Applied Element Modelling of Warping Effects in Thin-Walled C-Shaped Steel Sections

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Article Applied Element Modelling of Warping Effects in Thin-Walled C-Shaped Steel Sections

Giammaria Gabbianelli

Department of Science, Technology and Society, University School for Advanced Studies IUSS Pavia, Piazza Della Vittoria 15, 27100 Pavia, Italy; [email protected]

Abstract: The Applied Element Method (AEM) is a relatively recent numerical technique, originally conceived for simulating the large displacement nonlinear response of reinforced concrete, masonry and steel structures, and successful applications have been presented by various researchers. Recently, AEM was used to model the mechanical behaviour of steel storage pallet racks, i.e., particular cold- formed steel structures typically employed for storing goods and materials. Such systems are often subjected to peculiar displacements and stresses due to warping effects, which are inherent and often govern their behaviour, increasing the peak strength and ultimate displacement demand. This phenomenon has not been studied through AEM yet; hence, this work investigates the capabilities of AEM in simulating the warping effects in typical steel rack members, i.e., thin-walled C-shaped sections. Preliminary results and comparison against established modelling approaches indicate that AEM can accurately simulate this phenomenon, both in terms of displacements and stresses.

Keywords: Applied Element Method (AEM); ﬁnite element method; warping effects; thin-walled

sections; cold-formed steel sections

Citation: Gabbianelli, G. Applied Element Modelling of Warping 1. Introduction Effects in Thin-Walled C-Shaped Steel Sections. Buildings 2021, 11, 328. The modelling and analysis of multiple types of structures have been largely carried https://doi.org/10.3390/ out using the Finite Element Method (FEM), a pivotal method, adopted worldwide, to buildings11080328 evaluate buildings’ performance before they reach significant damages and the separation of structural elements. Despite its large capabilities, the FEM is not suitable to simulate the Academic Editors: Jiho Moon and response of structures in the case that, for example, a floor fails and hits the underlying Francisco López Almansa slab. Indeed, FEM assumes that nodes and elements have to remain connected to each other; otherwise, singularities in the global stiffness matrix occur. On the contrary, the Received: 13 June 2021 Discrete Element Method (DEM) is more appropriate in the case of post-failure situations Accepted: 27 July 2021 since it permits element separation. However, within a DEM framework, the development Published: 29 July 2021 of structural models can be a complex task, and high computational efforts are often demanded [1]. Moreover, since generally DEM software is based on explicit numerical Publisher’s Note: MDPI stays neutral methods, the modelling of static or quasi-static simulations is computationally inefficient, with regard to jurisdictional claims in requiring dynamic relaxation schemes and suitable calibration of the damping factors [2]. published maps and institutional affil- Considering the advantages and shortcomings of the FEM and DEM methods, the iations. Applied Element Method (AEM) was firstly proposed by Meguro and Tagel-Din [3]. This numerical approach aims to provide a tool that is able to perform the assessment and evaluation of buildings’ performance in pre- and post-failure scenarios. AEM combines the best features of both FEM and DEM, providing a new efficient and complete tool for Copyright: © 2021 by the author. structural analysis. When using the AEM, structures and structural items/components are Licensee MDPI, Basel, Switzerland. all modelled as an aggregation of rigid block elements, which are connected by themselves This article is an open access article with springs in the normal and tangential directions (Figure1). The springs simulate distributed under the terms and the stresses and strains of a certain area of the elements connected, and their stiffness is conditions of the Creative Commons computed as in Equation (1): Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ E · d · t G · d · t K = ; K = (1) 4.0/). n a s a

Buildings 2021, 11, 328. https://doi.org/10.3390/buildings11080328 https://www.mdpi.com/journal/buildings Buildings 2021, 11, x FOR PEER REVIEW 2 of 18

selves with springs in the normal and tangential directions (Figure 1). The springs simulate the stresses and strains of a certain area of the elements connected, and their stiffness is computed as in Equation (1): Buildings 2021, 11, 328 2 of 17 𝐸∙𝑑∙𝑡 𝐺∙𝑑∙𝑡 𝐾 = ; 𝐾 = (1) 𝑎 𝑎

wherewhereE Eand andG Gare are the the elastic elastic and and shear shear moduli, moduli, respectively; respectively;d dis is the the distance distance between between the the springs;springs;a ais is the the length length of of the the representative representative area; area; and andt ist is the the thickness thickness of of the the element. element.

FigureFigure 1. 1.Contact Contact points, points, spring spring distribution, distribution, normal normal and and shear shear stiffness stiffness connecting connecting two two elements. elements.

ItIt should should be be noted noted that that the the degrees degrees of freedomof freedom are are assumed assumed to be to at be the at centroid the centroid of the of blocks;the blocks; thus, thus, as a consequence,as a conseque thence, number the number of springs of springs does does not inﬂuencenot influence the dimensionthe dimension of theof globalthe global stiffness stiffness matrix. matrix. Summing Summing up up the the contribution contribution of of each each spring spring and and taking taking into into accountaccount the the relativerelative coordinatecoordinate of the spring spring contact contact point point with with respect respect to tothe the centroid, centroid, the thefinal ﬁnal element element stiffness stiffness matrix matrix can can be be comput computed.ed. For For the sake ofof simplicity,simplicity, the the element element stiffnessstiffness matrix matrix of of a singlea single spring, spring, for thefor casethe case of two-dimensional of two-dimensional (2D) analysis, (2D) analysis, is reported is re- inported Equation in Equation (2): (2):   Kn 𝐾 −Kn·dy1 −Kn Kn·dy2 𝐾 𝐾 ∙𝑑 𝐾 𝐾 ∙𝑑  Ks ⎡ Ks·dx1 −Ks −Ks·dx2  ⎤  𝐾2 2 𝐾 ∙𝑑 𝐾 𝐾 ∙𝑑   −Kn·d Ks·d ⎢ Ks·d + Kn·d Kn·d −Ks·d −Kn·d ·d − Ks·d ·d  ⎥  y1 x1 x1 y1 y1 x1 y1 y2 x1 x2  KE = ⎢𝐾 ∙𝑑 𝐾 ∙𝑑 𝐾 ∙𝑑 𝐾 ∙𝑑 𝐾 ∙𝑑 𝐾 ∙𝑑 𝐾 ∙𝑑 ∙𝑑 𝐾 ∙𝑑 ∙𝑑(2)⎥(2)  − = · − ·   Kn ⎢ 𝐾 Kn dy1 𝐾 ∙𝑑 Kn 𝐾 Kn dy2 𝐾 ∙𝑑  ⎥  ⎢  ⎥ −Ks −Ks·dx1 Ks Ks·dx2  ⎢ 𝐾 𝐾 ∙𝑑 𝐾 𝐾 ∙𝑑  ⎥ · − · − · · − · · − · · · 2 + · 2 Kn dy2 Ks dx⎣2 𝐾 ∙𝑑Kn d𝐾y1 d∙𝑑y2 K𝐾s d∙𝑑x1d∙𝑑x2 𝐾K n∙𝑑dy2∙𝑑 K𝐾s dx∙𝑑2 𝐾 ∙𝑑Ks dx2 K𝐾n ∙𝑑dx2 𝐾 ∙𝑑 ⎦

