Machine Learning Approaches for the Reliable Design of Steel Joints
Abstract
Abstract This thesis presents a framework for developing machine-learned prediction models design of steel connections, with a focus on maintaining the reliability level required by Eurocode design procedures. The work responds to the increasing use of finite element analysis (FEA) in structural joint design, where traditional analytical or empirical design equations are often insufficient to describe nonlinear and interaction-dependent behavior. The study outlines the motivation to replace simplified analytical expressions with data-driven models trained on validated numerical simulations, while preserving reliability in line with the requirements of structural engineering design practice. A column web in transverse compression is selected as a representative example to illustrate the proposed workflow. The procedure begins with automated generation of a large dataset through parametric numerical design calculations using IDEA StatiCa Connection, where more than 9~000 FE models were computed. The dataset covers a broad range of geometric and material parameters, and each simulation was evaluated according to a newly developed equivalent plastic strain limit criterion. The definition of this strain limit is one of the central contributions of the thesis, as it directly influences the calculated resistance and the consistency of the resulting machine learning models. A detailed discussion is provided on the identification of a reliable equivalent plastic strain limit for steel structural components. The thesis compares recommendations given in different parts of Eurocode~3 and highlights the inconsistencies that currently exist. A new plastic strain limit proposal is derived based on numerous physical tests, numerical simulations and numerical design calculations for the net section failure of plates with notches and bolt holes, which is deemed to be the governing case. The proposed limit is clear and simple; it enables reproducible resistance determination in numerical design calculations and serves as a solid basis for subsequent data-driven modeling. The thesis further introduces a complete process for the verification, validation, and reliability assessment of machine learning models in structural engineering applications. The process is based on the principles of EN~1990 and the draft standard FprEN~1993-1-14, ensuring that predictive models meet equivalent reliability levels to traditional design methods. Neural networks were selected as the primary modeling technique due to their ability to approximate complex nonlinear relationships. Using Python-based automation and systematic hyperparameter tuning, models were trained to predict resistances with a coefficient of variation below 2~\% compared with FEA results. The study also investigates the influence of dataset coverage, feature scaling, and model regularization on prediction accuracy and reliability. The results demonstrate that when trained on sufficiently rich and consistent data, machine learning models can replicate the results of detailed numerical simulations with no bias and negligible coefficient of variation. However, the reliability of such models remains limited to the range of input parameters represented in the training dataset. The thesis emphasizes that careful dataset generation, transparent preprocessing, and continuous verification are prerequisites for trustworthy machine learning applications in structural engineering. A discussion is provided on extending the range of validity through geometric similarity scaling. The thesis concludes with recommendations for how machine learning can be integrated into structural design practice while keeping the same level of safety and transparency required by engineering standards. Several directions for future research are suggested: (i) improving the quality of finite element models used for training, including possible expansion through geometric similarity scaling or the use of simplified models with coarser meshes trained to produce high-quality results; (ii) enhancing reliability assessment by considering objectively safer FEA modifications, such as reducing the plastic strain limit or neglecting strain hardening; and (iii) creating well-documented datasets of experimental studies and numerical simulations, and encouraging researchers to share their results -- a practice increasingly important in today’s data-driven research environment. The developed framework helps to connect advanced numerical simulations with practical design procedures and opens the path toward the next generation of efficient and reliable design tools.
Full text
BRNO UNIVERSITY OF TECHNOLOGY VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ FACULTY OF CIVIL ENGINEERING FAKULTA STAVEBNÍ INSTITUTE OF METAL AND TIMBER STRUCTURES ÚSTAV KOVOVÝCH A DŘEVĚNÝCH KONSTRUKCÍ MACHINE LEARNING APPROACHES FOR THE RELIABLE DESIGN OF STEEL JOINTS METODY STROJOVÉHO UČENÍ PRO SPOLEHLIVÝ NÁVRH STYČNÍKŮ OCELOVÝCH KONSTRUKCÍ HABILITATION THESIS HABILITAČNÍ PRÁCE AUTHOR AUTOR PRÁCE Ing. Martin Vild, Ph.D. BRNO 2025
Abstract: This thesis presents a framework for developing machine-learned prediction models design of steel connections, with a focus on maintaining the reliability level required by Eurocode design procedures. The work responds to the increasing use of finite element analysis (FEA) in structural joint design, where traditional analytical or empirical design equations are often insufficient to describe nonlinear and interaction-dependent behavior. The study outlines the motivation to replace simplified analytical expressions with data-driven models trained on validated numerical simulations, while preserving reliability in line with the requirements of structural engineering design practice. A column web in transverse compression is selected as a representative example to illustrate the proposed workflow. The procedure begins with automated generation of a large dataset through parametric numerical design calculations using IDEA StatiCa Connection, where more than 9 000 FE models were computed. The dataset covers a broad range of geometric and material parameters, and each simulation was evaluated according to a newly developed equivalent plastic strain limit criterion. The definition of this strain limit is one of the central contributions of the thesis, as it directly influences the calculated resistance and the consistency of the resulting machine learning models. A detailed discussion is provided on the identification of a reliable equivalent plastic strain limit for steel structural components. The thesis compares recommendations given in different parts of Eurocode 3 and highlights the inconsistencies that currently exist. A new plastic strain limit proposal is derived based on numerous physical tests, numerical simulations and numerical design calculations for the net section failure of plates with notches and bolt holes, which is deemed to be the governing case. The proposed limit is clear and simple; it enables reproducible resistance determination in numerical design calculations and serves as a solid basis for subsequent datadriven modeling. The thesis further introduces a complete process for the verification, validation, and reliability assessment of machine learning models in structural engineering applications. The process is based on the principles of EN 1990 and the draft standard FprEN 1993-1-14, ensuring that predictive models meet equivalent reliability levels to traditional design methods. Neural networks were selected as the primary modeling technique due to their ability to approximate complex nonlinear relationships. Using Python-based automation and systematic hyperparameter tuning, models were trained to predict resistances with a coefficient of variation below 2 % compared with FEA results. The study also investigates the influence of dataset coverage, feature scaling, and model regularization on prediction accuracy and reliability. The results demonstrate that when trained on sufficiently rich and consistent data, machine learning models can replicate the results of detailed numerical simulations with no bias and negligible coefficient of variation. However, the reliability of such models remains limited to the range of input parameters represented in the training dataset. The thesis emphasizes that careful dataset generation, transparent preprocessing, and continuous verification are prerequisites for trustworthy machine learning applications in structural engineering. A discussion is provided on extending the range of validity through geometric similarity scaling. The thesis concludes with recommendations for how machine learning can be integrated into structural design practice while keeping the same level of safety and transparency required by engineering standards. Several directions for future research are suggested: (i) improving the quality of finite element models used for training, including possible expansion through geometric similarity scaling or the use of simplified models with coarser meshes trained to produce high-quality results; (ii) enhancing reliability assessment by considering objectively safer FEA modifications, such as reducing the plastic strain limit or neglecting strain hardening; and (iii) creating well-documented datasets of experimental studies and numerical simulations, and encouraging researchers to share their results – a practice increasingly important in today’s data-driven research environment. The developed framework helps to connect advanced numerical simulations with practical design procedures and opens the path toward the next generation of efficient and reliable design tools. Keywords: finite element analysis; steel connections; numerical design calculation; plastic strain limit; Eurocode 3; reliability analysis; neural networks; machine learning; structural engineering; IDEA StatiCa
Bibliographic Citation of the Thesis: Vild, Martin: Machine Learning Approaches for the Reliable Design of Steel Joints. Habilitation thesis. Brno University of Technology, Faculty of Civil Engineering, Institute of Metal and Timber Structures, Brno, Czech Republic, 2025.
© Martin Vild 2025 Brno University of Technology Faculty of Civil Engineering Veveˇr´ı 331/95, 602 00 Brno, Czech Republic
4 Acknowledgments I express my deepest gratitude to my family – my wife Gabriela, my two sons Jaremi´aˇs and Kili´an, and daughter Judit – for allowing me to work long hours. I can see the deep truth in the viral post: ”Twenty years from now, the only people who will remember that you worked late are your kids.” Thank you to my children for teaching me the patience I never thought myself capable of, and to my wife for her unwavering patience with me despite my many flaws. I also thank my parents for raising me and supporting me in every step I have taken. I would never have climbed the ladder in my professional life without my mentors, Lubom´ır ˇ Sabatka and Miroslav Bajer, who constantly challenged me with deadlines and thought-provoking questions. Luboˇs loves the saying ”People only become great with great tasks.” I am grateful to my superiors, Juraj ˇ Sabatka and Milan ˇ Smak, for allowing me flexibility in my everyday responsibilities so that I could focus on my research and thesis. My thanks also go to Petr ˇ Cervinka, who replaced me as Product Owner at IDEA StatiCa, taking over the burdens of day-to-day management and freeing my hands to concentrate on research. I thank Frantiˇsek Wald for including me in his remarkable research group and for enabling me to see the broader landscape of European steel research at the highest level. I also thank Andreas M¨uller and Andreas Taras for guiding me to this topic of machine learning; showing me how useful this tool can be. My gratitude extends to Michael Koneˇcn´y and Jarom´ır Kabel´aˇc for guiding me through the intricacies and pitfalls of machine learning. My thanks go to our staff of the Institute of Metal and Timber Structures, particularly Alena Sklen´aˇrov´a for her daily assistance in navigating the administrative maze, and Jiˇr´ı Vesel´y, Jiˇr´ı Fischer, and Miloslav Ledina from our laboratory for conducting numerous experiments. I also acknowledge Ivan Bal´azs, the driving force behind the column web panel in compression experiments and related papers, and Ondˇrej Peˇsek, who developed the ANSYS numerical models. I am grateful to the members of the Laboratory of Numerical Structural Design – Mykola Lastovetskyi for his unparalleled enthusiasm, gym sessions, ice cream deliveries, and Python discussions, and Amina Hajdarevi´c for her thorough literature review on web panel in transverse compression and speedy review of this thesis, during which she revealed how many missing English articles I had overlooked. All of these people take selfishly without any trace of envy my success as their own. The core of this thesis lies in the programming of neural networks. I have only very basic programming skills in Python, I am no programmer at all. The developing of codes relied almost entirely on large language models (LLMs), such as ChatGPT or Microsoft Copilot (both using OpenAI’s GPT-4, GPT-4 Turbo, GPT-4o, GPT-4o mini or GPT-5). I have utilized LLMs also in writing itself. However, all the codes and texts were reviewed and modified, and I take full responsibility for them. The experimental works at Brno University of Technology (BUT) would not be possible without the financial support of projects Inter Excellence, LUAUS23114, supported by the Ministry of Education, Youth and Sports of the Czech Republic, and internal BUT grant FAST-S-25-8879, Advanced methods for the design of load-bearing structural elements made of composite materials and joints of steel structures.
Contents Acknowledgments 4 1 Introduction 7 1.1 Problemstatement.................................... 7 1.2 FiniteElementAnalysis ................................. 9 1.2.1 FailureCriteria.................................. 10 1.3 Reliability......................................... 11 1.4 MachineLearning..................................... 12 2 Methods 13 2.1 Design codes and state-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Web panel in transverse compression . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2 Net section failure in tension . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Experiments on web in transverse compression . . . . . . . . . . . . . . . . . . . . 23 2.3 Finiteelementanalysis.................................. 25 2.3.1 Meshsensitivity ................................. 25 2.3.2 Failurecriteria .................................. 27 2.4 Datageneration...................................... 28 2.5 Neuralnetwork...................................... 29 2.5.1 Neural network architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.2 Input and output scaler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.3 Optimizer..................................... 32 2.5.4 Learningrate................................... 32 2.5.5 Activationfunction................................ 33 2.5.6 Lossfunction................................... 35 2.5.7 Final model for web in compression . . . . . . . . . . . . . . . . . . . . . . 36 2.6 Reliability......................................... 37 2.6.1 Semi-probabilistic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6.2 Direct reliability approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 Determination of reliable strain limit 43 3.1 Explanationofconcept.................................. 43 3.2 Experimental investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.1 Brno University of Technology . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.2 Czech Technical University in Prague . . . . . . . . . . . . . . . . . . . . . 56 3.3 Numericalmodels..................................... 57 3.3.1 Damagemodel .................................. 57 3.3.2 Solidmodel.................................... 57 3.3.3 Shellmodel.................................... 58 3.4 Numerical-analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4.1 Materialfactor .................................. 59 3.4.2 Geometry reduction factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.3 Uncertaintyfactor ................................ 60 3.5 MonteCarlosimulations................................. 60 3.5.1 Designresistance................................. 60 3.5.2 Partial safety factor γM2 ............................. 62 3.6 Plastic strain limit for numerical design calculation . . . . . . . . . . . . . . . . . . 62 5
6CONTENTS 3.7 Outlook .......................................... 64 4 Application 65 4.1 Numericalmodel ..................................... 65 4.2 Datasetcreation ..................................... 68 4.3 Datasetscaling ...................................... 70 4.4 Datasetsize........................................ 73 4.4.1 Activelearning.................................. 73 4.5 Predictions of different datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.1 Full dataset without filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.2 IDEA StatiCa dataset created by nested for loops . . . . . . . . . . . . . . 77 4.5.3 Dataset created in ANSYS software . . . . . . . . . . . . . . . . . . . . . . 78 4.6 Reliability......................................... 84 4.6.1 Eurocodereliability ............................... 84 4.6.2 Reliability of new method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.6.3 Reliability of neural network . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.6.4 Comparison of methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.7 Explainability of DNN predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5 Conclusion and future outlook 95
Chapter 1 Introduction The aim of this thesis is to develop a framework for creating machine-learned predictions for components of steel connections. It does not attempt to provide these precise predictions or formulas, as their derivation will likely require several more years of research. The thesis shows a workflow and highlights the critical areas that should be addressed on an example of a column web in a transverse compression component. The process includes the generation of data for machine learning, whose quality must be better than the current code formulas, the establishment of a neural network, and the reliability assessment of predictions. It is assumed that the data will be generated by finite element analyses. The thesis focuses on two main areas, which are aligned with the projects and research papers by the author: Failure criterion of plastic strain limit for numerical design calculation of steel connections by finite element analysis. The process of data generation by numerical design calculation and machine learning, with recommendations for the best performance. 1.1 Problem statement There are two methods in the use of current design codes, such as EN 1993-1-8 [1]: Analytical methods Curve-fitting method Analytical methods are mathematical equations derived from physical phenomena based on predefined assumptions. In contrast, curve-fitting methods identify the variables that influence the results but do not explicitly define the underlying assumptions or dependencies. These methods are typically calibrated to a specific set of experiments or validated numerical simulations and are commonly used, e.g., in the design of hollow section joints. The curve-fitting methods are not popular in the scientific community because the mechanisms and theory behind them are not apparent. The curve-fitting methods are only as good as the input data, i.e., the experiments, in which regard the curve-fitting method is similar to machine learning. Although there was an attempt to generalize the design of hollow section joints using the component method [2], this effort was never completed. There are two major drawbacks of these traditional methods: 1. The phenomenon is too complex and cannot be correctly described by simple formulas. The assumptions and dependencies of variables are no longer apparent from the formulas. 2. The assumptions, e.g., first-order analysis, may be insufficient to achieve a good fit with the real-world behavior. Therefore, in many cases of the design of steel joints, the formulas are too complex, and regular engineers no longer recognize the mechanics and theory behind the design formulas. Often, there is no background to the code development, such as EN 1993-1-8, and scientific papers are not 7
