Numerical and theoretical investigation of bolted sleeve connections with rectangular hollow sections

Abstract This article presents a theoretical and numerical study of bolted sleeve connections with rectangular hollow sections (RHS) under axial tension and compression. The geometric form of a hollow section provides resistance for high axial loads, torsion and combines effects, spreading its utilization in truss systems. In this context, the sleeve connections proposed explore these characteristics of RHS and offer an attractive aesthetic appearance for the continuity of elements. The bolted sleeve connection with RHS is formed by two outer tubes connected by an inner tube and staggered bolts. Herein, a parametric study was developed for identification of the failure modes in the connection. Finite element models with different geometric parameters and number of bolts were created in commercial software. The width, depth and thickness of RHS tubes and diameter of bolts were variated. In the theoretical/numerical/parametric results, the yielding gross section failure, the fracture through the effective net area failure and the bearing failure were observed. These failure modes occurred in both outer and inner tubes. The load results were compared to determine the resistance capacity of sleeve connections. The theoretical formulations were evaluated for representation of the ultimate load of the failure modes.


Engenharia Civil
Steel tubular profiles are used on a large scale in some parts of the world, including Europe, Southeast Asia, Australia and North America.These profiles offer a good solution for structural elements with possibly wide spans.Tubular profiles are lightweight and more economical due to their high strength and low weight.These profiles are widely used in trussed structural systems because the axial loads are predominant in these types of structural systems (Wardenier et al., 2010).
For logistical, technical and manufacturing reasons, the tubes are divided into structures with a length shorter than that required for certain structures.Thus, a mechanism is needed to connect the tubes to allow their continuity from the structure.
In this context, there is an increase in investigations and studies related to the design and behavior of connections in tubular profiles that facilitate the structure assembly processes (Araujo et al., 2016;Luo et al., 2016;Xie et al., 2019;Xing et al., 2020).
Recently, Roquete et al. (2021) presented an extensive numerical, theoretical and experimental study for sleeve connections with steel CHS and staggered bolts.Five failure modes were described and evaluated, the yielding gross cross-section failure (YGCS), the fracture through the effective net cross-section failure (FNCS), the bearing failure (BF), the bolt bending (BB) and bolt shear (SB).

Introduction
The numerical model by the Finite Element Method was established using the commercial software ANSYS 22R1 (2021).The finite element SHELL 181 was attributed to represent the RHS tubes.It is a homogeneous structural element with 4-node and six degrees of freedom at each node.The finite element SOLID 186 was adopted to represent the bolts.This element is defined by 20-node with three degrees of freedom per node.
The contact pair CONTA175 and TARGE170 was used to represent the contact between the bolts and the transverse section of the RHS tube in the holes.The same finite elements were used by Oliveira et al. (2020), Roquete et al. (2021) and Roquete et al. (2022) for analysis of bolted sleeve connections with CHS.
The free mesh was adopted for all elements of the connection.For the RHS tubes, the mesh size was equal to 6 millimeters and the bolts equal to 5 millimeters.The behavior of the material characteristics was represented using the model of multilinear elastic material (Multilinear Isotropic Hardening), as presented by Salmon and Johnson (1990).
The boundary conditions were es-tablished: the nodes at one end of the outer tubes were constrained to displacement and rotation in all directions; At the other end a displacement of 12.5 millimeters was applied in axial direction to simulate the axial tension and compression.A displacement incremental control was applied.The Newton-Raphson iterative method was adopted for the solution.
The final configuration of the FE model was illustrated in Figure 3.The geometrical and material properties of the numerical models developed will be presented in the next item.
A parametric study is essential for the progress of research on steel connections.Despite the absence of experimental results to validate the FE numerical model, the finite element characteristics similar to those adopted by Roquete et al. (2021) were used.Thus, the present study is an important contribution for assessment of the behavior and failure modes of the sleeve connection with RHS.
In the study, 62 FE models were elaborated with a variation of the geometry and the number of bolts, divided into 31 models under axial tension and 31 models under compression.The FE models with similar properties were evaluated under axial tension and compression.The geometric properties of the sleeve connection models are presented in Table 1.In all models, the inner tube thickness was equal to 5mm to reduce the number of parameters that influence the connection.The bolt diameter (d b ) equal to 15 mm and hole diameter defined by adding a 1.5 mm was adopted in all models.The hole distance from the edge of the tube and the distance from hole to hole were defined by the normative standard ABNT NBR 8800 (2008).
The mechanical characteristics of the models were: for the tubes, the yield stress equal to 250 MPa and ultimate tensile strength of 360MPa; for the bolt the yield stress equal to 635 MPa and ultimate tensile strength of 825MPa.The nominal Young's modulus was equal 200 GPa and Poisson's ratio of 0.30.This article aims to study through theoretical and numerical analysis the behavior of 62 sleeve connections with aligned bolts and RHS under axial tension and compression.A numerical parametric study was developed varying the height and the width of the inner and outer tubes and the number of bolts.The results of FE models were compared with theoretical equations that determines the resistance capacity of the connection to verify the representability of equations.
However, the evaluation of sleeve connections with RHS was not performed and is the object of this re-search.The bolted sleeve connection with RHS is composed of two outer tubes with rectangular sections con-nected by staggered bolts to an inner tube with a smaller height and width, Figure 2.     *In all numerical models, the inner tube thickness (t IT ) was equal to 5 mm.

