EUROSTEEL 2017, September 13–15, 2017, Copenhagen, Denmark Flange buckling resistance of trapezoidal web girders Experimental and numerical study Bence Jáger*,a, Balázs Kövesdia, László Dunaia a Budapest University of Technology and Economics, Dept. of Structural Engineering, Hungary jager.bence@epito.bme.hu, kovesdi.balazs@epito.bme.hu, ldunai@epito.bme.hu ABSTRACT Girders with trapezoidally corrugated web are widely used structural elements in the civil engineering praxis due to their numerous advantages. However, there is a relatively small number of previous investigations focusing on the determination of the flange buckling resistance of the corrugated web girders. Previous experimental and numerical investigations confirmed that the flange buckling resistance model provided by the EN1993-1-5 [2] predicts often unsafe results. Therefore, the current paper collects all the previous proposals on the flange buckling resistance and introduces an experimental research program investigating the flange buckling behaviour of trapezoidally corrugated web girders. Based on the test results an advanced FE model is developed, validated and presented. An intensive numerical parametric study is performed on this model to investigate the buckling coefficient and the relationship between the relative slenderness and the reduction factor for corrugated web girders. Based on the experimental and numerical results the current paper introduces an enhanced design method for the determination of the flange buckling resistance. This design method consists of proposal to determine the critical load amplifier of the flange outstand and it also contains a new buckling curve which considers the specialties of the structural behaviour coming from the corrugation profile. Keywords: corrugated web, trapezoidal corrugation, bending moment resistance, flange buckling 1 INTRODUCTION Research on steel girders with corrugated web was started in 1956 by NACA [1] for wings of airplanes. After that the application of the corrugated web girder has been spread in the civil engineering praxis as well, especially in the field of bridges. Numerous researchers investigate the special structural behavior of this girder type, however, just a small number of research activities were focusing on the flange buckling behavior and on the bending moment resistance of girders with thin flanges. The current paper collects all the previous proposals for the determination of the flange buckling resistance of corrugated web girders. In the international literature there are some proposals available for flat web I-girders and for unstiffened plated elements which are also discussed and their applicability for corrugated web girders are studied. Several papers deal with the determination of the elastic buckling coefficient of flat web girders by considering the flange-toweb thickness ratio [7-8] what is usually neglected in the design codes. For trapezoidally corrugated web girders the existing proposals [2, 4–6] gives different upper limits for the theoretically [3] calculated buckling coefficient considering the buckling shape of the compressed flange subpanel bounded by the inclined folds of the web. For the relationship between the relative slenderness and reduction factor of outstand plated elements the EN1993-1-5 [2] recommends to use the same Winter-curve based formula for trapezoidally corrugated web girders as for flat web girders. Furthermore, additional formulas called reduced stress based effective equations were developed for unstiffened outstand elements by Bambach and Rasmussen [9] in 2004 considering the stress gradient in the flange. For trapezoidal web girders Koichi and Masahiro [6] proposed similar equation for the effective width in 2006. These aforementioned formulas are collected in Section 2 and evaluated based on the numerical parametric study. © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin · ce/papers 1 (2017), No. 2 & 3 https://doi.org/10.1002/cepa.465 wileyonlinelibrary.com/journal/cepa 4088 4089 | In the frame of the current research program 16 laboratory tests are performed at the Budapest University of Technology and Economics, Department of Structural Engineering in Hungary investigating the flange buckling behavior of trapezoidally corrugated web girders. To investigate the flange