Degree of bending of concrete-filled rectangular hollow section K-joints under balanced-axial loadings

Abstract

It has been found that the fatigue life of tubular joints is not only determined by the hot spot stress, but also by the stress distribution through the tube thickness represented as the degree of bending (DoB). Consequently, the DoB value should be determined to improve the accuracy of fatigue assessment for both stress-life curve and fracture mechanics methods. Currently, no DoB parametric formula is available for concrete-filled rectangular hollow section (CFRHS) K-joints, despite their wide use in bridge engineering. Therefore, a robust finite element (FE) analysis was carried out to calculate the DoB of CFRHS K-joints under balanced-axial loading. The FE model was developed and verified against a test result to ensure accuracy. A comprehensive parametric study including 190 models, was conducted to establish the relationships between the DoBs and four specific variables. Based on the numerical results, design equations to predict DoBs for CFRHS K-joints were proposed through multiple regression analysis. A reduction of 37.17% was discovered in the DoB, resulting in a decrease of 66.85% in the fatigue life. Inclusively, the CFRHS K-joints with same hot spot stresses, may have completely different fatigue lives due to the different DoBs.

Graphical abstract

Keywords

fatigue assessment / K-joint / design equations / degree of bending / fracture mechanics

1 Introduction

Concrete-filled steel tubular (CFST) bridges can fully utilize material characteristics of steel and concrete, leading to an excellent structural performance and aesthetically appealing shape [1]. In China, the application of CFST truss beam bridges has become increasingly popular, with more than 400 CFST arch bridges (with a span greater than 50 m) having been built in the past 30 years. However, a significant increase in the tensile stress range in the joints is observed and is prone to fatigue failure, when considering the use of CFST truss structures from the arch system, to the beam system. Additionally, considering harsh service environment, heavy traffic loadings, and geometric defects of joints, the fatigue problem in CFST joints has become more prominent [2].

The hot spot stress (HSS) S-N curve method is the normal practice to evaluate the fatigue behavior of CFST joints, and can reflect the concentrated stress at the brace-chord weld intersection. Xu et al. [3], Tong et al. [4], Zheng et al. [5], Musa et al. [6], Kim et al. [7], and Jiang et al. [8–13] conducted several tests and numerical studies on HSS of concrete-filled circular hollow section (CFCHS) and concrete-filled rectangular hollow section (CFRHS) joints. They proposed equations to predict the stress concentration factor (SCF). It was found that the SCFs were significantly decreased after concrete filling, thus prolonging the fatigue life. Li et al. [14] reported that the fatigue life of concrete-filled joints is longer than that of hollow section joints, and using the codes of hollow section joints would underestimate its fatigue life. Conversely, test studies carried out by Connolly [15], Shen and Choo [16], showed that filling the chord with concrete would shorten the fatigue life, despite the reductions in SCFs. The reason for this phenomenon was that the SCF was derived from the stress distribution on the outer wall of the tube instead of through the tube thickness. However, the latter is the key factor in determining the stress intensity factor (SIF) at the crack tip, and would control the crack growth rate at the joints [16]. Wei et al. [17] reported that the use of SIF in fracture mechanics to evaluate the fatigue life of the joints, can effectively estimate the remaining fatigue life and has broad application prospects. Qian et al. [18] proposed that the crack propagation life of CFCHS joints accounts for 50%–90% of the total life, and should not be ignored. Thus, the influence of the through-the-thickness stress must be considered to improve the accuracy of fatigue life estimation.

