Overview of FEM-Based Resistance Models for Local Buckling of Welded Steel Box Section Columns

Abstract

The local buckling behavior of welded square box section columns subjected to pure compression is investigated. Local buckling represents a crucial failure mode in thin-walled structures, exerting a significant impact on their overall stability and load bearing capacity. The primary objective of this research is to perform an extensive literature review considering the theoretical background of buckling phenomena and encompassing key findings and methodologies reported in previous studies. Additionally, the development and validation of a novel numerical model is presented, capable of accurately predicting the ultimate buckling capacity. Two different calculation methods are applied in the present study: (i) a numerical model using equivalent geometric imperfections to cover the residual stresses and out-of-straightness of plates, (ii) realistic geometric imperfections combined with an assumed residual stress pattern which has an experimental-based background. The objective of the numerical investigation is to investigate the accuracy of the numerical model by using different residual stress and imperfection patterns taken from the international literature. Many test results are collected from the international literature, to which the computational results are compared, and the effect of the residual stresses and geometric imperfections are analyzed. Based on the numerical analysis, the accuracy of the imperfection models is assessed and the imperfection model leading to the most accurate resistance is determined. The calculated buckling capacities are also compared to analytical design approaches, in which accuracy is also analyzed and evaluated. The current investigation proved the buckling curve developed by Schillo gives the most accurate results to the numerically calculated buckling resistance.

1. Introduction

Welded box section columns are used in many fields of the civil engineering industry; therefore, their appropriate design is crucial from economic and safety point of view. Welded box sections are commonly used in bridges, especially in steel arch bridges and for bracing systems. They are also commonly used for steel trusses in bridges and/or in buildings. In cases where steel sections are subjected to pure axial load, the plate slenderness emerges as a significant factor influencing the failure mode of steel member. It plays a crucial role in distinguishing between plastic failure and potential stability failures. A local buckling phenomenon limits the load-bearing capacity of the thin-walled sections based on the critical load. Additionally, stability problems are greatly influenced by residual stress patterns generated during the construction process and the geometric imperfections of individual plates and the entire structural element. The current design methods for calculating the reduction factor in local buckling have been subject to evaluation in academic research programs, revealing that they tend to yield outcomes that overestimate the load-bearing resistance. For instance, Winter’s formula demonstrates an overestimation for plate slenderness values greater than 0.9, while for stocky plates with a plate slenderness below 0.9, it underestimates the values [1,2]. Similarly, the effective width approach tends to underestimate the resistance of square welded box section columns to local buckling within the stocky range, while it overestimates the resistance as slenderness increases [3]. Furthermore, the equivalent imperfection magnitude recommended in the EN1993-1-5 [4] provides conservative values for the ultimate load in thin-walled sections, especially in cases where stability interaction problems arise due to the consideration of residual stresses being double-counted. These findings and further research results show the need to improve the computational methods to design thin-walled cross-sections, while also maintaining their practicality and widespread applicability. In the context of local buckling, imperfections have a significant influence on the load-bearing capacity. Their magnitude and shape performs a pivotal role in determining whether the plate behaves in a column-like or plate-like manner, consequently affecting the ultimate buckling load capacity of the cross-section. Previous studies have demonstrated that high-strength welded steel box sections have a higher buckling load resistance than sections made using low-strength steel, even when the magnitude of residual stresses is higher [1,5]. In essence, it is the ratio of residual stress to yield stress that has more influence over the decrement of the local buckling load-bearing capacity than the magnitude of residual stresses themselves. Considering the substantial quantity of experimental results necessary for a comprehensive assessment of buckling phenomena, an expensive approach, the most suitable method of investigation involves the development of computational models that are properly verified and validated using existing experimental data. This approach opens a broad spectrum of possibilities for controlling the parameters that influence the buckling resistance. The aims of this paper are (i) to analyze and evaluate several residual stress and imperfection models to determine which combination of these gives the most accurate numerical model compared to test results, and (ii) to investigate which of the analytical resistance models give the best approach to the most accurate numerical results.

2. Literature Review

2.1. Pure Local Buckling

Our comprehensive literature review encompassed mainly experimental research programs focusing on local buckling, but also other research topics that deal with the local buckling resistance of welded box sections, such as the interaction behavior between local and overall buckling, concrete-refilled box sections or the incorporation of additional stiffeners along the plates of the section. Most of the research focuses on local buckling in thin-walled elements, comparing the results with the values obtained using EN1993-1-5 [4], with the objective of enhancing the calculation process, since the standard is limited in its applicability to materials with a yield strength below 420 MPa. For materials exceeding these values, the uncertainties are often addressed by simply adjusting the partial factors, ignoring the potential advantage of utilizing high-strength materials. The studies that present findings from experimental tests focusing exclusively on local buckling phenomena are as follows:

• Clarin [1] examined a total of 48 experiments using materials with three different yield stresses. The findings indicated that the absolute magnitudes of residual tensile stresses were higher in high-strength plates compared to low-strength plates. However, the residual-stress-to-yield-stress ratio was lower for high-strength materials.
• Schillo and Feldmann [3] studied 34 stub column specimens with yield strengths of 500, 700, and 960 MPa, which were tested to compare the results with the EN1993-1-5 [4] design procedure. This work was supported by finite element analysis (FEA) that considered equivalent imperfections and realistic imperfections. The work presented in [6] introduces a buckling curve proposal that addresses the discrepancy between values obtained from EN1993-1-5 [4] and those derived from experimental tests, specifically when dealing with plate slenderness exceeding 1.55. Given the similarity in methodology between their research and the approach employed in this work, the buckling curve of Schillo will be further considered in the results.
• Hämäläinen et al. [5] conducted a study involving 12 specimens made from ultra-high-strength materials, with a yield strength exceeding 1100 MPa. The outcomes from these experiments were used to propose a simplification of factors designed to account for uncertainties specific to this type of material.
• Rasmussen and Hancock [7] used 18 specimens, which included cruciform, box, and I-sections, to investigate the effects of additional restraints along one or two edges and the residual stresses resulting from the welding process.
• In the work by Shi et al. [8], four box sections made of 460 MPa steel were tested, and the results were compared with FEA models that accounted for initial imperfections. The conclusion drawn was that the residual-stress-to-yield-strength ratio holds significant importance in the design moment of thin-walled structures.

2.2. Interaction between Local and Overall Buckling

To assess the interaction behavior between local and global stability failure modes, compression tests were made considering both centric and eccentric loading situations to ensure pure axial compression and/or the interaction between compression force and the bending moment. In most experiments, pinned–pinned support conditions were analyzed. The studies conducted by different researchers in this context are as follows:

• Chiew et al. [9] conducted tests on 20 welded box section columns to validate the applicability of empirical equations.
• In the research carried out by Usami et al. [10], 24 specimens were tested to verify an enhancement of an equation published by the same authors.
• Nishino et al. [11] examined eight specimens to verify theoretical values concerning elastic–plastic buckling in plates.
• The study by Schillo [6] aimed to investigate the interaction buckling phenomenon for welded box section columns made of different steel grades involving tests on 34 specimens.
• Liew and Shanmugam [12] conducted experiments on 28 large-scale square and rectangular sections with yield stresses ranging between 235 and 355 MPa. The research considered residual stresses and local and overall imperfections, and aimed to propose a load resistance equation.

