Lateral-torsional buckling of girders with class 4 web: Investigation of coupled instability in EC3-based design approach

Abstract

This article focuses on the lateral-torsional buckling resistance of girders with slender, class 4 cross-sections with a research aim to check the accuracy of the design resistance model of EN1993-1-1 and EN1993-1-5 on the coupled instability of lateral-torsional buckling and local plate buckling resistances. The current Eurocode-based design method considers in the effective cross-sectional resistance calculation that yield strength is reached in the extreme fibre of the cross-section, and the reduction factor $(ρ_{p})$ related to local plate buckling is calculated based on this assumption. However, if lateral-torsional buckling occurs, maximum stress in the web can be significantly smaller at the ultimate limit state which is not considered in the effective cross-sectional resistance calculation. On the other side, EN1993-1-1 proposes to consider the effective bending moment resistance in the relative slenderness calculation of lateral-torsional buckling, which is in contradiction with the general definition of the relative slenderness ratio $({\bar{λ}}_{LT})$ , which should refer to the plastic resistance divided by the critical load of the structure. This article aims to check if the current Eurocode-based design rules need improvement and to check the effect of the above-mentioned specific issues on the calculated lateral-torsional buckling resistance. An extensive numerical research programme is executed to check and compare the lateral-torsional buckling resistance of class 3 (as reference) and class 4 cross-sections, and results are compared to Eurocode-based design models.

Keywords

class 4 coupled instability lateral-torsional buckling numerical analysis plate buckling

Introduction

Slender I-sections are commonly applied for long-span beams of industrial halls, where mainly flanges provide the bending resistance and web has a relatively small thickness providing only hinged support to the flanges. These cross-sections are sensitive for local web buckling; however, the dominant failure mode of the entire girder is lateral-torsional buckling (LTB). Therefore, these girders should be designed for coupled instabilities, for the interaction of local plate buckling of the web and global LTB of the entire girder. The current design rules of EN1993-1-1 (2005) are based on a consistent derivation of the Ayrton–Perry formulation (Taras, Greiner and Unterweger, 2013; Taras and Greiner, 2010) developed for compact and semi-compact cross-sections (class 1, 2 and 3 cross-sections), and these design rules are applied for class 4 sections as well (Couto et al., 2015). According to EN1993-1-1 design proposal, the relative slenderness ratio related to LTB should be modified for class 4 sections by the effective section factor (β) according to equations (1) and (2)

{\bar{λ}}_{LT} = \sqrt{\frac{W_{eff} \cdot f_{y}}{M_{cr}}} = \sqrt{\frac{β \cdot W_{el} \cdot f_{y}}{M_{cr}}}

(1)

where β is the effective section factor, given as

β = \frac{W_{eff}}{W_{el}}

(2)

where $W_{eff}$ is the effective cross-section modulus, $W_{el}$ is the elastic cross-section modulus, $f_{y}$ is the yield strength of the steel material, and $M_{cr}$ is the critical bending moment related to LTB.

This definition seems to be in contradiction with the general definition of the overall slenderness ratio defined by equation (3) according to EN1993-1-6 (2006), therefore its application should be investigated and its accuracy has to be proved

\bar{λ} = \sqrt{\frac{R_{pl}}{R_{cr}}}

(3)

where $R_{cr}$ is the lowest load amplifier (eigenvalue) on the defined design loads to reach the elastic critical load (linear buckling analysis) of the structure and $R_{pl}$ is the lowest load amplifier on the design loads to reach the plastic reference resistance of the complete structure.

On the other side, the calculation method of the local plate buckling resistance according to EN1993-1-5 (2006) is developed for slender cross-section under bending (without LTB). However, it is used for the coupled instability check without any modification, where applicability also needs revision.

Despite these girders are commonly used in design praxis, there is a relatively small number of previous investigations focusing on the LTB resistance of girders with class 4 cross-sections. Therefore, the current research aims to check the applicability of the Eurocode (EC)-based plate buckling resistance calculation method and the LTB resistance check for girders with slender cross-sections and to investigate the consequences on the above-mentioned specialties on the resistance check. Significant numerical research programme has been conducted by Couto et al. (2015) on the LTB resistance of slender girders. These numerical investigations highlighted that the reduction factor for LTB of girders with class 4 cross-sections can be significantly larger compared to girders with stockier webs. The difference between the predicted and the numerically computed resistances are clearly proportional with the effective section factor (β), and the conclusion based on their results was that the design curves of the EN1993-1-1 could be improved in order to take into account the interaction between local and lateral-torsional buckling failure modes. The previous results also showed that the ‘general case’ of the LTB resistance calculation method of the EN1993-1-1 is more safe-sided when compared to the numerical results, but it is too conservative for high slenderness ranges. However, the comparison with the numerical results for the ‘specific case’ given in EN1993-1-1 shows that it is adequate for high slenderness ratios but unsafe for smaller slenderness ratios.

Investigations were continued by Couto and Vila Real (2019), and the influence of residual stresses, plate imperfections and global imperfections has been consistently investigated in a numerical parametric study. Different residual stress patters are applied and the resistances are compared. The research results proved that the LTB resistance of girders with class 4 sections is sensitive for the type and magnitude of the applied residual stress pattern, which can have impact on the applicable buckling curve.

