Influence of patch load length on plate girders. Part II: Numerical research

Journal of Constructional Steel Research 158 (2019) 213–229

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Journal of Constructional Steel Research

Influence of patch load length on plate girders. Part II: Numerical research S. Kovacevic a,⁎, N. Markovic b, D. Sumarac b, R. Salatic b a b

School of Mechanical and Materials Engineering, Washington State University, Pullman 99164, USA Faculty of Civil Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia

a r t i c l e

i n f o

Article history: Received 16 October 2018 Received in revised form 12 March 2019 Accepted 22 March 2019 Available online xxxx Keywords: Patch load length Finite element analysis Ultimate strength Plate girder Stiffened web

a b s t r a c t The companion paper Part I [1] deals with the experimental campaign of longitudinally stiffened plate girders subjected exclusively to patch loading. The current paper focuses on the numerical research and parametric study of the influence of patch load length and initial geometrical imperfections on the ultimate strength of longitudinally stiffened plate girders. In order to assess the patch load resistance, a geometrically and materially nonlinear finite element analysis has been performed. For a better verification with the experimental results, the finite element model includes the experimentally measured initial geometrical imperfections and material properties based on laboratory tests. The verification of the numerical model has been obtained through the comparison of the numerically and experimentally attained results for the ultimate loads and elastoplastic behavior of the girders. It has been shown that the numerical and experimental results are in perfect agreement, which enabled a fruitful background for parametric analysis, in which different initial geometrical imperfections have been used to ameliorate understandings about their influence on the ultimate strength under different patch load lengths. Conclusively, it may be stated that initial geometrical imperfections can play a decisive role, especially for longitudinally stiffened girders. Initial geometrical imperfections of stiffened girders that correspond to deformed shape at the collapse (collapse-affine imperfections), especially in the zone where the load is applied, will give the most unfavorable ultimate strengths. For the considered geometry in the present paper, the third buckling mode of longitudinally stiffened girders corresponds to the deformed shape and the lowest ultimate strengths are obtained. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The influence of the length of patch load on the ultimate strength and behavior of longitudinally stiffened I-section girders, experimentally tested, has been thoroughly analyzed in the companion paper Part I [1]. The research is continued in the current paper focusing on the numerical modeling, verification of the numerical model through the comparison between the experimental and numerical results, and parametric study, considering different initial geometrical imperfections. The driving force for this research was the fact that papers dealing with different patch load lengths are barely present in the literature. In order to investigate the patch loading resistance with different patch load lengths, a geometrically and materially nonlinear finite element analysis has been performed. Nowadays, numerical analysis techniques are widely used in research involving structural steel and in analyses and designs of steel structures and elements. The main criterion for a trustworthy and accurate numerical analysis is the concurrence between numerical and experimental results. Therefore, adequate and safe modeling of ⁎ Corresponding author. E-mail address: [email protected] (S. Kovacevic).

https://doi.org/10.1016/j.jcsr.2019.03.025 0143-974X/© 2019 Elsevier Ltd. All rights reserved.

engineering problems presents an important step in the process of research and design. Also, the current European design standard Eurocode 3 Part 1–5 [2] allows the usage of numerical analyses for the design of plate steel structures. Thus, it is obvious that the numerical approach and modeling technique represents a basic tool in the process of research and design. Scarce work has been found in the literature for the influence of patch load length on the ultimate strength of longitudinally stiffened plate girders. In Part I of this paper we have shown a detailed literature review emphasizing experimental tests conducted on longitudinally stiffened girders and proved that experimental results are very rare, while in the present Part II we will show that numerical investigations considering different patch load lengths on longitudinally stiffened plate girders are also barely present in literature. Also, the present Part II is exclusively devoted to numerical modeling and parametric study, in which different initial geometrical imperfections were varied. In addition, we will limit our focus only to longitudinally stiffened girders loaded centrically with web panels without web openings (cut-outs). For more general reviews including plate girders subjected to different types of loading, papers [3,4] are recommended. There exists an appreciable amount of literature on the influence of initial geometrical imperfections on the patch load resistance of

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longitudinally stiffened girders subjected to patch loading but the patch load length ss or the ratios ss/a and ss/hw (where a is the web plate length and hw is the height of the web plate) were mostly constant or not changed with greater length rates. For instance, extensive parametric studies carried out on longitudinally stiffened plate girders examining the influence of position of a longitudinal stiffener using different initial geometrical imperfection shapes was performed by Graciano and co-workers [5–11]. In all these analyses the patch load length was kept constant. Another comprehensive numerical research conducted on longitudinally stiffened plate girders was managed by Chacon, varying the web thickness, flange yield strength, longitudinal stiffener thickness and position of the longitudinal stiffener [12,13] but again, the patch load length was held constant. Similarly, Kuhlmann and Seitz [14] performed an imperfection sensitivity study considering different shapes of initial geometrical imperfections and their influence on the collapse load of longitudinally stiffened plate girders under a constant patch load length. On the other hand, Graciano et al. in [15,16] applied different patch load lengths in order to get more information about elastic buckling coefficients varying the relative position of the longitudinal stiffener, its flexural rigidity, flange-to-web thickness ratio and aspect ratio a/hw. A similar linear buckling and post-buckling analysis considering multiple longitudinally stiffened webs have been materialized by Loaiza et al. [17,18], respectively. Moreover, elastic buckling analysis performed on longitudinally stiffened and unstiffened web panels under opposite patch loading considering three different patch load lengths was conducted by Mezghanni et al. [19]. A parametric study performed by Kuhlmann and Seitz primarily investigating the influence of stiffener position on the magnification factor under three different patch load lengths is present in [20,21]. In addition, Maiorana et al. [22] used different values of patch loading length considering primary unstiffened webs with and without perforations. However, three different lengths of patch loading were applied to stiffened webs without perforations. The main goal of the study was to juxtapose ultimate loads with critical ones for a single panel in a multi-panel beam with the whole beam. Additionally, an extensive finite element study including 366 numerical simulations focusing on longitudinally stiffened webs has been accomplished by Davaine [23–26]. The study was mainly based on deep webs, e.g. from 2 to 5 m, and different patch load lengths were varied according to the web plate height ss/hw = 0.2 ÷ 1 considering also different positions of longitudinal stiffener. In addition, a parametric study including 900 models and varying different parameters, i.e. geometry of the web panel (a, hw, tw), flange size (bf, tf), patch loading length ss, location and number of longitudinal stiffeners and their size, has been conducted by Kövesdi [27]. The patch load length was varied from ss/a = 0.2 to ss/a = 0.8. The number of different patch load lengths is not specified but extracting data from the graphs it is concluded that the author used six different patch load lengths. Just recently, four different patch load lengths were incorporated into a numerical analysis performed by Loaiza et al. [28] in which the main parameters that were varied are the aspect ratio a/hw and web panel thickness tw. All aforementioned references are based on the finite element method which is the most popular computational tool in this field due to its wide acceptance and versatility, and it has been successfully applied in many papers regarding the ultimate strength of plate girders under different loading conditions. On the other hand, there are a few papers assessing the ultimate capacity of plate girders under patch loading or under combined action of patch loading and bending using a different numerical technique, i.e. in [29,30] the finite difference method has been used for the determination of ultimate loads or [31] for critical loads, in [32,33] the dynamic relaxation method is adopted as a numerical analysis method. Additionally, in [34] the finite strip method has been employed as a computational tool in order to obtain critical loads of longitudinally stiffened webs under patch loading. In all these references, the patch load length was constant or varied in a small range.

