Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Wakjira, Tadesse Gemeda
Doctor of Philosophy - PhD
University of British Columbia
This doctoral dissertation presents a novel framework for sustainable design, structural modelling, and performance-based design of ultra-high-performance concrete (UHPC) utilizing state-of-the-art artificial intelligence techniques. Initially, a novel framework for strength prediction and multi-objective optimization (MOO) of UHPC mixture was introduced using advanced machine learning (ML) and MOO algorithms. A total of 19 objective functions are considered, including cost, uniaxial compressive strength, and 17 environmental impact metrics that comprehensively evaluate the environmental sustainability of the UHPC mixture. Addressing the need for an accurate stress-strain constitutive model of confined UHPC, this dissertation introduces a hybrid ML model, augmented by a state-of-the-art conditional tabular generative adversarial network and Optuna to accurately predict the peak and ultimate stress-strain responses of UHPC confined with normal strength steel (NSS) or high-strength steel (HSS) reinforcement. Furthermore, hybrid ML-based predictive models for a complete stress-strain response curve of UHPC confined with NSS or HSS spirals are proposed. Besides, analytical equations are developed to predict the peak and ultimate stress-strain responses of confined UHPC. Utilizing these models and the uncertainties of the optimized UHPC along with other sources of uncertainties, drift capacity limit states are proposed for UHPC columns based on explainable ML models across four different damage states. The efficacy of the developed model is assessed using a range of statistical metrics, including the composite fitness score, which demonstrated a high predictive accuracy of the drift limit states. Moreover, the application of the proposed drift limit states in the performance-based design of UHPC columns is investigated. Lastly, the dissertation introduces a multivariate seismic fragility assessment approach for UHPC bridges using a hybrid ML model.
ADVANCED AI-POWERED COMPREHENSIVE FRAMEWORK FOR MODELLING, ANALYSIS, ASSESSMENT, AND PERFORMANCE-BASED DESIGN OF ULTRA-HIGH-PERFORMANCE CONCRETE ELEMENTS by Tadesse Gemeda Wakjira A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE COLLEGE OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) April 2024 © Tadesse Gemeda Wakjira, 2024 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, the dissertation entitled: Advanced AI-Powered Comprehensive Framework for Modelling, Analysis, Assessment, and Performance-Based Design of Ultra-High-Performance Concrete Elements submitted by Tadesse Gemeda Wakjira in partial fulfillment of the requirements of the degree of Doctor of Philosophy. Dr. M. Shahria Alam, School of Engineering Supervisor Dr. Rehan Sadiq, School of Engineering Supervisory Committee Member Dr. Abbas Milani, School of Engineering Supervisory Committee Member Dr. Mohammad H. Zarifi, School of Engineering University Examiner Dr. Mohamed A. Moustafa, University of Nevada, Reno External Examiner iii ABSTRACT This doctoral dissertation presents a novel framework for sustainable design, structural modelling, and performance-based design of ultra-high-performance concrete (UHPC) utilizing state-of-the-art artificial intelligence techniques. Initially, a novel framework for strength prediction and multi-objective optimization (MOO) of UHPC mixture was introduced using advanced machine learning (ML) and MOO algorithms. A total of 19 objective functions are considered, including cost, uniaxial compressive strength, and 17 environmental impact metrics that comprehensively evaluate the environmental sustainability of the UHPC mixture. Addressing the need for an accurate stress-strain constitutive model of confined UHPC, this dissertation introduces a hybrid ML model, augmented by a state-of-the-art conditional tabular generative adversarial network and Optuna to accurately predict the peak and ultimate stress-strain responses of UHPC confined with normal strength steel (NSS) or high-strength steel (HSS) reinforcement. Furthermore, hybrid ML-based predictive models for a complete stress-strain response curve of UHPC confined with NSS or HSS spirals are proposed. Besides, analytical equations are developed to predict the peak and ultimate stress-strain responses of confined UHPC. Utilizing these models and the uncertainties of the optimized UHPC along with other sources of uncertainties, drift capacity limit states are proposed for UHPC columns based on explainable ML models across four different damage states. The efficacy of the developed model is assessed using a range of statistical metrics, including the composite fitness score, which demonstrated a high predictive accuracy of the drift limit states. Moreover, the application of the proposed drift limit states in the performance-based design of UHPC columns is investigated. Lastly, the dissertation introduces a multivariate seismic fragility assessment approach for UHPC bridges using a hybrid ML model. iv LAY SUMMARY This doctoral dissertation presents a novel method for designing structures with a focus on earthquake resistance using a special kind of concrete referred to as ultra-high-performance concrete (UHPC). The study employs advanced artificial intelligence and machine learning algorithms to predict the strength of this concrete and optimize its composition in a cost-effective and eco-friendly way. A key part of the research involves developing a model to understand how UHPC reinforced with different types of steel behaves. This model helps in designing structures that are safer and more resilient to earthquakes. This research is not only a significant step forward in building safer structures but also balances cost and environmental sustainability, which makes it a holistic approach to modern construction challenges. v PREFACE The research in this dissertation is conducted by Tadesse Gemeda Wakjira at the University of British Columbia under the supervision of Dr. M. Shahria Alam. Various sections of this dissertation have been published or are submitted for publication, as detailed below. In each case, Wakjira took on the primary role of conceptualization, data collection, visualizations, formal analysis, software, writing, and composition of the papers. Dr. M. Shahria Alam supervised the research and reviewed/edited the papers. Journal papers 1. A part of Chapter 2 has been presented in the following publication: â¢ Wakjira T., Kutty A., Alam MS. (2024). A novel framework for developing environmentally sustainable and cost-effective ultra-high-performance concrete (UHPC) using advanced machine learning and multi-objective optimization techniques. Construction and Building Materials, Elsevier, Vol. 416, 135114, https://doi.org/10.1016/j.conbuildmat.2024.135114. 2. A part of Chapter 3 has been presented in the following publication: â¢ Wakjira T., Alam MS. (2024). Peak and ultimate stress-strain model of confined ultra-high-performance concrete (UHPC) using hybrid machine learning model with conditional tabular generative adversarial network. Applied Soft Computing, Elsevier, Vol. 154, 111353, https://doi.org/10.1016/j.asoc.2024.111353. 3. A part of Chapter 4 has been presented in the following publication: â¢ Wakjira T., Abushanab A., Alam MS. (2024). Hybrid machine learning model and predictive equations for compressive stress-strain constitutive vi modelling of confined ultra-high-performance concrete (UHPC) with normal-strength steel and high-strength steel spirals. Engineering Structures, Elsevier, Vol. 304, 117633, https://doi.org/10.1016/j.engstruct.2024.117633. 4. A part of Chapter 5 has been presented in the following paper: â¢ Wakjira T. and Alam MS. (2024). Performance-based seismic design of ultra-high-performance concrete (UHPC) bridge columns with design example â powered by explainable machine learning model. Engineering Structures, Elsevier [Under review]. 5. A part of Chapter 6 will be presented in the following paper: â¢ Wakjira T. and Alam MS. (2024). Multivariate fragility assessment of ultra-high-performance concrete (UHPC) bridges using hybrid machine learning model. Engineering Structures, Elsevier. Presentations 1. A part of Chapter 2 is presented at the 2023 Fall ACI Concrete Convention in Boston, MA, USA (October 2023) during the session titled Innovative Applications of UHPC in Columns, Part 2 of 2. Presentation title: Development of Eco-Friendly and Cost-Effective Ultra-High-Performance Concrete (UHPC): Leveraging Metaheuristic Optimization Algorithms and Machine Learning in Multi-Objective Design 2. A part of Chapter 3 is presented at the 2023 Fall ACI Concrete Convention in Boston, MA, USA (October 2023) during the session titled: Innovative Applications of UHPC in Columns, Part 1 of 2. Presentation title: Modelling of Peak and Ultimate Stress-Strain States in Confined Ultra-High-Performance Concrete (UHPC) Using Novel Hybrid Machine Learning Approach with Conditional Tabular Generative Adversarial Network vii TABLE OF CONTENTS ABSTRACT ................................................................................................................. iii LAY SUMMARY ......................................................................................................... iv PREFACE ...................................................................................................................... v LIST OF TABLES ....................................................................................................... xii LIST OF FIGURES .................................................................................................... xiv ACKNOWLEDGEMENTS ..................................................................................... xviii DEDICATION ............................................................................................................ xix CHAPTER 1: INTRODUCTION .................................................................................. 1 1.1. Emergence of UHPC as an Advanced Material ................................... 1 1.2. UHPC Mixture Strength Prediction ..................................................... 1 1.3. Sustainability Implications and MOO of UHPC Mixture .................... 3 1.4. UHPC in Structural Applications and Confinement Models ............... 5 1.5. Performance-Based Seismic Design of UHPC Structures ................... 7 1.6. Research Objectives ............................................................................. 9 1.7. Dissertation Organization ................................................................... 10 CHAPTER 2: A NOVEL FRAMEWORK FOR DEVELOPING ENVIRONMENTALLY SUSTAINABLE AND COST-EFFECTIVE ULTRA-HIGH-PERFORMANCE CONCRETE (UHPC) USING ADVANCED MACHINE LEARNING AND MULTI-OBJECTIVE OPTIMIZATION TECHNIQUES ........... 12 2.1. Introduction ........................................................................................ 12 2.2. Methodology ...................................................................................... 17 viii 2.2.1. Research data .............................................................................. 18 2.2.2. Overview of machine learning models ........................................ 20 2.2.3. Tuning hyperparameters.............................................................. 24 2.2.4. ML performance measures .......................................................... 26 2.2.5. Multi-objective optimization ........................................................ 27 2.2.6. Approaches to decision-making in multi-objective optimization scenarios ................................................................................................... 33 2.3. Results and Discussion ....................................................................... 35 2.3.1. Predictive performance of the models ......................................... 35 2.3.2. Multi-objective optimization results ............................................ 41 2.3.3. FAI-OSUSCONCRET: Fast, accurate, and intelligent prediction tool and multi-objective optimization of sustainable UHPC .................... 44 2.4. Summary ............................................................................................ 46 CHAPTER 3: PEAK AND ULTIMATE STRESS-STRAIN MODEL OF CONFINED ULTRA-HIGH-PERFORMANCE CONCRETE (UHPC) USING HYBRID MACHINE LEARNING MODEL WITH CONDITIONAL TABULAR GENERATIVE ADVERSARIAL NETWORK .......................................................... 49 3.1. Introduction ........................................................................................ 49 3.2. Methodology ...................................................................................... 52 3.2.1. Data Collection............................................................................ 52 3.2.2. Conditional tabular generative adversarial network .................. 54 3.2.3. Hybrid machine learning model .................................................. 56 ix 3.2.4. Hyperparameter optimization...................................................... 59 3.3. Results and Discussion ....................................................................... 60 3.3.1. Optuna-optimized CTGAN sampling for confined UHPC .......... 60 3.3.2. Optuna-optimized, CTGAN-augmented hybrid ML model for accurate stress-strain responses prediction of confined UHPC ............... 65 3.4. Summary ............................................................................................ 77 CHAPTER 4: HYBRID MACHINE LEARNING MODEL AND PREDICTIVE EQUATIONS FOR COMPRESSIVE STRESS-STRAIN CONSTITUTIVE MODELLING OF CONFINED ULTRA-HIGH-PERFORMANCE CONCRETE (UHPC) WITH NORMAL-STRENGTH STEEL AND HIGH-STRENGTH STEEL SPIRALS...................................................................................................................... 80 4.1. Introduction ........................................................................................ 80 4.2. Methodology ...................................................................................... 83 4.2.1. Numerical model.......................................................................... 84 4.2.2. Development and pre-processing of confined UHPC stress-strain response database ..................................................................................... 94 4.2.3. Hybrid ML model......................................................................... 97 4.3. Results and Discussion ....................................................................... 98 4.3.1. Results of hyperparameter optimization ...................................... 98 4.3.2. Stress-strain response prediction results ................................... 101 4.4. Predictive Equations ......................................................................... 108 4.5. Summary .......................................................................................... 113 x CHAPTER 5: PERFORMANCE-BASED SEISMIC DESIGN OF ULTRA-HIGH-PERFORMANCE CONCRETE (UHPC) BRIDGE COLUMNS WITH DESIGN EXAMPLE â POWERED BY EXPLAINABLE MACHINE LEARNING MODEL.................................................................................................................................... 117 5.1. Introduction ...................................................................................... 117 5.2. Methodology .................................................................................... 121 5.2.1. Definition of damage limit states ............................................... 122 5.2.2. Bridge pier database ................................................................. 124 5.2.3. Development of ML powered drift capacity limit states ............ 129 5.2.4. Comprehensive evaluation of model performance in modelling drift limit states ....................................................................................... 129 5.2.5. Model exPlainability .................................................................. 130 5.3. Results and Discussion ..................................................................... 132 5.3.1. Optimization results ................................................................... 132 5.3.2. Drift ratio limit state prediction results ..................................... 133 5.3.3. ExPlainable drift capacity model: Overcoming the black box challenge with SHAP .............................................................................. 135 5.3.4. Deployment of the model and design example .......................... 140 5.3.5. Uncertainty analysis of drift ratio at different limit states ........ 142 5.3.6. Seismic fragility assessment of UHPC bridge columns ............. 145 5.4. Summary .......................................................................................... 149 xi CHAPTER 6: MULTIVARIATE FRAGILITY ASSESSMENT OF UHPC BRIDGES USING HYBRID MACHINE LEARNING MODEL ............................................... 152 6.1. Introduction ...................................................................................... 152 6.2. Methodology .................................................................................... 155 6.3. Case Study ........................................................................................ 159 6.3.1. Bridge description and ground motion characteristics ............. 159 6.3.2. Modelling of seismic response ................................................... 161 6.3.3. Results of PSD model and seismic fragility ............................... 163 6.4. Summary .......................................................................................... 168 CHAPTER 7: SUMMARY AND RECOMMENDATION FOR FUTURE WORK 171 7.1. Summary and Conclusions ............................................................... 171 7.2. Recommendation for Future Work .................................................. 175 BIBLIOGRAPHY ...................................................................................................... 177 xii LIST OF TABLES Table 2.1: Statistics of ultra-high-performance concrete datasets (Abellán-García, 2022) ............................................................................................................................ 20 Table 2.2: Specific gravity and cost of constituent raw materials (VaitkeviÄius et al., 2014; Yalçınkaya and Yazıcı, 2017; Meng et al., 2018; Abellán, 2020) ..................... 31 Table 2.3: Overview of the midpoint impact categories and related units .................. 32 Table 2.4: Environmental mid-point impacts calculated per the ReCiPe 2016 framework .................................................................................................................... 33 Table 2.5: Optimized hyperparameter values for various machine learning models .. 36 Table 2.6: Performance evaluation of the ML models with statistical performance measures ....................................................................................................................... 39 Table 2.7: Single objective optimization results .......................................................... 43 Table 3.1 Optuna Optimized Hyperparameters of CTGAN ........................................ 61 Table 3.2 Optuna-optimized hybrid ML models for the peak and ultimate stress-strain responses ...................................................................................................................... 66 Table 3.3 Statistical performance metrics .................................................................... 67 Table 3.4 Performance evaluation of the hybrid ML models ...................................... 69 Table 3.5 Optuna-optimized hyperparameters of single and ensemble models .......... 71 Table 3.6 Performance measures of the single and ensemble models ......................... 74 Table 3.7 Modelsâ performance in predicting the stress-strain responses ................... 76 Table 4.1: Analytical equations utilized for modelling the unconfined UHPC ........... 87 Table 4.2: Experimental versus predicted peak strength of confined UHPC .............. 92 Table 4.3: Optuna-optimized hybrid ML model .......................................................... 99 Table 4.4: Performance of Optuna-optimized hybrid ML model in predicting the stress-strain response of confined UHPC at various stages across the curve. ..................... 102 xiii Table 4.5: Canonical correlation analysis (CCA) score ............................................. 103 Table 5.1 Selected performance criteria .................................................................... 123 Table 5.2 Uncertainty parameters of UHPC bridge columns and their distribution (Mangalathu et al., 2018). .......................................................................................... 126 Table 5.3 Hyperparameters ranges and their optimal values determined through Optuna with 10-fold cross-validation ..................................................................................... 133 Table 5.4 Performance of the model .......................................................................... 135 Table 5.5 Mean absolute SHAP values of the factors and their corresponding percentage contributions .............................................................................................................. 139 Table 5.6 PBSD of UHPC bridge bent of viaduct ..................................................... 142 Table 5.7 Drift ratio capacity uncertainty parameters at different damage states ..... 144 Table 5.8 Characteristics of selected ground motions ............................................... 146 Table 6.1 Drift ratio capacity uncertainty parameters for UHPC bridge piers .......... 158 Table 6.2 Uncertainty of bridge characteristics considered in this study (Mangalathu et al., 2018; Nielson, 2005; Tavares et al., 2012) .......................................................... 160 Table 6.3 Performance measures of PSD prediction for UHPC bridge columnsâ drift demand ....................................................................................................................... 164 xiv LIST OF FIGURES Figure 2.1. Flowchart illustrating the methodology adopted in this study. ................. 18 Figure 2.2. Proposed super learner model. .................................................................. 21 Figure 2.3. K-fold cross-validation. ............................................................................. 25 Figure 2.4. Experimental versus predicted 28-day uniaxial compressive strength of UHPC. .......................................................................................................................... 38 Figure 2.5. Performance metrics of the ML models based on the training and test datasets. ........................................................................................................................ 39 Figure 2.6. Violin plots of residual distributions for test dataset. ................................ 40 Figure 2.7. Parallel line plots of the optimized sustainable UHPC mix. ..................... 42 Figure 2.8. Distribution of optimized UHPC mixtures (type: normal distribution). ... 42 Figure 2.9. User-friendly FAI-OSUSCONCRET software tool for strength prediction and MOO of sustainable UHPC mixture tailored to project-specific requirements. ... 45 Figure 3.1: Methodology flowchart. ............................................................................ 52 Figure 3.2: Details of the experimental dataset (C: Circular, R: Rectangular). .......... 54 Figure 3.3: Illustration of GAN structure (Akiba, Sano, Yanase, Ohta, & Koyama, 2019). ........................................................................................................................... 55 Figure 3.4: Distributions of original and synthesized data for modelling peak stress. 63 Figure 3.5: Distributions of original and synthesized data for modelling peak strain. 63 Figure 3.6: Distributions of original and synthesized data for modelling ultimate stress....................................................................................................................................... 64 Figure 3.7: Distributions of original and synthesized data for modelling ultimate strain....................................................................................................................................... 64 Figure 3.8: Comparison of the predicted and actual values for (a) the peak stress and (b) the peak strain based on the hybrid ML model. ..................................................... 68 xv Figure 3.9: Comparison of predicted and actual values for (a) the ultimate stress and (b) the ultimate strain based on the hybrid ML model. ................................................ 70 Figure 3.10: Comparisons of the predicted and actual peak stress based on various single and ensemble models......................................................................................... 71 Figure 3.11: Comparison of the predicted and actual peak strain values using various single and ensemble models......................................................................................... 72 Figure 3.12: Comparative analysis of various models' performance in predicting the peak stress-strain response. .......................................................................................... 72 Figure 3.13: Comparison of the predicted and actual ultimate stress values using various single and ensemble models. ........................................................................... 73 Figure 3.14: Comparison of the predicted and actual ultimate strain values using various single and ensemble models. ........................................................................... 73 Figure 3.15: Comparative analysis of the performance of various models in their prediction of the ultimate stress-strain response. ......................................................... 74 Figure 3.16: ConUHPC-HybridOptML-CTGU: A graphical user interface-based software tool for the developed hybrid ML model optimized with Optuna and aided with CTGAN. ............................................................................................................... 77 Figure 4.1: Methodology flowchart. ............................................................................ 83 Figure 4.2: Details of the developed finite element model: (a) loading plates and concrete, (b) spiral reinforcement, and (c) loading and support sets. .......................... 84 Figure 4.3: Experimental versus analytical stress-strain responses of unconfined UHPC. .......................................................................................................................... 88 Figure 4.4: Concrete compressive strength results of FE model C1-SF2-SS2 (Naeimi & Moustafa, 2021a) with different mesh sizes: (a) stress-strain curves and (b) peak compressive strength for each mesh size. .................................................................... 91 xvi Figure 4.5: Comparison between the numerical and experimental stress-strain curves (Note: EXP = experimental and FE = finite element). ................................................ 93 Figure 4.6: Discretization of stress-strain response curve. .......................................... 97 Figure 4.7: Predicted versus actual stress-strain response of UHPC confined with NSS spirals. ........................................................................................................................ 105 Figure 4.8: Predicted versus actual stress-strain response of UHPC confined with HSS spirals. ........................................................................................................................ 106 Figure 4.9: Predicted versus experimental stress-strain curves of Specimen (a) 120FM2-35 (Sugano et al., 2007), (b) 160FM2-55 (Sugano et al., 2007), (c) 120FM2-35 (Sugano et al., 2007), and (d) C1-SF2-SS2 (Naeimi & Moustafa, 2021b). .......... 107 Figure 4.10: HAI-CONCise-UHPC interactive software tool for the developed hybrid ML-based stress-strain constitutive model of confined UHPC. ................................ 108 Figure 4.11: Scatter plots of predicted versus actual stress-strain responses at peak and ultimate states using the proposed equations. ............................................................ 112 Figure 4.12: Prediction evaluation of the proposed equations based on experimental results of confined UHPC. ......................................................................................... 113 Figure 5.1: Schematic representation of the performance-based seismic design workflow. ................................................................................................................... 118 Figure 5.2: Methodology adopted in this study. ........................................................ 121 Figure 5.3: Bridge pier model. .................................................................................. 127 Figure 5.4: Predicted versus experimental hysteretic response of UHPC bridge piers..................................................................................................................................... 129 Figure 5.5: Actual versus predicted drift ratio at various damage states. .................. 134 Figure 5.6: SHAP summary plot for drift ratio prediction at various damage states.137 Figure 5.7: Importance ranking of input parameters at various damage states. ........ 138 xvii Figure 5.8: Graphical user interface of the deployed ML-based predictive tool for drift ratio limit states in UHPC bridge columns. ............................................................... 140 Figure 5.9: Lognormal distribution estimation of drift ratio limit states. .................. 144 Figure 5.10: PSD model for (a) Specimen 1, (b) Specimen 2, (c) Specimen 3, and (d) Specimen 4. ................................................................................................................ 147 Figure 5.11: Results of fragility curves. .................................................................... 148 Figure 5.12: Comparative analysis of fragility curves of UHPC columns at different damage states. ............................................................................................................ 148 Figure 6.1: Hybrid ML-based multivariate seismic fragility assessment of UHPC bridges. ....................................................................................................................... 158 Figure 6.2: Numerical model of the selected bridge. ................................................ 162 Figure 6.3: PSD of UHPC column drift demand. ...................................................... 164 Figure 6.4: PSD for column drift based on conventional method. ............................ 165 Figure 6.5: Predicted versus actual PSD for column drift based on the conventional linear regression and proposed hybrid ML model. .................................................... 165 Figure 6.6: PSD of UHPC column drift ratio based on different ML models........... 166 Figure 6.7: SHAP summary plot. .............................................................................. 167 Figure 6.8: Fragility curves generated based on hybrid ML model. ......................... 168 xviii ACKNOWLEDGEMENTS First and foremost, I would like to express my profound gratitude to the Almighty God for guiding me through this journey of knowledge and discovery. My PhD study has been a transformative experience enriched not only by academic learning but also by the support and encouragement from those around me. I am deeply thankful to my supervisor, Dr. M. Shahria Alam, for his exceptional mentorship and guidance. Working under his supervision has been both an honor and a privilege. His expertise, humility, and unwavering support have been instrumental in my academic growth, and for this, I am eternally grateful. I am also grateful to the members of my supervisory committee, Dr. Rehan Sadiq and Dr. Abbas Milani, for their invaluable contributions in reviewing my dissertation. Their insights and feedback have significantly enhanced the quality of my work. My appreciation also extends to the University of British Columbia for its financial support through the University Graduate Fellowship. This funding has been crucial in facilitating my research and academic endeavors. Lastly, I would like to express my profound gratitude to my family for their endless love, support, and encouragement. xix DEDICATION This Dissertation is dedicated to My Mother, Adilee Tefera Tadesse 1 CHAPTER 1: INTRODUCTION 1.1. Emergence of UHPC as an Advanced Material Ultra-high-performance concrete (UHPC) is a highly promising construction material for promoting future sustainability and resilience of infrastructure, thanks to its exceptional mechanical properties and durability (Amran et al., 2022; Du et al., 2021). According to the Canadian Highway Bridge Design Code (CAN/CSA S6-19, 2019), UHPC is defined as a cementitious composite material that exhibits superior strength and durability compared to high-performance concrete. Furthermore, UHPC can lead to cost-efficient construction by reducing the size of structural cross-sections, conserving resources, and decreasing labor and installation costs (Du et al., 2021; Tahwia, Hamido, & Elemam, 2023). In addition, the high ductility and toughness properties of UHPC make it a promising material for the construction of earthquake-resistant structures (Krahl, de Miranda Saleme Gidrão, & Carrazedo, 2018). The provision of transverse reinforcement and fibers can further augment the ductility of UHPC structures, effectively enhancing their performance (Y. Chang, Shi, Cao, & Qiu, 2023; Naeimi & Moustafa, 2021b; Hyun Oh Shin, Yoon, Cook, & Mitchell, 2016; Xue et al., 2020). 