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.
RESUMEN
Las pruebas experimentales muestran que la mayoría de vigas en I de acero cargadas excéntricamente tienen un modo de colapso muy diferente del de las vigas cargadas centradas. En cuanto a la práctica de ingeniería, la diferencia más importante entre los modos de colapso es la carga final. La reducción de la carga última con un aumento de la excentricidad de carga es evidente en algunas vigas cargadas excéntricamente, mientras que bajo ciertas circunstancias, para una determinada combinación de parámetros influyentes, las vigas cargadas excéntricamente se comportan como si se carga en el plano del alma, con ninguna reducción significativa en la carga última debido a la carga excéntrica. Se analizan dos métodos para la determinación de la carga máxima: expresiones matemáticas empíricas y redes neuronales artificiales. Los resultados de ambos procedimientos se comparan. Se dan recomendaciones para su aplicación en la práctica.
KeywordsSteel I-girdereccentric patch loadultimate loadempirical expressionANN forecast modelPalabras claveVigas de acero en Icarga excéntricacarga últimaexpresión empíricamodelo de pronóstico ANNINTRODUCTION
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 elastic-plastic 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.
Patch loaded I-girder – centric and eccentric patch loading.
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/bf. Apart from this parameter, the influence of girder geometry parameters should be studied. Girder dimensions, primarily web and flange thicknesses, tw and tf, as well as ratio tf /tw are of important influence. Other ratios, like bf /tf, a/tw, hw /tw 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 capacity loss in the case of centric load is due to web buckling and local stability loss (centric collapse mode, Figure 2a). In the case of eccentric load, girders lose carrying capacity due to local elastic-plastic bending (eccentric collapse mode, Figure 2b). Three different collapse modes are observed in experimentally tested eccentrically patch loaded steel I-girders: eccentric (1)(2)(5)(6)(7)(8), centric (1)(2)(5)(6)(7)(8) and mixed (8) collapse mode. Mixed collapse mode, having characteristics of both, centric and eccentric collapse modes, may appear in two variants: as centric-mixed or as eccentric-mixed collapse mode, depending on dominant collapse mode characteristics (8).
Collapse modes typical for centric and eccentric patch loading – centric and eccentric collapse mode.
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 girders with eccentric collapse mode. For a certain girder geometry, even the smallest load eccentricity (e = 5 mm, i.e. e/bf = 1/30 in tested girders (8); series EB VIII, EB XII and EB XVIII, Table 1) reduced ultimate load over 40 %. However, eccentrically loaded girders under certain circumstances, for a certain combination of influential parameters, behave as if loaded in the web plane, with no or no significant reduction in ultimate load due to load eccentricity. Even for the highest load eccentricity (e = 25 mm, i.e. e/bf = 1/6 in tested girders (8); series EB I and EB XI, Table 1), girders of certain geometry behaved as if there is no eccentricity.
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.
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 tf /tw and varies linearly with e/bf , 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 ≤ tf/tw ≤ 4 and e/bf ≤ 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 tf/tw and e/bf out of range given in (3). Quality of its results is also dependant on other parameters (bf /tf, a/tw, c/a, a/hw) 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/bf and, same as in the original expression, dependent on the most influential geometry parameter tf /tw.
Additional requirements for values of R calculated by Equation [3] are: 0 < R ≤ 1 and R = 1 for e/bf = 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 ≤ tf /tw ≤ 5, (0) 1/30 ≤ e/bf ≤ 1/5, 45 ≤ a/tw ≤ 233, 10 ≤ bf /tf ≤ 50, c/a = 0.071 or 0.214, a/hw = 1 (1)(2)(3)(5)(6)(7)(8)(9). Range of FEM (ANSYS) data used for the derivation of [3] differs: 1 ≤ tf/tw ≤ 4, (0) 1/25 ≤ e/bf ≤ 1/6.25, 117 ≤ a/tw ≤ 233, 6.25 ≤ bf /tf ≤ 25, 0.036 ≤ c/a ≤ 0.071, 1 ≤ a/hw ≤ 2; with constants: hw = 700 mm, tw = 6 mm, bf = 150 mm, c = 50 mm and variables: e (0, 6, 12, 18, 24 mm), tf (6, 12, 18, 24 mm), a (700, 910, 1050, 1260, 1400 mm) (8)(9).
