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Structural damage assessment using identification techniques

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Structural damage assessment using identification techniques
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Soeiro, Francisco José da Cunha Pires, 1949-
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English
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xiv, 179 leaves : ill., photos ; 29 cm.

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Subjects / Keywords:
Analytical models ( jstor )
Damage assessment ( jstor )
Error rates ( jstor )
Laminates ( jstor )
Mathematical variables ( jstor )
Modeling ( jstor )
Property damage ( jstor )
Stiffness ( jstor )
Stiffness matrix ( jstor )
Trusses ( jstor )
Aerospace Engineering, Mechanics, and Engineering Science thesis Ph. D ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics, and Engineering Science -- UF ( lcsh )
Structural analysis (Engineering) ( lcsh )
Structural failures ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 175-177).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Francisco José da Cunha Pires.

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STRUCTURAL DAMAGE ASSESSMENT
USING IDENTIFICATION TECHNIQUES











BY

FRANCISCO JOSE DA CUNHA PIRES SOEIRO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1990

























To my parents


Adelino Pires Soeiro and


Benilde da Cunha Soeiro













ACKNOWLEDGMENTS


I would like to express my sincere gratitude to

Dr. P. Hajela for his continuous direction, support and encouragement since I started my doctoral studies. I am particularly grateful to him for the introduction to the subjects of structural optimization techniques, structural system identification and structural damage assessment, which is largely responsible for my current interest in the field. His diligence and competence have always impressed me and are examples to be followed.

I would like to acknowledge Dr. F. Fagundo and Dr. D.

Zimmermann, members of my supervisory committee, for helping me during the course of my research. We had valuable discussions that contributed significantly to the improvement and progress of my work. I would like to express my sincere thanks to Dr. C. T. Sun and Dr. C. C. Hsu, also members of my supervisory committee, for their support and encouragement.

Special thanks are owed to Dr. D. A. Jenkins and the staff of the laboratory of structures of the Department of Aerospace Engineering, Mechanics and Engineering Science, for helping me in the experimental work. Their experience


iii








and competence were fundamental for the completion of that important phase of my research.

I would like to express my thanks to the Brazilian Army for giving me the opportunity and providing me the support to come to the United States and pursue doctoral studies.

Finally, I would like to extend my sincerest gratitude to my wife, Maria Luiza, and my sons Renato and Junior. They have been a continuing source of inspiration, support and encouragement.














TABLE OF CONTENTS

page

ACKNOWLEDGEMENTS......................***.***** iii

LIST OF TABLES................... ....... ................. vii

LIST OF FIGURES....................... ............. ..... x

ABSTRACT... ............................... ............. xiii

CHAPTERS

1 INTRODUCTION .................................... 1

1.1 Motivation and Objectives................... 1
1.2 Literature Survey.......................... 4
1.3 Scope of Present Work....... ................ 10

2 SYSTEM IDENTIFICATION TECHNIQUES................ 13

2.1 The Problem of System Identification........ 13
2.2 Typical Formulations for Structural System
Identification.............................. . 21
2.3 Structural Damage Assessment............... 24

3 DAMAGE ASSESSMENT IN STRUCTURAL SYSTEMS.......... 28

3.1 Introduction ................................. 28
3.2 Output Error Approach in Damage Detection... 29
3.2.1 Use of Eigenmodes as Measured
Structural Response.................. 29
3.2.2 Use of Static Displacements as Measured
Structural Response.................. 32
3.2.3 Use of a Combination of Eigenmodes and Static Displacements as Structural Response . ......................... 44
3.2.4 A Rational Method for Structural
Damage Detection..................... 51
3.3 Equation Error Approach in Damage Detection. 60
3.4 Design Space Corresponding to the Studied
formulations................................ 65










4 COMPUTATIONAL EFFICIENCY ASPECTS IN STRUCTURAL
DAMAGE ASSESSMENT.............................. 76

4.1 Introduction................................ 76
4.2 Reduced Dimensionality Models ............... 77
4.3 Substructuring............................... .. 83
4.4 Approximation Concepts ...................... 87

5 DAMAGE DETECTION IN COMPOSITE STRUCTURES......... 94

5.1 Introduction............................... 94
5.2 The Stiffness-Reduction Method.............. 96
5.2.1 Use of Static Displacements as Measured
Structural Response.................. 96
5.2.2 Use of Eigenmodes as Measured
Structural Response.................. 115

6 EXPERIMENTAL INVESTIGATION OF DAMAGED
STRUCTURES ..................................... 145

6.1 Introduction............................... 145
6.2 Experimental Setup ......................... 147
6.3 Analytical Model Identification............. 156
6.4 Discussion of Results....................... 159

7 CONCLUSIONS AND RECOMMENDATIONS.................. 169

REFERENCES ............................................. 175

BIOGRAPHICAL SKETCH................................... 178














LIST OF TABLES


Table Page

3.1 Results for the seven bar planar truss ........... 34

3.2 Results obtained for the seven bar planar truss,
using an increased number of eigenmodes.......... 35

3.3 Results for the twenty-five bar planar truss,
using four eigenmodes ............................ 37

3.4 Results for the twenty-five bar truss using
static displacements............................. 40

3.5 Results for the fifteen bar planar truss using
static response ................................. 42

3.6 Results for the semimonocoque wing box
structure........................................ 46

3.7 Results for the idealized helicopter
fuselage model ............................................ 48

3.8 Results for the fifteen bar planar truss using
a combination of static and modal response....... 50

3.9 Equal stress load distribution for the fifteen
bar truss (stress level = 13.78 kPa) ............. 56

3.10 Member stresses in the fifteen bar truss for the
loading shown in Table 3.9 ....................... 57

3.11 Results for the fifteen bar truss obtained
using the output error approach .................. 58

3.12 Approximation of the displacement field using
an increasing number of eigenmodes............... 62

3.13 Results for the fifteen bar truss obtained using
output error approach. Increasing number of
weighted eigenmodes used to model the
displacement field obtained under the loading
shown in Table 3.9................................. 63


vii








3.14 Results for the fifteen bar planar truss using
the equation error approach ...................... 67

3.15 Results for the three design variable membranerod structure for two sets of initial designs.... 75

4.1 Damage detection as seen in the equivalent
beam model............................. ...... 84

4.2 Results for the forty-eight bar space truss.
Only members in 2md and 4y bays are treated
as design variables....................... ......... 85

4.3 Results for the twenty-five bar planar truss
using dominant variables method .................. 92

4.4 Results for the fifteen bar planar truss using
four eigenmodes as measured response and the
dominant variables method........................ 93

5.1 Dominance assessment for equation (5.17)......... 105

5.2 Dominance assessment of A11 in (5.24)............ 109

5.3 Mechanical properties of the material used
in the examples.................................. 122

5.4 Results for the [0,903], laminate subjected to
extensional forces............................... 123

5.5 Results for the [0,90], laminate subjected to
extensional forces................................. 125

5.6 Results for the [0,90], laminate subjected to
extensional load and with very low A
in element 1...................................... 126

5.7 Results for the [0,90], laminate subjected to
a pure shear force .............................. 127

5.8 Results for the antisymmetric [0,90] laminate
subjected to a bending load ..................... 129

5.9 Results for the [0,�45,90],, 24 element laminate
subjected to bending forces....................... 131

5.10 Results for the [0,�45,90],, 24 element laminate
subjected to an extensional force applied
at the free end.................................. 134
5.11 Results for the [0,�45,90],, 24 element laminate
subjected to bending moments ..................... 135

viii








5.12 Results for the 4 layer laminate using a 3-D
element for damage detection ..................... 137

5.13 Results for the [0,903], laminate using three
extensional eigenmodes............................ 139

5.14 Results for the [0,9031, laminate using three
bending eigenmodes............................... 140

5.15 Results for the [0,�45,90],, 24 element, simply
supported laminate, using three eigenmodes....... 142

5.16 Results for the [0,�45,90],, 24 element,
cantilevered laminate using eigenmodes........... 144

6.1 Measured response compared with results
predicted by the original model (Fig. 6.3)....... 162

6.2 Identified material properties for the truss
elements of the model described by Fig. 6.7...... 165
6.3 Predicted response by the new model compared
with the measured data .......................... 166

6.4 Results for the case with one damaged element.... 167 6.5 Results for the case with two damaged elements... 168














LIST OF FIGURES


Figure Page

2.1 Block diagram representation of the system
identification problem........................... 16

2.2 An n-parameter linear system .................... 18

3.1 Flowchart of the damage assessment procedure..... 30

3.2 A seven bar planar truss. Dimensions in
centimeters........................................ 33

3.3 A twenty-five bar planar truss. Dimensions in
centimeters...................................... 36

3.4 A fifteen bar planar truss structure
(Pi, i=1,2,3 represent three loading cases).
Dimensions in centimeters ........................ 41

3.5 A semimonocoque wing box structure (P1 and P2
represent two loading cases). Dimensions in
centimeters ...................................... 45
3.6 Skeleton structure of a helicopter tail.
Dimensions in centimeters.......................... 47

3.7 Two bar planar truss. Dimensions in centimeters.. 68

3.8 Design space for the two bar truss obtained
in the output error approach ..................... 69

3.9 Design space for the two bar truss obtained
in the equation error approach................... 70

3.10 Design space for the two bar truss obtained
in the output error approach, and using a
reduced set of measurements ...................... 71

3.11 Design space for the two bar truss obtained
in the equation error approach, and using a
reduced set of measurements...................... 72








3.12 Three design variable rod-membrane structure.
Dimensions in centimeters........................ 73

3.13 Contours of equal objective function values
for the three design variable membrane-rod
structure, and using the equation error
approach........................................ 74

4.1 A forty-eight bar space truss. Dimensions in
centimeters ..................................... 79

4.2 Equivalent beam model for the forty-eight bar
space truss. Dimensions in centimeters .......... 80

4.3 Flowchart illustrating an implementation of the
dominant design variable strategy................ 91

5.1 Composite laminate subjected to extensional
forces. .......................................... 119

5.2 Composite laminate subjected to bending loads.... 120

5.3 Two dimensional finite element model of an
eight layer E-glass/epoxy laminate subjected to
extensional force. Dimensions in centimeters..... 121

5.4 Two dimensional finite element model for the
[0,90], laminate subjected to pure shear loads
Dimensions in centimeters........................ 124

5.5 Two dimensional finite element model for the
antisymmetric [0,90] E-glass/epoxy laminate
subjected to bending loads. Dimensions in
centimeters....................................... 128

5.6 Two dimensional finite element model of a 24
element laminate subjected to bending forces.
Dimensions in centimeters........................ 130

5.7 Two dimensional finite element model of a 24
element cantilevered laminate subjected to
extensional forces. Dimensions in centimeters.... 132

5.8 Two dimensional finite element model of a 24
element cantilevered laminate subjected to
bending moments. Dimensions in centimeters....... 133

5.9 3-D finite element model of the region of
damage of a 4 layer composite structure.
Each layer corresponds to one element.
Dimensions in centimeters........................ 136








5.10 Two dimensional finite element model of a [0,9031,
eight layer E-glass/epoxy laminate. Dimensions
in centimeters.................................... 138

5.11 Two dimensional finite element model of a 24
element laminate. Dimensions in centimeters..... 141

5.12 First mode shape of a cantilevered composite
beam................................................. 143

5.13 Curve resulting from the superposition of the
first three mode shapes of a cantilevered
composite beam................................... 143
6.1 Original idealized model for the twelve bar
planar truss. Dimensions in centimeters......... 149 6.2 The twelve bar planar truss....................... 150

6.3 Modified idealized model for the twelve bar
planar truss. Dimensions in centimeters......... 151 6.4 Close-up view of a joint in the truss structure.. 152 6.5 The truss mounted in the MTS machine............. 154

6.6 Arrangement of dial indicators to characterize
the behavior of a joint........................... 155

6.7 New model for the twelve bar truss. Dimensions
in centimeters................................... 163

6.8 Isolated view of the model representing
a joint .......................................... 164


xii














Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
STRUCTURAL DAMAGE ASSESSMENT
USING IDENTIFICATION TECHNIQUES By

Francisco Jos4 da Cunha Pires Soeiro May, 1990

Chairman: Dr. Prabhat Hajela Major Department: Aerospace Engineering, Mechanics and Engineering Science

The present investigation describes an approach for

structural damage assessment that has its basis in methods of system identification. Response of a damaged structure differs from predictions obtained from an analytical model of the original structure, where the analytical model is typically a finite element representation. Output error and equation error methods of system identification are employed to determine changes in the analytical model necessary to minimize differences between the measured and predicted response. Structural damage is represented by changes in element stiffness matrices resulting from variations in geometry or material properties of the structure during damage. Measurements of static deflections and vibration modes are used in the identification procedure. A rational


xiii








basis for the proper selection of eigenmodes and loading conditions for the identification process is developed. Representative structural systems made of isotropic as well as orthotropic materials are solved to validate the approach. An experimental setup consisting of a simple planar isotropic truss structure was constructed to demonstrate the applicability of the approach to actual structures. The computational efficiency aspects of the problem are also examined. This includes a study of methods to reduce the number of design variables in the resultant optimization problem.


xiv













CHAPTER 1
INTRODUCTION

1.1 Motivation and Objectives

Structural systems in a variety of applications including aerospace vehicles, automobiles, and civil engineering structures such as tall buildings, bridges and offshore platforms, accumulate damage during their service life. In several situations, such damage may not be visually observable. From the standpoint of both safety and performance, it is desirable to monitor the occurrence, location, and extent of such damage.

System identification methods, which may be classified in a general category of nondestructive evaluation techniques, can be employed for this purpose. Using experimental data, such as eigenmodes and static displacements, and an analytical structural model, parameters of the structure such as its mass, stiffness and damping characteristics can be identified. The approach developed in this investigation is one where the structural properties of the analytical model are varied to minimize the difference between the analytically predicted and empirically measured response. In this process, the number of equations describing the system is typically different










from the number of unknowns, and the problem reduces to obtaining the best solution from the available data.

The genesis of identification methods can be traced to the model determination and model correction problems. The underlying philosophy in these efforts was that a reasonably close analytical model of the structural system was available, and that deviations in the analytical response from the measured response could be used to implement corrections in the model to account for these variations. This resulted in the adoption of a standard strategy wherein the change in the analytical model was minimized to obtain a match between analytical and empirical data. For the class of problems for which it was intended, the method has enjoyed a fair degree of success. A large body of identification work has been based on measurements of eigenmodes of the structure, and this is philosophically the right approach as eigenmodes reflect the global behavior of the structure. However, some reservations about this approach do exist, as these modes are sensitive to the physical boundary conditions that exist on the experimental model, and are not accounted for in the analytical model. It is therefore important to account for these differences and provide a corrected analytical model for the identification procedure.

The problem of damage assessment in structures by identification methods is similar to the one described










above. However, the approach of minimizing changes in the analytical model is no longer applicable, as significant variations can be introduced locally as a result of damage in a structural component. In the present work, damage in a structure is represented by a reduction in the elastic stiffness properties of the material, and these are designated as design variables of the optimization problem. Both static structural deflections and eigenmodes are used in the identification process. The use of static structural displacements as the measured response is a departure from the standard practice of using vibration modes. Since these displacements are reflective of the applied loading, an auxiliary problem in their use is one of determining the load conditions which are best suited for a global model identification.

Hence, the primary objective of this investigation is

to develop a methodology based on identification techniques, which would allow for detection of damage in structural systems. The focus of the effort resided in the proposition and validation of methods that are best suited for structural damage detection. Hence, the problems for which the proposed methods were implemented were simplistic, isotropic or composite structural systems. However, considerations of computational efficiency were also examined, resulting in the development of approximate










strategies that can be implemented in more realistic, largescale structures.

1.2 Literature Survey

The process of correlating a measured response with that obtained from an analytical model, and to use the difference to determine a modified analytical model that would better predict the response, can be broadly categorized in the realm of system identification methods. This subject is not new, and several studies pertinent to the field can be found in the literature. It is important to describe some of these studies in more detail, as they are fundamental to an understanding of how this technique is applicable in damage assessment strategies.

Berman and Flannelly [1] proposed an approach in which the parameters of an assumed discrete model were modified to obtain analytical predictions which were in agreement with measured normal modes. This assumed model represented the dynamics at some preselected points of interest in the structure. To this extent, the model may be regarded as incomplete. This approach can be included in the category of identification techniques where experimental data is employed directly to construct a mathematical model without recourse to an a priori model.

Baruch and Bar-Itzhack developed a technique to

optimally correct measured modes [2]. Usually, and for a










variety of reasons, the measured modes do not satisfy the theoretical requirement of weighted orthogonality. The corrected modes are closest to the measured ones in a weighted Euclidean sense. The analytical mass matrix is assumed to be accurately known and is considered the basis to correct both the measured modes and the stiffness matrix. To identify the stiffness matrix, an a priori analytical model is constructed. The corrected matrix is one which deviates the least from this analytical model, where the deviation is measured in a least squares sense. The corrected matrix should also satisfy the orthogonality conditions with respect to the corrected modes. Berman [34] used a similar approach to modify the mass matrix in order to be consistent with the measured modes, where the latter are assumed to be correct. Along the same line Baruch [5-7] developed methods of "Reference Basis," where depending on the degree of confidence in the analytical model available, either the mass, or the stiffness matrix, can be used as a reference to correct the other one. Experimental data can be a reduced set of modes or even a selected number of static displacements.

Chen and Garba [8] proposed a matrix perturbation

method where the sensitivity of eigenvalues and eigenvectors with respect to the model parameters are required. The sensitivity matrix is used to estimate the parameters. A prior knowledge of the analytical expressions for the mass








6

and stiffness matrices is not necessary. The method is also based on modal tests.

Berman and Nagy [9] developed a method for improvement of large analytical mass and stiffness matrices based on measured normal modes and natural frequencies. The method directly identifies, without iteration, a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements.

Kabe [10] introduced a procedure where, in addition to modal data, structural connectivity information is used to adjust deficient stiffness matrices. The physical configuration of the analytical model is preserved and the adjusted model reproduces exactly the modes used in the identification process. Zero entries of the stiffness matrix are maintained as zero throughout the process, thus reducing the number of components to be identified.

All the methods described above use modal data as

measured response in the identification problem. A work by Sanayei and Nelson [11] was the only paper that could be found in the literature where only static response was used as measured data to identify the stiffness matrix. Forces and displacements, measured on a subset of the total degrees of freedom of the structure were used in this study. A "measured" stiffness matrix was determined by applying, one at a time, an unit load at each of the considered degree of freedom, and the difference between this matrix and the










analytically predicted matrix was minimized using the least squares method. This study also included the effect of errors in force and displacements measurements.

A survey of the principal existing structural system

identification methods is presented by Hanagud, Meyyappa and Craig [12]. The terminology employed in the present work is based on this reference.

The use of system identification techniques to detect

damage in a structural assembly has been recently attempted, but with limited success. Chen and Garba [13] discuss the use of eigenmodes to determine changes in the structural stiffness matrix, with the underlying assumption that the mass matrix does not change with damage. The minimum deviation approach was employed, wherein the Euclidean norm of the matrix representing the perturbation of the analytical stiffness matrix due to the structural damage was minimized. The detection of damage was then attempted by examining the resultant entries of the perturbed matrix. This approach was only marginally successful as it is difficult to identify damage from changes in the stiffness matrix. Entries in the stiffness matrix depend on the elastic and geometric properties of the structure as well as on the element connectivity. Hence, a given element of the matrix may have contributions from several members sharing the same node, making it difficult to identify the precise location of damage. Furthermore, the minimum deviation










approach used by the authors results in small deviations from an a priori model. In this kind of a problem, the a priori model is the original stiffness matrix corresponding to the undamaged structure. In the event of severe damage which may be located in different members of the structure, the changes in the stiffness matrix may be quite significant, increasing the possibility of obtaining poor results. These problems were clearly evidenced by the results obtained in the study.

Smith and Hendricks [14] evaluate two different

identification methods with respect to their applicability for damage detection in large space structures. The first system identification method was the one developed by Kabe [10], and is described above. The other method, due to White and Maytum [15], is based on matrix perturbation techniques and energy distribution analysis. Both methods use eigenmodes as experimentally measured data, and were employed to detect damage by examining the changes in the stiffness matrix corresponding to the damaged structure from the one corresponding to the undamaged structure. Similar difficulties are reported in their work. The entries of the stiffness matrix corresponding to the damaged members do show considerable variations. However, entries corresponding to undamaged members are also affected, thereby making the damage detection process more uncertain. The analysis of changes in the stiffness matrix is typically










cumbersome, may not always yield correct answers, and does not permit the determination of the extent of damage.

All the papers on damage detection referenced in the preceding discussion relate to isotropic structures. No previous effort describing the use of identification methods to assess damage in composite structures could be found in the literature. There is, however, a significant body of information dealing with nondestructive evaluation of composites, and various methods have been developed for this purpose. Some of these, including the ultrasonic C-scan, Xray radiography, and fiber optic crack detection systems, are very good at qualitatively locating damage in fiber reinforced structures. However, they lack the quantitative information necessary to predict stress redistribution and residual life of the structure. Towards this end, methods using vibration analysis [16-18], damping measurements [19], and stiffness-reduction mechanisms [20-22] have been proposed. The stiffness-reduction approach is particularly important, and comprises the basis for the method proposed in this work.

As a laminated composite develops damage in one of the known mechanisms such as transverse cracking, delamination, fiber breakage, or fiber-matrix debonding, a reduction in stiffness is observed. Highsmith and Reifsnider [20] present an investigation on degradation of mechanical response of composite materials resulting from transverse







10

cracking. O'Brien and Reifsnider [21] study the influence of fiber breakage and fiber-matrix debonding on the stiffness of boron/epoxy laminates, subjected to a fatigue loading. O'Brien [22] describes the onset and growth of delamination in terms of stiffness loss and relates both to strain energy release. In all these studies, the stiffnesses of the composite laminate are measured in typical specimens of composite materials. Various tests, each with a different test setup, are conducted to detect changes in different laminate stiffnesses such as E,, Er, and G,,.

1.3 Scope of Present Work

The present dissertation is organized into seven

chapters. In Chapter 1, a clear statement of the problem studied, as well as the methodology employed, is presented. A survey of the most pertinent publications related to this work is also presented.

Chapter 2 defines system identification and gives the typical formulations employed for structural system identification purposes. These formulations are presented in a general fashion and are developed in greater detail in subsequent chapters. The actual implementation of identification techniques for the purpose of detecting damage is described in the last section of this chapter.

In Chapter 3, the approach used in this investigation

is developed and applied to representative structural models










to assess its validity. The output error and the equation error approaches of system identification are applied to such structures using eigenmodes, static displacements, or a combination of both as measured response. A rational method to select a convenient static loading to better detect damage in a structure, and the use of modes to represent the displacement field obtained with the application of that loading, are also explored. The results are discussed at the end of each section.

Chapter 4 deals with the computational efficiency

aspects of the developed approach. Methods to reduce the number of design variables in the resultant optimization problem, in particular the use of a reduced dimensionality model and substructuring techniques are presented. Some approximation techniques that use gradient information to reduce the problem dimensionality are also proposed.

Chapter 5 extends the proposed approach to composite

structures. The stiffness-reduction method is described and the approach applied to simple composite structures. Results obtained in the implementation of the approach are presented at the end of the chapter.

Chapter 6 reports the experimental work performed to demonstrate the applicability of the approach in actual structures. The design of the structure, the experimental setup, and the results obtained, are discussed. Shortcomings in the proposed strategy are also presented.








12

Chapter 7 summarizes the principal conclusions of the present study, including recommendations for further study in this area.













CHAPTER 2
SYSTEM IDENTIFICATION TECHNIQUES


1.1 The Problem of System Identification

One approach of determining the veracity of

analytical models is by comparing the response predicted by the model with that observed in tests or during operation. Although measurements are in themselves imprecise due to the presence of errors in the equipment or uncertainties in the data acquisition techniques used, reasonable bounds can be imposed within which the experimental data are expected to lie. The difference between the measured and analytical data may be large enough to be considered unacceptable. In this case, if there is sufficient confidence in the experimental data, identification methods can be invoked to improve or correct the analytical model. This is the basic problem studied in this work. For structural damage assessment, the same problem is seen from a different point of view as discussed later in this chapter.

System identification involves a broad set of

problems. Generally the problem of system identification is referred to as the determination of a mathematical model for a system or a process by observing its input-output relationships. The system model sought is the mathematical









equation that relates the input to the output at all times. In order to obtain such a model, the system is subjected to a variety of inputs and its response to these inputs are observed. The input-output data are then processed to yield a maximum likelihood model. System identification problems or inverse problems can be found in various applications in science, medicine, and engineering. In such problems, the fundamental properties of a system are to be determined from the observed behavior of that system.

Two categories of identification problems based on an a priori knowledge of the system can be readily identified. The first one is a complete identification of the model, a process which is initiated with little information about the basic properties of the system such as whether it is linear or nonlinear, memoryless or with memory, and so on. Usually some kind of assumptions have to be made before any meaningful solution can be attempted. The second category can be referred to as a partial identification problem where basic characteristics of the system such as linearity and bandwidth are known. This problem is easier to solve than the one involving a complete identification, and problem described at the beginning of this section, dealing with the veracity of an analytical model, belongs in this category. Most of research activity in identification is confined to this task, where the form of the model equations is already known. Hsia [23] classifies this problem as parameter










identification since the model already exists and just its parameters are to be modified to make the results predicted by the model agree with the measured response. This type of problem is also referred to by different researchers as model improvement [8-9], model adjustment [10], or model correction [4-6]. In the present dissertation, the terminology used by Hanagud et al. [12] is followed. According to them, all the cases described above are classified as system identification problems.

The system identification problem can be represented by a block diagram shown in Fig. 2.1. The procedure for carrying out the identification can be divided into the following steps:

1. Specify and parameterize a class of mathematical models that represents the system to be identified.

2. Apply an appropriately chosen test signal to the system and record the input-output data.

3. Perform the parameter identification to select the model in the specified class that best fits the statistical data.

4. Perform a validation test to see if the model chosen adequately represents the system with respect to final identification objectives.

5. If the validation test is passed, the procedure ends. Otherwise, another class of models must be selected and steps (2) through (4) performed until a validated model is obtained.






















DISTURBANCE


OUTPUT


MEASURING INSTRUMENT


MEASURABLE
INPUT


MEASURABLE
OUTPUT


Figure 2.1 Block diagram representation of the system
identification problem (Ref. 23).


INPUT










If the model is already known, only steps (2) through

(4) are performed.

For parameter estimation there are a number of wellknown techniques such as methods of maximum likelihood, least-squares, cross-correlation, and stochastic approximation. A variety of other optimization methods have also been proposed. In the present work, the method of least-squares has been consistently used to construct the objective function required in the resultant optimization problems. Its simplicity and applicability to a wide range of situations are the reasons for the choice. It also exhibits statistical properties as good as those of the maximum likelihood method for most practical situations [23]. The weighted least-squares method can include a degree of confidence in the measured data. The leastsquares technique provides a mathematical procedure by which a model can achieve a best fit to experimental data in the sense of minimum-error-squares. If a variable y is related linearly to a set of n variables x = (xl,x2,...,x) and P = (P1IP2,*.,Pn) is a set of constant parameters, then the following expression can be written.


