The direct strength method for combined bending and web crippling of second-generation trapezoidal steel sheeting

Second-generation trapezoidal sheeting, characterised by longitudinal stiffeners in webs and flanges, is loaded near a support by a concentrated force and a bending moment. Currently, design codes predict related failure by: (a) determining the ultimate bending moment via the effective width approach or the Direct Strength Method (DSM); (b) finding the web crippling load via a curve-fitted formula; and (c) using an interaction rule to take into account the load combination. However, the effective width approach is quite complex to use for many longitudinal stiffeners, and the accuracy of the design codes is subject to improvement. Moreover, nowadays the DSM provides a consistent and well-established method to predict ultimate loads for cold-formed steel structures. Therefore, in this paper the application of the DSM for combined bending and web crippling of second-generation sheeting is investigated. First, to create a set of numerical experiments, an internationally representative set of second-generation trapezoidal sheeting types is found, and these types are used to design numerical three-point bending experiments with relevant span lengths and load bearing plate widths. Then, finite element models are developed and verified, and used to predict the buckling, yield, and ultimate loads for the set of numerical experiments. Additionally, all simulations are also carried out for pure bending, and (Interior One Flange) IOF and (Interior Two Flange) ITF web crippling cases. With the resulting data, an explicit DSM approach is developed, fitted to the data of the three-point bending simulations, which predicts the ultimate load for combined actions directly. Hereafter, also an interaction DSM approach is studied, which first predicts the ultimate bending moment by the DSM (by fitting the pure bending simulations), then the web crippling load by the DSM (by fitting either the IOF or ITF simulations), and then uses a classic interaction rule for the load combination. The explicit and implicit DSM approaches perform equally well, with a Coefficient of Variation (CoV) equal to 0.13. As most commercially available sheeting has been incorporated, and the DSM approaches allow for sections with an arbitrarily number of stiffeners in the web (different from the current design codes), it is recommended to include the DSM in future code revisions. The interaction DSM approach resembles the current design rules most and may therefore be preferred; however, the explicit approach is more direct and certainly deserves consideration too.