where Kn and Ks are the normal and shear stiffnesses, as per Equation (1), while dx1, dx2, dy1 where Kn and Ks are the normal and shear stiffnesses, as per Equation (1), while dx1, dx2, y2 dyand1 and d d yare2 are the the distances distances of of the the contact contact points points wi withth respect to theirtheir relativerelative centroid, centroid, as as depicteddepicted in in Figure Figure1. 1. In In the the study study of of Meguro Meguro and and Tagel-Din Tagel-Din [ 3 [3],], the the theory theory and and formulation formulation ofof AEM AEM for for small small deformation deformation analysis analysis considering considering the the effects effects of of Poisson’s Poisson’s ratio ratio was was pre- pre- sented.sented. Further Further considerations considerations regarding regarding the the effect effect of of the the number number of of springs springs and and blocks blocks werewere also also made. made. Consequently, Consequently, the the AEM AEM was was validated validated for for the the case case of of large large deformations deformations analysesanalyses under under dynamic dynamic loading loading conditions conditions [4]. [4]. It is It worth is worth noting noting that that an AEM an AEM formulation formula- doestion notdoes need not aneed geometric a geometric stiffness stiffness matrix, matr entailingix, entailing a simpler a simpler numerical numerical procedure procedure in comparisonin comparison with with the cumbersomethe cumbersome one adoptedone adopted by FEM. by FEM. Ensuing Ensuing research research studies studies investi- in- gatedvestigated the accuracy the accuracy of AEM of formulationAEM formulation in the in case the of case reinforced of reinforced concrete concrete structures structures with nonlinear constitutive material laws applied at the springs [5,6]. The outcomes demon- strated again the feasibility of AEM since it was possible to accurately estimate the failure behaviour, including crack initiation and propagation, both with monotonic and cyclic loads. Subsequently, a new, improved AEM approach was developed [7,8], namely IAEM (Improved Applied Element Method), where a smaller number of elements in modelling can be applied, saving time and resources. Buildings 2021, 11, 328 3 of 17

The accuracy of AEM was verified also in the case of thin plate elements, in which the effect of the Poisson’s ratio plays a key role [9,10]. Other researchers instead [11] introduced an extension for the multi-layered IAEM to model bonded prestressed concrete structures. In addition, a Gaussian-based AEM was also developed in order to obtain a low computational cost and a higher accuracy for progressive collapse analyses [12]. Recent applications of AEM addressed also the seismic behaviour of unreinforced masonry structures [2,13–18], and blast loads on masonry [19] and masonry infill walls [20]. The AEM was adopted for several progressive collapse assessments for precast prestressed reinforced concrete beams [21,22] and steel moment frames structures [23,24]. Moreover, simulations for the assessment of failure modes of reinforced concrete walls subjected to monotonic loads [25] were compared with experimental tests. Another successful application [26,27] was related to the investigation of the poten- tial reason for the Morandi bridge failure, which occurred in Genoa on 14 August 2018, while in [28], the collapse of the Tsuyagawa bridge, damaged by the Tohoku tsunami, was investigated. The AEM framework was also adopted in research studies focused on the robustness of steel storage pallet racks under accidental loading situations, such as collision with a forklift truck [29]. The outcomes highlighted that the most accurate results were obtained through AEM, if compared with the notional upright removal that is traditionally performed with FEM software. Even though the AEM was introduced decades ago, several aspects should still be investigated. Considering specifically the AEM application [29] to steel storage pallet racks, analysts may wonder whether all structure-specific aspects are well captured or not in the AEM framework. As can be found in the literature, steel storage pallet racks can suffer greatly from warping [30] and, jointly, second order effects [31,32]; these aspects could be an additional threat for steel storage rack safety since partial collapse mechanisms will be more likely to occur [33], thus increasing further the already complex task of researchers to understand and prevent failures and damages [34]. Although second order effects can be well simulated both with FEM and AEM formulations, warping needs more attention. In the case of FEM, the traditional beam-column element with six degrees of freedom (6DOFs) per node should be replaced with an advanced beam–column element with seven degrees of freedom (7DOFs) per node [30–32], in which the seventh DOF represents the warping. The warping effects lead to an overall higher flexibility, measurable with lower critical loads and higher fundamental periods, and also to an increase in the stresses, as can be gathered by the outcomes presented in Bernuzzi et al. [30]. Note that such an increase of stresses produces an additional internal force in the element, namely bimoment. In addition to this, rack uprights are often open cross-sections, in which the distance between the shear centre and the centroid is not negligible; the 7DOFs formulation permits to capture this distance, if any. As highlighted by Bernuzzi et al. [30], neglecting such an aspect can lead to unconservative results in terms of displacements, rotations and stresses. Furthermore, even if steel storage pallet rack members are one of the utmost examples of how impactful warping can be, the theory of thin-walled beams [35] is still general and applicable to any member, such as the commonly adopted steel I-shaped sections. Albeit that the FEM formulation with 7DOFs is the reference option for rack structures to capture the warping deformations, it cannot help researchers and engineers when investigating the response for particular cases, such as impact loadings, blast loadings and progressive collapse, if the most common implicit solver is adopted. Together with FEM formulations using an explicit solver [36–41], AEM, with its features, could be a novel option for investigations in these two research fields. The abovementioned literature [14,21–24,26–30] highlights the need for further investigations on both aspects (impact loads and warping effects). As a consequence, a formulation that is able to predict reliable results, integrating the two characteristics, would be advantageous. With the above in mind, this paper aims at investigating the actual capabilities of the AEM formulation in capturing warping effects to enable its applicability in the analysis of steel storage pallet racks. Buildings 2021, 11, 328 4 of 17