1.2. UHPC Mixture Strength Prediction An accurate predictive model for the compressive strength of UHPC is essential for the advancement of safe, economical, and sustainable construction practices. However, the relationship between the characteristics of UHPC and its mixture content is notably complex and non-linear, which makes it difficult to model using traditional statistical methods. Adding to this complexity is the absence of a standardized composition for UHPC. Unlike conventional concrete, which typically involves simpler 2 formulations, UHPC includes a diverse array of components such as fibers, SCMs like fly ash or slag, and sometimes even nanoparticles (Yoo, Oh, & Banthia, 2022). This diversity in mixture compositions introduces additional layers of complexity in modelling the mechanical properties of UHPC and optimizing its mixture. In this context, machine learning (ML) techniques provide an effective approach for addressing the inherent complexities in modelling UHPC, owing to their ability to capture complex nonlinear relationships between response variables and predictors. In recent years, the growth of artificial intelligence and computational power has led to a significant rise in the use of ML techniques within the civil engineering domain (Mangalathu, Hwang, Choi, & Jeon, 2019; Mangalathu & Jeon, 2020; Zaker Esteghamati & Flint, 2021). Moreover, various ML techniques have been explored for compressive strength prediction of different concrete types (Behnood, Behnood, Modiri Gharehveran, & Alyamac, 2017; Chopra, Sharma, Kumar, & Chopra, 2018; Mamun, Wenzlick, Sathanur, Hawk, & Devanathan, 2021; J. Zhang et al., 2019; J. Zhang, Li, & Wang, 2020a, 2020b). Despite the substantial research on the application of ML algorithms for predicting the compressive strength of conventional concrete, their effectiveness has not been well studied for UHPC (Abellán-García, 2021, 2022; Abuodeh, Abdalla, & Hawileh, 2020; Nguyen, Abellán-García, Lee, Garcia-Castano, & Vo, 2022). Additionally, some existing research developed ML models based on a limited number of UHPC datasets (Abuodeh et al., 2020; Dingqiang et al., 2020), which restricts the generalizability of the developed models. For example, Abuodeh et al. (Abuodeh et al., 2020) developed deep machine-learning techniques using only 110 concrete mixes, of which only 11 concrete mix samples contained quartz powder in the dataset. Similarly, Ghafari et al. (Ghafari, Bandarabadi, Costa, & Júlio, 2015) developed an artificial neural network (ANN) model for predicting the properties of 3 UHPC using experimental results of only 53 concrete mixes, both of which restrict the generalizability of the developed models. Furthermore, in some cases, concrete with lower compressive strength has been used in developing the models (Marani, Jamali, & Nehdi, 2020; G. Sun, Du, Shan, Shi, & Qu, 2022), despite UHPC typically having a compressive strength of 100 MPa or higher (Xue et al., 2020; Zhou, Lu, Song, & Lee, 2018). Recently, Abellán-García (Abellán-García, 2022) developed ML-based models for predicting the compressive strength of UHPC based on a database of 931 UHPC mixes with a total of 17 input parameters such as metakaolin, GGBFS, fly ash, recycled glass powder, rice husk ash, and cement. However, a detailed examination of the database reveals numerous issues, such as significant gaps in experimental data and pronounced data imbalance. For example, metakaolin is only present in 27 mixes, rice husk ash in 59, fly ash in 146, and ground granulated blast furnace slag (GGBFS) in 129, while data for recycled glass powder is absent in 733 mixes. This discrepancy highlights the substantial limitations of the adopted database, which contains significant gaps in experimental data and demonstrates data imbalance. These issues should have been recognized and addressed, possibly by excluding outliers during the pre-processing phase. Consequently, the ML models developed from this flawed database inherently suffer from these limitations, thereby compromising their reliability. Therefore, there is a need to establish an accurate compressive strength prediction model for UHPC. 1.3. Sustainability Implications and MOO of UHPC Mixture Building on the need for accurate models to predict the compressive strength of UHPC mixture, it is equally important to assess the sustainability metrics of its mixture. In the context of the global sustainability crisis, the production of concrete is a major contributor to greenhouse gas (GHG) emissions, with global cement production alone 4 responsible for approximately 8% of the total CO2 emissions (IEA, 2022). The consumption of concrete is also on the rise, with the use of ordinary Portland cement (OPC) concrete contributing to global cement production which reached 4.1 billion metric tons in 2022 (U.S. Geological Survey, 2023). This consumption is expected to increase to approximately 6 billion metric tons by 2050 (Kazemian & Shafei, 2023), which presents significant sustainability challenges in terms of resource depletion and GHG emissions among others. In this regard, the role of construction materials like UHPC takes on added significance by offering a pathway for more sustainable construction by virtue of its enhanced mechanical properties, which can reduce the volume of material required and extend the lifecycle of structures. However, producing UHPC can be costly, due to the use of high-cost components such as superplasticizer and silica fume, as well as a high amount of cement. The incorporation of readily available SCMs, such as slag and fly ash, for partial or full substitution of cement and silica fume, has the potential to significantly reduce the material costs of UHPC (Abellán-García, 2021). Note that fly ash has been becoming scarce in North America because of stricter environmental regulations on burning coal for energy production. It is important to note that UHPC production also has environmental impacts, and a comprehensive evaluation is necessary to understand its overall environmental footprint (Abellán-García, 2021). Therefore, it is crucial to optimize the mixture design of UHPC by taking into account multiple objectives such as cost, strength, and environmental sustainability. Additionally, the quality and consistency of UHPC production are dependent on the appropriate selection of materials and an efficient mixture design approach. Typically, traditional concrete mixture design is conducted using single objective optimization, such as compressive strength alone. This approach typically involves costly and labor-5 intensive trial tests, which are difficult to perform when optimizing for multiple objectives (Liu et al., 2019; Z. Sun, Yang, Hang, Wang, & Yang, 2022; Tahwia et al., 2023). Additionally, the optimization of UHPC considering sustainability aspects is an under-researched area that can lead to the development of more sustainable and environmentally friendly concrete products. Therefore, efficient concrete mixture design should balance quality, economic efficiency, and environmental sustainability. Despite considerable research on concrete optimization, an exploration into multi-objective optimization of UHPC has remained relatively limited. Moreover, to date, studies on the sustainability aspects of UHPC mixtures have primarily focused on CO2 emissions, ignoring other potential environmental impacts. The accuracy and comprehensiveness of environmental sustainability assessments of UHPC mixtures are thereby limited. Moreover, to date, studies on the sustainability aspects of UHPC mixtures have primarily focused on CO2 emissions, ignoring other potential environmental impacts. This limits the accuracy and comprehensiveness of the overall environmental sustainability of UHPC mixtures. 1.4. UHPC in Structural Applications and Confinement Models Shifting to its structural applications, considerable research is being conducted on the potential applications of UHPC in new construction (Aboukifa & Moustafa, 2021; Elsayed et al., 2022; Xue et al., 2020) as well as in the strengthening of existing structures (Bertola, Schiltz, Denarié, & Brühwiler, 2021; Y. Huang, Grünewald, Schlangen, & LukoviÄ, 2022; B. Shrestha, Gheni, Abdulazeez, & Elgawady, 2020). As UHPC applications increase, it becomes crucial to establish an accurate confinement model to enable reliable modelling and design of UHPC structures and their wider adoption, particularly in seismic regions. Previous studies have shown that the stress-strain model of UHPC significantly differs from that of conventional concrete 6 (Hosinieh, Aoude, Cook, & Mitchell, 2015; Naeimi & Moustafa, 2021a). For example, Hosinieh et al. (Hosinieh et al., 2015) investigated the potential of using existing confinement models designed for fiber-reinforced concrete (FRC) and high-strength concrete (HSC) to predict the axial response of UHPC columns. This investigation was based on the analysis of six experimentally tested UHPC columns. Their findings showed that these models are inadequate for UHPC and underscore the importance of developing confinement models specifically tailored for UHPC. Similarly, Naeimi and Moustafa (Naeimi & Moustafa, 2021a) assessed the confinement models of HSC, normal strength concrete (NSC), and FRC in their ability to predict the uniaxial compressive behaviour of UHPC; and concluded that these models are inadequate in accurately representing the behaviour of confined UHPC. Nevertheless, existing research on the stress-strain constitutive models of confined UHPC is limited, despite the significance of these models in numerical simulations of UHPC structures (Naeimi & Moustafa, 2021a). Furthermore, the compressive strength and the corresponding axial strain at both peak and ultimate states of confined UHPC are critical parameters for the accurate numerical modelling of UHPC structures. Nevertheless, no models were developed to predict the ultimate stress and corresponding strains as well as the peak stress of confined UHPC, while the existing model for predicting peak strain response (Naeimi & Moustafa, 2021a) was empirically developed based on small datasets of confined UHPC, thus inherently limiting its accuracy and general applicability. Moreover, in determining the peak strain of confined UHPC with fibers, this model (Naeimi & Moustafa, 2021a) requires the test results of unconfined UHPC without fibers, which further limits its applications. In addition, a prior study (Naeimi & Moustafa, 2021a) reported complex interactions between steel spirals and fibers in UHPC, and the significant influence of such 7 interactions on the behaviour of confined UHPC, which has been neglected in current models. To better understand this interaction, Naeimi and Moustafa (Naeimi & Moustafa, 2021a) conducted an investigation examining the effects of varying steel fibersâ volumetric ratios of (4%, 2%, and 0%) and steel spiral confinements (2%, 4%, and 8%) on the stress-strain response of confined UHPC. The results revealed that while the incorporation of 2% steel fibers enhanced peak strength, increasing the fiber ratio beyond 2% did not necessarily result in an increase in the compressive strength and was influenced by the presence and volumetric ratio of steel spirals (Naeimi & Moustafa, 2021a). Increasing the fiber volumetric ratio in specimens with moderate spiral volumetric ratios (4%) reduced peak strength (Naeimi & Moustafa, 2021a). The study (Naeimi & Moustafa, 2021a) underscores the importance of accounting for the interaction between steel fibers and spirals in UHPC confinement. In addition to such complex interaction, the accuracy of the stress-strain models for confined UHPC is hindered by various factors, including the nonlinear behaviour of confined UHPC. 1.5. Performance-Based Seismic Design of UHPC Structures Performance-based design is a methodology that expresses structural design criteria through the attainment of specified performance objectives. It allows stakeholders to make informed decisions by evaluating a structureâs seismic performance against various performance objectives and seismic hazard levels. The primary objective is to ensure that a structureâs response does not exceed prescribed limiting values, known as limit states. Thus, a crucial aspect of performance-based design is the development of reliable limit states or performance objectives. The 2019 version of the Canadian Highway Bridge Design Code (CHBDC) (CAN/CSA S6-19, 2019) outlines four such performance levels, namely, immediate, service limited, service disruption, and life-safety performance, which correspond to minimal, 8 repairable, extensive, and probable replacement damage states, respectively. While extensive research and guidelines exist for establishing limit states in conventional concrete structures reinforced with normal strength steel (NSS) (Ghannoum & Matamoros, 2004; Lehman, Moehle, Mahin, Calderone, & Henry, 2004), they are not applicable to UHPC structures. Currently, no studies have focused on formulating limit states specifically for UHPC structures, a research gap that this dissertation aims to address. Furthermore, it is also vital to investigate the seismic vulnerability of UHPC structures through fragility analysis. Fragility curves are typically established for specific types of bridges and structural systems, based on a chosen classification system (Mangalathu, Heo, & Jeon, 2018). However, this method may undermine the reliability of seismic vulnerability assessments when applied to a diverse range of bridges that vary significantly within a particular class as it does not consider vital bridge attributes, such as geometries, which can significantly influence the seismic performance of bridges. Additionally, classification criteria may be subjective or vary by region. Therefore, the development of efficient, bridge-specific fragility assessments is critical for accurate evaluation of seismic performance. Moreover, developing a practical and effective framework for the probabilistic performance-based seismic design (PBSD) model (Ang & De Leon, 2005), which is integral to deriving analytical fragility curves presents significant challenges (Soleimani & Hajializadeh, 2022). Limitations of classical regression techniques in developing PBSD models include fixed functional forms, inability to capture nonlinearity, and prior assumptions of lognormal distribution (Soleimani, 2022). The inherent inflexibility of the conventional unidimensional PBSD model makes it challenging to accommodate diverse sources of uncertainties, which could potentially 9 undermine the reliability of the estimated demands (Soleimani, 2022). Moreover, the use of linear regression, which is a commonly used method for PBSD model development (Billah, Alam, & Bhuiyan, 2013), often yields models that inadequately represent the seismic response of structures, particularly when multiple independent variables are involved, underscoring the need for more flexible and reliable modelling methods in fragility analysis. As a result, various ML techniques have been explored to improve the fragility assessment of buildings (Kiani, Camp, & Pezeshk, 2019; Mitropoulou & Papadrakakis, 2011) and conventional bridges (C. Huang & Huang, 2020; Kameshwar & Padgett, 2018; MO et al., 2023; Pang, Dang, & Yuan, 2014; Wang, Zhang, Dai, & Shen, 2022; Xie, Ebad Sichani, Padgett, & DesRoches, 2020). However, the full potential of ML models in enhancing the fragility assessment of bridges remains to be fully explored despite their widespread use in various fields. Moreover, a recent comprehensive review by Soleimani and Hajializadeh (Soleimani & Hajializadeh, 2022) underscores the limited extent of research conducted on the application of ML algorithms for seismic fragility analysis of bridges. Furthermore, the scope of existing studies is generally limited as they typically concentrate on a limited set of input features, favor certain ML models without thoroughly investigating the potential of other ML models, and rely on particular assumptions (Soleimani & Hajializadeh, 2022). Moreover, the literature lacks research specifically addressing the seismic vulnerability assessment of UHPC bridges. 1.6. Research Objectives The overall objective of this dissertation is to develop a comprehensive framework that enables sustainable design, modelling, analysis, assessment, and performance-based design using state-of-the-art artificial intelligence techniques. 10 To accomplish this, this dissertation aims to: â¢ Develop FAI-OSUSCONCRET (Fast, Accurate, and Intelligent Optimization of Sustainable UHPC), a novel framework for strength prediction and multi-objective optimization of sustainable UHPC. â¢ Develop the peak and ultimate stress-strain prediction models for UHPC confined with NSS or HSS to enable accurate modelling of UHPC structures. â¢ Propose complete stress-strain prediction models for confined UHPC. â¢ Establish performance-based drift limit states for PBSD of UHPC columns. â¢ Develop multivariate seismic fragility assessment of UHPC highway bridges using hybrid ML model. 1.7. Dissertation Organization This dissertation is structured into seven chapters, with this initial chapter offering a brief background, motivation, and research objectives. The subsequent chapters detail the research as follows: In Chapter 2, the capability of the advanced super learner ML algorithm is integrated with a metaheuristic multi-objective optimization algorithm to establish a framework for predicting the strength and optimizing the mixture of UHPC while considering 19 different objective functions such as uniaxial compressive strength, cost, and various environmental sustainability metrics. This investigation aims to develop a cost-effective and environmentally friendly UHPC mixture that can be tailored to specific project requirements. Chapter 3 proposed peak and ultimate stress-strain response prediction models for confined UHPC using a novel hybrid ML model augmented with conditional tabular generative adversarial network (CTGAN). 11 Chapter 4 proposed a predictive stress-strain constitutive model for the complete stress-strain response of UHPC confined with either NSS or HSS spirals using hybrid ML model and numerical simulation. In Chapter 5, drift limit state models for UHPC columns reinforced with NSS HSS are established, in order to enable the PBSD of UHPC columns. This chapter includes a design example demonstrating the practical application of these proposed limit states in designing bridge columns. Additionally, this chapter aims to investigate the effect of various design parameters including aspect ratio, axial load ratio, section geometry, and material properties of UHPC and reinforcement steel on the drift limit states of bridge piers. Chapter 6 aims to propose multivariate seismic fragility assessment of UHPC highway bridges considering different sources of uncertainties. For this purpose, multivariate probabilistic seismic demand models are generated considering four damage states. Chapter 7 concludes the dissertation by summarizing its key findings and conclusions. Additionally, this chapter outlines potential avenues for future research building upon the work presented in this dissertation. 12 CHAPTER 2: A NOVEL FRAMEWORK FOR DEVELOPING ENVIRONMENTALLY SUSTAINABLE AND COST-EFFECTIVE ULTRA-HIGH-PERFORMANCE CONCRETE (UHPC) USING ADVANCED MACHINE LEARNING AND MULTI-OBJECTIVE OPTIMIZATION TECHNIQUES 2.1. Introduction The production of concrete is the main source of GHG emissions, with global cement production alone responsible for approximately 8% of the total CO2 emissions (IEA, 2022). The consumption of concrete is also on the rise, with the use of OPC concrete contributing to the global cement production that reached 4.1 billion metric tons in 2022 (U.S. Geological Survey, 2023), and is expected to increase to approximately 6 billion metric tons by 2050 (Kazemian & Shafei, 2023). This increase presents significant sustainability challenges in terms of energy consumption, resource depletion, and GHG emissions among others. One possible solution to these challenges is the use of ultra-high-performance concrete (UHPC). UHPC has high compressive strength and durability, and it can lead to cost-efficient construction by reducing the size of structural cross-sections, conserving resources, and decreasing labor and installation costs (Du et al., 2021; Tahwia et al., 2023). Due to its exceptional properties (Abokifa & Moustafa, 2021b), considerable research is being conducted on the application of UHPC in new construction (Aboukifa & Moustafa, 2021; Elsayed et al., 2022; Xue et al., 2020) as well as in the rehabilitation and strengthening of existing structures (Bertola et al., 2021; Y. Huang et al., 2022; B. Shrestha et al., 2020). However, producing UHPC can be costly, due to the use of high-cost components such as superplasticizers and silica fume, as well as a high amount of cement. The incorporation of readily available supplementary cementitious materials (SCMs), such 13 as slag and fly ash, for partial or full substitution of silica fume and cement, has the potential to significantly reduce the material costs of UHPC (Abellán-García, 2021). Note that fly ash has been becoming scarce in North America because of stricter environmental regulations on burning coal for energy production. It is important to note that UHPC production also has environmental impacts, and a comprehensive evaluation is necessary to understand its overall environmental footprint (Abellán-García, 2021). To decrease the environmental impact of UHPC production and improve sustainability, it is crucial to optimize its mixture design by taking into account multiple objectives such as cost, strength, and environmental sustainability. Additionally, the quality and consistency of UHPC production are dependent on the appropriate selection of materials and an efficient mix design approach. Typically, traditional concrete mixture design is conducted using single objective optimization, such as compressive strength alone. This approach typically involves costly and labor-intensive trial tests, which are difficult to perform when optimizing for multiple objectives (Liu et al., 2019; Z. Sun et al., 2022; Tahwia et al., 2023). Additionally, the optimization of UHPC considering sustainability aspects is an under-researched area that presents opportunities for developing more environmentally friendly concrete products. Therefore, efficient concrete mixture design should balance quality, economic efficiency, and environmental sustainability. Despite the substantial amount of research conducted on concrete optimization, relatively little research has been done on the multi-objective optimization (MOO) of concrete (Opricovic & Tzeng, 2004). To this end, this study aims to develop FAI-OSUSCONCRET (Fast, Accurate, and Intelligent Optimization of Sustainable UHPC), a novel framework for strength prediction and MOO of UHPC. In recent years, the growth of artificial intelligence and computational power has led to a significant rise in the use of machine learning (ML) 14 algorithms within the civil engineering domain (Mangalathu et al., 2019; Mangalathu & Jeon, 2020; Zaker Esteghamati & Flint, 2021). Moreover, various ML algorithms have been explored in the literature for the compressive strength prediction of diverse concrete types (Behnood et al., 2017; Chopra et al., 2018; Mamun et al., 2021; J. Zhang et al., 2019, 2020a, 2020b). Despite the substantial research on the use of ML algorithms for compressive strength prediction of conventional concrete, their effectiveness has not been well studied for UHPC (Abellán-García, 2021, 2022; Abuodeh et al., 2020; Nguyen et al., 2022). Additionally, some existing research developed ML models based on a limited number of UHPC datasets (Abuodeh et al., 2020; Dingqiang et al., 2020), which restricts the generalizability of the developed models. For example, Abuodeh et al. (Abuodeh et al., 2020) developed deep machine-learning techniques using only 110 concrete mixes, of which only 11 concrete mix samples contained quartz powder in the dataset. Similarly, Ghafari et al. (Ghafari et al., 2015) developed an ANN model for predicting the properties of UHPC utilizing the experimental results of only 53 concrete mixes, both of which restrict the generalizability of the developed models. Furthermore, in some cases, concrete with lower compressive strength has been used in developing the models (Marani et al., 2020; G. Sun et al., 2022), despite UHPC typically having a compressive strength of 100 MPa or higher (Xue et al., 2020; Zhou et al., 2018). Recently, Abellán-García (Abellán-García, 2022) developed ML-based models for predicting the compressive strength of UHPC based on a database of 931 UHPC mixes with a total of 17 input parameters such as metakaolin, rice husk ash, GGBFS, fly ash, recycled glass powder, and cement. However, a detailed examination of the database reveals numerous issues, such as significant gaps in experimental data and pronounced data imbalance. For example, metakaolin is only present in 27 mixes, rice husk ash in 59, fly ash in 146, 15 and GGBFS in 129, while data for recycled glass powder is absent in 733 mixes. This discrepancy highlights the substantial limitations of the adopted database, which contains significant gaps in experimental data and demonstrates data imbalance. These issues should have been recognized and addressed, possibly by excluding outliers during the pre-processing phase. Consequently, the ML models developed from this flawed database inherently suffer from these limitations, thereby compromising their reliability. Moreover, the utilization of ML-aided MOO of UHPC remains an under-explored area of research (Mahjoubi, Barhemat, Meng, & Bao, 2023). Recently, Mahjoubi et al., (2023) (Mahjoubi et al., 2023) performed MOO of UHPC. However, the study had some limitations such as the small sample size of concrete mix samples that included fly ash or slag, only 108 samples were used for predicting the porosity of UHPC, and only 9 samples had slag among those 108 samples (Mahjoubi & Bao, 2021; Mahjoubi et al., 2023). Furthermore, only 6 samples had limestone powder among the 146 databases used for developing ML models for flexural strength prediction of UHPC. A small and imbalanced dataset can limit the generalizability of ML models and lead to unreliable results, and using generative models to synthesize data may not fully address these issues. Thus, the optimization of UHPC using this model does not yield accurate and reliable results. Moreover, to date, studies on the sustainability aspects of UHPC mixtures have primarily focused on CO2 emissions, ignoring other potential environmental impacts. This limits the accuracy and comprehensiveness of the overall environmental sustainability of UHPC mixtures. Therefore, the current study aims to develop FAI-OSUSCONCRET, a novel framework for strength prediction and MOO of cost-effective and environmentally sustainable UHPC that can be tailored to a particular project specification and 16 requirements. A novel super learner ML model has been developed by integrating tree- and boosting ensemble models trained on a comprehensive dataset of UHPC mixtures, to address the limitations of prior works. Grid search and K-fold cross-validation are employed to optimize the ML models. Furthermore, the study employs a MOO algorithm that simultaneously accounts for cost-effectiveness, environmental sustainability, and compressive strength, providing a more comprehensive view of sustainable construction practices. The developed super learner model is used as one of the objective functions in the MOO problem, which is solved using the Unified Non-dominated Sorting Genetic Algorithm (UNSGA) metaheuristic algorithm (Seada & Deb, 2014). A total of 19 objective functions that comprehensively assess the cost, strength, and environmental sustainability of the UHPC are considered in the optimization. A notable feature is the inclusion of 17 mid-point environmental impact categories for the assessment of UHPC mixturesâ environmental sustainability, for the first time. These impact categories include climate change, depletion of fossil fuels, human toxicity, freshwater eutrophication, ionizing radiation, marine eutrophication, natural land transformation, depletion of metals, urban land occupation, marine ecotoxicity, ozone depletion, terrestrial acidification, photochemical oxidant formation, particle matter formation, terrestrial ecotoxicity, and water depletion. This broad spectrum consideration moves beyond the traditional focus on CO2 emissions, providing a more accurate and complete evaluation of UHPCâs environmental footprint. The results of the MOO are a set of non-dominated solutions referred to as Pareto fronts. The Technique for Order of Preference by Similarity to the Ideal Solution (TOPSIS) is used to determine the optimal UHPC mixture that balances cost efficiency, strength, and reduced environmental impact and meets the specific requirements of the project. Another significant limitation in existing research lies in the lack of practical 17 application for developed ML models. To bridge this gap, a graphical user interface (GUI)-based software tool is developed for the practical application of the developed framework, which facilitates accurate prediction and MOO of UHPC mixtures tailored to project-specific requirements. 2.2. Methodology The current study leverages the power of the super learner ML model and metaheuristic MOO technique to establish a framework for predicting the strength and optimizing the mixture of UHPC while considering uniaxial compressive strength, cost, and environmental sustainability under highly nonlinear constraints. The methodology followed in this study is summarized in Figure 2.1. In Stage 1, an efficient super learner ML model was developed for the 28-day uniaxial compressive strength prediction of UHPC. A variety of tree-based and boosting ensemble models were evaluated as base models for the super ensemble, and the performance of the super learner model was benchmarked against other popular tree-based and boosting models. The super learner model was then utilized as one of the objective functions in the optimization procedure, as indicated in Figure 2.1. In Stage 2, an MOO algorithm was employed to optimize the concrete mixture by considering 19 objective functions and one equality constraint. A comprehensive background on the ML models and MOO algorithms used in this study is presented in this section. 18 Figure 2.1. Flowchart illustrating the methodology adopted in this study. 2.2.1. Research data In this study, a large database of UHPC was obtained from (Abellán-García, 2022). The raw database, however, had various issues such as significant gaps in 19 experimental data and pronounced data imbalance. For instance, the component metakaolin was present only in 27 mixtures and rice husk ash in 59. To address these problems, a comprehensive pre-processing phase was conducted, which involved the systematic removal of outliers and the exclusion of data entries with missing variables to ensure the robustness and reliability of the dataset. This resulted in a final dataset of 540 UHPC samples. The input variables consisted of 10 design factors, including the content of cement (Cc), the content of silica fume (CSF), the content of fly ash (CFLA), the content of blast furnace slag (CS), the content of limestone powder (CLP), the content of water (CW), the content of superplasticizer (CSP), the content of quartz powder (CQP), the maximum size of aggregate (SA), and total aggregate content (CA), as listed in Table 2.1 along with their statistical details. The values of the input variables are in the following ranges: 0.06164 â 0.613651 m3 for Cc, 0 â 0.205569 m3/m3 for CSF, 0 â 0.249772 m3/m3 for CFLA, 0 â 0.246552 m3/m3 for CS, 0 â 0.190256 m3/m3 for CLP, 0.049 â 0.387 m3/m3 for CW, 0.002572 â 0.304738 m3/m3 for CSP, 0 â 0.234812 m3/m3 for CQP, 0 â 16000 μm for SA, and 0 â 0.693638 m3/m3 for CA, as listed in Table 2.1. It is worth noting that, while curing methods can influence the properties of UHPC, the surveyed literature for the database utilized standard curing conditions, without any specialized treatments like heating or autoclaving (Abellán-García, 2022). Consequently, the curing method is not considered as an input feature in aligment with the methodology of prior studies (Abellán-García, 2022). To overcome the limitations posed by the low learning rates of the models at extreme values, the input parameters are normalized to a range of 0 to 1. The normalized data is then randomly divided into a training set and a test set, which comprised 80% and 20% of the complete dataset, respectively. 20 Table 2.1: Statistics of ultra-high-performance concrete datasets (Abellán-García, 2022) S.N. Input parameters Average STD Minimum Quartiles Maximum Name Notation Unit 25% 50% 75% 1 Cement content Cc m3/m3 0.281 0.099 0.062 0.210 0.253 0.317 0.614 2 Silica fume content CSF m3/m3 0.065 0.046 0.000 0.024 0.070 0.099 0.206 3 Fly ash content CFLA m3/m3 0.024 0.053 0.000 0.000 0.000 0.000 0.250 4 Blast furnace slag content CS m3/m3 0.004 0.021 0.000 0.000 0.000 0.000 0.247 5 Limestone powder content CLP m3/m3 0.015 0.042 0.000 0.000 0.000 0.000 0.190 6 Water content CW m3/m3 0.215 0.056 0.049 0.177 0.202 0.231 0.387 7 Superplasticizer content CSP m3/m3 0.028 0.021 0.003 0.017 0.024 0.040 0.305 8 Quartz powder content CQP m3/m3 0.039 0.062 0.000 0.000 0.000 0.080 0.235 9 Maximum aggregate size SA μm 1923 2537 0.000 500 800 2360 16000 10 Total aggregate content CA m3/m3 0.333 0.178 0.000 0.309 0.367 0.451 0.694 STD: Standard deviation 2.2.2. Overview of machine learning models 2.2.2.1. Super learner model The super learner method integrates various ML models (base learners) to create a single, stronger model (Laan, Polley, & Hubbard, 2007). Figure 2.2 shows the proposed super learner model. The optimal combination of base learners is determined through K-fold (K = 10 in this study) cross-validation, where the algorithm assigns different weights to each base learner. The following steps were taken in implementing the super learner method in this study (Figure 2.2): a) Split the training data into 10-folds. b) Select a set of powerful models as candidate base learners, which are optimized individually before being utilized in the super learner framework. 21 c) Evaluate the performance of each candidate base learner through 10-fold cross-validation. d) Create a matrix of predictions made during the validation process, and refine the model by fitting it to the entire training dataset. e) Fit a meta-model that predicts the response variable utilizing the matrix of predictions as input. This study utilized different tree-based models as potential base models for the super learner algorithm, with a linear support vector regression (SVR) used as the meta-model, as can be seen in Figure 2.2. Figure 2.2. Proposed super learner model. 2.2.2.2. Decision trees The decision tree (DT) algorithm is a type of supervised ML algorithm that has a structure resembling a flowchart (Sutton, 2005). The highest node in the tree, known as the root node, represents the most crucial feature, while internal nodes specify the attribute tests. The leaf nodes, at the bottom of the tree, hold the value of the target variable and are used to make predictions. The algorithm builds the tree by dividing the N-dimensional input space âð into ð½ disjoint regions {ð 1, â¦ , ð ð½} through recursive GBM GBM GBM GBM 22 application of a set of predefined rules, with region having the same prediction for all inputs within it. The tree can be expressed as a function â(ð¥) given by Eq. (2.1) as: â(ð¥) = âððð¼(ð¥ðð ð)ð½ð=0 (2.1) where the disjoint region ð ð are assigned to each leaf of the tree and ðð represents the prediction value within the region ð ð. The indicator function ð¼(ð¥ðð ð) is equal to one when ð¥ â ð ð. 2.2.2.3. Boosting ensemble: AdaBoost, GBM, and XGB Boosting ensembles is a highly popular and effective type of ensemble model, where the base learners, typically DTs, are arranged sequentially to form a strong final model (Freund & Schapire, 1997). Consider a training example (ð¿, ð) = {(ðð, ðð)}ð=1ð with N samples, the task of the adaptive boosting (AdaBoost) model is to train T base or weak learners sequentially and then integrate them into a single model ð¹(ð), as shown in Eq. (2.2): ð¹(ð) = âðº{ð¤ð¡ðð¡(ð)}ðð¡=1 (2.2) where (ðð, ðð) is the ðð¡â sample, with ðð and ðð being the input and output variables. The ð¡ð¡â learner, ðð¡(ð), is assigned a weight, represented as ð¤ð¡, and the combination of these learners is governed by the rule denoted as ðº(Â·). The initial base learner is fitted on the training examples by assigning equal weights of {ð¤1,ð = 1/ð, âð} to each instance (Shrestha & Solomatine, 2006). Subsequently, each new learner is incorporated by assigning a higher weight to incorrectly predicted observations. The weight allocation at iteration ð¡ is revised using Eq. (2.3): 23 ð¤ð¡+1,ð =ð¤ð¡,ðð½ð¡1âð¿ð¡,ðâ ð¤ð¡,ðð½ð¡1âð¿ð¡,ððð=1 (2.3) where the distribution updating factor ð½ð¡ is within the range of [0, 1] and ð¿?Ì ? is the average loss function, as given in Eqs. (2.4) and (2.5), respectively (Drucker, 1997). ð½ð¡ =ð¿?Ì ?1 â ð¿?Ì ? (2.4) ð¿?Ì ? = âð·ð¡,ððð=1ð¿ð¡,ð (2.5) where ð¿ð¡ represents the loss function. A linear loss function is used in this study as given by Eq. (2.6): ð¿ð¡,ð =|ðð â ðð¡,ð(ðð)|max|ðð â ðð¡,ð(ðð)|, ð = 1,â¦ ,ð (2.6) Similar to AdaBoost, a gradient boosting machine (GBM) combines multiple base learners to create a robust model. Unlike the AdaBoost model, GBM employs a gradient descent approach to overcome the limitations of previous models. The GBM algorithm initializes the model with a constant value, as shown in Eq. (2.7) (Friedman, 2001): ð¹ð(ð) = arg minð¾âð¿(ðð, ð¾)ðð=1 (2.7) At tth step, the GBM fit base learner âð¡(ð) to pseudo-residuals (negative gradient descent) rð,ð¡ of the previous learner, Eq. (2.8), and updates the model, as given in Eq. (2.9). rð,ð¡ = â[ðð¿(ðð , ð¹(ðð) )ðð¹(ðð)]ð¹(ð)=ð¹ð¡â1(ð) , for ð = 1,â¦ ,ð (2.8) ð¹ð¡(ð) = ð¹ð¡â1(ð) + ð¾ð¡âð¡(ð), for ð¡ = 1,â¦ , ð (2.9) 24 where ð is the number of base learners and the multiplier ð¾ð¡ is determined by finding the optimal solution to a single variable optimization as shown in Eq. (2.10): ð¾ð¡ = arg minð¾âð¿(ðð, ð¹tâ1(ðð) + ð¾âð¡(ðð))ðð=1 (2.10) where ð¾ â [0, 1] is the learning rate (Friedman, 2001). Extreme gradient boosting (XGB) is a variant of the boosting model that has been developed to provide a more efficient implementation of GBM (Chen & Guestrin, 2016). One of the key features of XGB is the addition of a regularization component in its objective function, which helps to mitigate overfitting and simplify the modelâs complexity (Chen & Guestrin, 2016). This is demonstrated by Eq. (2.11). The incorporation of this regularization term allows XGB to produce more robust models that have improved generalization abilities and make more accurate predictions (Chen & Guestrin, 2016). ððð(ð) = âð¿(ðð, ?