Stress Concentration Factors of CHS-to-CFRHS Y-Joints Under Axial Tension Loading

Abstract

A CHS-to-CFRHS Y-joint that consists of a circular hollow section (CHS) brace and a concrete-filled rectangular hollow section (CFRHS) chord by welding has a simple and smooth weld profile that saves time and cost for the fabrication of CHS-to-CFRHS Y-joints and leads to a superior fatigue performance, compared with other welded tubular joints. This investigation presented an analysis of the stress concentration factors (SCFs) of CHS-to-CFRHS Y-joints subjected to axial tension loading of the brace. First, a finite element (FE) modelling method, which was validated with the experimental results cited in the reference, was utilised to establish the FE models of CHS-to-CFRHS Y-joints. Then, a parametric analysis was conducted to investigate the influences of the significant non-dimensional geometric parameters on the SCFs of CHS-to-CFRHS Y-joints. It is found that the intersection angle of the brace and chord has an important influence on the magnitudes of the SCF values. An increase in the intersection angle of the brace and chord will increase the values of the SCFs at the 60° location and saddle. The values of the SCFs at the 60° location and saddle reach the maximum value when the intersection angle of the brace and chord reaches 90°. Furthermore, on the basis of the large database of the SCF results, empirical design equations were established to calculate the SCFs at the crown toe, 60° location and saddle via multiple regression analysis. A safety factor was applied to the empirical design equations to ensure safe and reliable results of SCF calculations for the fatigue design of CHS-to-CFRHS Y-joints in a composite truss structure. Ultimately, a comparative analysis of SCFs was conducted with the FE models of welded tubular joints with rectangular hollow section (RHS) chords and CFRHS chords. The results reveal that infilling concrete in the chord leads to a reduction in SCFs along the weld profile of more than 11% on average, and the peak SCF decreases by more than 15%.

1. Introduction

Composite truss structures connected by welded tubular joints with concrete-filled steel tube (CFST) chords are emerging engineering structures that have been increasingly used in engineering projects, such as large spatial structure buildings, truss arch bridges, truss girder bridges, transmission towers, foundation structures of wind turbines and other engineering structures, because of their excellent mechanical properties [1,2]. Numerous studies have indicated that infilling concrete in the chord of a welded tubular joint can improve the bearing capacity, fatigue behaviour and fire endurance over those of its empty counterpart [3,4,5]. Previous investigations have focused primarily on welded tubular joints with hollow section chords, such as welded tubular joints with circular hollow section (CHS) braces and CHS chords (CHS-to-CHS joints) (Figure 1a), welded tubular joints with rectangular hollow section (RHS) braces and RHS chords (RHS-to-RHS joints) (Figure 1b) and welded tubular joints with CHS braces and RHS chords (CHS-to-RHS joints). On the basis of the studies on CHS-to-CHS joints, RHS-to-RHS joints and CHS-to-RHS joints, numerous investigation results, such as design parameters, construction measures and empirical design equations, have been adopted in related design standards [6,7]. However, studies on welded tubular joints with CFST chords, such as welded tubular joints with CHS braces and concrete-filled circular hollow section (CFCHS) chords (CHS-to-CFCHS joints), welded tubular joints with RHS braces and concrete-filled rectangular hollow section (CFRHS) chords (RHS-to-CFRHS joints) and welded tubular joints with CHS braces and CFRHS chords (CHS-to-CFRHS joints) (Figure 1c), are relatively limited. To date, numerous investigations on welded tubular joints with CFST chords have been carried out in recent years, and many significant results have been obtained in studies on bearing capacity [8,9,10,11,12,13,14,15], stiffness [16] and fatigue behaviour [17,18,19,20,21]; however, investigations on welded tubular joints with CFST chords are still incomplete and need improvement.