Based on this detailed literature overview, one can see that papers primarily concerned with the analysis of the ultimate strength of longitudinally stiffened plate girders under different patch load lengths are barely present. Usually, researchers directed their study including different parameters that have an influence on the ultimate strength, i.e. thickness of the web plate, position of the longitudinal stiffener and its rigidity, flange-to-web thickness ratio, aspect ratio and yield strength of the flange and web. Therefore, the aim of this paper is describing a geometrically and materially nonlinear behavior, and ultimate strength of longitudinally stiffened plate girders specifically concentrated on different patch load lengths. The paper is organized as follows. The next section introduces a nonlinear finite element model which has been employed in the parametric study. As a computational and simulation tool, the commercial multipurpose finite element software Abaqus was used [35]. The presented numerical model has been verified by comparison with our experimental results [1] and evoked further directions of the research. The numerical analysis regarding the behavior and ultimate strength of plate girders, including different patch load lengths and initial geometrical imperfections, is performed in the third section and the results are thoroughly discussed. At the end section, conclusions along with recommendations for further research are pointed out. 2. Nonlinear finite element model This section presents a nonlinear numerical model used in this paper. Numerical modeling technique is often employed to effectively expand limited experimental tests and used to scrutinize the influence of relevant parameters connected with a problem. In this research, the computer simulations are performed by the finite element method, which has been proven to be a powerful tool for modeling the postbuckling behavior of plate girders under patch loading. The finite element model is calibrated by comparison with own experimental results presented in [1] and parametric studies are conducted. The nonlinear computations were performed using the commercial multi-purpose finite element software Abaqus [35]. All patch loading resistances of the girders were determined using geometrically and materially nonlinear analysis to fully capture the post-buckling behavior. In order to properly and efficiently trace the complex nonlinear path of the load-displacement response of the girders, which generally can exhibit a decrease in load and/or displacement as the solution evolves, the modified Riks' method [36] has been used in the finite element analysis. This method is an incremental-iterative procedure and it is suitable for predicting unstable, geometrically nonlinear collapse of a structure including nonlinear materials. 2.1. Geometrical and material properties The numerical models are created taking into account the geometrical and material properties of the plate girders showed in Table 1 (see Fig. 1 for general notation). Mechanical behavior of the material is established by means of uniaxial tension tests which determine the engineering stress and strain. In order to include material nonlinearity in Abaqus, the engineering stress-strain diagrams are transformed into the true stress-true strain relationship, σtrue = σeng(1 + εeng) and εln = ln(1 + εeng). The steel was modeled as an isotropic material with a von Mises yield surface and with an isotropic work hardening assumption. The nonlinear stress-strain relationship was idealized by a multi-linear stress-strain curve assuming hardening up to the ultimate strength of the material. After the ultimate stress was reached, an indefinitely ductile plateau was assumed. For simplicity and to provide relatively general conclusions, one stress-strain curve for the web plate and one for the flange, transversal and longitudinal stiffeners was used in all presented simulations, Fig. 2. They represent a mean curve from the behavior of all uniaxial tests for the webs and flanges. Additionally, in order to define the

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Table 1 Geometrical and material characteristics of the experimentally tested girders. No.

Girder Label

1 A15 2 A14 3 A12 4 A4 5 A1 6 A3 7 A17 8 A11 9 A5 10 A6 11 A2 12 A7 13 A13 14 A16 Total average stresses

a [mm]

hw [mm]

tw [mm]

bf [mm]

tf [mm]

b1 [mm]

ss [mm]

hs [mm]

ts [mm]

fyw [MPa]

fyf [MPa]

fuw [MPa]

fuf [MPa]

500 500 500 500 500 500 500 500 500 500 500 500 500 500

500 500 500 500 500 500 500 500 500 500 500 500 500 500

4 4 4 4 4 4 4 4 4 4 4 4 4 4

120 120 120 120 120 120 120 120 120 120 120 120 120 120

8 8 8 8 8 8 8 8 8 8 8 8 8 8

– 100 – 100 – 100 100 – 100 100 – 100 – 100

0 0 25 25 50 50 75 100 100 125 150 150 150 150

– 30 – 30 – 30 30 – 30 30 – 30 – 30

– 8 – 8 – 8 8 – 8 8 – 8 – 8

327.50 320.50 323.50 320.50 305.00 324.00 318.00 305.00 327.50 332.00 323.50 318.00 324.00 332.00 321.50

321.00 315.00 321.00 315.00 n.a. 325.00 317.00 n.a. 321.00 320.00 321.00 317.00 325.00 320.00 319.83

443.00 441.50 445.00 441.50 434.00 437.50 434.00 434.00 443.00 444.00 445.00 434.00 437.50 444.00 439.86

468.00 476.00 466.00 476.00 n.a. 474.00 475.00 n.a. 468.00 475.00 466.00 475.00 474.00 475.00 472.33

n.a. = not available.

