I‐beam‐to‐SHS‐column moment resisting joints using passing‐through plates under equal bendings

Conventional I‐beam‐to‐SHS‐column moment resisting connections are generally stiffened with external diaphragms in order to limit excessive deformation of the tube‐wall. Passing through plates welded to the tube is an alternative solution that can develop significant strength and stiffness. This solution has been applied to CHS column and initial findings from RFCS project LASTEICON were encouraging. RHS columns are also widely used in practice due to simplicity of connections manufacturing. The new RFCS project LASTTS aims to investigate the application of this technique of passing through plates to SHS columns using laser cutting technology.

2 Finite-element model 2. 1 Material properties The mechanical properties of steel have been defined according to an elastic-plastic behaviour with kinematic hardening, adopting a Young's modulus of 210 GPa and a Poisson's ratio of 0.3. The stress-strain curve is multi-linear, based on the true stress -true strain curve obtained from coupon tests during LASTEICON project (see Figure  1). The yield stress and ultimate tensile stress are equal to 346.45 MPa and 510 MPa, respectively. The same mechanical properties have been associated to all model parts.

Joint modelling
Joints have been modelled with ANSYS 2021 R2 and analysed with non-linear static analysis. Sparse direct equation solver is used with automatic Newton-Raphson option. Furthermore, geometrical non-linearities are taken into account. 20-node solid hexahedral brick elements (SOLID186) are used. Each having three degrees of freedom per node: translations in the nodal x, y, and z directions. The general mesh layout adopts a fine mesh around the joint panel to better capture surface undulations, as well as stress and strain distributions. Moving away from the joint region, a gradual increase of element size is applied, resulting in a coarser mesh. Numerical modelling requires that at least three through-thickness elements to be used for the plates and two for the chord, otherwise results may prove excessive flaw.
To establish a connection between the SHS column and the passing elements, the two parts share the same nodes and elements at junction zones. The same principle is adopted to bond the IPE to the passing-through plates.
Regarding the boundary conditions, the hinges are modelled by suppressing the corresponding degrees of freedom to all nodes at both column end-sections. The cantilever beams have been restrained laterally from each side of the column in order to prevent lateral torsional buckling.
Simulations are performed in a displacement-controlled manner, acting on beam edges on both sides located at 2.5 m distance from column axis.  Before performing a non-linear analysis, a linear static simulation is launched first, followed by a linear buckling analysis. An imperfection homothetic to the buckling modes corresponding to passing-through plates buckling has been introduced. The imperfection is equal to sc /200 according to

Primary investigation
In this section, the behaviour of a SHS350x10 column crossed by flange and web passing plates of 10 and 8 mm thickness respectively is compared to an experimental test [6] performed on a CHS355.6x10 column instead of a SHS350x10.
Under equal hogging bending moments, failure is due to the development of plastic hinges in the compressed passing through plate portion inside the tube-wall. The bottom flange plate buckling marks the peak of the joint behaviour, followed by the web plate buckling. The post-peak behaviour is mainly influenced by the tube wall yielding. Once local buckling takes place in the passing plates, the proportion of transverse force transmitted to the tube-wall increases. The transverse stiffness of SHS being lower than a CHS (due to the absence of arch stiffening effect), the post-buckling is characterised by a smooth decrease of the loading (see Figure 4). The initial stiffness is not influenced by column cross-section shape.