As truss structures connected by welded tubular joints are subjected to long-term cyclic loads such as wind and vehicle loads, these welded tubular joints easily suffer from fatigue failure because there are high-stress concentrations near the weld [22]. Consequently, fatigue research is crucial for ensuring the reliable fatigue performance of welded tubular joints. Some investigations have shown that infilling concrete in the hollow section chord can reduce the stress concentrations of welded tubular joints by suppressing the wall deflection of the rectangular section chord and the ovalisation of the circular section chord [23]. Many investigations have considered the influence of infilling concrete in the chord on the stress concentration factor (SCF), which is the ratio of the hot spot stress (HSS) to the nominal stress; this factor is assessed with the HSS method, which is widely employed in fatigue research. Jiang et al. [23,24] investigated SCFs for RHS-to-CFRHS X-joints under axial loading and for RHS-to-CFRHS T-joints subjected to various loading conditions, and the results demonstrated that the SCFs in welded tubular joints were reduced by more than 10% by infilling concrete in the chord. Mashiri et al. [25,26,27,28,29] conducted numerous investigations on RHS-to-CFRHS T-joints and CHS-to-CFCHS T-joints under various loading conditions. Their research demonstrated that the peak SCF of T-joints decreased drastically when concrete was infilled into the chord, and the equations utilised to calculate the SCFs of welded tubular joints with hollow section chords were not appropriate for welded tubular joints with CFST chords. Similar conclusions were drawn by Chen et al. [30], Zheng et al. [31] and Zhao et al. [32].

Previous fatigue investigations on welded tubular joints with CFST chords have focused mainly on CHS-to-CFCHS joints and RHS-to-CFRHS joints, and few investigations have been conducted on CHS-to-CFRHS joints, which have many advantages. Compared with that of CHS-to-CFCHS joints, CHS-to-CFRHS joints have a simpler weld profile, which can reduce the time and cost of fabricating welded tubular joints [33,34,35]. Compared with RHS-to-CFRHS joints, which have identical geometric parameters and loading conditions, CHS-to-CFRHS joints have lower SCFs and superior fatigue performance [33,34,35]. Compared with CHS-to-RHS joints, which are traditional structural forms among welded tubular joints in truss structures, CHS-to-CFRHS joints have longer fatigue lives and superior structural performance [36,37]. In conclusion, owing to the superior structural performance of CHS-to-CFRHS joints, research on CHS-to-CFRHS joints is meaningful. In the study of CHS-to-CFRHS joints, one of the key aspects is fatigue research, as welded tubular joints easily suffer from fatigue failure. Tong et al. [38,39] investigated CHS-to-CFRHS T-joints subjected to various loading conditions via finite element analysis and the experimental method and derived empirical design equations to calculate the SCFs of CHS-to-CFRHS T-joints. Their investigations revealed that the maximum SCF value in CHS-to-CFRHS T-joints subjected to axial tension loading of the brace is approximately 67% of that of CHS-to-RHS T-joints on average. However, their investigations of CHS-to-CFRHS joints have focused mainly on T-joint, which is only a special type of Y-joints. Research on the influence of the intersection angle of brace and chord on the stress concentrations of CHS-to-CFRHS joints is lacking. Additionally, the design equations proposed in their investigations for SCF calculations are only applicable for CHS-to-CFRHS T-joints and are not suitable for CHS-to-CFRHS Y-joints, which are commonly used in truss structures. As a result, the present paper studies the effects of the intersection angle of brace and chord on the SCFs of CHS-to-CFRHS Y-joints and establishes the general design equations, which can be used to calculate the SCFs, of CHS-to-CFRHS Y-joints with various intersection angle of brace and chord.

In this work, the stress concentrations of CHS-to-CFRHS Y-joints subjected to axial tension loading of the brace are investigated. First, a finite element (FE) modelling method, which is validated with the experimental results cited in the reference, is used to establish 320 FE models of CHS-to-CFRHS Y-joints. Then, a parametric analysis is carried out to determine the influences of the non-dimensional geometric parameters on the SCFs of CHS-to-CFRHS Y-joints. On the basis of the large database of the SCF results, a multiple regression analysis method is used to derive the empirical design equations for SCF calculations at the crown toe, 60° location and saddle of CHS-to-CFRHS Y-joints. Ultimately, a comparative analysis of the SCFs between the CHS-to-CFRHS Y-joints and the CHS-to-RHS Y-joints is carried out to determine the influence of concrete infilled into chords on the SCFs along the weld profile and the peak SCF of welded tubular joints.