Improvement of expression [3] in comparison with the expression [2] is illustrated in Figures 3, 4, 5, 6, 7, where Rexp (ratio of experimental ultimate loads of eccentrically and centrically loaded girder, according to equation [1]), RFEM (ratio of FEM ultimate loads of eccentrically and centrically loaded girder, according to equation [1]) and Remp (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 RFEM /Remp – e/bf curves with slightly different numerical values, for each flange thickness, tf. Trend of curves proximity to horizontal line corresponding to value of 1 (Rexp/Remp ≈ 1 and RFEM/Remp ≈ 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.
Comparison of experimental and empirical values of reduction factor, <italic>R</italic>, calculated by equations <xref ref-type="disp-formula" rid="form0002">[2]</xref> and <xref ref-type="disp-formula" rid="form0003">[3]</xref>, for experimental data from 1988.
Comparison of experimental and empirical values of reduction factor, <italic>R</italic>, calculated by equations <xref ref-type="disp-formula" rid="form0002">[2]</xref> and <xref ref-type="disp-formula" rid="form0003">[3]</xref>, for experimental data from 1998.
Comparison of experimental and empirical values of reduction factor, <italic>R</italic>, calculated by equations <xref ref-type="disp-formula" rid="form0002">[2]</xref> and <xref ref-type="disp-formula" rid="form0003">[3]</xref>, for experimental data from 2001.
Comparison of experimental and empirical values of reduction factor, <italic>R</italic>, calculated by equations <xref ref-type="disp-formula" rid="form0002">[2]</xref> and <xref ref-type="disp-formula" rid="form0003">[3]</xref>, for experimental data from 2007 – girders with <italic>t</italic>
<sub><italic>w</italic></sub> = 3 mm.
Comparison of experimental and empirical values of reduction factor, <italic>R</italic>, calculated by equations <xref ref-type="disp-formula" rid="form0002">[2]</xref> and <xref ref-type="disp-formula" rid="form0003">[3]</xref>, for experimental data from 2007 – girders with <italic>t</italic>
<sub><italic>w</italic></sub> = 4 mm.
Comparison of experimental and empirical values of reduction factor, <italic>R</italic>, calculated by equations <xref ref-type="disp-formula" rid="form0002">[2]</xref> and <xref ref-type="disp-formula" rid="form0003">[3]</xref>, for experimental data from 2007 – girders with <italic>t</italic>
<sub><italic>w</italic></sub> = 5 and 10 mm.
Comparison of experimental and empirical values of reduction factor, <italic>R</italic>, calculated by equations <xref ref-type="disp-formula" rid="form0002">[2]</xref> and <xref ref-type="disp-formula" rid="form0003">[3]</xref>, for experimental data from 2007 – girders with <italic>t</italic>
<sub><italic>w</italic></sub> = 6 mm.
Comparison of FEM and empirical values of <italic>R</italic>, calculated by Equations <xref ref-type="disp-formula" rid="form0002">[2]</xref> and <xref ref-type="disp-formula" rid="form0003">[3]</xref> – girders with <italic>a</italic> = 700 mm.
ANN FORECAST MODELSArtificial 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, Pu, 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 1998, 2001 and 2007: with dimensional (e.g. e, tf, tw) and dimensionless (e.g. e/bf, e/tf, e/tw, bf /tf, tf /tw, hw /tw) geometry inputs. Several types of network architecture were constructed: with one or two hidden levels of neurons; with different number of neurons – 1 to 20 – in each level, depending on inputs number and number of training data. The same computer software (8)(9)(10)(11) was used for training of all 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, tw, tf, σ0.2, w – web yielding stress, σ0.2, f – flange yielding stress) and one output (Pu) 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 bf, a and hw: bf = 150 mm, a = 700 mm and hw = 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).