elastic behavior of the girders, an elastic modulus of 205 GPa and Poisson's ratio of 0.30 is employed. 2.2. Finite element mesh A general-purpose fully-integrated four-node quadrilateral shell element S4 from the Abaqus element library was used for modeling the web, flanges, transversal and longitudinal stiffeners. This first-order shell element has 6 DOFs per node (three displacements and three rotations) and it is applicable for most thick and thin shell applications and permitting large strains. They give realistic results and it is unnecessary to model plate girders using computationally expensive solid continuum elements or more specialized finite element formulations. Reliability and accuracy of numerical results with the accepted element type were previously proved in other similar studies. In order to include the same loading conditions as in the experiments [1], the rigid loading blocks are modeled as a separate structural element and the applied loads are transferred through these blocks onto the upper flange. The width of the load blocks is the same as the upper flange while the length varies from 0 to 150 mm. However, for the purpose of the parametric study, the load length will be extended up to 250 mm. For all load blocks except for the half-round bar (ss = 0 mm–crane wheel loading), a four-node 3D bilinear rigid quadrilateral element R3D4 was used. In order to generate a more regular finite element mesh with exclusively quadrilateral elements, all girders, and load blocks are meshed using a structured meshing technique with quad and hex element shape control. Special attention is paid in modeling the half-round bar ss = 0 mm since there is no finite loading length. According to the experimental measurement, the radius of the half-round bar was 25 mm and it was modeled as such. Only for this loading block,

a general-purpose fully integrated linear brick element C3D8 was employed. It was also meshed with hexahedral elements and sweep meshing technique is adopted. The girders and load blocks were modeled in full size and the finite element mesh for three representative models is shown in Fig. 3. The structural elements (web plate, flanges, and transversal stiffeners) are merged in order to define a whole girder. To take into account the influence of the longitudinal stiffener, a coupling constraint, restricting the motion of the stiffener to the connection line between the web plate and the stiffener, has been applied. In this case, all translational and rotational degrees of freedom are constrained. In order to avoid overlapping at the junction where the flange plates are welded to the web plate, the shell nodes of the flanges are offset so that they are located at the top or bottom surface of the shell instead of the mid-plane. On the other hand, the web plate, transversal, and longitudinal stiffener surface are modeled as the mid-plane. Furthermore, in order to achieve structural interaction between the load blocks and the upper flange, an equation-based constraint was used. The central node on the top side of the load blocks is defined as a master node (reference node) while the corresponding nodes on the upper flange are defined as slave nodes. All degrees of freedom for the slave nodes are constrained to follow the master node which enables an equal displacement along and across the upper flange. As mentioned above, the half-round bar was modeled in a different way. In this case, the interaction is defined by a tie-based constraint. The nodes on the bottom side of the half-round bar are tied with the nodes on the upper flange where all degrees of freedom are tied. For the contact discretization method, a surface-to-surface method was used. In addition, an h-refinement (reduction in the element sizes) convergence study has been performed for longitudinally stiffened and unstiffened girders, Fig. 4. One can instantaneously see that the last two subsequent mesh refinements (from 5 mm to 1.5 mm) do not change the results substantially, i.e. the relative difference for all four girders is around 0.7% but the computational costs are exceedingly increased. Therefore, an element size of 5 mm has been accepted for both groups of girders. The unstiffened and stiffened girders approximately contain 28,000 and 30,000 finite elements, respectively, where the load blocks contain from 120 to 1200 finite elements depending on the patch load length. 2.3. Initial imperfections

Fig. 1. Longitudinally stiffened plate girder. Notation.

Initial geometrical imperfections are defined by initially imperfect plates following a certain shape. They can be introduced into numerical modeling in different ways, i.e. using experimental data (experimentally measured imperfections), using mode shapes obtained from an eigenvalue buckling analysis or considering imperfections as a

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Fig. 2. Engineering stress-strain curves obtained by tensile test vs. curves used in the simulations for: (a) web plate; (b) flanges, transversal and longitudinal stiffeners.

two-dimensional random field. For a better validation between the numerical model and the experimental results, the experimentally measured initial geometrical web imperfections (precise shapes of initial imperfections) were used. The contour plots of the used initial geometrical imperfections can be found in [1]. Special interest is devoted to studying the influence of the shape of initial geometrical imperfections in the parametric study presented in the next section. Papers dealing with measured initial geometrical imperfections considering solely patch loading as a load case are very rare. For instance, papers [37,38] are dealing with the experimentally measured initial geometrical imperfections for longitudinally unstiffened, while papers [14,21,22] for stiffened plate girders. The modeling technique for the measured initial geometrical web imperfection is illustrated in Fig. 5. Modeling of these imperfections represents a challenge and includes the following steps. Firstly, all web imperfections are recorded as 3D points (x, y, z coordinates) before each test using a uniform-spaced grid pattern (50 × 50 mm). Secondly, the point-wise web imperfections are then imported into the commercial 3D computer graphics and computer-aided design (CAD) software Rhinoceros [39] in order to define a NURBS surface. The surface is sketched from the grid of points that lie on the surface using second order interpolation functions in both directions. After that, the developed surface is exported as an ACIS SAT file and, finally, imported into Abaqus. Contrary to the webs, all other elements of the girders, i.e. the flanges, transversal and horizontal stiffeners, were modeled as perfect straight surfaces.

The available pre-processor in Abaqus was employed for assembling all elements in order to define a whole girder and to mesh the model. Therefore, the meshing step is performed on an imperfect specimen. However, one can introduce the initial geometrical imperfection using the opposite way, e.g. performing the meshing step on a perfect specimen and changing the coordinates of all nodes according to measured imperfections [40]. On the other hand, structural imperfections, which are characterized by a residual stress pattern and can be differently idealized based on different design codes, were not considered since they do not play a decisive role as reported in [37,41,42] for longitudinally unstiffened plate girders. Also, any flaws concerning unintended rotation of the loaded flange by the load application were not included. 2.4. Loads and boundary conditions To implement the real loading conditions, like in the laboratory tests, the patch load was transferred onto the girders by a loading block and using the master node at the top surface of the block. The loading blocks have width bf and length ss and they were modeled as rigid regions to ensure the rigid behavior perpendicular to the flange surface. The load block for an even distribution of the load over the entire load length (special load block for ss = 150 mm, see Fig. 4 in [1]), was idealized with 4 independent rigid blocks, cf. Fig. 3. This load configuration was only used for comparison with the experimental results (girders A13

Fig. 3. Finite element mesh for three representative models. Rigid blocks for 0 b ss ≤ 150 mm (top-left), 4 independent rigid blocks for ss = 150 mm (top-middle) and half-round bar for ss = 0 mm (top-right). For the setup configuration of these blocks in the experiment see Fig. 4 in [1].

