Mechanical Modelling of the Strength and Stiffness of Circular Hollow Section Tube under Localised Transverse Compression and Tension

The component method is a powerful tool for designing and modelling steel beam-to-column connections. Its widespread use is ensured by several formulations currently included in Eurocode 3 part 1.8 for welded and bolted joints. However, the recent use of 3D Laser Cutting Technology (3D-LCT) in the construction market has enlarged the range of solutions, allowing the realisation of tubular columns with passing-through elements. Given the recent development, no design formulations are currently provided for this typology. At this moment, only a few research studies have developed to fill this knowledge gap. At the University of Salerno, since some years, research efforts are ongoing to characterise the flexural strength of connections between Circular Hollow Section columns and passing double-tee beams, suggesting methodologies to predict the behaviour of the resistance and stiffness of this typology and some of its elementary joint components. Within this framework, this paper aims to examine the strength and stiffness of one of the main components of this joint, which was never examined previously, that is the so-called tube under localised transverse tension/compression. Design formulations are derived from a parametric study carried out through numerical simulations of several geometric configurations.


Introduction
One of the most common ways to design seismic-resistant steel structures is by adopting Moment Resisting Frames (MRFs). They are systems conceived to withstand horizontal loads through the flexural behaviour of structural members (i.e., beams, columns and connections). As highlighted by Astaneh [1], several typologies of MRFs can be individuated in practice, differing for the three-dimensional layout, the connection typology, the ductility class, the selection of the dissipative zone.
Concerning the spatial distribution, MRFs can be classified as space frames and perimeter frames. In the first case, 3D frames withstand vertical and horizontal loads, which is very effective from the structural point of view because it reduces the beam and column sizes, limiting, at the same time, the forces in the nodal components (welds and bolts). Nevertheless, this approach requires a significant manufacturing effort since several complex beam-to-column connections have to be fabricated. Conversely, perimeter-framed buildings are characterised by MRFs located on the sides, while the remaining frames exhibit pendular behaviour. In this case, only few connections are designed to sustain the horizontal loads with low manufacturing costs. Nevertheless, typically deeper beams and columns are required.
Although technical requirements can inspire the choice of adopting one of the MRF typologies previously summarised, the selection also depends on the construction tradition of different countries. For instance, one-way perimeter MRFs are widely exploited in Europe and the United States due to their ability to clearly identify and design the seismic and gravity frames and limit the use of expensive joints only in MRFs. Conversely, space frames are preferred, for instance, in the Japanese tradition. The relevant consequence of this choice is the need to select circular or square tubular columns since they are able to provide strength and stiffness along both the main axes. Even though this solution requires a relevant number of joints, it has the beneficial effect of optimising the design of the beams limiting their depth and, consequently, the oversizing of brittle nodal components. Nevertheless, the main drawback of this practice is the need to manufacture complex beamto-column connections [2][3][4][5][6][7][8][9][10], usually characterised by the use of collar plates and stiffeners.
In this framework, the recent introduction of 3D Laser Cutting Technology (3D-LCT) in Civil Engineering provides a way to simplify the fabrication of complex configurations and exploit tubular columns more easily in the European construction market. An example of this simplification is the possibility to easily realise Circular Hollow Section (CHS) columns with passing-through double-tee beam connections [11,12] (Figure 1). However, the novelty of this solution represents a limit to its application in practice because no design equations are included in common standards. In fact, the current draft of Eurocode 8 does not provide design equations for these typologies or extends formulas which result in being overconservative [11][12][13].
Although technical requirements can inspire the choice of adopting one of the MRF typologies previously summarised, the selection also depends on the construction tradi tion of different countries. For instance, one-way perimeter MRFs are widely exploited in Europe and the United States due to their ability to clearly identify and design the seismi and gravity frames and limit the use of expensive joints only in MRFs. Conversely, spac frames are preferred, for instance, in the Japanese tradition. The relevant consequence o this choice is the need to select circular or square tubular columns since they are able to provide strength and stiffness along both the main axes. Even though this solution re quires a relevant number of joints, it has the beneficial effect of optimising the design o the beams limiting their depth and, consequently, the oversizing of brittle nodal compo nents. Nevertheless, the main drawback of this practice is the need to manufacture com plex beam-to-column connections [2][3][4][5][6][7][8][9][10], usually characterised by the use of collar plate and stiffeners.