2.3. Further Related Research Activities

To avoid local buckling in thin-walled members, one viable alternative involves increasing the plate stiffness, adding stiffeners along the longitudinal axis, as presented in [13] or [14]. These options effectively prevent the inward buckle pattern, compelling the structure to shift to higher buckling modes while concurrently bolstering its load-bearing capacity. Several experimental tests have been conducted to assess the effectiveness of these techniques in improving the local buckling load resistance. In a study by Ge and Usami [14], four specimens served as reference values to investigate the ductility and large energy absorption capacity achieved through the use of confined concrete. Furthermore, Bridge and O’Shea [2] tested 11 specimens to evaluate the applicability of the existing design guidelines. The creation of a computational–numerical model for analyzing the buckling failure mode of welded thin-walled members has been dealt with in previous works. However, they were primarily focused on interaction of buckling failure modes and making adjustments to equations and partial factors available in the existing building codes. The following is a list of relevant papers using FEA instead of physical tests to assess the local buckling failure mode. The findings of these papers were discussed in the validation and calibration phase of the computational model developed in this research.

• The research conducted by Vieira et al. [15] studied the interaction between the flanges and the web of a hollow section columns, considering the independent buckling behavior in each plate. To achieve this, they performed a parametric study to derive a local buckling coefficient for the combined effects of axial compression and bending actions. The numerical tools employed in their investigation included GBTUL 2.0 and CUFSM 3.13.
• Degée et al. [16] explored the influence of imperfections, such as residual stresses resulting from the welding process and initial global bow imperfection. Their work was supported by a combination of experimental and numerical investigations, leading to a proposal for modifying the non-dimensional slenderness equation used in global buckling verification. Additionally, they proposed an alternative to the current European buckling curve for welded box section columns.
• In the study by Johansson et al. [17], the effective width of the plate in thin-walled sections was investigated, and an evaluation of Winter’s formula was conducted. This research was carried out with the aid of FEA.
• Pircher et al. [18] presented their research findings on the influence of the fabrication process on buckling phenomena in thin-walled steel sections. Their aim was to propose a buckling curve specific to welded thin-walled members.
• Kövesdi and Somodi [19] studied and modified the Ayrton–Perry-type buckling formula to implement the effects of residual stresses and geometrical imperfections in the design procedure. This study was based on theoretical reviews and numerical analyses.

Welded steel box section columns, constructed from four flat plates, are typically produced using one of the plate arrangements shown in Figure 1. The weld position influences the pattern and magnitude of residual stresses, stemming from the heat generated during the welding process. As a result, differences in residual stress distribution and intensity are observed among plates A, C, B, and D for types 2 and 3. In contrast, type 1 exhibits a uniform residual stress distribution and intensity across all plates. In this research program, all studies on welded square box sections were compiled in order to present a general proposal.

Figure 1. Manufacturing possibilities for welded box sections (b is outer width, b₀ is inner width) [20].

Figure 1. Manufacturing possibilities for welded box sections (b is outer width, b₀ is inner width) [20].

The data collected from experimental research programs are summarized in

Table 1. Summary table of experimental data.

Source	Total Number of Tests	Failure by Local Buckling	Section Type	fy [MPa]	Loading Type
Clarin, M. (2004) [1].	49	24	1	441.3 471 772.7 794 1335 1350.7	Quasi-static Deformation-controlled
Russell Q. et al. (1998) [2].	11	6	2	282	Quasi-static Displacement-controlled
Schillo, N. and Feldmann, M. (2015) [3].	32	8	2	561 610 760 991	Quasi-static Displacement-controlled
Hämäläinen, O et al. (2018) [5].	12	12	1	1100 1121	no data
Schillo, N. (2017) [6].	26	16	2	562 610 760 980 991	Quasi-static Displacement-controlled
Rasmussen, K and Hancock, G. (1992) [7].	6	3	1	670	Quasi-static Displacement-controlled
Shi, G. et al. (2014) [8].	4	4	2	460	Displacement-controlled
Chiew. et al. (1987) [9].	13	13	3	248.6 253.6 261.2	Quasi-static Displacement-controlled Load-controlled
Usami, T. (1981) [10].	13	12	3	743.8	Load-controlled
Nishino, F, and Ueda. (1967) [11].	8	8	2	273.1 266.1 714.3 799.8	Displacement-controlled
Liew, J. et al. (1989) [12].	6	6	3	268 293 353	Static Displacement-controlled
Ge, H. et al. (1992) [14].	3	3	3	266	Cyclic Load-controlled Displacement-controlled
Dwight, J., and Bradfield, C. (1975) [21].	17	9	no data 1	250 275 312 403	no data 1

¹ Experimental values of Dwight, J., and Bradfield, C. (1975) [21] were taken from the Schillo, N. (2017) [6] research program.

. The total number of tests was determined exclusively from experiments conducted on welded square sections without any additional stiffeners and subjected to pure compressive load.

Altogether, 124 unique experimental results were collected for basic square cross-sections; these are used and evaluated in Section 5. For 22 of the collected experiments, the actual residual stress patterns were also measured and published. These experimental data are used for the following:

• To validate the numerical model;
• To evaluate the already existing models;
• To develop a new proposal for improved imperfections and a residual stress pattern.

The current research program focuses on the square welded box section columns, where welding seams are in the four corners of the cross-section. An interesting research program on parallelogram-shaped hollow-section girders was reported by He et al. in [22] and one on octagonally shaped tubular steel columns under uniaxial compression was reported by Zhu et al. [23].

There are also numerous research reports on the structural behavior of cold-formed hollow section columns, which have a significantly different manufacturing process, indicating different residual stress pattern and geometric imperfection magnitudes to the members. Comprehensive studies on this topic can be found in [24,25,26,27,28] in this topic, including the material properties and residual stress patterns, buckling capacity against local buckling and global buckling. The manufacturing method of the cold-formed sections can be different and several previous investigations have showed that the manufacturing process has a significant effect on the residual stress distribution and intensity. There are two common manufacturing methods for the cold-formed hollow sections, called in the international literature “direct forming” and “continuous forming—indirect way from circular-to-square rolling”. The direct forming process includes the following manufacturing steps: (i) roll forming a strip directly to an open section with the desired rectangular shape, and (ii) joining the edges of the open section by welding. The “continuous forming” includes (i) roll-forming the straight strip first into a circular open tube, (ii) joining the edges of the open tube by welding, and (iii) flattening the tube walls to form the desired rectangular shape [29]. The current investigation does not cover cold-formed sections, only welded sections. Further investigations dealing with cold-formed sections can be found in [30,31,32,33]. Further interesting investigations regarding plate buckling problems can be found in [34,35,36].

3. Finite Element Model

3.1. General Details

The numerical model was created using commercial finite element software Ansys 2023 R2 [37]. A full-shell model was developed using thin shell elements called SHELL181. This element type is suitable for analyzing the behavior of the thick plates, having the capability to work in a non-linear context. The applied element is a four-node thin element with 6 DOF’s on each node, 3 translational and 3 rotational. The purpose of the numerical model is to determine the local buckling resistance of box section columns; therefore, the overall flexural buckling failure mode should be avoided in the calculation. To achieve this, the general geometry of the modeled section is set in a way that the length of the specimens is equal to three times the longer width of the section or to be equal as the given value of the corresponding research paper where the specimen is taken from.