buckling resistance an advanced numerical model is developed and validated based on the current experimental investigations. By using the current test results and the advanced numerical model the magnitude of the equivalent geometric imperfection proposed by EN1993-1-5 [2] for flange twisting is studied. The EN1993-1-5 [2] prescribes a rotational imperfection magnitude equal to 1/50 for flange twisting. The applicability of this imperfection value is investigated and validated in the current research program. On the validated numerical model parametric study is executed to determine the buckling coefficient (kσ) of the large outstand part of the compressed flange. The relationship between the relative slenderness ratio and the reduction factor for outstand compression element is also investigated by geometrical and material nonlinear imperfect analysis (GMNIA). Based on the numerical results an enhanced design equation is proposed for the determination of the buckling coefficient (kσ) and the buckling curve (λ-ρ) for flange buckling of trapezoidally corrugated web girders considering the corrugation profile and the quality of the web-to-flange juncture. The layout of the tested girders, the applied dimensions and notations used in the current paper are shown in Fig. 1. Fig. 1. Used notations for girders with corrugated webs 2 LITERETURE REVIEW 2.1 Buckling coefficient (kσ) It is commonly accepted by researchers [2, 3, 5-6, 10], that for trapezoidally corrugated web girders the theoretically derived equation shown by Eq. (1) can be applied with adequate safety. However, this design equation does not consider the rotational support effect of the trapezoidal web and the non-uniform stress distribution in the flange, which may have significant effect on the flange buckling resistance [1]. c k 0.43 f a 2 (1) where cf is the larger flange outstand width, a=a1+2a4 is the estimated buckling wave length, where a1 and a4 are the length of the parallel web fold and the length of the longitudinal projection of the inclined web fold [2, 10], as shown in Fig. 1. Different researchers prescribe different limit values for this theoretically considered buckling coefficient. The DASt-Richtlinie 015 [4] and the EN19931-5 [2] prescribe a maximum value of 0.6. Sayed-Ahmed [5] proposed a limit of 0.7 while Koichi and Masahiro [6] proposed a value of 1.28. Johnson [7] proposed a new formula for the buckling coefficient based on test results of unstiffened flat web girders in 1985. The provided buckling coefficient takes the moderate rotational restraint of the web into account according to Eq. (2). © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017) | k 4 hw t w 86.5 t w 0.43 hw 0.5 (2) where hw and tw are the web depth and thickness. Similar formula has been derived by Park et al. [8] for flat web girders in 2016. The bending resistance of plated girders with longitudinal stiffeners was numerically studied focusing on the supporting effect of the flange and web plates. Based on the results of Park et al. a modified buckling coefficient was proposed in form of Eq. (3) where the web-to-flange thickness ratio is also considered. cf tf k 3.0 hw t w 0.6 cf t 0.43 25.5 w hw t f 0.6 (3) where tf is the compressed flange thickness, bf is the flange width and cf=bf/2. 2.2 Buckling curve (λ-ρ) In the main part of the previously developed design proposals for unstiffened plated elements, the relative slenderness may be calculated according to Eq. (4). p cf /tf f yf 28.4 k 235MPa (4) where fyf is the yield strength of the flange material. In the international literature different types of design curves can be found regarding plate buckling. In the current EN1993-1-5 [2] the Wintercurve based formula is implemented in the form of Eq. (5) where the relative slenderness limit (plateau length) is equal to 0.748 derived for buckling coefficient (kσ) equal to 0.43. c f ,eff p 0.188 (5) 1.0 cf p2 For unstiffened plated elements new equation was developed by Bambach and Rasmussen [9] in 2004 in form of Eq. (6) for stress gradients between 0≤ѱ≤1. The equation was developed by curve fitting based on test results. c f ,eff 0.2 3 (6) 1.0 cf p0.75 Based on the experimental and numerical investigations of Koichi and Masahiro [6] a design relationship was suggested in the form of Eq. (7) with no limitation in the