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Dynamic behavioir of base cracked gravity dam by the way of the hybrid frequency time domain procedure Hui, Pak K. 1995

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D Y N A M I C B E H A V I O R OF B A S E C R A C K E D G R A V I T Y D A M B Y T H E W A Y OF T H E H Y B R I D F R E Q U E N C Y T I M E D O M A I N P R O C E D U R E by Pak K. Hui B. A. Sc., The University of British Columbia, 1992 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Civil Engineering We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A September 1995 © PakK. Hui, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Afr^'&O ^C/^C^C The University of British Columbia Vancouver, Canada Date > ' 3 DE-6 (2/88) Abstract Due to deterioration, past loading conditions, and shrinkage, concrete tensile cracks may form along the dam-foundation interface within a typical concrete gravity dam monolith. These cracks render the assumption that a dam remains linear elastic during dynamic response invalid and requires a nonlinear analysis procedure. In this research, the response of the cracked dam system with the effects of a flexible foundation and hydrodynamic effects is studied using the Hybrid Frequency Time Domain (HFTD) analysis procedure. This nonlinear dynamic analysis procedure preserves the frequency dependent stiffness and damping in the foundation medium when modeled as a visco-elastic halfplane. Similarly, the HFTD procedure honours the frequency dependent hydrodynamic mass and damping of the reservoir medium when the hydrodynamic effects are modeled with a two dimensional wave equation. The nonlinear behavior of the crack opening and closing process is directly accounted for by applying the required restoring or contact force along the crack interface. The result of the study shows that dams with crack lengths less than 25% of the total base length have nominal differences in both magnitude and frequency of vibration when compared to the same uncracked system. For crack lengths in excess of 25%, larger response amplifications than those from the uncracked response were observed. The fundamental effective system frequency of vibration of the cracked dam system is found to be bounded by fundamental frequencies of the uncracked and cracked pseudo-linear systems of the dam. Large amplifications of the maximum dam top response obtained from the cracked system in comparison to the uncracked system are expected for systems near the dominant frequency of excitation. However, for systems with properties that are far removed from the dominant frequency of excitation, little, ii Abstract if any, amplification of the peak response is evident for all crack lengths. A simplified approach of representing the cracked dam system as a SDOF system with bi-linear stiffness proves to be an efficient and accurate method of estimating the cracked dam displacement and velocity response. The speed of the procedure allows for preliminary assessments to be made prior to dedicating valuable time to a full nonlinear analysis. In addition, this procedure can be used to produce cracked response spectra which can accurately predict the amplifications expected in the maximum relative displacement and velocity in a cracked dam system as compared to an uncracked dam system. iii Table of Contents Abstract ii Table Of Contents iv List Of Figures vii List Of Tables xi Acknowledgements xii SECTION 1. Introduction And Overview 1 1.1 Introduction: 1 1.2 Objectives Of Study 5 1.3 Scope Of Investigation 6 1.4 Organization Of The Thesis 7 SECTION 2. General Theory Of H F D T Approach And The Idealization Of Dam System 9 2.1 Introduction 9 2.2 Linear Dynamic Analysis In Frequency Domain 9 2.3 Hybrid Frequency Time Domain Analysis For Nonlinear Structures 12 2.4 Chapter Summary 15 SECTION 3. H F D T As Applied To Cracked Dam System 16 3.1 Idealization Of Reservoir-Dam-Foundation System 16 3.1.1 Gravity Dam Monolith 16 3.1.2 Visco-Elastic Half-Plane Foundation Medium 17 3.1.3 HydrodynamicEffects 19 3.1.4 Crack Identification 21 3.2 Identifying Cracked Pseudo-Linear System 22 3.3 Modal Decomposition 23 3.4 Sensitivity Of Response To Number Of Ritz Vectors 26 3.5 Calculation And Application Of Nonlinear Restoring Force 29 3.5.1 Crack Open Crack Close Criterion 30 iv Table of Contents 3.5.2 Tolerance Level 31 3.5.3 Selection Of Spring Stiffness 31 3.5.4 Application Of Restoring Force 32 3.5.5 Effectiveness Of Restoring Force On Generalized Response 35 3.6 Implementation Of HFTD 38 3.6.1 Choice Of Time Step 41 3.6.2 Time Segmentation Approach 42 3.7 Numerical Errors In The Analysis 44 3.8 Computer Program: C R K - D A M 49 3.9 Chapter Summary 49 SECTION 4. Effective System Frequency Identification 51 4.1 Introduction 51 4.2 System Identification Through Response Of White Noise Excitation 52 4.2.1 White Noise Excitation 52 4.2.2 Dam Model 54 4.2.3 Analysis Parameters 55 4.2.4 Results of Analysis 56 4.2.5 Limitations Of ESF Identification 61 4.3 Effective System Frequency Identification Through Sine Sweep 64 4.3.1 Description Of Dam Model 65 4.3.2 Excitation And Analysis Parameters 65 4.3.3 Case Study: 61.0 m. (200 ft.) 25% And 50% Cracked Dam Subjected To 3 Hz Excitation 66 4.3.4 Spectral Response 67 4.3.5 Secondary Resonance Phenomena 68 4.4 Chapter Summary 74 Table of Contents SECTION 5. SDOF Bi-Linear Stiffness Approximation 76 5.1 Approximation With SDOF Bi-Linear Stiffness System 76 5.2 Performance Of SDOF Bi-Linear Stiffness Approximation 81 5.3 Chapter Summary 85 SECTION 6. Cracked Dam Response To Earthquake Excitation 87 6.1 Introduction 87 6.2 Response Of Cracked Dam System On Rigid Foundation And No Reservoir 88 6.2.1 Model Specifications 88 6.2.2 Free-Field Excitation 89 6.2.3 Analysis Parameters 91 6.2.4 Case Study: 61.0 m. (200 ft.) Dam Subjected To Loma Prieta 91 6.2.5 Case Study: 152.4 m. (500 ft.) Dam Subjected To Loma Prieta 95 6.2.6 Spectral Response Of The Cracked Dam System 97 6.2.7 Conclusions.... 100 6.3 SDOF Bi-Linear Stiffness Approximation And The Cracked Spectra 102 6.3.1 SDOF Bi-Linear Stiffness Approximation Analysis Properties 102 6.3.2 Comparison Of Approximate Method With HFTD Result 104 6.3.3 Verifying Approximate Cracked Spectra with San Fernando Record 107 6.3.4 Conclusions 110 6.4 Effects Of Flexible Foundation On The Response Of Cracked Dam System 112 6.4.1 Result Of HFTD Analysis 114 6.5 Cracked Dam Response With Reservoir Effects 128 6.5.1 Result Of 61.0 m. Dam-Reservoir System Subjected to Loma Prieta 119 6.5.2 Result Of 152.4 m Dam-Reservoir System Subjected to Loma Prieta 126 6.6 Chapter Summary 132 SECTION 7. Conclusions And Recommendations 136 APPENDIX. List of References 144 vi List of Figures Figure 3.1: Reservoir-Dam-Foundation Model and its Associated Static and Dynamic Forces 17 Figure 3.2: Crack Identification Along Dam-Foundation Interface 22 Figure 3.3: Decomposition of Nonlinear Cracked Dam System to Pseudo-Linear System and Restoring Forces 23 Figure 3.4: Sensitivity of Number of Ritz Vectors used in the Analysis of Dam Top Response of 91.5 m. (300 ft.) Dam Section with 50% Crack Subjected to Loma Prieta Scaled to 5g .'. 27 Figure 3.5: Application of Restoring Force on Vertical Crack Opening Displacement (VCOD) of Crack Interface Nodes 29 Figure 3.6: Applied Impulse Load and Response 33 Figure 3.7: Application of Restoring Force Pair and its Effect on V C O D 35 Figure 3.8: Modal Response of First Uncracked and Cracked Modes in the Dam Top Response and Restoring Force at Crack Mouth of 61.0 m. (200 ft.) Dam Section with 50% Crack Subjected to Sinusoidal Excitation of 10Hz 37 Figure 3.9: Flowchart of the HFTD Procedure as Applied to the Cracked Dam System 39 Figure 3.10: Aliasing Effect on the Impulse Pair Response of Restoring Force Applied on Crack Interface 49 Figure 4.1: White Noise Free-Field Excitation 53 Figure 4.2: Power Spectral Density (PSD) of White Noise Excitation. Time Sampled at 0.01 sec 53 Figure 4.3: 61.0 m. (200 ft.) Cracked Dam Finite Element Model and Material Properties 54 Figure 4.4: Dam Top Displacement of 61.0 m. (200 ft.) Dam System with 0%, 25%, and 50% Crack Lengths Subjected to White Noise Excitation 57 Figure 4.5: PSD of First ESF of Dam Top Response of 61.0 m. (200 ft.) Dam Subjected with 0%, 25%, and 50% Crack Lengths Subjected to White Noise Excitation 57 vii List of Figures Figure 4.6: Mid Height Displacement of 61.0 m. (200 ft.) Dam System with 0%, 25%, and 50% Crack Lengths Subjected to White Noise Excitation 58 Figure 4.7: PSD of Second ESF of Mid Height Response of 61.0 m. (200 ft.) Dam with 0%, 25%, and 50% Crack Lengths Subjected to White Noise Excitation 58 Figure 4.8: Variation of First and Second ESF with Various Crack Lengths for 61.0 m. (200 ft.) Cracked Dam 60 Figure 4.9: Dam Top Displacement and PSD of 61.0 m. (200 ft.) Dam with 50% Crack Subjected to White Noise Plus - lg . V C O D for Crack Mouth 63 Figure 4.10: Dam Top Relative Displacement Response of 61.0 m. (200 ft.) Dam Section with 25% and 50% Crack Lengths Subjected to 3 Hz Sinusoidal Excitation 66 Figure 4.11: Spectral Response of 61.0 m. (200 ft.) Dam on Rigid Foundation and no Reservoir Subjected to Sinusoidal Excitation 67 Figure 4.12: Comparison of Resonance Behavior at 3 Hz. and 7 Hz 70 Figure 4.13: SDOF Representation of 61.0 m. (200 ft.) 50% Cracked Dam System 70 Figure 4.14: Occurrence of Secondary Resonance Phenomena in SDOF Bi-Linear Stiffness System 72 Figure 5.1: SDOF Bi-Linear Representation of 91.5 m. (300 ft.) 50% Cracked Dam System 82 Figure 5.2: Comparison Between the Dam Top Relative Displacement Response of 91.5 m. (300 ft.) 50% Cracked Dam System Subjected to Loma Prieta Scaled to 5g's as Obtained from HFTD Procedure and the SDOF Bi-Linear Approximate Method 82 Figure 5.3: Comparison Between the Dam Top Relative Velocity Response of 91.5 m. . (300 ft.) 50% Cracked Dam System Subjected to Loma Prieta Scaled to 5g's as Obtained from HFTD Procedure and the SDOF Bi-Linear Approximate Method 83 Figure 5.4: Comparison Between the Dam Top Absolute Acceleration Response of 91.5 m. (300 ft.) 50% Cracked Dam System Subjected to Loma Prieta Scaled to 5g's as Obtained from HFTD Procedure and the SDOF Bi-Linear Approximate Method 83 vii i List of Figures Figure 5.5: Comparison Between the V C O D Response at Crack Mouth of 91.5 m. (300 ft.) 50% Cracked Dam System Subjected to Loma Prieta Scaled to 5g's as Obtained from HFTD Procedure and the SDOF Bi-Linear Approximate Method 84 Figure 6.1: Typical Dam Configuration and Finite Element Mesh 88 Figure 6.2: Accelerograms for Loma Prieta and San Fernando Scaled to 2.5g 90 Figure 6.3. Power Spectral Density (PSD) Function for Loma Prieta and San Fernando Accelerograms Horizontal Component Scaled to 2.5g's 90 Figure 6.4: Response of 61.0 m. (200 ft). Cracked Dam on Rigid Foundation and No Reservoir Subjected to Loma Prieta Scaled to 2.5g's 93 Figure 6.5: Response of 152.4 m. (500 ft). Cracked Dam on Rigid Foundation and No Reservoir Subjected to Loma Prieta Scaled to 2.5g's 96 Figure 6.6: Normalized Maximum Response Values for Cracked Dam System Subjected to Loma Prieta Scaled to 2.5g with 5% Viscous Damping 99 Figure 6.7: Ratios of Modal Parameters for the Uncracked and Cracked Phase of Dam System. Crack Lengths of 0%, 25%, and 50% are used for the SDOF B i -Linear Stiffness Approximate Method 103 Figure 6.8: Comparison of Approximate Cracked Spectra from SDOF Bi-Linear Stiffness Method and Spectral Result Amplifications from HFTD with Respect to Uncracked Spectral Response for Loma Prieta Scaled to 2.5g for 5% damping 106 Figure 6.9: Comparison of Approximate Cracked Spectra from SDOF Bi-Linear Stiffness Method and Spectral Result Amplifications from HFTD with Respect to Uncracked Spectral Response for San Fernando Scaled to 2.5g with 5% damping 108 Figure 6.10: Dam Top Displacement of 61.0 m. Cracked Dam System with Flexible Foundation of E f = 27600 MPa and E f = 13800 MPa Excited by Loma Prieta Record scaled to 2.5g 115 Figure 6.11: Vertical Crack Edge Displacements at Heel of 61.0 m. Dam with 50% Crack on Flexible Foundation of Ef = 27600 MPa Excited by Loma Prieta Record Scaled to 2.5g 115 ix List of Figures Figure 6.12: . Cracked Spectral for 61.0 m. Dam with 0%, 25%, and 50% Crack on Flexible Foundation Excited by Loma Prieta Record Scaled to 2.5g with 5% Damping 117 Figure 6.13: Dam Response of 61.0 m. Cracked Dam System with reservoir Subjected to Loma Prieta Scaled to 2.5g 121 Figure 6.14: Variation of Fundamental Period of Coupled Dam-Reservoir System with the Fundamental Frequency Ratio of the Reservoir and the Dam Only (Chopra, 1968) 122 Figure 6.15: The Influence of Hydrodynamic Pressure on the Absolute Value of Horizontal Dam Top Acceleration (Chopra, 1968) 124 Figure 6.16: Dam Top Response of 152.4 m. Cracked Dam System with Reservoir Subjected to Loma Prieta Scaled 2.5g 127 Figure 6.17: Approximate Cracked Spectra with Amplified 152.4 m. Reservoir-Dam Spectral Ordinate for Loma Prieta Scaled to 2.5g 130 List of Tables Table 4.1: Modal Frequency of Pseudo-Linear Systems of 61.0 m. (200 ft.) Dam 56 Table 4.2: First and Second ESF for 61.0 m. (200 ft.) Cracked Dam System 59 Table 4.3: Natural Frequencies of Ritz Vectors for 61.0 m. (200 ft.) Cracked Dam System [Hz] 65 Table 4.4: Result of ESF Analysis Via Sine Sweep 68 Table 6.1: Spectral Response of Cracked Dam System on Rigid Foundation and No Reservoir of 30.5 m. through 243.9 m. with Crack Lengths of 0%, 25%, and 50%, Subjected to Loma Prieta Scaled to 2.5g 98 Table 6.2: Modal Properties of Cracked Dam System for SDOF Bi-Linear Stiffness Approximation 102 Table 6.3: Maximum Response Result of Dam Sections on Rigid Foundation and No Reservoir Subjected to San Fernando Scaled to 2.5g 107 Table 6.4: Material Properties for Foundation Medium 112 Table 6.5: Modal Properties of 61.0 m. (200 ft.) Cracked Dam Section on Flexible . Foundation 113 Table 6.6: Spectral Response of 61.0 m. Cracked Dam System on Flexible Foundation with E f = 27600 MPa and E f = 13800 MPa 116 Table 6.7: Spectral Response of 61.0 m. (200 ft.) Reservoir-Dam System to Loma Prieta 2.5g 119 Table 6.8: Fundamental Frequency for 61.0 m. Coupled Cracked Reservoir -Dam System 123 Table 6.9: Fundamental Frequency for 152.4 m. Coupled Cracked Reservoir -Dam System 126 Table 6.10: Maximum Response of 152.4 m. Reservoir-Dam System Subjected to Loma Prieta for 2.5g 129 xi Acknowledgement The author wishes to thank Dr. D. Anderson and Dr. C. Ventura for their advice and helpful suggestions during the course of this research. In addition, sincere gratitude is extended to Mr. Tibor Pataky, Mr. Ben Fan, Mr. Charles Holder, Mr. Roland Hui, and the remaining members of B.C. Hydro's Engineering Division for their assistance and guidance during the author's stay at B.C. Hydro in partial fulfilment of the Professional Partnership Program. The author also wishes to thank the Natural Science and Engineering Research Council for its financial support throughout the first two years of the research. Sincerest and deepest thanks to the author's mother and father for their emotional support and re-assurance. And finally, a special thanks to the author's best friend, Debbie Yeung, for her endless hours of proof-reading and for her patience throughout the course of the study. Xll Section 1: Introduction and Overview 1. Introduction and Overview 1.1 Introduction: There are few man-made structures that are built to meet the demands and safety requirements for which dams are designed. However, due to deterioration of concrete, past loading conditions, and prolong static loads, tensile cracks near the heel of the dam along planes of weakness can develop (Hall, 1986). As a result, the dynamic behaviour of the cracked dam under earthquake loading conditions no longer adheres to the assumption of a linear elastic response. There have been several studies conducted on the behaviour and formation of the these tensile cracks along the base of the dam. The majority of this work focuses on devising finite element constitutive models utilizing fracture mechanics principles to predict the cracking behaviour of concrete and its propagation. Discrete crack models (Srikeraud and Bachmann 1986) and smear crack models (Pal 1974) have been used to approximate the crack initiation and the topology of the crack path. More sophisticated crack models, such as crack band width models (Vargas-Loli and Fenves 1989) attempt to include the effects of reservoir-dam interaction and water cavitation into the global response of the cracked dam system. In all these studies, the main focus has been on the determination of crack initiation, and crack patterns, rather than on the effect they have on the global response of the dam structure. The main drawbacks to these studies are that the responses are very sensitive to the concrete fracture 1 Section 1: Introduction and Overview properties used in the crack models. These fracture properties are difficult to obtain with any reasonable amount of reliability, especially for a complex composite material such, as concrete. Therefore, the conclusions drawn from these studies must be accepted with caution. To eliminate the uncertainties involved with fracture mechanics, the dynamic analysis of the dam system can be performed by applying certain assumptions such that the response of the dam is isolated without concern of the crack behaviour. By assuming a dam system with a non-propagating, pre-existing crack, the effects of crack initiations and crack topography are eliminated from the analysis. Further, by assuming this crack exists along a horizontal dam-foundation interface, the location of the crack and the crack path are no longer of concern. The complexity involved with fracture mechanics and the stresses at the crack tip are now removed so that a better understanding of the response of the cracked dam system can be obtained. These assumptions are not without justification. Many of the older existing dam structures are plagued with problems with concrete deterioration due to alkali-silica reactions within the concrete matrix, which eventually leads to spalling, heaving, and joint displacement within the dam. In addition, the weaker planes along construction joint lifts make for ideal conditions for horizontal cracks to form even under normal operating conditions (Walters, 1973). A study performed by Tinawi and Guizani (1991) investigated the earthquake response of a cracked dam system of fixed length. In this study the crack was assumed to be of sufficient width such that the upper crack surface does not make contact with the lower crack surface during 2 Section 1: Introduction and Overview dynamic response. Therefore, the analysis neglected the nonlinear crack opening and closing behaviour and the investigation was carried forth as a linear dynamic analysis. This is unrealistic since typical crack width found near dam bases are often very small and are not discrete cracks. The tensile cracks near the dam base are typically zones of microcracks. Collectively, these zones reduce the tensile strength of the mass concrete and effectively render this region ineffective in transmitting tensile stresses. It is only for analytical convenience, that this region is considered as a discrete crack. Upon closure of this crack, the compression stresses will be transmitted across this zone. Therefore, to assume that the crack does not close neglects the interaction of the open phase and the closed phase of the crack in the global response of the dam system. Since it is not possible to determine the state of the crack (i.e. open or close) without prior knowledge of the response of the crack surface, a nonlinear analysis must be performed. The study by Tinawi and Guizani also neglected reservoir and flexible foundation effects, which may play a vital role in the crack behaviour. The reservoir subdomain in a dam-reservoir interaction, including the effects of compressibility, is typically modelled by a 2-D wave equation. The flexibility of the foundation medium can also be modelled by a visco-elastic halfplane. Solutions to both of these formulation lead to complex value frequency dependent properties which are easily solved in the frequency domain. However, traditional nonlinear analysis requires the use of time stepping techniques restricting the solution to the time domain. This conflict between frequency domain and time domain usually requires that a sacrifice be made in the modelling technique of the flexible foundation and the hydrodynamic effect. 3 Section 1: Introduction and Overview An alternative approach is to use the Hybrid Frequency Time Domain (HFTD) analysis procedure (Dabre and Wolf, 1988). The HFTD addresses this problem directly and combines the solution procedure of a frequency domain analysis with the time stepping required in the nonlinear analysis. HFTD has been used successfully in nonlinear dam analysis involving nonlinear concrete constitutive relationships (Fenves and Chavez, 1990), as well as, dam base sliding (Chavez and Fenves, 1994). However, the procedure can be easily extended to include any type of nonlinear mechanism, including contact type problems, such as the cracked dam system. The focus of this study is on the performance of a typical reservoir-dam-foundation system with a fixed predefined crack along the dam-foundation interface. The main objective is to obtain a better understanding of the global dynamic behaviour of the nonlinear system. Shifts in the natural frequency and amplifications in the response due to the opening of the crack will be investigated. This will be achieved by subjecting dams of various sizes and crack lengths with sinusoidal excitations as well as earthquake records. The analysis will be performed using the HFTD procedure to account for the nonlinear crack opening behaviour while preserving the frequency dependent properties of the foundation and reservoir. In addition, a simplified method is proposed to approximate the response of the cracked dam system that is subject to strong motion excitation. With this approximate method, a screening procedure can be developed that can rapidly assess the possible dam response prior to a full nonlinear analysis. 4 Section 1: Introduction and Overview 1.2 Objectives of Study The main objectives of this study are: 1) To establish a procedure based on the principles of HFTD method to analyse the nonlinear behaviour of a cracked dam system. The analysis procedure will include provisions for a flexible foundation as modelled by a elastic half-plane, as well as, a reservoir domain including hydrodynamic effects as modelled by a 2-D compressible wave equation. 2) To determine the effective system frequency of vibration of a cracked dam system. By applying sine sweep techniques and spectral analysis of the response, an effective system frequency of the cracked dam system can be obtained. This will lay the foundation for understanding the dynamic response of the cracked dam system. 3) To obtain a better understanding of the global response of a cracked dam system as compared to an uncracked linear dam system when subjected to earthquake and sinusoidal excitations. The displacement, velocity, and acceleration time history at key points on the dam domain will be studied, as well as, the maximum response attained during the response. 4) To develop a simplified procedure to quickly estimate the response of a cracked dam system. Such an approximate method can be used to generate cracked response spectra 5 Section 1: Introduction and Overview for a given earthquake record and to screen potential candidates for a more detailed analysis. 1.3 Scope of Investigation Because of the complexity of the problem, the investigation of the cracked dam system will be subjected to the following limitations and assumptions: • Only global response quantities such as relative displacement, relative velocity, and absolute acceleration will be considered. Local effects, such as stress and strain near the crack tip are better studied by more refined finite element models, and are beyond the scope of this investigation. • Only one horizontal crack originating from the heel of the dam at the dam-foundation interface is assumed to exist. Multiple cracks and cracks at other dam locations are not considered. • Uplift pressures and water effects within the cracks are complex and misunderstood processes and will not be considered. • Cavitation effects between the upstream dam surface and the reservoir have been found to be negligible in the dynamic response of a gravity dam (El-Aidi and Hall, 1989). These effects will not be considered here. • Other nonlinear processes, such as material yielding are not included in this study. • Crack length of 0%, 12.5%, 25%, 37.5%, and 50% of the total base length will be investigated. Crack lengths beyond 50% of the base length are consider to be unstable and unrealistic; therefore, they are excluded from this study. • Dam instability and base sliding due to reduction of the effective base length are ignored. 6 Section 1: Introduction and Overview • Only horizontal excitation will be investigated. Vertical motions are ignored. • 3-D effects such as cross canyon motions, and interaction of adjacent gravity blocks are ignored. Only in-plane motions will be considered. 1.4 Organization of the Thesis The thesis will be organized in the following manner: Section 1: Introduction and Overview Section 2: General Theory of HFTD Approach. The general theory of the hybrid frequency time domain analysis will be discussed in reference to general nonlinear dynamic systems. The basic principles of frequency domain analysis for linear systems will also be briefly reviewed. Section 3: HFTD as Applied to a Cracked Dam System The details of the HFTD as applied to the cracked dam system will be discussed. The idealization of the reservoir-dam-foundation systems will be outlined and assumptions stated. Implementations issues and source of analysis errors will also be discussed. Section 4: Effective System Frequency Identification. The effective system frequency of a 61.7 m (200 ft.) cracked dam system will be studied. The system frequency will be obtained by two methods: (1) spectral analysis of response due to white noise excitation; and (2) resonance frequency through sine sweep. 7 Section 1: Introduction and Overview Section 5: SDOF Bi-Linear Stiffness Approximation. An approximate method to simplify the cracked dam system is formulated and its accuracy is evaluated by comparing the approximate response with the response obtained from the FIFTD procedure. Section 6: Cracked Dam System Response to Earthquake Motions. The response of the cracked dam system is evaluated by subjecting it to two earthquake records. The response will be compared to those obtained for an uncracked dam system. Various dam sizes will be used in order to sample a range of dynamic systems such that a response spectra can be obtained. The spectra response will be used to verify the approximate SDOF bi-linear method. The influence of flexible foundation and reservoir will be separately evaluated. Section 7: Conclusions and Recommendations. A summary of findings and conclusions from all previous chapters are provided. Recommendations for future research are also suggested. 