Uncertainty Modeling in Structural Engineering: Application to Dynamic Analysis

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UNCERTAINTY MODELING IN STRUCTURAL ENGINEERING: APPLICATION TO DYNAMIC ANALYSIS

Presentation by:

V. ANIL KUMAR

Roll No. 0109-11109

V Semester

M.E. Structural Eng.

Under the guidance of

Dr. R. Ramesh Reddy

Professor Dept. of Civil Engineering

DEPARTMENT OF CIVIL ENGINEERING

UNIVERSITY COLLEGE OF ENGINEERING (AUTONOMOUS) OSMANIA UNIVERSITY, HYDERABAD

CERTIFICATECERTIFICATECERTIFICATECERTIFICATE

This is to certify that, this is a bonafide record of the seminar presentation entitled “Uncertainty Modeling In Structural Engineering: Application to Dynamic Analysis” carried out by Mr. V. ANIL KUMAR bearing Roll no. 0109-11109, of V Semester, M. E., Structural Engineering (CEEP), during the academic year 2010-2011 in partial fulfillment of academic requirements.

Guide

Dr. R. Ramesh Reddy

Professor,

Dept. of Civil Engineering

External Examiner

CONTENTS

Page No.

CERTIFICATE

SYNOPSIS 01

INTRODUCTION 01

INTRODUCTION TO STRUCTURAL DYNAMICS 02

PRINCIPLES OF DYNAMIC ACTION 03-08

SOURCES OF UNCERTAINTY 08-09

CATEGORIES OF UNCERTAINTY 09

UNCERTAINTY ANALYSIS 09-18

NEW AND MODIFIED METHODS 18

FUZZY FINITE ELEMENT METHOD (FFEM) 19-20

INTERVAL RESPONSE SPECTRUM ANALYSIS (IRSA) 21-23

CONCLUDING REMARKS 23

REFERENCES 24

Uncertainty Modeling In Structural Engineering: Application to Dynamic Analysis

ABSTRACT

The general terms that frequently heard in structural analysis or in fact in every analysis procedure are: uncertainty, randomness, fuzzy, ambiguity and stochastic (non-deterministic) process. The term uncertainty can be defined as unknowingness of a situation or a prediction that is going to occur in a specific process or phenomena. As per H. J. Zimmermann (ELITE) uncertainty can be defined as “a situation in which a person does not have the quantitatively and qualitatively appropriate information to describe, prescribe or predict deterministically and numerically a system, its behavior or other characteristic.” There are lot more advancements and investigations for finding out uncertainties and predicting a certain result in all fields like engineering, medicine, psychology etc. The scope of the topic includes the generally used modeling techniques for solving the problems having uncertain parameters in Dynamic Analysis of Structures. Already research is going on in this direction. Until now the methods like Fuzzy-Finite Element Method, Interval Analysis, Propagation theories, Perturbation techniques are successfully applied for modeling problems in static analysis. I am attempting to study and apply Fuzzy-Finite Element method using Interval based method to Dynamic Analysis of Structures.

As the Civil Engineering is itself subjected to lot of assumptions and limitations for solving various problems, it is essential for proofs based on validity of such assumptions are true and to how much extent. There are many methods dealing with problems of probability and stochastic processes which are both linear (static and dynamic) and nonlinear (static and dynamic) in behavior.

The main objective of Dynamic Analysis is to ascertain the Dynamic Equilibrium of structures under the action of various types of dynamic loads with the available data such as material properties, geometric configuration. It is difficult to attain the exact solution for any dynamic structure due to the uncertainties involved in the analysis process such as uncertainty in modeling, uncertainty in loads both static and dynamic, uncertainty in material properties such as Young’s Modulus of Elasticity, Poisson’s Ratio etc.. Fuzzy logic is used very effectively in solving problems of uncertainty in structural engineering to a more reliable degree of accuracy. The development of various software tools like ANSYS, MATLAB, ABACUS, MS-NASTRAN etc., by introduction of advanced methods in problem solving made these tasks of solving uncertainties simpler and easier. The structures were analyzed for static equilibrium by introducing Fuzzy Finite Element concept in Analysis by various researchers and effectively applied for solving multiple uncertainties. Until now Fuzzy Finite Element approach is used for solving static analysis of structures subjected to multiple uncertainties, dynamic analysis with single uncertainty. The dissertation makes an attempt to solve problems in dynamic analysis of structures subjected to multiple uncertainties.

of 24 Uncertainty Modeling In Structural Engineering:

Application to Dynamic Analysis by A.K.VISHWA

Uncertainty Modeling In Structural Engineering:

Application to Dynamic Analysis Anil Kumar Vishwanathula, Roll No: 0109-11109, V Semester,

M.E. Structural Eng (CEEP), University College of Engineering, Osmania University

Synopsis

The advancement in Fuzzy Finite Element Analysis simplified the structural analysis process in recent years in

the area of structural engineering for solving uncertainty. Many methodologies were developed to solve uncertainties in

analysis and design processes with help of new mathematical algorithms and simulations using software. This caused a

need for study in new directions for improving the precision and accuracy in analysis models. The mathematical

modeling of structural engineering phenomena changed significantly in the recent years. Various methodologies and

approaches were discussed in solving uncertain parameters in analysis process. The need for fuzzy finite Element

approaches is reviewed. The structures were analyzed for static equilibrium by introducing Fuzzy Finite Element

concept in Analysis by various researchers and effectively applied for solving multiple uncertainties. Until now Fuzzy

Finite Element approach is used for solving static analysis of structures subjected to multiple uncertainties, dynamic

analysis with single uncertainty. The dissertation makes an attempt to find solutions to the problems in dynamic

analysis of structures subjected to multiple uncertainties.

INTRODUCTION

Mathematical Modeling

The term mathematical modeling of a structure can be defined as the process of developing mathematical equations representing any physical phenomena in a structure. This involves lot of assumptions, derivations and calculations to maintain the equilibrium of any structure or object to retain its original geometrical arrangement by limiting the displacements simulating the actual process.