Y = Pix1 + P2X2 + -.. + pnXn


(2.1)








18

Here pi are the unknowns to be estimated by observing y and x at different times. A block diagram representation of the problem is shown in Fig. 2.2.


x
- Parameters
P P - P
ny


Figure 2.2 An n-parameter linear system.

Assuming that a sequence of m observations has been made on both y and x at times t,, t2,..., t,, and denoting the measured data as y(i) and x1(i), xz(i),..., xn(i), i = 1, 2,..., m, one can relate these data by the following set of m linear equations.


y(i) = p1x1(i) + P2x2(i) +...+ pnX n(i) i=1,2,...,m (2.2)

The system of equations given by (2.2) can be expressed in a matrix notation as follows: y =X P (2.3)


where,



y(1) xz(1) ... Xn(1) pi y(2) x1(2) ... xn(2) p2
y= X= P


* . . .
.y (m) xx (m) . .. xn (m) .Pn










To estimate the n parameters pi, it is necessary that m a n. If m = n, P can be solved uniquely from (2.3) as follows:


S= X-1 y (2.4)


provided that X-1, the inverse of the square matrix X, exists; P denotes the estimate for P. When m > n, it is generally not possible to determine a set of Pi which exactly satisfies all m equations (2.2). This may be due to the fact that the data may be complicated by random measurement noise, error in the model, or a combination of both. The alternative then is to determine P on the basis of least-error-squares.

The error vector e = (el, 2,.., em)T is defined as follows.


e = y - X P (2.5)


The objective function to be minimized, denoted by J is given by the following expression.


m
J = E = E (2.6) i=1


Substituting (2.5) into (2.6), the objective function assumes the following form.


J = (y - X P)T (y - X P)


(2.7)








20

Differentiating J with respect to P and equating the result to zero yields the estimate of P that extremizes J, as given by the following expressions.


S = -2 X' y + 2 XT X p = 0 (2.8)
8 P _=


XT X p = XT y (2.9) p = (X' X) -I X' y (2.10)


This result is referred to as the least-squares estimator of P. It is important to note that the expression is based on a criterion that weights every error ei equally. This formulation can be generalized to allow each error term to be weighted differently. If W is a desired weighting matrix then the weighted least-squares estimator is given by the following expression.


p = (X' W X)- X' W y (2.11)

There are two modes in which identification can be

accomplished. One is off-line identification, in which a record of input-output data is first observed and the model parameters then estimated based on the entire data record. In on-line identification, the parameter estimates are recursively calculated for every data set. The new data obtained is used to update the existing estimates. If the process of data acquisition is very fast, it becomes










possible to develop an on-line real-time identification technique.

Identification techniques may be classified in many

different ways. Such classification is typically based on the type of data used, on the type of system being identified, or on the type of formulation employed. Three of the more important formulations used in identification of structural systems are discussed in the following section. These are, the equation error approach, the output error approach and the minimum deviation approach.

2.2 TypvDical Formulations for Structural System Identification

In the equation error approach, equations describing the system response are explicitly stated. The system parameters, which are typically coefficients in such equations, are then selected to minimize the error in satisfying the system equations with a set of measured input-output data. Consider a linear differential equation represented in a functional form as follows.


f(p,11p2,x,t)=g(t) (2.12) Here, Pi and P2 are considered as the unknown system parameters, and x(t) and its derivatives represent the system response at a time t; g(t) is the forcing function. The system response and the loading is explicitly measured over some characteristic time period T. An objective










function, which is the measure of residual errors in the system equations for given values of the parameters, is formulated as follows.

T
F = { (f(p1IP2,X1,t) - g.(t))2 dt ) (2.13)


This function is then extremized by differentiating with respect to each system parameter, and equating to zero. The approach results in the same number of equations as coefficients to be determined, and is therefore regarded as a direct method. In (2.13), subscript m denotes measured response, and T denotes the characteristic period over which measurements are made.

The output error approach selects some system

characteristic response as the entity for which a match between the analytical prediction and experimental measurements is considered to reflect a good analytical model. An objective function is formulated that is typically an averaged least-squares measure as follows.


T
F = (xM(t) - x(t))2 dt (2.14)


The analytical model from which x(t) is obtained, contains system parameters which are adjusted to minimize the function F. In structural dynamics identification problems, system eigenmodes are generally selected as the










characteristic response quantities used to identify the model.

The minimum deviation approach is frequently used in structural identification problems. In this approach, deviation of the system parameters from initial assumed values is minimized, subject to the constraints that the system equations be satisfied. An illustration of this approach in structural applications is in the determination of changes in the elastic stiffness matrix [2]. In such applications, the mass matrix is assumed to be accurately defined. A weighted matrix norm of the difference between the a priori and corrected stiffness matrices is minimized in the identification process. This norm can be written as

F = I M-'2( K- K ) M-112 1 (2.15)

where M is the mass matrix, K is the desired stiffness matrix, and K is the a priori stiffness matrix. This minimization is subject to the constraint that the modified stiffness matrix remain symmetric. An incomplete set of eigenmodes are measured, and these are required to satisfy the eigenvalue equation and be orthogonal to the modified stiffness matrix. This results in the following equality constraints:

K4 = 2 (2.16)


K= KT


(2.17)











K = Mon2 (2.18)


In the above equations, 4 is an n x p modal matrix and 02 is a diagonal matrix of eigenvalues for the p measured modes; n is the number of degrees of freedom for the structural dynamic system. The constraints are incorporated into the objective function by means of Lagrange multipliers, and the application of the optimality condition (2.8) yields a close form expression for the corrected stiffness matrix as follows.


K = K - KR44TM - M44K + M4 K4 TM + M402 ~TM (2.19)

2.3 Structural Damage Assessment

In a finite element formulation, the characteristics of the structure are defined in terms of the stiffness, damping and mass matrices K, C, and M, respectively. Any variations in these matrices such as may be introduced by damage, would affect the dynamic response characteristics of the structure. A change in the stiffness matrix alone would influence the static displacement response. In the present work, simulated measurements of structure eigenmodes and static deflections under prescribed loads, were used for identification of structural damage.

The analytical model describing the eigenvalue problem for an undamped system can be stated in terms of the system










matrices defined above, the i-th eigenvalue wi2, and the corresponding eigenmode qj as follows.

( K - w 2 M ) qi = 0 (2.20) Matrices K and M may be adjusted to minimize the differences between the experimentally observed eigenmodes and values obtained from the analytical model described above. In using static deflections for identification purposes, the analytical model is even simpler, involving only the system stiffness matrix K as follows.

K x = P (2.21) Here x is a vector of displacements under applied static loads P. Although it is very clear from the previous equations that a variation in the system matrices results in a changed response, it is more important from a damage assessment standpoint to relate these differences to changes in specific elements of the system matrices. Since internal structural damage does not generally result in a loss of material, one can assume the mass matrix to be a constant. Typically, the stiffness matrix can be expressed in terms of the sectional properties of the various elements such as the cross-sectional area A, and the bending and polar moments of inertia I and J. There is also a dependence on element dimensions denoted by t and L, and on the extensional and








26

shear moduli E and G, respectively. These dependencies may be stated in a functional form as follows.

KiJ = Kij(A,I,J,L,t,E, G) (2.22) In the present work, the net changes in these quantities due to damage is lumped into a single coefficient di for each element, that is used to multiply the extensional modulus E. for that particular element. These di's constitute the design variables for the optimization problem. If the measured and analytically determined static displacements or eigenmodes are denoted by Ym and Y,, respectively, the optimization problem can be formulated as determining the vector of design variables di (and hence the analytical stiffness matrix) that minimize the scalar objective representing the difference between the analytical and experimental response, and stated as follows.

Z E (Y. -Yaij)2 (2.23) ij

Here i represents the degree of freedom and j denotes a static loading condition or a particular eigenmode. This minimization requires that Ya be obtained from the eigenvalue problem or the load deflection equations, using the K matrix that must be identified. Lower and upper bounds of 0 and 1 were established for the design variables di. This is the output error formulation of the damage assessment problem. Most of the work developed in the










present investigation was based on this approach. It has the advantage of comparing only two vectors. These vectors may contain any system response, including a combination of different types of variables.

The equation error approach was also used but only for the static load problem. In this case, the governing equation is very simple, with only one output vector. This results in a very good approach for damage detection purposes, which, in certain applications, presents some advantages over the output error approach. This will be discussed in a subsequent chapter. The error function to be minimized in this case is given as follows.

( z Kijxj - f1 )2 (2.24) i j

The minimum deviation approach for damage detection has been used by different researchers as discussed in the literature survey. However, this approach leads to a solution close to the a priori analytical model which may not be true when the structure contains severe damage. Because of this fact and the difficulties reported in the reviewed papers, this approach was not attempted in the present effort.













CHAPTER 3
DAMAGE ASSESSMENT IN STRUCTURAL SYSTEMS


3.1 Introduction

The method for damage assessment presented in section

2.3 was applied to a series of representative structural models with simulated damage in some components. Both the output error approach and the equation error approach were used to identify this damage, and these results are presented in subsequent sections of this chapter. As described in the previous chapter, these approaches result in the formulation of an unconstrained minimization problem. In the present work, the Broydon-Fletcher-Goldfarb-Shanno variable metric method [24-25] was used for function minimization. This approach has been shown to be efficient in the solution of large unconstrained optimization problems where function gradients are available as finite difference approximation [26]. A finite element analysis program EAL (Engineering Analysis Language) was used for response analysis [27]. The equality constraints of the optimization problem are actually implicit. They are represented by the equilibrium equations in static structural analysis or equations describing the eigenvalue problem in a dynamic problem, and are satisfied automatically when the response










analysis is performed. For function minimization, the general purpose optimization program for engineering design ADS (Automated Design Synthesis) was employed [28]. The numerical procedure was implemented on a VAX 11-750 system and the flow between the various processors was controlled in the Command Language feature of DEC systems. The flowchart of this process is shown in Fig. 3.1. The simulated measured data was the finite element solution obtained for the damaged structure corrupted by a random noise signal.


3.2 Output Error Approach in Damage Detection


3.2.1 Use of Eigenmodes as Measured Structural Response

In the limited literature available in the field of

identification based damage assessment, eigenmodes have been traditionally used as the measured response. They closely characterize the global behavior of the structure and can be applied to dynamic systems, including unrestrained space structures. Using this approach, the problem can be stated as finding a vector of design variables d (and hence the analytical stiffness matrix) that minimizes the quantity Z (Y.1 -YaiJ)2 (3.1) ij

where i represent the degree of freedom, and j denotes the particular eigenmode. The measured and analytically determined eigenmodes are denoted by Ym and Ya,





















FINITE ELEMENT ANALYSIS


t


ASSUMED FEN MODEL
OF THE STRUCTURE


ANALYTICAL RESPONSE (Nodes or displacements)


OPTIHIZER EXPERIMENTAL DATA


OPTIMIZER OUTPUT (New set of di's)


Figure 3.1


Flowchart of the damage assessment procedure.










respectively. This minimization requires that Y, be obtained from the eigenvalue problem,


( K - wi2 M ) i = 0 (3.2) where Ya = i ,and K and M are the system stiffness and mass matrices, respectively; wi2 is the i-th eigenvalue, and 0i is the corresponding eigenmode. In this expression, K is the matrix to be identified and is dependent upon the design variables di, for which lower and upper bounds of 0 and 1 were established. These lower and upper bounds represent the limits of a completely damaged member and an undamaged member.

The approach was applied to a series of test problems and results are summarized as follows. The first case is that of a seven bar planar truss structure shown in Fig.

3.2. It is a statically determinate truss, and all structural members have a cross sectional area of 14 cm2. For this structure damage in members 3 and 5 was introduced by reducing their Young's moduli to 40% of the original value. The first three eigenmodes were employed in the formulation of the objective function for damage detection. Table 3.1 shows the results obtained in this exercise, and from which the location of the damaged elements can be easily identified. Furthermore, the extent of that damage is also assessed with reasonable accuracy. The same structure was damaged in element 5 by reducing the Young's










modulus to 10% of its original value. This is in fact, a very severe damage for a statically determinate structure. Table 3.2 shows the effect of including a progressively larger number of eigenmodes on the results for damage detection. If the number of measured eigenmodes were less than or equal to four, the method was unable to detect damage.

The second example chosen was a larger, statically

indeterminate truss structure shown in Fig. 3.3. The cross sectional area of its members was also selected as 14 cm2. For this twenty-five planar truss, damage was simulated by removing member 11 of the structure, which entails setting the value of the Young's modulus for this member to zero. The results obtained in this exercise are presented in Table

3.3.

3.2.2 Use of Static Displacements as Measured
Structural Response

The use of static structural displacements as the

measured response is a departure from the standard practice of using eigenmodes alone for the identification problem. In the previous section, it was indicated that when eigenmodes alone were used for identification, the location and extent of damage predicted by the optimization approach was dependent on the number of modes used to match the measured and the predicted response. Higher modes are often difficult to measure, and only the first few modes are










































ure 3.2 A seven bar planar truss.
Dimensions in centimeters.


(


Fig





















Table 3.1 Results for the seven bar planar truss


Design variables (di) Element using three eigenmodes
No.
Value Exact Sol 1 0.98518 1.00000 2 0.86645 1.00000 3 0.36214 0.40000 4 0.87894 1.00000 5 0.35088 0.40000 6 0.96098 1.00000 7 0.87421 1.00000





















Table 3.2 Results obtained for the seven bar planar
truss, using an increased number of
eigenmodes


Design variables ( di )
Element using eigenmode response
No.
4 modes 6 modes 7 modes Exact sol


1 1.0000 0.6790 0.8706 1.0000 2 0.4205 0.8882 0.8690 1.0000 3 0.9251 0.8944 0.8892 1.0000 4 0.8614 0.9022 0.8677 1.0000 5 0.7650 0.0890 0.0860 0.1000 6 1.0000 0.7862 0.8714 1.0000 7 0.4629 0.8085 0.8780 1.0000


































4 -15252.4 4152--M -4 -152.4 - 441-152.4 l I






Figure 3.3 A twenty-five bar planar truss.
Dimensions in centimeters.


















Table 3.3


Results for the twenty-five bar planar truss using four eigenmodes


Design variables (di)
Element using four eigenmodes
No.
Value Exact Sol

1 0.92536 1.00000 2 0.93169 1.00000 3 0.93320 1.00000 4 0.91196 1.00000 5 0.96094 1.00000 6 0.87899 1.00000 7 0.96327 1.00000 8 1.00000 1.00000 9 0.87905 1.00000 10 0.98613 1.00000 11 0.00000 0.00000 12 0.89017 1.00000 13 0.85975 1.00000 14 0.91797 1.00000 15 1.00000 1.00000 16 0.85562 1.00000 17 0.90889 1.00000 18 0.92100 1.00000 19 0.98278 1.00000 20 1.00000 1.00000 21 1.00000 1.00000 22 0.93679 1.00000 23 0.93863 1.00000 24 0.94524 1.00000 25 1.00000 1.00000










typically available in actual practice. Static displacements provide a good alternative to the use of eigenmodes. The damage assessment problem using this approach can be stated in a manner similar to the one using eigenmodes alone. The objective function to be minimized is given by (3.1), where Ym and Ya are the measured and analytically predicted static displacements, respectively. The Ya vector, which was previously taken to be the eigenmode obtained from the eigenvalue problem expressed by (3.2), is now determined as the displacement vector from the equilibrium equations given by K x = P (3.3) where K is the stiffness matrix to be identified, P is the loading vector, and x is the response vector containing static displacements. Examples illustrating the applicability of this method are presented next.

The twenty-five bar truss structure studied in the previous section was selected as the first example in an approach using static displacements. A vertical unit load was applied to the free-end nodes 11 and 12. The results of damage detection are presented in Table 3.4, showing clearly as in the case where eigenmodes were used, the location and the extent of damage. The more stressed adjacent members also showed some damage as they are directly influenced by the damaged one. This behavior is typically observed when










static displacements are used for damage detection. The type and magnitude of the applied loading is a very important factor in determining the success of the damage detection strategy when static displacements are employed. Another example is the fifteen bar planar truss of Fig. 3.4. It is a statically indeterminate truss with member cross sectional areas of 14 cm2, and was damaged in members 1 and 13. The results are summarized in Table 3.5. Member 1 is the most stressed member in the structure, and critical to the overall structural integrity. That is the reason why it is easily identified. The stiffness of the diagonal member 13 is brought into play by load case 3, which when excluded from consideration, results in an incomplete detection of damage. A more rational approach to avoid problems such as incomplete damage detection due to poorly defined load set, is to choose a load condition that results in an equal stress distribution in each of the members. This will be discussed later in this chapter.

Another representative example is that of semimonocoque wing box structure of Fig. 3.5 consisting of axial rod elements and membrane elements. Membrane element 2 was damaged by reducing its Young's modulus to 10% of the original value. Table 3.6 summarizes the results of this example. It is quite evident from these results, that the application of a torque to the box structure which requires





















Table 3.4 Results for the twenty-five bar truss
using static displacements


Design variables (di) Element using static response
No.
Value Exact Sol.


1.00000 1.00000 1.00000 1.00000 0.99410 0.74186 1.00000 1.00000 1.00000 0.98792 0.01776 0.91020 0.99999 0.94951 0.99196 0.73150 0.90617 0.95281 0.98696 1.00000 0.96812 0.99026 0.96977 0.95638 0.99999


1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000


I _____________________________ I ________________________



























P


I- 152.4 -, 152,4 - 1 11 P2


Figure 3.4 A fifteen bar planar truss structure
(Pi, i=1,2,3 represent three loading
cases). Dimensions in centimeters.














Table 3.5 Results for the fifteen bar planar
truss using static response


Element Design Variables ( di ) No. static response

(Truss Load 1 Load 1 Load 1 Exact member only & 2 2 & 3 sol.


0.0266 1.0000 1.0000 0.7560 1.0000 0.7230 0.7358 1.0000 0.8841 1.0000 0.9686 0.8668 1.0000 1.0000 1.0000


0.0257 0.9999 1.0000 0.8366 0.9999 0.7823 0.9847 1.0000 0.9015 1.0000 0.8235 0.9933 1.0000 1.0000 1.0000


0.0218 0.9999 1.0000 0.8540 0.9730 0.7817 0.9999 1.0000 0.8962 0.9698 0.7849 0.9930 0.0868 0.7566 0.9396


0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 1.0000 1.0000


I ____________ L L _________ L _________










the membranes to participate more equitably in the load bearing process, improves the results for damage assessment.

In a large structure, it is often difficult to obtain the displacements for all degrees of freedom. Typically, only a few dominant ones are measured, and these dominant displacements can be effectively used for damage detection. The objective function in this case becomes:


E E (Ymij -Yaij )2 (3.4) ij

where i represent the degree of freedom where displacement was measured, and j denotes the particular load case.

As an example, a model of a helicopter fuselage

skeleton (Fig. 3.6) with 48 degrees of freedom was used. Damage was identified on the basis of measurements of twelve horizontal displacement components. Member 1 was damaged by reducing the Young's modulus to 10% of its original value. Table 3.7 summarizes the results for this example. The damaged member was clearly detected, as was the extent of damage. This method may not work in some cases, as the measured information may not be enough to identify the location of damage. Typically, this approach works for detection of damage in very stressed members, where the objective function shows a strong sensitivity to the variation of the design variables.










3.2.3 Use of Combination of Eigenmodes and Static
Displacements as Measured Structural Response

The approach developed in the present work also allows the use of a combination of static displacements and eigenmodes as the measured structural response. The examples discussed in previous sections clearly demonstrated that a given static load distribution affects some structural components more significantly than others. If the damaged member is one that does not play an important role in the load bearing process for the set of applied loads, the use of static displacements in damage detection would yield erroneous results. The use of the combination of eigenmodes and static displacements is very easy to implement in the context of the proposed approach. The objective function, which is essentially a scalar quantity is written as a sum of two contributions, arising from the use of eigenmodes and static displacement response, respectively. The implicit equality constraints corresponding to these contributions are simultaneously satisfied. The fifteen bar planar truss of Fig. 3.4 with damage in members 1 and 13 is a good example to show the usefulness of this method. Four eigenmodes were initially used. The effect of adding static displacements corresponding to simple extensional load to the set of measured modes, shows a distinct improvement in locating the global damage. These results are summarized in Table 3.8.








































1.0 6.35





I
r F,




Figure 3.5 A semimonocoque wing box structure (P,
and P2 represent two loading cases).
Dimensions in centimeters.














Table 3.6 Results for the semimonocoque wing box structure


Design Variables ( di )
Element using static response
No. Connectivity
(panel) (nodes) Load 1 Load 1 Exact only & 2 sol.

1 1-7-8-2 1.0000 1.0000 1.0000 2 2-8-9-3 0.1240 0.1020 0.1000 3 7-13-14-8 0.9601 1.0000 1.0000 4 8-14-15-9 0.7720 1.0000 1.0000 5 13-19-20-14 0.9585 1.0000 1.0000 6 14-20-21-15 0.9837 0.9999 1.0000 7 4-10-11-5 0.9736 1.0000 1.0000 8 5-11-12-6 1.0000 1.0000 1.0000 9 10-16-17-11 0.9954 1.0000 1.0000 10 11-17-18-12 0.8713 1.0000 1.0000 11 16-22-23-17 0.9979 1.0000 1.0000 12 17-23-24-18 0.8984 1.0000 1.0000 13 1-4-10-7 0.9999 1.0000 1.0000 14 7-10-16-13 0.9998 1.0000 1.0000 15 13-16-22-19 0.9981 1.0000 1.0000 16 3-6-12-9 0.9926 0.9702 1.0000 17 9-12-18-15 1.0000 1.0000 1.0000 18 15-18-24-21 0.9855 0.9999 1.0000








































- Io00 --- LoO--- -'00Figure 3.6 Skeleton structure of a helicopter tail.
Dimensions in centimeters.













Table 3.7 Results for the idealized helicopter
fuselage model


Element Nodes Design variables (di) Element NodesValue Exact Sol. No. Value Exact Sol.


1
2
3
4
5
6
7
8
9
10 11
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
29 30
31 32 33 34 35 36 37 38 39


1-5 1-6 1-8 2-5 2-6 2-7 3-6 3-7 3-8 4-5
4-7 4-8 5-6 5-8 5-9 5-10 5-12 6-7 6-9 6-10 6-11 7-8 7-10
7-11 7-12
8-9 8-11 8-12 9-10 9-12 9-13 9-14 9-14 10-11
10-13 10-14 10-15 11-12
11-14


0.10498 1.00000 1.00000 0.93199 1.00000 1.00000 0.99781 1.00000 0.99781 0.93199 1.00000 1.00000 1.00000 1.00000 0.97783 0.99372 0.99368 1.00000 0.98869 0.99906 1.00000 1.00000 0.99157 1.00000 0.99157 0.98858 1.00000 0.99991 0.99995 0.99995 1.00000 0.98484 0.98484 1.00000 1.00000 0.96628 1.00000 1.00000 0.98508


I _________________________ L


0.10000
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000


























Table 3.7 continued


Design variables (di)
Element Nodes
No. Value Exact Sol.

40 11-15 1.00000 1.00000 41 11-16 0.98506 1.00000 42 12-13 1.00000 1.00000 43 12-15 1.00000 1.00000 44 12-16 0.96631 1.00000 45 13-14 0.99994 1.00000 46 13-16 0.99992 1.00000 47 14-15 0.99996 1.00000 48 15-16 0.99992 1.00000














Table 3.8 Results for the fifteen bar planar truss
using a combination of static and modal
response


Element Design Variables ( di )
No. using static and modal response

(Truss First four Modes Exact member) eigenmodes +load 1 sol.


0.0000 0.7461 0.7571 0.7748 0.9934 0.8107 0.7517 0.9825 0.6603 0.7640 0.8116 0.7763 0.0006 0.7557 0.7623


0.0069 1.0000 0.9761 0.9503 1.0000 0.7515 0.9216 1.0000 0.8977 1.0000 0.9930 1.0000 0.0000 0.9433 1.0000


0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 1.0000 1.0000


I ____________ I ___________ ________










3.2.4 A Rational Method for Structural Damage Detection

As stated in section 3.2.2, the use of the output error approach with measured static displacements data can result in misleading results during damage detection. The displacement field is largely determined by the applied loading, and there might be considerable differences in magnitude of displacements at the various degrees of freedom. As a consequence of this, elements that undergo larger elastic displacements and are therefore more stressed, are easier to identify in the event of damage. Damage in lightly stressed elements, or in those that largely translate as rigid members during loading, is more difficult to identify. Although this is acceptable from the standpoint of safety, damage in the less critical member is likely to go undetected. In this section, a rational method to circumvent this problem is proposed. 3.2.4.1 Equal stress load distribution

In light of the foregoing discussion, it is possible to assert that a static loading that produces equally stressed members is the one most likely to yield an accurate assessment of damage in the system. In a truss structure, such a load condition may be determined by the use of a superposition method. Consider the statically indeterminate, fifteen bar planar truss shown in Fig. 3.4. This structure has twelve degrees of freedom. Stresses in each member of this truss can be determined under the action










of a unit load applied at each degree of freedom, one at a time. The total stress in each member is then obtained as follows:

12
a = Z Sij cj , i=1,15 (3.5) j-1

where Sij is the stress generated in member i due to a unit load applied to the degree of freedom j and cy are unknown coefficients. The stress vector can be represented in matrix notation by the following expression.


a = S c (3.6) Each component of the stress vector can be set to a desired value, and the coefficient vector which would achieve such a stress distribution can be determined. If the structure is statically determinate, the problem has a unique solution, and can be written as follows.


c = S-1 a (3.7) For the fifteen bar truss of Fig. 3.4, the number of equations is greater than the unknowns, and there is no exact solution to the equal stress load distribution problem. An approximate solution to this problem may be obtained by minimizing a Euclidean norm of the difference between the right and left hand sides of (3.6), following the procedure described in section 2.1. The final










expression for c is similar to (2.10) and is given as follows.


c = (STS)- ST a (3.8)

For the fifteen bar truss structure defined above, a stress level of 13.78 kPa was specified for each element. This stress level was obtained from the application of nominal loads to the structure. The desired coefficients, and hence the load distribution to achieve such a stress distribution are presented in Tables 3.9 and 3.10. This structure was damaged in two places, a critical horizontal element (member 1) at the point of load transfer to the boundary, and a less critical element (member 13) in an outer bay. The Young's moduli of the damaged members were reduced to 10% of their original values. An extensional force P1 was applied at the free end of the structure as shown in Fig. 3.4 . The output error approach failed to detect damage in member 13 as seen in Table 3.11. The use of the equal stress loading presented in Table 3.9 allowed the detection of damage in both members as shown in Table 3.11.

It is quite evident that such load conditions may prove quite unrealizable in actual practice, especially for complex structures with several degrees of freedom. Furthermore, the application of static loads would be impractical in unconstrained space structures, large civil engineering structures, and for the purposes of real-time










monitoring of structural integrity in aeronautical structures. When used in an appropriate manner, the first few eigenmodes of the system still present the best global information to quantify the extent and location of damage. The following section describes an approach which combines the obvious benefits of activating all members by an equal stress distribution, with the actual use of measured eigenmodes in the identification problem.

