buildings 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 simu- late 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 ∙∙ ∙∙ = ; = (1) 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): 2 3 Kn −Kn·dy1 −Kn Kn·dy2 ∙ ∙ 6 Ks ⎡ Ks·dx1 −Ks −Ks·dx2 7 ⎤ 6 2 2 ∙ ∙ 7 6 −Kn·d Ks·d ⎢ Ks·d + Kn·d Kn·d −Ks·d −Kn·d ·d − Ks·d ·d 7 ⎥ 6 y1 x1 x1 y1 y1 x1 y1 y2 x1 x2 7 KE = ⎢ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙(2)⎥(2) 6 − = · − · 7 6 Kn ⎢ Kn dy1 ∙ Kn Kn dy2 ∙ 7 ⎥ 6 ⎢ 7 ⎥ −Ks −Ks·dx1 Ks Ks·dx2 4 ⎢ ∙ ∙ 5 ⎥ · − · − · · − · · − · · · 2 + · 2 Kn dy2 Ks dx⎣2 ∙Kn dy1 d∙y2 Ks d∙x1d∙x2 K n∙dy2∙ Ks dx∙2 ∙Ks dx2 Kn ∙dx2 ∙ ⎦ 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