Ì?ð)ðð=1+ âðº(ðð¡)ðð¡=1 (2.11) where ð¿(Â·) is the training loss between the predicted (?Ì?) and actual (ð) values of the response, and ðº(Â·) represents the regularization term, outlined in Eq. (2.12): ðº(ðð¡) = ð¾ð +12ðâðð¡â2 (2.12) where ð¾ and ð are the penalty coefficients, ð represents the number of leave nodes in the tree, and ðð¡ denotes the leaf weights. 2.2.3. Tuning hyperparameters Hyperparameters are key factors in an ML model that determine its behaviour and are set by the programmer before training. Optimizing these hyperparameters is a critical step in building a successful ML model. One popular approach for 25 hyperparameter tuning is grid search, which searches exhaustively for the optimal values of each hyperparameter, considering all possible combinations to yield the best pair (L. Yang & Shami, 2020). To avoid overfitting, a commonly used technique called K-fold cross-validation is employed. This method involves splitting the training data into K equal, non-overlapping sets, with the model being trained using K-1 subsets and evaluated on the remaining subset. This process is repeated K times, with each subset serving as the validation set once. The ultimate evaluation of the model is established by taking the mean of the performance scores from the K validation subsets. A 10-fold cross-validation strategy (K = 10), is employed in this study, as illustrated in Figure 2.3. Figure 2.3. K-fold cross-validation. 26 2.2.4. ML performance measures The accuracy of ML models is evaluated through five distinct statistical measurements, which consist of average absolute percentage error, square root of the mean squared error, coefficient of determination, index of agreement, and Kling-Gupta efficiency. These metrics are mathematically expressed as follows (see Eq. (2.13) â Eq. (2.17)): â¢ Mean absolute percentage error, MAPE: ðð´ðð¸ =1ðâ |ðð â ?Ì?ððð|ðð=1 (2.13) â¢ Root mean squared error, RMSE: ð ððð¸ = â1ðâ(ðð â ?Ì?ð)2ðð=1 (2.14) â¢ Coefficient of determination, ð 2: ð 2 = 1 ââ (ðð â ?Ì?ð)2ðð=1â (ðð â ?Ì ?)2ðð=1 (2.15) â¢ Index of agreement, IA: ð¼ð´ = 1 ââ (ðð â ?Ì?ð)2ðð=1â (|ðð â ?Ì ?| + |?Ì?ð â ?Ì ?|)2ðð=1 (2.16) â¢ Kling-Gupta efficiency, KGE: ð¾ðºð¸ = 1 â â(ð â 1)2 + (ð¼ â 1)2 + (ð½ â 1)2 (2.17) where ?Ì ? denotes the average value of the response variable Y, ð is the linear correlation between ðð and ?Ì?ð, ð½ is the bias, ð¼ denotes a measure of variability. In evaluating predictive models, the highest values of R2, IA, and KGE, as well as the lowest values of MAPE and RMSE, are considered to be indicative of the best 27 predictive model. An ideal model holds values of R2, IA, and KGE equal to 1.0, and values of MAPE and RMSE equal to 0. The goal is for the values of the performance indices to be as close to their ideal values as possible. 2.2.5. Multi-objective optimization Multi-objective optimization refers to a type of optimization problem in which multiple objective functions need to be simultaneously optimized (maximized or minimized), taking into consideration the constraints defined in the problem. The mathematical representation is as follows (see Eq. (2.18) and Eq. (2.19)) (Dhiman & Kumar, 2018): Min/max ðð(ð), ð = 1,2, â¦ , ð (2.18)Subject to: ðð(ð) â¤ 0, ð = 1,â¦ , ð âð(ð) = 0, ð = 1, â¦ , D ð¥â(ð¿) â¤ ð¥â â¤ ð¥â(ð), â = 1, â¦ ,ð» (2.19) where the integer ð â¥ 2 is the number of optimization functions, represented as ðð(ð). The problem also includes ð inequality constraints and D equality constraints, expressed as ðð(ð) and âð(ð), respectively. H is the number of input parameters, each with lower and upper bounds defined as ð¥â(ð¿) and ð¥â(ð), respectively. In MOO, the goal is to optimize multiple conflicting and incommensurable objectives at the same time. In contrast to single-objective optimization, which locates a solution by evaluating the objective function, there is no single feasible solution in MOO that optimizes all objectives simultaneously. Instead, a collection of alternative solutions, referred to as Pareto optimal solutions, is often obtained (Abdullah, David, & Alice, 2006). These solutions are considered optimal because they surpass all other alternatives in the search space when taking all objectives into account. Pareto optimal 28 solutions represent the optimal trade-off between multiple objectives, as no objective can be enhanced without sacrificing at least one other objective (Abdullah et al., 2006). A viable solution ð¥1ââ â â ð is considered to Pareto dominate another solution ð¥2ââââ âð, if two conditions are satisfied as in Eq. (2.20) and Eq. (2.21) (Abdullah et al., 2006): (i) It is not worse than the solution ð¥2ââââ in any objective: ðð(ð¥1ââ â) â¤ ðð(ð¥2ââââ ), âð â {1,2, â¦ , ð} (2.20)(ii) In a minimum of one objective, the solution ð¥1ââ â is strictly better than the solution ð¥2ââââ : ðð(ð¥1ââ â) < ðð(ð¥2ââââ ), âð â {1,2, â¦ , ð} (2.21) The collection of Pareto optimal solutions, also known as the non-dominated solution set of the feasible solution space, is often called Pareto optimal set or Pareto set (PS) (Abdullah et al., 2006), as described by Eq. (2.22). The boundary defined by the set of all solutions mapped from the PS is referred to as the Pareto front (PF), as outlined in Eq. (2.23). ðð = {ðâ â ðð¥|âð â ðð¥: ð¹(ð) âº ð¹(ðâ)} (2.22)ðð¹ = {ð¹(ð) â ðð¦|ð â ð} (2.23) where â represents nonexistence, ð¹(Â·) is the objective function with P objectives, ðâ is Pareto optimal solution, and ðð¥ and ðð¦ are vector sets. In this study, the MOO consisted of 19 objective functions and an equality constraint, with the first objective function addressing the strength, the second objective function considering the cost, and the remaining 17 objective functions comprehensively evaluating the environmental sustainability of the UHPC (based on 17 mid-point impact categories modelled using the ReCiPe 2016 method (Huijbregts et al., 2017)). The objective functions are outlined as follows: 29 a) Uniaxial compressive strength: The primary objective is to maximize the uniaxial compressive strength of concrete at 28 days, which is estimated using a super learner ML model. b) Cost: The second objective aims to minimize material costs for economic benefits. c) Environmental sustainability: Objectives three through nineteen strive to minimize the environmental footprint of the mix for improved sustainability, given the composition of concrete constituents to be known. For this purpose, the harmonized characterization factors at the mid-point from ReCiPe 2016 (Huijbregts et al., 2017) are utilized. The use of harmonized mid-point characterization gives a stronger relation to the environmental impacts. The optimization problem can be mathematically formulated as presented in Eqs. (2.24) to (2.27), to maximize the strength of UHPC in Eq. (2.24), minimize the cost in Eq. (2.25), and minimize the environmental impacts across each of the impact categories in Eq. (2.26), subject to the equality constraint outlined in Eq. (2.27). Max ððâ² = ML model function (ð¶ð, ð¶ðð¹ , ð¶ð¹ð¿ð´, ð¶ð, ð¶ð¿ð, ð¶ð, ð¶ðð, ð¶ðð, ðð´, ð¶ð´) (2.24)Min ð¶ðð ð¡ = âð¶ðð ð¡ððð=1 (2.25) Min Sð = âððð , for ð = 1,2, â¦ ,ð» ðð=1 (2.26) ðð¢ððððð¡ ð¡ð: âð¶ððð=1= 1 (2.27) where ð¶ is the content of constituent materials in m3 per 1 m3 of concrete, and ððð denotes the jth environmental impact for the ith parameter, and H is the total number 30 of environmental impact categories (in this case, H = 17). It is worth noting that the maximum aggregate size is not considered in any objective function except for the first one, resulting in N = 9 in the above objective functions. The cost was calculated as the product of the volume of the material, its unit weight, and the cost per unit kilogram of the material. The environmental impact for each material was determined by multiplying the volume of the material, its unit weight, and the respective characterization factor for each material per unit kilogram of the material obtained from Eco-invent v3.7 Life Cycle Inventory database (Ecoinvent Association, 2023) using the ReCiPe 2016 harmonized life cycle impact assessment method at the mid-point (Huijbregts et al., 2017). The specific gravity and cost per unit kilogram of each constituent material are listed in Table 2.2 based on previous studies (VaitkeviÄius et al., 2014; Yalçınkaya and Yazıcı, 2017; Meng et al., 2018; Abellán, 2020). Table 2.3 provides the environmental impact categories, while the environmental impacts across each impact category for each constituent material are presented in Table 2.4. These impact categories include climate change, depletion of fossil fuels, freshwater eutrophication, human toxicity, ionizing radiation, marine eutrophication, urban land occupation, marine ecotoxicity, photochemical oxidant formation, terrestrial ecotoxicity, natural land transformation, ozone depletion, depletion of metals, terrestrial acidification, particle matter formation, and water depletion. The inclusion of these 17 mid-point impact categories provides a comprehensive overview of the environmental impacts of the optimized UHPC mixture. The equality constraint represents a limiting scenario, which renders a significant portion of the search space infeasible. To address this, the equality constraint in Eq. (2.27) is transformed into an inequality constraint as follows: 31 (âð¶ððð=1â 1)2â ð â¤ 0 (2.28) The constant ð is introduced to smooth the constraint (ð = 10â5 is used in this study). This study utilizes the unified non-dominated sorting genetic algorithm (U-NSGA) (Seada & Deb, 2014), a popular evolutionary optimization method, to solve MOO problems. The performance of the algorithm is evaluated using a hypervolume indicator. This indicator calculates the volume of the dominated region in the objective space occupied by a set of solutions relative to a reference point, which represents the worst possible values for all objectives. Hence, finding the optimal PF involves maximizing the hypervolume indicator. Table 2.2: Specific gravity and cost of constituent raw materials (VaitkeviÄius et al., 2014; Yalçınkaya and Yazıcı, 2017; Meng et al., 2018; Abellán, 2020) Variables Cost ($ USD/Ton) Specific gravity Cement 82 3.14 Fly ash 40 2.70 Silica fume 800 2.20 Ground granulated blast-furnace slag 100 2.90 Limestone powder 120 2.73 Water 1.00 1.00 Superplasticizer 3400 1.05 Quartz powder 800 2.67 32 Table 2.3: Overview of the midpoint impact categories and related units Mid-point impact category Mid-point characterization factor Unit Climate Change Global warming potential (GWP) kg CO2-Equivalents Freshwater Eutrophication Freshwater eutrophication potential (FEP) kg P-Eq to freshwater Freshwater Ecotoxicity Freshwater ecotoxicity potential (FETP) kg 1,4-DCB-Equivalents Human Toxicity Human toxicity potential (HTPnc) kg 1,4-DCB-Equivalents to urban air Fossil Depletion Fossil Depletion potential (FDP) kg oil-Equivalents Ionizing Radiation Ionizing radiation potential (IRP) kBq Co-60-Equivalents to air Marine Ecotoxicity Marine ecotoxicity potential (METP) kg 1,4-DCB-Equivalents to marine water Marine Eutrophication Marine eutrophication potential (MEP) kg N-Equivalents Metal Depletion Metal depletion potential (MDP) kg Fe-Equivalents Natural Land Transformation Natural land transformation potential (NLTP) m2 Terrestrial Ecotoxicity Terrestrial ecotoxicity potential (TETPinf) kg 1,4-DCB-Equivalents Ozone Depletion Ozone depletion potential (ODPinf) kg CFC-11-Equivalents Particulate Matter Formation Particulate matter formation potential (PMFP) kg PM10-Equivalents Urban Land Occupation Urban land occupation potential (ULOP) m2 Photochemical Oxidant Formation Photochemical oxidant formation potential (POFP) kg NMVOC Terrestrial Acidification Terrestrial acidification potential (TAP500) kg SO2-Equivalents Water Depletion Water depletion potential (WDP) m3 33 Table 2.4: Environmental mid-point impacts calculated per the ReCiPe 2016 framework Name of the impact Environmental mid-point impacts Cement Silica fume Fly ash Limestone powder Superplasticizer Quartz powder Climate change 0.22642 0.00305 0.59124 0.66042 1.1011 0.03457 Ionising radiation 0.00838 0.00023 0.01523 0.01548 0.07574 0.0012 Freshwater ecotoxicity 0.00164 3.05e-05 0.00352 0.00365 0.03749 0.00029 Fossil depletion 0.0351 0.00114 0.06756 0.07385 0.561 0.00938 Freshwater eutrophication 3.8e-05 3.24e-07 7.36e-5 7.69e-05 0.0003 7.48e-06 Marine eutrophication 0.00024 7.13e-06 0.00049 0.00056 0.00088 7.03e-05 Human toxicity 1.5118 0.02303 2.6878 2.8118 14.998 0.39606 Metal depletion 0.00401 0.00017 0.0086 0.00948 0.08771 0.00074 Marine ecotoxicity 1.5107 0.02683 2.7873 2.9132 21.564 0.3611 Photochemical oxidant formation 0.00068 2.19e-05 0.00138 0.00157 0.00316 0.00021 Urban land occupation 0.01421 0.00021 0.01009 0.01727 0.01049 0.01319 Natural land transformation 7.97e-5 1.12e-06 4.33e-5 -4.65e-06 0.00019 7.82e-05 Terrestrial ecotoxicity 0.00021 1.75e-05 0.0002 0.00023 0.00146 7.09e-05 Particulate matter formation 0.00033 8.00e-06 0.00061 0.00068 0.0018 9.57e-05 Ozone depletion 8.15e-9 4.87e-10 1.62e-8 1.85e-08 1.57e-07 1.77e-09 Terrestrial acidification 0.00075 1.76e-05 0.00148 0.00165 0.00416 0.00025 Water depletion 0.00487 3.44e-06 0.00076 0.00084 0.00385 0.00592 2.2.6. Approaches to decision-making in multi-objective optimization scenarios In the realm of MOO, it is impossible to determine a single optimal solution due to the inherent nature of trade-offs among multiple conflicting objectives. However, multi-criteria decision-making (MCDM) techniques enable the identification of an 34 appropriate solution from the PF. Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), a technique that prioritizes solutions based on their proximity to the ideal solution, is commonly employed for this purpose (Triantaphyllou, 2010). The TOPSIS identifies the optimal solution by selecting the alternative that is nearest to the positive ideal solution (best case condition) and furthermost from the negative ideal solution (worst case condition) within the PF. The methodology of TOPSIS comprises the following steps (Triantaphyllou, 2010): 1) Generation of an evaluation or decision matrix, ð¥ðð, derived from the PFs. 2) Normalization of the matrix to form a normalized matrix. 3) Assignment of weights to each objective function. 4) Calculation of the weighted normalized decision matrix as a product of the value of the element in the normalized matrix and the corresponding weight of each objective function. 5) Determination of the positive ideal solution (ð´ð) and negative ideal solution (ð´ð¤) based on the following equations (Triantaphyllou, 2010): ð´ð = {â©ððð(ð¡ðð|ð = 1,2, â¦ ,ð)|ð â ð½ââª, â©ððð¥(ð¡ðð|ð = 1,2, â¦ ,ð)|ð â ð½+âª}â¡ {ð¡ðð|ð = 1,2, â¦ , ð} ((2.29) ð´ð¤ = {â©ððð¥(ð¡ðð|ð = 1,2, â¦ ,ð)|ð â ð½ââª, â©ððð(ð¡ðð|ð = 1,2, â¦ ,ð)|ð â ð½+âª}â¡ {ð¡ð¤ð|ð = 1,2, â¦ , ð} ((2.30) where n denotes the number of objective functions, i represents a solution point in the PF, ð½+ = {ð = 1,2, â¦ , ð|ð} represents the set of criteria or objective functions having a positive impact, and ð½â = {ð = 1,2, â¦ , ð|ð} represents the set of criteria with negative impact, commonly referred to as cost criteria. 35 6) Determination of the Euclidean distance for target alternative ð in relation to the best and worst conditions based on the following equations (Triantaphyllou, 2010): ððð¤ = ââ(ð¡ðð â ð¡ð¤ð)2ðð=1, ð = 1, 2, â¦ . ,ð (2.31) ððð = ââ(ð¡ðð â ð¡ðð)2ðð=1, ð = 1, 2, â¦ . ,ð (2.32) where ð¡ð¤ð and ð¡ðð are the worst and best values, respectively, of the jth objective, as determined in Step (5), and ððð¤ and ððð represent the Euclidean distance between the target alternative ð and the worst and the best conditions, respectively. 7) Computation of the similarity to the worst condition, ððð¤ based on the following equation (Triantaphyllou, 2010): ð ðð¤ =ððð¤ððð¤ + ððð, 0 â¤ ð ðð¤ â¤ 1, ð = 1, 2, â¦ . ,ð (2.33) A high ð ðð¤ implies that the alternative solution is closer to the best condition, while a value of zero represents the solution with the worst condition. 2.3. Results and Discussion 2.3.1. Predictive performance of the models The hyperparameters of DT, AdaBoost, GBM, and XGB are tuned and validated through grid search and 10-fold cross-validation taking RMSE as the evaluation index, and results are presented in Table 2.5. 36 Table 2.5: Optimized hyperparameter values for various machine learning models Model Hyperparameter Optimized value Decision tree Maximum features 10 Maximum depth of the tree 7 Adaptive boosting Base learner Decision tree Maximum depth of the tree 8 Number of estimators (DTs) 35 Maximum features for base learner 4 Learning rate 0.2 Gradient boosting machine Number of estimators 100 Maximum depth of the tree 7 Maximum features 3 Learning rate 0.1 Extreme gradient boosting Number of estimators 79 Maximum depth of the tree 8 Learning rate 0.2 Column sample by tree 1.0 Column sample by level 0.9 Column sample by tree 1.0 For the DT model, the optimal maximum depth of the tree was found to be 7, and the optimal number of maximum features was 10. In the case of AdaBoost, the base learner used was a DT, with 35 estimators (DTs). The optimal learning rate was 0.2, and the optimal maximum tree depth and input features for the base learner were 8 and 4, respectively. For the GBM, the optimal number of estimators was 100, with a maximum depth of the tree of 7, a learning rate of 0.1, and optimal maximum features of 3. For the XGB model, the optimal number of estimators was 79, with a maximum depth of the tree of 8 and a learning rate of 0.2. The optimal values for column sample by node, column sample by level, and column sample by the tree were 1.0, 0.9, and 1.0, respectively. The optimized super learner model integrates the fine-tuned DT, GBM, and XGB through the implementation of SVR with the linear kernel as a meta-model, which was determined through a preliminary study. 37 Figure 2.4(aâd) show the scatter plots displaying the comparison of experimental compressive strength (ðð,ðð¥ð) and predicted compressive strength (ðð,ðððð) based on DT, AdaBoost, GBM, XGB, and super learner models, respectively. The results demonstrate that all models except DT exhibit a reasonable level of prediction accuracy for the uniaxial compressive strength of UHPC (R2 â¥ 86%). This emphasizes the advantages of utilizing ensemble and advanced ML models for accurately predicting the uniaxial compressive strength of UHPC, in contrast to single models like DT. Figure 2.4a highlights the limitations of the DT model in terms of its ability to produce accurate predictions. It is evident from this figure that a substantial number of the samples are either overestimated or underestimated to a significant degree. This result suggests that the model may be oversimplified, failing to capture the complex relationships between input variables and the uniaxial compressive strength of UHPC. In contrast, the AdaBoost, GBM, and XGB models, which are tree-based boosting ensemble models, showed remarkable improvement in their prediction accuracy as compared to the DT model. Among these models, XGB showed the best prediction capability compared to AdaBoost and GBM, as shown in Figure 2.4(b-d), with the highest R2 of 89% for the test dataset. The super learner model further enhanced the prediction capabilities of the models by leveraging a combination of various ML algorithms to optimize the predictions. This model obtained an R2 of 95% and 91% for the training and test datasets, respectively. 38 Figure 2.4. Experimental versus predicted 28-day uniaxial compressive strength of UHPC. Table 2.6 provides the statistical results of the performance of the models. Besides, to further demonstrate the efficacy of the developed models, the performance measures are plotted in Figure 2.5(a and b) for the training and test sets, respectively. Among the investigated boosting ensemble models, XGB showed the best performance 39 on the test set with the highest R2 value of 96% and the least RMSE and MAPE of 9.57 MPa and 5.41%, respectively. Also, on the training datasets, the XGB model obtained R2, MAPE, and RMSE values of 96%, 2.75%, and 5.96 MPa, respectively. The performance of the super learner model is significantly improved, with the smallest errors (MAPE and RMSE) and the highest R2. Although XGB has the highest R2 and lowest error in the training dataset, it exhibits a lower R2 and higher error on the test dataset compared to the super learner model. This indicates that the super learner model possesses a superior generalization ability, making it more effective in handling unseen data. Table 2.6: Performance evaluation of the ML models with statistical performance measures Models Training dataset Test dataset MAE (MPa) MAPE (%) RMSE (MPa) ð 2 (%) MAE (MPa) MAPE (%) RMSE (MPa) ð 2 (%) DT 7.71 5.74 10.9 87 10.42 7.39 15.1 73 AdaBoost 6.01 4.49 8.32 92 7.97 5.78 10.9 86 GBM 6.50 4.68 9.01 91 7.60 5.56 9.83 88 XGB 3.87 2.75 5.96 96 7.37 5.41 9.57 89 Super learner 4.84 3.38 6.88 95 6.93 5.01 8.88 91 Figure 2.5. Performance metrics of the ML models based on the training and test datasets. 40 Furthermore, the distribution of residuals for the test dataset in Figure 2.6(a-e) is represented as violin plots in which the x-axis denotes the density of the data, while the y-axis shows the residuals. As can be observed in this figure, reveals that the DT model has the largest residual range of 111.18, a minimum of -77.24, and a maximum of 39.94, indicating lower prediction accuracy. In contrast, ensemble models exhibit smaller residual ranges and minimum and maximum values, suggesting improved accuracy. Specifically, AdaBoost has a 62.86 residual range, a -37.85 minimum, and a 25.1 maximum; GBM has a 51.71 range, a -27.72 minimum, and a 23.99 maximum; XGB has a 55.64 range, a -31.70 minimum, and a 23.94 maximum. The super learner model stands out with the smallest residual range of 45.81, a -25.96 minimum, and a 19.85 maximum, which highlights its superior generalization and predictive performance on the test dataset. The super learnerâs tighter, more symmetric distribution around zero indicates its superior predictive capabilities compared to other models. Figure 2.6. Violin plots of residual distributions for test dataset. From the results presented, it can be concluded that the super learner model 41 provides the best predictive performance and generalization ability. Therefore, it is utilized as an objective function in the MOO process in the subsequent section. 2.3.2. Multi-objective optimization results A thorough evaluation and optimization of all the 19 objectives that are relevant to the uniaxial compressive strength, cost, and environmental sustainability of the mixture are carried out to achieve an economically feasible and environmentally sustainable UHPC mixture, thereby facilitate the design of sustainable and resilient UHPC structures that meet specific project requirements. The super learner ML model was utilized as the objective function for modelling the uniaxial compressive strength of UHPC in all cases. The MOO resulted in a total of 400 non-dominated optimal mixture proportions (nondominated solutions), as shown in Figure 2.7 using a parallel coordinate plot. Moreover, Figure 2.8 shows the distribution of the optimized sustainable UHPC mixtures, which is determined to best fit normal distribution with a mean of 181.95 MPa and standard deviation of 27.43 MPa based on statistical distribution test. The results of the MOO in Figure 2.7 demonstrate the trade-off among uniaxial compressive strength, cost, and environmental sustainability, and showcase the broad spectrum of solutions that can be obtained for minimizing the environmental impact. This trade-off is a common challenge in MOO problems and highlights the importance of balancing conflicting objectives. This figure also offers valuable insights into the ideal proportions of each constituent material needed to produce the most environmentally sustainable and economical UHPC. For example, the figure illustrates that the optimal cement content in the solutions ranges from 0.06166 m3 to 0.15 m3 and 0.275 m3 to 0.3971 m3 per 1 m3 of UHPC. Using cement content outside these ranges or within the range of 0.15 m3 to 0.275 m3 per 1 m3 of UHPC results in a non-optimal UHPC mixture. 42 Figure 2.7. Parallel line plots of the optimized sustainable UHPC mix. Figure 2.8. Distribution of optimized UHPC mixtures (type: normal distribution). Furthermore, the diverse concrete mixtures represented in the Pareto fronts displayed in Figure 2.7 provide a useful tool for designers and decision-makers to select the optimal UHPC mixture that meets the projectâs specific requirements. The results in Figure 2.7 also emphasize the limitations of determining a unique optimal solution in MOO problems. Despite these limitations, using MCDM techniques such as TOPSIS, it is possible to identify a suitable solution from the Pareto front, which offers a more comprehensive approach to optimizing multiple objectives, as discussed in Section 2.2.6. In TOPSIS, the closer the score is to 1, the more similar the alternative is to the ideal solution and is therefore considered better. The importance weight of 43 each objective function can be established based on the specifications and requirements of the project. By using these weights, MCDM techniques can be applied to identify the project-specific optimal concrete mix design for UHPC among the Pareto fronts. As an example, Table 2.7 presents the single objective optimal solution, which is obtained by disregarding the weight of all objectives except for one of the 19 objective functions in each case. In the subsequent study, the optimized UHPC mixtures will be experimentally investigated. Table 2.7: Single objective optimization results Optimized objective Mixture content (m3/m3) fc (MPa) Cc (m3/m3) CSF (m3/m3) CFLA (m3/m3) CS (m3/m3) CLP (m3/m3) CW (m3/m3) CSP (m3/m3) CQP (m3/m3) SA (μm) CA (m3/m3) Compressive strength 0.2860 0.1510 0.0426 0.0659 0.0366 0.1716 0.0107 0.0944 949.8961 0.1413 236 Cost 0.0781 0.0000 0.0222 0.0669 0.0291 0.2267 0.0026 0.0000 637.9978 0.5725 135 Climate Change 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Fossil Depletion 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Freshwater Ecotoxicity 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Freshwater Eutrophication 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Human Toxicity 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Ionising Radiation 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Marine Ecotoxicity 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Marine Eutrophication 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Metal Depletion 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Natural Land Transformation 0.0635 0.0374 0.0004 0.0728 0.0430 0.2235 0.0027 0.0003 928.1161 0.5542 130 Ozone Depletion 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Particulate Matter Formation 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Photochemical Oxidant Formation 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Terrestrial Acidification 0.0627 0.1361 0.0030 0.0727 0.0033 0.1804 0.0031 0.0139 678.2653 0.5223 174 Terrestrial Ecotoxicity 0.0635 0.1358 0.0020 0.0679 0.0075 0.1783 0.0027 0.0006 636.7333 0.5443 174 Urban Land Occupation 0.0635 0.1358 0.0020 0.0679 0.0075 0.1783 0.0027 0.0006 636.7333 0.5443 174 Water Depletion 0.0635 0.1358 0.0020 0.0679 0.0075 0.1783 0.0027 0.0006 636.7333 0.5443 174 44 2.3.3. FAI-OSUSCONCRET: Fast, accurate, and intelligent prediction tool and multi-objective optimization of sustainable UHPC For the practical application of the ML predictive model and MOO algorithm developed in this study for UHPC, a user-friendly GUI-based software called FAI-OSUSCONCRET has been created. FAI-OSUSCONCRET (Fast, Accurate, and Intelligent prediction and multi-objective Optimization of SUStainable ultra-high-performance CONCRETe) offers fast, accurate, and intelligent predictions and multi-objective optimizations tailored to specific project requirements, and enables the attainment of a UHPC mixture that meets the specific requirements of any project and UHPC application. The GUI of FAI-OSUSCONCRET is illustrated in Figure 2.9 and can be accessed via the URL: https://github.com/twakjira/Optimization-of-Sustainable-UHPC. The optimal solution is determined based on the projectâs specifications for UHPC strength, cost, and environmental impact. The user can choose from two approaches, as indicated by Option 1 and Option 2 in the software (Figure 2.9). Option 1 allows the user to define a target compressive strength, and the software determines the optimized sustainable and economical UHPC mixture. Simply input the desired 28-day compressive strength, and the software will provide the optimized mixture. 45 Figure 2.9. User-friendly FAI-OSUSCONCRET software tool for strength prediction and MOO of sustainable UHPC mixture tailored to project-specific requirements. Option 2 allows the user to input weight for each of the 19 objective functions, and the software uses TOPSIS to provide the optimized UHPC mixture based on the user-defined weights. The weights must add up to one, and each weight should be separated by a comma. The weight of each objective function can be specified by the user based on the projectâs requirements for UHPC strength, cost, and environmental sustainability. The software will then provide the optimal UHPC mixture that perfectly meets the projectâs specifications. For instance, as shown in Figure 2.9, Option 2 is used by assigning equal weight to each of the 19 objective functions, and the software provides the optimized UHPC mix. In addition to optimization capabilities, it also offers a prediction of the 28-day compressive strength for UHPC concrete mixtures. To use this feature, simply select the âPredict 28-day Compressive Strengthâ button, enter the values of the input parameters (mixture content (in m3/m3) and ðð´ (in μm)) separated 46 by commas in the prompted window, and the software will provide the predicted 28-day compressive strength using the developed super learner model. Thus, the FAI-OSUSCONCRET is a valuable tool for both strength prediction and MOO of sustainable UHPC mixtures. It offers a reliable combination of accuracy, speed, and intelligence, and provides results that are aligned with the requirements of any UHPC project. 2.4. Summary This study introduced a novel framework for strength prediction and MOO of cost-effective and environmentally sustainable UHPC. The framework integrates a super learner ML model, metaheuristic MOO algorithms, and MCDM technique to formulate UHPC mixtures that fulfill the unique specifications and requirements of a particular project. Various tree- and boosting-based ensemble models are optimized and integrated into a super learner model for strength prediction. The super learner model is then utilized as one of the objective functions in a MOO problem, which comprises a total of 19 objective functions and one equality constraint. A notable feature is the inclusion of 17 mid-point environmental impact categories under the ReCiPe 2016 framework for a comprehensive environmental sustainability assessment of UHPC mixtures, for the first time. This broad spectrum consideration moves beyond the traditional focus on CO2 emissions, and provides a more accurate and complete evaluation of UHPCâs environmental footprint. The following conclusions can be drawn from the findings of the current study: â¢ The developed tree- and boosting-based ML ensemble models demonstrated a reasonable level of accuracy in predicting the 28-day uniaxial compressive strength of UHPC. 47 â¢ Among the models evaluated for the super learner, the XGB model emerged as the most accurate model, closely followed by the GBM, while the decision tree showed the least predictive accuracy. â¢ The super learner model outperformed all the other models with its highest predictive accuracy and generalization ability. This is evident from its highest R2 value of 91%, lowest error values (MAE = 6.93 MPa, RMSE = 8.88 MPa, and MAPE = 5.01%), and smallest range of residuals centered around zero for the test dataset. Hence, it can be concluded that the developed super learner model is a highly effective and reliable tool for determining the compressive strength of UHPC. â¢ The MOO resulted in a spectrum of optimal solutions that provide a range of concrete mixtures to choose from, which allows the decision-makers to select the ideal UHPC mix based on the project specifications and requirements using MCDM (in this study, the technique used was TOPSIS). In addition, the project specific most optimal solution can also be selected by specifying the desired compressive strength value using the GUI-based software tool developed in this study. The multi-objective optimization and FAI-OSUSCONCRET software tool developed in this study aims to facilitate the development of an economically viable and environmentally sustainable UHPC mixture that meets specific project specifications and requirements. This framework serves as a comprehensive design guide to enable informed decision-making before the initiation of construction. It is important to note that the FAI-OSUSCONCRET applies to a range of UHPC constituent materials that have been considered in this study, based on the available 48 experimental database. However, further research is recommended to assess additional performance objectives, such as slump, flexural strength, and durability of UHPC, given the availability of a large experimental database. Additionally, it is recommended that subsequent experimental research efforts focus on the validation of the optimized mix proportions identified in this study. Moreover, it is worth noting that different parameters such as moisture, particle size distribution, packing density, and temperature can influence the mix design. Therefore, future studies are recommended to consider the influence of these factors on the optimized mixture with the availability of experimental database. 