The current investigation aims to investigate the applicability of the slenderness calculation method for beams with class 4 web (while the flanges remain compact) and the comparison of the LTB design method between class 3 and class 4 sections (in the latter case, only the web belongs to class 4). Different models are worked out for the investigation to analyse the specific failure modes separately and their interactions as well. The numerical models take only equivalent geometric imperfections into consideration because of the characteristics of the analysed design method. Despite it is well known that the manufacturing processes (which appear in the form of residual stresses) have visible influence on the LTB behaviour of welded I-sections and there is large variety of related residual stress patterns, EN1993-1-1 design procedure disregards the differences in the welding processes and considers all welded cross-sections under the same buckling curve. For this reason, the results of the current numerical simulation are approximate in terms of the reduction factor, since models do not contain actual imperfections and residual stresses. However, they provide appropriate precision according to the research aims and are used to derive tendencies that are valid, regardless of the genre of the residual stresses.

The results highlight that the calculation method of EN1993-1-1 for coupled instability (interaction of plate buckling and LTB) is on the safe side for the analysed beam configurations. This statement is not proved and well confirmed in previous investigations yet. The theory which is applied to compute the LTB resistance of I-girders with slender web is correct; however, the magnitude of EN1993-1-1-based reduction factor $(χ_{LT})$ needs further improvement, which latter is not the subject of this article. Merely, the accuracy of the theory is investigated by experimentally validated numerical simulation series.

Current EN1993-1-1-based design provisions

However, since limited research has been undertaken on this subject, not only the safety level of existing design rules is unclear but also the influence of residual stresses, plate imperfections and global imperfections have not yet been consistently addressed for such cases. The article studies such influence by employing material and geometrical nonlinear, shell-element-based finite element (FE) analysis. Short overview is given below on the EC’s design process (both general and specific methods). The LTB resistance $M_{b, Rd}$ is calculated as the product of the cross-sectional resistance $(M_{Rd})$ and reduction factor $(χ_{LT})$ according to equation (4)

M_{b, Rd} = χ_{LT} \cdot M_{Rd} = χ_{LT} \cdot \frac{W_{y} \cdot f_{y}}{γ_{M 1}}

(4)

where $W_{y} = W_{eff}$ in case of class 4 section and $χ_{LT}$ is the reduction factor for LTB.

According to the general method’s description for compact cross-sections, the reduction factor can be calculated according to equations (5)–(7)

χ_{LT} = \frac{1}{Φ_{LT} + \sqrt{Φ_{LT}^{2} - {\bar{λ}}_{LT}^{2}}} \leq 1

(5)

Φ_{LT} = 0.5 \cdot [1 + α_{LT} \cdot ({\bar{λ}}_{LT} - 0.2) + {\bar{λ}}_{LT}^{2}]

(6)

{\bar{λ}}_{LT} = \sqrt{\frac{W_{y} \cdot f_{y}}{M_{cr}}} = \sqrt{\frac{M_{pl}}{M_{cr}}} (relative buckling slenderness)

(7)

where $α_{LT}$ is the imperfection factor (according to Table 6.1 of EN1993-1-1) and $M_{cr}$ is the elastic critical moment of beam in bending.

According to EN1993-1-1, to calculate the stability resistance of bended steel members, those buckling curves are applied which are derived for columns in compression but calibrated for LTB phenomenon. The reduction factor calculation method for rolled and equivalent welded cross-sections is given in the form of equations (8) and (9) (so called specific case)

χ_{LT} = \frac{1}{Φ_{LT} + \sqrt{Φ_{LT}^{2} - 0.75 \cdot {\bar{λ}}_{LT}^{2}}} \leq 1

(8)

Φ_{LT} = 0.5 \cdot [1 + α_{LT} \cdot ({\bar{λ}}_{LT} - 0.4) + 0.75 \cdot {\bar{λ}}_{LT}^{2}]

(9)

Research strategy and model development

Numerical models are built in ANSYS (2016) using four-node shell elements (SHELL181) with 6 degrees of freedom (DOFs) at each node (translations in the x, y and z directions, rotations about the x, y and z axes). This element type is suitable for analysing thin to moderately thick shell structures, and it is well-suited for linear, large rotation and/or large strain nonlinear applications. Three types of beam models are worked out to analyse different phenomena: (a) pure local plate buckling, (b) pure LTB, and (c) interaction of LTB and plate buckling. Only double-symmetric cross-sections are considered in the analysis with class 4 webs, while flanges belong to higher classes (3, 2 or 1).

The purpose of the three models is as follows. The pure buckling model studies short beams that have cross-sections with class 4 web and aims to demonstrate that the developed model is able to follow the plate buckling phenomenon properly. Furthermore, the load-carrying capacity of the models is compared to the calculation for plate buckling resistance proposed by EC and conclusions are drawn. The pure LTB model studies beams with cross-sections having class 3 webs. The constraints are chosen to ensure that only LTB can occur on the girder. The model has similar aims than the pure buckling model, but it serves as compare basis for the interaction models. The interaction models study beams having cross-sections with class 4 webs. The development of the interaction model is divided into two parts. The first interaction model is built with the exact same constraints than the pure LTB model. Comparing its results to the results of the pure LTB model, it aims to demonstrate that the behaviour of the model does not change regardless of the slenderness of the cross-section as long as plate buckling is disabled. In the second interaction model, plate buckling is enabled so that LTB and local plate buckling can occur in the same time. The difference between the two interaction models (with disabled vs enabled plate buckling) shows the direct effect of plate buckling on the structural behaviour.

