LICENTIATE T H E S I S

Department of Civil, Environmental and Natural Resources EngineeringDivision of Structural and Construction Engineering

ISSN 1402-1757ISBN 978-91-7583-053-7 (print)ISBN 978-91-7583-054-4 (pdf)

Luleå University of Technology 2014

Tarek E

drees Saaed Structural Identification of Civil E

ngineering Structures

ISSN: 1402-1757 ISBN 978-91-7583-XXX-X Se i listan och fyll i siffror där kryssen är

Structural Identification of Civil Engineering Structures

Tarek Edrees Saaed

Structural Identification of CivilEngineering Structures

Licentiate Thesis

Tarek Edrees Saaed

Supervisors:

Prof. Jan-Erik Jonasson

Associate Prof. George Nikolakopoulos

November 2014

Department of Civil, Environmental and Natural Resources EngineeringDivision of Structural and Construction Engineering

Lulea University of TechnologySE-971 87 LULEAwww.ltu.se/sbn

Printed by Luleå University of Technology, Graphic Production 2014

ISSN 1402-1757 ISBN 978-91-7583-053-7 (print)ISBN 978-91-7583-054-4 (pdf)

Luleå 2014

www.ltu.se

Structural Identification of Civil Engineering StructuresTarek Edrees SaaedDivision of Structural and Construction EngineeringDepartment of Civil, Environmental and Natural Resources EngineeringLulea University of Technology

Academic dissertation

As duly authorized by the Board of the Faculty of Technology at LuleaUniversity for the Degree of Licentiate of Engineering degree will be

publicly defended:

Hall F1031, Lulea University of TechnologyTuesday, November 20, 2014, 10:00

Discussant: Dr. Andreas Andersson

Principal Supervisor: Prof. Jan-Erik Jonasson, Lulea University of Technology

Assistant supervisor: Associate Prof. George Nikolakopoulos, Lulea University ofTechnology

Lulea University of TechnologyLulea 2014www.ltu.se/sbn

This work is gratefully dedicated to:My Mother: your sincere care was beyond any measures,

The Memory of My Father: your memory will not be forgotten,My Brothers and Sisters: your trust is unbelievable,

My Wife: your support is incredible,And My Son and Daughters: your love is fantastic.

Acknowledgments

The research presented in this licentiate thesis was carried out in the research groupof Concrete Structures at the Department of Civil, Environmental and Natural ResourcesEngineering, Lulea University of Technology.

Primarily, I would like to thank my supervisors Associate Professor George Nikolakopou-los and Prof. Jan-Erik Jonasson for their continuously support and encouragement,advice, marvelous suggestions and the continuous daily follow up from George.

I would like to extend my gratitude to Dr. Dimitar Mihaylov and Professor Savka Dinevafor their helpful discussions concerning article A. Thanks are also given to Professor Nad-hir Al-Ansari for his help and support.

Additionally, I am thankful for the time I had with the staff and all colleagues at thedivision of Structural Engineering and to the staff of Lulea university library for theirwonderful support.

I gratefully acknowledge the Department of Civil Engineering, Al-Mosul University andthe Ministry of Higher Education and Scientific Research in Iraq for the PhD scholarshipprovided by them.

Many thanks are also due to all the Iraqi PhD students and all of my colleagues andfriends at Lulea university of Technology for their support and happy time that we had.

Finally, I would like to thank my mother, my wife, my wonderful kids (Abdullah, Dhuha,Kawther, Seedra) for their love, patience, endless support and encouragement during mystudy period.

Tarek, November 2014

Abstract

The assumptions encountered during the analysis and design of civil engineering struc-tures lead to a difference in the structural behavior between calculations based modelsand real structures. Moreover, the recent approach in civil engineering nowadays is torely on the performance-based design approaches, which give more importance for dura-bility, serviceability limit states, and maintenance.Structural identification (St-Id) approach was utilized to bridge the gap between the realstructure and the model. The St-Id procedure can be utilized to evaluate the struc-tures health, damage detection, and efficiency. Despite the enormous developments inparametric time-domain identification methods, their relative merits and performance ascorrelated to the vibrating structures are still incomplete due to the lack of comparativestudies under various test conditions and the lack of extended applications and verifica-tion of these methods with real-life data.This licentiate thesis focuses on the applications of the parametric models and non-parametric models of the System Identification approach to assist in a better under-standing of their potentials, while proposing a novel strategy by combining this approachwith the utilization of the Singular Value Decomposition (SVD) and the Complex ModeIndicator Function (CMIF) curves based techniques in the damage detection of struc-tures.In this work, the problems of identification of the vertical frequencies of the top storeyin a multi-storey building prefabricated from reinforced concrete in Stockholm, and theexistence of damage and damage locations for a bench mark steel frame are investi-gated. Moreover, the non-parametric structural identification approach to investigatethe amount of variations in the modal characteristics (frequency, damping, and modesshapes) for a railway steel bridge will be presented.

Contents

Part I

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aim and objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Main Research questions . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Specific Research questions . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Scientific approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Structural Identification 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Classification of System Identification Methods . . . . . . . . . . . . . . . 102.4 Parametric model structures . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Model Structures using Deterministic Input . . . . . . . . . . . . 132.4.1.1 ARMA Models . . . . . . . . . . . . . . . . . . . . . . . 132.4.1.2 State-Space Models . . . . . . . . . . . . . . . . . . . . . 13

2.4.2 Model Structures using Stochastic Input . . . . . . . . . . . . . . 142.5 Non-parametric model structures . . . . . . . . . . . . . . . . . . . . . . 14

3 Extended summary of appended papers 163.1 Paper One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Paper Two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Paper Three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Paper Four . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Discussion and conclusions 234.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Addressing research questions . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.1 Main Research questions . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.2 Specific Research questions . . . . . . . . . . . . . . . . . . . . . . 244.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.5 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Bibliography 29

Part II

Paper A1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 Comfort levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Paper B 601 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.1 ARMAX model structure and estimation . . . . . . . . . . . . . . 632.2 Transfer function estimation . . . . . . . . . . . . . . . . . . . . . 672.3 Modal parameters estimation . . . . . . . . . . . . . . . . . . . . 68

3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.1 Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.3 Modes Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Paper C 841 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.1 Estimation of the Txy Transfer Function . . . . . . . . . . . . . . 882.2 Modal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Description of Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . 904 Evaluation of Modal Characteristics . . . . . . . . . . . . . . . . . . . . . 925 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Paper D 1021 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022 Parametric model structures . . . . . . . . . . . . . . . . . . . . . . . . . 1033 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.1 Regular Steel Frame . . . . . . . . . . . . . . . . . . . . . . . . . 1063.2 Irregular Steel Frame . . . . . . . . . . . . . . . . . . . . . . . . . 107

4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

i

List of Figures

1.1 Possible applications for the System Identification concept . . . . . . . . 3

2.1 System Identification applications in civil engineering . . . . . . . . . . . 82.2 Kinds of mathematical models . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Classification of System Identification mathematical models . . . . . . . 112.4 A dynamic system with input u(t), output y(t) and disturbance v(t) . . . 122.5 Classification of the parametric models . . . . . . . . . . . . . . . . . . . 122.6 Signal flows for model structures . . . . . . . . . . . . . . . . . . . . . . . 142.7 Methods for the non-parametric models’ computation . . . . . . . . . . . 152.8 A dynamic system with input u(t) and output y(t) . . . . . . . . . . . . 15

3.1 Location of the disturbing vibrations in the building . . . . . . . . . . . . 173.2 The ambient vibration test of the building . . . . . . . . . . . . . . . . . 173.3 The trailer used for forced vibration test of the building . . . . . . . . . . 183.4 Forced vibration test of the building using harmonic loads (counterweight)

of about 300 kg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Steel frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 Mounting an accelerator at the bridge . . . . . . . . . . . . . . . . . . . . 203.7 Aby river railway steel bridge in its original location during train passing 213.8 Aby river railway steel bridge in its temporary location during train passing 21

ii

iii

Part I

The thesis

CHAPTER 1

Introduction

1.1 Background

Most of the civil engineering structures are exposed to vibrational loads throughout theirservice conditions. Owing to the existence of resonance phenomena in these structures,large deformations can result from small forces that might excite one or more of thesestructures resonances. In many cases, such as excessive deformations can cause discom-fort, damage or even a complete failure of the structures. A precise model of the dynamicsof the structure is essential to predict or modify the response of a structure. A famousmathematical model so-called modal model is usually used to achieve this mission.In this model, the behavior of a linear time-invariant system is expressed as: 1) a linearcombination of contributions from the different resonance modes of the structure, 2) themodal parameters, i.e., the damped natural frequency, damping, mode shapes and themodal participation factors are used to describe each mode. The determination of theseparameters is strongly affected by material properties, the geometry and boundary con-ditions of the structure. These parameters are necessary to construct the stiffness andmass matrices from which the modal parameters can be obtained [1–3]. It is very difficultto precisely estimate these parameters for real-life civil engineering structures.Furthermore, the analysis and design procedures of buildings during the last decades werebased mainly on a basic and uncompromising model of structures, For instance; manydesigners model the buildings as plane frames. These simplified procedures performedsuccessfully when used efficiently and produced economic and safe designs. However,these procedures were unable to precisely describe the actual behavior of the real struc-tures. In spite of that, the available modeling tools offer the ability to simulate the three-dimensional performance of the real structures, but the credible behavior of structuresstills needs more than just a refined model. Many examples showed that the differencebetween the simulated and measured responses may be reach up to 500 % and 100 %for local and global responses, respectively because the refined finite-element model of astructure is still affected by the approximation and finite-element assumptions.Moreover, the recent approach in civil engineering nowadays is to rely on the performance-

1

based design approaches, which give more importance for durability, serviceability limitstates, and maintenance.Structural identification (St-Id) approach was utilized to bridge the gap between the realstructure and the model. Basically, the St-Id is the procedure of constructing / updat-ing the finite-element model (physics-based model) from its measured dynamic/staticresponse, which can be utilized to evaluate the structures health, damage detection, andefficiency. St-Id is a transformation of the system identification concept, which is widelyused in electrical and control engineering to obtain a non-physics based model (state-space, differential and/or difference equations) of a dynamic system from its measuredresponse [4]. Figure 1.1 displays different possible applications for the System Identifica-tion concept.

1.2 Aim and objective

Despite the enormous developments in parametric time-domain identification methods,their relative merits and performance as correlated to the vibrating structures are stillincomplete. The reason for this limited knowledge is due to the lack of comparative stud-ies under various test conditions and the lack of extended applications and verification ofthese methods with real life-data.Thus, the main contribution of this Licentiate thesis is to assist in a better understandingof the potentials of the parametric and non-parametric structural identification approach,while proposing a novel strategy by combining this approach with the utilization of theSingular Value Decomposition (SVD) and the Complex Mode Indicator Function (CMIF)curves based techniques in the damage detection of structures.In this work, specific attention is given to the ARMAX parametric models due to itssuperior performance in comparison to the other ARMA models utilized so far in therelated literature.

1.3 Research questions

1.3.1 Main Research questions

RQ1: Implement the parametric and non-parametric structural identification approachto investigate the modal characteristics of civil engineering structures based on the dif-ferent kinds of excitations.

RQ2: Utilize the parametric and non-parametric structural identification approach fordamage detection of civil engineering structures.

2

Computer industry Aircraft industry

Historical tower Building tower

Bridges Oil platform

Figure 1.1: Possible applications for the System Identification concept

3

1.3.2 Specific Research questions

Paper One

RQ1: Utilize the structural identification approach to investigate the frequencies of abuilding based on the three kinds of vibration measurements.

RQ2: What is the reason for the disturbing vibrations in the building?

RQ3: Is the system identification concept suitable to identify the noise levels in an ir-regular multi-storey reinforced concrete building?

RQ4: Which one of the parametric models is the best to predict the mathematical mod-els of vibration’s propagation in the building?

RQ5: Which one of the three kinds of vibration tests gives better results for this building?

RQ6: What is the recommended solution to reduce the level of noise and vibrations inthis building?

Paper Two

RQ1: Propose a novel strategy for damage detection of structures by combining thisARMAX parametric model with the SVD and the CMIF curves based techniques.

RQ2: Is the linear parametric model ARMAX a robust scheme to construct the mobilitymatrix (H)?

RQ3: Are the frequency and modal damping indices suitable to detect the existence ofdamage in the structure?

RQ4: What damage detection index can be used to identify and localize the damage inthe structure?

Paper Three

RQ1: Utilize of the non-parametric structural identification approach to investigate theamount of variations in the modal characteristics of a railway steel bridge based on theambient vibration measurements due to train’s traffic.

RQ2: What is the rate of change in the damping ratio from healthy to collapsed bridge?

RQ3: Is the time-invariant transfer function Txy a powerful model to construct the mo-bility matrix (H)?

4

RQ4: Is the mode shape damage detection index suitable to identify and localize thedamage in the bridge?

RQ5: Is the Singular Value Decomposition (SVD) method applicable to identify howmany significant eigenvalues exist?

Paper Four

RQ1: Theoretical comparison between the different black box linear parametric models(namely, TF, ARX, ARMAX, OE, BJ) to identify the first 10th natural frequencies for abuilding’s frames.

1.4 Scientific approach

The research outlined in this thesis is the outcome of a traditional sequence of steps thatneed to be taken in order to achieve a Licentiate thesis at Lulea University of Technology.The available vibration measurements from the three kinds of vibration tests (providedby Skanska Sweden AB Technology) were utilized to study the vibrations’ propagation ina multi-storey building prefabricated from reinforced concrete in Stockholm. Then, fiveblack box linear parametric time-domain models of structural identification approach,specifically: ARX, ARMAX, BJ, OE and State Space Models have been implementedfor the identification of the vertical frequency in the top storey of the building using thetests’ results (Paper one). Another comparison between the same parametric models hasbeen conducted by comparing their results with Abaqus 6.12 frequency analysis results,and the results showed good agreement (Paper Four).Then, a novel strategy was proposed by combining the ARMAX model with the SVD,and the CMIF curves based techniques for the damage detection of structures, and thusclarifying to what extent damages in a multi-story steel building can be identified byevaluating the changes in the modal parameters (Paper Two). Furthermore, the non-parametric structural identification approach was investigated by a case study. Thus, theFrequency Response Functions (FRF) computed from the quotient of the cross powerspectral density and the power spectral density has been utilized for the purposes ofdamage detection and localization for a railway steel bridge (Paper Three). Finally, thefindings of all articles have been condensed into an extended summary, supported bypeer-reviewed journal and conference papers presented in part 2.

1.5 Limitations

The research has been limited for specific kinds of structural identification methods. So,the efficiency of the identification process was limited by their potentials. More detailsabout the available methods can be found in [5–8].

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1.6 Structure of the thesis

Part 1: It includes an extended summary, which spans from the introduction of thestructural identification approach to the final conclusions and recommendations for fur-ther research. The chapters of Part 1 are briefly described as follow:

Chapter 1: Works as a general introduction, highlight the main and specific researchquestions, and the research method.Chapter 2: Present the theory of structural identification approach.Chapter 3: It includes the extended summaries of the appended papers. The aim is tomake the thesis stand alone and provide a brief summary of all articles within the thesisitself and give them in details as appendices.Chapter 4: Highlight the outcome of the thesis by addressing the research questions,general discussion and conclusions, and future research.

Part 2: It includes the following four appended papers:

Paper One: Comfort level identification for irregular multi-storey building.Paper Two: Identification of building damage using ARMAX model: a parametricstudy.Paper Three: Investigation of changes in modal characteristics before and after damageof a railway bridge: A case study.Paper Four: A comparative study on the identification of building natural frequenciesbased on parametric models.

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CHAPTER 2

Structural Identification

2.1 Introduction

The finite-element method and experimental modal analysis have been intensively usedfor a long time as a tool for the weakness assessment and rehabilitation for civil engineer-ing structures. However, it has been increasingly recognized that FE model developedfrom design drawings has many probable errors sources like: discretization, geometric,numerical computation, shape function, geometry of various finite elements, insufficientrepresentation of structural systems, boundary and continuity conditions, and materialproperties and their variation. Moreover, the ever-growing complexity of structures andthe use of new construction materials impose more limitations on finite-element model’saccuracy.So, the need for Test-validated finite element models is crucial in order to secure the re-quired performance and reliability. Furthermore, owing to the cost considerations, urgentneed to sufficient and reliable methods to evaluate the real conditions of aged infrastruc-ture in order to take the optimal decisions concerning the rehabilitation of this kind ofstructures [9, 10]. Moreover, the general trend in civil engineering nowadays is to evolvefrom specification-based to performance-based engineering due to many reasons [11], forinstance, extreme loading events like earthquakes, hurricanes, and floods call for contin-uous improvement of design methods and procedures.The System Identification concept, which can be simply defined as a modeling of dynamicsystems from experimental data was originated after the Second World War for aircraftand spacecraft industry by Kennedy [12]. The sixties and seventies witness the start ofsystem identification for civil engineering due to the construction of many large structuresand the growing demand to measure the dynamic properties of these structures. One ofthe main motivated factors in the early of seventies is the interest to understand andconsider the dynamic performance of oil offshore structures during design stage [13].Comprehensive information about the early start of system identification and the firstcontributions until 1971 was presented by Astrom and Eykhoff [14] who discussed thor-oughly the methods for identification of linear, non-linear systems and real time identi-

7

fication methods. One of the important contributions of system identification was doneby Hart and Yao [15] who firstly, introduced the concept to the engineering mechanics’researchers and then introduced it to structural engineers by presenting and formulatingthe problem of system identification. Furthermore, they proposed the probable applica-tion of different testing procedures and the potential practical implementation of systemidentification method for damage detection [16].Imai et al. [17] reviewed the fundamentals and methods of parameter identification forlinear and nonlinear behavior in structural dynamics until 1989. According to Cat-bas et al. [4], the system identification can be used to fulfill the different investigationgoals (see Figure 2.1): (1) Design verification and construction planning, (2) a meansof measurement-based delivery of a design-build contract, (3) document as-is structuralcharacteristics to serve as a baseline for assessing any future changes, (4) Load-capacityrating for inventory or special permits, (5) Evaluate possible performance deficiency’scauses, (6) evaluate reliability and vulnerability, (7) designing structural modificationand retrofit or hardening, (8) Health and performance monitoring, (9) Asset manage-ment based on benefit/cost, (10) to help the civil engineers for better understanding ofhow actual structural systems are loaded.Nowadays, applications of system identification in civil engineering are widely spread

Structural

Identification

Understanding

of how actualstructural

systems are

loaded

Load-capacity

rating forinventory

Documentas-is structuralcharacteristics

Means ofmeasurement-

based delivery

Design

verification andconstruction

planning

Structuralcontrol

Asset

managementbased on

benefit/cost

Healthand

performance

monitoring

Designing structural

modification

and retrofit

Evaluatereliability and

vulnerability

Evaluatepossible

performance

deficiency’fscauses

Figure 2.1: System Identification applications in civil engineering

8

especially in the field of damage detection. For instance, Hilbert-Huang transform wasused to detect damage in benchmark buildings [18] and for experimental identificationof bridge health under ambient vibrations [19, 20]. Minami et al. [21] utilized an ARXmodel for system identification of super high-rise buildings using limited vibrations data.Loh et al. [22] implemented the recursive stochastic subspace identification method toidentify the time-varying dynamic properties of the mid-story isolation building utiliz-ing ambient vibration test data while the recursive subspace identification method wasused for same purpose utilizing the earthquake response data. Kampas and Makris [23]applied the Parameter Estimating Method (PEM) to identify the modal characteristics(damping and frequency) of a bridge, compared the results with the previous studies, andthey concluded that the linear models are able to fit the measured response.Valuable information about system identification and its applications in damage detectionare presented by Nagarajaiah [24]. Moreover, Nagarajaiah and coworkers develop a newinteraction matrix formulation and input error formulation, which is important to detectthe presence of damage in structural member up to level 4 (discover the extent of dam-age). A comprehensive overview of system identification principles, recent developmentsand typical case studies for successful applications of system identification to constructedbuildings around the world are well documented in Catbas et al. [4]. Despite enormousdevelopments in parametric time-domain identification methods but their relative meritsand performance as correlated to the vibrating structures still incomplete. The reasonfor this limited knowledge is due to the lack of comparative studies under various testconditions [25] and the lack of extended applications and verification of these methodswith real life data.