49 CHAPTER 3: PEAK AND ULTIMATE STRESS-STRAIN MODEL OF CONFINED ULTRA-HIGH-PERFORMANCE CONCRETE (UHPC) USING HYBRID MACHINE LEARNING MODEL WITH CONDITIONAL TABULAR GENERATIVE ADVERSARIAL NETWORK 3.1. Introduction It is crucial to establish an accurate UHPC confinement model to enable reliable modelling and design of UHPC structures and their wider adoption, particularly in seismic regions. Previous studies have shown that the stress-strain response of UHPC significantly differs from that of conventional concrete (Hosinieh et al., 2015; Naeimi & Moustafa, 2021a). For example, Hosinieh et al. (Hosinieh et al., 2015) investigated the potential application of existing confinement models designed for FRC and HSC in predicting the axial response of UHPC columns based on the analysis of six experimentally tested UHPC columns. Their findings showed that these models are inadequate for UHPC and underscore the importance of developing confinement models tailored explicitly for UHPC. Similarly, Naeimi and Moustafa (Naeimi & Moustafa, 2021a) assessed the confinement models of different concrete types in their ability to predict the uniaxial compressive response of confined UHPC. Based on the results, they concluded that these models are inadequate in accurately representing the behaviour of confined UHPC. Nevertheless, only limited research has focused on the development of stress-strain models for confined UHPC, despite their significance in modelling UHPC structures. Moreover, the existing models are mainly derived empirically using a very limited experimental database, which limits their accuracy and applicability. In addition, a prior study (Naeimi & Moustafa, 2021a) reported complex interactions between fibers in UHPC and steel spirals or transverse reinforcement and the significant 50 influence of such interactions on the behaviour of confined UHPC, which has been neglected in current models. To better understand this interaction, Naeimi and Moustafa (Naeimi & Moustafa, 2021a) conducted an investigation examining the effects of varying steel fibersâ volumetric ratios of (4%, 2%, and 0%) and steel spiral confinements (2%, 4%, and 8%) on the stress-strain response of confined UHPC. The results revealed that while the incorporation of 2% steel fibers enhanced peak strength, increasing the fiber ratio beyond 2% did not necessarily result in an increase in the compressive strength and was influenced by the presence and volumetric ratio of steel spirals (Naeimi & Moustafa, 2021a). For instance, increasing the fiber volumetric ratio in specimens with moderate spiral volumetric ratios (4%) reduced peak strength (Naeimi & Moustafa, 2021a). Thus, the study (Naeimi & Moustafa, 2021a) underscored the importance of considering the interaction between steel fibers and spirals in UHPC confinement. In addition to the complex interaction between steel fibers and confinement steel reinforcement, the accuracy of the stress-strain models for confined UHPC is hindered by various factors, including the complex nonlinear behaviour of confined UHPC. Advanced computational models, specifically machine learning (ML), have emerged as effective solutions for modelling such complex nonlinear problems. As a result, it has recently gained prominence as an efficient approach for modelling such complex problems (Cakiroglu, Islam, BekdaÅ, Isikdag, & Mangalathu, 2022; Luo & Paal, 2018, 2022; Rahman, Ahmed, Khan, Islam, & Mangalathu, 2021; Wakjira, Rahmzadeh, Alam, & Tremblay, 2022; Zaker Esteghamati & Flint, 2021). Therefore, ML can be used to develop a reliable predictive model for the stress-strain response of confined UHPC. However, the limited availability of experimental data presents a challenge in developing a generalized model for the stress-strain 51 behaviour of confined UHPC. In ML, the term âgeneralizationâ signifies a modelâs ability to accurately process and adapt to new or unseen data assumed to be from the same statistical distribution used to train it. To address the issue of limited datasets, data augmentation techniques such as Generative Adversarial Networks (GANs) can be employed. GANs, particularly the conditional tabular GAN (CTGAN) (Xu, Skoularidou, Cuesta-Infante, & Veeramachaneni, 2019), have proven to be a state-of-the-art solution for generating synthetic tabular data that resembles real data. However, only a few studies have explored the use of CTGAN and other variants of GAN for data augmentation in the field of civil engineering (Almustafa & Nehdi, 2022; Marani & Nehdi, 2022; Marani, Zhang, & Nehdi, 2023). Almustafa and Nehdi (Almustafa & Nehdi, 2022) used CTGAN to generate synthetic data for predicting the maximum displacement of steel FRC beams subjected to far-field blast loading using a limited database of 117 experimental results. In another study, Marani and Nehdi (Marani & Nehdi, 2022) used another variant of GAN, namely, tabular GAN (TGAN), to synthesize a database of the shear strength of concrete beams reinforced with fiber-reinforced polymer. Similarly, Marani et al. (Marani et al., 2023) used TGAN to generate synthetic data to aid the optimization of concrete that incorporates microencapsulated phase change materials. The results of these studies confirmed the efficacy of GAN and its variants in successfully generating synthetic data that resembles the original dataset. Despite its efficiency, to the best of the authorsâ knowledge, no study has applied GAN or its variants for data augmentation in modelling the stress-strain behaviour of confined UHPC to overcome the problem of a limited experimental database. Therefore, this study aims to develop a hybrid ML model enhanced with the next-generation optimization framework, Optuna, optimized CTGAN to accurately 52 predict the peak and ultimate stress-strain responses of confined UHPC, in order to aid accurate modelling of UHPC structures. For this purpose, a database of UHPC confined with normal or high-strength steel is first compiled. Next, the collected experimental data is utilized to generate a large set of synthetic confined UHPC samples using CTGAN optimized with Optuna. With this augmented dataset, an Optuna-optimized hybrid ML model is developed to model the stress-strain responses of confined UHPC at both peak and ultimate states. The prediction accuracy of the developed hybrid ML model is compared against a range of widely used ML models. Moreover, a user-friendly software tool is developed for the practical implementation of the developed model. 3.2. Methodology Figure 3.1 shows a simplified flowchart that illustrates the methodology adopted in this study, which will be discussed in the following subsections. Figure 3.1: Methodology flowchart. 3.2.1. Data Collection As shown in Figure 3.1, the initial step involves compiling a comprehensive experimental database of stress-strain responses for confined UHPC. A thorough literature review was conducted to collect experimental data on UHPC confined with 53 NSS or HSS reinforcement. The database was assembled from 22 published articles (Aboukifa, Moustafa, Itani, & Naeimi, 2019; W. Chang, Hao, & Zheng, 2020; W. Chang, Zheng, & Chen, 2021; W. Chang, Zheng, & Hao, 2022; Y. Chang, Shi, Hou, & Lu, 2021; Deng & Yao, 2020a, 2020b; Ding, Zhou, Wei, Huang, & Tian, 2022; Hosinieh et al., 2015; Hung & Yen, 2021; Kristombu Baduge, Mendis, Ngo, Portella, & Nguyen, 2018; Li, Deng, & Yao, 2020a; Ma, Ma, Xing, & Liu, 2021; Naeimi & Moustafa, 2021b; Shan et al., 2022; H. O. Shin, Yoon, Cook, & Mitchell, 2015; H. O. Shin, Yoon, Lee, Cook, & Mitchell, 2015; Hyun Oh Shin et al., 2016; Hyun Oh Shin, Min, & Mitchell, 2017, 2018; Sugano, Kimura, & Shirai, 2007; X. Yang, Zohrevand, & Mirmiran, 2016), incorporating 320 UHPC specimens confined with NSS or HSS reinforcement. Nonetheless, some studies did not provide the complete stress-strain response at peak and ultimate states, leading to incomplete data. Additionally, a few cases reported confined UHPC compressive strength lower than the unconfined compressive strength, which were consequently excluded from the database as outliers. The refined database comprised 211 confined UHPC specimens for peak stress (ððð) and 254 specimens for the corresponding peak strain (ððð), as well as 145 specimens for ultimate stress (ððð¢) and 116 specimens for the corresponding ultimate strain (ððð¢). To overcome the challenges of limited data, a state-of-the-art CTGAN technique optimized with Optuna was employed to generate synthetic confined UHPC samples. The input features comprise five parameters identified through a literature review. These include cross-sectional shape (rectangular or circular), transverse reinforcement ratio (ðð ð£), yield strength of transverse reinforcement (ðð ð£), compressive strength of unconfined UHPC (ððð), volume fraction of fibers (ð£ð), and effective lateral confinement of transverse reinforcement (ðð), which is determined as per the Mander 54 et al. model (Mander, Priestley, & Park, 1989). Figure 3.2 provides information on the input parameters through a 6 ï´ 6 matrix representation. Within this matrix, the diagonal elements contain histograms that illustrate the individual parameter distributions, while the lower triangular matrix demonstrates the pairwise scatter plots. Figure 3.2: Details of the experimental dataset (C: Circular, R: Rectangular). 3.2.2. Conditional tabular generative adversarial network Advanced synthetic data generators, such as GAN (Goodfellow et al., 2020) can offer an innovative approach to create more significant and enhanced synthetic datasets. Developed by Goodfellow et al. (Goodfellow et al., 2020), these algorithms are capable of generating synthetic data that maintains the quality of the real data by learning its 55 complex distribution. GANs comprise two neural networks, namely, the generator and the discriminator, as shown in Figure 3.3. The generator produces synthetic data and aims to create data that can deceive the discriminator, while the discriminator tries to accurately differentiate between real and synthetic data. GANs are trained to minimize the difference between real and synthetic data distributions. The objective of a GAN is to achieve a point where the generator produces synthetic data that is indistinguishable from the real data that the discriminator cannot differentiate between the two. Once this is achieved, the GAN is considered to have successfully learned the underlying distribution of the data and can be used to generate new high-quality data. Figure 3.3: Illustration of GAN structure (Akiba, Sano, Yanase, Ohta, & Koyama, 2019). Xu and Veeramachaneni (Xu & Veeramachaneni, 2018) introduced a variant of GAN called TGAN, specifically tailored for generating synthetic tabular data while modelling complex distributions. Building on this, Xu et al. (Xu et al., 2019) later developed CTGAN as an enhancement to TGAN. Unlike TGAN, which employs long short-term memory neural networks for GAN development, CTGAN is constructed using fully connected neural networks, which enables it to provide a more robust and 56 flexible framework that facilitates the generation of complex column distributions. The CTGAN training procedure resembles that of the standard GAN, with the generator and discriminator trained alternately. However, CTGAN (Xu et al., 2019) learns complex data distributions by integrating additional information into the generator and discriminator to enable data generation conditioned on specific response variables, which is impossible with a standard GAN. Utilizing a conditional generator, CTGAN effectively tackles the complexities of modelling probability distributions and generating authentic synthetic data for tabular data sets. The efficacy of CTGAN has been assessed on a range of simulated and real-world datasets, and the results demonstrated its capability to produce high-quality synthetic data that closely resembles the distribution of real data (Xu et al., 2019). Moreover, it has been shown to surpass the performance of several existing deep learning approaches, including traditional GANs, TGANs, and Bayesian networks (Xu et al., 2019). Therefore, this study leverages the power of CTGAN to generate high quality realistic synthetic data, for the first time, addressing the problem of limited datasets in confined UHPC stress-strain models. The CTGAN algorithm was fed with the collected experimental peak and ultimate stress-strain responses database to produce a large number of synthetic confined UHPC training data for ML models. 3.2.3. Hybrid machine learning model The hybrid ML model developed in this study is inspired by the error reduction strategy inherent in boosting algorithms. The learning procedure for the proposed hybrid ML model can be outlined as follows: 57 Algorithm: Hybrid ML Model for Improved Prediction Accuracy Require: J = total number of secondary models, ðð|ð=1ð½ 1. Input: Training dataset, primary ensemble model, pool of candidate secondary models (ðð|ð=1ð½). 2. Initialize the primary ensemble model. Compute the predictions of the primary ensemble model (?Ì?ððð ð ) and corresponding residuals, ð ððð ð = ð(ð) â ?Ì?ððð ð , for ð = 1, 2, â¦ ,ð. 3. Initialize j = 1, where J is the total number of secondary models required for enhanced prediction accuracy. 4. Model selection loop (j = 1 to J): While ð = 1 to ð, where P is the total number of iterations, do: a. Sample ð sets of secondary models from a pool of candidate models with diverse learning capabilities. b. Optimize the hyperparameters of the models as well as the weight of each model using ð ððð ð as the training set and the next-generation optimization technique Optuna. c. Train the selected models with optimized hyperparameters. d. Compute the predictions of model j, ?Ì?ðð. e. Evaluate the prediction accuracy using the root mean squared error (RMSE) f. if accuracy improved: Update ?Ì?âð¦ððððð = ?Ì?ððð ð + â ðð?Ì?ðððð=1 , where ðð =ð¤ðâ ð¤ððð=1. Increment p by 1. else: Return to step (a) with a different model type and/or number from the candidate pool. g. end if. end while. Output: Hybrid ML model with improved prediction accuracy. The training process in this model commences with the training of a primary ensemble model, with the extreme gradient boosting (XGB) model selected for this study based on a preliminary investigation, although other ensemble models could be utilized. The residuals from the primary model are then utilized to train subsequent secondary models to overcome the limitations of the primary ensemble model. Each of these secondary models is trained independently with weights assigned to them using Optuna. By aggregating the predictions from all models, the hybrid ML model yields a robust predictive framework for the stress-strain response of confined UHPC at both 58 peak and ultimate states. The proposed hybrid ML model is benchmarked against popular single and ensemble models with varying complexity levels. These include two single models (k-nearest neighbors and SVR) and two ensemble models (gradient boosting machine and XGB). The k-nearest neighbors (KNN) algorithm is designed to predict the relationship between input and response variables by considering the K nearest neighborsâ observations in its dataset. The value of K, which is critical to the algorithmâs performance, should be fine-tuned. An SVR is a widely utilized ML algorithm that generates predictions using structural risk optimization. This approach proves beneficial when dealing with complex nonlinear problems with moderate or small training sets. An SVR utilizes a kernel function to project the original data into a higher-dimensional space, which allows for a linear separation. For a training set comprised of N samples, each with an input vector ð¥ð and a corresponding response variable ð¦ð, the SVR solution can be expressed via a kernel function ð¾(ð¥ð, ð¥), a regularization parameter C, and a bias term b. The solution is determined by Eq. (3.1), with the multipliers ð¼ð and ð¼ðâ² being associated with the lower and upper support vectors (SVs), which are subsets of the training samples. ð(ð¥) = â(ð¼ð â ð¼ðâ)ð¾(ð¥ð, ð¥) + ððð ðð (3.1) Boosting ensemble is an effective type of ensemble model that sequentially combines base learners, typically decision trees (DTs), to form an improved model (Freund & Schapire, 1997). A background on the boosting algorithms is presented in Section 2.2.2.3. 59 3.2.4. Hyperparameter optimization In the context of ML, hyperparameters, which are pivotal elements that define the characteristics and learning process of a model, require careful optimization. This optimization is vital for developing an effective ML model, as it significantly influences the modelâs performance and accuracy. This study employs Optuna (Akiba et al., 2019), a next-generation hyperparameter optimization framework, to optimize the hyperparameters of the models under investigation. Optuna is recognized for its robustness and unique features, which makes it a powerful tool for hyperparameter optimization (Akiba et al., 2019). Its âdefine-by-runâ approach, inherited from deep-learning frameworks, enables dynamic construction of the hyperparameter search space. Additionally, Optuna stands out for its capacity to efficiently handle high-dimensional and non-convex parameter spaces, which is a typical challenge in ML model optimization. This efficiency is facilitated by its âpruningâ feature, which stops the training of trials that are deemed unpromising at an early stage, thus saving computational resources and speeding up the optimization process. By integrating Optuna, this study aims to achieve efficient and effective optimization of model hyperparameters essential for optimal predictive performance. Further details on this framework can be found elsewhere (Akiba et al., 2019). Furthermore, to address the issue of overfitting, a widely used 10-fold cross-validation is integrated with Optuna. This method splits the training dataset into ten mutually exclusive and approximately equal-sized subsets. For each iteration of the validation process, the model is trained on nine subsets and evaluated on the tenth. This process is repeated ten times so that each subset serves as the validation set once. The overall performance of the model is then calculated as the average of its performance on the ten validation sets. 60 3.3. Results and Discussion 3.3.1. Optuna-optimized CTGAN sampling for confined UHPC The hyperparameters for the CTGAN models are optimized using Optuna. These hyperparameters include embedding dimension, number of layers, number of neurons in each layer for the generator and discriminator, separate learning and decay rates for each component, batch size, and number of epochs. The embedding dimension determines the size of the latent space, which is the random noise vector that serves as the input for the generator. The batch size specifies the number of training examples used in a single iteration during the training process. The discriminator steps parameter controls the number of updates made to the discriminator for each generator update, thus maintaining a balanced development of both components during training. The learning rate, set individually for both the generator and discriminator, determines the speed and quality of learning, while the decay rate regulates the adjustment of the learning rate over time, thereby ensuring an efficient gradient descent. Lastly, the number of epochs defines the total number of passes the model makes through the entire training dataset, which is a major determinant of the modelâs ability to learn and generalize. Table 3.1 presents the optimal values of the hyperparameters of CTGAN for the peak and ultimate stress-strain responses. For instance, the optimized CTGAN model for ððð features an embedding dimension of 120, two hidden layers for the generator with 510 and 240 neurons, three hidden layers for the discriminator with 560, 160, and 320 neurons, learning rates of 0.000111 and 0.001899 for the generator and discriminator, respectively, weight decay rates of 0.000032 and 0.000942 for the generator and discriminator, respectively, a discriminator step of 1, a batch size of 140, and 244 epochs. 61 Table 3.1 Optuna Optimized Hyperparameters of CTGAN Hyperparameters Stress-strain responses Peak stress Peak strain Ultimate stress Ultimate strain Embedding dimension 120 220 150 60 Generator Number of hidden layers 2 4 3 2 Number of neurons in each layer (510, 240) (270, 170, 50, 60) (410, 160, 60) (370, 320) Learning rate 0.0001108 0.000108 0.0001449 0.0001155 Decay rate 3.1777e-5 0.0009967 3.3138e-5 0.0001441 Discriminator Number of hidden layers 3 4 3 3 Number of neurons in each layer (560, 160, 320) (370, 400, 60, 490) (280, 200, 450) (340, 540, 550) Learning rate 0.001899 0.0014487 0.0061366 0.003367 Decay rate 0.0009416 0.0004468 8.6049e-5 2.0479e-6 Steps 1 2 1 2 Batch size 140 270 120 170 Number of epochs 244 175 207 272 As listed in Table 3.1, all optimized CTGAN models exhibit moderately complex architectures to enable the generator capture complex patterns within the data, while the discriminator concentrates on distinguishing differences between generated and actual samples. A balance between convergence speed and the risk of overshooting the optimal solution is achieved through careful selection of learning rates for both components, with weight decay rates optimized for model regularization and preventing overfitting. Optimal values for batch size and the number of discriminator steps enhance the training process, and the chosen number of epochs provides sufficient time for the generator and discriminator to learn. These hyperparameter configurations play a crucial role in the overall performance and stability of the CTGAN models in generating synthetic confined UHPC samples that closely resemble the original dataset. The optimized CTGAN models are employed to generate 500 synthetic datasets in each case, conditioned on the experimental stress-strain responses of confined UHPC at peak and ultimate states. This value was chosen for its ability to enhance the diversity of the augmented dataset without disproportionately dominating the original data, as 62 evidenced in the preliminary investigation, where it significantly improved the modelâs learning capacity. The quality of the generated datasets was tested by comparing their distributions with that of the original dataset. Figure 3.4 through Figure 3.7 compare the empirical cumulative distribution functions (ECDFs) for peak and ultimate stress-strain responses derived from the original and synthesized datasets. These figures clearly illustrate that the distribution of each input variable as well as response variables in the synthesized confined UHPC samples closely resembles that of the original one. This result indicates that the optimized CTGAN has successfully captured the distribution of the experimental stress-strain responses at both peak and ultimate states. Consequently, the synthetic data samples it generated are representative of the original stress-strain response dataset.63 Figure 3.4: Distributions of original and synthesized data for modelling peak stress. Figure 3.5: Distributions of original and synthesized data for modelling peak strain. 64 Figure 3.6: Distributions of original and synthesized data for modelling ultimate stress. Figure 3.7: Distributions of original and synthesized data for modelling ultimate strain. 65 3.3.2. Optuna-optimized, CTGAN-augmented hybrid ML model for accurate stress-strain responses prediction of confined UHPC The proposed hybrid ML model, enhanced with CTGAN, is investigated to model the peak and ultimate stress-strain responses of both NSS and HSS confined UHPC. The hybrid ML model is optimized with the help of the next-generation optimization framework, Optuna, using RMSE as the evaluation metric. Based on a preliminary investigation, XGB is selected as a primary model in the hybrid ML model. The quantity and variety of the secondary models, along with their weight in the hybrid ML model are optimized using Optuna. As presented in Table 3.2, three different secondary models are used in the hybrid ML model. While detailed descriptions of each hyperparameter are available elsewhere, a brief overview of the hyperparameters for the primary model, XGB, is presented here. Key among these hyperparameters is the number of estimators, which determines the number of DTs in the XGB ensemble model. The depth of this individual DT is controlled by the maximum depth parameter, which allows the model to capture more complex relationships within the data. The extent to which each DT influences the boosting process is determined by the learning rate, with a lower learning rate potentially requiring more iterations for the model to converge. Besides, randomness is introduced into the model using the subsamples parameter, which determines the fraction of samples used in each DT, thereby reducing the chances of overfitting. The L2 regularization term, expressed as âreg_lambdaâ in XGB, mitigates overfitting by adding a penalty to the modelâs complexity. Conversely, âalphaâ represents the L1 regularization term, which encourages sparsity in the model by pushing the coefficients of less significant features toward zero. These hyperparameters, alongside those of the secondary models and their corresponding weights, are optimized using Optuna and the results are presented in Table 3.2. 66 Table 3.2 Optuna-optimized hybrid ML models for the peak and ultimate stress-strain responses Hybrid ML model Stress-strain responses Peak stress Peak strain Ultimate stress Ultimate strain Primary model Model type XGB XGB XGB XGB Hyperparameters Number of estimators 256 482 436 407 Maximum depth 7 5 4 9 Learning rate 0.05057 0.03888 0.09983 0.11680 Subsample 0.51107 0.93618 0.92711 0.59608 Column Subsampling by tree 0.84412 0.85477 0.92813 0.90391 Alpha 0.01464 0.03622 0.01997 0.62800 Reg_lambda 0.91186 0.24931 0.04377 0.33742 Minimum child weight 3 4 8 6 Secondary models Model type GBM GBM GBM GBM Model weight 0.673 0.486 0.013 0.387 Hyperparameters Number of estimators 118 88 477 127 Maximum depth 10 28 21 5 Minimum samples split 3 2 4 9 Minimum sample leaf 1 1 8 5 Learning rate 0.05094 0.26827 4.74e-5 0.16810 Subsample 0.75992 0.48916 0.46973 0.67312 Model type SVR SVR SVR SVR Model weight 0.126 0.024 0.413 0.086 Hyperparameters C 18.4326 785.42 93.238 75.2955 Epsilon 0.00043 0.0255 0.0195 0.00837 ð¾ 0.11947 3.5326 1.9997 11.1114 Kernel type RBF RBF RBF RBF Model type KNN KNN KNN KNN Model weight 0.201 0.490 0.575 0.528 Hyperparameters Leaf size 7 9 6 12 K 1 4 1 29 RBF: Radial basis function 67 The performance of the optimized hybrid ML model in predicting the stress-strain response of confined UHPC is evaluated using four statistical performance measures listed in Table 3.3. Based on these four metrics, the results of the hybrid ML model evaluations are listed in Table 3.4 for both peak and ultimate stress-strain responses and are further discussed in the subsequent subsection. Table 3.3 Statistical performance metrics Performance metrics Formulation Ideal value Root mean squared error (RMSE) ð ððð¸ = â1ðâ(ð¦ð â ð¦?Ì?)2ðð=1 RMSE = 0 Mean absolute percentage error (MAPE) ðð´ðð¸ =1ðâ |ð¦ð â ð¦?Ì?ð¦ð|ðð=1 MAPE = 0 Mean absolute error (MAE) ðð´ð¸ =1ðâ |ð¦ð â ð¦?Ì?|ðð=1 MAE = 0 Coefficient of determination (R2) ð 2 = 1 ââ (ð¦ð â ð¦?Ì?)2ðð=1â (ð¦ð â ?Ì ?)2ðð=1 R2 = 100% Multivariate RMSE (MRMSE) MRMSE = ââ1ððâ (ð¦ð,ðð â ð¦ð,ð?Ì? )2ððð=1ðð=1 MRMSE =0 ð¦ð = actual response value of ith sample; ð¦?Ì? = predicted response value of ith sample; ?Ì ? = 1ðâ ð¦ððð=1 ; m is the number of response variables; ð¦ð,ðð = normalized actual response value of ith sample of kth response variable; ð¦ð,ð?Ì? = normalized predicted response variable of ith sample of kth response variable 3.3.2.1. Predictive performance of the hybrid ML model Peak stress and the corresponding strain of confined UHPC are crucial parameters in modelling UHPC structures. Scatter plots of the actual (ððð,ðð) versus predicted (ððð,ðð) peak stress based on the proposed hybrid ML model is shown in Figure 3.8a, while Figure 3.8b shows the actual (ððð,ðð) versus predicted (ððð,ðð) strain at peak. A 45° solid line (y = x) in these plots represents the perfect match between the predicted and actual values of stress-strain responses at peak. The coefficient of determination for the training and test datasets is also shown in each plot. As can be observed in Figure 3.8a and Figure 3.8b, the proposed hybrid ML model exhibits high 68 predictive performance with regard to the peak stress and corresponding strain of confined UHPC. The scatter plots of actual versus predicted peak stress in Figure 3.8a show a clear concentration along the line of a perfect fit, which shows the modelâs excellent prediction capabilities. As listed in Table 3.4, the model yielded an R2 of 99.68%, RMSE of 2.504 MPa, MAE of 1.230 MPa, and MAPE of 0.643% for the training dataset. On the test dataset, the model achieved an R2 of 95.94, RMSE of 9.105 MPa, MAE of 6.681 MPa, and MAPE of 3.556%. When predicting peak strain, the model maintains its high performance, as shown in Figure 3.8b and Table 3.4, with an R2 of 98.96%, RMSE of 0.131â°, MAE of 0.057â°, and MAPE of 1.039% for the training dataset, along with an R2 of 93.01%, RMSE of 0.334â°, MAE of 0.257â°, and MAPE of 4.852% for the test dataset. Figure 3.8: Comparison of the predicted and actual values for (a) the peak stress and (b) the peak strain based on the hybrid ML model. 69 Table 3.4 Performance evaluation of the hybrid ML models Stress-strain response Training dataset Test dataset 1 RMSE (MPa/â°) 1 MAE (MPa/â°) MAPE (%) R2 (%) 1 RMSE (MPa/â°) 1 MAE (MPa/â°) MAPE (%) R2 (%) Peak state ððð 2.504 1.230 0.643 99.68 9.105 6.681 3.556 95.94 ððð 0.131 0.057 1.039 98.96 0.334 0.257 4.852 93.01 Ultimate state ððð¢ 3.017 1.870 1.143 99.67 9.609 7.438 4.624 97.22 ððð¢ 0.502 0.260 3.102 97.91 0.636 0.000 6.271 96.52 1 MPa for ððð and ððð¢ and â° for ððð and ððð¢ Furthermore, MRMSE is used to simultaneously evaluate the prediction capabilities of the model in predicting the stress-strain response based on the normalized dataset of (ððð,ðð, ððð,ðð) versus (ððð,ðð, ððð,ðð) for the stress-strain response at peak state as well as (ððð¢,ðð, ððð¢,ðð) versus (ððð¢,ðð, ððð¢,ðð) for the stress-strain responses at the ultimate state. The model resulted in a very small MRMSE value of 0.0115 (training set) and 0.0634 (test set) for the peak stress-strain response and 0.0274 (training dataset) and 0.0395 (test dataset) for the ultimate stress-strain response. Similarly, the proposed hybrid ML model has demonstrated a high level of accuracy in predicting the ultimate stress-strain response of confined UHPC. The scatter plots in Figure 3.9(a and b), which compare the predicted and actual ultimate stress and ultimate strain, respectively, reveal a significant agreement between the predicted and actual values. Furthermore, as presented in Table 3.4, the model achieved high R2 values of 97.22% and 99.67% for predicting the ultimate stress on the test and training datasets, respectively. Additionally, the model exhibited low RMSE, MAPE, and MAE values of 3.017 MPa, 1.143%, and 1.870 MPa, respectively, on the training dataset, and 9.609 MPa, 4.624%, and 7.438 MPa, respectively on the test dataset, suggesting a close agreement between the predicted and actual ððð¢. Similarly, in predicting the ultimate strain, the model achieved a high R2 value of 96.52% on the test 70 dataset and 97.91% on the training dataset. These findings substantiate the predictive efficiency of the proposed hybrid ML model and highlight its ability to capture the complex relationship between the stress-strain responses of confined UHPC and the predictors. Figure 3.9: Comparison of predicted and actual values for (a) the ultimate stress and (b) the ultimate strain based on the hybrid ML model. 3.3.2.2. Comparison with single and ensemble models To further demonstrate the predictive efficacy of the proposed hybrid ML model, a comparison was made with four widely used single and ensemble models. The hyperparameters of the single and ensemble models are optimized using Optuna, and the results are presented in Table 3.5. Figure 3.10 and Figure 3.11 show the scatter plots of predicted versus actual stress and strain responses of confined UHPC at peak state, respectively, based on the optimized models. In addition, the spider plots in Figure 3.12 illustrate the predictive performance of the models on the test dataset using the four performance metrics. 71 Table 3.5 Optuna-optimized hyperparameters of single and ensemble models Models Hyperparameters Stress-strain responses Peak stress Peak strain Ultimate stress Ultimate strain Single models KNN Leaf size 5 46 44 68 K 16 12 7 6 SVR C 3.54167 0.78852 0.44781 2.11895 Epsilon 0.03503 0.03828 0.06882 0.02723 Kernel type RBF RBF RBF RBF Ensemble models GBM Number of estimators 440 385 359 400 Maximum depth 8 6 4 8 Minimum samples split 4 3 4 5 Minimum sample leaf 5 4 5 5 Learning rate 0.01929 0.0338 0.0788 0.01549 Subsample 0.51298 0.5778 0.6383 0.52442 XGB Number of estimators 408 482 436 361 Maximum depth 7 5 4 7 Learning rate 0.02074 0.03888 0.09983 0.03828 Subsample 0.60066 0.93618 0.92711 0.91356 Column Subsampling by tree 0.92152 0.85477 0.92813 0.81034 Alpha 0.00080 0.03622 0.01997 0.14771 Reg_lambda 2.25332 0.24931 0.04377 4.4e-05 Minimum child weight 5 4 8 10 1 Radial basis function Figure 3.10: Comparisons of the predicted and actual peak stress based on various single and ensemble models. 72 Figure 3.11: Comparison of the predicted and actual peak strain values using various single and ensemble models. As can be seen in Figure 3.10(a and b) and Figure 3.11(a and b), the single models were unable to capture the complex nonlinear relationship between the peak stress-strain responses and their predictors. This inadequacy necessitated the use of more advanced and complex ML models, as can also be observed in Table 3.6. The ensemble models, specifically GBM and XGB, demonstrated significantly improved performance over the individual models when predicting the peak stress-strain responses, as seen in Figure 3.10 and Figure 3.11, Table 3.6, and the spider plots in Figure 3.12. (a) Peak stress (b) Peak strain Figure 3.12: Comparative analysis of various models' performance in predicting the peak stress-strain response. 73 Similar results can be observed in Figure 3.13 and Figure 3.14, which show the predicted versus actual ultimate stress and strain responses, respectively, as well as the performance metrics in Table 3.6 and Figure 3.15. Among all models, the hybrid ML model emerged as the most effective one, surpassing both single and ensemble models in predicting the strain-strain responses at both peak and ultimate states, as shown in Figure 3.12 and Figure 3.15, as well as in Table 3.6. With the lowest RMSE, MAPE, and MAE values and the highest ð 2 scores across both the training and test datasets, the hybrid ML model demonstrated superior predictive performance in predicting the peak and ultimate stress-strain responses of confined UHPC. Figure 3.13: Comparison of the predicted and actual ultimate stress values using various single and ensemble models. Figure 3.14: Comparison of the predicted and actual ultimate strain values using various single and ensemble models. 74 Table 3.6 Performance measures of the single and ensemble models Stress-strain response Model Predictive performance Training dataset Test dataset 1 RMSE (MPa/â°) 1 MAE (MPa/â°) MAPE (%) R2 (%) 1 RMSE (MPa/â°) 1 MAE (MPa/â°) MAPE (%) R2 (%) ððð KNN 18.74 13.69 7.40 82.07 19.66 14.44 7.842 81.06 SVR 14.73 10.35 5.60 88.92 18.42 11.73 6.355 83.38 GBM 6.849 3.251 1.650 97.61 11.15 7.407 3.847 93.91 XGB 6.653 3.118 1.592 97.74 11.01 7.363 3.887 94.07 Hybrid 2.504 1.230 0.643 99.68 9.105 6.681 3.556 95.94 ððð KNN 0.854 0.583 10.81 55.57 0.816 0.608 10.85 58.22 SVR 0.806 0.501 9.088 60.44 0.761 0.539 9.731 63.71 GBM 0.182 0.097 1.801 97.99 0.412 0.319 5.931 89.37 XGB 0.182 0.098 1.832 97.99 0.360 0.272 5.114 91.86 Hybrid 0.131 0.057 1.039 98.96 0.334 0.257 4.852 93.01 ððð¢ KNN 29.81 21.54 12.60 67.72 33.24 24.80 14.06 66.70 SVR 25.92 18.92 11.33 75.59 26.71 20.04 11.58 78.49 GBM 4.669 3.016 1.807 99.21 12.46 9.121 5.338 95.32 XGB 3.622 2.164 1.292 99.52 10.09 7.348 4.498 96.93 Hybrid 3.017 1.870 1.143 99.67 9.609 7.438 4.624 97.22 ððð¢ KNN 1.785 1.170 13.65 73.48 2.332 0.002 17.59 53.25 SVR 1.537 0.960 11.18 80.33 1.963 0.001 15.38 66.89 GBM 0.571 0.273 3.099 97.28 0.739 0.001 6.564 95.31 XGB 0.573 0.292 3.446 97.27 0.717 0.001 6.612 95.58 Hybrid 0.502 0.260 3.102 97.91 0.636 0.000 6.271 96.52 1 MPa for ððð and ððð¢ and â° for ððð and ððð¢ (a) Ultimate stress (b) Ultimate strain Figure 3.15: Comparative analysis of the performance of various models in their prediction of the ultimate stress-strain response. For a more thorough evaluation of the modelâs performance to simultaneously predict both stress and corresponding strain at peak and ultimate states, canonical correlation analysis (CCA) (Wegelin, 2000) is utilized. Canonical correlation analysis is a statistical technique used to investigate the correlation between two sets of 75 multivariate variables (Wegelin, 2000). In the context of this study, the objective of CCA is to examine the correlation between the predicted and actual stress-strain responses, specifically, (ððð,ðð, ððð,ðð) versus (ððð,ðð, ððð,ðð) for stress-strain responses at peak state and (ððð¢,ðð, ððð¢,ðð) versus (ððð¢,ðð, ððð¢,ðð) for stress-strain responses at ultimate state. With the aid of CCA, the linear relationship between the experimental and predicted stress-strain responses can be examined. This method finds the maximum correlation between two sets of variables, in this case, the predicted and actual stress-strain values. The CCA score can be determined using Eq. (3.2): ð¶ð¶ð´ ð ðððð = ððð¥(ðððð(ððð, ððð)) (3.2) where X represents the matrix of predicted stress and strain values, while Y represents the matrix of actual stress and strain values. The vectors a and b are the linear coefficients optimized to maximize the correlation between the predicted and actual sets of these two variables. A score close to 1.0 indicates a strong correlation between the predicted stress-strain values and the corresponding actual values. A CCA score of one signifies a perfect correlation between the predicted and actual stress-strain responses. Table 3.7 lists the values of CCA scores and MRMSE for all models. As listed in this table, the hybrid ML model achieved the lowest MRMSE and the highest CCA score across both the training and test datasets for both the peak stress-strain response (ððð, ððð) and ultimate stress-strain response (ððð¢, ððð¢). This performance confirmed that the hybrid ML model is the most accurate model and had the strongest linear relationship between the predicted and actual stress-strain responses. 