Static Analysis of Uncertain Structures

Static Analysis of Uncertain StructuresStatic Analysis of Uncertain Structures apply Interval Eigenvalue Decomposition1Mehdi Modargons and 2Robert L. Mullen1Department of genteel and Environmental Engineering Tufts University Medford, MA, 02155 2Department of Civil Engineering Case Western Reserve University Cleveland, OH, 44106 Abstract Static digest is an essential procedure to design a structure. Using dormant analysis, the structures response to the applied external compresss is obtained. This response includes internal forces/moments and internal stresses that is used in the design process. However, the mechanical characteristics of the structure possess changefulties which alter the structures response. maven rule to quantify the presence of these uncertainties is breakup or un cognize-but- leap variables.In this work a new mode is developed to obtain the limit on structures inactive response using legal sepa proportionalityn eigenvalue decomposition of the stiffn ess hyaloplasm. The bounds of eigenvalues atomic number 18 obtained using monotonic behavior of eigenvalues for a symmetric matrix subjected to non-negative definite psychological disorders. Moreover, the bounds of eigenvectors are obtained using affray of invariant subspaces for symmetric matrices. Comparisons with other legal separation finite constituent issue methods are confronted. Using this method, it has shown that obtaining the bound on static response of an uncertain structure does not require a combinatorial or Monte-Carlo simulation procedure.Keywords Statics, Analysis, Interval, Uncertainty 2008 by authors. Printed in USA. REC 2008 Modares and MullenIn design of structures, the procedure of the structure must be guaranteed over its lifetime. Moreover, static analysis is a fundamental procedure for designing reliable structure that are subjected to static or quasi-static forces induced by various(a) loading conditions and patterns.However, in accepted proced ures for static analysis of structural dodges, the existence of uncertainty in either mechanical properties of the constitution or the characteristics of forcing function is generally not considered. These uncertainties can be attributed to physical imperfections, modeling inaccuracies and system complexities.Although, in a design process, uncertainty is accounted for by a combination of load amplification and strength reduction factors that are based on probabilistic models of historic data, consideration of the effects of uncertainty has been removed from current static analysis of structural systems.In this work, a new method is developed to perform static analysis of a structural system in the presence of uncertainty in the systems mechanical properties as well as uncertainty in the magnitude of loads. The presence of these uncertainties is quantified using interval or unknownbut-bounded variables.This method obtains the bounds on structures static response using interval eige nvalue decomposition of the stiffness matrix. The bounds of eigenvalues are obtained using the concept of monotonic behavior of eigenvalues for a symmetric matrix subjected to non-negative definite perturbations. Furthermore, the bounds of eigenvectors are obtained using perturbation of invariant subspaces for symmetric matrices. Using this method, it has shown that obtaining the bound on static response of an uncertain structure does not require a combinatorial or MonteCarlo simulation procedure.The equation of sense of balance for a multiple degree of freedom structure is delineate as a unidimensional system of equations asKU=P (1)where, Kis the stiffness matrix, Uis the vector of unknown nodal displacements, and P is the vector of nodal forces. The termination to this system of equation isU = K1P (2)The concept of interval numbers has been originally applied in the error analysis associated with digital computing. Quantification of the uncertainties introduced by gruffness of real numbers in numerical methods was the primary application of interval methods (Moore 1966).A real interval is a closed set defined by extreme values as (Figure 1)l ,zu =z zl z zu (3)Z = zx = a,bFigure 1. An interval variable.In this work, the symbol () represents an interval quantity. One recitation of an interval number is a random variable whose probability density function is unknown but non-zero only in the range of interval.Another rendition of an interval number includes intervals of confidence for -cuts of fuzzy sets. The interval theatrical performance transforms the point values in the settled system to inclusive set values in the system with bounded uncertainty.Considering the presence of interval uncertainty in stiffness and force properties, the system of equilibrium equations, Eq.