Newton–Raphson method is used in the nonlinear analysis. The convergence of the solution is checked on the basis of the Euclidean norm of unbalanced force vector by applying a tolerance factor of 0.05%. Automatic time stepping is used which cuts a time step size into half whenever equilibrium iterations fail to converge. If the halved time step fails to converge, bisection will again cut the time step size and restart, continuing the process until convergence is achieved or until the minimum time step size is reached. Regular mesh is applied on the models with a maximum edge size of 25 mm. This size is proven to bring accurate results on the basis of initial mesh sensitivity analysis.

Steel material is characterized by multilinear material model associated with isotropic hardening rule, where Young’s modulus is 210,000 N/mm², yield strength is 355 N/mm² and the ultimate strength is 510 N/mm². The geometric imperfections vary by model types, hence detailed later in the model specifications in the related paragraphs.

Pure buckling model

The plate buckling phenomenon is analysed on a short beam model, where the length of the beam is double the height of the cross-section: $L = 2 \cdot h$ , where $h = h_{w} + (t_{uf} + t_{lf}) / 2$ . A total of 50 different cross-sections are analysed. The web height $(h_{w})$ varied between 450 and 1300 mm and its thickness $(t_{w})$ between 4 and 10 mm, making sure that the actual plate-to-thickness ratio belongs to class 4. The flange width $(b_{f})$ varied between 150 and 350 mm, its thickness $t_{f}$ between 8 and 30 mm, and flanges always belong to higher classes (1, 2, and 3). The range of reduction factors $(ρ_{w})$ of the class 4 webs is between 0.54 and 0.8, and the relative plate slenderness of the web $({\bar{λ}}_{p, w})$ is set between 0.87 and 1.72.

Applied boundary conditions are shown in Figure 1(a). End cross-sections are supported at the centre of gravity against translations (x, y and z at one end, x and y at the other end) and constraints are applied to avoid lateral displacement of the flanges at the ends. Furthermore, to simulate an infinitely rigid diaphragm at the beam’s ends, the DOFs of the nodes at both end cross-sections are connected to its own centre of gravity by constraint equations. To allow only the development of plate buckling, the uppermost and lowermost edges of the web are constrained against lateral displacement.

Figure 1.

(a) Boundary conditions and (b) local plate imperfection.

In the nonlinear analysis, local plate imperfection is applied with an amplitude of h/200 in sinusoidal shape according to Figure 1(b). Since residual stresses are not considered in the model, the maximum value for equivalent geometric imperfection is applied based on the proposal of EN1993-1-5. The beam is subjected to bending by end-moments, applied as force pairs ( $F = M / h$ , where h is the distance between the flanges). Relative slenderness $λ_{rel, p}$ is determined from linear analysis and the bending resistance $(M_{C, ANSYS})$ is taken from GMNI analysis (geometrically and materially nonlinear analysis with imperfections).

Pure LTB model

Pure LTB phenomenon is analysed on a long beam model. A total of 13 different cross-sections are chosen from the previously investigated 50. Modification is made, however, on the web thickness to get them into class 3. Beam length is 4, 6, 8 and 12 times the cross-sectional depth. A total of 52 simulations allowed to cover a relative slenderness range from 0.4 to 3.0. The layout of the ends is modified so that the fictional diaphragm is replaced by an actual endplate with $t_{s} = 30 mm$ , which is proven to effectively represent the required stiffness.

Applied boundary conditions are shown in Figure 2. The cross-section is supported at the centre of gravity at both ends against translation (x, y and z at one end, x and y at the other end) and constraints are applied to avoid lateral displacement of the web at the ends (Figure 2(a)). Additional constraints are applied at all internal cross-sections: they are connected by constraint equations to their own centre of gravity and the rotation around the longitudinal axis is coupled in order to exclude plate buckling and perform pure LTB (Figure 2(b)).

Figure 2.

Geometry and boundary conditions at (a) ends and (b) all internal cross-sections.

According to the model purpose, only bow imperfection is applied with L/400 magnitude according to Figure 10(a). Consistent with the EC-based design procedure, the genre of residual stress is neglected but its effect is included by applying larger bow imperfection in accordance with EN1993-1-1 proposal. To determine the value of the equivalent geometric imperfections, primary model is built in accordance with an experimentally validated model by Dunai et al. (2015) including residual stresses (distribution by Taras and Greiner, 2010) and bow imperfection with L/1000 magnitude. Parametric study is executed on simply supported beams with class 3 webs and class 1–2 flanges, covering relative slenderness range of 0.4 to 3.0. Results of GMNI analyses are compared to buckling curves c and d of specific case (EN1993-1-1). Conclusion is derived that the equivalent geometric imperfection related to curve c can be adopted to create a model which behaves in an equivalent manner with the experimentally validated model. EN1993-1-1 recommends $e_{0, d} = L / 200$ as initial equivalent geometric imperfection and this value is reduced by $k = 0.5$ (according to section 5.3.4 member imperfections of EN1993-1-1) resulting in L/400 as bow imperfection for the nonlinear analyses. To verify the model’s behaviour, the same exact beam set is analysed by the model with equivalent imperfections only, than with the primary model. The results show good agreement with the primary model’s results, thus L/400 is proved to be appropriate to replace the combination of L/1000 bow imperfection and residual stresses. Loads are applied on the same way than on the pure buckling model. Relative slenderness $λ_{rel, LT}$ is determined from linear analysis, and the bending resistance $(M_{b, ANSYS})$ is taken from GMNI analysis for all beams.