14 CHAPTER 2. METHODS component is described in Eurocode EN 1993-1-8:2005, Cl. 6.2.6.2 [1], EN 1993-1-8:2024, Cl. A.6 [28], or AISC 360-22, Cl. J.10 [18]. The state-of-the-art is described in detail in papers [24], [30] and it will be shortly also introduced here. Figure 2.1: Examples of web panel in transverse compression active component: welded (a) and bolted (b) beam-to-column connection, roof connection (c), and load near the unstiffened end(d) [30] The elastic behavior could be mathematically derived, but the phenomenon of yielding is actually complicated and parameters are set empirically from a limited number of experiments [31]. In EN 1993-1-8 [28], the CWC resistance is obtained as the minimum of web yielding and web buckling expressions based on an effective width into which the load is dispersed; the formulation does not explicitly account for proximity to a free member end. EN 1993-1-5 [32] adopts an effectivelength concept that embeds plate-buckling effects and distinguishes between loading at one flange, at both flanges, and near an unstiffened end. While broadly safe, parts of these models rely on empirical parameters calibrated over limited slenderness ranges, and simplifications in the assumed boundary conditions (primarily simply supported web-to-flange) may lead to conservatism. [33, 34] The American Specification AISC 360-22 [18] treats web local yielding with expressions equivalent to Eurocode away from member ends, and provides separate formulations for web buckling (for load acting from both sides) and crippling (for load acting from one side only). The resistances are halved when the load is applied near the unstiffened end. For web local yielding, the boundary is at a distance from the member end that is less than or equal to the full nominal depth of the member, h. For web local crippling and compression buckling, the boundary is at a distance from the member end that is less than h/2. Recent experimental campaigns and validated GMNIA studies have sharpened the understanding of the actual mechanisms [24]. For loading away from member ends, plastic strains concentrate at the flange–web transition and, for typical rolled IPE sections, local buckling seldom governs the resistance when realistic fixity at the junction is present. Interaction with axial force can, however, significantly diminish column web panel resistance—an effect not fully captured by the current EN 1993-1-8 [28] reduction factor kwc. Balazs [30] addressed a case still weakly codified: transverse compression applied close to an unstiffened end. Combining full-scale tests on IPE 200 specimens with GMNIA across IPE 100–600, the end-distance effect and its dependence on web slenderness was quantified. When the load is very close to the unstiffened end (e.g., e≈0.1h), the resistance may reduce to about 60% of the far-from-end case, and buckling becomes more likely due to the loss of restraint at the free edge. An analytical procedure consistent with current design philosophies is proposed to capture these effects. This phenomenon is therefore ideal as a test case for finite element analysis and subsequent machine learning predictions. Only the procedure of FprEN 1993-1-8:2024, Cl. A.5 [28] is described below, because the other code implementations are very similar and older. The design resistance is the minimum of the yielding and buckling resistances: Fc,wc,Rd =ω kwc beff,w twc fy,wc γM0 (2.1) Fc,wc,Rd ≤ω kwc ρ beff,w twc fy,wc γM1 (2.2)
2.1. DESIGN CODES AND STATE-OF-THE-ART 15 The spreading angle for the determination of beff,w starts at the weld outer edge and is 1:1 through an end plate and 2.5:1 through the flange and web-to-flange radius (of hot-rolled) or leg size of fillet weld (of welded) of the compressed member; see Fig. 2.2. Figure 2.2: Spreading angle through the rolled transversely compressed member [30] Reduction factor ωdepends on the ratios of bending moments of the beams on the opposite side of the column, the transformation parameter βthat can be taken from Cl. 7.2.3 [28]. In the dataset, the forces coming from the opposite sides will be equal, and therefore, ω= 1 as can be seen from Tab. A.1. [28] Further variables are twc, column web thickness, fy,wc, column web yield strength, and γM0= 1.0, partial safety factor. Buckling failure mode is determined by the buckling reduction factor ρ. Buckling of the web in transverse compression is based on the following assumptions: Web is pinned into the flange [35] The effective width, beff,w is irrelevant for the calculation of critical force The web plate is assumed to be infinitely long. Or in other words, the end is far enough that it does not affect the buckling resistance nor the critical force. If any stiffeners are nearby, but not directly under the point of compression, their contribution is disregarded. The buckling curve was modified in the second generation of Eurocode [28] to reflect Winter curve [36], which actually made buckling even more likely to govern the design. ρ= 1.0,if ¯ λp≤0.673 ¯ λp−0.22 ¯ λ2 p ,if ¯ λp>0.673 (2.3) Relative slenderness is calculated: ¯ λp= 0.932 sω kwc beff,w twc fy,wc E t2 wc (2.4) The following paragraphs are not included in the code [28], but they are included in the paper in preparation [30]. There an analytical approach is presented based on the optimization algorithms and engineering judgment. This serves as a very good comparison to machine-learning. In [30], the author and his colleagues have shown how this design method could be adapted to better reflect the numerical simulations performed with ANSYS [23] in [37]. Spreading angle Firstly, the results of materially nonlinear analysis (MNA) at 5% plastic strain, which is assumed to simulate the design yielding resistance, were used to obtain the effective width, i.e., the spreading angle set to 2.5:1 in codes. Two methods were used:
16 CHAPTER 2. METHODS 1. Division of numerical resistance by web thickness and yield strength beff,w =FRk,MNA twc ·fy,wc (2.5) 2. Distribution of stresses normal to the transverse load at the web directly below the rounding. Only the positive stresses were taken into account in the equivalent area averaging; see Fig. 2.3. Figure 2.3: Determination of the effective width beff,w using normal stress σz[30] The comparison of effective widths for IPE cross-sections determined using the current Eurocode approach with the spreading angle 2.5:1 (EC3), division of numerical resistance (FRk,MNA), and the distribution of normal stress (σz) is shown in Fig. 2.4. 100 200 300 400 500 600 IPE 0 50 100 150 200 250 Effective width beff , w [mm] EC3 FRk , MNA z Figure 2.4: Effective width determined by Eurocode and numerical methods It can be seen that the approach using σzis the safest, which is probably caused by a more complicated stress distribution including also shear stresses and σzcannot be directly compared to yield stress. Fig. 2.5 shows the reliability graphs. These contain number of specimens n, bias b (see Eq. (2.32)), and variation coefficient Vδ(see Eq. 2.36) for quick first order reliability method (FORM) assessment. FRk,MNA approach can be directly compared to EC approach where the
2.1. DESIGN CODES AND STATE-OF-THE-ART 17 spreading angle 2.5:1 through the flange and radius shows good estimation with IPE 300 and larger cross-sections and for small cross-sections (IPE 100 and IPE 200) is rather optimistic. In the subsequent calculations, the spreading angle of 2.5:1 is retained. Future work could focus on the source of this issue and perhaps a separate spreading angle through the flange and through the radius. 100 150 200 250 beff , w by FRk , MNA 50 100 150 200 250 beff , w by EC3 FRk , MNA n = 6 b = 1.026 V = 0.046 45 line +/-10% error 50 100 150 200 250 beff , w by z 50 100 150 200 250 beff , w by EC3 z n = 6 b = 1.127 V = 0.037 45 line +/-10% error Figure 2.5: Reliability graphs of the effective width determination by EC3 in comparison to division of numerical resistance FRk,MNA (left) and distribution of normal stress σz(right) [30] The resistance determined by MNA decreases as the distance towards the unstiffened end is getting closer. This could be expressed by decreasing the effective width as calculated by Eq. (2.5) by ratio re: re=beff,w,e beff,w (2.6) where: beff,w,e – effective width for load near the unstiffened end beff,w – effective width for load unaffected by the unstiffened end From the numerical models, it can be observed that the yield resistance FRk is not affected by the unstiffened end if the load is further from the edge than x=e/h = 0.45, i.e., the ratio re= 1.0 for x≥0.45. For the cases closer than that, it seems that a parabola fits well with the datapoints; see Fig. 2.6. Due to this, the coefficient of variation Vδis rather high, 7.2 %, but for higher loads, the differences are low. It can be assumed that the fitted parabola simulating the effect of unstiffened end proximity is in good agreement with the numerical data. re=−2.1·(x−0.45)2+ 1 (2.7) The reliability graph in Fig. 2.6 shows the most significant deviations at low loads, meaning for cross-sections IPE 100 and IPE 200. The difference originates primarily from the assumed distribution angle 2.5:1, which diverges by 10 % and 5 % for IPE 100 and IPE 200, respectively. Degree of fixity of web to flange transition The critical force for the plate of infinite length supported at two opposing edges was derived by Timoshenko [38]: Fcr =kcr ·π2·E·t3 wc 12 ·(1 −ν2)·dwc (2.8)
18 CHAPTER 2. METHODS 0.1 0.2 0.3 0.4 0.5 x = e / h [-] 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 re ( x ) [-] Function: re ( x ) = 2.10 ( x 0.45)2+ 1 Fitted Parabola Vertex at x = 0.45 f(x) = 1 Mean with min/max range 0 200 400 600 800 1000 FRk , Calc [kN] 200 400 600 800 1000 FRk , FEA [kN] Plastic strength n = 36 b = 0.975 V = 0.072 45 line +/-10% error Figure 2.6: Reduction of effective width in relation to end distance; reliability graph of FRk estimated by fitted formula [30] kcr = 4/π, for simply supported edges 14/π, for fixed edges (2.9) where: E= 210000 MPa – steel modulus of elasticity ν= 0.3 – Poisson coefficient dwc =h−2·tfc −2·r– web height between radii (for hot-rolled) or edges of fillet welds (for welded) kcr – factor for the degree of fixity of web plate into flange plates [35, 39] The values of Fcr are available from LBA, and since all other parameters are known, kcr can be calculated directly. For the general case where the unstiffened end is sufficiently far away and influences neither the yielding nor the buckling strengths of the web in transverse compression (see 1.0·hin Fig. 2.7), the degree of fixity is on average 3.619, which is closer to the fixed condition (14/π = 4.456) than to the pinned condition assumed in the code [28] (4/π = 1.273); see Fig. 2.7 and Eq. (2.9). For the cases of load near the unstiffened end, the critical force Fcr calculated by LBA is decreasing. This can be explained by the fact that the web plate is not infinitely long but its end is getting closer. The buckling behavior is thus more appropriate for the plate supported on three sides rather than all four sides. The degree of fixity of the web to the flanges likely remains the same, but for simplicity, the same factor kcr is used to account for this decrease in the critical force Fcr. Very conveniently, the values of kcr show small variation across all IPE cross-sections; the average coefficient of variation is 0.070. The averages of kcr across all the IPE sections seem to follow a bilinear trend when plotted against relative distance towards the end x=e/h, where eis the distance between the load application and the unstiffened end and his the cross-section depth. By minimizing mean squared error, the following bilinear function was constructed: kcr(x) = min (3.6,5.7·x) = min 3.6,5.7·e h(2.10) Fig. 2.8 proves that the critical force calculated by the proposed formulas closely fit the numerical solution by the LBA.
2.1. DESIGN CODES AND STATE-OF-THE-ART 19 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 kcr [-] 0.1⋅h 0.2⋅h 0.3⋅h 0.4⋅h 0.5⋅h 1.0⋅h Simply supported web Fixed web IPE 100 IPE 200 IPE 300 IPE 400 IPE 500 IPE 600 Figure 2.7: The degree of fixity kcr for varying distance to the unstiffened end [30] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 0.2 0.4 0.6 0.8 1 kcr [-] x = e/h [-] Data points Fitted linear function 500 1000 1500 2000 2500 Fcr , Calc [kN] 500 1000 1500 2000 2500 Fcr , FEA [kN] Critical force n = 36 b = 1.003 V = 0.029 45 line +/-10% error Figure 2.8: Fitted bilinear curve for kcr as a function of xand comparison of critical forces determined by analytical calculation and by LBA [30] Relative slenderness The code equation (see Eq. (2.4)) for relative slenderness can be rewritten according to EN 19931-5, Annex B [32] to: ¯ λp=rαult,k αcr =rFc,wc,Rd Fcr (2.11) Finite element analysis directly provides the αult,k or Fc,wc,Rd by MNA with failure criterium such as εpl = 5% and αcr or Fcr =F·αcr by LBA, where Fis the applied load. By the assumption of spreading angle 2.5:1 and above-mentioned equations, the compressive resistance and critical force may also be calculated. The comparison is in Fig. 2.9 showing good approximation with Vδ= 4.8% and higher deviations at low values of relative slenderness, where the difference is not extremely important. Buckling curves Buckling curves were the most challenging to determine due to the limited amount of data points for various relative distances of the compressive force to the unstiffened end x. Especially for the basic case of x= 1, the relative slenderness ¯ λdid not fall below 0.7. More significant reduction
20 CHAPTER 2. METHODS 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Relative slenderness Calculated 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Relative slenderness FEA Relative slenderness n = 36 b = 0.988 V = 0.048 45 line +/-10% error Figure 2.9: Relative slenderness calculation [30] due to buckling occurred only for load close to the unstiffened end; see Fig. 2.10. This makes any optimization difficult and unreliable; therefore, engineering judgment was employed extensively. Buckling curves in the form specified in EN 1993-1-5, Annex B [32] were chosen: ρ=1 ϕp+qϕ2 p−¯ λp (2.12) ϕp=1 2·1 + αp·(¯ λp−¯ λp0) + ¯ λp(2.13) By optimization algorithms, using the Differential Evolution function in scipy.optimize package in Python [40], the parameters of αpand ¯ λp0were sought. The datapoints were the reduction factor ρ, calculated as the ratio of GMNIA to MNA resistances, and relative slenderness calculated by Eq. (2.11) from MNA and LBA. The optimization algorithm was able to minimize the error, but the optimized buckling curves made little sense; see Fig. 2.10. The imperfection factors αp were wildly different, which should not happen for the same hot-rolled sections. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p 0.5 0.6 0.7 0.8 0.9 1.0 Optimized p 0 and p x = 0.1, p 0 = 0.436, p = 0.262 x = 0.2, p 0 = 0.312, p = 0.199 x = 0.3, p 0 = 0.404, p = 0.144 x = 0.4, p 0 = 0.519, p = 0.055 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p 0.5 0.6 0.7 0.8 0.9 1.0 Optimized p 0 and fixed p x = 0.1, p 0 = 0.372, p = 0.210 x = 0.2, p 0 = 0.332, p = 0.210 x = 0.3, p 0 = 0.493, p = 0.210 x = 0.4, p 0 = 0.642, p = 0.210 Figure 2.10: Optimized buckling curves [30]
2.1. DESIGN CODES AND STATE-OF-THE-ART 21 The second step was to fix the imperfection factor αpto 0.21, which seems reasonable for hotrolled IPE sections and is close to the average for distances x= 0.1,0.2, and 0.3. It was assumed that the distance x= 0.4 consists of too few points, the reduction ρdoes not go below 0.9, and the imperfection factor for the red curve is meaningless. The error was slightly larger, but still within acceptable limits. However, it was assumed that as the distance xis decreasing, the plate edge is getting closer and the buckling reduction should be more prominent. This is true for most cases, but strangely, the ¯ λp0= 0.372 for x= 0.1 and ¯ λp0= 0.332 (smaller!) for larger distance x= 0.2. This was assumed as a numerical error, and the final buckling curves were chosen with αp= 0.21 and ¯ λp0 according to Eq. (2.14), although the error increased for x= 0.1 and 0.2; see Fig. 2.11. λp0= min (0.673,0.223 + 0.9·x) = min 0.673,0.223 + 0.9·e h(2.14) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 p [-] 0.5 0.6 0.7 0.8 0.9 1.0 [-] Calculated p 0 and fixed p x = 0.1, p 0 = 0.313, p = 0.210 x = 0.2, p 0 = 0.403, p = 0.210 x = 0.3, p 0 = 0.493, p = 0.210 x = 0.4, p 0 = 0.583, p = 0.210 x = 0.5, p 0 = 0.673, p = 0.210 Figure 2.11: Buckling curves according to EN 1993-1-5, Annex B [32] with calculated parameters [30] Buckling resistance for the whole set The buckling resistance using above-mentioned equations were compared to the whole set of FEA models made in ANSYS [37] as well as the experiments performed at Brno University of Technology in 2023 [30]. Note that the numerical models utilize nominal geometrical and material properties, design equivalent geometric imperfections are applied, and therefore they may be considered as numerical design calculations proving design resistances [5]. For the experiments, measured geometry, yield strength, and modulus of elasticity was used in the above-mentioned formulas. In total, 10 experiments and 153 numerical design calculations are compared; see Fig. 2.12. Several issues are found with the proposed approach: Welded sections have much higher resistances, which indicates the spreading angle 1:2.5 is not constant through the flange and through the fillet radius or fillet weld. The reduction in EN 1993-1-8 – Eq. (A.25) [28] for compressive force in the column is insufficient. The calculations are to the unsafe side. Solving these issues is a topic for future research.
22 CHAPTER 2. METHODS 0 200 400 600 800 1000 1200 1400 Fc , wc , Calc [kN] 0 200 400 600 800 1000 1200 1400 Fc , wc , GMNIA or Experiment [kN] FRk , GMNIA n = 153 b = 1.001 V = 0.160 Experiment n = 10 b = 1.067 V = 0.083 45 line +/-10% error Figure 2.12: Reliability graph of the proposed method to all ANSYS numerical design calculations [37] and BUT experiments 2.1.2 Net section failure in tension Net section failure of plates loaded in tension is considered as the most dangerous for finding a reliable plastic strain limit for numerical design calculations. Therefore, the approaches in design codes are briefly summarized in this section. EN 1993-1-1 [4] requires verification of members loaded in tension in Clause 6.2.3 by the following equations: NEd Nt,Rd ≤1.0 (2.15) Nt,Rd = min Npl,Rd =A·fy γM0 , Nu,Rd =0.9·Anet ·fu γM2(2.16) FprEN 1993-1-1 [12] in Clause 8.2.3 modifies the net section failure as follows: Nu,Rd =k·Anet ·fu γM2 (2.17) where: k= 1.0 for sections with smooth holes (i.e. holes without notches), for example holes fabricated by drilling or water jet cutting; k= 0.9 for sections with rough holes (i.e. holes with notches), for example holes fabricated by punching or flame cutting; k= 0.9 for structures subjected to fatigue. So for standard cases, the net section resistance is 10% higher than in the first generation of Eurocodes. Note that both generations of Eurocode consider only one possibility of shear lag effect – in an angle cross-section connected by only one flange.
