## Abstract

Direct resistance check by applying advanced numerical models is getting increasingly used for the design of steel slender plated structures. This method has to take into account the same uncertainties as traditional analytical design calculations and should ensure the Eurocode-based prescribed safety level. The application of the model factor gives the possibility to account for the model-related uncertainties. The current study focuses on the determination of the model factor for one specific failure mode, the patch loading resistance. Numerical model has been developed and validated based on laboratory test results. To evaluate the model uncertainties, physically possible modeling differences are introduced, and their effects are evaluated on the resistance. The final aim of the study is to determine the model factor for the analyzed girder type and failure mode based on statistical evaluation.

## 1 Introduction

### 1.1 Patch loading resistance of steel plated structures

With the increased use of numerical models in the civil engineering design praxis, a new alternative in the design of steel structures is possible called as direct resistance check. Nowadays, this is an increasingly common and emerging method used in the structural engineering practice especially for the steel structural design; but its application for reinforced concrete structures also increases. In this case, a geometrically and materially nonlinear analysis is performed, imperfections are directly considered in the model, and the calculation result of the analysis is the nonlinear load-deformation path, from which the characteristic and design values of the load carrying capacity can be determined. This design method should consider the same uncertainties as traditional analytical calculations and provide the same level of safety. In the EN 1993-1-5:2006 [1], on the resistance side, in addition to the material uncertainty, the uncertainties of the numerical model must also be considered. To cover it, the model factor is introduced in prEN 1993-1-14:2020 [2], which determination is a specific task, to be made for all failure modes and design methods. In this study one specific failure mode is investigated, the patch loading resistance of slender plated structures and the model factor is determined for this specific design situation. The patch loading failure can often be a common failure mode in slender steel structures at support region or during launching of a bridge. Notations on the analyzed girder geometry used in the study are shown in Fig. 1. It is a common geometry, welded stiffened plates are widely used in the civil engineering practice [3].

The relative stiffness of the longitudinal rib can be calculated by using Eq. (2), where *I*_{sl} is the inertia of the rib to the midline of the web, and the notation of the geometric parameters as it is shown in Fig. 1. If the stiffener stiffness is less than this value, the failure mode may be global buckling of the entire web or an interaction of global and local buckling. The three different failure modes are illustrated in Fig. 2.

*k*

_{F}of the EN 1993-1-5:2006 [1] design equations have been changed to Eq. (3) and the calculation method of

*l*

_{y}is changed to Eq. (4) for the case of girders having equally placed open-section longitudinal stiffeners on the web. This design method will be the bases of the evaluation within the current study,

### 1.2 Laboratory experiments at BUTE

Eight large-scale specimens are tested in the laboratory of the Department of Structural Engineering, Budapest University of Technology and Economics (BUTE), between 2015 and 2016. The aim of the test was to determine the patch loading resistance of steel girders with stiffened webs having two or three uniformly placed longitudinal stiffeners. Four different specimen geometries are tested; the test details are described by Mecséri et al. [4]. There were specimens tested without longitudinal stiffeners as reference tests and there were three different stiffener configurations checked, representing different relative stiffener stiffness values. Each type of the specimens is loaded by load-introduction lengths of 100 and 200 mm. The web panel, totaling 1,000 mm in length, features rigid connection plates at both ends. The flanges always have a dimension of 150–10 mm, the web height is 500 mm with a thickness of 4 mm. For specimens equipped with two longitudinal stiffeners, the sub-panel depth is 165 mm, accompanied by stiffeners sized at 40-4 mm. Specimens with three longitudinal stiffeners have a sub-panel depth of 123 mm, with stiffener sizes of 40-4 mm and 60-4 mm, respectively. These stiffeners are positioned on only one side of the web panel. The girders are simply supported and supported against lateral torsional buckling at the supports and at the load introduction location. The results of this laboratory test program are used to assess the model uncertainties within the current study.