To this end, a custom AEM-based C# code capable of performing linear static analysis in a three-dimensional (3D) environment was developed. A representative warping-affected thin-walled C-shaped section was selected as the case study and subjected to a shear load in order to induce warping displacement and bimoment stresses. Finally, the AEM outcomes were compared with those of a FEM software [42] that features both traditional 6DOFs and reﬁned 7DOFs formulations.

2. Description of the Numerical Models To investigate the effectiveness of the AEM in simulating warping effects, a parametric analysis was performed. In particular, an open C-shaped steel section was selected, varying its web and ﬂanges thickness between 2 mm (Section #1), 4 mm (Section #2) and 8 mm (Section #3). This choice relies upon the fact that, as is well known from the literature, thin-walled open cross-sections are the most prone to exhibiting warping effects. The scope of sequencing the thickness in such a range is to cover a relatively wide part of the market supply for storage racking systems of this particular shape. The shared properties of the C-shaped sections are the height (h), equal to 100 mm and the width (w), equal to 80 mm. The sections are reported in Figure2, while additional information related to the geometric characteristics is provided in Table1. For all the models, a value of elastic modulus E = 210000 N/mm2 was assumed. It should be noted that plastic deformations are not considered since the aim of this study is to evaluate the applicability of AEM in capturing warping deformations, which can occur already in the elastic range, rather than computing the actual bearing capacity of the beams under investigation. In addition, thin- Buildings 2021, 11, x FOR PEER REVIEW 5 of 18 walled, open cross-sections often belong to Class 3 or Class 4, according to Eurocode 3 [43]; therefore, plastic deformations cannot be reached, due to the occurrence of local buckling.

FigureFigure 2. Selected2. Selected C-shaped C-shaped sections sections and their and discretisation. their discretisation.

Table 1. Geometrical properties of the selected sections.

Section #1 Section #2 Section #3 Height (mm) 100 100 100 Width (mm) 80 80 80 Thickness (mm) 2 4 8 Area (mm2) 520 1040 2080 Moment of Inertia, y (mm4) 9,666,666.67 1,933,333.33 3,866,666.67 Moment of Inertia, z (mm4) 367,589.74 735,179.49 1,470,358.97 Torsional inertia (mm4) 693.33 5546.67 44,373.33 Warping constant (mm6) 647,356,321.84 1,294,712,643.68 2,589,425,287.36 Shear centre, y (mm) −57.72 −57.72 −57.72

Table 1. Geometrical properties of the selected sections.

As per the structural scheme, a cantilevered beam was selected to maximise the warping effects. A constant shear load of 10 kN was applied to the free end of the beam. It can be noted that in order to apply the load at the centroid of the section, the load was distributed along each block element composing the free end, in which each load was scaled proportionally to its influence area. The direction of the load is the same as the asymmetric axes of the section (i.e., parallel to the web lengthwise); in this way, a coupled flexural–torsional behaviour is expected. Moreover, several beam lengths were used, ranging from 100 mm to 1000 mm. Furthermore, the parametric study also comprised several numerical discretisation schemes. Firstly, the influence of the refinement of the mesh along the length, i.e., in the longitudinal direction of the beam, was investigated. In this direction, the beam was Buildings 2021, 11, x FOR PEER REVIEW 6 of 18 subdivided into 20, 60, 100, 140 and 180 block elements. As an additional parameter to be investigated, the discretisation along the section was taken and varied as described hereinafter. The web and flanges were subdivided into 10, 20 and 30 block elements. With providingthese modelling insight approaches, into the numerical a large numbermodelling of analysesdiscretisation (450 in influence total) were for performed,a broad variety pro- ofviding cases. insight For comparative into the numerical purposes, modelling the same discretisation beams were also influence modelled for a with broad a varietyfinite ele- of mentcases. software For comparative [42] capable purposes, of performing the same analyses beams were with also both modelled 6DOFs and with 7DOFs a finite formula- element tions.software Experimental [42] capable tests of performingcould also be analyses added to with better both appraise 6DOFs the and accuracy 7DOFs formulations. of the differ- entExperimental numerical methods tests could adopted. also be However, added to bettersince the appraise 7DOFs the formulation accuracy ofis a the well-estab- different lishednumerical approach methods [35,44–46] adopted. and However, several experiment since the 7DOFsal full-scale formulation tests on is steel a well-established storage racks [47–49]approach have [35 ,confirmed44–46] and its several accuracy, experimental the 7DOFs full-scale formulation tests on was steel taken storage as the racks reference [47–49] methodhave confirmed for the numerical its accuracy, investigation the 7DOFs formulationproposed in wasthis takenstudy. as the reference method for the numericalThe numerical investigation analyses proposedfocused on in the this evalua study.tion of the vertical displacement of the free endThe of numerical the beamanalyses and the normal focused stresses, on the evaluation which develop of the at vertical the fixed displacement end. Indeed, of due the tofree the end coupling of the beam of flexural and the and normal torsional stresses, behaviour, which develop the free at end the not fixed only end. translates Indeed, duebut alsoto the rotates coupling about of the flexural shear and centre, torsional increasing behaviour, consequently, the free end the not vert onlyical translatesdisplacement. but also As anrotates example, about the the deformation shear centre, predicted increasing by consequently,adopting the AEM the vertical model displacement.is represented Asin an Figureexample, 3. the deformation predicted by adopting the AEM model is represented in Figure3.