The through-the-thickness stress for the location of the weld toe consists of three components: bending stress, membrane stress, and nonlinear peak stress. Among them, the nonlinear peak stress determined by the geometry of the weld, makes measurement in the design stage unrealistic [8]. In addition, it has been demonstrated that the nonlinear peak stress has little effect on the deep crack according to literature [19], so the through-the-thickness stress can be simplified to a linear combination of the bending stress, and membrane stress, as shown in Fig.1. The degree of bending (DoB) can be used to characterize the stress distribution through the tube thickness as follows:

(1)DoB=\frac{{\sigma }_{\text{B}}}{{\sigma }_{\text{T}}}=\frac{{\sigma }_{\text{B}}}{{\sigma }_{\text{B}}+{\sigma }_{\text{M}}},

where σ_T is the total stress, σ_B is the bending stress, and σ_M is the membrane stress.

A literature review showed that numerous studies have focused on the influence of DoBs on the fatigue life. Lee and Bowness [20], Berge et al. [21], Ahmadi et al. [22-24], and Nie et al. [25], carried out tests and numerical studies on the DoB of CHS joints, and proposed the prediction equation. Shen and Choo [16] conducted numerical studies on the DoB of CFCHS T-joint and found that the DoB of CHS T-joints would decrease after filling the chord with concrete. In general, some researchers have focused on CHS and CFCHS joints, however, the understanding of the DoB on CFRHS joints is very limited despite being widely used in bridges. Compared with CFCHS joints, CFRHS joints are easy to ensure the welding quality by cutting the plates end straight [10], so better fatigue behavior can be expected.

To address this research gap, the finite elements (FE) were developed to study the DoB of CFRHS K-joints. A specimen of the CFRHS truss beam was tested to verify the FE model. A comprehensive parametric analysis was then carried out including a total of 190 FE models. This paper presents the relationships between the DoBs and four variables: brace-to-chord width ratio (β), chord width-to-thickness ratio (2γ), brace-to-chord thickness ratio (τ), and the angle between braces and chord (θ), as shown in Fig.2. A design equation to predict the DoBs of the CFRHS K-joints was proposed using multiple regression analysis. Additionally, the maximum SCFs, and DoBs, between the CHS and CFRHS K-joints were compared. The application of the proposed DoB equation is evident in S-N curve methods and fracture mechanics methods.

2 Finite element model development and verification

2.1 Material properties

The software package ABAQUS was used to develop the FE model of CFRHS K-joints, and an elastic analysis using the experimental material properties was conducted. The chord and brace were made of grade Q345B steel. The material coupon tests were carried out according to GB/T 228.1-2010 [26]. Steel material properties are shown in Tab.1, which satisfied the Chinese specification JTG D64-2015 [27]. The concrete in the chord was made of C50. The 150 mm × 150 mm × 150 mm standard specimens of concrete cube were subjected to compressive strength tests according to GB/T 50081-2002 [28]. The concrete mix ratio and material properties are shown in Tab.2, which satisfied the Chinese specification JTG 3360-2018 [29]. The material properties of the joints in the FE model were based on the coupon tests.

2.2 Weld profile

It has been found that an accurate simulation of the weld profile has a significant influence on the stress field near the weld [11]. Therefore, the weld profile at the junction of the brace-chord intersection, as specified in the Chinese specification GB 50661-2011 was adopted [30]. The details of the weld profiles are presented in Tab.3. As shown in Fig.3, h_e is the weld thickness, and h_L is the weld length. b and ψ refer to the root gap, and dihedral angle between the brace and chord surface, respectively.

2.3 Element type and mesh scheme

Fig.4 shows the FE model for the CFRHS K-joint. A 20-node quadratic 3D-solid element with reduced integration (C3D20R) in the ABAQUS library was used for braces, chords, and weld profiles. To improve the calculation efficiency, only half of the model was developed considering the symmetry of the entire joint. The structured mesh division technology in ABAQUS was adopted for mesh generation. A fine mesh with a size of approximately 1 mm was used near the intersection of brace and chord to increase the reliability of the results. To reduce the time requirement, a coarse mesh with a size of approximately 10 mm was used for regions far away from the brace-chord intersection.