The research reviewed by Julie Gainsburg4 offers a mix of possible pictures of mathematical modeling activity—from creating original models for solving problems to selecting or adapting received, general models to avoiding models altogether in favor of situation- specific routines and representations—but no consensus about which is most accurate, when, and for whom. Even the literature making the strongest claims about use of models nevertheless provides little description of the nature of those models and the activities and skills their use requires.

Various authors (e.g., Bissell & Dillon, 2000; Edwards & Hamson, 1989; Lesh & Doerr, 2003; Pollak, 1997) have theoretically outlined the mathematical modeling process with considerable similarity. The following (cyclical) set of steps synthesizes their versions:

1. Identify the real-world phenomenon

2. Simplify or idealize the phenomenon



3. Express the idealized phenomenon mathematically (i.e., “mathematize”)

4. Perform the mathematical manipulations (i.e., “solve” the model)

5. Interpret the mathematical solution in real-world terms

6. Test the interpretation against reality

It is essential in engineering to have certain methods and models to describe the exact physical phenomena while analyzing any structure to counter its imbalance due to the action of various loads. Structural engineering is a vast subject that deals with analysis and design of various structures and the dissertation essentially focuses on modeling multiple uncertainties in dynamic analysis of structures at basic level. As of present day the study is mainly focused on single and multiple uncertainty solution to static problems and single uncertainty for dynamic problems. Now this dissertation attempts to study solving multiple uncertainties in dynamic analysis.

INTRODUCTION TO STRUCTURAL DYNAMICS

Fundamental objective8

As the term itself defines ‘dynamic’ loading is time-varying in magnitude, direction and position. Two basic approaches are available for evaluating structural response to dynamic loads: deterministic, non-deterministic. The choice of method depends on the definition of loading. If the time-variation of loading is fully known the loading is termed as prescribed dynamic loading and the analysis of response is known as deterministic analysis. If the time-variation of dynamic loading is not known fully but can be defined in a statistical sense the loading is termed random dynamic loading and the analysis of such loading response is called nondeterministic analysis.

Generally in problems of dynamic analysis, the structural response to dynamic loading expressed basically in terms of displacements. Thus a deterministic analysis leads directly to the displacement time-histories corresponding to prescribed dynamic loading history. The second phase of the analysis defines the other related response quantities like stresses, strains, internal forces. A nondeterministic analysis provides only statistical information of the displacements resulting from the statistical defined loading and the corresponding information of structural response then generated with help of independent nondeterministic analysis procedures.

There are two types of prescribed dynamic loadings viz. periodic and non-periodic. The periodic loading exhibits same time variation successively for large number of cycles (ex: sinusoidal variation termed as simple harmonic). Any periodic loading can be represented by Fourier analysis as the sum of the series of simple harmonic components. Non-periodic loadings may be either short-duration impulsive loading (blast) or long duration general loading (earthquake).

Characteristics of Dynamic Problem8

There are two differences that are to be noted by a dynamic problem. The first is the time-varying nature of the dynamic problem. As both the loading and response vary with time, it is evident that a dynamic problem does not have a single solution as a static problem does and a succession of solutions



must be established corresponding to all times of interest in the response history. Thus a dynamic analysis is more complex and time consuming than a static analysis. The second and more fundamental distinction between static and dynamic problems is, the displacements of a structural system resulting from application of a dynamic loading depend not only upon the loading but also upon the inertial forces in the structural system which opposes the accelerations producing them. Thus corresponding internal moments and shear forces must equilibrate not only the externally applied load bust also the inertial forces resulting from the accelerations of the system.

PRINCIPLES OF DYNAMIC ACTION 5

Free Vibration

The equilibrium equations for the free vibration of an undamped multiple degree of freedom system are defined as a set of linear homogeneous second-order ordinary differential equations as:

�� 0 … … … … �1.1�

Assuming a harmonic motion for the temporal displacement�U �φ e��, Eq. (1.1) is transformed to a set of linear homogeneous algebraic equations as:

�� 0 … … … … �1.2� Or

�� … … … … �1.3� Eq.(1.2) is known as a generalized eigenvalue problem between the stiffness and mass matrices of the

system.

The values of (ω) are the natural circular frequencies and the vectors {φ} are the corresponding mode shapes.

Solution to Eigenvalue Problem

For non-trivial solutions, the determinant of ([K] ( ²) [M]) must be zero. This leads to a scalar equation, known as the characteristic equation, whose roots are the system’s natural circular frequencies of the system (ω).

Substituting each value of circular frequency in Eq.(1.2) yields a corresponding eigenvector or mode shape that is defined to an arbitrary multiplicative constant. The modal matrix [{φ₁} ... {φn}] spans the N-dimensional linear vector space.

This means that the eigenvectors [{φ₁} ... {φn}] form a complete basis, i.e., any vector such as the vector of dynamic response of a multiple degree of freedom (MDOF) system, {U(t)} , can be expressed as a linear combination of the mode shapes:

��%� ��& '&�%� � �� '��%� � ( � ��) ')�%� *�� '+�%�)

+,&… … … … �1.4�

in which, the terms y (t) n are modal coordinates and therefore, {U(t)} is defined in modal coordinate space, since the values of {φ } are independent of time for linear systems, Eq. (1.3).



Furthermore, the temporal derivatives of total response can be expressed as:

.�/ �%�0 ��& '/&�%� � �� '/��%� � ( � ��) '/)�%� *�� '/+�%�)+,& … … … … �1.5�ω

.�� %�0 ��& '�&�%� � �� '��%� � ( � ��) '�)�%� *�� '�+�%�)+,& … … … … �1.6�ω

which are also defined in modal coordinate space.

Orthogonality of Modes

Considering the generalized eigenvalue problem for the mth and nth circular frequencies and corresponding mode shapes: ��K� � �ω4��M��φ4 �0 … … … … �1.7� ��K� � �ω7��M��φ7 �0 … … … … �1.8�

Pre-multiplying Eq.(1.7) and Eq.(1.8) by �φ7 9 and �φ4 9 , respectively: �φ7 9�K��φ4 � �ω4��φ7 9�M��φ4 �0 … … … … �1.9� �φ4 9�K��φ7 � �ω7��φ4 9�M��φ7 �0 … … … … �1.10� Then, transposing Eq (1.10) and invoking the symmetric property of the [K] and [M] matrices yields: �φ7 9�K��φ4 � �ω7��φ7 9�M��φ4 �0 … … … … �1.11� Subtracting Eq.(1.11) from Eq.(1.i) yields: ;�ω4�� ω7��<�φ7 9�M��φ4 �0 … … … … �1.12�

For any (m m), if�ω4�� = �ω7��: �φ7 9�M��φ4 �0 … … … … �1.13� �φ7 9�K��φ4 �0 … … … … �1.14� Eqs.(1.13,1.14) express the characteristic of “orthogonality” of mode shapes with respect to mass and

stiffness matrices, respectively.