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Fig. 4. Convergence plot for the ultimate load for: (a) unstiffened girders; (b) stiffened girders.

and A16) while for the parametric study the load length ss = 150 mm was modeled as one rigid block. A compressive displacement loading is applied by moving the master node towards the upper flange. A small displacement of 0.5 mm was applied and then the analysis is continued by performing an incremental load-deflection analysis using the modified Riks' method in order to trace the nonlinear path of the girders. The supports were designed as simply supported with symmetrically double-sided transverse stiffeners above them. These nodes are only constrained in the vertical direction. A node at the center of each support is only constrained in the direction perpendicular to the web plane, while a node at the center of the girder is only constrained in direction of the girder axis. For the master node, all degrees of freedom are restricted except the vertical direction. Remark: It is worth to note that some authors used the rigid boundary conditions (kinematic boundary constraints at girders' ends) instead of vertical stiffeners. However, based on the parametric analysis given in [43], the authors showed that the application of this approach is only justified for small patch load lengths (ss/a ≤ 0.25). Moreover, they pointed out that a better match between experimental and numerical results is obtained including transverse stiffeners. On the basis of this analysis and since our experimental program includes patch load lengths from ss/a = 0 to ss/a = 0.3, which will be extended in the parametric analysis up to ss/a = 0.5, we will model transverse stiffeners in all our models. Another interesting point regarding the boundary conditions is the usage of symmetry and modeling one-half of girders. Some authors used the symmetry boundary conditions and modeled one-half of girders, which represents a powerful tool for decreasing computational time and resources. However, in our numerical model, the symmetry boundary conditions are not included since the measured initial geometrical imperfections are highly nonsymmetric and the whole girder is modeled. In addition, the symmetry boundary conditions incorporated into an eigenvalue buckling analysis will entirely generate

symmetrical buckling modes. On the other hand, modeling the whole girders could also give nonsymmetrical ones. For instance, in our case, the third buckling mode for ss = 250 mm is not a symmetrical mode. Therefore, care should be taken in order to use this approach in an automated parametric study. 2.5. Model validation It was stated in the introduction that the central aim of this paper is a numerical investigation of the ultimate strength and behavior of plate girders under different patch load lengths. To clarify and examine the influence of patch load length on longitudinally stiffened girders, a parametric study by means of finite element analysis will be carried out. In order to perform a numerical-based parametric study and to investigate the problem further, it is necessary to conduct calibration of the numerical model so that the finite element analysis gives a reasonable resemblance of the experiments. We will juxtapose the patch loading resistance of all the experimentally tested girders with the above described numerical model, followed by the comparison of elastoplastic behavior between two investigations. Table 2 shows a comparison summary of the numerically (FFEA) and experimentally (Fexp) obtained patch loading resistance while a graphical juxtaposition is portrayed in Fig. 6. The present numerical model exhibits a remarkably close agreement with the experimental results with an average error of 0.83% and 2.60% for the unstiffened and stiffened girders, respectively. It is noteworthy to observe that for smaller patch load lengths the experimental values of the ultimate load are slightly greater than the numerical ones. In view of the foregoing analysis, it should be mentioned that all numerical simulations included the same material characteristics (cf. Fig. 2) and small discrepancies between the results are expected. As a further comparison, the elastoplastic behavior of the experimentally tested girders and numerical simulations will be addressed. As reported in Part I [1], the elastoplastic behavior for girder A1, A2,

Fig. 5. Modeling technique for measured initial geometrical imperfections.

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Table 2 Experimentally and numerically obtained results of the patch loading resistance [kN]. Unstiffened girders

A15 ss = 0 mm

A12 ss = 25 mm

A1 ss = 50 mm

A11 ss = 100 mm

A2 ss = 150 mm

A13 ss = 150 mm (even distr.)

Fexp FFEA Fexp/FFEA

143.30 142.73 1.00

154.60 144.32 1.07

165.00 166.02 0.99

199.00 204.18 0.97

215.00 224.53 0.96

230.00 242.35 0.95

Stiffened girders A14 ss = 0 mm A4 ss = 25 mm A3 ss = 50 mm A17 ss = 75 mm A5 ss = 100 mm A6 ss = 125 mm A7 ss = 150 mm A16 ss = 150 mm (even distr.) Fexp FFEA Fexp/FFEA

165.90 148.09 1.12

180.00 156.89 1.15

183.00 174.58 1.05

194.30 193.09 1.01

A3, and A7 is discussed in detail and it will be compared here with the present numerical model by means of von Mises stress contour plots at different levels of the ultimate load. The von Mises stress is calculated as σvM = (3/2 S:S)1/2, where S is the deviatoric stress tensor. As can be seen in Figs. 7 and 8, the yielding starts at about 50% of the maximum load for patch load length ss = 50 mm while for patch load length ss = 150 mm the plastification starts at 85% of the ultimate load, which is in full compliance with the discussion addressed in Part I. Furthermore, these figures also prove that stresses in the lower half of the girders are far below the yield stress which justifies the usage of the girders in the repeated tests, i.e. loaded on the opposite flange. One should bear in mind that the values showed in the legend of these figures are averaged at nodes since the von Mises stress is not a nodal field output. The values are stored at the Gauss integration points and for the purpose of visual detection of the plastification, they are extrapolated at nodes using a 75% averaging threshold (Figs. 9 and 10). Additionally, the vertical displacement of the load cell was not recorded in the experimental campaign but rather an out-of-plane deflection of the web plate (at a specific point on the web plate or the whole middle line web profile) or the vertical displacement of the loaded flange at points eccentrically placed with respect to the web plane. However, the load-displacement response for the vertical displacement and normalized capacity (normalized with respect to the girder's ultimate load), where the vertical displacement represents the displacement of the loading node, can be extracted from the finite element simulations, Figs. 11–13. These plots also support the findings from the experiments and show a linear behavior up to at least 80% of the ultimate load. In addition, plotting the load-displacement response for longitudinally stiffened and unstiffened girders on the same scale clearly shows the difference in the behavior of stiffened and unstiffened webs. One can instantaneously see that non-linearities for the longitudinally stiffened girders occur for much higher loads and that the behavior after the ultimate strength is reached can be the same as for unstiffened webs for small patch load lengths, while for greater patch load lengths the load decreases considerably faster for the longitudinally stiffened girders.

Fig. 6. Comparison of the ultimate strengths between the longitudinally unstiffened and stiffened plate girders obtained experimentally and numerically.

