Methods for determination of ultimate load of eccentrically patch loaded steel I-girders

Experimental tests show that most eccentrically patch loaded steel I-girders have a collapse mode quite different from that of centrically loaded girders. Concerning engineering practice, the most important difference between collapse modes is in ultimate load. The reduction in ultimate load with an increase in load eccentricity is obvious in some eccentrically loaded girders. Under certain circumstances, for a certain combination of influential parameters, eccentrically loaded girders behave as if loaded in the web plane, with no or no significant reduction in ultimate load due to load eccentricity. Dealing with such a big number of mutually dependant parameters that influence collapse mode and level of ultimate load reduction due to load eccentricity, still without theoretical formulation of collapse mechanism, two approaches for ultimate load determination are analysed: empirical mathematical expressions and artificial neural networks forecast models. Results of two procedures are compared. Recommendations for application in engineering practice are given.


INTRODUCTION
Patch loading acts locally, over a small area or length of a structural element.It is a common situation in structural engineering that local compressive load affects the flange of steel I-girder so that the web is compressed in the region below the applied load.Local stresses in the vicinity of load might cause local instability that may provoke element carrying capacity loss and, consequently, collapse of the whole structure.This is rather complex and challenging issue of extremely evident elasticplastic stresses and deformations.Apart from that, geometrical nonlinearity is noticeable even at the lowest loading level.
Patch loaded girders (Figure 1) are widely used and present in different structures, including crane girders loaded by crane wheels or bridge girders during launching.Although some eccentricity of load relative to the web plane is unavoidable in engineering practice, rather modest amount of worldwide research work has treated this issue in comparison with the amount of worldwide research work that have treated centric patch loading.While over 35 experimental researches (with more than 750 tested samples) dealt with I-girders patch loaded in the web plane, influence of load eccentricity was analysed in only eight experimental studies (with less than 200 tested samples) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10).Experimental analysis of eccentrically patch loaded girders started at the University of Maine in late 1980s (1) (2) (3).At the same time some tests were done at the Institute of Theoretical and Applied Mechanics, Czech Academy of Sciences (4).Ten years later a new series of experiments (1998,2001,2007) were initiated at the University of Montenegro (5) (6) (7) (8) (9) (10).The newest testing, with only four tested girders, was done at the University of Navarra, upon the initiative of the University of Granada, in 2009 (10).
Experimental work was followed by finite element method (FEM) modelling, by means of various computer software (1) (2) (3) (4) (5) (6) (7) (8) (9) (10).While over 30 mathematical expressions (mostly based on collapse mechanism) for centric ultimate load might be found in literature, only few empirical expressions for eccentric ultimate load have been proposed (3) (8) (9) (10).Artificial neural networks, being suitable for multi-parameter analysis, are also used nowadays for collapse mode and ultimate load estimation (8).However, mathematical model for ultimate load calculation based on collapse mechanism, or some other theoretical approach, has not yet been proposed.
Since the first experimental investigations, in 1980s, it has been evident that numerous parameters influence the behaviour, collapse mode and ultimate load of eccentrically patch loaded steel I-girders: geometric parameters (girder's dimensions and their dimensionless ratios), load eccentricity and its relations with girder dimensions, as well as the manner of load application (line or laterally distributed load).Dominant parameter is the load eccentricity, e, or ratio e/b f .Apart from this parameter, the influence of girder geometry parameters should be studied.Girder dimensions, primarily web and flange thicknesses, t w and t f , as well as ratio t f /t w are of important influence.Other ratios, like b f /t f , a/t w , h w /t w should be considered as well.Attention should also be paid to the load length, c, or ratio c/a and to the load application manner.Hence, not only that influential parameters are numerous, but they are also mutually dependant and related and therefore should not be considered separately.Combinations of certain parameters should be carefully analysed.
Even 1980s tests shown and all later experiments confirmed that most eccentrically patch loaded steel I-girders (but not all and not always!)have a collapse mode quite different from that of centrically loaded girders (Figure 2).Carrying varies linearly with e/b f , equation [2] (3).Application of this formulation is limited to girders with dimensions from the range of experimental data used for expression derivation (3): 1 ≤ t f /t w ≤ 4 and e/b f ≤ 1/6.
Recent experimental work (from 1998, 2001 and 2007), with wider range of test data (5) (6) (7) (8) (9), indicates that the original expression for R should be modified.Not only that its application does not provide valid results for girders with parameters t f /t w and e/b f out of range given in (3).Quality of its results is also dependant on other parameters (b f /t f , a/t w , c/a, a/h w ) that should be considered as limitation factors for each empirical expression application.
Join research work of teams from the University of Montenegro and Granada University resulted in several modifications of original expression for reduction factor (8) (9) (10).New, improved expressions are also empirical, obtained by regression analysis based on all available experimental data (until 2007) as well as on results of finite element modelling (FEM) by ANSYS.One of these expressions, which has very good match with the wide range of experimental and numerical data, is defined by equation [3] (8) (9).The reduction factor, R, is considered to be a quadratic function of the most relevant parameter e/b f and, same as in the original expression, dependent on the most influential geometry parameter t f /t w .
Additional requirements for values of R calculated by Equation [3] are: 0 < R ≤ 1 and R = 1 for e/b f = 0.
Same as any other empirical expression, equation [3] should be used only in the range of data used for regression analysis.Experimental data used for the derivation of [3] are in the following range: 1 ≤ t f /t w ≤ 5, (0) With such a big number of mutually dependant parameters that influence collapse mode and level of ultimate load reduction due to load eccentricity, still without formulation of collapse mechanism, two approaches for ultimate load determination are suitable: empirical mathematical expressions (3) (8) (9) (10) and artificial neural networks (ANN) forecast models (8).Both methods are based on experimental and/or FEM experience, their application is limited to cases from experimental and/or FEM data domain and every future experimental and/or FEM testing should be followed by their revision and adjusting in order to improve their accuracy.