2. FE Model and Verification

2.1. FE Model

As shown in Figure 2, a CHS-to-CFRHS Y-joint consists of a CHS brace and a CFRHS chord. The stress concentrations of CHS-to-CFRHS Y-joints, which were subjected to unit axial tension loading of the brace, were studied via the numerical simulation method in this research, and the FE models of the CHS-to-CFRHS Y-joints shown in Figure 3 were established in the FE software ABAQUS 6.14. To avoid the influence of end constraints, for the FE models of the CHS-to-CFRHS Y-joints, the ratio of length to width was established as six for the chord and the brace [40]. The geometric parameters of the FE models of the CHS-to-CFRHS Y-joints were as follows: the chord members had the same width of 400 mm, with a variable wall thickness ranging from 12.5 mm to 32 mm. The brace members had widths ranging from 80 mm to 320 mm, with wall thicknesses varying from 3.125 mm to 32 mm.

The Young’s modulus (Es) of the steel tube was set to Es = 206 GPa; the Poisson’s ratio (ν_s) of the steel tube was set to ν_s = 0.3 [41]. The weld metal was set to have material properties identical to those of steel tubes. The Young’s modulus of the infilled concrete was assumed to be 34,500 MPa; the Poisson’s ratio of the infilled concrete was assumed to be 0.2 [42]. Twenty-node solid quadratic brick elements with reduced integration (C3D20R) were adopted for all the FE models of the CHS-to-CFRHS Y-joints. A refined mesh was utilised in the FE model to obtain SCFs at the hot spot locations around the weld. To reduce the computing time, the mesh was gradually coarsened at locations far from the weld. Fixed hinged boundary conditions were applied at both ends of the chord, as shown in Figure 3c. The free end of the brace was set to the unit axial tension loading condition, ensuring an equivalence relation between the values of the HSS and the SCF.

Setting an appropriate mesh size is important for ensuring the computational efficiency and accuracy of the results. The meshes used in the FE models of the CHS-to-CFRHS Y-joints include fine mesh parts, coarse mesh parts, and transition parts, as shown in Figure 3. The wall thickness mesh on both the chord and the brace of the FE models was divided into three layers [35,39]. The different mesh densities between the outer surface of the concrete infilled into the chord and the inner surface of the chord have little influence on the SCFs in the FE models of the CHS-to-CFRHS Y-joints [23]. Therefore, to reduce the computing time, a coarse mesh was utilised for the infilled concrete.

2.2. Weld Details

The geometric dimensions of the intersecting weld, which is a full penetration weld, comply with the regulations of the American Welding Society (AWS) standard [43], as shown in Figure 3b. The weld profile at the junction of the brace–chord intersection, the detailed parameters of which are presented in

Table 1. Details of the weld profile according to the AWS specification.

Parameter		Detail A Ψ = 180–135°	Detail B Ψ = 150–50°	Detail C Ψ = 75–30°	Detail D Ψ = 40–15°
End preparation (ω)
	max.		90°	Needed to obtain required Φ
	min.		10° or 45° for Ψ > 105°	10°
Joint included angle (Φ)
	max.	90°	60° for Ψ ≤ 105°	40°; If more use Detail B
	min.	45°	37–1/2°; If less use Detail C	1/2 Ψ
Completed weld
	tw	≥tb	≥tb for Ψ > 90° ≥tb/sin Ψ for Ψ < 90°	≥tb/sin Ψ but need not exceed 1.75 tb	≥2tb
	L	≥tb/sin Ψ but need not exceed 1.75 tb		Weld may be built up to meet this

, was adopted as specified in the AWS standard. The weld thickness and weld length are represented by t_w and L, respectively, and the dihedral angle between the brace and chord is denoted by Φ, as illustrated in Figure 4.

2.3. Steel–Concrete Interface of the CFST Chord of the CHS-to-CFRHS Y-Joint

In the CFST chord of the CHS-to-CFRHS Y-joint, as the adhesion between the outer surface of the infilled concrete and the inner surface of the steel tube is very small, the interface between the steel surface and the concrete surface should separate under interfacial tension in the normal direction in the FE analysis. In the FE models of the CHS-to-CFRHS Y-joints, the contact relationship of the CFST chord includes a tangential friction contact that is defined by the Coulomb friction stress–slip relationship and a normal contact that is defined as a “hard contact” between the steel surface and the concrete surface. Previous studies have shown that a friction coefficient with a value between 0.2 and 0.6 can be adopted in FE analysis [39]. Other studies have also shown that the friction coefficient has little effect on SCFs [44]. Consequently, the friction coefficient was set to 0.35 in the FE models of the CHS-to-CFRHS Y-joints in this research [39].

