STATIC ANALYSIS OF UNCERTAIN STRUCTURES USING INTERVAL EIGENVALUE DECOMPOSITION Mehdi Modares Tufts University Robert L. Mullen Case Western Reserve University

Static Analysis An essential procedure to design a structure subjected to a system of loads.

Static Analysis (Cont.) In conventional static analyses of a structure, the existence of any uncertainty present in the structure’s geometric or material characteristics as well as loads is not considered.

Uncertainty in Transportation Systems Attributed to: Structure’s Physical Imperfections Inaccuracies in Determination of Loads Modeling Complexities of Load-Structure Interaction For reliable design, the presence of uncertainty must be included in analysis procedures

Research Objective To introduce a computationally efficient finite-element-based method for linear static analysis of a structure with uncertain properties expressed as interval quantities.

Presentation Outline Fundamentals of engineering uncertainty analysis Introduction of method of Interval Static Analysis (ISA) Numerical problem Work conclusions

Engineering Uncertainty Analysis Formulation Modifications on the representation of the system characteristics due to presence of uncertainty Computation Development of schemes capable of considering the uncertainty throughout the solution process

Uncertainty Analysis Schemes Considerations: Consistent with the system’s physical behavior Computationally feasible

Paradigms of Uncertainty Analysis Stochastic Analysis : Random variables Fuzzy Analysis : Fuzzy variables Interval Analysis : Interval variables

Interval Variable A real interval is a set of the form: Archimedes ( B.C.)

Deterministic Linear Static Analysis The equilibrium defined as a linear system of equation as: : Global elastic stiffness matrix : Displacement vector : External force vector

Interval Static Analysis The equilibrium defined as a linear system of equation as: : Interval elastic stiffness matrix (presence of uncertainty) : Displacement vector : External force vector

Prior Works In Interval Static Analysis Muhanna and Mullen (2001) Neumaier and Pownuk (2007)

Stiffness Eigenvalue Decomposition Considering an eigenvalue decomposition of the stiffness matrix, the stiffness is: : Matrix of eigenvalues : Matrix of eigenvectors Equivalently:

Stiffness Matrix Inversion The stiffness matrix inverse is obtained as: Equivalently the inversion is:

Static Response The static response (the solution to the linear system if equation is: Substitution of the decomposed stiffness matrix inverse, the response is:

Interval Eigenvalue Decomposition Considering an interval eigenvalue decomposition of the stiffness matrix due to the presence of uncertainty, the interval stiffness is: : Interval eigenvalues : Interval eigenvectors

Interval Static Response The interval static response is: In which, the bounds on the eigenvalues and eigenvectors of interval system must be obtained.

1. Interval Stiffness Matrix The deterministic structure global stiffness is a linear summation of the element contributions to the global stiffness matrix: [L i ] : Element Boolean connectivity matrix [k i ] : Element stiffness matrix

Non-Deterministic System with uncertainty in the stiffness characteristics Non-deterministic element stiffness matrix: : Interval of uncertainty Non-deterministic global stiffness matrix:

Interval Stiffness Matrix Non-deterministic global stiffness matrix: Deterministic element stiffness contribution: Structure’s global interval stiffness matrix:

2. Interval Eigenvalue Problem Interval eigenpair problem using the interval global stiffness matrix Central (pseudo-deterministic) and Radial (perturbation) stiffness matrices:

Updated Interval Eigenvalue Problem due to uncertainty in stiffness Updated interval eigenvalue problem using central and radial stiffness matrices: Problem Mathematical Interpretation: An eigenvalue problem on a central stiffness matrix when it is subjected to a perturbation of radial stiffness matrix (linear summation of non-negative definite matrices). (Constraint imposed by problem’s physical behavior )

3. Solution for Eigenvalues Classical linear eigenpair problem for a symmetric matrix : Real eigenvalues corresponding eigenvectors Rayleigh quotient (ratio of quadratics):

Minimization of Rayleigh Quotient with no constraint Performing an unconstrained minimization : The smallest eigenvalue is the solution to the unconstrained minimization on the scalar-valued function of Rayleigh quotient.