Interaction model

Modelling at this level is divided into two parts to support a step-by-step approach from pure LTB to the coupled (LTB plate buckling) phenomenon. The cross-sections and the beam lengths are not changed compared to the previous model. First, an analogous model is built with the pure LTB model with one modification: the web sizes are reduced so that they become class 4. The internal constraints on the internal cross-sections are kept constant, to be able to derive the change in the model’s response by changing the non-compact web to a slender one. The cross-section on this model belongs to class 4, but the plate buckling is disabled in the model yet. The rotational capability about the longitudinal axis (ROTZ) of the nodes of each cross-section is connected to their centre of gravity. In this way, section distortion and local plate buckling cannot occur while this internal constraint does not bother the girder’s response for LTB. In the next step, the internal cross-section constraints are removed and plate buckling is enabled. The comparison of the two model results shows the effect of plate buckling on the structural behaviour and load-carrying capacity. Both models have bow and plate imperfections but do not include residual stresses. For this reason, the equivalent geometric imperfection is applied with magnitude of h/200 and L/400, respectively (EN1993-1-1, 2005). Relative slenderness $λ_{rel, LT}$ is determined from linear analysis and the bending resistance ( $M_{b, el, ANSYS}$ and $M_{b, eff, ANSYS}$ ) is taken from GMNI analysis.

Model development and validation

The aim of the model development is to create a model that is able to follow the coupled failure mode coming from LTB and local plate buckling. First, the accuracy of the models with one specific failure mode is validated. Then, the two models are coupled to form the final interaction models.

Validation of the pure LTB model

Numerical model validation is performed by experiments of Kubo and Fukumoto (1988), who examined thin-walled welded I-sections. Specimens were tested by subjecting simply supported beams to a midspan concentrated force. The test setup and its results are summarized in Figure 3(a) and (c). Beam B2B-C1 is chosen for model verification, since it failed for LTB only, and no local buckling was observed before the ultimate moment was reached. Numerical model is built considering the experimental geometry with L/400 initial bow imperfection. The material nonlinearity is taken into account in accordance with the provided coupon test results. Numerical simulation is performed under both load and displacement control. Load–displacement curves are plotted in a diagram where the ordinate shows non-dimensional bending moment at midspan, $M / M_{y}$ , where $M = P \cdot L / 4$ (P is the applied load, L is the span of the beam and $M_{y}$ is the yield moment of the full section).

Figure 3.

Validation of pure LTB model: (a) experimental test setup, (b) ultimate deformation of the FE model, (c) FEM vs experiment: load–displacement and (d) experimental results.

Both models showed LTB failure; the ultimate deformation is shown in Figure 3(b). Comparing the ultimate moments, the experimental value is 37.17 kNm, while the load- and displacement-controlled models give 35.85 and 35.68 kNm, respectively. The deviation of the ultimate moments compared to the experimental value is 3.6% in case of load-controlled model and 4% in case of displacement-controlled model. The initial stiffness of the models is identical and gives good agreement with the experimental curve, and the character of the curves also agrees well (Figure 3(d)). The load-controlled model terminates at the peak of the load–displacement path, while the displacement-controlled model shows also the descending phase. By comparing the numerical load–displacement curves, it can also be concluded that the load-controlled model gives accurate result and trustful prediction on both initial stiffness and on the load-carrying capacity. In accordance with the research aims, the load-controlled model is proved to be appropriate and further used for the investigations.

Validation of the pure buckling model

The pure buckling model is validated by an experimentally calibrated model developed by Dunai et al. (2015). The basis experiments aimed to investigate local plate buckling; its layout is shown in Figure 4(a). The beam is divided into three parts. The beam parts are connected with full strength bolted connections, so that the inner part is exchangeable and a new specimen can be fixed instead of it for the next test round. The two longer parts on the side remain in elastic stage during the tests. Failure is concentrated exclusively at the inner 1-m-wide part.

Figure 4.

(a) Experimental layout; (b) experimental and numerical load–displacement diagram (Dunai et al., 2015).

Figure 4(b) shows the load–displacement curves of the experimental beam and of its calibrated global numerical model. The correspondence of two parameters are analysed to prove the appropriateness of the global model: initial stiffness and load bearing capacity. The initial stiffness of the experimental curve is 7.41 and 7.58 kN/mm for the model’s curve. The load-bearing capacity was 154.87 and 156.96 kN in the experiment and on the model, respectively. The descending part of the global model’s load–displacement curve cannot be followed because the simulation is load controlled (this part of the curve is, however, not needed to judge the model’s applicability). Based on the comparison of the parameters, the global model is proved to be appropriate to use it as basis of further model calibration.