2.4. Extracting SCFs from the FE Models

The HSS method is widely utilised for assessing the fatigue of welded tubular joints via SCFs and is supported by the International Institute of Welding (IIW) [45], the International Committee for Research and Technical Support for Hollow Section Structures (CIDECT) [46], the American Petroleum Institute (API) [47], the AWS [43] and Det Norske Veritas (DNV) [48]. The SCF values of the CHS-to-CFRHS joints were determined via the HSS method in this research. For CHS-to-CFRHS joints under axial tension loading in the brace, previous research has demonstrated that the SCF distribution in the brace is significantly nonlinear, but the nonlinearity of the SCF distribution in the chord is not obvious [39]. In this research, quadratic extrapolation was used to obtain the HSS for CHS-to-CFRHS Y-joints at the locations close to the weld. The Lagrange quadratic interpolation method was employed to obtain the extrapolation equation, as shown in Equation (1):

$${\sigma }_{\mathrm{HSS}}={\displaystyle \frac{t_2·t_3·{\sigma }_1}{(t_1-t_2)·(t_1-t_3)}}+{\displaystyle \frac{t_1·t_3·{\sigma }_2}{(t_2-t_1)·(t_2-t_3)}}+{\displaystyle \frac{t_1·t_2·{\sigma }_3}{(t_3-t_1)·(t_3-t_2)}}$$

(1)

where σ_i and t_j (i, j = 1, 2, 3) are the stresses of the extrapolation points and the distances from the extrapolation points to the weld, respectively. Following the International Institute of Welding (IIW), three extrapolation points are shown in Figure 5a. With reference to Xiao et al. [1] and Zhao et al. [49], three points are arranged at distances t₁ = 0.4 t, t₂ = 0.9 t and t₃ = 1.4 t away from the weld, as shown in Figure 5b. The HSS can be derived as shown in Equation (2) [49]:

$${\sigma }_{\mathrm{HSS}}=2.52·{\sigma }_{0.4t}-2.24·{\sigma }_{0.9t}+0.72·{\sigma }_{1.4t}$$

(2)

2.5. Verification of the FE Analysis Method

As welded tubular T-joint is a particular type of welded tubular Y-joint, the FE analysis method can be validated with the experimental results of the CHS-to-CFRHS T-joints in Tong et al. [39]. Eight FE models of the CHS-to-CFRHS T-joints, which were subjected to axial tension loading of the brace, with geometric parameters similar to those of the test samples in Tong et al. [39] and weld details designed to meet AWS [43] specifications, were established to verify the correctness of the FE analysis method, as shown in Figure 6. The experimental and calculated SCF values at the key hot spot locations are compared, as shown in Figure 7. Tong et al. [39] obtained the strain concentration factor (SNCF) from experimental results. The SNCF values were converted to SCF values according to the ratio of SCF to SNCF (1.1 on the chord side and 1.2 on the brace side) [39,46]. For specimen T8, the width of the brace was close to the width of the chord, so stress extrapolation was difficult to apply to obtain the HSS in the chord. Consequently, specimen T8 was excluded from the comparative analysis between the FE results and the experimental results of the CHS-to-CFRHS T-joints. The majority of the FE results of the SCF values were consistent with the experimental results. As shown in

Table 2. Comparative analysis of the FE results and the experimental results.

SCFFE/SCFText	Specimen T1	Specimen T2	Specimen T3	Specimen T4	Specimen T5	Specimen T6	Specimen T7
Mean	1.10	1.09	1.11	1.09	1.04	1.11	1.13
COVs	0.12	0.09	0.13	0.09	0.05	0.12	0.16

, the mean values of the SCF_FE-to-SCF_Text ratios at key hot spot locations are between 1.04 and 1.13, and the coefficients of variation (COVs) at key hot spot locations are less than 0.16. These results demonstrate the accuracy and reliability of the FE analysis method. A few of the numerical simulation results exhibited significant deviations from the corresponding experimental results. The geometry parameters of the weld profile in the FE models differed from the actual weld profile of the sample, resulting in this deviation.