FigureFigure 3. Deformation of of the beam element under shea shearr force with the AEM model (longitudinal discretisation equal to 180 and transversal discretisation of 10). It can be noted the typical coupled discretisation equal to 180 and transversal discretisation of 10). It can be noted the typical coupled flexural–torsional behaviour. ﬂexural–torsional behaviour. Such a behaviour can be accurately captured with a 7DOFs formulation. On the contrary, a traditional 6DOFs formulation will simulate only the flexural behaviour; therefore, only the translational displacement will be provided. Moreover, as already mentioned, the warping in the section causes additional normal stresses, which lead to the onset of the bimoment internal force. Additionally, in this case, the 6DOFs formulation cannot estimate such effects.

3. Numerical Analyses As mentioned above, 450 analyses were performed, varying the section thickness, the beam length and the beam and section discretisation. The outcomes are reported in the following subparagraphs in terms of vertical displacement, normal stresses and bimoment.

3.1. Vertical Displacement The first key parameter for evaluating the AEM efficiency in capturing the warping effect is the vertical displacement of the centroid at the free end of the beam. As C-shaped sections rotate about their shear centre instead of their centroid, when a shear load is applied, a coupled flexural–torsional behaviour is generally observed. The torsional rotation about the shear centre increases the total vertical displacement. Therefore, the outcomes of analyses performed with a 6 and 7DOFs formulations will inevitably be different since the traditional 6DOFs formulation is not able, by definition, to simulate the torsional rotation in a suitable manner. Since the AEM formulation has been broadly validated in terms of flexural displacement, the results herein presented will focus on the estimation of the torsional rotation. Buildings 2021, 11, 328 6 of 17

Such a behaviour can be accurately captured with a 7DOFs formulation. On the contrary, a traditional 6DOFs formulation will simulate only the ﬂexural behaviour; therefore, only the translational displacement will be provided. Moreover, as already mentioned, the warping in the section causes additional normal stresses, which lead to the onset of the bimoment internal force. Additionally, in this case, the 6DOFs formulation cannot estimate such effects.

3. Numerical Analyses As mentioned above, 450 analyses were performed, varying the section thickness, the beam length and the beam and section discretisation. The outcomes are reported in the Buildings 2021, 11, x FOR PEER REVIEWfollowing subparagraphs in terms of vertical displacement, normal stresses and bimoment.7 of 18

3.1. Vertical Displacement The first key parameter for evaluating the AEM efficiency in capturing the warping effectThe is the set verticalof sub-plots displacement presented of in the Figure centroid 4 shows at the the free vertical end of displacement the beam. As ofC-shaped Section #1sections (100 × 80 rotate × 2). aboutThe presented their shear curves centre exem insteadplify the of theirobtained centroid, displacement when a when shear increas- load is ingapplied, the beam a coupled length,flexural–torsional from 100 mm to 1000 behaviour mm, for is the generally different observed. formulations The (6DOFs torsional in thickerrotation red, about AEM the in shear colour centre scale increases red-to-blue the and total 7DOFs vertical in displacement.thicker blue) and Therefore, for the dif- the ferentoutcomes mesh of longitudinal analyses performed discretisation with opti a 6ons and for 7DOFs the AEM formulations model (as will illustrated inevitably in the be legend).different since the traditional 6DOFs formulation is not able, by definition, to simulate the torsionalIn Figure rotation 4a, the inanalysis a suitable outcomes manner. are Since depicted the AEM for the formulation section discretisation has been broadly of 10 blockvalidated elements in terms for ofeach flexural section displacement, part (i.e., web the and results flanges). herein As presentedcan be seen, will when focus a length on the ofestimation 100 mm ofis theconsidered, torsional for rotation. any longitudinal discretisation level, the AEM formulation providesThe sethigher of sub-plots results, i.e., presented −0.02 mm, in Figure −0.1 4mm shows and the −3.11 vertical mm for displacement the 6DOFs, of 7DOFs Section and #1 AEM(100 × (discr80 × 2).= 180), The presentedrespectively. curves Contrastingl exemplifyy, the when obtained a length displacement of 1000 mm when is increasingtaken, the displacementsthe beam length, are from −16.42 100 mm, mm − to86.57 1000 mm mm, and for − the54.14 different mm for formulations the 6DOFs, (6DOFs7DOFs and in thicker AEM (discrred, AEM = 180), in colour showing scale a stiffer red-to-blue response and 7DOFsof the AEM in thicker in comparison blue) and for with the the different 7DOFs mesh formulation.longitudinal discretisation options for the AEM model (as illustrated in the legend).

Figure 4. VerticalVertical displacement–length relation for Section #1 with the different discretisationdiscretisation conﬁgurations.configurations.