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 Pu,ann(e) for tw = 5 mm and different values of tf = 5 ÷ 15 mm, all at fixed c = 50 mm, σ0.2, w = 28 kN/cm2 and σ0.2, f = 28 kN/cm2. Some of these values of tf have been tested experimentally (tf = 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 tf values. Appropriate graphical presentation of Pu,ann(e) for fixed value of tf and different values of tw might also be created, assuming fixed values of c, σ0.2, w and σ0.2, f , Figure 9. Similar estimations of collapse load and its relations with the web thickness or dimensionless parameter tf/tw, i.e. Pu,ann(tw) or Pu,ann(tf/tw), 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 network training database. Herein that is range of experimental data from 2001 and 2007, Table 1: 1 ≤ tf/tw ≤ 5, 1/30 ≤ e/bf ≤ 1/6, 70 ≤ a/tw ≤ 233, 10 ≤ bf /tf ≤ 50, c/a = 0.071, a/hw = 1; with constants: a = hw = 700 mm, bf = 150 mm, c = 50 mm and variables: e (0, 5, 10, 15, 20, 25 mm), tf (3, 4, 6, 8, 9, 10, 12, 15 mm), tw (3, 4, 5, 6, 8, 10 mm) (8).
Estimation of collapse load <italic>P</italic>
<sub><italic>u,ann </italic></sub>
<italic>(e)</italic> for <italic>t</italic>
<sub><italic>w</italic></sub> = 5 mm and <italic>t</italic>
<sub><italic>f </italic></sub>= 5 ÷ 15 mm, at <italic>c</italic> = 50 mm, <italic>σ</italic>
<sub><italic>0.2, w</italic></sub> = 28 kN/cm<sup>2</sup> and <italic>σ</italic>
<sub><italic>0.2, f </italic></sub>= 28 <italic>kN</italic>/<italic>cm</italic>2,by means of artificial neural network <italic>“c50 – load – 2 – 10”</italic> (square dots present corresponding experimental data for <italic>t</italic>
<sub><italic>f </italic></sub>= 6, 8, 10 and 12 mm).
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 σ0.2, w and σ0.2, f , that may explain slight discrepancy of experimental and ANN results. In Figure 8, experimental data with tw = 5 mm are inserted: series EB V (tf = 10 mm), EB VII (tf = 12 mm), EB XVI (tf = 6 mm) and EB XVII (tf = 8 mm). In Figure 9, experimental data with tf = 10 mm are inserted: series EB V (tw = 5 mm), EB VI (tw = 10 mm) and EB XV (tw = 4 mm).
Estimation of collapse load <italic>P</italic>
<sub><italic>u,ann </italic></sub>
<italic>(e)</italic> for <italic>t</italic>
<sub><italic>f</italic></sub> = 10 mm and <italic>t</italic>
<sub><italic>w </italic></sub>= 3 ÷ 10 mm, at <italic>c</italic> = 50 mm, <italic>σ</italic>
<sub><italic>0.2, w</italic></sub> = 28 kN/cm<sup>2</sup> and <italic>σ</italic>
<sub><italic>0.2, f </italic></sub>= 28 kN/cm<sup>2</sup>,by means of artificial neural network <italic>“c50 – load – 2 – 10”</italic> (square dots present corresponding experimental data for <italic>t</italic>
<sub><italic>w</italic></sub> = 4, 5 and 10 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 girders have dimensions: bf = 150 mm, a = 700 mm and hw = 700 mm. Load length is c = 50 mm. Variables are: e, tf and tw, as well as dimensionless parameters (e.g. e/bf, e/tf, e/tw, tf /tw, bf /tf, hw /tw etc). Real mechanical characteristics of material σ0.2, w and σ0.2, f , obtained by tensile test, are taken into account.
Summary of experimental and numerical results – girder characteristics (<italic>t</italic>
<sub><italic>w</italic></sub>, <italic>t</italic>
<sub><italic>f</italic></sub>, <italic>e</italic>, σ<sub><italic>0.2, w</italic></sub> and σ<sub><italic>0.2, f </italic></sub>), experimental (<italic>P</italic>
<sub><italic>u,exp</italic></sub>) and numerical (<italic>P</italic>
<sub><italic>u,num</italic></sub>: <italic>P</italic>
<sub><italic>u,emp</italic></sub>, by equation <xref ref-type="disp-formula" rid="form0003">[3]</xref>, or <italic>P</italic>
<sub><italic>u,ann</italic></sub>, by <italic>“c50 – load – 2 – 10”</italic>) ultimate loads and comparison parameters (Δ<sub><italic>num</italic></sub> = |<italic>P</italic>
<sub><italic>u,num</italic></sub> – <italic>P</italic>
<sub><italic>u,exp</italic></sub>
<sub> </sub>|/<italic>P</italic>
<sub><italic>u,exp</italic></sub>; <italic>X</italic>
<sub><italic>num</italic></sub> = <italic>P</italic>
<sub><italic>u,num</italic></sub>
<sub> </sub>/<italic>P</italic>
<sub><italic>u,exp</italic></sub>).