Introduction
Cold-formed thin-walled trapezoidal steel sheeting is widely used in the building industry for cladding (walls) and sheeting (roofs), for among others its high strength and low weight. Trapezoidal sheeting can be classified into three categories: (i) first-generation (unstiffened); (ii) second generation (longitudinally stiffened); and (iii) third generation (longitudinally and transversely stiffened), see Fig. 1.
Trapezoidal steel sheeting is often used over multiple spans, resulting in a concentrated load and a bending moment above the intermediate supports, see Fig. 2. For first-generation trapezoidal steel sheeting, prediction of the related failure load above the support can be carried out by several methods: (i) via the current design rules, based on partly    3. Interaction rule to predict failure by a combined bending moment and concentrated load [2]. span /4), with span equal to the span length, the interaction rule can be used to calculate the load F for which the ultimate load is reached. 1 and 2 are fitting factors, found by the curve-fitting of experimental results.
(ii) The ultimate failure model is an alternative to the above procedure. It has a fully theoretical basis (thus it provides insight), and performs equally well or better than the current design rules. First the elastic web crippling stiffness is predicted via an energy approach, so the relationship between the concentrated force and the reduction of section height above the support is known. This prediction includes the determination of the out-of-plane deflection of a part of the flange directly above the support, this part also being loaded in compression due the bending moment. The idea of the ultimate failure model is that the out-of-plane deflection due to the concentrated load is used as the initial imperfection of the part of the flange (the plate), and Marguerre's equations [4] are used to predict this plate behaviour due the compressive forces of the bending moment up to first yield. As such, the model naturally includes the combination of loads. As mentioned earlier the model performs well, and it also describes the behaviour of sheeting for changing parameters correctly, as verified with finite element models [5]. However, it was deemed too complex for practice, and thus in subsequent projects it was simplified by using a two-strip/fictitious strain method-itself an insightful improvement to the effective width method-instead of Marguerre's equations [6].
(iii) The DSM provides a consistent and well-established method to predict ultimate loads for cold-formed steel structures [7]. It is based on the observed relationship between the ultimate load (e.g. for bending ) and the buckling load (e.g. ), both normalised by a so-called yield load (e.g. ), as shown in Fig. 4 and Eq. (1).
Once this relationship is established by finding the 1 and 2 values, via experiments or simulations, the ultimate load can be predicted by finding the buckling load, often by the use of the finite-strip method, and calculating the yield load, via a simple mechanical model. As formalised in the North American Specification for the Design of Cold-Formed Steel Structural Members AISI S100-16 [8], the DSM is applicable for single load cases like normal, shear, and bending forces. For load combinations, an interaction rule must be used. Several developments are underway to enable the DSM also for load combinations, and this can be carried out by two different approaches [9]: On one hand, the so-called explicit DSM approach predicts directly the combined ultimate load, using the combined buckling and combined yield loads. On the other hand, an interaction DSM approach can be developed, which predicts the ultimate load for each load case first by a dedicated DSM equation, and then an interaction rule, see Fig. 3, is used to take into account the load combination. For first-generation sheeting under combined bending moment and web-crippling, explicit and interaction DSM approaches have been developed [3]. It was concluded that both approaches are viable, but the interaction approach performs better than the explicit approach, and is a potential candidate for inclusion in the current design codes. Within the interaction approach, the DSM for pure bending performed well, whereas the DSM performance for IOF web crippling was less convincing. Therefore, Interior Two Flange (ITF) simulations were carried out, and the subsequent ITF buckling loads in combination with IOF yield loads and IOF ultimate loads were shown to increase the performance of the IOF web crippling DSM equations. There are several issues related to the current design rules. The effective width method, used within the design codes to determine the ultimate bending moment, does not take interaction into account between the flanges and the web. Furthermore, it cannot distinguish the possibly several buckling modes and types (local, distortional, and global) occurring at the same time. And with respect to accuracy, in 2000 it was found that for first-generation trapezoidal steel sheeting, loads predicted by the design codes showed differences between −40% and +10%, and the related design rules have not changed fundamentally since then [1].
For second-generation sheeting, the above issues for the design rules still apply, and additionally the longitudinal stiffeners in webs and flanges make the calculation of the effective cross-section, as needed for the effective width method, increasingly difficult. A possible alternative for the design rules could then be the ultimate failure model, however, Marguerre's equations or the two-strip model cannot be easily extended for application to longitudinally stiffened plates. As the DSM does not have these disadvantages, it is of interest to investigate whether the DSM can be used for second-generation sheeting. Even more so because the DSM showed to be successful for the predication of the ultimate load of first-generation sheeting [3]. Nevertheless, in this research on first-generation sheeting, it was shown that two issues need to be addressed, and likely this will also be the case for second-generation sheeting: (i) the type of yield load to be used is unclear; and (ii) for an interaction approach, the DSM performs less well for web crippling than for pure bending.
With respect to the type of yield load, different strategies exist [3,9,10]. For slender structures, which often fail by means of a yield line mechanism, the yield load can be defined by: (a) the load at which first plasticity occurs, after a stage of either (i) linear or (ii) nonlinear geometrically described elastic behaviour, in other words for a (i) fully or (ii) partly effective (buckled) cross-section; (b) a rigid-plastic mechanism initiation load, which predicts the onset of a yield line mechanism without taking into account elastic behaviour; (c) a first-order elastoplastic mechanism initiation load, which uses the intersection of a geometrical first-order elastic line (so fully effective cross-sections) and a plastic curve; and (d) a second-order elasto-plastic mechanism initiation load, similar to (c) but using a geometrical second-order elastic curve, resulting in partly effective cross-sections. Note that yield line mechanisms exist that show a significant higher ultimate load than their mechanism initiation load, but to predict their ultimate load often requires the complex modelling of the mechanism over its deformation trajectory, which is not further elaborated here. Furthermore, the DSM concept is based on predicting the ultimate strength via the buckling load (representing instabilities, i.e. geometrical nonlinearities, without taking into account strength) and the yield load (representing the strength, not taking into account instabilities). As such, for the yield load also (e) a sort of squash load can be used, e.g. for a column the cross-sectional area multiplied by the yield stress, and for a beam under bending the full plastic moment. In both cases no yield-line mechanisms are assumed to occur, and the cross-section is thought to be fully effective. Whereas for single load cases the yield load can be predicted in various ways, for load combinations this is much more complex. For combined bending moment and web crippling, in firstgeneration sheeting several yield-line mechanisms can occur, all very complex to model. For second-generation sheeting an even higher level of complexity can be expected, and models are not available. Also, a squash load cannot be defined for combined actions. Therefore, in this paper yield load (a) will be used: the load at which first plasticity occurs.
With respect to the DSM for web crippling, several research projects have been carried out. ITF and ETF experiments on hollow flange channel beams were conducted, for which the current DSM equations in the Australian/New Zealand Standard Cold-Formed Steel Structures AS/NSZ 4600 [11] and the AISI S100-16 [8] were shown to be unconservative, and new equations were proposed [12]. The DSM has also been developed for ITF and ETF web crippling of a variety of crosssections [13,14], where Generalised Beam Theory (GBT) was used for finding the buckling loads and a yield line model predicted the yield load. A similar approach was followed for EOF and IOF web crippling of lipped channels, however, a FE model predicted first yield, and other options for the yield load were tried, resulting in a preference for a rigid-plastic mechanism initiation load [10].
In order to study and model the behaviour of second-generation sheeting, in this case with the DSM, always reference should be made to real experiments. While for the first-generation a tremendous amount of experiments and simulations have been carried out (see e.g. an overview in [1]), for the second-generation, which is much more used in practice, surprisingly few experiments exist. A limited number of experiments were carried out in 2002, mainly to explore the differences in behaviour between first and second-generation sheeting [15]. The main conclusion was that the ultimate failure model could not be used for second-generation sheeting without significant adjustments. For that reason, it was investigated whether the compressed flange of second-generation sheeting could again be modelled as a plate, just as for the first-generation. Several finite element models for the compressed flange, with increasing complexity, were tested. However, even the most complex model of the compressed plate was not able to capture the behaviour of the complete sheeting, although large imperfections-so large that they mimic the action of the concentrated load-were not tried [16].
For second generation trapezoidal steel sheeting, the cross-section shape can be varied by adopting different stiffener geometries, dimensions, and distributions. In [17], research is presented to find optimised cross-section shapes to resist flexural buckling, resulting in the presentation of key design variables. It could be determined that the geometry of the intermediate stiffeners (characterised by the number of folded corners) is much more relevant for resisting buckling than the stiffeners' widths and the angles between web and flange. Finally, in [18,19], Yield Line Theory (YLT) was applied for first and second generation steel sheeting under pure compression or bending.
In this paper the application of the DSM to combined bending and web crippling of second-generation sheeting is investigated [20]. First existing experiments [15], which can verify the finite element models, to be developed, are presented in more detail in Section 2. Then to create a set of numerical experiments, in Section 3 an internationally representative set of second-generation trapezoidal sheeting types is presented, and these types are used to design a set of numerical experiments where sheet-sections are subjected to a three-point bending test with relevant span lengths and load bearing plate widths. In Section 4, for this finite element models are developed and verified with existing experiments and simulations, and used to predict the buckling, yield, and ultimate loads for the set of numerical experiments. Additionally, the simulations are also carried out for pure bending, and IOF and ITF web crippling. With the data obtained, an explicit DSM approach is developed, fitted to the data of the three-point bending simulations, which predicts the ultimate load for combined actions directly. Hereafter, also an interaction DSM approach is studied, which first predicts the ultimate bending moment by the DSM (by fitting the pure bending simulations), then the web crippling load, also by the DSM (by fitting either the IOF or ITF simulations), and then uses an interaction rule for combined actions. Both DSM approaches are introduced in Section 5. After comparing the two approaches with each other and with the current design rules in Section 6, conclusions and recommendations are given in Sections 7 and 8 respectively.  [15].