Forced Vibration

The equation of motion for forced vibration of an undamped MDOF system is defined as: �� >�%� … … … … �1.15� Expressing displacements and their time derivatives in modal coordinate space:

*��+ '�+�%�)+,& � *��+ '+�%�)

+,& �>�%� … … … … �1.16� Premultiplying each term in Eq.(1.16) by �φ7 9:

*��+ ?��+ '�+�%�)+,& � *��+ ?��+ '+�%�)

+,& ��+ ?�>�%� … … … … �1.17�



Invoking orthogonality, Eq.(1.16) is reduced to a set of N uncoupled modal equations as:

��+ ?��+ '�+�%� � ��+ ?��+ '+�%� ��+ ?�>�%� … … … … �1.18�

or:

��+��+ '�+�%� � �+��+ '+�%� �>+�%� … … … … �1.19�

where, �M7� �φ7 9�M��φ7 , �K7� �φ7 9�K��φ7 , �P7�t� �φ7 9�P�t� are generalized modal mass, generalized modal stiffness and generalized modal force respectively.

Dividing by modal mass Mn and adding the assumed modal damping ratio, �ξ7�, Eq.(1.19) becomes:

'�+�%� � �2D+�+�'+/ �%� � ��+��'+�%� >+�%��+ … … … … �1.20�

Proportional Excitation

If loading is proportional �P�t� �P p�t�) , meaning the applied forces have the same time variation defined by p(t) (such as ground motion), Eq.(1.20) can be expressed as: y� 7�t� � �2ξ7ω7�y7/ �t� � �ω7��y7�t� �φ7 9�P M7 ;p�t�< … … … … �1.21�

Defining a modal participation factor, Γ7 , as: Γ7 �φ7 9�P M7 �φ7 9�P �φ7 9�M��φ7 … … … … 1.22

Also defining a scaled generalized modal coordinate: D7�t� y7�t�Γ7 … … … 1.23

Eq.(1.20) is rewritten in terms of the scaled modal coordinate D7�t� as: D� 7�t� � �2ξ7ω7�D7/ �t� � �ω7��D7�t� P�t� … … … … �1.24�

Therefore, using modal decomposition, the equation of motion for an N-DOF system is uncoupled to N equations of motion of generalized single degree of freedom (SDOF) systems.

Response History Analysis (RHA)

In response history analysis (RHA), N uncoupled SDOF modal equations, Eq.(1.24), are solved for the modal coordinates ;D7�t�<, and then, by superposing the modal responses, the total displacement response of the system is obtained as:

��%� *;Dn�t�<Γ+�I+ )+,& … … … … �1.25�ω

in which the “time history” of the total response is obtained by the summation of modal responses as products of time history of modal coordinates (D (t)) n , modal participation factors �Γ+�, and modal displacements (mode shapes) �I+ . Moreover, the time history of any load effect, R(t) , may be expressed as:

J�%� *;KL�%�<M+�J+ )+,& … … … … �1.26�



in which, �J+ is a static modal load effect.

Response Spectrum Analysis (RSA)

In response spectrum analysis (RSA), for each uncoupled generalized SDOF modal equation, Eq.(2.32), the maximum modal coordinate ;K+,NOP<is obtained using the response spectrum of the external excitation p(t) and assumed modal damping ξ7 (Figure 1.1).

Response spectra are found by obtaining the maximum dynamic amplification (maximum ratio of dynamic to static responses) for a set of natural frequencies.

Figure 1.1: A generic response spectrum for an external excitation p(t)

Therefore, the modal response is obtained as: .�+,NOP0 ;Dn,max<Γ+�I+ … … … … �1.27�

Superposition of modal maxima

The total response is obtained using superposition of modal maxima. The superposition can be performed by summation of absolute values of modal responses.

�U4TU *VU7,4TUVW7,& … … … … �1.28�

which, provides a conservative estimate of the maximum response. As an approximation, the method of Square Root of Sum of Squares (SRSS) of modal maxima can be used when natural frequencies are distinct (Rosenblueth 1959):

�U4TU X*.U�7,4TU0W7,& … … … … �1.29�

Also, the method of complete quadratic combination (CQC) can be used.

Ground Excitation- Response Spectrum Analysis

The equation of motion for an undamped MDOF system subjected to ground excitation (support motion) from an earthquake is:

��.�Y� 0 � �� 0 … … … … �1.30�

where; .�Y� 0 is the vector of absolute acceleration. The vector {U} is defined as the relative displacement vector, defined as:

�� Y � �Z ;�[< … … … … �1.31�



where; �Z ;�[< is the vector of rigid body pseudo-static displaced shape due to horizontal ground

motion. Substituting Eq.(1.31) in Eq.(1.30) yields:

��.�� 0 � �� Z ;��[< … … … … �1.32�

As before, solving the linear eigenvalue problem, defining the response in modal coordinate space, uncoupling and adding assumed modal damping yields:

'�+�%� � �2D+�+�'+/ �%� � ��+��'+�%� ��+ ?�Z ��+ ?��+ ��[ … … … … �1.33�

Defining the modal participation factor, Γ7 , as:

Γ7 .�L0\�Z .�L0\��.�L0 … … … … �1.34� Also, defining the scaled generalized modal coordinate Dn (t) = yn (t) / n , Eq.(2.40) may be

rewritten in terms of the scaled modal coordinate (Dn (t)) as:

K� +�%� � �2D+�+�K+/ �%� � ��+��K+�%� ��[ … … … … �1.35�

Performing response spectrum analysis for ground excitation, for each uncoupled generalized SDOF modal equation, Eq.(1.35), the maximum modal response is obtained using earthquake response spectra such as the Newmark Blume Kapur (NBK) design spectra.