No
Girder
tw [mm]
tf [mm]
e [mm]
σ0.2, w [kN/cm2]
σ0.2, f [kN/cm2]
Pu,exp [kN]
Pu,emp [kN]
Δemp [%]
Xemp
Pu,ann [kN]
Δann [%]
Xann
1
EB I - 1
3.03
14.75
0
32.73
26.87
133
133
0.0
1.00
136
2.3
1.02
2
EB I - 2
3.03
14.72
5
32.73
26.87
128
133
3.9
1.04
129
0.8
1.01
3
EB I - 3
3.03
14.77
10
32.73
26.87
127
133
4.7
1.05
131
3.1
1.03
4
EB I - 4
3.03
14.74
15
32.73
26.87
135
133
1.5
0.99
134
0.7
0.99
5
EB I - 5
3.04
14.81
20
32.73
26.87
134
133
0.7
0.99
134
0.0
1.00
6
EB I - 6
3.04
14.80
25
32.73
26.87
124
123
0.5
0.99
125
0.8
1.01
7
EB II - 1
5.95
14.71
0
30.93
26.87
340
340
0.0
1.00
339
0.3
1.00
8
EB II - 2
5.95
14.78
5
30.93
26.87
320
322
0.5
1.00
325
1.6
1.02
9
EB II - 3
5.97
14.73
10
30.93
26.87
326
296
9.1
0.91
311
4.6
0.95
10
EB II - 4
5.97
14.71
15
30.93
26.87
296
269
9.0
0.91
280
5.4
0.95
11
EB II - 5
6.00
14.78
20
30.93
26.87
243
241
0.9
0.99
238
2.1
0.98
12
EB II - 6
6.00
14.81
25
30.93
26.87
197
210
6.8
1.07
192
2.5
0.97
13
EB III - 1
5.95
14.78
0
30.93
26.87
342
342
0.0
1.00
339
0.9
0.99
14
EB III - 2
5.95
14.80
5
30.93
26.87
321
324
0.9
1.01
324
0.9
1.01
15
EB III - 3
6.01
14.81
10
30.93
26.87
301
298
1.0
0.99
314
4.3
1.04
16
EB III - 4
6.01
14.79
15
30.93
26.87
267
271
1.3
1.01
282
5.6
1.06
17
EB III - 5
6.02
14.78
20
30.93
26.87
228
242
6.1
1.06
239
4.8
1.05
18
EB III - 6
6.02
14.81
25
30.93
26.87
187
211
12.7
1.13
193
3.2
1.03
19
EB IV - 1
7.98
14.83
0
26.24
26.87
401
401
0.0
1.00
495
23.4
1.23
20
EB IV - 2
7.98
14.77
5
26.24
26.87
418
349
16.6
0.83
420
0.5
1.00
21
EB IV - 3
7.98
14.78
10
26.24
26.87
394
300
23.9
0.76
363
7.9
0.92
22
EB IV - 4
7.98
14.77
15
26.24
26.87
301
259
14.1
0.86
303
0.7
1.01
23
EB IV - 5
7.97
14.77
20
26.24
26.87
245
224
8.6
0.91
244
0.4
1.00
24
EB IV - 6
7.97
14.77
25
26.24
26.87
209
198
5.2
0.95
195
6.7
0.93
25
EB V - 1
5.08
10.28
0
27.09
30.97
229
229
0.0
1.00
232
1.3
1.01
26
EB V - 2
5.08
10.35
5
27.09
30.97
212
205
3.4
0.97
212
0.0
1.00
27
EB V - 3
5.10
10.26
10
27.09
30.97
197
179
9.0
0.91
196
0.5
0.99
28
EB V - 4
5.10
10.26
15
27.09
30.97
175
157
10.2
0.90
176
0.6
1.01
29
EB V - 5
5.12
10.25
20
27.09
30.97
153
138
9.8
0.90
154
0.7
1.01
30
EB V - 6
5.12
10.29
25
27.09
30.97
129
122
5.5
0.95
134
3.9
1.04
31
EB VI - 1
10.33
10.29
0
30.97
30.97
720
720
0.0
1.00
701
2.6
0.97
32
EB VI - 2
10.33
10.27
5
30.97
30.97
575
