Fast probabilistic design procedure for axially compressed composite cylinders

doi:10.1016/j.compstruct.2011.06.017

Composite Structures

Volume 93, Issue 12, November 2011, Pages 3140-3149

https://doi.org/10.1016/j.compstruct.2011.06.017 Get rights and content

Abstract

A semi-analytic probabilistic procedure is presented that enables a fast prediction of the stochastic distribution of buckling load of cylindrical shells caused by the scattering of geometry, wall-thickness, material properties and loading imperfection. Compared to Monte Carlo simulations the semi-analytic method requires significantly fewer buckling load calculations giving equally accurate results. Knowing the distribution function of buckling load, a level of reliability is chosen and a design load is defined, which is lower than all test results and less conservative than NASA SP-8007.

Highlights

► A probabilistic design approach for composite cylindrical shells is given. ► The proposed semi-analytic approach is as accurate as Monte Carlo simulations. ► Probabilistic design allows a less conservative design than currently used guidelines. ► The reliability of several alternative design approaches is assessed.

Introduction

Thin-walled cylindrical shells tend to buckle under axial compression, where for circular shells there are significant differences between analytically determined buckling loads and experimental test results. Koiter [1] declared initial geometric imperfections to be the main reason for this considerable gap. Since then, most of the investigations on the effect of imperfections focused on geometric deviations from the perfect shell, which is why this kind of imperfection is often referred to as traditional imperfection and consequently, all other kind of imperfections, like loading imperfections, are called non-traditional imperfections (see e.g. Refs. [2], [3]). Until today, the most frequently used guideline to handle the effect of imperfections is NASA SP-8007 [4], which has been developed based on a multitude of test results, obtained in the sixties. As it turned out, the lower bound given by this purely empirical based guideline is very conservative in many cases. Furthermore, NASA SP-8007 was developed for isotropic and orthotropic shells, but not for composite structures. Thus, several attempts to develop mechanically motivated design methods have been followed over the last decades.

An efficient design is only possible, if knowledge about possibly occurring imperfections exists and if this knowledge is used within the design process. This can be reached by probabilistic methods, which enable predicting the stochastic distribution of buckling load. Once the distribution of buckling load is known, a lower bound can be defined by choosing a level of reliability. Arbocz et al. [2], [5], [6] showed several ways to vary geometric imperfections within first-order second-moment analyses and Monte Carlo simulations in order to obtain a probabilistic based lower bound of buckling load. Thereby, conservative assumptions are introduced in order to cover the effect of non-traditional imperfections. If a probabilistic analysis shall work without conservative assumptions, it is not sufficient to consider only geometric imperfections as randomly distributed, which was shown by Kriegesmann et al. [7] for a set of thin-walled cylinders. Degenhardt et al. [8] found knockdown factors, which are less conservative than NASA SP-8007, by executing Monte Carlo simulations with non-traditional imperfections. The geometric imperfections are not varied in these investigations and hence, no general distribution of buckling load was obtained. Rolfes et al. [9] executed a numerical probabilistic analysis for the same set of cylinders in which geometric imperfections as well as non-traditional imperfections are taken into account in order to estimate the distribution function. The results deliver a design load, which is lower than the minimal test result and less conservative than NASA SP-8007. The applied procedure is general and can be transferred to other shells, but it requires a lot of computational cost.

In the present paper, a probabilistic analysis presented in for the same shells as given in [8], [9] and a semi-analytic probabilistic procedure is introduced, which requires much less buckling load calculations than Monte Carlo simulations. Then, the procedure is transferred to cylindrical shells with different laminate setups. Furthermore, the semi-analytic method is used to quantify the influence of each random parameter on the scattering of buckling load.

Section snippets

Data basis

The shells that are considered in this paper have been manufactured, measured and tested at the German Aerospace Centre in Braunschweig (see Refs. [8], [10]). Two sets of shells are regarded, where all shells considered have a nominal radius of 250 mm, a nominal length of 510 mm and a nominal wall-thickness of 0.5 mm. The shells of set #1 have been investigated by Hühne et al. [10]. They have the identifier Z07–Z12 and different laminate setups as listed in Table 1. The ten shells of set #2 have

Numerical determination of buckling loads

For calculating buckling loads a multitude of analytical, semi-analytical and numerical codes has been developed in the past. For a broad overview the reader should consult Refs. [14], [15]. The numerical calculations of buckling loads within the probabilistic analysis are executed using the Finite Element code ABAQUS. A four node shell element with reduced integration is used. Because the shell edges were clamped in the experiments, this boundary condition has also been applied in the

Probabilistic analysis

General objective of probabilistic analyses is determining the distribution function F_g of an objective function g(x), where g is a function of realizations x of the random vector X with the probability density function (PDF) f_X. This goal can be achieved by solving the integral (4) $F_{g} (g (x)) = \int_{x : g (x) ⩽ g} f_{X} (x) d x$

In general, the objective function g(x) is not given analytically and hence, (4) has to be solved approximately, which can be done by semi-analytical and numerical approaches that are

Conclusions and outlook

A semi-analytic probabilistic procedure has been presented, which can be transferred to an arbitrary application. It requires the numerical estimation of derivatives, at which the step size has an influence on the accuracy of the probabilistic method. It has been shown that a step size of Δz_i = 1.5 σ_i robustly leads to the most accurate results for the given set of shells. Using the incomplete second order approach the distribution of the objective function can be estimated with the same accuracy

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Citation Excerpt :
The problems that analysis, experiment, and design of these composite shells are following to be resolved. For example, the National Aeronautics and Space Administration (NASA) [1–5] and the German Aerospace Center [6–10] have performed plenty of corresponding studies. In this paper, there was mainly investigated the imperfection sensitivity and design optimization of variable stiffness (VS) composite cylinders in aerospace applications.
Variable stiffness (VS) composite cylinders have attracted much more attention in the aerospace industry, due to their significantly improved mechanical properties. Most of the existing studies on the optimization of VS composite cylinders are based on linear buckling analysis, without considering the imperfection sensitivity and ignoring the potential of the non-linear post-buckling bearing of cylindrical shells. Therefore, the imperfection sensitivity analysis of VS composite cylinders under bending load is performed in the non-linear post-buckling regime. It is illustrated that the optimized result based on the linear buckling analysis is not robust, if imperfections are taken into account. Moreover, an optimization framework considering imperfection sensitivity is established, wherein the Worst Multiple Perturbation Load Approach (WMPLA) is used to calculate the lower-bound limit moment of composite cylinders. The critical moment of the optimized result with the worst imperfections has an improvement of 13.81%, and the knockdown factor (KDF) is improved by 27.27%. It is demonstrated to be necessary to consider the imperfection sensitivity based on the non-linear post-buckling analysis during the optimization of VS composite cylinders, and the considerable performance with improved anti-imperfection ability can be realized by tailoring the fiber orientation of VS cylinders.

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Fast probabilistic design procedure for axially compressed composite cylinders

Abstract

Highlights

Introduction

Section snippets

Data basis

Numerical determination of buckling loads

Probabilistic analysis

Conclusions and outlook

Thin-Wall Struct

Int J Non-Linear Mech

Compos Struct

Thin-Wall Struct

Int J Press Vess Pip

Toward a probabilistic preliminary design criterion for buckling critical composite shells

AIAA J

Collapse of axially compressed cylindrical shells with random imperfections

AIAA J

Probabilistic design of axially compressed composite cylinders with geometric and loading imperfections

Int J Str Stab Dyn