Therefore, the maximum modal coordinate is obtained as:

D7,4TU S^�ω7, ξ7� … … … … �1.36� The total response is obtained using superposition of modal maxima. The superposition is performed

by considering Square Root of Sum of Squares (SRSS) of modal maxima:

�U4TU X* D�7,4TUΓ7��φ7� W7,& … … … … �1.37�

Response Spectrum Analysis Summary

Response spectrum analysis to compute the dynamic response of a MDOF to external forces and ground excitation can be summarized as a sequence of steps as:

1. Define the structural properties.

• Determine the stiffness matrix [K] and mass matrix [M] . • Assume the modal damping ratio ξ7.



2. Perform a generalized eigenvalue problem between the stiffness and mass matrices.

• Determine natural circular frequencies � �+ �. • Determine mode shapes��+ .

3. Compute the maximum modal response.

• Determine the maximum modal coordinate ;D7,4TU< using the excitation response spectrum for the corresponding natural circular frequency and modal damping ratio.

• Determine the modal participation factor Γ7. • Compute the maximum modal response as a product of maximum modal coordinate, modal

participation factor and mode shape.

4. Combine the contributions of all maximum modal responses to determine the maximum total response using SRSS or other combination methods.

Limitations to RSA

In the presence of uncertainty in the structure’s physical or geometrical parameters, the deterministic structural dynamic analysis cannot be performed and hence, a new method must be developed to incorporate an uncertainty analysis into the conventional response spectrum analysis.

SOURCES OF UNCERTAINTY

The sources of uncertainty in modeling an analysis problem may be listed as follows:

1. Uncertainty in Geometry: Actual geometry may differ from that specified

2. Uncertainty in Material Properties: Actual material properties may be different from that specified. Uncertainty in stiffness, mass

3. Uncertainty in loading: Actual loads may differ from those assumed; Actual loads may be distributed in a manner different from assumed.

4. Uncertainty in Boundary Conditions.

5. Uncertainty in mathematical modeling of dynamic problem

6. The assumptions and simplifications inherent in any analysis may result in calculated load effects – moments, shears, etc., - different from those that, in fact act in the structure

7. The actual structural behavior may differ from that assumed, owing to imperfect knowledge

CAUSES OF UNCERTAINTY10

The uncertainties that a structural engineer encounters during a design come from a range of sources. The following five sources of uncertainty cover the vast majority of examples.



• Time: There is uncertainty in predicting the future (e.g., how much snow load will our structure experience?) and uncertainty in knowing the past (e.g., what was the concrete strength in the old building we need to renovate?);

• Statistical limits: We never can get enough data. (I took some cores from the old building and tested the concrete. Do these test values truly represent the concrete strength?);

• Model limits: The structural model used in the analysis and design leaves out or simplifies many aspects of the structure, and it is possible that the model is not conceptually correct;

• Randomness: The structural properties (e.g., modulus of elasticity, concrete strength_ are not a single number but vary over some range. The properties are random variables);

• Human error: It is possible that an error was made during the design or the construction.

None of these five causes of uncertainty separate uncertainties cleanly into aleatory or epistemic. Generally there are aspects of both in each of the five causes. But, the five causes allow us to categorize the uncertainties that we will encounter subsequently, and whether an uncertainty is aleatory or epistemic will allow us to focus on which uncertainties can be reduced through probabilistic techniques and which must be dealt with in other ways.

CATEGORIES OF UNCERTAINTY 5

The concept of uncertainty can be divided into two major categories:

• Aleatory: The system has an intrinsic random or stochastic nature and it is not predictable.

• Epistemic: The uncertainty induced by the lack of knowledge and it is predictable.

Example of aleatory uncertainty is the behavior of photons in quantum mechanics where there is no hidden variable in the model or missing information.

Epistemic systems have uncertainty that may be reduced upon additional information. Uncertainty in the stiffness of a structural member may be reduced by measurement of the element behavior.

Aleatory uncertainty assumes that an underlying probability density function (PDF) exists and is the square of the wave function in quantum mechanics and also, the PDF is a fundamental property of the system.

In most engineering systems, the PDF is obtained from historic data and represents both epistemic and aleatory uncertainties. Thus, the precise form of a PDF can only be assumed. On the other hand, interval methods play an important role in quantifying epistemic uncertainty.



UNCERTAINTY ANALYSIS

There are basically two methods to solve the problems of Uncertainty analysis, viz.:

1. Deterministic Analysis

2. Non-deterministic Analysis

These methods are discussed as follows.

1. Deterministic analysis5

In deterministic analysis of physical systems, defining the system’s characteristics as point quantities, using conventional deterministic algebraic values, is sufficient to model the system and perform the analysis (Figure 1.2).

a x = a

Figure 1.2 A deterministic algebraic variable

In order to perform uncertainty analysis on a physical system, the uncertainty present in the system’s physical characteristics must be fully mathematically quantified.

2. Non-Deterministic analyses3,5

Presently, these are methods to consider uncertainties in non-deterministic structural analysis are:

i. Stochastic analysis

ii. Fuzzy analysis

iii. Interval analysis

iv. Monte Carlo simulation Method

v. Perturbation Method

These methods were briefly discussed in preceding subheadings.

i. Stochastic Analysis5

In stochastic analysis, the theory of probability which was developed based on aleatory uncertainty. Extensions have been made such as “degree of belief” probability on subjective probability which includes epistemic effects.

The stochastic approach to uncertain problems is to model the structural parameters as random quantities (Pascal 1654). Therefore, all information about the structural parameters is provided by the probability density functions. This probability density function is then used to determine an estimate of the system’s behavior.



Random Variable

A random quantity, used in stochastic analysis, is defined by a deterministic function that yields the probability of existence of the random variable in a given subset of the real space (See Figure 1.3), (See Equation a):

Figure 1.3: Probability Density Function of a random quantity

_P�`� >��a b `�� c d�a�eaO

�∞… … … … `

in which, F (a) x is cumulative probability distribution function evaluated for random variable (a) and f (x) is the corresponding probability density function.

ii. Fuzzy Analysis5

In fuzzy analysis, the theory of possibility for fuzzy sets is used which assumes epistemic uncertainty.