553
3.9
0.96
579
0.7
1.01
33
EB VI - 3
10.23
10.21
10
30.97
30.97
365
414
13.5
1.13
430
17.8
1.18
34
EB VI - 4
10.23
10.22
15
30.97
30.97
313
313
0.1
1.00
342
9.3
1.09
35
EB VI - 5
10.26
10.24
20
30.97
30.97
275
245
11.0
0.89
276
0.4
1.00
36
EB VI - 6
10.26
10.23
25
30.97
30.97
220
213
3.4
0.97
217
1.4
0.99
37
EB VII - 1
4.98
12.20
0
27.09
28.88
230
230
0.0
1.00
227
1.3
0.99
38
EB VII - 2
4.98
12.21
5
27.09
28.88
225
217
3.6
0.96
220
2.2
0.98
39
EB VII - 3
5.11
12.19
10
27.09
28.88
212
197
7.0
0.93
218
2.8
1.03
40
EB VII - 4
5.11
12.19
15
27.09
28.88
180
179
0.7
0.99
196
8.9
1.09
41
EB VII - 5
5.02
12.17
20
27.09
28.88
170
162
5.0
0.95
167
1.8
0.98
42
EB VII - 6
5.02
12.19
25
27.09
28.88
149
141
5.2
0.95
144
3.4
0.97
43
EB VIII - 1
2.98
3.05
0
27.44
27.44
79
79
0.0
1.00
72
8.9
0.91
44
EB VIII - 2
2.98
2.98
5
27.44
27.44
44
60
37.3
1.37
52
18.2
1.18
45
EB VIII - 3
2.98
2.99
10
27.44
27.44
37
45
22.0
1.22
40
8.1
1.08
46
EB VIII - 4
2.98
3.05
15
27.44
27.44
29
34
18.1
1.18
34
17.2
1.17
47
EB VIII - 5
3.08
2.97
20
27.44
27.44
23
26
12.0
1.12
31
34.8
1.35
48
EB VIII - 6
3.08
3.00
25
27.44
27.44
20
23
13.8
1.14
29
45.0
1.45
49
EB IX - 1
3.01
5.69
0
27.44
28.73(*)
95
95
0.0
1.00
105
10.5
1.11
50
EB IX - 2
3.01
5.77
5
27.44
28.73(*)
80
83
4.4
1.04
88
10.0
1.10
51
EB IX - 3
3.00
5.72
10
27.44
28.73(*)
69
72
4.6
1.05
72
4.3
1.04
52
EB IX - 4
3.00
5.70
15
27.44
28.73(*)
57
63
9.7
1.10
57
0.0
1.00
53
EB IX - 5
3.26
5.76
20
27.44
28.73(*)
47
51
9.3
1.09
49
4.3
1.04
54
EB IX - 6
3.26
5.68
25
27.44
28.73(*)
39
45
14.8
1.15
41
5.1
1.05
55
EB X - 1
2.98
8.92
0
27.44
28.20
102
102
0.0
1.00
110
7.8
1.08
56
EB X - 2
2.98
9.04
5
27.44
28.20
105
102
2.9
0.97
104
1.0
0.99
57
EB X - 3
3.02
8.92
10
27.44
28.20
107
99
7.5
0.92
97
9.3
0.91
58
EB X - 4
3.02
8.97
15
27.44
28.20
90
93
3.5
1.04
88
2.2
0.98
59
EB X - 5
2.97
8.93
20
27.44
28.20
85
86
1.0
1.01
76
10.6
0.89
60
EB X - 6
2.97
8.89
25
27.44
28.20
70
74
6.2
1.06
66
5.7
0.94
61
EB XI - 1
2.97
12.17
0
27.44
28.88
116
116
0.0
1.00
112
3.4
0.97
62
EB XI - 2
2.97
12.12
5
27.44
28.88
113
116
2.7
1.03
109
3.5
0.96
63
EB XI - 3
3.05
12.17
10
27.44
28.88
115
116
0.9
1.01
114
0.9
0.99
64
EB XI - 4
3.05
12.18
15
27.44
28.88
110
116
5.5
1.05
114
3.6
1.04
65
EB XI - 5
3.03
12.17
20
27.44
28.88
105
116
10.5
1.10
110
4.8
1.05
66
EB XI - 6