Table 1
Classification of tested (first and) second generation sheet-sections [15].

Setup
Kaspers [21] reports 5 × 2 experiments on second-generation trapezoidal steel sheet-sections, subjected to combined bending and concentrated load, Hofmeyer et al. [15]. In practice, sheeting is oriented as shown in Fig. 5 at the top, and thus at an interior support, due to bending, the bottom flange is under compression and the top flange is under tension. In Kaspers' experiments, sheet-sections were used as shown at the bottom of Fig. 5: only a representative part of the sheeting was tested, with boundary conditions at the longitudinal edges such that the restraints of the remaining part of the sheeting were taken into account. Also, the sheet-section was turned upside down, such that the sheeting bottom flange became the sheet-section top flange, so it was under compression in a three-point bending configuration. Kaspers classified the sheet-sections following the number and type of stiffeners in the web and compressed flange, see Table 1, and each section indicated bold in the table was tested twice. A load bearing plate (its width specified by the manufacturer) was fixed to a hydraulic jack, which loaded the sheet-sections deformation-controlled with a constant speed of 1 mm/s downwards. The span length for the experiments was determined by the manufacturer advised maximum and varied between 1100 and 2800 mm. An overview of the experimental setup is shown in Fig. 6.
The sheet-sections were supported on both sides by a rolling support, made by support bars and strips. To prevent spreading of the webs, strips were fixed to the two half flanges (under tension) with a maximum in-between distance of 250 mm. To avoid sway, near both supports a strip was fixed between the support strip and the flange under compression.
Five variables were measured during the experiments. The web crippling deformation, which is the difference between the height of the original cross-section and the height of the deformed crosssection, was measured with a measurement strip and two displacement indicators. Furthermore, the beam deflection at midspan was measured by a displacement indicator, measuring the distance between the same measurement strip and a support connection beam. The rotation of the reaction support bars was obtained using two displacement indicators at each support. The strain in the top flange in longitudinal direction and the load applied by the load-bearing plate were measured too. Additionally, the post-failure modes were observed.

Results
Post-failure modes were defined by the occurring yield line patterns and the characteristics of the load-deformation graphs: The experiments showed two of the three distinct post-failure modes that were already found for first-generation sheeting: yield-arc (A) and yield-eye (E), as shown in Fig. 7. With respect to the load-deformation behaviour, second-generation trapezoidal sections showed load dips, see Fig. 8, which was not the case for the first-generation. Sheet-sections with stiffeners in the web showed these dips only after the ultimate load, whereas sections with stiffeners in the flange saw these dips always before the ultimate load, and occasionally also after the ultimate load.
In this section, these existing experiments are presented only briefly, for they are meant for verification of the finite element models in Section 4. More details can be found in [21] and [15].

Numerical experiments
To develop a set of numerical experiments that represents the situation in practice, commercially available second-generation trapezoidal steel sheeting is listed in Table 2, as available in four countries: (i) The Netherlands (manufacturers SAB, ArcelorMittal); (ii) Australia (Lysagth, Fielders); (iii) the United Kingdom (TTP, Accord steel cladding); and (iv) The United States (Verco decking). These sheeting types are categorised with regards to the number of stiffeners in the web and flange (see Table 1 for this categorisation), where in the case of cross-sections from different manufacturers with a similar geometry and classification only one type of sheeting is added to the list. The recommended multi span length as listed in the table has been retrieved from the manufacturers' documentation. By conversion of the multi span lengths to span lengths for an equivalent three-point bending simulation, and allowing for variation, eight different span lengths are chosen for the simulations. The sheeting types with the highest recommended multi span length received the highest span lengths for the simulations. The load bearing plate length ( ) varies between 40 and 160 mm. For each span length several load bearing plate lengths are present (see Table 3). The extreme cases, upper right corner and lower left corner of Table 3, are excluded because they are unlikely to occur in practice.
Given the cross-section properties of the selected sheeting, the chosen span lengths and load bearing plate widths, a set of numerical experiments has been designed as shown in Table 3. In this table, every simulation is defined by a number, this number referring to a sheeting type (and thus its cross-section dimensions) via Table 2. This sheeting type will be simulated with a span length in three-point bending and a load bearing plate width as given in Table 3. The variables and indicate the number of stiffeners in the flange and web respectively.