The buckling resistance was determined by using geometrical and material nonlinear analysis using equivalent geometric imperfections (GMNIA). A full Newton–Raphson approach was used in the nonlinear analysis with 0.1% convergence tolerance of the residual force-based Euclidian norm.

3.2. Material Model

Mechanical properties govern the behavior of the material in a computational model. In the case of overall buckling phenomenon in welded sections, the material law has a clear influence on the buckling resistance as it was shown in [38]. For general stability problem, material behavior takes a special importance as it defines the shape of the behavior curve and the ultimate load capacity. Post-buckling resistance may be expected for the elastic buckling of plates, though not for elastic-plastic buckling [11]. Since the data collected cover studies around the world, material composition of the steel and its treatment will not follow a uniform standard. To standardize the computational analysis, the material’s constitutive law will be determined based on the yield strength of the steel used. For normal-strength steel with a yield strength below 500 MPa, the material model proposed by the prEN 1993-1-14 [39] is used. For high-strength steel with a yield strength equal or exceeding 500 MPa, the Ramberg–Osgood-type material model was used following the proposal of prEN 1993-1-14 [39].

3.2.1. Material Model Used for Normal-Strength Steel

The material model proposed by the prEN 1993-1-14 [39] is a quad-linear material model that captures the yield plateau and the strain-hardening of the NSS until the ultimate strength is reached. The general shape of the curve is presented in Figure 2. This model was developed by Gardner et al. [40] through a combination of experimental and numerical analyses, and its validation was performed in accordance with the European standards. The parameters required to establish the strain–stress relationship for materials with varying yield strengths are presented in Equations (1)–(6).

Figure 2. Material model characteristic used for normal-strength steel [39].

Figure 2. Material model characteristic used for normal-strength steel [39].

$$\mathrm{\epsilon }=\left\{\begin{array}{c}{\mathrm{E}}_{\mathrm{\epsilon }} \mathrm{for} \mathrm{\epsilon }\le {\mathrm{\epsilon }}_{\mathrm{y}}\\ \\ {\mathrm{f}}_{\mathrm{y}} \mathrm{for} {\mathrm{\epsilon }}_{\mathrm{y}}\le \mathrm{\epsilon }\le {\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}\\ \\ {\mathrm{f}}_{\mathrm{y}}+{\mathrm{E}}_{\mathrm{s}\mathrm{h}}\left(\mathrm{\epsilon }+{\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}\right) \mathrm{for} {\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}\le \mathrm{\epsilon }\le {\mathrm{C}}_1{\mathrm{\epsilon }}_{\mathrm{u}}\\ \\ {\mathrm{f}}_{\mathrm{C}1\mathrm{\epsilon }\mathrm{u}}+\frac{{\mathrm{f}}_{\mathrm{u}-}{\mathrm{f}}_{\mathrm{C}1\mathrm{\epsilon }\mathrm{u}}}{{\mathrm{\epsilon }}_{\mathrm{u}}-{\mathrm{C}}_1{\mathrm{\epsilon }}_{\mathrm{u}}}\left(\mathrm{\epsilon }-{\mathrm{C}}_1{\mathrm{\epsilon }}_{\mathrm{u}}\right) \mathrm{for} {\mathrm{C}}_1{\mathrm{\epsilon }}_{\mathrm{u}}\le \mathrm{\epsilon }\le {\mathrm{\epsilon }}_{\mathrm{u}}\\ \end{array}\right.$$

(1)

$${\mathrm{E}}_{\mathrm{s}\mathrm{h}}=\frac{{\mathrm{f}}_{\mathrm{u}}-{\mathrm{f}}_{\mathrm{y}}}{{\mathrm{C}}_2{\mathrm{\epsilon }}_{\mathrm{u}}-{\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}}$$

(2)

$${\mathrm{C}}_1=\frac{{\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}+0.25({\mathrm{\epsilon }}_{\mathrm{u}}-{\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}})}{{\mathrm{\epsilon }}_{\mathrm{u}}}$$

(3)

$${\mathrm{C}}_2=\frac{{\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}+0.4\left({\mathrm{\epsilon }}_{\mathrm{u}}-{\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}\right)}{{\mathrm{\epsilon }}_{\mathrm{u}}}$$

(4)

$${\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}=0.1\frac{{\mathrm{f}}_{\mathrm{y}}}{{\mathrm{f}}_{\mathrm{u}}}-0.055 \mathrm{f}\mathrm{o}\mathrm{r} 0.015\le {\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}\le 0.03$$

(5)

$${\mathrm{\epsilon }}_{\mathrm{u}}=0.6\left(1-\frac{{\mathrm{f}}_{\mathrm{y}}}{{\mathrm{f}}_{\mathrm{u}}}\right) \mathrm{b}\mathrm{u}\mathrm{t} {\mathrm{\epsilon }}_{\mathrm{u}}\ge 0.06$$

(6)

where

• ${\mathrm{C}}_1$ “cut-off” strain to avoid over-prediction;
• ${\mathrm{C}}_2$ coefficient to compute strain hardening modulus;
• ${\mathrm{\epsilon }}_{\mathrm{s}\mathrm{h}}$ strain-hardening;
• ${\mathrm{\epsilon }}_{\mathrm{u}}$ ultimate strain;
• ${\mathrm{E}}_{\mathrm{s}\mathrm{h}}$ strain hardening modulus.

3.2.2. Ramberg–Osgood-Type Material Model Used for High Strength Steel (HSS)

To determine the buckling capacity of high strength steel structures, Ramberg–Osgood-type material model is applied, which is commonly accepted to represent the material characteristics of HSS. The specialty of HSS material is that usually it has no clear yield plateau; after the first yielding occurs, the plastic deformations increase by increasing stresses until the ultimate stresses are reached and fracture of the material occurs. The Ramberg–Osgood-type material model considers both elastic and plastic deformation in all material behaviors. In low-stress states, plastic deformation is minimal in comparison to elastic deformation. However, as the stress level increases, plastic deformation plays a prominent role in the material’s behavior. To define the mathematical relationship of this model, the following parameters are required. The first value is the yield offset $\left(\mathsf{\alpha }\times \frac{{\mathsf{\sigma }}_0}{\mathrm{E}}\right)$. This strain appears at the beginning of the material’s behavior, where the elastic deformation is dominant. The original author’s research determined this to be 0.2%. The second is the exponential value n; this value fits the relationship to the experimental data, and the literature recommends values greater than 5 [41]. This parameter can be obtained by taking the logarithm of the strain at any two points and dividing it by the logarithm of the stresses at those same points in various coupon tests [42]; this equation is presented in (7):

$$\mathrm{n}=\frac{\log \left(\frac{{\mathrm{\epsilon }}_1}{{\mathrm{\epsilon }}_2}\right)}{\log \left(\frac{{\mathsf{\sigma }}_1}{{\mathsf{\sigma }}_2}\right)}$$

(7)

The mathematical expression for the Ramberg–Osgood material model is shown in Equation (8). The general shape of the stress–strain curve using this model is shown in Figure 3:

$$\mathrm{\epsilon }=\frac{\mathsf{\sigma }}{\mathrm{E}}+0.002{\left(\frac{\mathsf{\sigma }}{{\mathrm{f}}_{\mathrm{y}}}\right)}^{\mathrm{n}}$$

(8)

where

• ${\mathrm{f}}_{\mathrm{y}} \mathrm{y}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{d} \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}$ [N/mm²];
• E Young´s modulus of steel [N/mm²];
• $\mathrm{\epsilon } \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}[\mathrm{m}\mathrm{m}/\mathrm{m}\mathrm{m}]$;
• $\mathsf{\sigma }$ stress [N/mm²].

