Plane Thermoelastic Waves in Infinite Half-Space Caused FACTA UNIVERSITATIS Series: Mechanical Engineering https://doi.org/10.22190/FUME240110026S © 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper ANN AND HYBRID MODELLING OF ECCENTRICALLY PATCH LOADED STEEL I-GIRDERS Biljana Šćepanović1, Snežana Rutešić1, Olga Mijušković1, Luisa María Gil-Martín2, Enrique Hernández-Montes2 1University of Montenegro, Podgorica, Montenegro 2University of Granada, Granada, Spain Abstract. Majority of eccentrically patch loaded girders collapse at lower load level than their geometrical equivalents loaded in the web plane, due to different collapse mechanism. However, some eccentrically patch loaded girders, even with significant load eccentricity, behave as if loaded in the web plane, having the same collapse mode and ultimate load as in the case of centric load. Hence, the ultimate strength of eccentrically patch loaded steel I-girders should be found out in two phases: firstly, girder collapse mode should be estimated; secondly, depending on expected collapse mode, ultimate load should be calculated appropriately. Both tasks are demanding, with plenty of mutually dependent influential parameters. Artificial neural network (ANN) modelling, being suitable for multi-parameter analysis, is quite convenient method for studying eccentrically patch loaded steel I-girders in both mentioned tasks, as confirmed through the examples elaborated in the paper. Not only that it is valuable on its own, as a standalone technique, but also in combination with and as support to other methods (hybrid modelling). In comparison of five procedures for ultimate load determination (empirical expression, mechanical model, two versions of refined mechanical model, one of them by ANN, and standalone ANN forecast model), ANN modelling, providing high quality results, qualified in top-two procedures for ultimate load, regardless of collapse mode type. It is also proved that certain shortages of mechanical model may be overcome by its coupling with ANN modelling. Such hybrid modelling provided remarkably more accurate results than original mechanical model. Study confirmed need of revision of mechanical model. As the first step, new, more demanding constraints of mechanical model validity are proposed in this paper. Key words: Eccentric patch load, Collapse mode, Ultimate load, Artificial neural network, Hybrid modelling Received: January 10, 2024 / Accepted March 15, 2024 Corresponding author: Biljana Šćepanović University of Montenegro, Faculty of Civ. Eng, Bul. Džordža Vašingtona bb, 81000 Podgorica, Montenegro E-mail: biljanas@ucg.ac.me, biljazs38@gmail.com 2 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES 1. INTRODUCTION The term “patch load” designates the load acting over a small area or length of a structural element, as a concentrated or locally distributed load, Fig. 1. A typical situation in structural engineering practice is when compressive patch load affects the flange of I- profile so that the web is locally pressed in the zone under the loading. Local compression might provoke local instability that may result in element carrying capacity loss and, consequently, collapse of the whole structure. Patch load examples are numerous and present in different types and segments of structures, such as crane girders loaded by crane wheels, bridge girders during erection by launching, rafters supporting secondary girders, column-beam joints, ship structures subjected to truck wheel or steel coil, etc. [1-3]. A comprehensive research work has been done worldwide, tackling different aspects of patch loaded steel I-girders [4,5]. The main research topic is the ideal case of patch load acting in the web plane, i.e. centric patch load, Fig. 1a. Numerous experimental studies provide solid amount of data as a base for further statistical, numerical (primarily by finite element method (FEM)) and analytical modelling, enabling proposals of different plastic collapse mechanisms, their mathematical formulations and adjoining expressions for ultimate load calculation. Increase in web (buckling) stability and girder resistance to patch load may be achieved by: (a) increase in web thickness; (b) longitudinal (web) stiffeners [6]; (c) transverse stiffeners [7]; (d) simultaneous transversal and longitudinal stiffening [8]; (e) delta stiffeners, i.e. triangular cell loaded flange [9,10]; (f) use of corrugated instead of flat web plates [11,12]. Not all these measures are fully efficient in case of load that changes its position (e.g. cranes, bridges, ships, etc.). Furthermore, production of such girders may be cost/time demanding. Hence, the rationalisation of patch loaded I-girders with unstiffened, flat web plate is still in focus of research. Additionally, investigation attention in this domain is still devoted to specific influential parameters and circumstances, such as: initial conditions (geometry imperfections and residual stresses) [13,14]; patch load length [14,15]; interaction of patch loading with bending and/or shear [16], etc. In line with needs of environmental protection and cost-effectiveness, hybrid steel I-girders (with different mechanical properties of flange/web steel) [17], as well as options of substitution for mild steel (aluminium [18], stainless steel [19], high-strength steel I- girders [20]) or its upgrade by cladding (I-girders having mild steel web, cladded by titanium [21] or stainless steel [22]) have been studied recently. A girder behaviour under patch loading at elevated temperatures or in fire conditions is analysed [23]. A piece of research findings concerning centric patch loading is implemented in European design standards, Eurocodes – EN1993-1-5 [24]. However, review and further adjustments of standards are needed, so to avoid too low underestimation of ultimate load, as well as to cover more diverse cases regarding girder geometry and load arrangement. In the engineering reality, a certain eccentricity of patch load relative to the web plane is unavoidable, i.e. eccentric patch load is more realistic case, Fig. 1b. This case is among issues which are not currently incorporated in Eurocodes, with the appropriate calculation procedure. In some circumstances, the load eccentricity may be treated as an imperfection that will not cause significant change in girder’s behaviour and carrying capacity compared to the centric load case. However, in other situations the load eccentricity should not be neglected, since it may be the reason for dangerous decrease in ultimate load (even up to 75%), as a result of girder’s behaviour being completely different from the one of centrically loaded girder. ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 3 Fig. 1 Patch loaded steel I-girder: (a) centric patch load; (b) eccentric patch load [25] Compared with the case of centric patch load, rather modest amount of research has been devoted to the case of eccentric patch load. While over 40 experimental researches (with more than 800 tested samples) dealt with centrically patch loaded I-girders, influence of load eccentricity was analysed in only eight experimental studies (with less than 200 tested samples). Majority of tests (135 specimens) were done at the University of Montenegro, in three series of experiments, in: 1998, reported by Lučić [26]; 2001, reported by Lučić and Šćepanović [27]; and 2007, reported by Šćepanović et al. [28]. The latest testing, with only four tested girders, was done at the University of Navarra, upon the initiative of the University of Granada, in 2009, reported by Gil-Martín et al. [29]. Experimental work was followed by non-linear finite element method (FEM) modelling [28,29]. While over 30 mathematical expressions (mostly based on collapse mechanism) for centric ultimate load might be found in scientific literature and design norms, only one mathematical/mechanical model for eccentric ultimate load calculation, based on the plastic collapse mechanism, is published by Graciano and Uribe-Henao [30]. Prior to the development of this mathematical/mechanical model, a few empirical expressions for eccentric ultimate load have been proposed, last modifications by Šćepanović et al. [28] and Gil-Martín et al. [29]. Design norms still do not encompass eccentric patch load. Arabzadeh and Varmazyari [9] analysed influence of delta stiffeners in eccentrically patch loaded girders, by means of FEM modelling. Inaam and Upadhyay [11], as well as Maiorana, Poh’sié and Emechebe [12] touched on an issue of eccentric patch load in girders with corrugated web, in their FEM based analyses. Experimental research demonstrated that majority of eccentrically patch loaded girders collapse at lower load level than their geometrical equivalents loaded in the web plane. Lover ultimate load is a consequence of different girder’s behaviour, i.e. different collapse mechanism or collapse mode, in case of eccentric load (flange warping/twisting, followed by web bending) compared with the case of centric load (web buckling, followed by flange “sinking”), Fig. 2. 4 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES Fig. 2 Collapse modes typical for centric and eccentric patch load: (a) centric collapse mode; (b) eccentric collapse mode [25] However, some eccentrically patch loaded girders, even with significant load eccentricity, behave as if loaded in the web plane, having the same collapse mode and ultimate load as in the case of centric load. Hence, the ultimate strength of eccentrically patch loaded steel I-girders should be found out in two steps: at first, girder collapse mode should be estimated; secondly, depending on expected collapse mode, ultimate load should be determined appropriately. Obviously, it is rather complicated task to formulate unique mechanical model (i.e. collapse mechanism and ultimate load expression based on it) that may be applied to all eccentrically patch loaded girders. Numerous parameters and their combinations determine behaviour, i.e. collapse mode and ultimate load of eccentrically patch loaded girders. The most prominent one is the load eccentricity, e, Fig. 1b. Girder dimensions, Fig. 1, primarily plate thicknesses, tw and tf, as well as dimensionless geometry parameters, such as tf /tw, hw /tw, etc. are also of a great significance. A change of only one girder dimension results in the variation of several influential parameters, what makes the analysis rather complex. Various artificial intelligence (AI) methods and techniques are highly suitable for multi-parameter analysis, particularly for solving compound stochastic problems, characterised with plenty of uncertainties and inter-dependencies. Artificial neural networks (ANN), machine learning (ML), deep learning (DL), etc. are nowadays fairly present in different areas of structural engineering, with high potential to complement, alternate or even replace traditionally used empirical modelling, based on statistics methods, and FEM based numerical modelling, as summarised in comprehensive reviews of Salehi and Burgueño [31], Sun et al. [32], Wang et al. [33], Tapeh and Naser [34]. Widening field and growing number of examples of successful ANN applying in design, construction, protection, monitoring and maintenance of diverse structures, such as reported by Chojaczyk et al. [35], Markovic et al. [36], Zarezadeh et al. [37], include the topic of patch loading, too. ANN modelling/forecasting, as a sophisticated AI calculation and modelling method, is quite convenient approach for studying centrically patch loaded I-girders, as presented by Kurtoglu et al. [18], Mai et al. [38], Kumar et al. [39]. In case of eccentric patch load, with all its complexity and intricacy, ANN modelling is even more welcome and valuable, ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 5 as demonstrated by Šćepanović et al. [25]. However, this technique has not been used to its full potential, which is evident in both phases of eccentrically patch loaded steel I- girders analysis: collapse mode estimation and ultimate load determination. ANN modelling may be successfully used on its own, but also as an advantageous tool coupled with other methods (i.e. hybrid modelling), e.g. with the only one existing mechanical model of collapse mechanism or with some of only few existing empirical expressions for ultimate load calculation. Benefits of ANN modelling, as well as hybrid modelling are elaborated in the paper, through the examples. 2. COLLAPSE MODES OF ECCENTRICALLY PATCH LOADED STEEL I-GIRDERS Eccentrically patch loaded steel I-girders mostly lose carrying capacity due to local elastic-plastic bending, with emphasised flange torsion and almost undeformed web, as in Fig. 3, but not necessarily. As already mentioned, under certain circumstances, eccentrically loaded girders may lose carrying capacity the same way as if there is no load eccentricity. Three different types of collapse mode, each with several variants, are observed in experimentally tested eccentrically patch loaded steel I-girders: eccentric (E, Fig. 3), centric (C, Fig. 4) and mixed (M, Fig. 5) collapse mode. Flange torsion, web deformation, position and shape of yielding lines (primarily one in web, but also one in flange), as well as their development level, in case of eccentric collapse mode, vary, depending on girder geometry and load eccentricity, as displayed in Fig. 3. Fig. 3 Eccentric collapse mode of eccentrically patch loaded I-girders (E) – variants, depending on girder geometry: (a) tw < tf & bf /tf ≥ 12.5; (b) tw = tf & bf /tf ≥ 12.5; (c) bf /tf = 10 (thick flange) 6 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES Fig. 4 Centric collapse mode of eccentrically patch loaded I-girders (C) – variants: (a) with two yielding lines (and one buckle) in web; (b) with three yielding lines (and two buckles) in web Fig. 5 Mixed collapse mode of eccentrically patch loaded I-girders (M) – variants: (a) centric-mixed collapse mode (gentle torsion of flange, centric collapse of web); (b) eccentric-mixed collapse mode (pronounced torsion of flange, undefined web collapse) Based on the experimental data, collapse mode identification criteria summarised in Table 1 have been established. These criteria correlate dimensionless geometry parameters tf /tw and a/tw = hw /tw with the dimensionless load eccentricity e/bf , e/tf or e/tw. Such correlations provide high level of reliability in collapse mode identification. Combining (checking) several of these criteria increases the reliability level. It has to be kept in mind that data from experiments done at the University of Montenegro [26-28] were used for this analysis, i.e. girders dimensions, load eccentricity and load length in range: (0) 5 mm  e  30 mm, 3 mm  tw  10 mm, 3 mm  tf  15 mm, a = hw = 700 mm, bf = 150 mm, c = 50 or 150 mm; and dimensionless geometry parameters in range: (0) 1/30  e/bf  1/5, (0) 0.33  e/tf  8.33, (0) 0.50  e/tw  8.33, 1  tf /tw  5, 10  bf /tf  50, 70  a/tw = hw /tw  233, a/hw = 1, c/a = 0.071 or 0.214. Hence, these are the limitations of high-level validity area for criteria from Table 1. Girders fulfilling criteria in dark grey fields of Table 1 have eccentric collapse mode (E), with the reduced ultimate load. Girders fulfilling criteria in light grey fields of Table 1 ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 7 might have any collapse mode (E, M, C) and these criteria imply that such girders should be carefully analysed each separately, with its specific characteristics, considering several influential parameters and their combinations, as well as paying particular attention to the initial deformation. Girders fulfilling criteria in white (i.e. no highlighted) fields of Table 1 will behave as if there is no load eccentricity (C collapse mode), having same ultimate load as in case of centric patch load. Primary influential parameter tf /tw enables high quality identification of collapse mode in correlation with each one of the three dimensionless eccentricities. However, web slenderness a/tw = hw /tw, not encompassing flange thickness tf, correlated with any of the three eccentricities does not provide comprehensive, exact collapse mode determination. As a complement to criteria from Table 1, but also in case of its standalone use, ANN modelling proved to be confidential method for collapse mode identification in case of girder geometry when all collapse modes are possible, as illustrated in Figs. 6 and 7. Several networks of various architecture were created and trained, by means of ANN software developed for the needs of research at the University of Montenegro. Two ANN forecast models for collapse mode are chosen for representation herein. Both ANN models displayed at Figs. 6 and 7 have good match with experimental data, particularly considering “pure” eccentric and centric collapse modes (E, C). Tolerable discrepancy may be observed in domain of mixed collapse modes. Trend of ANN models in that domain is completely in accordance with experimental findings. Better fitting of individual points may be achieved by further ANN modelling, which is generally endless activity. Another software or selection of training/verification dataset, as well as fine tuning of ANN models (e.g. different choice of numerals for collapse modes in this specific software etc.) will provide different forecasts. Note: In order to apply available ANN software, collapse mode types needed to be assigned numerals and they are chosen in interval [-1, 1], as follows: eccentric collapse mode E = -1; mixed collapse modes -1 < M <1; centric collapse mode C = 1. Table 1 Criteria for identification of collapse mode in eccentrically patch loaded I-girders dimensionless load eccentricity correlated dimensionless parameter criterion collapse mode* e/bf hw /tw hw /tw < 1050 · e/bf + 35 E hw /tw ≥ 1050 · e/bf + 35 E, M, C tf /tw tf /tw < 15 · e/bf + 0.5 E 15 · e/bf + 0.5 ≤ tf /tw ≤ 15 · e/bf + 1.5 E, M, C tf /tw > 15 · e/bf + 1.5 C e/tf hw /tw hw /tw ≤ 85 · e/tf + 60 E 85 · e/tf + 25 ≤ hw /tw ≤ 85 · e/tf + 105 M** hw /tw ≥ 85 · e/tf + 70 C tf /tw tf /tw < e/tf + 0.5 E e/tf + 0.5 ≤ tf /tw ≤ e/tf + 1.7 E, M, C tf /tw > e/tf + 1.7 C e/tw tf /tw tf /tw < 0.3 · e/tw + 0.8 E 0.3 · e/tw + 0.8 ≤ tf /tw ≤ 0.3 · e/tw + 1.8 E, M, C tf /tw > 0.3 · e/tw + 1.8 C * E = eccentric, M = mixed, C = centric collapse mode ** criterion is not sufficiently precise, detailed analysis of each girder, separately, is necessary 8 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES Fig. 6 Illustration of collapse mode identification by means of ANN modelling and according to criteria from Table 1, compared to experimental data, for girders geometry: tw = 5 mm, tf = 12 mm, a = hw = 700 mm, bf = 150 mm, c = 50 mm Fig. 7 Illustration of collapse mode identification by means of ANN modelling and according to criteria from Table 1, compared to experimental data, for girders geometry: tw = 10 mm, tf = 10 mm, a = hw = 700 mm, bf = 150 mm, c = 50 mm ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 9 Figs. 6 and 7 exhibit very good match of ANN modelling of collapse mode with criteria from Table 1, not only in outer zones of exclusively centric or eccentric collapse modes (C or E), but also in the middle zone, where all types of collapse mode are possible (C, M, E). Some girder dimensions or dimensionless parameters, analysed separately, without correlating them with other parameter(s), may be used as a rough identifiers of collapse mode. Girders having thin flange (tf ≤ 6 mm) or thick web (tw ≥ 8 mm, e.g. Fig. 7) or low web slenderness (hw /tw < 100, e.g. Fig. 7) or ratio tf /tw < 2 (e.g. Fig. 7) are affected even with the lowest eccentricity, as confirmed in Fig. 7. On the other side, girders with the ratio tf /tw > 3 have centric collapse mode even at the highest load eccentricity. Similarly to these conclusions, drawn from the experimental testing, the accuracy limitations of mechanical model, by Graciano and Uribe-Henao [30], are set up by means of load eccentricity, e, and tf /tw ratio: the mechanical model describing eccentric collapse mode should be used when e ≥ 10 mm and tf /tw ≤ 2.5. It may be concluded that two sets of limits for collapse modes are in a proper agreement, as confirmed in Figs. 6 and 7. Obviously, apart from the load eccentricity, e, collapse mode depends on both plate thicknesses, tf and tw, as well as on their relation, so that parameter tf /tw has the key-role in collapse mode determination, regardless of being analysed in co-relation with the other parameters or separately, only by itself. 