3.03
12.19
25
27.44
28.88
115
100
12.9
0.87
105
8.7
0.91
67
EB XII - 1
3.88
3.88
0
28.50
28.50
120
120
0.0
1.00
110
8.3
0.92
68
EB XII - 2
3.88
3.86
5
28.50
28.50
70
92
31.0
1.31
80
14.3
1.14
69
EB XII - 3
3.91
3.93
10
28.50
28.50
50
69
37.3
1.37
60
20.0
1.20
70
EB XII - 4
3.91
3.95
15
28.50
28.50
45
52
14.9
1.15
46
2.2
1.02
71
EB XII - 5
3.95
3.92
20
28.50
28.50
40
40
0.1
1.00
38
5.0
0.95
72
EB XII - 6
3.95
3.93
25
28.50
28.50
35
35
0.5
1.01
34
2.9
0.97
73
EB XIII - 1
3.87
5.69
0
28.50
28.73(*)
125
125
0.0
1.00
133
6.4
1.06
74
EB XIII - 2
3.87
5.69
5
28.50
28.73(*)
110
103
6.4
0.94
107
2.7
0.97
75
EB XIII - 3
3.87
5.69
10
28.50
28.73(*)
86
84
2.5
0.97
85
1.2
0.99
76
EB XIII - 4
3.87
5.71
15
28.50
28.73(*)
68
69
1.7
1.02
66
2.9
0.97
77
EB XIII - 5
3.87
5.80
20
28.50
28.73(*)
50
59
17.9
1.18
54
8.0
1.08
78
EB XIII - 6
3.87
5.67
25
28.50
28.73(*)
45
51
13.7
1.14
44
2.2
0.98
79
EB XIV - 1
3.89
8.07
0
28.50
30.03
140
140
0.0
1.00
150
7.1
1.07
80
EB XIV - 2
3.89
8.14
5
28.50
30.03
129
126
2.3
0.98
136
5.4
1.05
81
EB XIV - 3
3.93
8.22
10
28.50
30.03
130
112
13.9
0.86
125
3.8
0.96
82
EB XIV - 4
3.93
8.21
15
28.50
30.03
100
99
1.3
0.99
110
10.0
1.10
83
EB XIV - 5
3.91
8.08
20
28.50
30.03
86
86
0.4
1.00
92
7.0
1.07
84
EB XIV - 6
3.91
8.19
25
28.50
30.03
75
77
2.4
1.02
80
6.7
1.07
85
EB XV - 1
3.91
10.13
0
28.50
30.97
155
155
0.0
1.00
153
1.3
0.99
86
EB XV - 2
3.91
10.19
5
28.50
30.97
148
149
0.7
1.01
146
1.4
0.99
87
EB XV - 3
3.93
10.22
10
28.50
30.97
140
139
0.4
1.00
143
2.1
1.02
88
EB XV - 4
3.93
10.34
15
28.50
30.97
138
129
6.3
0.94
137
0.7
0.99
89
EB XV - 5
3.90
10.19
20
28.50
30.97
128
115
9.8
0.90
123
3.9
0.96
90
EB XV - 6
3.90
10.17
25
28.50
30.97
115
101
12.3
0.88
111
3.5
0.97
91
EB XVI - 1
5.01
5.75
0
27.09
28.73(*)
187
187
0.0
1.00
172
8.0
0.92
92
EB XVI - 2
5.01
5.74
5
27.09
28.73(*)
130
146
12.6
1.13
135
3.8
1.04
93
EB XVI - 3
4.95
5.71
10
27.09
28.73(*)
105
113
7.8
1.08
103
1.9
0.98
94
EB XVI - 4
4.95
5.72
15
27.09
28.73(*)
74
88
19.5
1.20
79
6.8
1.07
95
EB XVI - 5
4.97
5.70
20
27.09
28.73(*)
59
70
19.4
1.19
61
3.4
1.03
96
EB XVI - 6
4.97
5.62
25
27.09
28.73(*)
55
61
11.4
1.11
49
10.9
0.89
97
EB XVII - 1
5.04
8.14
0
27.09
30.03
209
209
0.0
1.00
212
1.4
1.01
98
EB XVII - 2
5.04
8.21
5
27.09
30.03
200
176
11.8
0.88
186
7.0
0.93
99
EB XVII - 3
4.98