Figure 3. General shape of Ramberg–Osgood-type material model based on Ramberg and W. R. Osgood [41].

Figure 3. General shape of Ramberg–Osgood-type material model based on Ramberg and W. R. Osgood [41].

3.3. Boundary Conditions

To represent a pure compression loading state in the computational model, the creation of master nodes was necessary for the load application and to define the constraint conditions. These master nodes are connected by rigid bodies to all the nodes of the end cross-sections. The arrangement of master and slave nodes and the constraints are shown in Figure 4. The same configuration was used at the opposite end of the model where the loading was applied. The compression load was applied on the model as a point load acting on the master node, which distributed the load among the plates of the cross-section uniformly. The ultimate load is usually determined by load-driven analysis within the current research program, but as cross-checks, some of the analyzed specimens are also tested by displacement load. The two calculation results led to the same ultimate load within an average of 0.1% difference.

3.4. Imperfections and Residual Stresses

In a geometric and material non-linear analysis using imperfections (GMNIA), the imperfections that arise in steel members coming from the manufacturing process should be considered. The method for implementing imperfections in a computational model depends on the complexity of the buckling phenomenon under consideration. The design code gives standardized values for equivalent geometric imperfections to address all the possible imperfections [4]. However, several investigations suggest that the use of geometric imperfections and residual stresses in combination may provide a more accurate representation of the local buckling resistance of welded box section columns [43]. Both will be analyzed in the current research program.

3.4.1. Equivalent Geometric Imperfections

This approach combines the effect of the geometric imperfections and residual stresses into a pure geometric imperfection with increased amplitude. The shape of the equivalent imperfection is usually chosen to be the first eigenmode shape, obtained by an initial linear buckling analysis (LBA). The magnitude of imperfection is taken from the Annex C of EN1993-1-5 [4] which gives, for the case of local buckling, an equivalent imperfection of b/200, with b being the internal length of the plate. An extensive study of the equivalent imperfection features such as the effect of the chosen eigenmode, number of half-sine waves along whole model length, the direction of the initial buckling shape (inward or outward) was presented by Radwan and Kövesdi [42] in a detailed manner.

3.4.2. Residual Stress Distribution

There are some investigations that have defined different residual stress distribution patterns, but all of them point out that the fabrication process generates tensile stress in the corners of the box section and in the welding cord vicinity, and compressive stress along the midspan on the plates. The maximum magnitude for both tensile and compressive stress depends on welding features such as the welding size and type, voltage used, and heat applied. The ECCS Manual [44] provides a feasible model for the residual stress pattern to be applicable for welded box sections. But it does not consider the yield strength of the material or the plate width to establish the lengths of residual stress zones; however, test results showed its importance. Other code-based residual stress models are presented in Swedish [45] and Chinese [46] design codes. Previous research results also present a wide variety of distribution; some focus on a specific yield material. Ban et al. [47], and Khan et al. [48] examined box sections built up with 460 and 690 MPa steel material, respectively. The unique work that presented a distribution for a wide range of yield stress values was made by Somodi and Kövesdi [20], which considers yield stresses from 235 to 960 MPa. These stress distribution models are shown in Figure 5.

The residual stress distribution pattern presented in the ECCS manual [44] provides specific values for the compressive stress developed in the plates due to the welding process, taking into account the width-to-thickness plate ratio and welding type. These values are outlined in

Table 2. Residual stresses for welded box sections from ECCS model.

H/t	Welding Type	σrt/fy	σrc/fy	Tensile Zone	Transition Zone
10	-----	1	−0.6	0	from equilibrium
20	Heavy weld	1	−0.85	3t	3t
20	Light weld	1	−0.29	1.5t	1.5t
40	Heavy weld	1	−0.29	3t	3t
40	Light weld	1	−0.13	1.5t	1.5t

. In the case of the Somodi and Kövesdi RS distribution model [20], the compressive stress generated in the plates is associated with the dimensions of the plate, and it is distinguished based on plate thickness. These relationships are expressed in Equation (9). In the Khan et al. model [48], the compressive residual strength is determined by a unique equation applicable to any plate geometry, considering the yield stress of the base material. This equation is presented in (10).

The residual stress distribution must be in equilibrium, which means the tensile force developed in the welding zone should be equal to the compressive force developed in the other parts of the cross-section. The magnitude of the tension force is directly related to the temperature reached in the welding seam, the yield stress of the material and the subsequent treatment [49]. The computational model created for this research incorporates axial residual stress. The initial residual stress pattern along the transversal direction of a plate is shown in Figure 6 as an example for a box section with a 82.5 mm width and 3.1 mm plate thickness, composed of steel having 441 MPa yield strength, which represents one of the experimental tests carried out by Clarin [1].

$${\mathsf{\sigma }}_{\mathrm{r}\mathrm{c}}=\left\{\begin{array}{c}70-21\mathrm{t}+{\mathrm{t}}^2-\left(2900-3600\left(\mathrm{t}-5\right)\right)\times {\left(\frac{\mathrm{b}}{\mathrm{t}}\right)}^{-1}, \mathrm{t}>5\mathrm{m}\mathrm{m}\\ \\ 70-21\mathrm{t}+{\mathrm{t}}^2-\left(2900-290\left(\mathrm{t}-5\right)\right)\times {\left(\frac{\mathrm{b}}{\mathrm{t}}\right)}^{-1}, \mathrm{t}<5\mathrm{m}\mathrm{m}\end{array}\right.$$

(9)

$${\mathsf{\sigma }}_{\mathrm{r}\mathrm{c}}={-\mathrm{f}}_{\mathrm{y}}\times 3.6607\times {\left(\frac{\mathrm{b}-\mathrm{t}}{\mathrm{t}}\right)}^{-0.924}$$

(10)

3.4.3. Geometrical Imperfection

If residual stress is considered in the computational model, the initial geometrical imperfection of the element should be considered in a realistic manner. The values of this imperfection can be taken from different academic sources, such as from Radwan and Kövesdi [42], where the plate slenderness and yield stress are taken into account to determine the imperfection value; this work considered different local imperfection values for normal- and high-strength steel using Equations (12) and (13), respectively. Somodi et al. [50] studied the partial factor to the application of the Winter-type buckling curve in accordance with the safety requirements of Eurocode and made a differentiation between normal-strength steel and high-strength steel. In their studies, the imperfection based on Equation (11) was considered, defined based on experimental measurements. Since two imperfection models from the same authors will be used, this model is managed under the name “2021 model”. Hassan et al. [51] assessed welded thin-walled hollow sections under a cyclic axial load. Equation (14) shows the proposed expression as a function of the section size. Another resource for the proposed value of local imperfection is the BSI manual [52], which presents unique value of b/125, Equation (15). Additionally, one more expression is used that is in research under publication at the moment of writing this work. It considers local imperfection as a function of the b/t ratio, taking its value as an average of b/68t Equation (16), and an upper limit of b/32t. This model is referred later under the name of “2023 model”. In the given equations parameter b refers to the plate width, t refers to the thickness, and e to the imperfection magnitude. f_y is the yield strength of the material, f_cr refers to the critical buckling stress, and λ_p refers to the relative slenderness of the plate defined by EN 1993-1-5 [4].