This experiment and its calibrated numerical model serve as good basis to develop and verify the numerical model for pure plate buckling. The inner beam part is analogous with the pure buckling model in its layout and loading. The aim is to create a new local model on the basis of the global model which can follow the same behaviour than the inner part of the global model.

The model is built in ANSYS and from the same four-node-shell elements than the other models in the current research. The mesh size is set to $e = 50 mm$ . The experimentally calibrated model is shown in Figure 5(a) and the local model that meant to behave in an analogous way than the inner part of the before-mentioned global model is shown in Figure 5(b). The cross-section of both models is the same: the flange is 150 m wide and 10 mm thick, the web is 500 mm high and 4 mm thick. The length of the local model and the inner part of the global model is 1.0 m. The material model is adopted from coupon test measurements for both web and flange plates. Geometric imperfections are applied on both models in the nonlinear calculation.

Figure 5.

(a) Experimentally calibrated global model and (b) local model for verification.

Plate imperfections are determined on the global model on the basis of relevant eigenforms, where maximal lateral displacement is scaled up to agree with the value of the equivalent geometric imperfection (h/200). The initial geometry of the model is overwritten with this deformed shape, and nonlinear calculation is performed on the geometrically imperfect model. In case of the local model, a decision was to be made. The eigenform-type imperfection is practical if the failure mode is obvious and the number of numerical calculations is not large. This specific case would allow us to apply the plate imperfections in the same manner. The next model level, however, where coupled instability has to be supported and large number of simulations are to be made, cannot accommodate the eigenform-type plate imperfection. Since we cannot predict which one of the eigenforms would show coupled instability, automated simulation series cannot be made but manual selection of the relevant eigenform should be performed for plate imperfection. For the above-mentioned reason, the calibrated local model is built with initially imperfect geometry for the nonlinear calculation. The web plate contains sinusoidal plate imperfection with magnitude of h/200.

Global imperfection is not applied since LTB cannot occur. In course of the experimental investigations, proper lateral constraints ensured that LTB is excluded and global model is built accordingly. The model’s behaviour is compared on the basis of deformation and load-carrying capacity (more details on the model verification can be found in the MSc Thesis of Fejes, 2018). The load is applied on both models as concentrated loads, as four-point bending on the global model, and as force pairs at the ends on the local model.

The whole load–displacement paths of the local and global models are not comparable since the deformation of the middle section on the global model (which is the subject of the validation) includes not only the deformations from plate buckling, like the local model does, but also the global deformations from four-point-bending. Only the relative displacement of the midspan is compared against each other. As it can be seen in Figure 6, the deformations of the global and local models are consistent with the loading. The relative deflection of the middle part of the global model is 1.478 mm (the global deflection of the beam is deducted), and the deflection of the mid cross-section on the local model is 1.498 mm.

Figure 6.

Ultimate deformation of the (a) global and (b) local model.

A total of 10 eigenforms are considered in the stability analysis. As it can be seen in Figure 7, the relevant buckling shapes show good agreement on the models. During the nonlinear analysis, $F = 180 kN$ load is applied on the global model and $180 kN \cdot 3.5 m$ end-moments on the local model. The load-carrying capacity comes back to 155.94 kN on the global model and 166.815 kN on the local model, and both show good agreement with the experimental load-carrying capacity which is 154.87 kN.

Figure 7.

Buckling shape of the (a) global and (b) local model.

From the comparison between the experimentally verified global model and the local model, it can be stated that the differences in the ultimate loads, deformations and stresses are negligible. Based on the analyses, the local model is further adapted to analyse plate buckling and is proven to provide accurate results.

Evaluation of numerical results

Results of the numerical models are presented by comparing them to relevant EC-based design method. Most commonly the relative slenderness of the cross-section/girder and the load-carrying capacities (bending resistance, buckling resistance) are investigated. In case of long beam models (LTB, interaction models), the reduction factor for LTB is analysed and compared against the EC’s buckling curves. The buckling curves are used here as references to derive tendencies and serve as guidelines for evaluation.

The accuracy of the load-controlled models’ results is further tested by building a displacement-controlled reference model for each beam configuration. The ultimate moments of the model pairs are compared. All results in case of every model type show good agreement in terms of limit load, and the deviation of the load- and displacement-controlled models stays within maximum ±1.5%.

Results related to local web plate buckling

The local buckling resistances are calculated by GMNI simulations. The moment resistance of the models is derived from the last converged loadstep in the simulation. The accuracy of the results is extensively tested: (a) it is validated by two independent experiments and (b) by comparing its results with consistent displacement-controlled models. The minimum loadstep is equal to 5 × 10⁻⁴ times the applied load M, which is defined as (1.2 ÷ 1.6) times the theoretical buckling resistance of the girder, starting with 1.2 and gradually increased if the buckling resistance from the model reached the magnitude of the applied load. Furthermore, the ultimate deformations are checked individually. The relative slenderness of the girder is approximated by the relative slenderness of the web, and its value ${\bar{λ}}_{w, EC}$ is calculated according to equation (10). The relative slenderness is also derived from linear calculations made on the model. In the stability analysis, the first 10 eigenforms are computed and ${\bar{λ}}_{p, ANSYS}$ is calculated for the comparison according to equation (11) using the first eigenmode related to local web plate buckling

{\bar{λ}}_{w, EC} = \frac{c_{w} / t_{w}}{28.4 \cdot ε \cdot \sqrt{k_{σ}}}

(10)

where $c_{w}$ and $t_{w}$ are the appropriate width and thickness of the web, respectively

ε = \sqrt{\frac{235}{f_{y}}}

$k_{σ}$ is the buckling factor according to EN1993-1-5

{\bar{λ}}_{w, ANSYS} = \sqrt{\frac{α_{ult}}{α_{cr}}} = \sqrt{\frac{σ_{max} / f_{y}}{α_{cr}}}

(11)

where $σ_{max}$ is the maximum extreme fibre stress from linear calculation, which issues from actual loading at midspan, $α_{cr}$ is the factor by which the design loads would have to be increased to cause elastic instability in a global mode.