76 Table 3.7 Modelsâ performance in predicting the stress-strain responses Stress-strain response Models Training dataset Test dataset 1 MRMSE CCA score 1 MRMSE CCA score Peak state (ððð, ððð) KNN 0.1382 0.667 0.1782 0.695 SVR 0.1428 0.728 0.2094 0.723 GBM 0.0229 0.978 0.1058 0.917 XGB 0.0231 0.979 0.0998 0.932 Hybrid ML 0.0116 0.993 0.0634 0.946 Ultimate state (ððð¢, ððð¢) KNN 0.1132 0.689 0.1384 0.577 SVR 0.1119 0.769 0.1352 0.722 GBM 0.0418 0.983 0.0626 0.956 XGB 0.0366 0.984 0.0428 0.964 Hybrid ML 0.0274 0.988 0.0395 0.970 1 determined based on normalized dataset The enhanced performance of the hybrid ML model can be attributed to its synthesis of different model types, which enabled it to capitalize on the strengths of each individual model while minimizing its weaknesses. For the practical implementation of the proposed framework, a software tool named ConUHPC-HybridOptML-CTGU is developed. This tool features a graphical user interface and utilizes the hybrid ML model optimized with the next-generation optimization framework Optuna and enhanced by CTGAN for predicting the stress-strain response of UHPC confined with either NSS or HSS reinforcement. A visual representation of ConUHPC-HybridOptML-CTGU is provided in Figure 3.16. The developed tool, along with the database containing input features as well as the predicted and actual values of the response variables, is accessible via the following link: https://github.com/twakjira/ConUHPC-HybridOptML-CTGU. 77 Figure 3.16: ConUHPC-HybridOptML-CTGU: A graphical user interface-based software tool for the developed hybrid ML model optimized with Optuna and aided with CTGAN. 3.4. Summary Despite the growing interest in ultra-high-performance concrete due to its desirable properties, a comprehensive stress-strain characterization for confined UHPC remains a challenge. This limitation hinders its efficient design and broader adoption, especially in seismic-prone areas. Accordingly, the current study proposed a hybrid ML model augmented with a state-of-the-art CTGAN and next-generation optimization framework, Optuna, for accurately predicting the peak and ultimate stress-strain responses of UHPC confined with either NSS or HSS reinforcement. Utilizing a limited experimental dataset of confined UHPC stress-strain responses compiled based on a thorough literature review, the Optuna-optimized CTGAN generated synthetic 78 confined UHPC samples. The augmented data is then used to develop a hybrid ML model for predicting peak and ultimate stress-strain responses of confined UHPC. The predictive performance of the developed hybrid ML model is compared with widely used single and ensemble models. The following conclusions can be drawn from the results of the current study: â¢ The comparison of the distributions for the synthetic data generated by Optuna-optimized CTGAN and the original experimental data revealed a close resemblance. This resemblance underscores the effectiveness of CTGAN in addressing the limited database issue related to the stress-strain responses of UHPC confined with NSS or HSS reinforcement. â¢ The proposed hybrid ML model, augmented with CTGAN and optimized using Optuna, effectively and accurately predicted the stress-strain responses of confined UHPC at both peak and ultimate states. â¢ A comparative analysis of the proposed hybrid ML model with other models revealed its superior performance with the lowest RMSE, MAE, and MAPE, and highest R2 scores, which demonstrated the potential of the proposed approach as a robust and powerful predictive tool for the peak and ultimate stress-strain responses of confined UHPC. â¢ Simple single models could not capture the complex nonlinear relationship between the stress-strain responses of confined UHPC and input variables. In contrast, advanced ensemble models demonstrated significantly better predictive capabilities, with XGB outperforming the others. â¢ The results show that the proposed hybrid ML model yielded the lowest MRMSE and the highest CCA score consistently across both the training and test datasets 79 for both (ððð, ððð) and (ððð¢, ððð¢) responses. Therefore, it can be inferred that it is the most accurate model and demonstrates the strongest linear relationship between the predicted and observed stress-strain responses for both peak and ultimate states. â¢ The results confirmed that the proposed framework can accurately predict the stress-strain responses of confined UHPC, irrespective of whether it is confined with NSS or HSS reinforcement. To enable practical implementation, the ConUHPC-HybridOptML-CTGU software tool has been developed, which enables fast, accurate, and intelligent predictions of peak and ultimate stress-strain responses of confined UHPC. 80 CHAPTER 4: HYBRID MACHINE LEARNING MODEL AND PREDICTIVE EQUATIONS FOR COMPRESSIVE STRESS-STRAIN CONSTITUTIVE MODELLING OF CONFINED ULTRA-HIGH-PERFORMANCE CONCRETE (UHPC) WITH NORMAL-STRENGTH STEEL AND HIGH-STRENGTH STEEL SPIRALS 4.1. Introduction Given its exceptional mechanical properties and durability (Abokifa & Moustafa, 2021b), a growing body of research has focused on the various structural applications of UHPC. These include its application in new construction (Aboukifa & Moustafa, 2021; Abushanab, Alnahhal, Sohail, et al., 2021; Elsayed et al., 2022; Xue et al., 2020), as well as in the rehabilitation and strengthening of existing structures (Bertola et al., 2021; Y. Huang et al., 2022; B. Shrestha et al., 2020) as a more durable and resilient alternative to conventional construction materials. The performance and durability of UHPC can be optimized through the addition of fibers to the mix (Y. Chang et al., 2023; Naeimi & Moustafa, 2021b; Hyun Oh Shin et al., 2016; Xue et al., 2020). This is due to the bridging effect provided by the fibers, as well as the reinforcing effect they have on the overall integrity of the concrete mix (Du et al., 2021). Moreover, confinement with properly detailed steel spirals is key to enhancing the performance of UHPC and preventing brittle responses. Accordingly, various studies have focused on experimental evaluation of the stress-strain behaviour of UHPC when confined with steel spirals (W. Chang et al., 2021; Li et al., 2020a; Hyun Oh Shin et al., 2018). However, despite the favorable mechanical characteristics of UHPC including high compressive strength, toughness, deformation capacity, and durability (Amran et al., 2022; Du et al., 2021), there is a notable research gap that hinders its broader application, particularly the lack of a reliable stress-strain 81 constitutive model for confined UHPC. Such models are essential for the accurate computational modelling and design of UHPC structures. While various stress-strain models for NSC and HSC exist, they have been found inadequate for predicting the behaviour of confined UHPC under axial load, as demonstrated by previous studies (Hosinieh et al., 2015; Naeimi & Moustafa, 2021a). A notable gap exists in the current body of research on the development of stress-strain models tailored to confined UHPC (Naeimi & Moustafa, 2021a). Addressing this gap, a recent study by Naeimi and Moustafa (Naeimi & Moustafa, 2021a) introduced a stress-strain model for confined UHPC. However, while this model provides an essential foundation for understanding the behaviour of confined UHPC, it is tailored to a specific UHPC mix, which limits its broader applicability. Furthermore, the model incorporates only two fiber volume fractions (2% and 4%) and three volumetric ratios of steel spirals (2%, 4%, and 8%). Additionally, the model considers a single grade of spiral reinforcement with a yield strength of 483 MPa. Furthermore, to use this model (Naeimi & Moustafa, 2021a) to predict the behaviour of confined UHPC with fibers, an auxiliary test is required to determine the compressive strength of unconfined UHPC without fibers. Given these limitations, more comprehensive research is needed to develop a reliable stress-strain constitutive model for confined UHPC based on a wider range of UHPC mixes, transverse reinforcement, and volume fractions of fibers. Yet, the complex composition of UHPC, which comprises a diverse array of components such as fibers, SCMs like fly ash or slag, and sometimes even nanoparticles, presents significant challenges in the development of accurate predictive models. Furthermore, the literature reports complex interactions between fibers in UHPC and spiral reinforcement (Naeimi & Moustafa, 2021a), which adds to the 82 complexity of accurately predicting the stress-strain responses of confined UHPC with fibers (Naeimi & Moustafa, 2021a). Thus, the accuracy of stress-strain models for confined UHPC is influenced not only by its inherent nonlinear behaviour but also by the complex interaction between steel fibers and confinement steel reinforcement. Given these challenges, there is a clear need for advanced modelling techniques that can effectively capture the complex dynamics of UHPC. In this context, artificial intelligence, particularly, ML has recently emerged as a powerful tool for accurately modelling such complex nonlinear problems without the necessity of comprehensive prior knowledge of the underlying physical or mathematical principles (Cakiroglu et al., 2022; Luo & Paal, 2018, 2022; Rahman et al., 2021; Rajneesh et al., 2023; Wakjira et al., 2022; Zaker Esteghamati & Flint, 2021). Therefore, the current study aims to develop a reliable stress-strain constitutive model for confined UHPC with normal strength steel (NSS) or high strength steel (HSS) spirals using hybrid ML model optimized with Optuna. A major challenge in the application of ML models in the field of civil engineering lies in transforming such models into practical tools so that they can have wider applications. Accordingly, the developed hybrid ML-based stress-strain constitutive model is deployed into an end-to-end interactive software tool to facilitate its practical application. Furthermore, predictive equations are developed for the stress and strain responses of confined UHPC at peak and ultimate states. Therefore, the novelty of this study is threefold: (1) the development of a novel hybrid ML-based stress-strain constitutive model for UHPC confined with NSS or HSS spirals, (2) the practical deployment of this model through a user-friendly software tool, and (3) the development of predictive equations for peak and ultimate stress-strain responses of UHPC confined with NSS or HSS spirals, which address a critical gap in the existing literature. The developed model can be used to 83 predict the stress-strain responses of UHPC confined with NSS or HSS spirals. Moreover, such models are useful in the accurate design and modelling of UHPC structures. 4.2. Methodology Figure 4.1 outlines the schematic framework of the methodology implemented in this study. Figure 4.1: Methodology flowchart. Due to the limited availability of experimental data on the stress-strain response of confined UHPC with NSS or HSS spirals, a well-calibrated and validated numerical model developed in Abaqus software (ABAQUS, 2014) is utilized to generate datasets on stress-strain responses of confined UHPC. This dataset is randomly split into the training and test datasets, which account for 80% and 20% of the complete dataset, 84 respectively. The training dataset is used to develop a hybrid ML-based stress-strain constitutive model for confined UHPC with NSS or HSS spirals. In contrast, the test dataset, comprising unseen data, is used to test the predictive performance of the model. This section provides details on the development and validation of the numerical model, the development and pre-processing of the stress-strain database, an introduction to the hybrid ML model, methodologies for hyperparameter optimization, and performance evaluation metrics. 4.2.1. Numerical model As can be seen in Figure 4.1, the first step involves the development and validation of the numerical model. Figure 4.2 shows the details of the developed numerical model. The adopted element type, geometric modelling, and mesh convergence criteria are discussed in the following subsections. (a) (b) (c) Figure 4.2: Details of the developed finite element model: (a) loading plates and concrete, (b) spiral reinforcement, and (c) loading and support sets. 4.2.1.1. Element and material modelling Concrete is considered a quasi-brittle material, that exhibits both brittleness, typically leading to crack initiation and propagation, and deformation capacity, which is evident in its ability to undergo plastic deformation when subjected to loads. Concrete 85 damage refers to the development and propagation of cracks, while plastic deformation is apparent in volumetric expansion, stiffness deterioration, and strain-softening (Abushanab, Alnahhal, & Farraj, 2021; Cicekli, Voyiadjis, & Al-rub, 2007). A comprehensive understanding of these phenomena and the inherent nonlinearity of concrete is essential to develop a robust constitutive model to simulate concrete behaviour. In this study, the unconfined UHPC behaviour was simulated in Abaqus software using concrete damaged plasticity (CDP) (ABAQUS, 2014). The CDP is a constitutive model that can simulate the damage and plastic deformation of concrete under various types of loading (e.g., dynamic, cyclic, and monotonic). It also has the capability of simulating the non-linear softening response of concrete under increasing strain. In addition, the CDP model can capture the compression and tension failure of an element (Abushanab, Alnahhal, & Farraj, 2021). The failure mechanism in the CDP model is defined by the compressive (ðð) and tensile (ðð¡) damage parameters, whose values range between 0 and 1. A value of 0 indicates undamaged concrete, while a value of 1 signifies completely damaged concrete. The value of ðð was determined as per Eq. (4.1), while the values of plastic strain and inelastaic strain are determined as per Eq. (4.2) and Eq. (4.3), respectively, (ABAQUS, 2014). ðð = 1 âððððð (4.1) ðððð = ðð âððð¸ (4.2) ðððð = ðððð âðð(1 â ðð)ððð¸ (4.3) where ðð and ðð are the compressive strength and corresponding strain of concrete at a specific point, ð¸ is the elastic modulus of concrete, ðððð is the concrete 86 plastic strain, ðððð is the concrete inelastic strain, and ððð is the 28-day compressive strength of unconfined concrete. UHPC was modelled as 8-node linear brick (C3D8R) element using methodologies and analytical models proposed in previous studies (Lu et al., 2021; Popovics, 1973; J. Yang, Chen, & Nuti, 2021; S. S. Zhang, Wang, Lin, Yu, & Fernando, 2023). The ascending stress-strain relationship of UHPC was based on the model proposed by Popovics (Popovics, 1973), while the descending stress-strain relationship was modelled according to Zhang et al. (S. S. Zhang et al., 2023). The elastic modulus was determined using the equation proposed by Yang et al. (J. Yang et al., 2021) and the strain at peak compressive strength of UHPC was modelled as per Lu et al. (Lu et al., 2021). Mathematical expressions for these models are listed in Table 4.1. Figure 4.3 compares the experimental and analytically derived stress-strain curves of UHPC with different volume fractions of steel fibers. The comparison was made to validate the equations used in this study in simulating the stress-strain behaviour of UHPC. As shown in Figure 4.3, a strong correlation is demonstrated between the experimental and analytically derived curves at the hardening and softening stages, peak stress and corresponding strain, and residual stress and corresponding strain. This result implies that the sets of equations proposed to estimate the stress-strain responses of UHPC are valid and can be used in the CDP model in Abaqus software. 87 Table 4.1: Analytical equations utilized for modelling the unconfined UHPC Reference Formula Eq No. Ascending branch of the stress-strain response of UHPC Popovics (Popovics, 1973) ððððð=(ðð/ððð) Ã ðð â 1 + (ðð/ððð)ð (4.4) ð =ð¸ð¸ â (ððð/ððð) (4.5) ððð is the concrete compressive strength ððð is the strain at ððð Descending branch of the stress-strain response of UHPC Zhang et al. (S. S. Zhang et al., 2023) ðð = ððððð +ððð â ððððð 1 + ð Ã (ðð/ððð â 1)2 (4.6) ð =21 + 100ð£ð (4.7) ððððð = ððð (9ð£ð + 4.7 (ððððð)) ð£ð is the volume fraction of steel fibers ððððð is the residual compressive strength ðð is the confinement pressure (4.8) Elastic modulus Yang et al. (J. Yang et al., 2021) ð¸ = 3.77âððð (4.9) Strain at peak ðð Lu et al. (Lu et al., 2021) ððð = (13.2ððð + 1926) Ã 10â6 (4.10) 88 (a) C1-SF0-U (Naeimi & Moustafa, 2021a) (b) C1-SF2-U (Naeimi & Moustafa, 2021a) (c) C1-SF4-U (Naeimi & Moustafa, 2021a) (d) P-2.5 (Li, Deng, & Yao, 2020b) (e) N-P (Shan et al., 2022) Figure 4.3: Experimental versus analytical stress-strain responses of unconfined UHPC. 0501001502000 0.005 0.01Stress (MPa)Strain (mm/mm)Experimental Analytical0501001502000 0.02 0.04Stress (MPa)Strain (mm/mm)Experimental Analytical0501001502002500 0.005 0.01 0.015Stress (MPa)Strain (mm/mm)Experimental Analytical0204060801001200 0.002 0.004 0.006Stress (MPa)Strain (mm/mm)Experimental Analytical0501001500 0.005 0.01Stress (MPa)Strain (mm/mm)Experimental Analytical89 The steel spiral was modelled as an 8-node linear brick (C3D8R) revolution element. The material properties of the steel spiral were modelled using a bi-linear model that accounts for stress and strain at yield and ultimate stages. In particular, two NSS with average yield strengths of 470 and 580 MPa and two HSS with average yield strengths of 708.2 and 834.3 MPa were utilized for the finite element (FE) models. Furthermore, the loading and supporting plates were modelled using 4-node 3-D bilinear rigid quadrilateral (R3D4) elements. 4.2.1.2. Geometric modelling The developed FE models have identical loading and boundary conditions and geometrical configuration of 150 Ã 300 mm UHPC cylinders, 160 mm diameter loading and support plates, 3.4 mm diameter spiral reinforcement, and 5 mm clear cover from all sides. The adopted geometry of the UHPC cylinders was selected to ensure conformity with ASTM C39/C39M-20 (ASTM C39, 2012). In addition, as can be observed in Figure 4.2, the loads were applied using an external plate positioned on the top of the cylindrical UHPC sample, with a loading point positioned below the plate. This assembly was preferred over the conventional way of applying the loads directly on the sample, as it simulated the actual experimental set and avoided the stress concentration on the top of the cylinders (Abushanab, Alnahhal, & Farraj, 2021). Moreover, the loading plate was restrained from the rotational and translational movements in all directions except the direction of the load. The top plate was used to measure the strain in the UHPC. Furthermore, a 160 mm plate was provided on the bottom of the cylinder with a node on the top of the plate, as shown in Figure 4.2(c). This plate was restrained from movement in all directions and the node on the plate was used to record the reaction force due to the transition of the cylinders. Additionally, the spiral reinforcement was positioned in the middle of the cylinder with a 5 mm clear 90 cover from all sides and a full embedment constraint with the UHPC. 4.2.1.3. Mesh convergence The mesh convergence step in FE modelling is of great importance in ensuring the efficiency and accuracy of the developed FE models. This process involves refining the mesh sizes systematically to select an optimized model that approaches the desired convergence precision in minimal computational time. In this study, the mesh convergence study was carried out on specimen C1-SF2-SS2 with a compressive strength of 198 MPa, which was tested by Naeimi and Moustafa (Naeimi & Moustafa, 2021a), using a range of mesh sizes between 10 and 60 mm. The selective criteria were the peak compressive strength and computational time. The predicted compressive strength results of the different mesh-size models are presented in Figure 4.4. As shown in Figure 4.4(a), increasing the mesh size from 10 to 60 mm narrowed the numerical stress-strain curves and decreased the peak compressive strength compared to the experimental stress-strain values. The reduced accuracy of FE models with higher mesh sizes is attributed to the fewer elements in the FE models with higher mesh sizes, which restraints the FE modelsâ capability of analyzing stress and strain, and thus averaging the responses at larger regions fails to capture the fine-scale responses. It can also be observed from Figure 4.4(b) that the FE models with mesh sizes of 30, 40, 50, and 60 mm achieved 5.59%, 6.50, 7.05%, and 7.95% lower ðð compared to the corresponding experimental results, respectively. Moreover, the FE models with mesh sizes of 10 and 20 mm achieved approximately similar results with a variation of less than 5% for both models. Nevertheless, the computational time of the 10 mm mesh-size model was significantly higher than that of the 20 mm mesh-size model. Thus, the FE models in this study were developed with a mesh size of 20 mm. Furthermore, it should be emphasized that loading and support plates and spiral reinforcement were assigned a 91 similar mesh size (20 mm) to ensure an equal sharing of the nodes by the different parts, avoid mesh mismatch, and allow for smooth transfer of stresses between different components. (a) (b) Figure 4.4: Concrete compressive strength results of FE model C1-SF2-SS2 (Naeimi & Moustafa, 2021a) with different mesh sizes: (a) stress-strain curves and (b) peak compressive strength for each mesh size. 4.2.1.4. Validation of the numerical model The developed numerical model was validated against experimental results 0204060801001201401601802000 0.002 0.004 0.006 0.008 0.01Stress (MPa)Strain (mm/mm)EXPFE (Mesh = 10 mm)FE (Mesh = 20 mm)FE (Mesh = 30 mm)FE (Mesh = 40 mm)FE (Mesh = 50 mm)FE (Mesh = 60 mm)188.19188.13186.63184.85183.77181.98180182184186188190102030405060f'co(MPa)Mesh size (mm)92 from multiple studies, including those by Naeimi and Moustafa (Naeimi & Moustafa, 2021a), Li et al. (Li et al., 2020b), and Shan et al. (Shan et al., 2022). This approach aims to confirm the effectiveness of the model in handling variations in compressive strengths of unconfined UHPC, fiber volumetric ratios (ð£ð), yield strengths of spiral (ðð¦), and spiral reinforcement ratios (ðð ). The characteristics of the selected experimental specimens that are chosen to represent variability in terms of ððð, ð£ð, ðð¦, and ðð , are listed in Table 4.2. A comparison between the stress-strain curves generated by the numerical model and their experimental counterparts is shown in Figure 4.5. The results of the comparative analysis reveal a strong correlation between the predicted and experimental stress-strain curves. Table 4.2: Experimental versus predicted peak strength of confined UHPC Reference Specimen ID vf (%) ðð (%) fy (MPa) ððð,ðð/ððð,ðð¥ð ððð,ðð/ððð,ðð¥ð ððð,ðð/ððð,ðð¥ð ððð,ðð/ððð,ðð¥ð Naeimi and Moustafa (Naeimi & Moustafa, 2021a) C1-SF2-SS2 2 2 483 0.95 0.99 1.01 1.04 Li et al. (Li et al., 2020b) H60-1.5 1.5 4.03 723 0.93 1.18 1.07 0.87 Li et al. (Li et al., 2020b) H60-0.5 0.5 4.03 723 1.04 1.04 1.15 1.02 Shan et al. (Shan et al., 2022) H5-20 0 7.62 935 1.01 1.00 1.06 0.97 Shan et al. (Shan et al., 2022) H10-20 0 7.63 935 0.94 1.10 1.01 0.96 Average 0.97 1.06 1.06 0.97 Standard deviation 0.05 0.08 0.06 0.07 Coefficient of variation (%) 4.96 7.43 5.39 6.93 Note: ððð,ðð¥ð= experimental peak compressive strength; ððð,ðð= predicted peak compressive strength; ððð,ðð¥ð= experimental strain at peak compressive strength; ððð,ðð= predicted strain at peak compressive strength; ððð,ðð¥ð= experimental residual compressive strength; ððð,ðð= predicted residual compressive strength; ððð,ðð¥ð= experimental strain at residual compressive strength; and ððð,ðð= predicted strain at residual compressive strength. 93 (a) C1-SF2-SS2 (Naeimi & Moustafa, 2021a) (b) H60-1.5 (Li et al., 2020b) (c) H60-0.5 (Li et al., 2020b) (d) H10-20 (Shan et al., 2022) (e) H5-20 (Shan et al., 2022) Figure 4.5: Comparison between the numerical and experimental stress-strain curves (Note: EXP = experimental and FE = finite element). 0501001502002500 0.005 0.01Stress (MPa)Strain (mm/mm)EXP FE0204060801001200 0.005 0.01 0.015Stress (MPa)Strain (mm/mm)EXP FE0204060801001200 0.01 0.02Stress (MPa)Strain (mm/mm)EXP FE0501001502002503000 0.01 0.02 0.03Stress (MPa)Strain (mm/mm)EXP FE0501001502002503000 0.02 0.04Stress (MPa)Strain (mm/mm)EXP FE94 Additionally, Table 4.2 presents a comparison between the numerically predicted and experimental peak and residual stresses and corresponding strains. As listed in this table, the predicted-to-experimental peak compressive strength (ððð,ðð/ððð,ðð¥ð) ratios vary between 0.93 and 1.04 with an average of 0.97, standard deviation of 0.05, and coefficient of variation (COV) of 4.96%. Similarly, the predicted-to-experimental residual compressive strength (ððð,ðð/ððð,ðð¥ð) ratios are between 1.01 and 1.15 with an average, standard deviation, and COV of 1.06, 0.06, and 5.39%, respectively. In terms of strains, the ratios at peak compressive strength (ððð,ðð/ððð,ðð¥ð) range from 0.99 to 1.18 with an average, standard deviation, and COV of 1.06, 0.08, and 7.43%, respectively. For residual compressive strain (ððð,ðð/ððð,ðð¥ð), these ratios range from 0.87 to 1.04 with an average of 0.97, standard deviation of 0.07, and COV of 6.93%. The close agreements between the predicted and experimental values across different samples indicate that the developed numerical model is capable of accurately predicting the stress-strain behaviour of confined UHPC at both peak and residual stages. Consequently, it is used for modelling the stress-strain response of confined UHPC. 4.2.2. Development and pre-processing of confined UHPC stress-strain response database The developed numerical model is used to generate a comprehensive database for the stress-strain response of confined UHPC, accounting for influential factors such as the compressive strength of unconfined UHPC, fiber volumetric ratio, spiral reinforcement ratio, and yield strength of spiral. It is important to note that the impact of fibers in UHPC is influenced by various factors, including the shape, diameter, length, hybridization, and aspect ratio of the fibers. However, while recognizing the 95 significance of these diverse attributes, the current study focuses explicitly on fiber content as the primary variable in alignment with previous research that aims to comprehensively understand the influence of fiber on the stress-strain response of confined UHPC (Naeimi & Moustafa, 2021b). The distribution data for the compressive strength of unconfined UHPC is obtained from optimized sustainable UHPC in Chapter 2, which has a normal distribution with a mean compressive strength of 181.95 MPa and a standard deviation of 27.43 MPa. Latin hypercube sampling (LHS) is utilized to produce a dataset comprising 200 distinct values of ððð due to its advantages over standard Monte Carlo sampling (MCS). Particularly for complex nonlinear systems, LHS provides a superior alternative to standard MCS by offering more uniform and efficient exploration of the parameter space, with reduced sample sizes and faster convergence rates (Giovanis & Papadopoulos, 2015; A. M. J. Olsson & Sandberg, 2002). The remaining factors (ð£ð, ðð¦, and ðð ) are varied within specified ranges yielding a database that captures a diverse range of material properties. The volumetric ratio of fibers is set to vary between 1% and 3% at intervals of 0.5%. Four bar grades are considered for the spiral reinforcement, with yield strengths of 470 MPa, 580 MPa, 708 MPa, and 834 MPa; thus comprising both NSS and HSS spiral reinforcement. Spiral reinforcements with a 3.4 mm bar diameter are used at spacings ranging from 7 mm to 40 mm (7, 8, 10, 12, 15, 20, 25, 30, 35, and 40 mm), which represent a spiral volumetric reinforcement ratio that ranges from 0.641% to 3.66%. Each unique set of these factors is paired with one of the 200 predetermined values of ððð to form 200 distinct confined UHPC samples, but an equal number of samples for each specific value of three factors. These samples are subsequently simulated using the validated numerical model and subjected to uniaxial compressive loading, thereby 96 generating the corresponding stress-strain curves for each sample. In the subsequent stage, the stress-strain response of the confined UHPC sample is discretized into well-defined segments. As shown in Figure 4.6, each stress-strain curve is represented by 17 strategically chosen points. Of these, six points (ð1 through ð6 in Figure 4.6) precede the peak state and are situated at increments of 10%, ranging from 40% to 90% of the peak strain value (strain corresponding to peak stress). Between the peak and ultimate states, nine additional points (ð8 through ð16 in Figure 4.6) are specified, excluding the ultimate point. These nine post-peak points are situated at locations determined by adding increments of 10% to 90% of the difference between the ultimate and peak strain values to the peak strain itself, as detailed in Figure 4.6. These increments increase in steps by 10%, as shown in Figure 4.6. The response between the origin to the first point (40% peak strain) is approximated by a straight line, while a smooth curve is fitted to determine the response curve between the 17. Each of these 17 points provides both stress and strain values, yet the strains at the peak and ultimate points remain unknown, while the strains at all other points are calculated based on these peak and ultimate strains, thereby resulting in a total of 19 response variables for each sample. ð7 through ð17 in Figure 4.6 represent the peak and ultimate points, respectively. The database, therefore, consists of four input features and 19 response variables. It is worth noting that the ultimate stress and ultimate strain represent the values corresponding to the ultimate state, which is defined in this study as the point on the stress-strain curve where the stress drops by 20% after the peak (Skyrianou, Koutas, & Papakonstantinou, 2022). 97 Figure 4.6: Discretization of stress-strain response curve. To address the issue of suboptimal learning rates at extreme values, the developed dataset is normalized to a range between [0, 1], as shown below: ðð =ððððð â ðððððððð¥ â ðððð (4.11) here, ððððð refers to the actual value of the variable before normalization and ðð is its corresponding normalized value. ðððð¥ and ðððð denote the highest and lowest values of the variable within the dataset, respectively. 4.2.3. Hybrid ML model In this study, a hybrid ML model is developed to accurately predict the stress-strain response of confined UHPC. As discussed in Section 3.2.3, the model architecture is hierarchical, initiated by a primary ensemble model that makes the primary predictions. To refine these initial predictions, supplementary ML models are introduced in a second layer. These models are trained on the residual errors produced by the primary model, essentially focusing on capturing patterns missed by the initial model. Leveraging the advanced optimization capabilities of the next-generation optimization framework, Optuna, the model adjusts the weights of these secondary models, thereby optimizing their contribution to the overall predictive performance. 98 Based on preliminary investigation, XGB is used as the primary model for the foundational layer of the hybrid ML model. While other ensemble models could also serve this purpose, XGB stands out for its built-in regularization features and strong performance in preventing overfitting (Chen & Guestrin, 2016). When considering secondary models, the focus was on ML algorithms that can naturally support multi-output predictions. Among these, the GBM is a strong candidate. Another multi-output machine learning model is random forest (RF). It is an ensemble of decision trees that averages the predictions from individual trees (in the case of regression-based problems) for a more stable and accurate prediction (Svetnik et al., 2003). It is expressed as a sum of weighted individual tree predictions: ð ð¹(ð) =1ðð¡ðððð â ðð(ð)ðð¡ðððð ð=1 (4.12) Critical hyperparameters include the number of trees (ðð¡ðððð ) and maximum depth. The hyperparameters of both the primary and the secondary models, as well as the weights assigned to the secondary models, are simultaneously optimized using Optuna (Akiba et al., 2019), with a 10-fold cross-validation technique employed to ensure robust performance and generalizability. The details of Optuna and cross-validation techniques are discussed in Section 3.2.4. 4.3. Results and Discussion 4.3.1. Results of hyperparameter optimization As discussed earlier, the hybrid ML model is optimized with Optuna coupled with 10-fold cross-validation, using RMSE as the evaluation metric. The 19 response variables are categorized into five distinct groups. The first group addresses the stress response before peak strain, the second focuses on stress at peak strain, the third 99 involves the stress response between peak and ultimate strains including the stress at ultimate strain, and the fourth and fifth focus on the strain at peak and ultimate states, respectively. For each group, the optimized hyperparameters are distinct, which reveals differing complexities of the response variables. A variety of hyperparameters have been fine-tuned for each model, and the results are presented in Table 4.3. Table 4.3: Optuna-optimized hybrid ML model Description Optimized hyperparameters Group 1 2 3 4 5 Primary model Model XGB Number of estimators 170 174 178 193 101 Maximum depth 2 5 3 2 6 Learning rate 0.087 0.078 0.095 0.073 0.072 Subsample 0.299 0.268 0.295 0.285 0.286 Column sample by tree 0.946 0.973 0.942 0.792 0.976 Secondary models Model GBM Number of estimators 89 67 99 69 92 Maximum depth 9 7 10 3 6 Learning rate 0.077 0.075 0.082 0.078 0.064 Subsample 0.226 0.247 0.153 0.118 0.286 Weight 0.964 0.985 0.924 0.925 0.529 Model RF Number of estimators 38 41 19 14 54 Maximum depth 3 2 5 5 7 Weight 0.036 0.015 0.076 0.075 0.471 In the hybrid ML model for Group 1, the main XGB model employs 170 estimators with a maximum depth of 2 and a learning rate of 0.087. Furthermore, the subsample rate is configured at 0.299, and the column sample by tree is set at 0.946. These additional hyperparameters help in preventing the modelâs tendency to overfit. The GBM model in this group is weighted at 0.964 and uses 89 estimators with a maximum depth of 9 and a learning rate of 0.077. The subsample for GBM is set at 0.226. The RF model utilizes 38 estimators and has a maximum depth of 3 making it a stabilizing factor in the hybrid ML model. In the second group, the XGB model is defined with 174 estimators, a maximum depth of 5, and a learning rate of 0.078. Additionally, it is given a subsample rate of 0.268 and a column sample by tree of 0.973. 