3. SCF Distribution of CHS-to-CFRHS Y-Joints

To determine the key hot spot locations in the CHS-to-CFRHS Y-joints under axial tension loading in the brace, a comprehensive investigation of the SCF distribution of CHS-to-CFRHS Y-joint was carried out by utilising the single variable method and the parametric analysis method. Previous research has shown that the non-dimensional geometric parameters determined by the standard of CIDECT Design Guide No. 8 [46] have the main influence on SCFs. Consequently, the key hot spot locations in the CHS-to-CFRHS Y-joint can be determined by investigating the variation in the SCF distribution for the CHS-to-CFRHS Y-joint under different single non-dimensional geometric parameter changes. In the CIDECT Design Guide No. 8 [46], the non-dimensional geometric parameters of the SCFs of the welded tubular joints that are intended for the RHS joints and CHS joints but also apply to the CHS-to-CFRHS joints are as follows: the ratio of the width of the brace to the width of the chord (β = b₁/b₀), the ratio of the width of the chord to the wall thickness of the chord (2γ = b₀/t₀), the ratio of the wall thickness of the brace to the wall thickness of the chord (τ = t₁/t₀) and the intersection angle of brace and chord (θ), as shown in Figure 2b. A total of 320 FE models of the CHS-to-CFRHS joints were established with the non-dimensional geometric parameters β = 0.2, 0.4, 0.6, and 0.8; γ = 6.25, 10, 12.5 and 16; τ = 0.25, 0.5, 0.75 and 1; and θ = 30°, 45°, 60°, 75° and 90° to study the influence of the non-dimensional geometric parameters on the SCF distribution along the weld path, as shown in Figure 8.

3.1. Influence of β on the SCF Distribution

The influence of β on the SCF distribution of CHS-to-CFRHS Y-joints with identical non-dimensional geometric parameters of γ = 12.5, τ = 0.5 and θ = 45° was investigated by setting different parameters β = 0.2, 0.4, 0.6 and 0.8, as illustrated in Figure 9. When the value of β is between 0.4 and 0.6, the SCFs on both the chord and the brace side reach the maximum values for the given non-dimensional geometric parameters γ, τ and θ. On the brace side, the SCF values generally reach the maximum values at the saddle, whereas on the chord side, the SCF values reach the maximum values at the crown toe or 60° location.

3.2. Influence of γ on the SCF Distribution

The influence of γ on the SCF distribution of CHS-to-CFRHS Y-joints with identical non-dimensional geometric parameters of β = 0.4, τ = 0.75 and θ = 45° was investigated by setting different parameters γ = 6.25, 10, 12.5 and 16, as illustrated in Figure 10. The SCFs on both the chord and the brace side exhibit a similar trend: the SCFs increase at all locations along the weld path with increasing γ. On the brace side, the SCF values generally reach the maximum values at the saddle, whereas on the chord side, the SCF values reach the maximum values at the crown toe or 60° location.

3.3. Influence of τ on the SCF Distribution

The influence of τ on the SCF distribution of CHS-to-CFRHS Y-joints with identical non-dimensional geometric parameters of β = 0.6, γ = 10 and θ = 45° was investigated by setting different parameters τ = 0.25, 0.5, 0.75 and 1, as illustrated in Figure 11. On the chord side, the SCFs at all locations along the weld path increase as the value of τ increases, but the SCFs on the brace side do not exhibit a clear regularity. The maximum SCF values on the brace side generally occur at the saddle for most cases, whereas on the chord side, the SCF values may reach the maximum values at the crown toe or 60° location.

3.4. Influence of θ on the SCF Distribution

The influence of θ on the SCF distribution of CHS-to-CFRHS Y-joints with identical non-dimensional geometric parameters of β = 0.6, γ = 12.5 and τ = 0.75 was investigated by setting different parameters θ = 30, 45, 60, 75 and 90, as illustrated in Figure 12. The maximum SCF values increase as the value of θ increases on both the chord and the brace side. The maximum SCF values on the brace side occur at the saddle, whereas on the chord side, the SCF values may reach the maximum values at the crown toe, 60° location or saddle.

4. SCFs at Key Hot Spot Locations on CHS-to-CFRHS Y-Joints

On the basis of the previous analysis of the influences of the significant non-dimensional geometric parameters on the SCF distribution, the maximum values of the SCFs of the CHS-to-CFRHS Y-joints subjected to axial tension in the brace are most likely to occur at the crown toe, 60° location or saddle. To determine empirical equations for SCFs at key hot spot locations (the crown toe, 60° location and saddle), it is necessary to investigate the influence of the significant non-dimensional geometric parameters (β, τ, γ and θ) on the SCFs at these locations of CHS-to-CFRHS Y-joints.

