology is faster than samples’ generation because the Hermite-Chaos polynomials are able to give the statistical measure without samples’ generation, leading to a time efficient numerical procedure, mainly, in the study of the nonlinear dynamic response of cylindrical shells.

#### 2. Problem Formulation

##### 2.1. Deterministic Shell Equations

Consider a simply supported cylindrical shell with radius , thickness , and length with an initial geometric imperfection described by a specified function , made of an elastic material with Young’s modulus , Poisson coefficient , and density . The geometry, coordinate system , and displacements are shown in Figure 1. The shell is subjected to an axial compressive load, , applied at both ends, namely, and .