A convergence test was performed before the parametric analysis to verify the FE results. A mesh sensitivity analysis determined that the layers of mesh through the tube thickness, could significantly affect the accuracy of DoB. The number of layers through the wall thickness should not be too small, otherwise the calculation results could be wrong or erroneous. However, the number of layers should not be too large, or it leads to a computational inefficient model. A rational layer can satisfy both the requirements of calculation accuracy and economy. Fig.5 and Tab.4 show the results of the mesh sensitivity analysis with 1, 2, 3, 4, 5, and 6 layers. When the number of layers is greater than 4, the DoB value tend to be stable. Therefore, a four-layer mesh was selected to develop models for the consideration of computation speed.

2.4 Steel-concrete interaction, boundary and loading conditions

As shown in Fig.4, the mesh of the concrete and steel tubes was matched to ensure the accuracy of the calculation results, which meant that the pair of contact nodes on the interface had the exact same coordinates. The friction must also be considered for the contact point between the chords and concrete infills. Based on the literature about the steel-concrete interface [31–33], the friction coefficient ranged from 0.1 to 0.6. To consider the effect of friction on the shear force transmission under load for CFRHS K-joint, Jiang et al. [10] established a contact model with different friction coefficients. It was found that the friction coefficient of 0.3 can effectively simulate the shear between the steel-concrete interface. Therefore, the Coulomb friction model was employed with a frictional coefficient of 0.3, in this paper. In the normal direction, a “hard contact” model was employed at the steel-concrete interface, which indicated that the chord and concrete infill could not penetrate each other.

Fig.6 shows the loading and boundary conditions, of CFRHS K-joints. The balanced axial force was applied for CFRHS K-joints. Both ends of the chord were hinge-supported to ensure no bending moment existed in the chord, and ends of the two braces were applied with tension and compression with nominal stresses of 1 MPa, respectively. In this way, the extrapolated HSS has the same value with the SCF.

2.5 Determination of degree of bending values

To determine the DoB values, the bending and the membrane stresses should first be obtained, according to Eq. (1), as follows:

(2){\sigma }_{\text{M}}=\frac{{\sigma }_{\text{O}}+{\sigma }_{\text{I}}}{2},

(3){\sigma }_{\text{B}}=\frac{{\sigma }_{\text{O}}-{\sigma }_{\text{I}}}{2},

where, σ_O is the HSS at the outer surface, and σ_I is the HSS at the inner surface.

Based on Eqs. (1)–(3), the DoB could be further expressed as follows:

(4)DoB=\frac{1}{2}\left(1-\frac{{\sigma }_{\text{I}}}{{\sigma }_{\text{O}}}\right).

2.6 Extraction of the hot spot stress

It is recommended by IIW-XV-E [31], that the HSS of RHS joints could be derived using the quadratic extrapolation method. Three points are arranged at the location of 0.4t, 0.9t, and 1.4t away from the weld toe as specified in Tab.5 and Fig.7. In this way, the HSS could be derived as follows:

(5){\sigma }_{\text{h}}=2.52\cdot {\sigma }_{0.4t}-2.24\cdot {\sigma }_{0.9t}+0.72\cdot {\sigma }_{1.4t}.

The HSS is the stress perpendicular to the weld toe [8]. So, all the measured stresses in the extrapolated region can be modified to perpendicular as follows:

(6){\sigma }_{\mathrm{\perp }}={\sigma }_x{l}_{\text{1}}^{\text{2}}+{\sigma }_y{m}_{\text{1}}^{\text{2}}+{\sigma }_z{n}_{\text{1}}^{\text{2}}+2\left({\tau }_{xy}{l}_{\text{1}}{m}_{\text{1}}+{\tau }_{yz}{m}_{\text{1}}{n}_{\text{1}}+{\tau }_{zx}{n}_{\text{1}}{l}_{\text{1}}\right),

where σ_i and τ_j (i, j = x, y, z) are the stress components, l₁, m₁, and n₁ are vectors that can be converted to the target coordinate system.