2.2 System Identification

The mathematical models are generally used to describe the dynamic systems. The dif-ferential equations are used to define such a model in continuous time systems whilethe difference equations are used in case of discrete time systems. Two approaches arefrequently used to create mathematical models: Physical modeling and System identi-fication. The physical modeling utilizes the physical principles and laws like Newton2law of motion to create these mathematical models. In spite of the good accuracy of thephysical model but it is not suitable for experimental modeling purposes because it isdifficult to measure all the degrees of freedom of the system and the physical model is incontinuous time while the measurements are obtained as a discrete time sample.Moreover, noise from unknown excitation sources or/and measurement’s noise should betaken into consideration in order to have a better representation of the vibrating structure.On the other hand, the System Identification approach is used to develop the mathemat-ical models in the case of limited physical information about the dynamic system. Thus;this model will be able to represent and replicate the behavior of the system based onthe possible previous knowledge and utilizing the input/output data. The accuracy ofsystem identification models depends upon the type of proposed usage [26].There are three structure types of this model (see Figure 2.2): (1) White box model,when the structure is selected based on full physical insight, (2) Black box model, whenthe structure is selected without regard to physical insight, but the selected structure

9

belongs to one of the kinds that proved fruitful performance in the past, (3) Grey boxmodel when some physical insight is available [27, 28].

System Identification

Models

White box model

(full physical insight)

Black box model

(without any physical insight)

Grey box model

(some physical insight)

Physical Models

Mathematical Models

Figure 2.2: Kinds of mathematical models

2.3 Classification of System Identification Methods

One of the important requirements of system identification procedure is to select a spe-cific mathematical model structure and then estimating the parameters of the selectedmodel utilizing the measurements data [26]. There are different classifications for themathematical models of dynamic systems. Generally, system identification methods canbe categorized according to the analysis domain into two main groups: (a) Time Domainrefereeing to the parameter estimation methods utilizing the time histories of the outputsignals which include a number of methodologies like state estimation using a Kalmanfilter, Maximum likelihood, recursive least squares, Recursive instrumental variable, andstochastic analysis and modeling, and (b) Frequency-Domain which includes parameter’sestimation methods based on signal Fourier transform.Anyway, it is always possible to transform signals between these two domains which maygive indication that this division is not real but actually, the practical implementationproved the existence of some differences between the two domains methods. For example,the time-domain methods are signal based; so, they are suitable for use with output onlymodal identification and they are the best for dealing with noisy data and free from mostof the signal processing errors like leakage, and better conditioned than the frequency-domain counterpart due to the effect of the powers of frequencies in frequency domainequations. In contrast, averaging is easier and more efficient for noisy measurement con-ditions in frequency-domain [24,29,30].Moreover, they can be classified as: (1) Univariate model (Single input-single output)andMultivariable models (Single input-several output), (2) Linear models-nonlinear mod-els, (3) Deterministic models-stochastic models, (4) Time invariant models-time varyingmodels, (5) Lumped models-distributed parameter models, (6) Discrete time models-continuous time models, (7) Time domain models-frequency domain models, (8) Para-metric models-nonparametric models [28, 31]. Figure 2.3 depicts these kinds of mathe-matical models of the dynamic systems. The last category is related to the subject of the

10

current thesis, so more information about the last kind of classification will be given inthe following sections.

Lumped

&Distributed

Linear&

Nonlinear

Time invariant &

Time varying

Time domain&

Frequency domain

Univariate&

Multivariable

Discrete time&

Continuous time

Deterministic

&Stochastic

Parametric&

Nonparametric

MathematicalModels of

Dynamic Systems

Figure 2.3: Classification of System Identification mathematical models

2.4 Parametric model structures

Power spectral density function (PSD) is usually used to determine the strength of thevariations (energy) as a function of frequency. More precisely, PSD is a very useful toolto identify oscillatory signals in time series data and specify their amplitude because itdisplays at which frequencies variations are strong and at which frequencies variationsare weak. Generally, there are two main methods of power spectral density estimation:parametric and non-parametric.Parametric methods usually adopt some kind of signal model prior to calculation of thepower spectral density estimate. Thus, it is supposed that some knowledge of the signalis known in advance. In this case, the mathematical models are assumed to be composedof set parameters to be estimated by system identification. This mathematical model

11

takes the form of differential equation in case of a linear and time-invariant continuous-time system while the corresponding discrete-time is in the form of difference equation.Figure 2.4 displays a typical dynamic system subjected to input u(t) and the responseof the system is described by the output y(t) which is affected by disturbance v(t). It isclear from this figure that the output is a combination of input and disturbance. It isworth mentioning that the disturbance cannot be controlled, and even the input may beunknown and uncontrollable in some kind of systems.So, according to whether the excitation of the structural system is measured or not, andexcitation type (stationary, impulse or step), Andersen [28] divided the parametric modelstructures into two main categories: 1)Model Structures using Deterministic Input, 2)Model Structures using Stochastic Input, and established the model structure suitable forthe second category which is called the Auto-Regressive Moving average Vector (ARMAV)model. Figure 2.5 presents the classification of the parametric models.

Systemy(t) : outputu(t) : input

v(t) : disturbance

+

Figure 2.4: A dynamic system with input u(t), output y(t) and disturbance v(t)

Parametric Models

Deterministic Input Stochastic Input

ARMA Models

ARX ARMAX Output Error Box-JenkinsTransfer-

Function

State-Space

Models

Distributed

Parameter ModelsARMAV model

Deterministic

state-space

Stochastic

state-space

Figure 2.5: Classification of the parametric models

12

2.4.1 Model Structures using Deterministic Input

This approach is used when the input is measured and in this case the parametric modelwill have a deterministic term as well as a stochastic term that defines the unknowndisturbance. The Auto-Regressive Moving Average with eXternal input (ARMAX) givenby Eq.2.1 below is the general input/output model structure for modeling of linear andtime-invariant dynamic systems excited by deterministic input.

y(t)=G(q)u(t)+H(q)e(t) (2.1)

whereG(q) and H(q) are the transfer functions of the deterministic part and the stochasticpart, while e(t) represents the stochastic input corresponding to the noise and predictionerrors.

Three different approaches are available to solve Eq.2.1 in terms of θ as shown in thefollowing subsequent sections:

2.4.1.1 ARMA Models

The direct way of parameterizing G and H is to consider them as rational functions andutilize the parameters as the numerator and denominator coefficients. This approach canbe fulfilled using different techniques, which are generally known as black-box models.These models include: ARX model, ARMAX model, Output Error Model Structure(OE), Box-Jenkins Model (BJ), and Transfer-Function Model (see Figure 2.5). Thecorresponding signal flows for these models are depicted in Figure 2.6. Full derivationsof these models can be located in the references [27, 31].

2.4.1.2 State-Space Models

The major difference between ARMA models and state-space models is that, ARMAmodels only describe the input-output behaviour of the system, while the internal struc-ture of a system is described also in state-space representation [28, 30]. In this case, therelationship between the input, the output, and noise is provided by a system of first-order differential or difference equations utilizing an auxiliary state vector x(t). Generally,the dynamic behavior of a general Multi-Degree-Of-Freedom (MDOF)structure can bedescribed by the following set of linear, second order differential equations written inmatrix form Eq.2.2:

[M ]{q(t)}+[C]{q(t)}+[K]{q(t)}={f(t)} (2.2)

[M ], [C] and [K] refer to the mass, damping and stiffness matrices; {q(t)}, {q(t)} and{q(t)} are the acceleration, velocity and displacement vector respectively, while {f(t)} isthe forcing vector. In this matrix equation (Eq.2.2), it was assumed that the structure islinear, time invariant (i.e. [M ], [C] and [K] are constant), observable system with viscousor proportional damping. So, the equations of motion coupled in this formulation can bedecoupled by solving an eigen problem. Thus, the superposition of the eigen solutionscan be used to obtain the solution [30, 32].This second order equation of motion (Eq.2.2) is converted into two first order equations(state equation and observation equation) using the state space models.

13

(a)ARXmodel

(b)ARMAXmodel

(c)OE model

(d)BJ model

(e)TF model

Figure 2.6: Signal flows for model structures

2.4.2 Model Structures using Stochastic Input

When the input is unknown, then the input and disturbance will be defined by a singlestochastic term. In this case, the ARMAX model cannot be used and instead the suitablemodel structure will be the Auto-Regressive Moving Average (ARMA) given in Eq.5.2which uses one transfer function H(q) to describe both the system dynamical propertiesand the noise.

y(t) = H(q)e(t) (2.3)

The model structure is called an Auto-Regressive Moving average Vector (ARMAV)model.

2.5 Non-parametric model structures

Non-parametric methods are used when little information about the signal is availablein advance. They usually have less computational complexity than parametric models.Even of application simplicity of the nonparametric methods but their accuracy is limited,

14

and the parametric methods should be used when it is required to obtain an accuratemodel of the system [31]. In this case, there are many methods for computing PSD andthe description is done using curves, functional relationships or tables using one of thefollowing analysis methods depicted in Figure 2.7: (1) Transient analysis, (2) Frequencyanalysis, (3) Correlation analysis, (4) Spectral analysis.When the excitation is transient (impulse or step excitation), Transient analysis is used

Methods for

Non-parametricModels

Frequency analysisSpectral analysisTransient analysis

Tuy

Correlation analysis

Figure 2.7: Methods for the non-parametric models’ computation

to identify the dynamic behavior of the system. While Frequency analysis is used withdeterministic, periodic or pseudo-random and periodic excitations. This method involvestransforming the measured excitation and corresponding system response to the frequencydomain, then the frequency response function is calculated as the ratio of the transformedexcitation and response. The correlation and spectral analysis are used with a stationarystochastically (white-noise input) excited system. In these methods, the excitation andthe system response can be described either by the correlation functions in time domainor the spectral densities in frequency-domain [28,31]. All of these methods are based onFourier transform methods (FFT) due to its speed and reliability.In this thesis, The transfer function is computed utilizing Matlab ”tfestimate” function,which is uses the Welch method (pwelch) to estimate the cross-power spectral density ofthe input x and output y (Pyu) and the PSD Puu of u. Figure 2.8 presents a typicaldynamic system, subjected to input u(t), the response of the system is described by theoutput y(t), while it is worth mentioning that this non parametric model Txy does nottake under consideration the disturbances of the system. for this system, the transfer

Figure 2.8: A dynamic system with input u(t) and output y(t)

function can be obtained from the quotient of the cross power spectral density Pyu of uand y and the power spectral density Puu of u as given in Eq.2.4 below:

Tuy=Pyu

Puu

(2.4)

15

CHAPTER 3

Extended summary of appended papers

An extended summary of the journal- and/or conference papers is presented here. Thepapers can be found in part II of this thesis.

3.1 Paper One

Comfort Level Identification for Irregular Multi-storey Building Subjectedto Vibrations.Aauthors: Tarek Edrees Saaed, George Nikolakopoulos, Jan-Erik JonassonJournal paper accepted for publication in the Automation in Construction Journal inDecember 2013

Summary: This article concludes that the ARMAX model and Output Error modelstructure showed excellent performance to predict the mathematical models of vibra-tion’s propagation in the building (see Figure 3.1) compared with the other models forthe three types of tests. Concerning the test type, the measurements of the ambientvibration test for this building and using these kinds of parametric models was unsuc-cessful to obtain a prediction for the dynamic behavior of the structures, while the forcedvibration test of the building (see Figure 3.2, Figure 3.3, and Figure 3.4 ) showed betterperformance but it is still unable to capture the structure behavior. Tests such as theforced vibration test on the rails can be used to excite the structures and obtain sat-isfactory results. All the test types and model structures used were able to identify aconcentration in the vertical frequency within the range of (7.5 - 12.5) Hz.As a general conclusion, it can be stated that the reason for the building tent’s complainthas been the resonance between the generated low frequencies and the human body partsfrequencies. In addition, all the values of floor acceleration have been within the accept-able limits, which confirm the previous theoretical and experimental studies performed

16

Disturbingvibrations

Figure 3.1: Location of the disturbing vibrations in the building

Figure 3.2: The ambient vibration test of the building

17

Figure 3.3: The trailer used for forced vibration test of the building

Figure 3.4: Forced vibration test of the building using harmonic loads (counterweight) ofabout 300 kg

on the building. The authors highly recommend reducing the level of noise and vibra-tions at the source using track dampers and other means for track’s vibration reductioninstead of using the vibration isolation devices at the foundation level of the building orthe viscous damping system inside the building.

3.2 Paper Two

Identification of Building Damage Using ARMAX Model: A parametricstudy.Aauthors: Tarek Edrees Saaed, George Nikolakopoulos, Jan-Erik JonassonJournal paper submitted to the Mechanical Systems and Signal Processing Journal inMarch 2014

18

Summary: In this article, the Structural Identification approach is used to identify

X

Y

Z

Figure 3.5: Steel frame

the existence of damage and damage locations for a bench mark steel frame depictedin Figure 3.5. The black box linear parametric models called Auto-Regressive MovingAverage with eXternal input (ARMAX) were utilized for the construction of the Fre-quency Response Functions (FRF), based on simulation results. Two damage scenarioswere assumed for damages in the frame. The efficiency of the estimated models andtheir suitability for describing the dynamical behavior of the frame were proven. In thesequel, the magnitudes’ part were utilized to construct the mobility matrix (H), whilethe phase’s part were utilized to plot the modes shapes of the frame. The Singular ValueDecomposition (SVD) method was adopted to identify how many significant eigenval-ues exist and plot the Complex Mode Indicator Function (CMIF) for the complete frame.Three damage indices were adopted to evaluate the state of damage in the frame, namely:frequency, modal damping, and modes shapes. The results indicated that the linear para-metric model ARMAX is a robust scheme to construct the mobility matrix (H), whilethe identified frame’s natural frequencies are very close to the theoretical ones, obtainedfrom Abaqus’s frequency analysis. Additionally, the frequency and modal damping in-dices were very successful to indicate the existence of damage in the frame, while themode shape index can detect the existence of damage and identify the frame damagelocations.

19

3.3 Paper Three

Investigation of changes in modal characteristics before and after damageof a railway bridge: A case study.Aauthors: Tarek Edrees Saaed, George Nikolakopoulos, Niklas Grip, Jan-Erik JonassonJournal paper submitted to the Structural Health Monitoring Journal in August 2014

Summary: This study presents the results of the modal identification of the Aby alvrailway steel bridge in Sweden. The ambient response of the bridge was recorded (seeFigure 3.6) for the functional bridge and after it was tested to failure as it is depictedin Figure 3.7 and Figure 3.8 respectively. The aim of this research effort is to imple-

Figure 3.6: Mounting an accelerator at the bridge

ment the structural identification approach to investigate the amount of variations in themodal characteristics (frequency, damping, and modes shapes) for the case study, whilethis has been achieved by the comparison between the modal characteristics for the func-tional bridge and for the same bridge after failure. The Frequency Response Functions(FRF) were obtained from the identified transfer functions, which were computed fromthe quotient of the cross power spectral density and the power spectral density. In thesequel, the magnitude part of the FRF has been utilized to construct the mobility matrix(H), while the phase’s part was utilized for plotting the mode shapes of the bridge. TheSingular Value Decomposition (SVD) method has been adopted to identify how manysignificant eigenvalues exist by plotting the Complex Mode Indication Function (CMIF)for the bridge before and after the collapse. The obtained results depicted that the lin-ear, time-invariant transfer function Txy is a powerful model to construct the mobilitymatrix (H). Furthermore, the proposed procedure is reliable and can be utilized furtherand efficiently for the purposes of damage detection and localization. The rate of change

20

Figure 3.7: Aby river railway steel bridge in its original location during train passing

Figure 3.8: Aby river railway steel bridge in its temporary location during train passing

in the damping ratio from healthy to collapsed bridge was about (206%), and this ratecould be regarded as an index for the existence of a serious damage in steel bridges.

21

3.4 Paper Four

A Comparative Study on the Identification of Building Natural FrequenciesBased on Parametric Models.Aauthors: Tarek Edrees Saaed, George Nikolakopoulos, Jan-Erik JonassonPresented at the 33rd IASTED International Conference on Modelling, Identification andContrl, MIC 2014, Austria, p.p.1-6, February 2014

Summary: The analysis and design of civil engineering structures is a complex problem,which is based on many assumptions to simplify these operations. This in turn, leads toa difference in the structural behavior between calculations based models and real struc-tures. Structural identification was proposed by many researchers as a tool to reduce thisdifference between models and actual structures. Moreover, Parametric models and non-parametric models were used intensively for system identification by many researchers.In this research effort, the system identification concept is utilized to identify the natu-ral frequencies for a steel building’s frames. Different black box linear parametric modelssuch as Transfer Function model (TF), Auto-Regressive model with eXternal input model(ARX), Auto-Regressive Moving Average with eXternal input (ARMAX) model, OutputError model structure (OE), and Box-Jenkins model (BJ) were examined for identifyingthe first 10th natural frequencies for the building’s frames, based on simulation results.Abaqus 6.12 finite-element software was utilized to perform the time history analysis forthe examples and the obtained responses at one point of the roofs (assumed as a sen-sor) were further processed by the parametric models to obtain the building’s naturalfrequencies based on the Abaqus time history analysis results (assumed as a measure-ments). After that, Abaqus 6.12 was utlized again to perform another analysis, which iscalled frequency analysis to obtain the building’s natural frequencies and mode shapesbased on the stiffness and mass (not the measurements) of the buildings. The resultsshowed that the linear parametric models TF, ARX, ARMAX, OE, and BJ are robustto identify the natural frequencies of building and they are recommend for future work.

22

CHAPTER 4

Discussion and conclusions

4.1 Introduction

The main purpose behind the current licentiate thesis is to acquire a general understand-ing in the field of Structural Identification. Then, utilize this knowledge into two areas:1) the damage detection of civil engineering structures, 2) the area of structural controlof civil engineering structures in order to enhance their performance under the effect ofdifferent kinds of dynamic loads.

4.2 Addressing research questions

4.2.1 Main Research questions

RQ1: Implement the parametric and non-parametric structural identification approachto investigate the modal characteristics of civil engineering structures based on the dif-ferent kinds of excitations.Answer: The system identification concept has been used successfully applied to inves-tigate the modal characteristics of civil engineering structures.

RQ2: Utilize the parametric and non-parametric structural identification approach fordamage detection of civil engineering structures.Answer: The system identification concept has been efficient for damage detection andlocalization in civil engineering structures.

23

4.2.2 Specific Research questions

Paper One

RQ1: Utilize of the structural identification approach to investigate the frequencies of abuilding based on the three kinds of vibration measurements.Answer: The system identification concept has been used successfully to investigate thefrequencies of the building.

RQ2: What is the reason for the disturbing vibrations in the building?Answer: The reason for the disturbing vibrations is the concentration in the verticalfrequencies within the range of (7.5 - 12.5) Hz for the 10th floor only, which is close tothe frequencies of human body parts.

RQ3: Is the system identification concept suitable to identify the noise levels in an ir-regular multi-storey reinforced concrete building?Answer: Yes, the system identification concept has been used successfully to identifythe noise levels in an irregular multi-storey reinforced concrete building and its resultsshowed good agreement with the results obtained by the classical methods of vibrationpropagation calculations.

RQ4: Which one of the parametric models is the best to predict the mathematical mod-els of vibration’s propagation in the building?Answer: The ARMAX model and the Output Error model have indicated an excellentperformance to predict the mathematical models of vibration’s propagation in the build-ing.