The ultimate deformation is similar in case of all analysed cross-sections as shown in Figure 8 and reflects the shape of the applied plate imperfection. Plate buckling occurs at the upper part of the web, and the upper flange shows buckling waves as well.

Figure 8.

Deformation at ultimate stage in the GMNI analysis.

The comparison between the relative slenderness and the load-carrying capacities is show in Figure 9. The hand-calculated values are plotted on the horizontal axis of the diagrams, and the numerical results are on the vertical axis. To make the comparison easier, $f : y = x$ line is added to the diagram, which would show the identical solution.

Figure 9.

(a) Relative slenderness and (b) moment resistance.

It can be seen that the model always gives lower relative slenderness ratios (note that the hand calculation is an approximated value), which come from the rotational supporting effect of the web, which is neglected in the standardized calculation method of the EN1993-1-5. Similar trends are presented by Kövesdi (2019) for global buckling of stiffened plates. The ultimate moment taken from the GMNI simulation is slightly higher than the calculated moment resistance based on EN1993-1-5. The maximum difference is 11%, minimum is 0.06% and average is 5.3% based on 50 calculations. Based on the simulation results, it can be concluded that the developed model gives a reliable response and is able to follow the plate buckling phenomenon. Furthermore, it shows good agreement with the standardized calculation method.

Results related to LTB

A total of 52 models are built using 13 class 3 cross-sections, each combined with 4 different lengths. For the comparison, $M_{b, Rd, EC}$ is the design buckling resistance and is calculated according to equations (12) and (13). Elastic section modulus is used to compute the standardized buckling resistance since the analysis involves class 3 cross-sections. The buckling resistance of the models $M_{b, el, ANSYS}$ is derived from the last converged loadstep in the simulation. The relative slenderness of the girder ${\bar{λ}}_{LT, EC}$ is calculated according to equation (12) for the comparison with the relative slenderness ${\bar{λ}}_{LT, ANSYS}$ , which is derived from the results of the model in the same way than in case of pure plate buckling

{\bar{λ}}_{LT, EC} = \sqrt{\frac{α_{ult}}{α_{cr}}} = \sqrt{\frac{M_{el, Rd}}{M_{cr}}}

(12)

where $M_{el, Rd}$ is the elastic bending moment resistance of the cross-section and $M_{cr}$ is the elastic critical moment for lateral-torsional buckling

M_{b, Rd, EC} = χ_{LT} \cdot \frac{M_{Rk, el}}{γ_{M 1}} = χ_{LT} \cdot \frac{W_{el} \cdot f_{y}}{γ_{M 1}}

(13)

The ultimate deformation of the model agrees with the expectations and shows LTB (Figure 10(b)). In the comparison between the relative slenderness results, the same tendency is found than in case of pure buckling (Figure 11(a)). The presentation of the results is the same than before, the hand-calculated values are plotted on the horizontal axis of the diagrams and the numerical results are plotted on the vertical axis. The simulations brought slightly lower relative slenderness values where the difference is less important than in the previous model’s case but the hand calculation in this time is more accurate (the hand-calculated value is always higher by an average of approximately 5.2%).

Figure 10.

(a) Global imperfection and (b) ultimate deformation on the pure LTB model.

Figure 11.

(a) Relative slenderness and (b) buckling resistance on the pure LTB model.

For the evaluation of the buckling resistance, there are two possibilities. The results of the numerical model can be compared against buckling resistance values which are calculated according to ‘general case’ or ‘specific case’ of EN1993-1-1. The general case adopts the exact same buckling curves than are developed for column flexural buckling. However, the buckling curves of specific case are directly calibrated to LTB. Choice has to be made about which one to consider in the further comparisons. As a first attempt, both methods are considered addressing curve d, as required for class 4 sections. Figure 11(b) shows that the results of the numerical model show better agreement with the buckling resistances by specific case (GC stands for general case and SC for specific case on the diagram). The $χ_{LT, ANSYS}$ values are determined from equation (14)

χ_{LT, ANSYS} = \frac{M_{b, el, ANSYS}}{M_{Rk, el}}

(14)

where $M_{b, el, ANSYS}$ is the buckling resistance based on numerical simulation and $M_{Rk, el}$ is the elastic characteristic moment resistance.

It is observed that $χ_{LT, ANSYS}$ values tend to pack close and in vast majority above curve d of specific case. Based on the observations, the specific case is chosen further as reference for the evaluation. It is found that the numerical results show an average difference of about 15.2% (the maximum is approximately 38.8% and the minimum is approximately −3.3%) compared to hand calculation.