2.7 Validation of finite element

2.7.1 Overview of the test

Until now, no test results of DoB in CFRHS K-joints are available because it is impossible to directly measure the through-the-thickness stress. However, the DoB can be derived by measuring the HSSs on the inner and outer walls of the tube according to Eq. (4). It is too difficult to arrange the strain gauges on the inner wall of the tube, in the practical sense, owing to the concrete infill. Therefore, to validate the FE models, a specimen of the CFRHS truss beam was tested and the HSSs on the outer wall of the tube at the brace-chord intersection were measured. Additionally, Ahmadi et al. [22–24] and Nie et al. [25] could not measure the stress on the inner wall and verified the model by the stress on the outer wall. The accuracy of the DoB value was ensured by controlling the number of layers through the tube thickness in Subsection 2.3.

Fig.8(a) and Fig.8(b) show the geometry of the specimen. The span and height of the truss beam are 5.3 and 6.6 m, respectively. The upper and lower chords are both CFRHS members with cross sections of b₀ × h₀ × t₀ = 120 mm × 60 mm × 6 mm, and b₀ × h₀ × t₀ = 120 mm × 100 mm × 8 mm, respectively. The vertical and diagonal braces are RHS members with cross sections of b₁ × h₁ × t₁ = 120 mm × 80 mm × 8 mm, and b₁ × h₁ × t₁ = 120 mm × 60 mm × 6 mm, respectively. The chord and brace tubes were linked together through welds specified by GB 50661-2011 [30].

Fig.9 shows the test setup. One end of the CFRHS truss beam was fixed-supported, and the other end was movable-supported. In addition, lateral displacement limit devices were placed on both sides of the truss. A concentrated force of 100 kN was vertically applied to the middle of truss by a servo-hydraulic actuator with real-time monitoring of truss mid-span deformation through displacement meter during loading [32,33]. The CFRHS truss beam was tested twice under the same concentrated force to avoid error in the strain gauges.

Fig.8(c) shows the arrangement of the strain gauges on the outer wall of the d joint during the static tests. Lines A, B, C, D, and E are the hot spot positions specified IIW-XV-E [31] in RHS joints. The strain gauge installed starts are 4 mm away from the weld toe and perpendicular to the weld. This satisfied the recommendation of IIW-XV-E (Tab.3) [31]. Finally, the strains measured from gauges could be transformed into stresses based on the generalized Hooke’s law as follows [8]:

(7)\sigma =1.1E\epsilon .

2.7.2 Test results and finite elements verification

Fig.10(a) is the 3D solid FE model of the test truss. A fine mesh with a size of approximately 1 mm was used near the d-joint, and a coarse mesh with a size of approximately 10 mm was used for the area away from the d-joint.

Fig.10(b) shows the comparison of load-displacement curves between the FE and test results. When the load P is 100 kN, the truss is in the linear elastic stage. The FE value is consistent with the measured value, and the maximum difference is only 5%. Fig.10(c) illustrates the contrast of HSS between the FE and test results. It can be observed that the calculated results using the developed FE model are in clear agreement with the test results, and the average ratio of the FE to the test results is 0.98. It could be accepted that the developed FE model is reliable for calculating the HSS, and for predicting the DoB of the CFRHS joints.

3 Parametric study

3.1 Parameter design

A total of 190 models were established based on the verified FE models to evaluate the influence of geometric parameters on the DoB of CFRHS K-joints under a balanced axial force. According to the CIDECT guidelines, the non-dimensional geometric parameters are within the following ranges: 0.4 ≤ β ≤ 1; 0.25 ≤ τ ≤ 1; 10 ≤ 2γ ≤ 35; 30° ≤ θ ≤ 60°. The specific parameters are listed in Tab.6. The following assumptions were made.

1) The center lines of the two braces intersect the chord center line without geometric eccentricity of the joints (e = 0).

2) The lengths of two braces are equal.

3) The angles between the two braces and chord are equal (θ₁ = θ₂).