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.

Fuzzy Subset

Considering E as a referential set inℜ, an ordinary subset A of the referential set is defined by its characteristic function �x� A µ as: ga h i j �a� h �0,1 k µ… … … … … �1. a�

which, exhibits whether or not, an element of E belongs to the ordinary subset A. For the same referential set E, a fuzzy subset A is defined by its characteristic function, membership function�x� A µ , as:



ga h i j �a� h �0,1�k l … … … … … �1. m�

A fuzzy number is defined by its membership function whose domain is ℜ while its range is bounded between [0, 1]. The domain of the membership function is known as the interval of confidence and the range is known as the level of presumption.

Therefore, each level of presumption α (α-cut membership, αh [0, 1]) has a unique interval of confidence [Aa = aa, ba ], which is a monotonic decreasing function of α (see Figure 1.4), (Eqs. (1.c, 1.d)): gn&, n�h �0,1�, �n&on�� p knq r kns … … … … … �1. t� Or gn&, n�h �0,1�, �n&on�� p �`nq , mnq� r �`ns , mns� … … … … … �1. e�

Figure 1.4: Membership function of a fuzzy quantity

iii. Interval Analysis5

In interval analysis, the theory of convex (interval) sets is used which assumes epistemic or aleatory uncertainties (such as Dempster-Shafer bounds that are epistemic bounds on aleatory probability functions).

The interval approach to the uncertain problems is to model the structural parameters as interval quantities. In this method, uncertainty in the elements is viewed by a closed set-representation of element parameters that can vary within intervals between extreme values. Then, structural analysis is performed using interval operations.

Interval (Convex) Number

A real interval is a closed set defined by extreme values as (Figure 1.5): wx �yz , y{� �y h ||yz b y b y{ … … … … … �2. `�



a b

a~ �`, m� Figure 1.5: An interval quantity

One interpretation of an interval number is a random variable whose probability density function is unknown but non-zero only in the range of interval.

Another interpretation of an interval number includes intervals of confidence for α -cuts of fuzzy sets. This interval representation transforms the point values in the deterministic system to inclusive set values in the system with bounded uncertainty.

Interval Arithmetic Operations

Interval arithmetic is a computational tool that can be used to represent uncertainty as:

1. A set of probability density functions.

2. In Dempster-Shafer models for epistemic probability.

3. α - cuts in fuzzy sets.

In this work, the symbol (~) represents an interval quantity.

Considering X� �a, b� and Y� �c, d� as two interval numbers, the basic interval arithmetic operations are:

Addition: �x � �x �` � t, m � e� … … … … … �2. m� Subtraction: �x � �x �` � t, m � e� … … … … … �2. t� Multiplication by scalar: � � �x ��L��`, �m�, �à��`, �m�� … … … … … �2. e� Multiplication: �x � �x ��L�`t, è, mt, me�, �à�`t, è, mt, me�� … … … … … �2. �� Properties of Interval Multiplication:

Associative: �x � �x �x � �x … … … … … �2. d� Commutative:



�x � ;�x � wx< ��x � �x� � wx … … … … … �2. �� Distributive:

�x � ;�x � wx< r �x � �x � �x � wx … … … … … �2. �� Therefore, the distributive property of interval multiplication is weaker than that in conventional

algebra and it is one possible cause of loss of sharpness in interval operations.

Division:

�x�x �`, m� � �1

e , 1t� , 0 � �t, e� … … … … … �2. ��

Interval Vector (2-D):

�x ��x�x� ��`, m�

�t, e�� … … … … … �2. �� which represents a “box” in 2-D space as the enclosure (Figure 1.6).

Figure 1.6: An interval vector

Transformation of Interval to Perturbation

Perturbation methods often use small change in a parameter, ε. To express interval problems in terms of perturbation, an interval perturbation, ε = [−1, 1], is introduced so that a general interval is written as summation of center and radial values.

Considering Zx �l, u� as an interval number, the median and radius can be defined as:

w� �� 2 � … … … … … �2. ��

wx �¡� �� 2 � … … … … … �2. ��



So, wxcan be redefined as:

wx w� � wx … … … … … �2. �� where, the interval number is shown as its median subjected to a perturbation of radius by which, the

result encompasses the range of the interval between the extreme values.

Functional Dependency of Interval Operations

Considering X� ��2,2� and Y� ��2,2� as two independent interval numbers, the functional dependent interval multiplication results in:

�x � �x �0,4� In contrast, the functional independent interval multiplication results in:

�x � �x ��4,4� Sharpness Considerations in Engineering

In interval operations, the functional dependency of intervals must be considered in order to attain sharper results. In fact, the issue of sharpness and overestimation in interval bounds is the key limitation in the application of interval methods. Naïve implementation of interval arithmetic algorithms (substituting interval operations for their scalar equivalence) will yield bounds that are not useful for engineering design.

Therefore, there is a need to develop algorithms to calculate sharp or nearly sharp bounds to the underlying set theoretic interval problems.

For instance, the calculation of exact sharp bounds to the interval system of equations resulting from linear static analysis using the finite element method has been proved to be computationally combinatorial problem. However, even the 2n combinations of upper and lower bounds do not always yield the bounds.

In problems with narrow intervals associated with truncation errors, the naïve implementation of interval arithmetic will yield acceptable bounds. However, for wider intervals representing uncertainty in parameters, the naïve method will overestimate the bounds by several orders of magnitude.

Successful applications of the interval method in the linear static problem have required the development of new algorithms that are computationally feasible yet still provide nearly sharp bounds (Muhanna and Mullen 2003).

iv. The Monte Carlo Simulation Method,9,3

Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in simulating physical and mathematical systems. Because of their reliance on repeated computation of random or pseudo-random numbers, these methods are most suited to calculation by a computer and tend to be used when it is infeasible or impossible to compute an exact result with a deterministic algorithm. This method is also used to complement the theoretical derivations.