### 1.3 Model factor determination method

The uncertainties inherent in the numerical model can be managed through either stochastic analysis or statistical assessment or by employing partial factor in the design process. To streamline design tasks, the concept of the model factor *γ*_{FE} was introduced in prEN 1993-1-14:2020 [2]. This factor establishes a partial safety factor specifically addressing the uncertainty associated with the numerical model utilized in the investigations. According to prEN 1993-1-14:2020 [2], the model factor aims to account for uncertainties related to the numerical model and the type of analysis conducted. It is of paramount importance to underscore that the model factor does not override or replace any other existing partial safety factors. Instead, it functions in tandem with them, providing an additional layer of safety assurance. Specifically, the model factor is designed to address and account for uncertainties inherent to the numerical model itself. The model factor plays a complementary role, ensuring a more comprehensive safety evaluation by focusing on aspects that might otherwise remain unaddressed. As per the design specifications in prEN 1993-1-14:2020 [2], the model factor's determination involves comparing the numerical calculation (*R*_{check}) against known test results (*R*_{test,known}) or against established resistances obtained through approved methods (*R*_{k,known}). In this study, *R*_{known} is the analytically determined resistance value (characteristic value), to which the results of numerical calculations are compared.

*R*

_{test,known}or

*R*

_{k,known}) is available, and numerical analyses are performed for each case (

*R*

_{check}), the determination of the model factor can involve utilizing statistical assessment in line with EN 1990:2002 Annex D [10]. This process involves establishing the model factor by assessing the ratio (

*R*

_{k,known}/

*R*

_{check}or

*R*

_{test,known}/

*R*

_{check}) for each case. Subsequently, the mean value (

*m*

_{X}) and the coefficient of variation (

*V*

_{X}) for these analyzed ratios are computed. Through statistical evaluation of these ratios, the model factor can be derived as Eq. (5). For deriving the characteristic resistance from the results of the numerical analysis, Eq. (6) is applied, where

*R*

_{GMNIA}represents the resistance obtained from the deterministic Geometrical and Material Nonlinear Imperfect Analysis (GMNIA),

*m*

_{X}is the mean value of the ratio of the measured (or known) to computed results for

*n*samples;

*k*

_{n}is the characteristic fractile factor according to EN 1990:2002 [12], Annex D, Table D.1Table D.1 (data row corresponding to

*V*

_{X}unknown should be used, in this study it is taken to 1.64),

*V*

_{X}is the coefficient of variation of the ratio of the measured (or known) to computed results for

*n*samples.

The model factor is a relatively new design parameter, which determination is clearly defined in design standards as presented above. However, the calculated value of the model factor for different failure modes is still missing from the international literature for the steel structural design. In the case of concrete structural design, there are predefined values to determine the failure mode dependent model factor. Similar research is completely missing from the international research for steel structures. The current paper tries to solve this problem and investigates the calculation method of the model factor for one specific failure mode.

## 2 Research strategy and numerical modeling

### 2.1 Numerical model development

An advanced finite element model is developed using ANSYS 19.2 software [13]. The numerical model is validated by laboratory experiments, it is proved that the failure mode is the same in physical and virtual experiments and the load capacity values are recovered with sufficient accuracy described in detail by Mecséri [4]. A verification of the model is also performed, which is practically a mesh sensitivity analysis. The Finite Element (FE) model uses four-node thin shell elements, namely Shell 181 of the ANSYS software. Figure 3 shows the meshed numerical model together with the boundary conditions. The girder is always supported at both ends against displacement and the end plates against lateral displacement along the complete height. Loads are defined in the middle of the specimens uniformly distributed over the load introduction length (*s*_{s}).