2.2. EXPERIMENTS ON WEB IN TRANSVERSE COMPRESSION 23 AISC 360-22 [18] in Section D.2 determines the tensile strength as the minimum of tensile yielding (corresponding to gross section yielding) and tensile rupture (corresponding to net section failure): Pn= min {ϕt,y ·A·fy, ϕt,u ·Anet ·fu}(2.18) where: A– gross section area Anet – effective net area specified in Section D.3 fy– yield strength fu– ultimate strength ϕt,y = 0.9 – resistance factor for tensile yielding ϕt,u = 0.75 – resistance factor for tensile rupture Interestingly, AISC 360 covers in Table D3.1 (see Fig. 3.3) a large variety of cases where shear lag effect decreases the net section area. Note that the ratio between safety assigned to gross section yielding and net section failure is different according to each code. A·fy Anet·fu·0.9 0.75 for EN 1993-1-1, A·fy Anet·fu·1.25 0.9for FprEN 1993-1-1, A·fy Anet·fu·1.25 1.0for AISC 360. (2.19) Provided that the ratio of fu/fyremains constant, with increasing weakening, net section failure begins to govern first according to the current EN 1993-1-1, then according to the second generation FprEN 1993-1-1, and lastly according to AISC 360. 2.2 Experiments on web in transverse compression This section describes the physical experiments performed at BUT and elsewhere. The simplest setup is to apply the load from both sides against an I-section; see Fig. 2.13. Applying load from one side is complicated, because the web panel in shear or the member in bending may fail first. The geometry for such testing is therefore limited. The overview of experimental research is given in Tab. 2.1. Most tests are for double-sided concentrated load [30, 24, 44, 35]. Note that the four specimens included in [24] are the same as Figure 2.13: Experiments performed at BUT – test setup (left) and web buckling after significant yielding (right) [24]
30 CHAPTER 2. METHODS complex, non-linear problems. In recent years, their application in architecture, engineering, and construction has grown significantly, driven by improved hardware (notably GPUs), algorithmic advancements, and accessible software frameworks. A single artificial neuron combines input features xnwith trainable weights Wn, adds a bias bNN , and applies a non-linear activation function: y(x) = a·XWn·xn+bNN (2.22) Networks consist of multiple interconnected neurons organized in layers, allowing the model to learn hierarchical feature representations. During training, weights and biases are adjusted to minimize prediction error, quantified by a loss function such as mean squared error (MSE) or mean absolute error (MAE). This optimization typically relies on gradient-based algorithms such as backpropagation and stochastic gradient descent. Data preprocessing is critical for effective training. Normalization or standardization ensures that all features contribute equally by removing differences in scale and distribution; see Tab. 2.4. The outputs for training should be also scaled, so that large values do not cause mathematical issues during the training process. The output of a neural network may subsequently be passed through another activation function (e.g., sigmoid for binary classification or softmax for multiclass classification), depending on the task. In this study, no activation function was used for the output. 2.5.1 Neural network architecture In this work, deep neural networks (DNNs) are trained to predict the non-linear resistance of component column web panel in transverse compression and tensile rupture of weakened plates. The model architecture and hyperparameters follow prior studies by Andreas M¨uller [62], [63], [64] and following investigations in the scope of the project Machine Learning-based Design Optimization of Steel Connections (MADESCO) in a cooperation of IDEA StatiCa, ETH Z¨urich and CTU Prague. In the following, the effects of the input and output scalers, activation functions, optimizers, learning rate, and the width and depth of the neural network are investigated. As a dataset, a knee welded joint loaded by a combination of bending moment and shear force was chosen. This dataset contains over 800 thousand analyzed models, but for this exercise, only 5 % of the available data was used. The graphs were done primarily by Tamer C¸akır at ETH. The DNN architecture, in terms of the number of neurons and layers, was varied in each of the following investigations. The differences between wider DNN (with more neurons) and leaner DNN (fewer neurons in one layer) are not great. The wider DNN typically achieves better error in the form of R2than the leaner DNN. On the other hand, it should be noted that the more neurons, the longer the training takes. Overall, the variation in parameters (scalers, optimizers, learning rate, activation functions) shows that the correct selection of DNN architecture is very important. Some parameters perform much worse than others. These comparisons indicate that the DNN architecture must be tested for a particular problem, and various options should be considered. 2.5.2 Input and output scaler Prior to model training, all input features may be subjected to dimensional scaling to cross-section depth of e.g. 200 mm using similitude theory [65], [66]. For example, IPE 300 with cross-section depth of 300 mm is scaled by L= 200/300 = 2/3. Then, features are scaled as follows: Lengths ∼L1 Areas and forces ∼L2 Volumes, masses, and bending moments ∼L3 Constants, stresses, and strains ∼L0(invariant under scaling) Stiffness (force/displacement) ∼L1 This is called a Similitude scaling, and it utilizes the physical similarity of the model and the same stress distribution as in the scaled model. This allows drastically reducing the number of cases for training and potentially increasing the range of validity. Training data are generated by
2.5. NEURAL NETWORK 31 physical testing, numerical simulations or numerical design calculations, which are the most timeand resource-intensive. When using physical experiments, some experimental variations may get lost and it can be beneficial to add scaled data to the original dataset. When using numerical simulations or numerical design calculations, similitude theory should be perfectly valid and the original dataset may be replaced by the scaled dataset. Next, all input features should be scaled according to one of the following methods. This preprocessing step ensures that all features contribute equally to the learning process, avoids dominance by features with larger magnitude, and enhances convergence during optimization [58]. Figure 2.17: Comparison of input and output scalers for different widths and depths of neural network Table 2.4: Overview of common data scaling and transformation methods Method Formula Standardization x′ i=xi−µ σ, µ =1 nPn j=1 xj, σ =r1 nPn j=1(xj−µ)2 Mean Normalization x′ i=xi−µ xmax −xmin , µ =1 nPn j=1 xj Min–Max Scaling x′ i=xi−xmin xmax −xmin Robust Scaling x′ i=xi−median(x) Q3−Q1 , Q1= 25th percentile, Q3= 75th percentile Logarithmic Transform x′ i= log(xi+ε), ε > 0 Fig. 2.17 shows a comparison of the different scalers; see Tab. 2.4. In this case, the outputs were normalized to the bending and shear resistance of the connected beam. Note that the results may vary significantly when the outputs, such as load resistance, are not normalized. The performance of all four tested DNNs is very poor with the log scaler, where both inputs and outputs are transformed into a logarithmic scale. The performance of the other scalers is relatively similar, with the best choice varying depending on the DNN width and depth.
32 CHAPTER 2. METHODS Figure 2.18: Comparison of different optimizers 2.5.3 Optimizer Fig. 2.18 shows a preliminary comparison of the optimizers. A thorough evaluation would require tuning additional hyperparameters for each selected method. Adagrad clearly performs worst in this comparison, while RMSProp achieves the highest R2. However, RMSProp exhibits significant fluctuations during the early stages of training, whereas Adam and AdamW remain much smoother. The overview of optimizers used historically and in current machine-learning tasks is in Tab. 2.5. Table 2.5: Overview of selected gradient-based optimizersa Optimizer Update rule / principle Remarks SGD wt=wt−1−η∇LtSimple stochastic gradient descent; sensitive to learning rate; may oscillate in narrow valleys. SGD + Momentum vt=βvt−1+∇Lt wt=wt−1−ηvt Accelerates convergence in relevant directions; dampens oscillations; introduces momentum hyperparameter β. Adagrad [67] Gt,i =Gt−1,i + (∇iLt)2 wt,i =wt−1,i −η pGt,i +ϵ∇iLt Effective for sparse data; learning rates decrease monotonically, may become too small. RMSProp [68] vt,i =βvt−1,i + (1 −β)(∇iLt)2 wt,i =wt−1,i −η √vt,i +ϵ∇iLt Mitigates Adagrad’s diminishing learning rate; works well in non-stationary settings; requires decay tuning. Adam [69] mt=β1mt−1+ (1 −β1)∇Lt vt=β2vt−1+ (1 −β2)(∇Lt)2 ˆmt=mt 1−βt 1 ,ˆvt=vt 1−βt 2 wt=wt−1−η √ˆvt+ϵˆmt Combines momentum and adaptive scaling; fast and robust; may generalize worse than SGD with momentum. AdamW [70] Same as Adam, but with decoupled weight decay (regularization applied separately from the gradient step). Improves generalization; widely used in large-scale models such as Transformers. aNotation: wt– parameter vector at iteration t;∇Lt– gradient of the loss at iteration t;∇iLt– gradient w.r.t. parameter wi;η– learning rate; ϵ– small constant to prevent division by zero; Gt,i – accumulated squared gradients (Adagrad); vt,i – exponential moving average of squared gradients (RMSProp, Adam); mt– exponential moving average of gradients (Adam); ˆmt,ˆvt– bias-corrected estimates (Adam); β, β1, β2– decay rates for moving averages; vt– momentum term in SGD + momentum. 2.5.4 Learning rate Fig. 2.19 shows the DNN performance for varying learning rates. Very high learning rates may overshoot the global minima and fail to converge to a good solution, while very low learning
2.5. NEURAL NETWORK 33 Figure 2.19: Comparison of learning rates rates converge too slowly. The results indicate that particularly large learning rates should be avoided. Interestingly, Adam should dynamically adjust the learning rate based on estimates of first and second moments of the gradients. It slows down the learning rate when nearing a minimum, which helps not overshooting the minimum and reducing bouncing around the minimum. However, an additional (manual) adaptive learning rate that decreases as the loss function becomes smaller provides the best results. PyTorch provides a built-in scheduler for this purpose, called ReduceLROnPlateau. 2.5.5 Activation function Figure 2.20: Activation functions in the range [−1,1] Fig. 2.21 compares the model performance across different activation functions. Among these,
34 CHAPTER 2. METHODS Figure 2.21: Comparisons of activation functions the ReLU function is currently the most widely used due to its simplicity and computational efficiency. An overview of commonly implemented activation functions available in frameworks such as PyTorch is provided in Tab. 2.6 while Fig. 2.20 illustrates their behavior over the input domain [−1,1]. Interestingly, different activation functions converge to a stable error R2.The GELU function achieves the highest performance with a final error of R2≈0.99 whereas some functions plateau below R2≈0.96. This difference is substantial in terms of prediction quality. A more comprehensive investigation would require evaluating combinations of activation functions across hidden layers and output layers to fully understand their impact. Figure 2.22: Optimized model The optimized model shown in Fig. 2.22 is able to quickly achieve very high prediction quality with a final error R2≈0.99. In this optimized model, RMSProp optimizer, GELU activation function, adaptive learning rate using ReduceLROnPlateau in-built scheduler and a wide neural network with the initial width of 128 neurons progressively decreasing across 5 hidden layers to one output. The optimization considers only the resulting error R2without any regard for the speed of learning process. That is because the calculation of dataset takes days or even months while the learning process is typically in minutes.
2.5. NEURAL NETWORK 35 Table 2.6: Overview of Activation Functions Activation Function Formula Range Key Characteristics ReLU [71] f(x) = max(0, x) [0,∞) Simple, computationally efficient; can suffer from the ”dying ReLU” problem. Leaky ReLU f(x) = max(αx, x) (−∞,∞) Allows a small, non-zero gradient when the unit is inactive and not saturating. Tanh f(x) = ex−e−x ex+e−x(−1,1) Smooth, zero-centered output; suffers from vanishing gradients for large inputs. ELU f(x) = (xif x > 0 α(ex−1) if x≤0(−α, ∞) Smooth and differentiable; aims to push mean activations closer to zero, mitigating vanishing gradients. GELU [72] f(x) = x·Φ(x) (−∞,∞) Smooth, non-monotonic; approximates identity for large positive inputs and zero for large negative inputs. SiLU (Swish) f(x) = x·σ(x) (−∞,∞) Self-gated activation function; combines properties of sigmoid and linear functions. aNotation: α– scaling factor for negative inputs (Leaky ReLU, ELU); Φ(x) – cumulative distribution function of the standard normal distribution; σ(x) – sigmoid function. 2.5.6 Loss function Minimizing the loss function encourages the model to make predictions close to the actual target values. The most commonly used loss function for regression problems is the Mean Squared Error (MSE): LMSE =1 n n X i=1 (yi−ˆyi)2(2.23) The disadvantage is that MSE penalizes large errors more severely. The prediction tends to lean towards the outputs with high magnitudes. This skews the results when the dataset contains members or connecting elements of different sizes. For example, the results with IPE 600 crosssection are more relevant than the results with IPE 100 cross-section. There are several approaches to mitigate this issue. The outputs, e.g., component load resistance, can be transformed into a logarithmic scale. Furthermore, different loss functions may be used: Mean Absolute Error (MEA) LMEA =1 n n X i=1 |yi−ˆyi|(2.24) Huber loss [73] Lδ(y, ˆy) = (1 2(y−ˆy)2if |y−ˆy| ≤ δ δ|y−ˆy|− 1 2δotherwise (2.25) Relative mean squared error (RMSE) LRMSE =1 n n X i=1 (yi−ˆyi) yi2 (2.26)
36 CHAPTER 2. METHODS Log-Cosh LLog-Cosh =1 N N X i=1 log (cosh (yi−ˆyi)) (2.27) 10 310 210 1100101102103 Error Magnitude | y y | (log scale) 10 9 10 7 10 5 10 3 10 1 101 103 105 Loss Value (log scale) Relative MSE (y=2) Relative MSE (y=10) Relative MSE (y=100) MSE ( x 2) MAE ( x ) Huber ( =1.0) Log-Cosh Figure 2.23: The overview of different loss functions and their output with increasing error The comparison of loss values in relation to error magnitude is shown in Fig. 2.23. For effective optimization, larger errors must be penalized more heavily. MSE increases quadratically with error, i.e., in log-log scale is with the steady slope. MAE increases only linearly, i.e., the slope in log-log scale is smaller than for MSE. Huber loss function provides a transition from MSE to MAE at error δ, blending quadratic and linear behavior around this point. The Log-Cosh loss function behaves similarly to the Huber loss with δ= 1, producing nearly the same curve with only slight differences near the transition region. Note that using RMSE loss function prevents use of some types of scalers (Mean normalization, Standardization, and Robust scaling) for output. The output values yiand ˆyicannot be equal to zero or close to zero. This would produce a mathematical error. 2.5.7 Final model for web in compression In the investigation of column web in transverse compression, these parameters of neural network as a feedforward architecture were adopted: 8 input features that include geometric and material properties, 128 neurons, 64 neurons, 32 neurons, 16 neurons and 1 output layer – plastic resistance Fpl,Rd or αcr. The model was implemented using torch.nn.Sequential. Similirity scaler is applied – all inputs and outputs are scaled to the reference beam depth h= 200 mm. StandardScaler from scikit-learn [74] for the inputs. Custom scaler for the output so that the values are between 0 and 10. yscaled =y ymax ·10 (2.28) Adam optimizer, because it is more advanced and does not suffer from the initial fluctuations. Its performance scored similarly as the first, RMSProp. Learning rate with and initial value of 0.01 and the use of scheduler with the factor of 0.9 (learning rate is reduced only by 10 % loss function is not decreasing for 20 epochs) and minimum learning rate of 0.0001.
2.6. RELIABILITY 37 To avoid unnecessary training cycles and prevent overfitting, early stopping was applied. Training was halted if the coefficient of variation (CoV) on the validation set failed to improve over 2 000 consecutive epochs. A minimum of 3 000 epochs was enforced to allow sufficient training. A maximum of 10 000 epochs are used. GELU activation functions in hidden layers and no activation function at the output layer. Loss function implemented as Relative Mean Squared Error. The purpose of this loss function is for each data point, no matter the size of member, to add the same weight into the training process. To evaluate the generalization capability of the model, the dataset is randomly split into training and testing subsets. The training set is used to fit the model, while the testing set provides an unbiased evaluation of prediction accuracy. An 80:20 split ratio was employed using train test split from scikit-learn [74]. The split is utilized to: Prevent overfitting to the training data, Allow estimation of model performance on unseen data, Simulate real-world deployment conditions. The following metrics were used to assess model performance: Bias: Measures systematic deviation of predictions from true values; see Eq. (2.32). Coefficient of Variation (CoV): Measures relative variability in predictions. CoV is defined as: CoV = std(y/ˆy) mean(y/ˆy)(2.29) Relative Error: Used to identify outliers with poor prediction accuracy. Predicted vs. true values were visualized using scatter plots, including ±10% error bounds. Matplotlib [75] was used for plotting. Any outliers with relative error >10%) were extracted and analyzed for patterns or data quality issues. The final trained model was serialized and saved as a .pth file using torch.save, enabling future inference without retraining. Note that only weights and biases of the model are not enough. Input and output scaling parameters and DNN options must also be preserved to maintain prediction consistency. 2.6 Reliability Eurocode EN 1990 [6] allows two types of reliability verification: Semi-probabilistic approach using partial safety factors Reliability-based design and assessment 2.6.1 Semi-probabilistic approach This chapter provides step-by-step guidance for the reliability assessment according to Eurocode standards and is mostly built upon SAFEBRICTILE deliverable D1.1 [15], which was not published. First, it is necessary to split the reliability of actions and resistances, which greatly simplifies the task. Eurocode assumes factors for resistance αR= 0.8 and for loads αE= 0.7. The probability that the actual resistance R is smaller than the design resistance Rdis then: P(R≤Rd) = Φ (−αR·β) = Φ (−0.8·3.8) = Φ (−3.04) (2.30) In other words, 1183 samples should fail out of a million. Note that the target reliability is the same for all types of failure modes; member yielding or bolt or weld fracture, with possibly higher consequences, keep the same reliability target. Also, all reliability classes (RC 1, RC 2,
38 CHAPTER 2. METHODS RC 3) are defined by the same resistance reliability target αR·β= 3.04. The reliability differentiation is achieved by either through increases of the load-sided partial factors or through increased supervision and fabrication inspection levels [15]. Then a set of experiments should be specified, and experimental resistance re,i determined. This may seem like an easy task, but the load resistance is often an ambiguous term that needs to be clarified. For the example of tensile rupture, the load resistance is assumed to be at maximum load. Then the numerical design model is created for each experiment and the numerical resistances rt,i are obtained. The geometrical and material properties are set as measured in the experiment. The numerical model should be subjected to a validation and verification process as described in FprEN 1993-1-14:2024 – Chapter 7 [5]. The next step is to determine the basic variables that affect the results. These variables are, for example, the plate thickness or the material yield strength. The mean values and coefficient of variation for the basis for calibration of partial factors are now codified in FprEN 1993-11:2017 – Table E.1 [6]. Interestingly, the variability in dimensional properties does not match the manufacturing tolerances, e.g. in EN 10029:2010 [7], but it outlines the assumptions, upon which the reliability of Eurocodes is built. Next, the mean value of bias bshould be calculated: b= n X i=1 re,irt,i/ n X i=1 r2 t,i (2.31) Where nis the number of specimens. There is a problem with this bias when dataset contains varying sizes of members or connection elements, e.g., bolts or welds. This bias is affected disproportionately more by large members and elements. A more appropriate formula is the simple mean: b=1 n n X i=1 re,i rt,i (2.32) FprEN 1993-1-14 – Annex A [5] states that bshould be in the range of 0.8<b<1.25. SAFEBRICTILE [14] requires even more restrictive range of 0.85 < b < 1.15. However, for numerical design calculation models with bilinear material model and small plastic strain limit, it is expected to reach high values of b, e.g. 1.3. Such higher values should be expected when neglecting strain-hardening and should not be the reason to terminate the reliability assessment due to inaccuracy. Then the coefficient of variation of the error Vδshould be calculated by the following equations: δi=re,i b·rt,i (2.33) ∆i= ln (δi) (2.34) s2 ∆=1 n−1 n X i=1 ∆i−¯ ∆2(2.35) Vδ=qexp (s2 ∆)−1 (2.36) Where ¯ ∆ is the average of ∆i. The next step is to calculate the coefficient of variation Vrt for the basic input variables. For that, the geometrical and material properties should be changed in these numerical models to mean values according to Table E.1 [12]. The load resistances of these numerical models are rt,i(Xm). Furthermore, the numerical models with slightly changed mean values should be calculated. For example, if the basic variables are plate thickness and yield strength, a set of numerical models with changed plate thickness (load resistances rt,i(∆t)) and another set of numerical models with changed yield strength should be calculated (load resistances rt,i(∆fy)). The change should be small but meaningful, and it should not completely change the failure mode. The authors suggest using values close to the standard deviation of the basic variable σ.
