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 DYNAMIC ANALYSIS OF STRUCTURES WITH INTERVAL UNCERTAINTY pdf

DYNAMIC ANALYSIS OF STRUCTURES WITH INTERVAL UNCERTAINTY

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.
A new method is developed for the dynamic analysis of the response spectrum of a structural system with interval uncertainty. This method based on finite interval elements is able to obtain the limits on the dynamic response of a structure with interval uncertainty. The present method is performed using a theoretical formulation (range) to quantify the uncertainty present in the parameters of the structure, such as material properties and cross-sectional geometry. Independent and / or dependent variations are considered for each element of the structure. At each stage of the analysis, the existence of variation is considered as the presence of perturbation in a pseudo-deterministic system. Given this consideration, first, a problem of linear range eigenvalues ​​is performed using the concept of monotonous behavior of eigenvalues ​​for symmetric matrices subjected to non-negative defined perturbations. Then, using the procedures for the perturbation of invariant matrices sub-spaces, the limits of the directional (tilt) deviation of each mode shape are obtained. Next, the interval response spectrum analysis is performed considering the effects of the input variation in terms of the total response of the structure, which includes modal maximum coordinates, modal participation factors and mode shapes.
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