theoretical buckling coefficient kσ for plates being simply supported at the three edges (between 0.43 and 1.28 [3]). c f ,eff cf 0 .7 p 0.64 1 .0 (7) The DASt-Richtlinie 015 [4] proposes effective width for the compressed flange of flat and corrugated web girders according to Eqs. (8a)-(8b). b f ,eff 25.8 t f 240 b f for flat web girders f yf (8a) b f ,eff 30.7 t f 240 b f for trapezoidally corrugated web girders f yf (8b) where bf,eff is the effective width of the compressed flange. In addition Johnson and Cafolla [11] verified that the average flange outstand (bf/2) can be used for the calculation of the relative slenderness ratio of corrugated web girders, if Eq. (9) satisfies. This formula characterizes how large is the enclosing effect of the web from the whole flange width. In case of flat web girders R is equal to zero. a a 4 a3 (9) R 1 0.14 a1 2a 4 bcf © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017) 4090 4091 | 3 NUMERICAL MODEL DEVELOPMENT AND VALIDATION 3.1 Applied analysis method Advanced numerical model is developed by the authors in ANSYS 15.0 [12] finite element program. The model is based on a full shell model using eight-node-thin (SHELL281) shell elements. Two analysis types are applied in the current study, geometrical nonlinear buckling analysis (GNBA) for the determination of the critical load amplifier and geometrical and material nonlinear imperfect analysis (GMNIA) for the determination of the ultimate resistances. The applied numerical model can handle the application of equivalent geometric imperfections using the first eigenmode shape. Figure 2 presents the developed geometrical model with the boundary and loading conditions. The numerical model is simply supported and the compressed flange is constrained against lateral displacement to eliminate lateral torsional buckling. The bending moments are applied in the flanges through force pairs at both ends. Convergence study shows that 4 elements along the fold lengths could be acceptable in average in the numerical model. A linear elastic-hardening plastic material model with von Mises yield criterion is used in the numerical model. The material model behaves linear elastic up to the yield stress (fy) by obeying Hook’s law with Young’s modulus equal to 210000 MPa. The yield plateau is modelled up to 1% strains with a small increase in the stresses. By exceeding the yield strength, the material model has an isotropic hardening behavior with a reduced modulus until it reaches the ultimate strength (fu). From this point the material is assumed to behave perfectly plastic. In the validation procedure of the FE model the measured yield and ultimate strengths are implemented for all the tested girders, however, in the numerical parametric study the nominal yield and ultimate strengths are applied relevant for S355 steel grade. Fig. 2. Geometric model with boundary and loading conditions 3.2 Model validation For the model validation a recently performed experimental research program is applied. 16 large scale simply supported specimens are tested under four-point-bending to investigate the local flange buckling phenomenon and to determine the bending resistance of the investigated girders. Ten different girder geometries having four different trapezoidal profiles (denoted by TP) are investigated as shown in Table 1. The geometrical properties of the tested girders are summarized in Tables 1-2; the applied notations are given in Fig. 1. Table 1. No. TP1 TP2 TP3 TP4 Geometry of the investigated corrugation profiles α [°] 45 45 30 30 a1 [mm] 97 145 88 134 a2 [mm] 97 145 88 134 a3 [mm] 69 103 44 67 a4 [mm] 69 103 76 116 © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017) | The nominal web depth of all the tested girders is 500 mm. The widths of the parallel and inclined folds of the web are varied between 88 – 145 mm using different corrugation angles equal to 30°and 45°. The applied steel material is S355. The measured geometrical and material properties of the test specimens are summarized in Table 2 (column #2 - #8). The first column of Table 2 refers to the specimen numbers, which contains the applied corrugation profile (e.g. TP1 in 1TP1-1 specimen) denoted according to Table 1. For the specimen types 1-6 always two test series are executed on the same geometry to be able to evaluate