3.2.4.2 Modal superposition approach

The equal stress load distribution obtained in the previous section is significant for damage detection purposes as it favors no particular member. However, this type of loading is difficult to apply in a realistic structure. Instead, the approach adopted is one in which an unknown displacement field is assumed for this loading. The measured eigenmodes of the damaged structure are used as the basis vectors to expand the displacement field as



yi = Ck 40 , i=l,n (3.9) k-1

where 4ik is the i-th component of the k-th eigenmode, m is the number of measured modes, n is the number of degrees of freedom of the structure, and ck are the unknown coefficients in the series expansion. Substituting into the equilibrium equation and premultiplying both sides by 41, the following expression is obtained.











ciK + ...+ ciO + . .+ cm. # iK~m = if (3.10) Orthogonality of the eigenmodes is invoked by requiring



T 2 (3.11) 01 K 4 = 0

2
where w are the eigenvalues of the structure. This yields an expression for ci as follows.


ci = f/~ (3.12) The objective function for the output error approach is then reformulated as the minimization of Z (yiY.a)2 (3.13)
i

where y, is the analytically predicted response under the equal stress load distribution, and y, is a function of the measured response i'" obtained from (3.9). The orthogonality conditions (3.11) are only valid if the eigenmodes are orthonormal with respect to the mass matrix. Hence, the measured modal matrix must be first orthonormalized with respect to the mass matrix to permit their use as a base for the displacement field [2]. The procedure is initiated by first normalizing each measured mode 0i with respect to M as follows:




















Table 3.9


Equal stress load distribution for the fifteen bar truss (stress level = 13.78 kPa)


Node d.o.f Load (N)

3 x -0.79
3
y 45.95
4 x -0.79
4
y -45.95
5 x 0.06
5
y 46.68
6 x 0.06
6
y -46.68 7 x 32.96 y 32.96
8x 32.96
y -32.96






















Table 3.10


Member stresses in the fifteen bar truss for the loading shown in Table 3.9


Member Stress (kPa)

1 15.02 2 15.02 3 11.23 4 11.23 5 14.88 6 13.64 7 13.64 8 13.98 9 13.98 10 13.64 11 13.78 12 13.78 13 13.71 14 13.71 15 13.78



















Table 3.11 Results of the fifteen bar truss obtained
using the output error approach


Load type
Element Exact
number Extensional Equal stress solution
force load

1 0.157 0.098 0.100 2 0.999 0.900 1.000 3 0.993 1.000 1.000 4 0.613 1.000 1.000 5 1.000 0.949 1.000 6 0.646 1.000 1.000 7 0.811 0.997 1.000 8 0.999 0.962 1.000 9 0.732 0.877 1.000 10 1.000 0.881 1.000 11 1.000 0.941 1.000 12 0.892 0.971 1.000 13 0.918 0.134 0.100 14 0.956 1.000 1.000 15 1.000 0.900 1.000











-T - 1/2
S= 4t (40 M O)-1 (3.14) where 4 is the normalized mode. All n measured modes normalized as in (3.14) are put together into an n x m matrix 4, where m is the total number of degrees of freedom in the structure. The matrix 4 must further satisfy the orthogonality condition

ST M = I (3.15)


for it to be used in the identification process. This is facilitated by assuming the existence of an n x m matrix X, and minimizing the weighted Euclidean norm,


C = M1(2 ( - ) 1I (3.16)

subject to the constraint as follows:


XT MX = I (3.17) The constraint (3.17) can be incorporated into the objective function of (3.16) by the use of Lagrange multipliers,


S= e + A (XTM X - I) (3.18) where A is the vector of Lagrange multipliers. The partial differentiation of (3.18) with respect to X, and equating the result to zero in a procedure similar to the one described earlier in section 2.1, yields the expression for the orthonormalized measured mode shapes.











X = 0 (4T M 4)112 (3.19) The approximation to the displacement field based on a limited number of measured eigenmodes is generally good. For the fifteen bar truss discussed earlier, the displacement field for equal stress loading and its approximation in terms of the measured eigenmodes, are shown in Table 3.12. The same example, discussed earlier in this chapter, was used to demonstrate the concept of using measured eigenmodes to simulate deflections under the equal stress load distribution. An increasing number of modes were superposed to obtain the displacement field. Table

3.13 clearly shows the need for at least four eigenmodes to detect all damaged members. The overall results improved with the inclusion of more modes in the superposition.


3.3 Ecuation Error Approach in Damae Detection

The equation error approach discussed in Section 2.2 was also implemented for damage assessment purposes. In this case, only static displacements were used as structural response for reasons of simplicity afforded by the governing equilibrium equation for static problems. This approach avoids an explicit decomposition of the stiffness matrix in the solution process. This feature appears to produce a better conditioned numerical space for obtaining solutions of the optimization problem. The Euclidean norm of the








61

difference between the right and the left hand sides of the equilibrium equation was minimized as follows.


( Z Kix - ft )2 (3.20)
i j

This expression is valid only for the case where all displacement degrees of freedom of the damaged structure are known. In the case where only a few displacements are measured, the system of equilibrium equations can be partitioned as follows:


= (3.21)
K12 K22


where ul denotes the measured subset, u2 are the unknown displacement components, and K11, K12 and K22 are submatrices of the stiffness matrix. The unknown displacements can be eliminated to obtain a condensed equation.


(K11-K12K22-1K2 )ul = f1-K12K22-1f2 (3.22) The objective function is then constructed in a manner similar to the case where all displacement components are known. The design variables for the optimization are the same as in the output error approach, and yield a new stiffness matrix. The optimum design variables result in a stiffness matrix that satisfies the equilibrium equations. It is worthwhile to note that the use of a reduced set of




















Table 3.12 Approximation of the displacement field using an increasing number of eigenmodes.
The square Euclidean norm e 112 measures the
error



x and y components of displacements in cm
Node d. o. f
Node d.o.f 2 3 4 5 Exact
modes modes modes modes solution

x 1.69 1.67 1.75 1.77 1.63
3
y -0.85 -0.97 -0.85 -0.91 -0.65
4 x 0.26 0.25 0.18 0.15 0.16
y -0.51 -0.62 -0.50 -0.56 -0.81
5 x 1.96 1.91 1.96 2.01 1.93
y -2.04 -2.12 -2.22 -2.17 -1.97
6 x 0.62 0.66 0.55 0.47 0.46
y -2.00 -2.10 -2.26 -2.24 -2.43 7 x 2.09 2.02 2.08 2.22 2.39 y -3.13 -3.00 -2.85 -2.82 -2.69 8 x 0.70 0.86 0.83 0.72 0.92 y -3.14 -3.02 -2.92 -2.92 -3.14

l e2 0.6919 0.5710 0.4295 0.3305


















Table 3.13


Results for the fifteen bar truss obtained using output error approach. Increasing number of weighted eigenmodes used to model the displacement field obtained under the loading shown in Table 3.9


Element Design Variables (di) number 2 modes 3 modes 4 modes 5 modes Exact sol

1 0.100 0.120 0.100 0.099 0.100 2 0.689 0.994 0.886 0.960 1.000 3 1.000 1.000 1.000 1.000 1.000 4 0.979 0.831 1.000 0.991 1.000 5 1.000 0.999 1.000 1.000 1.000 6 0.891 1.000 1.000 1.000 1.000 7 0.718 0.572 0.825 0.911 1.000 8 1.000 0.999 1.000 1.000 1.000 9 1.000 1.000 1.000 1.000 1.000 10 1.000 1.000 1.000 1.000 1.000 11 1.000 1.000 1.000 1.000 1.000 12 0.999 0.999 1.000 1.000 1.000 13 0.741 0.615 0.219 0.128 0.100 14 1.000 1.000 1.000 1.000 1.000 15 1.000 1.000 1.000 1.000 1.000








64

measurements requires a matrix inversion. Consequently, the advantage of the equation error approach over the output error is limited in such situations. The equation error approach is particularly useful in problems where the region of damage is known, and can result in a significant reduction in the number of design variables. For this case, the equilibrium equations can once again be written in the partitioned form shown in (3.21). Here, the submatrix K1 corresponds to the substructure containing damage. The first row of this partitioned system yields


K11u, + K12U2 = fl (3.23) which allows the damage detection problem to be reduced to determining the design variables di for the submatrix K11 so as to satisfy (3.23).

The equation error approach was used to detect damage in each of the structures examined by the output error approach in Section 3.2 . The method yielded good results, and in some cases, outperformed the output error approach. An impressive example of this performance is the detection of damage in the fifteen bar truss discussed earlier, and for which the correct results were obtained in the output error formulation only when member 13 was forced to participate in the load bearing process by the load case 3. The equation error approach yielded good results for an arbitrarily chosen extensional end load on the structure,










such as load case 1, shown in Fig. 3.4. The results for this case are presented in Table 3.14. This difference can be largely attributed to the scaling of the design space that naturally emerges from the two formulations. The use of appropriately weighted eigenmodes or special load functions in the output error approach scales the design space so that it is easier for the numerical optimization technique to locate the optimum design. The equation error formulation does not require an explicit decomposition of the stiffness matrix, and for simple structures, presents an almost quadratic design space.

3.4 Design Spaces Corresponding to the Studied Formulations
The design space for the two described approach can be visualized by studying the simple two bar truss of Fig. 3.7, in which damage was simulated by reducing the Young's moduli of members 1 and 2 to 50% and 40% of the original values, respectively. Fig. 3.8 and 3.9 show the design spaces for the two approaches. The equation error approach presents a slightly more convex design space with a better defined optimum. For larger structures this might be of significance in correctly locating the damage. If a reduced set of measured displacements is employed in damage detection, the design space for both methods present several local minima that may lead the optimizer to the








66

wrong solution. Figures 3.10 and 3.11 illustrate the design space behavior for this case.

An example of a damage detection problem in which the design space has local optima is obtained for the three design variable membrane-rod structure of Fig. 3.12. Contours of equal objective function values obtained in the equation error approach, with damage in members 1 (rod) and

3 (membrane), are shown in Fig. 3.13 . This damage was simulated by reducing the respective Young's moduli by 40% and 50% of the original values. As seen in Fig. 3.13, the design space offers the possibility of convergence to local optima, and such a convergence would be dictated by the choice of starting design variables. Table 3.15 shows the optimum solutions for two different initial designs.





















Table 3.14 Results for the fifteen bar planar truss
using the equation error approach


Element Design Variables Exact
number di solution

1 0.099 0.100 2 1.000 1.000 3 0.853 1.000 4 0.949 1.000 5 0.912 1.000 6 0.872 1.000 7 0.978 1.000 8 0.965 1.000 9 0.996 1.000 10 1.000 1.000 11 0.926 1.000 12 0.977 1.000 13 0.153 0.100 14 0.986 1.000 15 0.970 1.000



















I.o 12.7 2 i I.O 1 2, 7 - I


Q


X


'0


0


Figure 3.7 Two bar planar truss. Dimensions in
centimeters.











































.0-


I ' I ' I ' I ' I ' i ' I ' I ' I .2 .4 .6 .8 1


E(1)


Figure 3.8 Design space for the two bar truss
obtained in the output error approach.


1



.8







.4-


rr


U I


[
































.8









o 6-


U I I # I . I


.2
.2


2- .01.0

..



.05



1.0 --


. . . . . . .........................................................."Il". 0". . - - . 0 .


.4 .6 .8 1


E(1)



Figure 3.9 Design space for the two bar truss
obtained in the equation error approach.
































.8-0.
0 01



0.01 10.0.6

0-. OL 5.0.

0.01 3.o .4 - 0.


3. 0
. 0
10.0


100.00
..2


8 .2 .4 .6 .8

E(1)



Figure 3.10 Design space for the two bar truss
obtained in the output error approach,
and using a reduced set of measurements.






























0
0.


0.1-.

0.05
4.005.0.1-


0.5
1.0-


I p I I


.2


.4


.6
E(1)


.8 1


Figure 3.11 Design space for the two bar truss
obtained in the equation error approach, and using a reduced set of measurements.


1



. 8




.6




.4




.2-




















4 25.4 0 I


Membrane


Figure 3.12 Three design variable membrane
structure. Dimensions in centimeters.























Obj-0 .005


E(3) Obj-0.0005
I



,.


Figure 3.13 Contours of equal objective function
values for the three design variable
membrane-rod structure, and using the
equation error approach.


E(3)


E(j7






















Table 3.15


Results for the three design variable membrane-rod structure for two sets of initial designs


Element Initial Design Exact number design variables solution

1 1.0 0.375 0.400 2 1.0 0.933 1.000 3 1.0 0.789 0.500

1 0.6 0.411 0.400 2 1.0 0.929 1.000 3 0.6 0.521 0.500













CHAPTER 4
COMPUTATIONAL EFFICIENCY ASPECTS IN STRUCTURAL DAMAGE ASSESSMENT


4.1 Introduction

The focus of previous chapters was the proposition and validation of an approach for damage assessment using identification techniques. The examples presented in those chapters dealt with simplistic structures, and were intended to show the key findings of the investigation. However, if a more realistic structural system is to be examined for damage, the associated computational effort would increase considerably. According to the proposed approach, there are as many design variables as the number of elements in the finite element model of the structure. This results in significant computational costs when using a gradient-based nonlinear programming algorithm for function minimization. Two distinct strategies for reducing the number of design variables were studied in the present work. One was based on the use of reduced dimensionality models, where equivalent structures with fewer degrees of freedom were constructed to represent the actual structure. The second approach was one in which only a dominant subset of the design variables was considered at any stage of the error








77

minimization problem. These two approaches are discussed in subsequent sections of this chapter.

4.2 Reduced Dimensionality Models

The construction of a reduced dimensionality equivalent continuum model depends primarily on the type and shape of the structure under consideration. Beams are a natural approximation for trusses, allowing the substitution of the actual structure by one with fewer number of degrees of freedom. Use of equivalent structures to facilitate the analysis and quickly determine the response characteristics of the structures such as natural frequencies and vibration modes is commonly adopted in engineering practice. Aerospace structures such as mast trusses and precision trusses can be represented by equivalent beams. Consider a truss model of a mast structure shown in Fig. 4.1, subjected to a tensile load (PA=1 N), a transverse load (PT=1 N), and a torsional load (T=l N-m). The Young's modulus of each rod was considered as unity, and a nominal value of 10.47 cm2 was assumed for all cross sectional areas. An equivalent beam model (Fig. 4.2) with an independent axial, bending and torsional stiffness for each section, was obtained to simulate the deflection characteristics of the truss structure. Each section of the beam corresponds to a bay in the truss. The reduced dimensionality beam model with only

4 design variables, was first used to determine the










approximate location of damage. The model of the truss structure had a total of 48 design variables. The identification problem was then solved for the beam model, using interpolated displacements obtained from measurements on the damaged truss. These interpolations were necessary due to the difference in the number of degrees of freedom for the two models. In such an analysis, the beam section corresponding to the bay of the truss containing the damaged members is first identified. This is a preliminary identification problem with a fewer number of design variables. Once the region of damage is identified, the actual structure is then considered. The design variables for this secondary identification problem are the parameters di corresponding to the members of the damaged bay(s). If only one member is damaged as in the example of Fig. 4.1, the problem reduces to one with 15 design variables.

The procedure described above was applied to the truss of Fig. 4.1. Damage was introduced in members number 12 (2nd bay) and 40 (4th bay) by reducing the respective Young's moduli to 10% of the original values. The first step was to calculate the response of the undamaged structure to the applied loads. This information determines the geometric characteristics of the equivalent beam (Fig. 4.2). For example, to determine the cross sectional areas of the different elements in the equivalent beam, the extensional displacements are used. Based on these displacements, an

































10
Ll

1 /

7

9

41/
6 1 ,







3



Figure 4.1 A forty-eight bar space truss. Dimensions
in centimeters.








































Figure 4.2


Equivalent beam model for the forty-eight bar space truss. Dimensions in centimeters.


41l4 - 1oo p1- ---1woo w1oo oo-----i0.1








81

averaged length variation AL for each bay can be calculated. This yields the value of the equivalent area S of one particular bay, given by the following expression.


Ps L
S = (4.1) E AL

Here, E is the Young's modulus and may be assumed as unity; P, is the resultant normal force acting on the bay. Likewise, bending moment of inertia for the equivalent cantilever beam can be calculated from the following expression,


Fx2
I = (x- 3L) (4.2)
6 Ey

where F is the resultant transverse force applied at the free end of the truss, x is the distance from the support of any node in the beam, L is the length of the beam, and y is the transverse displacement of a node in the beam obtained from the truss response. The Young's modulus E can again be taken as unity. This expression will usually give different values for the moment of inertias corresponding to the different elements of the beam, since the displacements used are the ones obtained from the truss. One can take the average of all such computed values, and use it as the approximate moment of inertia of the equivalent beam. For








82

more precise values, the following differential equation can be solved for each element of the beam,


d2y M
d (4.3)
dx2 El I


where M is the bending moment acting on that element. The solution of this equation at each element will satisfy the continuity conditions of a beam and the displacement field determined from the analysis of the truss. This way, an exact moment of inertia can be calculated for each element of the beam. To calculate the polar moment of inertia J at each element of the equivalent beam, the following expression was used.


T L
J = (4.4) G AO8

In this expression G is the shear modulus and can be taken as unity; T is the torque applied to the element; L is the element length, and A0 is the difference between the rotations at the ends of an element. These angles are calculated from the transverse displacements obtained when a torque is applied at the tip of the truss. The above properties are independent, and represent a beam with a behavior close to the original truss.

The second step involved the solution of the equivalent beam with only four design variables. The measured response










of this structure corresponds to interpolated or averaged displacements obtained from the damaged structure. In Table

4.1, the results show considerable change in the final values of the design variables corresponding to the 2nd and 4th bays.

Finally, the actual structure was solved considering as design variables the parameters di corresponding to the members of the damaged bays, as detected from the equivalent beam model. The final results for the actual structure are summarized in Table 4.2.


4.3 Substructuring
The last step of the problem discussed in the previous section is also amenable to substructuring techniques. Such an approach allows each bay to be treated separately. The system of equilibrium equations can be partitioned as follows.


K11 K12 u1 f 1= (4.5)
K12 K22 2 f2

The submatrix K11 corresponds to the substructure containing damage. It is always possible to isolate the damaged portion of the whole structure in submatrix K11 by rearranging the system of equations (4.5) using properties of permutation. The first row of this partitioned system yields the following expression.
























Table 4.1


Damage detection as seen in the equivalent beam model


Design variables (di) Element
No. Lower Upper bound Value bound

1 0.0000 0.9021 1.0000 2 0.0000 0.5849 1.0000 3 0.0000 0.9537 1.0000 4 0.0000 0.5890 1.0000













Table 4.2


Results for the forty-eight bar space truss. Only members in 2nd and 4 bays are treated as design variables


Design variables (di)
Element Nodes
No. Value Exact sol

2nd bay

10 4-5 1.00000 1.00000 11 4-6 1.00000 1.00000 12 4-7 0.10589 0.10000 13 4-8 0.99999 1.00000 14 4-9 0.99999 1.00000 15 5-6 1.00000 1.00000 16 5-7 0.89710 1.00000 17 5-8 1.00000 1.00000 18 5-9 1.00000 1.00000 19 6-7 0.89694 1.00000 20 6-8 1.00000 1.00000 21 6-9 1.00000 1.00000 22 7-8 1.00000 1.00000 23 7-9 1.00000 1.00000 27 8-9 1.00000 1.00000

4h bay

34 10-11 1.00000 1.00000 35 10-12 0.96474 1.00000 36 10-13 1.00000 1.00000 37 10-14 0.99816 1.00000 38 10-15 1.00000 1.00000 39 11-12 0.97694 1.00000 40 11-13 0.11129 0.10000 41 11-14 1.00000 1.00000 42 11-15 0.93723 1.00000 43 12-13 0.93341 1.00000 44 12-14 1.00000 1.00000 45 12-15 1.00000 1.00000 46 13-14 1.00000 1.00000 47 13-15 0.98714 1.00000 48 14-15 0.99159 1.00000











Kzu1 + K12u2 = f1 (4.6) The damage detection problem is reduced to determining the design variables di for the submatrix K11. The equation error approach is particularly useful in this approach.The error vector c for this formulation is defined as follows.


S= K11u, + K12u2 - f1 (4.7) The objective function is then constructed as the squared Euclidean norm of the error vector and the procedure described in section 3.3 is adopted, yielding the following expression.

Z ( K u' + Z Ki uk - f )2 i=j (4.8)
i j k

Here i includes all degrees of freedom of the substructure where damage is located, and j+k equals the total number of degrees of freedom in the structure.

In the space truss example of the previous section where two bays were detected as containing damage, a separate solution for each bay is possible as they do not share any common degree of freedom. If the damaged bays had any common degrees of freedom, then all the design variables corresponding to the two bays would have to be considered simultaneously. In this case some of the entries in the stiffness matrix would have the contribution of the material properties of members belonging to the two damaged bays and




Full Text

PAGE 1

STRUCTURAL DAMAGE ASSESSMENT USING IDENTIFICATION TECHNIQUES BY FRANCISCO JOSE DA CUNHA PIRES SOEIRO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1990

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To my parents Adeline Pires Soeiro and Benilde da Cunha Soeiro

PAGE 3

ACKNOWLEDGMENTS I would like to express my sincere gratitude to Dr. P. Hajela for his continuous direction, support and encouragement since I started my doctoral studies. I am particularly grateful to him for the introduction to the subjects of structural optimization techniques, stiructural system identification and structural damage assessment, which is largely responsible for my current interest in the field. His diligence and competence have always impressed me and are examples to be followed. I would like to acknowledge Dr. F. Fagundo and Dr. D. Zimmermann, members of my supervisory committee, for helping me during the course of my research. We had valuable discussions that contributed significantly to the improvement and progress of my work. I would like to express my sincere thanks to Dr. C. T. Sun and Dr. C. C. Hsu, also members of my supervisory committee, for their support and encouragement. Special thanks are owed to Dr. D. A. Jenkins and the staff of the laboratory of structures of the Department of Aerospace Engineering, Mechanics and Engineering Science, for helping me in the experimental work. Their experience iii

PAGE 4

and competence were fundamental for the completion of that important phase of my research. I would like to express my thanks to the Brazilian Army for giving me the opportunity and providing me the support to come to the United States and pursue doctoral studies. Finally, I would like to extend my sincerest gratitude to my wife, Maria Luiza, and my sons Renato and Junior. They have been a continuing source of inspiration, support and encouragement. iv

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TABLE OF CONTENTS page ACKNOWLEDGEMENTS iii LIST OF TABLES vii LIST OF FIGURES X ABSTRACT xiii CHAPTERS 1 INTRODUCTION 1 1.1 Motivation and Objectives 1 1.2 Literature Survey 4 1.3 Scope of Present Work 10 2 SYSTEM IDENTIFICATION TECHNIQUES 13 2.1 The Problem of System Identification 13 2 . 2 Typical Formulations for Structural System Identification 21 2.3 Structural Damage Assessment 24 3 DAMAGE ASSESSMENT IN STRUCTURAL SYSTEMS 28 .^^ 3.1 Introduction 28 3.2 Output Error Approach in Damage Detection... 29 3.2.1 Use of Eigenmodes as Measured Structural Response 29 3.2.2 Use of Static Displacements as Measured Structural Response 32 \ 3.2.3 Use of a Combination of Eigenmodes and Static Displacements as Structural ; Response 44 3.2.4 A Rational Method for Structural Damage Detection 51 3.3 Equation Error Approach in Damage Detection. 60 3.4 Design Space Corresponding to the Studied formulations 65 V

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4 COMPUTATIONAL EFFICIENCY ASPECTS IN STRUCTURAL DAMAGE ASSESSMENT 76 4.1 Introduction 76 4.2 Reduced Dimensionality Models 77 4.3 Substructuring 83 4.4 Approximation Concepts 87 5 DAMAGE DETECTION IN COMPOSITE STRUCTURES 94 5.1 Introduction 94 5.2 The Stiffness-Reduction Method 96 5.2.1 Use of Static Displacements as Measured Structural Response 96 5.2.2 Use of Eigenmodes as Measured Structural Response 115 6 EXPERIMENTAL INVESTIGATION OF DAMAGED STRUCTURES 145 6.1 Introduction 145 6.2 Experimental Setup 147 6.3 Analytical Model Identification 156 6.4 Discussion of Results 159 7 CONCLUSIONS AND RECOMMENDATIONS 169 REFERENCES T. . . . 175 BIOGRAPHICAL SKETCH 178 vi

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LIST OF TABLES Table Page 3.1 Results for the seven bar planar truss 34 3.2 Results obtained for the seven bar planar truss, using an increased number of eigenmodes 35 3.3 Results for the twenty-five bar planar truss, using four eigenmodes 37 3.4 Results for the twenty-five bar truss using static displacements 40 3.5 Results for the fifteen bar planar truss using static response 42 3 . 6 Results for the semimonocoque wing box structure 46 3.7 Results for the idealized helicopter fuselage model 48 3.8 Results for the fifteen bar planar truss using a combination of static and modal response 50 3.9 Equal stress load distribution for the fifteen bar truss (stress level = 13.78 kPa) 56 3.10 Member stresses in the fifteen bar truss for the loading shown in Table 3.9 57 3.11 Results for the fifteen bar truss obtained using the output error approach 58 3 . 12 Approximation of the displacement field using an increasing number of eigenmodes 62 3.13 Results for the fifteen bar truss obtained using output error approach. Increasing number of weighted eigenmodes used to model the displacement field obtained under the loading shown in Table 3.9 63 vii

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3.14 Results for the fifteen bar planar truss using the equation error approach 67 3.15 Results for the three design variable membranerod structure for two sets of initial designs.... 75 4.1 Damage detection as seen in the equivalent beam model 4.2 Results for the forty-eight bar space truss. Only members in 2""^ and 4^'' bays are treated as design variables 85 4.3 Results for the twenty-five bar planar truss using dominant variables method 92 4.4 Results for the fifteen bar planar truss using four eigenmodes as measured response and the dominant variables method 93 5.1 Dominance assessment for equation (5.17) 105 5.2 Dominance assessment of in (5.24) 109 5.3 Mechanical properties of the material used in the examples 122 5.4 Results for the t0,903]s laminate subjected to extensional forces 123 5.5 Results for the [0,90], laminate subjected to extensional forces 125 5.6 Results for the [0,90], laminate subjected to extensional load and with very low Age in element 1 126 5.7 Results for the [0,90]^ laminate subjected to a pure shear force 127 5.8 Results for the antisymmetric [0,90] laminate subjected to a bending load 129 5.9 Results for the [0,±45,90]8, 24 element laminate subjected to bending forces 131 5.10 Results for the [0,145,90],, 24 element laminate subjected to an extensional force applied at the free end 134 5.11 Results for the [0,±45,90],, 24 element laminate subjected to bending moments 135 viii

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5.12 Results for the 4 layer laminate using a 3-D element for damage detection 137 5.13 Results for the [0,903]s laminate using three extensional eigenmodes 139 5.14 Results for the [0,903], laminate using three bending eigenmodes 140 5.15 Results for the [0,±45,90]3, 24 element, simply supported laminate, using three eigenmodes 142 5.16 Results for the [0,±45,90]3, 24 element, cantilevered laminate using eigenmodes 144 6.1 Measured response compared with results predicted by the original model (Fig. 6.3) 162 6.2 Identified material properties for the truss elements of the model described by Fig. 6.7 165 6.3 Predicted response by the new model compared with the measured data 166 6.4 Results for the case with one damaged element.... 167 6.5 Results for the case with two damaged elements... 168 ix