8.12
10
27.09
30.03
145
147
1.3
1.01
159
9.7
1.10
100
EB XVII - 4
4.98
8.13
15
27.09
30.03
130
123
5.0
0.95
136
4.6
1.05
101
EB XVII - 5
4.96
8.08
20
27.09
30.03
98
105
7.6
1.08
112
14.3
1.14
102
EB XVII - 6
4.96
8.12
25
27.09
30.03
83
94
12.8
1.13
93
12.0
1.12
103
EB XVIII - 1
5.75
5.73
0
28.73(*)
28.73(*)
335
335
0.0
1.00
217
35.2
0.65
104
EB XVIII - 2
5.75
5.75
5
28.73(*)
28.73(*)
200
257
28.6
1.29
166
17.0
0.83
105
EB XVIII - 3
5.99
5.71
10
45.83
28.73(*)
130
189
45.3
1.45
129
0.8
0.99
106
EB XVIII - 4
5.99
5.73
15
45.83
28.73(*)
104
140
34.3
1.34
99
4.8
0.95
107
EB XVIII - 5
6.04
5.73
20
45.83
28.73(*)
88
109
23.6
1.24
82
6.8
0.93
108
EB XVIII - 6
6.04
5.72
25
45.83
28.73(*)
69
94
36.9
1.37
70
1.4
1.01
109
EB XIX - 1
5.99
8.83
0
45.83
28.20
330
330
0.0
1.00
330
0.0
1.00
110
EB XIX - 2
5.99
8.96
5
45.83
28.20
285
274
3.8
0.96
260
8.8
0.91
111
EB XIX - 3
6.07
8.97
10
45.83
28.20
217
224
3.4
1.03
204
6.0
0.94
112
EB XIX - 4
6.07
8.97
15
45.83
28.20
155
185
19.5
1.19
158
1.9
1.02
113
EB XIX - 5
5.97
8.95
20
45.83
28.20
125
155
24.3
1.24
125
0.0
1.00
114
EB XIX - 6
5.97
8.87
25
45.83
28.20
107
138
29.2
1.29
104
2.8
0.97
115
EB XX - 1
5.67
12.14
0
28.73(*)
28.88
395
395
0.0
1.00
302
23.5
0.76
116
EB XX - 2
5.67
12.13
5
28.73(*)
28.88
365
358
1.9
0.98
279
23.6
0.76
117
EB XX - 3
6.00
12.21
10
45.83
28.88
311
311
0.1
1.00
310
0.3
1.00
118
EB XX - 4
6.00
12.16
15
45.83
28.88
235
273
16.2
1.16
253
7.7
1.08
119
EB XX - 5
5.99
12.22
20
45.83
28.88
202
241
19.5
1.20
205
1.5
1.01
120
EB XX - 6
5.99
12.17
25
45.83
28.88
165
212
28.7
1.29
164
0.6
0.99
(* – Values are not obtained by tensile test, but estimated as average of σ0.2 values for other plate thicknesses, determined by tensile tests.
blue highlight
– discrepancy > 11 %, i.e. 11 % < Δnum ≤ 20 %;
yellow highlight
– discrepancy > 20 %, i.e. 20% < Δnum ≤ 30 %;
red highlight
– discrepancy Δnum >30 %;
grey highlight
– significant discrepancy for centric load.
Statistical indicators of methods quality are summarised in Table 2. Graphical interpretation or results dispersions, i.e. discrepancy between experimental and numerical results for both methods is shown in Figure 10.
Statistical parameters as indicators of quality of numerical methods for determination of <italic>P</italic>
<sub><italic>u</italic></sub> .