Finite element simulations
In this section, finite element models are presented and verified against the existing experiments and other simulations, and used to predict the buckling, yield, and ultimate loads for the numerical experiments in Section 3. Additionally, the simulations are also carried out for pure bending, and IOF and ITF web crippling cases.

Sheeting geometry
To make the modelling of all the different second-generation sheeting types efficient in the finite element method, a parametric model has been defined in Python [22]. The Python script is based on a model of the cross-section with stiffeners as presented in Fig. 9 (due to symmetry only half top and bottom flanges are shown). The model can handle a maximum of three stiffeners for each flange or web. For a stiffener in the top flange, a more detailed drawing is shown on the left, and similar  detailed drawings have been made for a stiffener in the web, and in the bottom flange (see at the bottom right corner). Fewer stiffeners are enabled by setting the non-existing stiffeners' dimensions to zero. The many variables in Fig. 9 are self-explanatory, are not needed in the following sections, and thus are not defined here further. Fig. 10 shows an overview of the finite element simulation approach. The (FEM) C NL model (Combined bending and web crippling, Non-Linear analysis) simulates the three-point bending tests as presented in Section 3, and provides the (ultimate load) and (yield load), to be explained. The C MOD model (Combined bending and web crippling, MODal analysis) generates the (buckling load). As such, an explicit DSM-C equation can be derived using the , and loads and will yield a prediction for the ultimate load ;

Simulation approach
. Similar to the C NL and C MOD models, finite element models are made for pure bending (models B NL and B MOD ), and IOF or ITF web crippling (W NL and W MOD ). The data generated by these models can be used to develop an interaction DSM approach by DSM equations for pure bending (DSM-B) and web crippling (DSM-W), which subsequently can be used in combination with an interaction rule. In the end, both the explicit and interaction DSM approaches can be evaluated by their Coefficient of Variation (CoV) to , which has been obtained by the NL model.

FEM setup
The finite element models are a further development of existing models for first-generation sheeting [3], and have been developed in ABAQUS CAE 6.14 [22]. As asymmetric failure behaviour (in length direction) is possible-e.g. as for the yield-eye post-failure mode for  Simulation # (see Table 3) first-generation sheeting-a half of the sheet-section is modelled, as shown in Fig. 11. The C NL model consists of two parts: the sheet-section modelled by four node shell elements with reduced integration and hourglass control, and the load bearing plate made by rigid shell elements. Frictional interaction exists between the sheet-section (being the socalled slave surface) and the load-bearing plate (the master surface), formulated by standard ''surface-to-surface contact'' with a coefficient of friction equal to 0.3, however this coefficient is not critical. Fig. 12 gives a detailed view of the boundary conditions. First of all, symmetry is ensured by symmetry boundary conditions along the symmetry line (Fig. 11). Note that these conditions automatically avoid sway, and thus strips against sway (as present in the experiments) are not needed. Secondly, to model sheet-section and not hat-section behaviour, strips preventing spreading of the webs are modelled by constraining lines of element nodes in the tension flange for U x , with a maximum distance in-between the lines of 250 mm. The support strips are first modelled by reference points, which are constrained for all degrees of freedom, except for U z (translation in the z-direction) and R x (rotation about the x-axis). Subsequently, the element nodes in the cross-section above the support strips are tied to the reference points. All nodes in the crosssection are selected, and not only those in the flange under tension, to simulate the experiments (for which the strip against sway resulted in similar behaviour), and to avoid local failure near the supports. Finally, the load-bearing plate is constrained in all directions, except for U y (the translation in y-direction), which is subject to a prescribed displacement of −25 mm.  Mesh densities are very similar to the models for first-generation [3]. The finest mesh is located at mid-span. Here the mesh contains elements sized 1 × 1 mm for the compressed flange, the compressed flange corner, and the web, Fig. 13. The mesh gradually changes to a coarse mesh made from 8 × 8 mm elements. The flat parts of each stiffener are meshed equally sized as for that mesh region, and the load-bearing plate was meshed by elements 2 × 2 mm. Note that the corners of the stiffeners are not modelled, different from the web-flange corners, and a motivation for this is given in the next section on verification.
For the finite element model for pure bending, also to be presented, a global mesh size of 3 × 3 mm is used, following the constant stress gradients over the length, due to the constant bending moment. In these simulations, the compressed flange corner was meshed with at least three elements along the circumference, which was increased to five elements for corner radii larger than 6 mm.
With respect to the material model, the steel is modelled by a von Mises yield surface with associated plastic flow and isotropic hardening. The yield stress and hardening are defined by input of the points of the true stress-true strain curves. For S320 steel ( = 320 N/mm 2 , see Table 2), measured values from [21] were used as shown in Fig. 14. For S220, S345 and S550 the curves were scaled based on the S320 values, keeping the linear elastic curve the same.
For the solution, an implicit dynamic procedure is used, in which the total applied deformation of −25 mm takes place within 1 s. Further solution settings are taken from the previous models [3], and for these models it was already shown that results are quasi-static for 1 s, so the dynamic procedure can be used. The C MOD model is strongly based on the C NL model, for which the load-bearing plate is replaced by two unit loads at the junction between the compressed flange and its corner, at an in-between distance equal to the load-bearing plate width. The W NL model (for IOF web crippling) is also derived from the C NL model, by simply adjusting the span length to a value equal to the load bearing width + 3 ( = the web height), following the requirements in the AISI S100-16 [8]. Naturally then, the W MOD model equals the C MOD model with again this adjusted span length. The B NL model for pure bending moment starts with the C NL model too, but here the load bearing plate is replaced by a prescribed rotation R x of the supports. Also the mesh is modified, as mentioned above. Finally, the B MOD model is derived from the C MOD model by the same modifications. Note that no imperfections are used for the C NL and W NL models, for the lateral loads introduced at the corner edges provide large eccentricities that have shown to dominate possible plate imperfections. However, for the B NL model a (small) imperfection is needed to trigger non-linear buckling and the subsequent localised plastic yield line mechanisms. For this imperfection the first positive Eigenmode from the B MOD model is used for the shape, and scaled to a maximum amplitude equal to 1/1000 of the compressed flange width. The C NL simulations are focused on interaction failure, as mentioned, and the loads at the corners provide large eccentricities, leading to cross-sectional deformations showing significantly larger imperfections to the bending moment than manufacturing imperfections. As this effect is part of the interaction (rule), here it is chosen to make the imperfection for bending in the B NL model small, as to only trigger buckling and localised yielding. Note however, that [18,19] have shown that for bending using 1/200 of the compression flange width leads to accurate ultimate loads when compared to experimental tests, and that for that case there is a sensitivity of the results for the imperfection magnitude. Nevertheless, for first generation sheeting, imperfections of 1/1000 still led to a correlation of 0.962 of simulations and Eurocode predictions [3].
For the determination of the first yield load in the B models, a geometrically linear analysis is conducted, as required by the AISI S100-16, up to the first occurrence of plastic dissipation, taking into account the whole model, and measured both at the membrane surfaces and the outer surfaces of the shell elements. This is different for the C and W models, for which the yield load is determined in a similar fashion, but during a geometrically and material non-linear analysis. However, an alternative was researched too, see Section 5.2.