2.6. RELIABILITY 39 V2 r,t,i,∆t="rt,i Xm−rt,i (∆t) ∆t·σ(t)#2 (2.37) V2 r,t,i,∆fy="rt,i Xm−rt,i (∆fy) ∆fy·σ(fy)#2 (2.38) V2 rt,i =V2 rt,i,∆t+V2 rt,i,∆fy/r2 t,i(Xm) (2.39) Vrt = n X i=1 qV2 rt,i (2.40) It is probable that many numerical models will have more basic variables. In that case, the formulas should be expanded accordingly. The values of Vrt depend mainly on the relationship of the result to the basic variables. If the relationship is linear, such as for tensile rupture, the expected values of Vrt are about 5%. For phenomena, where relationship is quadratic or even on a power of three, such as plate bending or buckling, the values of Vrt will be higher. It is difficult to establish Vrt specifically for every experimental set, and Vrt may be set based on previous experience; e.g. first generation of Eurocode was built mainly with Vrt = 7%. Finally, the two errors, Vδand Vrt are added together to the coefficient of variation: Vr=qV2 δ+V2 rt (2.41) Now variation coefficients Qcan be calculated either for individual Vrt,i for each specimen, or global Vrt: Qrt,i =qln V2 rt,i + 1,or for global Vrt :Qrt =qln (V2 rt + 1) (2.42) Qδ=qln (V2 δ+ 1) (2.43) Qi=qln V2 δ+V2 rt,i + 1,or for global Vrt :Q=pln(Vr+ 1) (2.44) The design resistance is calculated according to the number of specimens n(for global Vrt, replace Qrt,i for Qrt and Qifor Q): rd,i = b·rt,i Xmexp −kd,∞· Q2 rt,i Qi!−kd,n ·Q2 δ Qi−0.5·Q2 i!,for n≤100 b·rt,i Xmexp −kd,∞·Qi−0.5·Q2 i,for n > 100 (2.45) Where kd,n and kd,∞are from EN 1990 – Table D2 [6], and FprEN 1993-1-14 – A3(4) [5] suggests that Vxunknown should be used. Often, the number of specimens is higher than 30 and interpolation between 30 and infinity should be made. Authors suggest treating 100 specimens as infinity, i.e., kd,100 =kd,∞= 3.04 and interpolate between 30 and 100. Finally, the partial safety factor can be calculated for each model and then the average: γM,i =rnom,i/rd,i (2.46) γM=1 n n X i=1 γM,i (2.47) This partial safety factor should be below an acceptable limit, otherwise, it should be applied to each numerical design calculation using these model settings. SAFEBRICTILE [15] recommends acceptable levels of γMaccording to current Eurocode practice based on coefficient of variation Vr; see Table 2.7. These acceptance limits are within the target reliability according to Equation (1). In case γMis below acceptance limit, it does not need to be applied. Furthermore, coefficient of variation Vδmay be reduced by two options:
46 CHAPTER 3. DETERMINATION OF RELIABLE STRAIN LIMIT Numerical Design Calculation Numerical Simulation Experiment NN Proven by reliability study Necking initiation Figure 3.4: Damage initiation at numerical simulation with probability of failure Pf(−β·αR) corresponds to reliable load resistance Reliable resistance by numerical simulation means the damage initiation (reaching ultimate strain εuat any location) of the simulation where only 0.1183 % of simulations provide the lower load at the damage initiation. Such a simulation is at the design resistance level with a probability of failure: Pf(−β·αR) = Pf(−3.8·0.8) = 1.183 ·10−3(3.1) Consequently, the maximum plastic strain at the numerical design calculation equal to this reliable load resistance is the design plastic strain limit. Based on the maximum possible weakening in EN 1993-1-8 – Tab. 3.3 [1] (see Fig. 3.5) and knowledge of fatigue, the most critical variants were selected; see Fig. 3.6. The stress concentration factor at elastic stage is determined from numerical simulations or for simpler cases according to Murakami [89]: Figure 3.5: Maximum weakening in structural plates according to EN 1993-1-8 [1] SCFinit =σmax σnom = 1 + 2rh R(3.2) where: σmax – maximum normal stress near the stress concentrator σnom – uniformly distributed normal stress far from the stress concentrator h– width of weakening R– radius of weakening The initial triaxility parameter ηinit is determined from finite element analysis; see Fig. 3.7.
3.2. EXPERIMENTAL INVESTIGATION 47 Figure 3.6: Selected geometries with stress concentration factor and initial triaxiality parameter Figure 3.7: Triaxiality parameter determined by ANSYS at elastic stage 3.2 Experimental investigation 3.2.1 Brno University of Technology The first set of experiments was performed using Labortech 6.1000.1 testing machine at AdMaS research center at the Faculty of Civil Engineering, Brno University of Technology in July 2024. A total of 18 specimens were tested. The plates were laser-cut and holes drilled. Three geometries (see Fig. 3.8) and two steel grades (S235 and S355; see Tab. 3.1) varied; each geometry of a steel grade was tested three times; see Tab. 3.2. The nominal dimensions of length Ö width Ö thickness were 500 Ö 100 Ö 6 mm, respectively. The thickness of each specimen was measured at multiple locations and the average calculated. Notably, specimens made of S235 were thinner, at the edge of tolerance limits. Only the force and displacement over the original gauge length of 220 mm, with bolt holes in the middle, were measured. The speed of loading was 2.8 mm/min. Table 3.1: Material properties of used steel from tensile coupon tests [10] Steel grade E[MPa] fy[MPa] fu[MPa] Strain at fu[%] Strain at fracture [%] S235 209067 279 405 18.7 33.9 S355 202167 383 537 14.9 26.6
48 CHAPTER 3. DETERMINATION OF RELIABLE STRAIN LIMIT Figure 3.8: Geometry of three specimen types for BUT experimental set; hole layout 2 Ö d20 – type V5 (left), 4 Ö d20 – type V6 (middle), 2 Ö d26 – type V11 (right) Table 3.2: Geometric and material properties of specimens Specimen Holes Steel grade Thickness [mm] d0[mm] Aneta[mm2]Ag[mm2] S1 2×d20 S235 5.50 20.00 330.0 550.0 S2 2×d20 S235 5.49 20.00 329.3 548.8 S3 2×d20 S235 5.54 20.00 332.3 553.8 S4 2×d20 S355 5.95 20.10 355.8 595.0 S5 2×d20 S355 5.95 20.23 354.3 595.0 S6 2×d20 S355 5.83 20.15 347.8 582.5 S7 4×d20 S235 5.76 20.05 345.2 576.3 S8 4×d20 S235 5.88 20.08 351.6 587.5 S9 4×d20 S235 5.81 20.10 347.6 581.3 S10 4×d20 S355 6.26 20.15 373.9 626.3 S11 4×d20 S355 6.19 19.83 373.4 618.8 S12 4×d20 S355 6.06 20.10 362.5 606.3 S13 2×d26 S235 5.83 26.10 387.8 582.5 S14 2×d26 S235 5.91 26.00 394.8 591.3 S15 2×d26 S235 5.88 26.03 392.0 587.5 S16 2×d26 S355 6.26 26.00 418.2 626.3 S17 2×d26 S355 6.25 26.03 417.0 625.0 S18 2×d26 S355 6.13 26.08 408.1 612.5 aCalculated according to EN 1993-1-1, Cl. 6.2.2.2 [4]. The photographs of failed specimens are in Fig 3.9. Specimens S7–S12 with four bolt holes seemingly failed in the same way as specimens S1–S6 with two bolt holes, but the deformation at failure was considerably larger. There was a significant strain achieved near the single hole before the tensile rupture at two bolt holes occurs. The load-deformation curves are plotted in Fig 3.10. The results are very consistent. Note that the specimens with shear plane (S13–S18) achieved the peak load just before tensile rupture, but the shear plane ultimately failed at very large deformations. Eurocode reliability EN 1993-1-1 [4] provides guidance on the calculation of the net area in Cl. 6.2.2.2 and tensile rupture resistance in Cl. 6.2.3. In the conducted experiments, tensile rupture always governed over tensile yielding. To maintain consistent reliability, the ratio of experimental resistance to calculated resistance should be constant, but this ratio changes dramatically in Tab. 3.3. Results of specimen triplets are averaged, and measured material properties are used in calculations. Ntstands for simple multiplication of Anet and fu, Interestingly, specimens S1–S6 reached even greater resistances by 10%, indicating that the normal stress exceeded the nominal ultimate strength. This can be explained by stress biaxiality. Due to stress flow and Poisson effect,
3.2. EXPERIMENTAL INVESTIGATION 49 Figure 3.9: Photographs of ruptured plates with centered and staggered bot holes 0 50 100 150 200 250 0 2 4 6 8 10 12 14 Force [kN] Deformation [mm] S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 Figure 3.10: Results of specimens S1–S18 tested at Brno University of Technology the steel at bolt holes is in biaxial tension and the normal strength is increased according to von Mises theorem. For specimens S7–S12, the effect was similar but lessened by the occurrence of another hole, which disrupted the stress flow. Nt,Rd is calculated according to Cl. 6.2.3. The formulas are highly conservative for specimens S1–S12, overestimating the resistance on average by 50%. The overestimation for specimens S13– S18 is significantly lower, only 30%. Clearly, the reliability for only tensile rupture and combined tensile and shear rupture is different. These findings are supported by another research [86]. The reason for very high safety margin was an assumption of fatigue fracture at the bolt hole. However, this is inconsistent; the assumption is not kept for combined tensile and shear rupture. A revision of tensile rupture calculation would be beneficial, but the formulas stay the same also in the next generation of Eurocode [12]. On the other hand, the reliability of both steel grades, S235 and S355, seem very consistent.
50 CHAPTER 3. DETERMINATION OF RELIABLE STRAIN LIMIT Table 3.3: Comparison of analytical, design, and experimental tensile resistances Specimen S1–S3 S4–S6 S7–S9 S10–S12 S13–S15 S16–S18 Anet [mm2] 330.5 352.6 348.1 369.9 391.6 414.4 Ag[mm2] 550.8 590.8 581.7 617.1 587.1 621.3 Nt[kN] 133.9 189.4 141.0 198.7 158.6 222.6 Nt,Rd [kN] 96.4 136.3 101.5 143.0 114.2 160.2 Ny,Rd [kN] 153.7 226.3 162.3 236.3 163.8 237.9 Nt,exp [kN] 147.3 208.0 146.8 209.3 147.6 208.9 Ny,exp [kN] 102.3 132.1 101.9 138.3 103.7 134.5 Nt,exp/Nt[%] 110.1 109.9 104.1 105.4 93.1 93.9 Nt,exp/Nt,Rd [%] 152.9 152.6 144.6 146.4 129.3 130.4 60 100 20 6 230 40 230 100 R20 R20 100 R5 15 5 5 R5 100 R2 18 2 2 R2 18 100 5 R5 30 5 6 and 8 R80 100 20 250 250 2020 500 60 20 2060 R5 20 2060 30 3040 15 250 250 55 500 25 2550 20 15 30 20 15 R10 R10 R10 R10 100 6 250 250 500 25 2550 4545 20 15 30 20 15 R10 R10 R10 R10 R10 R10 250 250 500 250 250 500 55 20 2060 22 250 250 500 2020 250 250 R80 60 250 250 500 V1 V2 V3 V3b V7 V8 V9 ROLLING DIRECTION 50 50 100 R5 R5 V10 500 100 6 250 250 500 25 2550 20 15 30 20 15 R10 R10 V5 100 6 250 250 500 25 2525 4545 R10 R10 R10 R10 25 V6 100 6 250 250 500 32 32 2626 R13 R13 36 V11 6 and 8 6 and 8 6 and 8 6 and 8 6 and 8 45° 25 4010 10 220 220 60 Figure 3.11: Geometries of the second set of specimens tested at BUT
3.2. EXPERIMENTAL INVESTIGATION 51 The second set of experiments was performed again at the AdMaS center at BUT using the same test machine. The experiments proceeded in the same manner as the first set, only the loading rate was decreased to 1 mm per minute. The measurements and testing were performed by Amina Hajdarevi´c, Michal ˇ Strba, Martin Vild, Mykola Lastovetskyi, Petr Danˇek, and Pavel Schmid. This is a very large set of experiments containing 11 geometries, two thicknesses (see Fig. 3.11), and four different nominal materials: S 235 (Note that the delivered specimens consisted of two distinct materials, labeled based on visual surface texture: S for smooth and R for rough. Tensile coupon tests were conducted for both materials.), S 355, S 460, and S 700. Every type with the same geometry, type, and thickness was manufactured and tested three times. In total, 204 specimens were tested. However, due to some issues with the test machine, several specimens had to be discarded. Figure 3.12: Load-deformation curves for specimen V6 and nominal thickness of 6 mm for all steel grades Examples of load-deformation curves for geometry type V6 with four bolt holes are shown in Fig. 3.12. The yield point is marked by circle. It was determined by fitting the initial stiffness, shifting it by 0.0005×gauge length equal to 220 mm and finding the intersection with load-deformation curve. The peak point is marked by square. Ultimate strength was likely achieved at slightly smaller load, but it cannot be experimentally recognized. The fracture is marked by a cross, and was determined by the gradient of −50 kN/mm, but for many cases manually overridden by the engineering judgment and alignment to the DIC measurements. Nevertheless, the real fracture initiation will be found later with the usage of numerical simulations. Fracture initiates where the numerical simulation with the true stress-strain material model begins to diverge – the experiment starts descending faster than the numerical simulation. The data can be visualized in the form of histograms. All geometry types and thicknesses are grouped together. The average engineering stress (calculated as the peak load divided by original measured area) at the peak load is shown in Fig. 3.13. Note that some specimens failed in pure tension, others in pure shear (V10) and others in combined tension and shear. The average peak stress is greatly affected by this. S 700 steel grade achieved by far the highest average stresses; the differences between other steel grades are not so apparent. Both S 235 and S 355 were very likely much stronger than their nominal strengths. The deformations at the ultimate load are shown in Fig. 3.14 and are very relevant to the
52 CHAPTER 3. DETERMINATION OF RELIABLE STRAIN LIMIT 200 400 600 800 1000 Peak stress [MPa] 0 5 10 15 20 25 30 35 40 Frequency All specimens V1 mean = 629 V2 mean = 673 V3 mean = 655 V3B mean = 636 V5 mean = 577 V6 mean = 436 V7 mean = 595 V8 mean = 581 V9 mean = 589 V10 mean = 140 V11 mean = 564 Figure 3.13: Histogram of peak stresses achieved for all specimens 2 4 6 8 10 12 Peak deformation [mm] 0 5 10 15 20 Frequency All specimens V1 mean = 7.463 V2 mean = 5.905 V3 mean = 4.343 V3B mean = 3.305 V5 mean = 3.511 V6 mean = 4.593 V7 mean = 3.823 V8 mean = 7.669 V9 mean = 8.120 V10 mean = 9.303 V11 mean = 5.272 Figure 3.14: Histogram of deformations achieved at peak load research of plastic strain limit in numerical design calculation. The deformations were measured at the gauge length of 220 mm by LVDT, therefore some very small portion of deformation is attributed to the elastic deformation of plates. The smallest deformations at peak load were generally achieved by plates made of S 700 and by geometries with the highest stress concentration factor (geometry V3b); see Tab. 3.4. Deformation at fracture determined by a gradient or manually measured at 220 mm gauge length is shown in a histogram in Fig. 3.15. The order of geometry types by fracture deformation is similar to that of deformations at the peak load. The exception is geometry type V10 loaded by pure shear, where all the specimens failed very shortly after reaching the peak load. Most of the specimen types and materials were measured by digital image correlation by Kirill Golubiatnikov and Vojtˇech Stanˇc´ık. Several outputs are in Fig. 3.16 and 3.17. From DIC measurements, several preliminary findings could be drawn: The peak strains in the plastic region were in the center of the weakened specimen and the fracture originated there for all steel grades for the geometries with low stress concentration factor, i.e. type V1 and V8.