225.00 227.13 0.99

259.00 251.78 1.03

255.00 281.25 0.91

244.60 254.71 0.96

Conclusively, it may be stated that the numerical and experimental results, as well as post-buckling behavior, are in perfect agreement and that the present numerical model can be used to reach a deeper insight into the behavior and ultimate strength of plate girders subjected to patch loading. The presented finite element modeling technique and verification of the model with the experimental results enabled a fruitful background for parametric analysis in which a large number of numerical tests will be developed.

3. Parametric study Since the presented numerical model has been proved to be valid and accurate for describing the behavior and ultimate strength of the experimental tests, a further set of numerical analyses is performed to investigate the influence of patch load length. The parametric study is formulated with the aim of examining the influence of patch load length considering different shapes of initial geometrical imperfections which was impossible through the experimental program. To get a better visual picture of the ultimate capacity of longitudinally stiffened girders, the patch load length was varied from 0 mm to 250 mm (ss/hw = ss/a = 0–0.5) while the geometrical characteristics (Table 1) and material properties (Fig. 2) were kept constant. It is well-known that this position of the longitudinal stiffener (b1 = 0.2hw) is not the most efficient for patch load resistance [9,10,20,21,44–46]. It is recommended by design codes as the optimum location to increase the ultimate strength of plate girders subjected to in-plane bending moments, which is the most present load case for practical purposes. Therefore, considering the position of longitudinal stiffeners only to patch loading may give an uneconomical design. In this section, the results will be presented along with detailed discussions and conclusions. The present numerical base contains 360 runs, and all numerically obtained ultimate loads for longitudinally unstiffened and stiffened girders for all considered cases are listed in Tables 3 and 4, respectively. Firstly, the experimentally measured initial geometrical imperfections of all tested girders are considered and used for both longitudinally unstiffened and stiffened plate girders. These shapes are given in the Appendix of the companion paper Part I [1] and it will not be repeated here. In order to exclude the amplitude size, the initial geometrical imperfections are normalized so that the maximum out-of-plane deformation in the upper half of a girder is hw/100 = 5 mm, which is within the allowable tolerance in the Eurocode 3 Part 1–5 [2]. The patch load resistance using normalized initial geometrical imperfections from the girders experimentally tested as unstiffened (A15, A12, A1, A11, A2, and A13) is displayed from Figs. 14 to 16 while Figs. 17 to 20 present the ultimate capacity using initial geometrical imperfections from the girders experimentally tested as stiffened (A14, A4, A3, A17, A5, A6, A7, A16). One can instantaneously notice that for smaller patch load lengths, the influence of the longitudinal stiffener is negligible and a certain threshold (patch load length) exists after which an appreciable strengthening effect (Fstiffened/Funstiffened) can be achieved.

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Fig. 7. von Mises stress contour plots [MPa] for unstiffened girder A1 (ss = 50 mm) at different levels of the ultimate load. The contour plots at the top represent stresses in the shell surface facing the reader while the bottom plots represent the other surface.

Secondly, the most used initial geometrical imperfections represent buckling modes based on a linear buckling analysis since they can be very easily incorporated into the finite element analysis. Many researchers in this field have used this approach since they did not have experimentally measured imperfections, or they did not use them in the finite element analysis. In the linear buckling analysis, no imperfections were included; the web plate was perfectly plain. Generally, researchers [6,8,14,37,47], have used the first three buckling modes and/or their combinations, so the same approach will be used in this paper. The first three buckling modes for five different patch load lengths for longitudinally stiffened webs are presented in Fig. 21. In order to juxtapose these results with the previous ones, the same amplitude of outof-plane deflection has been accepted. The attained results are graphically shown in Figs. 23 and 24 while numerical values are tabulated in Tables 3 and 4. Additionally, another way to define initial geometrical imperfections is considering a twodimensional random field, usually defined with sine [20,44–46] or cosine function [9,10,41,48], in both directions (longitudinal and transversal). Using these functions, the initial geometrical imperfections in this numerical research are defined as   πx πy wðx; yÞ ¼ w0 sin sin a hw

ð1Þ

and wðx; yÞ ¼

w0 4



    2πx 2πy 1− cos 1− cos ; a hw

ð2Þ

respectively. In Eqs. (1) and (2) w 0 is a required amplitude. A schematic view showing the difference between these two imperfections is given in Fig. 22. The results considering the sine or cosine function for different patch load lengths are plotted in Fig. 25. 4. Discussion 4.1. Experimentally measured imperfections Considering the experimentally measured initial geometrical imperfections, two groups of analysis were established. The first group included the initial geometrical imperfections from the girders that were originally unstiffened in the experiment, cf. Figs. 14–16. According to these results, one can immediately see that the influence of the longitudinal stiffener for ss/hw ≤ 0.15 (ss = 75 mm) is negligible since it increases the ultimate capacity of b5%. On the other hand, for larger patch load lengths the longitudinal stiffener increases the patch load resistance significantly, from 30 to 40% for ss/hw = 0.5 (ss = 250 mm). The second group involved the initial geometrical imperfections from the girders that were initially stiffened in the experiment, cf. Figs. 17–20.

Fig. 8. von Mises stress contour plots [MPa] for stiffened girder A3 (ss = 50 mm) at different levels of the ultimate load. The contour plots at the top represent stresses in the shell surface facing the reader (longitudinal stiffener's side) while the bottom plots represent the other surface.

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Fig. 9. von Mises stress contour plots [MPa] for unstiffened girder A2 (ss = 150 mm) at different levels of the ultimate load. The contour plots at the top represent stresses in the shell surface facing the reader while the bottom plots represent the other surface.

Again, a similar conclusion can be given, and the same threshold ss/hw ≤ 0.15 is valid. However, for some initial imperfections, the threshold has been shifted even up to ss/hw = 0.30 (ss = 150 mm) and for larger patch load lengths for longitudinally stiffened girders, the ultimate strength is again increased, but notably less than for the first group of girders, from 15 to 30% for ss/hw = 0.5 (ss = 250 mm). Only in one case, an appreciable strengthening effect (Fstiffened/Funstiffened) of 40% has been achieved for ss/hw = 0.5. Fig. 26 recaps these conclusions. Using the numerically obtained results for the experimentally measured imperfections listed in Tables 3 and 4 for longitudinally unstiffened and stiffened girders respectively, a more detailed quantitative analysis of the influence of initial geometrical imperfections can be handled. Isolating results for the first (geometrical imperfection from the experimentally unstiffened girders) and second group of girders (geometrical imperfection from the experimentally stiffened girders), it can be shown that the shape of initial geometrical imperfections can play a decisive role for both longitudinally stiffened and unstiffened girders since it affects the ultimate loads in a range of N15%, as portrayed in Fig. 27. This is more pronounced for larger patch load lengths while its impact for ss/hw b 0.10 (ss = 50 mm) is inappreciable (b10%). 4.2. Buckling mode-based imperfections Considering the initial geometrical imperfections defined from an eigenvalues analysis we can confirm the statement from other