Monte Carlo simulation methods are especially useful in studying systems with a large number of coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model). More broadly, Monte Carlo methods are useful for modeling phenomena with significant uncertainty in inputs, such as the calculation of risk in business. These methods are also widely used in mathematics: a classic use is for the evaluation of definite integrals, particularly multidimensional integrals with complicated boundary conditions. It is a widely successful method in risk analysis when compared with alternative methods or human intuition. When Monte Carlo simulations have been applied in space exploration and oil exploration, actual observations of failures, cost overruns and schedule overruns are routinely better predicted by the simulations than by human intuition or alternative "soft" methods.

The term "Monte Carlo method" was coined in the 1940s by physicists working on nuclear weapon projects in the Los Alamos National Laboratory, after the famous Monte Carlo casino, a gambling venue based on random-number generation.

One of its major advantages is that accurate solutions can be obtained for problems whose deterministic solution is known. Since it is completely general this method is frequently used to calibrate and validate other methods. The main disadvantage is that it is time consuming, but among all numerical methods that rely on n-point evaluations in a d-dimensional space to produce an approximate solution, the Monte Carlo method has an absolute error estimation that decreases as n−1/2, while in the absence of an exploitable special structure all others methods have an error estimation that decreases as n−1/d at best.

Since each realization is independent of the others Monte Carlo simulations can be easily parallelized.

v. Perturbation Method1,3

The simple pendulum is a excellent paradigm for studying the nonlinear behavior of non-equilibrium systems. Moreover, a sequence of coupled pendula will provide a natural setting for introducing the Sine Gordon nonlinear Partial Differential Equation (PDE), an integrable nonlinear PDE, from which another integrable pde, the Nonlinear Schr¨odinger (NLS) equation, can be derived as a small amplitude approximation.

These provide a nice illustration of how the techniques of weakly nonlinear analysis are developed systematically.

Exact solutions are available to the simple pendulum problem for comparison. We will see that the nonlinear dependence of the frequency of oscillation of the pendulum on its amplitude of oscillation is a crucial signature of nonlinear behavior. In fact, this naturally suggests two weakly nonlinear paradigms: the anharmonic oscillator, a weakly nonlinear oscillator and the Van der Pol oscillator, an essentially linear oscillator with a nonlinear damping. These two models will illustrate the use of singular perturbation methods to derive uniformly valid perturbation corrections to the basic oscillator amplitude and frequency. The principal idea is to expand the oscillator amplitude in an asymptotic series, allowing for sufficient flexibility to avoid unbounded (secular) growth of the correction term to the amplitude at each order in the perturbation expansion. We will first see how a regular perturbation expansion lacks this flexibility.



Singular perturbation expansions are extremely powerful analytic tools for studying a whole class of nonlinear problems. They will form the basis for deriving many of the hierarchy of soliton equations (the Nonlinear Schr¨odinger equation (NLS), in particular) and universal order parameter equations (of Complex Ginzburg-Landau type) valid near a bifurcation point in spatially extended systems. As these perturbation theories involve expansions in a small parameter, one might be left with the impression that they are of limited utility. Remarkably, in many instances the results prove accurate even for values of the parameter approaching

The perturbation method, which is equivalent to a lower-order Taylor expansion, has been widely used for its tractability and computational time-saving. It expresses the structural matrices and response in terms of a lower-order polynomial function with respect to the parameters centered at the mean values, i.e., makes an approximation of the response surface. Introducing the following notation

Φ£ Φ;µUq , µUs , … , µU¥< … … … … … �3. a�

Φ�¦ ∂Φ∂x ;µUq , µUs , … , µU¥< … … … … … �3. b�

Φ� ¨¦ ¦ ∂�Φ∂x� ∂x¨ ;µUq , µUs , … , µU¥< … … … … … �3. c� For any variable Φ which depends of the parameters x1, x2, …, xn then the stiffness and matrices are

expanded as

£ � * ©ª«©+

©,& � 12 *+©,& * © ¬ª ª«©

+¬,& «¬ � ( … … … … �3. e�

� �£ � * �©ª«©+

©,& � 12 *+©,& * �© ¬ª ª «©

+¬,& «¬ � ( … … … … �3. ��

Where, «© a© � lP. In the same manner, for the eigenvalue problem

� � ®��I 0 … … … … �3. d�

The resulting eigenvalues λ and eigenvectors φ are expanded as

® ®£ � * ®©ª«©+

©,& � 12 *+©,& * ®© ¬ª ª «©

+¬,& «¬ � ( … … … … �3. ��

I I£ � * I©ª«©+

©,& � 12 *+©,& * I© ¬ª ª«©

+¬,& «¬ � ( … … … … �3. ��

The mean and variances of the eigenvalues are



i�®� ®£ � 12 *

+

©,&* ®© ¬ª ª+¬,& ¯°±�a© , a¬� � ( … … … … �3. ��

�`Z�®� *+©,& * ®© ª +

¬,& ®¬ ª ¯°±�a© , a¬� � ( … … … … �3. ��

It can be shown that λI is given by

®©ª ²£³;©ª � ®£�©ª<I£²£³MI£ … … … … �3. k�

Where λ0 and φ0 are obtained by the mean eiganvalue problem

�£ � ®£�£�I£ 0 … … … … �3. �� The natural frequencies can be expanded as

� √®

l¶&� � 12 l¶�&��® � l¶� � 128 l¶�·��® � l¶�� ( … … … … �3. ��

Therefore considering up to the first order term, their mean and standard deviation can be obtained from equations 3.i and 3.j as

l¸ ¹l¶ … … … … �3. L�

º¸ 12¹l¶ º¶ … … … … �3. °� It is required that the random variables involved in the analysis do not deviate much from their

expected values. If the coefficient of variation is not very small the solution cannot ever be improved by using a finer mesh. Of greater importance than the magnitude of variability of the original random variables is how appropriate the response surface is. The response quantity should be chosen in such a way that these quantities are not highly nonlinear in respect to the random variables, e.g., a frequency response function instead of a response in time domain.

If the number of random variables is large, such as in problems involving random fields, or if a high-order expansion is used, then the calculation effort becomes prohibitive.

NEW AND MODIFIED METHODS

The preceding subheadings discuss the recently introduced modified methods for Structural Dynamic analysis problems for use in the dissertation.



FUZZY FINITE ELEMENT METHOD 7

This approach first successfully implemented by Rafi L. Muhanna a professor at University of Maryland at College Park, USA for material and geometric uncertainty in trusses and plane stress problems, and for material uncertainty in beams.