3.2. EXPERIMENTAL INVESTIGATION 53 Table 3.4: Mean values of deformation at peak load for individual geometry types Geometry Mean deformation [mm] V1 7.463 V2 5.905 V3 4.343 V3b 3.305 V5 3.511 V6 4.593 V7 3.823 V8 7.669 V9 8.120 V10 9.303 V11 5.272 4 6 8 10 12 14 16 18 20 Fracture deformation [mm] 0 5 10 15 20 25 Frequency All specimens V1 mean = 11.888 V2 mean = 9.036 V3 mean = 5.808 V3B mean = 4.098 V5 mean = 5.590 V6 mean = 6.607 V7 mean = 4.745 V8 mean = 13.450 V9 mean = 10.622 V10 mean = 10.015 V11 mean = 7.560 Figure 3.15: Deformation at fracture (significant load gradient drop) The peak strains and fracture origin were located at the weakening edge for higher SCFinit, i.e. geometry types V2, V3, V3B. For V5 (two bolt holes) and V6 (four staggered holes), the strain peaks and fracture origins were located at the inner edge of the bolt holes, but interestingly, this switched to the outer edge of the bolt holes for V9 (six bolt holes). The stress concentration factor appears to govern the maximum strains measured by DIC immediately before fracture. The smallest strains occurred for geometry type V3B (SCFinit = 9.944; see Eq. (3.2)) around 0.1. Note that such weakening geometry is outside the scope of structural engineering detailing rules where the minimum notch radius should be 5 mm (EN 1090-2 – Cl. 6.7 [11]). The highest strains were reached for V1 (around 0.3) and for V8 (up to 0.5). The difference between steel grades is noticeable but not great. For several geometries, the strain at fracture initiation as measured by DIC was actually lower for S 355 than for S 700. Typically, the strains at fracture initiation were for S 700 lower by about 30 % than for other steel grades. The fracture area was captured by an electronic microscope by Karel Vaˇreka, a Ph.D. student at CEITEC at BUT. A comparison of micrographs with the same magnification of steel grades
54 CHAPTER 3. DETERMINATION OF RELIABLE STRAIN LIMIT Figure 3.16: Specimen V1 made of S235 steel grade captured by DIC Figure 3.17: Specimen V8 made of S460 steel grade captured by DIC S 355 and S 700 is shown in Fig. 3.18, 3.19, and 3.20. From these images, the following observations can be made: S 355 (left): –At small and medium magnifications, the fracture surface is irregular with multiple indentations and uneven topography. No clear layering is visible; the surface appears more homogeneous. –The fracture surface at large magnification is rough and irregular. Numerous dimples and microvoid coalescence are visible, which indicates a ductile fracture morphology. S 700 (right):
3.2. EXPERIMENTAL INVESTIGATION 55 Figure 3.18: Small magnification of fracture surfaces: Significant delamination is observed at each S 700 specimen Figure 3.19: Medium magnification of fracture surfaces –At small and medium magnifications, the cross-sectional view is with distinct layers. There are visible fracture separations between layers, suggesting a more complex mechanism, involving delamination. The structure looks more organized compared to S 355. –The fracture surface at large magnification is smoother and more layered. There are fewer dimples and voids compared to S355, which suggests a less ductile (more brittle) fracture morphology. This theme is repeated in all cases. Steel grade S 700 shows a distinctly different fracture pattern with delamination, while mild steel (S 235 to S 460) fractures with large uniform surfaces at an angle about 45◦from the longitudinal axis. These surfaces are switched suddenly several times; see Fig. 3.18. The experimental measurements, DIC data, and photographs will be subjected to further inves-
62 CHAPTER 3. DETERMINATION OF RELIABLE STRAIN LIMIT ratios fu/fy. Geometry type V7 (the highest SCFinit = 5.76, above-average triaxiality factor ηinit = 0.52; see Fig. 3.6) shows consistently the lowest design resistance of all geometry types. Otherwise, no clear correlation with stress concentration or triaxiality was found. 3.5.2 Partial safety factor γM2 The same procedure as for design resistance was used for the partial safety factor γM2. They were determined as the ratio of nominal resistance Anet ·futo the real possible resistance NR= f(M, G, U). The nominal resistances are calculated from the material curves in Fig. 3.25. The highest 0.1184 % of partial safety factors were discarded. Figure 3.29: Partial safety factors for net section failure for each steel grade and geometry type The maximum partial safety factor γM2 = 1.22 for geometry type V3. This confirms the Eurocode γM2 = 1.25 and is consistent with recommended γM2 = 1.23 from the literature [85, 86]. 3.6 Plastic strain limit for numerical design calculation Numerical design calculations should directly provide design resistance when nominal geometry and nominal material properties are used as an input. Different solid elements (20-node and 8node), shell elements (8-node and 4-node) and mesh densities (1.0×1.0tn, 0.5×0.5tn, 0.25 × 0.25tn, 0.2×0.2tn, where tnis the nominal thickness) were tested. Material nonlinear analysis was performed (small deflection theory). Section 3.5.1 found the reliable design resistance from a representative set of numerical-analytical simulations (see the right side of Fig. 3.4), these numerical design calculations allow finding the reliable plastic strain limit εlim (see the left side of Fig. 3.4). Note that in some cases, the numerical design calculation did not reach the reliable design resistance NR,d. In such instances, a conversion based on the deformation energy ratio [98] was applied; see steel grade S235 in Fig. 3.30. Such cases are not critical, i.e. causing minimal plastic strain limit, and will not be further discussed here. Numerical design calculation with the reliable plastic strain limit εlim provides at most the design resistance NR,d determined by Monte Carlo simulations. The reliable plastic strain limit is suggested in a form of multiplication of ultimate strain εu; see Eq. (2.20). The ultimate strains are εu= 20.8 %,16.5 %,and 8.9 % for steel grades S 235, S 355, and S 460, respectively. A reduction factor γX=εlim/εuis sought. Fig. 3.30 and 3.31 show that the element type nor the mesh density are relevant for the reliable reduction factor γX; see also Fig. 2.14. Surprisingly, the geometry types with low initial stress concentration factor determined the smallest γX. This is due to high thickness tolerances and critical material models with fu/fy= 1.1 ratios that cause low design resistance NR,d, and at the same time the strains in numerical design calculation being evenly distributed across the net section. The recommended plastic strain limit for numerical design calculation is: εlim =γX·εu= 0.25 ·εu(3.5)
3.6. PLASTIC STRAIN LIMIT FOR NUMERICAL DESIGN CALCULATION 63 Figure 3.30: Plastic strain limit and reduction factor γXfor geometry type V1 Figure 3.31: Plastic strain limits εlim and reduction factors γXfor all geometries, element types, and mesh densities The numerical design calculations with this plastic strain limit provide in the vast majority of samples smaller design resistance than the reliable design resistance NR,d. The experimental load resistances of specimens failing in net section [90, 93, 85, 94, 96, 97] were plotted against the numerical design calculations with the proposed equivalent plastic strain limit according to Eq. (3.5) in Fig. 3.32.
64 CHAPTER 3. DETERMINATION OF RELIABLE STRAIN LIMIT Figure 3.32: Reliability graph: comparison of numerical design calculations with the proposed plastic strain limit εlim with the experiments This proposed equivalent plastic strain limit must precede the values of fracture strain for structural steel from the available literature, e.g. [99]. This requirement is satisfied. 3.7 Outlook The proposed equivalent plastic strain limit is simple to use and provides similar values to 5 % for steel grades S 235, S 275, and S 355. The equivalent plastic strain limit will gradually decrease for higher steel grades – decreasing fu,nom/fy,nom. The author and colleagues from CTU will attempt to push this proposal into FprEN 1993-1-14 [5] and FprEN 1993-1-8 [28]. In next months, the BUT experiments will be included in the experimental database of net section failures and numerical design calculation models will be analyzed. The reliability of numerical design calculations with the proposed plastic strain limit for all specimens will be determined using the procedure in Section 2.6. Furthermore, numerical simulations will be built in Abaqus [100] for the BUT specimens to find the plastic strain at fracture initiation. This plastic strain must be higher than the suggested plastic strain limit for the numerical design calculation and will illustrate the safety of the proposal.
Chapter 4 Application In this chapter, the complete workflow is illustrated using the example of a column web panel in transverse compression. The current design methods, along with possible improvements, were presented in Section 2.1, followed by experimental research by the author and others in Section 2.2. The numerical model is described, and the process of a dataset creation is shown. The performance of the neural network with various settings is investigated. Finally, the reliabilities of code formulas, the new proposed method, and machine-learned predictions are evaluated. 4.1 Numerical model A numerical model in IDEA StatiCa Connection was used because through API, thousands of models can be generated and solved within a reasonable time. The trade-off is the model precision. The in-house-made four-node shell finite elements [101] are used for all plates. The rolled Isection is made of three plates, where the nodes of flange are directly connected to the nodes of the web. The number of elements dividing the flange along the cross-section therefore has to be even. The meshing follows the default settings of IDEA StatiCa Connection. The biggest member cross-section size his divided between 12 elements; see Fig. 4.1. The biggest side of plates is divided into 20 elements. The stiffeners are exempt from this rule for plates and instead follow the meshing rule for cross-sections. The minimum element size is set to 8 mm and maximum size to 50 mm. The software attempts to create elements as close to a square shape as possible. Figure 4.1: Biggest member cross-section size his height or width, whichever is bigger The model in IDEA StatiCa is created by a series of manufacturing operations; see Fig. 4.2. The top and bottom blocks are solid steel with a cross-section of 200×200 and a length of 1.25×h of shell elements plus 4×has condensed beam element. These blocks are intended to be as rigid as possible. Between the blocks and the I-section, spreading plates are placed with the cross-section 65
66 CHAPTER 4. APPLICATION Figure 4.2: Model in IDEA StatiCa Connection height of 30 mm and width equal to the cross-section width of the I-section. The thickness of these plates is one of the examined variables. Both blocks and the spreading plates are modeled with elastic material (fy= 4 000 MPa). Figure 4.3: The model of butt welds in IDEA StatiCa Connection The transition between the blocks, spreading plate, and I-section is modeled using butt welds, represented as links between the nodes with weights proportional to the weld spreading area, calculated as 0.6·tp, where tpis the edge plate thickness; see Fig. 4.3. For each model, materially nonlinear analysis (MNA) and linear buckling analysis (LBA) is solved; see Fig. 4.4. The failure criteria in this dataset do not follow the IDEA StatiCa defaults of 5% plastic strain limit for structural steel and 1% for high-strength steel. Instead, the conclusions
4.1. NUMERICAL MODEL 67 Figure 4.4: Buckling shape (left) and plastic strain (right) of current research into plastic strain limit shown in Chapter 3 are used, i.e., the plastic strain limit is calculated according to Eq. (2.21). The Stop at limit strain function is utilized, which means the load between the converging step (plastic strain is below the set limit) and the diverging step (plastic strain is above the set limit) is halved eight times. The achieved plastic strain at the last converged step is below but may still be quite far away from the set plastic strain limit, e.g. εpl = 3.52 % < εlim = 4.13%. Furthermore, it should be noted that the scale in the software’s 3D scene shows values averaged in nodes, which are typically even smaller. The values presented in the Check tables show values averaged in elements that are more suitable for strain check. Mesh sensitivity has already been observed by Pesek [37]. IDEA StatiCa model of the web panel in transverse compression is strongly dependent on mesh size; see Fig. 4.5 for an example of IPE 300 cross-section made of S 355 steel grade. Web height was divided into 6, 8, 12, 16, and 20 elements. The highest resistance is achieved with the coarsest mesh and the lowest resistance with the finest mesh. The theoretical plastic resistance at infinite mesh determined by MNA is Fpl,Rd = 224 kN. Accepting the 5 % error, 1.05 ·224 = 235 kN is the qualified plastic resistance. However, even the finest tested mesh with 20 elements per cross-section height reaches 262 kN. The theoretical critical force at infinite mesh determined by LBA is Fcr =αcr ·Fpl,Rd = 459 kN. Accepting the 5 % error, 1.05 ·459 = 482 kN is the qualified critical force. Again, even the finest tested mesh with 20 elements per cross-section height reaches 501 kN. The default value of 12 elements per cross-section height causes an error of 33 % in plastic resistance and 12 % in critical force compared to the infinite mesh. On the other hand, the model is simplified, the fillet radius is not modeled, nor does the fillet weld add any stiffness to the force distribution. These simplifications decrease the resistance against transverse compression. The
68 CHAPTER 4. APPLICATION y = 856.49x + 223.52 y = 723.78x + 458.9 0 100 200 300 400 500 600 700 0 0.05 0.1 0.15 0.2 Fpl,Rd or Fcr [kN] 1/N F_pl,Rd F_cr Linear (F_pl,Rd) Linear (F_cr) Figure 4.5: Mesh sensitivity for a web panel in transverse compression model – plastic resistance Fpl,Rd and critical force Fcr default value of 12 elements per cross-section height was found to be a good balance for hot-rolled cross-section and this value is used in the dataset creation. 4.2 Dataset creation The dataset is created as follows: The whole range of IPE, HEA, HEB (European), and W (USA) sections is used totalling 432 cross-sections. This approach was chosen to maintain the realistic ratios of cross-section dimensions; see Fig. 4.6. b / h tw / tf r / tf tf / h 0.5 1.0 1.5 2.0 2.5 min-max mean Figure 4.6: Histogram of cross-section geometry ratios European steel grades were chosen in this study; see Tab. 4.1. Relative distance from the unstiffened end x=e/h is generated in a log-spaced grid between 0.05 and 1. Log-spaced grid is chosen because it was observed that the load resistance changes rapidly with a small change near the unstiffened end but is stabilized with increasing distance x. Therefore, it is beneficial to have a denser population of data points in the region of large load resistance gradients – with small xnear the unstiffened end.
4.2. DATASET CREATION 69 Table 4.1: Steel grades and corresponding mechanical properties Steel Grade fy[MPa] fu[MPa] εuεlim S 235 235 360 20.8% 5.21% S 275 275 430 21.6% 5.41% S 355 355 490 16.5% 4.13% S 450 440 550 12.0% 3.00% S 420 MH/MLH 420 520 11.5% 2.88% S 460 MH/MLH 460 540 8.9% 2.22% S 620 Q/QL/QL1 620 700 6.9% 1.71% S 690 Q/QL/QL1 690 770 6.2% 1.56% S 500 MC 500 550 6.0% 1.50% S 550 MC 550 600 6.0% 1.50% S 650 MC 600 650 6.0% 1.50% S 700 MC (tle8 mm) 700 750 6.0% 1.50% S 700 MC (t>8 mm) 680 750 6.0% 1.50% Relative spreading plate thickness a=tp/tfis uniformly randomly distributed between a∈[1,5]. In this case, it is beneficial to have all values evenly populated. The dataset was selected to directly provide the design load resistance when using nominal material and geometric properties. It should be noted that the choice of properties would differ if the objective were to generate millions of simulations for a Monte Carlo analysis aimed at a reliability-based design approach. In such a case, it would be beneficial to treat yield strength and ultimate strength as random variables interlinked with each other, e.g. fu∈[1.1fy,1.6fy], because a uniform distribution is preferable to a step-wise distribution of feature values. The whole dataset is shown in histograms in Fig. 4.7. The variations in beam sizes can be seen. The small number of datapoints in the regions of very thick flanges (tf>80 mm) and very thick webs (tw>55 mm) will likely cause issues with predictions of neural network. It is useful to filter out such datapoints and reduce the range of validity. The relative end distance xalso has a decreasing trend, but that is by design and should not cause any troubles. The outputs – plastic resistance Fpl,Rd and especially Fcr may reach extremely high values, which cannot directly be used as targets for training a neural network, as such models work best with targets normalized to ranges [-1, 1] and of comparable order of magnitude. Therefore, some form of normalization needs to be devised for these variables. It is strictly needed to scale these outputs down and most likely also necessary to filter out the extremely high values. A viable alternative is using αcr – the multiplier of Fpl,Rd to obtain critical force Fcr. This value is basically already scaled by Fpl,Rd. Another option is to scale Fpl,Rd against some cross-sectional property, e.g. the web thickness multiplied by the flange thickness. The Python code utilizing IDEA StatiCa Connection API [47] that was used to model and calculate the dataset is published at a Github repository [102] along with the generated dataset. In total, 9 241 models were solved and the load resistance Fpl at strain limit as determined by MNA, critical force Fcr and αcr as determined by LBA were noted. The main limitations of this dataset are as follows: Buckling resistance is not provided and must be further calculated by analytical equations from the results of MNA and LBA Shell models are utilized, the effect of the fillet between the web and the flange is totally neglected. Range of validity is determined by the dataset. Each feature is divided into 30 bins, and bins containing fewer than 5 samples, along with all subsequent bins, are removed. Sparsely populated regions of the feature space would likely lead to poor predictions and degrade the performance of the training process. The remaining dataset defines the range of validity shown in Tab. 4.2. The range of validity of geometrical features may be exceeded provided the ratios plotted in Fig. 4.6 are maintained if the Similitude scaler is utilized.
70 CHAPTER 4. APPLICATION 250 500 750 1000 0 200 400 600 Beam depth [mm] 100 200 300 400 0 500 1000 Beam width [mm] 0 50 100 0 500 1000 1500 Flange thickness [mm] 0 25 50 75 100 0 500 1000 1500 Web thickness [mm] 10 20 30 0 500 1000 Fillet radius [mm] 0.25 0.50 0.75 1.00 0 500 1000 1500 Relative end distance x [-] 12345 0 100 200 300 Relative plate width a [-] 400 600 0 500 1000 1500 fy [MPa] 0.02 0.03 0.04 0.05 0 2000 4000 lim [-] 0 20000 40000 0 2000 4000 6000 Fpl , Rd [kN] 0 200000 400000 0 2500 5000 7500 F_cr [kN] 0 20 40 0 2000 4000 cr [-] Figure 4.7: Histogram of all features with outputs Table 4.2: Minimum and maximum values of input and output features after filtering Feature Minimum Maximum Depth h[mm] 80.0 1137.9 Width b[mm] 46.0 447.0 Flange thickness tf[mm] 4.9 89.9 Web thickness tw[mm] 3.8 51.3 Fillet radius rw[mm] 5.0 30.0 Relative end distance x0.05 1.0 Relative plate width a1.0 5.0 Yield strength fy235 690 Plastic strain limit εlim 0.015 0.054 Load resistance Fpl,Rd [kN] 34 13549 4.3 Dataset scaling Dataset scaling may be difficult to grasp only by engineering judgment. How could the predictions be better when only different units (e.g. MN instead of kN for the output of load resistance) are used? The neural networks are numerical optimizers; therefore, all inputs should have the same scale and outputs should be in a reasonable range. Large values lead to large gradients and therefore unstable training; small values lead to slow training. The benchmark case without any scaler is shown in Fig. 4.8. The predictions of this model are terrible clearly indicating that scalers are necessary and greatly improve model performance.