In this framework, the recent introduction of 3D Laser Cutting Technology (3D-LCT in Civil Engineering provides a way to simplify the fabrication of complex configuration and exploit tubular columns more easily in the European construction market. An exam ple of this simplification is the possibility to easily realise Circular Hollow Section (CHS columns with passing-through double-tee beam connections [11,12] (Figure 1). However the novelty of this solution represents a limit to its application in practice because no de sign equations are included in common standards. In fact, the current draft of Eurocode does not provide design equations for these typologies or extends formulas which resul in being overconservative [11][12][13]. Recently, research works have started to fill this knowledge gap. For instance, th LASTEICON research project [14][15][16][17], financed by the European Commission, has studied the technical aspects related to the use of the 3D-LCT in Civil Engineering. Furthermore attention has been devoted to defining design equations for exploiting the CHS column and through I-beam joints in the construction market. In order to complement this work some studies are ongoing at the University of Salerno. In particular, experimental, numer ical and theoretical activities have been carried out to extend the component method to these typologies [18][19][20][21]. With this goal, the main components of this joint have been iden tified in [22] (Figure 2): pct/pcc are the components representative of the behaviour in tension and compres sion of the attachment between the tubular profile and the flange plates of the beam; ttt/ttc are sources of deformability ascribed to the possible transverse tension or com pression actions transmitted by the flanges to the CHS and induced by the beam's rotation cs and bws represent the parts of the tubular profile and the beam web subjected to the horizontal shear. Recently, research works have started to fill this knowledge gap. For instance, the LASTEICON research project [14][15][16][17], financed by the European Commission, has studied the technical aspects related to the use of the 3D-LCT in Civil Engineering. Furthermore, attention has been devoted to defining design equations for exploiting the CHS column and through I-beam joints in the construction market. In order to complement this work, some studies are ongoing at the University of Salerno. In particular, experimental, numerical and theoretical activities have been carried out to extend the component method to these typologies [18][19][20][21]. With this goal, the main components of this joint have been identified in [22] (Figure 2): pct/pcc are the components representative of the behaviour in tension and compression of the attachment between the tubular profile and the flange plates of the beam; ttt/ttc are sources of deformability ascribed to the possible transverse tension or compression actions transmitted by the flanges to the CHS and induced by the beam's rotation; cs and bws represent the parts of the tubular profile and the beam web subjected to the horizontal shear.
where in Equations (3) In Equations (1) and (2) β = b 1 d 0 , γ = d 0 2t 0 and τ = t 1 t 0 where b 1 and t 1 are the plate's width and thickness, while d 0 and t 0 are the CHS diameter and thickness, respectively.
The shear components cs and bws have been characterised by extending the same formulas already proposed by Eurocode 3 part 1.8 [5]: where in Equations (3)-(6), A v,cs is the shear area of the tubular profile (A v,cs = 2A/π), A v,bws is the shear area of the beam web (A v,bws = d 0 t bw ), t bw is the web thickness of the beam, γ M0 = 1 is the partial safety factor, f y is the yield stress, β v is the transformation parameter equal to β v = 1 − z L c , z is the lever arm (distance between the centerlines of the beam flanges) and L c is the column length.
Consequently, to achieve a complete knowledge of the whole joint behaviour between CHS columns and passing through I-beams with the component method, only the study of the sources of deformability ttt/ttc ( Figure 3) is still needed. Therefore, the main objective of this paper is the characterisation of this component which is representative of the column parts which are subjected locally to transverse tension or compression actions transmitted by the beam flanges. beam, = 1 is the partial safety factor, is the yield stress, is the transformation parameter equal to = 1 − , is the lever arm (distance between the centerlines of the beam flanges) and is the column length. Consequently, to achieve a complete knowledge of the whole joint behaviour between CHS columns and passing through I-beams with the component method, only the study of the sources of deformability ttt/ttc ( Figure 3) is still needed. Therefore, the main objective of this paper is the characterisation of this component which is representative of the column parts which are subjected locally to transverse tension or compression actions transmitted by the beam flanges. The paper is structured as follows: in paragraph 2, the methodology of the work is briefly reported, focusing on the numerical implementation of the analysed connection in Abaqus [23]. The current investigation intends to define strength and stiffness formulations of the examined component through a parametric analysis of 31 different geometrical configurations of connections whose behaviour is numerically simulated. Paragraph 3 concentrates on the main results of the simulations and provides the required strength and stiffness equations of the component employing regression analyses of results obtained from the parametric study.