4.1. Influence of β on SCFs at Key Hot Spot Locations

The influences of β on SCFs at the crown toe, 60° location and saddle are illustrated in Figure 13. The SCF value initially increases, followed by a slight decrease at all the key hot spot locations (the crown toe, 60° location and saddle) on both the chord and the brace side. The peak SCF appears between β = 0.4 and 0.6 at all the key hot spot locations. Consequently, the width of the brace should be avoided to be set to half of the width of the chord to avoid high stress concentrations that lead to poor fatigue performance of CHS-to-CFRHS Y-joints.

4.2. Influence of γ on SCFs at Key Hot Spot Locations

The influences of γ on SCFs at the crown toe, 60° location and saddle are illustrated in Figure 14. The values of the SCFs exhibit a constant linear increase at all the key hot spot locations on both the chord and the brace side. Therefore, increasing the thickness of the chord can reduce the stress concentrations and improve the fatigue performance of CHS-to-CFRHS Y-joints.

4.3. Influence of τ on SCFs at Key Hot Spot Locations

The influences of τ on SCFs at the crown toe, 60° location and saddle are illustrated in Figure 15. The values of the SCFs on the chord side show a constant linear increase at all the key hot spot locations, whereas the values of the SCFs on the brace side initially increase and then slightly decrease at the crown toe.

4.4. Influence of θ on SCFs at Key Hot Spot Locations

The influences of θ on SCFs at the crown toe, 60° location and saddle are illustrated in Figure 16. The values of the SCFs at the 60° location and saddle increase as the value of θ increases, reaching the maximum values when the value of θ reaches 90°, on both the chord and the brace side. The results indicate that the value of θ has an important influence on the magnitude of the SCF values.

5. Proposed Design Equations

5.1. Multiple Regression Analysis

On the basis of the correlation between the SCFs of CHS-to-CFRHS Y-joints at key hot spot locations (crown toe, 60° location and saddle) and the significant non-dimensional geometric parameters (β, τ, γ and θ), referring to the equations for RHS-to-RHS T-joints (CIDECT [46]), RHS-to-CFRHS T- and X-joints (Jiang et al. [23,24]), CHS-to-RHS T-joints (Tong et al. [50]), CHS-to-CFRHS T-joints (Tong et al. [39]) and RHS-to-CFRHS Y-joints (Jiang et al. [21]), Equation (3) was selected to obtain the empirical design equations to calculate the SCFs of CHS-to-CFRHS Y-joints via multiple regression analysis.

$$SCF=(c_1+c_2·\beta +c_3·{\beta }^2)·{(2\gamma )}^{c_4+c_5·\beta +c_6·{\beta }^2}·{\tau }^{c_7+c_8·\beta }·{(\sin \theta )}^{c_9}$$

(3)

where the values of c₁ to c₉ are determined via multiple regression analysis. The empirical design equations were obtained to calculate the SCFs of CHS-to-CFRHS Y-joints at the crown toe, 60° location and saddle, as shown in Equations (4)–(6) (brace side) and Equations (7)–(9) (chord side). The range of validity for CHS-to-CFRHS Y-joints in the empirical design equations is as follows: 0.2 ≤ β ≤ 0.8, 6.25 ≤ γ ≤ 16, 0.25 ≤ τ ≤ 1.0 and 30° ≤ θ ≤ 90°.

• At the crown toe in the brace:

$$SC{F}_{\mathrm{brace}, \mathrm{crown} \mathrm{toe}}=(-0.00420+0.439\beta -0.143{\beta }^2)·{(2\gamma )}^{1.093+0.104\beta -0.698{\beta }^2}·{\tau }^{-0.0248-0.242\beta }·{(\sin \theta )}^{0.376}$$

(4)

• At the 60° location in the brace:

$$SC{F}_{\mathrm{brace}, 60° \mathrm{location}}=(-0.00777+0.374\beta +0.188{\beta }^2)·{(2\gamma )}^{1.247+0.221\beta -0.812{\beta }^2}·{\tau }^{-0.0477+0.445\beta }·{(\sin \theta )}^{1.423}$$

(5)

• At the saddle in the brace:

$$SC{F}_{\mathrm{brace}, \mathrm{saddle}}=(-0.0122+0.645\beta -0.302{\beta }^2)·{(2\gamma )}^{1.086+0.669\beta -0.808{\beta }^2}·{\tau }^{-0.101+0.733\beta }·{(\sin \theta )}^{1.754}$$