4.3. DATASET SCALING 71 0 2000 4000 6000 8000 10000 Epoch 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 Coefficient of Variation CoV Learning Rate 0.000 0.002 0.004 0.006 0.008 0.010 Learning Rate CoV and Learning Rate over Epochs (a) CoV and learning rate progress (b) Reliability graph Figure 4.8: Model performance without any scaler Similitude scaling is explained in Section 2.5.2. Its usage is shown in the following graph in Fig. 4.9 where cross-sectional dimensions are varying and the outputs in the form of yield resistance and critical force can be predicted by scaling. The dimensions of cross-section – beam depth h, beam width b, flange thickness tf, and web thickness tware all multiplied by a scaling factor. The plastic resistance Fpl,Rd and critical force Fcr are multiplied by the scaling factor on the power of two. Figure 4.9: Scaling dimensions perfectly proportionally allows prediction of plastic resistance and critical force It can be observed that for the factor up to 3, the scaling prediction fits the finite element analysis nearly perfectly. However, for larger factor, the prediction and calculation diverge. The reasons why similitude scaling may not work well are as follows: Yield strength of some grades is thickness-dependent – thicker plates may be assigned a lower yield strength. For some standards, such as EN 10025 [55], the yield strength for thickness above 40 mm is reduced. For other standards, such as EN 10149 [103], the yield strength is reduced for thickness above 8 mm [12]. Not all elements in the model are scaled – in the model for the column web in transverse
78 CHAPTER 4. APPLICATION 0.90 0.95 1.00 1.05 r FEA/ r DNN 0 2 4 6 8 10 12 14 Density Normal fit b = 0.983 V = 0.044 Lognormal fit (a) With Similitude scaling 0.85 0.90 0.95 1.00 1.05 r FEA/ r DNN 0 5 10 15 20 Density Normal fit b = 0.984 V = 0.044 Lognormal fit (b) Without Similitude scaling Figure 4.20: Histogram of dataset created by for cycles The DNN trained on this dataset is actually able to capture this numerical error. That is not desirable and is a sign of overfitting. Interestingly, this numerical error increases with increasing cross-section size. The relative errors calculated as Fmin/Fmax in the range x= [0.4,1.0] are 99.01%, 98.77%, 97.65%, 97.02%, 97.06%, 95.21% for IPE 100 to IPE 600, respectively. Because this numerical error reaches up to 5%, achieving higher levels of accuracy by DNN prediction does not seem feasible. This finding is relevant also for other datasets created by IDEA StatiCa Connection. -6% -4% -2% 0% 2% 4% 6% 8% 10% 12% 14% 16% 0 0.2 0.4 0.6 0.8 1 Relative error ( rt–re)/rt Relative distance x[-] Figure 4.21: The plot of relative prediction error to relative distance for IPE 600 Against expectations, Fig. 4.19 and 4.20 show that the DNN has troubles to perfectly imitate the behavior of dataset. Fig. 4.19 shows clear bands of results. The DNN overpredicts the plastic resistance for small relative distance x. The relative errors are the smallest for IPE 200 and IPE 300, but increase up to 14 % for IPE 400, 500, and 600. The relative error of prediction without the Similitude scaler plotted against relative distance xfor IPE 600 is shown in Fig. 4.21. There is not much of a difference between the usage of the Similitude scaler. The training set was intentionally created with high density of samples with small relative distance xexactly to prevent this issue; see Fig. 4.7. It is an unpleasant surprise that the predictions are not capturing the load decrease near the unstiffened end for some cross-sections well enough. On the other hand, the variation coefficient Vδ= 0.044 for both DNNs is small. 4.5.3 Dataset created in ANSYS software The aim of this section is to investigate the differences between the predictions trained on a crude numerical design calculation in IDEA StatiCa compared to the finely-meshed solid models created
4.5. PREDICTIONS OF DIFFERENT DATASETS 79 in ANSYS by Ondˇrej Peˇsek [37]. Note that the same ANSYS models were used in papers [25, 26, 24, 30] and their responses have been validated on physical experiments. Figure 4.22: Web and flange is divided between 4 solid elements across their thickness; beam length is equal to 4·h[37] 0 50 100 150 200 250 300 0 2 4 6 8 10 12 14 16 18 20 22 load [kN] vertical displacement [mm] T1 T2 T7 T8 numerical analysis, imperfection dw/200 numerical analysis, imperfection dw/400 numerical analysis, imperfection dw/1000 numerical analysis, imperfection dw/2000 Figure 4.23: Validation of ANSYS model – transverse load-displacement curves for IPE 200 [24] The mesh and von Mises stress for the base model of IPE 400 are shown in Fig. 4.22. The model is validated by using the measured true stress-strain curve and low geometric imperfections. The basic case of a web in transverse compression is validated on BUT tests described in [24]; see Fig. 4.23. Models with axial force are validated on tests by Kuhlmann [45] and with load near the unstiffened end on other BUT tests [30]. The models with the end plate have not been validated yet, but due to their simplicity, no issues are expected. In the following comparisons, ANSYS models utilize bilinear stress-strain curve with insignificant strain-hardening of E/1000 and plastic strain limit of εlim as suggested by EN 1993-1-5 [32]. Equivalent geometric imperfections equal to dw/200 as suggested by FprEN 1993-1-14 [5], where dwis the depth of straight portion of column web (clear depth). The dataset created in ANSYS brings two additional parameters for several models: (i) axial force in the member in the range n=NEd/Npl,Rd ∈[0.1,0.9], (ii) end plate with the thickness in the range tep/tf∈[0.5,2]. The presence of an axial force cannot be captured by the trained model. The end plate thickness was transformed into the relative plate width ausing the Eurocode assumption, where the distribution slope is 1:1, i.e. amod =a+ 2 ∗tep. Furthermore, two welded sections are introduced; one is small and thick-webbed, which may not be an issue, but the other is with the depth of h= 550 mm and web thickness of tw= 10 mm. Such high web slenderness is unusual among the hot-rolled sections. Also note that for the DNN predictions, relative distance xwas set to 1 (the highest value in the range of validity), although in ANSYS models, it was x= 2. It is assumed that any models with the distance over x > 1 provide the same load resistance as x= 1. Plastic resistance Fpl,Rd by MNA is investigated first. The histograms in Fig. 4.24 show high scatter, which requires further investigation. Understandably, the DNN cannot accurately predict the additional parameters.
80 CHAPTER 4. APPLICATION 0.8 1.0 1.2 1.4 1.6 r FEA/ r DNN 0 1 2 3 4 5 Density Normal fit b = 1.099 V = 0.152 Lognormal fit (a) With Similitude scaling 1.00 1.25 1.50 1.75 2.00 r FEA/ r DNN 0 1 2 3 4 Density Normal fit b = 1.165 V = 0.180 Lognormal fit (b) Without Similitude scaling Figure 4.24: Histogram of dataset created by validated ANSYS models – Fpl,Rd 0 200 400 600 800 1000 1200 1400 1600 Predicted [kN] 0 200 400 600 800 1000 1200 1400 1600 FEA [kN] Standard b =1.076, V =0.080 N b =1.070, V =0.166 End plate b =1.383, V =0.187 45 line (a) With Similitude scaling 0 200 400 600 800 1000 1200 1400 1600 Predicted [kN] 0 200 400 600 800 1000 1200 1400 1600 FEA [kN] Standard b =1.064, V =0.066 N b =0.976, V =0.107 End plate b =1.266, V =0.160 45 line (b) Without Similitude scaling Figure 4.25: Reliability graphs of dataset created by validated ANSYS models – plastic resistance by MNA, Fpl,Rd Fig. 4.25 shows reliability graph in several batches: Standard: Contains IPE and welded sections without any additional parameters. N: Contains the same cross-sections but loaded also by axial force n=NEd/Npl,Rd = {0.1,0.3,0.5,0.7,0.9} End plate: Contains the same cross-sections but with the transverse load acting through an end plate with the relative thickness of tep/tf={0.5,0.75,1.0,1.5,2.0} The Standard batch shows good alignment of DNN predictions towards ANSYS models. IDEA StatiCa models appear to be well calibrated. The exception is the welded section with high web slenderness (re= 902 kN). The DNN with the Similitude scaler grossly underestimated its resistance (rt= 545 kN), whereas DNN without the Similitude scaler overestimated it (re= 981 kN). This point disrupts the variation coefficient Vδfor Similitude scaler. The N batch shows that the predicted values are constant, equal to the standard case of n= 0. The load resistance determined by ANSYS progressively decreases with increasing axial load ratio n. Therefore, many points fall to the unsafe side, below 45◦line. Again, the welded section disrupts DNN results with the Similitude scaler, pushing bias beven above one.
4.5. PREDICTIONS OF DIFFERENT DATASETS 81 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1 2 3 4 5 Fpl,Rd (a) / Fpl,Rd (a=1) a= tp/tf IDEA StatiCa (a) IDEA StatiCa (End plate) ANSYS Figure 4.26: Load resistances by IDEA StatiCa and ANSYS for varying plate width aand added end plate thickness The End plate batch is the most interesting. The DNN predictions are extremely safe. There may be two reasons for this discrepancy; either the spreading angle 1:1 is not correct, or IDEA StatiCa Connection does not predict well high values of relative plate thickness a. This can be quickly verified by several calculations with IPE 300 cross-section in IDEA StatiCa Connection; see Fig. 4.26. The ratio of IDEA StatiCa resistance Fpl,Rd(a= 5) is only 8 % higher than Fpl,Rd(a= 1). On the other hand, ANSYS with end plate tep/tf= 2 carries 56 % more. An additional IDEA StatiCa model with an added end plate was created and plotted into the same graph. The shell model with end plate follows the same trend as ANSYS model. Critical buckling factor αcr by LBA is shown next. 0.6 0.8 1.0 1.2 r FEA/ r DNN 0 1 2 3 4 Density Normal fit b = 0.882 V = 0.232 Lognormal fit (a) With Similitude scaling 0.6 0.8 1.0 1.2 1.4 r FEA/ r DNN 0 1 2 3 4 5 Density Normal fit b = 0.888 V = 0.233 Lognormal fit (b) Without Similitude scaling Figure 4.27: Histogram of dataset created by validated ANSYS models – αcr Fig. 4.27 shows the histograms of αcr,F EA to αcr,DNN ratios. The bias bis smaller than one, meaning DNN predicts on average higher critical factor than determined by ANSYS, which is unsafe. Furthermore, the scatter is very large with Vδ= 0.233. Note that αcr =Fcr/Fpl,Rd, and Fpl,Rd is already not determined correctly and this error propagates also into αcr. Fig. 4.28 utilizes the same division into batches as Fig. 4.25. Standard batch shows good agreement of DNN predictions with ANSYS models. Batches N and End plate are not correctly captured. From the above-mentioned graphs, it can be seen that: (i) End plate thickness cannot be simulated by increased bearing plate thickness ain the IDEA StatiCa Connection model and therefore in trained DNNs, (b) additional parameter of normal force significantly disrupts the alignment of predictions to the ground truth (ANSYS model) despite Eurocode assumption that
82 CHAPTER 4. APPLICATION 100101 Predicted [-] 100 101 FEA [-] Standard b =1.015, V =0.064 N b =0.915, V =0.156 End plate b =0.658, V =0.167 45 line (a) With Similitude scaling 100101 Predicted [-] 100 101 FEA [-] Standard b =1.029, V =0.057 N b =0.926, V =0.147 End plate b =0.649, V =0.138 45 line (b) Without Similitude scaling Figure 4.28: Reliability graphs of dataset created by validated ANSYS models – buckling factor αcr (log scale) up to n= 0.7, no reduction is necessary, and (iii) unknown welded cross-section may cause serious issues, although all the geometric parameters are within the range of validity; see Tab. 4.2. Buckling resistance Fb,Rd can be calculated using geometrically and materially nonlinear analysis with imperfections (GMNIA) or estimated using General method in EN 1993-1-1 – Cl. 6.3.4 [4] or EN 1993-1-5 – Annex B [32]. Web buckling resistance was determined in ANSYS by GMNIA, while only predictions of Fpl,Rd and αcr were trained on MNA and LBA in IDEA StatiCa Connection. 0.7 0.8 0.9 1.0 1.1 r FEA/ r DNN 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Density Normal fit b = 1.005 V = 0.070 Lognormal fit 0 250 500 750 1000 1250 Predicted [kN] 0 200 400 600 800 1000 1200 1400 FEA [kN] r FEA/ r DNN n =153 b =1.005 V =0.070 45 line ±10% error Figure 4.29: The comparison of General method (rt) and GMNIA (re) using only ANSYS data At first, the test is performed to determine whether the General method using Eurocode buckling curves and the new proposal described in Section 2.1.1 is feasible; see Fig. 4.29. In some cases, the General method provides nearly the same buckling resistance factor, ρ– for standard cases including load near unstiffened end. However, the buckling resistance factor decreases with increasing axial load; for n= 0.9, the drop is down to 70 % and the General method does not follow the decreasing trend nearly as much – only down to ∼92 %. Although, it should be mentioned
4.5. PREDICTIONS OF DIFFERENT DATASETS 83 that for n= 0.7, the drop is not nearly as dramatic, and for n≤0.5, it is not noticeable. The end plate thickness is predicted better, but on the other hand, smaller buckling resistance is predicted. In sum, the General method in combination with the Eurocode buckling curve for the web panel in transverse compression and new proposed method may be used with the exception of extreme axial forces (n≥0.7). 0 200 400 600 800 1000 1200 Predicted [kN] 0 200 400 600 800 1000 1200 FEA [kN] Standard b =1.085, V =0.095 N b =1.036, V =0.268 End plate b =1.344, V =0.193 45 line (a) With Similitude scaling 0 200 400 600 800 1000 1200 Predicted [kN] 0 200 400 600 800 1000 1200 FEA [kN] Standard b =1.072, V =0.064 N b =0.933, V =0.213 End plate b =1.212, V =0.140 45 line (b) Without Similitude scaling Figure 4.30: Reliability graphs of dataset created by validated ANSYS models – GMNIA calculation in ANSYS vs General method using predicted Fpl,Rd and αcr Fig. 4.30 shows the comparison of buckling resistance Fb,Rd determined directly by GMNIA in ANSYS and the General method using the predicted plastic resistance Fpl,Rd and buckling factor αcr by DNN. The results for Standard cases and cases with low axial force have good alignment except for the welded section. It is concerning that such a poor prediction is obtained for a relatively standard welded crosssection. The welded cross-section has a large web slenderness – ratio of (h−2·tf)/tw. However, all the individual dimensions and this ratio was included in the training set – the range of validity of this ratio is (h−2·tf)/tw∈[3.19,62.52]. What is apparently outside the range of validity is the fillet radius. 545 400 500 600 700 800 900 1000 1100 0 10 20 30 Predicted Fpl,Rd [kN] Fillet radius [mm] Figure 4.31: Predictions of plastic resistance by DNN with the Similitude scaler for varying fillet radius Fig. 4.31 shows the predictions of the same welded cross-section with only one varying geometric parameter – fillet radius rw. As shown earlier, the calculations of IDEA StatiCa remain constant
84 CHAPTER 4. APPLICATION with varying rw. Also, the SHAP graphs (see Section 4.7) show that rwdoes not affect the predictions nearly at all. However, going outside the range of validity with the scaled rw,scaled = 6/550 ·200 = 2.18 mm <3.48 mm (minimum rw,scaled in the training dataset) by the Similitude scaler, the predictions are wildly modified. 4.6 Reliability This section presents the reliability of the current approach of the second generation Eurocode [28], the reliability of the proposed model by Bal´azs et al. [30] with an increased degree of fixity between the flange and web, better corresponding to the numerical models and also with a possibility to provide design resistance close to the unstiffened end (see Section 2.1.1), and finally the reliability of predictions made by the neural network described in the previous sections of this chapter. The procedure for reliability assessment is described in Section 2.6. 4.6.1 Eurocode reliability The experimental set of 91 specimens has to be reduced – specimens with load applied close to the unstiffened end tested by Bal´azs [30] are removed. A total of 85 specimens remain to be calculated according to Eurocode procedures. The ratios of experimental results to Eurocode predictions using measured material and geometrical (if available) properties are shown in Fig. 4.32 and 4.33. No experiment failed at a lower than calculated load with measured properties. It can be seen that some specimens failed at extremely large ratios, demonstrating the Eurocode prediction is highly conservative. Taking into account all tests, the bias calculated as the average of ratios is b= 1.462, the coefficient of variability of prediction errors is rather high, Vδ= 0.199, and by assuming variation of the basic variables Vrt = 0.07, as was used in Eurocode development, the coefficient of variation Vr= 0.211 is obtained. Theoretically, Vrt could be calculated more precisely, but with high Vδthis does not make any sense. The resulting safety factor is γM= 1.128, which is below the acceptance limit 1.15. 200 400 600 800 1000 1200 Measured [kN] 200 400 600 800 1000 1200 Experimental [kN] Experimental vs Measured n =85 b =1.462 V =0.199 45 line +/-10% error Figure 4.32: Reliability graph of all specimens – comparison of experiments to calculations with measured properties The histogram shows a log-normal rather than the normal distribution assumed in Eurocode reliability calculations. The tail-fitting should be applied. In Fig. 4.34, where the specimens with the highest Experimental/Measured ratios, re/re, are deleted one-by-one and γMis calculated for each step, the best γMis achieved at 43 removed specimens.
4.6. RELIABILITY 85 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Experimental / Measured 0.0 0.5 1.0 1.5 2.0 2.5 Density Normal = 1.46, = 0.31 Lognormal fit Figure 4.33: Experimental resistance at peak load vs. Eurocode calculation [28] 0 10 20 30 40 50 Tail fitting iteration 0.950 0.975 1.000 1.025 1.050 1.075 1.100 M M progression Figure 4.34: Tail fitting with safety factor evaluated for each iteration The resulting parameters of reliability assessment for the dataset of 44 removed specimens with the highest re/reare as follows: b= 1.228, Vδ= 0.071, Vr= 0.100, γM= 0.937, and the acceptance limit is 1.075. That means Eurocode formulas are well-calibrated even with a sufficient safety margin. However, half of the specimens removed by tail-fitting fail at a much higher load in the experiment than in the EC calculation. The formulas may clearly be improved. 200 400 600 800 1000 1200 Measured [kN] 200 400 600 800 1000 1200 Experimental [kN] Experimental vs Measured n =91 b =1.243 V =0.181 45 line +/-10% error Figure 4.35: Reliability graph of all specimens calculated by new proposal
86 CHAPTER 4. APPLICATION 4.6.2 Reliability of new method The new method proposed in Bal´azs [30] and shown in the second part of Section 2.1.1 allows the calculation of all 91 specimens including those with concentrated load near the unstiffened end. The reliability graph in Fig. 4.35 and the histogram in Fig. 4.36 show that the coefficient of variation is smaller. On the other hand, multiple specimens failed with a lower experimental resistance than determined by calculation, which indicates that the safety factor γM>1.0 must be applied. 0.8 1.0 1.2 1.4 1.6 Experimental / Measured 0.0 0.5 1.0 1.5 2.0 2.5 Density Normal = 1.24, = 0.22 Lognormal fit Figure 4.36: Experimental resistance at peak load vs. calculation according to the new proposal [30] Taking into account all specimens results into the following parameters of reliability assessment: b= 1.243 (significantly lower than EC), Vδ= 0.181 (slightly lower than EC), Vr= 0.194, γM= 1.226, and the acceptance limit is 1.15. 0 10 20 30 40 50 Tail fitting iteration 1.16 1.17 1.18 1.19 1.20 1.21 1.22 M M progression Figure 4.37: Tail fitting with safety factor evaluated for each iteration The histogram in Fig. 4.36 again indicates that tail-fitting may be beneficial. According to Fig. 4.37, removing 35 specimens results in the smallest safety factor. The resulting parameters of reliability assessment for the dataset of 35 removed specimens with the highest re/reare as follows: b= 1.098, Vδ= 0.119, Vr= 0.138, γM= 1.163, and the acceptance limit is 1.104. The safety factor is above the acceptance limit and the safety factor has to be applied. It can be calculated as the ratio of γMand the acceptance limit, i.e., the final γM= 1.163/1.104 = 1.053. However, some safety should be added for the potential compressive force in the column, which is typical. The Eurocode formulation decreases the resistance of the column web panel in transverse compression component only for loads at 70 % of the column plastic axial resistance and higher. The experiments by Kuhlmann and K¨uhnemund [45] suggest that the component resistance decreases even at smaller axial loads. Therefore, a higher safety factor, about γM≈1.15 should be applied for this new procedure. Alternatively, a safer assumption of resistance reduction for axial force in the column may be adopted, e.g. by Corman [105].