researchers that the first buckling mode will not give the lowest ultimate strength for longitudinally stiffened girders, [6–8,11]. Furthermore, the first buckling mode gives the same threshold ss/hw ≤ 0.15 (ss = 75 mm) like in the previous case for the experimentally measured initial geometrical imperfections and the difference between the patch load resistance of longitudinally stiffened and unstiffened girders before the threshold is negligible (b5%). In addition, an interesting phenomenon can be noticed in the analysis of buckling modes. Firstly, the lowest ultimate strength for longitudinally stiffened girders is obtained for the third buckling mode and it is lower than the ultimate capacity for unstiffened girders for ss/hw ≤ 0.25 (ss = 125 mm). The authors in [6–8,11] also concluded that the third buckling mode gives the lowest patch load resistance for longitudinally stiffened girders for small patch load length (ss = 0.04hw) and for stiffener positions b1/hw ≤ 0.25 considering different imperfection amplitudes (hw/1000-hw/100). Secondly, the combination of modes (1st + 2nd + 3rd) again gives smaller ultimate loads for longitudinally stiffened than for unstiffened girders for ss/hw ≤ 0.30 (ss = 150 mm). Next, a similar observation can be noticed for the second buckling mode which, again, gives lower ultimate loads for longitudinally stiffened than for unstiffened girders only for very small patch load lengths ss/hw ≤ 0.05 (ss = 25 mm). This can be clarified with the shape of the second and third buckling mode for longitudinally stiffened girders. As can be seen in Fig. 21, the buckle for these modes of longitudinally

Fig. 10. von Mises stress contour plots [MPa] for stiffened girder A7 (ss = 150 mm) at different levels of the ultimate load. The contour plots at the top represent stresses in the shell surface facing the reader (longitudinal stiffener's side) while the bottom plots represent the other surface.

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Fig. 11. Load-displacement response for unstiffened and stiffened girder under the patch load length of: (a) ss = 0 mm; (b) ss = 25 mm.

Fig. 12. Load-displacement response for unstiffened and stiffened girder under the patch load length of: (a) ss = 50 mm; (b) ss = 100 mm.

stiffened girders occurs between the longitudinal stiffener and loaded flange and its magnitude decreases with increasing the patch load length. As a corollary, this shape of initial geometrical imperfection is much more unfavorable than the second and third buckling mode of unstiffened girders. On the other hand, both the third and combination of modes of longitudinally stiffened girders give the same increase of 20% for the ultimate loads for ss/ h w = 0.50 (s s = 250 mm) while the maximum increase of 35% is obtained for the second buckling mode. Fig. 28a summarizes these conclusions. 4.3. Function-based imperfections Describing the initial geometrical imperfections with sine function in both directions (Eq. (1)) proves the aforementioned conclusions.

Again, there is a threshold before which the influence of the longitudinal stiffener has a negligible impact on the ultimate capacity. In this case, for patch load lengths ss/hw ≤ 0.05 (ss = 25 mm), an increase of 5% in the ultimate loads of longitudinally stiffened girders was noticed while the maximum increase was 45% for ss/hw ≤ 0.50 (ss = 250 mm), see Fig. 28b. On the other hand, it is interesting to note that the initial geometrical imperfections defined by cosine function (Eq. (2)) have the same threshold ss/hw = 0.30 (ss = 150 mm) as some experimentally measured initial geometrical imperfections of the second analyzed group of imperfections, cf. Fig. 26b. More precisely, this initial geometrical imperfection is in correspondence with one of girder A16. In both cases, the same strengthening effect (Fstiffened/Funstiffened) of 15% has been recorded, as graphically presented in Fig. 28b. A direct comparison of the obtained results with the available literature is not feasible since different initial geometrical imperfections

Fig. 13. Load-displacement response for unstiffened and stiffened girder under the patch load length of: (a) ss = 150 mm; (b) ss = 150 mm (distribution block).

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Table 3 Numerically obtained patch load resistance for longitudinally unstiffened plate girders using the experimentally obtained initial geometrical imperfections, buckling modes, sine and cosine functions [kN]. Shape of initial geometrical imperfection

A1 A2 A3 A4 A5 A6 A7 A11 A12 A13 A14 A15 A16 A17 1st buckling mode 2nd buckling mode 3rd buckling mode 1st + 2nd + 3rd mode Sine function Cosine function

ss/hw 0

0.05

0.1

0.15

0.2

0.25

0.3

0.4

0.5

142.68 142.82 138.56 146.35 148.78 147.98 150.20 144.68 145.25 146.29 143.04 140.44 144.45 142.25 150.24 149.01 149.91 159.31 153.35 144.62

146.20 146.13 141.11 149.72 152.14 151.26 153.75 148.05 148.63 149.82 146.25 143.70 147.43 145.62 146.58 146.03 146.76 155.39 157.05 147.74

164.84 165.14 153.76 168.66 170.06 168.80 173.03 166.99 167.69 169.54 164.46 161.28 163.95 165.08 163.14 158.65 160.64 171.52 172.60 163.63

179.82 185.47 169.04 189.82 189.67 188.01 193.79 187.97 188.79 191.29 185.00 174.98 182.47 182.13 182.19 171.09 174.07 188.59 185.72 181.12

193.07 200.17 184.79 209.02 212.71 210.10 218.06 204.18 204.54 208.57 205.45 187.94 203.88 195.38 203.05 183.37 188.19 210.22 201.24 202.51

206.87 214.03 200.78 223.95 234.51 236.13 235.40 218.70 219.60 224.66 219.94 200.93 228.99 208.98 223.99 195.32 201.97 236.69 216.32 227.12

220.42 227.86 218.58 239.14 250.64 260.58 253.55 233.23 234.73 240.63 233.65 213.76 253.59 222.29 237.50 206.99 215.48 262.29 230.80 253.78

248.12 255.51 258.27 268.11 280.64 291.62 287.54 261.76 264.58 271.73 260.53 239.91 283.58 248.79 263.25 230.98 243.50 292.35 259.49 285.54