Materials and Methods
The main aim of the current work is the strength and stiffness characterisation of the components ttt/ttc belonging to CHS to passing-through double-tee beam connection, representing the possible transverse tension/compression on the tubular section generated by the rotation of the beam flanges. The need to investigate these components derives from preliminary monotonic and cyclic tests performed on beam-to-column sub-assemblies at the University of Salerno [11], highlighting that the column parts close to the flange-tocolumn attachment may experience buckling or tearing phenomena if they are respectively loaded in compression or tension ( Figure 3).
The study of ttt/ttc components was carried out through a parametric analysis concerning 31 geometrical configurations of specimens representative of the connections between CHS columns and flanges of double-tee profiles. The mechanical model of the considered configurations refers to an axially loaded cantilever scheme shown in Figure 4. In particular, to study the behaviour of the analysed components in tension and compression, only the sides of the plate close to the fixed constraint were welded to the column; instead, contact properties were applied for the remaining sides indicated as Free sides in Figure 4. The paper is structured as follows: in paragraph 2, the methodology of the work is briefly reported, focusing on the numerical implementation of the analysed connection in Abaqus [23]. The current investigation intends to define strength and stiffness formulations of the examined component through a parametric analysis of 31 different geometrical configurations of connections whose behaviour is numerically simulated. Paragraph 3 concentrates on the main results of the simulations and provides the required strength and stiffness equations of the component employing regression analyses of results obtained from the parametric study.

Materials and Methods
The main aim of the current work is the strength and stiffness characterisation of the components ttt/ttc belonging to CHS to passing-through double-tee beam connection, representing the possible transverse tension/compression on the tubular section generated by the rotation of the beam flanges. The need to investigate these components derives from preliminary monotonic and cyclic tests performed on beam-to-column sub-assemblies at the University of Salerno [11], highlighting that the column parts close to the flange-to-column attachment may experience buckling or tearing phenomena if they are respectively loaded in compression or tension ( Figure 3).
The study of ttt/ttc components was carried out through a parametric analysis concerning 31 geometrical configurations of specimens representative of the connections between CHS columns and flanges of double-tee profiles. The mechanical model of the considered configurations refers to an axially loaded cantilever scheme shown in Figure 4. In particular, to study the behaviour of the analysed components in tension and compression, only the sides of the plate close to the fixed constraint were welded to the column; instead, contact properties were applied for the remaining sides indicated as Free sides in Figure 4.  Starting from the results provided in [11], it was possible to observe that the main parameters affecting the response of the analysed component are and . Consequently, the 31 cases were appropriately selected by varying between 0.44 and 0.72, with ranges between 13.69 and 39.51 (Table 1).  Starting from the results provided in [11], it was possible to observe that the main parameters affecting the response of the analysed component are β and γ. Consequently, the 31 cases were appropriately selected by varying β between 0.44 and 0.72, with γ ranges between 13.69 and 39.51 (Table 1). The analyses were conducted using the Finite Element (FE) software Abaqus 6.17 [23]. In particular, the adopted scheme refers to the specimens tested in [11], which are characterised by a CHS profile with a length equal to 500 mm, restrained at one end with a fixed constraint while the other end is free. The hole to pass through the plate is modelled without tolerance, and Tie constraints ensure the connection between the plate and the tube to avoid modelling the welds. The material properties associated with the plate and the CHS refer to a nominal S355JR steel modelled with a quadri-linear constitutive law, as proposed by Faella et al. [24], with an elastic modulus equal to 210 GPa and a Poisson's ratio of 0.30. The damage evolution of the material properties was included in the analysis by implementing an equivalent plastic strain at fracture equal to 2.4 mm, complying with the recommendations provided by [25,26]. The plates and the tubular profiles were meshed by adopting C3D8-type (8-node linear brick) elements with a 5 mm size. The simulations were carried out employing a static solver and applying increasing displacements along the plate face in the direction of the longitudinal axis of the column, as reported in Figure 5. The analyses were conducted using the Finite Element (FE) software Abaqus 6.17 [23]. In particular, the adopted scheme refers to the specimens tested in [11], which are characterised by a CHS profile with a length equal to 500 mm, restrained at one end with a fixed constraint while the other end is free. The hole to pass through the plate is modelled without tolerance, and Tie constraints ensure the connection between the plate and the tube to avoid modelling the welds. The material properties associated with the plate and the CHS refer to a nominal S355JR steel modelled with a quadri-linear constitutive law, as proposed by Faella et al. [24], with an elastic modulus equal to 210 GPa and a Poisson's ratio of 0.30. The damage evolution of the material properties was included in the analysis by implementing an equivalent plastic strain at fracture equal to 2.4 mm, complying with the recommendations provided by [25,26]. The plates and the tubular profiles were meshed by adopting C3D8-type (8-node linear brick) elements with a 5 mm size. The simulations were carried out employing a static solver and applying increasing displacements along the plate face in the direction of the longitudinal axis of the column, as reported in Figure 5. The developed FE model is the same one validated in [22] against the results of experimental activities carried out on the components pcc/pct, with the only difference in the way of application of the displacement condition; consequently, no repetitive information is provided in this manuscript.
It is worth highlighting that referring to the component in compression, the monotonic simulation was anticipated by a buckling analysis intended to define the shape of the imperfection to be applied to the specimen. Therefore, the buckling mode was The developed FE model is the same one validated in [22] against the results of experimental activities carried out on the components pcc/pct, with the only difference in the way of application of the displacement condition; consequently, no repetitive information is provided in this manuscript.
It is worth highlighting that referring to the component in compression, the monotonic simulation was anticipated by a buckling analysis intended to define the shape of the imperfection to be applied to the specimen. Therefore, the buckling mode was amplified according to the requirements of Eurocode 3 part 1.5 [27] and EN 10034 [28] with a scale factor reproducing an imperfection on the tube equal to the 3% of the nominal internal diameter ( Figure 6). amplified according to the requirements of Eurocode 3 part 1.5 [27] and EN 10034 [28 with a scale factor reproducing an imperfection on the tube equal to the 3% of the nomina internal diameter ( Figure 6). The choice of applying displacements along the longitudinal direction only withou also the rotation of the plate is ascribed to the mechanical behaviour observed during th experimental campaigns carried out in [11]. In fact, as it has also been confirmed by nu merical simulations (Figure 7) that the part of the passing-through plate in the tubula profile is connected to the other nodal components and acts as a rigid plate; consequently the actions transmitted by the beam become loads acting orthogonally to the plate. More over, the rotation of the flanges in the column is lower than the rotation exhibited at th beam-to-column attachment, justifying the fact that this effect can be ignored.