Sensitivity analysis
A total of sixteen additional joints have been studied varying the column size and thickness as well as the passing though plates thicknesses. The I-beam profile corresponds to an IPE400. The flange plate width and web plate depth are equal to 180 and 320 mm, respectively. Regarding the length, the passing flange measures 970 mm and the passing web size is of 550 mm for SHS350 and 500 mm for configurations with SHS300. The investigated geometries are presented in Table 1 as well as the initial rotational stiffness of the joint and the ultimate bending moment. The rotational stiffness is assessed by extracting horizontal displacement right at the junction between the passing flanges and the tube-wall. The vertical load-displacement curves are depicted for sets 1 and 4 in Figure 6. For passing plates of 10/8, 12/10 and 15/10mm, the failure is due to passing-flange buckling followed by passing-web buckling. A clear peak load can be observed (see Figure 6-a). The post-peak behaviour depends on the SHS dimensions.
However, for passing plates of 20/12mm thickness, the failure starts outside the joint node, by beam flange buckling in compression, followed by passing-through flange buckling (see Figure 6-b). With a SHS350x10mm, the passing-through web buckled shortly after passing-through flange failure. This latter phenomenon was not observed for the rest of configurations in set 4.
Besides, what emerges from the results is that the passing through plates thicknesses have a primary role in the bending resistance of the joint. Shifting from 10/8mm to 12/10mm passing plates improved the joint resistance by 32 to 39%. The gains are more pronounced when going from 10/8mm to 15/10mm passing plates, with an enhancement of 71 to 85%. Regarding the stiffness, the increase is less significant with 8.6% among studied configurations between 10/8mm and 12/10mm passing plates.  The SHS side size also influences the joint behavior. A decrease of the tube size enhances the resistance as the buckling length decreases. The contribution of the column thickness is more limited as the tube transverse stiffness is lower comparatively to the passing-through plate's stiffness.

3
Analytical model

Bending resistance
The analytical model proposed to determine the bending resistance and initial rotational stiffness is based on assumptions similar to those suggested by Couchaux et al. [6] for joints composed of CHS column but adapted to SHS one's. The failure mode corresponds to buckling of flange plates inside the tube. The stiffness of the different components is considered for the evaluation of force distributions within the joint. The components are: • Flange plates in compression/tension, • Web plate in bending, • Tube-wall in transverse tension/compression.
Using rotation compatibility and force equilibrium based on the simplified model presented in Figure 7, the bending resistance is: Angle θ is represented in Figure 8, and can be calculated as follows: The multiplier 2 in equ.(4) accounts for both lateral walls.
The buckling resistance of the flange passing plate, Ffp,I,u, is calculated according to Eurocode 3 part 1-1: With fp  the reduction factor for relevant buckling curve.
fy,fp corresponds to the flange plate yield strength and Afp is its area.
Here, the calculation was performed with respect to imperfection curve "c" according to EC3 part 1-1 requirements for a solid rectangular section.
The plate is assumed clamped at both edges; the effective buckling length is thus: The Euler elastic critical buckling load is finally:

Initial rotational Stiffness
Based on the spring model represented in Figure 7-c, and by equating the rotation of the passing plates and the tube face at junction, the initial rotational stiffness can be expressed as: Where the lever arm equals: With hb the height of the I-beam and tfp the thickness of the passing flange plate.
Moreover, the initial rotational stiffness can be re-written in a better convenient form: With Sj,wp the bending stiffness of the passing-web plate.

Comparison to numerical results
The bending resistance and initial rotational stiffness calculated analytically and numerically are presented in Table  2. The results are in good agreement particularly concerning the bending resistance with a mean value of 0,98 and a standard deviation of 4%. The analytical model is able to capture the beneficial effects of increase of passing plate and tube thickness's as well as the decrease of SHS side. The precision concerning the initial rotational decrease with a mean value of 1,12 but the results are satisfactory.

Conclusions
The present paper summarises the numerical and analytical investigations performed on the behaviour of I-beamto-SHS column moment resisting joints using passingthrough under equal bending moments. Passing-through plates to SHS-column full joint has been studied using 20nodes SOLID186 elements in ANSYS under gravitational loading. Connection numerical strength capacity and initial stiffness were then compared to experimental test results from LASTEICON project and analytical predictions from current literature.
The comparison between CHS and SHS profiles shows that the global behaviour is similar for both section shapes. The failure mode remains unchanged and is governed by the buckling of both, the passing-flange in compression and the passing-through web in bending. However, the postbuckling behaviour is different between SHS and CHS as a consequence of a different flexibility in bending. The parametric study highlighted that the passing through plates thicknesses play a primary role in determining both the resistance and stiffness of the assembly. Increasing the flange plate thickness to 20 mm allowed to observe the failure on the beam. Varying tube geometry also contributes to connection strength but have no clear impact on joint stiffness. As a matter of fact, the side size directly interferes with the failure mechanism since it influences the buckling length of the passing-plates. A thicker column also means that fewer forces are transmitted to the internal portion of the passing-plates. Finally, the comparison of the analytical model predictions against numerical results leads to satisfactory strength and stiffness matchings.
The proposed analytical and numerical models should be validated by comparison to experimental tests.