RQ5: Which one of the three kinds of vibration tests gives better results for this build-ing?Answer: The forced vibration test on the rails gives the best results.

RQ6: What is the recommended solution to reduce the level of noise and vibrations inthis building?Answer: The recommended solution is to reduce the level of noise and vibrations at thesource instead of the building using track dampers and other means for track’s vibrationreduction.

Paper Two

RQ1: Propose a novel strategy for damage detection of structures by combining thisARMAX parametric model with the SVD and the CMIF curves based techniques.Answer: The proposed strategy showed efficient performance to describe the dynamicalbehavior of the test structure.

RQ2: Is the linear parametric model ARMAX a robust scheme to construct the mobilitymatrix (H)?

24

Answer: Yes, the results indicated that the linear parametric model ARMAX is a ro-bust scheme to construct the mobility matrix (H).

RQ3: Are the frequency and modal damping indices suitable to detect the existence ofdamage in the structure?Answer: Yes, the results indicated that the frequency and modal damping indices arevery successful at indicate the existence of damage in the frame.

RQ4: What damage detection index can be used to identify and localize the damage inthe structure?Answer: The mode shape damage detection index can be used successfully to identifyand localize the damage in the structure.

Paper Three

RQ1: Utilize the non-parametric structural identification approach to investigate theamount of variations in the modal characteristics of a railway steel bridge based on theambient vibration measurements due to train’s traffic.Answer: The system identification concept has been used successfully to investigate theamount of variations in the modal characteristics of a railway steel bridge.

RQ2: What is the rate of change in the damping ratio from healthy to collapsed bridge?Answer: The rate of change in the damping ratio from healthy to collapsed bridge isabout (206%).

RQ3: Is the time-invariant transfer function Txy a powerful model to construct the mo-bility matrix (H)?Answer: Yes, the results indicated that the time-invariant transfer function Txy is arobust scheme to construct the mobility matrix (H).

RQ4: Is the mode shape damage detection index suitable to identify and localize thedamage in the bridge?Answer: Yes, the modes shape for the collapsed bridge (calculated from the collapsedbridge’s measurements) showed good agreement with the actual damage shape and loca-tions.

RQ5: Is the Singular Value Decomposition (SVD) method applicable to identify howmany significant eigenvalues exist?Answer: Yes, the Singular Value Decomposition (SVD) method is applicable to identifyhow many significant eigenvalues exist by plotting the Complex Mode Indication Func-tion (CMIF) for the bridge before and after the collapse.

25

Paper Four

RQ1:Theoretical comparison between the different black box linear parametric models(namely, TF, ARX, ARMAX, OE, BJ) to identify the first 10th natural frequencies for abuilding’s frames.Answer: The results showed that the linear parametric models TF, ARX, ARMAX,OE, and BJ are robust to identify the natural frequencies of building, and the resultsshowed good agreement with Abaqus 6.12 frequency analysis results.

4.3 Contributions

Structural identification of civil engineering structures is a broad subject of research, andit is still an open research area. The main contribution of this Licentiate thesis is toassist in a better understanding of the potentials of the parametric and non-parametricstructural identification approach, while proposing a novel strategy by combining thisapproach (the ARMAX model and Txy model) with the utilization of the SVD, andthe CMIF curves based techniques in the damage detection of structures. The currentlicentiate thesis includes four articles and their contributions to the field of structuralidentification are clarified as follows:

1. Paper One: Comfort level identification for irregular multi-storey build-ing Subjected to Vibrations.In this article, the system identification concept has been used to measure the noiselevels in an irregular multi-storey reinforced concrete building in Stockholm. Theblack box linear parametric models, as transfer-function models (ARX, ARMAX,BJ, OE) and State Space Models have been utilized to identify the comfort levelin the top storey of an irregular (i.e., long and narrow wedge-shaped) prefabricatedfrom reinforced concrete building using three kinds of vibration tests, namely theambient vibration test and two types of forced vibration tests, while a comparisonamong them has also been made. The novelty of this research is to use the differentavailable vibration measurements to contribute in the study of the relative meritsand performance of the well-established identification methods as correlated to thevibrating structures.

2. Paper Two: Identification of building damage using ARMAX model:a parametric study.The contribution of this article is to assist in a better understanding of the poten-tials of using ARMAX modelling, while proposing a novel strategy by combiningthis approach with the utilization of the SVD, and the CMIF curves based tech-niques in the damage detection of structures, and thus clarifying to what extentdamages in a multi-story steel building can be identified by evaluating the changesin the modal parameters. The novel proposed methodology, for evaluating the dam-age detection, is based on the following procedures: a) implementing an ARMAXmodel for predicting the Frequency Response Functions (FRF), b) constructing the

26

mobility matrix (H) from the predicted FRF and utilizing them for damage identi-fication and localization. The proposed method is similar to the Frequency DomainDecomposition (FDD), which is usually encountered in the field of structural identi-fication. The difference is FDD using the power spectral density matrix for singularvalue decomposition, while in this paper, the ARMAX model is used to constructthe FRF between the input and output.

3. Paper Three: Investigation of changes in modal characteristics beforeand after damage of a railway bridge: A case study.In this article, the structural identification approach is utilized to investigate theamount of variations in the modal characteristics (frequency, damping, and modeshapes) and the existence of damage and damage locations for a railway steel bridgebased on the ambient vibration measurements due to trains’ traffic, before and af-ter its known early stage of failure. This bridge is located in the north of Sweden,and such an application and study are quite important, especially when propos-ing novel approaches in the damage detection and experimentally evaluating them,which according to the authors’ viewpoint is not possible every time. During theanalysis, the percentage of change in the modal damping as an indication for theassessment of damage existence for other railway steel bridges, will be also derived.The transfer functions obtained from the quotient of the cross PSD, and the PSDare utilized to obtain the Frequency Response Functions (FRF) based on ambientvibration measurements. Due to the rectangular shape of mobility matrix (H),the Singular Value Decomposition (SVD) method will be adopted to identify howmany significant eigenvalues exist and plot the Complex Mode Indicator Function(CMIF) for the whole bridge.

4. Paper Four: A comparative study on the identification of building nat-ural frequencies based on parametric models.This conference article serves as pilot study to evaluate the performance of dif-ferent black box linear parametric models such as Transfer Function model (TF),Auto-Regressive model with eXternal input model (ARX), Auto-Regressive Mov-ing Average with eXternal input (ARMAX) model, Output Error model structure(OE), and Box-Jenkins model (BJ).

4.4 Concluding remarks

The main conclusion from this Licentiate thesis is that the proposed strategy based oncombining the ARMAX model and Txy model with the utilization of the SVD, and theCMIF curves based techniques have shown a very efficient performance in the field ofdamage detection of civil engineering structures. Another important conclusion is thatthe ARMAX model and Output Error model structure showed excellent performance topredict the mathematical models of vibration’s propagation in the buildings. Moreover,tests such as the forced vibration test on the rails can be used to excite the structures

27

and obtain satisfactory results.Furthermore, the sharp increases in the amplitudes of the CMIF curves revealed the ex-istence of damage in a structure, while the high percentage of change in modal dampingcan clearly indicate the existence of a serious damage in that structure.

4.5 Future research

For the future research, it is recommended to improve the proposed methodology by in-cluding the detection of damage up to levels 3 and 4 (size of damage and actual safety ofthe structure respectively). Moreover, it is recommended to consider the modal partici-pation factors for each mode to overcome the main drawback of the proposed strategy,which is its limited capability to handle the closely spaced modes by specifying the contri-bution from each mode. Furthermore, during the current research, the gained knowledgeof the structural identification approach will be utilized to design a semi-active controllerfor civil engineering applications.

28

CHAPTER 5

Bibliography

29

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32

Part II

The appended papers

Paper One

Comfort Level Identification for Irregular

Multi-storey Building Subjected to Vibrations

Aauthors:Tarek Edrees Saaed, George Nikolakopoulos, Jan-Erik Jonasson

Journal paper accepted for publication in the Automation in Construction Journal

(edited to fit the format of the thesis - content has not been altered)

Comfort Level Identification for Irregular Multi-storey BuildingSubjected to Vibrations

Tarek Edrees Saaed, George Nikolakopoulos, Jan-Erik Jonasson

Keywords: Structural system identification, Comfort level, Vibrations mitigation, Struc-tural dynamic, Time frequency methods.

Abstract

In this article, the System Identification approach is being used to identify the vertical fre-quencies of the top storey in a multi-storey building prefabricated from reinforced concrete inStockholm. Before building construction, detailed investigation indicated that the building willnot be affected by train vibrations from the nearby railway yard. After building completion,disturbing vibrations were observed in the building. Three measurement types namely: ambientvibration test, forced vibration test on the rails, and forced vibration test had been performedby different vibration measuring laboratories in order to specify the probable reasons for thesevibrations. Five methods of structural identification approach, specifically: ARX, ARMAX,BJ, OE and State Space Models have been implemented for the identification process in thisstudy using the tests’ results. All the test types and model structures utilized identified a con-centration in the vertical frequencies within the range of (7.5 - 12.5) Hz for the 10th floor only,which is close to the frequencies of human body parts. Furthermore, the article concludes thatthe ARMAX model and the Output Error model have indicated an excellent performance topredict the mathematical models of vibration’s propagation in the building, when comparedwith other models used from the three types of tests. In addition, the results of the aforemen-tioned system identification methods implemented for this study have indicated that there areno other reasons for the disturbing vibrations still observed in the building. Furthermore, theresults confirmed the correctness of the previous theoretical and experimental results obtainedby different specialists who stated that the values of floor acceleration are within the acceptablelimits, and the probable reason for any disturbance is the resonance between the generated lowfrequencies and the human body parts frequencies. Moreover, the authors highly recommendreducing the level of noise and vibrations at the source instead of the building using trackdampers and other means for track’s vibration reduction.

1 Introduction

Mechanical vibrations in structures generated by many sources such as motion of different typesof heavy machines, railway systems, construction activities, earthquakes and storms have seriouseffects on these structures. For example, trains traffic on tracks results in noise and vibrations,which are usually regarded as one and the similar discipline since both are resulted from theoscillations of rails and wheels during trains movement on the track [1]. At lower frequencies,the excitation energy will transmit through the ground in the form of vibrations. At the end,

40

these vibrations will be attenuated in the human body, and its energy will be absorbed by thedifferent human bodys organs. While for the higher frequencies; the energy will expand throughthe air in the form of noise. Finally, both the vibrations and noise will cause a discomfort tothe buildings inhabitants [2].Actually, one of the main features of the modern life is the high noise level all around the world,and thus it is very important to take the necessary precautions to avoid the existence of suchnoise in the buildings due to its effects on the mental and physical well-being of humans. Forinstance, the percentage of people who are living in noisy environments, with sound pressurelevels more than 65 dB and subjected to its effects; only in Europe is about 10 % [3]. A self-answered questionnaire was carried out by [4] to evaluate the peoples subjective responses tonoise in a hospital in Spain revealed that the noise levels affected even the recovery of patientsin the hospital. Furthermore, a study performed by [5] showed that more than 25% of theinhabitants of Messina city in Italy are extremely disturbed by noise resulted from road traffic.Thus, in order to successfully protect the people from the noise pollution, the need is urgentfor advanced well-aimed measures.A State-of-the-Art study conducted by [6] called the attention to the deficiency of standardsin the estimation of ground borne noise and conflict in the handling of low-frequency vibrationin other standards. The spread of noise in Klaipeda city in Lithuania was examined by [7] inorder to identify the source of bigger noise areas in the city and evaluate the measured noiselevels against the permissible standards. The performance requirements for comfort levels,safety, serviceability, and security were evaluated using the fuzzy probabilistic optimizationapproach for a building by [8]. Moreover, a deterministic model depends on the Monte Carlosimulation method was used by [9] to estimate the magnitude and frequency of the noise levelsproduced by construction equipment and their effects on an eight-storey parking garage inLondon. Furthermore, a three-dimensional numerical model was presented by [10] to estimatethe vibrations in the free field from excitation caused by trains traffic in the subways usingthe finite-element method. Also, the wave number finite and boundary element methods weresuccessfully used by [11] to predict the disturbance to buildings inhabitants from trains traffic.

The research efforts on utilizing the system identification approach to evaluate the noiselevels in buildings until now have been very limited. Nowadays, the most applications of thesystem identification concept, in the field of civil engineering, are mainly in the area of damagedetection [12–19]. However, system identification methods have promising capabilities and canbe used to predict mathematical models to define the dynamics of many aspects in a building,like temperature, humidity and CO2 level of a room, which are parameters that can be veryhelpful to perform a sequential control of the Heating, the Ventilation and the Air Conditioning(HVAC) [20]. Moreover, system identification methods have been utilized successfully in me-chanical engineering to study the contribution of noise sources, determining the transmissionpaths by defining the transfer function between the sources and responses and developing amathematical model that defines the dynamics of the noise propagation [21].Despite the enormous developments in parametric time-domain identification methods, theirrelative merits and performance as correlated to the vibrating structures are still incomplete.The reason for this limited knowledge is due to the lack of comparative studies under varioustest conditions [22] and the lack of extended applications and verification of these methods withreal life data.In this article, the system identification concept has been used to measure the noise levels inan irregular multi-storey reinforced concrete building. The black box linear parametric mod-els, as transfer-function models (ARX, ARMAX, BJ, OE) and State Space Models have beenutilized to identify the comfort level in the top storey of an irregular (i.e., long and narrowwedge-shaped) prefabricated from reinforced concrete building using three kinds of vibration

41

tests, namely the ambient vibration test and two types of forced vibration tests, while a com-parison among them has also been made. The novelty of this research is to use the availablevibration measurements to contribute in the study of the relative merits and performance of thewell-established identification methods as correlated to the vibrating structures. Moreover, thisarticle investigates the potentials of the identification methods usually encountered in electricaland mechanical engineering applications in order to benefit from their capabilities and widentheir applications in the civil engineering field.So, the system identification approach was used to calculate the noise levels in the structuredue to different kinds of excitation to validate the results obtained by the classical method ofvibration propagation calculations already conducted by some specialist and to explore whetherthere are other reasons for the disturbing vibrations still observed in the building. This article isorganized as it follows: In Section 2, a short review about comfort levels will be presented. Sec-tion 3 will establish the mathematical models for the dynamic systems utilized in the presentedcomfort level identification processes. In the sequel, the case study and the measurements con-ducted will be depicted in Section 4, while section 5 outlines the Methodology adopted in thisstudy. The results of the system identification approaches will be presented in Section 6, whilethe conclusion will be drawn in Section 7.

2 Comfort levels

Actually, most of the buildings are exposed to ambient vertical vibrations in the range between(1-80) Hz [23], while the horizontal frequency for high-rise buildings lays in the range of (0.1-15)Hz [24]. In general, the evaluation of these noise levels is a complex task, while the methodsspecified by the related standard regulations are tricky and involve complicated calculations [25].The scientific methods that have been appeared in the field of predicting noise in structures canbe classified generally into the following three categories, based on the frequency content: 1)Finite-Element Analysis (FEA) which is used in the case of low frequencies [26], 2) StatisticalEnergy Analysis (SEA), which is utilized for high frequencies [27], and 3) Hybrid FEA-SEA [28].In addition to these methods, there have been various empirical and semi-empirical methods asit can be investigated in the following references [25, 29,30].From a different consideration, human body parts have distinct natural frequencies. However,for vertical vibrations with frequencies less than 2 Hz, the human body may be consideredas one mass, while in case of high frequencies the body different parts have their distinctivefrequencies, and it can be defined as a lumped mass model [31].In such a body consideration, vertical vibrations in the range between 4-8 Hz are the mostimportant because they may cause resonance in some of the internal parts of the body, whileTable 1 displays some indicative human body parts’ frequencies when subjected to vertical vi-brations [32].

Acceptable limits of human exposure to vibrations are dependent on the time of day, thenature of activities in the place, and the direction utilized by vibrations to enter the humanbody as it is presented in Figure 1 [23].

Overall, weighted Root-Mean-Square acceleration (RMS) in each orthogonal axis is used toevaluate the vibrations levels. Acceptable values for continuous and impulsive vibration accel-eration according to the British standards BS 6472 are being presented in Table 2 [33].

42

Table 1: Human body parts frequencies when subjected to vertical vibrations [32]

Body part name Natural frequencies (Hz)

Abdominal mass 4-8Arm 5-10Chest wall 5-10Eyeball 20-90Hand 30-50Head (axial mode) 20-30Legs-flexing knees 2Legs-rigid posture 20Shoulder 4-5Spinal column (axial mode) 10-12

Figure 1: Orthogonal axes for assessment of human exposure to vibration

3 System Identification

In general two approaches are frequently used to create mathematical models of processes: Phys-ical modeling and System identification. The physical modeling utilizes the physical principlesand laws like Newton second law of motion to create these mathematical models. In spite ofthe good accuracy of the physical model obtained by this approach, it is not suitable for exper-imental modeling purposes because it is difficult to measure all the degrees of freedom of thesystem and the physical model is in continuous time, while the measurements are obtained indiscrete time. Moreover, noise from unknown excitation sources or/and measurement’s shouldbe taken into consideration in order to have a better representation of the vibrating structure.On the other hand, the System Identification approach is used to develop the mathematicalmodels in the case of limited physical information about the dynamic system. Thus, this modelwill be able to represent and replicate the behavior of the system based on the possible previousknowledge and by utilizing the obtained input/output data [35].

A typical dynamic system is depicted on Figure 2, subjected to the input u(t) and the re-sponse of the system is described by the output y(t), which is affected by disturbance v(t). Theinput u(t) is produced by the environment and affects the system. This input is assumed toinclude any known or measured excitation. While, the output y(t) is produced by the systemitself and affects the environment. All contributions to the output y(t) that are not producedby the linear system need to be part of the disturbance v(t) and for this reason, its contribu-tion in the dynamic system was added after the system block (see Figure 2). The disturbancev(t) might comprise non-linear and/or unmodelled system behavior, persistent excitation, theoutput from any non-measured inputs, and the measurement noise. It is worth mentioning thatthe disturbance cannot be controlled and even the input may be unknown and uncontrollable insome kind of systems. According to whether the excitation of the structural system is measuredor not and the excitation type (stationary, impulse or step), the parametric model structures

43

Table 2: Preferred and maximum weighted RMS values for continuous and impulsivevibration acceleration (m/s2) 180 Hz [33,34]

Location Assessment period Preferred values Maximum values

Z-axis X-& Y-axes Z-axis X-& Y-axes1. Continuous vibration

Critical areas Day-or night-time 0.0050 0.0036 0.0100 0.0072

ResidencesDaytimeNight-time

0.0100.007

0.00710.005

0.0200.014

0.0140.010

Offices, schools,educationalinstitutionsand places ofworship

Day-or night-time 0.0200 0.0140 0.0400 0.0280

Workshops Day-or night-time 0.04 0.029 0.080 0.058

2. Impulsive vibrationCritical areas Day-or night-time 0.0050 0.0036 0.0100 0.0072

ResidencesDaytimeNight-time

0.3000.100

0.2100.071

0.6000.200

0.4200.140

Offices, schools,educationalinstitutionsand places ofworship

Day-or night-time 0.6400 0.4600 1.2800 0.9200

Workshops Day-or night-time 0.6400 0.4600 1.2800 0.9200

can be divided into the two main categories [36]: model structures for stochastic input andmodel structures for deterministic input, while the last model will be adopted in this work.