Results and Discussion
This section discusses the numerical results of the parametric analysis mentioned above. Monotonic tests were carried out, and the values of stiffness and resistance wer defined. In particular, the resistance of the analysed component was evaluated when th first element of the model achieved a principal plastic tensile deformation equal to the 5% Conversely, the stiffness was assessed when the force experienced by the connections wa equal to 2/3 the plastic resistance. The main outcomes are summarised in Table 2, wher The choice of applying displacements along the longitudinal direction only without also the rotation of the plate is ascribed to the mechanical behaviour observed during the experimental campaigns carried out in [11]. In fact, as it has also been confirmed by numerical simulations (Figure 7) that the part of the passing-through plate in the tubular profile is connected to the other nodal components and acts as a rigid plate; consequently, the actions transmitted by the beam become loads acting orthogonally to the plate. Moreover, the rotation of the flanges in the column is lower than the rotation exhibited at the beam-to-column attachment, justifying the fact that this effect can be ignored. amplified according to the requirements of Eurocode 3 part 1.5 [27] and EN 10034 [28 with a scale factor reproducing an imperfection on the tube equal to the 3% of the nomina internal diameter (Figure 6). The choice of applying displacements along the longitudinal direction only withou also the rotation of the plate is ascribed to the mechanical behaviour observed during th experimental campaigns carried out in [11]. In fact, as it has also been confirmed by nu merical simulations (Figure 7) that the part of the passing-through plate in the tubula profile is connected to the other nodal components and acts as a rigid plate; consequently the actions transmitted by the beam become loads acting orthogonally to the plate. More over, the rotation of the flanges in the column is lower than the rotation exhibited at th beam-to-column attachment, justifying the fact that this effect can be ignored.

Results and Discussion
This section discusses the numerical results of the parametric analysis mentione above. Monotonic tests were carried out, and the values of stiffness and resistance wer defined. In particular, the resistance of the analysed component was evaluated when th first element of the model achieved a principal plastic tensile deformation equal to the 5% Conversely, the stiffness was assessed when the force experienced by the connections wa equal to 2/3 the plastic resistance. The main outcomes are summarised in Table 2, wher

Results and Discussion
This section discusses the numerical results of the parametric analysis mentioned above. Monotonic tests were carried out, and the values of stiffness and resistance were defined. In particular, the resistance of the analysed component was evaluated when the first element of the model achieved a principal plastic tensile deformation equal to the 5%. Conversely, the stiffness was assessed when the force experienced by the connections was equal to 2/3 the plastic resistance. The main outcomes are summarised in Table 2, where the strength in compression (F ttc ), tension (F ttt ) and stiffness (k ttt = k ttc = k ttt/ttc ) are reported. As an example, images concerning the von Mises stress and principal plastic defor mation distributions for Case 10 are shown in Figures 8-11.