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i^. LIST OF FIGURES Figure Page 2 . 1 Block diagram representation of the system identification problem 16 2.2 An n-parameter linear system 18 3.1 Flowchart of the damage assessment procedure 30 3.2 A seven bar planar truss. Dimensions in centimeters 33 3.3 A twenty-five bar planar truss. Dimensions in centimeters 36 3.4 A fifteen bar planar truss structure (Pi, i=l,2,3 represent three loading cases). Dimensions in centimeters 41 3.5 A semimonocoque wing box structure (Pi and P2 represent two loading cases) . Dimensions in centimeters 45 3.6 Skeleton structure of a helicopter tail. Dimensions in centimeters 47 3.7 Two bar planar truss. Dimensions in centimeters.. 68 3.8 Design space for the two bar truss obtained in the output error approach 69 3 . 9 Design space for the two bar truss obtained in the equation error approach 70 3.10 Design space for the two bar truss obtained in the output error approach, and using a reduced set of measurements 71 3.11 Design space for the two bar truss obtained in the equation error approach, and using a reduced set of measurements 72 X

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3.12 Three design variable rod-membrane structure. Dimensions in centimeters 73 3.13 Contours of equal objective function values for the three design variable membrane-rod structure, and using the equation error approach 74 4.1 A forty-eight bar space truss. Dimensions in centimeters 79 4.2 Equivalent beam model for the forty-eight bar space truss. Dimensions in centimeters 80 4.3 Flowchart illustrating an implementation of the dominant design variable strategy 91 5.1 Composite laminate subjected to extensional forces 119 5.2 Composite laminate subjected to bending loads.... 120 5.3 Two dimensional finite element model of an eight layer E-glass/epoxy laminate subjected to extensional force. Dimensions in centimeters 121 5.4 Two dimensional finite element model for the [0,90]s laminate subjected to pure shear loads Dimensions in centimeters 124 5.5 Two dimensional finite element model for the antisymmetric [0,90] E-glass/epoxy laminate subjected to bending loads. Dimensions in centimeters 128 5.6 Two dimensional finite element model of a 24 element laminate subjected to bending forces. Dimensions in centimeters 130 5.7 Two dimensional finite element model of a 24 element cantilevered laminate subjected to extensional forces. Dimensions in centimeters.... 132 5.8 Two dimensional finite element model of a 24 element cantilevered laminate subjected to bending moments. Dimensions in centimeters 133 5.9 3-D finite element model of the region of damage of a 4 layer composite structure. Each layer corresponds to one element. Dimensions in centimeters 136 xi

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5.10 Two dimensional finite element model of a [0,903], eight layer E-glass/epoxy laminate. Dimensions in centimeters 138 5.11 Two dimensional finite element model of a 24 element laminate. Dimensions in centimeters 141 5.12 First mode shape of a cantilevered composite beam 5.13 Curve resulting from the superposition of the first three mode shapes of a cantilevered composite beam 143 6.1 Original idealized model for the twelve bar planar truss. Dimensions in centimeters 149 6.2 The twelve bar planar truss 150 6.3 Modified idealized model for the twelve bar planar truss. Dimensions in centimeters 151 6.4 Close-up view of a joint in the truss structure.. 152 6.5 The truss mounted in the MTS machine 154 6.6 Arrangement of dial indicators to characterize the behavior of a joint 155 6.7 New model for the twelve bar truss. Dimensions in centimeters 163 6.8 Isolated view of the model representing a joint 164 xii

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRUCTURAL DAMAGE ASSESSMENT USING IDENTIFICATION TECHNIQUES By Francisco Jose da Cunha Pires Soeiro May, 1990 Chairman: Dr. Prabhat Hajela Major Department: Aerospace Engineering, Mechanics and Engineering Science The present investigation describes an approach for structural damage assessment that has its basis in methods of system identification. Response of a damaged structure differs from predictions obtained from an analytical model of the original structure, where the analytical model is typically a finite element representation. Output error and equation error methods of system identification are employed to determine changes in the analytical model necessary to minimize differences between the measured and predicted response. Structural damage is represented by changes in element stiffness matrices resulting from variations in geometry or material properties of the structure during damage. Measurements of static deflections and vibration modes are used in the identification procedure. A rational xiii

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basis for the proper selection of eigenmodes and loading conditions for the identification process is developed. Representative structural systems made of isotropic as well as orthotropic materials are solved to validate the approach. An experimental setup consisting of a simple planar isotropic truss structure was constructed to demonstrate the applicability of the approach to actual structures. The computational efficiency aspects of the problem are also examined. This includes a study of methods to reduce the number of design variables in the resultant optimization problem. xiv

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CHAPTER 1 INTRODUCTION 1.1 Motivation and Objectives Structural systems in a variety of applications including aerospace vehicles, automobiles, and civil engineering structures such as tall buildings, bridges and offshore platforms, accumulate damage during their service life. In several situations, such damage may not be visually observable. From the standpoint of both safety and performance, it is desirable to monitor the occurrence, location, and extent of such damage. System identification methods, which may be classified in a general category of nondestructive evaluation techniques, can be employed for this purpose. Using experimental data, such as eigenmodes and static displacements, and an analytical structural model, parameters of the structure such as its mass, stiffness and damping characteristics can be identified. The approach developed in this investigation is one where the structural properties of the analytical model are varied to minimize the difference between the analytically predicted and empirically measured response. In this process, the number of equations describing the system is typically different

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2 from the number of unknowns, and the problem reduces to obtaining the best solution from the available data. The genesis of identification methods can be traced to the model determination and model correction problems. The underlying philosophy in these efforts was that a reasonably close analytical model of the structural system was available, and that deviations in the analytical response from the measured response could be used to implement corrections in the model to account for these variations. This resulted in the adoption of a standard strategy wherein the change in the analytical model was minimized to obtain a match between analytical and empirical data. For the class of problems for which it was intended, the method has enjoyed a fair degree of success. A large body of identification work has been based on measurements of eigenmodes of the structure, and this is philosophically the right approach as eigenmodes reflect the global behavior of the structure. However, some reservations about this approach do exist, as these modes are sensitive to the physical boundary conditions that exist on the experimental model, and are not accounted for in the analytical model. It is therefore important to account for these differences and provide a corrected analytical model for the identification procedure. The problem of damage assessment in structures by identification methods is similar to the one described

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3 above. However, the approach of minimizing changes in the analytical model is no longer applicable, as significant variations can be introduced locally as a result of damage in a structural component. In the present work, damage in a structure is represented by a reduction in the elastic stiffness properties of the material, and these are designated as design variables of the optimization problem. Both static structural deflections and eigenmodes are used in the identification process. The use of static structural displacements as the measured response is a departure from the standard practice of using vibration modes. Since these displacements are reflective of the applied loading, an auxiliary problem in their use is one of determining the load conditions which are best suited for a global model identification. Hence, the primary objective of this investigation is to develop a methodology based on identification techniques, which would allow for detection of damage in structural systems. The focus of the effort resided in the proposition and validation of methods that are best suited for structural damage detection. Hence, the problems for which the proposed methods were implemented were simplistic, isotropic or composite structural systems. However, considerations of computational efficiency were also examined, resulting in the development of approximate

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4 strategies that can be implemented in more realistic, largescale structures. 1.2 Literature Survey The process of correlating a measured response with that obtained from an analytical model, and to use the difference to determine a modified analytical model that would better predict the response, can be broadly categorized in the realm of system identification methods. This subject is not new, and several studies pertinent to the field can be found in the literature. It is important to describe some of these studies in more detail, as they are fundamental to an understanding of how this technique is applicable in damage assessment strategies. Herman and Flannelly [1] proposed an approach in which the parameters of an assumed discrete model were modified to obtain analytical predictions which were in agreement with measured normal modes. This assumed model represented the dynamics at some preselected points of interest in the structure. To this extent, the model may be regarded as incomplete. This approach can be included in the category of identification techniques where experimental data is employed directly to construct a mathematical model without recourse to an a priori model. Baruch and Bar-Itzhack developed a technique to optimally correct measured modes [2]. Usually, and for a

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variety of reasons, the measured modes do not satisfy the theoretical requirement of weighted orthogonality. The corrected modes are closest to the measured ones in a weighted Euclidean sense. The analytical mass matrix is assumed to be accurately known and is considered the basis to correct both the measured modes and the stiffness matrix. To identify the stiffness matrix, an a priori analytical model is constructed. The corrected matrix is one which deviates the least from this analytical model, where the deviation is measured in a least squares sense. The corrected matrix should also satisfy the orthogonality conditions with respect to the corrected modes. Berman [34] used a similar approach to modify the mass matrix in order to be consistent with the measured modes, where the latter are assumed to be correct. Along the same line Baruch [5-7] developed methods of "Reference Basis," where depending on the degree of confidence in the analytical model available, either the mass, or the stiffness matrix, can be used as a reference to correct the other one. Experimental data can be a reduced set of modes or even a selected number of static displacements. Chen and Garba [8] proposed a matrix perturbation method where the sensitivity of eigenvalues and eigenvectors with respect to the model parameters are required. The sensitivity matrix is used to estimate the parameters. A prior knowledge of the analytical expressions for the mass

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and stiffness matrices is not necessary. The method is also based on modal tests. Berman and Nagy [9] developed a method for improvement of large analytical mass and stiffness matrices based on measured normal modes and natural frequencies. The method directly identifies, without iteration, a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements. Kabe [10] introduced a procedure where, in addition to modal data, structural connectivity information is used to adjust deficient stiffness matrices. The physical configuration of the analytical model is preserved and the adjusted model reproduces exactly the modes used in the identification process. Zero entries of the stiffness matrix are maintained as zero throughout the process, thus reducing the number of components to be identified. All the methods described above use modal data as measured response in the identification problem. A work by Sanayei and Nelson [11] was the only paper that could be found in the literature where only static response was used as measured data to identify the stiffness matrix. Forces and displacements, measured on a subset of the total degrees of freedom of the structure were used in this study. A "measured" stiffness matrix was determined by applying, one at a time, an unit load at each of the considered degree of freedom, and the difference between this matrix and the

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7 analytically predicted matrix was minimized using the least squares method. This study also included the effect of errors in force and displacements measurements. A survey of the principal existing structural system identification methods is presented by Hanagud, Meyyappa and Craig [12]. The terminology employed in the present work is based on this reference. The use of system identification techniques to detect damage in a structural assembly has been recently attempted, but with limited success. Chen and Garba [13] discuss the use of eigenmodes to determine changes in the structural stiffness matrix, with the underlying assumption that the mass matrix does not change with damage. The minimum deviation approach was employed, wherein the Euclidean norm of the matrix representing the perturbation of the analytical stiffness matrix due to the structural damage was minimized. The detection of damage was then attempted by examining the resultant entries of the perturbed matrix. This approach was only marginally successful as it is difficult to identify damage from changes in the stiffness matrix. Entries in the stiffness matrix depend on the elastic and geometric properties of the structure as well as on the element connectivity. Hence, a given element of the matrix may have contributions from several members sharing the same node, making it difficult to identify the precise location of damage. Furthermore, the minimum deviation

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8 approach used by the authors results in small deviations from an a priori model. In this kind of a problem, the a priori model is the original stiffness matrix corresponding to the undamaged structure. In the event of severe damage which may be located in different members of the structure, the changes in the stiffness matrix may be quite significant, increasing the possibility of obtaining poor results. These problems were clearly evidenced by the results obtained in the study. Smith and Hendricks [14] evaluate two different identification methods with respect to their applicability for damage detection in large space structures. The first system identification method was the one developed by Kabe [10], and is described above. The other method, due to White and Maytum [15], is based on matrix perturbation techniques and energy distribution analysis. Both methods use eigenmodes as experimentally measured data, and were employed to detect damage by examining the changes in the stiffness matrix corresponding to the damaged structure from the one corresponding to the undamaged structure. Similar difficulties are reported in their work. The entries of the stiffness matrix corresponding to the damaged members do show considerable variations. However, entries corresponding to undamaged members are also affected, thereby making the damage detection process more uncertain. The analysis of changes in the stiffness matrix is typically

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9 cumbersome, may not always yield correct answers, and does not permit the determination of the extent of damage. All the papers on damage detection referenced in the preceding discussion relate to isotropic structures. No previous effort describing the use of identification methods to assess damage in composite structures could be found in the literature. There is, however, a significant body of information dealing with nondestructive evaluation of composites, and various methods have been developed for this purpose. Some of these, including the ultrasonic C-scan, Xray radiography, and fiber optic crack detection systems, are very good at qualitatively locating damage in fiber reinforced structures. However, they lack the quantitative information necessary to predict stress redistribution and residual life of the structure. Towards this end, methods using vibration analysis [16-18], damping measurements [19], and stiffness-reduction mechanisms [20-22] have been proposed. The stiffness-reduction approach is particularly important, and comprises the basis for the method proposed in this work. As a laminated composite develops damage in one of the known mechanisms such as transverse cracking, delamination, fiber breakage, or fiber-matrix debonding, a reduction in stiffness is observed. Highsmith and Reifsnider [20] present an investigation on degradation of mechanical response of composite materials resulting from transverse

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10 cracking. O'Brien and Reif snider [21] study the influence of fiber breakage and fiber-matrix debonding on the stiffness of boron/ epoxy laminates, subjected to a fatigue loading. O'Brien [22] describes the onset and growth of delamination in terms of stiffness loss and relates both to strain energy release. In all these studies, the stiffnesses of the composite laminate are measured in typical specimens of composite materials. Various tests, each with a different test setup, are conducted to detect changes in different laminate stiffnesses such as £„, Eyy and G,^. 1.3 Scope of Present Work The present dissertation is organized into seven chapters. In Chapter 1, a clear statement of the problem studied, as well as the methodology employed, is presented. A survey of the most pertinent publications related to this work is also presented. Chapter 2 defines system identification and gives the typical formulations employed for structural system identification purposes. These formulations are presented in a general fashion and are developed in greater detail in subsequent chapters. The actual implementation of identification techniques for the purpose of detecting damage is described in the last section of this chapter. In Chapter 3 , the approach used in this investigation is developed and applied to representative structural models

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to assess its validity. The output error and the equation error approaches of system identification are applied to such structures using eigenmodes, static displacements, or a combination of both as measured response. A rational method to select a convenient static loading to better detect damage in a structure, and the use of modes to represent the displacement field obtained with the application of that loading, are also explored. The results are discussed at the end of each section. Chapter 4 deals with the computational efficiency aspects of the developed approach. Methods to reduce the number of design variables in the resultant optimization problem, in particular the use of a reduced dimensionality model and substructuring techniques are presented. Some approximation techniques that use gradient information to reduce the problem dimensionality are also proposed. Chapter 5 extends the proposed approach to composite structures. The stiffness-reduction method is described and the approach applied to simple composite structures. Results obtained in the implementation of the approach are presented at the end of the chapter. Chapter 6 reports the experimental work performed to demonstrate the applicability of the approach in actual structures. The design of the structure, the experimental setup, and the results obtained, are discussed. Shortcomings in the proposed strategy are also presented.

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Chapter 7 summarizes the principal conclusions of the present study, including recommendations for further study in this area.

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CHAPTER 2 SYSTEM IDENTIFICATION TECHNIQUES 1.1 The Problem of System Identification One approach of determining the veracity of analytical models is by comparing the response predicted by the model with that observed in tests or during operation. Although measurements are in themselves imprecise due to the presence of errors in the equipment or uncertainties in the data acquisition techniques used, reasonable bounds can be imposed within which the experimental data are expected to lie. The difference between the measured and analytical data may be large enough to be considered unacceptable. In this case, if there is sufficient confidence in the experimental data, identification methods can be invoked to improve or correct the analytical model. This is the basic problem studied in this work. For structural damage assessment, the same problem is seen from a different point of view as discussed later in this chapter. System identification involves a broad set of problems. Generally the problem of system identification is referred to as the determination of a mathematical model for a system or a process by observing its input-output relationships. The system model sought is the mathematical 13

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14 equation that relates the input to the output at all times. In order to obtain such a model, the system is subjected to a variety of inputs and its response to these inputs are observed. The input-output data are then processed to yield a maximum likelihood model. System identification problems or inverse problems can be found in various applications in science, medicine, and engineering. In such problems, the fundamental properties of a system are to be determined from the observed behavior of that system. Two categories of identification problems based on an a priori knowledge of the system can be readily identified. The first one is a complete identification of the model, a process which is initiated with little information about the basic properties of the system such as whether it is linear or nonlinear, memoryless or with memory, and so on. Usually some kind of assumptions have to be made before any meaningful solution can be attempted. The second category can be referred to as a partial identification problem where basic characteristics of the system such as linearity and bandwidth are known. This problem is easier to solve than the one involving a complete identification, and problem described at the beginning of this section, dealing with the veracity of an analytical model, belongs in this category. Most of research activity in identification is confined to this task, where the form of the model equations is already known. Hsia [23] classifies this problem as parameter

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15 identification since the model already exists and just its parameters are to be modified to make the results predicted by the model agree with the measured response. This type of problem is also referred to by different researchers as model improvement [8-9], model adjustment [10], or model correction [4-6]. In the present dissertation, the terminology used by Hanagud et al. [12] is followed. According to them, all the cases described above are classified as system identification problems. The system identification problem can be represented by a block diagram shown in Fig. 2.1. The procedure for carrying out the identification can be divided into the following steps: 1. Specify and parameterize a class of mathematical models that represents the system to be identified. 2. Apply an appropriately chosen test signal to the system and record the input-output data. 3. Perform the parameter identification to select the model in the specified class that best fits the statistical data. 4. Perform a validation test to see if the model chosen adequately represents the system with respect to final identification objectives. 5. If the validation test is passed, the procedure ends. Otherwise, another class of models must be selected and steps (2) through (4) performed until a validated model is obtained.

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DISTURBANCE I INPUT SYSTEM TO BE IDENTIFIED OUTPDT NOISE NOISE MEASURIHG INSTRUMENT HEASURABLE INPUT IDENTIFICATION TECHNIQUE MEASUEIABLE OUTPUT SYSTEM MODEL Figure 2 . 1 Block diagram representation of the system identification problem (Ref . 23) .

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17 If the model is already known, only steps (2) through (4) are performed. For parameter estimation there are a nximber of wellknown techniques such as methods of maximum likelihood, least-squares, cross-correlation, and stochastic approximation. A variety of other optimization methods have also been proposed. In the present work, the method of least-squares has been consistently used to construct the objective function required in the resultant optimization problems. Its simplicity and applicability to a wide range of situations are the reasons for the choice. It also exhibits statistical properties as good as those of the maximum likelihood method for most practical situations [23]. The weighted least-squares method can include a degree of confidence in the measured data. The leastsquares technique provides a mathematical procedure by which a model can achieve a best fit to experimental data in the sense of minimum-error-squares. If a variable y is related linearly to a set of n variables x = (Xi,X2, . . . ,x„) and P = (Pi/P2» • • • #Pn) is a set of constant parameters, then the following expression can be written. Y = PiXi + P2X2 + . . . + p„x„ (2.1)

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18 Here are the unknowns to be estimated by observing y and X at different times. A block diagram representation of the problem is shown in Fig. 2.2. X Parameters Figure 2.2 An n-parameter linear system. Assuming that a sequence of m observations has been made on both y and x at times t^, tg, . . . , t„/ and denoting the measured data as y(i) and Xi(i), XzCi),..., Xn(i) , i = 1, 2,..., m, one can relate these data by the following set of m linear equations. y(i) = PiXi(i) + P2X2(i) +...+ p„x„(i) i=l,2,...,m (2.2) The system of equations given by (2.2) can be expressed in a matrix notation as follows: y = X P (2.3) where , y = y(i) y(2) X = Xi(l) Xi(2) x„(l) x„(2) P = Pi P2 . y(m) . . Xi(m) x„(m)

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19 To estimate the n parameters p^, it is necessary that m > n. If m = n, P can be solved uniquely from (2.3) as follows: P = X'' y (2.4) provided that X'S the inverse of the square matrix X, exists; P denotes the estimate for P. When m > n, it is generally not possible to determine a set of p^ which exactly satisfies all m equations (2.2). This may be due to the fact that the data may be complicated by random measurement noise, error in the model, or a combination of both. The alternative then is to determine P on the basis of least-error-squares. The error vector e = {e^, • • • / «in)' is defined as follows. £ = y X P (2.5) The objective function to be minimized, denoted by J is given by the following expression. m J = E €i = e^£ (2.6) i=l Substituting (2.5) into (2.6), the objective function assumes the following form. J = (y X P)My X P) (2.7)

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20 Differentiating J with respect to P and equating the result to zero yields the estimate of P that extremizes J, as given by the following expressions. a J = -2 y + 2 X P = 0 (2.8) a p p=p X^ X P = X^ y (2.9) P = (X^ X)'^ X^ y (2.10) This result is referred to as the least-squares estimator of P. It is important to note that the expression is based on a criterion that weights every error equally. This formulation can be generalized to allow each error term to be weighted differently. If W is a desired weighting matrix then the weighted least-squares estimator is given by the following expression. P„ = (X^ W X)'^ X^ W y (2.11) There are two modes in which identification can be accomplished. One is off-line identification, in which a record of input-output data is first observed and the model parameters then estimated based on the entire data record. In on-line identification, the parameter estimates are recursively calculated for every data set. The new data obtained is used to update the existing estimates. If the process of data acquisition is very fast, it becomes

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possible to develop an on-line real-time identification technique. Identification techniques may be classified in many different ways. Such classification is typically based on the type of data used, on the type of system being identified, or on the type of formulation employed. Three of the more important formulations used in identification of structural systems are discussed in the following section. These are, the equation error approach, the output error approach and the minimum deviation approach. 2 . 2 T ypical Formulations for Structural System Identification In the equation error approach, equations describing the system response are explicitly stated. The system parameters, which are typically coefficients in such equations, are then selected to minimize the error in satisfying the system equations with a set of measured input-output data. Consider a linear differential equation represented in a functional form as follows. f (Pi/P2/X,t)=g(t) (2.12) Here, Pi and pg are considered as the unknown system parameters, and x(t) and its derivatives represent the system response at a time t; g(t) is the forcing function. The system response and the loading is explicitly measured over some characteristic time period T. An objective

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22 function, which is the measure of residual errors in the system equations for given values of the parameters, is formulated as follows. F = {|Tf(Pi,P2,x«>/t) g„(t))' dt } (2.13) This function is then extremized by differentiating with respect to each system parameter, and equating to zero. The approach results in the same number of equations as coefficients to be determined, and is therefore regarded as a direct method. In (2.13), subscript m denotes measured response, and T denotes the characteristic period over which measurements are made. The output error approach selects some system characteristic response as the entity for which a match between the analytical prediction and experimental measurements is considered to reflect a good analytical model. An objective function is formulated that is typically an averaged least-squares measure as follows. F = (x„{t) x(t))2 dt (2.14) The analytical model from which x(t) is obtained, contains system parameters which are adjusted to minimize the function F. In structural dynamics identification problems, system eigenmodes are generally selected as the

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23 characteristic response quantities used to identify the model . The minimum deviation approach is frequently used in structural identification problems. In this approach, deviation of the system parameters from initial assumed values is minimized, subject to the constraints that the system equations be satisfied. An illustration of this approach in structural applications is in the determination of changes in the elastic stiffness matrix [2]. In such applications, the mass matrix is assumed to be accurately defined. A weighted matrix norm of the difference between the a priori and corrected stiffness matrices is minimized in the identification process. This norm can be written as where M is the mass matrix, K is the desired stiffness matrix, and K is the a priori stiffness matrix. This minimization is subject to the constraint that the modified stiffness matrix remain symmetric. An incomplete set of eigenmodes are measured, and these are required to satisfy the eigenvalue equation and be orthogonal to the modified stiffness matrix. This results in the following equality constraints: F = M"^'^( K K ) M -1/2 (2.15) (2.16) K = (2.17)

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24 = K4>(1^ (2.18) In the above equations, 0 is an n x p modal matrix and is a diagonal matrix of eigenvalues for the p measured modes; n is the number of degrees of freedom for the structural dynamic system. The constraints are incorporated into the objective function by means of Lagrange multipliers, and the application of the optimality condition (2.8) yields a close form expression for the corrected stiffness matrix as follows. K = K K^^^M M^^'^ + M^(^'^<^^^M + M<^n^(^^M (2.19) 2 . 3 Structural Damage Assessment In a finite element formulation, the characteristics of the structure are defined in tezrms of the stiffness, damping and mass matrices K, C, and M, respectively. Any variations in these matrices such as may be introduced by damage, would affect the dynamic response characteristics of the structure. A change in the stiffness matrix alone would influence the static displacement response. In the present work, simulated measurements of structure eigenmodes and static deflections under prescribed loads, were used for identification of structural damage. The analytical model describing the eigenvalue problem for an undamped system can be stated in terms of the system

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25 matrices defined above, the i-th eigenvalue Wi^, and the corresponding eigenmode qi as follows. ( K M ) qi = 0 (2.20) Matrices K and M may be adjusted to minimize the differences between the experimentally observed eigenmodes and values obtained from the analytical model described above. In using static deflections for identification purposes, the analytical model is even simpler, involving only the system stiffness matrix K as follows. K X = P (2.21) Here x is a vector of displacements under applied static loads P. Although it is very clear from the previous equations that a variation in the system matrices results in a changed response, it is more important from a damage assessment standpoint to relate these differences to changes in specific elements of the system matrices. Since internal structural damage does not generally result in a loss of material, one can assume the mass matrix to be a constant. Typically, the stiffness matrix can be expressed in terms of the sectional properties of the various elements such as the cross-sectional area A, and the bending and polar moments of inertia I and J. There is also a dependence on element dimensions denoted by t and L, and on the extensional and

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26 shear moduli E and G, respectively. These dependencies may be stated in a functional form as follows. Kij = Kij(A,I,J,L,t,E,G) (2.22) In the present work, the net changes in these quantities due to damage is lumped into a single coefficient d^ for each element, that is used to multiply the extensional modulus Ei for that particular element. These d^'s constitute the design variables for the optimization problem. If the measured and analytically determined static displacements or eigenmodes are denoted by and Y^, respectively, the optimization problem can be formulated as determining the vector of design variables d^ (and hence the analytical stiffness matrix) that minimize the scalar objective representing the difference between the analytical and experimental response, and stated as follows. S E (Y„'J -Ya'^)^ (2.23) i j Here i represents the degree of freedom and j denotes a static loading condition or a particular eigenmode. This minimization requires that be obtained from the eigenvalue problem or the load deflection equations, using the K matrix that must be identified. Lower and upper bounds of 0 and 1 were established for the design variables di. This is the output error formulation of the damage assessment problem. Most of the work developed in the

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27 present investigation was based on this approach. It has the advantage of comparing only two vectors. These vectors may contain any system response, including a combination of different types of variables. The equation error approach was also used but only for the static load problem. In this case, the governing equation is very simple, with only one output vector. This results in a very good approach for damage detection purposes, which, in certain applications, presents some advantages over the output error approach. This will be discussed in a subsequent chapter. The error function to be minimized in this case is given as follows. S ( S KijXj f, (2.24) i j The minimum deviation approach for damage detection has been used by different researchers as discussed in the literature survey. However, this approach leads to a solution close to the a priori analytical model which may not be true when the structure contains severe damage. Because of this fact and the difficulties reported in the reviewed papers, this approach was not attempted in the present effort.