4) without considering the influence of tube height-to-width ratio (h/b) on DoB, thus the chord is assumed to be a square section.

5) The gap joints (g′ > 0) are normally adopted in bridge engineering. Therefore, overlapped joints (g′ < 0) are not considered.

6) The width b₁ of the brace is the dimension along the transverse direction of the joint, that is, the x coordinate direction in Fig.4.

The geometric dimensions of the joints were based on the CFRHS joint of the Huang-Yan Bridge [8] in China. The width and height of the chord were both 400 mm, the thickness of the chord ranged from 11 to 20 mm, the width and height of the brace ranged from 160 to 400 mm, and the thickness of the brace ranged from 3 to 40 mm. The length of the brace was 1500 mm. To eliminate the influence of end restraints on stress distributions at the brace-chord intersection according to the Saint Vernet principle, the length of the brace should be greater than 6 times the chord diameter. Therefore, the length of the chord was set to 3000 mm. For certain parameter designs, there were overlapping joints (g' < 0) with the following parameters: β = 0.55, 2γ = 35, τ = 0.5, θ = 60°, g = –23, which are mutually exclusive to the assumed conditions, so they were ignored. Thus, there were 190 effective models exclusive of the joints that did not satisfy the assumptions mentioned above.

3.2 Extraction of degrees of bending from the finite element model

A typical model was selected to present the steps of DoB extraction. Firstly, the stress field of joints was obtained, and the HSSs at the side of weld toes and weld heels were extracted. Fig.11 shows the stress diagram for the CFRHS K-joint (β = 0.4, 2γ = 10, τ = 0.75, θ = 45°) under a balanced axial force. Secondly, the location to extract the DoB was determined. The position of the maximum HSS (HSS_max) is vital for accurate fatigue assessment [20], as it presents the most likely location for fatigue failure occurrence. Therefore, this study only analyzed the DoB at the HSS_max point. Finally, the DoB was calculated in combination with Eq. (4) in Subsection 2.5. Fig.11 shows the distribution of HSS at the weld toe and weld heel of the joint. The HSS_max point was located at the D point of the weld toe in the tensile brace, so DoB = 0.88 of D point was extracted as the analysis data.

However, the location of HSS_max was affected by the change of joint geometric parameters. The fatigue life predictions were commonly connected with cracks in the chord member, and the HSS_max always occurred on the chord [34]. To provide an accurate prediction of DoB on the chord, this study only analyzed Lines B, C, and D. Fig.12 shows the influence of four variables on the HSS_max location. As shown in Fig.12(a), with the increase of β, the proportion of HSS_max at point D increases, while that at point C decreases. When β = 0.85, there is almost no HSS_max at point C. As shown in Fig.12(b), with the increase of 2γ, the proportion of HSS_max at point C increases, while that at point B decreases. As shown in Fig.12(c), the proportion of HSS_max at point B increases, and that at point C decreases with the increase of τ. As shown in Fig.12(d), with the increase of θ, the proportion of HSS_max at point C increases, while those at points B and D decrease. In addition, the HSS_max at points B and D are almost equal in proportion to the change of θ.

3.3 Effects of the concrete on degrees of bending

To evaluate the influence of concrete on DoB, Fig.13 shows the comparison of DoB between CFRHS and RHS K-joint in the parametric case of β = 0.4, 2γ = 10, τ = 0.75, and θ = 45°. It can be found that after filling the chord with concrete, σ_O, σ_I, and σ_M decreases, however, the DoB increases. For tubular joints with a defect, a higher DoB is related to a smaller crack driving force, which means that the fatigue performance of RHS joints is improved after filling the chord with concrete. As shown in Fig.11, the deformation of the tension brace was greater than that of the compression brace. The reason for this phenomenon is that the infilled concrete significantly limits the vertical deformation of the brace under compression, but the brace under tension is less restricted. The action of concrete leads to the redistribution of three-dimensional stress field of joints, making it necessary to study the DoB of the joints after concrete filling.