Verification
Verification of the finite element models was carried out by comparing the results of the C NL model with the experiments as presented in Section 3, and with previous simulations of these experiments by Vervoort [16]. Fig. 15 presents the Pearson product-moment correlation coefficient , and the average and Coefficient of Variation (CoV) of the Fig. 11. FEM setup, C NL model.     ultimate loads as found for the experiments , and as predicted by the C NL model (left graph), and similar for the ultimate load as predicted by the previous simulations , and the C NL model (right graph).
On the left of Fig. 15, the ultimate loads of the simulations do not agree as well as desired with the experiments. The cause is difficult to find, as only limited documentation of the experiments exists. However, the figure on the right shows good agreement between different finite element models (different programs, different operators, different models), which indicates a high probability that the finite element model provides reasonable outcomes. Differences with the experiments are likely due to uncertainties with respect to detailed information on material properties, actual imperfections, dimensions, test rig elasticity, etc. Also, the load-beam deflection, load-web crippling deformation, and support-web crippling deformation graphs, and the post-failure modes as found with the C NL model can be compared to the experiments and previous simulations. In the previous simulations, two solution strategies were tried: Explicit Dynamic (ED) and Implicit Dynamic (ID). For all these above aspects, useful resemblances were found, as shown in Table 4, and Figs. 16 and 17, which verifies the new finite element model C NL to be used in this paper.

Influence of the modelling of stiffener corners
To determine the influence of the modelling of the stiffeners' corners, three different C NL simulations are carried out. For the first simulation, each corner, also of each stiffener, is modelled. In a second simulation, only the corners between the flanges and the web are modelled, and for the third simulation no corners are modelled at all, e.g. the web and flange are flat plates joined together. The three different simulations are carried out for two sheeting types: (i) W2-F2 ArcelorMittal 135/310-3T (simulation 25, see Table 3), see Fig. 18, which is a frequently occurring type in the simulations, and (ii) W1-F2 Lysagth Klip-lok classic 700 (simulation 60, see Table 3), which has unusually shaped stiffeners, see Fig. 19.
Comparing the results of the three C NL simulations by means of the load-beam deflection, load-web crippling deformation, and support rotation-web crippling deformation graphs, it can be concluded that the corners between web and flange need to be modelled, however, the corners of the stiffeners have a neglectable influence. This is shown for the W2-F2 ArcelorMittal 135/310-3T in Fig. 20, with similar conclusions for the W1-F2 Lysagth Klip-lok classic 700.

Results
If the C NL finite element model is used to carry out the numerical experiments, as given in Table 3, all the post-failure modes that occur are the same as the three post-failure modes that were found for first-generation trapezoidal steel sheeting. Fig. 21 shows these three post-failure modes for typical simulations: For simulation 1 (see Table 3) the yield-arc post failure mode shows up; the rolling post failure mode is obtained for e.g. simulation 5, and the yield-eye post failure mode occurs among others in simulation 15. Corresponding load-web crippling deformation diagrams are presented in Fig. 22, and again the characteristics of these curves are similar to those of first-generation sheeting: a yield-arc mode decreases in strength quickly after the   ultimate load; the rolling mode has plastic capacity with irregularities due to the yield lines moving across elements (integration points), and the yield-eye mode shows a seemingly decreasing web-crippling deformation due to asymmetric behaviour and the measurement of the web-crippling deformation exactly in the middle, see for more details [1].
Using the finite element models as presented in Fig. 10 for the numerical experiments in Table 3, the ultimate load, the yield load, and the first positive Eigenvalue (buckling load) are determined and represented in Table 5. The yield load is predicted by the first occurrence of plastic dissipation, taking into account the whole model, and measured both at the membrane surfaces and the outer surfaces of the shell elements. For the pure bending simulations, the yield load (moment) is determined in a similar fashion, but using a linear elastic simulation (by switching off geometrical nonlinearities), in accordance with the DSM guidelines in AISI-S100-16 [8]. This data will be used in the next section to develop the DSM approaches.