CHS Tube under Localised Transverse Compression (ttc)
It is worth highlighting that the specimens were adequately selected to define sets o cases characterised by the same values of and . Consequently, the grouping of case with the same characteristics ( Figure 12) allowed to highlight that the strength under lo calised transverse compression depends on parameter according to an exponential law with power varying between 0.56 and 0.80. Similarly, for the parameter the exponen varies between −1.11 and −1.16.

CHS Tube under Localised Transverse Compression (ttc)
It is worth highlighting that the specimens were adequately selected to define sets o cases characterised by the same values of and . Consequently, the grouping of case with the same characteristics ( Figure 12) allowed to highlight that the strength under lo calised transverse compression depends on parameter according to an exponential law with power varying between 0.56 and 0.80. Similarly, for the parameter the exponen varies between −1.11 and −1.16.

CHS Tube under Localised Transverse Compression (ttc)
It is worth highlighting that the specimens were adequately selected to define sets of cases characterised by the same values of and . Consequently, the grouping of cases with the same characteristics ( Figure 12) allowed to highlight that the strength under lo calised transverse compression depends on parameter according to an exponential law with power varying between 0.56 and 0.80. Similarly, for the parameter the exponen varies between −1.11 and −1.16.

CHS Tube under Localised Transverse Compression (ttc)
It is worth highlighting that the specimens were adequately selected to define sets of cases characterised by the same values of β and γ. Consequently, the grouping of cases with the same characteristics ( Figure 12 To define a formula for the prediction of the resistance of this component, a regression study was performed referring to the dimensionless parameters F ttc / b 1 t 0 f y versus β and γ. The regression analysis led to Equation (7).
The reliability of the proposed formulation is demonstrated in Table 3 through the comparison between the numerical outcomes and the analytical proposal. In most cases, the prediction is accurate, with a coefficient of variation equal to 10.4%.

CHS Tube under Localised Transverse Tension (ttt)
The same approach was applied to characterise the resistance of the tube under localised transverse tension, F ttt .
The strength of this component depends on β according to an exponential law with power varying between 0.82 and 0.91, while referring to γ, the power is about -1.02 ( Figure 13). For this reason, a regression analysis was performed considering the dimensionless parameters F ttt / b 1 t 0 f y , β and γ, obtaining Equation (8).
The accuracy of the proposed formulation is demonstrated against the results of the numerical simulations, as shown in Table 4. In most cases, the prediction is accurate, even though a coefficient of variation equal to 6.6% was observed. The accuracy of the proposed formulation is demonstrated against the results of the numerical simulations, as shown in Table 4. In most cases, the prediction is accurate, even though a coefficient of variation equal to 6.6% was observed.
(e) (f) (g) Figure 13. Results of the parametric analysis referring to .

Stiffness Formulation
Considering that the equations adopted to define the resistance of the analysed components are reported in the simple formulation F/b 1 t 0 f y = β constant1 γ constant2 , it was chosen to adapt this equation also to characterise the stiffness (k) according to Equation (9): A regression study was carried out starting from the numerical outcomes of the parametric analysis, leading to Equation (10): The accuracy of the derived equation is proven in Table 5, where it is shown that the mean of the ratios between predicted results and FE outcomes is equal to 1.00, while the coefficient of variation is about 14.4%. This equation is to be considered valid both for the case of compression and tension.

Conclusions
The present work characterised the mechanical modelling of the stiffness and strength of two of the main components of joints with Circular Hollow Section columns and passingthrough double-tee beams. In particular, the analysed sources of deformability are activated by the rotation of the beam flanges and generate local compressive (ttc) and tensile (ttt) actions on the tubular profile. The first step of the study consisted in a proper selection of 31 geometrical configurations of CHS columns with passing-through plates whose behaviour was assessed with numerical simulations with Finite Element software. Monotonic simulations were carried out imposing an increasing displacement of the plates in the direction of the tubular axis. For each analysed case, the strength and stiffness were derived. The results showed that, generally, the strength and stiffness of the analysed components have an exponential dependence on the parameters β and γ. Consequently, regression analyses were carried out to propose design formulations which proved their accuracy since the scatters with the numerical outcomes are characterised by coefficients of variation equal to 10.4%, 6.6% and 14.4% in terms of strength in compression, tension and stiffness, respectively.
Future developments of the current research will consist in collecting information about all the nodal components involved in characterising the flexural strength of CHS columns and passing-through I-beams so that a component modelling of the joint can be proposed.