$$\frac{\mathrm{b}}{\mathrm{e}}=\max \left(\frac{-0.0513\times \left(\frac{\mathrm{b}}{\mathrm{t}}\right)+2.6131}{1000},\frac{0.8176}{1000}\right)$$

(11)

$$\frac{{\mathrm{b}}_{\mathrm{N}\mathrm{S}\mathrm{S}}}{\mathrm{e}}=\left\{\begin{array}{c}675-0.53{\mathrm{f}}_{\mathrm{y}}, \overline{{\mathrm{\lambda }}_{\mathrm{p}}}<1.2\\ \\ \left(2.65{\mathrm{f}}_{\mathrm{y}}-3010.5\right)\overline{{\mathrm{\lambda }}_{\mathrm{p}}}+\left(-3.71{\mathrm{f}}_{\mathrm{y}}+4287.6\right), 1.2\le \overline{{\mathrm{\lambda }}_{\mathrm{p}}} \le 1.4\\ \\ \frac{200}{{\overline{{\mathrm{\lambda }}_{\mathrm{p}}}}^3}, \overline{{\mathrm{\lambda }}_{\mathrm{p}}} \ge 1.4\end{array}\right.$$

(12)

$$\frac{{\mathrm{b}}_{\mathrm{H}\mathrm{S}\mathrm{S}}}{\mathrm{e}}=\left\{\begin{array}{c}424, \overline{{\mathrm{\lambda }}_{\mathrm{p}}}<1\\ \\ 420 {\overline{{\mathrm{\lambda }}_{\mathrm{p}}}}^{-5.5}+4 \overline{{\mathrm{\lambda }}_{\mathrm{p}}} \ge 1\end{array}\right.$$

(13)

$$\frac{\mathrm{b}}{\mathrm{e}}=0.034\times \mathrm{t}{\left(\frac{\mathrm{f}\mathrm{y}}{{\mathrm{f}}_{\mathrm{c}\mathrm{r}}}\right)}^{0.5}$$

(14)

$$\frac{\mathrm{b}}{\mathrm{e}}=125$$

(15)

$$\frac{\mathrm{b}}{\mathrm{e}}=68\times \mathrm{t}$$

(16)

Figure 7 shows an example of the general shape of square section, which includes initial geometrical imperfection in an over-scaled manner.

4. Verification and Validation of the Numerical Model

4.1. Model Verification—Mesh Sensitivity Analysis

During the verification stage, it should be proven that the numerical solution provides a good approximation of the exact mathematical solution. Firstly, the accurate mesh size of the computation model was determined. Convergence analysis ensures a mesh size at which the outcomes are not affected [53]. The system response quantity (RSQ) used for the convergence study was the axial reaction force generated in the model during a displacement-controlled analysis. Given the model’s elementary geometry composed of four flat plates, h-type refinement was applied on the whole geometry. The criteria to consider appropriate mesh quality is considered with an error under 1%. This is set as a requirement to consider the numerical model and the computational result reliable and independent on the mesh quality and FE size. For one specific cross-section geometry it was reached by the fifth iteration, as shown in Figure 8. The ultimate resistance at each iteration alongside the meshing size is shown in Figure 8, where h is the depth of the analyzed cross-section. Error was calculated using Equation (17).

$$\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\_\mathrm{i}=\frac{\mathrm{F}{\mathrm{u}}^{\mathrm{i}}-\mathrm{F}{\mathrm{u}}^{\mathrm{i}-1}}{\mathrm{F}{\mathrm{u}}^{\mathrm{i}}}\times 100$$

(17)

where

• $\mathrm{F}{\mathrm{u}}^{\mathrm{i}}:$ Ultimate local buckling resistance at “i” iteration;
• $\mathrm{F}{\mathrm{u}}^{\mathrm{i}-1}:$ Ultimate local buckling resistance at “i − 1” iteration.

Figure 8. Convergence analysis for 200 × 5 mm square section (red dots indicates the individual iteration steps).

Figure 8. Convergence analysis for 200 × 5 mm square section (red dots indicates the individual iteration steps).

4.2. Model Validation and Evaluation of Imperfection and RS Models

4.2.1. Evaluation Strategy

Factors directly related to the decrease in load-carrying capacity of thin-walled welded elements include the shape and magnitude of the residual stress pattern and a realistic geometric imperfection. To test all the models found in academic research and conduct subsequent evaluations, a pre-validation process was undertaken based on specimens that provided measurements of actual residual stresses. From the collected research, 22 test specimens were utilized, which gave the validation domain within the current research program. For these specimens not only the ultimate load and material properties, but also the residual stresses and geometric imperfections were also known from measurement results. Throughout the validation process, the ratio of ultimate experimental resistance to numerical analysis outcomes is assessed. Models are considered within the limitations of their respective research. Subsequent adjustments were made using the academic literature and building codes to develop an adequate and general expression for the local buckling phenomenon. Various scenarios were tested to achieve the best combination of residual stress distribution pattern and realistic geometrical imperfections. The strategy of the validation process can be summarized as follows:

• Testing different geometrical imperfection models (using the same residual stress model for all cases, called ECCS_Eq model—see details later)—Section 4.2.2;
• Modifying the already existing geometrical imperfection models to develop more general models that can more accurately reproduce experimental resistances—Section 4.2.2;
• Testing the different residual stress models (using the best two geometrical models based on previous step)—Section 4.2.3;
• Modifying the already existing residual stress models to develop general model that can more accurately reproduce experimental resistances—Section 4.2.4;
• Execute the final validation step using the best geometrical imperfection and residual stress models but using the actual measured residual stress values published in the corresponding papers (so at this step only the shape of the residual stress pattern according to the model is used, but the actual compressive and tensile stresses are taken from the measurements)—Section 4.2.5.

4.2.2. Evaluation of Realistic Geometric Imperfection Models

To obtain a geometrical imperfection model, an initial comparison was made using the residual stress model of ECCS, ensuring force equilibrium in the plate. This modified model addresses stress equilibrium in the plates, maintaining a ratio of 1:1 between the lengths of the maximum tensile residual stress and the transition zone from the tensile zone to the compression zone. The comparison of realistic geometrical imperfections was conducted using values from academic works and design standards. Initial results of load resistances are presented in

Table 3. Resistance values with different imperfection models—in [kN].

Ref.	$\overline{{\mathbf{\lambda }}_{\mathbf{p}}}$	Experiment	b/125 [52]	2021 Model [50]	2023 Model	Hassan et al. [51].	Radwan et al. [42].	2021_mod
[1]	0.47	5460	5290	5357	5357	5357	5357	5357
	0.84	8923	8072	8746	8605	8761	8589	8844
	1.20	7035	6831	6919	6863	7049	6991	7018
	1.56	7579	7891	7500	7510	7888	7891	7924
[2]	0.72	182	151	174	150	164	171	177
	1.09	184	176	173	171	178	178	177
	1.45	184	197	176	179	197	197	196
	1.80	211	214	173	178	211	214	211
[3]	0.83	1833	1446	1677	1560	1492	1613	1687
[6]	0.80	506	426	480	437	439	454	470
	0.86	1280	1070	1206	1126	1098	1159	1215
	0.93	1651	1311	1403	1353	1297	1372	1403
[11]	0.83	1608	1417	1572	1506	1491	1500	1557
	1.18	1510	1580	1510	1516	1587	1587	1588
	1.40	3101	3008	2972	2975	3010	3008	3004
	0.77	2909	2530	2942	2834	2435	2830	2947
[21]	0.65	381	329	378	342	354	368	379
	0.98	433	383	395	382	390	389	390
	1.15	442	401	389	389	403	402	404
	1.23	150	139	132	131	139	139	139
	1.44	160	144	129	130	144	144	143
	1.64	159	153	151	152	151	153	151

; the statistical evaluation of the results is presented in

Table 4. Comparison of realistic geometric imperfection models.