In the work presented by Rafi L. Muhanna, Fuzzy finite element calculations, in level of presumption form, are implemented by first developing element stiffness matrices in parametric form (with length, modulus, and applied traction as parameters) using a standard non-fuzzy formulation. Uncertainty is introduced by assigning an interval value for parameters. The element stiffness and load matrices are assembled and the final system of interval equations is solved using an implementation of Hanson's Algorithm (Hanson 1965). This system of interval equations can be written as:

��z , �{��z, �{� �»z , »{� … … … … … �4. `�

This procedure has been used to solve several matrix structural and continuum problems with uncertainty for material properties, geometry, and loading (Muhanna and Mullen 1995; Mullen and Muhanna 1996; Muhanna and Mullen 1998).

In conventional finite element formulations, the nodal load is given by

» »¼ � »½ … … … … … �4. m� Where pc is the vector of concentrated load and pb is the nodal load contribution from an element and

has the form:

»¼ ¾¿? c À?z m�a�ea … … … … … �4. t�

Which is the assamblage of

»© c À?z m�a�ea … … … … … �4. e�

Where pi is the generalized nodal load for node i, b(x) is the applied traction and Ni is the shape function for node i.

To achieve sharp results, two different approaches are applied. The first is used in the load uncertainty case and the second is for material and geometric uncertainty.

In the uncertain load case, the function b(x) is allowed to be fuzzy. The evaluation of fuzzy integral in equation (4.a) will be done in a Riemann sense. In order to correctly obtain inclusive interval values for pi, attention must be paid to the sign of the terms Ni and b(x). If the shape function is negative, the upper and lower limits of the interval value of b(x) must be interchanged. For linear displacement truss elements and cubic Euler-Bernoulli beam elements, the shape functions maintain the same sign over the length of an element. However, in the case of a distributed moment on a beam, the appropriate shape functions change sign over the length of the element. The order of multiplication of fuzzy number has a strong influence on the sharpness of the results. In order to maintain sharp results for the displacements, stress,



and reaction calculations, the use of a fuzzy value should be delayed as much as possible. For the case of load uncertainty the sharpness of the interval solution (exact in the case of independent element loads) can be achieved by multiplying all non-interval values first, and the last multiplication involves the interval quantities. The load vector will be presented in the following form: > �_ … … … … … �4. ��

Where F(mx1) is the interval element load vector and m is the number of elements. Matrix M is such matrixes that ensure the proper conversion of interval element load into interval nodal one.

The system of interval equations for the interval nodal displacements can be written as: � ��&��_ … … … … … �4. d�

However, in the case of material and geometric uncertainty, the problem is more complicated. Obtaining a guaranteed enclosure of the solution is achievable, but the central issue is how much the solution is sharp? Sharpness of the solution is a relative matter; we try to compare with the exact solution that can be obtained by solving for all possible combinations. The non-sharpness can be attributed to dependency, order of operations and the failure of distributive law in interval arithmetic.

It is observed that when uncertainty is included in the stiffness matrix of a system (left-hand side of the equation system), the element-based local properties cannot be maintained and inhered to the final system after the assemblage. For example, if the modulus of elasticity, E is given an uncertain value in interval form, the solution of the interval system k u = p will be different from the solution of the same system after factoring out the interval constant E in any of the following forms: i��Á�� » … … … … … �4. �� Â� �Ã/i�» … … … … … �4. ��

To overcome this difficulty in treatment of geometric and material uncertainty, a new approach is developed. An equivalent system, which includes the geometric and material uncertainty in a form of an equivalent load uncertainty, is developed. This approach is based on element-by-element technique that maintains the element-based local properties inherent to the final solution.

First the deterministic system is solved for forces in the form: _Å,½ �Å,½¿�½�&_½ … … … … … �4. ��

On the element level the equilibrium equation will take the form: �Å�Å _Å,½ … … … … … �4. ��

If material uncertainty is introduced in the form E =δ Ec. Equation 9 could be written as: _Å Æ1 ÇÈ É�Å,½¿�½�&_½ … … … … … �4. ��

After assemblage the following equivalent system, with interval load vector, is obtained: �� _ … … … … … �4. ��

In the above equations the subscripts e and c denote the element and center (deterministic) values respectively.



INTERVAL RESPONSE SPECTRUM ANALYSIS (IRSA) 5

This method was introduced by Mehdi Modarreszadeh in 2005 for solving uncertainty in Dynamic Analysis with interval uncertainty.

First, IRSA defines the uncertainty in the system’s parameters as closed intervals; therefore, the imprecise property can vary within the intervals between extreme values (bounds). Then, having the uncertain parameters represented by interval variables for each element, the interval global stiffness and mass matrices of MDOF system are assembled. This assemblage is performed such that the element physical characteristics and the matrix mathematical properties are preserved.

Then an interval generalized eigenvalue problem between the interval stiffness and mass matrices is established. From this interval eigenvalue problem, two solutions of interest are obtained:

1. Bounds on variation of circular natural frequencies (Interval natural frequencies) 2. Bounds on directional deviation of mode shapes (Interval mode shapes)

Then, the interval modal coordinate and the maximum modal coordinate are determined using the excitation response spectrum evaluated for the corresponding interval of natural circular frequency and assumed modal damping ratio. Then, the interval modal participation factor is computed. Dependency or independency of variations in interval modal participation factor is considered. Following this, the maximum modal response is computed as a maximum of the product of the maximum modal coordinate, the interval modal participation factor and the interval mode shape.

Finally, the contributions of all maximum modal responses are combined to determine the maximum total response using SRSS or other combination methods.

Interval Representation of Uncertainty

The presence of uncertainty in a structure’s physical or geometrical property can be depicted by a closed interval. Considering q~ as a structure’s uncertain parameter:

Ê~ ��, �� … … … … … �6. `�

in which, l and u are the lower and upper bounds of the uncertain parameter, respectively.

Interval Stiffness Matrix

The structure’s deterministic global stiffness matrix can be viewed as a linear summation of the element contributions to the global stiffness matrix,

�� *�¿©��©�+

©,&�¿©�? … … … … … �6. m�

where, [L i ] is the element Boolean connectivity matrix and [K i] is the element stiffness matrix in the global coordinate system (a geometric second-order tensor transformation may be required from the element local coordinates to the structure’s global coordinates).