100 The GBM for this group holds a weight of 0.985 and has settings of 67 estimators, a maximum depth of 7, and a learning rate of 0.075, alongside a subsampling rate of 0.247. The RF model for this group employs 41 estimators with a maximum depth of 2. In Group 3, the XGB model used 178 estimators with a maximum depth of 3 and a learning rate of 0.095 supplemented by a subsampling rate of 0.295 and a column sample by tree of 0.942. The GBM had a weight of 0.924, with 99 estimators, a maximum depth of 10, and a learning rate of 0.082. Its subsample rate is set at 0.153. The optimized RF model utilizes 19 estimators and a maximum depth of 5. Group 4âs XGB model features 193 estimators, a maximum depth of 2, and a learning rate of 0.073. Additional hyperparameters include a subsampling rate of 0.285 and a column sample by tree of 0.792. The GBM model is weighted at 0.925 and uses 69 estimators, a maximum depth of 3, a learning rate of 0.078, and a subsample rate of 0.118. The RF model employs 14 estimators and a maximum depth of 5. Lastly, in Group 5, the XGB model comprises 101 estimators, a maximum depth of 6, and a learning rate of 0.072 with the subsample rate set at 0.286 and the column sample by tree of 0.976. The GBM is characterized by a weight of 0.529 and uses 92 estimators, a maximum depth of 6, a learning rate of 0.064, and a subsampling rate of 0.286. The RF model is configured with 54 estimators and a maximum depth of 7. Across the different groups, the primary XGB models have learning rates that span from 0.072 to 0.095, while GBM models possess learning rates that lie between 0.064 and 0.082. On the other hand, RF models consistently utilize a fewer number of estimators, with their count ranging from 14 to 54 and they typically adopt shallower depths, specifically between 2 and 7. This simpler configuration by RF models offers a balanced approach, ensuring the models donât overly adapt to the training data. Therefore, while XGB and GBM aim to capture detailed data patterns, RF models 101 anchor the hybrid ML model by offering stability and making sure that the ensemble provides both reliable and accurate predictions across all groups. 4.3.2. Stress-strain response prediction results The optimized hybrid ML-based stress-strain constitutive model is used to predict the stress-strain response curves of UHPC confined with either NSS or HSS spirals. The effectiveness of the constitutive model is investigated through a detailed evaluation of various statistical performance metrics discussed earlier, and results are summarized in Table 4.4 for each group on both the training and test datasets, as well as the complete dataset. Each group represents a subset of the stress-strain response curve, and the MAE, RMSE, and R² are calculated for all the 19 response variables in these groups. This detailed evaluation helps to understand how well the developed constitutive model performs at different stages along the stress-strain curve. For instance, for Group 1, which addresses the stress response before the peak state, the R² values for the training dataset consistently surpass 0.995, while for the test dataset, they remain above 0.941. Moreover, the model showed low errors (RMSE and MAE) on both the training and test datasets, as well as the complete dataset. These results highlight the efficacy of the model in capturing the initial stress variations along the curve up to the peak state. Group 2 is aimed to predict the peak stress (ððð), and the evaluation of the model on the training dataset reveals an R² of 0.999, RMSE of 1.08 MPa, and MAE of 0.810 MPa, while the evaluation on the test dataset showed an R² of 0.967, RMSE of 7.59 MPa, and MAE of 5.333 MPa. These results highlight the modelâs accuracy in predicting the peak stress.102 Table 4.4: Performance of Optuna-optimized hybrid ML model in predicting the stress-strain response of confined UHPC at various stages across the curve. Group Response variable Training dataset Test dataset Complete dataset R2 RMSE MAE R2 RMSE MAE R2 RMSE MAE 1 Stress at ð1 0.995 2.110 1.541 0.941 8.896 5.499 0.980 4.429 2.345 Stress at ð2 0.996 2.330 1.700 0.961 8.430 6.307 0.986 4.331 2.635 Stress at ð3 0.997 2.052 1.564 0.967 8.226 5.897 0.989 4.135 2.444 Stress at ð4 0.998 1.851 1.415 0.974 7.354 5.622 0.991 3.703 2.269 Stress at ð5 0.998 1.452 1.137 0.980 6.259 5.011 0.993 3.104 1.923 Stress at ð6 0.999 1.099 0.840 0.986 5.112 4.047 0.995 2.504 1.491 2 ððð 0.999 1.080 0.810 0.967 7.590 5.333 0.991 3.553 1.729 3 Stress at ð8 0.999 1.275 1.007 0.973 6.858 4.974 0.992 3.293 1.812 Stress at ð9 0.998 1.553 1.198 0.969 7.229 5.222 0.990 3.540 2.015 Stress at ð10 0.998 1.555 1.242 0.970 7.039 5.043 0.990 3.462 2.014 Stress at ð11 0.998 1.552 1.219 0.965 7.624 5.472 0.989 3.704 2.083 Stress at ð12 0.998 1.582 1.221 0.957 8.445 6.264 0.986 4.059 2.245 Stress at ð13 0.998 1.634 1.239 0.959 8.333 6.083 0.987 4.028 2.223 Stress at ð14 0.998 1.606 1.208 0.957 8.584 6.351 0.986 4.125 2.252 Stress at ð15 0.998 1.501 1.161 0.960 8.351 6.025 0.987 3.994 2.149 Stress at ð16 0.998 1.513 1.179 0.960 8.532 6.531 0.987 4.075 2.266 ððð¢ 0.998 1.500 1.174 0.954 9.240 6.299 0.985 4.373 2.215 4 ððð 0.964 0.02% 0.01% 0.905 0.03% 0.02% 0.949 0.02% 0.01% 5 ððð¢ 0.992 0.03% 0.02% 0.896 0.13% 0.09% 0.971 0.06% 0.04% 103 Considering Group 3, which examines stress between peak and ultimate states including ððð¢, the errors for the training dataset ranged between 1.007 MPa and 1.242 MPa for MAE, while for the test data, they ranged from 4.974 MPa to 6.299 MPa. Besides, both training and test datasets exhibit consistently high R² values and low RMSE values, underscoring the robustness of the model in predicting stress variation between the peak and ultimate states as well as ððð¢. Consistent patterns of low MAE, low RMSE, and high R² were obtained in Groups 4 and 5 (see Table 4.4) on both the training and test datasets, which confirm the reliability and predictive ability of the model across all groups and all the 19 response variables in varied states for both the training and test datasets. Special attention is given to the capability of the proposed constitutive model to simultaneously predict the stress and strain responses at peak and ultimate states, as listed in Table 4.5. For the peak state represented by (ððð, ððð), the model achieved a CCA score of 98% on the training dataset, 94% on the test dataset, and 97% on the complete dataset (Table 4.5). Similarly, for the ultimate state defined by (ððð¢, ððð¢), the model exhibited a high CCA score of 99% for the training dataset, 93% for the test dataset, and 98% for the complete dataset, as listed in Table 4.5. These consistently high CCA scores, being above 90% for both peak and ultimate states, confirm the accuracy and reliability of the proposed hybrid ML model in capturing both stress and strain responses at these critical states. Table 4.5: Canonical correlation analysis (CCA) score State Response variables Training dataset Test dataset Complete dataset Peak (ððð, ððð) 98.2 94.1 97.0 Ultimate (ððð¢, ððð¢) 99.4 93.3 97.9 104 Given the extensive database used in this study, representative confined UHPC samples selected from the test dataset are used to illustrate the predictive performance of the model. These selected samples cover a wide range of material properties and confinement conditions, as demonstrated in Figure 4.7(a-f) for UHPC confined with NSS spirals and Figure 4.8(a-f) for UHPC confined with HSS spirals. The results in these figures clearly demonstrate a high level of model accuracy, and the effectiveness of the model in reliably capturing the stress-strain response curve of confined UHPC, regardless of whether it is confined with NSS or HSS spirals. In addition to evaluation using an unseen or test dataset, the proposed model has also been assessed against experimental results for the stress-strain response of confined UHPC reported in the literature (Naeimi & Moustafa, 2021b; Sugano et al., 2007). The result of this evaluation is illustrated in Figure 4.9(aâd). As can be seen in these figures, the stress-strain responses predicted by the proposed hybrid ML model align well with the corresponding experimental results. Therefore, it can be concluded that the proposed hybrid ML model is effective in predicting the stress-strain response curve for UHPC confined with NSS or HSS spirals. 105 Figure 4.7: Predicted versus actual stress-strain response of UHPC confined with NSS spirals. 106 Figure 4.8: Predicted versus actual stress-strain response of UHPC confined with HSS spirals. 107 Figure 4.9: Predicted versus experimental stress-strain curves of Specimen (a) 120FM2-35 (Sugano et al., 2007), (b) 160FM2-55 (Sugano et al., 2007), (c) 120FM2-35 (Sugano et al., 2007), and (d) C1-SF2-SS2 (Naeimi & Moustafa, 2021b). For practical application of the developed hybrid ML model, an end-to-end interactive software tool named HAI-CONCise-UHPC: Hybrid Artificial Intelligence-based CONstitutive model for Confined UHPC is developed. HAI-CONCise-UHPC provides a rapid, accurate, and intelligent means of predicting the stress-strain response of UHPC confined with either NSS or HSS spirals and can be accessed at https://github.com/twakjira/HAI-CONCise-UHPC. The graphical user interface of the developed software tool is illustrated in Figure 4.10 with an example. The stress-strain response can be predicted using the âPredict and Plotâ button. Moreover, the predicted stress-strain response data can be exported to Excel using the âExport to Excelâ button. However, it is important to note that predictions should be generated before exporting by clicking the âPredict and Plotâ button after inputting the required parameters for the 108 input factors. Figure 4.10: HAI-CONCise-UHPC interactive software tool for the developed hybrid ML-based stress-strain constitutive model of confined UHPC. 4.4. Predictive Equations In addition to the hybrid ML-based stress-strain constitutive model, predictive equations are developed for determining the stress and strain responses at two critical states, particularly peak and ultimate states. The structure of these equations is determined based on the literature review (Mander et al., 1989; Paulay & Priestley, 1992) and integrates the effects of the fiber in terms of its volume fraction, as shown in Eqs. (4.13)â(4.16). For the prediction of peak compressive strength, two alternative equation forms are considered, as presented in Eqs. (4.13a) and (4.13b). The first equation is consistent with the confinement model developed by Mander et al. (Mander 109 et al., 1989) for conventional concrete but extends it to account for the impact of fiber volume fraction. The second equation for peak compressive strength undergoes modification to optimize a specific constant (ð6) in the equation, to enhance its accuracy compared to the use of a constant value of 1.0, as is the case in Eq. (4.13a). Similarly, two alternative equation forms are considered for the calculation of peak compressive strain. The first method is in line with the model by Mander et al. (Mander et al., 1989), wherein the second term is derived from the peak compressive strength of the confined concrete, Eq. (4.15a). The second alternative calculates peak compressive strain directly from the input parameters, intending to minimize the risk of errors that might arise from the estimation of the peak compressive strength, Eq. (4.15b). The accuracy of the first approach depends on the correct determination of peak compressive strength, where errors in this estimate could affect the entire model. Therefore, the second approach aims to offer a more direct and potentially more accurate equation. ððð = ð1ð£ðð2ðððð3 (1.0 + ð5â1 +ð7(ðð¦ðð )ð8ð£ðð9ðððð10+ ð11(ðð¦ðð )ð12ð£ðð13ðððð14) (in Mpa) (4.13a) ððð = ð1ð£ðð2ðððð3 (ð4 + ð5âð6 +ð7(ðð¦ðð )ð8ð£ðð9ðððð10+ ð11(ðð¦ðð )ð12ð£ðð13ðððð14) (in MPa) (4.20b) ððð¢ = ð1ð£ðð2ðððð3 (ð4 + ð5âð6 +ð7(ðð¦ðð )ð8ð£ðð9ðððð10+ ð11(ðð¦ðð )ð12ð£ðð13ðððð14) (in Mpa) (4.14) ððð = ð1ð£ðð2ðððð3 (ð4 + ð5 (ð6ðððð7ð£ðð8ðððð9+ ð10)) (4.15a) ððð = ð1ð£ðð2ðððð3 (ð4 + ð5 (ð6(ðð¦ðð )ð7ð£ðð8ðððð9+ ð10)) (4.22b) ððð¢ = ð1ð£ðð2ðððð3 (1 + ð4(ðð¦ðð )ð5ð£ðð6ðððð7) (4.16) here, ð£ð and ðð are given in percentage, while ððð, ðð¦, and ððð are given in MPa. 110 Genetic algorithm (GA) is utilized as an optimization technique to fine-tune key parameters in the above equations including constants, coefficients, and exponents. The purpose of employing GA is to systematically adjust these parameters and find the optimal values that enable the equation to most accurately predict the peak and ultimate stress-strain responses of UHPC confined with NSS or HSS spirals. This algorithm is inspired by the principles of natural selection, as originally outlined by Charles Darwin, and focuses on the survival of the fittest solutions (Goldberg, 1989). The process begins by creating an initial pool of random solutions often referred to as âchromosomes,â which contain various combinations of âgenesâ or parameters (Goldberg, 1989). In each iteration of the algorithm, known as a âgeneration,â the best-performing solutions are identified and combined to produce new, potentially better-performing solutions. This is done through a series of mathematical operations modelled on biological processes like selection, crossover, and mutation (Goldberg, 1989). With each successive generation, the quality of the solutions typically improves, and this iterative process continues until an optimal solution is reached or no significant improvements are observed in the solutions. Following the application of GA for optimization, the optimized equations are presented in Eqs. (4.17)-(4.20). Subsequent analyses, illustrated in Figure 4.11(aâd), compare the predicted and actual values of stress and strain at both peak and ultimate states. As can be observed in these figures, the scatter plots are closely clustered around the line ð¦ = ð¥, which indicates an excellent match between the predicted and actual values of peak and ultimate stresses, as well as their corresponding strains. Therefore, the optimized equations provide a satisfactory level of accuracy in predicting the stress and the corresponding strain at these critical points. 111 ððð = 2.62ð£ð0.397ððð0.791 (0.021 + 0.058â0.978 + 6.1(ðð¦ðð )0.136ð£ð0.339ððð0.549+ 0.366(ðð¦ðð )0.182ð£ð0.423ððð0.003) (4.17) ððð¢ = 2.768ððð0.649ð£ðâ0.097 (1 + 1.495â0.917 + 26.05(ðð¦ðð )â0.378ð£ðâ0.199ððð0.355+ 0.000406(ðð¦ðð )0.733ð£ðâ0.584ðððâ0.62) (4.18) ððð(%) = 0.1ð£ð0.263ððð1.326 (0.002 + 1.23(ðð¦ðð )0.260ð£ð0.141ððð1.401) (4.19) ððð¢(%) = 0.082 + 0.619ðð 0.599ðð¦0.023ð£ð0.629ððð0.018 ðð 0.096+ 0.152ð£ð0.167ðð¦0.013ððð0.013 (4.20) In the above equations, ð£ð and ðð are given in percentage, while ððð and ðð¦ are given in MPa. ðð stands for the yield strain of the spiral reinforcement. 112 (a) Peak stress (b) Peak strain (c) Ultimate stress (d) Ultimate strain Figure 4.11: Scatter plots of predicted versus actual stress-strain responses at peak and ultimate states using the proposed equations. To further substantiate the effectiveness of the proposed equations, comparisons are made with experimental results in the literature involving UHPC confined with spirals in which both the stress and strain responses at both peak and ultimate states are provided. Figure 4.12 shows the ratio of the predicted-to-experimental values of both the stress and strain at both peak and ultimate states (Shan et al., 2022; Hyun Oh Shin et al., 2018). From Figure 4.12, it is evident that, except for the ultimate strain of one specimen, the equations are generally accurate. Specifically, the predicted-to-experimental ratios ranged between 0.84 and 1.22 for peak stress and between 0.62 and 1.10 for its corresponding strain. For the ultimate stress, the ratio lies between 0.81 and 1.16, while for its associated strain, it varies from 0.53 to 1.00 except for one specimen 113 having a ratio of 1.92. Moreover, the mean of the predicted-to-experimental ratio was 0.87±0.11 for peak strain and 0.85±0.26 for ultimate strains, while it was 1.01±0.11 for peak stress and 0.99±0.10 for ultimate stress. Thus, the mean ratios being close to unity for most predictions together with the low standard deviations underscore the accuracy and reliability of the proposed equations in predicting the stress and strain responses at both the peak and ultimate states. Figure 4.12: Prediction evaluation of the proposed equations based on experimental results of confined UHPC. 4.5. Summary While ultra-high-performance concrete has gained considerable focus for its advanced material properties and durability, there remains a notable research gap in establishing reliable stress-strain constitutive models for confined UHPC. Such models are pivotal for the accurate modelling and design of UHPC structures. Addressing this critical gap, the current study focuses on developing a reliable stress-strain constitutive model for confined UHPC. For this purpose, a novel hybrid ML-based constitutive model for confined UHPC with either NSS or HSS spirals is introduced. The proposed 114 hybrid ML model integrates one primary model and two secondary models aimed to enhance its accuracy and robustness. The next generation optimization framework known as Optuna is used to optimize the hyperparameters of the primary and secondary models as well as the weight of each secondary model. A large database of stress-strain curves for confined UHPC is generated using a rigorously calibrated and validated numerical model, which then served as the training data for the hybrid ML-based constitutive model. Additionally, predictive equations are developed for the stress and strain responses at both peak and ultimate states. The main findings from this study are summarized as follows: â¢ The developed numerical model effectively predicts the stress-strain response of confined UHPC with NSS or HSS spirals. LHS is used to generate a database of confined UHPC samples, taking into account four critical factors; namely, compressive strength of unconfined UHPC, yield strength of spirals, reinforcement ratio of spirals, and volume fraction of fibers. Using this numerical model, the response variables, specifically, the stress-strain curves as well as the stress and corresponding strains at both peak and ultimate states, are then determined. â¢ The hybrid ML-based stress-strain constitutive model optimized with Optuna showed high predictive accuracy for stress-strain curves of confined UHPC, regardless of the grade of the steel spiral used. The efficiency of the optimized hybrid ML model is rigorously evaluated using statistical metrics such as RMSE, MAE, and R2 in predicting the stress and strain responses of confined UHPC at various stages across the curve. Moreover, special attention was given to the ability of the proposed constitutive model to simultaneously predict the stress and strain responses at peak and ultimate states evaluated through CCA 115 score. The results of the evaluation showed the high prediction accuracy and reliability of the proposed constitutive model, as evidenced by low RMSE and MAE and high R2 and CCA scores. â¢ In addition to its evaluation using unseen or test datasets, the proposed hybrid ML-based constitutive model has been further assessed against experimental stress-strain responses of confined UHPC. The results of this evaluation demonstrated the efficacy of the proposed hybrid ML-based constitutive model in accurately predicting the stress-strain response of confined UHPC with NSS or HSS spirals. â¢ The predictive equations, which are formulated using GA, have been evaluated for their ability to predict peak compressive stress and the associated peak compressive strain. These equations achieved high R² values of 0.95 and 0.85, which indicate their strong predictive capabilities. In addition, the equations for ultimate stress and corresponding ultimate strain also yielded highly accurate results. Besides, the proposed equations maintained a high level of accuracy when tested against independent experimental data of confined UHPC. â¢ To facilitate the application of the developed hybrid ML-based stress-strain constitutive model, an end-to-end interactive software tool named HAI-CONCise-UHPC: Hybrid Artificial Intelligence-based CONstitutive model for Confined UHPC has been developed. HAI-CONCise-UHPC not only allows for fast and accurate predictions of the stress-strain curves for confined UHPC but also provides a user-friendly environment. Overall, the hybrid ML-based stress-strain constitutive model has been proven effective and reliable in predicting the stress-strain response of confined UHPC with 116 either NSS or HSS spirals. Additionally, the equations developed through the use of GA have demonstrated their efficacy in predicting the peak and ultimate stress and their corresponding strains. Nonetheless, this study is limited to the uniaxial stress-strain analysis of confined UHPC with steel spirals, and hence room for further advancement exists. Future studies should address other UHPC material characteristics such as biaxial and triaxial stress-strain behaviour, strain rate effect, fracture energy, and damage behaviour. Moreover, a comprehensive assessment of the fiber effect considering not only fiber content but also fiber diameter, aspect ratio, and length is recommended to fully understand their combined influence on the compressive and flexural strengths of UHPC. Potential areas for improvement also include the integration of the developed models into established numerical software platforms, such as OpenSEES to enhance accuracy in the modelling of UHPC structural members. 117 CHAPTER 5: PERFORMANCE-BASED SEISMIC DESIGN OF ULTRA-HIGH-PERFORMANCE CONCRETE (UHPC) BRIDGE COLUMNS WITH DESIGN EXAMPLE â POWERED BY EXPLAINABLE MACHINE LEARNING MODEL 5.1. Introduction Highway bridges constitute a vital aspect of transportation infrastructure, which necessitate their utmost structural resilience to maintain their functionality. Bridge columns play a pivotal role in determining the overall performance of bridge systems, and among all the bridge components, these elements are particularly vulnerable to seismic excitations. In this context, innovative materials and advanced design methodologies are vital to improve the seismic resilience of bridges. UHPC has emerged as a potential advanced material in enhancing bridge structural performance due to its favorable properties such as high strength and enhanced durability (Abokifa & Moustafa, 2021b, 2021a; Krahl et al., 2018). Despite the clear benefits of UHPC, the absence of earthquake-specific design guidelines for UHPC structures hinders their widespread use in areas susceptible to earthquakes. Complementing the benefits of UHPC as an advanced material is the performance-based design methodology, which expresses structural design criteria through the attainment of specified performance objectives. In addition, the PBSD methodology not only allows for achieving specific performance levels but also enables the quantification of associated risks and economic implications. By integrating these quantifications, stakeholders can make informed decisions regarding infrastructure investment and strategies for risk mitigation, which enhances the resilience of structural designs against seismic events. Extensive research has been conducted to establish performance limits states in terms of engineering demand parameters (EDPs), such as strain, drift, rotation, etc for conventional concrete 118 structures (Aldabagh & Alam, 2022; Aldabagh, Hossain, & Alam, 2022; Ghannoum & Matamoros, 2004; Lehman et al., 2004; Reza, Alam, & Tesfamariam, 2014; Todorov & Muntasir Billah, 2022). Moreover, the shift towards this design methodology is evident in its incorporation in various design provisions including AASHTO (AASHTO, 2011) and CSA-S6-14 (CSA, 2014). As shown in Figure 5.1, the PBSD approach starts by defining a set of performance objectives by engaging stakeholders (owners, designers, building officials), followed by a conceptual bridge design. The conceptual design is then translated into a preliminary design, considering factors such as hazard levels and fundamental design requirements. The next step involves evaluating the bridgeâs performance through nonlinear time history analyses (NLTHAs). This analysis yields EDPs, drift ratio being the most commonly used EDP, which inform assessments of potential seismic impacts including structural damage, repair costs, expected downtime, and safety risks. The bridge design is iteratively refined and re-evaluated against these performance metrics until it meets the predefined objectives, as shown in Figure 5.1. Figure 5.1: Schematic representation of the performance-based seismic design workflow. 119 Nevertheless, this iterative nature of PBSD poses significant computational and labor demands, which discourages its widespread implementation. Consequently, to facilitate more practical implementation, extensive research has been conducted to establish simplified performance limit states in terms of EDPs, particularly, drift ratio for conventional concrete bridge columns (Aldabagh & Alam, 2022; Aldabagh et al., 2022). These limit states set for conventional concrete structures are not directly applicable to UHPC structures due to the unique response mechanisms inherent in each material. Thus, it is vital to develop EDP limit states specifically for UHPC structures. However, there is a dearth of studies that have focused on formulating EDP limits specifically for UHPC bridge columns. Thus, this study aims to address this gap by establishing drift ratio limit states for UHPC bridge columns by leveraging the power of explainable ML model. Unlike conventional statistical approaches and empirical models that often fail to accurately capture the relationship between EDP limit states and their associated predictors, data-driven ML models offer substantial advantages, particularly owing to their ability to accurately determine this relationship without a deep understanding of the systemâs underlying physics. Consequently, these models are emerging as valuable tools in structural and earthquake engineering (Mangalathu & Jeon, 2019; Soleimani, 2022; Todorov & Muntasir Billah, 2022; Wakjira et al., 2022). The increasing integration of ML models in civil engineering research represents a promising trend toward data-driven solutions. However, a critical limitation remains in the translation of these models into practical tools. This oversight limits the practical applicability of both the findings and the models developed in the literature. Thus, to fully realize the potential of ML capabilities in the civil engineering domain, the focus must shift from the theoretical development of the models to the formulation of user-friendly tools that 120 practitioners can employ. Furthermore, there is an equally important need to offer insights into the problem under consideration from the developed models. This process, termed model explainability, facilitates an understanding of key relationships within the data that the models utilize, and investigate the influence of each input feature or factor on predictions. Certain ML models, such as linear regression and decision trees, are transparent in their structure, and thus enables easy interpretation. Conversely, more complex models like ensemble models are âblack boxesâ in nature. To address this challenge, a range of interpretation methods have been proposed in the literature. Some of these methods are model-specific, engineered to interpret distinct ML models. In contrast, other methods, designated as model-agnostic are versatile and can be applied to any ML model. In this context, Shapley Additive exPlanation (SHAP) is a powerful model agnostic ML interpretation technique (Lundberg & Lee, 2017). Therefore, this study aims to bridge the aforementioned research gaps by introducing explainable ML-powered drift ratio limit states for UHPC bridge columns. Given the scarcity of experimental databases, akin to methodologies applied for conventional concrete (Aldabagh & Alam, 2022; Aldabagh et al., 2022), this study employs numerical models to create a comprehensive database considering various sources of uncertainties for UHPC bridge columns. The developed database is then used to introduce ML-based capacity drift limit states for UHPC bridge columns. Additionally, the study investigates the influence of various design parameters on the drift limit states, such as axial load ratio, aspect ratio, section geometry, and the material properties of UHPC and reinforcement steel, using the SHAP approach to provide insights and enhance the explainability of the developed model. Moreover, to ensure the practical applicability of the developed model, it is deployed into a user-friendly GUI software 121 tool. This software tool is designed to enable fast, accurate, and intelligent drift limit state prediction for the UHPC bridge column. An illustrative design example is also provided to demonstrate the practical application of the proposed model in the PBSD of UHPC bridge columns. In addition, capacity uncertainty parameters are proposed to enable the fragility assessment of UHPC columns. 5.2. Methodology Figure 5.2 shows the methodology adopted for this study. Initially, a set of damage criteria is established to define four damage levels for UHPC bridge columns. A comprehensive database of UHPC bridge columns is then developed. Subsequently, an explainable ML powered predictive model is formulated to establish a correlation between EDP and various input parameters that define column geometry, material properties, and internal reinforcement. Figure 5.2: Methodology adopted in this study. 122 5.2.1. Definition of damage limit states In this study, following the PBSD provisions of the Canadian highway bridge design code (CAN/CSA S6-19, 2019), the post-seismic functionality of a bridge is categorized into four performance levels; namely, immediate, service limited, service disruption, and life safety performance. These performance levels correspond to minimal, repairable, extensive, and probable replacement damage states, respectively (CAN/CSA S6-19, 2019). As listed in Table 5.1, the first damage state, DS1, is defined by the first yielding of longitudinal bars, while repairable damage (DS2) is characterized by the onset of spalling in the concrete cover (Aldabagh & Alam, 2022). Though the inclusion of fibers in UHPC is known to mitigate and postpone concrete spalling (Hung & Yen, 2021), the strain corresponding to the spalling initiation has been conservatively set at 0.006. 123 Table 5.1 Selected performance criteria Damage State Damage category Limit state description Damage description DS1 Slight or minimal damage Yielding of longitudinal bars The column exhibits minor damage, which do not compromise its structural integrity or load-bearing capacity. DS2 Moderate/repairable damage Presence of hairline to fine cracks; initial signs of concrete spalling. The column has sustained moderate damage. However, it remains structurally sound, and damage can be repaired without significant reconstruction or element replacement. DS3 Extensive damage with service disruption Presence of wide, through-thickness cracks The column has experienced severe damage, which may jeopardize its load-bearing function. Extensive repairs may be required, which could include partial deconstruction or reinforcement to restore serviceability. DS4 Complete damage, likely requiring replacement Extensive spalling of concrete; core concrete crushing; buckling of longitudinal reinforcing bars The column has suffered critical damage, leading to a substantial reduction in its lateral load resistance. The extent of deterioration is such that structural repair is impractical, and column replacement is recommended. 124 Progressing to extensive damage, DS3, the presence of significant cracks and substantial spalling of the concrete becomes a prominent feature (Aldabagh et al., 2022). Within this damage state, the allowance for widespread spalling is premised on the anticipation that normal service operations will be suspended, thus providing an opportunity to conduct the necessary repair work (CAN/CSA S6-19, 2019). This critical state is reached under one of two conditions (CAN/CSA S6-19, 2019): (a) when the compressive strains in the concrete core approach 80% of the confined concreteâs ultimate compression strain (Éðð¢) to provide a margin of safety against crushing of the core concrete, a condition that often necessitates urgent and extensive intervention (Lehman et al., 2004), or (b) when the strain in the reinforcing steel reaches 0.05 (CAN/CSA S6-19, 2019). DS-4 represents the highest damage level, which is distinguished by one of two possible conditions: either the onset of concrete core crushing (the strain in core concrete reaches to Éðð¢) or buckling of the longitudinal reinforcement bars. The value of Éðð¢ is determined using equation developed in Chapter 4, Eq. (4.20). The strain corresponding to the buckling of the longitudinal bars is determined using the equation proposed by (Jason Chad Goodnight, Kowalsky, & Nau, 2016), as illustrated in Eq. (5.1). ððð = 0.03 + 700ðð ð¡ðð¦ð¸ð â 0.1ðððð´ð (5.1) In the above equations, the variables ðð and ðð¦ denote the compressive strength of the unconfined concrete and the yield strength of the reinforcement bars, respectively, and both are provided in MPa. The variable ð¸ð is the elastic modulus of the steel bar. Lastly, ð/ððð´ð signifies the axial load ratio. 5.2.2. Bridge pier database Following the definition of damage state for each performance level in terms of 125 strain limits, a Latin hypercube sampling (LHS) method is used to construct an extensive database of UHPC bridge column samples comprising 10,000 UHPC columns. The primary advantage of LHS over traditional sampling methods like Monte Carlo sampling (MCS) is its ability to provide a more uniform sampling of the parameter space, to ensure all portions of the parameter space are explored, unlike the purely random approach of MCS. This uniformity is particularly important when dealing with complex, nonlinear systems like bridge columns under seismic loading, where the response can be highly sensitive to specific parameter combinations. The bridge pier database is constructed considering the uncertainties stemming from a range of design factors that influence the seismic response of bridge columns. The seismic response of bridge columns is influenced by several critical factors. Among these, the aspect ratio (height-to-diameter ratio, L/D) of conventional bridge columns is one of the critical factors that significantly influence seismic behaviour (Lehman et al., 2004; Mostafa, Sanders, & Saiidi, 2004; Yeh, Mo, & Yang, 2001, 2002). The axial load ratio is another important factor that is extensively studied in the literature for conventional bridge columns (Esmaeily & Xiao, 2005; Jason Chad Goodnight et al., 2016; Lehman et al., 2004). The mechanical properties of steel reinforcement bars (Barcley & Kowalsky, 2020) and concrete, alongside the ðð ð¡ (Lehman et al., 2004; Mo & Nien, 2002) and longitudinal reinforcement ratio (ðð ð) (Lehman et al., 2004), are also critical. Additionally, the pier geometry plays a crucial role (Jason C Goodnight, Kowalsky, & Nau, 2013). Each of these factors must be carefully considered to accurately assess and enhance the seismic response of bridge columns. These factors, along with their distributions and associated parameters, are detailed in Table 5.2. As listed in this table, bridge columns with aspect ratio of 4.0 or higher are considered, implying flexural dominated column (Zhu, Elwood, & Haukaas, 126 2007). Utilizing LHS, 10,000 diverse sets of bridge column samples are generated. These bridge columns are subsequently subjected to static pushover loading to determine the drift ratios at the onset of each of the four damage states. Table 5.2 Uncertainty parameters of UHPC bridge columns and their distribution (Mangalathu et al., 2018). Parameters Distribution type Distribution parameters Mean Standard deviation Lower Upper Yield strength of steel bars, ðð¦ (MPa) Normal 475.7 37.9 400.0 551.6 Compressive strength of concrete, ðð (MPa) Normal 181.9 27.4 100 250 Volume fraction of UHPC fibers, ðð (%) Uniform - - 1.0 3.0 Longitudinal reinforcement ratio, ðð ð (%) Uniform - - 1.0 6.0 Volumetric ratio of spirals, ðð ð¡ (%) Uniform - - 0.4 3.0 Column height, L (m) Normal 7.13 1.15 3.62 14.1 Aspect ratio, ð¿/ð· Uniform - - 4 10 Axial load ratio, ð/ððð´ð (%) Uniform - - 5 20 *Uniform distribution is assumed for the cases where a distribution type is not available. In the present study, a numerical model of the bridge piers was developed using OpenSees software platform (Mckenna, 2011), as shown in Figure 5.3. The model employed displacement-based beam-column elements that are partitioned into six elements (Figure 5.3). 