(1), is modified as an interval system of equilibrium equation asKU=P (4)where, Kis the interval stiffness matrix, Uis the vector of unknown nodal displacements, and P is the vec tor of interval nodal forces. In development of interval stiffness matrix, the physical and mathematical characteristics of the stiffness matrix must be preserves.This system of interval equations is mainly solved using computationally iterative procedures (Muhanna et al 2007) and (Neumaier and Pownuk 2007). The present method proposes a computationally cost-efficient procedure with to the highest degree sharp results using interval eigenvalue decomposition of stiffness matrix.While the external force can also have uncertainties, in this work only capers with interval stiffness properties are addressed. However, for useable independent variations for both stiffness matrix and external force vector, the extension of the proposed work is straightforward.3.1. DETERMINISTIC EIGENVALUE corruptionThe deterministic symmetric stiffness matrix can be decomposed using matrix eigenvalue decomposition asK = T (5)where, is the matrix of eigenvectors, and is the diagonal matrix of eigenva lues. Equivalently,NK =iiiT (6)i=1where, the values of i is the eigenvalues and the vectorsiare their sameeigenvectors. Therefore, the eigenvalue decomposition of the inverse of the stiffness matrix isequivalently,K1 =1T (7)N 1TK 1 = ii (8)i=1 iSubstituting Eq.(8) in the solution for the deterministic linear system of equation, Eq.(2), the solution for response is shown asU= ( N 1 iiT )P (9)3.2. interval EIGENVALUE DECOMPOSITIONSimilarly, the solution to interval system of equilibrium equations, Eq.(4), isU= (N 1 iT )P (10) ii=1 i are their where, the values of i is the interval eigenvalues and, the vectors icorresponding interval eigenvectors that are to be determined.4.1. BACKGROUNDThe research in interval eigenvalue fuss began to emerge as its applicability in science and engineering was realized. Hollot and Bartlett (1987) studied the spectra of eigenvalues of an interval matrix family which are found to depend on the spectrum of its extreme sets. Dief (1991) presented a method for computing interval eigenvalues of an interval matrix based on an assumption of invariance properties of eigenvectors.In structural dynamics, Modares and Mullen (2004) have introduced a method for the solution of the interval eigenvalue problem which determines the exact bounds of the natural frequencies of a system using Interval Finite Element formulation.4.2. DEFINITIONThe eigenvalue problems for matrices containing interval values are known as the interval nn ) and A is a outgrowth of the eigenvalue problems. If A is an interval real matrix (Ainterval matrix (AA) , the interval eigenvalue problem is shown as4.2.1. Solution for EigenvaluesThe solution of reside to the real interval eigenvalue problem for bounds on apiece eigenvalue isdefined as an inclusive set of real values () such that for any member of the interval matrix, the eigenvalue solution to the problem is a member of the solution set. Therefore, the solution to the interval eigenvalue problem for each eigenvalue can be mathematically expressed asl ,u AA (AI)x = 0 (12)= 4.2.2. Solution for Eigenvectors The solution of interest to the real interval eigenvalue problem for bounds on each eigenvector is defined as an inclusive set of real values of vector x such that for any member of the interval matrix, the eigenvector solution to the problem is a member of the solution set. Thus, the solution to the interval eigenvalue problem for each eigenvector is4.3. INTERVAL STIFFNESS MATRIXThe systems worldwide stiffness can be viewed as a summation of the element contributions to the global stiffness matrixni=1where Li is the element Boolean connectivity matrix and Ki is the element stiffness matrix in the global coordinate system. Considering the presence of uncertainty in the stiffness properties, the non-deterministic element elastic stiffness matrix is expressed asin which, li ,ui is an interval number that pre-multiplies the deterministic element stiffness matrix. This procedure p reserves the physical and mathematical characteristics of the stiffness matrix.Therefore, the systems global stiffness matrix in the presence of any uncertainty is the linear summation of the contributions of non-deterministic interval element stiffness matrices,ui )Li Ki Li =i=1i=1in which, Ki is the deterministic element elastic stiffness contribution to the global stiffness matrix.4.4. INTERVAL EIGENVALUE PROBLEM FOR STATICSThe interval eigenvalue problem for a structure with stiffness properties expressed as interval values isK = () (17)Substituting Eq.