8 Section 2: General Theory of HFTD Approach and Idealization of Dam System 2. General Theory of HFDT Approach and the Idealization of Dam System 2.1 Introduction Nonlinear systems are traditionally solved in a time step integration technique in which the dynamic equation of motion is solved at each time step and nonlinear material variations updated depending on the state of the nonlinearity. However, restricting solution techniques to time domain eliminates the use of frequency dependent properties which are often found in half-space modelling of foundation and wave equation formulation for the impounded reservoir. The Hybrid Frequency Time Domain (HFTD) procedure was developed to overcome the linear limitation of the frequency domain analysis (Dabre and Wolf, 1988). This chapter will explain the HFTD procedure in its general form, and as it is applied to any nonlinear system. As a primer, the basic principles of frequency domain analysis will be reviewed. 2.2 Linear Dynamic Analysis in Frequency Domain Linear dynamic analysis can be separated into two categories: (1) analysis in the time domain; and (2) analysis in the frequency domain. In the time domain analysis, the equation of motion is separated into discrete time steps, and solved at each time step by numerically differentiating the response quantities. Through mathematical transformation methods, the time response quantities and equation of motion can be transformed into the frequency domain. The essence of the frequency domain is the use of Fourier Transform to represent a time function as a summation of a series of sinusoidal waves of varying amplitudes and frequencies. By applying the above theory to dynamic analysis, the excitation time history can be transformed to a collection of 9 Section 2: General Theory of HFTD Approach and Idealization of Dam System sine and cosine waves. The response to this excitation can be computed by superimposing the steady state response of each sinusoidal excitation transformed by the Fourier Transform. Considering a discrete multi-degree of freedom (MDOF) dynamic system, the time domain equation of motion is given by: W]{x'(t)} + [C]{x'(t)} + [K]{x(t)} = Fit) = -[M]{\}x"g(t) (2. where M , C, K are the global mass, damping, and stiffness matrices of the system respectively. The response quantities are given by the vector {x(t)} and its respective time derivatives. For base excited systems, the free-field base acceleration xg"(t) produces the inertia forces based on influence coefficient vector {I}. For usual base excited systems, this vector will have unit value corresponding the degree of freedom in the direction of the excitation. To transform Eqn. 2.1 to the frequency domain, a Fourier transform of the time dependent quantities is required. For practical purposes, the forcing and response functions are considered to be composed of discrete values at equal time intervals. Therefore frequency transforms must be evaluated via discrete Fourier Transforms methods. Considering the forcing and response time functions of length t = 0 to T 0 being sampled at N equal intervals of At from n = 0 to N - l . Similarly, the corresponding frequency transform function of frequency range from oo = 0 to 1/To are sampled at N equal intervals of Ml from k = 0 to N - l . Therefore, the Discrete Fourier Transform (DFT) of a time function g(kAt) and the 10 Section 2: General Theory of HFTD Approach and Idealization of Dam System Inverse Discrete Fourier Transform (IDFT) of frequency function G(nA£2 ) in their general forms may be given by: JV-l g(kAt)= ^T.G(nACi)ei(kAt)(nAn)AQ N-l V G(nAQ)=^g(kAt)e-KkAtXnAn)At k=0 where the total duration of response is given by T 0 = NAt and frequency increment AD. = 2n/T0. Applying Fourier transforms to the m* component of the response vector and free-field acceleration in Eqn (2.1), and letting t; = kAt and <x>j = nAQ, yield, ™ TPaY ~rnav — —ma* xm(0= Yxi^y^AQ; x'm(t)=ico] Zx(«,y f f l / iAQ; x"m(t)= -co/ Z x ^ K ^ A Q o)=0 Q)-0 a)=0 *;(0= ilx^y^AQ (2.3) <B=0 where X(oo) and Xg(co) are Fourier transform of x(t) and x"g(i). The maximum frequency a w is given by (N -l)^^j> • The equation of motion in the frequency domain can then be written as: l !T [ ( ^ / [ ^ X ( a > y ) } + « ^ (2.4) to=0 The matrix equation for each frequency, u)j is therefore: [-co;[M] + ico ,[C] + W]{X(^)} = {Z}{x'>,)} where: {L} = ~[M]{l} Solving for X(co) for all frequencies would yield the response in the frequency domain. The response in the time domain, x(t) can then be obtained by calculating the inverse Fourier transform (IDFT) of X(co). 11 Section 2: General Theory of HFTD Approach and Idealization of Dam System 2.3 Hybrid Frequency Time Domain Analysis for Nonlinear Structures Nonlinear dynamic systems possess properties which varies during the duration of the time response. These varying properties usually take the form of changes in stiffness or damping of a defined conservative system. The magnitude of the change is dependent on the current response of the system which is not known in advance; therefore, an iterative approach must be taken. This stiffness or damping change when combined with its corresponding response quantities produces an additional force in the equation of motion. This force can be viewed as a restoring force which corrects the response of the system to conform to the changes in the system properties. In time domain analysis, the dynamic response is calculated in a progressive time step manner. If the material properties are found necessary to change at the current time step, the property will be updated and the response at that time step recalculated until the force or energy balance in the equation of motion is satisfied. In the frequency domain analysis, this change in material property cannot be accommodated directly. The basic principle of superpositioning of the steady state response to each sinusoidal excitation of varying frequency restricts the frequency domain analysis to invariant linear systems. However, as stated earlier, these time varying forces produced by the change in the material properties can be viewed as an restoring force and moved to the right hand side of the equation of motion to be applied as an external force. 12 Section 2: General Theory of HFTD Approach and Idealization of Dam System As an example, for a system with nonlinear stiffness, the governing equation of motion in time domain is given by: [M]{x"(0}+[C]{x'(0} + [^  + ^ ]{^(0} = {^^(0 (2-6) where K„i is the stiffness variant which is dependent on the displacement response, x(t). By moving the nonlinear force [Kni]{x(t)}={R(t)} to the right hand side of the equation, Eqn. 2.6 becomes: [M]{x"(0} + [C]{x'(0} + [K]{x(t)} = {L}x'g\t) - [Knl }{x(t)} = {L}x^(t) + [Rf (t)} (2.7) where R/t) is the nonlinear force, which will be cumulatively updated until the force balance in the Eqn. 2.7 is satisfied to a tolerable level. Note that the left hand side is now a invariant linear sub-system of the original nonlinear system. This sub-system will be referred to as the pseudo-linear system of the nonlinear system. To apply the nonlinear procedure in the frequency domain, Eqn. 2.7 is transformed to: [-co2[M] + ico[C] + [K]]{X(co)} = {L}X"g(co) + {Tf(co)} (2.8) where R/<x>) is the Fourier transform of the nonlinear force R/t) determined in the time domain. Note that the left hand side of the Eqn. 2.8 now represents the pseudolinear system subjected to the nonlinear force, -ft/co) and the linear force {L}X g"(ui). Since, the nonlinear force Rj(t) is dependent on the response x(t), it is evident that to solve the equation of motion, an iterative procedure is required. Each /?* iteration determines the additional corrective force A/?/t)" required to satisfy Eqn. 2.7 and is added to R/t)" of the current iteration to form the corrective force time history for the next iteration (n+1) (see Eqn 2.9). 13 Section 2: General Theory of HFTD Approach and Idealization of Dam System Rf(t)n+X = Rf(t)n + ARf(t)n (2. The overall iterative procedure is outlined below: 1. The pseudo-linear response is determined in the frequency domain via Eqn. 2.8 for some initial value of Rf(co) 2. The response quantities are transformed into the time domain, and the response dependent additional corrective force AR/t) is evaluated in the time domain for the period of interest and is added to R0) to form the total corrective force for the next iteration (Eqn. 2.7 and Eqn. 2.9). 3. The nonlinear force time history, R/t) is then transformed to the frequency domain R/co), and the equation of motion of the entire system is recalculated and solved in the frequency domain via Eqn. 2.8. 4. Repeat Step 2 through 3 until each time step has satisfied a given convergence criteria. The convergence criteria in Step 4 can be any suitable response check (i.e. displacement or energy) to ensure an energy or force balance in the equation of motion Eqn. 2.7. As we have seen, the HFTD is the merger of the frequency domain solution and time stepping convergence. The HFTD procedure is especially important in dynamic systems involving frequency dependent properties that would otherwise be unsolveable in the time domain. Foundations modelled as elastic half planes and structures with impounded water regimes modelled as 2-D wave equation have such dynamic properties and can only be solved in the frequency domain. By using HFTD analysis, a new breed of nonlinear dynamic dependent structural systems can now be analysed. 14 Section 2: General Theory of HFTD Approach and Idealization of Dam System 2.4 Chapter Summary The HFTD procedure to solving nonlinear problems is a three step process: • Linearization of Problem: The nonlinear system is broken down into a pseudo-linear system and nonlinear effects are applied as restoring forces. The pseudo-linear system must possess all kinematics displacement expected during the nonlinear response of the system. • Solving a Linear Problem in Frequency domain: The pseudo-linear system, that is subjected to external applied force, and the restoring force are solved in the frequency domain. All the frequency dependent properties in the system are preserved, thus allowing frequency dependent properties to be included in the solution. • Convergence Criteria and Restoring Force: At each time step, the system is checked to determine whether the system is in its linear or nonlinear state. If the system is found to be in the nonlinear state, restoring forces are applied and the system is analysed with updated restoring forces. This is repeated until the response at that time step satisfies the required convergence criteria. If the system is found to be in a linear state, the time step will advance to the next time step. The above explanation of the procedure is in its general form and can be applied to any nonlinear system. In the next section, the HFTD will be explained in further detail as it is applied to the cracked dam system. 15 Section 3: HFTD as Applied to Cracked Dam System 3. HFDT as Applied to Cracked Dam System 3.1 Idealization of Reservoir-Dam-Foundation System In order to apply the HFTD analysis procedure a pseudo-linear system must first be identified. In the cracked dam model, the pseudolinear system will consist of several key components which form the reservoir-dam-foundation system. These components will be considered separately as sub-domains. The following is a description of each of the components. 3.1.1 Gravity Dam Monolith The gravity dam monolith is a two dimensional representation of a typical gravity dam cross section. The dam section is composed of four-noded parametric plain stress finite elements (Figure. 3.1). Concrete damping within the dam is considered as viscous damping. The upstream face of the dam is assumed to be vertical, and the base is horizontal to simplify the applied hydrodynamic and foundation models. Three dimensional effects from adjacent blocks will not be considered. This is because under severe earthquake loading, the shear stresses transmitted across the sides of adjacent dam blocks are relatively small compared to those induced by the inertia forces created by the free-field acceleration (Rea, Liaw, Chopra, 1975). Therefore, the dam block is considered to behave independently of the adjacent blocks. 16 Section 3: HFTD as Applied to Cracked Dam System Figure 3.1. Reservoir-dam-foundation model and its associated static and dynamic forces 3.1.2 Visco-Elastic Half-Plane Foundation Medium The foundation medium will be modelled as a visco-elastic half-plane. This formulation has frequency dependent stiffness properties which complements the framework of the HFTD procedure. As well, the half-plane representation has two main advantages over the traditional assemblage of foundation finite elements. Firstly, the size of the overall model does not increase since no additional finite elements are required to model the physical foundation. Only additional terms are combined to augment the dynamic stiffness of the system. Second, and more importantly, the half-plane formulation provides a means for the energy accumulated in the supported structure to dissipate and radiate into the semi-infinite halfspace. If finite elements were used to model the foundation, distortions and energy reflections through the foundation domain may be of concern. To eliminate some of these concerns, the extent of the assemblage of foundation elements must be large enough to reduce these effects; however, the size of the model 17 Section 3: HFTD as Applied to Cracked Dam System increases and efficiency of the solution technique is sacrificed. Special energy transmitting boundaries have been developed to limit the extent of foundation required (ANSYS, 1991). However, the horizontal boundaries defining the bottom of the foundation remain rigid, thus causing unwanted energy reflection especially in the vertical motions. Therefore, the visco-elastic halfplane representation of the foundation appears to be the more suitable choice. The use of this mathematical representation of the halfspace produces complex-value dynamic stiffness matrices when solved in the frequency domain (Dasgupta and Chopra, 1979). This stiffness is considered dynamic in the sense that the stiffness values vary with frequency. The half space is considered to be a homogeneous, isotropic, linear elastic material. The displacements and forces arising from the elastic half plane are active along a set of equally spaced foundation nodes. The coordinates of the foundation are coincident with the dam base nodes, and the foundation-dam interface is formed by constraining or slaving the dam base nodal degree of freedom (DOF) with the foundation nodal DOF. Note that the displacement between adjacent foundation nodes are linearly interpolated functions; therefore, they are compatible with the dam base elements. Details of the evaluation of the foundation stiffness are outlined in a study by Dasgupta and Chopra (1969). The contributions of the foundation are provided as an addition to the global stiffness and damping matrix at the foundation nodal degree of freedoms. Foundation damping is provided as a frequency independent constant hysteretic damping value. The equation of motion from Eqn. 2.8 in the frequency domain then becomes, 18 Section 3: HFTD as Applied to Cracked Dam System '-6)2[M]+i(co[C] + nSf (co)) + [K + Sf (co)]]{X(co)} = {L}{x » } + {Tf(co)} (3. where rj is the constant hysteretic damping value and Sf(co) is the complex-valued dynamic foundation stiffness. 3.1.3 Hydrodynamic Effects Several different models exist which represent the hydrodynamic effects of the impounded reservoir. The simplest model is the lumped mass representation (Zangar, 1952). With this approach, an empirically developed hydrodynamic pressure distribution is calculated based on the depth of reservoir, magnitude of excitation, and shape of the upstream face of the dam. Forces resulting from the pressure distribution are applied onto the upstream nodes as discrete masses. The drawback of this approach is that the empirical hydrodynamic pressure profile is based on the first mode response of a linear elastic dam block. Also the effects of water compressibility and possible resonance of the water reservoir with the response of the dam is neglected. With the expected nonlinear behaviour of the cracked dam system, the assumption of first linear mode response is violated and would lead to unsatisfactory results. Hydrodynamic effects can also be represented as an assemblage of fluid elements solved in the time domain (Lee and Tsai, 1990). These fluid elements are formulated to account for energy wave propagation; however, similar to the foundation assemblage, the size and expense of solving the model is increased with this approach. 19 Section 3: HFTD as Applied to Cracked Dam System To take advantage of the frequency domain used by the HFTD analysis, the hydrodynamic effects of the reservoir can be solved in the frequency domain by applying a two dimensional wave equation formulation. With this approach, the impounded reservoir is represented in two parts. The hydrostatic pressure is represented by equivalent nodal forces applied to the upstream face of the dam. Hydrodynamic effects developed from applied ground acceleration are modelled as a two dimensional wave equation. Considering the water as a linearly compressible fluid neglecting viscosity and irrotational flow, the wave equation assumes that the reservoir is serni-infrnitely long; therefore, the energy dissipated through the wave propagation is not reflected back towards the dam. By considering compressibility of the water, the resulting hydrodynamic pressures will be complex valued and frequency dependent. Following the formulation by Chopra (Chopra, 1968), the development of the hydrodynamic pressure can be decomposed into two complex-value force components: BI(GO) and B2(CJO). Each component is dependent on different motions of the dam system. The free-field ground motion which corresponds with the rigid body motion of the dam develops the hydrodynamic pressure Bi(oo). The flexible response of the dam upstream face produces the hydrodynamic pressure B2((o). Therefore, the equation of motion from Eqn. 2.8 including the hydrodynamic effect written in frequency domain becomes: [-co2 [M] + ico[C}+ [K]]{X(co)} = x;(o})[-[M]{l} + B, (co)\-X(co)B2 (co) +Rf(co) (3. 20 Section 3: HFTD as Applied to Cracked Dam System The force component B 2 which corresponds with the flexible response of the dam can be moved over to the left hand side of the equation and added to the properties of the dam system. By decomposing B 2 into its real and imaginary components, the real components will represent an added mass component to the dam mass, and the imaginary component as an added damping. The force component Bi, which represents the hydrodynamic pressure due to the rigid body motion remains on the right hand side of the equation as an applied force dependent on the ground motion excitation. Hence, the equation of motion becomes: [-co2 [M + Re[B2 (co)]] + ico[C + Im[B2 (co)]] + [K]]{X(co)} = X"g(co)[{L] + BL (co)] (3.3) Including the foundation stiffness from Eqn. 3.1, the total equation of motion in the frequency domain becomes: -co2 [M + RQ[B2 (CO)]]+i[co[C + Im[B2 (co)]] + rjSf (co)) + [K + Sf (co)]]{X(co)} = {L + B,(co)}Xffg(co) + \Rf(co)) The details of calculating the hydrodynamic force components, Bi(w) and B 2(w) terms are given in Chopra, 1968 and Fenves and Chopra, 1984. 3.1.4 Crack Identification The crack along the dam-foundation interface is identified by two free surfaces, namely the base of the dam and the foundation surface. The length of the crack is defined by a set of nodal 21 Section 3: HFTD as Applied to Cracked Dam System pairs (see Figure 3.2). Each of these cracked nodal pairs contains one dam base node and one foundation node; these nodes are unrestrained. In the uncracked portion of the dam-foundation interface, the dam base nodes are slaved to the foundation nodes and are constrained to move together in both horizontal and vertical motions. For the uncracked dam case, all base nodes and foundation nodes are slaved. Cracked Q A M Slaved Hodee Figure 3.2. Crack identification along dam-foundation interface. 3.2 Identifying Cracked Pseudo-linear System To apply the HFTD analysis procedure, the nonlinear system must be decomposed into two components: a pseudo-linear system and a set of time varying restoring forces (see Figure 3.3). This linear system must satisfy all kinematic motions expected throughout the duration of the dynamic response. For the cracked dam model, these motions include the opening of crack. Therefore, the pseudo-linear system will be the cracked dam model; i.e. the model without nodal constraints applied at the cracked nodes. The applied nonlinear or restoring forces will prevent the displacement overlap of the upper crack surface of the dam structure with the lower crack surface 22 Section 3: HFTD as Applied to Cracked Dam System of the foundation. Hence, the restoring force will be dependent solely on the displacement of the upper and lower crack surfaces. Figure 3.3. Decomposition of nonlinear cracked dam system to pseudo-linear system and restoring forces. 3.3 Modal Decomposition Up to this point, all equation of motions presented include all degrees of freedoms (DOF) present in the dam-foundation system. Solving this system will involve dealing with a set of 2*(N t o t e l) simultaneous equations where N t o t a i is the number of nodes in the finite element model. Each node will consist of two DOF. This is a tedious and computationally expensive task. By transforming the global set of DOF to a generalized set of co-ordinates via a modal transformation with a set of Ritz vectors, the computational demands can be significantly reduced. In modal transformation, it is essential that the set of Ritz vectors or displacement shapes, represent the full range of motion expected of the system. The logical choice of Ritz vectors for the cracked dam system includes the eigenvectors of the pseudo-linear cracked system to represent the crack open phase of the dam response. However, this alone with the application of the Peeudo-Wnear Cracked Dam System Nonlinear Cracked Dam System Restoring Force 0. 23 Section 3: HFTD as Applied to Cracked Dam System restoring force cannot adequately represent the range of motion during the crack closed phase of the dam response. This is unfortunate since this represents an orthogonal set of n modal vectors uncoupling the equation of motion resulting in n SDOF equations to be solved. To represent the response of the system in the uncracked phase, a set of eigenvectors derived from the uncracked dam system must also be included. Note that this combination of two sets of eigenvectors is no longer mutually orthogonal; thus, the reduced mass, stiffness, and damping matrices will contain non-zero off diagonal terms. Nevertheless, the goal of reducing the system to a manageable size problem has been achieved. If we let the matrix [\\r] represent the set of n Ritz vectors, each containing 2*N t otai degrees of freedom, and the vector {Z(t)} be the response of the generalized n coordinates, then the dam response vector {x(t)} is given by: {x(t)} =[¥]{Z(t)} (3.5) The Ritz vectors are composed of two orthogonal set of eigenvectors, mainly the uncracked dam system and cracked dam system. The eigenvectors of each orthogonal set are computed from the eigenvalue problem: Kucrk,crk ]{^}. = [ M u c r k c r k ]{^} ! (3.6) where \ is the eigenvalue of i * order, M u c rk, c rk is the structural mass matrix, and Kucrk,crk is the structural stiffness matrix plus static foundation stiffness. The subscript designations urck and crk of the mass and stiffness matrix indicate the matrix formulated for the uncracked and 24 Section 3: HFTD as Applied to Cracked Dam System cracked linear system for the purpose of calculating the orthogonal Ritz vector set only. For all subsequent formulations regarding the pseudo-linear system, the cracked stiffness, K c r k will be used. Partitioning the structural stiffness matrix into the dam DOF and base DOF, the combination with the foundation stiffness is given by: [K\ = [K]+\sf] (3.7) o ! o i 0 i ^(0)_ where the subset d represents degrees of freedom within the dam including the upper crack surface and/is degrees freedom of the base nodes slaved with the foundation. S/0) is the static foundation stiffness taken to be the dynamic foundation stiffness evaluated at oo=0. By combining the substitution of generalized co-ordinates and the equation of motion, the reduced system is given by: [M*]{Z'XO}+[C*]{ZXO}+[^ ]{Z(0} = {^}^(0+{^/(0}+{^;} (3.8) Therefore, the response will contain both static and dynamic components. The modal transformation to the generalized mass M * , damping C*, and stiffness K*, modal participation vector L*, the restoring force Rf*, and the applied static force F m g * are given below. [M*]=MR[M]M; [c1 = Mr[q[H; M = MTHH; KhkfM; {*/}=WTM; lV} = M>,} (3.9) K where: [K] df K fd K Sf. fr. 25 Section 3: HFTD as Applied to Cracked Dam System Note the static gravitational force {Fmg*} is included in the dynamic analysis to account for the gravitational force to prevent any crack opening. Similar to the transition of the equation of motion of a SDOF system from the time domain to the frequency domain outlined in Eqn. 2.1 and Eqn. 2.4, Eqn. 3.8 may be reduced to the frequency domain leading to: [ - ^ [ A T M C ] + [K\co)]]{Z(co)} = + {R/(co)} + {F:S(CO)} (3.10) where {Z(u))} is the frequency response of the generalized coordinate {Z(t)}. 