Systemy(t) : outputu(t) : input

v(t) : disturbance

+

Figure 2: A dynamic system with input u(t), output y(t) and disturbance v(t)

The model structures for deterministic input are used when the input is measured and inthis case the parametric model will have a deterministic term as well as a stochastic term thatdefines the unknown disturbance. The Auto-Regressive Moving Average with eXternal input(ARMAX) model, provided by Eq.5.1 below is the general input/output model structure forthe modeling of linear and time-invariant dynamic systems, excited by a deterministic input

44

and defined as:y(t)=G(q)u(t)+H(q)e(t) (5.1)

where G(q) and H(q) are the transfer functions of the deterministic part and the stochastic partrespectively, u(t) represents the input signal, y(t) is the output signal and e(t) is the white noise(equation error). The parameters in the transfer function in Eq.5.1 are determined during thesystem identification process. The vector θ is usually used to designate these parameters andthe system description given in Eq.5.1 can be rewritten in the following form:

y(t)=G(q,θ)u(t)+H(q,θ)e(t) (5.2)

There are two different approaches available to solve Eq.5.2 in terms of θ including: a) Transfer-function models and b) State-space models. The Transfer-function models approach can befulfilled using different techniques, which are generally known as black-box models. Thesemodels include: ARX model, ARMAX model, Output Error model Structure (OE), and Box-Jenkins model, while for the case of State-space models, the relationship between the input, theoutput, and the noise is provided by a system of first-order differential or difference equationsutilizing an auxiliary state vector x(t). More details about these methods and their derivationscan be found in [37,38].

4 Case Study

The building constructed in 2008 is located in central Stockholm between Railway yard in theeast and Barnhus Bridge in the west as presented in Figure 3. The building, which is prefabri-cated concrete structure, is a long and narrow wedge-shaped structure. The potential impact ofvibrations generated by the passage of trains on the building has been the subject of in-depthdiscussions and underwent several tests to ascertain the level of these vibrations and under thatthe final decision was made to establish the building because studies have indicated that thevibrations generated will not exceed the limits allowed by the comfort levels specifications.The doubts were correct, directly after building completion, disturbing vibrations were ob-

Barnhus -bron

Bangård

TorsgatanNorra

Bantorget

Vasastan

Figure 3: Building location [39]

served in the building. Specifically, the vibrations were concentrated on the level 10th in thearea on either side of the stabilizing lattice truss (line 58) shown in Figure 4. The vibrationswere judged to be related to train passing in the rail yard area as it was expected in the early

45

MP 1,3,5

MP 12

MP 20 (Plan 10)

MP 30 (Plan 5)

Line 58

MP 5

MP 3

MP 1

MP 20

MP 12,15,30

Torsgatan

Bangård

Section Plan

Figure 4: Location of measuring points in the plan and section [39]

planning stages.During planning stages and after completing the construction of the building, several differenttypes of measurements have been carried out by different vibration measuring laboratories tostudy the characteristics of these vibrations. One of the important results from these measure-ments is that the building has several resonance frequencies in the range of 5 to 10 Hz, whichmeans that ground vibrations can be transferred into the building and may be enhanced. Moredetails about the previous investigations are available in reference [39] and its appendices.

From these measurements, the three measurement types that have been conducted aftercompleting the construction of the building were: a) ambient vibration test, b) forced vibrationtest on the rails, and c) forced vibration test.Ambient vibration test: responses of the building have been recorded at a large number of trainpassages. Types of trains, tracks and direction have been systematically recorded in order tofind correlations between the disturbance in the building and the type of train caused them.In total, a set of 15 sensors were utilized for recording the responses of the embankment andthe building, 6 of the donors were placed in the embankment outside the building and near therailway, 3 for measuring responses in the vertical direction and 3 for measuring responses in thehorizontal direction across the track, while the other nine donors were placed in the buildingand measured responses in the vertical direction. Figure 4 presents the location of the Measure-ment Points (MP) on the yard (MP1, MP3 and MP5, sensors in the vertical direction) and aselection of sensors on the plan 10 (MP 20), level 8 (MP 15), level 5 (MP30) and plan 2 (MP12).

Forced vibration test on the rails: Transport Agency has developed a Support Stand thatcan excite a sinusoidal load at various frequencies. The trailer has been used to measuresimultaneously the responses of both the embankment and the building. The track was loadedwith two point loads (one on each rail) with a harmonic varying load for a few minutes on astationary state into existence. The load was varied at four different frequencies at each siteof action, while measurements were performed in a total of three measuring points on the twotracks 24 loads. The amplitude of the load on the track (consisting of two point loads) wasconstant over the loading, and ranged from about 20 to 30 kN between the various loads.Forced vibration test: to determine the building structure dynamic properties, the building

46

body has been excited with a dynamic load, while the response has been measured in severaldifferent points in the building. To minimize the interference in the measurements, the test wasconducted at the night when there is limited traffic. The building frame was excited with knownharmonic loads (counterweight) of about 300 kg with different frequencies (5 Hz to 10 Hz), andwith a sweeping sinusoidal load (4 Hz to 12 Hz). The exciter is biased so that the counterweightdoes not burden the floor. In this approach, some parts of the trusses between floor 3 and 4 wereuncovered and accessible (Figure 5) and they were considered to be appropriate to be loadedfor a simultaneous excitation over a large part of the building, while for practical reasons, trusswas chosen in line 61. The responses were measured by accelerometers on floors 2, 4, 8 and 10.The arrangement for all of the 15 sensors is depicted in Figure 5.

Floor Plan 4

y

x

Principle section Line 61

Z

y

Figure 5: Location of the forced vibration test [39]

5 Methodology

Three kinds of vibration tests have been used to identify the noise levels in the building, and acomparison between the obtained results have been conducted in order to investigate the effi-ciency of the examined approach in identifying the buildings vertical frequencies. Well-plannedand performed measurements are the basis for successful implementation of the system identi-fication process. It is very important to ensure that good-quality signals have been recorded.Furthermore, adequate measuring time period, sampling rate, number of sensors, and optimumlocation of sensors has been used. Valuable information about the optimum number of sensorsand their locations are given in [40, 41]. In the current study; the main target is to determinethe frequency of vertical vibrations at exactly specified location (i.e., level 10th in the area oneither side of line 58 as it is shown in Figure 4).For this reason, the sensors were directly allocated to the study area. More specifically, a multi-channel data logger with fifteen channels of accelerometers has been utilized for recording theresponses of the building at five occasions during November and December 2009 and with a

47

sampling rate of 1200 samples per second. A Matlab code was written to read the recordedtime history signals. The measurements were used without any filtration to prevent any biasinto the results due to the personal opinion of the researchers. Each one of the recorded signalswas split into two parts to use the first half for model estimation (assumed as input) and thesecond half for model validation (assumed as output). The Matlab function called merge wasimplemented for grouping up the input data sets into a single object in order to use them formodel estimation. The same operation was repeated for the output data sets in order to preparethe model validation set.In the sequel, another Matlab function named dtrend was utilized to subtract the mean valuesfrom each signal. In order to check for the correctness of the predicted models, the best-fitmethod, which gives an indication about the model efficiency to represent the main system dy-namics and whether the linear simulation is appropriate, has been used. Moreover, a comparisonbetween the grouped estimation data sets and one of the grouped validation sets (arbitrarilyselected) has been conducted to find the Best-Fit percentage. Finally, the five different time-domain identification methods, which have been presented in Section 3, namely: a) ARX (LeastSquares approach), b) ARMAX (Prediction Error Method approach), c) Output Error (Pre-diction Error Method approach), d) Box-Jenkins (Prediction Error Method approach), and e)State Space (subspace iteration technique-N4SID) were implemented for each vibration test.

6 Results

For the forced vibration test, 18 of 23-minutes vertical accelerations time-history records havebeen employed in the identification process. Samples of the original recorded data are presentedin Figure 6 noting that the excitation source is at the 4th floor. The Best Fit approach results,

0 5 10 15 20 25 30 35 40 45 50 55−0.01

0

0.01

Time (Seconds)

Acc

. (m

/s2 )

10th

Fl.

0 5 10 15 20 25 30 35 40 45 50 55−0.01

0

0.01

Time (Seconds)

Acc

. (m

/s2 )

8th

Fl.

0 5 10 15 20 25 30 35 40 45 50 55−0.05

0

0.05

Time (Seconds)

Acc

. (m

/s2 )

4th

Fl.

0 5 10 15 20 25 30 35 40 45 50 55−5

0

5x 10

−3

Time (Seconds)

Acc

. (m

/s2 )

2th

Fl.

Figure 6: Samples of the original recorded data for the forced vibration test inside thebuilding

for this kind of vibrations test, showed that all the predicted model structures used are unable to

48

capture the dynamic behavior of the building, while the ARMAX, OE, and BJ models produceapproximately equal performance and superior to the one obtained from the ARX and StateSpace models as it has been indicated in Figure 7. However, all the model structures excluding

Forced vibration Ambient Vibration Forced vibration on the rails0

10

20

30

40

50

60

Type of measurements

Ave

rage

Fit

%

14.2

21.7 21.920.2

12

1.643.49 3.3 3.85

0.82

45

51.5

46.9

52.850.5

ARX

ARMAX

BJ

OE

State Space

Figure 7: Comparison of the performance of the three tests using ARX, ARMAX, BJ,OE, and SS model approaches

the State Space models showed a concentration of vertical frequency values in the range of (7.5 -12.5) Hz for the 10th floor as it is clear from the Bode plots of the five predicted models (averageof eighteen measurements) in Figure 8. In addition, Figure 8 displays the frequency-responsefunction for one sample of the actual recorded data for reference only. While the responses atfloor eight and floor five depicted in Figure 9 and Figure 10 respectively, did not reveal anyconcentration in the vertical frequencies at these floors.Moreover, for the case of measurements for the ambient vibration test, 71 of 26-minutes verti-

cal accelerations time-history records have been utilized in the identification process. Samples ofthe original recorded data are presented in Figure 11, which is clearly revealed, a magnificationin the acceleration response from the 5th floor to the 10th floor.The obtained Best-Fit results from this test showed that all the model’s structures used wereunable to capture the dynamic behavior of the building but still the ARMAX, OE, and BJmodels produced approximately equal performance and at the same time better identificationcapability, when compared with the ones obtained by the ARX and the State Space modelsas it is depicted in Figure 7. Figure 12 displays the Average Bode plots for the models ofthis test (average of 71 measurements) for the 10th floor. In this case, all the models utilizedhave depicted a concentration in the vertical frequency values within the range of (7.5 - 12.5)Hz. In addition, Figure 12 displays the frequency-response function for one sample of the actualrecorded data for reference only. The responses at floor eight and floor five depicted in Figure 13and Figure 14 respectively, did not reveal any concentration in the vertical frequencies at thesefloors. Finally, in the forced vibration test on the rails, 34 of 39-minutes vertical accelerationstime-history records are used in the identification process. Samples of the original recorded dataare presented in Figure 11, which is clearly revealed, a magnification in the acceleration responsefrom the 5th floor to the 10th floor. The Best-Fit algorithm results have been clearly better thanthe results obtained by the previous two test cases, and it can be utilized to capture the dynamicbehavior of the building as it is depicted in Figure 7. In addition, the ARMAX, OE, and State

49

0 5 10 15 20 25 30 35 40 45 50-40

-30

-20

-10

0

10

20

Mag

nitu

de (

dB)

Frequency (Hz)

ARX

ARMAX

BJ

OE

State Space

Frequency-response

Figure 8: Average Bode plots of the five predicted models for the 10th floor using forcedvibration test

0 5 10 15 20 25 30 35 40 45 50-40

-30

-20

-10

0

10

20

30

Ma

gn

itu

de

(dB

)

Frequency (Hz)

ARX

ARMAX

BJ

OE

State Space

Frequency-response

Figure 9: Average Bode plots of the five predicted models for the 8th floor using forcedvibration test

Space models produce approximately an equal performance and a better one, when comparedto the ones obtained by the ARX and BJ models. From Figure 16, which displays the averageBode plot for the models of this test for the 10th floor , clearly there is a concentration in thevertical frequency within the same range of (7.5 - 12.5) Hz. In addition, Figure 16 displays thefrequency-response function for one sample of the actual recorded data for reference only. The

50

h

0 5 10 15 20 25 30 35 40 45 50-35

-30

-25

-20

-15

-10

-5

0

Mag

nit

ud

e (d

B)

Frequency (Hz)

ARX

ARMAX

BJ

OE

State Space

Frequency-response

Figure 10: Average Bode plots of the five predicted models for the 5th floor using forcedvibration test

0 10 20 30 40 50 60 70 80 90−0.05

0

0.05

Time (Seconds)

Acc.

(m/s2 )

10th Fl. (MP20)

0 10 20 30 40 50 60 70 80 90−0.02

0

0.02

Time (Seconds)

Acc.

(m/s2 )

8th Fl. (MP15)

0 10 20 30 40 50 60 70 80 90−0.01

0

0.01

Time (Seconds)

Acc.

(m/s2 )

5th Fl. (MP30)

0 10 20 30 40 50 60 70 80 90−0.2

0

0.2

Time (Seconds)

Acc.

(m/s2 )

Embankment (MP3)

Figure 11: Samples of the original recorded data for the ambient vibration test insideand outside the building

responses predicted by some models at floor eight and floor five as depicted in Figure 17 andFigure 18 respectively, revealed a concentration in the vertical frequencies close to the range of(7.5 - 12.5) Hz at these floors.

51

Frequency (Hz)0 5 10 15 20 25 30 35 40 45 50

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Ma

gn

itu

de (

dB

)

ARX

ARMAX

BJ

OE

State Space

Frequency-response

Figure 12: Average Bode plots of the five predicted models for the 10th floor using ambientvibration test

0 5 10 15 20 25 30 35 40 45 50-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Ma

gn

itu

de (d

B)

Frequency (Hz)

ARX

ARMAX

BJ

OE

State Space

Frequency-response

Figure 13: Average Bode plots of the five predicted models for the 8th floor using ambientvibration test

The summary of the predicted vertical frequency ranges and acceleration amplitude rangesof the five prediction models for the three test types is shown in Table 3.

52

0 5 10 15 20 25 30 35 40 45 50-70

-60

-50

-40

-30

-20

-10

0

Ma

gn

itu

de (d

B)

Frequency (Hz)

ARX

ARMAX

BJ

OE

State Space

Frequency-response

Figure 14: Average Bode plots of the five predicted models for the 5th floor using ambientvibration test

0 10 20 30 40 50 60−0.01

0

0.01

Time (Seconds)

Acc

. (m

/s2 )

10th Fl. (MP20)

0 10 20 30 40 50 60−5

0

5x 10−3

Time (Seconds)

Acc

. (m

/s2 )

8th Fl. (MP15)

0 10 20 30 40 50 60−5

0

5x 10−3

Time (Seconds)

Acc

. (m

/s2 )

5th Fl. (MP30)

0 10 20 30 40 50 60−0.01

0

0.01

Time (Seconds)

Acc

. (m

/s2 )

Embankment (MP3)

Figure 15: Samples of the original recorded data for the forced vibration test on the railsinside and outside the building

The absolute value of the maximum amplitude (floor acceleration) from this Table is 110.0dB (0.00011 m/s2), which is too much lower than the preferred and maximum overall weightedRoot-Mean-Square acceleration (RMS) values for continuous and impulsive vibration accelera-

53

0 5 10 15 20 25 30 35 40 45 50-60

-50

-40

-30

-20

-10

0

10

Ma

gn

itu

de

(dB

)

Frequency (Hz)

ARX

ARMAX

BJ

OE

State Space

Frequency-response

Figure 16: Average Bode plots of the five predicted models for the 10th floor using forcedvibration test on the rails

0 5 10 15 20 25 30 35 40 45 50-60

-50

-40

-30

-20

-10

0

Ma

gn

itu

de

(dB

)

Frequency (Hz)

ARX

ARMAX

BJ

OE

State Space

Frequency-response

Figure 17: Average Bode plots of the five predicted models for the 8th floor using forcedvibration test on the rails

tion during day or night times and for any building usage shown in Table 2. From the obtainedresults, it is possible to mention that the reason for building tent’s complaint is the generationof some low frequencies within the range of resonance of (7.5 - 12.5) Hz, which is very close to

54

0 5 10 15 20 25 30 35 40 45 50-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Ma

gn

itu

de

(dB

)

Frequency (Hz)

ARX

ARMAX

BJ

OE

State Space

Frequency-response

Figure 18: Average Bode plots of the five predicted models for the 5th floor using forcedvibration test on the rails

Table 3: Summary of the predicted vertical frequency ranges and acceleration amplituderanges of the five prediction models for the three test types

Test LocationModels Structures

ARX ARMAX OE BJ State SpaceMin. Max. Min. Max. Min. Max. Min. Max. Min. Max.

Forced ∗1 Freq. 0.0 43.9 0.0 50 8.59 50 0.0 42.9 0.0 50vibration ∗2 acc. -22.9 19.2 -25.4 40.5 -62.1 32.2 -69.4 24.3 -65.9 32

∗3 Av. Freq. 9.6 8.59 8.59 23.7 8.08∗4 Av. acc. -11.2 -0.816 14 -14.5 -14.8

Ambient ∗1 Freq. 0.0 20.0 0.0 13.1 0.0 46.0 0.0 46.5 0.0 50vibration ∗2 acc. -53.6 14.3 -31.6 10 -22.3 95.9 -26.3 110 -53.6 16.8

∗3 Av. Freq. 9.09 8.59 7.58 13.1 10.6∗4 Av. acc. -39.2 -22.4 0.614 -12.1 -52.3

Forced ∗1 Freq. 0.0 16.7 0.0 17.7 0.0 19.2 0.0 25.3 0.0 50vibration ∗2 acc. -33.6 7.58 -75.2 24.6 -22 36.6 -29.4 15.2 -28.2 38.9test on ∗3 Av. Freq. 9.09 8.59 8.08 3.54 8.59the rails ∗4 Av. acc. -3.21 -7.58 -21.7 0.625 9.16∗1 Frequency range in (Hz).∗2 Floor acceleration range in (dB).∗3 Average Frequency over all the measurements (Hz).∗4 Average Floor acceleration over all the measurements in (dB).

55

the human body parts frequencies shown in Table 1. Thus, the resonance is responsible for thedisturbance, while all the values of floor acceleration are within the acceptable limits.In order to eliminate this disturbing noise and vibrations in the structure, the authors highlyrecommend reducing the level of noise and vibrations at source instead of using the vibrationisolation devices at the foundation level of the building or the viscous damping system insidethe building, which are mainly used for reduction of the horizontal vibration’s component only.For instance, Track dampers and other means for track’s vibration reduction can be used inthis case.

7 Conclusions

This article concludes that the ARMAX model and Output Error model structure showed excel-lent performance to predict the mathematical models of vibration’s propagation in the buildingcompared with the other models for the three types of tests. Concerning the test type, themeasurements of the ambient vibration test for this building and using these kinds of paramet-ric models was unsuccessful to obtain a prediction for the dynamic behavior of the structures,while the forced vibration test of the building showed better performance but it is still unable tocapture the structure behavior. Tests such as the forced vibration test on the rails can be usedto excite the structures and obtain satisfactory results. All the test types and model structuresused were able to identify a concentration in the vertical frequency within the range of (7.5 -12.5) Hz.As a general conclusion, it can be stated that the reason for the building tent’s complaint hasbeen the resonance between the generated low frequencies and the human body parts frequen-cies. In addition, all the values of floor acceleration have been within the acceptable limits,which confirm the previous theoretical and experimental studies performed on the building.The authors highly recommend reducing the level of noise and vibrations at the source usingtrack dampers and other means for track’s vibration reduction instead of using the vibrationisolation devices at the foundation level of the building or the viscous damping system insidethe building.

Acknowledgements

The authors would like to thank Skanska Sweden AB Technology and the design manager Mr.Per-Ola Svahn for providing us with vibration measurements and allowing us to use them inour research. The first author would like to acknowledge the Ministry of Higher Education andScientific Research - University of Mosul-Iraq for the PhD studentship provided by them.

56

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59

Paper Two

Identification of Building Damage Using ARMAX

Model: A parametric study

Aauthors:Tarek Edrees Saaed, George Nikolakopoulos, Jan-Erik Jonasson

Journal paper submitted to the Mechanical Systems and Signal Processing Journal in March2014.