3. ULTIMATE LOAD OF ECCENTRICALLY PATCH LOADED STEEL I-GIRDERS From the engineering practice aspect, the most significant characteristic of eccentric and mixed collapse modes is decrease in ultimate load with the increase in load eccentricity. Experiments revealed that the ultimate load decrease of over 40% is possible even in cases of such a small load eccentricity of only 5 mm (i.e. e/bf = 1/30), while larger eccentricity, of 25 mm (i.e. e/bf = 1/6), may decrease ultimate load up to 75%. FEM modelling, based on non-linear analysis, may provide confident results for ultimate load, but it is not always comfortable to be used by engineers. For common situations in practice, simple calculation procedures are appreciated. Two types of such expressions for ultimate strength of eccentrically patch loaded girders may be found in literature: empirical expressions [28,29], obtained by statistical processing of primarily experimental, but also FEM results, and mathematical expression based on plastic collapse mechanism [30]. ANN modelling may be of great help in determination of ultimate load, whether ANN forecast models are used directly for ultimate load [25], or this tool is coupled with some of previously mentioned methods/expressions, for determination of certain parameters figuring in those expressions for ultimate load. 3.1 Expressions for Ultimate Load 3.1.1 Empirical Procedure – Expression for Reduction Coefficient Empirical procedure relates the ultimate load of eccentric collapse mode to the ultimate load of centric collapse mode, in girders of same geometry with and without load eccentricity, by introducing reduction coefficient, R, as: R = ultimate load of eccentrically loaded girder ultimate load of centrically loaded girder . (1) 10 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES The ultimate load of centrically loaded girder might be determined by some of numerous expressions proposed in literature, including design norms. Reliable reduction coefficient, R, will then provide ultimate load of eccentrically loaded girder with acceptable accuracy. For the further analysis in this paper, the following expression for reduction coefficient, proposed by Šćepanović et al. [28], will be used: 𝑅 = 𝑚 ⋅ ( 𝑒 𝑏𝑓 ) 2 + 𝑛 ⋅ ( 𝑒 𝑏𝑓 ) + 1.01 ≤ 1   𝑚 = −0.864 ⋅ ( 𝑡𝑓 𝑡𝑤 ) 2 − 14.40 ⋅ ( 𝑡𝑓 𝑡𝑤 ) + 38.00   𝑛 = −12.30 + 4.22 ⋅ ( 𝑡𝑓 𝑡𝑤 ) } . (2) Eq. (2) provides valid results for input in range of data used for its derivation (experimental data reported in [26-28] and FEM data reported in [28]), as follows: 1  tf /tw  5, (0) 1/30  e/bf  1/5, 45  a/tw  233, 6.25  bf /tf  50, 1  a/hw  2, 0.036  c/a  0.071 or c/a = 0.214. 3.1.2 Expression Based on Mechanical Model of Collapse Only one mathematical/analytical expression based on plastic collapse mechanism has been proposed so far, by Graciano and Uribe-Henao [30]: 𝑃𝑢 = 𝑀𝑓 ∙ (2𝛽ξ + 𝑐ξ + 𝑏𝑓−2𝑒 𝛽 ) + 𝑀𝑤 ∙ 2𝛽+𝑐 𝛼𝑐𝑜𝑠𝜃 𝑀𝑓 = 𝑓𝑦𝑓𝑡𝑓 2 4⁄ 𝑀𝑤 = 𝑓𝑦𝑤𝑡𝑤 2 4 ⁄ ξ = 𝑏𝑓 (𝑒(𝑏𝑓 − 2𝑒))⁄ 𝑐𝑜𝑠𝜃 = 3 8 𝑓𝑦𝑓𝑎(𝑐+0.7𝑎) 𝐺𝑏𝑓𝑡𝑓 𝛽 = √ 𝑀𝑓(𝑏𝑓 2⁄ −𝑒)∙𝛼𝑐𝑜𝑠𝜃 𝑀𝑤+𝑀𝑓ξ∙𝛼𝑐𝑜𝑠𝜃 �̃� = 1.9447𝑡𝑤 ( 𝑒 𝑏𝑓 ∙ ( 𝑡𝑤 𝑡𝑓 ) 3 ) −0.451 } , (3) where: Mf and Mw are plastic moments per unit length of yielding line in the flange and web, respectively; fyf and fyw are flange and web yield stresses, respectively; G = 81000 MPa is the shear modulus for steel; α, β, θ and ξ are geometrical parameters describing yielding lines and deformed girder geometry in the collapse mechanism. They differ in character, i.e. in method of obtaining. While ξ is used only as a symbol to shorten writing, α, β and θ have clear geometrical meaning explained in [30]. While β and θ are defined analytically, based on mechanical models, by mathematical calculation, α (= distance of the web yielding line from the loaded flange, Fig. 3) may be determined as ᾶ, using the expression from Eq. (3), obtained by regression analysis, based on the experimental data [30]. To provide accuracy of Eq. (3), Graciano and Uribe-Henao [30] set up following limits: e ≥ 10 mm and tf /tw ≤ 2.5. These limits and need of more precise definition of Eq. (3) validity area are going to be discussed herein. ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 11 3.2 ANN Modelling as Support for Ultimate Load Calculation In addition to its successful standalone application for direct determination of ultimate load [25], ANN modelling may be used as a very convenient and welcome tool combined with other methods in order to improve them and provide more precise values of ultimate load. Such coupling of ANN modelling with expressions for ultimate load may be characterised as hybrid modelling. It is going to be presented herein, at the example of refining Eq. (3) by means of ANN modelling. Experimental database from 1998 [26], 2001 [27] and 2007 [28], containing 135 test samples, has been used for various ANN training and ANN forecast models creation. Also, comparison of different methods for determination of ultimate load and/or necessary parameters has been done using these test data. Tables 2 and 3 comprise tested girders geometry data, experimental collapse mode type, α and Pu values obtained by different methods (all calculated and ANN determined Pu values are related to respective experimental values, so to provide easier data comparison): - experimental position of web yielding lines αexp, (Table 2); - experimental collapse loads Pu,exp, (Table 2); - ultimate loads Pu,ANN obtained by ANN forecast models (as Pu,ANN /Pu,exp), (Table 3); - ultimate loads Pu,Eqs(1-2) obtained by Eqs. (1) and (2) when Pu,centr = Pu,exp (e = 0) (as Pu,Eqs(1-2) /Pu,exp = REq(2) /Rexp), (Table 3); - ᾶ values, obtained by expression from Eq. (3), (Table 2); - ultimate loads Pu,Eq(3) obtained by Eq. (3), including expression for ᾶ (as Pu,Eq(3) /Pu,exp), (Table 2); - ultimate loads Pu,Eq(3)-exp obtained by expression for Pu from Eq. (3), but with αexp values (as Pu,Eq(3)-exp /Pu,exp) – experimentally refined Eq. (3), (Table 2); - αANN values, obtained by ANN modelling, (Table 2); - ultimate loads Pu,Eq(3)-ANN obtained by expression for Pu from Eq. (3), but with αANN values (as Pu,Eq(3)-ANN /Pu,exp) – hybrid modelling of Pu, (Table 2). Complete calculation, as well as creation of ANN models were done with real values of yield stresses fyf and fyw, determined experimentally (Table 3), and G = 81 GPa. Table 2 Summary of α values (αexp, ᾶ, αANN), experimental collapse loads (Pu,exp) and ultimate loads determined by original and refined Eq. (3) (Pu,Eq(3), Pu,Eq(3)-exp, Pu,Eq(3)-ANN), all related to Pu,exp, for experimental database