Model	b/125 [52]	2021 Model [50]	2023 Model	Hassan et al. [51].	Radwan et al. [42]	2021_mod
Mean value	1.095	1.070	1.100	1.076	1.052	1.039
Standard deviation	0.094	0.069	0.074	0.087	0.064	0.057
Min.	0.933	0.989	0.996	0.937	0.933	0.941
Max.	1.268	1.241	1.229	1.272	1.203	1.176

. The table shows the resistance ratios between the numerical model and the experiment, calculated using Equation (18):

$$\mathrm{R}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}=\frac{\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\left({\mathrm{F}}_{\mathrm{E}\mathrm{X}\mathrm{P}}\right)}{\mathrm{U}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\left({\mathrm{F}}_{\mathrm{F}\mathrm{E}\mathrm{M}}\right)}$$

(18)

The model with the mean resistances ratio closest to 1 was presented by Radwan and Kövesdi [42], since this model was developed for welded box sections facing local buckling failure and already incorporates upper bound following the BSI manual [52]. This model will be considered in the next validation step; the imperfection amplitude ranged from b/125 to b/533. The model with the second-best mean was presented by Hassan et al. [51].

The reason to choose these models was that based on the literature review, these proposals were the most recent models having significant research background (test-based and numerical model-based origins) according to the international literature.

The imperfection models referred to in Table 3 and Table 4 are introduced in Section 3.4.3. The given notations refer to the following equations: (i) b/125—Equation (15); (ii) 2021 model—Equation (11); (iii) 2023 model—Equation (16); (iv) Hassan et al. [51]—Equation (14); Radwan et al. [42]—Equations (12) and (13); 2021_mod—Equation (19).

Although it provided acceptable results, this model was excluded from the further evaluation process. This decision was made because the model was developed to be used in combination with a global imperfection factor. Additionally, investigation does not provide specifications about the hollow sections’ manufacturing process nor consider residual stress distribution; values ranged between b/91 and b/1056. The model with the third-best result was presented by Somodi et al. [50]; since this model was developed considering welded sections under pure axial loading process and stress patterns generated during the manufactured process, it will be considered in the next validation step. Values ranged between b/670 and b/1223. The model presented by Somodi et al. in 2023 (pending publication) considers hybrid girders manufactured by the welding process, taking only a type 3 plate arrangement (see Figure 1); it will not be considered due to the specificity of that research. Finally, the value from BSI manual [52] was considered as the upper bound. The models used to establish a realistic geometric imperfection should be adjusted to narrow the range of the values provided by the mathematical expressions. It is evident that the magnitude of the initial geometric imperfection parameter cannot take values too high, because of the manufacturing tolerances. Therefore, it is necessary to set an upper bound. Computational analysis of the data using this limit was performed; results are given in Table 4.

Since the model presented by Radwan and Kövesdi [42] considers a separated computing processes to set a value for NSS and HSS members, modification of them cannot be performed without a deep analysis of the process they used. The model presented by Somodi et al. [50] does not specify an upper limit value, but they do for the lower one. The adjustments of the model were to fix a new value for both the upper and lower bound. The actual lower limit was replaced by b/1000, and the upper limit was taken as b/125 from the BSI manual [52]. The rearrangement of the expression is shown in Equation (19). The new version of the model presented by Somodi et al. [50] was labeled as “2021_mod” and tested with the same data to compare the results to the original expression and to the other models in Table 4. This comparison was conducted to proceed with the validation of residual stress distribution models. Although the “2021_mod” model provides the best mean of the ultimate resistance ratio, this model and the model presented by Radwan and Kövesdi [42] will be tested further using different residual stress distribution patterns to check if the ultimate resistance ratio, between numerical and experimental results, remains close to 1.

$$\frac{\mathrm{e}}{\mathrm{b}}=\frac{-0.0513\times \left(\frac{\mathrm{b}}{\mathrm{t}}\right)+2.6131}{1000} \mathrm{with} \mathrm{the} \mathrm{assumption} \mathrm{of} 125\le \frac{\mathrm{e}}{\mathrm{b}}\le 1000$$

(19)

4.2.3. Evaluation of Residual Stress Models

The validation of residual stress distribution patterns considers three different models, which are the most acknowledged and researched models according to the international literature. The model presented by ECCS [44] establishes the maximum compressive residual stress according to the welding type and the ratio between the height of the section and the thickness of the plates. The second model is presented by Somodi et al. [20], where the influence of the grade of the steel on the local buckling resistance is studied. As a result, a new distribution of residual stresses is presented, considering the yield strength of the applied material. Third model is presented by Khan et al. [48], where the influence of the number of welding passes during manufacturing process is studied. The plots of residual stress distributions are presented in Figure 5. The results of an initial test of the residual stress distribution models are presented in

Table 5. Validation of RS distribution models using the Radwan and Kövesdi geometric imperfection model.

Model	ECCS [44] + Radwan et al. [42]	Somodi et al. + Radwan et al. [42]	Khan et al. [48] + Radwan et al. [42]	ECCS_Eq + Radwan et al. [42]	Somodi_mod + Radwan et al. [42]
Mean	1.097	1.003 1	0.976	1.052	1.032
Standard Deviation	0.068	0.065 1	0.079	0.064	0.087

¹ Values obtained by evaluating 36% of 22-specimen data pool, applying limitations of original RS model.

and

Table 6. Validation of RS distribution models using “2021_mod” geometric imperfection model.

Model	ECCS [44] + 2021_mod	Somodi et al. [20] + 2021_mod	Khan et al. [48] + 2021_mod	ECCS_Eq + 2021_mod	Somodi_mod + 2021_mod
Mean	1.084	0.976 1	0.947	1.039	1.003
Standard Deviation	0.078	0.056 1	0.080	0.057	0.056

¹ Values obtained by evaluating 36% of 22-specimen data pool, applying limitations of original RS model.

. These tables show results incorporating the geometric imperfection model presented by Radwan and Kövesdi [42] and the new model “2021_mod”, respectively. From the comparison between residual stress distribution patterns, the model presented by Somodi et al. [20] gives the best ultimate resistance ratio.

4.2.4. Improvement in Residual Stress Models

The residual stress distribution model with second-best ultimate resistances ratio is the ECCS [44] model, the lengths of the tensile and transition zone maintain the same-value relationship at any steel grade and plate dimension causing that the force equilibrium on the plate is not always fulfilled. Therefore, the aim of the next step is to obtain lengths of the zone, which ensure the equilibrium of the plates. The residual stress model of Khan et al. [48] will not be considered in further analysis due to the model was developed for 690 MPa steel box sections and the elementary residual stress distribution (see Figure 5) cannot allow convenient adjustments. To extend the applicability of the residual stress distribution model presented by Somodi et al. [20], the limitations presented in that research will be modified. Restrictions are presented below followed by how they were modified. The steel grade used in the research ranged from S235 to S960. The yield strength of the base material is used to set the length of the tensile zone, parameter “a” in Figure 9.