3.4 Effects of β on degrees of bending

Fig.14 shows the effects of β on DoBs. The figure was drawn by varying β ratios (β = 0.4, 0.55, 0.7, and 0.85) and keeping τ and θ ratios constant (τ = 0.25, θ = 30°). It can be found that with the increase of β, DoB decreases, which indicates that β has significant effect on DoB. Also, the downward trend of DoB slows down with the increase of 2γ under constant β. As β increases, the width of the chord decreases, resulting in a smaller cross-sectional area of the chord. Therefore, the membrane stress per unit area of the cross section of the chord becomes larger, resulting in a higher ratio of membrane stress to the total stress. It is demonstrated that a wider chord will lead to a lower value of DoB, which can also be explained in the fact that the axis and rotational stiffness of the chord increase with decreasing β ratios, which brings about higher bending stress due to balanced axial force.

3.5 Effects of γ on degrees of bending

Fig.15 shows the influence of parameter γ on the DoB under a balanced axial force. The figure was drawn by varying 2γ ratios (2γ = 10, 15, 20, 25, and 30) and keeping θ and β constant (β = 0.7, θ = 45°). The DoB increases with an increase in 2γ. Changes in the DoB were also significantly affected by 2γ. This indicates that the thickness of the chord has a significant effect on the DoB under constant chord width. With the increase of γ, the chord width increases, resulting in a larger cross section area of the chord. Thus, the membrane stress per unit area of the main cross section becomes smaller, resulting in a lower ratio of membrane stress to total stress. Furthermore, the wider the chord width, the greater the value of DoB. It is also attributed to the chord filling with concrete which makes the stress distribution of the chord more uniform for lower 2γ ratios.

3.6 Effects of τ on degrees of bending

Fig.16 depicts the influence of the parameter τ on the DoB under a balanced axial force. The figure was presented by varying τ ratios (τ = 0.25, 0.5, 0.75, and 1) and keeping θ and 2γ constant (θ = 30°, 2γ = 30). For a balanced axial force, the DoB decreases with an increase in τ. As τ increases, the chord thickness decreases, resulting in a smaller cross section area of the chord. Thus, the membrane stress per unit area of the chord cross section becomes larger, resulting in a higher ratio of membrane stress to total stress. It is demonstrated that a thinner chord will lead to a lower value of DoB. This also can be explained that the axis and rotational stiffness of the chord decreased with increasing τ ratios, which contributes to a smaller bending stress.

3.7 Effects of θ on degrees of bending

Fig.17 presents the influence of parameter θ on the DoB under a balanced axial force. The figure was plotted by changing the θ (θ = 30°, 45°, and 60°) and keeping β and τ ratios constant (β = 0.4, τ = 1). The DoB increases gradually with an increase in θ. The results show that the value of DoB fluctuates less with the change in parameter θ. Both the horizontal and vertical components of the axial force of the brace contribute to the distribution of the chord wall thickness stress field. As θ increases, the vertical force component becomes larger, and the horizontal force component becomes smaller. Therefore, the effects of θ on DoBs can be ignored due to the contribution of the horizontal and vertical component of the axial force to the bending stress would cancel each other out.

4 Design equations to prediction degrees of bending of CFRHS K-joints

4.1 Parametric equations

Considering design, the parametric equation to predict DoBs for CFRHS K-joints was proposed by using multiple regression analysis. Limited number of general equations are available for fitting. However, two fitting choices are available, namely Fourier expressions and polynomials. The latter is more consistent with the FE results, easier to design, and more consistent with existing equations. Therefore, the general equation of DoBs was selected based on previous studies [20] for CHS K-joints as follows:

(8)\begin{array}{rl}DoB=& c_1(c_2+c_3\beta -c_4\beta \gamma ){\beta }^{c_5}{\left(2\gamma \right)}^{c_6}{\tau }^{c_7}{\sin }^{c_8}\theta (g^{\prime }{)}^{c_9}\\ & \cdot (c_{10}+c_{11}{\beta }^{c_{12}}{g}^{\prime c_{13}}),\end{array}

where constants c₁ to c₁₃ should be determined by regression analysis.