EMPIRICAL MATHEMATICAL EXPRESSIONS
In the case of eccentric (or mixed-eccentric) collapse mode, ultimate load reduces as the load eccentricity increases (1) (2) (3) (5) (6) (7) (8) (9) (10).This decrease in the ultimate load might be expressed by a reduction factor, R, that relates the ultimate load of eccentrically loaded girder to the ultimate load of identical centrically loaded girder, equation [1] (3) (8) (9) (10).Ultimate load of centrically patch loaded girder might be calculated by one of numerous and very accurate existing mathematical expressions.Ultimate load of eccentrically patch loaded girder then might be easily calculated if the reduction is evaluated correctly and confidently.

R ultimate load of eccentrically loaded girder ultimate load of centrically loaded girder [1]
The first published expression for the ultimate load reduction factor, R, is based on experimental studies conducted in 1980s (1) (2) (3).Reduction factor is expressed in terms of two main geometric parameters: R is a function of t f /t w and of curves proximity to horizontal line corresponding to value of 1 (R exp /R emp ≈ 1 and R FEM /R emp ≈ 1), i.e. better match with experimental and FEM data, is much more obvious for equation [3] than for equation [2].
It has to be pointed out that every future experimental testing or FEM modelling should be followed by new revision of empirical expression for the ultimate load reduction factor in order to improve its accuracy.Apart from that, even with the existing experimental and FEM data base, this kind of mathematical modelling is almost endless, offering almost countless options -to choose different functions for R, different influential parameters, their forms and combinations or different methods of expression calibration.Presented expression by equation [3] is chosen and recommended as simple and reliable for application in engineering practice.
Improvement of expression [3] in comparison with the expression [2] is illustrated in Figures 3,4,5,6,7, where R exp (ratio of experimental ultimate loads of eccentrically and centrically loaded girder, according to equation [1]), R FEM (ratio of FEM ultimate loads of eccentrically and centrically loaded girder, according to equation [1]) and R emp (calculated by equations [2] and [3]) are experimental, FEM and empirical values of reduction factor, respectively.All experimental data with load length c = 50 mm from 1998 (Figure 4), 2001 (Figure 5) and 2007 (Figure 6), as well as relevant experimental data from 1988 (Figure 3) are included into graphical presentation.Only one set of FEM data, with span girder a = 700 mm, is graphically presented (Figure 7).Diagrams for higher values of girder span, a, are very similar to this one, having the same shape of R FEM /R emp -e/b f curves with slightly different numerical values, for each flange thickness, t f .Trend         The best evaluated models show high level of match with experimental data and prove to be acceptable for engineering practice.Particularly good results are obtained from network with two hidden levels, each with ten neurons ("c50 -load -2 -10") (8).Examples of collapse load forecast models of this network are presented in Figure 8 which illustrates estimation of collapse load and its relation with the load eccentricity P u,ann (e) for t w = 5 mm and different values of t f = 5 ÷ 15 mm, all at fixed c = 50 mm, σ 0.2, w = 28 kN/cm 2 and σ 0.2, f = 28 kN/ cm 2 .Some of these values of t f have been tested experimentally (t f = 6, 8, 10 and 12 mm).However, ANN models fill in the gaps for values that were not present in the experiment and also widen domain of t f values.Appropriate graphical presentation of P u,ann (e) for fixed value of t f and different values of t w might also be created, assuming fixed values of c, σ 0.2, w and σ 0.2, f , Figure 9. Similar estimations of collapse load and its relations with the web thickness or dimensionless parameter t f /t w , i.e.P u,ann (t w ) or P u,ann (t f /t w ), might be made, as well.Such diagrams proved to be interesting, leading to various and important conclusions.
It is important to point out that ANN forecast models provide reliable output only for input data from the domain of