Interaction model with disabled plate buckling

The interaction model with disabled plate buckling is different from the pure LTB model in two details: (a) webs are chosen to belong to class 4 and (b) plate imperfection is applied with h/200 value besides the bow imperfection with a magnitude of L/400. The values ${\bar{λ}}_{LT, EC}$ , ${\bar{λ}}_{LT, ANSYS}$ , $M_{b, e l^{*}, ANSYS}$ and $χ_{LT, ANSYS}$ are calculated in the same manner that in case of the pure buckling model. However, $M_{b, e l^{*}, ANSYS}$ is marked by superscript * to remind that the cross-section belongs to class 4 based on its $h_{w} / t_{w}$ ratio, though treated as class 3 in the calculation, since plate buckling is prevented by the internal constraints.

According to the expectations, structural behaviour does not show difference compared to the LTB model. Figure 12 shows the ultimate deformation of two correspondent models. The frame of the cross-sections and the length of the girder are exactly the same, but one is built with class 3 and the other with class 4 web. It can be observed that the two simulations result in similar LTB failure, and there are no significant differences.

Figure 12.

Ultimate deformation on (a) pure LTB model and on (b) interaction model with disabled pate buckling.

Then, the relative slenderness and the buckling resistance values are evaluated in the same manner than the results of the previous model and the comparisons are shown in Figure 13. The hand-calculated values are plotted on the horizontal axis of the diagrams, and the numerical results are plotted on the vertical axis. The relative slenderness ${\bar{λ}}_{LT, ANSYS}$ from the results of the model is always slightly below the calculated value ${\bar{λ}}_{LT, EC}$ . The relative slenderness values are closer to each other in the 0.4–1.5 range and the difference is more significant over 1.5. The buckling resistance $M_{b, e l^{*}, ANSYS}$ is always higher than the standardized value $M_{b, Rd, EC}$ . The minimum difference is 5.1%, the maximum difference is 31.5 and the average of the differences based on 42 simulations is 17.2%.

Figure 13.

(a) Relative slenderness and (b) buckling resistance of interaction model.

It can be seen in Figure 14, where the numerical results are presented together with the buckling curves c and d of specific case, that $χ_{LT, ANSYS}$ values follow the buckling curves. However, markers shift towards the lower relative slenderness values and in the same time show higher reduction factor values. However, it can be observed that the results of the LTB model and the interaction model with disabled local plate buckling lead to similar trends and could be characterized by the same buckling curve.

Figure 14.

Buckling curves d and c of specific case versus model results.

Interaction model with enabled plate buckling

In the next step of the modelling process, the internal constraints that prevent the model from plate buckling are removed. In the same time, class 4 properties are included in the calculation of ${\bar{λ}}_{LT, EC}$ , $M_{b, Rd, EC}$ and $χ_{LT, ANSYS}$ according to equations (15)–(17), respectively. The buckling resistance derived from the simulation is denoted by $M_{b, eff, ANSYS}$ . The calculation of ${\bar{λ}}_{LT, ANSYS}$ does not change

{\bar{λ}}_{LT, EC} = \sqrt{\frac{α_{ult}}{α_{cr}}} = \sqrt{\frac{M_{eff, Rd}}{M_{cr}}}

(15)

where $M_{eff, Rd}$ is the effective bending moment resistance of the cross-section and $M_{cr}$ is the elastic critical moment for lateral-torsional buckling

M_{b, Rd, EC} = χ_{LT} \cdot \frac{M_{eff, Rk}}{γ_{M 1}} = χ_{LT} \cdot \frac{W_{eff} \cdot f_{y}}{γ_{M 1}}

(16)

χ_{LT, ANSYS} = \frac{M_{b, el, ANSYS}}{M_{Rk, eff}} \cdot γ_{M 1}

(17)

where $M_{b, el, ANSYS}$ is the buckling resistance based on numerical simulation and $M_{Rk, eff}$ is the effective characteristic moment resistance.

Typical ultimate deformations are presented in Figure 15 for all beam lengths via the example of a cross-section having web (550-5) and flanges (180-10).

Figure 15.

Typical ultimate deformation for (a) $L = 4 \cdot h$ , (b) $L = 6 \cdot h$ , (c) $L = 8 \cdot h$ and (d) $L = 12 \cdot h$ lengths.

It can be seen that for shorter beam lengths, the plate buckling has a larger influence on the deformations, and for longer lengths, the effect of plate buckling is de-emphasized and it bears resemblance more with pure LTB type of deformation. Evaluation and comparison of the relative slenderness and the buckling resistance results are presented in Figure 16. Hand-calculated values are plotted on the horizontal axis, and numerical results are plotted on the vertical axis. This model shows less reliability in forecasting the relative slenderness; however, scattered results are obtained only in the range of 0.4–1.0, where the effect of local plate buckling is more important. The rest of the results still show good agreement with hand calculation (the average difference is 4.7% based on 42 simulations), though this is the first model where the numerical results can be higher than the EC-based calculation. The buckling resistance, however, still shows the same tendency than the previous models, and the results of the models show good agreement with the hand calculations with an average difference of 18.7% (the numerical models show higher buckling resistance).

Figure 16.