(edited to fit the format of the thesis - content has not been altered)

Identification of Building Damage Using ARMAXModel: A parametric study

Tarek Edrees Saaed, George Nikolakopoulos, Jan-Erik Jonasson

Keywords: Structural Identification; Damage Detection; Non-Destructive Evaluation; AR-MAX model; Structural Dynamic.

Abstract

In this article, the Structural Identification approach is used to identify the existence of damageand damage locations for a bench mark steel frame. The black box linear parametric modelscalled Auto-Regressive Moving Average with eXternal input (ARMAX) were utilized for theconstruction of the Frequency Response Functions (FRF), based on simulation results. Twodamage scenarios were assumed for damages in the frame. The efficiency of the estimatedmodels and their suitability for describing the dynamical behavior of the frame were proven.In the sequel, the magnitudes’ part were utilized to construct the mobility matrix (H), whilethe phase’s part were utilized to plot the modes shapes of the frame. The Singular ValueDecomposition (SVD) method was adopted to identify how many significant eigenvalues existand plot the Complex Mode Indicator Function (CMIF) for the complete frame. Three damageindices were adopted to evaluate the state of damage in the frame, namely: frequency, modaldamping, and modes shapes. The results indicated that the linear parametric model ARMAXis a robust scheme to construct the mobility matrix (H), while the identified frame’s naturalfrequencies are very close to the theoretical ones, obtained from Abaqus’s frequency analysis.Additionally, the frequency and modal damping indices were very successful to indicate theexistence of damage in the frame, while the mode shape index can detect the existence ofdamage and identify the frame damage locations.

1 Introduction

The Finite-Element (FE) and the experimental modal analysis methods have been intensivelyused for a long time as a tool for the weakness assessment and rehabilitation for civil engi-neering structures. However, it has been increasingly recognized that a FE model developedfrom design drawings, has many probable error sources like: discretization, geometric, numeri-cal computation, shape function, geometry of various finite elements, insufficient representationof structural systems, boundary and continuity conditions, and material properties and theirvariations [1]. Moreover, the ever-growing complexity of structures and the use of new construc-tion materials impose more limitations on the finite-element model’s accuracy. Many examplesshowed that the difference between the simulated and measured responses may be reach up to500% and 100% for local and global responses respectively, because the refined finite-elementmodel of a structure is still affected by the approximation and finite-element assumptions [2].Furthermore, owing to the cost considerations, there is an urgent need for sufficient and reliablemethods for evaluating the real conditions of aged infrastructures in order to take the optimal

60

decisions concerning their rehabilitation, and thus, the need for Test-validated finite-elementmodels is crucial in order to secure the required performance and reliability [1, 3].

Moreover, the general trend in civil engineering nowadays is to evolve from specification-based to performance-based engineering due to many reasons, such as the extreme loadingevents like earthquakes, hurricanes, and floods, which call for a continuous improvement of de-sign methods and procedures [4]. Comprehensive information about the early start of structuralidentification and the first contributions until 1971 were presented by Astrom and Eykhoff [5],who discussed thoroughly the methods for identification of linear, non-linear systems and realtime identification methods. One of the important contributions of system identification wasdone by Hart and Yao [6], who firstly, introduced the concept to the engineering mechanics’researchers and then introduced it to structural engineers by presenting and formulating theproblem of system identification. Furthermore, they proposed the probable application of dif-ferent testing procedures and the potential practical implementation of system identificationmethod for damage detection [7]. According to Catbas et al. [2], system identification canbe utilized to fulfil different investigation goals, for instance: (1) design verification and con-struction planning, (2) means of measurement-based delivery of a design-build contract, (3)document as-is structural characteristics to serve as a baseline for assessing any future changes,(4) Load-capacity rating for inventory or special permits, (5) evaluate possible performance de-ficiency’s causes, (6) evaluate reliability and vulnerability, (7) designing structural modificationand retrofit or hardening, (8) health and performance monitoring, (9) asset management basedon benefit/cost, and (10) to help the civil engineers for better understanding of how actualstructural systems are loaded.

Nowadays, applications of system identification in civil engineering are widely spread, espe-cially in the field of damage detection, while the overall aim of this article is to utilize the blackbox linear parametric model called Auto-Regressive Moving Average with eXternal input AR-MAX model for damage identification and localization. The importance of the ARMAX modelsemployed in the current study and in comparison with the ARMA models utilized so far in therelated literature, comes from the following facts: 1) the ARMAX model structure involves dis-turbance dynamics and have more flexibility in the handling of disturbance modeling than theother parametric stochastic time series models because they offer to model deterministic andstochastic parts of the system independently, 2) they are helpful when the dominating distur-bances enter early in the process (for instance, at the input), 3) the ARMAX model structurerelies not only on the present value of the input and output, but the history of both, and 4) itintroduces several different variants and techniques available to handle a variety of cases [8, 9].Thus, the contribution of this article is to assist in a better understanding of the potentials ofusing ARMAX modelling, while proposing a novel strategy by combining this approach withthe utilization of the SVD and the CMIF curves based techniques in the damage detection ofstructures, and thus clarifying to what extent damages in a multi-story steel building can beidentified by evaluating the changes in the modal parameters. The novel proposed methodology,for evaluating the damage detection, is based on the following procedures: a) implementing anARMAX model for predicting the Frequency Response Functions (FRF), b) constructing themobility matrix (H) from the predicted FRF and utilizing them for damage identification andlocalization.

For instance, Mitsuru et al. demonstrated the efficiency of using a neural network-basedapproach to detect damage in a steel building in Japan [10]. Capecchi & Vestroni studied themonitoring of structural systems by using frequency data and clarify when it is enough to mea-sure natural frequencies only or natural frequencies and modal shapes to detect damage [11].Xia et al. [12] outlined a statistical method with combined uncertain frequency and mode shapedata for structural damage identification, and their experimental tests proved the efficiency of

61

this method to detect damage existence. Li et al. [13] utilized empirical mode decompositionand wavelet analysis for damage detection of a 4-storey shear building model. Da Silav et al. [14]used fuzzy clustering method for classification of structural damage. Valuable information aboutsystem identification and its applications in damage detection are presented by Nagarajaiah.Furthermore, Nagarajaiah and coworkers developed a new interaction matrix formulation andinput error formulation to detect the presence of damage in the structural member up to level4 (discover the extent of damage) [15]. Hilbert-Huang transform was used to detect damage inbenchmark buildings [16] and for experimental identification of bridge health, under ambientvibrations [17, 18].

Additionally, Lee and Yun [19] estimated only the modal parameters using ARMAX mod-els. Vector ARMA models were utilized by Anderson [20] for identification of civil engineeringstructures. Bodeux and Golinval [21] used the Auto-Regressive Moving Average Vector (AR-MAV) model for system identification and damage detection of buildings. Xing and Mita [22]proposed a substructure approach based on ARMAX model that permits for the local damagedetection of a shear structure. Minami et al. [23] utilized an ARX model for system identifi-cation of super high-rise buildings using limited vibrations data. Loh et al. [24] implementedthe recursive stochastic subspace identification method to identify the time-varying dynamicproperties of the mid-story isolation building by utilizing ambient vibration test data, whilethe recursive subspace identification method was used for the same purpose by utilizing theearthquake response data. Kampas and Makris [25] applied the Parameter Estimating Method(PEM) to identify the modal characteristics (damping and frequency) of a bridge, compared theresults with the previous studies, and concluded that the linear models are able to fit the mea-sured responses. Furthermore, a comprehensive overview of the system identification principles,recent developments and typical case studies for successful applications of system identificationto constructed buildings around the world are well documented in [2].

Despite the enormous developments in parametric model’s identification methods, their rel-ative merits and performance as correlated to the vibrating structures are still incomplete. Thereason for this limited knowledge is due to the lack of comparative studies under various testconditions [26] and the lack of extended applications of these methods with real life data. In thepresent work, the Structural Identification approach is used to identify the existence of damageand damage locations for a bench mark steel building’s frame. The ARMAX modeling wasutilized for the construction of FRF based on simulation results. Abaqus 6.12 finite-elementsoftware was utilized to perform the time history analysis for the case under study and theobtained responses at 110 different locations (assumed as a sensors) correspond to the endsof columns and mid of beams were further processed by the parametric models to obtain thebuilding’s FRFs based on the Abaqus analysis results (assumed as measurements).

The damage in the frame was simulated by reducing the modulus of elasticity (E) for specificcolumns by 75% of the undamaged ones. Two damage scenarios were assumed for damages inthe frame. Damage scenario no.1 consist of increasingly destroying the columns of the groundfloor, while for the damage scenario no.2, the damage locations were changed by destroying col-umn no.1 in a floor, and this was repeated for five stories. The efficiency of the estimated modelsand their suitability for describing the dynamical behaviour of the frame were proved. Then,the magnitudes’ part, which represents the Frequency Response Function (FRF) were utilizedto construct the mobility matrix (H), while the phases’ part were used to plot the modes shapesof the frame. Due to the rectangular shape of (H), the singular value decomposition (SVD)method was adopted to identify how many signicant eigenvalues exist and plot the ComplexMode Indicator Function (CMIF) for the complete frame. Three damage indices were adoptedto evaluate the state of damage in the frame, namely: frequency, modal damping, and Modes

62

Shapes.

Overall, this article is structured as it follows. Section 2 outlines the article’s methodology,while in Section 3, a numerical example was provided to demonstrate the application of theARMAX model for damage detection. Finally, the conclusions were drawn in Section 4.

2 Methodology

There are a lot of categories of damage detection methods, according to the technique utilizedto identify the damage from the measured data. Valuable information about damage detec-tion methods, their levels, and damage indicators are available in references [10, 11, 27–33].Frequency’s changes method will be adopted in the current paper, since shifts in natural fre-quencies were widely used to detect damage in structures. This method mainly depends on theconcept that variations in the structural properties will result in alterations in the vibration fre-quencies and amplitudes. In spite of the significant practical limitations of this method [27,29],it has been utilized by many researchers to detect damage, especially in applications wherethe frequency shifts can be measured very precisely or there is an expectation of large levelsof damage, and thus in the current approach, it was assumed that these two conditions wereapplicable in the current research effort. Figure 1 depicts the overall methodology proposedthroughout this article for damage identification and localization.

First of all, a n-story shear frame is simulated, in the following scenarios and cases:

Ssj with sj=1,....,sn

where Ssj represents the damage scenario and sn is the number of damage scenarios considered.

Ccj , cj=1,....,cm

where Ccj is the damage cases and cm represents the number of damage cases considered foreach of damage scenarios. Moreover, u(t) is the excitation, y(t)i , i=1,....,k is the response ofthe building at the ki location, where y(t)i represents the output and k is the total number ofmeasurement points.

Then, the responses of the frame (assumed as measurements), due to each one of the damagecases, will be utilized to obtain the transfer functions of the frame by the ARMAX parametricmodel, while considering the frame as a linear system for all the measurement points (sensors),as it will be presented in the following subsection.

2.1 ARMAX model structure and estimation

A typical dynamic system is presented at Figure 2, subjected to input u(t) and the responseof the system to be described by the output y(t), which is affected by the disturbance v(t).It is worth mentioning that the disturbance cannot be controlled and even the input may beunknown and uncontrollable in some kind of systems. According to whether the excitation ofthe structural system is measured or not, and the excitation type (stationary, impulse or step),the parametric model structures can be divided into two main categories [20]: a) model struc-tures for stochastic input, and b) model structures for deterministic input. Eq. 1 below denotesthe general input/output model structure for modeling of linear and time-invariant dynamicsystems, excited by a deterministic input:

y(t)=G(q)u(t)+H(q)e(t) (1)

63

Figure 1: Proposed damage diagnosis scheme

64

System

y(t) : outputu(t) : input

v(t) : disturbance

+

Figure 2: A dynamic system with input u(t), output y(t) and disturbance v(t)

where G(q) and H(q) are the transfer functions of the deterministic part and the stochastic partrespectively, u(t) represents the input signal, y(t) is the output signal and e(t) is the white noise(equation error). The parameters in the transfer function in Eq. (1) are determined during thesystem identification process. The vector θ is usually utilized to designate these parameters andthe system description given in Eq. (1) can be rewritten in the following form:

y(t)=G(q,θ)u(t)+H(q,θ)e(t) (2)

The Auto-Regressive Moving Average with eXternal input ARMAX model is one of many dif-ferent approaches that are available to solve Eq. (2) in terms of θ as presented in the sequel.The ARMAX model used in this study is a Single Input Single Output (SISO) ARMAX-model,since only one source of excitation is used to excite the structure. Generally, the simple relationbetween the input and output is provided by the following linear difference equation:

y(t)+a1y(t− 1)+...+anay(t− na)=b1u(t− 1)+...+bnbu(t− nb)+e(t) (3)

Clearly, Eq.(3) has limited capability in defining the disturbance term since it describes thewhite noise as a discrete error. The ARMAX models overcome this problem by defining theerror as a Moving Average (MA) of white noise as presented in Eq.(4):

y(t)+a1y(t − 1)+...+anay(t − na)=b1u(t − 1)+...+bnbu(t − nb)+e(t)+c1e(t − 1)+...+cnce(−nc)

(4)

The vector θ of adjustable parameters can be now formulated in the following form:

θ=[a1 ... ana b1 ... bnbc1 ... cnc ]

T

while q in an equivalent polynomial form can be denoted as:

A(q)y(t)=B(q)u(t)+C(q)e(t) (5)

with the following polynomial definitions:

A(q)=1+a1q−1+...+ anaq

−na , B(q)=b1q−1+...+ bnb

q−nb , C(q)=1+c1q−1+...+ cncq

−nc

and na, nb, nc are the maximum orders of the corresponding polynomial, which are usuallydetermined by an extended trial-and-error process. In Eq.(5), the Moving Average (MA) partis given by C(q)e(t). while, Eq. (5) is equivalent to Eq. (2) with

G(q,θ)=(B(q))/(A(q)), H(q,θ)=(C(q))/(A(q))

There are several optimization methods available to obtain the optimal estimate of θ by minimiz-ing the disturbance. More details about these methods and their full derivations can be found

65

in [34–37]. In this work, Gauss-Newton method has been utilized to optimize the mean squarevalue of the prediction error when searching for the optimal ARMAX-model. This searchingprocess is iterative and might converge to a local minimum rather than a global minimum.For this reason, the validation of the estimated model is crucial, and it is allowed to use theestimated model if it passed the validation test. By determining the coefficients of the vectorθ, the transfer function of the building will be known, and thus the modal parameters of thesystem will be derived directly from the coefficients.

In this study, the correctness of the estimated models will be validated using the Matlab’sbest-fit method, according to the following equation [38], which provides an indication aboutthe estimated model’s efficiency to represent the main system dynamics (in time domain) andwhether the linear simulation is appropriate.

Fit = 100 ∗(1− ‖y(t)h−y(t)‖

‖y(t)−y(t)‖)

In this equation, y(t) represents the real output, y(t) is the sample mean, and y(t)h representsthe output obtained from the identified model. In the sequel, the magnitudes and phases ofeach one of the estimated transfer functions of the ARMAX models will be extracted.

Mag(i) , i=1,....,k

Phz(i) , i=1,....,k

where Mag(i) represents the magnitude part for each measurement point, Phz(i) is the phasepart for each measurement point, and k is the total number of measurement points. The mag-nitudes’ part, which represents the FRF will be utilized to construct the measurements’ matrix,i.e. mobility matrix (Hmag), while the phases’ part will be used to plot the mode shapes (PHZ)of the frame as shown in the sequel:

Hmag(nf ,nmf) = [Mag(1,1) · · ·Mag(1,nmf

) · · ·Mag(nf ,1) · · ·Mag(nf ,nmf)]

PHZ(nf ,nmf) = [phz(1,1) · · · phz(1,nmf

) · · · phz(nf ,1) · · · phz(nf ,nmf)]

where nf is the numbers of floors, and nmfis the number of measurement points per floor.

For instance, in the case study given in Section 3, each one of the element Mag, in the abovementioned equation, contains one FRF column vector of length 100 (as it resulted from theFRF estimation process). Hmag includes the FRF columns vectors, corresponding to all themeasurement points in each floor (11 measurement points) and for all the buildings floors start-ing from the zero level to the 10th level (11 levels), and thus the total dimension of Hmag is(100,121). The same thing is applicable for the PHZ equation. Since there are unique responsesfor each measurement location and single excitation, H will have a rectangular shape, thus theSVD method was adopted to identify how many significant eigenvalues exist and plot the CMIFcurve for the frame [39]:

CM=SVD(H)

where CM denotes the CMIF curve. The basic assumption for all the Single Degree Of Free-dom (SDOF) methods for modal analysis is that at the proximity of resonance, the FRF will bedominated by that vibration mode and the contributions of other vibration modes can be ne-glected. Based on this assumption, the FRF from a real structure with Multi Degree Of Freedom(MDOF) can be considered as a FRF from a SDOF system [15,40]. Moreover, the peak-picking

66

method can be utilized to identify the frames natural frequencies, since this method is able tosearch for peaks by stepping through the CMIF curves, and when the curves have reached amaximum, it will indicate a new frequency by utilizing the following equation:

| CM(ω) |max ⇒ ωr = ωpeak,

where ω represents the identified natural frequency for the mode number r. After that, thesearch for the next peak corresponding for the next mode will start and so on. Thus, themethod will take into accounts the damage influenced by higher vibration modes.

According to [41] the CMIF curves were normalized so that their magnitudes at zero fre-quencies are unity. Thus, the values of modal damping will be determined from the normalizedCMIF curves denoted as:

CM(nor) = CM/CM(ω=1)

2.2 Transfer function estimation

In this subsection, the transfer function in terms of the magnitude and the derivation of theamplification factor, utilized for modal damping calculations are presented. Based on the as-sumption of applicability of the SDOF behaviour [15,40] mentioned in the previous subsection,and according to Newton’s second law, the time domain equation of motion for a single degreeof freedom system can be given by [42]:

Mx(t) + Cx(t) + Kx(t)=f(t) (6)

where M, C, and K represent the mass, damping, and stiffness values correspondingly, whilex(t), x(t), and x(t) represent the acceleration, velocity, and displacement and the excitationforce is f(t). The equivalent frequency domain equation of motion can be obtained using theLaplace transform of Eq.(6) assuming all initial conditions are zero [43]:

[Ms2 + Cs + K]X(s)=F(s) (7)

Where X(s) is the displacement Laplace transform and F(s) is the force Laplace transform,while Eq. (7) can be rearranged as:

H(s)= X(s)F (s) =

[1

Ms2+Cs+K

]

The transfer function H(s) is a complex valued, so it has two parts; a magnitude and a phase.The Frequency Response Function (FRF) can be obtained by substituting the values of thetransfer function along the frequency axis (jω-axis) as in the sequel [44]:

[H(s)]s=jω=[H(ω)], or [H(ω)]=[

1−ω2M+jωC+K

]=

[1/M

−ω2+jωC/M+K/M

](8)

According to [42], C/M=2ξωn where ωn is the natural frequency (radians/sec), and ξ is thedamping ratio, while (8) becomes:

[H(ω)]= 1/M[

1−ω2+jω2ξωn+ω2

n

],

which can be formulated as:

[H(ω)]= 1/M[

1(ω2

n−ω2)+j(2ξωωn)

], [H(ω)]= 1/M

[1√

(ω2n−ω2)2+j(2ξωωn)2

](9)

67

Substituting for M = ω2n/K [42] in Eq.(9) yields the transfer function in terms of the magni-

tude:

[H(ω)]= [1/K]

[ω2n√

(ω2n−ω2)2+j(2ξωωn)2

](10)

2.3 Modal parameters estimation

In the presented work, the Magnification-Factor method was utilized to calculate the modaldamping. According to this method, the peak value for the magnitude of the frequency responsefunction happens when the denominator of Eq.(10) is minimum [41] or the derivative of Eq.(10)is set to zero as shown below:

ddω

[(ω2

n − ω2)2 + 4ξ2ω2nω

2]=0 (11)

The solution of Eq.(11) for ω, is called the resonant frequency ωr and is being calculated by:

ωr =√

1− 2ξ2ωn (12)

Finally, substituting Eq.(12) in Eq.(10) gives the magnitude of the frequency response functionat the resonant frequency given in the sequel, which is called the amplification factor Q.