Artificial neural networks (ANN) modelling method
ANN modelling method is based on the analogy with the human nervous system (8) (9) (10) (11).Artificial neuron imitates biological neuron.Artificial neural network (ANN), consisted of artificial neurons, is computational simulation of human neural network, consisted of biological neurons.Humans use their mind to make conclusions and decisions in certain situations based on the previous (similar) experience.ANN does not have human mind and experience that should be used to process input data and make appropriate conclusions/ decisions, i.e. output.In ANN modelling method human mind is replaced by mathematical functions (as much as such replacement is possible) and human experience is replaced by existing data base which is used for ANN training.By training and validation of ANN on some data base, forecast models are created in order to estimate output parameter(s) for certain set of input parameters that is not present in the data base, but that is in the range of data base.ANN modelling method is highly suitable for multi-parameter analysis.

ANN forecast models for collapse load
The basic idea is to estimate the collapse load, P u , as the only output parameter, depending on numerous input parameters (material characteristics, girder geometry and load eccentricity), as well as to asses applicability of ANN modelling method as a tool for collapse load determination in engineering practice and to compare it with more conventional method of empirical expressions (8).
Several types of forecast models were made using experimental data from   2. Graphical interpretation or results dispersions, i.e. discrepancy between experimental and numerical results for both methods is shown in Figure 10.
In both numerical procedures, high difference between experimental and numerical results is observed in girders with same or nearly same flange and web thickness (girders no.33, 35, 44-48, 68-70, 92-96, 104-108, Table 1 and Figure 10), particularly in case of small eccentricities, when experimental ultimate load highly departs from experimental centric ultimate load (girders no.44, 45, 68, 69, 104, 105, Table 1 and Figure 10).This is more emphasised in girders with thinner plates, as in series EB VIII, EB XII, Table 1.Furthermore, in both numerical procedures, high deviation from experimental results happens in girders whose mechanical characteristics of material (flange and web yielding/ ultimate stresses) are not precisely determined, but assumed, as in girders no.54, 77, 78, 92-96, 104-108, 116, Table 1 and Figure 10.The same happens in girders no.105-108, 112-114, 118-120, Table 1 and Figure 10, whose mechanical characteristics of material are determined by tensile test, but have significant discrepancy in comparison with average values of mechanical characteristics of material in other girders.Difference between experimental and numerical results due to mechanical characteristics of material is more prominent in empirical expression than in ANN forecast model.The explanation is in fact that mechanical characteristics of material do not figure in empirical expression, but are considered in ANN forecast model.Hence, the empirical expression does not account with the difference in mechanical characteristics of centrically and eccentrically loaded girders, as really happened in analysed girders and as was taken into account by ANN forecast models.In order to graphically present validation of ANN results, i.e. match with experimental data, in Figure 8 and 9 experimental results are given as separate dots, in colour of corresponding line which presents ANN forecast model results.Experimental data are those from Table 1, with real values of mechanical characteristics of material σ 0.2, w and σ 0.2, f , that may explain slight discrepancy of experimental and ANN results.In Figure 8, experimental data with t w = 5 mm are inserted: series EB V (t f = 10 mm), EB VII (t f = 12 mm), EB XVI (t f = 6 mm) and EB XVII (t f = 8 mm).In Figure 9, experimental data with t f = 10 mm are inserted: series EB V (t w = 5 mm), EB VI (t w = 10 mm) and EB XV (t w = 4 mm).
Future experimental work, as well as inclusion of existing FEM data in training/validation data base, providing wider range of data base, will help improving quality of ANN modelling results.Apart from that, even on the existing experimental data base, results might be improved -by different network architecture, by different choice of data for training and validation set or by means of another ANN software.Presented forecast models (Figure 8 and 9) are representatives of set of forecast models of ANN "c50 -load -2 -10" which provides simplicity and confident results for application in engineering practice.