(16) in Eq.(17)) = ()i=1This interval eigenvalue problem can be transformed to a pseudo-deterministic eigenvalue problem subjected to a matrix perturbation. Introducing the central and radial (perturbation) stiffness matrices asi 1 KR =i=n1 (i )(ui 2li )Ki , i =1,1 (20)Using Eqs. (19,20), the non-deterministic interval eigenpair problem, Eq.(18), becomesHence, the determination of bounds on eigenvalues and bounds on eigenvect ors of a stiffness matrix in the presence of uncertainty is mathematically interpreted as an eigenvalue problem on a central stiffness matrix (KC ) that is subjected to a radial perturbation stiffness matrix (KR ).This perturbation is in fact, a linear summation of non-negative definite deterministic element stiffness contribution matrices that are scaled with bounded real numbers(i ) .5. Solution 5.1. BOUNDS ON EIGENVALUESThe following concepts must be considered in order to bound the non-deterministic interval eigenvalue problem, Eq.(21). The classical linear eigenpair problem for a symmetric matrix iswith the solution of real eigenvalues (1 2 n ) and corresponding eigenvectors( x1, x2,, xn ). This equation can be transformed into a ratio of quadratics known as the Rayleigh quotientR(x) = (23)The Rayleigh quotient for a symmetric matrix is bounded between the smallest and the largest eigenvalues (Bellman 1960 and Strang 1976). (24)Thus, the first eigenvalue (1) can be obtained b y performing an unconstrained minimization on the scalar-valued function of Rayleigh quotient ( (25)xFor finding the next eigenvalues, the concept of maximin characterization can be used. This concept obtains the kth eigenvalue by imposing (k-1) constraints on the minimization of the Rayleigh quotient (Bellman 1960 and Strang 1976)k = maxminR(x)(subject to constrains(xT zi = 0),i =1,k 1,k 2 ) (26)5.1.1. starting the Eigenvalues for StaticsUsing the concepts of minimum and maximin characterizations of eigenvalues for symmetric matrices, the solution to the interval eigenvalue problem for the eigenvalues of a system with uncertainty in the stiffness characteristics (Eq.(21)) for the first eigenvalue can be shown asnxRnxT xfor the next eigenvaluesxT KxxT (K +K )x5.1.2. Deterministic Eigenvalue Problems for Bounding Eigenvalues in StaticsSubstituting and expanding the right-hand side terms of Eqs. (27,28)T K xui(li +ux(29)Since the matrix Ki is non-negative definite, the term () is n on-negative.Therefore, using the monotonic behavior of eigenvalues for symmetric matrices, the fastness bounds on the eigenvalues in Eqs.(19,20) are obtained by considering maximum values of interval coefficients of uncertainty (i = 1,1), ((i )max = 1), for all elements in the radial perturbation matrix.Similarly, the lower bounds on the eigenvalues are obtained by considering minimum values of those coefficients, ((i )min =1) , for all elements in the radial perturbation matrix. Also, it can be ascertained that any other element stiffness selected from the interval set will yield eigenvalues between the upper and lower bounds. This imonotonic behavior of eigenvalues can also be used for parameterization purposes.Using these concepts, the deterministic eigenvalue problems corresponding to the maximum and minimum eigenvalues are obtained (Modares and Mullen 2004) asnn5.2. BOUNDS ON EIGENVECTORS5.2.1. Invariant SubspaceThe subspace is defined to be an invariant subspace of matrix A ifA(32)Equivalently, if is an invariant subspace of Ann and also, columns of X1nm form a basis for, then there is a unique matrix L1mm such thatThe matrix L1 is the representation of A on with respect to the basis X1 and the eigenvalues of L1 are a subset of eigenvalues of A. Therefore, for the invariant subspace,(v,) is an eigenpair of L1 if and only if (X1v,) is an eigenpair of A.5.2.2. Theorem of Invariant SubspacesFor a real symmetric matrix A, considering the subspace with the linearly independent columns of X1 forming a basis for and the linearly independent columns of X2 bridgening the complementary subspace , then, is an invariant subspace of A iffTherefore, invoking this condition and postulating the definition of invariant subspaces, the symmetric matrix A can be trim down to a diagonalized form using a unitary similarity transformation asX1X2T AX1X2 = X1TTAAXX11X2where Li =Xi T AXi , i =1,2.5.2.3. Simple Invariant SubspaceX1T AX2 L1 X2T AX2= 00 L2(35)An invarian t subspace is simple if the eigenvalues of its representation L1 are distinct from other eigenvalues