Stability analysis of uncertain structures with uncertain-but-bounded parameters

This paper focus onstability analysis of uncertain structures with uncertain-but-bounded parameters. For uncertain structures, thestandardand geometry stiffness matrices are both uncertain. With non-negative decomposition of a matrix, the standardand geometry stiffness matrices are both expressed as the combination of the uncertain structural parameters. Then, the approach acquiring the critical buckling load vector of an uncertain structure is presented. Anumerical examplewas used to show the effects of the proposed method.


Introduction
The stability of a body may be defined as a measure of its tendency to persist in some state under a disturbing influence of an external agency which encourages the body to abandon that state [1].
The world is an uncertain world. Uncertainty is the absolute property while certainty is the relative property. Thus, it is necessary to take uncertainties into account for engineering design. In engineering field, uncertainties can be divided into three categories: physical aspect, material aspect and geometrical aspect.For physical aspect, for example, the mass and the density have some uncertainties.For material aspect, for instance, there are some uncertainties exist in Young's modulus, yield stress and Poisson ratio. For geometrical aspect, the measured dimensions such as length, width and height, are with some errors under common conditions. Deterministic stability theory [2] is not applicable to structures with uncertain parameters any longersince the predictions are not accurate and may lead to the dangerous structural design.
A good way to describe these uncertain structural parameters including uncertain loads is interval. An interval contains a lower bound and an upper bound which reflects more available information than a single point. The mathematical model describing this kind of uncertainties called non-probabilistic model.Qiuet al. [3][4][5][6][7], Ben-Haim and Elishakoff [8],Elishakoff et al. [9][10], Pantelides [11], Mullen and Muhanna [12] andChen and Yang [13], developed different interval analysis methods for analysing structures with uncertain-but-bounded parameters. In their interval analysis methods, only values of the lower and upper bounds for uncertain-but-bounded parameters are needed, the structural responses which are in the form of intervals can be finally obtained.
In this study, with the non-negative decomposition of the standard and geometry stiffness matrices of a structure, a novel approach for solving the critical buckling load of uncertain structures with uncertain-but-bounded parameters is proposed.

Stability problemof anuncertain structure
The buckling eigenvalue problem with uncertainties can be written as follows where K is the global standard stiffness matrix and g K is the global geometricstiffness matrix.
The structural parameter vector can be expressed as or , 1,2, , is the lower bound vector of the structural parameter vector band is the upper bound vector. Using interval mathematics and interval analysis theory [14],Eq.(2)should be rewritten as or , 1, 2, , where ( ) , , , 1, 2, , where I b is the interval structural parameter vector, , 1, 2, , , are the components of the interval vector. The eigenvalues in Eq. (1) with satisfying constrain conditions (2) can be expressed in set form as Then, the interval of the buckling eigenvalue I i  should be determinted ( ) , , , , 1, 2, , The i-th bucklingeigenvalue can be acquired through is an arbitraryi-dimensional subspace [15].
After acquiring the interval buckling eigenvalues , p is defined as the reference load vector, the minimum lower bound min  should be used for searching the corresponding critical buckling load vector cr p to ensure the structural safety.

The approach for determining the interval buckling eigenvalues
The non-negative decomposition of the standard and geometric stiffness matrices are required for determining the interval buckling eigenvalues.  3 It is assumed that the global standard stiffness matrix has k structural parameters 12 , , , k b b b and the global geometric stiffness matrix has mk − ( mk  ) structural parameters 12 , , , , the nonnegative decomposition of the both matrices are where j K and j  K are positive definite and the parameters j b are positive. This decomposition is called the non-negative decomposition of a matrix. Such decompositions arise naturally in practical engineering application. For instance, in finite element analysis of structures, j K and j  K can be considered as the elemental standard and geometric stiffness matrices with respect to the structural parameter j b .
Further, combine the parameter vectors of standard and geometric stiffness matrices as 12 ( , , , ) , Eq.(10)should be expressed as ( ) where the matrics 12 , , , k k m ++ K K K and 12 , , , k    K K K are null matrics. It is assumed that the parameter interval vector is given by I b . Then, the set of boundary vectorsof the interval vector I b of the uncertan parameters is ( )  : , , and or , 1, 2, , Since the set of boundary vectors are the extreme vectors of the parameter interval vector , the element number should be 2 m , where m is the length of the vector b.
With the non-negative decomposition of the standard and geometric stiffness matrices, the i-th buckling eigenvalue i  can be obtained. i  is contained in an interval [ , ]  Two cases for the plane frame with uncertain-but-bounded structural parameters will be discussed to compare the interval buckling eigenvalues obtained by the presented parameter vertex solution methodand the conventional stability theory.  It is found out that the critical buckling load acquired by the proposed parameter vertex solution method is smaller than that from the conventional stability theory. The results indicate that more conservative critical buckling loads can be obtained by the present method for stuctures with uncertan parameters.

Conclusions
In this paper, the stability analysis of uncertain structures with uncertain-but-bounded parameters is proposed. By using the non-negative decomposition of the standard and geometric stiffness matrices, the approach for determining the interval buckling eigenvalues is presented. The presented method is capable of dealing with the stability problem of uncertain structures.A numerical example of a simple portal frame was used to show the effectiveness of the presented method. Results indicate that the proposed approach is capable of providingus with safer critical buckling load than the conventional stability theory.
In conclusion, the presented method isa practical and effective for dealing with the stability problem of uncertan structures.