Q = 1

2ξ√

1−ξ2 (13)

The phases part, previously described in section (2.1), will be used to plot the mode shapes(PHZ) of the frame.

3 Simulation Results

In the current research, a regular building’s steel frame has been utilized as a case study. Theframe is for a ten story bench mark building 45.75m by 45.75m in plan and 40.82m in elevationwith one underground level. This bench mark building was proposed and designed by the SACproject for the Los Angeles, USA [45]. The lateral load-resisting system is composed from foursteel perimeter moment-resisting frames. The bays are 9.15 m on center, in both directions.The floor-to-floor height is 3.65 m for the underground floor, 5.49 m for the ground floor and3.96 m for the remaining eight stories. Figure 3 shows the building elevation. The lumpedseismic mass for each story was applied at the center of each level.

Table 1 displays seismic masses for each story and the steel sections used for the beams andcolumns. In this Table 1, the code W (d×wt) refers to the nominal size of wide flange structuralsteel section with, I or H shape. The nominal depth of the section in (inch) is referred by (d),while (wt) refers to the section weight per unit length (Ib/ft). The analysis was conductedbased on a pinned support condition assumption at the bottom of the underground floor, whichis also prevented from side movement as it is shown in Figure 3.

Abaqus 6.12 [46] finite-element software was utilized to perform the time history analysis forthe frame. The obtained responses at 110 different locations (assumed as a sensors) correspondsto the ends of columns and mid of beams were assumed as measurements. Quadratic elementstypes (B22) from Abaqus’s beam library were used for simulating the structural behavior ofbeams and columns. A horizontal ground acceleration in the form of a normally distributedGaussian white noise excitation with zero mean and a unitary variance was used to excite themodel. The amplitude of the Gaussian white-noise signal, in the time domain, was scaled by a

68

Figure 3: Building’s elevation

Table 1: Sections dimensions and properties of the building

Levels Beams Sections Columns Sections Seismic mass (kg)

Ground W36× 160 W14× 500 9.65× 105

1st W36× 160 W14× 500 1.01× 106

2nd W36× 160 W14× 500 9.89× 105

3rd W36× 135 W14× 455 9.89× 105

4th W36× 135 W14× 455 9.89× 105

5th W36× 135 W14× 370 9.89× 105

6th W36× 135 W14× 370 9.89× 105

7th W30× 99 W14× 283 9.89× 105

8th W27× 84 W14× 283 9.89× 105

9th W24× 68 W14× 257 9.00× 106

factor of 0.3g. The total simulation time was 60 seconds, with a simulation time step of 0.02seconds. The frequency bandwidths of the excitation and of the obtained signals have been25 Hz, which were equal to one-half of the sampling frequency. The damage in the frame wassimulated by reducing the Modulus of elasticity E for specific columns by 75% of the undamagedones. The value of E for the healthy columns was 203.9 GPa, while for the damaged columnswas 50.9 GPa. Two scenarios were assumed for damages in the frame, damage scenario no.1consist of increasingly destroying the columns of the ground floor, while for damage scenariono.2, the damage locations were changed by destroying column no.1 in a floor, and this wasrepeated for five stories. Table 2 summarizes the assumed damage scenarios.

The transfer function of the frame has been derived by utilizing the ARMAX parametricmodel and considering the frame as a linear system for all measurement points (sensors). Sincethe orders of the ARMAX model were determined by a trial-and-error process, many trials withdifferent combinations of the factors (na, nb, nc, and nk) were examined to obtain the highestpossible best-fit result, which is considered one of the best methods to guarantee that theestimated transfer function can effectively describe the dynamic behavior of the system. Thus,the orders of the polynomials na, nb, nc, and nk for the healthy, damage scenario no.1, anddamage scenario no.2 were 40, 40, 40, 1, 42, 42, 42, 1 and 40, 40, 40, 1 respectively. To validatethe correctness of the estimated models, the Matlab best-fit method has been implemented.

69

Table 2: Frame damage scenarios

Damage Scenario No.1 Damage Scenario No.2

C0: No damage case C0: No damage caseC1: Col. No.1 - ground floor C1: Col. No.1 - ground floorC2: Cols. No.(1 & 2) - ground floor C2: Col. No.1 - 1st floorC3: Cols. No.(1 & 2 & 3) - ground floor C3: Col. No.1 - 2nd floorC4: Cols. No.(1 & 2 & 3 & 4) - ground floor C4: Col. No.1 - 3rd floorC5: Cols. No.(1 & 2 & 3 & 4 & 5) - ground floor C5: Col. No.1 - 4th floorC6: Cols. No.(1 & 2 & 3 & 4 & 5 & 6) - ground floor

Figure 4 displays the best-fit results, which clearly prove the efficiency of the estimated modelsand their suitability for further usage.

C0 C1 C2 C3 C4 C5 C60

10

20

30

40

50

60

70

80

90

100

Damage Cases

Ave

rage

Bes

t−Fi

ts p

erce

ntag

e fo

r all

mea

sure

men

ts p

oint

s

89.685.8

82.8 80.8

90.286.8

89.6

C0 C1 C2 C3 C4 C50

10

20

30

40

50

60

70

80

90

100

Damage Cases

Ave

rage

Bes

t−Fi

ts p

erce

ntag

e fo

r all

mea

sure

men

ts p

oint

s

88.585.8

76.8

85.6 83.4 81.9

Damage Scenario No.1 Damage Scenario No.2

Figure 4: Efficiency of estimated ARMAX models in term of Best-Fit results

3.1 Frequency

The identified frequencies for all cases of damage scenarios are presented in the CMIF curves ofFigure 5 and Figure 6. The present work has been restricted to investigate the first five modes.The first five frame natural frequencies, obtained from Abaqus’s frequency analysis and basedon the frame mathematical model are: 0.4428, 1.1798, 2.0314, 3.0493 and 4.1986 Hz. The firstfive identified natural frequencies, based on the assumed measurements, for the healthy case ofthe frame are: 0.379, 1.14, 2.02, 3.03 and 4.17 Hz, which are very close to the frame naturalfrequencies obtained from Abaqus’s frequency analysis.

As it has been depicted in Figures 5 and 6, the comparison between the identified naturalfrequencies clearly indicated the existence of damage due to the sharp increase in the amplitudes(resonance) of the CMIF curves. For the damage scenario no.1, it is obvious from Figure 7 thatthe increase of damage caused a decrease (shift) in natural frequencies, while the change indamage locations for damage scenario no.2 did not affect the resonances absolutely as it wasdepicted in Figure 8. There are two possible reasons to explain the failure of the proposeddamage identification procedure for damage detection of damage scenario no.2. Firstly, due tothe professional design of this bench mark building, the redistribution of the stresses and forces

70

0 2 4 6 8 10 12 14100

200

300

400

500

600

700

800

Frequency (Hz)

Ampli

tude (

dB)

Healthy1 Col. Damaged2 Col. Damaged3 Col. Damaged4 Col. Damaged5 Col. Damaged6 Col. Damaged

Figure 5: Comparison of CMIF curves estimated from the mobility matrix (H) for alldamage cases of damage scenario no.1

0 2 4 6 8 10 12 14150

200

250

300

350

400

450

500

550

600

650

Frequency (Hz)

Ampli

tude (

dB)

Healthy1st Floor Damaged2nd Floor Damaged3rd Floor Damaged4th Floor Damaged5th Floor Damaged

Figure 6: Comparison of CMIF curves estimated from the mobility matrix (H) for alldamage cases of damage scenario no.2

in the frame overcome the damage. In this case, the different members of the frame workedeffectively to compensate for the deficiency of the frame in resisting the excitation due to thedamage existence at a specific member and at a specific floor. Secondly, the level of damage forscenario no.2 is not so large to be identified by the frequency’s changes method.

3.2 Damping

The damage in the frame can also be identified from the amplitudes of vibrations of CMIFcurves. The amplitudes of vibrations of the damage scenarios no.1 and no.2 are shown inFigure 9 and 10 respectively, where it is clearly indicated an increase in the vibration amplitudes

71

C0 C1 C2 C3 C4 C5 C60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Damage Cases

Freq

uenc

ies (H

z)

0.379

1.14

2.02

3.03

4.17

0.379

1.14

2.02

3.03

4.17

0.379

1.14

2.02

3.03

4.17

0.379

1.14

2.02

3.03

4.17

0.379

1.14

1.89

2.9

3.54

0.379

1.14

1.89

2.4

2.9

0.379

1.01

1.89

2.4

2.9

Mode 1Mode 2Mode 3Mode 4Mode 5

Figure 7: The 1st five natural frequencies identified in the damage scenario no.1

C0 C1 C2 C3 C4 C50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Damage Cases

Freq

uenc

ies (H

z)

0.379

1.14

2.02

3.03

4.17

0.379

1.14

2.02

3.03

4.17

0.379

1.14

2.02

3.03

4.17

0.379

1.14

2.02

3.03

4.17

0.379

1.14

2.02

3.03

4.17

0.379

1.14

2.02

3.03

4.17 Mode 1Mode 2Mode 3Mode 4Mode 5

Figure 8: The 1st five natural frequencies identified in the damage scenario no.2

as a general trend. This indicates also that the damping effects are considerably decreased sothat the amplitude can increase.

In order to find the values of modal damping, the CMIF curves were normalized so thattheir magnitudes at zero frequencies are unity. Afterwards, the modal damping values werecalculated using Eq. (15). The results of the damage scenarios no.1 are depicted in Figure 11.It is obvious that the general trend (apart of mode 3) is a decrease in the modal damping. Thereader should distinguish between modal damping and the overall damping of the structure,which exhibit marked non-linearity, for more details see [47,48]. This decrease in modal dampingreflects the increase in the damage’s size of the frame and it was vanished for some frequenciesof damage cases C4, C5, C6 and C7. In the case of the damage scenario no.2, Figure 12 showsthat there is a slight increase in the modal damping for the first two vibration modes, whilethere is a clear decrease in modal damping for the higher modes. It can be concluded that thegeneral trend for vibration amplitudes and modal damping can be considered as an index for

72

C0 C1 C2 C3 C4 C5 C60

100

200

300

400

500

600

700

800

Damage Cases

Ampli

tude (

dB)

333 35

244

839

536

3

333 35

244

839

536

3

473

530

453

440 46

6

417

544

359

352 36

5

473

470

431

419

333

500

411

504

322

455

717

455 465

371

488

Mode 1Mode 2Mode 3Mode 4Mode 5

Figure 9: Amplitudes of identified natural frequencies for the damage scenario no.1

C0 C1 C2 C3 C4 C50

100

200

300

400

500

600

700

Damage Cases

Ampli

tude (

dB)

347 36

445

540

537

2

463 49

155

1 579

636

372 39

450

846

947

0

407 42

052

4 545

585

410 42

558

6 608

468

426 44

363

849

963

9

Mode 1Mode 2Mode 3Mode 4Mode 5

Figure 10: Amplitudes of identified natural frequencies for the damage scenario no.2

detecting the increase or decrease in the damage size, but this effect will not provide any furtherinformation about damage locations.

3.3 Modes Shapes

Figures 13 and 14 depict the first two mode shapes of the frame for each damage case of damagescenario no.1 as they were identified from the measurements.

Visual inspection for these modes shapes reveals the locations of damage in the frame.Clearly, the deformed shape of the beams at the first floor level (assumed damage locations)tends to be more flattening as the damage increases from case C0 to case C6 for the two modeshapes. In other words, it is possible to expect damage locations at the points (sensors) wherethe difference in the phase is the least (for successive different monitoring cases). For the damage

73

C0 C1 C2 C3 C4 C5 C60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Damage Cases

Mod

al Da

mping

0.544

0.499

0.365

0.426

0.477

0.544

0.499

0.365

0.426

0.477

0.547

0.46

0.593

0.641

0.562

0.515

0.363

0 0 0

0.508

0.514

0.602

0.649

0

0.477

00.4

70

0.551

0.327

0.624

0.591

00.5

41

Mode 1Mode 2Mode 3Mode 4Mode 5

Figure 11: Modal damping for the damage scenario no.1

C0 C1 C2 C3 C4 C50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Damage Cases

Mod

al Da

mping

0.558

0.515

0.382

0.443

0.497

0.566

0.513

0.436

0.409

0.365

0.525

0.481

0.349 0.3

830.3

83

0.558

0.529

0.391

0.373

0.343

0.567

0.53

0.348

0.334

0.461

0.568

0.529

0.331

0.445

0.33

Mode 1Mode 2Mode 3Mode 4Mode 5

Figure 12: Modal damping for the damage scenario no.2

scenario no.2, shown in Figure 15 and Figure 16, even that it looks that the points of assumeddamage have the least differences in the phase, it is difficult to specify damage locations exactlydue to the small damage size in each floor.

4 Conclusions

In this article, the Auto-Regressive Moving Average with eXternal input ARMAX model wassuccessfully utilized for the construction of FRF, based on Abaqus’s simulation results. Theidentified frame’s natural frequencies were very close to the theoretical ones obtained fromAbaqus’s frequency analysis. Sharp increases in the amplitudes of the CMIF curves revealedthe existence of damage in the frame. Furthermore, it can be concluded that the general trend

74

for vibration amplitudes and modal damping can be considered as an index for detecting theincrease or decrease in the damage size, but it is not able to give any information about damagelocations, while visual inspection for modes shapes reveals the locations of damage in the frame.In the future, the authors are planning for further development and evaluation of the proposedmethodology to include the detection of damage up to levels 3 and 4. One recognizable drawbackis that this method does not manage closely spaced modes, while it is possible to expect damagelocations at the points (sensors) where the difference in the phase is the least (for successivedifferent monitoring cases).

Acknowledgements

The first author would like to acknowledge the Ministry of Higher Education and ScientificResearch - University of Mosul-Iraq for the PhD scholarship provided by them.

75

Mode shape no.1 -C0 Mode shape no.2 -C0

Mode shape no.1 -C1 Mode shape no.2 -C1

Mode shape no.1 -C2 Mode shape no.2 -C2

Mode shape no.1 -C3 Mode shape no.2 -C3

Figure 13: Vibration mode shapes of the frame in the case of damage scenario no.1

76

Mode shape no.1 -C4 Mode shape no.2 -C4

Mode shape no.1 -C5 Mode shape no.2 -C5

Mode shape no.1 -C6 Mode shape no.2 -C6

Figure 14: Vibration mode shapes of the frame in the case of damage scenario no.1

77

Mode shape no.1 -C0 Mode shape no.2 -C0

Mode shape no.1 -C1 Mode shape no.2 -C1

Mode shape no.1 -C2 Mode shape no.2 -C2

Figure 15: Vibration mode shapes of the frame in the case of damage scenario no.2

78

Mode shape no.1 -C3 Mode shape no.2 -C3

Mode shape no.1 -C4 Mode shape no.2 -C4

Mode shape no.1 -C5 Mode shape no.2 -C5

Figure 16: Vibration mode shapes of the frame in the case of damage scenario no.2

79

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83

Paper Three

Investigation of changes in modal characteristics

before and after damage of a railway

bridge: A case study

Aauthors:Tarek Edrees Saaed, George Nikolakopoulos, Niklas Grip, Jan-Erik Jonasson

Journal paper submitted to the Structural Health Monitoring Journal in August 2014.

(edited to fit the format of the thesis - content has not been altered)

Investigation of changes in modal characteristics before andafter damage of a railway bridge: A case study

Tarek Edrees Saaed, George Nikolakopoulos, Niklas Grip, Jan-Erik Jonasson

Keywords:Structural Identification, Damage Detection, Non-Destructive Evaluation, HealthMonitoring.

Abstract

This study presents the results of the modal identification of the Aby river railway steel bridgein Sweden. The ambient response of the bridge was recorded for the functional bridge and afterit was tested to failure. The aim of this research effort is to implement the structural identifi-cation approach to investigate the amount of variations in the modal characteristics (frequency,damping, and modes shapes) for the case study, while this has been achieved by the comparisonbetween the modal characteristics for the functional bridge and for the same bridge after failure.The Frequency Response Functions (FRF) were obtained from the identified transfer functions,which were computed from the quotient of the cross power spectral density and the power spec-tral density. In the sequel, the magnitude part of the FRF has been utilized to construct themobility matrix (H), while the phase’s part was utilized for plotting the mode shapes of thebridge. The Singular Value Decomposition (SVD) method has been adopted to identify howmany significant eigenvalues exist by plotting the Complex Mode Indication Function (CMIF)for the bridge before and after the collapse. The obtained results depicted that the linear,time-invariant transfer function Txy is a powerful model to construct the mobility matrix (H).Furthermore, the proposed procedure is reliable and can be utilized further and efficiently forthe purposes of damage detection and localization. The rate of change in the damping ratiofrom healthy to collapsed bridge was about (206%), and this rate could be regarded as an indexfor the existence of a serious damage in steel bridges.

1 Introduction

During the last decades, damage detection in a structure from changes in the modal characteris-tics (structural identification) has been subject to many studies from the structural engineeringcommunities, while it is well known that variations in the structure’s physical properties likestiffness, mass, and boundary conditions can affect the modal characteristics of the structure(frequency, damping, and modes shapes) [1,2]. Representative articles by [3–5] provide valuableinformation about the methods used for damage detection and localization in a structure basedon changes in its modal characteristics. From another point of view, one of the important con-tributions of system identification was done by Hart and Yao [6], who has initially introducedthe identification concept to the mechanical engineering researchers and afterwards to struc-tural engineers by presenting and formulating the problem of system identification in related

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use cases. Furthermore, in the related literature it has been proposed the probable applicationof different testing procedures and the potential practical implementation of system identifica-tion method for damage detection [7]. Catbas et al. [8] reported the state of the art of systemidentification, outlined its promising application in civil engineering and confirmed that one ofthe most powerful advantages of system identification is its efficiency for damage detection andfor evaluating possible performance deficiency causes.

In this research effort, the main novelty stems from the utilization of the structural identifi-cation approach to investigate the amount of variations in the modal characteristics (frequency,damping, and modes shapes) for a railway steel bridge before and after its known early stageof failure, based on the ambient vibration measurements due to trains traffic. The clear anddistinct main novelties of this article are the following: a) extracting the modal characteristicsfrom a typical real scale example, when it was functional and after its known early stage offailure, which is according to authors’ viewpoint not possible every time, b) possibility to verifythe proposed identification method’s results against an existing damaged bridge and validateits efficiency for damage detection and localization, c) obtain the percentage of change in modaldamping at early stage of failure, in order to be utilized as an indication for existence of a seri-ous damage, d) propose a methodology for deriving the Frequency Response Functions (FRF)from the quotient of the cross Power Spectral Density and the PSD, and e) utilise the mobilitymatrix (H) derived from FRF for identifying and localizing the damage in the presented casestudy.

Until now, many researchers applied the system identification approach to identify the dy-namic properties of bridges. For instance, Fan et al. [9] studied the effects of the damage locationand size on the structural behavior of an old reinforced concrete arch bridge by utilizing fieldmeasurements. Perez and Gonzalez [10] successfully applied the artificial neural networks forthe structural damage detection of a vehicular bridge. Matsumoto et al. [11] investigated therelation between structural changes and changes in the modal properties for a steel truss bridgein Japan. Brunell and Kim [12] investigated the effect of local damage on the behavior ofa small-scale steel bridge and concluded that the presence of local damage will significantlyaffect the serviceability of the bridge. In addition, Shu et al. [13] successfully used artificialneural networks for the damage detection of a bridge model. Lee et al. proposed a continu-ous relative wavelet entropy-based reference-free damage detection algorithm for truss bridgestructures and demonstrated the efficiency of the proposed method on a laboratory-size trussstructure [14], while Andersen and Brincker [15] found that frequencies and modal shapes for ahighway bridge changed significantly during damage. Gongkang et al. [16] successfully applieddata acquisition by a high-resolution camera to detect the existence and location of damage ina bridge model in the laboratory. Also, Park and Towashiraporn [17] proposed the response-surface statistical model for rapid damage assessment of railway bridges. Solla et al. [18] utilizedGround-Penetrating Radar (GPR) to evaluate internal damage of ancient masonry arch bridge.