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CHAPTER 3 DAMAGE ASSESSMENT IN STRUCTURAL SYSTEMS 3 . 1 Introduction The method for damage assessment presented in section 2.3 was applied to a series of representative structural models with simulated damage in some components. Both the output error approach and the equation error approach were used to identify this damage, and these results are presented in subsequent sections of this chapter. As described in the previous chapter, these approaches result in the formulation of an unconstrained minimization problem. In the present work, the Broydon-Fletcher-Goldf arb-Shanno variable metric method [24-25] was used for function minimization. This approach has been shown to be efficient in the solution of large unconstrained optimization problems where function gradients are available as finite difference approximation [26]. A finite element analysis program EAL (Engineering Analysis Language) was used for response analysis [27]. The equality constraints of the optimization problem are actually implicit. They are represented by the equilibrium equations in static structural analysis or equations describing the eigenvalue problem in a dynamic problem, and are satisfied automatically when the response 28

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analysis is performed. For function minimization, the general purpose optimization program for engineering design ADS (Automated Design Synthesis) was employed [28]. The numerical procedure was implemented on a VAX 11-750 system and the flow between the various processors was controlled in the Command Language feature of DEC systems. The flowchart of this process is shown in Fig. 3.1. The simulated measured data was the finite element solution obtained for the damaged structure corrupted by a random noise signal. 3 . 2 Output Error Approach in Damage Detection 3.2.1 Use of Eiqenmodes as Measured Structural Response In the limited literature available in the field of identification based damage assessment, eigenmodes have been traditionally used as the measured response. They closely characterize the global behavior of the structure and can be applied to dynamic systems, including unrestrained space structures. Using this approach, the problem can be stated as finding a vector of design variables d (and hence the analytical stiffness matrix) that minimizes the quantity i j where i represent the degree of freedom, and j denotes the particular eigenmode. The measured and analytically determined eigenmodes are denoted by Y„ and Y^,

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30 START i FINITE ELEMENT ANALYSIS ANALYTICAL RESPONSE CModes or displacements] I OPTIMIZER OPTIHIZER OUTPUT (Nev set of di' s) X RESULTS CONVERGED? ASSUMED FEM MODEL or THE STRUCTURE EXPERIMENTAL DATA Figure 3 . 1 Flowchart of the damage assessment procedure .

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31 respectively. This minimization requires that be obtained from the eigenvalue problem, ( K M ) = 0 (3.2) where = , and K and M are the system stiffness and mass matrices, respectively; is the i-th eigenvalue, and 4>^ is the corresponding eigenmode. In this expression, K is the matrix to be identified and is dependent upon the design variables d^, for which lower and upper bounds of 0 and 1 were established. These lower and upper bounds represent the limits of a completely damaged member and an undamaged member. The approach was applied to a series of test problems and results are summarized as follows. The first case is that of a seven bar planar truss structure shown in Fig. 3.2. It is a statically determinate truss, and all structural members have a cross sectional area of 14 cm^. For this structure damage in members 3 and 5 was introduced by reducing their Young's moduli to 40% of the original value. The first three eigenmodes were employed in the formulation of the objective function for damage detection. Table 3.1 shows the results obtained in this exercise, and from which the location of the damaged elements can be easily identified. Furthermore, the extent of that damage is also assessed with reasonable accuracy. The same structure was damaged in element 5 by reducing the Young's

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32 modulus to 10% of its original value. This is in fact, a very severe damage for a statically determinate structure. Table 3.2 shows the effect of including a progressively larger number of eigenmodes on the results for damage detection. If the number of measured eigenmodes were less than or equal to four, the method was unable to detect damage . The second example chosen was a larger, statically indeterminate truss structure shown in Fig. 3.3. The cross sectional area of its members was also selected as 14 cm^. For this twenty-five planar truss, damage was simulated by removing member 11 of the structure, which entails setting the value of the Young's modulus for this member to zero. The results obtained in this exercise are presented in Table 3.3. 3'2.2 Use of Static Displacements as Measured Structural Response The use of static structural displacements as the measured response is a departure from the standard practice of using eigenmodes alone for the identification problem. In the previous section, it was indicated that when eigenmodes alone were used for identification, the location and extent of damage predicted by the optimization approach was dependent on the number of modes used to match the measured and the predicted response. Higher modes are often difficult to measure, and only the first few modes are

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Figure 3,2 A seven bar planar truss. Dimensions in centimeters.

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34 Table 3 . 1 Results for the seven bar planar truss Element No. Design variables (d^) using three eigenmodes Value Exact Sol 1 0.98518 1. 00000 2 0.86645 1.00000 3 0. 36214 0.40000 4 0.87894 1. 00000 5 0.35088 0.40000 6 0.96098 1.00000 7 0.87421 1. 00000

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35 Table 3 . 2 Results obtained for the seven bar planar truss, using an increased number of eigenmodes Element No. Design variables ( d^ ) using eigenmode response 4 modes 6 modes 7 modes Exact sol 1 1. 0000 0.6790 0.8706 1.0000 2 0.4205 0.8882 0.8690 1. 0000 3 0.9251 0.8944 0.8892 1.0000 4 0.8614 0.9022 0.8677 1.0000 5 0.7650 0. 0890 0.0860 0. 1000 6 1. 0000 0.7862 0.8714 1.0000 7 0. 4629 0.8085 0.8780 1.0000

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Figure 3 . 3 A twenty-five bar planar truss. Dimensions in centimeters.

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Table 3.3 Results for the twenty-five bar planar truss using four eigenmodes No . Design variables (d^) using four eigenmodes value Exact Sol 1 n Q O K 1 ^ u • y Z D J o 1.00000 2 u • y J xoy 1 . 00000 3 u • y J J z u 1 r\ /\ /\ r\ r\ 1 . 00000 4 vj • J X xy D i . uuuoo 5 0.96094 1. 00000 6 0 . 87899 1. 00000 7 0.96327 1.00000 8 1.00000 1. 00000 0. 87905 1. 00000 10 0.98613 1.00000 11 0. 00000 0. 00000 12 0.89017 1.00000 13 0.85975 1. 00000 14 0.91797 1. 00000 15 1. 00000 1. 00000 16 0. 85562 1. 00000 17 0.90889 1. 00000 18 0.92100 1. 00000 19 0.98278 1. 00000 20 1.00000 1. 00000 21 1. 00000 1. 00000 22 0.93679 1. 00000 23 0. 93863 1. 00000 24 0.94524 1. 00000 25 1. 00000 1. 00000

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38 typically available in actual practice. Static displacements provide a good alternative to the use of eigenmodes. The damage assessment problem using this approach can be stated in a manner similar to the one using eigenmodes alone. The objective function to be minimized is given by (3.1), where Y„ and are the measured and analytically predicted static displacements, respectively. The Ya vector, which was previously taken to be the eigenmode obtained from the eigenvalue problem expressed by (3.2), is now determined as the displacement vector from the equilibrium equations given by K X = P (3.3) where K is the stiffness matrix to be identified, P is the loading vector, and x is the response vector containing static displacements. Examples illustrating the applicability of this method are presented next. The twentyfive bar truss structure studied in the previous section was selected as the first example in an approach using static displacements. A vertical unit load was applied to the free-end nodes 11 and 12. The results of damage detection are presented in Table 3.4, showing clearly as in the case where eigenmodes were used, the location and the extent of damage. The more stressed adjacent members also showed some damage as they are directly influenced by the damaged one. This behavior is typically observed when

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39 static displacements are used for damage detection. The type and magnitude of the applied loading is a very important factor in determining the success of the damage detection strategy when static displacements are employed. Another example is the fifteen bar planar truss of Fig. 3.4. It is a statically indeterminate truss with member cross sectional areas of 14 cm^, and was damaged in members 1 and 13. The results are summarized in Table 3.5. Member 1 is the most stressed member in the structure, and critical to the overall structural integrity. That is the reason why it is easily identified. The stiffness of the diagonal member 13 is brought into play by load case 3, which when excluded from consideration, results in an incomplete detection of damage. A more rational approach to avoid problems such as incomplete damage detection due to poorly defined load set, is to choose a load condition that results in an equal stress distribution in each of the members. This will be discussed later in this chapter. Another representative example is that of semimonocoque wing box structure of Fig. 3.5 consisting of axial rod elements and membrane elements. Membrane element 2 was damaged by reducing its Young's modulus to 10% of the original value. Table 3.6 summarizes the results of this example. It is quite evident from these results, that the application of a torque to the box structure which requires

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Table 3 . 4 Results for the twenty-five bar truss using static displacements Element No. Design variables (d^) using static response Value Exact Sol . 1 1. 00000 1 . 00000 2 1.00000 1.00000 3 J. • u u uu u 1 ri n n f\ n 4 1. 00000 1.00000 5 0.99410 1.00000 6 0.74186 1.00000 7 1. 00000 1. 00000 8 1. 00000 1.00000 9 1. 00000 1.00000 10 0.98792 1.00000 11 0.01776 0. 00000 12 0.91020 1. 00000 13 0.99999 1.00000 14 0.94951 1.00000 15 0.99196 1.00000 16 0.73150 1. 00000 17 0.90617 1.00000 18 0.95281 1. 00000 19 0.98696 1.00000 20 1. 00000 1. 00000 21 0. 96812 1. 00000 22 0. 99026 1.00000 23 0.96977 1.00000 24 0.95638 1. 00000 25 0.99999 1. 00000

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41 Figure 3.4 A fifteen bar planar truss structure (Pi/ i=l,2,3 represent three loading cases) . Dimensions in centimeters.

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Table 3.5 Results for the fifteen bar planar truss using static response Element No. Design Variables ( c static response ii ) (Truss member Load 1 only Load 1 & 2 Load 1 2 & 3 Exact sol . 1 0.0266 0. 0257 0.0218 0.0000 2 1.0000 0.9999 0.9999 1.0000 3 1.0000 1.0000 1.0000 1.0000 4 0.7560 0.8366 0.8540 1. 0000 5 1.0000 0.9999 0.9730 1.0000 6 0.7230 0.7823 0.7817 1.0000 7 0.7358 0.9847 0.9999 1. 0000 8 1. 0000 1. 0000 1. 0000 1. 0000 9 0. 8841 0.9015 0.8962 1.0000 10 1. 0000 1. 0000 0.9698 1. 0000 11 0.9686 0.8235 0.7849 1.0000 12 0.8668 0.9933 0.9930 1.0000 13 1. 0000 1. 0000 0. 0868 0. 0000 14 1. 0000 1. 0000 0.7566 1.0000 15 1. 0000 1. 0000 0.9396 1.0000

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43 the membranes to participate more equitably in the load bearing process, improves the results for damage assessment. In a large structure, it is often difficult to obtain the displacements for all degrees of freedom. Typically, only a few dominant ones are measured, and these dominant displacements can be effectively used for damage detection. The objective function in this case becomes: S E (Y„'^ -Y/^)' (3.4) i j where i represent the degree of freedom where displacement was measured, and j denotes the particular load case. As an example, a model of a helicopter fuselage skeleton (Fig. 3.6) with 48 degrees of freedom was used. Damage was identified on the basis of measurements of twelve horizontal displacement components. Member 1 was damaged by reducing the Young's modulus to 10% of its original value. Table 3.7 summarizes the results for this example. The damaged member was clearly detected, as was the extent of damage. This method may not work in some cases, as the measured information may not be enough to identify the location of damage. Typically, this approach works for detection of damage in very stressed members, where the objective function shows a strong sensitivity to the variation of the design variables.

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44 3.2.3 Use of Combination of Eigeninodes and Static Displacements as Measured Structural Response The approach developed in the present work also allows the use of a combination of static displacements and eigenmodes as the measured structural response. The examples discussed in previous sections clearly demonstrated that a given static load distribution affects some structural components more significantly than others. If the damaged member is one that does not play an important role in the load bearing process for the set of applied loads, the use of static displacements in damage detection would yield erroneous results. The use of the combination of eigenmodes and static displacements is very easy to implement in the context of the proposed approach. The objective function, which is essentially a scalar quantity is written as a sum of two contributions, arising from the use of eigenmodes and static displacement response, respectively. The implicit equality constraints corresponding to these contributions are simultaneously satisfied. The fifteen bar planar truss of Fig. 3.4 with damage in members 1 and 13 is a good example to show the usefulness of this method. Four eigenmodes were initially used. The effect of adding static displacements corresponding to simple extensional load to the set of measured modes, shows a distinct improvement in locating the global damage. These results are summarized in Table 3.8.

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45 Figure 3.5 A semimonocoque wing box structure (P and P2 represent two loading cases) . Dimensions in centimeters.

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46 Table 3 . 6 Results for the semimonocoque wing box structure Element No. (panel) Connectivity (nodes) Design Variables ( d^ ) using static response Load 1 only Load 1 & 2 Exact sol . 1 1-7-8-2 1.0000 1.0000 1.0000 2 2-8-9-3 0.1240 0.1020 0. 1000 3 7-13-14-8 0.9601 1.0000 1.0000 4 8-14-15-9 0.7720 1.0000 1.0000 5 13-19-20-14 0.9585 1. 0000 1.0000 6 14-20-21-15 0.9837 0.9999 1.0000 7 4-10-11-5 0.9736 1.0000 1.0000 8 5-11-12-6 1. 0000 1.0000 1.0000 9 10-16-17-11 0.9954 1. 0000 1.0000 10 11-17-18-12 0.8713 1.0000 1.0000 11 16-22-23-17 0.9979 1.0000 1.0000 12 17-23-24-18 0.8984 1.0000 1.0000 13 1-4-10-7 0.9999 1.0000 1.0000 14 7-10-16-13 0.9998 1.0000 1.0000 15 13-16-22-19 0.9981 1.0000 1.0000 16 3-6-12-9 0.9926 0.9702 1.0000 17 9-12-18-15 1.0000 1.0000 1.0000 18 15-18-24-21 0.9855 0.9999 1. 0000

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47 Figure 3.6 Skeleton structure of a helicopter tail. Dimensions in centimeters.

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Table 3.7 Results for the idealized helicopter fuselage model Element No. Nodes Design variables (d^) Value Exact Sol . 1 1-5 0 . 10498 0 . 10000 2 1-6 1. 00000 1 . 00000 3 1-8 1. 00000 1. 00000 4 2-5 0 . 93199 1 . 00000 5 2-6 1 . 00000 1.00000 6 2-7 1 . 00000 1 . 00000 7 3-6 0 . 99781 1 . 00000 8 3-7 1. 00000 1 . 00000 9 3-8 0 . 99781 1 . 00000 10 4-5 0 . 93199 1 . 00000 11 4-7 1. 00000 1 . 00000 12 4-8 1. 00000 1. 00000 4^ • vy vy \j \j \j 13 5-6 1. 00000 1 . 00000 • vy vy vy ^y \y 14 5-8 1. 00000 1 ooono ^ • v/ \y vy W Vy 15 5-9 0.97783 1 nnnnn 16 5-10 0.99372 1 onnnn 17 5-12 0 . 99368 1 nnnnn 18 6-7 1 . 00000 1 nnnnn 19 6-9 0 98869 1 nnnnn X . u u u u u 20 6-10 W . ^ ^ \J \J X • u u u u u 21 6-11 1 nnnnn i. . UUUUU 22 7-8 1 nnnnn L . UUUUO 23 7-10 U . 3 y J. D / 1 . 00000 24 7-11 1 nnnnn T f\ f\ f\ l\ f\ 1 . 00000 25 7-12 0.99157 1.00000 26 8-9 0. 98858 1. 00000 27 8-11 1.00000 1.00000 Z o Q 1 O 0. 99991 1. 00000 29 9-10 0.99995 1. 00000 30 9-12 0.99995 1.00000 31 9-13 1.00000 1.00000 32 9-14 0.98484 1. 00000 33 9-14 0.98484 1. 00000 34 10-11 1. 00000 1. 00000 35 10-13 1. 00000 1. 00000 36 10-14 0.96628 1. 00000 37 10-15 1. 00000 1. 00000 38 11-12 1. 00000 1. 00000 39 11-14 0.98508 1.00000

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Table 3 . 7 continued Element No. Nodes Design variables (d^) Value Exact Sol. 40 11-15 1.00000 1.00000 41 11-16 0.98506 1.00000 42 12-13 1.00000 1.00000 43 12-15 1.00000 1.00000 44 12-16 0.96631 1.00000 45 13-14 0.99994 1.00000 46 13-16 0.99992 1. 00000 47 14-15 0.99996 1. 00000 48 15-16 0.99992 1.00000

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50 Table 3.8 Results for the fifteen bar planar truss using a combination of static and modal response Element No. Design Variables ( d^ ) using static and modal response (Truss member) First four eigenmodes Modes +load 1 Exact sol . 1 0.0000 0.0069 0.0000 2 0.7461 1.0000 1.0000 3 0.7571 0.9761 1.0000 A 4 0.7748 0.9503 1.0000 C 0.9934 1. 0000 1.0000 c D 0.8107 0.7515 1.0000 1 0.7517 0.9216 1.0000 8 0.9825 1.0000 1.0000 9 0.6603 0.8977 1.0000 10 0.7640 1. 0000 1.0000 11 0.8116 0.9930 1.0000 12 0.7763 1.0000 1.0000 13 0. 0006 0.0000 0.0000 14 0.7557 0.9433 1.0000 15 0.7623 1.0000 1. 0000

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51 3.2.4 A Rational Method for Structural Damage Detection As stated in section 3.2.2, the use of the output error approach with measured static displacements data can result in misleading results during damage detection. The displacement field is largely determined by the applied loading, and there might be considerable differences in magnitude of displacements at the various degrees of freedom. As a consequence of this, elements that undergo larger elastic displacements and are therefore more stressed, are easier to identify in the event of damage. Damage in lightly stressed elements, or in those that largely translate as rigid members during loading, is more difficult to identify. Although this is acceptable from the standpoint of safety, damage in the less critical member is likely to go undetected. In this section, a rational method to circumvent this problem is proposed. 3.2.4.1 Equal stress load distribution In light of the foregoing discussion, it is possible to assert that a static loading that produces equally stressed members is the one most likely to yield an accurate assessment of damage in the system. In a truss structure, such a load condition may be determined by the use of a superposition method. Consider the statically indeterminate, fifteen bar planar truss shown in Fig. 3.4. This structure has twelve degrees of freedom. Stresses in each member of this truss can be determined under the action

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52 of a unit load applied at each degree of freedom, one at a time. The total stress in each member is then obtained as follows: 12 = S Sij Cj , i=l,15 (3.5) where S^j is the stress generated in member i due to a unit load applied to the degree of freedom j and Cj are unknown coefficients. The stress vector can be represented in matrix notation by the following expression. a = S C (3.6) Each component of the stress vector can be set to a desired value, and the coefficient vector which would achieve such a stress distribution can be determined. If the structure is statically determinate, the problem has a unique solution, and can be written as follows. c = S"^ a (3.7) For the fifteen bar truss of Fig. 3.4, the number of equations is greater than the unknowns, and there is no exact solution to the equal stress load distribution problem. An approximate solution to this problem may be obtained by minimizing a Euclidean norm of the difference between the right and left hand sides of (3.6), following the procedure described in section 2.1. The final

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53 expression for c is similar to (2.10) and is given as follows. c = (S^S)'^ a (3.8) For the fifteen bar truss structure defined above, a stress level of 13.78 kPa was specified for each element. This stress level was obtained from the application of nominal loads to the structure. The desired coefficients, and hence the load distribution to achieve such a stress distribution are presented in Tables 3 . 9 and 3.10. This structure was damaged in two places, a critical horizontal element (member 1) at the point of load transfer to the boundary, and a less critical element (member 13) in an outer bay. The Young's moduli of the damaged members were reduced to 10% of their original values. An extensional force was applied at the free end of the structure as shown in Fig. 3.4 . The output error approach failed to detect damage in member 13 as seen in Table 3.11. The use of the equal stress loading presented in Table 3.9 allowed the detection of damage in both members as shown in Table 3.11. It is quite evident that such load conditions may prove quite unrealizable in actual practice, especially for complex structures with several degrees of freedom. Furthermore, the application of static loads would be impractical in unconstrained space structures, large civil engineering structures, and for the purposes of real-time

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54 monitoring of structural integrity in aeronautical structures. When used in an appropriate manner, the first few eigenmodes of the system still present the best global information to quantify the extent and location of damage. The following section describes an approach which combines the obvious benefits of activating all members by an equal stress distribution, with the actual use of measured eigenmodes in the identification problem. 3.2.4.2 Modal superposition approach The equal stress load distribution obtained in the previous section is significant for damage detection purposes as it favors no particular member. However, this type of loading is difficult to apply in a realistic structure. Instead, the approach adopted is one in which an unknown displacement field is assumed for this loading. The measured eigenmodes of the damaged structure are used as the basis vectors to expand the displacement field as yi = s Ck , i=i,n (3.9) where ^^i. is the i-th component of the k-th eigenmode, m is the number of measured modes, n is the number of degrees of freedom of the structure, and c^ are the unknown coefficients in the series expansion. Substituting into the equilibrium equation and premultiplying both sides by (f>l, the following expression is obtained.

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55 Ci^Ik^i + ...+ Ci^lK^i + ... + C„^Ik^„ = ^If (3.10) Orthogonality of the eigenmodes is invoked by requiring 4>l K 4>i = 0 T _ 2 ^^'^^^ 4>L ^ i — where are the eigenvalues of the structure. This yields an expression for as follows. Ci = lf/u>l (3.12) The objective function for the output error approach is then reformulated as the minimization of S iyj--yj^)^ (3.13) i Where y, is the analytically predicted response under the equal stress load distribution, and y„ is a function of the measured response ^i'' obtained from (3.9). The orthogonality conditions (3.11) are only valid if the eigenmodes are orthonormal with respect to the mass matrix. Hence, the measured modal matrix must be first orthonormal i zed with respect to the mass matrix to permit their use as a base for the displacement field [2]. The procedure is initiated by first normalizing each measured mode with respect to M as follows:

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Table 3 . 9 Equal stress load distribution for the fifteen bar truss (stress level = 13.78 kPa) Node d. o. f Load (N) 3 X -0.79 y 45.95 4 X -0.79 y -45.95 5 X 0.06 y 46.68 6 X 0.06 y -46.68 7 X 32.96 y 32.96 8 X 32.96 y -32.96

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57 Table 3.10 Member stresses in the fifteen bar truss for the loading shown in Table 3.9 Member Stress (kPa) 1 15.02 2 15.02 3 11.23 4 11.23 5 14.88 6 13.64 7 13.64 8 13.98 9 13.98 10 13.64 11 13.78 12 13.78 13 13.71 14 13.71 15 13.78

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58 Table 3.11 Results of the fifteen bar truss obtained using the output error approach Element number Load type Exact solution Extensional force Equal stress load 1 0.157 0.098 0.100 2 0.999 0.900 1.000 3 0.993 1.000 1.000 4 0.613 1.000 1.000 5 1.000 0.949 1.000 6 0.646 1.000 1.000 7 0.811 0.997 1.000 8 0.999 0.962 1.000 9 0.732 0.877 1.000 10 1.000 0.881 1.000 11 1.000 0.941 1.000 12 0.892 0.971 1.000 13 0.918 0. 134 0.100 14 0.956 1.000 1.000 15 1.000 0.900 1.000 V

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59 4,, = (4>I M ^J"''^ (3.14) where is the normalized mode. All n measured modes normalized as in (3.14) are put together into an n x m matrix ^, where m is the total number of degrees of freedom in the structure. The matrix ^ must further satisfy the orthogonality condition M ^ = I (3.15) for it to be used in the identification process. This is facilitated by assuming the existence of an n x m matrix X, and minimizing the weighted Euclidean norm, € = II M^/=^(X ) II (3.16) subject to the constraint as follows; M X = I (3.17) The constraint (3.17) can be incorporated into the objective function of (3.16) by the use of Lagrange multipliers, ^ = £ + A (X^M X I) (3.18) where A is the vector of Lagrange multipliers. The partial differentiation of (3.18) with respect to X, and equating the result to zero in a procedure similar to the one described earlier in section 2.1, yields the expression for the orthonormalized measured mode shapes.

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60 X = ^ (4>^K 4>y'^'^ (3.19) The approximation to the displacement field based on a limited number of measured eigenmodes is generally good. For the fifteen bar truss discussed earlier, the displacement field for equal stress loading and its approximation in terms of the measured eigenmodes, are shovm in Table 3.12. The same example, discussed earlier in this chapter, was used to demonstrate the concept of using measured eigenmodes to simulate deflections under the equal stress load distribution. An increasing number of modes were superposed to obtain the displacement field. Table 3 . 13 clearly shows the need for at least four eigenmodes to detect all damaged members. The overall results improved with the inclusion of more modes in the superposition. 3 . 3 Equation Error Approach in Damage Detection The equation error approach discussed in Section 2.2 was also implemented for damage assessment purposes. In this case, only static displacements were used as structural response for reasons of simplicity afforded by the governing equilibrium equation for static problems. This approach avoids an explicit decomposition of the stiffness matrix in the solution process. This feature appears to produce a better conditioned numerical space for obtaining solutions of the optimization problem. The Euclidean norm of the

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61 difference between the right and the left hand sides of the equilibrium equation was minimized as follows. S ( S KijXj f, (3.20) i j This expression is valid only for the case where all displacement degrees of freedom of the damaged structure are known. In the case where only a few displacements are measured, the system of equilibrium equations can be partitioned as follows: Ki2 1 ^22 . f2 . where Ui denotes the measured subset, U2 are the unknown displacement components, and K^, K12 and K22 are submatrices of the stiffness matrix. The unknown displacements can be eliminated to obtain a condensed equation. (K11-K12K22 ^Ki2^)Ui = fi-Ki2K22 ^fa (3.22) The objective function is then constructed in a manner similar to the case where all displacement components are known. The design variables for the optimization are the same as in the output error approach, and yield a new stiffness matrix. The optimum design variables result in a stiffness matrix that satisfies the equilibrium equations. It is worthwhile to note that the use of a reduced set of

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62 Table 3.12 Approximation of the displacement field using an increasing number of eigenmodes. The square Euclidean norm ||€||^ measures the error X and ] ^ components of displacements in cm Node d.o. f Exact 2 3 4 5 modes modes modes modes solution 3 X 1.69 1.67 1.75 1.77 1.63 y -0.85 -0.97 -0.85 -0.91 -0.65 4 X 0.26 0.25 0.18 0.15 0.16 y -0.51 -0. 62 -0.50 -0.56 -0.81 5 X 1.96 1.91 1.96 2.01 1.93 y -2.04 -2.12 -2.22 -2.17 -1.97 6 X 0.62 0. 66 0.55 0.47 0.46 y -2.00 -2.10 -2.26 -2.24 -2.43 7 X 2.09 2.02 2.08 2.22 2.39 y -3.13 -3.00 -2.85 -2.82 -2.69 8 X 0.70 0.86 0.83 0.72 0.92 y -3.14 -3.02 -2.92 -2.92 -3.14 2 0.6919 0.5710 0.4295 0.3305

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63 Table 3.13 Results for the fifteen bar truss obtained using output error approach. Increasing number of weighted eigenmodes used to model the displacement field obtained under the loading shown in Table 3.9 Element number Design Variables (dj 2 modes 3 modes 4 modes 5 modes Exact sol 1 0.100 0.120 0. 100 0.099 0.100 2 0.689 0.994 0.886 0.960 1.000 3 1.000 1.000 1.000 1.000 1.000 4 0.979 0.831 1.000 0.991 1.000 5 1.000 0.999 1.000 1.000 1.000 6 0.891 1.000 1.000 1.000 1.000 7 0.718 0.572 0.825 0.911 1.000 8 1.000 0.999 1.000 1.000 1.000 9 1.000 1.000 1. 000 1.000 1.000 10 1.000 1.000 1.000 1.000 1.000 11 1.000 1.000 1.000 1.000 1.000 12 0.999 0.999 1.000 1.000 1.000 13 0.741 0. 615 0.219 0.128 0.100 14 1.000 1.000 1.000 1.000 1.000 15 1.000 1. 000 1.000 1.000 1.000

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64 measurements requires a matrix inversion. Consequently, the advantage of the equation error approach over the output error is limited in such situations. The equation error approach is particularly useful in problems where the region of damage is known, and can result in a significant reduction in the number of design variables. For this case, the equilibrium equations can once again be written in the partitioned form shown in (3.21). Here, the submatrix Kn corresponds to the substructure containing damage. The first row of this partitioned system yields KiiUi + K12U2 = fi (3.23) which allows the damage detection problem to be reduced to determining the design variables d^ for the submatrix so as to satisfy (3.23). The equation error approach was used to detect damage in each of the structures examined by the output error approach in Section 3.2 . The method yielded good results, and in some cases, outperformed the output error approach. An impressive example of this performance is the detection of damage in the fifteen bar truss discussed earlier, and for which the correct results were obtained in the output error formulation only when member 13 was forced to participate in the load bearing process by the load case 3. The equation error approach yielded good results for an arbitrarily chosen extensional end load on the structure.