Linear elastic and hardening plastic material model is used, so the material is linear elastic until reaching the yield strength by a Young modulus of 210000 MPa. The yield plateau is modeled to 1% plastic strains and from the end of the plateau it follows linear hardening until reaching the ultimate strength by 15%. In the study (except for validation), the yield strength, and ultimate strength of the steel are assumed to be characteristic values. The ultimate loads are determined by geometrical and material nonlinear analysis using equivalent geometric imperfections. EN 1993-1-5:2006, Annex C [1] gives guidance on the use of FE methods. It specifies that the numerical model should incorporate both initial global and local (plate) imperfections. It is required to select one type of imperfection as the primary imperfection, and the associated imperfections may be scaled down to 70% [14]. Based on Mecséri et al. [4] the effect of global imperfection is negligible, and therefore only sub-panel imperfection with amplitude *b*_{1}/200 is considered. Full Newton-Raphson approach is used in the nonlinear analysis with 0.1% convergence tolerance of the residual force based Euclidian norm.

### 2.2 Model uncertainties considered

During the numerical model development, several physically possible and reasonable modeling variations are introduced and their impact on the calculated patch loading resistance is analyzed. There are differences in the way forces/loads are applied. The concentrated load can be applied along a single line over the web, or with a small extension on either side of the web 20-20 mm wide, as it is shown in Fig. 4.

The designer can support these loaded nodes laterally and prevent them from being rotated around the longitudinal axis and can ignore this constraint. It is also investigated what happens if rotation Degree Of Freedom (DOF) of the nodes is connected along the length of the load application so that they can only twist together, as it is shown in Fig. 5. The modeling of these different loading and supporting conditions can have physical meaning, thus the load introduction plate always have a certain stiffness and support in lateral direction, which makes differences in its application in the numerical model, which gives significant uncertainty to the model.

The next uncertainty comes from imperfections. The standard allows using hand-defined imperfections and also to use the first eigenmode shape scaled up to the appropriate amplitude. In this study, only equivalent geometric imperfection is used, so the residual stresses and the geometric imperfection are taken into account combined. Figure 6a shows the scaled hand-defined imperfection shape, which is applied in two ways, one way for the wave to start in one direction (later referred to as “A”) and the other way for the wave to start in the other direction (later referred to as “B”). As a second imperfection definition option, the first eigenvalue is shown in Fig. 6b). For better visibility, highly over scaled imperfections are shown in Fig. 6. In the numerical parametric study, the maximum imperfection magnitude is set equal to *e*_{0} = *b*_{i}/200.

The next uncertainty comes from the possibilities of solver settings. Currently, there is no exact specification of which load steps should be used for the nonlinear calculation. So, it needs to be investigated how much effect this has on the load capacity if failure occurs at 85% or at 95% of the applied load and always define the same percentage of the applied load as the maximum and minimum load steps.

## 3 Results and discussion

### 3.1 Determination of the model factor

Using a combination of these uncertainties, five girders having different slenderness values are tested and the load-bearing capacity of each girder is determined separately for a total of 320 cases. The results showed that the largest effect is due to whether a force or displacement-controlled method is used. The latter consistently resulted in at least ∼9%, and in the most extreme case 20%, higher patch loading resistance.

The second largest effect is whether the nodes are supported laterally and prevented from rotating in the environment of the force application. If so, an additional load-bearing capacity of approximately 4–6% is obtained. Figure 7a shows all the results in comparison to the Eurocode buckling curve, and Fig. 7b indicates those, which come from force-controlled analysis.

The final objective of the study is to determine the appropriate model factor for a given beam type and failure mode based on the statistical evaluation of the numerical results. The load carrying capacity values are determined analytically using Kövesdi's improved analytical formula [6–7] described in section 1.1 and compared with the numerically obtained load capacity values. The ratio of the analytically calculated results to numerical result for all samples is determined. The model factor is determined based on EN 1990:2002 Annex D standard [12], as described in section 1.3. The calculation is performed in different groupings and the results are shown in Fig. 8. It is defined both per slenderness and collectively for three cases:

- i)including all results;
- ii)considering only the results of force-controlled calculations; and
- iii)considering only force-controlled calculations and the nodes of force introduction are supported as described in Section 2.2.