Nowadays, the vibration-based damage detection approach includes a promising set of meth-ods for damage detection and health monitoring of civil engineering structures. However, thehuge variance in their dynamic response under working loads and the large size of such structuresmake the successful application of these methods to real-life structures a daunting task [19]. Forinstance, in spite the fact that the damping is one of the crucial factors that controls the dy-namic behavior of a structure, it is at the same time difficult to be measured and is still subjectfor further researchers’ studies [20].

In order to derive the FRF, for their application in a modal analysis of a structure, measure-ments for both input force and response are essential [21]. Usually civil engineering structuresare large, and it is not easy to measure the input force for such structures. However, the outputonly response or ambient vibration measurements can be utilized to estimate reliable values for

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modal characteristics [22]. In the present work, the structural identification approach is utilizedto investigate the amount of variations in the modal characteristics (frequency, damping, andmode shapes) and the existence of damage and damage locations for a railway steel bridge basedon the ambient vibration measurements due to trains traffic, before and after its known earlystage of failure. This bridge is located in the north of Sweden, and such an application andstudy are quite important, especially when proposing novel approaches in the damage detectionand experimentally evaluating them, which according to the authors’ viewpoint is not possibleevery time. In the presented investigation, the responses of the bridge were measured whenthe bridge was functional in its original position and when it was collapsed in the temporaryposition as it will be analysed in the sequel.

In the established framework, the modal characteristics of the collapsed bridge will be com-pared against the results of the healthy bridge to evaluate the changes in the modal character-istics between a healthy bridge and a defected bridge at its known early stage of failure. Duringthe analysis, the percentage of change in the modal damping as an indication for the assessmentof damage existence for other railway steel bridges, will be also derived. The transfer functionsobtained from the quotient of the cross PSD and the PSD are utilized to obtain the FRF basedon ambient vibration measurements. Then, the magnitudes part, which represents the FRFwill be utilized to construct H, while the phases part will be utilized to plot the modes’ shapesof the bridge. Due to the rectangular shape of H, the Singular Value Decomposition (SVD)method will be adopted to identify how many significant eigenvalues exist and plot the ComplexMode Indicator Function (CMIF) for the whole bridge.

Overall, this article is structured as it follows. Section 2 outlines the utilized methodologyfor the bridge damage detection, while Section 3 provides a description of the case study. Section4 presents the evaluation of modal characteristics before and after failure, while the conclusionsare drawn in the final Section 5.

2 Methodology

There are a lot of categories of damage detection methods according to the technique utilizedto identify the damage from the measured data. Valuable information about damage detectionmethods, their levels, and damage indicators are available in the following references [2,3,23,24].However, the frequency changing methods will be adopted in the current article, since shiftsin natural frequencies were widely used to detect damage in structures. This method mainlydepends on the concept that variations in the structural properties will result in alterations inthe vibration frequencies and amplitudes. Figure 1 depicts the overall methodology proposedthroughout this article for damage identification and localization.

First of all, we consider the set of measurements (healthy and collapsed) denoted by:

Ccj , cj=1,....,cm

where Ccj is the measurements cases and cm represents the number of measurements casesconsidered. Moreover, u(t) is the excitation, y(t)i , i = 1, ...., k is the response of the bridge atthe ki location, where y(t)i represents the output and k is the total number of measurementpoints. In the sequel, the responses of the bridge due to each one of the measurement caseswill be utilized to obtain the transfer functions of the bridge by the Txy non-parametric model,while assuming without a loss of generality that the bridge can be fully described by a linearsystem.

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Figure 1: Proposed damage identification scheme

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2.1 Estimation of the Txy Transfer Function

A typical dynamic system is presented in Figure 2, subjected to input u(t), while the response ofthe system is described by the output y(t), while it is worth mentioning that this non parametricmodel Txy does not take under consideration the disturbances of the system.

Figure 2: A dynamic system with input u(t) and output y(t)

In this article, and since there are no input forces measured (only operating data from out-put measurements), the transfer function can be obtained from the quotient of the cross powerspectral density Pyu of u and y and the power spectral density Puu of u as given below:

Tuy=Pyu

Puu

The transfer functions can be calculated using Matlab ”tfestimate” function as it is givenin Eq. 1 below:

Txy=f(u(t), y(t), window type, noverlap, nfft, fs) (1)

Since the excitation is random and not periodic, the appropriate window type shall be ap-plied to decrease the effects of leakage in the recorded signals, however this method can notremove it completely. Moreover, each window type has its own advantages and disadvantages.According to [25], Hanning window is considered as a very good window type to obtain goodfrequency resolution and with a good spectral leakage suppression and fair amplitude accuracy.So, Hanning window with 1024 samples in each window will be utilized in this work. The pa-rameter noverlap determines the number of samples by which the windows overlap and it wasassumed to be 50% of samples number. While nfft specifies the number of the Fast FourierTransform FFT points used to calculate the transfer functions for each window and it controlsthe resolution of the frequency function obtained. During the experimentations, the samplingfrequency of measurements fs was set at 800 Hz. In the sequel each one of the transfer functionsmodels will be split into two parts: magnitudes and phases as it follows:

Mag(i) , i=1,....,k

Phz(i) , i=1,....,k

where Mag(i) represents the magnitude part for each measurement point, Phz(i) is the phasepart for each measurement point, and k is the total number of measurement points. The mag-nitudes’ part, which represents the FRF will be utilized to construct the measurements’ matrix,i.e. mobility matrix (H), while the phases’ part will be used to plot the mode shapes (PHZ) ofthe bridge as shown in the sequel.

H(k) =(Mag(1) · · ·Mag(k)

)

PHZ(k) =(Phz(1) · · ·Phz(k)

)where (k) is the total number of measurement points. Since there is a unique response for eachmeasurement location and a single excitation, the H will have a rectangular shape and thus

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the Singular Value Decomposition SVD method was adopted to identify how many significanteigenvalues exist and plot the Complex Mode Indicator Function CMIF curve for the bridge [26].

CM=SVD(H) (2)

where CM denotes the CMIF curve. The basic assumption for all the Single Degree Of Free-dom (SDOF) methods for modal analysis is that at the proximity of resonance, the FRF willbe dominated by that vibration mode and the contributions of other vibration modes can beneglected. Based on this assumption, the FRF from a real structure, with Multiple DegreesOf Freedom (MDOF), can be considered as a FRF from a SDOF system [27, 28]. Moreover,the Peak-Picking Method (PPM) can be utilized to identify the bridge’s natural frequenciessince this method is able to search for peaks by stepping through the CMIF curves, and whenthe curves have reached a maximum, it will indicate a new frequency by utilizing the followingequation:

| CM(ω) |max ⇒ ωr = ωpeak, where ω represents the identified natural frequency for themode number r.

Following the analysis presented in [29], the CMIF curves were normalized so that their magni-tudes (in dB) at zero frequency are unity. Thus, the values of modal damping will be determinedfrom the normalized CMIF curves denoted as:

CM(nor) = CM/CM(ω=1) (3)

where ω = 1 represents the frequency magnitude at zero frequency.

2.2 Modal Damping

Based on the assumption of applicability of SDOF behaviour [27,28] mentioned in the previoussubsection, and according to Newton’s second law, the time domain equation of motion for asingle degree of freedom system can be given by [30]:

Mx(t) + Cx(t) + Kx(t)=f(t) (4)

where M, C, and K represent the mass, damping, and stiffness values correspondingly, x(t),x(t), and x(t) represent the acceleration, velocity, and displacement, while the excitation forceis denoted by f(t). The equivalent frequency domain equation of motion can be obtained usingthe Laplace transform of Eq.(4) assuming all initial conditions are zero [31]:

[Ms2 + Cs + K]X(s)=F(s) (5)

where X(s) and F(s) is the displacement and force, while Eq. (5) can be rearranged as:

H(s)= X(s)F (s) =

[1

Ms2+Cs+K

]

The transfer function H(s) is complex valued, so it has two parts; a magnitude and a phase(real and imaginary). The Frequency Response Function (FRF) can be obtained by substitutingthe values of the transfer function along the frequency axis (jω-axis) as in the sequel [32]:

[H(s)]s=jω=[H(ω)], or [H(ω)]=[

1−ω2M+jωC+K

]=

[1/M

−ω2+jωC/M+K/M

](6)

According to [30], C/M=2ξωn where ωn is the natural frequency (radians/sec), and ξ is the

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damping ratio, while (6) becomes:

[H(ω)]= 1/M[

1−ω2+jω2ξωn+ω2

n

],

which can be formulated as:

[H(ω)]= 1/M[

1(ω2

n−ω2)+j(2ξωωn)

]or

[H(ω)]= 1/M

[1√

(ω2n−ω2)2+j(2ξωωn)2

](7)

Substituting for M = K/ω2n [30] in Eq.(7) yields the transfer function in terms of the magnitude:

[H(ω)]= [1/K]

[ω2n√

(ω2n−ω2)2+j(2ξωωn)2

](8)

In the present work, the Magnification-Factor method was utilized to calculate the modaldamping. According to this method, the peak value for the magnitude of the frequency responsefunction happens when the denominator of Eq.(8) is minimum [29] or the derivative of Eq.(8)is set to zero as shown below:

ddω

[(ω2

n − ω2)2 + 4ξ2ω2nω

2]=0 (9)

The solution of Eq.(9) for ω, is called the resonant frequency ωr and is being calculated by:

ωr =√

1− 2ξ2ωn (10)

Finally, substituting Eq.(10) in Eq.(8) gives the magnitude of the frequency response func-tion at the resonant frequency given in the sequel, which is called the amplification factor Q.

Q = 1

2ξ√

1−ξ2 (11)

3 Description of Case Study

This case study deals with a railway steel truss for a bridge over Aby river, which has beenconstructed in 1957 and is being presented in Figure 3. This bridge is located about 60 km westof Pitea, Sweden, while the bridge spans a small river with a span length of 33 m and a widthof 4.7 m.

The bridge was a riveted connection’s single span carrying a single track, as it has beenpresented in Figure 4, while this bridge was owned and managed by the Swedish TransportAdministration.

During a renewal of the embankment in 2012, it was decided to exchange the old railwaybridge with a new bridge that requires less maintenance. The redundant bridge was taken out ofservice and replaced by a new steel beam bridge in September 2012. The old truss was placedon temporary abutments on a road near its original location as it is shown in Figure 5 andbecame an excellent research subject.

On September 12 of 2013, the obsolete bridge was tested to failure. The reasons behindsuch a testing were: a) this bridge has four replications so the assessment outcome will be avery good indication for evaluating the service conditions of other bridges, b) it is also excitingto distinguish where the bridge breaks in order to avoid breakage at the weak points, and c) theSwedish Transport Administration is investigating the possibility of life extension and raising

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Figure 3: General view of the Aby river railway bridge in its original position [33]

Figure 4: Geometry of the bridge, plan and elevation [33]

the allowable axle load on existing railway bridges. The bridge, which was designed to resista weight of 25 tons per axle (or 8 tons per meter) would, according to designers’ calculations,collapse at a load of about 800 tons. However, in practice, it was capable of taking a considerablehigher loading of approximately 1200 tons [33, 34].

The main focus of the current study is to utilize the transfer functions Txy obtained from thequotient of the cross power spectral density and the power spectral density to identify the modalcharacteristics of the bridge, as it has been presented in Figure 1. The modal characteristicsof the collapsed bridge will be compared against the results of the healthy bridge to evaluatethe changes in the modal characteristics between a healthy bridge, and a defected bridge at itsknown early stage of failure and obtain the percentage of change in the modal damping. Theseresults will be utilized in the sequel as an indication for the assessment of damage existence forother railway steel bridges.

At this point it should stated that the comparison between the healthy bridge (in its originalposition) and the defected bridge (in the temporary position) is based on the fact that this bridgeis a simply supported steel truss bridge and the boundary conditions of this bridge are very

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The bridge on its temporary position When train passing

The bridge on its temporary abutment From inside the bridge

Figure 5: Views of the Aby alv railway bridge in its temporary position

clear, i.e.; besides the simple supports, only two continuous rails connect the bridge. These railshave some effect on the boundary condition of the bridge, but are so slender and their bendingrigidity is too small, especially when compared with such a large steel-truss bridge, while theirconstraint effect can be ignored from an engineering point of view. Moreover, several modelsfor this bridge were built in Abaqus [33], and in some of these models, rotational springs withdifferent magnitudes at the bridge ends (to simulate the boundary constraint effects of differentlengths of rails) have been included. After comparing these models, it was found that for thebridge under investigation, it is not necessary to include extra rail effect or rotational springsat the ends of the bridge, i.e., the simply supported bridge model is accurate enough, and theboundary conditions are close enough in the two cases to make it reasonable to compare thetwo systems.

4 Evaluation of Modal Characteristics

Two sets of ambient vibrations measurements were conducted for the bridge. The first set wasconducted when the bridge was functional and in its original position. After the final breakingof the bridge, the second set of vibration measurements were performed on the collapsed bridge.The HBM MGCPlus Data acquisition system with ML801/AP801 cards was used for dataacquisition that had a fixed input signal range of -10.5 to 10.5 V, with 64 bits for each sample;the 21 V dynamic ranges were divided into uniform intervals of length 1.1384*10-18 V. This

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is high enough a resolution also for very low amplitude vibrations. Six tri-axial accelerometersColibrys SF3000L, were used for the measurements. Finally the recorded data were sampled ata sampling frequency of fs= 800 Hz for all setups.

In each set of measurements, the bridge was instrumented with the accelerometers in differ-ent setups. Table 1 presents details about the arrangement of accelerometers in each setup andfor each set of measurements. For example, in each one of the six setups, the measurements atthe accelerometer no. 12 were considered as an input (since there are only operating data fromoutput measurements), while the measurements at the rest of the accelerometers have beenutilized as the outputs. For each setup, two accelerometers were utilized as reference sensorsand their numbers are depicted in the last column of the Table 1. The accelerometers weremounted on steel plates that were strongly fixed with fasteners on the bridge’s top and bottomchord.

Figure 6 displays an overview of the accelerometers’ locations in setup No.1 for the col-lapsed bridge. The horizontal accelerations time-history records of the bridge, in the directionperpendicular to the bridge longitudinal axis (X-direction in Figure 6), have been recorded forat least 35-minutes for each measurement point (depending on trains traffic).

Figure 6: Accelerometers’ layout for setup No.1

The CMIF were calculated according to Eq.(2) based on the flowchart depicted in Figure 1and the first 15 identified frequencies (although the first five frequencies are usually used) forthe healthy and collapsed bridge are presented in Figures 7 and 8 respectively.

Distinguishing the real vibration modes from pseudo ones in an ambient vibration test isa source of errors and a challenging task, while not every peak of the FRF from the ambientvibration measurements is related to a vibration mode of the structure [35]. Figure 9 displaysthe CMIF for the bridge before and after collapse, where it is clearly revealed the existenceof significant shift in the frequency response of the bridge, especially for the frequency range(0-5.46) Hz. Moreover, Table 2 displays the bridge natural frequencies extracted from the CMIFcurves before and after collapse.

In order to find the values of modal damping, the CMIF curves were normalized as it wasstated in Eq. (3), so that their magnitudes (should be in dB) at zero frequencies are unity,see Figure 1. Afterwards, the modal damping values were calculated using Eq. (11). Table 3presents the damping ratios for the healthy and collapsed bridge. The values ”> 1” in the thirdcolumn of this table indicate that the modal damping pass the hundred percent (see Eq. 11).

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Table 1: Arrangement of accelerometers in each set of measurements

Input accelerometer u(t) Output accelerometer y(t) Setup no. remarks

1 12 12 1 1st Reference acc.2 12 26 1 2nd Reference acc.3 12 10 14 12 14 15 12 28 16 12 30 17 12 12 2 1st Reference acc.8 12 26 2 2nd Reference acc.9 12 13 210 12 16 211 12 29 212 12 32 213 12 12 3 1st Reference acc.14 12 26 3 2nd Reference acc.15 12 5 316 12 8 317 12 21 318 12 24 319 12 12 4 1st Reference acc.20 12 26 4 2nd Reference acc.21 12 4 422 12 6 423 12 20 424 12 22 425 12 12 5 1st Reference acc.26 12 26 5 2nd Reference acc.27 12 2 528 12 9 529 12 17 530 12 18 531 12 12 6 1st Reference acc.32 12 26 6 2nd Reference acc.33 12 11 634 12 15 635 12 27 636 12 31 637 12 12 7 1st Reference acc.38 12 26 7 2nd Reference acc.39 12 1 740 12 3 741 12 7 742 12 25 743 12 12 8 1st Reference acc.44 12 26 8 2nd Reference acc.45 12 19 846 12 23 8

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0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200

Frequency (Hz)

Ampli

tude (

dB)

Figure 7: Identified natural frequencies for the healthy bridge

0 5 10 15 20 25 300

50

100

150

200

250

Frequency (Hz)

Ampli

tude (

dB)

Figure 8: Identified natural frequencies for the collapsed bridge

Table 2: Identified natural frequencies for the healthy and collapsed bridge

Nr Healthy Bridge Collapsed Bridge

1 1.4 1.62 3.6 2.33 5.6 5.04 6.7 6.95 8.9 8.86 9.8 10.07 11.4 11.68 12.9 12.49 13.6 13.210 16.5 14.911 17.6 16.612 18.4 18.413 19.6 19.914 20.1 22.015 22.6 23.4

95

0 5 10 15 20 25 300

50

100

150

200

250

Frequency (Hz)

Ampli

tude (

dB)

HealthyCollapsed

Figure 9: Comparison for the CMIF of the bridge before and after the collapse

It is obvious that there is a tremendous change in the damping ratios for the first 15 identifiedfrequencies and the average of the percentage of change in modal damping is 206%. Accordingto the authors’ opinion, the abnormal percentage of change in modal damping, between thehealthy and any other condition for a structure (like 56.1% in Table 3), can be regarded asa serious indicator for early stages of damage, while the high percentage of change in modaldamping (like 251.1% in Table 3) can clearly indicate the existence of damage in that structure.

Table 3: Calculated damping rations for the healthy and collapsed bridge

Nr Healthy Bridge Collapsed Bridge % of change

1 0.325 0.300 7.32 0.318 0.297 6.53 0.307 0.479 56.14 0.285 > 1 251.15 0.282 > 1 254.56 0.295 > 1 239.17 0.305 > 1 228.08 0.274 > 1 264.59 0.279 > 1 257.910 0.281 > 1 256.011 0.273 > 1 266.912 0.277 > 1 260.713 0.300 > 1 233.914 0.309 > 1 223.515 0.258 > 1 287.7

Average 206%

Figures 10 and 11 depict the first four modes shapes of the healthy and collapsed bridgerespectively, as they were identified from the measurements (PHZ) and according to the pro-cedure outlined in Figure 1. It is obvious that Figure 10 depicts the four scenarios for bridgepossible failures with some differences among them. The scenarios depicted in Figure 11 aremore homogenous and clearly indicate the locations of failure, which are very close to the actualdamage locations of the bridge presented in Figure 12.