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65 such as load case 1, shown in Fig. 3.4. The results for this case are presented in Table 3.14. This difference can be largely attributed to the scaling of the design space that naturally emerges from the two formulations. The use of appropriately weighted eigenmodes or special load functions in the output error approach scales the design space so that it is easier for the numerical optimization technique to locate the optimum design. The equation error formulation does not require an explicit decomposition of the stiffness matrix, and for simple structures, presents an almost quadratic design space. 3 . 4 Design Spaces Corresponding to the Studied Formulations The design space for the two described approach can be visualized by studying the simple two bar truss of Fig. 3.7, in which damage was simulated by reducing the Young's moduli of members 1 and 2 to 50% and 40% of the original values, respectively. Fig. 3.8 and 3.9 show the design spaces for the two approaches. The equation error approach presents a slightly more convex design space with a better defined optimum. For larger structures this might be of significance in correctly locating the damage. If a reduced set of measured displacements is employed in damage detection, the design space for both methods present several local minima that may lead the optimizer to the

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66 wrong solution. Figures 3.10 and 3.11 illustrate the design space behavior for this case. An example of a damage detection problem in which the design space has local optima is obtained for the three design variable membrane-rod structure of Fig. 3.12. Contours of equal objective function values obtained in the equation error approach, with damage in members 1 (rod) and 3 (membrane), are shown in Fig. 3.13 . This damage was simulated by reducing the respective Young's moduli by 40% and 50% of the original values. As seen in Fig. 3.13, the design space offers the possibility of convergence to local optima, and such a convergence would be dictated by the choice of starting design variables. Table 3.15 shows the optimum solutions for two different initial designs.

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67 Table 3.14 Results for the fifteen bar planar truss using the equation error approach Element Design Variables Exact nuinber di solution 1 0.099 0.100 2 1.000 1.000 3 0.853 1.000 4 0.949 1.000 5 0.912 1.000 6 0.872 1.000 7 0.978 1.000 8 0.965 1.000 9 0.996 1.000 10 1.000 1.000 11 0.926 1.000 12 0.977 1.000 13 0. 153 0.100 14 0.986 1.000 15 0.970 1.000

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68 Figure 3.7 Two bar planar truss, centimeters. Dimensions in

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t 69

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70

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71 Figure 3.10 Design space for the two bar truss obtained in the output error approach, and using a reduced set of measurements.

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.4 1^ .6 .8 E(l) Figure 3.11 Design space for the two bar truss obtained in the equation error approach, and using a reduced set of measurements.

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73 Figure 3.12 Three design variable membrane structure. Dimensions in centimeters.

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74 Figure 3.13 Contours of equal objective function values for the three design variable membrane-rod structure, and using the equation error approach.

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Table 3.15 Results for the three design variable membrane-rod structure for two sets of initial designs Element Initial Design Exact number design variables solution 1 1.0 0.375 0.400 2 1.0 0.933 1.000 3 1.0 0.789 0.500 1 0.6 0.411 0.400 2 1.0 0.929 1.000 3 0.6 0.521 0.500

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CHAPTER 4 COMPUTATIONAL EFFICIENCY ASPECTS IN STRUCTURAL DAMAGE ASSESSMENT 4 . 1 Introduction The focus of previous chapters was the proposition and validation of an approach for damage assessment using identification techniques. The examples presented in those chapters dealt with simplistic structures, and were intended to show the key findings of the investigation. However, if a more realistic structural system is to be examined for damage, the associated computational effort would increase considerably. According to the proposed approach, there are as many design variables as the number of elements in the finite element model of the structure. This results in significant computational costs when using a gradient-based nonlinear programming algorithm for function minimization. Two distinct strategies for reducing the number of design variables were studied in the present work. One was based on the use of reduced dimensionality models, where equivalent structures with fewer degrees of freedom were constructed to represent the actual structure. The second approach was one in which only a dominant subset of the design variables was considered at any stage of the error 76

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77 minimization problem. These two approaches are discussed in subsequent sections of this chapter. 4 . 2 Reduced Dimensionality Models The construction of a reduced dimensionality equivalent continuum model depends primarily on the type and shape of the structure under consideration. Beams are a natural approximation for trusses, allowing the substitution of the actual structure by one with fewer number of degrees of freedom. Use of equivalent structures to facilitate the analysis and quickly determine the response characteristics of the structures such as natural frequencies and vibration modes is commonly adopted in engineering practice. Aerospace structures such as mast trusses and precision trusses can be represented by equivalent beams. Consider a truss model of a mast structure shown in Fig. 4.1, subjected to a tensile load (Pa=1 N) , a transverse load (Pt=1 N) , and a torsional load (T=l N-m) . The Young's modulus of each rod was considered as unity, and a nominal value of 10.47 cm^ was assumed for all cross sectional areas. An equivalent beam model (Fig. 4.2) with an independent axial, bending and torsional stiffness for each section, was obtained to simulate the deflection characteristics of the truss structure. Each section of the beam corresponds to a bay in the truss. The reduced dimensionality beam model with only 4 design variables, was first used to determine the

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78 approximate location of damage. The model of the truss structure had a total of 48 design variables. The identification problem was then solved for the beam model, using interpolated displacements obtained from measurements on the damaged truss. These interpolations were necessary due to the difference in the number of degrees of freedom for the two models. In such an analysis, the beam section corresponding to the bay of the truss containing the damaged members is first identified. This is a preliminary identification problem with a fewer number of design variables. Once the region of damage is identified, the actual structure is then considered. The design variables for this secondary identification problem are the parameters di corresponding to the members of the damaged bay(s) . If only one member is damaged as in the example of Fig. 4.1, the problem reduces to one with 15 design variables. The procedure described above was applied to the truss of Fig. 4.1. Damage was introduced in members number 12 (2°"* bay) and 40 (4''*' bay) by reducing the respective Young's moduli to 10% of the original values. The first step was to calculate the response of the undamaged structure to the applied loads. This information determines the geometric characteristics of the equivalent beam (Fig. 4.2). For example, to determine the cross sectional areas of the different elements in the equivalent beam, the extensional displacements are used. Based on these displacements, an

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79

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80 Figure 4.2 Equivalent beam model for the forty-eight bar space truss. Dimensions in centimeters.

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81 averaged length variation AL for each bay can be calculated. This yields the value of the equivalent area S of one particular bay, given by the following expression. P„ L S (4.1) E AL Here, E is the Young's modulus and may be assumed as unity; Ph is the resultant normal force acting on the bay. Likewise, bending moment of inertia for the equivalent cantilever beam can be calculated from the following expression, F I = (X 3L) (4.2) 6 Ey where F is the resultant transverse force applied at the free end of the truss, x is the distance from the support of any node in the beam, L is the length of the beam, and y is the transverse displacement of a node in the beam obtained from the truss response. The Young's modulus E can again be taken as unity. This expression will usually give different values for the moment of inertias corresponding to the different elements of the beam, since the displacements used are the ones obtained from the truss. One can take the average of all such computed values, and use it as the approximate moment of inertia of the equivalent beam. For

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82 more precise values, the following differential equation can be solved for each element of the beam, d^y M — = (4.3) dx^ E I where M is the bending moment acting on that element. The solution of this equation at each element will satisfy the continuity conditions of a beam and the displacement field determined from the analysis of the truss. This way, an exact moment of inertia can be calculated for each element of the beam. To calculate the polar moment of inertia J at each element of the equivalent beam, the following expression was used. T L J = (4.4) G he In this expression G is the shear modulus and can be taken as unity; T is the torque applied to the element; L is the element length, and he is the difference between the rotations at the ends of an element. These angles are calculated from the transverse displacements obtained when a torque is applied at the tip of the truss. The above properties are independent, and represent a beam with a behavior close to the original truss. The second step involved the solution of the equivalent beam with only four design variables. The measured response

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83 of this structure corresponds to interpolated or averaged displacements obtained from the damaged structure. In Table 4.1, the results show considerable change in the final values of the design variables corresponding to the 2"*^ and 4'^ bays. Finally, the actual structure was solved considering as design variables the parameters d^ corresponding to the members of the damaged bays, as detected from the equivalent beam model. The final results for the actual structure are summarized in Table 4.2. 4 . 3 Substructurinq The last step of the problem discussed in the previous section is also amenable to substructuring techniques. Such an approach allows each bay to be treated separately. The system of equilibrium equations can be partitioned as follows. Ki2^ K22 The submatrix corresponds to the substructure containing damage. It is always possible to isolate the damaged portion of the whole structure in submatrix Kn by rearranging the system of equations (4.5) using properties of permutation. The first row of this partitioned system yields the following expression. * . U2 . :: (4.5)

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Table 4 . 1 Damage detection as seen in the equivalent beam model Element No. Design variables (di) Lower bound Value Upper bound 1 0.0000 0.9021 1.0000 2 0.0000 0.5849 1.0000 3 0.0000 0.9537 1.0000 4 0.0000 0.5890 1.0000

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85 Table 4 . 2 Results for the forty-eight bar space truss. Only members in 2"*^ and 4*^ bays are treated as design variables Design variables (d^) Element Nodes No. Value Exact sol 2"** bay 10 4-5 1.00000 1.00000 11 4-6 J. . UUUUU JL . UUUUU 12 4-7 0. 10589 0.10000 13 4-8 0.99999 1.00000 14 4-9 0.99999 1.00000 15 5-6 1.00000 1.00000 16 5-7 0.89710 1.00000 17 5-8 1.00000 1.00000 18 5-9 1.00000 1.00000 19 6-7 0.89694 1.00000 20 6-8 1.00000 1.00000 21 6-9 1.00000 1.00000 22 7-8 1.00000 1.00000 23 7-9 1.00000 1.00000 27 8-9 1.00000 1.00000 4*''' bay 34 10-11 1.00000 1.00000 35 10-12 0.96474 1.00000 36 10-13 1.00000 1.00000 37 10-14 0.99816 1.00000 38 10-15 1.00000 1.00000 39 11-12 0.97694 1.00000 40 11-13 0.11129 0. 10000 41 11-14 1.00000 1.00000 42 11-15 0.93723 1.00000 43 12-13 0.93341 1.00000 44 12-14 1.00000 1.00000 45 12-15 1. 00000 1.00000 46 13-14 1.00000 1. 00000 47 13-15 0.98714 1.00000 48 14-15 0.99159 1.00000

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86 KiiUi + K12U2 = fl (4.6) The damage detection problem is reduced to determining the design variables d^ for the submatrix K^. The equation error approach is particularly useful in this approach. The error vector e for this formulation is defined as follows. € = KiiUi + Ki2U2 fl (4.7) The objective function is then constructed as the squared Euclidean norm of the error vector and the procedure described in section 3.3 is adopted, yielding the following expression. 2 ( S K[i ui + S u^ f[ )2 i=j (4.8) i j k Here i includes all degrees of freedom of the substructure where damage is located, and j+k equals the total number of degrees of freedom in the structure. In the space truss example of the previous section where two bays were detected as containing damage, a separate solution for each bay is possible as they do not share any common degree of freedom. If the damaged bays had any common degrees of freedom, then all the design variables corresponding to the two bays would have to be considered simultaneously. In this case some of the entries in the stiffness matrix would have the contribution of the material properties of members belonging to the two damaged bays and

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87 no separation would be possible. It is also important to mention that this method, in order to be applicable, requires the knowledge of the displacements for all the degrees of freedom of the damaged structure, as indicated in (4.6). 4 . 4 Approximation Concepts As stated in earlier chapters, the mathematical formulation of damage assessment results in an unconstrained optimization problem. An efficient solution to this problem is most commonly obtained by one of the several gradient based approaches. Such methods require the derivative of the objective function (in this case the error function) with respect to each design variable. Computations of these gradients repetitively in the optimization process places severe demands upon the total computational requirements. In an effort to reduce this computational requirement, the approach adopted here used only a select number of dominant variables at any given stage of the optimization process, keeping other inactive members fixed at their current values. The dominant set of variables was revised after a fixed number of iterations. The selection of the dominant set was based on a measure which had the most influence in the computation of the search direction. For an unconstrained problem of the type under consideration, the objective function gradient emerges as the most obvious

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88 measure of the design variable dominance. In the present work, the Broydon-Fletcher-Goldf arb-Shanno variable metric was used for function minimization [24-25], Design variable dominance was assessed on the basis of the computed search direction given as follows: S** = -H FCX^) (4.9) where F(X'') is the gradient of the objective function at a particular iteration q, and H is a matrix initialized as the identity matrix and updated according to the BroydonFletcher-Goldf arb-Shanno formula, in order to give an approximation to the Hessian matrix. This set of dominant variables was used for a prescribed number of optimization cycles and then revised with a new assessment of dominance. These dominant design variables were determined as those for which the objective function had gradients equal or higher than 10% of the largest gradient. A flowchart for the present implementation is presented in Fig. 4.3. The number of cycles for which the optimizer was allowed to proceed with a given set of dominant variables proved to be somewhat crucial in avoiding convergence to a local optimum. If that convergence were attained, the method would in some situations be unable to proceed, even with a full set of design variables. Determination of the number of such cycles was one of the shortcomings of the method, as it is difficult to know a priori the optimum number of cycles that

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89 the optimizer should execute before dominance is reassessed. A number of five, based on experience with the several examined structures, was used and yielded good results in some cases. However, even this rule-of -thumb was inapplicable in other problems. Convergence of the iterative process was based on no improvement in the final values of the objective function over three consecutive cycles. In the above discussion, each cycle involves the assessment of variable dominance by the calculation of the gradients of the objective function with respect to all design variables, reduction of the problem by selecting the dominant variables, and solution of the optimization problem with these selected design variables. This procedure was implemented for the twenty-five bar planar truss shown in Fig. 3.3. Damage was introduced by assuming member 14 of the structure to have failed completely. Table 4.3 shows the results obtained, indicating a reduction in the number of function evaluations, and hence a reduced computational effort. It is also interesting to note an improvement in the final values of the design variables. Another example where this method did not work very well, and actually resulted in an increase in the number of function evaluations, was the fifteen bar planar truss shown in Fig. 3.4. For this problem, damage was introduced in members 1 and 13 and four eigenmodes were used as measured

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90 structural response. This is the same example discussed in section 3.2.3. Table 3.8 shows the results obtained using the proposed method. The structure was also solved using the full set of design variables and yielded better results with less computational cost. The results are summarized in Table 4.4. Despite the fact that this method generally worked for damage detection, it failed to conclusively establish any advantage in terms of reduced computational effort for most of the cases in which it was applied. The approach does, however, merit further study to assess its effectiveness in damage detection.

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91 Figure 4.3 Flowchart illustrating an implementation of the dominant design variable strategy.

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Table 4.3 Results for the twenty-five bar planar truss using the dominant variables approach Element Design Variables ( ) No . using static response (Truss Reduced Full Exact member) set set sol . J. 1 . 0000 1 . 0000 i. . UUUU 1 /\ /\ /\ /\ /-I Q >l O C U . 74o3 i. . UUUU 1 f\f\f\f\ 1 . uuuu X . UUUU X . uuuu A 'i X . uuuu J. . uuuu X . uuuu 1 o n n n J. . uuuu u . OU J
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93 Table 4.4 Results for the fifteen bar planar truss using four eigenmodes as measured response and the dominant variables approach Element No. Design Variables ( d^ ) using four eigenmodes (Truss member) Reduced set Full set Exact sol. 1 0.0000 0.0036 0.0000 2 0.7461 0.9397 1.0000 3 0.7571 0.8212 1.0000 4 0.7748 0.8104 1.0000 5 0.9934 0.9745 1.0000 6 0.8107 0.8934 1.0000 7 0.7517 0.8800 1.0000 8 0.9825 0.8387 1.0000 9 0.6603 0.6745 1.0000 10 0.7640 0.8503 1.0000 11 0.8116 0.8138 1.0000 12 0.7763 0.8526 1.0000 13 0.0006 0.0000 0.0000 14 0.7557 0.8107 1.0000 15 0.7623 0.8653 1.0000 Number of function eval . 693 483

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CHAPTER 5 DAMAGE DETECTION IN COMPOSITE STRUCTURES 5 . 1 Introduction The use of composites materials in aerospace and automotive structures has increased significantly in the past several years due the distinct advantages they present. These include light weight, high strength, and an ability to tailor the structure for specific performance requirements. As the number of designs employing composite materials increases, a demand for more rational procedures to describe and explain the mechanisms of damage in such structures also grows. The determination of residual strength, stiffness, and life of composite materials is important so as to implement corrective action before a structural collapse. This is particularly useful if the behavior of the structure can be monitored during its service life by means of measurements of the structural response using any nondestructive technique. In the present work, the stiffness-reduction method [20-22] was employed and the idea of correlating stiffness losses to any kind of damage was extended to an actual structure. Measured response quantities such as static displacements, strains, and eigenmodes of the structure were 94

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assumed to be available. System identification techniques were then employed to determine changes in the analytical model which would provide a match between the predicted and experimentally measured data. The formulation is similar to the one developed for isotropic structures, discussed in the previous chapters. In the study involving isotropic structures, only one design variable was necessary per element in a finite element model of the structure. In the present case, at least four design variables are required for each two dimensional finite element. This number would be even higher if the laminate were not symmetric. The optimization problem turns out to be more difficult to solve, and the method is typically unable to determine the exact changes in all material properties. However, not all properties are needed for use as design variables in the identification problem. There are some dominant properties that depend on the loading applied to the structure or on the eigenmode used in formulating the identification problem. Changes in the values of the dominant property defines the location of damage as well as its severity. Analytical models based on both 2-D and 3-D finite element formulation can be used. For a more precise characterization of damage such as transverse cracking or fiber breakage in certain layers, the two dimensional element does not work well. Such a model only captures the overall behavior of the laminate, as the strains or

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displacements obtained along the thickness constitute a set of linearly dependent quantities. One approach to overcome this drawback is to modify the design problem and assign to each lamina its own set of design variables. In addition to a significant increase in design variables and the attendant increase in computational requirements, this results in a relatively difficult optimization problem with nonconvex spaces. The solution is only possible in some very particular cases. Hence, a three dimensional element which gives more information on the behavior of the laminate is necessary. This is discussed in a subsequent section of this chapter. Some representative examples using static displacements and eigenmodes as measured data are also presented. 5.2 The Stiffness-Reduction Method 5.2.1 Use of Static Displacements as Measured Structural Response 5.2.1.1 Theoretical background In the classical lamination theory [29], the resultant forces and moments acting on a laminate are written as Al2 A16 Al2 A22 A26 A26 ^66 . V r 1 B12 B16 My B22 B26 M . B26 Bee . 7" ' xy 'xy + Bii B12 B16 B12 B22 B26 B16 B26 Bge Dii D12 ^12 ^22 ^26 ^16 ^26 D56 "xy } "xy J (5.1) (5.2)

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97 where the N's and M's are the resultant in-plane forces and resultant moments, per unit length, respectively; A is the laminate extensional stiffness matrix; B is the laminate coupling stiffness matrix; D is the laminate bending stiffness matrix; €°'s and 7°'s are laminate mid-plane strains and /c's are mid-plane curvatures. The components of the stiffness matrices are defined as follows: k-l 1 N Bij = — 2 (Qij)i, (zl z^i) (5.3) 1 k=l Dij = i (Qij)k (Zk Zk-i) k=i where Q^j are lamina transformed reduced stiffness components, k is an index for a given layer and z's are distances from the mid-plane of the laminate in the direction perpendicular to the plane of the laminate; N is the number of layers. For symmetric layups, the coupling stiffnesses Bi^ vanish. The displacements u and v, in the x and y directions, respectively, at any point z through the thickness, are obtained from the Kirchhof f-Love hypothesis, as follows.

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98 U = Un Z V = Vn Z a Wo ax a Wo ay (5.4) The middle surface displacements Uq, Vq and Wq are related to the middle surface strains and curvatures by the following expressions. auo ax avo ~a7" auo "ay" + avo ax (5.5) f "\ (5.6) a'wo T dye a^wo ay^ a'wo • axay The foregoing analysis describes the effect of stiffness coefficient changes on the structural response. The stiffness-reduction method for damage detection uses the changes in laminate stiffnesses, which are measured quantities, to determine damage in the structure. A distinct experimental set-up is necessary to determine the

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99 stiffness quantities. As an example, to determine the laminate extensional modulus E^, a force resultant is applied to a specimen. For such a load, (5.1) can be written as r N 1 " An Al2 A16 • 0 Al2 A22 A26 (5.7) . 0 , . Ai5 A26 Aee . from which the strain in x direction can be calculated as follows . fx = AnN, (5.8) Here A is the inverse of the laminate extensional stiffness matrix. Additionally the resultant force in x direction can be written as 't = (7^ dx = t (5.9) Where t is the thickness of the laminate and the stressstrain relation is given as follows. '^x = fx (5.10) The combination of (5.8), (5.9) and (5.10), yields the expression for the laminate extensional modulus.

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100 Likewise other laminate stiffnesses are determined, each one requiring a special test setup. In the present work, a different way to apply the stiffness-reduction method is proposed. The measured response are strains or displacements obtained from the actual structure. The design variables are the components of the stiffness matrices A, B, and D, which are a function of the measurable laminate properties as shown in (5.11). The changes in these quantities are represented by a coefficient d^ used to multiply the stiffness matrix components. The resultant optimization problem is similar to the one derived for isotropic structures and can be stated as follows. Find a vector d that minimizes the quantity N M 2 S i Y'J Yi' )2 (5.12) i=l j=l where i represents the degree of freedom, j denotes the static load condition for which the displacement is obtained, and Y„ and are, respectively, the measured and analytically determined strains or static displacements. Here, the components of the design variable vector d^ are ratios of the true and original values of the structure stiffness coefficients. The number of design variables per element in a finite element representation of a laminate is a function of the layup. For symmetric cross-ply laminates subjected to in-

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101 plane loads, four design variables are necessary per element. These are A^, A12, A22 and AgeThe other entries in the A matrix vanish. For an angle-ply laminate under inplane loads, that number increases to six as no term in the A matrix can be set to zero. For non-symmetric laminates, coupling between extension and bending exists, indicating that the components of the B matrix must be considered in the design variable set. In the most general case, eighteen design variables per element would be necessary in the problem formulation. For a realistic structural system, the number of design variables would be unmanageably large. Hence, strategies to limit the number of design variables must be devised. When using static displacements as the measured response in the identification problem, the loading applied to the structure is particularly important, as it determines the critical laminate property. Usually only one dominant property per damaged element will show some change when this approach is adopted. The others have little influence on the final response of the structure, and typically go undetected even though they might have undergone significant changes. As was shown in previous chapters, detection of damage is sensitive to the stress distribution in the structure. The equal stress distribution load method, for example, was devised to overcome the problem of damage detection in less stressed members in a truss structure. In this case there

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102 was only one material property per element and the stress considered was the principal stress a^. In an orthotropic material, the stress-strain relations in plane stress are given as follows: °\ " Qii Ql2 0 f "\ «1 Ql2 Q22 0 * €2 • 0 0 Q66 . . 7l2 > where Q^j are the reduced stiffnesses of the material. If, for example, an in-plane load is applied in direction 1, the dominant stress present is a-^ and the dominant property is Qii. This can be shown using the strain energy stored in such a plate element. The energy per unit volume U for a uniaxial state of stress is given by the following expression. *> 1 U = ai €1 (5.14) 2 The strain in the direction perpendicular to the load, since a2 is very small, can be obtained by the Poisson effect as follows. ^2 = "12 (5.15) Using (5.15) in (5.13) one can obtain a^, as follows. = Qii fi + Q12 €2 = Qu ei + 1^12 Q22 ^2 (5.16)

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103 Substituting (5.15) into (5.16) and the resultant relation into (5.14) the following expression for the energy U is obtained in terms of the material properties and the strain Table 5 . 1 presents representative values of the terms in the energy expression for a few select composite materials. These figures clearly demonstrate the importance of in the energy expression. Similarly, for pure shear loads the dominant stress is and the related property is Qge* The design variables in the optimization problem defined by (5.12) are directly related to the transformed reduced stiffnesses Q^j as given by (5.3), and these are a function of the reduced stiffnesses Q^j and the fiber orientation e, as shown in the following expressions. Qii = Qn oosU + 2(Qi2 + 2Q66) sin^^ cos^^ + Q^z sin*5 Qi2 = (Qii + Q22 4Q66) sin^^ cos^e + Qi2(sin''^ + cos*^) 1 U = ( Qii + "12 Q22 ) «1 (5.17) 2 Q22 = Qii sin*^ + 2(Qi2 + 2Q66) sln^e cos^e + Q22 sin''^ Q16 = (Qn Q12 2Q66) sin e cos^e + + (Q12 Q22 + 2Q66) sin^^ cos e (5.18) Q26 = (Qu Q12 2Q66) COS 8 sin^e + + (Q12 Q22 + 2Q66) cos^^ sin e Qee = (Qu + Q :22 2Qi2 2Q66) sin^^ cos^e + + Q66(sin''tf + cos*^)

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104 Changes in the reduced stiffnesses, for prescribed values of e, affect directly the design variables of (5.12), allowing the selection of a set of dominant variables to reduce the problem dimensionality. Similar to the case of isotropic structures where damage in members with lower stress levels would go undetected, here only the dominant stress components would be identified. Subsequent sections of this chapter explore the effect of various loadings on the laminate properties, and provides guidelines on reducing the number of design variables that are necessary in the identification problem. The procedure described above is useful in determining the location and severity of damage. However, it is not sufficient to define what type of damage the laminate contains. Certain types of damage caused by degrading layers, such as fiber breakage and transverse cracking can be determined by system identification methods provided that more information on the behavior of the laminate is available. A two dimensional shell element is not adequate, as all the information obtained refers to the mid-plane and strains or displacements in different layers are obtained on the basis of a linear variation through thickness. The strains in the different layers are given by the following expression: /• -1 fx X r > s 'I • + Z ' (5.19)

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105 Table 5.1 Dominance assessment for equation (5.17) Property Composite Glass/Epoxy Boron/Epoxy Graphite/Epoxy Q22/Q11 0.33 0. 100 0.020 0.25 0. 300 0.250 ^^12 Q22/Q1I 0.02 0.009 0.001

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106 Here, the number of linearly independent measurements remains the same while the number of design variables increases dramatically due to the contributions of each ply. Three dimensional brick elements can also be used in this problem with good results. However, measurements of laminate response through the thickness must be available. The solution to the optimization problem defined by (5.12) identifies the layer (s) in which damage is present, and also the constituent which presents a problem (fiber or matrix) . 5.2.1.2 Closed-form solution For simple composite structures such as beams subjected to extensional or bending loads, closed-form solutions can be derived based on the classical lamination theory and on the assumption that only one composite material property is able to define the location of damage. Consider the composite laminate of Fig. 5.1 subjected only to extensional loads. For each element of this structure the resultant normal force is and can be determined analytically. Again, (5.1) can be written as follows. 0 0 A A A 12 16 •11 A A; a' '26 22 12 A A^ '26 66 16 (5.20) From this expression, the mid-plane strain vector can be determined as follows.