3.4 Sensitivity of Response to Number of Ritz Vectors In a linear dynamic system, the majority of the response is generally governed by only the first few modes of vibration. This is also expected to hold true for the global response of the nonlinear cracked dam system. To determine the influence of the number of Ritz vectors on the accuracy of the response, a 91.5 m (300 ft) dam section with 50% crack length will be analysed with different numbers and combinations of Ritz vectors. Details of the analysis will be given in latter sections. Three different combinations of uncracked and cracked modal vectors are used to form the set of Ritz vectors: 2x1, 4x4, and 6x6, where the numbers respectively indicate the number of eigenvectors determined from uncracked and cracked dam systems. It should be noted that the system represented by 2x1 modes was composed of the first uncracked and cracked modal vectors and the third uncracked modal vector. It is important to include the third modal vector because the static gravity forces applied to the dam causing static displacement will also be 26 Section 3: HFTD as Applied to Cracked Dam System determined through modal decomposition and therefore, at least the first vertical mode of the uncracked dam should be present in the collection of Ritz vectors to account for static displacements. For the typical dam section, the first vertical mode is usually the third mode in the uncracked dam system. J I v 1 "1 I I 1 Mode 1:4.54 Hz Mode1:Z16Hz 2 2.5 3 3.5 4 4.5 5 2.5 3 35 * *-5 time [sec] Figure 3.4. Sensitivity of number of Ritz vectors used in the analysis of dam top response of 91.5 m. (300 ft.) dam section with 50% crack subjected to Loma Prieta scaled to 5g. 27 Section 3: HFTD as Applied to Cracked Dam System The dam section will be subjected to the horizontal component of the Loma Prieta 1989 earthquake scaled to 5g. This large scale factor is unrealistic; however, it is required for calibration purposes to ensure crack opening such that both uncracked and cracked modes will participate in the response. The horizontal relative displacement, relative velocity, and absolute acceleration of the dam top are provided in Figure 3.4. From the relative displacement response, all combinations of Ritz vectors provide very similar results. The maximum displacement from the 6x6 modes shows negligible differences when compared to the 4x4 modes, and it only differs by 3.9% for 2x1 modes. The maximum velocity from 6x6 is again only 13.0% greater than that obtained from 2x1, and only negligible differences result when compared to 4x4 modes. Again for maximum acceleration, the response from the 6x6 modes and 4x4 modes show only slight differences in their responses. Although the maximum acceleration obtained from the 6x6 modes is only 6.7% greater than 2x1 modes, a noticeable difference in the frequency content of the response in the 2x1 modes renders this unacceptable. This is due to the participation of the higher modes in the acceleration response which is absent with only 2x1 modes. Therefore, if acceleration and stresses are of concern in the investigation, higher number of modes is required in order to capture the contributions of the higher modes. The performance of the 4x4 modes Ritz vector set shows excellent accuracy without sacrificing the computational resources of the 6x6 modes. Therefore, the use of 4x4 modes will be used in all subsequent HFTD analysis. 28 Section 3: HFTD as Applied to Cracked Dam System 3.5 Calculation and Application of Nonlinear Restoring Force To account for the nonlinear effect of I the cracked dam system, a set of restoring forces must be determined and applied at the necessary time step. This force represents the contact force when the upper crack surface makes contact with the lower crack surface. In the pseudo-linear crack dam model where the crack surface is free to overlap, the restoring force will limit the amount of overlapping (see Figure 3.5). Between the upper and lower crack surfaces, it is assumed that a vertical spring exists between each nodal pair. The upper and lower crack nodes are allowed to slide horizontally relative to VC0D3 each other without restriction, thus the crack Figure 3.5: Application of restoring force on vertical surface is assumed to be frictionless in the crack opening displacement (VCOD) of crack interface nodes horizontal direction. The restoring force is applied iteratively until either the crack is closed or the crack is determined to remain open. The status of the crack is determined by the crack open/close criterion. 29 Section 3: HFTD as Applied to Cracked Dam System 3.5.1 Crack Open Crack Close Criterion The difference between vertical displacement of the upper crack surface and the vertical displacement of the lower crack surface is defined as the vertical crack open displacement (VCOD). If the VCOD is negative, which indicates that there is crack overlapping, then the restoring force is applied. The restoring force will be determined based on the VCOD and spring stiffness at each crack node pair location to prevent the overlapping. The restoring force will be applied iteratively until the absolute value VCOD is less than the specified tolerance level. When this is true, the crack is considered to be closed. If VCOD is positive, this indicates that there is a separation of the upper and lower crack surfaces. In this case, the crack at that location is considered to be open only if no restoring force is present. If restoring force is present, the force must be reduced iteratively based on the VCOD and spring stiffness until either restoring force is zero or the VCOD satisfies the tolerance level. Note that the restoring force cannot be negative since this will restrain the crack from opening. The crack open/close criterion can be summarized as follows: Crack Close = Crack Open \VCOD\ < ToleranceLevel Rf>0 VCOD > 0 I */=<> (3.7) Before the iteration process begins, the restoring forces are initialized by setting their values for all time steps to the static reaction force of the uncracked dam at the location of the crack nodes. This 30 Section 3: HFTD as Applied to Cracked Dam System will eliminate VCOD overlap during any small magnitude excitation. Iterated restoring forces are added to or subtracted from this initial value as dictated by the crack state. 3.5.2 Tolerance Level The specified tolerance level describes the level of accuracy used in specifying the amount of allowable V C O D overlapping within the dam-foundation crack interface. From experience, the global response of the dam is not sensitive to the overlapping tolerance. In general, the tolerance level is set at either 0.5% of the maximum expected dam top displacement, or 1% of the maximum VCOD. Note that a tolerance level too small will lead to slow convergence, thus leading to inefficient use of computing time. 3.5.3 Selection of Spring Stiffness The stiffness of the springs along the crack interface at each crack node pair are uncoupled and independent of each other. Selection of the magnitude of spring stiffness is critical in the efficiency of the HFTD procedure. Spring stiffness which is too small or too large can lead to slow convergence of the crack open/close criterion. A certain amount of trial and error is required to obtain the optimal stiffness value, but a suitable spring stiffness can be estimated by: KR = — 10-' C where KRf is the spring stiffness for crack node pair, A is the distance between base nodes, E is the lesser elastic modulus of the dam or foundation, and L is the height of the base element. This guideline originates from a nonlinear time step analysis procedure and suggests an increase in 31 Section 3: HFTD as Applied to Cracked Dam System magnitude to the spring stiffness, AEIL by one or two orders of magnitude (ANSYS, 1990). However, during the course of the study, it was found that an overly stiff spring resulted in a slow rate of convergence and in extreme cases caused divergence in the VCOD iterations. Small spring stiffness resulted in convergence however, the rate of convergence depended on the magnitude of the spring stiflhess. A detailed study of the effects of spring stiffness versus rate of convergence was beyond the scope of this investigation, but the recommendation given in Eqn. 3.8 appeared to give the best result and for the analysis performed in this study. Typically, convergence was reached within 3 to 4 iterations for the tolerance level recommended in Section 3.5.2. 3.5.4 Application of Restoring Force During the analysis procedure, the VCOD of each crack nodal pair is determined at each time step by the crack open/close criterion. If neither the crack open nor crack closed states are identified, the restoring force is updated incrementally as determined by the spring stiffness and the VCOD as given below: Ks x VCODm(nAt) = ARfm(nAt) (3. where Ks, VCODm, and ARfm are the restoring stiffness, VCOD, and the restoring force for the m* crack node pair along the crack interface evaluated at time nAt. The restoring force is then applied to both upper and lower crack nodes in equal magnitudes but opposing directions. The negative sign in the right hand side of Eqn. 3.9 provides the direction that is required to force the crack apart when the crack is overlapping (VCOD is negative), and to force the crack together (VCOD is positive) when the crack is opened by the restoring force. 32 Section 3: HFTD as Applied to Cracked Dam System The applied restoring forces are actually a set of impulse forces applied at time t. From SDOF dynamic theory, the response to a single impulse load induces only a change in velocity at the time of application. However, the goal is to induce a displacement at time t to reduce the crack interface overlap to a tolerable level. Therefore, a single impulse force applied at t is inadequate in satisfying the displacement criterion. Applied Unit Force R e s p o n s e Figure 3.6: Applied Impulse Load and Response In order to induce a displacement at time t, a doublet is required. A force doublet is defined as the first derivative of a impulsive load. This can be interpreted as a pair of impulsive loads of equal magnitude applied in opposing directions (see Figure 3.6). The response resulting 33 Section 3: HFTD as Applied to Cracked Dam System from a doublet can be mimicked by applying a pair of impulse forces; one at the current time step and one at one time step back as shown in Figure 3.6. This will invoke the necessary displacement to reduce the overlap at time t. Note that the application of the force doublet would not produce zero velocity at time of application. The main goal here is to minimize the crack overlap and the velocity is not of concern. However, the induced velocity will affect convergence at later time steps as the response due to the induced velocity propagate through time. It is speculated that the rate of convergence of the overlapping displacement criteria will benefit if a combination of a doublet and impulse forces is applied such that the response produces a displacement and no velocity at time of application. This method was not applied in this study but should be explored in future improvements. Another justification for applying a force at one time step behind the occurrence of contact is that the pair of forces bounds the exact time at which the contact occurs (see Figure 3.7). In a nonlinear time domain finite element analysis dealing with contact type problems, the time step can be readily adjusted to determine the exact time of contact. This is accomplished by subdividing the time steps into substeps such that the response of the contact surfaces calculated within the substeps yields a zero velocity (Cook, 1989). This will reduce the amount of numerical error causing period elongation and amplitude decay. In HFDT with the Fast Fourier Transform (FFT) to transform time domain to frequency domain, the time step must remain constant. Therefore, the best approximation of the exact time of contact is to bound the time of occurrence with two adjacent time steps (Figure 3.7). Although 34 Section 3: HFTD as Applied to Cracked Dam System this will produce some numerical error, it will be limited since the displacement caused by the trailing time step produces very small displacement changes. The error can be quantified by comparing the amount of energy applied to the dynamic system with the internal energy resulting from the spring, damping, and the kinetic energy of the responding system. Q O O > H 1 1 > I . crack in contact tolerance level iteration n=1 Q O O > H 1 1 Q x : 1-n=3 n=2 n=1 t-1 t After iteration n=3 Figure 3.7. Application of restoring force pair and its effect on VCOD 3.5.5 Effectiveness of Restoring Force on Generalized Response It is clear that the purpose of the restoring forces is to prevent overlapping of the upper and lower crack surfaces. Within the implementation of modal decomposition, the restoring forces serve a more practical role. As mentioned above the uncracked and cracked mode sets of Ritz vectors contribute to both phases of the crack dam system. When the crack is open, no restoring is applied and the contribution of the cracked mode set is dominant. When the crack is closed, the restoring force is applied to the cracked mode set and effectively nullifies the modal response of the cracked modes leaving uncracked modes to contribute to the response. 35 Section 3: HFTD as Applied to Cracked Dam System Figure 3.8 shows the effects of the restoring force on the first uncracked and cracked modal responses of the relative displacement at the dam top of a 50% cracked dam, 61.0 m (200 ft.) high subjected to a 10 FIz sinusoidal acceleration. The analysis used 4x4 modes (4 uncracked modes and 4 cracked modes) to reduce the system. Only the modal component of the first cracked and first uncracked modes are shown. It is clear that the contribution of each mode is quite distinct. When the crack is opened, as indicated by zero restoring force, the cracked mode mainly contribute to the response. When the crack is closed, indicated by nonzero restoring force, the contribution of the cracked mode is almost zero and the majority of the response is from the uncracked mode. Note also the higher frequency oscillations within each modal response. These are due to the modal interaction between these two non-orthogonal modes and modes of higher order. It is evident that the magnitudes of these interactions are quite small compared to the main modal response. Therefore, aside from the interaction, it is clearly demonstrated that the role of the restoring force is to control which mode set is to be active depending on the current crack state. 36 Section 3: HFTD as Applied to Cracked Dam System Time [sec] 4.E+04 O •B 2.E+04 2 1.E+04 Crack Open (b) Crack Open T ime [sec] Figure 3.8. Modal response of first uncracked and cracked modes in the dam top response (a) and restoring force at crack mouth (b) of 61.0 m (200 ft) dam section with 50% crack subjected to sinusoidal excitation of 10Hz. 37 Section 3: HFTD as Applied to Cracked Dam System 3.6 Implementation of HFTD The HFTD analysis procedure as applied to the cracked dam system can be summarized in the following flowchart diagram (Figure 3.9). There are five main steps in implementing the analysis procedure. Step 1: Input model data: All relevant model data, including the geometry and material properties of the cracked dam model, are specified and the global mass, stiffness, and damping matrices are constructed. Step 2: Construct Ritz vectors: Two linear models; uncracked dam system and pseudo-linear cracked dam system are constructed by applying the dam-foundation nodal constraint equation. For the uncracked dam system, all dam-foundation node pairs are constrained. For the cracked dam system, only the uncracked portion of the crack path is constrained. The eigenvalues and eigenvectors are computed for both systems forming fymok and (jw. For the uncracked linear system, static analysis is performed to determine the static base reaction force. The reaction force at the cracked nodes in the pseudo-linear system will be used to initialize the restoring force. 38 Section 3: HFTD as Applied to Cracked Dam System Flowchart for Hybrid Frequency Time Domain Analysis Procedure as Applied to Cracked Dam System Generate Data: Geometry Construct G loba l Materials Property Matr ices: Dynamic Paramters [M],[K],[C] Control Motion Step 1 Cracked P h a s Uncracked P h a s e Apply nodal constraints along uncracked portion of the dam-foundation interface and determine modeshapes , $cr Apply nodal constraints along entire dam-foundation interface and determine modeshapes , tjjuncrk C o m b i n e modeshapes from both phases to form Ritz vector transform matrix [cJ)cr]+[(t)Ucr]=:[cD] Step 2 Static Analys is : calculate base reaction force Initialize restoring force with base reaction force determined from uncracked static analys is, Rf(t) Step 3 R e d u c e [M],[C].[K] and force vector to a general ized system by [C>] Calculate F F T of control motion, Xg(t) and calculate dynamic stiffness matrix S(CD) and dynamic force vector, F ( Q ) Calculate f requency response £{a>)}=[S(m)]M{F(a>)+Rf((D)} for all CD Calculate IFFT of (Zfa)} and obtain response of general ized system, Calcu la te FFT(Rf(t)) to Rf(cD) Calcu la te V C O D at crack node location by expanding Z(t) by O Ca lcu la te new restoring force Rf(t) based on V C O D and restoring st i f fness Step 4 Expand response to full D O F and advance t ime step Step 5 Figure 3.9. Flowchart of the HFTD procedure as applied to the cracked dam system. 39 Section 3: HFTD as Applied to Cracked Dam System Step 3: Modal Decomposition and Dynamic Property Initialization: The eigenvectors from the linear uncracked and pseudo-linear cracked systems are combined to form Ritz vectors <P. The global matrices from the cracked dam system are reduced by the Ritz vectors. The global force vectors, including static gravitational force, base excitation, and restoring force are also reduced by the Ritz vectors. The dynamic stiflhess matrix is formed with the reduced dam properties, as well as, any contribution from hydrodynamic effects and foundation half-plane for each frequency. The inverse of the dynamic stiflhess matrix is calculated and stored for the iteration routine in step 4. The dynamic force vector is also formed by calculating the FFT of base excitation time history. Step 4: Iteration Routine and Crack Open/Close Criterion: The response of the reduced pseudolinear cracked system is calculated in the frequency domain by solving for the frequency response {Z(oo)}. The time response is determined by calculating the IFFT of {Z(co)} to obtain the generalized response {Z(t)} for the entire duration of the analysis, T 0. The vertical crack opening displacement (VCOD) is calculated by expanding {Z(t)} to the global DOF at the crack nodes only. The crack open/close criterion is applied at the next non-converged time step, t c+i, where t0 represents the last converged time step. If the criterion identifies the VCOD at tc+i as being neither crack open, nor crack close, then restoring force will be calculated and updated by adding to the previous restoring force. The FFT of the restoring force time history for the entire duration T 0 is calculated and re-introduced into the frequency response calculation. The process is repeated until the VCOD satisfies the criterion as either being crack open or crack close. 40 Section 3: HFTD as Applied to Cracked Dam System Step 5: Expand Converged Modal Response: If the criterion identifies the VCOD at tc+ias being either crack close or crack open, the response of the entire global DOF can be expanded from {Z(t)} and [O]. The time is increased and the process is repeated until the end of the analysis is reached. 3.6.1 Choice of Time Step Unlike time domain analysis, where time step integration techniques allows for sub-increments of time step such that the exact time of occurrence of the V C O D overlapping can be located, HFTD must be performed with a fixed constant time step. Therefore the choice of time step is important in accurately determining the time of applying the restoring force. The error can be measured by an energy balance check. Details of this are outlined in Section 3.7. A general guideline to determine the size of the time step is to use one that is half of the required time step needed to satisfy the frequency domain analysis procedure. To satisfy the frequency domain analysis, a time step is chosen such that the Nyquist or folding frequency is larger than the largest dominant frequency expected to be involved in the response. In practice, it is recommended that the Nyquist frequency be at least two times the maximum frequency of interest co highest- The effective frequency range is (0, co highest) and the results from (co highest, co Nyquist) are discarded since these may be affected by aliasing and leakage errors. By reducing the time step by a factor of two, the highest frequency content is 41 Section 3: HFTD as Applied to Cracked Dam System preserved and the error in applying the restoring force is minimized. This guideline is given where ©highest is the highest frequency in rad/sec expected to be in the system. 3.6.2 Time Segmentation Approach The convergence of the overlapping V C O D is found to progress in time with each successive iteration. To facilitate this process, it has been suggested that the total duration of analysis T„ is to be subdivided into n smaller time segments of T n such that the solution within each segment is calculated and converged before progressing to the next time segment (Dabre and Wolf, 1987). The segmentation approach is outlined as follows: Beginning with first segment: 1. The response for the pseudo-linear system is calculated in the frequency domain. 2. Crack state is determined and restoring force is calculated if required for all time steps within the current time segment 3. The response for the pseudo-linear system with an updated restoring force for the total analysis duration is re-analyzed in the frequency domain. Only the response within the current time segment and beyond is updated. Previously converged time segments remain unchanged. 4. Re-determine the crack state and restoring force if necessary until all time steps within the current segment has converged. If convergence is met then advance to next time segment and repeat process until end of analysis is reached. by: (3.10) 42 Section 3: HFTD as Applied to Cracked Dam System It is critical that in Step 4, that the previously converged response are not updated. Aliasing and leakage errors from the applied restoring force pair response can induce erroneous displacements at previously converged time steps. Details of these errors are given in Section 3.7. From experience, the subdivision of time duration of analysis into too many time segments may result in the inefficient use of computing time. Too few segments can lead to convergence instability. The optimal number of segments varies with the nature of the nonlinear problem and must be attained by trial and error. For the nonlinear cracked dam problem, 50 time segments were used in most of the analysis without concern of instability. 43 Section 3: HFTD as Applied to Cracked Dam System 3.7 Numerical Errors in the Analysis To ensure that accurate results are obtained in the HFTD analysis, an energy balance check is performed after the analysis. In general, for any dynamic system beginning at rest in its initial static position, the energy of the system and any externally applied energy at time (n+l)At should satisfy the following equation: W n + , « + W n + r p + T n + ] = W n + r ( 3 1 1 ) where W m t represents the elastic work done in deforming the system (i.e. strain energy), W d a m p represents the work dissipated through damping, and T represents the kinetic energy of the system. W 6 * is the work done by any externally applied load. With the crack dam system, W6*1 can be expressed as: W- =WMS +Wqcr (3.12) where W 3 * 8 is the induced inertia load applied through ground acceleration and W c r is the work done by the restoring force to prevent crack overlap. By evaluating each of these work terms through the trapezoidal rule and velocity through central difference method, we get: 44 Section 3: HFTD as Applied to Cracked Dam System { O p } = { O p } + i M l c 1 M 1 . 1 {«'},,,+[C*]{«'}I{II'}11J te)={^r}+iM[F-]w„+1M„+1+[;^]M>i] where: {u')n = — The above equations represent the total accumulated energy from t=0 to t=(n+l)At. The energy equation given in Eqn. 3.11 will not be fully satisfied. This imbalance is due to the errors in numerical integration, and the excessive tolerance level of convergence in the crack open/close criterion. The measure of the energy imbalance can be given by: (Wml + WdamV 4. 7* ^ _ TV1**1 E = — — — — * 100% (3 14) Energy imbalance, e, less than 5% indicates that the calculated response is stable and contains no significant errors (Cook, 1989). Any response with an energy imbalance exceeding 5% will indicate instability and inaccuracy in the calculations. It is recommended that the analysis be repeated with a smaller time step until either the energy imbalance is less than 5% or the difference in the response from the two analysis is negligible. Sources of these errors can be classified into two categories in a HFDT analysis. These sources of error are described below. 1) Similar to time domain analysis, the size of the time step can contribute to producing calculation errors, especially in contact type problems. The HFDT analysis is limited by an invariant time step and can only approximate the time of contact to a tolerance of ±At/2. If the time step is too large such that the time of contact is grossly misjudged, an 45 Section 3: HFTD as Applied to Cracked Dam System accumulation of numerical errors will cause amplitude decay and period elongation. To alleviate this problem, a smaller time step must be used. As well, the energy imbalance should be compared with that of the larger time step. 2) Aliasing effects resulting from discrete Fourier transforms can cause erroneous responses not induced by the externally applied loads. Aliasing error is a result of truncating the unit impulse response and/or forcing function when the Fourier transform of the forcing function and the response function are being convoluted. There are two consequences which are of concern. Firstly, the initial conditions, usually zero displacement and zero velocity, at the beginning of the analysis may be not be satisfied. This is due to the length of the excitation used in the analysis. Secondly, the applied restoring impulse force pair during the iteration procedure of the FJFTD analysis can produce erroneous displacement at times less than the current applied time step. Figure 3.10 illustrates the complications of aliasing errors interfering with the convergence of VCOD. At time t, the applied restoring force pair induces a displacement of magnitude, URXO). However the error e from the aliasing of the impulse response within the time duration T 0 will cause interference with Urf(0). This can lead to instability in the convergence if the aliasing errors affect the converging VCOD at the time of the applied restoring force. Both of these situations can be avoided by appending the end of the excitation with a series of zeros. This "quiet zone" can reduce the error by truncating the impulse response function at a 46 Section 3: HFTD as Applied to Cracked Dam System later time where the damping of the system would have decayed the response to an insignificant magnitude. An estimate of the length of quiet zone can be made by assuming that the impulse function of the restoring force only excite the first mode of the cracked system. The decay of the impulse response can be estimated by the SDOF response with the first cracked modal properties. Consider the usual linear transient SDOF impulse response: x(t) = e-**A(sincot + 0) (3.15) 4 = percentage of critical damping p = undamped natural frequency of system pd - damped natural frequency of system 6 = phase angle The decay of the response is governed by the exponential function term . For the amplitude of the response to decay to say 1% of the peak amplitude, the expression of the time is given by: e-** =0.01 ln(Q.Ql) (3.16) Therefore, for a pseudo-linear system of 5% viscous damping and with a natural frequency of 2hz, the total duration of the analysis must be greater than 7.329 sec in order to avoid aliasing which may taint the response caused by the contact force. Unfortunately, this method of adding a "quiet zone" increases the length of the analysis by increasing the number of frequencies in the frequency transform since AGO = 1/T0. To eliminate the 47 Section 3: HFTD as Applied to Cracked Dam System need for increasing the length of the quiet zone, corrective response methods have been successfully applied in linear frequency domain analysis (Veletsos and Ventura, 1984). These methods add corrective responses or append forces at the end of the response to correct the errors found in the initial conditions of the response due to aliasing. Thus the length of the analysis need only to be as long as the time span of interest. This significantly reduces the computational intensity especially when calculating the dynamic flexibility matrix [S^)]" 1 since the number of frequency increments are kept to a minimum. One drawback of the corrective response method is the requirement to know the extent of the errors at the initial condition prior to applying the appropriate corrective responses. Hence for each increment of loading, two analysis must be performed; one subjected to aliasing errors, and one with the correction responses. It is speculated that in the HFTD iterative framework, the use of corrective responses would increase the number of iteration required since each iteration would introduce a new set of dynamic forces. However, the speed and computational efficiencies of the corrective response approach as stated above may offset this disadvantage. At this time, a comparative study between both approaches has not been conducted and the computational efficiency of each method has not been assessed. For simplicity and research purposes, the HFTD implemented in this study used the "quiet zone" approach to reduce aliasing errors and the corrective response methods was not considered. Perhaps, in a future research, the corrective response techniques may be incorporated in the HFTD approach and a comprehensive comparative study of the efficiencies of both methods be conducted. 