from 1998, 2001 and 2007 tests [26-28] No. girder tw tf e co ll a p se m o d e αexp Pu,exp ᾶ Pu,Eq(3) Pu,exp Pu,Eq(3)-exp Pu,exp αANN Pu,Eq(3)-ANN Pu,exp [mm] [mm] [mm] [mm] [kN] [mm] [mm] 1 EB I – 1 3 15 0 C 133 2 EB I – 2 3 15 5 C 128 238.7 3.10 55.5 3.14 3 EB I – 3 3 15 10 C 127 174.6 2.01 55.4 2.04 4 EB I – 4 3 15 15 C 135 145.4 1.49 55.2 1.52 5 EB I – 5 3 15 20 C 134 127.7 1.28 54.9 1.31 6 EB I – 6 3 15 25 C 124 115.5 1.24 54.6 1.27 7 EB II/III - 1 6 15 0 C 341 8 EB II/III - 2 6 15 5 C 321 186.9 1.25 53.2 1.30 9 EB II/III - 3 6 15 10 C 313 136.7 0.84 52.9 0.88 10 EB II/III - 4 6 15 15 E 50 282 113.9 0.75 0.80 52.5 0.79 11 EB II/III - 5 6 15 20 E 50 235 100.0 0.78 0.83 52.1 0.83 12 EB II/III - 6 6 15 25 E 55 192 90.4 0.86 0.91 51.6 0.92 13 EB IV - 1 8 15 0 C 401 12 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES 14 EB IV - 2 8 15 5 C 418 168.8 0.97 52.7 1.03 15 EB IV - 3 8 15 10 E 45 394 123.5 0.68 0.75 52.3 0.73 16 EB IV - 4 8 15 15 E 60 301 102.9 0.72 0.77 51.9 0.79 17 EB IV - 5 8 15 20 E 50 245 90.4 0.78 0.86 51.5 0.85 18 EB IV - 6 8 15 25 E 45 209 81.7 0.84 0.94 51.0 0.91 19 EB V - 1 5 10 0 C 229 20 EB V - 2 5 10 5 CM 212 115.2 0.97 43.5 1.00 21 EB V - 3 5 10 10 E 40 197 84.2 0.68 0.71 42.8 0.71 22 EB V - 4 5 10 15 E 43 175 70.2 0.62 0.64 42.0 0.64 23 EB V - 5 5 10 20 E 47 153 61.6 0.62 0.63 41.1 0.64 24 EB V - 6 5 10 25 E 50 129 55.7 0.66 0.67 40.2 0.69 25 EB VI - 1 10 10 0 C 720 26 EB VI - 2 10 10 5 EM 575 90.2 0.39 24.0 0.48 27 EB VI - 3 10 10 10 E 22.5 365 66.0 0.44 0.59 23.3 0.58 28 EB VI - 4 10 10 15 E 22.5 313 54.9 0.45 0.60 22.6 0.60 29 EB VI - 5 10 10 20 E 22.5 275 48.3 0.47 0.63 22.0 0.64 30 EB VI - 6 10 10 25 E 22.5 220 43.6 0.57 0.75 21.5 0.76 31 EB VII - 1 5 12 0 C 230 32 EB VII - 2 5 12 5 C 225 147.4 1.23 50.1 1.26 33 EB VII - 3 5 12 10 CM 212 107.8 0.85 49.7 0.88 34 EB VII - 4 5 12 15 EM 180 89.8 0.80 49.2 0.83 35 EB VII - 5 5 12 20 E 25 170 78.9 0.73 0.83 48.7 0.76 36 EB VII - 6 5 12 25 E 25 149 71.3 0.75 0.86 48.0 0.78 37 EB VIII - 1 3 3 0 E/EM 79 38 EB VIII - 2 3 3 5 E 22.5 44 27.0 0.41 0.42 22.5 0.42 39 EB VIII - 3 3 3 10 E 22.5 37 19.8 0.35 0.34 22.0 0.35 40 EB VIII - 4 3 3 15 E 22.5 29 16.5 0.40 0.37 21.5 0.37 41 EB VIII - 5 3 3 20 E 22.5 23 14.5 0.47 0.41 21.1 0.42 42 EB VIII - 6 3 3 25 E 22.5 20 13.1 0.52 0.44 20.7 0.45 43 EB IX - 1 3 6 0 C 95 44 EB IX - 2 3 6 5 EM 80 69.1 0.86 33.1 0.88 45 EB IX - 3 3 6 10 E 40 69 50.5 0.66 0.67 32.2 0.67 46 EB IX - 4 3 6 15 E 25 57 42.1 0.64 0.67 31.3 0.66 47 EB IX - 5 3 6 20 E 25 47 37.0 0.68 0.71 30.4 0.69 48 EB IX - 6 3 6 25 E 25 39 33.4 0.75 0.78 29.5 0.76 49 EB X - 1 3 9 0 C 102 50 EB X - 2 3 9 5 C 105 119.6 1.44 25.5 1.48 51 EB X - 3 3 9 10 C 107 87.5 0.91 44.9 0.93 52 EB X - 4 3 9 15 CM 90 72.9 0.86 44.2 0.88 53 EB X - 5 3 9 20 E 50 85 64.0 0.78 0.79 43.4 0.80 54 EB X - 6 3 9 25 E 50 70 57.9 0.85 0.86 42.5 0.87 55 EB XI - 1 3 12 0 C 116 56 EB XI - 2 3 12 5 C 113 176.5 2.42 53.0 2.44 57 EB XI - 3 3 12 10 C 115 129.1 1.53 52.7 1.55 58 EB XI - 4 3 12 15 C 110 107.5 1.26 52.4 1.28 59 EB XI - 5 3 12 20 CM 105 94.5 1.13 52.1 1.15 60 EB XI - 6 3 12 25 CM 115 85.4 0.92 51.7 0.94 61 EB XII - 1 4 4 0 C/CM 120 62 EB XII - 2 4 4 5 E 25 70 36.1 0.47 0.49 22.4 0.50 63 EB XII - 3 4 4 10 E 25 50 26.4 0.48 0.48 21.9 0.50 64 EB XII - 4 4 4 15 E 25 45 22.0 0.47 0.45 21.4 0.47 65 EB XII - 5 4 4 20 E 25 40 19.3 0.49 0.45 21.0 0.48 66 EB XII - 6 4 4 25 E 25 35 17.5 0.54 0.48 20.6 0.51 67 EB XIII - 1 4 6 0 C 125 68 EB XIII - 2 4 6 5 EM 110 62.4 0.64 28.7 0.67 69 EB XIII - 3 4 6 10 E 25 86 45.7 0.55 0.59 27.8 0.58 70 EB XIII - 4 4 6 15 E 25 68 38.0 0.58 0.62 27.0 0.61 71 EB XIII - 5 4 6 20 E 25 50 33.4 0.70 0.74 26.3 0.73 72 EB XIII - 6 4 6 25 E 15 45 30.2 0.72 0.87 25.5 0.75 73 EB XIV - 1 4 8 0 C 140 74 EB XIV - 2 4 8 5 C 129 92.1 0.99 37.2 1.02 75 EB XIV - 3 4 8 10 C 130 67.4 0.65 36.3 0.67 76 EB XIV - 4 4 8 15 E 50 100 56.1 0.68 0.68 35.3 0.70 77 EB XIV - 5 4 8 20 E 37.5 86 49.3 0.69 0.71 34.3 0.71 ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 13 78 EB XIV - 6 4 8 25 E 37.5 75 44.6 0.72 0.73 33.3 0.74 79 EB XV - 1 4 10 0 C 155 80 EB XV - 2 4 10 5 C 148 124.6 1.39 44.8 1.41 81 EB XV - 3 4 10 10 C 140 91.1 0.95 44.2 0.97 82 EB XV - 4 4 10 15 CM 138 75.9 0.77 43.4 0.79 83 EB XV - 5 4 10 20 EM 128 66.7 0.71 42.7 0.74 84 EB XV - 6 4 10 25 EM 115 60.3 0.72 41.8 0.74 85 EB XVI - 1 5 6 0 C 187 86 EB XVI - 2 5 6 5 E 25 130 57.7 0.56 0.60 27.2 0.59 87 EB XVI - 3 5 6 10 E 25 105 42.2 0.48 0.52 26.5 0.51 88 EB XVI - 4 5 6 15 E 25 74 35.2 0.57 0.62 25.7 0.61 89 EB XVI - 5 5 6 20 E 25 59 30.9 0.66 0.69 25.0 0.69 90 EB XVI - 6 5 6 25 E 25 55 27.9 0.66 0.69 24.3 0.69 91 EB XVII - 1 5 8 0 C 209 92 EB XVII - 2 5 8 5 CM 200 85.1 0.65 35.5 0.68 93 EB XVII - 3 5 8 10 E 35 145 62.3 0.60 0.63 34.6 0.63 94 EB XVII - 4 5 8 15 E 32 130 51.9 0.54 0.58 33.6 0.57 95 EB XVII - 5 5 8 20 E 28 98 45.6 0.64 0.69 32.6 0.67 96 EB XVII - 6 5 8 25 E 25 83 41.2 0.69 0.76 31.6 0.73 97 EB XVIII - 1 6 6 0 C/CM 208 98 EB XVIII - 2 6 6 5 E 25 170 54.1 0.44 0.49 23.3 0.50 99 EB XVIII - 3 6 6 10 E 20 130 39.6 0.46 0.58 16.8 0.63 100 EB XVIII - 4 6 6 15 E 15 104 33.0 0.53 0.74 16.7 0.70 101 EB XVIII - 5 6 6 20 E 88 29.0 0.60 16.6 0.78 102 EB XVIII - 6 6 6 25 E 69 26.2 0.75 16.6 0.95 103 EB XIX - 1 6 9 0 C/CM 330 104 EB XIX - 2 6 9 5 E 20 285 93.6 0.56 0.69 19.6 0.69 105 EB XIX – 3 6 9 10 E 20 217 68.5 0.51 0.67 19.3 0.68 106 EB XIX – 4 6 9 15 E 20 155 57.0 0.61 0.82 19.0 0.84 107 EB XIX – 5 6 9 20 E 20 125 50.1 0.69 0.94 18.7 0.96 108 EB XIX – 6 6 9 25 E 107 45.3 0.76 18.5 1.06 109 EB XX – 1 6 12 0 C 300 110 EB XX – 2 6 12 5 EM 265 138.2 1.05 46.6 1.09 111 EB XX – 3 6 12 10 E 25 311 101.1 0.60 0.73 26.2 0.73 112 EB XX – 4 6 12 15 E 25 235 84.2 0.66 0.82 25.5 0.82 113 EB XX – 5 6 12 20 E 25 202 74.0 0.68 0.86 24.8 0.86 114 EB XX – 6 6 12 25 E 165 66.9 0.76 24.2 0.99 115 B I/II 5 5 10 0 C 259 116 EB XXI – 2 5 10 5 C 250 115.2 1.50 43.5 1.54 117 EB XXI – 1 5 10 10 CM 240 84.2 0.95 42.8 0.99 118 B III 1/1 5 10 15 E 75 202 70.2 0.80 0.79 40.7 0.84 119 B III ½ 5 10 20 E 70 171 61.6 0.81 0.80 39.8 0.86 120 B III 1/3 5 10 25 E 65 141 55.7 0.90 0.88 38.9 0.95 121 B III ¼ 5 10 30 E 60 122 51.3 0.98 0.96 37.9 1.03 122 B I/II 11 10 10 0 C 801 123 EB XXI – 4 10 10 5 E 22.5 790 90.2 0.51 0.65 24.0 0.64 124 EB XXI – 3 10 10 10 E 50 640 66.0 0.43 0.45 23.3 0.57 125 B III 2/1 10 10 15 E 50 387 54.9 0.55 0.56 25.5 0.73 126 B III 2/2 10 10 20 E 47.5 297 48.3 0.67 0.67 24.9 0.89 127 B III 2/3 10 10 25 E 45 254 43.6 0.75 0.74 24.2 1.00 128 B III 2/4 10 10 30 E 42.5 219 40.2 0.86 0.84 23.6 1.13 129 B I/II 6 5 12 0 C 266 130 EB XXI – 6 5 12 5 C 255 147.4 1.97 50.1 2.02 131 EB XXI – 5 5 12 10 C 255 107.8 1.19 49.7 1.23 132 B III 3/1 5 12 15 E 85 228 89.8 0.96 0.97 48.4 1.01 133 B III 3/2 5 12 20 E 70 177 78.9 1.06 1.07 47.8 1.12 134 B III 3/3 5 12 25 E 65 162 71.3 1.05 1.06 