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r e° 1 An Ai2 A16 -1 r N 1 Al2 A22 A26 0 • Ai6 A26 Aee . 0 107 (5.21) In an experimental procedure, the mid-plane strain €° and the mid-plane curvature for all the elements in the structure can be obtained from the following equations: e° + (t/2)K, = e upper (t/2)/c, = c lower (5.22) where t is the laminate thickness, and e"PP®' and e^""®"^ are strains measured on the upper and lower surfaces of each element of the structure. Usually these quantities are measured with strain-gages mounted at the upper and lower surface of the structure. The quantity e° obtained from tests can be equated to the value obtained from (5.21) as follows : = An N, (5.23) where is given by the following expression. An = All ~ A A12A66 A^gAse 12 . A22A55 A26 , + A A12A26 A22Aig 16 A22A66~A26 , (5.24) This expression can also be used to demonstrate the dominance of A^ in the term A^ and hence also in €°. For a few select composite materials with different representative

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108 stacking sequences, the terms of the denominator of (5.24) were calculated, and are summarized in Table 5.2. An expression for can be derived from (5.23) and (5.24), as follows: N, An = ^12 ^22^66 J A Al2A26~A22Al6 16 ^22^66 J (5.25) The comparison between the value of A^^ for each element in the damaged structure and the corresponding value in the undamaged structure indicates the location of damage. A similar procedure can be followed for the case of composite beams subjected to bending (Fig. 5.2). Now the force-displacement relations for each element are the following: (5.26) r K 1 Dl2 Die f . 0 > = Dl2 D22 D26 * . 0 ^ . Die D26 Dee . V The property is dominant in this case, and is used as the design variable for damage detection. Again, the strains on the upper and lower surfaces at each element of the structure are used as measured response, and the values of the mid-plane curvatures can be calculated by (5.22). From (5.26) the term is determined, yielding the following expression. Dii = + D Di2D6e DisDge 12 . D22Dg5 D26 _ D 16 D12D26 D22D16 . D22D66 D26 , (5.27)

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109 Table 5.2 Dominance assessment of in (5.24) Stacking sequence Composite Glass/Epoxy Boron/Epoxy Graphite/Epoxy Tl T2 Tl T2 Tl T2 Cross-ply symmetric 5 layers* 0. 0150 0.00 0.0022 0.00 0.0001 0.00 Cross-ply antisymm. 4 layers^ 0. 0150 0.00 0.0029 0.00 0.0001 0.00 Angle-ply symmetric 5 layers" 0.30 0. 006 0.50 0. 016 0.46 0.01 Angle-ply antisymm. 4 layers'* 0. 10 0.00 0.42 0.00 0.50 0.00 Tl = (Ai2/An) T2 = (Aie/Au) ^12^66 Aj^gAes L A22A56 A26 J ^12^26 ^22^16 I. A22A66~A26 J a : [0,90,0,90,0] b : [0,90,0,90] c : [-30,30,-30,30,-30] d : [-30,30,-30,30]

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110 The comparison between the properties of each element of the undamaged and the damaged structures indicates the location of damage, as in the case of extensional loads. No numerical results are included in this presentation to illustrate the use of this closed-form solution. However, as mentioned in earlier in this section, this method is applicable only to simple structures such as may be encountered in laboratory experimental setups. 5.1.2.3 Discussion of results The proposed method for damage detection was applied to a series of laminate panels which are representative of real structures such as fuselage panels, aircraft stabilizers, or automotive semi-elliptic springs. Fig. 5.3 shows a finite element model (FEM) of a typical laminate with 8 elements. Material properties corresponding to E-Glass/Epoxy were used in this example, and are listed in Table 5.3. Different stacking sequences were assumed in the implementation. These properties were determined in the laboratory for planned future experimental work in the area of damage assessment in composite materials. The finite element model (FEM) was once again used to simulate the measured response. Damage was simulated by reducing the laminate stiffnesses of a specific element and obtaining the corresponding FEM solution. This response was then corrupted by a random noise signal, and was then considered as the experimental data. The simulated reduction of laminate stiffnesses was

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Ill based on experimental results found in the literature [2021] as well as on damage introduced in certain layers simulating fiber breakage or matrix cracking. In this case the classical lamination theory was employed to calculate the resultant laminate stiffnesses. In the first example, a symmetric [0,903]s laminate was subjected to an in-plane extensional force (Fig. 5.3). Matrices B and D were not considered, and the non-zero laminate stiffnesses were A^, Ai2/ and Age. Hence, there were 4 design variables per element for a total of 32 design variables in the resultant optimization problem. Two cases of damage were simulated by reducing A^ and A12 of element number 1 to 60% and 30% of their original values, respectively. The results obtained from the identification problem are shown in Table 5.4. Only changes in A^ were detected although A12 has undergone a much larger change. The reason is that the applied load only activates the stiffness coefficient A^. The second example further supports this conclusion. A [0,90], laminate was damaged by reducing A^, A22 and Age of element number 1 to 80%, 85% and 85%, respectively, of their original value. Again, if an extensional load is applied, as shown in Fig. 5.3, only a change in A^ is detected. Even if Age is reduced to 30% of its original value, that change is not detected under the application of the extensional force alone. Now, if the load is changed to a pure shear as shown in Fig. 5.4, Aee becomes the dominant

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112 stiffness coefficient and changes in are no longer detected. These results are summarized in Tables 5.5, 5.6 and 5.7. This information is very important when more general loading cases such as out-of-plane loads are considered or when the structure is a general non-symmetric laminate. In such problems, the number of independent stiffnesses increases dramatically, rendering the solution of the identification problem extremely difficult. One can, however, use only those stiffness coefficients as design variables as are likely to be activated under the applied loading. Consider now an antisymmetric laminate [0,90] subjected to a vertical load as shown in Fig. 5.5. Matrices B and D as well as A have to be taken into account, as there are bending loads, and the stacking sequence yields coupling between bending and extension. Matrix cracking in the 90 degrees layers was assumed. The response of the laminate was determined by making the stiffnesses Q22, Q12 and Qge of those layers equal to zero. The resultant laminate stiffnesses calculated from (5.3) show significant changes. Four design variables per element were considered. Only changed as is to be expected on the basis of the load analysis described above. These results are shown in Table 5.8. Finally, to show the influence of the dominant stresses in the damage detection process, the 8 layered, [0, ±45, 90] 3 cantilevered plate with 24 elements was studied.

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First, this structure was subjected to a transverse load applied at the free end of the plate, as shown in Fig. 5.6. To simulate damage, the properties A^, Age, and Dgg of elements 1 and 22 were reduced to 60%, 80%, 65% and 75%, respectively, of their original values. As design variables, only the coefficient for each element was used, yielding a total of 24 design variables instead of 96 variables that would be used if all the 4 properties per element were considered. Table 5.9 summarizes the results of this exercise. Again only damage in element 1 which is close to the support and more stressed than element 22 was detected. In the second case, an extensional load is applied at the free end of the same structure, as shown in Fig. 5.7, and damage introduced in the elements 1 and 22 with property A^ reduced to 60% and 50% of the original ones, respectively. The other properties were assumed to have the same changes as in the previous case. The stresses are equally distributed along the length of the structure and the changes in stiffness in both damaged members have the same influence in the displacement field. In this case the damage detection is complete as summarized in Table 5.10. If a moment Mi is applied at the free end of this structure (Fig. 5.8), and property of members 1 and 22 are reduced to 60% and 50% of the original values, respectively, to simulate damage, again only damage in member 1 is detected, even though this load also yields an

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114 equal stress field over the entire structure. Damage in member 22 was only detected when another moment M2 was applied close to this member (Fig. 5.8), forcing the change in stiffness of member 22 to affect more significant changes in the obtained displacement field. These results are summarized in Table 5.11, and this problem is discussed in greater detail in the next section. The method discussed above may be a potential tool to detect the region of damage. It can also give an idea of the severity of damage by comparing the values of the stiffnesses in the undamaged structure and the ones determined by the proposed approach. However it does not yield information about the type of damage in the laminate and its depthwise location. It has been conclusively shown that the three principal damage modes, namely transverse cracking, delamination, and fiber breakage, produce a reduction in the laminate stiffnesses. However, this information alone is not sufficient even to classify the damage. In the present work, attempts to describe the behavior of the laminate through its thickness (by 2-D elements) for detecting possible layers with broken fibers or matrix cracks, met with limited success. Each layer was assigned four design variables corresponding to each lamina property. The experimental data available is practically the same as above but the number of design variables increases

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115 significantly. The approach works only in some very special cases. To overcome this problem, the laminate was modeled using a three dimensional element. Fiber breakage can be modeled by reducing the stiffness in a given layer by a prescribed amount, whereas matrix failure can be simulated by reducing Q22, Q12 arid Qgs to zero. The zeros are actually very small numbers to avoid numerical difficulties in a computer simulation. Figure 5.9 shows a [0,90]s four layered laminate subjected to extensional forces. Damage was simulated as matrix failure in the 90 degree plies. The results for this exercise are summarized in Table 5.12. A clear indication of damage is obtained from the proposed approach. 5.2.2 Use of Eiqenmodes as Measured Structural Response The method described in the previous section can be extended for the use of eigenmodes as measured structural response. The governing differential equations for the vibration problem will have a different form dependent upon the type of lamination. If an antisymmetric cross-ply laminate is considered, a bending-extension coupling exists, resulting in the following coupled differential equations [30]. AiiU,^ + AggU yy + (A12 + A66)v_^ BiiW „„ = 0 (A12 + A66)u,^ + AggV^, + A^Vyy BiiWyyy = 0 (5.28) Dii(W.„^, + W.yyyy) + 2(0^2 + 2 D ^ g ) W _ " -Bii(U_^ V yyy) + ^W ^fc = 0

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116 The above equations show that any variation in the system properties results in an altered response. Again one can correlate changed responses, and consequently changed stiffness properties, with damage. In a finite element formulation, the eigenvalue problem is stated as follows. ( K «i M ) qi = 0 (5.29) Here the stiffness matrix K includes all the material stiffnesses A^j, B^j and Dij. Similar to the previous cases, the optimization problem is represented by (5.12). In this case i represents the degree of freedom, j denotes the particular eigenmode, Y„ and are, respectively, the measured and analytically determined eigenmodes. The number of design variables per element is also a function of the layup, and in the general case, this number would be unmanageably large. Again some analysis is required to reduce the number of design variables per element, and an approach analogous to the static problem was adopted. In this case, the type of the measured mode will determine the dominant property. For example, if bending modes are considered in a beam made of composite material, then the property will be dominant and can be considered as the only design variable in the optimization problem. This will be demonstrated in the following examples. Consider the cantilevered plate of Fig. 5.10. The plate is an eight layered, [0,90333, E-glass/epoxy laminate

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117 with properties given in Table 5.3. Damage was simulated in element number 1 by reducing the values of A^, Age, and Dge to 60%, 80%, 65% and 75%, respectively, of their original values. In the first exercise the three lowest extensional eigenmodes were considered. The results are presented in Table 5.13 showing that only changes in A^ were detected. The reason for this is that A^ is the dominant property for extensional modes. If instead, the first three bending modes are picked for damage detection, then the property that shows the most significant changes is Dii, as shown in Table 5.14. The results obtained in previous examples show that only one design variable corresponding to the dominant material property can be considered for damage detection. This is particularly important for large structures where the number of design variables increases dramatically. To represent a structure with a larger number of elements, a simply supported plate with 24 elements is considered (Fig. 5.11). It is an eight layered, [0,±45,90], laminate, with material properties as given in Table 5.3. The experimental results were simulated by reducing the values of A^, A22, and D22 in element number 2 and 12 to 80%, 90%, 60% and 80%, respectively, of their original values. To detect damage, the first three bending modes were used. Based on previous analysis, only the property for each element was considered as design variable. The results are presented in

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118 Table 5.15, showing clearly the location and extent of damage. Consider now the structure of Fig. 5.11 used as a cantilever beam with damage simulated in members 1 and 22 by reducing the property to 60% and 50% of the original values, respectively. If only one eigenmode is used in the damage detection process, damage in member 22 is not detected. This mode has the same shape as the elastic curve obtained when a moment is applied at the free end of the beam (Fig. 5.12) and, as it was shown in the previous section, failed to detect damage in the outboard section. When three eigenmodes were used, the complete damage characterization was obtained. The elastic deformation corresponding to the superposition of the first three modes is shown in Fig. 5.13, and exhibits a significant change in curvature close to element 22. Such a deformation would be obtained by the loading shown in Fig. 5.8, where a concentrated moment is applied in the region close to the damage section. Hence, the use of the three eigenmodes results in a successful identification of damage, much as in the case of the static load application shown in Fig. 5.8. The results of this exercise are presented in Table 5.16.

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119 gure 5.1 Composite laminate subjected to extensional forces.

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Figure 5 . 2 Composite laminate subjected to bending loads.

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121 Figure 5.3 Two dimensional finite element model of an 8 layer E-glass/epoxy laminate subjected to extensional force. Dimensions in centimeters.

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Table 5.3 Mechanical properties of the material used in the examples Properties of E-Glass/Epoxy El 36.00 GPa E2 11.00 GPa G12 3.00 GPa "12 0.28 Vf 0.50 P 1.90 g/cm^

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123 0) o u o (0 c 0 •H (0 c +J X 0) o +J -o a) -p o 0) •r-> to 0) +J (0 c •H (0 f— 1 O 0) p o (0 0) Oi in rH (0 Eh 0) C +J c Q) rH -P U iH o o o O (0 O o o o o X to • • • • iH rH 1 r-% 1 >i -P 1 >-i C Q) Q) O C\ o\ o Q) -H Ci< o (Ti o "0 'w 0 • • • • H -H >-l rH O o rH 4-> Q^ +J O rH O o o O (0 0 O o o O X to • • • • M H rH rH rH 1 T3 +J C Q) U Q) -H Q) 0^ o o^ o T3 £li o o H -H 0 • « • • P Sh O rH o rH -P O i eg -P 1 >H COO) H O o H o p a -p c •H E (0 I to <«H 0} <4H to -H -P to ^ ^ ^ ^ u 0) 3 c -p c 0) e Q) rH w 4J 0 rH o o o O (0 0 o o o O X to • • • • H H H H 00 1 T3 M COO) o o o O Q Q Q L-J ••-4 l_i rH _L_1 O. o O o o 0 rH o O O o rH M-l >1 w u; Ui 0) -i-i n. Q Q ^f UJ Lj rn -P U rH o o o o fl3 0 o o o o V rA TtI W rn rn rn rn +J 1 73 C 0) (U o o o o o o 'w 0 • • • • H -H o H rH H P a P 0 rH o O O O (0 0 o O O O X to • • • • u H H H H in >1 1 -0 u c a> 0) Oi O O TJ 0 • • • • H -rH >^ o O H o p ft -p «3 C •H e rO I (0 (tH 0)
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124 Figure 5.4 Two dimensional finite element model for the [0,90] 3 laminate subjected to pure shear loads. Dimensions in centimeters. ft* ! .

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125 u x> c -p c 0) g (1) 0) -p (0 c •H e (0 p (0 o X m w p I T) M C 0) Q) 0) -H ft TJ <*-i O H -H ^ P ft -P O rH (0 o T3 <*-! ft 4J O U ft (0 o X 01 C 0) -O 1 0) d) ft O U ft O H (C O X W M P I T( ^ C 0) 0) Q) -H ft T) O H -H M P ft I (0 (/] (1) -p o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o CO o o in 00 00 o o o o ^ ^ ^ ^ u 0) G 4J C 0) 0) rH M 0 rH o O o O (0 0 o o o O • rv] rH rH rH CO >1 -P 1 t3 >-i C fl) fl) »-l W W o fll -iH O. Q o b~l -iH Lj r 1 ' 1 rH rH rH o o o o 0 rH o o o o (0 0 • X w rH rH rH w I COO) o o o o fl) -rH Qi o o Q Q 13 4h 0 M -r-l ^ rH rH rH +J ft O rH o o o o <0 0 o o o X 10 rH rH rH rH >. VO P 1 73 U COO) o o o o 0) -H ft O O O o TJ 0 • • • • H -rl ^ H H rH rH P ft 4J 0 rH O O O O (0 0 O O O O X 10 • • • • w H H H H If) >1 p 1 73 U C 0) 0) O O O 0) -H ft O O O 73 0 • • • • H -H ^ rH o H p ft 0) p c •H (0 (0 t(H 0) •H -p u ^ ^ ^ j6

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126 O (0 c o •H c Q) -P !«) Q) C (0 o p T3 0) -P O 0) •r-i Si W H 0) jj (1) g o CTl a) p o u a) > o W -P -P -H rH > 3 W T3 0) c in (1) rH X( (0 Eh E 3 C P C 0) 0) rH M 0) -p (0 c •H -P O rH (0 O X (0 >i P I TS U COO) -O IH o H -rH }-l p a p O rH (C O x w I T! C 0) 0) -H 73 U >i +J Q) ft O ft 4J O rH (0 O X (0 w 0) -O <)-( >1 -p 0) Q) ft O ft +J O rH (0 O X M H >i -P I -d u c H ft I VI (t-l (1) 1-1 w -H -P o o o o o o o o o o o o o o o o o o o o o o o o CTl o o o o o o o o o o o o 0^ o o o 00 o o IT) CO o CO o o o o 4^ ^ ^ ^ u 0) Xi +J c Q) e 0) rH w CO in 0) P (0 c O rH (0 o X (0 o o O O o o o o >1 -p I -o u c 0) a) 0) -H H -H ft o u ft o o o o o o o o p O rH (0 o X (0 o o o o o o o o >1 p I 73 U C (U 0) Q) -H ft T( <»H O H -rl -P ft o o o O O O O O O rH m o X U) o o o o o o o o I TJ U COO) Q) -H ft
i -P 1 T3 c a) 0) o o CO CTl o o •O i +J 1 T5 M C 0) 0) CT^ O O in 0) -H a O O o^ TJ MH 0 • • • H -H M o H H o -p a 0) 4J (0 C •H (0 J (0
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128 Figure 5.5 Two dimensional finite element model for the antisymmetric [0,90] E-glass/epoxy laminate subjected to bending loads. Dimensions in centimeters.

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129 U 0) 3 C -P c e 1 I TJ -P C (1) M Q) -H Q) T3 1 -p C Q) Q) 0) -H a H -H -p o u ft p U rH (0 o u I T3 U C 0) 0) 0) -H ft TJ 1 -p 1 Tl 1 \J C 01 fl) 0 -rl o. Q Q frt 0) -iH fl. Q u 1 n\ Q (J 1-1 o >— 1 /-> ri ft -P /—V V— (u U \j l>s v/J rn _j 1 rn >1 4-> 1 TJ U n\ CU 0^ 0^ 0> Q) tH 0^ fry (L J X5 HH u rH •ri t 1 « 0 1 1 P p 0 rH 0 0 0 0 1 +J 1 TJ C Q) 0) 0 a> a> 0 0) -H D4 0 0 T3
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130 Figure 5.6 Two dimensional finite element model of a 24 element laminate subjected to bending forces. Dimensions in centimeters.

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131 Table 5.9 Results for the [0,±45,90]3, 24 element laminate subjected to bending forces Element No. Design variables (d^) using static response Tdpnti fi pc\ Va 1 iip 1-1 A CI w kJ\JA. 1 2 1 nn J. . u u 3 u • o o 1 nn X . uu 4 0.93 1.00 5 0.97 1.00 6 1.00 1.00 7 1.00 1.00 8 1.00 1.00 9 1.00 1.00 10 1.00 1.00 11 1. 00 1.00 12 1.00 1.00 13 1.00 1.00 14 1. 00 1.00 15 0.97 1.00 16 0.95 1. 00 17 0.94 1.00 18 1.00 1.00 19 0.87 1.00 20 0.90 1.00 21 0.98 1.00 22 1.00 0.65 23 0.93 1. 00 24 1.00 1.00

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132 Two dimensional finite element model of a 24 element cantilevered laminate subjected to extensional forces. Dimensions in centimeters .

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133 Figure 5.8 Two dimensional finite element model of a 24 element cantilevered laminate subjected to bending moments. Dimensions in centimeters.

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134 Table 5.10 Results for the [0,±45,90]3, 24 element laminate subjected to an extensional force applied at the free end V 1 ^Tn^aTrf" No Design variables (d^) using static response riXaCu bOl 1 0.64 0. 60 2 0. 94 1. 00 3 1.00 1.00 4 1.00 1.00 5 1.00 1.00 6 1.00 1.00 7 1.00 1.00 8 1.00 1.00 9 1.00 1.00 10 1.00 1.00 11 1. 00 1.00 12 0.96 1.00 13 0.99 1.00 14 1.00 1. 00 15 0.97 1.00 16 0.97 1. 00 17 0.98 1.00 18 0.99 1.00 19 1. 00 1. 00 20 1.00 1.00 21 0.87 1. 00 22 0.60 0.50 23 0.78 1.00 24 0.93 1.00

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135 Table 5.11 Results for the [0, ±45, 90] 5, 24 element laminate subjected to bending moments iiiiement. NO . Design variables (d^) using static response Load 1 Load 2 Exact Sol T X 0 . 62 0 . 67 0. 60 2 0.80 0 . 86 1. 00 "I J 0.90 0 . 96 1. 00 A 0.96 1. 00 1.00 R 1.00 1.00 1.00 f 1.00 1.00 1.00 7 1.00 1.00 1.00 8 1. 00 1.00 1.00 9 1. GO 1.00 1.00 10 1.00 1.00 1.00 11 1. 00 1.00 1.00 12 1.00 1. 00 1.00 13 1.00 1. 00 1.00 14 1.00 1.00 1.00 15 1.00 1.00 1.00 16 1. 00 1.00 1.00 17 0.91 1.00 1.00 18 1.00 0.99 1.00 19 0.94 1.00 1.00 20 0.93 1.00 1.00 21 0.85 0.98 1.00 22 0.83 0.55 0. 50 23 0.76 0.70 1.00 24 0.99 0.98 1.00 a : One bending moment applied at the free end b : The same as load 1 plus a bending moment applied closed to member 22

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136 Figure 5.9 3-D finite element model of the region of damage of a 4 layer composite structure. Each layer corresponds to one element. Dimensions in centimeters.

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137 Table 5.12 Results for the 4 layer laminate using a 3-D element for damage detection Layer no. Lamina stiffnesses Identified property Exact solution 1 Qii 0.99 1.00 Ql2/ Q22 / Qee 1. 00 1.00 2 Qii 1.00 1.00 Ql2/ Q22/ Qee 0.01 0.01 3 Qu 1.00 1.00 Q22/ Qee 0.01 0.01 4 Qu 0.99 1.00 Q22 / Qee 1. 00 1.00

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138 Figure 5.10 Two dimensional finite element model of a [0, 90333, 8 layer E-glass/epoxy laminate. Dimensions in centimeters.

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139 (0 c o •H c a) 4J X 0) Q) 0) -p c •H M 3 (U +J (0 c •H e 10 o O t/i (4-1 0) (/I -P iH to o g c 0) CP H in 0) rH nj 0) g 3 C -P C e iH 0) -P (0 c -H g O rH «3 O X w >1 -p I T3 ^ C 0) 0) Q) -H a O t-i o H -H M -p O rH (0 o M I T) -P c o o o o o o o o o o in o o CO in in H O O ^ ^ rff a'B (1) g d c +J c 0) g Q) rH W P 0 rH o O o o <0 0 o O o o X u • • • • M H H H H 00 >i P 1 T3 ^ C 0) Q) -H a o O 73 1 1 T) V^ C 0) 0) o O o 0) -H a o a\ O o 73 <»H 0 • • H -H M i-H o rH rH p a -p 0 rH O o O o lO 0 O o O o X (0 • • • • H rH H H >1 vo -p 1 T3 >H C 0) (1) O a\ O a) -H a O O T3 <*H O • • • • H -H M H o H o -p a -p U rH o O o (0 o o o o o X u • • • • H H H H in >. p 1 TJ M O in tn o C 0) 0) O o Q) -H a • • • • T3 <»H O H o o rH -p a 0) +J c •H g (|H
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f 140 C •H Tl c 0) 0) 0) CP c •H 01 0) 4J (C c •H (0 w »— 1 n O Q) -P u O U] 1 1 73 U C Q) Q) ON o\ ON 0) -H & ON ON ON "O m o 1— 1 -H V-( o o o o 1) Q. O rH o o o o (0 o o o o o X w M H H H H 1 4J C 0) ^-1 o> ON 0) -H Q) a\ ON ON ON Dj • • • H -H O o o o o -P U P U rH o o o o (0 0 o o o o • • • u H H H H >1 CM p C ON n ON (Tl ON ON ON TJ i 1 73 ^ cr> as H ON C 0) 0) Oi VO ON • • • • T3 <)H O o o o o H -H >H p a 0) p c •H e (0 I 01 Mh 0) 01 01 0) c •H p 01 ^ J8 o« u 0) XI g 3 C -P Q) g Q) rH W P O rH o o o o (0 0 O o o o X 01 • • • • M H H H H CO >i P 1 tJ U C 0) 0) ON ON ON ON 0) -rH a o» ON ON ON 73
. +j 1 73 u C 0) 0) Oi Ol n ON 0) -H a ON ON ON ON 73 HH o • • • • H -H o O o o -P a P 0 rH o o o o (0 0 o o o o X 01 • • • • u H H H H VO •p 1 73 u C 0) 0) ON ON ON 0) -H a ON ON ON ON 73 <1H 0 • • • • H -H vh o O o o -P a -P V-/ 1^ /-\ (0 0 o o o o X 01 • • • • M H H H H in >i +j 1 T3 u ON ON ON C (U 0) ON ON ON ON 0) -rl a • • • • 73 "w 0 O O o H -H u P a 0) 4J (0 C •H g (0 I 01 (
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141 Figure 5.11 Two dimensional finite element model of a 24 element laminate. Dimensions in centimeters.