(6)

• At the crown toe in the chord:

$$SC{F}_{\mathrm{chord}, \mathrm{crown} \mathrm{toe}}=(0.232-2.007\beta +7.770{\beta }^2)·{(2\gamma )}^{1.857-2.905\beta +0.984{\beta }^2}·{\tau }^{0.775+0.241\beta }·{(\sin \theta )}^{0.725}$$

(7)

• At the 60° location in the chord:

$$SC{F}_{\mathrm{chord}, 60° \mathrm{location}}=(-0.0517+1.232\beta -1.236{\beta }^2)·{(2\gamma )}^{1.537-0.595\beta +0.343{\beta }^2}·{\tau }^{0.851+0.504\beta }·{(\sin \theta )}^{1.571}$$

(8)

• At the saddle in the chord:

$$SC{F}_{\mathrm{chord}, \mathrm{saddle}}=(0.141+0.150\beta -0.319{\beta }^2)·{(2\gamma )}^{1.256+1.078\beta -0.869{\beta }^2}·{\tau }^{0.876+0.543\beta }·{(\sin \theta )}^{2.088}$$

(9)

5.2. Verification of the Design Equations

The ratio of the SCFs calculated via the empirical design equations (SCF_Equation) to the SCFs obtained via the FE models (SCF_FE) was used to validate the reliability of the empirical design equations, as shown in Figure 17. As shown in

Table 3. Comparative analysis of the SCFs calculated via the empirical design equations and the SCFs obtained via the FE models.

SCFEquation/SCFFE	At Crown Toe in Brace	At 60° Location in Brace	At Saddle in Brace	At Crown Toe in Chord	At 60° Location in Chord	At Saddle in Chord
Mean	0.97	0.98	0.97	1.00	1.01	0.99
COVs	0.18	0.16	0.16	0.13	0.12	0.14

, the mean values of the SCF_Equation-to-SCF_FE ratios at key hot spot locations are between 0.97 and 1.01, and the COVs at key hot spot locations are less than 0.18. These results demonstrate the accuracy and reliability of the empirical design equations. In the fatigue design of engineering structures, the final design equations for SCF calculations at key hot spot locations (the crown toe, 60° location and saddle) in CHS-to-CFRHS Y-joints on both the chord and the brace side should be modified by applying a safety factor of 1.2 [51] to ensure the safety of engineering structures, as shown in Equation (10). Additionally, the minimum value of the SCFs is specified to be no less than 2.0 to ensure that the results of the SCF calculations are conservative in the fatigue design of engineering structures [35].

$$SC{F}_{\mathrm{Design}}=1.2SC{F}_{\mathrm{Equation} \left(4\right) - \left(9\right)}$$

(10)

6. Comparative Analysis of the SCFs for CHS-to-CFRHS Y-Joints and CHS-to-RHS Y-Joints

A modelling method similar to that utilised for CHS-to-CFRHS Y-joint is employed for CHS-to-RHS Y-joint, with the sole difference being that the FE model of CHS-to-RHS Y-joint does not require the simulation of concrete infilled into the chord. Eight FE models of the CHS-to-CFRHS Y-joints and the CHS-to-RHS Y-joints with the identical non-dimensional geometric parameters of β = 0.4, γ = 10, τ = 0.5 and θ = 45°; β = 0.6, γ = 10, τ = 0.5 and θ = 45°; β = 0.4, γ = 10, τ = 0.75 and θ = 45°; and β = 0.4, γ = 10, τ = 0.5 and θ = 75° were established to determine the influence of infilling concrete on the SCFs along the weld profile for welded tubular Y-joints. The loading and boundary conditions of the FE models of the CHS-to-RHS Y-joints were the same as those of the FE models of the CHS-to-CFRHS Y-joints.