RESULTS COMPARISON AND CONCLUSION
Assessment of quality of two presented numerical methods for determination of ultimate load (empirical mathematical expressions, represented by Equation [3], Paragraph 2, and ANN forecast models, represented by models of network "c50 -load -2 -10", Paragraph 3) has been done by comparing their results with the experimental results.As summarised in Table 1, comparison is done for set of 120 experimental samples tested in 2001 and 2007.All tested  The fact is that presented methods are not exact solutions and deviation of their results is expected.As long as the deviation is up to the acceptable level, approximate solutions may have practical application.In addition, it has to be pointed out that both presented methods are artificial, in a way, purely mathematical procedures not entering the core of girder collapse problem, not explaining the girder collapse, its mechanism and real, physical happenings in girder in the moment of collapse, immediately before and after the collapse moment.The issue of collapse mode and difference between centric and eccentric collapse mode in eccentrically loaded girder is not tackled by any of these two methods.In both methods mathematical apparatus is applied without taking into consideration collapse mode.Hence, both procedures should be considered as a plain tool, useful in engineering practice as well as in scientific research, but without solving problem of definition of collapse mode in eccentrically patch loaded I-girders.Generally, both methods have satisfying and acceptable, reliable and confident results, as well as simple practical application.However, it is necessary to be aware and to take care about their limits and domains of reliable application.It is also important to keep in mind that equation [3] and ANN "c50 -load -2 -10" are not final solutions for P u determination.Both methods, ANN modelling and empirical expressions, might and should be improved by future experimental and numerical work.

Figure 2 .
Figure 2. Collapse modes typical for centric and eccentric patch loading -centric and eccentric collapse mode.

Figure 3 .
Figure 3.Comparison of experimental and empirical values of reduction factor, R, calculated by equations [2] and [3], for experimental data from 1988.

Figure 4 .
Figure 4. Comparison of experimental and empirical values of reduction factor, R, calculated by equations [2] and [3],for experimental data from 1998.

Figure 5 .
Figure 5.Comparison of experimental and empirical values of reduction factor, R, calculated by equations [2] and [3], for experimental data from 2001.

Figure 7 .
Figure 7.Comparison of FEM and empirical values of R, calculated by Equations [2] and [3] -girders with a = 700 mm.

Figure 9 .
Figure 9. Estimation of collapse load P u,ann (e) for t f = 10 mm and t w = 3 ÷ 10 mm, at c = 50 mm, σ 0.2, w = 28 kN/cm 2 and σ 0.2, f = 28 kN/cm 2 , by means of artificial neural network "c50 -load -2 -10" (square dots present corresponding experimental data for t w = 4, 5 and 10 mm) doi.org/10.3989/ic.13.076  7all created artificial neural networks.Comparison of different networks and forecast models was done in comparison data set, in order to evaluate which models provide the best forecast of collapse load.ANN models were made separately for girders with different load lengths (c = 50 or 150 mm -two load lengths used in experiments from 1998, 2001 and 2007).Herein only load length of c = 50 mm and models with five dimensional inputs (e, t w , t f , σ 0.2, w -web yielding stress, σ 0.2, f -flange yielding stress) and one output (P u ) are considered.The complete experimental data set for girders with load length c = 50 mm consisted of 120 girders tested in 2001 and 2007, all having same dimensions b f , a and h w : b f = 150 mm, a = 700 mm and h w = 700 mm.19 testes were exempted from the network training process and used as a comparison data set, i.e. as data for the evaluation of forecast models.The rest of 101 tests were divided in training data set (71 tests) and validation data set (30 tests).

Table 1 .
Summary of experimental and numerical results -girder characteristics (t w , t f , e, σ 0.2, w and σ 0.2, f ), experimental (P u,exp ) and numerical (P u,num : P u,emp , by equation [3], or P u,ann , by "c50 -load -2 -10") ultimate loads and comparison parameters (Δ num = |P u,num -P u,exp |/P u,exp ; X num = P u,num /P u,exp ).Values are not obtained by tensile test, but estimated as average of σ 0.2 values for other plate thicknesses, determined by tensile tests.

Table 2 .
Statistical parameters as indicators of quality of numerical methods for determination of P u .Values in brackets are for case when ANN forecast model is used only for P u,ann of eccentrically loaded girders, while P u,ann of centrically loaded girders are not forecasted, but considered to be equal to experimental values, i.e. for centrically loaded girders: P u,ann = P u,exp , Δ i,ann = 0 and X i,ann = 1, the same as for empirical expression.)Itmight be concluded that presented ANN forecast model has better statistical indicators, i.e. dispersion of results is lower than for presented empirical expression.However, presented ANN forecast model has narrower range of database used for its creation than presented empirical expression, i.e. empirical expression is formulated for wider domain of some input parameters (e/b f , a/t w , b f /t f , c/a, a/h w ).It is expected to have higher dispersion of results for wider domain of creation database.Hence, ANN forecast model provides more precise results, but empirical expression provides application in wider domain of input parameters.Recommendation for engineering practice would be to combine both methods.