47.0 1.10 135 B III ¾ 5 12 30 E 60 147 65.7 1.08 1.09 46.3 1.13 For all girders: a = hw = 700 mm, bf = 150 mm For samples Nos. 1-114: c = 50 mm For samples Nos. 115-135: c = 150 mm Series B I, B II, B III – 1998 tests [26] Series EB I - EB IV – 2001 tests [27] Series EB V - EB XXI – 2007 tests [28] 14 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES Table 3 Summary of material properties (fyf, fyw) and ultimate loads determined by Eqs. (1) and (2) and forecasted by ANN models (Pu,Eqs(1-2), Pu,ANN), both related to Pu,exp, for experimental database from 1998, 2001 and 2007 tests [26-28] No. series fyw fyf e [mm] [MPa] [MPa] 0 5 10 15 20 25 30 1-6 EB I 327.3 268.7 1.00 1.04 1.05 0.99 0.99 1.00 Pu,Eqs(1-2)/Pu,exp 1.02 1.01 1.03 0.99 1.00 1.01 Pu,ANN/Pu,exp 7-12 EB II/III 309.3 268.7 1.00 0.39 0.37 0.38 0.41 0.43 Pu,Eqs(1-2)/Pu,exp 0.99 1.01 1.00 1.00 1.01 1.00 Pu,ANN/Pu,exp 13-18 EB IV 262.4 268.7 1.00 0.84 0.77 0.87 0.93 0.96 Pu,Eqs(1-2)/Pu,exp 1.23 1.00 0.92 1.01 1.00 0.93 Pu,ANN/Pu,exp 19-24 EB V 270.9 309.7 1.00 0.96 0.90 0.89 0.89 0.93 Pu,Eqs(1-2)/Pu,exp 1.01 1.00 0.99 1.01 1.01 1.04 Pu,ANN/Pu,exp 25=30 EB VI 309.7 309.7 1.00 0.96 1.13 0.99 0.88 0.97 Pu,Eqs(1-2)/Pu,exp 0.97 1.01 1.18 1.09 1.00 0.99 Pu,ANN/Pu,exp 31-36 EB VII 270.9 288.8 1.00 0.96 0.93 0.99 0.94 0.93 Pu,Eqs(1-2)/Pu,exp 0.99 0.98 1.03 1.09 0.98 0.97 Pu,ANN/Pu,exp 37-42 EB VIII 274.4 274.4 1.00 1.38 1.22 1.17 1.16 1.16 Pu,Eqs(1-2)/Pu,exp 0.91 1.18 1.08 1.17 1.35 1.45 Pu,ANN/Pu,exp 43-48 EB IX 274.4 287.3 1.00 1.05 1.07 1.14 1.21 1.28 Pu,Eqs(1-2)/Pu,exp 1.11 1.10 1.04 1.00 1.04 1.05 Pu,ANN/Pu,exp 49-54 EB X 274.4 282.0 1.00 0.97 0.93 1.04 0.99 1.03 Pu,Eqs(1-2)/Pu,exp 1.08 0.99 0.91 0.98 0.89 0.94 Pu,ANN/Pu,exp 55-60 EB XI 274.4 288.8 1.00 1.03 1.01 1.05 1.10 0.85 Pu,Eqs(1-2)/Pu,exp 0.97 0.96 0.99 1.04 1.05 0.91 Pu,ANN/Pu,exp 61-66 EB XII 285.0 285.0 1.00 1.31 1.37 1.14 1.01 1.01 Pu,Eqs(1-2)/Pu,exp 0.92 1.14 1.20 1.02 0.95 0.97 Pu,ANN/Pu,exp 67-72 EB XIII 285.0 287.3 1.00 0.94 0.98 1.02 1.18 1.16 Pu,Eqs(1-2)/Pu,exp 1.06 0.97 0.99 0.97 1.08 0.98 Pu,ANN/Pu,exp 73-78 EB XIV 285.0 300.3 1.00 0.96 0.84 0.95 0.97 0.98 Pu,Eqs(1-2)/Pu,exp 1.07 1.05 0.96 1.10 1.07 1.07 Pu,ANN/Pu,exp 79-84 EB XV 285.0 309.7 1.00 0.99 0.97 0.90 0.87 0.84 Pu,Eqs(1-2)/Pu,exp 0.99 0.99 1.02 0.99 0.96 0.97 Pu,ANN/Pu,exp 85-90 EB XVI 270.9 287.3 1.00 1.14 1.09 1.22 1.24 1.17 Pu,Eqs(1-2)/Pu,exp 0.92 1.04 0.98 1.07 1.03 0.89 Pu,ANN/Pu,exp 91-96 EB XVII 270.9 300.3 1.00 0.88 1.00 0.94 1.06 1.11 Pu,Eqs(1-2)/Pu,exp 1.01 0.93 1.10 1.05 1.14 1.12 Pu,ANN/Pu,exp 97-98 EB XVIII 287.3 287.3 1.00 0.94 0.92 0.86 0.80 0.89 Pu,Eqs(1-2)/Pu,exp 99-102 458.3 287.3 1.04 0.98 0.99 0.95 0.93 1.01 Pu,ANN/Pu,exp 103-108 EB XIX 458.3 282.0 1.00 0.96 1.03 1.19 1.24 1.28 Pu,Eqs(1-2)/Pu,exp 1.00 0.91 0.94 1.02 1.00 0.97 Pu,ANN/Pu,exp 109-110 EB XX 287.3 288.8 1.00 1.00 0.75 0.87 0.89 0.96 Pu,Eqs(1-2)/Pu,exp 111-114 458.3 288.8 1.01 1.05 1.00 1.08 1.01 0.99 Pu,ANN/Pu,exp 115, 118-121 B I/II, III 1 286.5 274.7 1.00 0.92 0.84 0.87 0.90 0.97 0.99 Pu,Eqs(1-2)/Pu,exp 116-117 EB XXI 270.9 309.7 122, 125-128 B I/II, III 2 274.7 274.7 1.00 0.78 0.72 0.89 0.91 0.93 1.11 Pu,Eqs(1-2)/Pu,exp 123-124 EB XXI 309.7 309.7 129, 132-135 B I/II, III 3 286.5 267.4 1.00 0.98 0.90 0.91 1.04 0.99 0.93 Pu,Eqs(1-2)/Pu,exp 130-131 EB XXI 270.9 288.8 ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 15 Graphical presentation of data from Tables 2 and 3 is given separately for girders with eccentric or mixed collapse mode (E, M – 92 samples, Nos. 10-12, 15-18, 20-24, 26-30, 33-36, 38-42, 44-48, 52-54, 59, 60, 62-66, 68-72, 76-78, 82-84, 86-90, 92-96, 98-102, 104- 108, 110-114, 117-121, 123-128, 132-135, Figs. 8a and 9a) and for girders centric collapse mode (C – 43 samples, Nos. 1-9, 13, 14, 19, 25, 31, 32, 37, 43, 49-51, 55-58, 61, 67, 73- 75, 79-81, 85, 91, 97, 103, 109, 115, 116, 122, 129-131, Figs. 8b and 9b), having in mind sense(lessness) and/or (in)appropriateness of certain methods implementation, depending on collapse mode type of eccentrically patch loaded girder. Fig. 8 depicts α values: αexp, ᾶ and αANN. It is obvious that ᾶ values are excessively overestimated for the majority of test samples having eccentric or mixed collapse mode (E, M), Fig. 8a. On the other hand, αANN values are mostly in range of αexp. This is valid for samples with load length c = 50 mm (i.e. samples Nos. 1-114(117)). Situation is different for samples with load length c = 150 mm (i.e. samples Nos. 114(117)-135). A satisfying agreement of ᾶ and αexp may be noted in this zone, while certain discrepancy between αANN and αexp occurs in these samples. An acceptable match between ᾶ and αexp in girders with c = 150 mm, Fig. 8a, also stated by Graciano and Uribe-Henao [30], is expected, having in mind that expression for ᾶ is fitted to these experimental data (1998 [26] tests data were used in the regression analysis that provided the expression for ᾶ [30]). However, strong disagreement of ᾶ values with experimental data for c = 50 mm (2001 [27] and 2007 [28] tests), indicates that conditions for application of Eq. (3) should be revised. This mismatch is also expected and easily explainable. Experimental samples from 1998 [26], 2001 [27] and 2007 [28] have same dimensions a = hw = 700 mm, bf = 150 mm, but different plate thicknesses tf and tw, as well as their ratios, which immensely influence girder behaviour, so that three experimental campaigns complement each other in constituting comprehensive database. Considering this simultaneously with the fact of difference in load length, it is clear that the expression for ᾶ in Eq. (3) may not be assumed as universal and should be applied only for girders with geometry in range of data used for the regression analysis. Although parameter α from Eq. (3) does not have (the same) realistic meaning in girders with centric collapse mode (C), i.e. Eq. (3) is not intended for girders with centric collapse mode, illustration Fig. 8b is kept herein, not only to display significant difference between ᾶ and αANN values, but also to point out non-realistically high values of ᾶ. Actually, several samples in Fig. 8b have parameters e and tf /tw at the very border of domain suitable for use of Eq. (3), according to Graciano and Uribe-Henao [30]. Majority of samples from Fig. 8b confirm that Eq. (3) does not provide reliable results for girders with e < 10 mm or tf /tw > 2.5, in addition to the fact that it should not be applied for these girders at all, because they have completely different collapse mechanism. Consequently, it may be concluded that complete set of Eq. (3) may provide valid results for input in the following range (i.e. range of experimental data reported by Lučić [26], which were used for statistical data procession during Eq. (3) derivation [30]): 1  tf /tw  