48 Section 3: HFTD as Applied to Cracked Dam System 3.8 Computer Program: CRK-DAM To analyze the cracked dam system, a computer program with the algorithm outlined above has been developed using the HFTD procedure. Written in Microsoft Visual Basic 3.0 (Microsoft Corp. 1992) the application called C R K - D A M follows the linear analysis, flexible foundation, and hydrodynamic algorithms developed for EAGD-84 (Fenves and Chopra, 1984). Modifications were made to EAGD-84 to include the HFTD procedure. C R K - D A M includes a preprocessor for constructing a FE dam model, the analytical engine based on the HFTD procedure and Fast Fourier Transform methods, and a post processor to display the results. 3.9 Chapter Summary In applying HFTD to the cracked dam system, four key issues must addressed: • Discrete crack dam model: The dam model with the discrete crack along the dam foundation interface will be identified as the pseudo-linear system in the HFTD analysis. The pseudo-49 Section 3: HFTD as Applied to Cracked Dam System linear system must satisfy all necessary kinematic displacement requirements of the crack opening and closing motions expected in the cracked dam system. • Modal Decomposition: The problem is broken down into only several generalized co-ordinates. These are identified by a combination of eigenvectors which are obtained from the pseudo-linear cracked dam model and the linear uncracked dam model. This combination of eigenvectors from the two distinct linear models results in a nonorthogonal set of Ritz vectors. • Restoring Force Application: The nonlinear restoring force is applied to the crack surface to prevent any crack overlapping in the pseudo-linear system. The restoring force is determined from an assumed crack spring stiffness and the amount of overlapping VCOD at each crack interface node pair. The force is applied in pairs, one at the current time step and the other at a previous time step. This will allow the necessary displacement to be evoked at the current time and it will reduce the error in locating the exact time of crack contact to within ± At/2. • Crack Open/Close Criterion: The crack is considered to be closed if the VCOD (vertical crack open displacement) is within the tolerance level and the restoring is not zero. The crack is considered to be open if the VCOD is positive and there is no restoring force applied to the crack surface. Any errors in the analysis are identified by an energy balance check performed at the end of the analysis. An energy imbalance of greater than 5% indicates a possible inaccuracy in the result; thus, it is recommended that the problem be re-analysed with a smaller time step. 50 Section 4: System Frequency Identification of Cracked Dam System 4. Effective System Frequency Identification 4.1. Introduction The concept of the natural period of vibration is restricted to linear elastic dynamic systems only. For nonlinear systems, vibrational characteristics such as natural period of vibration or eigenvectors and eigenvalues do not exist. However, when any system is excited by an external source, certain frequencies in the response will dominate. These stronger frequencies will contain contributions from the prevalent frequency of the excitation, as well as, the dominant frequencies inherent in the system. The former frequency represents the effective system frequencies (ESF) of a dynamic system. For the cracked dam system, it will be demonstrated that the ESF is dependent not only on the length of crack, but also the loading condition to which the nonlinear system is subjected. The effective system frequency of a cracked nonlinear system may be identified by numerous methods. The simplest method is to monitor the response of the system when subjected to sinusoidal excitation of various frequencies until the resonance of the response is identified. This method is called "sine sweeping". Another method is to subject the system to either a broad band or white noise excitation, and measure the spectral strength of the response frequencies. These two general methods will be used in this study. 51 Section 4: System Frequency Identification of Cracked Dam System 4.2 System Identification Through Response of White Noise Excitation. To study the effective system frequency of a cracked dam system, a 61.0 m. (200 ft.) dam section set on rigid foundation and no reservoir will be subjected to a white noise excitation. Crack lengths ranging from 0% to 50% of the dam base will be considered. As with modal frequencies, there exists an infinite number of ESF's; however, only the first two ESF will be determined. The following will describe the components and details of the procedure. 4.2.1 White Noise Excitation As mentioned previously, the frequency content of the dynamic response will contain spectral peaks corresponding to either the resonant frequency of the system, or to the dominant frequencies of the excitation. In order to eliminate the spectral peaks contributed by the excitation, a pure white noise excitation can be applied. An ideal white noise excitation contains frequencies of equal magnitude spanning over a large frequency range. Such an excitation can be formulated by summing a finite series of sinusoidal excitations of equal magnitude and varying frequencies, and with a randomly generated phase angle. The formulation is given below. con = nAco ; A<y = ^n/j xg(t) = Mtsm(cont + en) where: = random phase angle with seed n ( 4 «=i M = magnitude of each sinusoidal excitation T = total duration of excitation To measure the strength of the frequency, the autospectrum or the power spectral density (PSD) of the time history will be calculated as follows: _ X(CJ) = component of FFT of response quantity PSD(o)) = X((o)X(co) where: - . (4. X(co) = complex conjugate of X(co) 52 Section 4: System Frequency Identification of Cracked Dam System Figure 4.1 shows a white noise excitation generated for a duration of 10.24 sec sampled at At=0.01 sec. The PSD of the excitation confirms the lack of any dominant frequencies, and is represented as a flat line across the frequency spectrum. A sudden dip in the PSD at 50Hz marks the folding or the Nyquist frequency. This is the highest frequency content that the excitation sampled at At=0.01 sec can contain. 80 -1 60 40 p* in e 20 c 0 0 *J CO k. -20 a o < -40 -60 -80 -100 Figure 4. 1. White noise free-field excitation 8E+6 000E+O Figure 4. 2. Power spectral density (PSD) of white noise excitation. Time sampled at 0.01 sec. 53 Section 4: System Frequency Identification of Cracked Dam System 4.2.2 Dam Model A 61.0 m. (200 ft.) high 2-D gravity dam section set on a rigid foundation and no reservoir will be used in the analysis. The F E M idealization of the section consists of 135 nodes and 104-4 noded 2 DOF quadrilateral parametric element. Crack lengths of 0%, 12.5%, 25%, 37.5%, and 50%, of the total dam base will be investigated. Dam concrete properties and the configuration of the dam structure are provided in Figure 4.3. 9.76 = 3 @ 3.25 Elastic Modulus Ec 27600 MPa Poission ratio 1) 0.2 Density P 2400 kg/m3 Viscous damping 0.05 for freq range 1Hz ~ 7Hz Note: all dimensions are in m Node 110 Figure 4. 3. 61.0 m. (200 ft.) cracked dam finite element model and material properties. Three key points along the upstream face of the dam will be monitored during the response. Node 1 will provide the dam top displacement where the response will be most significant. Node 64 will provide the response at the mid-height of the dam. Vertical motions at Node 118 will track the crack opening displacement at the crack mouth. 54 Section 4: System Frequency Identification of Cracked Dam System 4.2.3 Analysis Parameters A total response duration of 10.24 sec will be used in the analysis. A time step of 0.01 sec will satisfy the frequency domain analysis requirement by giving a Nyquist frequency of 50 Hz. Since the goal is to obtain the frequency content of the response, preserving the initial conditions of the displacement response is not of concern and the white noise excitation is not padded with zeros. Aliasing of the restoring impulse response is not expected to be of concern since the lowest frequency of 3.13 Hz for the 50% crack pseudo4inear dam system requires only 4.7 sec to decay to 1% of the expected amplitude of the response excited by the restoring impulse (see Eqn. 3.16). The accuracy of the spectral analysis is dependent on the frequency resolution of the frequency spectrum analysed by the PSD. This is further dependent on the length of time of the analysis. The following relationship which governs this is given by: where ACQ is the frequency resolution in hertz, T 0 is the total duration of the analysis, and n is the total number of time step At. This yields a frequency resolution of Aco = 0.0976Hz for the specified 10.24 sec of analysis. A more refined Aco can be attained by increasing the time duration of analysis; however, since the main goal is to determine how the ESF of the cracked dam system will degrade from the uncracked case, this frequency resolution is sufficient. In applying the modal transformation used in the HFTD analysis, 4 modes from both cracked and uncracked phases of the dam section will be used to construct a total of 8 modal vectors. The natural frequencies of each modal vector are listed in Table 4.1. (4.4) 55 Section 4: System Frequency Identification of Cracked Dam System Crack Length Mode 0% 12.5% 25% 37.5% 50% 1 6.37 6.05 5.24 4.20 3.13 2 14.4 13.75 12.78 11.84 10.92 3 18.67 18.47 17.75 16.60 15.19 4 25.67 24.92 24.24 23.20 21.37 Table 4. 1. Modal frequency [Hz] of pseudo-linear systems of 61.0 m. (200 ft.) dam. The maximum acceleration of the white noise excitation will be scaled to 4g's. This large acceleration is required to ensure the crack will open and close with each full oscillation. A magnitude too small may not impart enough inertial energy to overcome the gravitational force to cause the crack to open. Therefore, to eliminate the possibility of evoking only one crack state, a large acceleration is required. The implications of the magnitude and nature of the excitation will be discussed in Section (4.2.5). 4.2.4 Results of Analysis The response and the its associated PSD of the 0%, 25%, and 50% cracked dam section are given in Figures 4.4 through 4.7. The first ESF (effective system frequency) is deteirnined from the PSD of dam top horizontal displacement at Node 1, since the majority of the first effective system mode is concentrated at the top of the structure (Figures 4.4,4.5). This is indicative of the single curvature displacement shape inherit in the first mode of linear systems. The second ESF is determined from the horizontal displacement at mid-height of the dam (Node 127) so that the second mode double curvature can be captured (Figures 4.6,4.7). Deterioration of the second ESF signal strength is evident in Figure 4.7 especially for 50% crack case where the strength of the frequencies is more erratic and less smooth than it was for the uncracked case. Also note the 56 Section 4: System Frequency Identification of Cracked Dam System decrease in two orders of magnitude of PSD for the first mode versus the second mode indicates a much stronger presence of the fundamental modal response (Figures 4.4 to 4.7). Higher orders of ESF could not be obtained because either the signal strength of the higher frequency was not strong enough or the range of the spectral peaks was too broad to ascertain a single dominant frequency. 0.10 ui 5 -0.10 2.5 3.5 time [sec] 4.5 Figure 4.4. Dam top displacement of 61.0 m. (200 ft.) dam system with 0%, 25%, and 50% crack lengths subjected to white noise excitation 4E-05 3E-05 8 2E-05 O. 1E-05 4 0E+00 50% - 5.07Hz • I 25% - 5.76Hz .1 5 6 7 F r e q u e n c y [Hz] 10 Figure 4.5. PSD of first ESF of dam top response of 61.0 m. (200 ft.) dam with 0%, 25%, and 50% crack lengths subjected to white noise excitation 57 Section 4: System Frequency Identification of Cracked Dam System Figure 4.6. Mid height displacement of 61.0 m. (200 ft.) dam system with 0%, 25%, and 50% crack lengths subjected to white noise excitation 8E-08 6E-08 + w 4E-08 Q. 2E-08 + OE + 00 6 50% - 12.99Hz • - -• * • v * ' • f A • . ' • * [ 14 15 16 F r e q u e n c y [Hz] Figure 4.7. PSD of second ESF of mid height response of 61.0 m. (200 ft.) dam with 0%, 25%, and 50% crack lengths subjected to white noise excitation Table 4.2 shows the effective system frequency of the cracked system as determined from the spectral peaks of the displacement PSD's. Listed are the first and second cracked and uncracked 58 Section 4: System Frequency Identification of Cracked Dam System modal frequencies of the 61.0 m. dam, as well as, the ESF as determined by the spectral analysis. As expected, for the uncracked linear system, the spectral analysis yields an ESF close to the uncracked natural frequency. Any discrepancies are due to the frequency resolution and the damping since the resonance frequency identifies the damped natural frequency. However, for 5% damping this difference is negligible. Crack Length Frequency [Hz]: Mode 1 Frequency [Hz]: Mode 2 Uncracked Cracked ESF Uncracked Cracked ESF 0% 6.37 6.37 6.35 14.4 14.40 14.45 12.5% 6.37 6.05 6.15 14.4 13.75 14.16 25% 6.37 5.24 5.76 14.4 12.78 13.67 37.5% 6.37 4.20 5.47 14.4 11.84 13.28 50% 6.37 3.13 5.08 14.4 10.92 12.99 Table 4. 2. First and second ESF for 61.0 m. (200 ft.) cracked dam system Figure (4.8) shows the modal frequencies and the calculated ESF plotted against the crack length. Several important observations concerning the distribution of the ESF in relation to the uncracked and cracked pseudo-linear natural frequencies are worth noting. Firstly, the decrease of both the first and second ESF with increasing crack length are nominal for crack lengths up to 25%. For a 12.5% crack length, a decrease of 3.5% for first ESF with respect to the first uncracked frequency and 1.7% for second modal frequency were observed. Even for 25% crack length, only a decrease of 12% and 15% for the first and second ESF with respect to the uncracked natural frequencies are observed. The ESF of the cracked system for crack lengths exceeding 25% showed appreciable degradation from the uncracked natural frequency. Depreciation of 14.1% and 20.3% for 37.5% and 50% crack lengths are evident for the dam system analysed. 59 Section 4: System Frequency Identification of Cracked Dam System 7 9 Uncracked 1 st Mode IstESF "" ""--0 N X O c CD _J CT CD CO "D O Cracked 1 st Mode 3 -I— 0% 10% 20% 30% 40% 50% 15 14 13 12 11 + Uncracked 2nd Mode 2nd ESF Cracked 2nd Mode 10 0% 10% 20% 30% 40% 50% Crack Length [% of Base Length] Figure 4.8. Variation of first and second ESF with various crack lengths for 61.0 m. (200 ft.) cracked dam Secondly, as illustrated in Figure 4.8, the ESF is bounded by the cracked and uncracked natural frequencies of the system of respective order. This is to be expected, since the effective system can not be more stiff than the uncracked system nor can it be more flexible than the cracked system. Therefore, the systems frequencies must be bounded by the natural frequencies of the linear cracked and uncracked system. Note that this only applies to the fundamental ESF and can often be extended to the second ESF. Higher orders of ESF would not be applicable because of 60 Section 4: System Frequency Identification of Cracked Dam System the complex combinations of higher modes and possible modal coupling of higher frequencies will reduce the contributions of the modal coordinates in the vicinity of the crack. Thirdly, for the dam configuration used here, the fundamental modal frequency of the pseudo-linear system decays quite rapidly with increasing crack length. This is of significance since the range bounded by the uncracked and cracked natural frequencies directly govern the ESF of the system. For a 25% crack length, the cracked modal frequency of 5.24 Hz represents a decay of 17.7% whereas for a 50% crack length, the cracked modal frequency of 3.13 Hz is over 50% of the uncracked fundamental frequency of 6.37 Hz. 4.2.5 Limitations of ESF Identification The above results of the ESF are based on the response of a broad band excitation of 4g magnitude. As previously mentioned, the ESF is not a definitive property of a nonlinear system, rather it is dependent on the magnitude and frequency content of the applied excitation. For example, if a cracked dam system is excited by the same white noise excitation but with a much smaller magnitude, then the inertial forces may be insufficient to cause a crack opening and the dam will behave mainly as an uncracked dam system. The ESF determined through the frequency content of the response will be equal to the uncracked natural frequency. Conversely, if an excitation of large magnitude is applied, the resulting response would be ensured to contain contributions from the cracked phase of cracked dam system. It is expected that responses from a cracked system subjected to large magnitude excitations and responses from an uncracked system will form the upper and lower bounds of the possible range of responses. Hence the responses 61 Section 4: System Frequency Identification of Cracked Dam System from a moderate magnitude excitation are expected to lie within these boundaries; therefore, they can be interpolated. The nature of the excitation can also influence the ESF. For example, if the white noise excitation contained a constant acceleration component causing either the crack to remain open or closed, the resulting ESF can be biased towards either the cracked or uncracked natural frequency respectively. Figure 4.9 shows the response of the same 50% crack 61.0 m. dam excited by the same white noise excitation as used previously, superimposed with a constant acceleration of - lg . This constant acceleration effectively acts as a constant force causing the crack to open. This is similar to hydrostatic loading conditions applied on the upstream face of the dam causing tension at the heel of the dam and thus inducing the crack to open. The ESF from the spectral analysis of the dam top response results in a value of 3.61 Hz, which is much closer to the cracked natural frequency of 3.13 Hz. The response behaviour corresponds with this change. From the dam top displacement time history, the period of oscillation is much longer than that exhibited in Figure 4.4 without the constant force of- lg . The VCOD time history response also confirms this with the crack remaining open much of the time. 62 Section 4: System Frequency Identification of Cracked Dam System 0.08 |[ _ 0.04 c _ E JS o.oo _ _ Q CD -0.04 _ _ / ! A s A i\ A H A .A / \ \l P | v» V | 3.5 time [sec] 4.0E-05 3.5E-05 3.OE-05 2.5E-05 Q (fi 2.0E-05 CL 1.5E-05 1.OE-05 5.0E-08 1.0E-12 |50%-3.61 Hz Frequency [Hz] 30 £ 25 sz "3 o 20 O 15 2 o "co 10 Q O O 5 > A I M i\ 11 II n • i 'I S / ' 1 i \ ,0 i 1 i i ; 1 i A. f i i ! 1 1 \ i ; A IS i! i i /' i i ;! ! 1 ! i i i ! / 1 1 1 1 ! i ! ' 1 ! ! \ 1 \ \ i i i i ! i i _U—  1 U 1 3.0 " 4.0 time [sec] Figure 4. 9. Dam top displacement and PSD of 61.0 m. (200 ft.) dam with 50% crack subjected to white noise plus -lg. VCOD for crack mouth This finding shows that the effective system frequency is very dependent on the specified loading condition. In terms of the dam only a method of identifying properties of the crack dam system would be to specify the frequency range bounded by the first uncracked and cracked natural frequencies. It is within this range or frequency envelope that the ESF of the specified loading condition is represented. As previously shown in Figure 4.8, the ESF envelope of crack lengths less than 25% are quite narrow and regardless of the loading conditions, the ESF will not vary greatly. However, for crack lengths greater than 25%, the ESF envelope broadens. This makes it difficult to generalize the expected ESF and the behaviour of the cracked dam system without knowledge of the loading conditions. 63 Section 4: System Frequency Identification of Cracked Dam System 4.3 Effective System Frequency Identification through Sine Sweep Another method of obtaining the effective system frequency is to identify the frequency at which the steady state response of the system reaches its maximum amplitude or its resonance state. This can be achieved by subjecting the system to a series of sinusoidal excitation of varying frequencies. The frequency of the excitation at which resonance is achieved will be identified as the ESF of the system. This is a tedious process, since each sinusoidal excitation of a particular frequency represent one analysis; however, a better understanding of the response of the cracked nonlinear system can be achieved through this exercise. Only the first ESF will be determined using the sine sweep technique. The same 61.0 m. (200 ft.) dam model used in the Section 4.2 will be used here. The maximum relative horizontal displacement of the dam top in the steady state portion of the response will be noted and used to identify when resonance is reached. Unfortunately, during the development of the computer application C R K - D A M , an error in the parametric formulation of the elemental stiffness matrix was not identified until the sine sweep analysis was completed. This oversight caused deficient stiffness terms in the non-rectangular elements leading to a reduced global structural stiffness. Therefore, the resulting motion and frequency of oscillation do not correspond with the 61.0 m. (200 ft.) dam model used in the white noise technique. This error has been corrected; however, because of the tremendous expense and the time constraint involved in performing the sine sweep analysis, this study was not re-analysed. However, the sine sweep study would provide valuable information regarding the cracked system response and the ESF. 64 Section 4: System Frequency Identification of Cracked Dam System 4.3.1 Description of Dam Model The geometry of the 61.0 m. (200 ft.) dam model and material properties are identical to the model used in the previous Section 4.2. Because of the previously mentioned oversight in the computer formulation, the structural stiffness, natural frequency and modeshape of the uncracked and cracked phases will indicate some discrepancies. Only crack lengths of 0%, 25%, and 50% were investigated. Again, the first four uncracked and cracked modeshapes were used to form the displacement modal vector. See Table 4.3. Note that these frequences are differenet from those given using the correct stiffness matrix shown in Table 4.1. Crack Length Mode 0% 25% 50% 1 4.44 3.88 2.69 2 13.63 12.51 11.07 3 20.3 19.22 16.73 4 27.78 27.2 24.43 Table 4.3. Natural frequencies of Ritz vectors for 61.0 m. (200 ft.) cracked dam system [Hz] 4.3.2 Excitation and Analysis Parameters Sinusoidal excitation of frequencies ranging from 2 Hz to 10 Hz at varying increments was applied to the analysis of the dam. This frequency range encompasses the expected first ESF range as indicated by the first uncracked and cracked modal frequencies. A maximum duration of 10.24 sec at 0.01 sec increment was used in the analysis to distinguish the transient response from the steady state response. The sinusoidal excitation was scaled to 4g to ensure adequate crack opening. Quiet zones was added to the excitations of frequencies less than 5 Hz to avoid aliasing errors caused by the restoring impulse forces. 65 Section 4: System Frequency Identification of Cracked Dam System 4.3.3 Case Study: 61.0m (200 ft.) 25% and 50% Cracked Dam Subjected to 3 Hz Excitation Figure 4.10 shows the response of the 61.0 m. (200 ft.) dam with 25%, and 50% crack lengths subjected to 3 Hz sinusoidal acceleration scaled to 4g. Note that the response is identified by three distinct regions: (1) transient; (2) steady state; and (3) free vibration response. During the steady state response, it is clear that the maximum displacement is achieved during the crack open phase of response. Also note that the duration of which the crack opens is longer than when the crack is closed. The proportion of time during one full oscillation in which the crack is open as compared to when it is closed, reflects the relative values of the cracked and uncracked natural frequencies of the system. For the 25% crack, the distribution of time of each phase is almost the same, since the first uncracked natural frequency of 4.44 Hz and cracked natural frequency of 3.88 Hz only differs by 13%. Conversely, for the 50% crack, the cracked frequency of 2.69 Hz differs by nearly 50% of the uncracked frequency. 1.5 ~ 1.0 E (1) E 0.5 ai o ra a. in 5 0.0 <u > '*-t m a -0.5 -1.0 J f f !:. '• 25% 5 0% • " • '•'] •' ;• ; i: • •. !• « ••; ; i ' . •! • • : • ' i : ! • •' ' • i ' * •' » / * *• ' >• i, i. " II ", ' • i ; ?. . :: <';;; A ,* ii ;i ;• /Ml l\'t f' ll ('• t V 1 ! ! • ! 1 * y ' \ • JA" 'AI *f\ § * . • * \f{1\'f\ » • 1 • .' • • 1 • 'J\i •• : I \\ ii::" : : .':- ' ' • i . • • i i i !: : • 1 AAA/ \ A A A M. *. V •• i '.' ;. ; i . * • • •" * •' :• !.: r- ft I i/ 1 ?• :'. '<'< • * . * • i' .: ,. if !• •; * •.: ".* * • i • • • • # . • i i! >: ! ;. •• •' !