Direct strength method
With the data obtained in the previous section (Table 5), in this section first an explicit DSM approach is developed, fitted to the data of the three-point bending simulations, which predicts the ultimate load for the combined actions of bending moment and concentrated load directly. Hereafter, also an interaction DSM approach is studied, which first predicts the ultimate bending moment using the DSM (by fitting the pure bending simulations), then the web crippling load by again the DSM (by fitting either the IOF or ITF simulations), and then uses an interaction rule for the combined actions.

Explicit DSM approach
In general, to calibrate a DSM equation, three loads are required: (i) the ultimate load ( ); (ii) a linear-elastic buckling load ( ); and (iii) a yield load ( ). Note that for the variable P also M or F can be employed, as used in Table 5. For the format of a DSM equation, many different versions are given in literature. Here, Eqs. (2) to (7) are selected. Eqs. (2) to (5) are used in the AISI S100-16 [8]. Eqs. (2) and (3) are for members in compression, to determine the strength for yielding in combination with global buckling (AISI eq. E2-2 and E2-3, note that these equations are not listed as part of a DSM). Eq. (4) is listed as a DSM equation for members in compression (AISI eq. E3.2.1-2) and for members in flexure (AISI eq. F3.2.1-2), for the strength related to local buckling interacting with yielding and global buckling, and for distortional buckling (AISI eq. F4.1-2). Besides, Eq. (4) is also used for flexural members to determine the shear strength of webs (AISI eq. G2.2-2). Eq. (5) equals AISI eq. F2.1-4 and is for members in flexure, to predict the initiation of yielding strength combined with global buckling. Finally, Eq. (6) comes from [23], and Eq. (7) originates from [14], and both are used for web crippling.
The factors in Eqs.
(2) to (7) are determined such that the sum of the squared errors between the (yield load normalised) ultimate loads from the finite element model and the DSM equation is minimal. This is carried out by the GRG nonlinear solver in Microsoft Excel using ''Multistart'', which tries 100 different initial values to obtain more confidence regarding a global optimum. Some of these solutions were also verified with the evolutionary solver. All factors were bound to be larger or equal to 0, and smaller than 100. As a result, for the explicit DSM approach using , , and , the best performing equation is found to be equation (7), for which the average (Avg.) and Coefficient of Variation (CoV) of the simulation to DSM prediction ratio equal 1.00 and 0.13 respectively. Corresponding k-factors are found in Fig. 23. The (Pearson) correlation between the simulations and DSM predictions equals = 0.969. This result was verified by the evolutionary solver. Note, that although the presented equation is optimal, solutions with other k-factors may perform almost equally well. For instance, if a common value for 2 = 0.15 is set (so fixed during the optimisation), 3 will be 0.50 to obtain the same CoV and only a slightly lower correlation (0.966). In a similar vein, allowing the k-factors to be negative may give well performing equations too, however, with incomprehensible factors, like for instance 2 = − 0.51. This is important because kfactors in the DSM equations imply different levels of post-buckling, e.g. for the power of r ∕ ( 2 in Eq. (4)), a value of 1.0 is basically Euler, and lower values imply higher levels of post-buckling. However, as mentioned above, fitting the k-factors by statistical analysis alone does not take this into account, and this is a problem if (many) different combinations of k-factors yield (almost) the same minimum sum of squared errors. Therefore, during optimisation, it may be considered to fix certain k-factors to a value that makes sense with respect to the expected level of post-buckling. For the cases here, it was shown that this approach will not harm the performance of the equation, however, the precise value of the k-factor is then still subject to debate, as the level of post-buckling can never be determined exactly in a numerical way.

Explicit DSM approach with linear elastic determined yield loads
In the previous section, for the yield load, first plastic dissipation was used as obtained in a geometric and material non-linear simulation. As mentioned earlier, the yield load is determined by the first moment of plastic dissipation within the whole model, measured on the membrane surfaces and the outer surfaces of the shell elements. However, for the pure bending simulations, used in the next section for the interaction approach, the yield load (moment) is determined by a linear elastic simulation, in accordance with the DSM guidelines in AISI-S100-16 [8] and research on the DSM for first-generation sheeting [3]. As such, the question arises whether it is also useful for the explicit approach to apply a yield load determined by a linear elastic simulation. Table 5 shows that for simulations 4, 14, 31, and 36 the yield load exceeds the first positive Eigenvalue, which means instabilities may occur here before the yield load. Therefore, for all simulations a linear elastic determined yield load is obtained as follows: with u10% equal to 10% of the ultimate load; the yield stress; 10% the maximum Von Mises stress found in the sheeting at 10% of the ultimate load and _ the linear elastic determined yield load. Following the same approach as presented in Section 5.1, but now using the linear elastic determined yield load, results in virtually the same performance, see Fig. 24.