4.6. RELIABILITY 87 4.6.3 Reliability of neural network Reliability of predictions by DNN with and without the Similitude scaler is shown on a set of all experiments except for tests by Bougoffa [42, 41] that were welded with large patch loading (feature a) outside the range of validity; experimental dataset comprises 85 specimens. In both DNNs, only one specimen failed at a slightly lower load in the experiment than the DNN predictions with measured material and geometrical properties – Delft 5.1 [35] but only by 2% and 1% with and without the Similitude scaler, respectively. DNN was trained to predict plastic resistance, Fpl,R and buckling factor αcr. The final buckling resistance, Fb,R, which can be compared to the experimental resistance, must be calculated by the following procedure. The relative slenderness is calculated: ¯ λp=p1/αcr (4.1) For the load sufficiently far away from the unstiffened end (x > 0.5), the standard Eurocode [28] equation for the buckling reduction coefficient according to Eq.(2.3) is used. For the load near the unstiffened end, the new approach is used, i.e., Eq. (2.13) followed by Eq. (2.12) where λp0is calculated using Eq. 2.14. Finally, the buckling resistance of the web in transverse compression is calculated: Fb,R =ρ·Fpl,R (4.2) where ρis calculated according to Eq. (2.12) for x≤0.5 and according to Eq. (2.3) otherwise. Three buckling resistances are calculated as in previous reliability assessments: 1. Using measured material properties and measured geometric properties (if available) 2. Using nominal material and geometrical properties 3. Using mean material and geometrical properties Similitude scaler provides very conservative predictions, with bias b= 1.565 and relatively large scatter Vδ= 0.221; see Fig. 4.38. When considering all specimens, the partial safety factor γM= 1.100 is below the acceptance limit of 1.15. 200 400 600 800 1000 1200 Measured [kN] 200 400 600 800 1000 1200 Experimental [kN] Experimental vs Measured n =85 b =1.565 V =0.221 45 line +/-10% error Figure 4.38: Reliability graph of all specimens predicted by DNN with the Similitude scaler Fig. 4.39 shows again lognormal rather than normal distribution, and therefore the tail fitting is used. Fig. 4.40 shows that the best result is achieved after removing 26 specimens with the highest re/rtratio. The final bias b= 1.368 and variation coefficient Vδ= 0.112. Again, assuming
94 CHAPTER 4. APPLICATION Figure 4.47: SHAP scatter plot – SHAP impact of flange thickness and web thickness (Note that SHAP values are in a different scale in this graph) This trend suggests that thicker flanges generally lead to higher plastic resistance. The observed linearity is likely influenced by correlations with other geometric parameters in the dataset, such as web thickness, beam width, and overall depth, which tend to increase proportionally with flange thickness in the sampled beams. Such scatter plots are particularly useful to identify non-linear effects, thresholds, or saturation points in feature contributions, as well as potential interactions with other features. For instance, deviations from the linear trend may indicate regions where other parameters, material properties, or joint detailing start to dominate the resistance behavior. This enables more targeted interpretation of the model’s predictions and highlights areas where design optimization or additional experimental validation may be needed.
Chapter 5 Conclusion and future outlook In this thesis, a framework for developing a reliable machine-learned model was outlined. It covered all stages of the process, and potential challenges were always highlighted. It is expected that only numerical design calculations are able to provide enough data samples for the successful creation of machine-learned model; therefore, big attention is given to the process of modeling, verifying, and especially failure criteria, which is still unresolved within the development of the second generation of Eurocode. An example of a column web panel in transverse compression component that aligns with the latest research focus of the author is selected. Analytical methods were described, experimental results collected, and a large-scale dataset of numerical models was generated with IDEA StatiCa Connection. The process of verifying numerical design calculations was described with special attention on failure criterion C2 – a plastic strain limit. The procedure for reliability assessment using the current state-of-the-art was described and the reliability of four different methods was assessed using the experimental dataset. Machine learning is able to predict the training dataset with near-perfect accuracy, provided that a sufficient amount of data (several thousand samples) is available. The reliability of the machine-learned predictions is basically the same as that of samples used for training. There is a big potential in machine learning used in engineering practice for the following reasons: The precision of predictions is unparalleled. The coefficient of variation less than 2% is achievable. The speed of predictions is extremely fast, comparable to any calculation using code formulas. Machine-learned predictions may be used for a direct reliability approach using Monte Carlo simulations. The reliability of machine-learning predictions within a certain range of validity may be assessed in the same way as the reliability of any other approach. On the other hand, there are several critical issues that anyone using neural networks must keep in mind. The range of validity must be strictly kept. This applies not only to the boundaries of individual features but also to the ratios of feature values; for example, not only must the thickness of the web and flange lie within the range of validity, but also the ratio of web to flange thicknesses. The machine-learned model, if perfectly applied, copies the training dataset. If the training dataset has any shortcomings, they directly propagate into the machine-learned model. Dataset for training was created using IDEA StatiCa Connection, where thousands of models can be effortlessly calculated using Python programming language and IDEA StatiCa API. More than 9 000 models simulating the behavior of the column web in transverse compression component were calculated. The dataset comprises all hot-rolled Ior H-sections used in Europe and USA (IPE, HEA, HEB, and W), main steel grades used in Europe (S 235, S 275, S 355, S 450, S 420 MH/MLH, S 460 MH/MLH, S 620 Q/QL/QL1, S 690 Q/QL/QL1, S 500 MC, S 550 MC, S 600 MC, 95
96 CHAPTER 5. CONCLUSION AND FUTURE OUTLOOK S 650 MC, S 700 MC), varying distance to the unstiffened end, and varying width of plate inducing the transverse compression. IDEA StatiCa provides good approximation of load resistance for the standard cases, however, there were issues in the chosen strategy. The width of plate inducing the transverse compression has very little effect on the load resistance and wide plate cannot be used to simulate patch loading or loading through the end plate. For these reasons, the machine-learned predictions failed to improve the code formulas for web panel in transverse compression. However, after expanding the dataset by cases with end plate and with an axial force in the column, the variation coefficient of machine-learned predictions will become lower than code calculations. The geometrically and materially nonlinear analysis with imperfections, which would directly predict buckling resistance comparable to the experimental resistance, is only available in IDEA StatiCa Member without API. IDEA StatiCa Connection provides only materially nonlinear analysis and linear buckling analysis. The general method was utilized to estimate buckling resistance using the combination of both analyses. There are possibilities for creating the dataset with higher-quality models: Similitude scaler could be used for enlarging the training and testing dataset rather than being used in the input and output scaling. In principle, for every calculated numerical model, another hundred geometrically similar models could be created. Machine learning could also be involved in the creation of a dataset: –It was shown that the General method works well for some cases and badly in others. Machine learning could be utilized to predict the results of computationally demanding GMNIA from the results of MNA, LBA and other parameters, such as geometry and material. –Assume Abaqus or ANSYS models created with solid elements. There is very likely a correlation between models with coarse and fine mesh. Again, machine learning could predict the results of numerical models with mesh conforming to the mesh sensitivity conditions from the simple and quickly analyzed models with a coarse mesh. This way, models could be calculated in minutes instead of several hours. –Much as likely, there is a correlation between IDEA StatiCa shell models and detailed models finely meshed with solid elements. Approximating correctly the results of detailed models by machine learning could speed up the process of dataset creation even more – seconds per model. The machine-learned predictions achieve almost perfect alignment with the dataset used for training and testing – coefficient of variation equal to 0.0158 and 0.0134 for two different scaling strategies. This precision is only possible by optimizing the neural network architecture. Within the thesis, the effects of width (number of neurons in a layer) and depth (number of layers) were investigated together with learning rate, activation functions, optimizers, loss functions, and input and output scalers. The dataset size even for this simple model with only a few features (varying parameters) must be large – several thousand models. Each added model increases the prediction precision; however, this benefit diminishes with growing numbers. The coefficient of variation asymptotically approaches the numerical error of the finite element model. Active learning strategies were presented, but not applied. The performance of machine-learned predictions was tested on several other datasets – (i) full dataset including large cross-sections that were filtered out with the aim of observing model performance outside the range of validity, (ii) dataset created in a traditional manner using nested for cycles, which is suitable for human perception but may cause overfitting of neural networks, and (iii) dataset created in another software – ANSYS where the shortcomings of IDEA StatiCa shell models were identified. Building the numerical models with higher accuracy is an ideal task for Ph.D. students. These models should overcome the limitations of the numerical design calculations used in this thesis – missing geometrically nonlinear analysis with imperfections, neglected fillet radius, and insufficient end plate simulation. A sufficiently large database may serve as a training dataset for machine learning the foundations for improvements of current analytical models. Methods to enlarge the database by e.g. Similitude scaling or physically-informed methods could be investigated as well
97 as active learning. Such advancements may provide sufficiently large datasets for the creation of reliable machine learning models of structural components with much smaller deviations from numerical simulations and experiments. Reliability assessment in accordance with current Eurocodes [6], based on the SAFEBRICTILE project [15], was presented as a step-by-step procedure. This procedure can be applied to any method – analytical, numerical simulations, numerical design calculations, or machine-learned predictions. Reliability is a central topic of this thesis. It was evaluated for a Eurocode component, a new approach aimed at reducing Eurocode conservatism by increasing the degree of fixity between the web and the flange, and particularly for machine-learned predictions with and without the Similitude scaler. Achieving this is only possible through the collection of a large number of experimental results from the literature. Unfortunately, this is a challenging task, as many publications omit crucial information. There is room for improvement in verifying reported data and recovering missing information, such as measured geometrical properties. The reliability assessment procedure is suitable for sets of similar experiments involving similar materials. However, when multiple steel grades or different types of experiments are considered, the limitations of the method become apparent. For instance, a material model with a high plastic strain limit and significant strain hardening exhibits low variability between experimental and numerical resistances. Such a model may appear highly reliable, allowing the application of a small partial safety factor γM. Conversely, if the model is made objectively safer – by reducing the plastic strain limit or using a bilinear material model that neglects strain hardening – the variability increases substantially, particularly when different steel grades with varying yield-toultimate strength ratios are involved. Paradoxically, this more conservative model may show lower reliability than the model closely matching experimental data. Clearly, this is inconsistent, highlighting the need for improvements in the reliability assessment procedure. Data availability is more important today than ever. Collecting published data enables confident use of material models or assessment of manufacturing tolerances. Recent examples include the formulation of a characteristic material model by Yun and Gardner [7], distributions of material properties and thicknesses of plates and hot-rolled sections from the SAFEBRICTILE project [14], and bolt tests compiled by Ding and Elkadhy [107]. Published experimental data are essential for validating numerical models, which is of paramount importance. Just as machine learning models have a limited range of validity, modeling choices may perform well for one failure mode but poorly for another. Validation against a wide variety of experiments builds confidence and allows the development of sufficiently general solutions. The work by Ding and Elkadhy [107] in compiling bolt experiments highlights common issues with insufficient reporting in research papers and technical reports. Firstly, researchers often measure different and incompatible displacements. Secondly, published values are frequently postprocessed without clear explanation and presented only in graphs, making it difficult – or sometimes impossible – to extract key parameters such as initial stiffness. Out of potentially hundreds or thousands of bolt tests, only about 60 were found to be reported with sufficient detail. A potential solution is the development of unified databases of standardized tests, such as tensile coupon tests [10] or bolt tensile tests [108], which could gradually expand to include additional test types, such as connection component tests and end-plate connection tests. Such databases would encourage researchers to measure and report the necessary data while also providing guidance and shared experience for early-career researchers and laboratory personnel. Fields in which data sharing and proper documentation are standard—such as software engineering—demonstrate much higher productivity than civil engineering. Time is wasted on the literature reviews again and again. Deep neural networks are often considered black boxes, but in reality they are optimizers whose behavior can be interpreted. In this thesis, SHAP plots [106] were employed to facilitate this interpretation. These plots visualize the deviation of each prediction from the mean and allow assessment of the contribution of each input feature (parameter). Tools like SHAP enable a deeper understanding of neural network behavior. A critical aspect of using neural networks correctly is ensuring predictions remain within the range of validity. For machine-learned predictions to be safely adopted by the engineering com-
98 CHAPTER 5. CONCLUSION AND FUTURE OUTLOOK munity, engineers must be informed whether the input parameters fall within this valid range. Predictions outside the range of validity should not be provided, as they are likely to be unreliable or misleading. Neural networks are suitable for significant improvements in design formulas. The reliability of design resistances can be maintained while increasing the load resistance. Thus, significant savings in material are possible.
Bibliography [1] EN 1993-1-8:2005 – Eurocode 3: Design of steel structures – Part 1-8: Design of joints. CEN. Brussels, 2005. [2] K. Weynand et al. Proposal for implementation of the component method for tubular joints in EN 1993-1-8. Tech. rep. CIDECT, Oct. 2015. [3] Eurocode No.3 – Design of Steel Structures – Part 1 - General Rules and Rules for Buildings, Background Documentation D.04 – Statistical analysis of strength functions for welded Hsection joint with respect fo available experimental data. Tech. rep. Eurocode 3 Editorial Group, Nov. 1989. [4] EN 1993-1-1:2005 – Eurocode 3: Design of steel structures – Part 1-1: General rules and rules for buildings. CEN. Brussels, 2005. [5] FprEN 1993-1-14:2024 – Eurocode 3: Design of steel structures – Part 1-14: Design assisted by finite element analysis. CEN. Brussels, 2024. [6] EN 1990 – Eurocode – Basis of structural design. CEN. Brussels, 2002. [7] Xiang Yun and Leroy Gardner. “Stress-strain curves for hot-rolled steels”. In: Journal of Constructional Steel Research 133 (2017), pp. 36–46. issn: 0143-974X. doi:https://doi. org/10.1016/j.jcsr.2017.01.024.url:https://www.sciencedirect.com/science/ article/pii/S0143974X16305181. [8] Adam J. Sadowski, J. Michael Rotter, and Thomas Ummenhofer. “12.13: On recent characterisations of the post-yield properties of structural carbon steels”. In: ce/papers 1.2-3 (2017), pp. 3577–3583. doi:https://doi.org/10.1002/cepa.413. eprint: https:// onlinelibrary.wiley.com/doi/pdf/10.1002/cepa.413.url:https://onlinelibrary. wiley.com/doi/abs/10.1002/cepa.413. [9] Leroy Gardner and Xiang Yun. “Description of stress-strain curves for cold-formed steels”. In: Construction and Building Materials 189 (2018), pp. 527–538. issn: 0950-0618. doi: https : / / doi . org / 10 . 1016 / j . conbuildmat . 2018 . 08 . 195.url:https : / / www . sciencedirect.com/science/article/pii/S0950061818321317. [10] EN ISO 6892-1:2019 – Metallic materials – Tensile testing – Part 1: Method of test at room temperature. CEN. Brussels, 2019. [11] EN 1090 – Execution of steel structures and aluminium structures – Part2: Technical requirements for steel structures. CEN. Brussels, 2008. [12] FprEN 1993-1-1:2024 – Eurocode 3: Design of steel structures – Part 1-1: General rules and rules for buildings. CEN. Brussels, 2024. [13] F.S.K. Bijlaard, G. Sedlacek, and J.W.B. Stark. Procedure for the determinantion of design resistance from tests. Tech. rep. TNO Institute for Building Materials and Structures, June 1988. [14] L. da Silva et al. Standardization of Safety Assessment Procedures across Brittle to Ductile Failure Modes (SAFEBRICTILE). Tech. rep. Research Fund for Coal and Steel, June 2016. [15] A. Taras et al. Standardization of Safety Assessment Procedures across Brittle to Ductile Failure Modes (SAFEBRICTILE) – Deliverable D1.1: Guideline for Safety Assessment of Design Rules for Steel Structures in Line with EN 1990. Tech. rep. Research Fund for Coal and Steel, June 2016. [16] PyTorch Documentation. PyTorch Foundation. 2025. url:https://docs.pytorch.org/ docs/stable/index.html. 99
100 BIBLIOGRAPHY [17] OpenAI. ChatGPT. Large language model, GPT-5, accessed on 2025-08-29. 2025. url: https://chat.openai.com/. [18] American Institute of Steel Construction. Specification for Structural Steel Buildings (ANSI/AISC 360-22). Chicago, IL: AISC, 2022. url:https://www.aisc.org/products/ publication/standards/aisc-360/specification-for-structural-steel-buildingsansiaisc-360-16-download2/. [19] Frantiˇsek Wald et al. Component-Based Finite Element Design of Steel Connections. Czech Technical University in Prague, 2021. isbn: 9788001068625. url:https://payhip.com/b/ p0Hr. [20] Mark D. Denavit et al. Steel Connection Design by Inelastic Analysis. Hoboken, NJ: Wiley, 2024. isbn: 9781394222162. url:https://www.wiley.com/en-us/Steel+Connection+ Design+by+Inelastic+Analysis-p-9781394222155. [21] IDEA StatiCa. IDEA StatiCa Support Center – Verifications.https://www.ideastatica. com/support-center-verifications?product=steel. Accessed on 2025-08-29. 2025. [22] IDEA StatiCa. Web in Transverse Compression. 2024. url:https://www.ideastatica. com/support-center/web-in-transverse-compression (visited on 06/19/2025). [23] ANSYS, Inc. ANSYS Mechanical User’s Guide. Release 2024 R1. ANSYS, Inc. Canonsburg, PA, USA, 2024. url:https://www.ansys.com (visited on 06/19/2025). [24] Ivan Bal´azs et al. “Numerical and experimental verification of column web in transverse compression”. In: Thin-Walled Structures 203 (2024). Cited by: 0; All Open Access, Hybrid Gold Open Access. doi:10.1016/j.tws.2024.112267. [25] Ivan Bal´azs et al. “Geometrically and materially nonlinear analysis of selected aspects influencing resistance of column webs in transverse compression”. In: vol. 2928. 1. Cited by: 0. 2023. doi:10.1063/5.0170547. [26] Ivan Bal´azs et al. “Resistance of Column Webs Subjected to Transverse Compression at the Unstiffened End”. In: vol. 2950. 1. Cited by: 0. 2023. doi:10.1063/5.0180866. [27] Jorge Conde et al. “Welded beam-to-column steel joints: Assessment of European design rules”. In: Thin-Walled Structures 199 (2024). Cited by: 5; All Open Access, Green Open Access, Hybrid Gold Open Access. doi:10.1016/j.tws.2024.111844. [28] EN 1993-1-8:2024 – Eurocode 3: Design of steel structures – Part 1-8: Design of joints. CEN. Brussels, 2024. [29] V. H. Cochrane. “Rules for Rivet-Hole Deduction in Tension Members”. In: Engineering News-Record 89.20 (1922), pp. 847–848. [30] Ivan Bal´azs et al. “Experimental and Numerical Analysis of Column Web Subjected to Transverse Compression at the Unstiffened End”. Manuscript submitted for publication. 2025. [31] B. Johansson et al. Commentary and worked examples to EN 1993-1-5 “Plated structural elements”. Luxembourg: Office for Official Publications of the European Communities, 2007. [32] EN 1993-1-5:2006 – Eurocode 3: Design of steel structures – Part 1-5: Plated structural elements. CEN. Brussels, 2006. [33] Harald Unterweger and Andreas Taras. “Steifenlose Krafteinleitung bei Biegetr¨agern”. In: Stahlbau 84.6 (2015), pp. 435–448. doi:https://doi.org/10.1002/stab.201510279. eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/stab.201510279.url: https://onlinelibrary.wiley.com/doi/abs/10.1002/stab.201510279. [34] Tudor Golea, Jean-Pierre Jaspart, and Jean-Fran¸cois Demonceau. “Characterisation of the Behaviour of Beam-to-Column Steel Joints up to Failure”. In: Advanced Steel Construction 18.3 (2022), pp. 679–686. doi:10. 18057/ IJASC .2022 .18. 3. 5.url:https :/ /www. ascjournal.com/index.php?option=com_content&view=article&id=608:vol18no35&catid=153:vol18no3&Itemid=538. [35] J. M. Aribert, A. Lachal, and M. Moheissen. “Modelling and experimental investigation of plastic resistance and local buckling of H or I steel sections submitted to concentrated or partially distributed loading”. In: Contact Loading and Local Effects in Thin-Walled Plated and Shell Structures. IUTAM Symposium Prague. Berlin: Springer-Verlag, 1992, pp. 101– 110.