the reliability of the test results. The schematic drawing of the applied test arrangement is presented in Fig. 3. The damaged part of the specimens is localized to an internal removable panel subjected by pure bending moment with a length of 1050 mm. The span of the tested girders is 8 m, which has two external girder parts with a length of 3475 mm for each and an internal panel with a length of 1050 mm. The internal part of the girder is only changed between the tests to improve the productivity of the tests. The joints between the outer and inner parts are bolted connections, which are significantly over-designed to represent fixed and moment transmitting connection with full rigidity. The test specimens are simply supported at both ends; vertical stiffeners are placed at the support locations. The specimens are also supported laterally at the locations of the load introduction points to prevent the lateral torsional buckling failure mode. Fig. 3. Applied load and support conditions Figure 4 shows a typical flange buckling failure mode observed in the tests and a typical first eigenmode shape related to specimen 5TP2-1. The first eigenmode shape has a specific importance in the numerical model validation, since in the GMNI analysis the first eigenmode shape is applied as equivalent geometric imperfection with an imperfection rotation magnitude equal to 1/50 proposed by the EN1993-1-5 [2]. a) b) Fig. 4. a) Typical flange buckling failure (5TP2-1); b) Typical first eigenmode shape (5TP2-1) The measured load carrying capacities (2·Ftest), the calculated bending moment resistances (Mtest) and the relevant material properties are also summarized in columns #9 and #10 of the Table 2. The self-weight of the specimens is also considered in the moment resistance calculation process as uniformly distributed load having the intensity of 2.5 kN/m for the outer girders and 1.37-1.62 kN/m for the internal panels. The self-weight results in ~18 kNm bending moment calculated in the cross-sections of the bolted connection. The column #11 represents the computed FEM based © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017) 4092 4093 | bending moment resistances. The ratios of the FEM based and test based resistances are included in column #12. The comparison shows that the application of 1/50 as imperfection magnitude for flange buckling in the first eigenmode shape results in a conservative solution for girders having thicker flanges (12-14 mm) and/or thin web plates (3-4 mm). But all the results are on the safe side with a minimum and maximum differences equal to 2% and 15%, therefore, the application of an imperfection magnitude of 1/50 is proposed for girders with trapezoidally corrugated web calculating the flange buckling resistance under bending. Table 2. Number 1TP1-1 1TP1-2 2TP1-1 2TP1-2 3TP1-1 3TP1-2 4TP2-1 4TP2-2 5TP2-1 5TP2-2 6TP2-1 6TP2-2 7TP1 8TP2 9TP3 10TP4 4 tf [mm] 7.92 7.92 7.9 7.88 14.59 14.52 7.73 7.82 7.82 7.69 14.57 14.62 12.2 12.27 12.16 12.2 bf [mm] 250 249 250 250 250 250 249 250 250 248 250 250 250 246 247 250 Measured properties, capacities and FEM based resistances tw [mm] 2.88 2.93 5.97 5.97 3.01 2.84 2.99 2.93 5.97 5.95 2.99 2.96 3.84 4.05 4.04 3.89 fyf [MPa] 455 450 452 447 387 382 465 488 455 495 382 396 364 365 365 361 fyw [MPa] 410 364 406 383 363 418 376 366 390 392 373 364 474 450 457 457 fuf [MPa] 548 541 548 541 516 516 561 595 557 590 518 515 496 499 500 488 fuw [MPa] 555 511 530 507 514 566 510 511 508 516 503 510 584 584 584 560 2.Ftest [kN] 200.4 175.5 202.1 199.5 417.1 415.3 148.7 156.2 172.0 174.5 411.6 415.8 327.8 306.2 326.2 318.5 Mtest [kNm] 366.0 322.7 369.1 364.8 742.9 739.9 276.1 289.2 316.9 321.3 733.4 740.7 587.7 550.1 585.0 571.5 Mnum [kNm] 319.6 312.8 360.4 357.3 681.5 670.0 244.0 253.4 310.6 313.4 624.9 646.1 521.2 478.7 541.9 515.2 Mnum/ Mtest 0.87 0.97 0.98 0.98 0.92 0.91 0.88 0.88 0.98 0.98 0.85 0.87 0.89 0.87 0.93 0.90 NUMERICAL PARAMETRIC STUDY 4.1 Investigated parameter domain A numerical parametric study is performed using the validated numerical model. Two analysis types are applied; first bifurcation analysis (GNBA) in order to determine the critical load amplifier and a nonlinear analysis (GMNIA) to investigate the ultimate bending moment resistance of the trapezoidally corrugated web girders. The parameter domain is chosen in order to cover the average geometries used in the praxis and even the most extreme layouts as well. In the numerical analysis the following parameter domains are analysed: a1/a2: 0.33 – 0.67 – 1.0 – 1.5 – 2.0 – 3.0, α: 10⁰ – 20⁰ – 30⁰ – 40⁰ – 45⁰– 60⁰– 80⁰, R: 0.043 – 0.47, bf/a3: 1.0 – 15.6, cf/tf: 8 – 26, tf/tw: 1 – 6, hw/tw: 100 – 800, fy: 355 MPa. In the frame of the parametric study more than thousand different geometries are investigated. 