•' » ' ••" \l V V V VI i h 4 !i ii ii 5 i ' V H 1/ • ' '< ? i >• .i i ? ; *  I i 4 5 6 Time [sec] 10 Figure 4.10. Dam top relative displacement response of 61.0 m. (200 ft.) dam section with 25% and 50% crack lengths subjected to 3 Hz sinusoidal excitation 66 Section 4: System Frequency Identification of Cracked Dam System 4.3.4 Spectral Response The maximum displacement and acceleration amplitudes within the steady state response of each crack case subjected to various excitation frequencies are plotted in the Figure 4.11 with respect to the sinusoidal excitation frequency. Note that the excitation frequency used in the analysis is marked by a data point in each curve. From the plot, it is clear which frequency causes resonance in the cracked dam system. 10 c 1 d> E o to a in Q a o 0.1 E ra a 0.01 10000 o o in E C | 1000 «> 0) u u < > _ro & co c o N o X 10 100 25% -4.2 Hz 50% -3.61Hz 0% - 4.44Hz —i —t A , \ / \ \ / / / n \ / \ . c - J y \ 3 3% * — [ —4 25% 1(1% 4 6 Frequency [Hz] 4 6 Frequency [Hz] 10 —x—o% — a— 25% —ti 50% 10 Figure 4.11. Spectral response of 61.0 m. (200 ft.) dam on rigid foundation and no reservoir subjected to sinusoidal excitation; a) dam top horizontal relative displacement and b) dam top horizontal relative acceleration (b). 67 Section 4: System Frequency Identification of Cracked Dam System As expected, for the uncracked case, the resonance peak identifies the fundamental damped frequency of the 61.0 m. (200 ft.) dam system at 4.4 Hz. For the 25% and 50% cracked dam systems, the resonance frequencies are found, respectively, at 4.2 Hz and 3.6 Hz. As verified by the white noise study, the effective system frequency of vibration is identified within the range bounded by the fundamental uncracked and cracked modes. (See Table 4.4) It is also clear that the ESF of the cracked dam system shifts as the crack length increases. Very little shifting is evident in the 25% crack length; this is reflected by the narrow ESF range. Crack lengths of 50% produce a much larger ESF shift. Crack Length Frequency [Hz]: Mode 1 Uncracked Cracked ESF 0% 4.44 4.44 4.44 25% 4.44 3.88 4.2 50% 4.44 2.69 3.61 Table 4.4. Result of ESF analysis via sine sweep In addition, the peak steady state response for the uncracked system and cracked system are very similar for high frequency excitation beyond 8 Hz. This is because, at the higher frequencies, the higher modes of vibration are excited. These higher modes contain values of modal ordinates along the crack interface which are smaller than the ordinates in the lower modes which exhibit larger cracking opening motions. Thus, the participation of the higher modes is less affected by cracking, thus resulting in a response similar to that for the uncracked case. 4.3.5 Secondary Resonance Phenomena From Figure 4.11, it is noted that both 0% and 25% crack length dam systems produce a single resonance peak which identifies the first ESF. The 50% crack length 68 Section 4: System Frequency Identification of Cracked Dam System however, produces two resonance peaks; the first peak at 3.6 Hz which.corresponds with the first ESF, and the second peak at 7 Hz. The second peak is unexpected since it does not correspond with the second ESF of the system which is expected to lie between the range of 11.07 Hz and 13.63 Hz. The response behavior at the second resonance frequency is also peculiar. Figure 4.12 compares the response of the 50% crack length at frequencies near these two resonance frequencies. A steady state response generally oscillates at a frequency equal to the excitation frequency; this is demonstrated by the dam top displacement response excited at 3.0 Hz. However, when the 50% crack length system is excited at 7 Hz, a repeating oscillation occurs at only half of the exciting frequency; that is, the oscillatory pattern of displacement repeats itself at a frequency of approximately 3.5 Hz, rather than the expected 7 Hz. The observed secondary resonance phenomena occurs not only in a complex cracked dam system, but in a simple SDOF system with bi-linear stiffness. This simple model can mimic the stiffness variation of the cracked dam system as the crack opens and closes when the properties of the SDOF system are chosen to reflect the same characteristics of the first cracked and uncracked natural frequencies. Details and further investigation of this simplified model are given in Chapter 5. 69 Section 4: System Frequency Identification of Cracked Dam System Figure 4.12. Comparison of resonance behavior at 3 Hz and 7 Hz. The SDOF bi-linear system used to represent the 61.0 m. (200 ft.) dam systems with 50% crack length is shown in Figure 4.13. crack close (-) <* j *• (+) crack open m=l v _ / \ X V _ s \ / \ / \ I co,=4.44(2jt) £ = u ) / \ ' £=0), 0), =2.69(27C) \ i Figure 4.13. SDOF representation of 61.0 m. (200 ft.) 50% cracked dam system. 70 Section 4: System Frequency Identification of Cracked Dam System The first ESF of the nonlinear SDOF model occurs at 3.4 Hz and the second resonance at 6.5 Hz. The following study will focus on excitation frequencies near 6.5 Hz. Figure 4.14 shows the displacement time history of the SDOF bi-linear system subjected to a series of sinusoidal excitations ranging from 5.9 Hz to 7.9 Hz. All the displacement responses are plotted on the same scale for comparison. Beginning at an excitation of 5.9 Hz, the steady state response yields a maximum of 0.016 m during the crack open phase. The response oscillations are typical for any steady state response with the frequency of oscillations equal to the applied frequency of excitation. As the excitation frequency increases, two distinct patterns in the changes to the oscillations emerge. Firstly, the maximum displacements in both the crack open phase and crack close phase increase at alternating periods. Secondly, as a result of the alternating oscillations, the frequency of the repeating response is reduced to half of the excitation frequency. This increase in the maximum response and the alternating patterns continues until the second resonance frequency is reached at 6.5 Hz. A maximum displacement of 0.034 m is reached at this point. Increasing the excitation frequency beyond 6.5 Hz yields a decrease in response displacement and in the period of the repeating oscillations. This decrease continues until it reaches 7.9 Hz where the typical steady state response behavior resumes yielding a maximum steady state displacement of 0.009 m. 71 Section 4: System Frequency Identification of Cracked Dam System Section 4: System Frequency Identification of Cracked Dam System So far, there has been no explanation for the occurrence of this secondary resonance; however, as demonstrated by the SDOF bi-linear stiffness model, this phenomenon is not restricted to complex systems. It is known, however, that the secondary resonance only occurs when large nonlinearities are present. For the case of the 61.0 m. (200 ft.) cracked dam system, only crack lengths greater than 25% exhibited this behavior. Similarly, for the SDOF bi-linear approximation, only stiffness ratios between uncracked stiffness and cracked stiffness of 4 or greater experience this second resonance. Also, this resonance only affects the steady state displacement response and not the relative acceleration as seen in Figure 4.11 for the dam system analyzed. Therefore, the accelerations and forces in the structure are unaffected by this phenomena. In addition, the second resonance has only adverse effects on the steady state portion of the response. The maximum displacement in the transient portion of the response appears to be unaffected. This is evident in the gradual decrease in the transient portion of the displacement response of the SDOF approximation as the excitation frequency increases (Figure 4.14). This suggests that if the nonlinear systems are subjected to random excitation, such as an earthquake, where the steady state is difficult to achieve, the second resonance phenomena would have a minimal, if any, effect on the response of the system. This will be further discussed in Chapter 6 where the response of the cracked system when it is subjected to random earthquake type of excitation will be investigated. 73 Section 4: System Frequency Identification of Cracked Dam System 4.4 Chapter Summary The effective systems frequency of a 61.0 m. (200 ft.) cracked dam system was obtained by two methods: (1) spectral analysis of the response to a white noise excitation and (2) induction of resonance through a sine sweep technique. Both procedures produced an ESF for crack lengths ranging from 0% to 50% crack lengths. However, more importantly, the response behaviour of a cracked dam system was better understood. Several key observations are outlined below: • For nonlinear cracked dam systems, a single effective system frequency, does not exist, since the dominant frequency of vibration is dependent on the nature of the excitation, as well as, on the magnitude. A range of first mode system frequencies can be used to identify the effective system frequency of vibration expected. This range is bounded by the first uncracked and first cracked modal frequencies and will be referred to as the ESF envelope. • Dam systems with crack lengths greater than 25% produce responses which are dramatically different from the uncracked condition. This is reflected in the ESF and the steady state response. For systems with crack lengths of 25% or less, the ESF and amplitude do not differ significantly from the uncracked condition resulting in minor differences in the response. • The maximum displacement attained in the steady state response occurs in the crack open phase of the response. For high frequency excitation, the difference between the crack open phase and crack close phase of response diminishes. 74 Section 4: System Frequency Identification of Cracked Dam System • For 50% crack length, an second resonance frequency was exhibited during the sine sweep study. This second resonance frequency does not correspond with any expected ESF of the system, and it amplifies the displacement steady state response but not acceleration response. Higher order responses and responses in the transient response region do not seem to be affected. The cause of the secondary resonance phenomenon is unknown; however, it does occur in simple nonlinear SDOF systems with abrupt changes in stiffness. 75 Section 5: SDOF Bi-Linear Approximation 5. SDOF Bi-Linear Stiffness Approximation Section 4.3.1 introduced the SDOF bi-linear stiffness approximation of the cracked dam system. In this chapter, the mathematical development of the approximate method is described, and its accuracy is tested by comparing the response obtained from the rigorous HFTD analysis procedure. 5.1 Approximation with SDOF Bi-Linear Stiffness System The crack response behaviour can essentially be characterised by the opening and closing of the crack. It can be assumed that the onset of crack opening and closing essentially occurs simultaneously for all points along the crack surface. This occurrence indicates that the transition from the cracked phase to the uncracked phase of the response can be considered instantaneous, and that upon crack closing an sudden increase in the global stiflhess of the pseudo-linear system results. It will then be reasonable to consider the cracked dam system as a mass conservative system with a variable stiflhess which represents the cracked and the uncracked phases of the response. In addition, since the majority of the global response of the cracked dam system is contributed by the first cracked and uncracked modes, the simplified system can be approximated by a SDOF bi-linear stiflhess system. Consider the original equation of motion for the HFDT analysis procedure with full DOF of the system. [M]{x"} + [C]{x'} + [K]{x} = -[M]{/K + [Rf } (5.1) 76 Section 5: SDOF Bi-Linear Approximation where M , C, and K are full structural mass, damping and stiffness matrices, respectively, of the pseudo-linear system. Rf is the restoring force used in the HFTD analysis procedure to account for the crack opening and closing nonlinearity. Consider the cracked structure as composed of 2 non-orthogonal modes namely the first cracked {$crocked) and first uncracked {$ unlocked) modes of the pseudo-linear system. The response quantity {x} would be substituted by the generalized coordinate {u) in accordance to Eqns. 5.2 and 5.3. {x} = [<${«} = [$ VU2 (5.2) (5.3) where: [<J>] = [[^ 1 uncracked J 1 c^racked Jj The two components of {u}, U\ and u2 would then represent the modal response of the first uncracked and first cracked modes respectively. Combining the substitution of Eqn. 5.2 and the equation of motion given in Eqn. 5.1 leads to: [M]{M"} + [C]{M'} + [Z]{M} = {L)x"g +{Rf] (5.4) Partitioning the matrices into the two generalized coordinates, u\ and u2, leads to: (5.5) where the M,C K, matrices are the reduced mass, stiffness, and damping matrices of the system. Note that if the modal vector matrix [<I>] has been normalized with respect to the mass matrix so unit values form along the diagonal of the reduced mass matrix. The square of the natural frequencies of the two modes will also form the diagonal of the reduced stiffness matrix. The off diagonal terms of the reduced matrix are not zero since the Ritz vectors are non-orthogonal. However, for the time being, their contributions will be ignored. The L vector is the modal 1 i m2\ 'i/j u\ co2\ i k2X \Rf\ j . -- j -- ' + i -- -- r m2 ! l U2. _ C 1 2 1 C 2 2 _ u'2 _k\2 1 ^ 2 _ U2. [L2\ 77 Section 5: SDOF Bi-Linear Approximation participation factor for the ground acceleration. For the cracked system, L\ and L2 are similar in value and do not vary more than 5%; this will be later shown in Chapter 6. Therefore, for practical purposes, L; can be assumed to be equal to L2, indicating a single common excitation force for both degrees of freedom. The force vector Rf on the right hand side of the equation is the restoring force of the cracked system. The value of Rn is essentially zero, since it applies to the first uncracked mode where the nodes along the crack interface do not overlap the foundation; thus, they are not affected by the restoring force. If the crack is closed, then Rn is fully active and will eliminate the crack open displacement shape from the response. If the crack is open, then Rn is nearly zero and the cracked displacement shape can contribute to the response. Note that the uncracked displacement shape will also participate when the crack is open; however, the contribution to the response is minor. Thus, the role of the Rn is to govern which displacement shape, either first cracked mode or first uncracked mode, will participate in the response of the system. This was demonstrated earlier in Section 3.5.5. This binary action controlled by the restoring force effectively reduces the 2DOF system to a single DOF system with varying stiffness depending on the crack status. By ignoring the mass and stiffness coupling terms, the above system can be interpreted as a SDOF system with a unit mass value. The stiffness variation will be bilinear alternating between the C0i2 and CO22 depending on the crack status. The convention used here will consider the crack to be open if the generalized displacement is positive and closed if negative. For small damping values, the damping variation between crack open and close will be negligible, thus a constant damping value can be assumed based on the uncracked stiffness. The SDOF bi-linear approximation is therefore given as: 78 Section 5: SDOF Bi-Linear Approximation crack close (-) •* > (+) crack open m=l Vuncrk / \ V crack Q mU" + cU' + kU = Lx[ (5.6) m = 1 where: c - 2^kum = l^com k \ co2\ for U > 0 (crack open) (5.7) m [co 22 for U < 0 (crack close) The solution of the SDOF bi-linear stiffness system can be readily obtained by a time-step integration technique. To obtain the global response, the generalized coordinate, U, must be transformed back to its Hill set of DOF, {x}. From Eqn(5.2), the transformation involves combination of both generalized co-ordinates of the first cracked and uncracked. mode shapes. With the further simplification to a single bi-linear DOF system, only the generalized response that is greater than zero, U> 0, corresponds with the cracked displacement shape, and the generalized response that is less than zero, U < 0, corresponds with uncracked displacement shape. Therefore, the global response is given by: {^rJUfor U < 0 (5.8) = < 79 Section 5: SDOF Bi-Linear Approximation The relative velocity of the global response can be obtained by numerically differentiating the displacement response obtained from above. However,' because of the discontinuity at the point of transition of the cracked and uncracked displacement shape ordinates, the acceleration obtained from doubly differentiating the displacement will produce erroneous sharp spikes. Recognising that the acceleration time history is essentially in phase with the displacement time history for lightly damped systems, the relative global acceleration can be obtained by doubly differentiating the generalized displacement response, U" and then applying the appropriate displacement shape ordinates which correspond with the same criteria as outlined above. Thus: The simplification of the full cracked dam system to a single bi-linear stiffness system is limited by the assumptions as listed below. • The first mode of the cracked and uncracked pseudo-linear system of the structure are used in representing all displacement configurations of the response. Higher and more complex displacements shapes will not be adequately represented. However, as previously illustrated, the most significant response is mainly contributed by the first modes of the two models, especially for displacement and velocity responses. • The simplified bi-linear system does not consider the static effects of structural mass in reducing the crack opening. Thus, the SDOF system assumes a complete cycle of crack opening and closing for every oscillation. This is more appropriate to strong motion excitation where the dynamic forces are significant enough to cause frequent crack opening. • The coupling terms in the mass, stiffness, and damping were ignored. { ^ r J U ' f o r U < 0 (5--< 80 Section 5: SDOF Bi-Linear Approximation • Although the stiffiiess effects of an elastic foundation can be included in the displacement shapes and frequencies, the foundation damping cannot be included in the bi-linear SDOF approximation. The viscous damping in the SDOF system must represent a global damping value for both dam and foundation sub domains. • Reservoir effects, including hydrostatic forces and hydrodynamic effects, cannot be accounted for by the SDOF approximation. The only effect that can be approximated is the resulting added mass from the hydrodynamic pressure. This will account for the change in the natural period of the now coupled reservoir-dam system, but the hydrodynamic force which is significant near resonance of the coupled system is ignored. This will be discussed further in Section 6. 5.2 Performance of SDOF Bi-Linear Stiffness Approximation The accuracy of the bi-linear SDOF approximation was illustrated by a comparison with the results of a 91.5 m. (300 ft.) 50% cracked dam model on rigid foundation and no reservoir using the FfFTD analysis procedure. The dam top displacement, velocity and acceleration obtained from the HFTD analysis were compared with the approximate method. Four uncracked modes and 4 cracked modes were used to form the Ritz vectors in the HFTD analysis. The modal properties of the first uncracked and first cracked modes were used in the SDOF bilinear approximation. The modal properties for the SDOF system are given in Figure 5.1. 81 Section 5: SDOF Bi-Linear Approximation 7.32 crack close (-) 91.5 9.76 = 3 <8> 3.25 81.71 = 10 <S> 6.17 (+) cracA: opew m=7 - 0 -k=813.71 =4.54 Hz k=184.19 wcrk=2.16Hz HoAe 116 -7-7-7-7-7-7-7-7-7-7-< > Lxg"=-238.62xg" ^=9.5x1 a3 nodel(x) ^=5.8x10 tf-*=0 nodell8(y) ^=2.3x10 Figure 5.1. SDOF bi-linear representation of 91.5 111 (300 ft.) 50% crack dam system The dam was subjected to the Loma Prieta horizontal ground acceleration record scaled to 5 g's. Details of this record is given in Section 6.2.2. Since static gravity force effect tendes to prevent the crack opening movement a large excitation is required in order to overcome this effect. The comparison of the response at the dam top is illustrated in Figures 5.2 through 5.4. The vertical crack opening displacement (VCOD) at the crack mouth, Node 118, is given in Figure 5.5. -Modes 4x4 SDOF Approx 3.5 time [sec] Figure 5.2. Comparison between the dam top relative displacement response of 91.5 m. (300 ft.) 50% cracked dam system subjected to Loma Prieta scaled to 5g's as obtained from HFTD procedure and the SDOF bi-linear approximate method. 82 Section 5: SDOF Bi-Unear Approximation Figure 5.3. Comparison between the dam top relative velocity response of 91.5 m. (300 ft.) 50% cracked dam system subjected to Loma Prieta scaled to 5g's as obtained from HFTD procedure and the SDOF bi-linear approximate method. Figure 5.4. Comparison between the dam top absolute acceleration response of 91.5 m. (300 ft.) 50% cracked dam system subjected to Loma Prieta scaled to 5g's as obtained from HFTD procedure and the SDOF bi-linear approximate method. 83 Section 5: SDOF Bi-Linear Approximation 0.5 0.4 0.1 0.0 modes 4x4 Bi-Linear SDOF 1 V / l1 / V \ A if if 1 V Y 1 y V 1 v \ h I' \ ' /' \ ' /' If if 1 If JL \i V li li \\ \ In /i \ ' /' /' \' /' i y i j' y Ji 1 1.5 2 2.5 3 3.5 4 4.5 5 time [sec] Figure 5.5. Comparison between the VCOD response at crack mouth of 91.5 m. (300 ft.) 50% cracked dam system subjected to Loma Prieta scaled to 5g's as obtained from HFTD procedure and the SDOF bi-linear approximate method. Aside from the initial portion of the response where the ground excitation is not strong enough to cause the base crack to open (t <2.6 sec), the relative displacement and velocity time history responses from the HFTD procedure are very similar to the SDOF approximation. The maximum dam top displacement and velocity achieved by the HFTD analysis is only 2% and 5% greater respectively than the response obtained from the approximate method. However for maximum acceleration response, the rigorous HFTD analysis is 20% greater than the SDOF bi-linear approximation. This is to be expected since the contribution of the higher modes becomes prevalent in acceleration response and the participation of the higher modes is directly accounted for in the HFTD analysis and not in the SDOF approximation. The crack open displacement at the crack mouth at the rigid foundation-dam interface is given in Figure 5.5. Note that there is no negative vertical displacement of the upper crack node 84 Section 5: SDOF Bi-Linear Approximation (Node 118) and this is enforced in the SDOF approximation by the zero modal ordinate of the uncracked mode (i.e. (J)1^* = 0 for Node (118)). The benefits of this approximation is that it is inexpensive and reasonably accurate. Therefore, it acts as a tool for screening possible dam candidates for more expensive nonlinear analysis using full F E M modelling. As illustrated in the above comparison, the strong motion portion of the response is adequately approximated by the simplified method for a given excitation record. Thus, a response spectra of the nonlinear SDOF or cracked spectra representing various crack lengths can, with a reasonable amount of confidence, be generated quickly. Furthermore, the response of the simplified system is linearly scaleable. That is, the response varies linearly with magnitude of the input motion since gravitational forces are neglected in the approximation. However, superpositioning techniques cannot be applied since the combination of cracked and uncracked phases cannot be achieve algebraically. 5.3 Chapter Summary With the knowledge that the majority of the response is contributed by the fundamental cracked and uncracked modes, a SDOF bi-linear stiffness system with the same dynamic properties of the participating fundamental modes can often be used to represent the fiill cracked dam system with reasonable accuracy. This simplified system can be quickly solved in the time domain by numerical time integration techniques. 85 Section 5: SDOF Bi-Linear Approximation The accuracy of this approximate method has been shown in a comparison with the rigorous HFTD procedure using a 91.50 m. (300 ft.) dam subjected to Loma Prieta record scaled to 5 g's. The displacement and velocity are shown to be in excellent agreement but the approximate method underestimates the acceleration. This is due to the disregard of the participation of the higher modes by the SDOF bi-linear system and the added damping that assumes a constant damping ratio based on the uncracked stiffness. The possible use of the approximate method to generate a response spectra for cracked dam systems will be explored in the next chapter. 86 Section 6: Cracked Dam Response to Earthquake Excitation 6. Cracked Dam Response to Earthquake Excitation 6.1 Introduction Thus far the main focus has been on the frequency content of the response of the cracked dam system. This has provided valuable information on the principle mechanics of the nonlinear system. However, the ultimate goal is to obtain and understand the response behavior of the cracked system as compared to the uncracked system for earthquake ground motions. In this chapter, the cracked dam system will be subjected to earthquake excitations and the maximum global response will be monitored for different dam sizes. The use of variously sized dams reflects the selection of different fundamental periods such that a response spectra, or more appropriately, a cracked response spectra can be constructed with respect to the uncracked dam response. Generalizations can be made from the cracked spectra such that a screening process can be developed in which potential candidates that might require a more rigorous nonlinear analysis can be selected. The use of the SDOF bi-linear approximate method will prove to be a valuable tool for estimating this response. The validity of the method will also be tested. In order to isolate the influence of the foundation and reservoir subdomains on the cracked dam response, each will be investigated separately. In Section 6.2, the crack dam system on rigid foundation and no reservoir will be investigated with the HFTD procedure. In Section 6.3, the results of Section 6.2 will be used to calibrate the SDOF bi-linear approximate method. The influence of the flexible foundation will be analyzed with no 87 Section 6: Cracked Dam Response to Earthquake Excitation reservoir in Section 6.4. Finally, the effects of full reservoir conditions, without the effects of flexible foundation will be analyzed in Section 6.5. 6.2 Response Of Cracked Dam System On Rigid Foundation And No Reservoir In order to gain a better understanding of the dynamic response of the cracked dam system, the dam system will be analysed without the influence of its accompanying subdomains. The cracked dam system will be set on a rigid foundation, and no reservoir will be present. 6.2.1 Model Specifications The dimensions of the various cracked dam models will follow a standard configuration. The configuration consists of a vertical upstream face, horizontal base, constant back slope of 0.75, and roadway of 7.32 m. (24') on top of the dam. Seven different dam sizes are used in the analysis ranging from 30.5 m. to 243.9 m. in height. Only 0%, 25%, and 50% crack 7 . 