Nevertheless,In Figure4a, thethe analysis beneficial outcomes effect of are a depictedhigher longitudinal for the section discretisation discretisation can ofbe 10clearly block appraised elements forsince each the section displacement part (i.e., increases web and ﬂanges).from −17.60 As canmm be for seen, the whenAEM a(discr length = 20)of 100 to − mm54.14 is mm considered, in the case for of any AEM longitudinal (discr = 180), discretisation both considering level, the a length AEM formulationof 1000 mm. Similarprovides considerations higher results, can i.e., be− made0.02 mm,for the− 0.1counterparts mm and − with3.11 a mm section for thediscretisation 6DOFs, 7DOFs of 20 (Figureand AEM 4b) (discr and 30 = 180),(Figure respectively. 4c) block elements. Contrastingly, However, when it a should length ofbe 1000noted mm that is with taken, a refinedthe displacements section discretisation, are −16.42 the mm, outcomes−86.57 ar mme closer and −to54.14 the theoretical mm for the response 6DOFs, (7DOFs) 7DOFs sinceand AEM the displacement (discr = 180), values showing, for athe stiffer 1000 responsemm length, of theare AEM−67.51 inmm comparison and −71.25 with mm thefor the7DOFs section formulation. discretisation 20 and 30, respectively, for the AEM (discr = 180). ConcerningNevertheless, Section the beneﬁcial #2, the vert effectical of displacement–length a higher longitudinal relation discretisation is reported can bein clearlyFigure 5.appraised With respect since to the the displacement former section, increases the displacements from −17.60 for mm lower for lengths the AEM are (discr highly = 20)reduced; also, the overall trend of the curve benefits from such a section thickness increment. In particular, the displacements at L = 100 mm with a longitudinal discretisation of 180 are −0.82 mm, −1.16 mm and −1.25 mm for the section discretisation 10, 20 and 30, respectively, while the 6 and 7DOFs formulations provide −0.01 mm and −0.05 mm. Con- sidering the same comparison for a length L = 1000 mm, the displacements are −23.98 mm, −28.76 mm and −29.88 mm for the different section discretisation options, whereas the 6 and 7DOFs formulations result in −8.21 mm and −32.90 mm. Buildings 2021, 11, 328 7 of 17

to −54.14 mm in the case of AEM (discr = 180), both considering a length of 1000 mm. Similar considerations can be made for the counterparts with a section discretisation of 20 (Figure4b) and 30 (Figure4c) block elements. However, it should be noted that with a reﬁned section discretisation, the outcomes are closer to the theoretical response (7DOFs) since the displacement values, for the 1000 mm length, are −67.51 mm and −71.25 mm for the section discretisation 20 and 30, respectively, for the AEM (discr = 180). Concerning Section #2, the vertical displacement–length relation is reported in Figure5. With respect to the former section, the displacements for lower lengths are highly reduced; also, the overall trend of the curve beneﬁts from such a section thickness increment. In particular, the displacements at L = 100 mm with a longitudinal discretisation of 180 are −0.82 mm, −1.16 mm and −1.25 mm for the section discretisation 10, 20 and 30, respectively, while the 6 and 7DOFs formulations provide −0.01 mm and −0.05 mm. Consid- ering the same comparison for a length L = 1000 mm, the displacements are −23.98 mm, Buildings 2021, 11, x FOR PEER REVIEW 8 of 18 −28.76 mm and −29.88 mm for the different section discretisation options, whereas the 6 and 7DOFs formulations result in −8.21 mm and −32.90 mm.

FigureFigure 5. 5. VerticalVertical displacement–length displacement–length relation relation for Section #2 with the different discre discretisationtisation conﬁgurations.configurations.

Finally,Finally, Section Section #3 #3 shows an even closer match between the theoretical solution (7DOFs)(7DOFs) and and the the AEM, AEM, mainly mainly with with the the higher higher mesh mesh discretisation discretisation refinement reﬁnement (Figure (Figure 6).6). ForFor a a longitudinal longitudinal discretisation discretisation of 100 100 block block elements elements and and for for all all the the section section discretisation discretisation levels,levels, the the outcomes outcomes are are very very similar similar and and tend tend to to be be satisfactory satisfactory from from a a numerical numerical valida- valida- tiontion point point of of view. view. It It is worth noting that the initial displacements for the shorter length (L(L = = 100 100 mm) mm) areare reduced reduced to to −−0.160.16 mm, mm, −0.18−0.18 mm mm and and −0.18−0.18 mm mmfor AEM for AEM (discr (discr = 180) = and 180) alland the all section the section discretisation discretisation options, options, respectively, respectively, while while 6DOFs 6DOFs and 7DOFs and 7DOFs formulations formula- providetions provide 0 mm 0and mm −0.02 and −mm.0.02 Taking mm. Taking into account into account the larger the larger length length (L = 1000(L = mm), 1000 mm),a com-a parisoncomparison of the of same the same kind kind leads leads to values to values of −8.99 of − mm,8.99 mm,−9.91− mm9.91 and mm − and10.09− mm10.09 for mm AEM for andAEM −4.11 and mm−4.11 and mm −9.82 and mm−9.82 for mm6DOFs for and 6DOFs 7DOFs. and 7DOFs. For the sake of brevity, the displacement values for all the discretisation conﬁgurations and sections are reported in Table2, only for the longer length (i.e., L = 1000 mm).

Figure 6. Vertical displacement–length relation for Section #3 with the different discretisation configurations.

For the sake of brevity, the displacement values for all the discretisation configurations and sections are reported in Table 2, only for the longer length (i.e., L = 1000 mm).

Table 2. Centroid vertical displacement (mm) for the three considered sections with different configurations and discretisation levels (L = 1000 mm).

Section Long. Discretisation Section Discretisation AEM 6DOFs 7DOFs 20 −17.60 60 −34.34 100 10 −45.07 #1 −16.42 −86.57 140 −50.89 180 −54.14 20 20 −17.76 Buildings 2021, 11, x FOR PEER REVIEW 8 of 18

Figure 5. Vertical displacement–length relation for Section #2 with the different discretisation configurations.

Finally, Section #3 shows an even closer match between the theoretical solution (7DOFs) and the AEM, mainly with the higher mesh discretisation refinement (Figure 6). For a longitudinal discretisation of 100 block elements and for all the section discretisation levels, the outcomes are very similar and tend to be satisfactory from a numerical valida- tion point of view. It is worth noting that the initial displacements for the shorter length (L = 100 mm) are reduced to −0.16 mm, −0.18 mm and −0.18 mm for AEM (discr = 180) and all the section discretisation options, respectively, while 6DOFs and 7DOFs formulations Buildings 2021, 11, 328 provide 0 mm and −0.02 mm. Taking into account the larger length (L = 1000 mm), a 8com- of 17 parison of the same kind leads to values of −8.99 mm, −9.91 mm and −10.09 mm for AEM and −4.11 mm and −9.82 mm for 6DOFs and 7DOFs.