4.2 Evaluation of EN1993-1-5 proposal Figure 5 shows all the results of the GNB and GMNI analysis in comparison with the standard resistances. In Fig. 5a the vertical axis represents the ratio of the standard based (Eq. (1)) and the FEM based buckling coefficients derived according to Eq. (10). Ratio equal to 1 means that the hand calculated and the numerically computed values are identical. All the results are plotted in the function of the flange width-to-thickness ratio (cf/tf/ԑ) regarding the larger flange outstand. The results show that for class 4 flanges the current design method approximates the buckling coefficient with a large scatter and there is a large amount of numerical results on the unsafe side. In Fig. 5b all the GMNIA results are summarized. The diagram also contains the design curve of the © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017) | EN1993-1-5 [2] given by Eq. (5). The required reduction factor related to the effective width of the larger flange outstand is determined according to Eq. (11), while the buckling coefficient and the relative slenderness are determined according to Eq. (1) and Eq. (4). 2 cr , num 12 1 2 t f (10) k , num c E 2 f req N u ,num c f ,eff , 2 t f f yf (11) c f t f f yf where Nu,num is the ultimate normal force in the compressed flange obtained by the GMNI analysis and cf,eff,2 is the effective width of the small flange outstand according to Eq. (1) and Eqs. (4)-(5). It can be seen that several points are on the unsafe side. There is a contradiction in the calculation of the relative slenderness limit of 0.748 suggested by the standard since it is based on the buckling coefficient equal to 0.43 applicable only for flat web girders. These results prove that the standard’s buckling curve needs revision and development for girders with corrugated web. a) b) Fig. 5. a) GNBA results; b) GMNIA results 4.3 Investigation of the buckling coefficient (kσ) The test results prove that the web-to-flange juncture including the enclosing effect of trapezoidal web panel has significant effect on the load carrying capacity. Previous investigations pointed out that the buckling coefficient could be smaller or larger than 0.43 depending on the web-to-flange juncture and on the corrugation profile. To consider these effects, the buckling coefficient formula developed for plates (Eq. (1)) should be further improved based on the results of the numerical simulations. The development is performed according to the concept that the first term of the buckling coefficient (0.43) should consider the stress gradient and all the boundary conditions of the fictitious plate subjected by compression. The second term should consider the width-to-length ratio of the sub-plate bounded by the inclined folds. It means that only the first term should be modified by considering the previous proposals and numerical and experimental investigations. This concept is convenient and compatible with the previous proposals developed for flat web girders. By considering the experimental and numerical results and the proposal of Park et al. [8] (Eq. (3)) the equation given by Eq. (12) is developed by the authors to determine the buckling coefficient. t k 0.43 2.5 w tf 0.6 R cf a1 2 a 4 2 1.3 . (12) The derived equation is verified by the numerical results. The comparison is plotted in Fig. 6 where the dark points represent the results regarding the current proposal. It can be seen that safe side solutions are provided with significantly smaller coefficient of variation than the standard’s proposal shown in Fig. 5a. © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017) 4094 4095 | Fig. 6. Verification of the current proposal for the buckling coefficient (Eq. (12)) 4.4 Investigation of the buckling curve (λ-ρ) The numerical results proved that the original plate buckling curve is not suitable for corrugated web girders, it needs improvement. As a first step the theoretical limit value of the relative slenderness is determined by substituting cf/tf/ԑ=14 into Eq. (4) using the larger flange outstand. This limit value is derived and applied for the slenderness limit value in the form of Eq. (13) which represents the plateau length of the buckling curve. This limit value is equal to 0.751 for flat web girders if kσ is equal to 0.43. In case of corrugated web girders, the corrugation profile results in changes in the buckling coefficient which should be also considered in the buckling curve plateau length. 