127 Figure 5.3: Bridge pier model. The initial element, characterized by a height equivalent to the plastic hinge length (ð¿ð), incorporated 10 integration points. The remaining height of the pier is partitioned into five elements of equal height, each with five integration points. Given its enhanced predictive accuracy over alternative models, the probabilistic model proposed by Ning and Li (Ning & Li, 2016) is used to determine the value of ð¿ð, as follows: ð¿ð = (0.042 + 0.072ððð) ð¿ + 0.298ð» + 6.407ðð (5.2) where ð/ðð is the axial load ratio, ð¿ is the pier length, ð» is the pier sectional height, and ðð is the diameter of longitudinal reinforcement bars. The integration points with each element are represented with a fiber cross-section that accounts for both geometric nonlinearity and the spatial distribution of plasticity. These fiber cross-sections facilitate the assignment of material-specific properties at distinct locations within the memberâs cross-section. Each fiber section is further divided into concrete fibers, which comprise both cover (unconfined) and core 128 (confined) concrete fibers, as well as steel fibers. The uniaxial stress-strain response of both unconfined and confined concrete was modelled with the Concrete01 model (Mckenna, 2011). To simulate the behaviour of the longitudinal reinforcement, the model adopts the nonlinear Giuffrè-Menegotto-Pinto steel model along with the isotropic strain hardening rules of Filippou et al. (Filippou, Popov, & Bertero, 1983), designated as the uniaxial Steel02 model in OpenSees (Mckenna, 2011). The peak and ultimate stress-strain responses of confined UHPC were determined using the equations developed in Chapter 4. Additionally, the geometric nonlinearity is accounted by incorporating the P-delta effect into the bidge columns. The strain penetration effect is accounted for in the numerical model through the integration of a zero-length section element, which incorporates the hysteretic model developed by Zhao and Sritharan (Zhao & Sritharan, 2007), labeled as Bond SP01 in OpenSees, as shown in Figure 5.3. The numerical model is validated against UHPC columns tested by Aboukifa and Moustafa (Aboukifa & Moustafa, 2021). As shown in Figure 5.4, the developed numerical model exhibited a high level of accuracy in simulating the hysteresis behaviour of UHPC columns. The comparison reveals that the peak strength predictions by the model diverge from the experimentally obtained values by a narrow margin of no more than 8%. Furthermore, the discrepancies in initial stiffness between the numerical simulations and experimental results are primarily attributed to uncertainties in material characteristics, compounded by minor imperfections in the test specimens. Surface irregularities in the concrete samples, for instance, can lead to localized stress concentration and result in a conservative estimation of the initial stiffness. 129 (a) S1 (Aboukifa & Moustafa, 2021) (b) S3 (Aboukifa & Moustafa, 2021) Figure 5.4: Predicted versus experimental hysteretic response of UHPC bridge piers. 5.2.3. Development of ML powered drift capacity limit states The developed numerical model is used to determine the drift ratio of the 10,000 UHPC column samples generated by LHS at the four damage states. Following the development of the bridge column database, an advanced ML algorithm, particularly, XGB is utilized to model the drift ratio across the damage states. It is particularly noted for its efficiency and ability to handle large datasets, which is largely attributable to its underlying architecture and optimization techniques. The choice of XGB in this study is supported by a preliminary study that includes simple to complex ML models. The hyperparameters of the model are fine-tuned using Optuna (Akiba et al., 2019), a next-generation optimization framework, complemented by the 10-fold cross-validation technique. 5.2.4. Comprehensive evaluation of model performance in modelling drift limit states Several statistical performance metrics, each addressing different aspects of the modelâs accuracy and explanatory power, are used to comprehensively evaluate the 130 performance of the model developed in this study. These metrics include R2, RMSE, normalized RMSE (NRMSE), MAE, composite fitness score (CFS), and DX% metrics. The integration of ð 2 and NRMSE into a CFS in Eq. (5.3) offers a thorough assessment of a modelâs performance by capturing both the strength of the linear correlation (as indicated by ð 2) and the accuracy of the predictions (as reflected by NRMSE), which is critical for the development of a robust predictive model for drift limit states. CFS = ðð ððð¸ +1ð 2 (5.3) A CFS value of 1.0 signifies an ideal model, where the predictions perfectly match the observed data. This is characterized by the lowest possible prediction error (NRMSE = 0) and the highest coefficient of determination (ð 2 = 1.0). In addition, the DX% metric proposed by Guan et al. (Guan, Burton, Shokrabadi, & Yi, 2021) is used to further assess model performance. It quantifies the proportion of predictions for which the relative error falls within the specified threshold, such as 25% (ð·25%). This approach offers a practical, easily interpretable measure of accuracy, especially beneficial in scenarios where adhering to specific error limits is crucial (Guan et al., 2021). Let DX% be the percentage of data points where the predicted value is within X% of the actual value. This can be calculated by dividing the count of such data points by the total number of data points, as given in Eq. (5.4). DX% =1ð(ððð¢ðð¡ðð [|?Ì?ð â ð¦ðð¦ð| â¤ð100]) (5.4) where ð is the user-defined percentage threshold. 5.2.5. Model exPlainability A unified model-agnostic SHapley Additive exPlanations (SHAP) (Lundberg & Lee, 2017) framework is used in this study to explain the outputs of the proposed 131 model and identify the most significant factors that influence the drift ratio at various damage states. SHAP assigns a Shapley value to each factor, reflecting its contribution to the prediction (Lundberg & Lee, 2017). To calculate the Shapley value for a given factor, the modelâs output is computed with and without this factor across all possible combinations of other factors (Lundberg & Lee, 2017). This process requires examining the change in the prediction when a particular factor is added to a subset of factors, thus capturing its marginal contribution. The Shapley value for factor ð with respect to a prediction model ð and an input instance ð¥ is given by (Lundberg & Lee, 2017): ðð(ð, ð¥) = â|ð§â²|! (ð â |ð§â²| â 1)!ð![ðð¥(ð§â²ð{ð}) â ðð¥(ð§â²)]ðâ²âð¥{ð} (5.5) where ðð(ð, ð¥) is the Shapley value for factor ð, ðâ²ð{0,1}ð equals unity when the factor is observed and zero otherwise, |ð§â²| is the number of elements in the subset ðâ², ð is the total number of factors, ðð¥(ð§â²ð{ð}) is the prediction of the model when the factor set ðâ² is augmented with the factor ð, ðð¥(ð§â²) is the prediction of the model using the subset ðâ² without the factor ð. In the above equation, the sum runs over all subsets ðâ² of factors that do not include ð, while the factorial terms |ð§â²|! and (ð â |ð§â²| â 1)! along with ð! compute the number of ways factors can be ordered, which is essential for fair distribution among factors according to Shapley value theory. The overall sum accounts for the average contribution of factor ð to the change in prediction from all possible combinations of other factors within the model. This average is the Shapley value, and it quantifies the importance of factor ð. A higher absolute Shapley value indicates a greater influence of that factor on the predicted drift ratio. 132 5.3. Results and Discussion 5.3.1. Optimization results As previously discussed, Optuna, an advanced optimization framework, is utilized to conduct a thorough and efficient search of the hyperparameter space of the model. This process was complemented by 10-fold cross-validation, ensuring that the modelâs performance was rigorously evaluated across different subsets of the data. In the optimized XGB model used for predicting the drift ratio of UHPC bridge columns at various damage states, several key hyperparameters were fine tuned. Table 5.3 lists the range of each hyperparameter along with its optimized value. The number of trees, a critical hyperparameter, was set at 482, as listed in Table 5.3. This number is significant in determining the capacity of the model to identify diverse patterns within the data, which is essential for accurate drift capacity prediction. Complementing the number of trees, the learning rate was optimized to approximately 0.0216, and the maximum tree depth was set to 5. The learning rate controls how quickly the model adapts to the data, and a rate of 0.0216 helps in capturing the complex details in the data and avoids the pitfall of overfitting. Similarly, a maximum depth of 5 allows the model to explore complex relationships in the data without becoming excessively complex. This depth is particularly important in modelling the drift limit states, where the relationships between various parameters can be highly complex, but a too deep tree might start learning noise, detracting from the predictive accuracy of the model. 133 Table 5.3 Hyperparameters ranges and their optimal values determined through Optuna with 10-fold cross-validation Hyperparameter Range Optimized value Number of estimators (number of trees) [10, 1000] 482 Maximum depth [2,10] 5 Learning rate [0.01, 0.5] 0.0216 Sumsample fraction [0.1, 1.0] 0.6526 α (L1 regularization term) [0.00001, 1.0] 0.00175 ð (L2 regularization term) [0.00001, 1.0] 0.04222 The subsample fraction, which is the fraction of the training data to be used for each tree was optimized to about 0.6526. This hyperparameter introduces a degree of randomness into the model, which ensures that each tree in the model is built with a slightly different subset of the data. Moreover, the ð¼ and λ parameters are optimized to approximately 0.00175 and 0.04222, respectively. These hyperparameters help prevent overfitting by ensuring that the model makes decisions based on substantial data evidence and significant improvements in loss reduction. These carefully optimized hyperparameters collectively enhance the capability of the model to produce reliable and accurate predictions for drift capacity ratio at various damage states. 5.3.2. Drift ratio limit state prediction results Figure 5.5 shows the predictive performance of the proposed model for the drift capacity of UHPC columns at the four damage states in terms of the scatter plots of the predicted versus actual drift. For the first damage state, the scatter plot in Figure 5.5a shows a very close alignment between predicted and actual values, as indicated by a high ð 2 value of 98.1% for the training set and 96.1% for the test set. For DS2, the model maintains a high level of accuracy with ð 2 scores of 97.1% for the training set and 94.1% for the test set, as shown in Figure 5.5b. Similarly, the model demonstrated consistency in its high predictive accuracy for drift capacities at DS3 and DS4, as shown 134 in Figure 5.5(c and d), respectively. The evaluation of the model for both DS3 and DS4 reveal ð 2 values of 99.6% in the training, while on the test dataset these values were 98.8% (DS3) and 98.6% (DS4). The consistent high ð 2 scores across both training and testing phases underscore the robustness of the proposed model. Figure 5.5: Actual versus predicted drift ratio at various damage states. In addition to ð 2, the performance of the model is comprehensively assessed using RMSE, MAE, NRMSE, and CFS metrics, along with the proportion of predictions falling within 25% of the actual values (ð·25%) and the results are summarized in Table 5.4 for both the training and test datasets. For the drift at DS1, the low RMSE (0.021 in training and 0.030 in test), MAE (0.015 in training and 0.020 in test), and NRMSE (0.052 in training and 0.080 in test), together with the CFS (1.071 in training and 1.121 in test), underscore the high predictive performance of the model. 135 The model also showed exceptional consistency in prediction accuracy, with ð·25% at remarkable 99.7% in training and 97.9% in testing phases. While the drift at DS2 indicates a slight decrease in these performance metrics, the results remain robust. This trend suggests the varying efficiency of the model in handling different damage states. The prediction of drift capacity at DS3 and DS4 exhibited consistency in high predictive accuracy as shown by the RMSE, MAE, and NRMSE, with all metrics indicating low prediction errors and CFS values (1.055 and 1.059 in train, 1.090 and 1.097 in test) that signify a reliable model performance. The ð·25% remains exceptionally high for both DS3 and DS4, which highlights the efficacy of the model in predicting the drift limits of UHPC columns with high reliability across various damage states. Table 5.4 Performance of the model Response Variable Training set Test set R2 RMSE MAE NRMSE CFS ð·25% R2 RMSE MAE NRMSE CFS ð·25% DS1 0.981 0.021 0.015 0.052 1.071 99.7 0.961 0.030 0.020 0.080 1.121 97.9 DS2 0.971 0.023 0.016 0.061 1.091 98.9 0.941 0.032 0.020 0.094 1.157 96.6 DS3 0.996 0.011 0.008 0.052 1.055 99.8 0.988 0.019 0.012 0.078 1.090 99.3 DS4 0.996 0.012 0.008 0.055 1.059 99.7 0.986 0.020 0.013 0.083 1.097 99.1 5.3.3. ExPlainable drift capacity model: Overcoming the black box challenge with SHAP The application of the SHAP approach in this study addresses the often-cited âblack-boxâ nature of ML models. SHAP values offer a dual perspective on feature importance, detailing not only how strongly each factor influences the predictions of drift capacity across various damage states but also indicating the direction of its effects, whether it acts to increase or decrease the predicted values. The SHAP summary plots in Figure 5.6(aâd) quantify the influence of each factor on drift ratio predictions across different damage states. At DS1, the SHAP summary plot highlights ð¿/ð·, ð/ððð´ð, ðð ð, and ðð¦ as the primary contributors, as shown in Figure 5.6a. The presence 136 of ðð¦ indicates the sensitivity of the drift ratio at DS1 to the yielding behaviour of reinforcing steel at this damage state. These parameters (ð¿/ð·, ð/ððð´ð, ðð ð, and ðð¦) similarly showed significant influence at DS2, with ð· replacing ðð¦ as one of the top four contributors (Figure 5.6b). This shift highlights the varying sensitivity of drift ratio predictions to different factors across damage states. Furthermore, the results in Figure 5.6 showed that higher values of the top three parameters tend to increase the drift ratio. Conversely, ð· showed a mixed influence with a cluster of both positive and negative SHAP values, suggesting a non-linear relationship with drift ratio that may depend on the interaction with other factors, as can be seen in Figure 5.6b. At the higher damage states of DS3 and DS4, the ðð ð¡ becomes among the top four key parameters alongside ð»/ð·, ð/ððð´ð, and ðð ð, as can be observed in Figure 5.6(c and d). The high influence of ðð ð¡ in these states reflect the influence of confinement level on the post-cracking behaviour of the concrete. Higher values of ðð ð¡ lead to enhanced confinement of the concrete and effective control of crack widths, which could be critical in maintaining the structural integrity under severe damage levels, hence its increased influence on the drift prediction in these states.137 (a) DS1 (b) DS2 (c) DS3 (d) DS4 Figure 5.6: SHAP summary plot for drift ratio prediction at various damage states. 138 (a) DS1 (b) DS2 (c) DS3 (d) DS4 Figure 5.7: Importance ranking of input parameters at various damage states.139 For a comprehensive understanding of the importance of the factors in predicting the drift ratio of UHPC bridge columns at various damage states, the mean absolute SHAP values and their corresponding percentage contributions for all factors across different damage states are listed in Table 5.5. Moreover, Figure 5.7(aâd) rank the factors by their relative importance on the y-axis, with the x-axis showing the mean absolute SHAP values. For the drift ratio at DS1, the aspect ratio has the highest mean absolute SHAP value, and its percentage contribution is approximately 43.62% of the total SHAP impact, as shown in Figure 5.7a and Table 5.5. Table 5.5 Mean absolute SHAP values of the factors and their corresponding percentage contributions Input feature Mean absolute SHAP values Percentage contribution DS1 DS2 DS3 DS4 DS1 DS2 DS3 DS4 ð· 0.010 0.012 0.019 0.018 4.214 6.238 5.710 5.548 ð¿/ð· 0.102 0.085 0.077 0.070 43.62 44.38 23.23 21.31 ð/ððð´ð 0.051 0.047 0.049 0.045 21.73 24.71 14.77 13.70 ðð 0.006 0.003 0.037 0.042 2.544 1.501 11.22 12.70 ðð 0.006 0.004 0.003 0.002 2.776 2.011 0.778 0.651 ðð¦ 0.023 0.006 0.019 0.018 9.978 2.993 5.674 5.453 ðð ð 0.033 0.032 0.038 0.043 14.24 16.55 11.65 12.97 ðð ð¡ 0.002 0.003 0.089 0.091 0.904 1.617 26.98 27.66 The axial load ratio accounts for about 21.73% of the total influence on drift ratio predictions at DS1. As the damage state progresses to DS2, while the ð¿/ð· remains a major factor, its relative importance slightly increased to 44.38%, as shown in Figure 5.7b and Table 5.5. At the same damage state, the ð/ððð´ð increases in relative importance, making up 24.71% of the total SHAP impact (Table 5.5). This shift underscores the changing influence of structural parameters as damage progresses. The ðð ð¡, which is a key factor of post-cracking behaviour, showed a marked increase in its 140 mean absolute SHAP values and relative importance at the more severe damage states of DS3 and DS4. Its percentage contributions are 26.98% and 27.66% for DS3 and DS4, respectively, which underscore the critical role of concrete confinement in maintaining structural integrity during extreme seismic events. 5.3.4. Deployment of the model and design example The developed model has been integrated into an interactive software tool to facilitate its practical application. As illustrated in Figure 5.8, the tool features an intuitive graphical user interface, designed to simplify the user experience. This tool offers a rapid, accurate, and intelligent method for predicting the drift capacity of UHPC bridge columns across a spectrum of potential damage states. Figure 5.8: Graphical user interface of the deployed ML-based predictive tool for drift ratio limit states in UHPC bridge columns. 141 To show the application of the proposed ML-based model for the PBSD of UHPC bridges, a design example of a UHPC bridge is presented. The case study features a viaduct located in Vancouver, BC, Canada, which consists of a series of identical 50 m spans supported by single-column bents, integral to a three-lane freeway. Its design includes a foundation on stiff soil, classified as site soil class C. The superstructure imposes a uniform self-weight of 300 kN/m, which results in a tributary seismic weight of 15,000 kN applied at an elevation of 12.5 m, equivalent to the centroidal axis of the superstructureâs mass. For the analysis, it is assumed that the seismic excitation and the corresponding dynamic response of the viaduct are synchronous along its length. This assumption allows the behaviour of the viaduct to be modelled using a single-degree-of-freedom system that represents the response of all bents by a single representative column. The column had a diameter of 1.85 m and a clear height of 12.0 m, which corresponds to an aspect ratio of 6.5. The reinforcement of the bridge column employed CSA G30.18 400W rebar with a yield strength of 440 MPa. The concrete used was UHPC with a design compressive strength of 100 MPa with a volumetric fiber ratio of 3%, which results in an axial load ratio of 5.6%. The reinforcement ratio for the longitudinal bars was initially set at 1%, per CSA S6-19. Moreover, the transverse reinforcement involved 20M spirals spaced at 50 mm (transverse reinforcement ratio of 1.29%). The influence of foundation flexibility is outside the scope of the current study and thus it was not considered in the analysis. The bridge, given its significance as a lifeline route, necessitates adherence to two primary performance objectives under performance-based design guidelines. The first objective dictates minimal damage (DS1) under a seismic event with a 5% exceedance probability over a 50-year period (return period of 975-year), while the second allows for more extensive damage (DS3) under a more severe event with a 2% 142 exceedance probability over the same duration (return period of 2475-year) as per (CAN/CSA S6-19, 2019). Using site-specific uniform hazard spectra, the drift demands were calculated, while the drift capacity limit states were determined using the developed software tool, and the results are presented in Table 5.6. Table 5.6 PBSD of UHPC bridge bent of viaduct Iteration Drift ratio demand (%) Drift ratio capacity Evaluation of performance objective 975-years 2,475-year DS1 DS2 Objective 1 Objective 2 1 1.69 2.46 1.22 2.12 Not achieved Not achieved 2 1.58 2.30 1.30 2.30 Not achieved Achieved 3 1.49 2.17 1.42 2.50 Not achieved Achieved 4 1.41 2.05 1.51 2.61 Achieved Achieved Iterative design adjustments were made to the longitudinal reinforcement ratios to meet these objectives, as detailed in Table 5.6. During every iteration, the ratio of longitudinal reinforcement was increased by 1%, keeping all other variables constant. Table 5.6 illustrates that an increase in the longitudinal reinforcement ratio led to higher drift ratios at DS1 and DS3 (capacities), while also reducing the drift ratio demands by increasing the effective flexural rigidity. Ultimately, the final design, with a 4% longitudinal reinforcement ratio, successfully met the two performance objectives as outlined in Table 5.6. 5.3.5. Uncertainty analysis of drift ratio at different limit states In the realm of seismic fragility analysis, the probabilistic seismic demand (PSD) model or fragility function hinges on the accurate characterization of the seismic capacity of structures. In this analysis, the fragility function, denoted as ð¹ð(ð¥), is defined by the probability of a structure reaching or exceeding a given limit state during an earthquake. This probability can be expressed as: 143 ð¹ð(ð¥) = ð· (ln(ðð·/ðð¶)ð½ð ) (5.6) where ðð¶ and ðð· are the median estimate value for the structural capacity and seismic demand, respectively, while ð½ð is the dispersion parameter that reflects the uncertainties associated with seismic demand and structural capacity, as given below in terms of the logarithmic standard deviation demand (ð½ð·|ð¼ð) and capacity (ð½ð¶). ð½ð = âð½ð·|ð¼ð2 + ð½ð¶2 (5.7) The parameters ð½ð and ðð¶ are crucial for quantifying the variability and central tendency in structural responses to seismic events, respectively. Specifically, ð½ð, the logarithmic standard deviation of capacity, measures the spread or dispersion around the median capacity. ðð¶, on the other hand, signifies the median value of the structural capacity. While established literature offers a spectrum of ð½ð and ðð¶ values for conventional concrete bridge columns, such specific metrics have not been previously identified for UHPC bridge columns. This absence is notable, given the critical influence of these parameters on the seismic fragility analysis of UHPC bridges. To address this gap, in this study, the drift ratio, the most widely used EDP, obtained considering the uncertainty of several factors that affect the seismic response of UHPC bridge columns in the previous section are fitted to lognormal distribution, as shown in Figure 5.9. The adherence of the drift data to the lognormal distribution is evident from the high ð 2 values obtained for each limit state. These ð 2 values, 0.90 for DS1, 0.98 for DS2 and DS3, and 0.99 for DS4, confirm that the lognormal distribution provides a strong representation of the drift data, which ensures that the capacity estimates derived from it are reliable. The values of ð½ð and ðð¶ are calculated based on the fitted 144 probabilistic distribution curves. Specifically, ðð¶ value is 1.93 for DS1, 3.14 for DS2, 5.04 for DS3, and 6.19 for DS4. Correspondingly, ð½ð value is, 0.27 for DS1, 0.26 for DS2, 0.33 for DS3, and 0.36 for DS4. These values, summarized in Table 5.7 for all limit states, are critical for the fragility analysis of UHPC bridge columns. Figure 5.9: Lognormal distribution estimation of drift ratio limit states. Table 5.7 Drift ratio capacity uncertainty parameters at different damage states Damage state Parameters ðð¶ ð½ð¶ DS1 1.93 0.27 DS2 3.14 0.26 DS3 5.04 0.33 DS4 6.19 0.36 145 5.3.6. Seismic fragility assessment of UHPC bridge columns In this section, the developed drift capacity models and drift capacity uncertainty parameters are used to assess the seismic fragility of UHPC columns at various damage states. The bridges are assumed to be located in Soil Class-C, as categorized by the National Building Code of Canada (NBCC, 2020), which is equivalent to the Type-II soil profile according to (CSA, 2014). The analysis comprised four distinct UHPC bridge columns labelled as Specimen 1 through Specimen 4, each reinforced with varying grades of steel bars. Each of these four UHPC bridge column specimens had a diameter of 1.85 meters and an aspect ratio of 6.49. They are designed with an axial load ratio of 5%, and the UHPC used contains a volume fiber composition of 3%, along with a compressive strength of 150 MPa. The specimens are uniformly confined using 20M spirals spaced at intervals of 50 mm. The longitudinal reinforcement used 40 bars of 35M steel with yield strengths of 420 MPa, 440 MPa, 480 MPa, and 550 MPa for Specimen 1 through Specimen 4, respectively. A range of 20 near-fault ground motion records from the PEER strong motion database (PEER, 2020) were utilized for the fragility analysis of the columns. The characteristics of the selected ground motions are summarized in Table 5.8. These records, with original peak ground accelerations (PGAs) ranging from 0.11â0.85 g and within 10 km of the epicenter, represent the near-field seismic effects (CSA, 2014). The ground motions are spectrally matched to the uniform hazard design response spectra of the selected location, focusing on scenarios with a 2% probability of exceedance over a 50-year period, and were conducted following the NBC 2020 standards for soil class C with a damping ratio of 5%. The wavelets algorithm by Hancock et al. (Hancock et al., 2006), implemented through the SeismoMatch software (Seismosoft, 2024), was employed for this purpose. The ground motion records were scaled using factors from 146 0.25 to 3.0 in increments of 0.25. This scaling approach generated a comprehensive set of 240 varied ground motions. Thus, each specimen was subjected to the 240 ground motions applied in the longitudinal direction of the column. For dynamic analyses, Rayleigh damping is utilized to to model the energy dissipation in the column. Table 5.8 Characteristics of selected ground motions S.N. Earthquake Year Station Distance to fault (kN) Magnitude PGV (cm/s) PGA (g) 1 Chi-Chi 1999 CHY101 9.9 7.6 65 0.34 2 Northridge 1994 Newhall-Fire Sta 9.4 6.7 7 0.11 3 Imperial Valley 1940 El Centro Array #9 6.1 7 30.9 0.28 4 Kobe, Japan 1995 KJMA 1 6.9 91.1 0.83 5 Coalinga 1983 Oil City 9.5 5.2 13.6 0.37 6 Northridge 1994 Sepulveda Hospital 8.4 6.7 77.7 0.75 7 Imperial Valley 1979 Westmorland Fire 9.8 6.5 12 0.11 8 Northridge 1994 Arleta-Nordhoff Fire 8.7 6.7 41.1 0.35 9 Landers 1992 Lucerne 2.2 7.3 133.4 0.73 10 Loma Prieta 1989 Corralitos 3.9 6.9 56 0.65 11 Chi-Chi 1999 TCU050 9.5 7.6 36.7 0.15 12 Imperial Valley 1979 El Centro Array #5 4 6.5 48.9 0.53 13 Chi-Chi 1999 TCU067 0.6 7.6 92.1 0.5 14 Northridge 1994 Sylmar-Converter 5.2 6.7 121 0.85 15 Loma Prieta 1989 LG-Lexington 5 6.9 85.7 0.44 16 Coalinga 1983 Pleasant Valley P. P 8.4 6.4 39.4 0.3 17 Kobe, Japan 1995 Takatori 1.5 6.9 120.7 0.62 18 Loma Prieta 1989 Saratoga-Valley Coll 9.3 6.9 42.1 0.26 19 Kobe, Japan 1995 Port Island 3.3 6.9 90.7 0.35 20 Chi-Chi 1999 CHY080 2.7 7.6 106.8 0.81 In this study, PGA is adopted as the IM for demand modelling, since past studies indicate its validity for bridge fragility assessment (Mackie & StojadinoviÄ, 2005; Padgett & Desroches, 2008; Padgett, Nielson, & Desroches, 2008). Figure 5.10 shows the probabilistic seismic demand for each specimen. To determine the fragility curves, the drift capacity of the columns is determined with the aid of the developed model; while the value of ð½ð determined in Section 5.3.5 is employed. The resulting fragility curves presented in Figure 5.11 illustrate the seismic vulnerability of the four UHPC 147 column specimens with different reinforcement bar grades, evaluated across four damage states (DS1 to DS4). The probability of exceedance for all specimens increases with PGA, which is the expected behaviour for fragility curves. Notably, specimens with higher-grade reinforcement bars (with yield strengths of 480 MPa and 550 MPa) showed lower probabilities of exceedance, which implies better seismic resilience compared to those with lower-grade bars. This can be observed in Figure 5.12 which compares the fragility curves of the four specimens at various damage states. As can be seen in Figure 5.12(aâd), the UHPC column reinforced with higher bar grades showed less vulnerability. This trend is consistent across all damage states though the difference is higher in lower damage states. These results suggest that the use of higher-grade reinforcement can significantly improve the performance of UHPC columns under seismic loading. Figure 5.10: PSD model for (a) Specimen 1, (b) Specimen 2, (c) Specimen 3, and (d) Specimen 4. 148 Figure 5.11: Results of fragility curves. Figure 5.12: Comparative analysis of fragility curves of UHPC columns at different damage states. 149 5.4. Summary This study proposed explainable ML-powered performance limit states for UHPC bridge columns in terms of drift ratio. The performance of the proposed model is comprehensively evaluated using various performance metrics. Moreover, the influence of different factors that characterize the pier geometry, internal reinforcement, and material strengths on the drift limit states of UHPC columns is investigated with the aid of a unified SHAP approach. The application of SHAP in this study addresses the often-cited âblack-boxâ nature of ML models. The mean absolute Shapley values, coupled with their percentage contributions, allowed a comprehensive interpretation of the model predictions and investigated the influence of the factors on the drift ratio limit states across different damage states. The proposed model is deployed to a GUI-based software tool in order to enable its practical application. A design example is also presented to demonstrate the application of the proposed model in the PBSD of UHPC bridge columns. Additionally, the study proposed capacity uncertainty parameters ð½ð and ðð¶ for different damage states in order to enable seismic fragility assessment of UHPC structural members. Finally, the study conducts a seismic fragility assessment of UHPC bridges in Vancouver, BC, Canada using the proposed capacity model and ð½ð. The following conclusions can be drawn from the results of the current study: â¢ The proposed model exhibited strong predictive performance across varying damage states (DS1 to DS4). In lower damage states (DS1 and DS2), the model showed high R2 values of 0.961 and 0.941, respectively, in the test sets, complemented by RMSE values of 0.030 and 0.032, MAE about 0.020, and NRMSE values of 0.080 and 0.094. The consistency in high predictive performance in these damage states is further confirmed by CFS values 150 (1.121 and 1.157) and ð·25% metrics close to 98%, indicating the effectiveness of the model in capturing drift capacity at DS1 and DS2. â¢ Similarly, in severe damage states (DS3 and DS4), the model showed a consistently high predictive performance as evidenced by high R2 values of 0.988 and 0.986 on the test dataset for DS3 and DS4, respectively. Additionally, the model exhibits significantly lower error metrics with RMSE of 0.019 and 0.020, MAE of 0.012 and 0.013, and NRMSE values of 0.078 and 0.083 on the test dataset). The consistency and reliability of the model in these states are further evidenced by CFS of 1.090 to 1.097 and ð·25% metrics at approximately 99%. Overall, these metrics collectively underscore the reliability and accuracy of the model in predicting the drift capacity of UHPC columns across various damage states. Furthermore, this performance on an unseen dataset demonstrates the robustness and generalization ability of the model. â¢ The results of the SHAP analysis revealed the significance of the input features on drift across various damage states, with their relative importance shifting as damage progresses. Particularly, ð¿/ð·, exhibits the highest mean absolute Shapley values across all damage states, with a significant percentage contribution of 43.62%, 44.38%, 23.23%, and 21.31% at DS1, DS2, DS3, and DS4, respectively. However, it showed a higher significant influence and percentage contribution at lower damage states DS1 and DS2. Moreover, ð/ððð´ð and ðð ð showed the highest absolute Shapley values and percentage contribution for drift capacity at DS1 and DS2. As the damage states progress to DS3 and DS4, a shift occurs, with ðð ð¡ emerging as the key 151 factor, signifying the significance of the confinement level at severe damage states. â¢ The lognormal distribution fitting for the drift capacity of UHPC columns, evidenced by high ð 2 values, underscores its effectiveness in accurately capturing the seismic response across various limit states. The seismic capacity parameters for UHPC bridge columns are identified, revealing ðð values ranging from 1.93 to 6.19 and ð½ð values from 0.27 to 0.36 across limit states, significantly enriching the seismic fragility analysis of these structures. â¢ Fragility assessments of UHPC columns with varying steel bar grades indicate that columns reinforced with higher-grade bars are less vulnerable to seismic forces, though the influence of bar grade on seismic vulnerability reduces at more severe damage states. Furthermore, the practical applicability of the proposed model in performance based seismic design is exemplified by a case study involving a bridge viaduct in Vancouver, BC, Canada. In addition, the study performed the fragility assessment of UHPC bridge columns using the developed model and capacity parameters. Future research should focus on exploring the use of the model and the proposed ð½ð and ðð¶ values for different UHPC structures. Additionally, it is recommended to conduct bridge-specific fragility assessments for highway bridges constructed with UHPC. In these assessments, the capabilities of advanced ML models can be leveraged to conduct UHPC bridge-specific fragility analysis. Moreover, future research is suggested to investigate the influence of different sources of uncertainties on the seismic fragility of UHPC bridges. 