Several trials have been done to make the calculated results in agreement with the FE results, and the DoB parametric equation can be expressed as follows:

(9)\begin{array}{rl}DoB=& (0.80549-1.17278\beta +1.15409{\beta }^2){\left(2\gamma \right)}^{0.03701}\\ & \cdot (1.35404+0.048884\tau -0.04372{\tau }^2)\\ & \cdot {\sin }^{0.92119}\theta (g^{\prime }{)}^{0.24689},R^2=0.955.\end{array}

The value of R² was quite high, therefore, Eq. (9) satisfied accuracy requirements.

4.2 Assessment of the parametric equation

To evaluate the DoB parametric Eq. (9), the method described by Shao and Lie [35] was used. The relative error between the DoB values extracted from the FE analysis and Eq. (9) was shown as follows:

(10)e=\frac{Do{B}_{\text{P}}-Do{B}_{\text{R}}}{Do{B}_{\text{R}}}\times 100\mathrm{\%},

where, e is the relative error, DoB_P and DoB_R are DoBs predicted from the parametric equation and FE analysis, respectively. Fig.18 shows the relative errors for all 190 models. It can be found that all values of e are less than 5%, which indicates a good agreement between DoB_P and DoB_R.

The United Kingdom Department of Energy (DoE) [36] also suggested the judgment criteria for evaluating the accuracy of proposed equations. If (% P/R < 1.0) ≤ 25% and (% P/R < 0.8) ≤ 5%, it can be considered that the parametric equation can be accepted. P is the predicted DoB from the parametric equation and R is the extracted DoB from FE or HSS test. In Eq. (9), the (% P/R < 1.0) and (% P/R < 0.8) are 2.3% and 0%, which satisfies the above judgment criteria. It can be derived that Eq. (9) is accepted. In addition, the corresponding COV is 0.061, and the mean value of DoB_P-to-DoB_R is 1.002, which confirms a high reliability and accuracy of the parametric equation.

Although Eq. (9) is accepted, there are some value points below the line of Y = X, shown in Fig.18. This is understood as the predicted result, which is overestimated. The joints with a smaller DoB have a larger membrane stress through the wall thickness, which is considered dangerous. The predicted value of DoB when less than the extracted value, by the actual FE analysis, implies that the prediction of the residual fatigue life is conservative [23]. A lower-bound value, when derived from the mean minus three times the standard deviation is ensured, and 99.73% of the data is guaranteed to be within the safe range, the parametric equation can be considered conservative. The correction factor f is proposed as follows:

(11)f=\frac{Do{B}_{\text{D}}}{Do{B}_{\text{p}}},

where DoB_D is the design value of the DoB predicted from the final parametric equation.

It can be concluded that the correction factor f was 0.95. The final parametric equation for prediction the DoB of CFRHS K-joints can be expressed as follows:

(12)Do{B}_{\text{D}}=0.95\times DoB,

(13)\begin{array}{rl}Do{B}_{\text{D}}=& 0.92(0.80549-1.17278\beta +1.15409{\beta }^2)\\ & \cdot {\left(2\gamma \right)}^{0.03701}(1.35404+0.048884\tau -0.04372{\tau }^2)\\ & \cdot {\sin }^{0.92119}\theta (g^{\prime }{)}^{0.24689}.\end{array}

To Eq. (13), the (% P/R < 1.0) = 10% and (% P/R < 0.8) = 1.5% satisfies the above judgment criteria. Therefore, Eq. (13) is accepted.