Table 7. Width of tensile zone, value of the parameter “a”.

Steel Material	Original Model	Modified Model
fy < S235	--- 1	$\left(3.48-\frac{{\mathrm{f}}_{\mathrm{y}}}{240}\right)\mathrm{t}$         (20)
S235	2.5t	2.5t
S355–S460	2t	2t
S500	1.5t	1.5t
S700	0.75t	0.75t
S960	0	0
fy > S960	--- 1	0

¹ original model did not specify value for cases presented with a --- sign.

shows the computed expression to define it from the original model. To complete the table, it is necessary to consider steel grades out the studied range. For values below S235, Equation (20) is applied, which is an extrapolation of the values between S235 and S355. For values above S960, zero is considered, because the material composition does not allow to reach the yields strength in the vicinity of the welding toe. The adjustments are shown in Table 7.

The thickness range used in that research was between 4 and 12 mm, and the b/t ratio ranged from 9.7 to 62.5. Both parameters were related to the magnitude of compressive residual stress and the transition zone length (parameter “c”, as shown in Figure 9). An adjustment was made to avoid the limitation of the b/t ratio as it should restrict the applicability range for plate thickness at the same time. The intensity of compressive residual stress depends on the b/t ratio, as given in Equation (9). Therefore, the adjustment made is to fix limits to the compressive residual stress. These bounds are derived from ECCS [44] model, where limits are set according to the welding process and b/t ratio.

To choose an adequate value, analysis of the actual residual stress measures was conducted, which showed that the magnitude of the compressive residual stress ranged between 8% and 51% of the yield strength. Consequently, the first and second row of Table 2 were ignored, because in the practice the plates of welded box sections do not reach those values. To evaluate the proximity of the actual measured values to the limit from ECCS manual [44], the percentage of the measured values that are within a range between ±1 standard deviation was calculated. From the 22-specimen data pool, 54% were around the upper limit, and 40% were around the lower bound. The modified expression to compute the magnitude of compressive residual stress is presented in Equation (21):

$${\mathsf{\sigma }}_{\mathrm{r}\mathrm{c}}=\left\{\begin{array}{c}0.13 \mathrm{f}\mathrm{y}, \mathrm{i}\mathrm{f} {\mathsf{\sigma }}_{\mathrm{r}\mathrm{c}}<0.13 \mathrm{f}\mathrm{y}\\ \\ 70-21\mathrm{t}+{\mathrm{t}}^2-\left(2900-3600\left(\mathrm{t}-5\right)\right)\times {\left(\frac{\mathrm{b}}{\mathrm{t}}\right)}^{-1}, \mathrm{t}>5\mathrm{m}\mathrm{m}\\ \\ 70-21\mathrm{t}+{\mathrm{t}}^2-\left(2900-290\left(\mathrm{t}-5\right)\right)\times {\left(\frac{\mathrm{b}}{\mathrm{t}}\right)}^{-1}, \mathrm{t}<5\mathrm{m}\mathrm{m}\\ \\ 0.29 \mathrm{f}\mathrm{y}, \mathrm{i}\mathrm{f} {\mathsf{\sigma }}_{\mathrm{r}\mathrm{c}}>0.29 \mathrm{f}\mathrm{y}\end{array}\right.$$

(21)

The model created from the model presented by Somodi et al. [20] is referred later under the name “Somodi_mod”.

4.2.5. Validation Using the Improved Residual Stress Model

The 22-specimen data pool was tested with the new version of the residual stress distribution model. The results when using the initial imperfection model presented by Radwan and Kövesdi [42] are shown in Table 5, and the results from the new model for initial local imperfection developed in this work,“2021_mod”, are presented in Table 6. In the case of the ECCS model, the adjustment made was to ensure the residual stress equilibrium. As shown in Table 2, lengths of the tensile and transition zones maintained a ratio of 1 without considering the grade of the material or plate dimensions.

Table 8. Force equilibrium on plates applying ECCS residual stress model.

Specimen	Research Work	b/t	fy [MPa]	ECCS Model			ECCS_Eq Model
				a [mm]	b [mm]	Force-Remain [N]	a [mm]	b [mm]	Force-Remain [N]
W72-0b	Clarin, M. (2004) [1]	26.8	772.7	6.14	6.14	−6213.71	8.21	8.21	−4.4
#3	Russell et al. (1998) [2]	75.0	282.0	6.43	6.43	−6136.38	12.06	12.06	1.6
T2A-T2B	Nishino et al. (1967) [11]	25.5	714.3	9.75	9.75	−7309.76	12.39	12.39	−6
S.52.10	Liew et al. (1989) [12]	53.4	353.0	9.15	9.15	−4186.40	12.21	12.21	−3.1

shows the force equilibrium of each individual plate. For this comparison, experimental data collected were used, and for all of them, the magnitude of compressive residual stress was set to 0.29 fy. It is evident that the original model does not guarantee force equilibrium. Therefore, adjustment involved solving the equation system for force equilibrium. As specimens have a wide range of material yield strength and geometrical features, it is necessary to adjust the lengths of the zones one by one. Results are shown in Table 8. This modified model started from ECCS [44] model is referred under the name “ECCS_Eq”. A sample data pool was tested using this new model. Table 5 contains the results from the model for geometric imperfection presented by Radwan and Kövesdi [42], and Table 6 contains the results when applying the realistic imperfection model “2021_mod”.

Both models, “ECCS_Eq” and “Somodi_mod”, have more accurate results compared to the original versions. In order to take the definitive residual stress distribution model and realistic geometric imperfection model couple and to conclude the validation of the computational model, residual stress models were taken into consideration using the actual residual stress values presented in the academic investigations. The stress pattern was taken into consideration based on the corresponding residual stress models; the shape parameters needed for the residual stress model were computed using the actual input data. Results are shown in

Table 9. Comparison of imperfection and RS distribution models using measured RS values.

Model	ECCS_Eq + 2021_mod	Somodi_mod. + 2021 mod	ECCS_Eq + Radwan [42]	Somodi_mod + Radwan [42]
Mean	1.012	0.993	1.028	1.017
Standard Deviation	0.051	0.049	0.060	0.055

, which were obtained using all the specimens that present actual measures for a total of 22 specimens. The results show that the modified RS models always perform better than their original versions, and it can be seen that the 2021_mod is the geometrical imperfection model that provides the most accurate results if the RS values are considered based on the measured values. The accuracies were 1.012 and 0.993, respectively, for the ECCS_RS pattern and for Somodi´s RS pattern. An overview of the scenarios tested during in the validation process is presented in Figure 10.

Based on the result, the best residual stress distribution model is the “Somodi_mod” that was developed in this work, paired to the geometrical imperfection model “2021_mod” that was also developed in this research. The whole validation presented until this point was compared to the ultimate local buckling resistance. However, the computational model should be able to replicate the behavior of the specimens along the loading process as much as possible. The last validation step involves a comparison of load–deformation curves and academic works which include this information: these are the publications of Clarin [1], and Schillo [6]. Therefore, load–deformation curves were also compared, taking one from each research program. From first investigation, the specimen labeled S20- was chosen, with dimensions of 100 × 3.05 mm and a yield strength of 471 MPa. The comparison is shown in Figure 11. For the second investigation, the selected box section was S960SHS220X6, with dimensions of 220 × 6 mm and a nominal yield strength of 960 MPa. The comparison is shown in Figure 12. Both the tabular and the curve comparisons show a good agreement, so the numerical model used can be considered accurate.