0.493 (13) p , lim k If Eq. (13) is implemented into the proposed equation of the buckling curve of [6, 9], Eq. (14) is obtained. It can be observed that the reduction factor depends on the relative slenderness, on the buckling coefficient and on the index β, what should be calibrated based on the numerical results. c f ,eff cf p ,lim 0.493 k p p t 14 f cf 1.0 (14) Fig. 7. GMNIA results with the current proposal using β=1.0 Figure 7 presents the comparison of the FEM based reduction factors using Eq. (12) for the buckling coefficient determination and the surface represented by Eq. (14) using β=1.0. It can be seen that the FEM based results are in the close region of the design proposal, however, some points © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017) | are found to be too conservative. Therefore, a systematic parametric study is executed to determine the index β depending on the geometry of the corrugation profile. It is to be noted that the enclosing effect of trapezoidal web (R – Eq. (9)) cannot describe the supporting effect of the web alone since any α can belong to any R and vice versa. The parametric study reveals that the index β depends on three parameters such as the enclosing effect of the web (R), the corrugation angle of the inclined fold (α) and the flange-to-web thickness ratio (tf/tw) which clearly defines a specific profile for a given flange width. Figure 8a presents the effect of the flange-to-web thickness ratio for a geometry having fold lengths equal by a1=a2=200 mm in the function of increasing α (and R) represented by the upper and lower horizontal axes, respectively. It is shown that the larger is the tf/tw ratio, the larger is the index β and smaller is the resistance to flange buckling. In addition, it can be seen that the curves have a reverse parabola character which peak value is obtained by α=45⁰ representing the most unfavourable layout from flange buckling point of view. By increasing the corrugation angle from α=45⁰ the supporting effect of the web increases. By decreasing the corrugation angle from α=45⁰ the web contribution to the longitudinal load bearing increases. In the “accordion effect” based flange buckling resistance model the web is completely neglected from the bending moment resistance, however, through the index β it can be indirectly considered. In the case of α=0⁰ – representing flat web girder – the whole web panel practically takes part in the longitudinal load carrying and therefore the value of β can be decreased. a) b) Fig. 8. a) tendency and effect of tf/tw (a1=a2=200mm)on β; b) β – R relationship (tf/tw=4.8) Figure 8b presents the computed β values in the function of R having flange-to-web thickness ratio equal by 4.8. The black solid polygon represents the same polygon found in Fig. 8a. It can be seen that the index β linearly depends on the R if the corrugation angle is constant. However, the slope of the coloured lines must depend on the corrugation angle. Therefore, the slopes are calculated for each investigated corrugation angle and the relationship is determined by curve fitting. As a result, a closed formula with lower limit of 0.5 and upper limit of 1.0 is developed for the approximation of the index β in the form of Eqs. (15a)-(15b) considering the flange-to-web thickness ratio. a 1 5 R 4 5 R tg ( ) a3 0.45 0.06 where 0.5 1.0 tf (16a) (16b) tw The statistical evaluation of the current design and FEM based resistances are determined and evaluated regarding β equal to 1.0 (as a lower bound solution) and according to Eq. (15). It is shown that by using β=1.0 (Fig. 7) the average ratio is obtained to 7.3% on the safe side with coefficient of variation equal to 0.058. The maximum deviations on the safe