3 2 9.76 = 3 @ 3.25 *• Dam Concrete Elastic Modulus Ec 27 600 MPa Poission ratio V 0.2 Density P 2400 kg/m3 Viscous damping c 0.05 for freq range 1Hz ~ 7Hz (H-9.76) = \0® (H-r-r\ 9.76)/10 Mods 115 Figure 6.1. Typical dam configuration and finite element mesh Dam Height, H 30.5 m. (100') 61.0 m (200') 91.5 m. (300') 122.0 m. (400') 152.4 m. (500') .102.9 m. (600') 245.9 mA&OO) 88 Section 6. Cracked Dam Response to Earthquake Excitation lengths will be used in the study. The typical dam configuration and the material properties are given in Figure 6.1. 6.2.2 Free-Field Excitation The horizontal ground acceleration record of the October 17, 1987 Loma Prieta quake measured in the E-W direction at Corralitos - Eureka Canyon Road, and the February 9, 1971 San Fernando quake measured in the E-W direction at Caltech Seismological Lab, Pasadena, will be used as the control motion. The acceleration record as well as the PSD of both events are given in Figure 6.2 and 6.3. The main focus of the analysis will be on the response to the Loma Prieta record and verification of the results will be performed based on the San Fernando record. In order to better understand the mechanics of the response to these earthquake excitations, the dominant frequency of the earthquake motion is determined. The Loma Prieta record contains several strong signals within the frequency range of 1.5 FIz to 4 Hz, the strongest frequency of which centres about 1.5 Hz. However this frequency has a very narrow bandwidth. For a dynamic system to resonate at this frequency, the ESF of the system must be very close to 1.5 Hz. From Figure 6.3, it can be seen that it is more likely that the dominant frequency for Loma Prieta is closer to 3 Hz, because the signal strength spans over a broader bandwidth. For the San Fernando excitation, the resonant frequency is more easily identifiable since most of the signal strength is centred at about 4 Hz. To ensure crack opening in the majority of the response, the peak ground acceleration of the ground motion will 89 Section 6: Cracked Dam Response to Earthquake Excitation be normalized to 2.5g. This may be significantly larger than what a dam will experienced under more realistic conditions. However, difficulties in comparing the results may arise i f the excitation is insufficient to overcome the gravitational forces that prevent the crack from opening. Furthermore, the goal here is to compare the cracked response to the uncracked linear response; therefore the large magnitude of excitation is justified in order to explore an upper bound on crack opening effects. In reality where the excitation magnitude will be smaller, the response will lie within these two boundaries. Loma Prieta Horizontal E-W Component Scaled to 2.5 g 3 0 ^ 2 0 1 0 -2 0 -3 0 t\-r 4 5 6 t i m e [ s e c ] San Fernando Horizontal E-W Component Scaled to 2.5 g 2 0 1 0 0 -1 0 -2 0 -3 0 A A A V%1 t i m e [ s e c ] Figure 6.2. Accelerograms for Loma Prieta and San Fernando scaled to 2.5 g Loma Prieta San Fernando JW\A. Frequency [Hz] Frequency [Hz] Figure 6.3. Power spectral density (PSD) function for Loma Prieta and San Fernando accelerograms horizontal component scaled to 2.5 g's 90 Section 6: Cracked Dam Response to Earthquake Excitation Because of the forward and backward directional nature of the crack position, (i.e. crack opens with downstream direction and closes with upstream direction) the direction of the earthquake motion will have an influence. Thus both forward and backward pass for the applied ground motion is required to induce the maximum response of the nonlinear cracked system. This is achieved by simply reversing the sign of the scaling factor applied to the earthquake motion. 6.2.3 Analysis Parameters The total duration of the analysis will be 10.24 sec sampled at 0.01 sec. This will be composed of the first 8.0 sec of the earthquake motion with the remaining duration padded with a quiet zone. The O.Olsec time step would yield a maximum frequency content of 50 Hz. This is sufficient for capturing the main frequency of the excitation and the expected response. As well it will satisfy energy balance criteria required to minimize the iteration errors. To speed the iteration process, the time segmentation approach will be applied with 50 time segments. A constant viscous damping value of 5% of the critical damping specified over 1Hz to 7Hz is used to represent damping of the concrete dam. This damping is formulated by a linear combination of proportioned structural mass and stiffness matrix in accordance to Rayleigh damping. The frequency range is sufficient in maintaining a constant damping ratio over most of the fundamental frequencies of the dam systems tested. 6.2.4 Case Study: 61.0 m. (200 ft.) Dam subjected to Loma Prieta The cracked dam system clearly demonstrates the deterioration of the global stiffness and the lengthening of the effective period. Figure 6.4 shows the dam top relative displacement, 91 Section 6: Cracked Dam Response to Earthquake Excitation velocity, absolute acceleration, as well, the crack mouth displacement of a 61.0 m. (200 ft.) dam model subjected to Loma Prieta normalized to 2.5g on rigid foundation and no reservoir. Three crack conditions are used for the investigation, 0%, 25%, and 50% crack lengths. The mechanism in which the influence of crack opening is most evident is in the relative displacement time history. From the time period of 0.0 to 2.6 sec, the cracked dam system of all crack lengths have not exhibited any opening of the crack and the response of the cracked system shows similar behaviour to the uncracked system in both response magnitude and frequency of oscillation. This observation is validated by the VCOD remaining zero (Figure 6.4 d). After 2.6 sec, the inertia forces induced by the ground motion overcomes the gravity force of the dam mass and the crack opens. The dam top displacement of the cracked system now behaves with a decreased stiffness causing increase displacement and elongation of the effective system period. The initial onset of crack opening causes an increase in displacement; this is indicative of similar behaviour found with nonlinear materials such as material yielding and fracture mechanics (El-Aisi and Hall, 1989). Both systems of 25% and 50% crack lengths exhibit this behaviour. However, as indicated in previous sections, cracked dam systems containing base crack lengths of 25% or less do not significantly alter the ESF from that of the uncracked natural frequencies. Thus the amplification of response as compared to the uncracked dam system is not as significant as with larger crack lengths such as crack lengths of 50%. 92 Section 6: Cracked Dam Response to Earthquake Excitation 0.15 4.0 3.0 2.0 1.0 0.0 -1.0 -2.0 -3.0 2 250.0 200.0 150.0 100.0 50.0 0.0 -50.0 -100.0 -150.0 0 045 0 040 0 035 0 030 3D [m] 0 025 3D [m] 0 020 O > 0 015 0 010 0 005 0 000 -0 005 J? 50% - 3.93 i i A - ii'[ | r i 25% - 1.62 IjA 0% - 1 08 ' t \ * }/\ A k 7 \:\ Ki '• / \V f T ' A V '• 1 \l l i ^ { L A 1 * / : vV y \ i« * . \ V*. 3.5 tim e [sec] 3.5 tim e [sec] — cnn; 4 n n c 25% 71.3 n!". • • ^ : i 5 0% - 52 4 ^ ' 'A t AWT " '** i.' 1 2.5 3 3.5 4 4.5 5 tim e [sac] 50% - 0.044 : :. ' • ) i • • /I 25% - 0.01 ft ! • i • i A i ! ; ;A 1 ri '•• .< > f> . •J 1/;'. •/> A ,-. J Figure 6.4. Response of 61.0 m. (200 ft.) cracked dam system on rigid foundation and no reservoir subjected to Loma Prieta scaled to 2.5g's. a) Relative displacement at dam top, b) Relative velocity at dam top, c) Absolute acceleration at dam top, d) VCOD at crack mouth. 93 Section 6: Cracked Dam Response to Earthquake Excitation For the given excitation, peak displacement amplification (see Figure 6.4 a ) in excess of 2.5 times the peak displacement of the uncracked dam system can be seen for a 50% crack length. For a 25% crack length the amplification is about 1.3. Note that the occurrence of these peak displacements may not necessarily coincide in time since the dynamic characteristic of the cracked system has now been altered. Another observation is that the peak displacement of the cracked systems does not necessarily occur during the crack open phase of the response. Recall that in Section 2.4, peak displacement during the steady state response of the cracked system always occurred during the crack opening phase. However for a transient response, this generalization cannot be made. The relative velocity and absolute acceleration time history responses exhibit much of the same characteristics of the displacement time history response. Again, an appreciable amount of amplification in the peak velocity and acceleration of the cracked system as compared to the uncracked system is evident for the 50% crack. In addition, the oscillatory motion of these higher order response quantities shows frequency of higher modes which was not evident in the displacement response. It is recognized that the contributions of the higher modes are more significant in response quantities of a higher order, such as velocity and acceleration. The increase in the response of the cracked system can be attributed to one main factor. The fundamental period for the 61.0 m. (200 ft.) uncracked dam system is 0.1568 sec ( a w , = 6.35 Hz). When compared to the dominant frequency of the Loma Prieta excitation of approximately tfw = 3 Hz, a frequency ratio, Q= G)excit /costruct of 0.472 is obtained. Once the 94 Section 6: Cracked Dam Response to Earthquake Excitation dynamic inertia forces overcome the mass gravity forces maintaining crack closure, the effective system period will increase driving the frequency ratio closer to 1.0. In other words, the crack opening effectively softens the system causing a more favourable condition for resonance; hence the increase in the dynamic amplification as compared to the uncracked case. 6.2.5 Case Study: 152.4 m. (500 ft.) Dam Subjected to Loma Prieta To investigate the response of a more flexible cracked system relative to the ground motion, a 152.4 m. (500 ft.) dam model will be studied. The fundamental frequency of the uncracked phase is 2.84 Hz and the 50% cracked phase is 1.32 Hz. This is well removed from the dominant frequency of the ground motion at approximately 3 Hz. According to the argument outlined above for the 61.0 m. (200 ft.) dam, no amplification of the response should be expected when comparing it to the uncracked condition for this softer system. The dam top displacement velocity and absolute acceleration time history of 0%, 25%, and 50% crack lengths are given in the Figure 6.5. As with the 61.0 m. (200 ft.) dam, the response of all crack lengths are essentially identical up to 2.6 sec. It is not until beyond this point that dynamic inertia forces created by the ground excitation is large enough to overcome the mass gravitational force to cause crack opening. For the dam top relative displacement time history response, it is evident that upon the initial opening of the crack there is a slight increase followed by a significant decrease in the remaining cracked response of both 25% and 50% crack lengths. This initial increase in comparison with the uncracked condition is small, and it is caused by the full transfer of kinetic energy generated by the suffer uncracked state to full strain energy in the less stiff cracked phase. 95 Section 6: Cracked Dam Response to Earthquake Excitation After the initial crack opening event, the ESF decreases and moves away from the dominant frequency of the free-field excitation thus de-amplifying the response in comparison to the uncracked dam system. Q. Ui 5 > o o > = -5 i c o 300 250 200 150 100 50 0 -50 -100 -150 -200 -250 /•"H 50% -0.537 /\» \ /' / V 1 // • \^ / I / ' I\ 1 / \\ W- V // IN / •"' 1 Vv 1 /••'' \ '•• / ' : \ \ /S f ^ 1 j \ /'••'' \ IF' \ \ / > i r I ' / i \\ I \ ••'1 \l i \'-1 /•'/ \ 1 ••/ \ / )LH • 1 \ / ' 1 _t i \y M i \ \ll ] V k * \ ' /. \. •' 1 V i I \ji \ \}\> k f ' \ I y r I / y \ / <d 25 % - 0.564 V/ 0% - 0.591 3.5 t ime [sec] 5% -11.86 ••"•/ M /A / / fl 50%'77.55 r \ r \ V /• * V A ' 1 / V ;i\ /A /1• !iv / A / ' V / / '--n / \ \ i i v V r b 1 ' . r t t / ''•» W/^ , : ' \ N T A V I i \ A"i' \ H \ ~~\T~ v J \ V y V fr- • UA/v ^ \ / M / tt 0% -11.71 3.5 t ime [sec] 3.5 t ime [sec] 1 c n o / o JI o A D i i i\ : i i i i A ! i " A P r\ J ' A r t / • • / i V ' ' t - \ h r I:' \i i I ^ :: / \ft / 1 ' / \ r A 1 / \ / i -\! • / t ifi r - fe 1' / A: 1 Ii 1 1 f- A V\ J U / / |S 1 ) | . | V J ' | I1 \ : fcv"w\ • \ i —u—V V*'—7 v*\.— —-^ y J-)! r—r— ' \' : /> • \ i : X i •l- \ •• /: \A\'\ J I •/ • • * \' y? 1 \'W ' >• i I' if- '  i-\ \ / I \ 1 'I V y d 0% - 211.28 Figure 6.5. Response of 152.4 m. (500 ft.) cracked dam system on rigid foundation and no reservoir subjected to Loma Prieta scaled to 2.5g's. a) Relative displacement at dam top, b) Relative velocity at dam top, c) Absolute acceleration at dam top. 96 Section 6: Cracked Dam Response to Earthquake Excitation The observed de-amplification can also be seen in the relative velocity time history. Unlike the displacement response, contributions from higher modes are evident from the oscillatory nature of the velocity response. These higher modes become even more significant in the absolute acceleration time history response. Some of these higher modes are subjected to amplification from the dominant frequencies of the free-field excitation. Thus, the trend of de-amplification as seen in the displacement and velocity may not be applicable for the acceleration response. 6.2.6 Spectral Response of the Cracked Dam System The peak response of the 0%, 25%, and 50% cracked dam systems ranging in height from 30.5 m. to 243.9 m. are given in Table 6.1. This represents the maximum displacement, velocity, and acceleration of the dam system at the dam top for both forward and backward passes of the Loma Prieta excitation. From these peak response values, a response spectra can be constructed illustrating the effects of the crack lengths on the maximum response of cracked dam system. The peak response values are normalized with respect to the maximum input acceleration x gmax and the first uncracked frequency of the responding system, aw*. The normalizing factors are given below. S d = T ^ ~ Sv=j^— S . = - £ - (6.1) / a w 2 * m a x where Sd, Sv, and S a are the maximum response values and Sd, Sv, Sa are the normalized spectral values. 97 Section 6: Cracked Dam Response to Earthquake Excitation 0% Crack - Loma Prieta Scaled to 2.5g Dam First Mode Spectral Reponse Height [m/ft] Freq [Hz] Period [sec] S d 0 [m] Sv 0 [m/s] Sao [m/s2] 30.5/100 10.64 0.094 0.011 0.37 36 61.1/200 6.37 0.157 0.047 1.12 55 91.5/300 4.54 0.220 0.151 3.75 106 122.0/400 3.5 0.286 0.442 10.12 208 152.4/500 2.84 0.352 0.595 11.74 212 182.9/600 2.39 0.418 0.726 13.72 228 243.9/800 1.81 0.552 1.098 17.65 289 25% Crack - Loma Prieta Scaled to 2.5g Dam First Mode Spectral Response Amplification wrt 0% Height [m/ft] Freq [Hz] Period [sec] Sd [m] Sv [m/s] Sa [mis] Sd/Sdo Sv/Svo Sa/Sao 30.5/100 8.92 0.112 0.014 0.39 37 1.237 1.067 1.008 61.1/200 5.25 0.190 0.060 1.76 83 1.277 1.565 1.503 91.5/300 3.6 0.278 0.205 4.85 192 1.357 1.293 1.818 122.0/400 2.82 0.355 0.537 11.83 354 1.214 1.169 1.698 152.4/500 2.28 0.439 0.588 11.86 286 0.990 1.010 1.348 182.9/600 1.909 0.524 0.872 15.06 318 1.202 1.098 1.393 243.9/800 1.44 0.694 1.119 17.29 265 1.019 0.979 0.919 50% Crack - Loma Prieta Scaled to 2.5g Dam First Mode Spectral Response Amplification wrt 0% Height [m/ft] Freq [Hz] Period [sec] Sd [m] Sv [m/s] Sa [m/s ] Sd/Sdo Sv/Svo Sa/Sao 30.5/100 5.5 0.182 0.028 1.02 74 2.564 2.775 2.050 61.1/200 3.13 0.319 0.122 4.05 205 2.574 3.614 3.707 91.5/300 2.16 0.463 0.354 10.21 299 2.339 2.724 2.824 122.0/400 1.64 0.610 0.439 10.15 311 0.993 1.003 1.493 152.4/500 1.32 0.758 0.595 11.13 381 1.000 0.948 1.796 182.9/600 1.11 0.901 0.835 14.27 328 1.151 1.040 1.438 243.9/800 0.83 1.205 1.274 17.23 381 1.161 0.976 1.320 Table 6.1. Spectral response of cracked dam system on rigid foundation and no reservoir of 30.5m. through 243.9 m. with crack lengths of 0%, 25%, and 50%, subjected to Loma Prieta scaled to 2.5g Since the exact ESF of the cracked system is not known, the spectral response values of both uncracked and cracked dam systems are plotted against the fundamental uncracked period in Figure 6.6. By plotting in this manner, the cracked spectra curves can be viewed as an amplification to the uncracked spectra response curve. 98 Section 6: Cracked Dam Response to Earthquake Excitation 12 10 8 6 4 2 0 ( 14 12 10 8 6 4 2 0 0.0 0.0 16 14 12 10 S~. 8 6 4 2 0 0.1 i I O-. A . | " X '-A • V C / A ' J / X / | & w j A 0.1 0.2 0.3 0.4 0.5 0.6 / a -a. • // u ^ u — 6.1 'x c / t 0.2 0.3 0.4 0.5 0.6 A-••'\ ^ t * - . - > < A • i ' / I / -f— X X—u % — 25% • 50% c / H 1 j 0.0 0.1 0.2 0.3 0.4 0.5 Uncracked Period [sec] 0.6 Figure 6.6. Normalized maximum response values for cracked dam system subjected to Loma Prieta scaled to 2.5g with 5% viscous damping. From the response spectra of the uncracked and cracked dam systems, several key observations are clear. Significant amplifications in all response quantities of the 50% crack dam system are evident when compared to the uncracked spectral response. For the period range of 0.1 to 0.35 sec amplifications as large as 3.7 for the normalized spectral acceleration can be seen for the 61.0 m. dam case with a period of 0.157 sec. This is due to the resonant response of the 50% cracked dam system as the ESF of the system approaches the dominant frequency of the Loma Prieta excitation. However, for systems with an uncracked period greater than 0.35 sec (152.4 m, 182.9 m, and 243.9 m dam heights), very little, if any, amplification is present 99 Section 6: Cracked Dam Response to Earthquake Excitation in the displacement and velocity spectral values when compared to the cracked dam. However, amplifications are still evident for the spectral accelerations in this period range. This is mainly due to the participation of the higher modes which are stimulated by the dominant frequency components of the excitation. Crack lengths of 25% or less generate only nominal amounts of amplification in the displacement and velocity spectral values for all period ranges. This is expected since the ESF range of a 25% cracked dam system differs only slightly from the uncracked natural frequency, and the dynamic properties of the 25% cracked dam system are not significantly altered. However, amplifications in acceleration are still prominent for dam with 25% crack lengths. This was confirmed with the results presented in Section 4. 6.2.7 Conclusions The following points summarize the response behaviour of the cracked dam system subjected to the Loma Prieta excitation: • For crack lengths greater than 25%, the amplification in the response compared to the uncracked dam system is significant for large amplitude excitations. Crack lengths less than 25% do not differ significantly for relative displacement and velocity. Acceleration may be of concern if resonance with the excitation's dominant frequency is expected. • Larger dam systems where the fundamental uncracked frequency is well removed from the excitation frequency, little amplification of the spectral cracked displacement, and the velocity comparable to those of the linear uncracked spectral response are expected. • For larger dam systems where the fundamental uncracked frequency is well removed from the excitation frequency, the peak response is reached at the first onset of the crack opening. 100 Section 6: Cracked Dam Response to Earthquake Excitation Subsequent response is de-amplified as the ESF shifts away from dominant frequency of excitation. 101 Section 6: Cracked Dam Response to Earthquake Excitation 6.3 SDOF Bi-linear Stiffness Approximation and the Cracked Spectra As noted in Chapter 5, the simplified approach of approximating the complex behaviour of the cracked dam system by a SDOF bi-linear stiffness system is ideal for producing response spectra for each of the crack lengths in question. This accumulation of response spectra of different crack lengths will be known as "cracked spectra". With the results of the full dam models subjected to Loma Prieta obtained in Section 6.2, the cracked spectra calculated in this section with the SDOF bi-linear procedure can be used to determine the accuracy of the approximation. 6.3.1 SDOF Bi-linear Stiffness Analysis Properties The system properties required for the SDOF bi-linear system will be calibrated with the global properties of the cracked dam system as described in Section 6.2.1. Table 6.2 shows the uncracked and cracked fundamental periods and mode shape ordinates of the horizontal component of the dam top (Node 1). The modal participation factor of the fundamental cracked and uncracked modes are also listed. 25% Crack Length Dam First Modal Frequency [Hz] Wcrk/Wunc stiff1/stiff2 First Modal Ordinate Crack/Uncrk Participation Factor Height [m/ft] Cracked Uncracked Ratio Ratio Cracked Uncracked Ratio Uncrack Cracked 30.5/100 8.92 10.64 838.3E-3 702.8E-3 18.2E-3 20.2E-3 902.5E-3 84.25 90.37 61.1/200 5.24 6.37 822.6E-3 676.7E-3 11.3E-3 13.3E-3 845.6E-3 156.97 170.58 91.5/300 3.60 4.54 793.0E-3 628.8E-3 7.8E-3 9.5E-3 825.9E-3 238.62 256.93 122.0/400 2.82 3.50 805.7E-3 649.2E-3 5.9E-3 7.2E-3 820.1 E-3 321.90 344.15 152.4/500 2.28 2.84 802.8E-3 644.5E-3 4.7E-3 5.8E-3 818.7E-3 405.29 431.40 182.9/600 1.91 2.39 798.7E-3 637.9E-3 3.9E-3 4.8E-3 818.5E-3 488.54 518.66 243.9/800 1.44 1.81 795.6E-3 632.9E-3 2.9E-3 3.6E-3 819.2E-3 654.63 692.95 Average 808.1 E-3 653.3E-3 835.8E-3 50% Crack Length Dam First Modal Frequency [Hz] Wcrk/Wunc stiff1/stiff2 First Modal Ordinate Crack/Uncrk Participation Factor Height [m/ft] Uncracked Cracked Ratio Ratio Cracked Uncracked Ratio Uncrack Cracked 30.5/100 5.50 10.64 516.9E-3 267.2E-3 14.9E-3 20.2E-3 737.8E-3 84.25 88.52 61.1/200 3.13 6.37 491.4E-3 241.4E-3 8.5E-3 13.3E-3 638.7E-3 156.97 164.07 91.5/300 2.16 4.54 475.8E-3 226.4E-3 5.8E-3 9.5E-3 616.1 E-3 238.62 246.36 122.0/400 1.64 3.50 468.6E-3 219.6E-3 4.4E-3 7.2E-3 612.3E-3 321.90 323.45 152.4/500 1.32 2.84 464.8E-3 216.0E-3 3.5E-3 5.8E-3 612.8E-3 405.29 403.80 182.9/600 1.11 2.39 464.4E-3 215.7E-3 2.9E-3 4.8E-3 614.3E-3 488.54 484.26 243.9/800 0.83 1.81 458.6E-3 210.3E-3 2.2E-3 3.6E-3 617.0E-3 654.63 645.34 Average 477.2E-3 228.1 E-3 635.6E-3 Table 6.2. Modal properties of cracked dam system for SDOF bi-linear stiffness approximation 102 Section 6: Cracked Dam Response to Earthquake Excitation Since the purpose of the cracked spectra is to obtain the amplification of the cracked spectral response in comparison with the uncracked spectral response, rather than the exact values, the ratio of the uncracked versus cracked modal values are required. Average ratios of (0.838:1) for 25% and (0.639:1) for 50% crack length are used for the mode shape ordinate ratio at the dam top of the cracked to uncracked phase. Also, average ratios of (0.654:1) for 25% crack lengths and (0.23:1) for 50% crack lengths are used for the stiffness reduction of the fundamental cracked mode as compared to the fundamental uncracked mode. The cracked and uncracked modal participation factors do not differ more than 5% for the cases studied. Therefore, the force variation from the transition of phases is not of concern, and can be considered to the same at a ratio of 1:1. The SDOF bi-linear systems are represented in Figure 6.7: m=l m=l tytmcrk / \ tyuncrk W ™J ^ 0 . 8 3 6 ^ ^uncrk ® uncrk % ncrk S—N. T C T * = If kimcrk uncrk TT7TT7TT77-Crack Length = 0% , ' t r - © - r h i ' k^ 0.653k.* ' k = TT7-7-r7-rrrr Crack Length = 25% 0.228k. ////////// Crack Length = 50% Figure 6.7. Ratios of modal parameters for the uncracked and cracked phase of dam system. Crack lengths of 0%, 25%, and 50% are used for the SDOF bi-linear stiffness approximate method. With the above ratios, cracked spectra for 0%, 25%, and 50% crack lengths are generated for the Loma Prieta horizontal ground motion with 5% viscous damping with respect to the uncracked stiffness. Peak relative displacement, relative velocity, and absolute acceleration from both the forward and the backward pass of the ground motion was generated at 50 intervals between uncracked periods from 0.05sec to 2.0sec. The spectral response values were normalized with 103 Section 6: Cracked Dam Response to Earthquake Excitation respect to the maximum acceleration of the ground motion, and the uncracked period of the system as given by Eqn 6.1. Figure 6.8 shows the results plotted against the fundamental uncracked period of the system. To solve the nonlinear bi-linear stiffness equation of motion, a linear acceleration time step integration technique was used. A constant time step of 0.002 sec, which satisfies the necessary stability criterion and minimizes any energy error during transitions of stiffness phases were used in the time step integration. 6.3.2 Comparison of Approximate Method with H F T D Result Comparison is performed by plotting an amplified cracked spectral ordinate determined from the HFTD procedure to the uncracked natural frequency. The amplified HFTD cracked spectral ordinate was calculated based on the amplification factor from Table 6.1 applied to the SDOF uncracked response spectra ordinate normalized in accordance to Eqn. 6.1 (See Eqn. 6.2). By comparing the spectral response this way, the uncracked response from the HFTD procedure and the uncracked SDOF response are assumed to have the same value, thus forming a baseline in which the amplified cracked spectral ordinates can be compared. The relative displacement, relative velocity, and absolute acceleration approximate SDOF cracked spectra for the Loma Prieta record with 5% damping are given in Figure 6.8. The amplified cracked spectral response obtained from the HFTD procedure are also plotted. The 104 Section 6: Cracked Dam Response to Earthquake Excitation spectra generated from the bi-linear SDOF approximation are given as line plots and the spectra ordinates obtained from the HFDT analysis are given as data points. Sm - amplified m% SDOF cracked spectral ordinate. S / 7 ? = ratio of m% cracked spectral response to uncracked response = S / — Sm= ™/ S0 where from HFTD given in Table (6.1). / o 0 S0 = SDOF uncracked spectral response ordinate normalized with respect to Xgm a x in accordance to Eqn. (6.1). m = percentage of crack length to overall base length. The accuracy of the SDOF approximation in estimating the amplifications of the cracked spectral response is evident in the relative displacement and the relative velocity cracked spectra. The amplifications of the relative displacement and relative velocity spectral ordinate values which are obtained from the HFDT analysis match well with the approximation. However, the absolute acceleration cracked spectral values obtained from the approximation underestimate the values obtained from the HFTD analysis. This is because acceleration tends to amplify the contributions of the higher modes which are absent in the SDOF bi-linear procedure. The contributing higher modes are prevalent in more flexible systems where they are excited by the dominant frequency of excitation. For the stiffer systems, namely the 30.5 m., 61.0 m., and 91.5 m. dams, the higher modal frequencies are well removed from the excitation frequency, thus they are less amplified, and allow the absolute acceleration spectral ordinates to be well approximated by the bi-linear SDOF system. 