Minimization of Rayleigh Quotient with imposed constraints Imposing a single constraint on the minimization: x : Trial vector z : Arbitrary vector Performing a constrained minimization :

Constrained Minimizations on R.Q. to find the second eigenvalue Minimization of R(x) subject to a single constraint : choosing the {z} that maximizes the minimum yields the second smallest eigenvalue. For the next eigenvalues more constraints must be imposed.

Generalization of Constrained Minimizations on R.Q. to find the next eigenvalues Generalization of results for the Kth eigenvalue: Subjected to This principle is called the “ Maximin Characterization” of eigenvalues for symmetric matrices.

Matrix Non-Negative Definite Perturbation Symmetric matrix [A] subject to non-negative definite perturbation matrix [E ]: First eigenvalue : using unconstrained minimization: Next eigenvalues : using maximin characterization:

Monotonic Behavior of Eigenvalues All eigenvalues of a symmetric matrix subject to a non-negative perturbation monotonically increase from the eigenvalues of the exact matrix. Similarly, all eigenvalues of a symmetric matrix subjected to a non- positive perturbation monotonically decrease from the eigenvalues of exact matrix.

Bounds on Eigenvalues Using minimum and Maximin characterization of eigenvalues The first eigenvalue: The next eigenvalues:

Bounds on Eigenvalues Using monotonic behavior of eigenvalues Upper bound: Lower bound:

Bounding Interval Eigenvalue Problems Solution to interval eigenvalue problem correspond to the maximum and minimum eigenvalues: Two deterministic problems capable of bounding all eigenvalues (Modares and Mullen 2004)

4. Solution for Eigenvectors Invariant Subspaces The subspace is an invariant subspace of if: i.e., if is an invariant subspace of and, columns of form a basis for, then there is a unique matrix such that: matrix is the representation of with respect to.

Theorem of Invariant Subspaces For a real symmetric matrix : Matrices and forming bases for complementary subspaces and. Then, is an invariant subspace of if and only if:

Simple Invariant Subspaces An invariant subspace is simple if the eigenvalues of its representation are distinct from other eigenvalues of :

Perturbation of Invariant Subspaces Due to perturbation of matrix Considering the column spaces of and to span two complementary subspaces, the perturbed orthogonal subspaces are: in which [P ] is the matrix to be determined.

Perturbation Problem A perturbed matrix: Using theorem of invariant subspaces, is an invariant subspace of iff : Substitution and linearization: in the form of a Sylvester’s equation.

Sylvester’s Equation A Sylvester’s equation is in the form: Equivalently defining a linear operator [T ]: The uniqueness of the solution guaranteed when operator [T] is non- singular. This operator is non-singular when eigenvalues of [A] and [B] are distinct:

Perturbation of Eigenvectors Specialization of perturbation of invariant subspaces The perturbed first eigenvector: The perturbation problem: If the eigenvalues are simple the solution for [p] exists and is unique as:

Perturbation of Eigenvectors Solution The solution for the perturbed first eigenvector:

Perturbation of Eigenvectors Solution for static analysis The perturbation matrix: The solution:

Interval Static Response The interval static response is: To attain sharper results, the functional dependency of the intervals must be considered.

Numerical Example Present Method Interval Static Analysis Combinatorial Solution (NP-hard) Lower and upper values for each element (2 n )

Example Problem 2-D Statically indeterminate truss with material uncertainty

Results Vertical displacement of the top node Lower Bound Present Method Lower Bound Combination Method Upper Bound Combination Method Upper Bound Present Method Error % % 0.12

Conclusions An alternate finite-element based method for static analysis of structural systems with interval uncertainty is presented. This proposed method is simple to implement and computationally efficient. The method can potentially obtain the sharp bounds on the structure’s static response. While this methodology is shown for structural systems, its extension to various mechanics problems is straightforward. Future work Comparison to other existing schemes for efficiency, sharpness and stability

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