5. Evaluation of the Numerical Results

5.1. Comparison to Different Calculation Methods

The primary objective of this research is to investigate the ultimate resistance of local buckling failure of square-welded box sections subjected to pure compression and analyze the difference between the experimental and computed buckling resistances. Four approaches were studied: (i) analytical expression provided by EN1993-1-5 [4], (ii) alternative analytical calculation based on Annex B of EN 1993-1-5 [4], (iii) numerical model utilizing equivalent geometric imperfection (b/200) to cover the residual stresses and initial geometrical imperfection together, and (iv) advanced computation model which incorporates residual stress patterns and realistic geometric imperfections using the models analyzed in Section 4. The ultimate resistance provided by all calculation methods was compared to the experimental specimens available [1,3,5,6,7,8,9,10,11,12,14,21]. The results of the ultimate resistances ratios for 124 test results are shown in

Table 10. Ultimate loads ratio using all data collected.

Model	Advanced FEM with RS + imp. (Somodi_mod + 2021_mod)	FEM with Equivalent Imperfection (b/200)	EN 1993-1-5 Annex B [4]	EN 1993-1-5 [4] (Winter Curve)
Mean	1.022	0.967	1.058	0.946
Standard deviation	0.095	0.112	0.101	0.108
Maximum value	1.318	1.304	1.279	1.255
Minimum value	0.808	0.640	0.718	0.606

. The obtained data points are also plotted in function of relative slenderness in Figure 13. The vertical axis of the first four diagrams shows the results of the experiments and the horizontal axis shows the buckling resistance calculated by the four different design methods (two analytical and two numerical methods). The diagram e and f compares the standard-based resistances to the FEM-based values.

Comparing Figure 13a,b, results show that for slender sections, EN1993-1-5 [4] are always on the safe side and for stockier sections the calculated resistance is a good approximation of the average resistance. The advanced FEM effectively provides the expected value of the real resistance for every slenderness region within a ~10% range. The results shown in Figure 13c prove that the application of the Annex B curve is more on the safe side than the Winter-type buckling curve of the EN 1993-1-5 code [4] and the application of this buckling curve leads to a generally safe side resistance, especially for slender cross-sections. Figure 13d shows that the application of b/200 as equivalent geometric imperfections leads to safe side resistance for stocky plates and unsafe side values for slender columns. This observation proves that the value of the equivalent geometric imperfection needs to be revised and its accurate value should be slenderness-dependent. Comparing Figure 13e,f, results indicate the local buckling resistance determined by EN1993-1-5 [4] is conservative when it is compared to the resistance values obtained using an advanced computational model for the entire range of plate slenderness analyzed. Conversely, the computational approach outlined in EN1993-1-5 Annex B [4] demonstrates similar resistance to the values obtained by advanced computational models for stocky plates, while exhibiting a reduction in resistance capacity for slender plates.

Utilizing the data gathered throughout this project, Figure 14 displays the normal distribution of the ultimate local buckling resistance ratio from the analyzed methods for welded square box sections. The results show that GMNIA modeling incorporating geometric imperfection based on the Somodi_mod model (see Equation (19)), and the residual stress pattern based on 2021_mod model (see Equation (21)), provide the most accurate prediction for the load capacity, with a smaller standard deviation compared to the other methods analyzed.

5.2. Comparison to Analytical Buckling Curves

Using the final advanced computational model, an additional numerical parametric study was executed to obtain the most accurate local buckling resistances, which could be compared to the local buckling curves depending on the relative slenderness of the analyzed cross-section. The numerical parametric study was made on 100 virtual models, where the input data of the yield strength, cross-section size and plate thickness values were generated using Matlab’s built-in uniformly distributed random number generator. The parameter limits were set to the lowest and highest value used in the experimental tests, ranging from 280 to 1350 MPa, 70 to 680 mm, and 1 to 14 mm, respectively. Figure 15 illustrates the numerical results depending on the relative slenderness ratio (as presented on the horizontal axis). The reduction factor for local buckling (presented on the vertical axis) was back-calculated from the numerically calculated buckling resistances. The obtained numerical results were compared to different analytical buckling curves and the best-fit curve is also indicated on the graph. Three different analytical buckling curves were compared to the numerical results, which were (i) the local buckling curve of EN 1993-1-5 [4] using Winter´s formula; (ii) the local buckling curve of EN 1993-1-5 [27] using formula of Annex B; and (iii) the buckling curve proposal of Schillo [6]. The fitting process was performed using the software tool “ODRPACK” version 2.01, employing a weighted orthogonal distance regression method (ODR). Based on the obtained result, it can be seen that the best approach with results closest to the data points calculated by the most accurate numerical model was the buckling curve proposed by Schillo [6]. The figure also proves that the buckling curve given by Annex B of EN 1993-1-5 [4] leads to safe side resistances, especially for very slender plates. The results also prove that the Winter-type buckling curve would overestimate the buckling resistance for the welded box section columns, as proven by previous research results as well.

6. Conclusions

After an extensive literature review, 124 sets of experimental data on local buckling failure in square-welded box sections were collected. These test results formed the basis for a detailed verification and validation process of an advance computational model that incorporated the main factors contributing to the decrement of ultimate resistance in welded thin-walled sections. These factors include residual stress distribution patterns and geometric imperfections. Based on the comparison of the test results and the advanced numerical model, the accuracy of each residual stress and geometric imperfection models were evaluated from the local buckling resistance point of view. The best performing geometric imperfection model was developed in this research and is shown by Equation (19), building upon the proposal presented by Somodi et al. [50]. For residual stresses pattern generated during the manufacturing process the best performing model is the model presented by Somodi et al. [20].

After identifying the most accurate numerical model, a numerical parametric study was conducted to analyze the local buckling resistance of welded box section columns. The resistances were compared to different analytical design approaches, which were evaluated by the obtained results. Numerical calculations prove the following:

• The best analytical approach closest to the calculated data points obtained by the most accurate numerical model was the buckling curve proposed by Schillo [6];
• The buckling curve given by Annex B of EN 1993-1-5 [4] leads to safe side resistances, especially for very slender plates;
• The Winter-type buckling curve would overestimate the buckling resistance for the welded box section columns, as proven by previous research results as well.

Based on the current investigations, proposal can be given to designers to use the buckling curve given in the Annex B of EN 1993-1-5 [4] within the design process by calculating the local buckling resistance of slender welded box section columns. Other conclusions and design proposals is, if direct resistance check is used in the design process by calculating the local buckling resistance with numerical model, residual stresses and geometric imperfections should be selected properly; thus, they can have a significant impact on the obtained buckling resistance. The given overview can help to select the most appropriate models for the design purposes.

The current study analyzed only welded box section columns where the failure mode was pure local buckling, eliminating any global flexural buckling failure. All the analyzed cross-sections were sensitive to local plate buckling, so they belong to class 4 cross-sections according to EN 1993-1-1 [54]. Another limitation of the current research program is that cross-section widths varied between 70 and 700 mm and plate thicknesses between 1 and 14 mm were analyzed. This was the investigation range within the current study.