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0

5

10

15

20

25

30

35

02

46

0

2

4

6H

eigh

t (m

)

0

5

10

15

20

25

30

35

02

46

0

2

4

6

Hei

ght (

m)

Mode shape no.1 Mode shape no.2

0

5

10

15

20

25

30

35

02

46

0

2

4

6

Hei

ght (

m)

0

5

10

15

20

25

30

35

02

46

0

2

4

6

Hei

ght (

m)

Mode shape no.3 Mode shape no.4

Figure 10: Vibration mode shapes for the healthy bridge

0

5

10

15

20

25

30

35

02

46

0

2

4

6

Hei

ght (

m)

0

5

10

15

20

25

30

35

02

46

0

2

4

6

Hei

ght (

m)

Mode shape no.1 Mode shape no.2

0

5

10

15

20

25

30

35

02

46

0

2

4

6

Hei

ght (

m)

0

5

10

15

20

25

30

35

02

46

0

2

4

6

Hei

ght (

m)

Mode shape no.3 Mode shape no.4

Figure 11: Vibration mode shapes for the collapsed bridge

5 Conclusions

In this study, the linear, time-invariant transfer function Txy was successfully utilized for theconstruction of FRF, based on the ambient vibration measurements. The results presented inthis article indicates the possibility to identify and localize damages in steel railway bridges fromthe variations in the modal characteristics of the structure obtained from measured vibrationsdue to trains traffic. The comparison between frequencies, damping ratio, and mode shapes

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Damage in the bridge, General view Damage’s location

Damage in the bridge, side view Enlarged damage’s location

Figure 12: Actual deformed shape of the bridge after the collapse

for the healthy and collapsed bridge confirmed that damage had been existed. Moreover, theresults showed that modal characteristics for the bridge changed significantly during damage.The abnormal percentage of change in modal damping, between the healthy and any othercondition for a structure, can be regarded as a serious indicator for early stages of damage,while the high percentage of change in modal damping can clearly indicate the existence ofdamage in that structure. The average ratio of change in the damping ratio from the healthyto the collapsed bridge was about (206%) and this ratio could be regarded as an index for theexistence of a serious damage in steel bridges. Additionally, the modes, shape for the collapsedbridge (calculated from the collapsed bridge’s measurements) showed good agreement with theactual damage shape and locations.

Acknowledgements

The authors gratefully acknowledge Professor Lennart Elfgren for his valuable comments andhelp, the support from the Swedish Transport Administration and the European ResearchProject MAINLINE, http://www.mainline-project.eu/, in which the tests were carried out.The third author was funded by the Swedish Research Council Formas and by the developmentFund of the Swedish Construction Industry (SBUF).

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Paper Four

A Comparative Study on the Identification of

Building Natural Frequencies Based on

Parametric Models

Aauthors:Tarek Edrees Saaed, George Nikolakopoulos, Jan-Erik Jonasson

c© Proceedings of the 33rd IASTED International Conference on Modelling, Identification andContrl, MIC 2014, Austria, p.p.1-6, February 2014, DOI: 10.2316/P.2014.809-008, reprintedwith permission.

(edited to fit the format of the thesis - content has not been altered)

A Comparative Study on the Identification of Building NaturalFrequencies Based on Parametric Models

Tarek Edrees Saaed, George Nikolakopoulos, Jan-Erik Jonasson

Keywords: System identification, Structural health monitoring, Modal identification tech-niques, Structural behavior, Finite element method.

Abstract

The analysis and design of civil engineering structures is a complex problem, which is based onmany assumptions to simplify these operations. This in turn, leads to a difference in the struc-tural behavior between calculations based models and real structures. Structural identificationwas proposed by many researchers as a tool to reduce this difference between models and actualstructures. Moreover, Parametric models and non-parametric models were used intensively forsystem identification by many researchers. In this research effort, the system identification con-cept is utilized to identify the natural frequencies for a steel building’s frames. Different blackbox linear parametric models such as Transfer Function model (TF), Auto-Regressive modelwith eXternal input model (ARX), Auto-Regressive Moving Average with eXternal input (AR-MAX) model, Output Error model structure (OE), and Box-Jenkins model (BJ) were examinedfor identifying the first 10th natural frequencies for the building’s frames, based on simulationresults. Abaqus 6.12 finite-element software was utilized to perform the time history analysisfor the examples and the obtained responses at one point of the roofs (assumed as a sensor)were further processed by the parametric models to obtain the building’s natural frequenciesbased on the Abaqus time history analysis results (assumed as a measurements). After that,Abaqus 6.12 was utlized again to perform another analysis, which is called frequency analysisto obtain the building’s natural frequencies and mode shapes based on the stiffness and mass(not the measurements) of the buildings. The results showed that the linear parametric modelsTF, ARX, ARMAX, OE, and BJ are robust to identify the natural frequencies of building andthey are recommend for future work.

1 Introduction

The analysis and design procedures of buildings during the last decades were based mainly ona basic and uncompromising model of structures [1–3] For instance; many designers model thebuildings as plane frames [4]. These simplified procedures performed successfully when usedefficiently and produced economic and safe designs. However, these procedures were unableto describe the actual behavior of the real structures precisely [5]. Nowadays, and even withthe ability to simulate the three-dimensional performance of the real structures, the reliablebehavior of structures still needs more than just a refined model.

Structural Identification (St-Id) approach was utilized to bridge the gap between the realstructure and the model. Basically, the St-Id is the procedure of constructing/ updating the

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finite-element model (physics-based model) from its measured dynamic/static response, whichcan be utilized to evaluate the structures health, damage detection, efficiency and to obtaina non-physics based model (state-space, differential and/or difference equations) of a dynamicsystem from its measured response [6]. The use of a parametric model, even with the compu-tational effort required, is justified in order to get the more accurate estimate for the modalparameters, which will help to obtain a precise understanding of the dynamic behavior of thestructure [7]. Non-parametric models were used intensively for system identification by manyresearchers [8–10]. Moreover, Parametric model structures have been utilised for damage detec-tion in civil engineering structures. For instance, Bodeux and Golinval used the Auto-RegressiveMoving Average Vector (ARMAV) model for system identification and damage detection ofbuildings [11]. Vector ARMA models were utilized for identification of civil engineering struc-tures by [12, 13]. A comparison of four system identification methods namely, Peak-picking,poly reference, stochastic subspace, and prediction error were conducted by Andersen et al [14].Despite the enormous developments in parametric model’s identification methods, their rela-tive merits and performance as correlated to the vibrating structures are still incomplete. Thereason for this limited knowledge is due to the lack of comparative studies under various testconditions [15]. The aim of this research is to utilize the structural identification concept toidentify the natural frequencies for a 2D-frame of a multi-story steel building. Different blackbox linear parametric models such as Transfer Function model (TF), Auto-Regressive modelwith eXternal input model (ARX), the Auto-Regressive Moving Average with eXternal input(ARMAX) model, Output Error model structure (OE), and Box-Jenkins model (BJ) were ex-amined for identifying the first 10th natural frequencies for the frame based upon simulationresults. Abaqus 6.12 finite-element software was utilized to perform the time history analysis forthe building’s frame and the obtained response at one point of the roof (assumed as a sensor) wasfurther processed by the parametric models to obtain the building’s natural frequencies basedon the simulation results (assumed as a measurements). In the sequel, Abaqus 6.12 was utilizedagain to perform another analysis, which is called frequency analysis to obtain the building’snatural frequencies based on the stiffness and mass (not the measurements) of the building’sframe. A comparison between the results obtained from the two approaches was conducted toexplore the efficiency of the previously mentioned parametric model in predicting the building’snatural frequencies. Overall, this article is structured as it follows: Section 2 review the differentkinds of parametric model structures, while in Section 3 numerical examples were given to showthe differences in models’ prediction efficiency. Finally, the conclusions were drawn in Section4.

2 Parametric model structures

Mathematical models are generally used to describe the dynamic systems. These models canbe divided into two main categories of model structures: Non-parametric model structuresand parametric model structures [5]. Although the embedded application simplicity of thenonparametric methods, like the Fast Fourier Transform (FFT), the accuracy of these methodsis limited, and the parametric methods should be utilized when it is required to obtain anaccurate model for the system [12]. In this case, the mathematical models are assumed to becomposed of a set of parameters to be calculated by system identification. This mathematicalmodel takes the form of differential equation in case of a linear and time-invariant continuous-time system, while the corresponding discrete-time is in the form of difference equation.

Figure (1), displays a typical dynamic system subjected to input u(t) and the response ofthe system is described by the output y(t), which is affected by disturbance v(t). It is clear

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from Figure (1) that the output is a combination of the input and the disturbance. Moreover,

Systemy(t) : outputu(t) : input

v(t) : disturbance

+

Figure 1: A dynamic system with input u(t), output y(t) and disturbance v(t)

it is worth mentioning that the disturbance cannot be controlled, and even the input may beunknown and uncontrollable in some kind of systems. Then the basic complete description of alinear system, including the impulse response, additive disturbance, and the Probability DensityFunction (PDF) of the disturbance e(t) is provided by equation (1) [16]:

y(t)=G(q)u(t)+H(q)e(t) (1)

where: u(t) is the input signal, y(t) is the output signal, e(t) represents the stochastic inputcorresponding to the noise and prediction errors, G(q) is the transfer function of the determin-istic part of the system, and H(q) is the transfer function of the stochastic part of the system.The parameters in the transfer function of equation (1) are determined during the system iden-tification process. The vector θ is usually used to designate these parameters and the systemdescription given in equation (1) can be rewritten in the following form:

y(t)=G(q,θ)u(t)+H(q,θ)e(t) (2)

Transfer-function models can be utilized directly for parameterizing G and H of Eq.1 by con-sidering them as rational functions and use the parameters as the numerator and denominatorcoefficients. This approach can be fulfilled using different techniques, which are generally knownas black-box models. These models include: Transfer Function model (TF), Auto-Regressivemodel with eXternal input model (ARX), Auto-Regressive Moving Average with eXternal input(ARMAX) model, Output Error model structure (OE), and Box-Jenkins model (BJ). The maindifference between these models is how to parameterize the transfer functions (G and H ) ofEq.2.

The transfer functions for all the models will be presented in the following sequence, whilethe corresponding signal flows will be also depicted in Figure (2). Full derivations of thesemodels can be located in the references [17, 18].

• Transfer Function model: The effects of disturbance are ignored and are not takeninto consideration as presented in (Eq.3):

G(q,θ)=B(q)/A(q) (3)

• ARX model: Represents the simple relation between the input and output given by thelinear difference equation (Eq.4). Although that fact that the ARX model predictor ex-presses a linear regression, the disturbance passes through 1/(A(q)) due to mathematicalconsideration, which is not correct physically.

G(q,θ)=B(q)/A(q) , H(q,θ)=1/A(q) (4)

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(a)TF model

(b)ARXmodel

(c)ARMAXmodel

(d)OE model

(e)BJ model

Figure 2: Signal flows for model structures

• ARMAX model: The main problem with the ARX model (Eq.4) is the limited capabil-ity in defining the disturbance term. ARMAX model overcomes this problem by definingthe equation error as a Moving Average (MA) of disturbance as presented in the sequel:

G(q,θ)=B(q)/A(q), H(q,θ)=C(q)/A(q)

• Output Error model: In the previous two models (equation error model structures)the polynomial A was used as a common factor in the denominator for determining thetransfer functions G and H, while in the output error model structure, these transferfunctions are parameterized separately because it is more natural from a physical pointof view as shown below:

G(q,θ)=B(q)/F(q)

• Box-Jenkins model: This type of models is an extension of the output error model by

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describing the output error as an ARMA model and can be expressed in the followingform:

G(q,θ)=B(q)/F(q), H(q,θ)=C(q)/D(q)

3 Numerical Examples

For evaluating the presented identification methods, two different examples have been analyzedin this article. The first one is for a regular building’s frame and the other one is an irregularbuilding’s frame. Abaqus 6.12 finite-element software was utilized to perform the time historyanalysis for the examples and the obtained responses at one point of the roofs (assumed asa sensor) were further processed by the parametric models to obtain the building’s naturalfrequencies, based on the Abaqus time history analysis results (assumed as a measurements).After that, Abaqus 6.12 was utlized again to perform another analysis, which is called frequencyanalysis to obtain the building’s natural frequencies based on the stiffness and mass (not themeasurements) of the building’s frame. Quadratic elements types (B22) from Abaqu’s beamlibrary were used for simulating the structural behavior of beams and columns. A horizontalground acceleration in the form of a normally distributed Gaussian white noise excitation withzero mean and a unitary variance was used to excite the model. The amplitude of the Gaussianwhite-noise signal in the time domain was scaled by a factor of 0.3g. The total simulation timewas 60 second, with a simulation time step of 0.02 second. Matlab 2012b system identifica-tion toolbox was utilized to identify the frame natural frequencies from the roof accelerationresponses obtained from Abaqu’s time history analysis. As a check for the correctness of thepredicted models, the Matlab’s best-fit method, which gives a clue about the model efficiencyto represent the main system dynamics and whether the linear simulation is appropriate hasbeen utilized.

3.1 Regular Steel Frame

A ten story bench mark building 45.75m by 45.75m in plan, and 40.82m in elevation with oneunderground level was used for the first example. This bench mark building was proposedand designed by the SAC project for the Los Angeles, USA [19]. The lateral load-resistingsystem is composed from four steel perimeter moment-resisting frames. The bays are 9.15m oncenter, in both directions. The floor-to-floor height is 3.65m for the underground floor, 5.49mfor the ground floor, and 3.96m for the remaining eight stories. Figure (3) shows the buildingelevation.The lumped seismic mass for each story was applied at the center of each level.Table (1) displays seismic masses for each story and the steel sections used for the beams andcolumns. The analysis was conducted based on a pinned support condition assumption at thebottom of the underground floor, which is also prevented from side movement as it is shown inFigure (3). Figure (4) presents the first 5 mode shapes for the frame obtained from Abaqusfrequency analysis and the values for the first 10th frame natural frequencies were displayed inTable (2).The Best-Fit approach results showed that TF, ARMAX, OE, and BJ Models are superior inpredicting the model structures. The ARX model gives also good performance but less than thepreviously mentioned models. A comparison for the performance of the different models basedon their Best-Fits results was presented in Figure (5). The identified natural frequencies(from the assumed measurements) are depicted in Figure (6). Clearly, all of the models werevery successful in predicting the frame natural frequencies, and their results were very close to

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Figure 3: Regular steel frame elevation

Table 1: Sections dimensions and properties of the building

Levels Beams Sections Columns Sections Seismic mass (kg)

Ground W36 ∗ 160 W14 ∗ 500 9.65 ∗ 1051st W36 ∗ 160 W14 ∗ 500 1.01 ∗ 1062nd W36 ∗ 160 W14 ∗ 500 9.89 ∗ 1053rd W36 ∗ 135 W14 ∗ 455 9.89 ∗ 1054th W36 ∗ 135 W14 ∗ 455 9.89 ∗ 1055th W36 ∗ 135 W14 ∗ 370 9.89 ∗ 1056th W36 ∗ 135 W14 ∗ 370 9.89 ∗ 1057th W30 ∗ 99 W14 ∗ 283 9.89 ∗ 1058th W27 ∗ 84 W14 ∗ 283 9.89 ∗ 1059th W24 ∗ 68 W14 ∗ 257 9.00 ∗ 106

Table 2: The first 10th natural frequencies of the regular frame as obtained from Abaqusfrequency analysis

Modes 1 2 3 4 5 6 7 8 9 10

Freq. (Hz) 0.4428 1.1798 2.0314 3.0493 4.1986 4.8604 5.0690 5.1715 5.3331 5.4043

the natural frequencies obtained from Abaqu’s frequency analysis, which are based on buildings’mass and stiffness (see Table 2).

3.2 Irregular Steel Frame

The second example is a virtual irregular steel frame with 3 different spans of 9m, 7.5m, and10m as depicted in Figure (7). The floor-to-floor height is 4.0m for the first floor, 5.0m for thesecond floor, and 4.0m for the remaining seven stories. The lumped seismic mass for each storywas applied at the top of the columns at each level.

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Mode shape No.1 Mode shape No.2 Mode shape No.30.4428(Hz) 1.1798(Hz) 2.0314(Hz)

Mode shape No.4 Mode shape No.53.0493(Hz) 4.1986(Hz)

Figure 4: The first 5 mode shapes and natural frequencies for the building’s regular frameobtained from Abaqus frequency analysis

TF ARX ARMAX OE BJ0

10

20

30

40

50

60

70

80

90

100

Type of Model

Aver

age F

it %

99.7

89.8

99.799.799.7

Figure 5: Percentage of average fit of the models estimated responses to the measuredone

Table (3) displays seismic masses for each story and the steel sections used for the beams andcolumns. The analysis was conducted based on a fixed support condition assumption at thebottom of the first floor. Figure (8) presents the first 5 mode shapes for the frame obtainedfrom Abaqus frequency analysis. In addition, the values for the first 10th frame naturalfrequencies were displayed in Table (4).

Again, the Best-Fit approach results showed that TF, ARMAX, OE, and BJ Models aresuperior in predicting the model structures. The ARX model gives also good performance

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0 2 4 6 8 10 12−40

−30

−20

−10

0

10

20

30

40

50

Mag

nitu

de (d

B)

Frequency (Hz)

TF ARX ARMAX BJ OE

0.505

2.021.14

5.434.17

3.03

Figure 6: Identified frequencies for the regular frame

Figure 7: Irregular steel frame elevation

but its performance differs for the range of high frequency (above 7 Hz). A comparison for theperformance of the different models based on their Best-Fits results was presented in Figure (9).The identified natural frequencies (from the assumed measurements) are depicted in Figure (10).Clearly, all of the models were very successful in predicting the frame natural frequencies, andtheir results were very close to the natural frequencies obtained from Abaqu’s frequency analysis,which are based on buildings’ mass and stiffness (see Table 4).

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Table 3: Sections dimensions and properties of the building

Levels Beams Sections Columns Sections Seismic mass (kg)

1st W36 ∗ 160 W14 ∗ 500 3.21 ∗ 1052nd W36 ∗ 160 W14 ∗ 500 3.36 ∗ 1053rd W36 ∗ 160 W14 ∗ 500 3.29 ∗ 1054th W36 ∗ 135 W14 ∗ 455 3.29 ∗ 1055th W36 ∗ 135 W14 ∗ 455 3.29 ∗ 1056th W36 ∗ 135 W14 ∗ 370 3.29 ∗ 1057th W36 ∗ 135 W14 ∗ 370 3.29 ∗ 1058th W30 ∗ 99 W14 ∗ 283 3.29 ∗ 1059th W24 ∗ 68 W14 ∗ 257 1.78 ∗ 105

Mode shape No.1 Mode shape No.2 Mode shape No.30.525(Hz) 1.305(Hz) 2.174(Hz)

Mode shape No.4 Mode shape No.53.296(Hz) 4.391(Hz)

Figure 8: The first 5 mode shapes and natural frequencies for the irregular building’sframe obtained from Abaqus frequency analysis

4 Conclusion

The main conclusion of this article is that the linear parametric models like Transfer Functionmodel (TF), Auto-Regressive model with eXternal input model (ARX), Auto-Regressive Moving

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Table 4: The first 10th natural frequencies of the irregular frame as obtained from Abaqus

Modes 1 2 3 4 5 6 7 8 9 10

Freq. (Hz) 0.5251 1.3058 2.1744 3.2961 4.3911 5.8066 5.8530 6.3217 6.8079 6.9667

TF ARX ARMAX OE BJ0

10

20

30

40

50

60

70

80

90

100

Type of Model

Aver

age F

it %

97.397 97.397.3 98.2

Figure 9: Percentage of average fit of the models estimated responses to the measuredone

0 2 4 6 8 10 12−30

−20

−10

0

10

20

30

40

50

60

Mag

nitu

de (d

B)

Frequency (Hz)

TF ARX ARMAX BJ OE

0.5051.26

3.282.15

4.42

5.81

6.94

6.31

Figure 10: Identified frequencies for the irregular frame

Average with eXternal input (ARMAX) model, Output Error model structure (OE), and Box-Jenkins model (BJ) are robust to identify the natural frequencies of the buildings (regular andirregular) and they are recommended for future work.

Acknowledgements

The first author would like to acknowledge the Ministry of Higher Education and ScientificResearch - University of Mosul-Iraq for the PhD scholarship provided by them.

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