105 Section 6: Cracked Dam Response to Earthquake Excitation a 4 = 3 <D or T3 *, o 0% ( S D O F ) 2 5 % ( S D O F ) 5 0 % ( S D O F ) A 2 5 % ( H F T D ) |_ • 5 0 % ( H F T D ) i 0.5 1 Uncracked Period [sec] 1.5 £• 4 | o > CD 0) 0.5 Uncracked Period [sec] 1.5 4j (1) O o < 0) .a < •o 0) o z • A 0% ( S D O F ) 2 5 % ( S D O F ) 5 0 % ( S D O F ) A 2 5 % (HFTD) a 5 0 % ( H F T D ) 0.5 1.5 Uncracked Period [sec] Figure 6.8. Comparison between approximate cracked spectra from SDOF bi-linear stiffness method and spectral result amplifications from HFTD with respect to uncracked spectral response for Loma Prieta scaled to 2.5g for 5% damping. 106 Section 6: Cracked Dam Response to Earthquake Excitation Aside from the discrepancies in the absolute acceleration cracked spectra, the SDOF bi-linear approximation performs well in estimating the relative displacement and relative velocity cracked spectra for the cracked dam system. 6.3.3 Verifying Approximate Cracked Spectra with San Fernando Record To confirm the validity of the SDOF bi-linear stiffness approximation of the cracked spectra, the procedure performed in 6.3.2 will be repeated with an excitation based on the horizontal component of the San Fernando record as described in Section 6.2.2. Again the record will be scaled to 2.5g's, both backward and forward passes of the record will be applied, and the maximum of the two passes will represent the spectral response value. The same dam models and sizes will be used with the exceptions of 91.5 m. and 152.4 m. dam height. Due to time constraints, only the 0% and 50% crack lengths were used in this analysis. The resulting spectral response values from the HFTD analysis are given in Table 6.3. Figure 6.9 provides the comparison between the spectral response amplification for relative displacement, relative velocity, and absolute accelerations with that of the approximate SDOF bi-linear method. 0% Crack - San Fernando Scaled to 2.5g Dam First Mode Spectral Reponse Height [m/ft] Freq [Hz] Period [sec] Sd 0 [m] Svo [m/s] Sao [m/s2] 100/30.5 10.64 0.094 0.015 0.67 49 200/61.0 6.37 0.157 0.080 2.70 109 400/122.0 3.5 0.286 0.393 8.93 210 600/182.9 2.39 0.418 0.616 9.39 242 800/243.9 1.81 0.552 0.875 16.71 409 50% Crack - San Fernando Scaled to 2.5g Dam First Mode Spectral Response Amplification wrt 0% Height [m/ft] Freq [Hz] Period [sec] Sd [m] Sv [m/s] Sa [m/s2] Sd/Sdo Sv/Svo Sa/Sa0 100/30.5 5.5 0.182 0.025 1.11 72 1.700 1.659 1.469 200/61.0 3.13 0.319 0.131 4.63 193 1.629 1.716 1.760 400/122.0 1.64 0.610 0.415 8.96 357 1.054 1.003 1.698 600/182.9 1.11 0.901 0.631 12.01 294 1.025 1.279 1.216 800/243.9 0.83 1.205 1.247 14.45 433 1.425 0.865 1.060 Table 6.3. Maximum response result of dam sections on rigid foundation and no reservoir subjected to San Fernando scaled 2.5g. 107 Section 6: Cracked Dam Response to Earthquake Excitation 0 A 1 1 1 1 1 1 1 1 1 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Uncracked Period [sec] 0 -I 1 1 1 1 1 1 1 1 1 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Uncracked Period [sec] Figure 6.9. Comparison of approximate cracked spectra from SDOF bi-linear stiffness method and spectral result amplifications from HFTD with respect to uncracked spectral response for San Fernando scaled to 2.5g with 5% damping. 108 Section 6: Cracked Dam Response to Earthquake Excitation Again, the relative displacement and relative velocity cracked spectra for the 50% crack length provides a reasonable estimate of the spectral amplification expected from the full analysis. However, at lower frequencies (i.e. 30.5 m. and 61.0 m. dam) the approximate SDOF method overestimates the true response, and agreement is not as good as that found using the Loma Prieta record. This may be due to an insufficient magnitude of excitation causing less frequent crack opening. The main assumption in the approximate SDOF bi-linear method is that the simplified system undergoes a complete cycle of crack opening and closing for each cycle of oscillation. This is only true for the cracked dam system when analysed by the HFTD procedure if the gravitational forces of the dam mass are ignored, or if the magnitude of excitation is large enough to cause a sufficient number of crack opening events. The applied San Fernando record with a of maximum acceleration of 2.5g may not satisfy this requirement. Perhaps a larger magnitude should have been used in order to match the amplification of the cracked spectral values from the approximate method. Furthermore, the approximate cracked spectra for the San Fernando record shows regions of de-amplifications which was not as prominent in the approximate cracked spectra for Loma Prieta. These regions are most evident for the 50% crack case in the relative velocity and absolute acceleration cracked spectrums The evidence of de-amplification in the cracked spectral response was never confirmed by the HFTD analysis for either the San Fernando, or the Loma Prieta excitation. De-amplification is possible in flexible dam systems, however, recall from Section 6.2.5 109 Section 6: Cracked Dam Response to Earthquake Excitation that the resulting peak response is at least equal in magnitude to the uncracked response. Therefore, without further evaluation with other earthquake records, it is recommended that any de-amplifications indicated by the approximated cracked spectra be ignored. A maximum spectral response equal in magnitude to the uncracked response should be assumed. 6.3.4 Conclusions The simple and efficient SDOF bi-linear approximation of the cracked spectra accurately estimates the maximum response of a cracked dam system. By assessing the performance of a linear uncracked dam system either from a linear time history analysis or from a simple response spectra procedure, the approximated cracked spectra can provide a reasonable estimate of the maximum relative displacement and relative velocity cracked dam response for a specified crack length. However, the accuracy of the cracked spectra is limited to systems which must adhere to the assumptions applied to the bi-linear SDOF approximation. Since only the modal properties of the first uncracked and crack modes are reproduced in the SDOF system, reasonable accuracy in the estimating maximum absolute acceleration cannot be achieved because of the participation of the higher mode has been neglected. Furthermore, neglecting of static mass gravitational forces restricts the accuracy of the approximation to large magnitude excitations only. However, the response to moderate earthquake excitations are expected to be bounded between the estimates produced by this approach and the uncracked linear systems. 110 Section 6: Cracked Dam Response to Earthquake Excitation As with any response spectra approach, the objective of the cracked spectra is to measure the maximum response of the cracked dam system and to compare it to the maximum response of the uncracked dam system. The nature of the response spectra format is limited, and does not provide vital information of the time history response. Thus, post peak response and direction in which the response is achieved is unavailable with this format. Nonetheless, the speed and the accuracy of the SDOF bi-linear approximation makes it a valuable tool in screening potential candidates for more detailed analysis if cracking is determined to be of concern. 111 Section 6: Cracked Dam Response to Earthquake Excitation 6.4 Effects of Flexible Foundation on the Response of Cracked Dam System. All structures analysed thus far have been set on rigid foundations. The following analysis will investigate the response of the cracked dam system founded on a flexible medium. Only the 61.0 m. dam model will be used in this investigation with three crack lengths: 0%, 25%, and 50%. With the exception of the specified foundation properties, all dam material properties used in Section 6.2 for the case of rigid foundation will also be used here. Two different foundation elastic modulii will be investigated; mainly Ef=Ec and Bf^EJl where Ef is the elastic modulus of the foundation and E c is the elastic modulus of the dam concrete (see Table 6.4). Hysteretic damping of 10% is specified for the foundation. This is approximately equivalent to 5% viscous damping (Chopra et al., 1975). The horizontal component of the Loma Prieta ground motion as described in Section 6.2.2 scaled to 2.5g will be used to excite the dam structure. Foundation Properties Case 1: Case 2: Elastic Modulus Elastic Modulus Ef Ef 27600 MPa 13800 MPa Density Pf 2560 kg/m3 Hysteretic Damping r| 0.10 Table 6.4. Material properties for foundation medium Aside from the foundation damping, the dynamic mechanics of the cracked dam system on a rigid foundation and on a flexible foundation are similar. Therefore, the SDOF bi-linear approximation method for cracked spectra in Section 6.2 should also be valid for flexible foundations. 112 Section 6: Cracked Dam Response to Earthquake Excitation Flexible Foundation — £.1 D U U i v t r a p = 2560 kg/nT Stiffness Ratio Modal Frequency 0% Crack 25% Crack 50% Crack 25% Crack 50% Crack Mode 1 4.86 3.79 2.32 0.608 0.228 Mode 2 8.72 8.24 7.40 Mode 3 10.37 10.15 9.49 Mode 4 18.12 17.08 15.82 First Mode Ordinate (normalized wrt 0% model 1.000 0.882 0.754 Flexible Foundation E f = 13800 MPa p = 2560 kg/m Stiffness Ratio Modal Frequency 0% Crack 25% Crack 50% Crack 25% Crack 50% Crack Mode 1 4.01 3.14 1.97 0.613 0.241 Mode 2 6.46 6.15 5.65 Mode 3 9.02 8.78 8.19 Mode 4 16.28 15.17 14.36 First Mode Ordinate (normalized wrt 0% mode) 1.000 0.935 0.839 Table 6.5. Modal properties of 61.0 m. (200 ft.) cracked dam section on flexible foundation. Table 6.5 shows the modal properties of the cracked dam-foundation systems. Focusing on Case 1 where the foundation modulus is equivalent to the dam concrete modulus, the modal stiffness ratio of the fundamental cracked and uncracked modes for 25%, (0.608:1) and 50%, (0.228:1) are similar to the rigid foundation of (0.653:1) and (0.228:1) for 25% and 50% crack lengths respectively (see Table 6.2). Also, the dam top mode shape ordinate ratio of uncracked to crack is similar as it has values of (0.882:1) for 25% and (0.754:1) for 50% in comparison with (0.836:1) and (0.636:1), respectively, for a rigid foundation. Therefore, the same cracked spectra generated by the bi-linear SDOF approximation used in Section 6.2 can also be used here to predict the peak response from the Loma Prieta ground motion with flexible foundation. To verify the approximations, amplified cracked spectral values obtained by HFTD analysis will be compared. 113 Section 6: Cracked Dam Response to Earthquake Excitation 6.4.1 Result of H F T D Analysis The relative displacement of the dam top is given in Figure 6.10. The nature of the response is similar to that of the case of a rigid foundation. However, due to the reduced foundation stiffness, the ESF of the cracked systems has been reduced. Thus, the response at the dam top is indicative of that of a more flexible dam system on a rigid foundation. Furthermore, the rocking motion of the foundation produces vertical movement. These motions allow the lower crack surface to move with the upper crack surface reducing the tendency of crack opening. This is illustrated in the crack mouth time histories shown in Figure 6.11. This implies that the full system stiffness is available for longer duration and the ESF would be closer to the uncracked natural frequency than the cracked frequency. In smaller magnitude excitations where the inertial forces to cause crack opening are nominal, this may be important and crack opening may never occur. It is noted that for the 2.5g used in this investigation, sufficient crack opening was evident. 114 Section 6: Cracked Dam Response to Earthquake Excitation Figure 6.10. Dam top displacement of 61.0 m. cracked dam system with flexible foundation of Ef=27600 MPa and Ef=13800 MPa excited by Loma Prieta record scaled to 2.5 g. Ef=27600 VI Pa: 50% C rack I 1 \ \ U p p e r .' % i * c r a c k l ip 1 \ 1 1 ' 1 * ' r \ •' '• * .' -—s •' s~ \ •s • \ v : ;. L o w e r c r a c k 2 2.5 3 3.5 4 4.5 5 tim e [sec] Figure 6.11. Vertical crack edge displacements at heel of 61.0 m. dam with 50% crack on flexible foundation of Ef=27600 MPa excited by Loma Prieta record scaled to 2.5 g. 115 Section 6: Cracked Dam Response to Earthquake Excitation To verify that the cracked spectra generated by the SDOF bi-linear approximation method is valid for cracked dam systems on a flexible foundation, the spectral response quantities of the dam top will be plotted in relation to the spectral values of the uncracked dam system (see Eqn. 6.2). These spectral values are given in Table 6.6, and the normalized cracked spectra for the Loma Prieta horizontal ground component with the amplified HFTD cracked spectral response are compared in Figure 6.12. 0% Crack - 61.0 m. Dam on Flexible Foundation: Loma Prieta Scaled to 2.5g Foundation Modulus Ef First Mode Freq [Hz] Period [sec] Spectral Response S d 0 [m] Svo [m/s] S a 0 [m/s2] 13800 27600 4.01 0.249 4.86 0.206 0.114 2.55 72 0.076 1.74 57 25% Crack - 61.0 m. Dam on Flexible Foundation: Loma Prieta Scaled to 2.5g Foundation Modulus Ef First Mode Freq [Hz] Period [sec] Spectral Response Sd [m] Sv [m/s] Sa [m/s2] Amplification wrt 0% Sd/Sdo Sv/Svo Sa /Sa 0 13800 27600 3.14 0.318 3.79 0.264 0.160 3.54 194 0.120 2.98 109 1.400 1.389 2.706 1.578 1.710 1.914 50% Crack - 61.0 m. Dam on Flexible Foundation: Loma Prieta Scaled to 2.5g Foundation Modulus Ef First Mode Freq [Hz] Period [sec] Spectral Response Sd [m] Sv [m/s] Sa [m/s2] Amplification wrt 0% Sd/Sdo Sv/Svo Sa /Sa 0 13800 27600 1.97 0.508 2.32 0.431 0.213 4.45 170 0.193 5.09 218 1.867 1.749 2.379 2.538 2.920 3.829 Table 6.6. Spectral response of 61.0 m. cracked dam system on flexible foundation with Ef=27600 MPa and Ef=13800 MPa. 116 Section 6: Cracked Dam Response to Earthquake Excitation •x 5 a. •a re § 5 z 1 > o 3 9. 2 3 5 < a 1 4 < •o 3 E 2 o z 1 0 0.5 0.5 Sd-0% Sd-25% f Sd-50% n Ef= 27600 _ A _ E f = 13800 j . U A . ! / A | ' / \ - I i -I / \ ! ' M \ i ! 1.5 1.5 0.5 1 Uncracked Period [sec] 1.5 ! A I ! | I i A / S . • / \ I / \ / r W i / i / i J i • 9 i / i . / \ V / \ \ \ j . / I C ^ . D A A f •._ ' / \ Figure 6.12. Cracked spectra for 61.0 m. dam with 0%, 25%, and 50% crack on flexible foundation excited by Loma Prieta record scaled to 2.5 g with 5% damping. Similarly to the rigid foundation case, the spectral responses obtained from the rigorous HFTD are well approximated by the cracked spectra for relative displacement and relative velocity. The absolute acceleration is, however, underestimated by the cracked spectra. This is because the crack spectra does not account for the contribution of higher modes of which accelerations are sensitive. This limitation was illustrated earlier in the rigid foundation case. Therefore, with the exception of foundation damping, the consideration of a flexible foundation does not alter the assumptions and accuracy of the same cracked system on rigid foundation. 117 Section 6: Cracked Dam Response to Earthquake Excitation 6.5 Cracked Dam Response with Reservoir Effects Significant changes in the dam response are expected with the presence of a reservoir. Extensive studies have shown that the hydrodynamic pressures generated from the reservoir excited by the horizontal ground motions can cause a significant resonant behaviour in the response of a linear dam system (Chopra and Fenves, 1985). This is mainly due to the compressibility of water which results in strong resonant behaviour causing large hydrodynamic forces. It has also been shown that vertical components of the ground motion can also contribute a significant amount to the generation of hydrodynamic pressure resulting in an increase of dynamic response. If alluvium and silt accumulation on the reservoir floor are accounted for in the analysis, a portion of the dynamic pressure can be absorbed. In this investigation of the effects of hydrodynamic pressures on a cracked dam system having only horizontal ground motions will be considered, with no pressure absorption from silt layers. The 61.0 m. and 152.4 m. dam models with 0%, 25% and 50% crack lengths subjected to the Loma Prieta horizontal ground acceleration as described in Section 6.2.2 will be used in this study. The upstream reservoir level is assumed to be at 100% of the height of the dam and no tailwater will be present. The hydrodynamic effect was modelled with a 2-D wave equation as described in Section 3.3, and its effects were incorporated in the HFTD procedure. Again, the main focus will be in the comparative response of the crack dam system to that achieved by the uncracked dam. 118 Section 6: Cracked Dam Response to Earthquake Excitation 6.5.1 Result of 61.0 m. Dam-Reservoir System Subjected to Loma Prieta The relative displacement, velocity, and absolute acceleration time history response at the top of the 61.0 m. dam is given in Figure 6.13. As well, the VCOD time history at the crack mouth is also provided. As expected, the presence of a base crack amplifies the response in relation to the uncracked dam. However, a significant amplification in the 50% crack length dam system is evident. The maximum relative displacement, velocity, and absolute acceleration response are tabulated in Table 6.7. The peak response amplifications of the 50% cracked model are about 4 times greater than that found in the dry case. Sd [m] Sv [m/sec] Sa [m/sec2] 0% 0.073 1.95 62.2 25% 0.104 2.65 119.5 50% 0.308 6.92 265.9 Table 6.7. Spectral response of 61.0 m. (200 ft.) reservoir-dam system subjected to Loma Prieta 2.5g 119 Section 6: Cracked Dam Response to Earthquake Excitation 120 Section 6: Cracked Dam Response to Earthquake Excitation 0.15 0.10 g 0.05 o > 0.00 -0.05 i i H 50% - c / \ ).119 t \ i \ A ; 1 \ 1 \ \ ' 1 ' ! I ! ; >5% -0.023 , " / A A ; A A ; \ j 1 (d) 2.5 3.5 time [sec] 4.5 Figure 6.13. Dam response of 61.0 m. cracked dam system with reservoir subjected to Loma Prieta scaled 2.5g (reverse pass), a) Relative displacement, b) Relative velocity, c) Absolute acceleration at dam top, and d) VCOD at crack mouth. This large amplification can be attributed to the changes in the dynamic properties of the coupled reservoir-dam system. As mentioned in Section 2.1.1, aside from the hydrodynamic pressures applied, the 2-D wave model of the reservoir effectively increases the dynamic mass of the system and the stiffness is left unchanged. Thus, the natural period of the coupled system increases. A study conducted by Chopra (Chopra, 1968) concluded that the increase in the fundamental period of dam, with the presence of a full reservoir modelled by 2-D wave equation, is dependent on two factors: (1) the ratio of fundamental reservoir frequency to the fundamental frequency of the dry dam Q r = %J(ti h and (2) the ratio of reservoir depth to dam height. For practical purposes, the depth ratio in this study can be considered to be of unit value. The first fundamental period of the reservoir, is given by: Ar-7r where C = velocity of sound in water; taken to be 1440 m / s (6.3) H = height of reservoir 121 Section 6: Cracked Dam Response to Earthquake Excitation — 120 < a l i -CS o O 100 < H = height of reservoir Hs = height of dam X, = fundamental frequency of reservoir (Eqn. 6.3) co i = fundamental frequency of dry dam 4 0 f H/Hs = 1 INCOMPRESSIBLE WATER, H/Hs = 1 7 \ _ H/Hs * 1 2 3 Figure 6.14. Variation of fundamental period of coupled dam-reservoir system with the fundamental frequency ratio of the reservoir and the dam only (Chopra, 1968) Figure 6.14 shows the result of the parametric results in Chopra's studies. This figure illustrates the coupled natural period of the dam-reservoir system as it varies with the fundamental frequency ratio Q r . It suggests that for Q r < 2.0, the amplification of fundamental period of the coupled reservoir-dam system increases dramatically as Q r approaches 0. This is expected because as Q r becomes smaller, indicating that the dam is stiffer in comparison to the reservoir, the coupled frequency approaches the fundamental frequency of the dam alone, thus only a small decrease in the frequency. In terms of period, this represent a infinite increase in the fundamental period. Note that for systems with Q r > 2.0, indicating a flexible dam compared to the reservoir, the increase in the coupled period is independent of Q r and is given as a constant increase of 26%. For the 61.0 m. dam system with 0%, 25%, and 50% crack lengths, the effective system frequency (ESF) envelope, determined in Section 4.1.1.(from PSD), can be adjusted to account for the reservoir coupling effects by using Figure 6.14. The resulting coupled ESF envelope are given in Table 6.8. 122 Section 6: Cracked Dam Response to Earthquake Excitation Note that the coupled ESF envelope has decreased 61.0 m. (200") Dam-Reservoir System towards the dominant excitation frequencies of the Loma Prieta free-field motion (approximately 3.0 FIz) thus amplification of the response is expected. Moreover, the coupled ESF for the 50% cracked dam is expected to be within the range of 2.48 Hz < ESFeoupied < 4.83 Hz. With the applied Table 6.8. Fundamental frequency for 61.0 m. hydrostatic forces causing crack opening, it is more coupled cracked reservoir-dam system likely that the ESF of 50% crack length is closer to the lower end of the envelope of 2.48Hz.. This is very close to the dominant frequency of free-field excitation of 3 Hz where large amplifications are expected. The amplification of the response of the 25% crack length is also evident but the intensity is not as extreme as for the 50% crack length because the excitation frequency is outside the coupled ESF envelope for the 25% crack length system. Crack Length 0% 25% 50% 1st Modal Freq [Hz], 001 6.37 5.24 3.13 1st Reservoir Freq. [Hz], A r 6.21 6.21 6.21 Q =Ar/C0i 0.97 1.19 1.98 % Increase in T [Fig 6.14] 32% 28% 26% Coupled Freq [Hz], CDs 4.83 4.09 248 123 Section 6: Cracked Dam Response to Earthquake Excitation Chopra also clearly illustrates the effects of the hyarodynamic forces on the acceleration response. This is given in Figure 6.15. Here, the maximum horizontal acceleration at the dam top is plotted against the ratio of the excitation frequency, co, and the fundamental frequency, co b of the dry dam, given as Q. It is clear that the presence of the reservoir causes a shift in the resonance frequency of the dam system. More importantly, the resonant amplification is significantly greater than that of a dry dam. This is illustrated by the maximum acceleration attained in the dry dam case (curve 6) as compared to the full reservoir case (curve 1 through 4). Depending on Q r, a two-fold increase can be expected; however, it must be noted that this large amplification is restricted to a very narrow band of excitation frequencies. Therefore, for the hydrodynamic effects to cause a large amplification, the excitation frequency must be isolated within this narrow band. For excitation frequency ratios of f2> 1.5, the in influence of hydrodynamic pressures as little effect on the response. Figure 6.15. The influence of hydrodynamic pressure on the absolute value of horizontal dam top acceleration (Chopra, 1968). Q =GU/GU, 124 Section 6: Cracked Dam Response to Earthquake Excitation Another observation of the coupled reservoir-dam system is that the occurrences of crack opening have greatly increased over the dry dam case presented in Section 6.2. This is due to the hydrostatic pressure from the reservoir decreasing the compressive stresses along dam base near the upstream edge producing more favourable conditions for crack opening. This effect is similar to the application of the constant acceleration component to the white noise excitation as discussed in Section 4.3. Thus the magnitude of excitation required to opening a base cracking is significantly decreased. Figure 6.14 also reveals that for a more flexible dam, with excitation frequency ratio of Q,>2, the coupled reservoir-dam systems behaves similarly to that of a system considering reservoir fluid as incompressible (i.e. £2 r= °°). This is supported by the acceleration response given in Figure 6.15. As Q r —> °°, the acceleration response approaches that attained by the model considering incompressible fluid. Therefore, it is suggested that such a system can effectively be modelled using a constant added mass applied on to the upstream face. Therefore, the response of large dam sizes, especially with significantly long crack lengths in which the ESF is expected to be well removed from the dominant frequencies of excitation, (i.e. for Q, > 1.5), the response including hydrodynamic effects can be approximated by adding mass to increase the effective system period according to Figure 6.14. For such systems, the approximation made by the SDOF bi-linear procedure would be applicable. 125 Section 6: Cracked Dam Response to Earthquake Excitation 6.5.2 Result of 152.4 m. Dam-Reservoir System Subjected to Loma Prieta To test the theory previously outlined, a larger 152.4 m. dam-reservoir system will be studied. The same Loma Prieta free-field excitation scaled to 2.5g will be applied. The coupled ESF envelope and fundamental frequencies are given in Table 6.9. 152.4 m. (500') Dam-Reservoir System Crack Length 0% 25% 50% 1st Modal Freq [Hz], 00! 2.84 2.28 1.32 1st Reservoir Freq. [Hz], A r 2.48 2.48 2.48 Q =Ar/C01 2.19 1.09 1.88 % Increase in T [Fig 6.14] 40% 32% 26% Coupled Freq [Hz], 00s 2.03 1.73 1.05 Table 6.9. Fundamental frequency for 152.4 m. coupled cracked dam-reservoir systems The relative displacement, relative velocity, and absolute acceleration response of the 152.4 m. cracked and uncracked system are given in Figure 6.16. The peak response is provided in Table 6.10. Unlike the 61.0 m. dam in which the coupled 50% crack dam-reservoir system was subjected to large amplifications, the peak response of the 152.4 m. coupled system exhibits little amplification in displacement and velocity. Even for acceleration, only modest amounts of amplifications in comparision to the dry dam case are evident. 126 Section 6: Cracked Dam Response to Earthquake Excitation 1 00 0 80 0 60 0 40 C <U E 0 20 u (0 Q . 0 00 5 > -0 20 Rel -0 40 -0 60 -0 80 -1 00 / » . 50% - 0.90 5 ^ t ; / / \ tw iw tf \\\ V \ r" / A A A I ii ! / \ \ X/ V ' / \\ \ lAi - - A / V i i \\ A h i \ ': V. 1 \ J \ v V v; / \ i -Ji • ' 1 : \ \ i : \ V 1 M / w \ / \. | ' 1 1 \ 1 1 V/ • / \J > r 0% - 0 V/'--738 V ' y 2b% - 0.84b ! 2 2.5 3 3.5 4 4.5 5 T i m e [sec] /v. 0% - 10.49 / V /• i \ \ t A p 1 I; \ \\ I I t ! * / :\ \ i 1 / •' \ \ \y/A\\ /•' ^ '•' • / ' Hi.' / •> / / \ ; \ j \ 1 1 r \ \ 1 i \ / ' M t- / J < \ H W 10 0 50% - 1 0.76 ^ E 3.5 Tim e [sec] 2 2.5 3 3.5 4 4.5 5 Tim e [sec] Figure 6.16. Dam top response of 152.4 m. cracked dam system with reservoir subjected to Loma Prieta scaled to 2.5g. a) Relative displacement, b) relative velocity, and c) absolute acceleration for dam with 0%, 25%, and 50% crack lengths. 