(a) Relative slenderness and (b) buckling resistance of interaction model.

Results of the simulations are plotted together with the buckling curves c and d of specific case in Figure 17. It is observed that data points of those models that can develop plate buckling shift towards lower relative slenderness values but the points remain on the same curve. Clear tendency, however, cannot be drawn at this point because the results belong to 12 different cross-sections and the results of one single cross-section does not cover the whole analysed relative slenderness range. The clear explanation for the obtained trend is given in the following section.

Figure 17.

Buckling curves d and c of specific case versus model results.

Evaluation of results

The interaction models are analysed further to clarify the relationship and the differences between them and to determine the effect of the plate buckling on LTB in a qualitative manner. Four cross-sections are randomly chosen from the analysed cross-sections that are considered so far for the analysis according to Table 1. All cross-sections are paired up with 16 different lengths to cover the relative slenderness range of 0.4–1.9 (which is in the practical application interval) with an increment of 0.1. Both interaction models are built with same geometric features which resulted in a total of 128 simulations. To ease to distinguish between the models, the interaction model with disabled plate buckling is further referred to as ‘model a’ and the interaction model where plate buckling is enabled as ‘model b’.

Table 1.

Characteristics of analysed cross-sections.

No.	$h_{w}$ (mm)	$t_{w}$ (mm)	$Clas s_{w}$	$b_{f}$ (mm)	$t_{f}$ (mm)	$Clas s_{f}$
1	1300	9	4	300	16	3
2	600	5	4	250	12	3
3	950	7	4	180	16	1
4	1300	10	4	350	20	3

Cross-section No. 3 from Table 1 is chosen to present the results of the calculation and the results of the simulations are plotted against the SC buckling curves in Figure 18. Note that all model pairs brought the same tendencies, so the following statements are valid for the entire cross-section set. As it is detailed in Table 1, the flanges of the cross-sections belong to class 3 and 1 (semi-compact and plastic in other terms). Though class 2 (compact) is not considered in this analysis cycle as flange, the results are stated to be valid for every non slender flange–slender web pairs. For a clearer presentation, the results of ‘model a’ are linked with line and the results of ‘model b’ are shown only with markers. Buckling curve c is only plotted to help judging how realistic the results are and not to propose a new design curve or to change the currently proposed curve d. This question cannot be discussed without the detailed investigation of residual stresses, and statistical evaluation of the results, which is beyond the aim of this article and topic of ongoing research work of the authors.

Figure 18.

Buckling curves d and c of specific case versus model results.

Numerical calculations showed that ‘model a’ brings always larger resistance than ‘model b’; however, the relevant reduction factor is always larger on ‘model b’ due to the local plate buckling phenomenon. Figure 18 shows that all the results fit on almost the same curve regardless of the model type, and the shape of the described function is much like the buckling curves. The results of ‘model b’ has systematically lower ${\bar{λ}}_{LT}$ and higher $χ_{LT}$ for the same length than those values of ‘model a’. The higher relative slenderness value (which is used for the evaluation) comes directly from the calculation of EN1993-1-1: since $M_{eff, Rd} < M_{el, Rd}$ for the same cross-section and $M_{cr}$ is the same, ${\bar{λ}}_{LT, el} = \sqrt{M_{el, Rd} / M_{cr}} < {\bar{λ}}_{LT, eff} = \sqrt{M_{eff, Rd} / M_{cr}}$ . If we consider buckling curves in general, it is known that $χ_{LT}$ is higher for lower ${\bar{λ}}_{LT}$ values on the same curve due the nature of the function. The results of the models mimic the same phenomenon that it is expected from the hand calculations.

The presented results give answer to the question given at the beginning of the article, and it can be concluded that consideration of local buckling within the cross-section properties in the calculation of the relative slenderness is correct, and it leads to correct interpretation of the combination of the local and global buckling phenomena.

Conclusion

The presented research work focused on the LTB resistance of girders with slender, class 4 cross-sections with a research aim to check the accuracy of the design resistance model of EN1993-1-1 and EN1993-1-5 on the coupled instability of LTB and local plate buckling resistances. Extensive numerical research programme is carried out on a validated numerical model using shell elements. The numerical model validation is made based on two independent experiments, and the reliability of the model is numerically tested. The numerical model is verified for pure local and global buckling, separately, and the combined stability failure is analysed on the complex numerical model. The validation proved that the numerical model is able to determine the pure plate buckling and pure LTB resistances, separately, and it is reliable with large accuracy.

Results on the numerical parametric study made on the interaction model proved that the EC-based design method for combined LTB and plate buckling phenomena follows the numerically computed trends extremely well. It means that the consideration of the local buckling in the cross-section properties within the global slenderness calculation process is correct. The theory, which is applied to compute the slenderness and the reduction factor for LTB resistance of girders with slender web itself, does not need further adjustment based on the analysed parameter and imperfection range. The magnitude of the reduction factor proposed for girders with class 4 webs, however, needs further revision. Results highlighted that the executed numerical results are closer to the buckling curve c, than buckling curve d, of the EN1993-1-1, which calls the attention on an optional improvement possibility in the future. However, the EC-based buckling curves are derived based on statistical evaluation of test results, therefore further investigations are needed in this topic using residual stresses and more sophisticated statistical evaluation process.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Noémi Seres

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