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142 Table 5.15 Results for the [0,±45,90]3, 24 element, simply supported laminate, using three eigenmodes Element No. Design variables (d^) using three eigenmodes vaxue txacu bo± 1 X . uu 2 U . dO 3 0.94 1.00 4 0.93 1.00 5 0.93 1. 00 6 0.93 1.00 t 0.94 1.00 8 0.96 1.00 9 0.96 1.00 10 0.94 1.00 11 0.95 1.00 12 0.56 0.60 13 0.95 1.00 14 0.95 1.00 15 0.93 1.00 16 0.95 1.00 17 0.95 1.00 18 0.94 1.00 19 0.94 1.00 20 0.94 1. 00 21 0.94 1.00 22 0.94 1.00 23 0.95 1. 00 24 0.98 1.00

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143 Element Numher Figure 5.12 First mode shape of a cantilevered composite beam. Element Nxuntier Figure 5.13 Curve resulting from the superposition of the first three mode shapes of a cantilevered composite beam.

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144 Table 5.16 Results for the [0,±45,90]3, 24 element cantilevered laminate using eigenmodes Element No. Design variables (d^) using eigenmodes case X case z Exact Sol 1 u . / D A CO u . bo 0 . 60 2 U . o 4 0.95 1 . 00 3 0.90 0.99 1.00 4 0.99 0.99 1. 00 5 0.99 0.96 1.00 6 0.99 0.95 1.00 / 0.98 0.96 1. 00 8 0.98 0.95 1.00 9 1. 00 0.96 1.00 10 0.97 0.96 1.00 11 1. 00 0.96 1.00 12 0.99 0.97 1.00 13 1. 00 0.95 1.00 14 1.00 0.95 1. 00 15 1.00 0.95 1. 00 16 1.00 0.95 1. 00 17 0.99 0.95 1. 00 18 0.99 0.95 1. 00 19 0.99 0.94 1.00 20 0.99 1. 00 1.00 21 0.99 0.90 1.00 22 0.99 0.47 0. 50 23 0.99 0.92 1. 00 24 0.99 0.99 1.00 a: Only the first eigenmode b: First three eigenmodes

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CHAPTER 6 EXPERIMENTAL INVESTIGATION OF DAMAGED STRUCTURES 6.1 Introduction The proposed approach for damage detection requires that measurements of the static and dynamic response of the structure be available to correlate against the response obtained from the analytical model. In all theoretical developments outlined in previous chapters, the measured response was assumed to be available, and was actually obtained using a finite element approach and a model with parameters corresponding to the damaged structure. The results obtained from such an analytical model were additionally corrupted by a random noise signal so as to simulate measurement errors. All response obtained from the finite element model of the damaged structure was multiplied by a factor obtained as follows: Fi = 1 + (1 2Ri)L/100 (6.1) where R^ was a random number and L was an input error bound expressed as a percentage. The random numbers were uniformly distributed between 0 and 1, and can be obtained from any random number generation routine such as GGUBS available in the IMSL Library of FORTRAN subroutines for 145

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146 Mathematics and Statistics (IMSL, Inc., Houston Texas). The error bound L depends on the available precision of the measuring equipment and represents the expected value of the maximum deviation of the measured data from the exact response. This chapter describes an experimental procedure designed to validate the concepts presented in earlier chapters. The problem was limited to the use of static response data, and a twelve bar planar truss was selected as the test structure. The elements of the truss were made of steel, and damage in the structure was simulated by substituting members of the truss by rods made of aluminum and brass with smaller Young's moduli. The structure was subjected to prescribed extensional loads in a test machine. It is reasonable to expect that there would be significant differences in the response obtained from a physical model as described above, and the results obtained from an idealized finite element model. These differences can be largely attributed to problems in the physical structure such as out-of -plane bending, nonideal joints, and prestress in the members and joints due to poor control of the component dimensions. The expected discrepancy between the measured and analytically obtained results was indeed present, and required as the first step of the experimental work, an identification of the analytical model that would predict more correctly the response of the structure. The

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147 behavior of the joints was carefully studied and equivalent stiffness properties were determined. Following this step, and with an identified model available, the procedure of damage detection explained in previous chapters was applied in two distinct cases of simulated damage in the structure. The first example was one with a single damaged member. The second example included two damaged members with different extents of damage. Although the exact extent of damage was not predicted in both cases, the location and a more conservative estimate of the extent of damage was obtained. The experimental setup, the adopted procedure, and the results obtained are presented in subsequent sections of this chapter. 6.2 Experimental Setup The twelve bar planar truss was fabricated in the machine shop of the Department of Aerospace Engineering, Mechanics and Engineering Science. The geometrical dimensions of the structure were primarily determined by the capabilities of the available testing and fabrication equipment within the department. All structural dimensions, including fits and screw threads, were originally in English units, and where possible, have been converted to SI units. The idealized structure shown in Fig. 6.1 is a statically determinate truss with 3 identical bays. The individual rods of this structure were made of steel SAE W-1

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148 (drill rod) with 9.525 mm diameter. This corresponds to a cross sectional area of 71.225 mm^. The truss was loaded by two extensional forces of 4450 N applied at nodes 7 and 8 (Fig. 6.1). This load produces a maximum stress of 76.82 MPa in member 1. This stress level in steel assures that the truss deflection under the specified load is in the linear portion of the stress-strain curve. The actual structure is shown in Fig. 6.2. In order to be mounted on the testing machine, each end of the truss was connected to a transverse square aluminum bar in which a cylindrical piece was screwed in the center. This piece was pinned to the fixtures on the load frame of the testing machine. This setup introduced a constraint at the free end of the truss (nodes 7 and 8) , disallowing transverse movement in these nodes, and yielding a change in the original model, as shown in Fig. 6.3. Constraining these transverse displacements introduces a small prestress in the diagonal and horizontal rods which would otherwise have been unloaded. The joints were constructed with three U-shaped pieces and one inner square piece, screwed to the rods sharing the same node and fastened by a dowel pin of 9.52 mm diameter. These pieces were fabricated from AISI C-1018 steel. Additional details of the joints can be seen in Fig. 6.4. The threads were 3/8"-24 NF (9.52 mm nominal diameter and 24 threads per inch) and the holes in the fittings were 3/8" (9.52 mm), reamed for close fit.

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Figure 6.1 Original idealized model for the 12 bar planar truss. Dimensions in centimeters.

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151 Figure 6.3 Modified idealized model for the twelve bar planar truss. Dimensions in centimeters.

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152 Figure 6.4 Close-up view of a joint in the truss structure .

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153 The MTS machine (MTS Systems Corporation, Minneapolis, Minnesota) , available in the departmental Structures Laboratory was used for loading the truss structure. The controller for the testing machine is Model 445-11, and features three working modes; load, strain or stroke. For this experiment, only the load mode was used. The machine can apply tensile, compressive and torque loads. The maximum tensile load (100% of the full load) that the machine can apply is 100 kN. Limits of 10%, 20% and 50% of the full load can be preset on the machine. The maximum torque available is 1 kN.m, and limits similar to the tensile loads can be set on the machine. The truss was mounted vertically (Fig. 6.5) and an extensional load applied by a hydraulic piston. The load cell is located on the upper part of the load frame. The displacement of the load cell under the designated load was taken into consideration by subtracting this value from the measured displacement of each node. The displacements were measured using Teclock model AI921 travel dial indicators, with 1" (25.4 mm) of travel and 0.001" (0.0254 mm) graduation. Magnetic bases with fine adjustment, fixed on the side columns of the MTS machine, were used to mount the gages. This setup is shown in Fig. 6.6. In the initial implementation, six gages were used to obtain the displacement response of the structure. Due to the difficulty encountered in keeping each of the gages set

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154 Figure 6.5 The truss model mounted in the MTS machine.

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156 correctly at the same time, this approach was not very successful. Furthermore, the displacement of a node was not obtained precisely as the gages were placed on flat surfaces of the fittings and not on the pins. To further compound the problems, the fittings did not behave as predicted by the idealized models. To circumvent these problems, separate analysis of the elements in each bay of the truss structure was attempted. Gages were mounted at the ends of the vertical structural members of each bay to record the displacements under the applied loading. Several measurements were then taken to establish the repetitivity of the results until the entire structure was completely characterized. Figure 6.6 illustrates the instrumentation setup to characterize the behavior of node 6. The analytical model for this structure was identified from these measurements as described in greater detail in section 6.3, and then used in the damage assessment strategy described in previous chapters. 6.3 Analytical Model Identification The initial experimental results did not match the response predicted by the model shown in Fig. 6.3. Table 6.1 shows the discrepancy between select components of the two sets of displacement vector. A new model was then proposed to obtain better agreement between measured and predicted response. Figure 6.7 shows an assumed model with

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157 20 nodes and 24 elements. The behavior of the joints was modeled by beam elements attached to the original structural members. The rotational degrees of freedom for these elements were eliminated to allow connection to the primary axial-rod elements of the truss structure. A preliminary analysis of the measured response indicated that the joints were the main source of discrepancies between the measured response and the original model of Fig. 6.3. For the proposed model, instead of using the standard value of the steel Young's modulus found in the literature (207 GPa) , new values were determined for each element (both primary structural members and members modeling the joints) , based on the experimental data. The principal motivation for doing this was to correct the original analytical model, lumping all sources of discrepancies into a factor multiplying the material property of each element. The Young's moduli of the rods were calculated by the following expression: N L E = (6.2) S AL where N is the normal force applied at each member, L is the member length, AL is the length variation obtained from the tests, and S is the rod cross sectional area. The material properties of the elements representing joints 3, 4, 5, and 6 were determined in a slightly different manner. Figure

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158 6.6 shows how the deflection across each such joint was measured. The elastic deformations in the joint are represented by two small elements of length L/2 which are connected to the primary structural members. These small elements are also connected together by an ideal pin as shown in Fig. 6.7. An isolated view of one such joint, with the characteristic loads and lengths is shown in Fig. 6.8. The difference between the two displacements measured at each end of a joint is represented by the total length variation of the two elements which model the joint. If the length variations of these two elements, considered separately, are denoted by ALi and AL2, respectively, and the corresponding stiffnesses are assumed to be identical, one can write the following expressions. NiL/2 ALi = S E (6.3) N2L/2 AL2 = S E Here Ni and N2 are the normal forces acting at each end of a joint, as shown in Fig. 6.8, and S is the cross sectional area considered for simplicity to have the same value as those of the primary structural members. Adding up the two equations of (6.3), one obtains the following expression for the Young's moduli of the joints:

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159 (N1+N2) L/2 E = (6.4) S AL where AL is the total length variation of a joint obtained from the tests. The material properties corresponding to the joints 1, 2, 7, and 8 (Fig. 6.3) were determined by applying (6.2) and assuming that the cross sectional areas are the same of those of the rods. The Young's moduli of the diagonal and horizontal members were those of steel, as obtained in the related literature, and the cross sectional areas were assumed to be constant from pin to pin and equal to the cross sectional area of the rods. The modified material properties are shown in Table 6.2. One can notice that the joints were indeed responsible for the large discrepancies between the measured displacements and those obtained from the original analytical model. Table 6.3 shows the predicted response according to the new analytical model. These results show generally good agreement with the measured response. With such an analytical model, which better represents the true behavior of the structure, the damage detection approach of previous sections could be used. 6.4 Discussion of Results For damage detection, the output error approach was employed as described in Sec. 3.2.2. The measured response were the static displacement obtained for a select number of

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160 degrees of freedom. The identification problem could be stated as determining the components of a vector d (d^, i=l,..,12), to minimize the following objective function. 2 S (yJ^ -y/'^)' (6.5) i j The problem is characterized by a single load vector (i=l) , and 12 vertical displacements (j=12) measured at nodes 3, 4, 5, 6, 9, 10, 11, 12, 15, 16, 17 and 18 (Fig. 6.7). The first problem involved introduction of simulated damage in only one member. This was the rod connecting nodes 3 and 5, and damage was simulated by substituting the original steel rod by one made of aluminum with a Young's modulus of about one third that of the steel. The procedure of initializing the setup on a basis of element by element analysis described above, was repeated. Table 6.4 presents the results obtained, showing clearly the location of damage. The extent of damage, however, was not predicted as accurately as one would like. It is still regarded as a good indication of the severity of damage. The second experiment examined the case of two damaged members in the truss. The rod connecting nodes 3 and 5 of the first case was substituted by an aluminum rod. Additionally, the rod connecting nodes 10 and 12 was substituted by a brass rod with a Young's modulus of about one half that of steel. Once again the proposed approach

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161 worked rather well in detecting the location of damage and in giving a good estimate of the extent of damage. Results for this problem are summarized in Table 6.5.

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162 Table 6.1 Measured response compared with results predicted by the original model (Fig. 6. 3) Node No. Vertical Static Displacements (mm) Measured Predicted 1 0.00 0.00 2 0.00 0.00 3 0.1321 0.0396 4 0.1981 0.0686 5 0.2413 0.0851 6 0.3556 0.1313 7 0.3556 0.1361 8 0.4216 0. 1885

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Figure 6.8 Isolated view of the model representing a joint.

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165 Table 6.2 Identified material properties for the truss elements of model described by Fig. 6.7 Element No. Nodes Young's modulus (GPa) 1 3-5 212.9 2 4-6 189.5 3 9-11 205.3 4 10-12 192.2 5 15-17 189.5 6 16-18 190.8 7 1-3 14.5 8 2-4 18.6 9 5-7 59.9 10 6-8 25.5 11 7-9 59.9 12 8-10 25.5 13 11-13 41.9 14 12-14 38.6 15 13-15 41.9 16 14-16 38.6 17 17-19 37.9 18 18-20 37.9 Others* 207.0 Diagonal and horizontal members

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Table 6.3 Predicted response by the new model compared with the measured data Node No. Vertical Displacement (mm) Error (%) Measured Predicted 3 0. 0813 0.0805 1.0 4 0. 0838 0. 0800 4.8 5 0.1118 0.1115 0.3 6 0.1372 0.1331 3.1 9 0.1575 0.1529 3.0 10 0.2667 0.2515 6.1 11 0.2083 0.2007 3.8 12 0.3175 0.3023 5.0 15 0.2769 0.2642 4.8 16 0.3937 0.3734 5.4 17 0.3226 0.3125 3.2 18 0.4420 0.4268 3.6

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167 Table 6.4 Results for the case with one damaged element Element No. Nodes Design variables (d^) Value Exact Sol 1 3-5 0.23 0.33 2 4-6 0.85 1.00 3 9-11 0.82 1.00 4 10-12 0.83 1.00 5 15-17 0.89 1.00 6 16-18 0.98 1.00

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168 1 Table 6.5 Results for the case with two damaged elements Element No. Nodes Design variables (d^) Value Exact Sol 1 3-5 0.39 0.33 2 4-6 0.82 1.00 3 9-11 0.91 1.00 4 10-12 0.49 0.50 5 15-17 0.94 1.00 6 16-18 0.98 1.00

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CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS The present study investigated an approach for damage detection in structures that is closely related to methods of system identification. The approach results in the damage detection problem posed as one of function minimization, and for which iterative nonlinear optimization techniques were used. Both eigenmodes and static displacements were used as measured response, and damage was assumed to affect changes in the material stiffness properties. Scaling constants varying between zero and unity were used to scale the material stiffness properties to simulate varying degrees of damage, and were considered as design variables in the optimization problem. These constants were changed in the identification process until the measured response agreed with the results predicted by the analytical model, usually a finite element representation of the structure. The method was implemented for a series of trial problems with extremely encouraging results. An important finding which emerged from the present study is the sensitivity of stress distribution to the damage in the structure. If stress response could be 169

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170 measured directly, the problem would be easier to solve. However, the stresses are typically determined by measuring strains and using subsequently the constitutive equations of the structural material. Since the numerical optimization problem is based on an objective function that consists of the squared sum of the difference between the measured and analytically predicted response, it is quite possible that damage in areas which do not affect the overall response substantially would go undetected. Hence, a static deflection or an eigenmode that is not significantly affected by some local changes in stiffness, would not reveal that change, particularly in the presence of more dominant changes. In this sense, a rational approach to detect damage in a structure would be to use measured response that is equally affected by all components of the stiffness matrix. One possible approach of achieving this would be to apply a load that would stress equitably all elements in the structure. Since this type of loading is hypothetical and often difficult to practically realize, eigenmodes were used to represent the displacement field resulting from such a load, and this process is described in section 3.2.4. In the present work, the measured response that yielded the determination of damage with more accuracy and less computational effort was the static displacement vector obtained with the application of such a hypothetical load condition. Unless such a load condition can be applied

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171 to activate the damaged element, the use of a given number of eigenmodes typically worked better at locating damage than the static displacements due to an arbitrary load case. However, there are some advantages to be considered in using static displacements, and these are identified as: a) relatively easier to measure, and b) require lower computational costs in the identification problem. Another conclusion from the present work is that all modes that can be measured reliably should be included in the objective function. A larger number of modes provide a much better representation of the displacement field resulting from an equal stress load distribution. The equation error approach presents a sound alternative to the output error approach, and is more advantageous in some applications. Section 3.4 concluded that this approach generally provides a better scaled design space when all degrees of freedom are included in the measurement. On the other hand, the output error approach is more flexible, allowing the use of combination of different types of response as well as the use of a reduced set of measurements. The ability to work with such a reduced set is extremely useful when large structures are being examined. In general, the output error approach should be the preferred strategy, and should be used with all available measured responses.

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172 Two distinct strategies for reducing the number of design variables were studied in the present work. The approach using equivalent reduced dimensionality models proved to be more robust. The second approach in which only a dominant subset of the design variables was considered active based on the gradients of the objective function with respect to each design variable, proved to be unreliable due to switching in the dominant variable set. More frequent assessment of the dominance of the design variables adds to the computational requirements. The experimental work has shown that the method is able to detect damage in structures if the errors in the measurements are kept within certain bounds. The effect of errors in the measurements was not studied in the present work and is a logical candidate for a further work. The establishment of limits on measurement errors that are compatible with accuracy requirements in the parameter identification process would be the goal of such a study. For future experimental work, the use of a structure for which the response is more in accordance with the original finite element model, is recommended. For this purpose, existing trusses in the commercial market such as the Meroform Construction System M12, a modular truss manufactured by Mero Corporation and used at NASA Langley Research Center for scale models of Space Station [31], can be employed. For dynamic analysis this is particularly

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173 useful as the model correction in this problem would be more involved. Also, the use of more sophisticated electronic or optical measurement devices, which would increase the reliability of the measured data, should be considered in future work. The use of a reduced set of measurements needs further investigation. One of the advantages of the output error approach is the possibility of using only a few dominant measurements to detect damage. As described in section 3.2.2, this approach of working with a reduced set of measurements is not very reliable. A rational method to determine which degrees of freedom should be required to participate in the damage detection process for a particular structure would be a very useful contribution. The iterative optimization methods used in this work are susceptible to convergence to local optima, and the use of different approaches to circumvent the problem of nonconvexities in the design space, particularly the use of non-gradient methods [32], can be attempted in future work. The study of damage detection in composite structures was based only on simulated experimental data. Although the literature in stiffness-reduction techniques is supported by an extensive body of experimental work, the application of this method for damage detection in actual structures needs further validation. In further efforts, experimental analysis of actual composite structures can be performed.

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The study of the influence of damage on the damping characteristics of a structure holds some promise. Damping is a form of energy dissipation, and is sensitive to both microscopic or macroscopic damage in the structure [33]. The difficulties of using this parameter for damage detection in the context of the proposed approach arise from the fact that an analytical model that accurately predicts the damping factor of any structural element is not available. The study would be completely based on experimental data. Identification techniques can be used to derive empirical models for damage propagation in composite structures. The variations in stresses due to different types of damage such as delamination, fiber breakage or matrix cracking, including the extent of damage, could be used towards this end. For extensive damage where the structural topology is assumed to have changed, heuristic methods can be employed in addition to the procedural identification techniques.

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REFERENCES 1. Berman, A. and Flannelly, W. G. , "Theory of Incomplete Models of Dynamic Structures," AIAA Journal . Vol. 9, Aug. 1971, pp. 1481-1487. 2. Baruch, M. and Bar-Itzhack, I. Y., "Optimal Weighted Orthogonal izat ion of Measured Modes," AIAA Journal . Vol. 16, April 1978, pp. 346-351. 3. Berman, A., "Comments on Optimal Weighted Orthogonalization of Measured Modes," AIAA Journal . Vol. 17, Aug. 1979, pp. 927-928. 4. Berman, A., "Mass Matrix Correction Using an Incomplete Set of Measured Modes," AIAA Journal . Vol. 17, Nov. 1979, pp. 1147-1148. 5. Baruch, M. , "Optimization Procedure to Correct Stiffness and Flexibility Matrices Using Vibration Tests," AIAA Journal ^ Vol. 16, Nov. 1978, pp. 12081210. 6. Baruch, M. , "Optimal Correction of Mass and Stiffness Matrices Using Measured Modes," AIAA Journal . Vol. 20, Nov. 1982, pp. 1623-1626. 7. Baruch, M. , "Methods of Reference Basis for Identification of Linear Dynamic Structures," AIAA Journal . Vol. 22, April 1984, pp. 561-563. 8. Chen, J. C. and Garba, J. A., "Analytical Model Improvement Using Modal Test Results," AIAA Journal , Vol. 18, June 1980, pp. 684-690. 9. Berman, A. and Nagy, E. J., "Improvement of a Large Analytical Model Using Test Data," AIAA Journal |. Vol. 21, Aug. 1983, pp. 1168-1173. 10. Kabe, A. M. , "Stiffness Matrix Adjustment Using Mode Data," AIAA Journal . Vol. 23, Sep. 1985, pp. 1431-1436. 11. Sanayei, M. and Nelson, R. B. , "Identification of Structural Element Stiffness from Incomplete Static Test Data," SAE Technical Paper No. 861793, 1986. 175

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176 12. Hanagud, S., Meyyappa, M. and Craig, J. I., "Identification of Structural Dynamic Systems," Recent Trends in Aeroelasticity and Structural Dynamics . Hajela, P. (ed.). University Press, Gainesville, i Florida, 1987, pp. 120-141. 13. Chen, J. C. and Garba, J. A., "On Orbit Damage Assessment for Large Space Structures," Proceedings of the 28th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Monterey, California, April 1987. 14. Smith, S. W. and Hendricks, S. L. , "Evaluation of Two Methods for Damage Detection in Large Space Trusses," Proceedings of the 6th VPI&SU/AIAA Symposium on Dynamics and Control of Large Structures, Blacksburg, Virginia, June 1987. 15. White, C. W. and Maytum, B. D. , "Eigensolution Sensitivity to Parametric Model Perturbation," Shock and Vibration Bulletin . Bulletin 46, Aug. 1976, pp. 123-133. 16. Adams, R. D. , Cawley, P., Pye, C. J. and Stone, B. J., "A Vibration Technique for Non-Destructively Assessing the Integrity of Structures," Journal Mechanical Engineering Science . Vol. 20, No. 2, 1978, pp. 93-100. 17. Lifshitz, J. M. and Rotem, A., "Determination of Reinforcement Unbonding of Composites by a Vibration Technique," J. Composite Materials . Vol. 3, July 1969, pp. 412-423. 18. Cawley, P. and Adams, R. D. , "A Vibration Technique for Non-Destructive Testing of Fibre Composite Structures," J. Composite Materials . Vol. 13, April 1979, pp. 161175. 19. Lee, B. T. , "Measurements of Damping for Nondestructively Assessing the Integrity of Fibre Reinforced Composites," Ph.D. dissertation. University of Florida, Gainesville, 1985. 20. Highsmith, A. L. and Reif snider, K. L. , "StiffnessReduction Mechanisms in Composite Laminates," ASTM STP 775, 1982, pp. 103-117. 21. O'Brien, T. K. and Reifsnider, K. L. , "Fatigue Damage: Stiffness/Strength Comparisons for Composite Materials," Journal of Testing and Evaluation ^ Vol. 5, No. 5, Sep. 1977, pp. 384-393.

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177 22. O'Brien, T. K. , "Characterization of Delamination Onset and Growth in a Composite Laminate," ASTM STP 775, 1982, pp. 140-167. 23. Hsia, T. C, System Identification . Lexington Books, Lexington, Massachusetts, 1977. 24. Rao, S. S., Optimization Theory and Applications . Wiley Eastern, New Delhi, 1979. 25. Vanderplaats, G. N. , Numerical Optimization Techniques for Engineering Design; With Applications . McGraw-Hill Book Company, New York, 1984. 26. Fox, R. L. , Optimization Methods for Engineering Design , Add i son-Wesley Publishing Company, Reading, Massachusetts, 1971. 27. Whetstone, D. , "SPAR Reference Manual," NASA CR145098-1, Feb. 1977. 28. Vanderplaats, G. N. , "ADS A Fortran Program for Automated Design Synthesis," NASA CR-177985, Sep. 1985. 29. Jones, R. M. , Mechanics of Composite Materials . Hemisphere Publishing Company, New York, 1975. 30. Ashton, J. E. and Whitney, J. M. , Theory of Laminated Plates . Technomic Publishing Company, Westport, Connecticut, 1970. 31. Hallauer, W. L. and Lamberson, S. E., "A Laboratory Planar Truss for Structural Dynamics Testing," Experimental Techniques . Vol. 13, Sep. 1989, pp. 24-27. 32. Hajela, P., "Genetic Search An Approach to the Nonconvex Optimization Problem," Proceedings of the 30th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Mobile, Alabama, April 1989. 33. Sun, C. T., Wu, J. K. and Gibson, R. F. , "Prediction of Material Damping of Laminated Polymer Matrix Composites," Journal of Materials Science . Vol. 22, 1987, pp. 1006-1012.

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BIOGRAPHICAL SKETCH Francisco Jose da Cunha Pires Soeiro was born on September 4, 1949, in Rio de Janeiro, Brazil. He is married to Maria Luiza S. Soeiro and has two sons, Renato and Francisco, Jr. , 14 and 12 years old, respectively. He joined the Brazilian Army in 1968 entering the Agulhas Negras Military Academy as a cadet. He graduated in 1971 and was commissioned as lieutenant in the Army Ordnance Corps. In 1976 he entered the Military Institute of Engineering, located in Rio de Janeiro, and received the bachelor's degree in mechanical engineering in 1979. After working for two years in the industrial city of Sao Paulo with Army contractors participating in the process of development and fabrication of equipment for the Army, he was invited to be an instructor at the Military Institute of Engineering. He joined the faculty at this Institute in 1982 and taught undergraduate level courses in the field of strength of materials. He started his master's degree program in 1984 at the same Institute, defending his thesis on "Shells of Revolution Subjected to Nonsymmetric Loads" in June of 1986. After obtaining his Master's Degree in Mechanical Engineering he was selected by the Brazilian Army 178

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to pursue doctoral studies at the University of Florida in the Department of Aerospace Engineering, Mechanics and Engineering Sciences.

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1 I certify that I have read this study and that in my opinion it conforms to acceptable sta?idards of scholarly presentation and is fully adequate, iji scope and quality, as a dissertation for the Degree of Doctor of Philosophy. Prabhat Jlajeila, Chairman Associate PrUfessor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the Degree of Doctor of Philosophy. Chang-Tsan Sun Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the Degree of Doctor of Philosophy. Chen-Chi Hsu Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the Degree of Doctor of Philosophy. David Z iiiBtidrmann Assist^^>-^rof essor of Aerospace Engineering, Mechanics and Engineering Science

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the Degree of Doctor of Philosophy. Fe'rnando Fagund<6 / Associate Professor of Civil Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1990 Ac^Winfred M. Phillips ' Dean, College of Engineering Madelyn M. Lockhart Dean, Graduate School