2.4, 1/10  e/bf  1/5, 70  a/tw  140, 12.5  bf /tf  15, a/hw = 1, c/a = 0.214. Furthermore, Eq. (3), describing eccentric collapse mechanism, should be applied only in girders with eccentric collapse mode, regardless of the existence and level of load eccentricity, and this is, actually, the first precondition for application of Eq. (3). These constraints are more demanding and more comprehensive than those set up by Graciano and Uribe-Henao [30] (e/bf ≥ 1/15, in the context of available experimental data, and tf /tw ≤ 2.5), which are close to, but do not completely fit within newly proposed 16 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES conditions and do not cover all influential parameters. Hence, newly proposed constraints (in the previous paragraph) will provide higher reliability of Eq. (3) results. Discrepancy between αANN and αexp in samples Nos. 114(117)-135, Fig. 8a, is easily explained by the fact that ANN for αANN models were trained on experimental data in which c = 50 mm prevailed. Hence, those models may not provide perfect forecast for input with c = 150 mm. (a) (b) Fig. 8 α values in girders with: (a) eccentric or mixed collapse mode (E, M); (b) centric collapse mode (C) In Fig. 9, Pu values are compared: experimental values Pu,exp, values Pu,Eq(3) obtained by original Eq. (3), and values Pu,Eq(3)-exp and Pu,Eq(3)-ANN obtained by refined Eq. (3). Refinement of Eq. (3) is achieved by using αexp or αANN instead of ᾶ. ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 17 (a) (b) Fig. 9 Pu values in girders with: (a) eccentric or mixed collapse mode (E, M); (b) centric collapse mode (C) It is clearly noticeable that almost all values of ultimate load obtained by Eq. (3), in its original as well as refined form, are immensely underestimated in girders with eccentric or mixed collapse mode (E, M), Fig. 9a. Calculated values (Pu,Eq(3), Pu,Eq(3)-exp and Pu,Eq(3)-ANN) are in average around 71% of experimental values (Pu,exp), while some of them are as low as only 34% Pu,exp (Table 4, middle segment). Both percentages are too low and may not recommend application of Eq. (3) in the whole range of available experimental data. If only samples with load length c = 50 mm are analysed (i.e. samples Nos. 1-114(117), with E or M collapse mode), level of underestimation is slightly worse: calculated ultimate loads are in average around 68% Pu,exp, while minimal calculated value remains 34% Pu,exp. Concerning only samples with load length c = 150 mm (i.e. samples Nos. 114(117)-135, 18 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES with E or M collapse mode), both percentages of underestimation are more favourable, i.e. underestimation is less emphasized: calculated values of ultimate loads are in average around 86% Pu,exp, while minimal calculated value is 48% Pu,exp. Certain differentiation of these percentages, for c = 150 mm, in comparison to those reported by Graciano and Uribe- Henao [30] are generally due to differences in yield stresses fyf and fyw (real values are used herein, while Graciano and Uribe-Henao [30] used nominal values), but may also be due to the fact that more samples from this category are analysed herein (15 samples) than by Graciano and Uribe-Henao [30] (12 samples). Values Pu,Eq(3)-ANN are closest to Pu,exp. In average, they are 75% Pu,exp for the whole dataset from Fig. 9a (Table 4, middle segment), 71% Pu,exp for c = 50 mm and even 93% Pu,exp for c = 150 mm. This is significantly better match than for Pu,Eq(3) values, obtained by original Eq. (3), which are the most distant from Pu,exp: in average 68% Pu,exp for the whole dataset from Fig. 9a (Table 4, middle segment), 65% Pu,exp for c = 50 mm and 82% Pu,exp for c = 150 mm. The beneficial difference of even 11% (c = 150 mm) speaks in favour of Eq. (3) refinement, achieved by its coupling with ANN α modelling, Fig. 10. 92 samples, with eccentric or mixed collapse mode, are presented in Fig. 9a. Only two of them have ratio tf /tw > 2.5 and other twelve have load eccentricity e < 10 mm. Hence, even 85% (78/92) of samples satisfy preconditions for Eq. (3) defined by Graciano and Uribe-Henao [30]. Still, results of Eq. (3) are not appropriate for engineering practice. However, only 29% (27/92) of samples satisfy more realistic (and stricter) preconditions for Eq. (3), suggested herein. This is the main reason for described unacceptable disagreement of original Eq. (3) and experimental results. Thorough analysis of data from Table 2 reveals that exactly these 27 samples (15 samples with c = 50 mm, Nos: 22-24, 28- 30, 34-36, 106-108, 112-114; 12 samples with c = 150 mm, Nos: 118-121, 125-128, 132- 135, that were used for the derivation of Eq. (3)) have the best match of calculated and experimental values of ultimate load, with average calculated value of 81% Pu,exp (Table 4, lower segment), i.e. for 10% better than for the whole dataset from Fig. 9a (Table 4, middle segment). Again values Pu,Eq(3)-ANN have smallest deviation from Pu,exp. In average, they are 88% Pu,exp (Table 4, lower segment), i.e. for 13% better than for the whole dataset from Fig. 9a (Table 4, middle segment). Statistics enhancement is well pictured in Fig. 11. When samples with different load length are considered separately, results are also significantly better, for both categories. As assumed, samples with c = 150 mm have the best match of refined/hybrid calculated and experimental results: in average Pu,Eq(3)-ANN = 98% Pu,exp. Obvious shortage of Eq. (3), i.e. its expression for ᾶ, may be overcome by ANN modelling of α. With αANN values, instead of ᾶ, Pu values are remarkably more precise. Although calculated Pu values (by any version of Eq. (3)) might not be valid ultimate load of eccentrically loaded girders with centric collapse mode (no ultimate load reduction due to load eccentricity, therefore no need and no sense to calculate ultimate load by Eq. (3) or any other procedure related to eccentric load mode), Fig. 9b is displayed herein. It is interesting that, unlike the case of eccentric or mixed collapse mode from Fig. 9a, these “calculated ultimate loads” in Fig. 9b are mostly overestimated, significantly higher than Pu,exp, which confirms senselessness of these data. Again, even if it is not known that these samples have centric collapse mode, they are almost all out of validity domain of Eq. (3), according to Graciano and Uribe-Henao [30], so that use of Eq. (3) would be inadequate. However, it should be pointed out that Fig. 9b has neither practical purpose nor meaning, primarily because these samples have centric collapse mode, which is not described by Eq. (3), regardless of the boundaries for load eccentricity and girder geometry. ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 19 3.3 Comparison of Methods for Ultimate Load Determination Based on summaries from Tables 2 and 3, general comparison of following methods and procedures for ultimate load determination has been presented: - empirical procedure, defined by Eqs. (1) and (2) - Pu,Eqs(1-2); - mechanical model, mathematically defined by Eq. (3) in its original form - Pu,Eq(3); - mechanical model coupled with experimental data αexp, i.e. experimentally refined Eq. (3) - Pu,Eq(3)-exp; - mechanical model coupled with ANN modelling data αANN, i.e. ANN refined Eq. (3) (hybrid model) - Pu,Eq(3)-ANN; and - ANN forecast model - Pu,ANN. For the sake of comparison simplicity, all calculated (by Eqs. (1-3)) and forecasted (by ANN model) values of ultimate load (Pu,calc) are related to respective experimental collapse loads (Pu,exp). Statistical parameters (max, min and mean value, standard deviation and coefficient of variation) for ratios Pu,calc/Pu,exp are determined and summarised in Table 4. Statistics has been done initially for the complete database (135 samples – upper segment of Table 4), then only for samples with eccentric or mixed collapse mode (92 samples – middle segment of Table 4), and finally only for samples with eccentric or mixed collapse mode, satisfying preconditions for applying Eq. (3), as suggested in Chapter 3.2 (27 samples – lower segment of Table 4). Graphical presentation is enclosed for later two cases, Figs. 10 and 11. Table 4 Statistics parameters for five procedures of ultimate load determination Pu,Eqs(1-2) Pu,exp Pu,Eq(3) Pu,exp Pu,Eq(3)-exp Pu,exp Pu,Eq(3)-ANN Pu,exp Pu,ANN Pu,exp ALL GIRDERS (135 SAMPLES) Max 1.38# 3.10 ca lc . o n ly f o r g ir d er s w it h ec c. a n d m ix ed co ll ap se m o d e 3.14 1.45 Min 0.37 0.35 0.35 0.89 mean value 0.98 0.82 0.88 1.02 stand. dev. (S) 0.16 0.40 0.40 0.08* coeff. of var. (V) 0.17 0.49 0.45 0.08* ECCENTRIC OR MIXED COLLAPSE MODE – E, M (92 SAMPLES) Max 1.38# 1.13 1.09 1.15 1.45 Min 0.38 0.35 0.34 0.35 0.89 mean value 0.99 0.68 0.70 0.75 1.03 stand. dev. (S) 0.18 0.17 0.17 0.19 0.09* coeff. of var. (V) 0.18 0.25 0.24 0.25 0.09* VALIDITY DOMAIN FOR Eq. (3), CHAPTER 3.2 (27 SAMPLES) Max 1.28 1.08 1.09 1.13 1.09 Min 0.87 0.45 0.56 0.60 0.97 mean value 0.97 0.75 0.81 0.88 1.02 stand. dev. (S) 0.11 0.17 0.15 0.16 0.04 coeff. of var. (V) 0.11 0.22 0.18 0.18 0.04 Although displayed herein, data from dark grey fields in Table 4 should be exempted from further discussion, since they include ultimate loads obtained by Eq. (3) for girders for which this method is not intended – girders with centric collapse mode. Mean values are above 0.80 (much better than in the middle segment of Table 4), but these are fake information, as a consequence of improper use of Eq. (3), and it should be recognised. 20 B. ŠĆEPANOVIĆ, S. RUTEŠIĆ, O. MIJUŠKOVIĆ, L.M. GIL-MARTÍN, E. HERNÁNDEZ-MONTES Fig. 10 Ratios of calculated and experimental ultimate load (Pu,calc/Pu,exp) for girders with eccentric or mixed collapse mode (E, M) – set of 92 samples Fig. 11 Ratios of calculated and experimental ultimate load (Pu,calc/Pu,exp) in domain of Eq. (3) validity – set of 27 samples ANN and Hybrid Modelling of Eccentrically Patch Loaded Steel I-Girders 21 Statistical parameters of ultimate load obtained by empirical procedure (Pu,Eqs(1-2)) or forecasted by ANN models (Pu,ANN) do not depend on whether girders with centric collapse mode are included in sample set or not – statistical values are same for set of 135 samples, having various collapse modes (upper segment of Table 4), and for set of 92 samples with eccentric or mixed collapse mode only (middle segment of Table 4). This is due to the fact that both methods are derived from the complete experimental database (135 samples), by statistical “best fitting” techniques. Mean values of both methods are very close to 1, i.e. almost perfect (bolded numbers in Table 4; Fig. 10). Somewhat lower standard deviation and coefficient of variation may bring certain advantage to ANN modelling (* marked numbers in Table 4), judging only by statistics. In reality, the fact that empirical procedure approximates experimental values from the safe side (Fig. 10), also having lower max unsafe discrepancy (# marked numbers in Table 4), stands in favour of this method. Regarding the variants of Eq. (3), its underestimation of ultimate load (italic underlined numbers in Table 4) is not tolerable in engineering practice, despite its evident refinement by coupling with ANN modelling (bold italic underlined number in Table 4). This method is on the safe side (Figs. 10 and 11), but not simultaneously economical. Statistical parameters for Eq. (3) variants are notably enhanced when it is applied in a real domain of its validity (light grey fields, lower segment of Table 4; Fig. 11). However, even in this narrow domain, i.e. on dataset of only 27 samples, other methods (empirical procedure with reduction coefficient and ANN modelling) provide better statistics and are more favourable, which implies necessity of further revision of Eq. (3), considering complete existing experimental database, exactly as Graciano and Uribe-Henao [30] concluded. 4. CONCLUSION ANN modelling proved to be valuable in the analysis of eccentrically patch loaded steel I-girders. Well-structured and properly trained artificial neural networks provide reliable forecast models for different needs. They may be used as a self-contained tool for resolving both key-issues: identification of collapse mode type and determination of collapse load. In addition, ANN forecasting may be successfully combined with other methods, in order to complement and/or to refine them, by providing confident determination of certain parameters needed for the other method. Such procedure is denoted as hybrid modelling. Regarding collapse mode identification, whether used standalone or complemented with empirical criteria, ANN forecast models have very good results. As concerns ultimate load determination, pure ANN modelling provides nearly perfect results, regardless of collapse mode type, which positions this method in top-two procedures for ultimate load. This is of particular importance when girder geometry enables any (or more) collapse mode(s). Applying ANN modelling within mathematical model, as its support to determine specific influential parameter necessary for ultimate load calculation, is highly beneficial, fixes insufficiencies and significantly contributes to the enhancement mechanical model. Although coupling with ANN modelling considerably improved mechanical model for ultimate load and new, more demanding constraints of its validity are proposed, further revision of mechanical model should be done, based on all available experimental data. 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