As it is illustrated in Figs 7 and 8, it has a large effect whether the calculation is force or displacement controlled and it is therefore proposed to define and apply the model factor separately for the two cases.

## 4 Conclusion

In the current study, the uncertainty of numerical models is studied, and the model factor is determined for a specific beam type and failure mode. Highlighting only the most relevant results, the numerical study presented above leads to the following conclusions:

In this paper, the patch loading resistance is calculated for steel plated structures stiffened by strong open ribs using different load-introduction, type of equivalent geometric imperfections and solver settings;

A total of 72 different setting combinations are developed, considered with equal importance and calculated for 5 slenderness values;

The largest difference in the load carrying capacity values is depending on whether the calculation is force or displacement driven, so it is recommended to evaluate these separately;

Several groupings are defined based on the Eurocode proposal for the model factor value and it is proposed to set it to 1.05 for force-driven calculation in favor of safety.

## Acknowledgements

The research was financially supported by the New National Excellence Programme; the Authors would like to thank the Ministry for Innovation and Technology and the National Research, Development and Innovation Office. The research work is also connected to the Grant MTA-BME Lendület LP2021-06/2021 “Theory of new generation steel bridges” program of the Hungarian Academy of Sciences; the financial support is gratefully acknowledged.

## References

- [1]↑
*EN 1993-1-5:2006*,*Eurocode 3: Design of steel structures, Part 1-5: general Rules, Plated structural elements*, CEN, Brussels, 2006. - [2]↑
*prEN 1993-1-14:2020*,*Eurocode 3: Design of steel structures, Part 1-14: Design assisted by Finite element analysis*, CEN, Brussels, 2021. - [3]↑
Z. Virág, “Optimum design of stiffened plates,”

, vol. 1, no. 1, pp. 77–92, 2006.*Pollack Period.* - [4]↑
B. Kövesdi, B. J. Mecséri, and L. Dunai, “Imperfection analysis on the patch loading resistance of girders with open section longitudinal stiffeners,”

, vol. 123, pp. 195–205, 2018.*Thin-Walled Struct.* - [5]↑
B. Kövesdi, “Patch loading resistance of slender plate girders with longitudinal stiffeners,”

, vol. 140, pp. 237–246, 2018.*J. Constr. Steel Res.* - [6]↑
B. Kövesdi and L. Dunai, “Patch loading resistance of slender plate girders with multiple longitudinal stiffeners,”

, vol. 16, no. 1, pp. 16–30, 2023.*Steel Constr.* - [7]
R. Chacón, M. Bock and E. Real, “Longitudinally stiffened hybrid steel plate girders subjected to patch loading,”

, vol. 67, no. 9, pp. 1310–1324, 2011.*J. Constr. Steel Res.* - [8]↑
E. Mirambell, R. Chacón, U. Kuhlman, and B. Braun, “Statistical evaluation of the new resistance model for steel plate girders subjected to patch loading,”

, vol. 5, no. 2, pp. 10–15, 2012.*Steel Constr.* - [9]
B. Johansson and M. Veljkovic, “Review of plate buckling rules in EN1993-1-5,”

, vol. 2, no. 4, pp. 228–234, 2009.*Steel Constr.* - [10]↑
O. Lagerqvist, “

*Patch loading resistance of steel girders subjected to concentrated forces*,” Doctoral Tesis, Luleå University of Technology, Luleå, Sweden, 1995. - [11]
O. Lagerqvist and B. Johansson, “Resistance of I-girders to concentrated loads,”

, vol. 39, no. 2, pp. 87–119, 1996.*J. Constr. Steel Res.* - [14]↑
S. Nemer and F. Papp, “Influence of imperfections in the buckling resistance of steel beam-columns under fire,”

, vol. 16, no. 2, pp. 1–6, 2021.*Pollack Period.*