The comparison results of the SCFs along the weld profile between the CHS-to-RHS Y-joints and the CHS-to-CFRHS Y-joints are shown in Figure 18. The CHS-to-CFRHS Y-joints and CHS-to-RHS Y-joints exhibit the identical trends in the SCF distribution along the weld profile on both the chord and the brace side. Most of the cases exhibit a similar trend: the SCF value generally has an initial increase followed by a slight decrease for CHS-to-CFRHS Y-joints and CHS-to-RHS Y-joints. Infilling concrete in the chord leads to a reduction in SCFs at all locations along the weld profile, and the reduction in SCFs along the weld profile is more than 15% in the chord and 11% in the brace on average. The peak SCF of the welded tubular Y-joint is decreased by more than 15.6% on the chord side and 15.2% on the brace side by infilling concrete in the chord. As shown in Figure 19, the sample deformation results of the welded tubular Y-joints under axial tension loading of the brace with parameters of β = 0.4, γ = 10, τ = 0.75 and θ = 45° indicate that the maximum deformation at the chord top wall of the welded tubular Y-joint is decreased by 29.5% by infilling concrete in the chord. For the CHS-to-RHS Y-joints subjected to axial tension loading of the brace, the chord side walls exhibit inwards deformation, and the chord top wall exhibits outwards deformation. Infilling concrete in the chord can limit the inwards deformation of the chord side walls and decrease the outwards deformation of the chord top wall, leading to lower bending stresses in the chord top wall which cause lower HSS and reduce the stress concentrations. In conclusion, infilling concrete in the chord for welded tubular joints can decrease the SCFs along the weld profile and peak SCF and reduce the deformation of the chord walls.

7. Conclusions

In this work, the stress concentrations of CHS-to-CFRHS Y-joints subjected to axial tension loading of the brace were investigated. A comprehensive parametric analysis was carried out to determine the effects of the non-dimensional geometric parameters on the SCFs of CHS-to-CFRHS Y-joints. Then, on the basis of the large database of the SCF results obtained from the FE models of the CHS-to-CFRHS Y-joints, a multiple regression analysis method was utilised to derive the empirical design equations for SCF calculations at the crown toe, 60° location and saddle of CHS-to-CFRHS Y-joints. Ultimately, a comparative analysis of the SCFs between the CHS-to-CFRHS Y-joints and the CHS-to-RHS Y-joints was conducted to determine the effects of infilling concrete in the chord on the SCFs of welded tubular joints. The main conclusions of this research can be drawn as follows:

• When the CHS-to-CFRHS Y-joint is subjected to axial tension loading of the brace, on the chord side, the SCF values reach the maximum values at the crown toe, 60° location or saddle in most cases. On the brace side, the majority of the maximum SCF values are located at the saddle, with a minority at the crown toe.
• For the CHS-to-CFRHS Y-joint under axial tension loading of the brace, as the value of β increases, the values of the SCFs initially increase and then decrease, reaching the maximum values between β = 0.4 and 0.6. Therefore, the width of the brace of the CHS-to-CFRHS Y-joint should not be set to half of the width of the chord to avoid high stress concentrations that lead to poor fatigue performance. As the value of γ increases, the values of the SCFs exhibit a constant linear increase. As a result, increasing the thickness of the chord of the CHS-to-CFRHS Y-joint can reduce stress concentrations and improve the fatigue performance.
• The intersection angle of brace and chord has an important influence on the magnitudes of the SCF values on both the chord and the brace side of CHS-to-CFRHS Y-joints under axial tension loading of the brace. The values of the SCFs at the 60° location and saddle increase as the value of θ increases, reaching the maximum values when the value of θ reaches 90°, on both the chord and the brace side.
• The empirical design equations proposed in this research for SCF calculations at the crown toe, 60° location and saddle of CHS-to-CFRHS Y-joints under axial tension loading of the brace provide safe and reliable results. The SCFs calculated via the empirical design equations can be utilised to calculate the HSS and predict the fatigue life via the HSS method for the fatigue design of CHS-to-CFRHS Y-joints in a composite truss structure.
• Infilling concrete in the chord for welded tubular joints can decrease the SCFs along the weld profile and peak SCF. In the comparison of the SCFs along the weld profile and peak SCF between the CHS-to-CFRHS Y-joints and the CHS-to-RHS Y-joints, infilling concrete in the chord leads to a reduction in SCFs along the weld profile of more than 15% in the chord and 11% in the brace on average, and the peak SCF is decreased by more than 15.6% on the chord side and 15.2% on the brace side.
• For the CHS-to-CFRHS Y-joint under axial tension loading of the brace, the infilled concrete can limit the inwards deformation of the chord side walls and decrease the outwards deformation of the chord top wall, leading to lower HSS and reducing the stress concentrations.