127 Section 6: Cracked Dam Response to Earthquake Excitation The reason for this lack of amplification is mainly attributed to the excitation frequency ratios of the coupled ESF of the cracked reservoir-dam system. From the estimated coupled ESF envelope, all crack conditions are well removed from the dominate frequency of excitation of 3 Hz. Frequency ratios ranging from £2=1.22 for uncracked conditions to £2=2.36 for a 50% crack, suggest that the dam systems are too flexible to be affected by the hydrodynamic pressures and the dominant frequency of excitation. As mentioned earlier, for excitation frequency ratios £2 >1.5, the dam systems with and without hydrodynamic effects yield similar responses. Therefore, for the 152.4 m. dam systems, the amplifications that are exhibited by the response time histories are from the nonlinear crack behavior rather than the hydrodynamic effect. This suggests that for reservoir-dam systems with coupled ESF yielding excitation frequency ratios Q> 1.5, the amplification of the cracked system is unaffected by the hydrodynamic forces. Therefore, the reservoir effects can be replaced by applying hydrostatic forces and reducing the ESF of the dry dam system to the levels given in Figure 6.14. This can be verified by applying the cracked spectra approximation to estimate the dynamic response of such a reservoir-dam system. Recall that the SDOF bi-linear system does not directly account for the hydrodynamic forces; therefore, if the above conclusion is accurate, the SDOF approximation should produce reasonable results. In order to compare the dynamic response, the SDOF bi-linear system parameters must be equated to the coupled fundamental frequency of the uncracked and cracked dam system as outlined in Table 6.10, and the static displacement component of the response removed. The maximum response for the 152.4 m. reservoir-dam 128 Section 6: Cracked Dam Response to Earthquake Excitation system is tabulated below along with the static displacement and amplification of dynamic response with respect to the uncracked condition. % Cracked Coupled Uncracked Freq [Hz] Coupled Cracked Freq. [Hz] S d [m] S d static [m] Sdoynamio [m] S v [m/s] S a [m/s2] S d amplifi-cation S v amplifi-cation S a amplifi-cation 0% 2.03 2.03 0.738 0.0109 0.726 10.5 188.4 - - -25% 2.03 1.73 0.845 0.0109 0.832 10.0 272.3 1.147 0.953 1.445 50% 2.03 1.05 0.905 0.0109 0.893 10.8 282.3 1.231 1.026 1.498 Table 6.10. Maximum response of 152.4 m. reservoir-dam system subjected to Loma Prieta for 2.5g obtained from HFTD analysis (reverse pass). Figure 6.17 shows the amplified spectral response ordinate for the 152.4 m. reservoir-dam system plotted on the approximate Loma Prieta cracked spectra. The spectral ordinate is plotted against the uncracked coupled frequency of the dam system. As expected, for a flexible dam system where Q > 1.5, the hydrodynamic effect produces only negligible hydrodynamic forces. Therefore, the amplifications predicted by the SDOF bi-linear approximation are still applicable when they are based on the coupled reservoir-dam periods for systems with excitation frequency ratios Q >1.5. Stiffer dam systems with excitation frequency ratios Q <1.5, are susceptible to possible resonant behavior and significant hydrodynamic effects. In such cases, the approximate method is not applicable and a full analysis is recommended. 129 Section 6: Cracked Dam Response to Earthquake Excitation „ 5 a 4 Q. w 5 at = 3 JO a> / \ I \' \ \ \ ^ Sd-0% Sd-25% — — Sd-50% ' A \ 50% ' / ' ^ / / 25% 0% / / / / / J 0 0 5 1 5 2 I 3 1 \ 1 '{ t i i ' : K 1 1 A 25%,50% ' 0% i ' / i ' / /i —• -Sr^* e__-(a) (b) 0.5 1.5 5 5 I 4 13 5 4 o> u a 3 *-> ^ 3 tn < \ 1 ' \ ' A 1 / 50% ( ' A ' 25% i / A-' 1 I V s 1 / 0% I <j \ \ A ^ v \ V v V ^ ~ - — — (c) 0.5 1.5 Figure 6,17. Approximate cracked spectra with amplified 152.4 m. reservoir-dam spectral ordinate for Loma Prieta scaled to 2.5g. a) Normalized relative displacement, b) normalized relative velocity, and c) normalized absolute acceleration for systems with 0%, 25%, and 50% crack lengths. 130 Section 6: Cracked Dam Response to Earthquake Excitation Note that the conclusions drawn here for the reservoir effects are based on only two different reservoir-dam systems, namely 61.0 m. and 152.4 m high dams. Without further confirmation, the conclusions drawn here must be accepted with caution. However, these two dam systems are indicative of the two regimes expected to be encountered for most dam dimensions. Therefore, the general guidelines presented here provide an adequate means to make a sound decision as to which analysis procedure to undertake. 131 Section 6: Cracked Dam Response to Earthquake Excitation 6.6 Chapter Summary This section analyzed the response of cracked dams of various dam heights subjected to earthquake excitation. The effects of a flexible foundation and reservoir were also independently investigated. Crack lengths of 25% and 50% of each dam size were compared to the response of the uncracked dam condition. The peak displacement, velocity, and acceleration amplification with respect to the uncracked conditions were compared to those obtained by the SDOF bi-linear approximate method in the form of a cracked spectra. The findings and conclusions are summarized below: Cracked Dam on Rigid Foundation and No Reservoir • For dam systems with crack lengths less than 25%, the displacement, velocity, and acceleration response do not differ greatly from the uncracked dam system. For cracked lengths in excess of 25%, amplifications in response and dramatic changes in dynamic properties are evident. For systems with ESF envelopes encompassing the dominant frequencies of excitation, maximum amplifications of 3 to 4 times that of the maximum response attained from the uncracked case can be expected. For systems with an ESF envelope that are well removed from the dominant frequencies of excitation, little or no amplifications can be expected in the maximum displacement and velocity response as compared to the uncracked condition. This is applicable for both 25% and 50% crack lengths. Because of the contributions of the higher modes, general behaviour regarding the maximum absolute acceleration for the cracked systems cannot be made. 132 Section 6: Cracked Dam Response to Earthquake Excitation • Unlike the steady state response, the maximum displacement attained in a cracked system does not necessarily occur within the crack open phase of the response. Maximum acceleration, however, usually occurs within the crack close phase of the response. • For flexible cracked systems, the maximum response occurs immediately after the initial opening of the crack. Subsequent responses of the cracked system exhibit a de-amplification which is indicative of systems with material non-linearity after reaching first yield. • The use of the SDOF bi-linear stiffness method to generate a cracked spectra for estimating the amplification of the cracked response as compared to the uncracked response shows an agreement between the relative displacement and relative velocity. However, the approximate method underestimates the amplification of the maximum absolute acceleration experienced in the cracked system, because the contributions of the higher modes are ignored. Regardless, the cracked spectra from the SDOF bi-linear approximate method can be used as an efficient screening tool for potential candidates requiring detailed analysis performed by the HFTD procedure. • The approximate cracked spectra predicted possible de-amplification in the San Fernando record for maximum relative velocity and absolute acceleration for a 50% crack case as compared to the maximum uncracked response. However, there was no evidence of de-amplification of the maximum response from the HFTD analysis. Therefore, it is recommended that any de-amplification indicated by the approximate response be disregarded 133 Section 6: Cracked Dam Response to Earthquake Excitation and the maximum estimated value of the cracked response be equated to the maximum response of the uncracked system. Flexible Foundation and No Reservoir. • The findings outlined for a rigid foundation also apply to flexible foundations. The only marked difference is the added flexibility from the foundation stiffness and the added hysteretic damping that may decrease the tendency of crack opening, thus favouring uncracked response. • The added foundation flexibility reduced the ESF envelope of the cracked dam system. The maximum response amplifications can be estimated by the approximate SDOF bi-linear method using the modal properties of the dam-foundation system. The limitations of the approximate method also applies to the dam-foundation system. Rigid Foundation and Full Reservoir. • The 2-D wave equation and the applied hydrostatic force on the upstream face of the dam represents the effects of the reservoir-dam system. The 2-D wave equation which represents the hydrodynamic effect can be summarized as adding an additional mass, damping, and force applied onto the dam system. The added mass causes a decrease in the modal frequency of the coupled dam-reservoir system. This will effectively decrease the ESF envelope of the cracked dam-reservoir system. This decrease can be estimated by the guidelines given by Figure 6.14. The hydrodynamic pressure increases the response of the coupled system near the resonance 134 Section 6: Cracked Dam Response to Earthquake Excitation frequency of the coupled system. At frequencies well removed from the resonance frequency, Q > 1.5, the hydrodynamic force generated is negligible. • The hydrostatic pressures along the upstream face of the dam decreases the compressive stresses along the heel of the dam, thus creating a more favourable situation of crack opening. Therefore, the response of the cracked dam system tends to operate closer towards the low frequency end of the coupled ESF envelope. In addition, the hydrostatic forces facilitate crack opening and the requirement for a strong motion excitation is relaxed. Moderate magnitude excitation should be adequate in causing sufficient crack opening events. For the 61.0 m. (200 ft.) dam section studied, significant amounts of amplification is evident for the case of a 50% crack. This is due to the resonance of the coupled cracked reservoir-dam system with the dominant frequency of the applied Loma Prieta record. Possible occurrence of resonance can be identified by comparing the frequency ratio of the coupled ESF envelope to the dominant frequency of the excitation. If the frequency ratio is near CI =1, then large amplification in the response caused by the increase hydrodynamic pressures is possible. In such a situation, the approximate SDOF bi-linear method cannot be applied since the hydrodynamic force cannot be represented. If the frequency ratios are greater than Q >1.5, then the amplification is not expected to be large since the reservoir can now be considered incompressible and the hydrodynamic pressures are no longer significant. The SDOF approximation can be used in such a situation since the added mass can be directly accounted for. 135 Section 7: Conclusions and Recommendations 7. Conclusions and Recommendations A study of the dynamic performance of a gravity dam section with an assumed horizontal crack along the dam foundation interface of fixed length has been conducted using the Hybrid Frequency Time Domain (HFTD) analysis procedure. The 2-D dam section was idealized as an assemblage of finite elements set on a flexible foundation which was modelled as a visco-elastic halfplane. Hydrodynamic effects of the reservoir were represented by a 2-D wave equation, including effects of water compressibility. The nonlinear analysis of the crack opening and closing behaviour was performed by the HFTD procedure, which allows for dynamic frequency dependent properties of the foundation and reservoir representations to be maintained. The effective system frequency and dynamic response of the cracked dam system were compared to that of the uncracked dam system allowing for a better understanding of the effects caused by the presence and extent of a base crack. For linear systems, the effective system frequency (ESF) represents the natural frequency of the dynamic system. For nonlinear systems, such as the cracked dam, the ESF represents the frequency at which resonance of the nonlinear system is expected to occur. Unlike the linear system, it is not a definitive system property and it is dependent on the magnitude and nature of the excitation. For the cracked dam system, it is more appropriate to identify the system by an ESF envelope bounded by the first uncracked dam natural frequency and the first cracked pseudolinear dam natural frequency. It is within this range of frequencies that the effective cracked system is expected to operate. ESF for the cracked dam system was determined by two methods: (1) 136 Section 7: Conclusions and Recommendations spectral analysis of the response to a white noise excitation, and (2) induction of resonance through sine sweep technique. For crack lengths less than 25% of the dam base, the ESF range is narrow and does not significantly degrade from the uncracked natural frequency. For crack lengths beyond 25%, the ESF envelope broadens dramatically and the ESF falls between the uncracked natural frequency and 50% of the uncracked natural frequency. The effects of the increasing ESF envelope is not only evident in the frequency of vibration, but also the global response of the system. During the sine sweep study, a formation of a secondary resonance in the steady state portion of the response was evident at a frequency far removed from the expected primary and secondary ESF envelope of that system. The secondary resonance is characterized by a sudden increase in the maximum steady state displacement response and a reduction in the frequency of repeating oscillation by a factor of two. The maximum steady state acceleration is unaffected. This phenomena also occurs in simple nonlinear SDOF systems with abrupt changes in stiffness. The cause of the phenomena cannot be explained and warrants further study. However, it is speculated that this phenomena will not affect the response to a random excitation, such as an earthquake record, since a steady state response is unlikely to occur in such an environment. To study the response behaviour of cracked dam systems, dam sections of various sizes with crack lengths of 0%, 25%, and 50% of the base length were subjected to two different earthquake motions. Because of the fixed direction of the crack, the excitation record was applied 137 Section 7: Conclusions and Recommendations in two different directions (forwards and backwards) such that the maximum response could be attained. The maximum relative displacement, relative velocity, and absolute acceleration response at the dam top for each dam size and crack length were plotted against the fundamental uncracked natural frequency to form a cracked spectra. For the case of a dam section on rigid foundation and no reservoir, crack lengths of 25% or less produced only nominal amplification in the maximum cracked response. An increase of not more than 50% of that achieved by the uncracked response was evident in the result of this study. Conversely, crack lengths exceeding 25% produced amplifications of the uncracked response of as much as 200%. This is due to the increase in flexibility of the crack opening phase driving the ESF towards the dominant frequency of the excitation. However, for dam systems which are flexible relative to the excitation frequency, little or no amplification of the cracked response was evident for all crack lengths except for maximum absolute acceleration where resonance of the higher frequencies are possible. Besides the decrease in the values of the ESF envelope, the effects of flexible foundation with no reservoir did not cause a marked difference in the response behaviour of the cracked dam from that of the case of a rigid foundation. The increased flexibility of the foundation, however, may cause an increase in the tendency for the upper and lower crack surfaces to remain in contact. Therefore, the likelihood for the cracked dam system to operate in its uncracked phase is increased forcing the ESF to be closer towards the uncracked natural frequency of vibration. Also, the 138 Section 7: Conclusions and Recommendations magnitude of the excitation that is required to cause the crack open will increase with increasing foundation flexibility. When the reservoir effects were included in the analysis of the cracked dam system on a rigid foundation, significant changes in the response were noted. The hydrodynamic effects represented by the 2-D wave equation effectively increases the dynamic mass and damping of the dam system. The increase in mass decreased the ESF of the coupled dam-reservoir system. This can be estimated by applying methods outlined by Chopra (Chopra, 1968). If the excitation frequency is close to the coupled dam-reservoir ESF, large hydrodynamic forces can develop causing large amplifications in the cracked dam response. In the 61.0 m. (200 ft.) 50% crack dam section, amplifications as large as 400% of the uncracked response were observed. For dam-reservoir systems with coupled ESF that are much more flexible than the dominant excitation frequency such that the frequency ratio is fi>1.5, the effects of the hydrodynamic force is no longer of concern. Effectively, the coupled dam-reservoir system can be treated as a dry cracked dam with the increased mass and damping properties. In all cases studied, the maximum response was used to calibrate and verify the responses obtained from a SDOF bi-linear approximate method. This approximation represents the cracked dam system as a SDOF system with a bi-linear varying stiffness. The mass and stiffness values of this simplified system are selected to represent both the first uncracked natural frequency and the first cracked natural frequency of the pseudolinear cracked dam system. The nonlinear SDOF system was solved using a time-step integration technique. Results of this simplified method 139 Section 7: Conclusions and Recommendations produced approximate cracked spectra, and the amplification of the cracked spectral response compared to the uncracked condition from the HFTD analysis show that the maximum relative displacement and velocity amplifications were very accurate. As expected, absolute accelerations were underestimated by the approximate cracked spectra since the SDOF bi-linear stiffness approach neglects the contributions of higher modes. Any de-amplification indicated by the approximate cracked spectra should be viewed with caution since de-amplification was never exhibited in the HFTD results. Therefore it is recommended that the peak cracked spectral response estimated by the cracked spectra should not be less than the value indicated for the uncracked condition. Nonetheless, the SDOF bi-linear stiffness approximation provides a quick and efficient representation of the response of a cracked dam system. The generalization made in this study for the cracked dam system can provide valuable guidelines in assessing the stability of dams with known existing cracks or dams with damage incurred in a previous earthquake. The two procedures outlined in this report can provide two different levels of analysis. The use of the SDOF bi-linear stiflhess approximate method can be used as a screening process for potential candidates for a more detailed nonlinear analysis. If a comprehensive nonlinear analysis is required, then a more rigorous analysis involving the use of the HFTD procedure can be applied. Even with these tools, the findings of this study are not definitive. Many assumptions were made, and factors which were beyond the scope of this study may have a profound effect on the 140 Section 7: Conclusions and Recommendations response of the cracked dam system. The following is a list of topics which are recommended for future research. • Local Response: The scope outlined for this study was limited to the global response effects and excluded local effects such as stress and strain near the crack tip. High stresses near the crack tip may suggest propagation of the crack. Consideration of these effects would involve fracture mechanic principles and a much more refined and detailed finite element modelling. • Vertical Earthquake Motions: Vertical motions can reduce or increase the effect of gravitational forces of the concrete mass on crack response. In addition, previous research (Chopra, 1962) has indicated that the vertical motions of earthquake excitation can generate significant hydrodynamic pressures which are much larger than that generated by horizontal motions. Therefore, the effects of the vertical components of excitation may alter some of the conclusions of the crack dam behaviour made in this study. • Reservoir Effects on the Cracked Dam Response: Due to the time constraint of this study, cracked dam response with the effects of the reservoir were limited. Future research can include a more expanded study, including effects of different reservoir levels, presence of tailwater, and water pressure uplift effects. • Base Sliding Stability with Reduced Base Length: With the crack open, the effective base of the dam is reduced. Stability issues, such as sliding factor of safety and stresses at the crack tip 141 Section 7: Conclusions and Recommendations causing further crack lengthening, were not addressed in this study. Non-linear base sliding of gravity dams have been recently analysed by Chavez and Fenves (Chavez and Fenves, 1994) using an HFTD approach. This base sliding component can be easily incorporated into the base crack formulation used in this study. • Effects of Moderate Magnitude Excitations: All excitations used in this study were scaled to a peak ground acceleration of 2.5g or 5g, such that crack opening was ensured. Realistically, earthquake excitation is more commonly experienced in the order of 0.5g to lg. This raises questions of the threshold magnitude required to cause crack opening, as well as the amplifications, if any at all, expected with moderate magnitude excitations. • Hydrodynamic Pressures in Crack: When the crack opens during the dynamic response water may rush in. This would tend to keep the crack open and would tend to produce an increase in hydrostatic pressures within the crack as the crack attempts to close. Currently, there exists a great debate as to the existence of such a behaviour. One school of though believes that this build-up in hydrostatic pressure would be detrimental to the base stability of the dam. The opposing group argues that the crack widths are much to small to facilitate the intrusion of water and if water does rush into the cracks, the negative pressures developed when the crack opens would eliminate concerns of instability. In any case, if large pressures do exist within the crack interface, the response of the cracked dam system may be highly dependent on the magnitudes and dissipation of these pressures. 142 Section 7: Conclusions and Recommendations • Local Damping Effects at the Crack: The damping specified in this study was approximated as a global viscous damping. Realistically, the damage caused by the concrete cracking suggests that a high concentration of damping originates around this damage zone. This local damping principle has been applied to cracking models used in dam analysis (Barrett, Foadian, Rashid, 1991). The increased damping surrounding the crack can also be applied to the HFTD procedure for a fixed crack by assuming an increased damping factor when the crack is closed, and a reduced damping factor when the crack is open. • Efficiency of HFTD Procedure as applied to Cracked Dam Systems. Efforts should be made to increase the computational efficiency of the HFTD procedure. As suggested in Section 3.5.4, the application of the restoring pair can be calculated to enforce zero velocity as well as induce the required displacement at the crack nodes. As well, the use of corrective response to reduce aliasing errors and hence reduce the overall length of analysis should be explored. Both of these techniques may decrease the iteration required in the nonlinear analysis. 143 List of References Bracewell, Ronald N . , The Fourier Transform and its Application., McGraw-Hill Inc., New York, U.S.A., 1978. Chavez, J.W. and Fenves, G.L., "Earthquake Analysis And Response Of Concrete Gravity Dams Including Base Sliding", Report No. UCB/EERC-93/07, Earthquake Engineering Research Center, University of California, Berkeley, CA, 1993 Chakrabarti, P. and Chopra, A.K. , "Earthquake Response Of Gravity Dams Including Reservoir Interaction Effects," Report No. EERC 72-6, Earthquake Engineering Research Center, University of California, Berkeley, Calif, Dec. 1972. Chakrabarti, P. and Chopra, A.K. , "Hydrodynamic Effects In Earthquake Response Of Gravity Dams", Journal of the Structural Division, ASCE, Vol. 100, No. ST6, June, 1974 Chen, B., and Hung, T., "Dynamic Pressure Of Water And Sediment On Rigid Dam", Journal of Engineering Mechanics, ASCE, Vol.119, No. 21, 1990 Chopra, A.K. , "Earthquake Behavior Of Reservoir-Dam Systems", Journal of Engineering Mechanics Division, ASCE, Vol. 94, No. EM6, Dec, 1968, pp. 1475-1500. Chopra, A .K. , Chakrabarti, P., and Dasgupta, G. "Frequency Dependent Stiffness Matrices for Viscoelatic Halfplane Foundations," Report No. EERC 75-22, Earthquake Engineering Research Center, University of California, Berkeley, Calif, 1975. Clough, R.W., and Penzien, P. Dynamics of Structures. McGraw Hill, New York, U.S.A., 1975 Cook, R.D., Malkus, D.S., and Plesha, M.E. , Concepts and Applications of Finite Element Analysis. 3rd Ed., John Wiley & Sons, U.S.A., 1989 Darbre, G.R., "Seismic Analysis Of Non-Linearly Base-Isolated Soil-Structure Interacting Reactor Building By Way Of The Hybrid Frequency Time Domain Procedure", Earthquake Engineering and Structural Dynamics, 19,725-738, (1990) Darbre, G.R., "Application Of The Hybrid Frequency Time Domain Procedure To The Soil-Structure Analysis Of A Shear Building With Multi Nonlinearities", 5th Int. Conf. Soil Dynamics Earthquake Eng. Karlsruthe., (1991) 144 List of References Darbre, G.R. "Nonlinear Reservoir-Dam Interaction By Way Of The Hybrid Frequency-Time Domain Procedure", 2nd European Conference on Structural Dynamic, Trondheim, (1993) Darbre, G.R. and Wolf, J.P., "Criterion Of Stability And Implementation Issues Of Hybrid Frequency-Time-Domain- Procedure For Non-Linear Dynamic Analysis", Earthquake Engineering and Structural Dynamics, 16, 569-581 (1988) Dewey, R., Reich, R.W., and Saouma, V.E. , "Uplift Modelling for Fracture Mechanics Analysis of Concrete Gravity Dams", ASCE Structural Division (Draft), 1993. El-Aisi, B, and Hall, J.F., "Non-linear Earthquake Response of Concrete Gravity Dams Part 1: Modelling", Earthquake Engineering and Structural Dynamics, 18, 837-851 (1989) El-Aisi, B, and Hall, J.F., "Non-linear Earthquake Response of Concrete Gravity Dams Part 2: Behavior"Earthquake Engineering and Structural Dynamics, 18, 853-865 (1989) Fenves, G L . and Chavez, J.W., "Hybrid Frequency Time Domain Analysis Of Nonlinear Fluid-Structure Systems", Proceed. 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