280.56 285.03 300.94 298.28 311.12 322.04 322.37 292.12 296.66 304.31 289.25 268.02 314.33 276.94 291.11 257.71 360.12 337.10 289.96 316.08

Table 4 Normalized patch load resistance of longitudinally stiffened plate girders – Fstiffened/Funstiffened. Shape of initial geometrical imperfection

ss/hw 0

0.05

0.1

0.15

0.2

0.25

0.3

0.4

0.5

A1 A2 A3 A4 A5 A6 A7 A11 A12 A13 A14 A15 A16 A17 1st buckling mode 2nd buckling mode 3rd buckling mode 1st + 2nd + 3rd mode Sine function Cosine function

1.03 1.02 1.06 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.03 1.02 1.02 1.00 0.95 0.83 0.90 1.06 1.03

1.03 1.02 1.06 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.03 1.02 1.02 1.04 0.99 0.87 0.94 1.06 1.03

1.04 1.02 1.08 1.02 1.03 1.03 1.03 1.02 1.02 1.02 1.02 1.05 1.03 1.02 1.04 1.03 0.89 0.95 1.09 1.04

1.08 1.02 1.09 1.02 1.03 1.04 1.03 1.02 1.02 1.02 1.02 1.09 1.03 1.05 1.04 1.07 0.93 0.98 1.13 1.05

1.13 1.07 1.11 1.04 1.03 1.04 1.03 1.06 1.06 1.05 1.03 1.15 1.04 1.10 1.05 1.12 0.97 0.99 1.17 1.05

1.18 1.11 1.13 1.07 1.05 1.04 1.07 1.11 1.10 1.09 1.07 1.20 1.03 1.15 1.06 1.18 1.00 0.99 1.23 1.05

1.23 1.16 1.14 1.15 1.10 1.05 1.11 1.16 1.15 1.14 1.12 1.26 1.04 1.20 1.10 1.23 1.05 1.00 1.29 1.04

1.35 1.28 1.14 1.24 1.20 1.14 1.21 1.27 1.26 1.25 1.22 1.36 1.12 1.31 1.17 1.31 1.12 1.09 1.41 1.12

1.39 1.35 1.13 1.30 1.25 1.18 1.26 1.34 1.32 1.30 1.29 1.39 1.17 1.40 1.26 1.33 1.20 1.20 1.45 1.17

Fig. 14. Ultimate load for longitudinally stiffened and unstiffened girders using normalized initial geometrical imperfections from girder: (a) A1; (b) A2.

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Fig. 15. Ultimate load for longitudinally stiffened and unstiffened girders using normalized initial geometrical imperfections from girder: (a) A11; (b) A12.

Fig. 16. Ultimate load for longitudinally stiffened and unstiffened girders using normalized initial geometrical imperfections from girder: (a) A13; (b) A15.

Fig. 17. Ultimate load for longitudinally stiffened and unstiffened girders using normalized initial geometrical imperfections from girder: (a) A3; (b) A4.

under different patch load length are not considered. As shown, our analysis covers a wide spectrum of initial geometrical imperfections and patch load lengths. Moreover, for a complete comparison, all parameters should match, e.g. shape of initial geometrical imperfection, girder geometry, patch load length, etc., since they all affect the ultimate strength. However, our results can be partially juxtaposed with the results in [28] even though the girder geometry is different, but the relevant ratios can be used. For instance, taking a = hw = 3600 mm, tf = 35 mm, tw = 24 mm and b1 = 720 mm from [28], the following ratios

hw/tw = 150, tf/tw = 1.46, b1/hw = 0.2 and b1/tw = 30 (the closest ratios to our analysis hw/tw = 125, tf/tw = 2, b1/hw = 0.2 and b1/tw = 25, see Table 1) can be used for comparison. The authors in [28] considered a C-shape initial deformation (idealized with cosine function) which corresponds to the first buckling mode of longitudinally stiffened girders. They reported strengthening effects of 1.03, 1.04, 1.09 and 1.19 for patch load lengths ss/hw = 0.1, 0.2, 0.3 and 0.4, respectively. One can instantaneously see that these results and the results for the first buckling mode in Table 4 are in perfect agreement while a small deviation

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Fig. 18. Ultimate load for longitudinally stiffened and unstiffened girders using normalized initial geometrical imperfections from girder: (a) A5; (b) A6.

Fig. 19. Ultimate load for longitudinally stiffened and unstiffened girders using normalized initial geometrical imperfections from girder: (a) A7; (b) A14.

exists only for ss/hw = 0.4 for imperfections modeled as cosine function (see Eq. (2)). Furthermore, using the obtained results in [5,10] (a/hw = 1, b1 ≈ 0.2hw and ss/hw = 0.05) it can be seen that an S-shape initial deformation (similar to our second buckling mode, cf. Fig. 21) gives a lower ultimate load comparing to a C-shape initial deformation (described as cosine function). The authors in [5,10] concluded that the reduction in patch load resistance of longitudinally stiffened girders between the Sshape and C-shape initial geometrical imperfections is around 7%. Using our results from Table 4 for cosine function and the second buckling mode one can see that the ultimate strength is decreased by 5%. Thus, both analyses show that an S-shape imperfection is more

unfavorable than a C-shape deformation for patch load resistance of stiffened girders. A further juxtaposition with the other literature is not reasonable since the relevant parameters (a/hw, hw/tw, b1, b1/tw, etc.) are different. 4.4. Unfavorable geometrical imperfection for patch load resistance Additionally, in order to juxtapose the ultimate strength of longitudinally unstiffened and stiffened girders using buckling-mode based and function-based geometrical imperfections with the experimental ones, the minimum and maximum values for unstiffened and stiffened girders using the experimentally measured imperfections (cf. Fig. 27)

Fig. 20. Ultimate load for longitudinally stiffened and unstiffened girders using normalized initial geometrical imperfections from girder: (a) A16; (b) A17.