and unsafe sides are obtained to 27.4% and 3.0%, respectively. By applying Eq. (15) for the determination of the index β the results fit better the numerical calculations with an average ratio equal to 0.97 on the safe side and with a coefficient of variation equal to 0.027. The maximum deviations on the safe and unsafe sides are obtained to 13.4% and 3.0%, respectively. © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017) 4096 4097 | 5 SUMMARY In the current research program experimental and numerical investigation is executed on the flange buckling resistance of girders with trapezoidally corrugated webs. Numerical model is developed and validated based on the executed test results. It is shown that by using the flange buckling eigenmode as equivalent geometric imperfection in the nonlinear analysis the proposal of the EN1993-1-5 [2] for the imperfection magnitude is applicable with adequate safety. By evaluating the numerical results, the following conclusions are drawn: (i) The results of the GNB analysis show that the proposal of the EN1993-1-5 [2] is not suitable for girders with trapezoidally corrugated webs since it does not consider the web-to-flange juncture. (ii) New proposal is developed for the buckling coefficient considering the effect of the web-to-flange juncture in form of Eq. (12) with the theoretical upper limit of 1.3. The proposal considers the previous proposals in the field of flat and corrugated web girders and the evaluation of the current numerical results as well. (iii) Nonlinear analysis revealed that the proposal of the EN1993-1-5 [2] for the reduction factor is not applicable for girders with trapezoidally corrugated webs. Therefore, a new formulation is developed in form of Eq. (14). According to the FEM based resistances the index β of the buckling curve is derived and calibrated in form of Eq. (15). ACKNOWLEDGMENT The presented research program is part of the “BridgeBeam” R&D project No. GINOP-2.1.1-152015-00659; the financial support is gratefully acknowledged. Through the first author the paper was also supported by the ÚNKP-16-3-I. New National Excellence Program of the Ministry of Human Capacities and by the second author’s János Bolyai Research Scholarship of the Hungarian Academy of Sciences; all the financial supports are gratefully acknowledged. REFERENCES [1] NACA Technical note 3801. “Experimental investigation of the strength of multiweb beams with corrugated webs”. Washington, 1956 [2] EN 1993-1-5:2005, Eurocode 3: Design of steel structures, Part 1-5: Plated structural elements [3] Timoshenko S.P., Gere J.M. “Theory of elastic stability”. 2nd Edition, McGraw-Hill Publishing Co., New York, 1961 [4] DASt-Richtlinie 015. “Trager mit schlanken Stegen”. Stahlbau-Verlagsgesellshaft, Köln, 1990 [5] Sayed-Ahmed E.Y. “Design aspects of steel I-girders with corrugated steel webs”. Electronic Journal of Structural Engineering 7, pp. 27-40, 2007 [6] Koichi W., Masahiro K. “In-plane bending capacity of steel girders with corrugated web plates”. Journal of Structural Engineering (JSCE) 62, No. 2, pp. 323-336, 2006 [7] Johnson DL. “An investigation into the interaction of flanges and webs in wide flange shapes”. Proc. SSRC Annual Technical Session, Structural Stability Research Council, pp. 395–405, 1985 [8] Park Y.M., Lee K.-J., Choi B.H., Cho K.I. “Modified slenderness limits for bending resistance of longitudinally stiffened plate girders”. Journal of Constructional Steel Research 122, pp. 354-366, 2016 [9] Bambach M.R., Rasmussen K.J. “Effective widths of unstiffened elements with stress gradient”. Journal of Structural Engineering 130, No. 10, pp. 1611-1619, 2004 [10] Johansson B., Maquoi R., Sedlacek G., Müller C., Beg D. “Commentary and worked examples to EN1993-1-5: Plated Structural Elements”, pp. 152-167, 2007 [11] Johnson R.P., Cafolla J. “Local flange buckling in plate girders with corrugated webs”. Proceedings of the Institution of Civil Engineers, Structures and Buildings 122, No. 2, pp. 148–156, 1997 [12] ANSYS® v15.0, Canonsburg, Pennsylvania, USA. [13] EN 1993-1-1:2005, Eurocode 3: Design of steel structures, Part 1-1: General rules and rules for buildings. © Ernst & Sohn Verlag für Architektur und technische Wissenschaften GmbH & Co. KG, Berlin ∙ CE/papers (2017)

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