Considering the presence of uncertainty in the stiffness characteristics, the nondeterministic element stiffness matrix is expressed as:

�Ë� � ��© , �©��©� … … … … … �6. t� in which [ li, ui] is an interval number that pre-multiplies the deterministic element stiffness matrix.

Considering the variation as a multiplier outside of the stiffness matrix preserves the element physical characteristics such as real natural frequencies and rigid body modes as well as stiffness matrix properties such as symmetry and positive semi-definiteness. In terms of the physics of the system, this means that the stiffness within each element is unknown but bounded and has a unique value that can independently vary from the stiffness of other elements.

This parametric form must be used to preserve sharp interval bounds. The uncertainty in each element’s stiffness is assumed to be independent. For a substructure with an overall interval uncertainty, Eqs.(6.b,6.c) are used to assemble the substructure’s stiffness matrix.

For coupled elements, matrix decompositions can be used. For instance, in a beam-column, if functional independent values of axial and bending properties are uncertain, the axial and bending components can be additively decomposed as:

�Ë� � ��©, �©�OP©Oz��©�OP©Oz � ;��©, �©�¼Å+Ì©+[<�©�¼Å+Ì©+[ … … … … … �6. e�

Likewise, for continuum problems with functional independent uncertain properties at integration points, the contribution of each integration point can be assembled independently.

Interval Global Stiffness Matrix

The structure’s global stiffness matrix in the presence of any uncertainty is the linear summation of the contributions of non-deterministic interval element stiffness matrices:

�� *�¿©��©, �©��©� +

©,&�¿©�? … … … … … �6. ��ω

or: � ∑ ��©, �©��¿©��©�+©,& �¿©�? ∑ ��©, �©��Ð©�+©,& … … … … … �6. d� in which �KÐ�� is the deterministic element stiffness contribution to the global stiffness matrix.

Interval Mass Matrix

Similarly, the structure’s deterministic global mass matrix is viewed as a linear summation of the element contributions to the global mass matrix as:

�� *�¿©��©�+

©,&�¿©�? … … … … … �6. ��ω



where, �M�� is the element stiffness matrix in the global coordinate system.

Considering the presence of uncertainty in the mass properties, the nondeterministic element mass matrix is:

��ËÑ� ��©, �©��©� … … … … … �6. ��

in which �l�, u�� is an interval number that pre-multiplies the deterministic element mass matrix. Considering the variation as a multiplier outside of the mass matrix preserves the element physical properties. Analogous to the interval stiffness matrix, this procedure preserves the physical and mathematical characteristics of the mass matrix.

The structure’s global mass matrix in the presence of any uncertainty is the linear summation of the contributions of non-deterministic interval element mass matrices:

�� *�¿©��©, �©��©� +©,& �¿©�? … … … … … �6. ��ω

or: �� ∑ ��©, �©��¿©��©�+©,& �¿©�? ∑ ��©, �©��Ð©�+©,& … … … … … �6. �� in which ��Ð©� is the deterministic element mass contribution to the global mass matrix.

The above equations will be solved for interval Eigen values with bounds on frequencies and mode shapes and a solution is obtained finally.

Some conservative overestimation in dynamic response occurs because of linearization in formation of bounds of mode shapes and also, the dependency of intervals in the dynamic response formulation. These are the expected cause of loss of sharpness in the interval results. Due to this the method has to be improved for nonlinearity. The computational efficiency of the method makes IRSA an attractive method to introduce uncertainty into dynamic analysis.

CONCLUDING REMARKS

Solving multiple uncertainties in dynamic analysis problems makes the design of structures much easier and optimal solutions can be achieved due to increased accuracy in structural analysis process.

The following conclusions can be drawn from the seminar:

1. These methods can be effectively used in solving Seismic Response Analysis & Optimal Design of Structures under uncertain conditions of all parameters.

2. These methods can be effectively useful in analyzing structures subjected to aerodynamic forces like tall buildings where uncertainty is predominant.

3. These methods can be effectively useful in analyzing and design of marine structures.



REFERENCES

1. A. Aceves, N. Ercolani, C. Jones, J. Lega & J. Moloney, Lecture notes on “Introduction to

singular perturbation methods - Nonlinear oscillations”, for a summer school held in Cork,

Ireland, from 1994 to 1997.

2. Franklin Y. Cheng, Hongping Jiang and Kangyu Lou, Smart Structures: Innovative Systems for

Seismic Response Control, CRC Press, Boca Raton, London, Newyork.

3. José Fonseca, Cris Mares, Mike Friswell, John Mottershead, "Review of Parameter

Uncertainty Propagation Methods in Structural Dynamic Analysis", Proceedings of the

International Conference on Noise and Vibration Engineering (ISMA2002), Sep 2002, 1853-

1860.

4. Julie Gainsburg, Mathematical Thinking And Learning, 8(1), 3–36, Department of secondary

Education, California State University, Northridge

5. Mehdi Modarreszadeh , A doctoral thesis on “Dynamic Analysis of Structures with Interval

Uncertainty” , Department of Civil Engineering, Case Western Reserve University, August 2005

6. Nilson Arthur H., David Darwin, Charles W. Dolan, Design of Concrete Structures, Tata

McGraw Hill, 2004

7. Rafi L. Muhanna, Interval based Methods for Load and Stiffness Uncertainty in Computer Aided

Analysis, Department of Civil Engineering, University of Maryland at College Park, MD 20742,

USA, http://ecivwww.cwru.edu/civil/research/fuzzy.htm

8. Ray W. Clough, Joseph Penzien, Structural Dynamics, Computers & Structures, Inc., 1995

9. www.wikipedia.org, internet

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Structural Design And Construction, © ASCE / FEBRUARY 2008

11. Zimmermann H.-J., “Dynamic Fuzzy Data Analysis And Uncertainty Modeling In Engineering”,

European Laboratory for Intelligent Techniques Engineering (ELITE), Promenade 9, D-52076

Aachen

Uncertainty Modeling in Structural Engineering: Application to Dynamic Analysis

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