U. Nymanand P. J. Gustafsson Division of Structural Mechanics
Lund University, Lund, Sweden e–mail: ulf.nyman@byggmek.lth.se
ABSTRACT
Summary This paper is a merging of two categories of established methods, one for the analysis of stability and collapse of geometrically non-linear shells and one for the reliability analysis of structures using FORM. The formulation and implementation of a finite element procedure is described as well as the finite difference method in order to find the gradients of the limit state function. Numerical examples are performed on an in-plane loaded corrugated board panel involving uncertainties in geometrical imperfection, material properties and load.
Thin structures such as laminated shells has been extensively investigated by the use of the finite element method during the last decades. Numerous work is presented, where the aim is to formu-late a shell element which can be used for response analysis of general shell structures. Since the degenerated shell element was presented [1], this type of formulation has formed a basis for the development of procedures for as well small as finite shell deformations. Its popularity is due to the straightforward representation of geometry and kinematics, nevertheless including the effect of transverse shear deformations. However, the simple assumption of kinematics leads to deteror-iating performance when the thickness decreases, which is known as shear locking. To overcome this shortcoming work has been done in order to reduce the locking effect, e.g. reduced integration techniques, [2, 3], and formulations based on assumed strain, [4, 5].
In many applications it is from a lifetime and economic perspective important that a structure un-der consiun-deration possesses a suitable degree of safety, i.e. the structure should withstand loads under normal conditions, but nevertheless, it must not be exceedable dimensioned in order keep house with resources. One example, which is the focus of this work, is corrugated board packages as, for example, used for the distribution of consumer goods. Corrugated board is a material which to a large extent incorporates uncertainties, manifested as well in material properties, geometrical properties and load conditions under handling. In a reliability analysis, the variables affecting the performance of the structure, called basic variables, are depicted probabilistic measures, i.e. mean and variance. The outcome of the analysis is the share of structures that will fail encountering cer-tain load conditions. The purpose of this paper is to merge two categories of established methods, one for the analysis of stability and collapse of geometrically non-linear shells and one for the reliability analysis of structures using FORM.
The different techniques excisting for reliability analysis can be categorized as either exact or approximate, where in the latter case, some error is inferred from a simplified representation of variation of stochastic variables. Among the exact methods are multifold integration and Monte Carlo simulation techniques. Examples of approximate techniques are methods involving response surface fitting and FORM/SORM (First/Second Order Reliability Methods. The exact methods are numerically intensive and in the analysis of structures, e.g. by the finite element method, the computational cost may be prohibitively large. This is certainly expressed for problems which are numerically intensive in the deterministic case, for example as in non-linear finite element analysis.
In this work, FORM is used together with a geometrically non-linear finite element procedure for the collapse analysis of in-plane loaded shells.
In using FORM, a limit state function is expressed in terms of the structural resistance and the load.
This limit state function represents an-dimensional surface in the basic variable space. The limit surface can then be mapped into the standard uncorrelated normal space of the basic variables, as proposed in [6]. The idea in FORM is to approximate the limit state surface by a tangent hyper-plane at the design point, which is the point at the limit surface closest to the origin. At this point the frequency function of the standard normal variables is most dense, i.e. it provides the largest contribution to the probability content. The design point can be found by an iterative minimization procedure, e.g. as described in [7], and the distance from the origin to the design point is referred to as the reliability index, which provides a first order measure of the probability of failure. Typically, the convergence of the iterative procedure is very fast, even though the basic problem involve strong non-linear properties. In the figure below an example of the convergence of the limit state function is shown, wheregis the difference between the failure stress and evaluated stress. The failure probability level for the postbuckling problem is in this case 410 4. The application of reliability methods to the finite element method is currently subject for intensive research activities and examples of work done in this area [8, 9, 10, 11, 12].
The iterative procedure for finding the design point involves finding the gradient of the limit state function. The gradient can be found either analytically, as was done in [10] for the case of truss elements and four node plane elements, or numerically. The advantage of an analytically derived gradient is of course the computational fastness, nevertheless, at the expense of the generality in the code. Herein, a finite difference technique is used in order to find the gradient. A Newton Raphson procedure is used for calculation of the response at the current values of the stochastic variables.
This point is subsequently used as a start point for the new values of variables and equilibrium iterations are anew performed. It can be noted that all the variables in the non-linear finite element equation are functions of the stochastic variables. For example, the strain-displacement matrix varies as a function of the geometrical imperfections.
The structure analyzed in this work is an in-plane loaded corrugated board panel. The analysis is part of a larger project devoted to reliability design of corrugated board packages. Failure is assumed to take place in either of the facings due to material failure or local buckling and the failure criterion presented in [13] is used for the analysis. The stability behaviour of the panel is analyzed by several numerical examples. The variable uncertainties studied are for example the magnitude of the geometrical imperfection of the panel, material properties such as strength and stiffness, and load magnitude.
1 2 3 4 5 6 7 8
−2 0 2 4 6 8 10 12x 106
g
ITERATION NR
Figure 1: Convergence of the limit state function.
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