Parameter
empirical expressionEquation [3]
ANN “c50 – load – 2 – 10” forecast model
8.49 %
6.06 % (4.77 %)(#)
1.04
1.01 (1.01)
12.34 %
9.47 % (8.08 %)
11.86 %
9.37 % (7.07 %)
(# - Values in brackets are for case when ANN forecast model is used only for Pu,ann of eccentrically loaded girders, while Pu,ann of centrically loaded girders are not forecasted, but considered to be equal to experimental values, i.e. for centrically loaded girders: Pu,ann = Pu,exp , Δi,ann = 0 and Xi,ann = 1, the same as for empirical expression.)
Ratios <italic>P</italic>
<sub><italic>u,emp </italic></sub>
<italic>/P</italic>
<sub><italic>u,exp </italic></sub>and <italic>P</italic>
<sub><italic>u,ann </italic></sub>
<italic>/P</italic>
<sub><italic>u,exp </italic></sub>for tested girders and their average values <italic>X</italic>
<sub><italic>av,emp </italic></sub>and <italic>X</italic>
<sub><italic>av,ann</italic></sub>
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.
Significant differences (>10 %) between experimental and primarily empirical values in girders no. 18, 20-22, 66, 81, 90, 98, 101, 102, Table 1 and Figure 10, which are not due to upper elaborated reasons, might be explained either by unpredictable and unexpected behaviour of experimentally tested girders, by unavoidable experimental variations (associated with human factors, limitations of available equipment and testing procedure, inherent variability in materials, dimensions etc) or by imperfections of proposed numerical procedures. The same may be said for reasonable results discrepancies of less than 10 %, in all the rest girders.
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.
It might 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/bf, a/tw, bf /tf, c/a, a/hw). 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.
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 Pu determination. Both methods, ANN modelling and empirical expressions, might and should be improved by future experimental and numerical work.
ACKNOWLEDGMENTS
Special thanks to Coimbra Group and CGHS (Coimbra Group Hospitality Scheme) programme which enabled cooperation and join research work of scientists from the University of Montenegro and the University of Granada.
REFERENCESElgaalyM.NunanW.Behavior of rolled section web under eccentric edge compressive loadsJournal of Structural Engineering1989115715611578doi: http://dx.doi.org/10.1061/(ASCE)0733-9445(1989)115:7(1561)ElgaalyM.SalkarR.K.Behavior of webs under eccentric compressive edge loadsIUTAM SymposiumPrague1990GalambosT.V.Guide to stability design criteria for metal structures1998John Wiley & Sons, Inc.New YorkDrdackyM.On two particular problems of plate girders webs under partial edge loadsJournal of Constructional Steel Research1991203183190doi: http://dx.doi.org/10.1016/0143-974X(91)90031-ULučićD.Experimental Research: Thin-Walled I Girders Subjected to Centric and Eccentric Patch LoadingExperimental Researches20011University of Montenegro - Faculty of Civil EngineeringPodgoricaLučićD.ŠćepanovićB.Experimental Investigation on Locally Pressed I-beams Subjected to Eccentric Patch LoadingJournal of Constructional Steel Research2004603-5525534doi: http://dx.doi.org/10.1016/S0143-974X(03)00127-5ŠćepanovićB.I-girders under eccentric local loading – experimental-theoretical analysis2002University of BelgradeBelgradeMaster thesisŠćepanovićB.Analysis of eccentrically locally loaded steel I-girders2009University of Montenegro and University of GranadaPodgoricaDoctoral thesisŠćepanovićB.Gil-MartínL.M.Hernández MontesE.AschheimM.LučićD.Ultimate strength of I-girders under eccentric patch loading: Derivation of a new strength reduction coefficientEngineering Structures200931714031413doi: http://dx.doi.org/10.1016/j.engstruct.2009.02.005Gil MartínL.M.ŠćepanovićB.Hernández-MontesE.AschheimM.LučićD.Eccentrically Patch Loaded Steel I-Girders: Influence of Patch Length on Ultimate StrengthJournal of Constructional Steel Research2010665716722doi: http://dx.doi.org/10.1016/j.jcsr.2009.12.012KneževićM.Risk management during realisation of the civil engineering projects2004University of BelgradeBelgradeDoctoral thesis