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Fig. 21. Buckling modes for longitudinally stiffened girders for different patch load lengths: (a) 1st mode; (b) 2nd mode; (c) 3rd mode.

are obtained. It turns out that the ultimate load for longitudinally unstiffened girders considering the experimentally measured imperfections are bounded by the ultimate load using buckling mode-based geometrical imperfections (the second buckling mode and the combination of modes), Fig. 29a. A reduction in the ultimate capacity between these two extreme limits of 13% and 40% is obtained for patch load length ss/ hw = 0 and ss/hw = 0.5, respectively. On the other hand, the ultimate load for longitudinally stiffened girders considering the experimentally measured imperfections are bounded by the ultimate capacity using buckling mode-based and function-based geometrical imperfections (the third buckling mode

and sine function), Fig. 29b. A reduction in the ultimate strength between these two extreme limits of 30% and 27% is achieved for patch load length ss/hw = 0 and ss/hw = 0.5, respectively. Based on this figure, we can see that the lower band (the third buckling mode) is valid for all patch load lengths ss/hw ≤ 0.4. The reason that the lowest ultimate capacities for longitudinally stiffened girders are obtained for the third buckling mode is the fact that this buckling mode corresponds to the deformed shape at the collapse load, cf. Fig. 30. As it can be seen, the pronounced deformation is dominant between the longitudinal stiffener and loaded flange and it leads to the lowest ultimate resistance. One can argue that these ultimate loads

Fig. 22. a) Schematic view of initial shape imperfections modeled using sine and cosine functions; b) midline profile along web panel length/height for sine and cosine function.

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Fig. 23. Ultimate load for longitudinally stiffened and unstiffened girders using initial geometrical imperfections from buckling mode: (a) 1st mode; (b) 2nd mode.

Fig. 24. Ultimate load for longitudinally stiffened and unstiffened girders using initial geometrical imperfections from buckling mode: (a) 3rd mode; (b) combination of the 1st, 2nd and 3rd mode.

are underestimated and that these initial geometrical imperfections should be scaled differently but our analysis shows that if these geometrical imperfections are allowed, they will lead to notably smaller patch load resistances. Conclusively it may be stated that initial geometrical imperfections can play a decisive role, especially for longitudinally stiffened girders. The initial geometrical imperfections of longitudinally stiffened girders that correspond to the deformed shape at collapse (collapse-affine imperfections), especially in the zone where the load is applied, will give the most unfavorable ultimate strengths. Therefore, it is less important how initial geometrical imperfections are defined (buckling-mode or function-based) but where the deformation is more pronounced. For

this particular case presented in this paper, it turns out that the third buckling mode of longitudinally stiffened girders corresponds to the deformed shape at the collapse and the lowest patch load resistance is obtained. 5. Conclusion Possibilities for numerical modeling of I-section steel plate girders subjected to patch loading were demonstrated in this work. The patch loading resistance was determined using the finite element analysis. The accepted numerical model is compared with the experimental results and it is shown that a perfect agreement between the numerically

Fig. 25. Ultimate load for longitudinally stiffened and unstiffened girders using initial geometrical imperfections as: (a) sine function; (b) cosine function.

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Fig. 26. Comparison of the ultimate strengths between longitudinally stiffened Fstiffened and unstiffened Funstiffened girders for the experimentally obtained initial geometrical imperfections: (a) experimentally unstiffened girders; (b) experimentally stiffened girders.

Fig. 27. Ultimate capacity for: (a) unstiffened girders using geometrical imperfections of the experimentally unstiffened girders; (b) stiffened girders using geometrical imperfections of the experimentally stiffened girders.

Fig. 28. Comparison of the ultimate strengths between longitudinally stiffened Fstiffened and unstiffened Funstiffened girders using as geometrical imperfections: (a) buckling modes; (b) sine and cosine function.

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Fig. 29. Ultimate capacity using different initial geometrical imperfections for: (a) unstiffened girders; (b) stiffened girders.

Fig. 30. The deformed shape at the collapse from the experiment (left [mm]) and the third buckling mode (right) for girder A5 (ss = 100 mm).

and experimentally obtained results was achieved. Based on that, the parametric analysis varying initial geometrical imperfections for different patch load lengths, focusing mainly on the experimentally measured imperfections, was performed. From the presented parametric study, several remarks can be pointed out. The main finding from this analysis shows that for smaller patch load lengths the influence of longitudinal stiffener has a negligible impact on the ultimate strength of longitudinally stiffened girders. Even more, it can be disregarded from the analysis inasmuch as it increases the patch load resistance of b5%. When a specific patch load length (threshold) is reached, an appreciable strengthening effect can be obtained. This can be easily defined with a bilinear curve (represented as a magnification factor) in which a separation point is the obtained threshold. Apparently, this threshold is a function of initial geometrical imperfections and different values are obtained for the experimentally measured imperfections, buckling modes, sine and cosine function basedimperfections. This proves our hypothesis, stated in the companion paper Part I [1], that the patch load resistance of longitudinally stiffened girders under smaller patch load lengths follows the ultimate strength of unstiffened girders. However, in order to propose an expression for this, a further set of studies is required since the present analysis is exclusively based on the same geometrical and material characteristics. Additionally, it is shown that initial geometric imperfections can play a decisive role. In this analysis, a focal point was on the experimentally measured initial geometrical imperfections since other researchers usually considered other types of initial geometrical imperfections. Also, measured initial geometrical imperfections are more realistic than other

assumed function-based imperfections. However, since these imperfections are not known in advance, it is highly required for the practical purposes to have more information about their influence. According to our parametric study, the lowest ultimate strengths for longitudinally stiffened girders are obtained using initial geometrical imperfections that correspond to deformed shape at the collapse (collapse-affine imperfections). As mentioned previously, we have restricted our attention to geometry used in the experiment and for more general conclusions, an additional parametric analysis is highly indispensable. Conclusively, it may be stated that further analyses are required in order to attain more information about the influence of initial geometrical imperfections considering different patch load lengths. Our succeeding work will consider the effects of girder geometry, material characteristics and aspect ratio of the web plate, both experimentally and numerically. References [1] N. Markovic, S. Kovacevic, Influence of patch load length on plate girders. Part I: experimental research, J. Constr. Steel Res. 157 (2019) 207–228. [2] Eurocode 3, EN 1993-1-5. Design of Steel Structures. Part 1–5: Plated Structural Elements, CEN, 2006. [3] Q. Hasan, W.W. Badaruzzaman, A.W. Al-Zand, A.A. Mutalib, The state of the art of steel and steel concrete composite straight plate girder bridges, Thin-Walled Struct. 119 (2017) 988–1020. [4] C.D. Tetougueni, E. Maiorana, P. Zampieri, C. Pellegrino, Plate girders behaviour under in-plane loading: a review, Eng. Fail. Anal. 95 (2019) 332–358. [5] C. Graciano, Longitudinally stiffened steel plate girder webs under patch loading, Department of Structural Engineering. Steel and Timber Structures, Chalmers University of Technology, Göteborg, Sweden, 2001.

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