Interaction DSM approach
The interaction DSM approach first predicts the ultimate bending moment (by fitting the pure bending simulations), then the web crippling load (in this section by IOF simulations), and then uses an interaction rule for combined actions. For the pure bending simulations, using the same approach as in Section 5.1, all equations resulted in a CoV lower than 0.20 and similar correlations, all higher than 0.992. For consistency with the previous section, and the fact that Eq.       for web crippling [24]. For the data in this paper, Eq. (7) resulted in a CoV = 0.20 and a correlation = 0.803 with the k-factors: 1 = 1.73, 2 = 0.08 and 3 = 0.71. This is certainly useful performance, and the last k-factor (0.71) fits nicely within the window of values found by Nguyen et al. [24], which ranged from 0.4 to 1.0. For the remaining part of the paper, for IOF web crippling the best performing Eq. (5) will be used, but Eq. (7) should certainly not be discarded as a future alternative for code consistency.
The somewhat high CoV for web crippling is investigated further by conducting ITF web crippling simulations. This may be useful, for ITF simulations do not include a (small) bending moment, and their Eigenmodes are related to buckling of the web, and not buckling of the flange (as in IOF simulations). For the derivation of the DSM-W equation, now for the Eigenvalue either the IOF or ITF value can be used, and in a similar fashion, the IOF or ITF yield load value can be applied. Resulting correlations between the ultimate loads of the (IOF!) simulations and the DSM-W equation are found in Table 6. All variants perform worse than the original intended approach. This is surprising, as ITF buckling is determined by buckling of the web, whereas IOF buckling is also determined by buckling of the compressed flange due to the (small) bending moment. As the failure of IOF web crippling is based on failure of the web, not the flange, using an ITF based Eigenvalue would have to be seemed a logic choice. For the last step in the interaction DSM approach, the standard format of an interaction rule for combined bending and web crippling is given by Eq. (9). In this equation, F is the actual load to be checked for web crippling against , and M the actual bending moment to be checked against the ultimate bending moment . The allowed ratio between concentrated load and bending moment is considered by the factors a and b. For a three-point bending test with concentrated load ; (IR stands for Interaction Rule), the bending moment M in Eq. (9) can be rewritten as 1/4 ; ( -) and the concentrated load F can be replaced by load ; , resulting in Eq. (10).
The ultimate concentrated loads from the W NL model and the ultimate bending moments from the B NL model are used as input to calibrate the factors a and b. This is carried out by a least squares approach in Microsoft Excel: The GRG nonlinear solver varies the factors a and b for a minimal sum of the squares of the residuals between (the ultimate load of the C NL model) and ; , see Fig. 27. Fig. 27 also represents the results of the finite element simulations in the interaction space, in which the loads , and are obtained from the finite element simulations, F is equal to and the M is 1/4 ( -). The interaction rule is displayed by the solid (blue) line. Note that as it starts at the left almost at ∕ = 1, the last inequality in Eq. (9) is not utilised.
With the interaction rule derived, the and values as predicted by the DSM-W and DSM-B equations can be inserted in the interaction rule, resulting in the ; as predicted by the interaction DSM approach. For this, a CoV of 0.13 is obtained between ; and . Fig. 28 presents the results (red markers), including the results of the explicit DSM approach from Section 5.1 (black markers). In Fig. 28 the results of the interaction DSM approach are given in the interaction space. Note that the markers form another diagonal line as the line used by the codes, as factors a and b of the interaction rule were derived specifically for the simulations here.

Current design rules
To assess the developed DSM approaches further, they can also be compared with the current design rules. As mentioned earlier, these design rules (a) determine the ultimate bending moment via the effective width approach or the Direct Strength Method (DSM); (b) find the web crippling load via a curve-fitted formula; and (c) use an interaction rule to take into account the load combination. Here, for (a) the ultimate bending moment is used as obtained in the simulations, as the effective width approach may be quite complex to use for secondgeneration trapezoidal steel sheeting. For (b), Eurocode EN 1993 − 1 − 3 [25] only provides equations for sheeting with at most 1 stiffener in the web, see Eqs. (11) to (13) and Fig. 29. Note that for a stiffener, Eq. (11) is to be multiplied by the result of Eq. (12), which should not exceed the limit value given in Eq. (13). Some variables are given in Fig. 29, and further is a numerical factor, equals the yield strength, E is  Young's Modulus, r the corner radius, t is the plate thickness, the load bearing plate width, and the angle between web and flange ( in the figure). Here, for sheeting with 2 and 3 stiffeners in the web, it is suggested to use for the minimum and maximum ''out-of-plane'' distances and the minimum and maximum ''out-of-plane'' distances taking all stiffeners into account. For , it is suggested to use the distance towards the first stiffener, probably the most critical one for web crippling.
, = 1,45 − 0, 05 * ∕ , ≤ 0, 95 + 35000 * 2 * ∕( 2 * ) For IOF web crippling, using the equations and the suggestions for more than 1 stiffener, the correlation between the ultimate loads of the simulations and the predicted loads by the Eurocode ; 3 are presented in Fig. 30 for sections with 0 or 1 stiffener, and in Fig. 31 for two or three stiffeners in the web. Striking is that Eurocode results are better for 1 or 2 stiffeners than without. For three stiffeners only 2 cases exist, so conclusions are not useful.   Eurocode interaction rule, as shown in Figs. 32 and 33 for 0 and 1, and 2 and 3 stiffeners respectively. Although the CoV is higher for 2 stiffeners than for the other groups, overall the CoV is useable. The results of the Eurocode calculation are also presented in the interaction space in Figs. 32 and 33 for 0 and 1, and 2 and 3 stiffeners respectively. The loads and are obtained from the Eurocode calculations and is retrieved from the finite element simulations. The force F is equal to and the bending moment M is 1/4 ( -). The North American Specification for the Design of Cold-Formed Steel Structural Members AISI S100-16 [8] only provides web crippling equations for sheeting without stiffeners in the web. Fig. 34 shows S100-16 predictions for the web crippling strength compared to the simulations of sheeting without stiffeners in the web. Performance with a CoV = 0.28 is better than the Eurocode predictions (CoV = 0.32, see Fig. 30). Also, the interaction rule of the S100-16 can be used, as shown in Fig. 35. Again, good results are obtained, similar to the Eurocode. The results of the AISI calculation are also presented in the interaction space in Fig. 35, in which the loads and are obtained from the AISI calculation and is retrieved from the finite element simulations. The force F is equal to and the moment M equals 1/4 ( -).