FigureFigure 6. 6. VerticalVertical displacement–length relation for Section #3 with the different discretisationdiscretisation conﬁgurations.configurations.

For the sake of brevity, the displacement values for all the discretisation configura- Table 2. Centroid vertical displacement (mm) for the three considered sections with different tions and sections are reported in Table 2, only for the longer length (i.e., L = 1000 mm). conﬁgurations and discretisation levels (L = 1000 mm).

Table 2. Centroid vertical displacement (mm) for the three considered sections with different con- Section Long. Section AEM 6DOFs 7DOFs figurations and discretisatDiscretisationion levels (L =Discretisation 1000 mm). 20 −17.60 Section Long. Discretisation60 Section Discretisation− 34.34 AEM 6DOFs 7DOFs 20 100 10 −45.07−17.60 140 −50.89 60 180 −54.14−34.34 100 20 10 −17.76−45.07 #1 60 −36.98 −16.42 −86.57 #1 140 100 20 −52.03−50.89 −16.42 −86.57 180 140 −61.58−54.14 180 −67.51 20 20 20 −17.81−17.76 60 −37.58 100 30 −53.74 140 −64.41 180 −71.25 20 −8.46 60 −15.95 100 10 −20.44 140 −22.74 180 −23.98 20 −8.54 60 −17.05 #2 100 20 −23.15 −8.21 −32.90 140 −26.70 180 −28.76 20 −8.52 60 −17.20 100 30 −23.69 140 −27.58 180 −29.88 20 −3.92 60 −6.74 100 10 −8.08 140 −8.69 180 −8.99 20 −3.92 60 −7.02 #3 100 20 −8.69 −4.11 −9.82 140 −9.49 180 −9.91 20 −3.91 60 −7.04 100 30 −8.79 140 −9.64 180 −10.09 Buildings 2021, 11, 328 9 of 17

3.2. Normal Stress Distribution and Bimoment The second key parameter under investigation is the normal stress developed in the ﬁxed end of the cantilever beam and the resulting internal bimoment force. To do so, two different approaches were adopted distinguishing the cases of 7DOFs and the AEM framework. In the case of 7DOFs, the software provides the internal bimoment force acting at the restrained end of the cantilever. As a consequence, to also obtain the stresses acting at the various locations of the section, it is necessary to compute the sectorial coordinates of each selected section point. Then, Equation (3) was adopted:

B σi = · ω(s)i (3) Iw

where B is the bimoment, Iw is the warping constant, ω(s) is the sectorial coordinate and σ is the normal stress. The pedix i denotes the ith point. On the contrary, in the case of the AEM, the outputs provided by the C# code are the stresses acting at each spring. As such, the procedure works exactly in the opposite direction than the previous approach. Indeed, the bimoment is computed, summing up the stress contribution of each spring times the sectorial coordinate of each spring location since, from a theoretical point of view, the bimoment is computed following Equation (4). Z B = σ · ω(s) · dA (4) A where A is the area of the section. In the case of the 6DOFs formulation, the warping and, hence, the bimoment, cannot be computed; accordingly, the stress distribution is affected only by the developed flexural moment. It can be seen in Figure7a that the stress distribution, for Section #1 and the 6DOFs case, is constant along the flanges, while it assumes a linear variation along the web, with peaks at the ends of the web and a null value in the middle. On the contrary, if the 7DOFs formulation is considered (Figure7b), a linear variation can be appraised both along the flanges and the web. It can be noted also that the bimoment increases considerably the acting stresses since the maximum peaks, in absolute terms, are 517 MPa and 1819 MPa for the 6DOFs and 7DOFs formulations, respectively. Regarding the numerical simulation with the AEM, it can be pointed out that the stress distribution is well captured since in all the distribution cases (Figure7c–f), a linear variation can be observed. However, satisfactory results can be appreciated only in the case of a longitudinal discretisation consisting of 180 block elements; considering the absolute peak value, a stress value equal to 1613 MPa is computed, which should be compared with the 1819 MPa of the 7DOFs formulation. In any case, all the discretisation configurations of the AEM models lead to more accurate and conservative results if they are compared with a traditional 6DOFs formulation. Focusing on the internal forces (illustrated in Figure7), the results in terms of the flexural moment are accurate for every discretisation configuration. Regarding the bimoment, instead, following the same considerations highlighted for the displacement and stress distributions, quite an accurate value can be observed only in the case of a longitudinal discretisation of 180 block elements (i.e., B = −0.4018 kNm2 and B = −0.5091 kNm2 for AEM and 7DOFs formulations, respectively). It is also worth mentioning that the percentagewise differences are up to 61% for the longitudinal discretisation of 60 blocks and about 21% for the longitudinal discretisation of 180 blocks. In the case of Section #2 and Section #3, the normal stress distributions are reported in Figures8 and9. Buildings 2021, 11, 328 10 of 17 Buildings 2021, 11, x FOR PEER REVIEW 11 of 18

Figure 7. Normal stress distribution for Section #1, with a length L = 1000 mm and a section discreti- Figure 7. Normal stress distribution for Section #1, with a length L = 1000 mm and a section discretisation of 30 blocks for sation of 30 blocks for each section part. each section part. However, satisfactory results can be appreciated only in the case of a longitudinal discretisation consisting of 180 block elements; considering the absolute peak value, a stress value equal to 1613 MPa is computed, which should be compared with the 1819 MPa of the 7DOFs formulation. In any case, all the discretisation configurations of the AEM models lead to more accurate and conservative results if they are compared with a traditional 6DOFs formulation. Focusing on the internal forces (illustrated in Figure 7), the results in terms of the flexural moment are accurate for every discretisation configuration. Regarding the bimoment, instead, following the same considerations highlighted for the displacement and stress distributions, quite an accurate value can be observed only in the case of a longitudinal discretisation of 180 block elements (i.e., B = −0.4018 kNm2 and B = −0.5091 kNm2 for AEM and 7DOFs formulations, respectively). It is also worth mentioning that the percentagewise differences are up to 61% for the longitudinal discretisation of 60 blocks and about 21% for the longitudinal discretisation of 180 blocks. Buildings 2021, 11, x FOR PEER REVIEW 12 of 18

Buildings 2021, 11, 328 11 of 17 In the case of Section #2 and Section #3, the normal stress distributions are reported in Figures 8 and 9.