152 CHAPTER 6: MULTIVARIATE FRAGILITY ASSESSMENT OF UHPC BRIDGES USING HYBRID MACHINE LEARNING MODEL 6.1. Introduction Despite its robust properties, bridges made with UHPC are not exempt from the threats posed by earthquakes, which can lead to significant human and economic consequences. This reality underscores the importance of seismic vulnerability assessments in UHPC bridges. Such assessments play a key role by providing data for proactive risk reduction and prioritizing structural upgrades or retrofit (Padgett & DesRoches, 2009; Padgett, Desroches, & Nilsson, 2010). A fundamental part of these assessments is seismic fragility analysis, which is a probabilistic approach that estimates the likelihood of a structure or its components meeting or exceeding a particular limit during an earthquake of a specified intensity measure (IM) (Choi, DesRoches, & Nielson, 2004; Nielson & Desroches, 2007; Salkhordeh, Govahi, & Mirtaheri, 2021). This approach ensures that bridges are not only constructed with advanced materials but are also designed to endure the challenges posed by natural disasters. Numerous techniques such as analytical, empirical, experimental, and hybrid techniques have been introduced for seismic fragility assessment of civil infrastructures with the analytical methods being the most frequently used approach (Billah & Alam, 2015; M, MQ, J, & T, 2000; P, A, & K.M, 2002; Rossetto & Elnashai, 2003). However, these methods suffer from either poor resolution or the need for excessive time and computational resources (Choi et al., 2004). The most widely used analytical fragility analysis approach, for instance, often necessitates substantial computational resources through nonlinear time history analysis (NLTHA) to compute the EDPs of concern, such as drift. In addition, the conventional approach for fragility analysis assumes that 153 structures with similar configurations will suffer comparable damage levels under a given excitation. Consequently, fragility curves are typically established for specific types of bridges and structural systems based on a chosen classification system (Mangalathu et al., 2018). However, this method may undermine the reliability of loss assessment when applied to a diverse range of bridges that vary significantly within a particular class as it does not consider vital bridge attributes, such as geometries. Additionally, classification criteria may be subjective or vary by region (Mangalathu et al., 2018). Moreover, developing a practical and effective framework for PSD models (Ang & De Leon, 2005), which are integral to deriving analytical fragility curves, presents significant challenges (Soleimani & Hajializadeh, 2022). Limitations of classical regression techniques in developing PSD models include fixed functional forms, inability to capture nonlinearity, and prior assumptions of lognormal distribution (Soleimani, 2022). The inherent inflexibility of the conventional unidimensional PSD model makes it challenging to accommodate diverse sources of uncertainties such as bridge geometrics and material characteristics, which could potentially undermine the reliability of the estimated demands (Soleimani, 2022). Furthermore, the predefined distribution may not be suitable for all EDPs and bridge classes, as the distribution of demands is typically unknown in prior (Soleimani & Hajializadeh, 2022). The emergence of advanced ML techniques driven by advances in computer technology provides an efficient alternative to conventional statistical procedures owing to their efficacy in managing complex data. ML techniques facilitate the development of more accurate fragility functions. As a result, various ML techniques have been explored to improve the fragility assessment of buildings (Kiani et al., 2019; Mitropoulou & Papadrakakis, 2011) and conventional bridges (C. Huang & Huang, 154 2020; Kameshwar & Padgett, 2018; MO et al., 2023; Pang et al., 2014; Wang et al., 2022; Xie et al., 2020). For instance, (Lagaros & Fragiadakis, 2007) and (Mitropoulou & Papadrakakis, 2011) utilized ANN for the fragility assessment of steel frames and buildings, respectively. In addition, (Mangalathu et al., 2018) used ANN for the development of PSD models and fragility functions of conventional bridges. Pang et al. (Pang et al., 2014) and (MO et al., 2023) utilized ANN and support vector machine, respectively, for the fragility assessment of conventional highway bridges. However, the full potential of ML models in enhancing the fragility assessment of bridges remains to be fully explored despite their widespread use in various fields. Moreover, a recent comprehensive review by (Soleimani & Hajializadeh, 2022) underscores the limited extent of research conducted on the application of ML techniques for seismic fragility analysis of bridges. Furthermore, the scope of existing studies is generally limited as they typically concentrate on a limited set of input features and favor certain ML models without thoroughly investigating the potential of other ML models (Soleimani & Hajializadeh, 2022). Besides, there is currently a lack of research on the bridge-specific multivariate seismic fragility assessment of UHPC bridges. Therefore, this study introduces a novel framework for multivariate and bridge-specific fragility assessment of UHPC bridges based on the hybrid ML model optimized with Optuna. Different parameters that characterize the uncertainties in geometry, material, and ground motion are considered. The efficacy of the hybrid ML model is then assessed against a conventional PSD model as well as a diverse spectrum of ML models of varying complexity levels. Furthermore, this research investigates the influence of these factors on the PSD for UHPC bridge columns using the SHAP approach (Lundberg & Lee, 2017). 155 6.2. Methodology Seismic fragility defines the conditional probability of a structure exceeding a predetermined limit state when subjected to various ground motion intensity levels (Tavares, Padgett, & Paultre, 2012). The fragility curve, in this context, represents the conditional probability of failure. Here, the term failure does not exclusively denote a complete structural collapse; rather, it implies a scenario in which the structural demand surpasses a predefined limit of capacity. Seismic fragility models employ this principle to calculate the conditional probability of exceeding a given damage state (DS), following an earthquake event of a specific intensity measure level (IM = α), as defined in Eq. (6.1): ðð(ð¼) = ð(ð·ð > ð·ð |ð¼ð = ð¼) (6.1) A critical step in the fragility curve development of bridges involves implementing the PSD model to probabilistically compare the demands and capacities of different bridge components. This model estimates the likelihood of meeting or surpassing a specified demand level under a given intensity of ground motion. The conventional PSD model is based on a single parameter; namely, the ground motion intensity measure, as given in Eq. (6.2) (Cornell, Jalayer, Hamburger, & Foutch, 2002), where ðð· denotes the median value of the EDP: ln (ðð·) = ð + ðð(ð¼ð) (6.2) In this study, the drift ratio, which is the most widely used engineering damage parameter, is used as a key indicator of seismic demand of bridge column considering four distinct damage states; namely, slight (minimal damage), moderate (damage that can be repaired), extensive (extensive damage), and complete (probable replacement). Figure 6.1 summarizes the methodology adopted in this study to develop a novel hybrid 156 ML-based framework for multivariate and bridge-specific fragility assessment of UHPC bridges. The proposed methodology involves the following steps: (a) Generation of database: This step involves the identification of important input parameters that influence the seismic vulnerability of UHPC bridges, the identification of the distribution of each parameter, and the generation of a database of M bridge samples using the LHS technique. Each bridge sample is then paired with specific ground motion based on the site where the bridge is located to form ð database of input parameters (ððð¾), as given in (6.3), where K is the number of input parameters. In this study, the bridges are assumed to be located in Vancouver, British Columbia, Canada. ððð¾ =[ ð11 ð21 â¦ ðð1 ð12 ð22 â¦ ðð2â® â® â®ððð¾ ððð¾ â¦ ððð¾] (6.3) (1) In the next step, a 3D finite element model of highway bridges is developed for this purpose. A NLTHA is then conducted for each bridge sample to determine the seismic demand, which is used as a response variable in the next step. Furthermore, to determine the probability of exceedance at each damage state, it is necessary to integrate the fragility functions with respect to the ground motion IMs. In this study, PGA is adopted as the IM for demand modelling. This choice is supported by previous research, which has demonstrated the effectiveness of PGA in developing fragility curves for bridges (Mackie & StojadinoviÄ, 2005; Padgett & Desroches, 2008; Padgett et al., 2008). 157 (b) PSD model development using hybrid ML: In this step, the input parameters are paired with each seismic demand to form a database for training different ML models including hybrid ML model and other widely used ML models. The ML models are trained and tested on 80% and 20% of the complete dataset, respectively. (c) Evaluation of the model: The proposed model is then evaluated and compared against a wide range of ML models varying in their degree of complexity as well as the conventional single parameter based PSD model. (d) Fragility curve generation: In this step, a new set of bridge samples (106 in this study) are generated using LHS. The seismic demands of each sample are then determined using the optimized hybrid ML model. Once seismic demand estimates are obtained, fragility analysis necessitates capacity estimates that serve as thresholds at each limit state. The distribution and parameters of this distribution for the capacity of UHPC bridge columns are obtained from the results of Chapter 5, as listed in Table 6.1. The seismic demand is then compared with the structural capacity of the bridges to determine whether the corresponding damage states have been reached and a value of 0 and 1 are assigned to represent the reached damage state or no, respectively. The fragility curve is then constructed using logistic regression based on the probability of failure at different intensity measures. 158 Figure 6.1: Hybrid ML-based multivariate seismic fragility assessment of UHPC bridges. Table 6.1 Drift ratio capacity uncertainty parameters for UHPC bridge piers Damage state Parameters ðð¶ ð½ð¶ DS1 1.93 0.27 DS2 3.14 0.26 DS3 5.04 0.33 DS4 6.19 0.36 The application of the hybrid ML-based fragility model in this study enables the generation of multivariate and bridge-specific fragility curves at different damage states. The fragility curves derived from the methodology described above offer critical information for efficient seismic vulnerability assessment of UHPC bridges and enable informed decision-making in bridge design and retrofitting. Details of the hybrid ML model are discussed in Chapter 5. 159 6.3. Case Study 6.3.1. Bridge description and ground motion characteristics Concrete box girder bridges, which constitute the majority of the bridge inventory in British Columbia, Canada (Siddiquee & Shahria Alam, 2017) were selected to demonstrate the proposed hybrid ML-based procedures in this study. As discussed earlier, the analysis considers multiple sources of uncertainty of bridge attributes, as presented in Table 6.2, along with uncertainties in ground motion. The distribution and parameters of UHPCâs compressive strength were established based on the recently optimized sustainable UHPC developed presented in Chapter 2. For the remaining parameters, their distribution and the characteristics of their distribution were identified through a thorough literature review (Mangalathu et al., 2018; Nielson, 2005; Tavares et al., 2012) with characteristics representative of those found in British Columbia, Canada. Highway bridge columns with an aspect ratio that ranged between 4 and 10, thus representing flexural dominated bridge piers (Zhu et al., 2007), are considered in this study. The diameter of the bridge column (ð·ð) is determined as a ratio of the pier height and aspect ratio. LHS is employed to generate 1520 statistically significant samples of bridges using the parametersâ distributions in Table 6.2. The LHS offers a method for efficiently sampling from the variable distributions (Mckay, Beckman, & Conover, 1979; A. Olsson, Sandberg, & Dahlblom, 2003) and guarantees that each derived bridge class sample set accurately reflects the range of all parameters. 160 Table 6.2 Uncertainty of bridge characteristics considered in this study (Mangalathu et al., 2018; Nielson, 2005; Tavares et al., 2012) Input features Distribution type Distribution parameters* ð/ð¿ ð/ð Span length, S (m) Normal 41.15 10.61 Column height, ð»ð (m) Lognormal 7.13 1.15 Compressive strength of UHPC, ððð»ðð¶ (MPa) Normal 181.95 27.43 Volume fraction of fibers, ðð (%) Uniform 1.0 3.0 Transverse reinforcement ratio, ðð¡ (%) Uniform 0.4 1.3 Longitudinal reinforcement ratio, ðð (%) Uniform 1.0 3.0 Yield strength of reinforcement bars, ðð¦ (MPa) Normal 475.7 37.9 Width of the deck, ð¤ð (m) Normal 12.8 0.61 Deck mass multiplication factor, ðð Uniform 0.9 1.1 Foundation translational stiffness ratio, ð¾ðð (%) Uniform 0.5 1.5 Foundation rotational stiffness ratio, ð¾ðð¡ (%) Uniform 0.5 1.5 Damping ratio, ðð (%) Normal 4.5 1.25 * ð: Mean for normal distribution; ð: standard deviation for normal distribution; ð¿: lower value for uniform distribution; and ð: upper value for uniform distribution Twenty near-fault ground motion records from the PEER strong motion database (PEER, 2020) were selected for this study. The characteristics of the selected ground motion are presented in Table 5.8 of Chapter 5. These records feature original PGAs ranging from 0.11 to 0.85 g and are located within 10 km of the epicenter, thus are representative of near-field seismic effects (CSA, 2014). The selected ground motions were spectrally matched to the uniform hazard design response spectra of Vancouver, British Columbia, Canada, as discussed in Chapter 5. Additionally, the ground motion records are scaled with a scaling factor ranging from 0.25 to 4.0 at 0.05 161 increment, resulting in a total of 1520 varied ground motions. Each bridge sample was randomly associated with a distinct ground motion. Subsequently, a 3D model of the bridges was developed in OpenSees and NLTHA was conducted on all the combinations of bridge samples and the maximum column drift demand was recorded at the end of shaking for each excitation. 6.3.2. Modelling of seismic response In the process of analytical fragility evaluation, fragility curves are established through the execution of numerical simulations, which generate a set of seismic demand data samples that are subsequently used for training PSD models. In this study, the 3D numerical model of the bridge was developed using OpenSees (Mckenna, 2011). Figure 6.2 presents the schematic representation of the numerical model. The bridge piers were modelled using displacement-based beam-column elements. Each fiber section was divided into concrete fibers (cover and core fibers) and steel fibers. The longitudinal reinforcement is modelled using the uniaxial Steel02 model in OpenSEES. The uniaxial stress-strain response of both unconfined and confined concrete was simulated using the OpenSees Concrete01 material model. Thus, the tensile capacity of the concrete is not considered due to the lack of reliable model to determine the tension response of of UHPC. For the confined concrete, its properties at the peak and ultimate states were determined based on the stress-strain constitutive model developed in Section 4.4 of Chapter 4. Additionally, the geometric nonlinearity is accounted by incorporating the P-delta effect into the bidge columns. For dynamic analyses, Rayleigh damping is utilized to account for the first and second vibration modes. 162 Figure 6.2: Numerical model of the selected bridge. Furthermore, bridge decks are modelled using elastic beam-column elements to reflect the expectation that they remain elastic during seismic events. Consequently, the effective stiffness of prestressed concrete decks is represented by their gross stiffness without accounting for potential concrete cracking. Both diaphragms and expansion joints in the decks, as depicted in transverse deck elements (see Figure 6.2), are simulated using elastic beam-column elements. These elements are designed to be rigid and massless. To facilitate the transfer of moments and forces between adjacent components, the deck elements are connected to the columns using rigid elements. As shown in Figure 6.2, expansion joints between the deck and abutment comprised elements such as elastomeric bearings (transverse and longitudinal), a transverse shear key, and a component to simulate pounding between the deck and abutment. Elastomeric bearings in the expansion joints are modelled as elastic-perfectly plastic elements. Pounding between the deck and abutment is represented using zero-length elements following Muthukumar and DesRoches (Muthukumar & DesRoches, 2006) 163 compression-only bilinear material model with a gap. The transverse shear key is simulated with a symmetric trilinear behaviour and a gap. Bridge foundations are modelled using lumped linear translational and rotational springs with zero-length elements assigned at the center of the footings, as illustrated in Figure 6.2. 6.3.3. Results of PSD model and seismic fragility Different single and ensemble models such as KNN, SVR, GBM, and XGB are investigated for use as the base learner for hybrid ML model. Based on the results of preliminary investigations, XGB is used as the primary model, while GBM, KNN, and SVR are used as the secondary models in the proposed hybrid ML model. The hyperparameters of both the primary and secondary models as well as the weight of each secondary model are optimized with the use of Optuna in conjunction with 10-fold cross validation. Figure 6.3 shows the predicted versus actual PSD for column drift ratio. Moreover, Table 6.3 presents the predictive performance of the model in terms of the different statistical performance metrics. As can be observed in Figure 6.3, the proposed hybrid ML model achieved a satisfactory level of prediction accuracy across both training and test datasets. Notably, the model demonstrated a strong ability to generalize to unseen data as evidenced by the consistency in performance metrics between the training and testing phases. For instance, it achieved a remarkably low RMSE of 0.39% in the training dataset and 0.64% in the test dataset, which signifies a minimal deviation from the actual values, as listed in Table 6.3. Furthermore, the achieved MAE values of 0.25% for training and 0.47% for test datasets reinforce the prediction accuracy of the model, which highlights its consistent performance across different data sets (Table 6.3). Moreover, it showed a high coefficient of determination for both training and test datasets, as can be seen in Figure 6.3 and Table 6.3. 164 Figure 6.3: PSD of UHPC column drift demand. Table 6.3 Performance measures of PSD prediction for UHPC bridge columnsâ drift demand Model Training dataset Test dataset RMSE (%) MAE (%) R2 (%) RMSE (%) MAE (%) R2 (%) KNN 1.54 1.15 41.46 1.44 1.18 40.05 SVR 0.65 0.29 89.64 0.93 0.68 74.86 GBM 0.09 0.07 99.81 0.79 0.56 82.12 XGB 0.55 0.35 92.44 0.68 0.49 86.65 Hybrid 0.39 0.25 96.17 0.64 0.47 88.10 Based on the conventional linear regression of the PSD model, the resulting equation is shown in Figure 6.4. This figure also shows the resulting equation in which ð¦ = ln (ð¿) and ð¥ = ln(ð¼ð). Moreover, Figure 6.5 compares the traditional PSD model with the proposed hybrid ML model in terms of the scatter plots of the predicted versus actual PSD for column drift demand. As can be seen in this figure, the conventional single variable (i.e., IM) based PSD model is inadequate, while the proposed hybrid ML-based multivariable PSD model showed a higher degree of accuracy. 165 Figure 6.4: PSD for column drift based on conventional method. Figure 6.5: Predicted versus actual PSD for column drift based on the conventional linear regression and proposed hybrid ML model. The accuracy of the proposed hybrid ML model is further investigated by comparing it to other ML models; namely, KNN, SVR, GBM, and XGB. It is observed that the proposed hybrid ML model has the highest PSD predictive performance with the lowest error metrics and highest R2 in comparison to other ML models, as listed in Table 6.3. Among the base models, the XGB model showed the highest prediction accuracy, while the KNN model showed the least prediction accuracy, as can be observed in Table 6.3. This can further be observed in Figure 6.6 which compares the 166 predicted versus actual values of the result of the PSD for column drift demand. As can be seen in this figure, the GBM showed a clear overfitting, while the other models showed poor performance across both the training and test datasets. However, the proposed hybrid ML model has superior prediction capability and generalization ability across unseen data as it leverages the strengths of individual models while mitigating their weaknesses in order to produce enhanced overall performance. Figure 6.6: PSD of UHPC column drift ratio based on different ML models. The SHAP method is used in this study to enable the investigation of the influence of different input features on the PSD for column drift ratio. Figure 6.7 shows a SHAP summary plot in which the input factors are arranged on the y-axis by their level of impact on the PSD prediction with the most influential factor at the top. Moreover, the magnitude of the influence of each input is presented in the x-axis through the SHAP values. As can be seen in Figure 6.7, the ground motion intensity has the highest significant influence on the PSD for column drift. This result is in agreement with the previous study for conventional concrete bridges (Mangalathu et al., 2018). The bridge pier column diameter is found to be the second most influential parameter followed by the span length and damping ratio. In addition to the order of the significant effect of the input features, the SHAP summary plot shows the direction of their effect on the predicted PSD. For example, an increase in PGA and bridge span 167 increased the drift demand, whereas a larger column diameter reduced the drift demand. In addition, the input features such as the damping ratio, concrete compressive strength, and longitudinal reinforcement ratio are shown to have a mitigating effect on drift demand, with their increase leading to reduced predicted drift demand. Figure 6.7: SHAP summary plot. Given its high predictive capability, the developed hybrid ML model is utilized to develop the fragility curves based on the methodology discussed earlier. The resulting fragility curves for the bridge column are shown in Figure 6.8 for all four damage states. The developed multivariate fragility curves can then be used to determine the vulnerability of UHPC bridges and potential losses resulting from 168 earthquakes. However, further studies are required to investigate the scalability and applicability of the proposed method for different UHPC bridge types other than continuous box-girder bridges considered in this study. Figure 6.8: Fragility curves generated based on hybrid ML model. 6.4. Summary An efficient fragility assessment approach is essential to enable accurate seismic vulnerability assessment of UHPC bridges. However, the conventional seismic fragility assessment approach may not accurately reflect the varied nature of bridges within a specific category as it disregards critical bridge characteristics like bridge geometries. Moreover, there is a lack of seismic fragility assessment for UHPC bridges. Therefore, this study introduces a multivariate and bridge-specific seismic fragility assessment method for UHPC bridges at various damage states. This method incorporates a hybrid ML model that accounts for uncertainties in bridge characteristics and ground motion. The proposed methodology involves a systematic identification and selection of the critical design parameters and their uncertainties, generation of the UHPC bridge database using the LHS technique considering different sources of uncertainties, 169 determination of demand parameters at different damage states, generation of PSD models using hybrid ML model, generation of capacity and demand database using the proposed PSD models and uncertainty in the capacity, and generation of fragility curves. The effectiveness of the proposed approach is benchmarked against the conventional single parameter based PSD model. Furthermore, the predictive performance of the proposed hybrid ML model is evaluated in comparison with widely used ML models in existing literature. Additionally, the SHAP method is employed to explore the impact of various factors on the PSD for the drift demand of the bridge column. The following conclusions can be drawn from the results of the current study: â¢ Among the models investigated for use as base learner for the hybrid ML model, the XGB emerged as the most effective model in predicting the PSD for column drift demand. â¢ The hybrid ML model achieved satisfactory accuracy in estimating the PSD for column drift ratios with a high coefficient of determination (ð 2 = 88.1%) and low error metrics on the test set. â¢ Compared to traditional single parameter based PSD models, the hybrid ML model exhibited significantly higher prediction accuracy. The conventional PSD model showed inadequate prediction accuracy. These results highlight the limitations of single parameter conventional PSD models in capturing drift demand complexities. â¢ The results of SHAP analysis on the box-girder bridge used as a case study revealed ground motion intensity measure (PGA in this study) as the most 170 influential factor on PSD for UHPC column drift demand followed by the bridge span and column diameter. â¢ The proposed hybrid ML based approach enables efficient and bridge specific fragility assessment of UHPC bridges. The developed multi-variate fragility assessment approach facilitates seismic vulnerability assessment and potential earthquake-induced losses for UHPC bridges, thus supporting informed decision-making in bridge design and retrofitting. Nevertheless, additional research is necessary to explore the applicability of the proposed method across a broader range of UHPC bridge types beyond the box-girder bridges examined in this study. 171 CHAPTER 7: SUMMARY AND RECOMMENDATION FOR FUTURE WORK 7.1. Summary and Conclusions In this dissertation, an AI-powered comprehensive framework is proposed to aid sustainable design, structural modelling, and performance-based design of UHPC structures. The main objectives of this study are: (a) develop a novel framework for strength prediction and MOO of economically feasible and environmentally sustainable UHPC that can be tailed to specific project requirements; (b) develop stress-strain constitutive models of UHPC confined with either NSS or HSS at two critical states (i.e., peak and ultimate states) to enable modelling of UHPC structures; (c) develop complete stress-strain response prediction model for confined UHPC; (d) develop PBSD of UHPC columns at different limit states; (e) propose multivariate fragility assessment method for UHPC bridges. Accordingly, five different studies were performed, each addressing the above main objectives. In the first part of the dissertation, a novel framework is introduced for compressive strength prediction and MOO of sustainable UHPC considering a total of 19 objective functions. This approach integrates an advanced ML model, with metaheuristic MOO algorithm and MCDM technique. The MOO problem is set with 19 objective functions. The first and second objective functions address the compressive strength and cost of the UHPC mixture, while the remaining 17 objective functions deal with the environmental impact of the mixture. A total of 17 mid-point environmental impact categories are used to comprehensively evaluate the environmental sustainability of the UHPC mix. The resulting impacts from the mid-point indicators were calculated using the Eco-invent v3.7 Life Cycle Inventory database. The results showed that the super learner model accurately predicted the 172 uniaxial compressive strength of UHPC. The MOO resulted in Pareto fronts, demonstrating the trade-off among the uniaxial compressive strength, cost, and environmental sustainability of the mix and a broad range of solutions that can be obtained for the 19 objectives. The study provides a useful tool for designers and decision-makers to select the optimal UHPC mixture that meets specific project requirements. Finally, for the practical application of the ML predictive model and MOO algorithm for UHPC, a graphical user interface-based software tool, FAI-OSUSCONCRET, was developed. This software tool offers fast, accurate, and intelligent predictions and multi-objective optimizations tailored to specific project requirements, thus resulting in a UHPC mixture that perfectly meets project needs. The results of this part of the study serve as a basis for the subsequent studies. Addressing the lack of a reliable model for predicting the stress-strain response of confined UHPC, the second part of the study introduced a framework that implements ML models augmented by a state-of-the-art CTGAN and Optuna to accurately predict the peak and ultimate axial stress-strain responses of UHPC confined with either NSS or HSS reinforcement. The Optuna-optimized CTGAN is employed to address the issue of limited data by generating synthetic datasets of hypothetical confined UHPC specimens. A comprehensive database of confined UHPC stress-strain responses was compiled from existing literature and used to condition the CTGAN. The augmented database is then leveraged to develop a hybrid ML model that integrates extreme gradient boosting, gradient boosting machine, SVR, and K-nearest neighbors for predicting peak and ultimate stress-strain responses of confined UHPC. The predictive accuracy of the proposed hybrid ML model is evaluated and compared with a diverse set of ML models of varying complexity, and the results demonstrate its superior performance in predicting the peak and ultimate stress-strain response of 173 confined UHPC. Furthermore, a graphical user interface of the proposed model is developed to facilitate its practical implementation and provide a rapid, intelligent, and accurate prediction of the stress-strain response of confined UHPC at both peak and ultimate states. The third part of the study extends beyond the peak and ultimate states and introduces a novel hybrid ML-based complete stress-strain response prediction model for UHPC confined with NSS or HSS spirals. Given the limited experimental data available, a numerical model was developed in Abaqus and validated against experimental results of confined UHPC. Subsequently, it was employed to generate a comprehensive dataset of confined UHPC stress-strain responses, which were then used to train the hybrid ML model. Hyperparameters of the hybrid ML model are subsequently optimized using Optuna along with a cross-validation technique. The optimized hybrid ML model exhibited exceptional accuracy in predicting the complete stress-strain response of confined UHPC, achieving high R2 values ranging from 0.905 to 0.999 and maintaining low error metrics across varying stress levels and datasets. Moreover, the capability of the model in simultaneously predicting both stress and strain responses was further validated through CCA, and the results demonstrated robust predictive performance of the model with CCA scores between 93.3% and 99.4%. To facilitate the practical implementation of the model, an end-to-end interactive software tool is developed. In addition, this study proposes predictive equations for the peak and ultimate stress-strain responses of confined UHPC. The evaluation of the developed models demonstrates their efficacy in accurately predicting the stress-strain responses of confined UHPC, as evidenced by the R2 values, which range from 83â91%. The fourth part aims to develop an explainable ML powered predictive model 174 for the drift ratio limit states of UHPC bridge columns across four damage states. To achieve this, a fiber-based numerical model was developed and then validated against large-scale experimental results of UHPC columns tested under reverse cyclic loading. The LHS is employed to generate a comprehensive database of UHPC bridge columns (10,000 UHPC columns), considering uncertainties across a spectrum of design factors, including column geometry, material characteristics, reinforcement ratio, and axial load ratio. The drift ratios at the initiation of four damage states are determined using the numerical model. Subsequently, the power of ML algorithm is leveraged to introduce drift ratio limit states for UHPC columns, capturing the onset of four different damage states. The efficacy of the developed drift capacity model is assessed using a range of statistical metrics, including the composite fitness score, DX% metrics, R2, RMSE, normalized RMSE, and MAE, which demonstrated a high predictive accuracy of the drift limit states on both the training and unseen datasets. Furthermore, a model agnostic SHAP approach is used to explain the output of the model and examine the influence of various design factors on drift ratio across the damage states. Notably, the axial load ratio and aspect ratio are found to be major factors in determining the drift ratios across the damage states. The developed model is deployed into a software tool in order to enable its practical application. Additionally, the implementation of the developed model in the context of PBSD is illustrated through a design example. Furthermore, the study proposed capacity uncertainty parameters in order to aid the fragility assessment of UHPC columns. Finally, the proposed model along with capacity uncertainty parameters are used for the fragility assessment of UHPC columns reinforced with varying steel bar grades. The results demonstrate the less vulnerability of UHPC columns reinforced with higher bar grade, yet the effect of bar grade lessens at severe damage levels. 175 In the final part of the study, multivariate and bridge-specific fragility assessment approach for the UHPC highway is proposed using hybrid ML model. For this purpose, multi-parameter PBSD models are generated considering four damage states. The capability of the hybrid ML model was investigated by comparing its performance against widely used ML models in the literature. Moreover, the study employs the SHAP approach to investigate the effects of different factors on the PBSD of UHPC columns. In contrast to the conventional approach, which relies on a single parameter, the proposed hybrid ML-based approach accounts for the uncertainties associated with various bridge characteristics, thereby enabling bridge-specific fragility assessment of UHPC bridges. The results of a case study focused on a box-girder bridge, one of the most commonly used bridge types in Canada, reveal that ground motion intensity measure is the most influential factor on the PBSD for bridge columns, followed by column diameter and bridge span length. Moreover, the accuracy of the proposed hybrid ML-based approach is compared with that of the conventional PBSD model and resulted in a higher predictive performance. 7.2. Recommendation for Future Work The following future research is suggested to enable widespread applications of UHPC in seismic prone regions: â¢ Future research should consider additional performance objectives, such as slump, flexural strength, and durability of UHPC, given the availability of a large experimental database. Moreover, the mechanical properties and durability of the optimum UHPC mixtures identified in this study should be experimentally investigated. â¢ Apart from the factors considered in this research, different parameters such as 176 moisture, particle size distribution, packing density, and temperature can influence the performance of UHPC. Therefore, future studies are recommended to consider the influence of these factors for optimization of sustainable UHPC, upon the availability of an extensive experimental database. â¢ The constitute stress-strain model developed in this study should be continually validated as more experimental data on the stress-strain response of confined UHPC becomes available to increase its range of applications. â¢ Given the availability of additional experimental data specifically on the tensile behavior of UHPC, it is recommended to develop constitutive models that incorporate the tensile response of UHPC in the modelling of UHPC structures. â¢ The integration of the developed constitutive model with advanced numerical software, such as OpenSEES, is also recommended to enable efficient and accurate modelling of UHPC structures. â¢ The complete stress-strain response prediction models developed in this research are primarily for circular section columns. 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Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Alam, M. Shahria
Advanced AI-powered comprehensive framework for modelling, analysis, assessment, and performance-based design of ultra-high-performance concrete elements
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