Conclusions
In this paper, explicit and interaction DSM approaches have been developed for combined bending and web crippling of secondgeneration trapezoidal steel sheeting. A set of numerical experiments was developed, based on commercially available sheeting in the USA, UK, Australia, and The Netherlands. Then, finite element models were presented, and verified against existing experiments and existing simulations. A study on the influence of the modelling of the stiffener corners showed that rounding the stiffener corners has a very limited effect on the results and can be omitted. This resulted in data for the development of the DSM approaches, namely Eigenvalues, yield loads, and ultimate loads for combined bending and web crippling, pure bending moment, and IOF and ITF web crippling.
An explicit DSM approach was developed by fitting the first positive Eigenvalues, the yield loads, and the ultimate loads found by the simulations for combined bending and web crippling. Several equation types were tried, and the DSM equation as used for the AISI, with an extra k-factor, was found to perform the best, with a CoV equal to 0.13. The initially found k-factors (for all three factors variable) did yield the smallest sum of squared errors, however, only slightly higher sums were found if some k-factors were fixed at commonly used values. For the yield load, the load for first plastic dissipation in the simulations  was used, even if instabilities occurred in an earlier stage. Therefore, also a yield load was tried based on an extrapolated maximum Von Mises stress in the linear elastic stage. The alternative yield load did not change the results, probably also because only 4 out of 78 simulations showed instabilities before the yield load.
For the interaction DSM approach, separate DSM equations were developed for pure bending moment and IOF web crippling. For pure bending, all equation types performed equally well. For IOF web crippling, among the equation types, Eq. (5) performed the best. This equation (5) equals an AISI equation for members in flexure, to predict the initiation of yielding strength combined with global buckling and is not listed as a DSM equation. However, note that as an alternative, DSM based AISI equation (4) performed similar. Even Eq. (5) did not perform very well for 0 stiffeners in the web (CoV = 0.19), however, still better than the current design rules for web crippling (Eurocode3: CoV = 0.32 and AISI S100-12: CoV = 0.28). In an attempt to improve the results, ITF Eigenvalues and yield loads were tried, with the idea that ITF buckling and yielding is determined by buckling and yielding of the web, whereas IOF buckling and yielding is also determined by buckling and yielding of the compressed flange, due to the (small) bending moment. As the failure of IOF web crippling is based on failure of the web, not the flange, using ITF based Eigenvalues and yield loads could be tried. However, ITF Eigenvalues and/or yield loads did not improve the CoV values. Finally, predictions of the implicit DSM approach for combined actions, using the equations above and as well DSM equations for pure bending moment and an interaction rule, performed well for the numerical experiments, with a CoV equal to 0.13, but for sections without stiffeners in the web, not better than Eurocode3 (CoV = 0.09) and AISI S100-12 (CoV = 0.15).
The fact that (a) most commercially available sheeting has been incorporated; (b) that both the explicit and implicit DSM approaches perform similarly well as the current design rules; and (c) that the DSM approaches include sections with an arbitrarily number of stiffeners in the web, it is recommended to include the DSM in the design rules. Although for the interaction approach the DSM does not perform flawlessly for web crippling, this is neither the case for the current design rules, which in some cases even perform worse.
The optimal k-factors for the strength expressions are not unique, and as a result, explicit statements about the level of post-buckling based on these coefficients may not be rigorously proven.

Recommendations and future research
Not many existing experiments were found for second-generation sheeting, however, these are needed to further verify the finite element simulations. Also, the set of numerical experiments should be further extended with other sheeting types, and even more variety in span lengths and load bearing plate widths.
The definition and calculation of the yield load should be further investigated. For this study, the load is used at which first plastic dissipation occurs-either via a linear elastic or geometrically and material nonlinear simulation-but at least theoretically the yield load can also be defined by a first-order or second-order elasto-plastic mechanism initiation, or ultimate load, or by a rigid-plastic mechanism initiation load [10]. Note, however, that this is easily possible for pure bending moment, difficult for web crippling, and almost impossible for combined bending moment and web crippling. For the latter case, mechanisms are complex, and an easy ''squash'' load like for a column under compression, or a fully effective beam under bending, seems completely impossible. Nevertheless, if the DSM is to be used without a linear finite element simulation (because that will deliver the yield load used here) further investigations are needed to present a practically derivable yield load. As such, a simplified ultimate failure model [2] could serve a role. Alternatively, the use of a linear finite element model is still far less problematic than geometrically and material nonlinear simulations, and with the advent of finite element simulations in practice, should still be an option to consider too.