FigureFigure 8. Normal 8. Normal stress distribution stress distribution for Section for #2, Section with a length#2, with L =a 1000length mm L and= 1000 a section mm and discretisation a section discreti- of 30 blocks for each sectionsation part.of 30 blocks for each section part. Buildings 2021, 11, 328 12 of 17 Buildings 2021, 11, x FOR PEER REVIEW 13 of 18

Figure 9. Normal stress distribution for Section #3, with a length L = 1000 mm and a section discreti- Figure 9. Normal stress distribution for Section #3, with a length L = 1000 mm and a section sation of 30 blocks for each section part. discretisation of 30 blocks for each section part. FollowingFollowing thethe samesame method of interpreting interpreting th thee results, results, it it can can be be noted noted that, that, overall, overall, thethe AEM AEM simulationssimulations andand thethe 7DOFs7DOFs formulationformulation become become closer. closer. Indeed, Indeed, the the maximum maximum absoluteabsolute stress stress peakpeak forfor SectionSection #2 (discretisation 180, 180, Figure Figure 8f)8f) is equal equal to to 749 749 MPa, MPa, while while forfor thethe theoreticaltheoretical solution, solution, the the stress stress is is752 752 MPa. MPa. The The counterparts counterparts for Section for Section #3 are #3 302 are 302MPa MPa (Figure (Figure 9f)9 andf) and 271 271 MPa MPa (Figure (Figure 9b).9b). Additionally, Additionally, in in terms terms of of the the bimoment, bimoment, the the AEMAEM isis ableable toto accuratelyaccurately capturecapture the warping effects. effects. Section Section #2 #2 and and Section Section #3 #3 show show a a bimomentbimoment equal to to −−0.38560.3856 kNm kNm2 and2 and −0.2222−0.2222 kNm kNm2, respectively;2, respectively; comparing comparing such a result such a resultwith the with best the results best results of the ofAEM, the AEM,a very asati verysfactory satisfactory agreement agreement is obtained is obtained since the since com- the computedputed bimoment bimoment forces forces are − are0.3507−0.3507 kNm kNm2 and 2−0.2341and − 0.2341kNm2. kNm2.

4.4. DiscussionDiscussion ofof thethe Results InIn commoncommon practice,practice, the warping effects effects are are often often ignored ignored because because only only a afew few com- com- mercialmercial softwaresoftware cancan predictpredict such a phenomenon.phenomenon. The The AEM AEM is is a apromising promising methodology methodology that can help engineers and researchers to assess the response of complex structures, incor- porating structure-speciﬁc behavioural aspects in the simulation. The results shown in the Buildings 2021, 11, x FOR PEER REVIEW 14 of 18

Buildings 2021, 11, 328 13 of 17 that can help engineers and researchers to assess the response of complex structures, in- corporating structure-specific behavioural aspects in the simulation. The results shown in the previous sectionprevious enable section a first enable evaluation a ﬁrst evaluation of warping of effects warping in effectsthe case in that the casean AEM that an AEM formu- formulation is employed.lation is employed. As clearly As shown clearly in shown previous in previoussections, sections,and also andherein also summa- herein summarised in rised in Figure Figure10, the 10 AEM, the AEMcan take can takeinto intoaccount account the thewarping warping effects effects when when a asuitable suitable discretisation, discretisation, bothboth in in the the longitudinal longitudinal direction direction and and at the at the section-level, section-level, is applied. is applied. Figure Figure 10 depicts, 10 depicts, for thefor case the case of 1000 of 1000 mm mm of length, of length, the thestress stress peaks peaks obtained obtained with with 6DOFs 6DOFs and and AEM, both AEM, both normalisednormalised to the to stress the stress peak peak derived derived from fromthe 7DOFs the 7DOFs formulation. formulation. As a gen- As a general trend eral trend for thesefor theseparticular particular cases, cases, satisfac satisfactorytory results results are obtained are obtained when when the section the section dis- discretisation cretisation reachesreaches at least at least 20 blocks, 20 blocks, while while the longitudinal the longitudinal one onereaches reaches 140 blocks. 140 blocks. This This considera- consideration highlightstion highlights the fact the that fact an thatAEM an formulation AEM formulation can be adopted can be adoptedif detailed if mod- detailed models are els are developed.developed. If not, simila If not,r, albeit similar, always albeit slightly always bette slightlyr, the better, outcomes the outcomes of traditional of traditional 6DOFs 6DOFs formulationformulation are expected. are expected. In any case, In anythe advantageous case, the advantageous aspects of aspectsthe AEM, of e.g., the AEM, e.g., the the capability ofcapability simulating of simulatingthe post-collaps the post-collapsee response, still response, remain, still and remain, users/analysts and users/analysts can can certainly makecertainly use of make all of use those of allAEM-specific of those AEM-speciﬁc features. features.