BIBLIOGRAPHY 101 [36] George Winter. “Strength of Thin Steel Compression Flanges”. In: Transactions of the American Society of Civil Engineers 112.1 (1947), pp. 527–554. doi:10.1061/TACEAT. 0006092. eprint: https://ascelibrary.org/doi/pdf/10.1061/TACEAT.0006092.url: https://ascelibrary.org/doi/abs/10.1061/TACEAT.0006092. [37] Ondˇrej Peˇsek, Ivan Bal´azs, and Martin Hor´aˇcek. Verification of numerical model of I-beam in IDEA StatiCa Steel. Tech. rep. Institute of Metal and Timber Structures, Faculty of Civil Engineering, Brno University of Technology, Oct. 2022. [38] S. P. Timoshenko and J. M. Gere. Theory of Elastic Stability. 2nd. New York: McGraw-Hill, 1985. [39] Jean-Pierre Jaspart, Adrien Corman, and Jean-Fran¸cois Demonceau. “Mechanical Properties of the Component “Column Web in Compression” in Steel Beam-to-column Joints”. In: ce/papers 5.4 (2022), pp. 242–250. doi:https : / / doi . org / 10 . 1002 / cepa . 1752. eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/cepa.1752.url:https: //onlinelibrary.wiley.com/doi/abs/10.1002/cepa.1752. [40] Pauli Virtanen et al. “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python”. In: Nature Methods 17.3 (2020), pp. 261–272. doi:10.1038/s41592-019-0686-2. url:https://doi.org/10.1038/s41592-019-0686-2. [41] Sabra Bougoffa et al. “Full Length Transverse Stiffener Under Compression”. In: ce/papers 5.4 (2022), pp. 967–973. doi:https : / / doi . org / 10 . 1002 / cepa . 1841. eprint: https : / / onlinelibrary . wiley . com / doi / pdf / 10 . 1002 / cepa . 1841.url:https : //onlinelibrary.wiley.com/doi/abs/10.1002/cepa.1841. [42] Sabra Bougoffa et al. “Compression zone with partial stiffeners in beam-to-column steel connections”. In: Engineering Structures 229 (2021), p. 111674. issn: 0141-0296. doi:https: //doi.org/10.1016/j.engstruct.2020.111674.url:https://www.sciencedirect. com/science/article/pii/S0141029620342759. [43] Javier Alvarez Rodilla and Keith Kowalkowski. “Determination of Capacities of Eccentric Stiffeners Part 1: Experimental Studies”. In: Engineering Journal 58.2 (2021), pp. 79–98. doi:10.62913/engj.v58i2.1176.url:https://doi.org/10.62913/engj.v58i2.1176. [44] L. De Mita, V. Piluso, and G. Rizzano. “Theoretical and Experimental Analysis of Column Web in Compression”. In: The Open Construction and Building Technology Journal 2 (2008), pp. 294–303. issn: 1874-8368. doi:10.2174/1874836800802010294.url:https: //doi.org/10.2174/1874836800802010294. [45] Ulrike Kuhlmann and Frank K¨uhnemund. Proposal of a new design resistance of the joint component ”column web in compression”. Universit¨at Stuttgart, Institut f¨ur Konstruktion und Entwurf, 2001. [46] B. Bose. “Design Resistance of Unstiffened Column Web Subject to Transverse Compression in Beam-to-Column Joints”. In: Journal of Constructional Steel Research 45.1 (1998), pp. 1– 15. issn: 0143-974X. doi:10.1016/S0143-974X(97)00065-5.url:https://doi.org/10. 1016/S0143-974X(97)00065-5. [47] IDEA StatiCa. IDEA StatiCa Developer Site: API Documentation and Resources. Accessed on 2025-08-29. 2025. url:https://developer.ideastatica.com/. [48] CEN/TR 1993-1-141:2024 – Eurocode 3: Design of steel structures – Part 1-141: Background and Explanations on EN 1993-1-14 Design assisted by finite element analysis. CEN. Brussels, 2024. [49] Luis Sim˜oes da Silva et al. Moment Resistance of Strong-Axis Open-Section Beam-toColumn Welded Steel Joints. Tech. rep. ISISE UC – Institute for Sustainability and Innovation in Structural Engineering, June 2024. [50] Lu´ıs Sim˜oes da Silva et al. “FE-based assessment of the resistance of steel joints using strain averaging”. In: Thin-Walled Structures (2025), p. 114038. issn: 0263-8231. doi:https: //doi.org/10.1016/j.tws.2025.114038.url:https://www.sciencedirect.com/ science/article/pii/S0263823125011279.
102 BIBLIOGRAPHY [51] F. Walport, L. Gardner, and D.A. Nethercot. “Equivalent bow imperfections for use in design by second order inelastic analysis”. In: Structures 26 (2020), pp. 670–685. issn: 2352-0124. doi:https: / /doi. org/10 .1016/ j.istruc . 2020.03 .065.url:https: //www.sciencedirect.com/science/article/pii/S2352012420301478. [52] Jaap Wardenier et al. Design Guide for Circular Hollow Section (CHS) Joints under Predominantly Static Loading. Construction with Hollow Steel Sections, Guide No. 1. Ed. by Comit´e International pour le D´eveloppement et l’´ Etude de la Construction Tubulaire (CIDECT). 2nd ed. Stuttgart, Germany: LSS Verlag, 2008. isbn: 978-3-938817-03-2. [53] Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. 2nd. Springer, 2009. isbn: 978-0-387-84858-7. [54] European Committee for Standardization. EN 10365:2017 - Hot rolled steel channels, I and H sections – Dimensions and tolerances. CEN, Brussels. 2017. [55] European Committee for Standardization. EN 10025-2:2019 - Hot rolled products of structural steels – Part 2: Technical delivery conditions for non-alloy structural steels. CEN, Brussels. 2019. [56] IDEA StatiCa Connection Library.https : / / connectionlibrary . ideastatica . com/. Accessed: 2025-08-06; Cloud-based database of over 1 000 000 real-world steel connection designs. IDEA StatiCa, 2025. [57] Adam Paszke et al. “PyTorch: An imperative style, high-performance deep learning library”. In: Advances in Neural Information Processing Systems 32 (2019). [58] Christopher M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006. [59] Warren S. McCulloch and Walter Pitts. “A Logical Calculus of the Ideas Immanent in Nervous Activity”. In: The Bulletin of Mathematical Biophysics 5 (1943), pp. 115–133. doi: 10.1007/BF02478259. [60] Donald O. Hebb. The Organization of Behavior: A Neuropsychological Theory. New York: Wiley, 1949. [61] Marvin Minsky and Seymour A. Papert. Perceptrons: An Introduction to Computational Geometry. Reissue of the 1988 Expanded Edition with a new foreword by L´eon Bottou. Cambridge, MA: MIT Press, 2017. isbn: 9780262534772. [62] A. M¨uller and A. Taras. “Prediction of the local buckling strength and load-displacement behaviour of SHS and RHS members using Deep Neural Networks (DNN)–Introduction to the Deep Neural Network Direct Stiffness Method (DNN-DSM)”. In: Steel Construction 15.S1 (2022). Special Issue, pp. 78–90. [63] A. M¨uller, A. Taras, and M. A. Kraus. “Scientific Machine and Deep Learning Investigations of the Local Buckling Behaviour of Hollow Sections”. In: ce/papers 5.4 (2022), pp. 1034– 1042. doi:10.1002/cepa.1848.url:https://doi.org/10.1002/cepa.1848. [64] A. M¨uller and A. Taras. “Prediction of the deformation and local buckling behavior of structural systems using the Deep Neural Network Direct Stiffness Method (DNN-DSM)”. In: Proceedings of the Annual Stability Conference, Structural Stability Research Council. Online. Charlotte, North Carolina, 2023, p. 18. [65] E. Buckingham. “On physically similar systems; illustrations of the use of dimensional equations”. In: Physical Review 4.4 (1914), pp. 345–376. doi:10.1103/PhysRev.4.345. [66] Zdenek P. Bazant and Jordi Planas. Fracture and Size Effect in Concrete and Other Quasibrittle Materials. Boca Raton, FL: CRC Press, 1998. isbn: 084938284X. [67] John Duchi, Elad Hazan, and Yoram Singer. “Adaptive subgradient methods for online learning and stochastic optimization”. In: Journal of Machine Learning Research 12.7 (2011), pp. 2121–2159. [68] Geoffrey Hinton. Lecture 6e rmsprop: Divide the gradient by a running average of its recent magnitude. Coursera: Neural Networks for Machine Learning. Lecture slides. 2012. [69] Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization.https: //arxiv.org/pdf/1412.6980. arXiv preprint arXiv:1412.6980. 2014. [70] Ilya Loshchilov and Frank Hutter. “Decoupled Weight Decay Regularization”. In: International Conference on Learning Representations (ICLR). 2019.
BIBLIOGRAPHY 103 [71] Vinod Nair and Geoffrey E Hinton. “Rectified linear units improve restricted Boltzmann machines”. In: Proceedings of the 27th International Conference on Machine Learning (ICML). 2010, pp. 807–814. [72] Dan Hendrycks and Kevin Gimpel. “Gaussian Error Linear Units (GELUs)”. In: arXiv preprint arXiv:1606.08415 (2016). [73] Peter J. Huber. “Robust estimation of a location parameter”. In: Annals of Mathematical Statistics 35.1 (1964), pp. 73–101. doi:10.1214/aoms/1177703732. [74] Fabian Pedregosa et al. “Scikit-learn: Machine learning in Python”. In: Journal of Machine Learning Research 12 (2011), pp. 2825–2830. [75] John D Hunter. “Matplotlib: A 2D graphics environment”. In: Computing in Science & Engineering 9.3 (2007), pp. 90–95. [76] Robert E. Melchers and Andr´e T. Beck. “Integration and Simulation Methods”. In: Structural Reliability Analysis and Prediction. John Wiley and Sons, Ltd, 2017. Chap. 3, pp. 63– 93. isbn: 9781119266105. doi:10.1002/9781119266105.ch3. [77] Filip Ljubinkovi´c Dr et al. “Towards Stochastic Characterization of Welded Beam-to-Column Steel Joints Using Artificial Neural Networks”. In: Structural Engineering International 0.0 (2025), pp. 1–17. doi:10.1080/10168664.2025.2526010. eprint: https://doi.org/ 10.1080/10168664.2025.2526010.url:https://doi.org/10.1080/10168664.2025. 2526010. [78] David Lehk´y, Luk´aˇs Nov´ak, and Drahom´ır Nov´ak. “Surrogate Modeling for Stochastic Assessment of Engineering Structures”. In: Machine Learning, Optimization, and Data Science. Ed. by Giuseppe Nicosia et al. Cham: Springer Nature Switzerland, 2023, pp. 388–401. isbn: 978-3-031-25891-6. [79] David Lehk´y and Martina ˇ Somod´ıkov´a. “Reliability calculation of time-consuming problems using a small-sample artificial neural network-based response surface method”. In: Neural Computing and Applications 28.6 (2017), pp. 1249–1263. doi:10.1007/s00521-016-24853.url:https://link.springer.com/article/10.1007/s00521-016-2485-3. [80] European Commission et al. Modern plastic design for steel structures. Publications Office, 2010. doi:10.2777/91747. [81] Yingbin Bao and Tomasz Wierzbicki. “On fracture locus in the equivalent strain and stress triaxiality space”. In: International Journal of Mechanical Sciences 46.1 (2004), pp. 81– 98. issn: 0020-7403. doi:https://doi.org/10.1016/j.ijmecsci.2004.02.006.url: https://www.sciencedirect.com/science/article/pii/S0020740304000360. [82] Marvin Nahrmann and Anton Matzenmiller. “Modelling of nonlocal damage and failure in ductile steel sheets under multiaxial loading”. In: International Journal of Solids and Structures 232 (2021), p. 111166. doi:10.1016/j.ijsolstr.2021.111166.url:https: //doi.org/10.1016/j.ijsolstr.2021.111166. [83] Helen Bartsch et al. “Damage mechanics based failure prediction for structures under seismic action”. In: Journal of Constructional Steel Research 173 (2020), p. 106264. issn: 0143974X. doi:https://doi.org/10.1016/j.jcsr.2020.106264.url:https://www. sciencedirect.com/science/article/pii/S0143974X20308166. [84] Franc Sinur and Darko Beg. “Reliability analysis of net cross-section resistance with accidental eccentricity of holes”. In: International Journal of Steel Structures 9.2 (2009), pp. 153– 160. doi:10.1007/BF03249490.url:https://doi.org/10.1007/BF03249490. [85] H.H. Snijder, R.W.A. Dekker, and P.A. Teeuwen. “13.01: Net cross-section failure of steel plates at bolt holes: Numerical work and statistical assessment of design rules”. In: ce/papers 1.2-3 (2017), pp. 3679–3688. doi:https://doi.org/10.1002/cepa.424. eprint: https:// onlinelibrary.wiley.com/doi/pdf/10.1002/cepa.424.url:https://onlinelibrary. wiley.com/doi/abs/10.1002/cepa.424. [86] Hubertus H. Snijder et al. “Safety Assessment for Capacity Design of Bolted Steel Connections in Tension”. In: ce/papers 5.4 (2022), pp. 282–289. doi:https://doi.org/10.1002/ cepa.1757. eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/cepa.1757. url:https://onlinelibrary.wiley.com/doi/abs/10.1002/cepa.1757.
110 LIST OF FIGURES
List of Tables 2.1 Overview of experimental programs on column web in compression . . . . . . . . . 24 2.2 Summarization of resistance ratios (MNA, GMNIA) and critical loads (LBA) [37] . 25 2.3 Strain limits according to Eurocode 3 parts (A. Taras, personal communication, Sep.20,2025)....................................... 28 2.4 Overview of common data scaling and transformation methods . . . . . . . . . . . 31 2.5 Overview of selected gradient-based optimizers . . . . . . . . . . . . . . . . . . . . 32 2.6 Overview of Activation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.7 Recommended acceptance limits of γM[15] ...................... 40 3.1 Material properties of used steel from tensile coupon tests [10] . . . . . . . . . . . . 47 3.2 Geometric and material properties of specimens . . . . . . . . . . . . . . . . . . . . 48 3.3 Comparison of analytical, design, and experimental tensile resistances . . . . . . . 50 3.4 Mean values of deformation at peak load for individual geometry types . . . . . . . 53 3.5 Values of parameters p1and p2for geometry types V1–V9 [91] . . . . . . . . . . . 60 4.1 Steel grades and corresponding mechanical properties . . . . . . . . . . . . . . . . 69 4.2 Minimum and maximum values of input and output features after filtering . . . . . 70 4.3 Bias, CoV, and the worst predictions for scalers with and without Similitude scaling forfiltereddataset .................................... 76 4.4 Comparison of statistical parameters before and after tail fitting . . . . . . . . . . 89 4.5 Ratio re/rtfor experimental dataset – All approaches . . . . . . . . . . . . . . . . 90 111