of A. Thus, using the reduced form of A with respect to the unitary matrixX1X2, is a simple invariant subspace if the eigenvalues of L1 and L2 are distinct5.2.4. Perturbed EigenvectorConsidering the column spaces of X1 and X2 to span two complementary simple invariant subspaces, the perturbed orthogonal subspaces are defined asX1 =X1+X 2 P(37)X 2 =X 2X1PT(38)in which P is a matrix to be determined.Thus, each perturbed subspace is defined as a summation of the exact subspace and the contribution of the complementary subspace. Considering a symmetric perturbationE , the perturbed matrix is defined asApplying the theorem of invariant subspaces for perturbed matrix and perturbed subspaces, and linearizing repayable to a small perturbation compared to the unperturbed matrix, Eq.(34) is rewritten asThis perturbation problem is an equation for unknown P in the form of a Sylvesters equati on in which, the singularity of the solution is guaranteed by the existence of simple perturbed invariant subspaces.Finally, specializing the result for one eigenvector and solving the above equation, the perturbed eigenvector is (Stewart and Sun 1990)x1 = x1+X 2 (1IL2 )1X 2 T Ex15.2.5 Bounding Eigenvectors for Statics For the perturbed eigenvalue problem for statics, Eq.(21), the error matrix is(41)nu E = KR = ((i )( i li )Ki )(42)i=12Using the error matrix in eigenvector perturbation equation for the first eigenvector, Eq.(33) the perturbed eigenvector isin which, 1is the first eigenvector, (1) is the first eigenvalue, 2 is the matrix of remaining eigenvectors and 2 is the diagonal matrix of remaining eigenvalues obtained from the deterministic eigenvalue problem. Eq.(30,31 and 43) is used to calculate the bounds on interval eigenvalues and interval eigenvectors in the response equation, Eq.(9).In order to attain sharper results, the functional dependency of intervals in dire ct interval multiplications in Eq.(9) is considered. Also, remark intervals are subdivided and the union of responses of subset results is obtained.6. Numerical Example Problem The bounds on the static response for a 2-D statically indeterminate truss with interval uncertainty present in the modulus of elasticity of each element are determined (Figure 2). The crosssectional area A, the length for horizontal and vertical members L , the Youngs moduli E for allelements are E = (0.99,1.01)E .Figure 2. The structure of 2-D trussThe problem is solved using the method presented in this work. The functional dependency of intervals in the response equation is considered. A hundred-segment subsection of input intervals is performed and the union of responses is obtained. For comparison, an exact combinatorial analysis has performed which considers lower and upper values of uncertainty for each element i.e. solving (2n = 210 =1024 ) deterministic problems.The static analysis results obtained by the present method and the brute force combination solution for the vertical displacement of the top nodes in are summarized Table (1).Lower BoundPresent MethodLower BoundCombination Method Upper BoundCombination Method Upper BoundPresent Method delusion%U PL AE -1.6265-1.6244-1.5859-1.5838% 0.12Table1. Bounds on Vertical Displacement of Top NodesThe results show that the proposed robust method yields nearly sharp results in a computationally efficient manner as well as preserving the systems physics.4.Conclusions A finite-element based method for static analysis of structural systems with interval uncertainty in mechanical properties is presented.This method proposes an interval eigenvalue decomposition of stiffness matrix. By obtaining the exact bounds on the eigenvalues and nearly sharp bounds on the eigenvectors, the proposed method is capable to obtain the nearly sharp bounds on the structures static response.Some conservative overestimation in response occurs that can be attributed to the linearization in formation of bounds of eigenvectors and also, the functional dependency of intervals in the dynamic response formulation.This method is computationally feasible and it shows that the bounds on the static response can be obtained without combinatorial or Monte-Carlo simulation procedures.This computational efficiency of the proposed method makes it attractive to introduce uncertainty into structural static analysis and design. While this methodology is shown for structural systems, its extension to various mechanics problems is straightforward.References Bellman, R. Introduction to Matrix Analysis, McGraw-Hill, New York 1960.Dief, A., Advanced Matrix theory for Scientists and Engineers, pp.262-281. 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