Figure 10. FigureDistribution 10. Distribution of the normalised of the normalised stress peaks stress to peaks stress to peaksstress basedpeaks based on the on 7DOFs the 7DOFs formulation. formula- The results tion. The results correspond to the case with a length of 1000 mm, varying, in the case of AEM, the correspond to the case with a length of 1000 mm, varying, in the case of AEM, the section discretisation (SD10, SD20 and section discretisation (SD10, SD20 and SD30) and longitudinal discretisation (LD20, LD60, LD100, SD30) and longitudinal discretisation (LD20, LD60, LD100, LD140 and LD180). LD140 and LD180). Moreover, it should be noted that, for example, if the case of Section #1 (Figure4) is considered, a higher displacement is captured for any discretisation for the small-length cases. Buildings 2021, 11, x FOR PEER REVIEW 15 of 18

Buildings 2021, 11, 328 14 of 17 Moreover, it should be noted that, for example, if the case of Section #1 (Figure 4) is considered, a higher displacement is captured for any discretisation for the small-length Thiscases. particular This particular phenomenon phenomenon is represented is represen in Figureted in 11 Figure in which 11 in the which case oftheSection case of #1 Sectionwith a#1 length with ofa 100length mm of is 100 depicted. mm is Asdepicted. can be seenAs can therein, be seen the increasetherein, the in the increase displacement in the dis- of theplacement section centroidof the section is due centroid to the local is due deformation to the local of deformation the ﬂanges of of the the C-section flanges of and, the inC- fact,section the localand, deformationsin fact, the local cannot deformations be simulated cannot with 6DOFsbe simulated or 7DOFs with formulations. 6DOFs or 7DOFs Such aformulations. phenomenon Such is more a phenomenon pronounced is for more Section pronounced #1 and tends for Section to disappear #1 and for tends Sections to disap- #2 andpear #3, for where Sections the #2 section and #3, thickness where the is increased. section thickness Clearly, is the increased. slenderness Clearly, of the the web slender- and ﬂangesness of playsthe web a fundamental and flanges role;plays reducing a fundamenta the slendernessl role; reducing (i.e., increasing the slenderness the thickness) (i.e., in- leadscreasing to stiffer the thickness) responses leads at the to sectional stiffer response level. Thiss at interesting the sectional aspect level. opens This the interesting possibility as- ofpect adopting opens AEMthe possibility for predicting of adopting distortional AEM and for localpredicting buckling distortional modes of and failure. local However, buckling furthermodes researchof failure. is However, needed in further this direction. research is needed in this direction.

Figure 11. Local deformation of the flanges for Section #1, length = 100 mm, section discretisation 30 Figure 11. Local deformation of the ﬂanges for Section #1, length = 100 mm, section discretisation 30 and longitudinal discretisation of 60. and longitudinal discretisation of 60.

5.5. Conclusions Conclusions ThisThis study study exploredexplored thethe capabilitiescapabilities ofof thethe Applied Element Method (AEM) (AEM) in in simu- sim- ulatinglating the the warping warping effects effects in in thin-walled thin-walled C-shaped C-shaped steel steel sections. sections. The The AEM AEM has has been been ex- extensivelytensively validated validated for for several several structural structural typology typology and and materials. materials. However, However, no no investiga- investi- gationstions have have been been conducted conducted in inthe the field field of ofthin-walled thin-walled sections, sections, particularly particularly concerning concerning the thewarping warping effects. effects. With With the the aim aim of of contributing contributing to to this this gap, anan AEM-basedAEM-based program—program— throughthrough C# C# programming—was programming—was developed developed to to conduct conduct static static analysis. analysis. A A cantilever cantilever beam beam andand a a C-shaped C-shaped section section were were selected selected as as the the case case study. study. A A parametric parametric analysis analysis was was then then performed,performed, varyingvarying bothboth thethe geometricalgeometrical properties,properties, i.e.,i.e., thethe beambeam lengthlength andand section section thickness,thickness,and and the the discretisation discretisation configuration configuration adopted adopted in in the the AEM AEM models. models. In In order order to to exciteexcite the the warping warping effects, effects, i.e., i.e., the the torsional torsional rotation rotation and theand warping the warping displacement, displacement, a shear a loadshear acting load acting along thealong asymmetric the asymmetric axis was axis applied. was applied. For theFor purposesthe purposes of comparison of comparison of theof the numerical numerical efficiency, efficiency, the the displacement displacement along along the the load load direction, direction, the the bimoment bimoment and and thethe normal normal stress stress distribution distribution were were selected selected as as the the key key parameters. parameters. The The results results that that were were obtainedobtained enable enable the the following following concluding concluding remarks remarks to to be be made: made: -- TheThe AEMAEM frameworkframework predictspredicts thethe warpingwarping effects with sufficientsufficient accuracy if a a suita- suit- ableble discretisation level (at least 2020 andand 140140 blocksblocks forfor thethe section section and and longitudinal longitudinal discretisation,discretisation, respectively)respectively) isis adopted.adopted. ThisThis resultresult suggestssuggests thatthat ifif thethe scope scope of of the the analysisanalysis is is only only to to capture capture warping warping effects, effects, the the FEM FEM approach approach with with 7DOFs 7DOFs is is still still the the bestbest optionoption inin terms of the results results and and co computationalmputational onus. onus. On On the the other other hand, hand, in the in thecase case of impact of impact loads, loads, blast blast loads loads or collapse or collapse analyses, analyses, only FEM, only with FEM, explicit with explicit solvers, solvers, and AEM can be adopted. In addition, the AEM framework was proven to be satisfactorily accurate when simulating warping effects. Further investigations are needed to compare impact loads, blast loads and collapse analyses, adopting Buildings 2021, 11, 328 15 of 17

FEM with an explicit solver and AEM in order to assess the advantages of the two frameworks in terms of accuracy and computational effort; - Considering the academic and research environment, the AEM framework is currently adequate to analyse full steel storage pallet rack models; moreover, it permits to analyse also model portions and to replicate experimental tests and structure- specific behavioural aspects, such as beam-to-column connections, base–plate connections, pallet–structure interactions, upright holes and perforations influence, forklift hits, etc.; - Future work might include the modelling of distortional mechanisms and local buckling of thin-walled sections. Currently, these phenomena can be accurately captured adopting plate/solid elements in the case of a finite element method, or by using the strip method. The advantage of simulating all the peculiarities affecting thin- walled elements in a unique tool is, nevertheless, very useful for researchers and engineering practitioners.

Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: The authors would like to thank the editor and the three anonymous reviewers who kindly reviewed the paper providing significant comments and suggestions. Conflicts of Interest: The author declares no conflict of interest.

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