Parametric study
The fracture through the effective net cross-section failure (FNCS), bearing failure (BF) and yielding cross-section failure (YGCS) are the failure modes evaluated.
The FNCS and BF were characterized when an excessive material deformation was observed.The failure was identified when the deformation of von Mises reaches values higher than the ultimate rupture deformation.The YGCS occurred at moment of the peak load, when other failure modes have not been identified.
Currently, the sleeve connection design is not included in normative standards.The failure modes verified in this article are based design formulation presented by Oliveira (2019) and Roquete et al. (2021).
In addition, the ultimate limit state of the tubes and bolts was evaluated according to ABNT NBR 8800 (2008).
The yielding gross cross-section failure is characterized the R HS reaches the plastic deformation, exceeding the elastic deformation limit.The YGCS theoretical loading (Pteo) was calculated according to Eq. ( 1) and ABNT NBR 8800 (2008).
Table 2 presents the results obtained by the FE models and by the theoretical formulations.In addition, the failure modes and the location of occurrence are shown.Among the 31 FE models analyzed under compression, 23 failed by BF and 8 by YGCS.While for the 31 FE models evaluated under axial tension, 15 presented BF and 16 presented FNCS.The BF occurred in both outer tubes and inner tubes.The FNCS and YGCS was limited to the inner tubes.
Comparing the FE models and theoretical results (P num /P teo ) for models under axial tension, the mean value of the P num /P teo ratio was equal to 0.87 and coef-ficient of variation (CoV) of 0.241.For models evaluated under compression, the mean value of the P num /P teo ratio was equal to 0.83 and CoV of 0.218.Both mean value results indicate conservative numerical results compared to theoretical formulations.
Figure 4 shows the von Mises stress distribution in the models representing the three different failure modes identified.Figure 4a) represents the FNCS in the holes in the region central of the inner tube in the FE model M3. Figure 4b) illustrate the BF in the holes of the outer tube in the FE model M6 under axial tension.Finally, in Figure 4c), YGCS was observed in the central region of the inner tube of the FE model M4 under compression.
However, it is necessary to prevent failure modes to occur in the inner tube of the connection, due to the impossibility of visual control, as recommended by Roquete et al. (2021).Thus, the models that presented failures in the inner tube were disregarded.The bearing failure was the only failure mode observed in the outer tubes.In this context, the mean value of the P num /P teo ratio increased to 0.97 with CoV of 0.22, for models under axial tension and compression, Figure 5. Therefore, Eq. 4 can satisfactorily represent the ultimate load of the bearing failure in the outer tubes.
N c,Rk = P teo = A g x f y for: n > 4:l c = l f1 + (n-1) l f2 N Rk = P teo = 2.4d b x t x f u for: n > 4:l c = l f1 + 3l f2 Where: A g -the cross-sectional area of RHS tubes; f y -yield tensile stress of tubes; The fracture through the effective net cross-section is described as the steel necking followed by the rupture of the element along the edge of the hole.The FNCS theoretical loading (Pteo) for sleeve connection with RHS was calculated according to Eq. ( 2) and (3), and Roquete et al. (2021) and ABNT NBR 8800 (2008).The theoretical values representing the ultimate load resistance of bearing failure were given by Eq. ( 4), and ABNT NBR 8800 (2008).
Where: A e -effective net area of tubes; A n -net area of tubes; f u -ultimate tensile strength; C t -reduction factor due to holes presence; d f -hole diameter; t -tube thickness; e c -eccentricity of connection; Figure 6 shows the influence of the increase in the number of bolts on the resistance capacity of the sleeve connection through load-displacement curves.A high increase in the strength when the number of bolts was increased from 2 to 3 bolts was observed.This increase was also identified when the number of bolts was from 3 to 4. However, no significant gain in resistance capacity was observed when the sleeve connection was from 4 to 5 bolts.This behavior was also identified by Roquete et al. (2021) and Roquete et al. (2022) for sleeve connections with CHS.This article presented a theoretical and numerical study for the sleeve connection with RHS under axial tension and compression.A total of 62 models were evaluated: 31 models under axial tension and 31 under compression.The FE models allowed to verify the influence of the geometric parameters and number of bolts on the behavior of the connection.The fracture through the effective net cross-section failure (FNCS), bearing failure (BF) and yielding gross cross-section failure (YGCS) were the failure modes identified.
The numerical and theoretical failure load of the sleeve connection were compared.The mean value of the P num /P teo was equal to 0.87 and CoV of 0.241 for connections under axial tension.And P num /P teo of 0.83 and CoV of 0.218 for connections under compression.The models with inner tube failure were discarded due to difficulty in visual control.The BF was the only failure mode that occurred in the outer tubes.Thus, the mean value of the Pnum/Pteo increased to 0.97 and the theoretical formulation is adequate to represent the BF in outer tubes.
The influence of the number of bolts on the resistance capacity was presented using the load-displacement curves.The increase in the resistance capacity as the number of bolts increased was observed.However, for 4 and 5 bolts, the ultimate load was similar.It is suggested for future studies that an analysis be made of the optimal combination of number of loads per load gain in relation to the increase in the cost of the connection.In addition, experimental analyzes of the sleeve connection with RHS is recommended.
l c -the effective length of the connection; l f1 -distance between the center of the hole and the edge of the tube distance between the center of the hole and the center of the adjacent holed; l f2 -distance between the center of the hole and the center of the adjacent hole; d -width of the RHS tube; b -height of the RHS tube; and n -number of bolts connecting an outer tube to the inner tube.Where: t -tube thickness; d b -bolt diameter; f u -ultimate tensile strength;

Figure 6 -
Figure 6 -Influence of number of bolts in the sleeve connection: a) under axial tension; b) under compression.4.Conclusions

Table 1 -
Geometric data of the parametric study.