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DYNAMIC ANALYSIS OF STRUCTURES WITH INTERVAL UNCERTAINTY

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ABSTRACT

A new method for dynamic response spectrum analysis of a structural system with

interval uncertainty is developed. This interval finite-element-based method is capable of

obtaining the bounds on dynamic response of a structure with interval uncertainty. The

proposed method is the first known method of dynamic response spectrum analysis of a

structure that allows for the presence of any physically allowable interval uncertainty in

the structure’s geometric or material characteristics and externally applied loads other

than Monte-Carlo simulation. The present method is performed using a set-theoretic

(interval) formulation to quantify the uncertainty present in the structure’s parameters

such as material properties. Independent variations for each element of the structure are

considered. At each stage of analysis, the existence of variation is considered as presence

of the perturbation in a pseudo-deterministic system. Having this consideration, first, a

linear interval eigenvalue problem is performed using the concept of monotonic behavior

of eigenvalues for symmetric matrices subjected to non-negative definite perturbation

which leads to a computationally efficient procedure to determine the bounds on a

structure’s natural frequencies. Then, using the procedures for perturbation of invariant

subspaces of matrices, the bounds on directional deviation (inclination) of each mode

shape are obtained.

INTRODUCTION

However, throughout conventional dynamic response spectrum analysis, the

possible existence of any uncertainty present in the structure’s geometric and/or material

characteristics is not considered. In the design process, the presence of uncertainty is

accounted for by considering a combination of load amplification and strength reduction

factors that are obtained by modeling of historic data. However, the impact of presence of

uncertainty on a design is not considered in the current deterministic dynamic response

spectrum analysis. In the presence of uncertainty in the geometric and/or material

properties of the system, an uncertainty analysis must be performed to obtain bounds on

the structure’s response.

Dissertation Overview

In chapter II, the analytical procedure for deterministic dynamic analysis is

presented. Chapter III is devoted to fundamentals of uncertainty analyses with emphasis

on the interval method. In chapter IV, matrix perturbation theories for eigenvalues and

eigenvectors are discussed. Chapter V introduces the method of interval response

spectrum analysis. In chapter VI, the bounds on variations of natural frequencies and

mode shapes are obtained. Chapter VII is devoted to determination of the bounds on the

total response of the structure. In chapter VIII, exemplars and numerical results are

presented. Chapter IX is devoted to observations and conclusions.

Structural Dynamics Historical Background

Modern theories of structural dynamics were introduced mostly in mid 20th

century. M. A. Biot (1932) introduced the concept of earthquake response spectra and G.

W. Housner (1941) was instrumental in the widespread acceptance of this concept as a

practical means of characterizing ground motions and their effects on structures. N. M.

Newmark (1952) introduced computational methods for structural dynamics and

earthquake engineering. In 1959, he developed a family of time-stepping methods based

on variation of acceleration over a time-step.

A. W. Anderson (1952) developed methods for considering the effects of lateral

forces on structures induced by earthquake and wind and C. T. Looney (1954) studied

the behavior of structures subjected to forced vibrations. Also, D. E. Hudson (1956)

developed techniques for response spectrum analysis in engineering seismology. A.

Veletsos (1957) determined natural frequencies of continuous flexural members.

Moreover, he investigated the deformation of non-linear systems due to dynamic loads.

E. Rosenblueth (1959) introduced methods for combining modal responses and

characterizing earthquake analysis.

Background

In structural engineering, design of an engineered system requires that the

performance of the system is guaranteed over its lifetime. However, the parameters for

designing a reliable structure possess physical and geometrical uncertainties. The

presence of uncertainty can be attributed to physical imperfections, model inaccuracies

and system complexities. Moreover, neither the initial conditions, nor external forces,

nor the constitutive parameters can be perfectly described. Therefore, in order to design a

reliable structure, the possible uncertainties in the system must be included in the analysis

procedures.

Fuzzy Analysis

The fuzzy approach to the uncertain problems is to model the structural

parameters as fuzzy quantities (Lotfi-zadeh 1965). In conventional set theories, either an

element belongs or doesn’t belong to set. However, fuzzy sets have a membership

function that allows for “partial membership” in the set. Using this method, structural

parameters are quantified by fuzzy sets. Following fuzzifying the parameters, structural

analysis is performed using fuzzy operations.

CONCLUSIONS

• A finite-element based method for dynamic analysis of structures with interval

uncertainty in structure’s stiffness or mass properties is presented.

• In the presence of any interval uncertainty in the characteristics of structural

elements, the proposed method of interval response spectrum analysis (IRSA) is

capable to obtain the nearly sharp bounds on the structure’s dynamic response.

• IRSA is computationally feasible and it shows that the bounds on the dynamic

response can be obtained without combinatorial or Monte-Carlo simulation

procedures.

• The solutions to only two non-interval eigenvalue problems are sufficient to bound

the natural frequencies of the structure. Based on the given mathematical proof, the

obtained bounds on natural frequencies are exact and sharp.

• Computation time for the algorithm increases between linear to quadratic with

increasing the number of degrees of freedom.