4.3 Comparison of degrees of bending between CFRHS K-joints and CHS K-Joints

The effect of the concrete on fatigue behavior was further evaluated by a comparison of CFHS joints and hollow section (HS) joints. Up until now, only SCFs and DoBs research has been completed for CFRHS and CHS K-joints. Morgan and Lee [34] provided the parametric equations for SCF_max and DoB for CHS K-joints. Jiang et al. [10] provided the parametric equations for the SCF_max for CFRHS K-joints. Therefore, a comparison of the CFRHS and CHS K-joints was made based on the parameters shown in Tab.6.

Fig.19 shows the comparison of the SCF_max values between CHS and CFRHS K-joints under a balanced-axial force. The mean value of the SCF_CFRHS-to-SCF_CHS is 2.011. This indicates an increase of 101.1% in SCF_max for CFRHS K-joints, compared to CHS K-joints. Fig.20 shows the comparison of DoB values between the CHS and CFRHS K-joints under a balanced-axial force. The mean value of the DoB_CFRHS-to-DoB_CHS is 0.944. This indicates a 6.4% reduction in DoB for CFRHS K-joints, compared to CHS K-joints. This relationship may be due to the influence of the concrete core, which increases the stiffness of the chord and restricts the deformation of the chord tubular shape, leading to more bending stress.

Additionally, a reduction of 3.1% in DoB and 101.1% growth of SCF for CFRHS K-joints, compared to CHS K-joints, are seen. Therefore, using only SCF to evaluate the fatigue behavior without considering the influence of the DoB would result in errors.

5 Application of degree of bending

The application of the proposed DoB equation, improves the accuracy of the commonly used fatigue assessment in S-N curve methods and fracture mechanics methods.

5.1 S-N curve method

The HSS S-N curve method uses SCF as the main variable to evaluate the fatigue details, but it does not consider the through-the-thickness stress. To improve the S−N curve of HSS accuracy, a correction factor can be introduced to solve the problem of low DoB results leading to unsafe evaluation of fatigue life. When the DoB of the joint is lower than the critical value DoB₀, the result of the fatigue life determined by the HSS S−N curve is corrected, and the modified formula is as follows:

(14)N=N_0{\left(\frac{DoB}{Do{B}_0}\right)}^{\alpha },

where DoB is the predicted values by Eq. (13). N₀ is the fatigue life obtained by the S-N curve method of HSS, and N is the revised value considering the stress distribution through the tube wall thickness. The α and DoB₀ were determined experimentally and were related to the geometry and material of the joint and weld.

5.2 Fracture mechanics method

In bridge engineering, the fatigue of weld joints in fatigue detail, is usually high cycle fatigue whose stress is mainly linear elastic. Therefore, this paper proposes a fatigue assessment process for weld joints based on linear elastic fracture mechanics, which can be divided as follows.

1) The stress amplitudes of the weld joints are calculated according to the geometry and load of the weld joints. The initial crack depth a_0, and crack length c₀, as well as the crack depth limits a_c, and crack length limits c_c, were given as input conditions.

2) The SIF and the SIF ranges ΔK are calculated.

In the linear elastic fracture mechanics assessment process, the residual fatigue life of the cracked joint was evaluated according to the SIF at the crack tip, which controls the rate of crack growth. The commonly used crack propagation law, is the Paris’ law [37]:

(15)\frac{\text{d}a}{\text{d}N}=C(\mathrm{\Delta }K{)}^m,

where according to Aaghaakouchak et al. [38], ΔK can be expressed as follows:

(16)\begin{array}{r}\mathrm{\Delta }K=\mathrm{\Delta }{\sigma }_{\text{h}}\left[M{k}_{\text{m}}{Y}_{\text{m}}\left(1-DoB\right)+M{k}_{\text{b}}{Y}_{\text{b}}DoB\left(1-a/t\right)\right]\sqrt{\text{π}a},\end{array}

where, C and m are the material constants. Mk_m and Mk_b are the weld-toe magnification factors corresponding to σ_M and σ_B, respectively. Y_m and Y_b are the plain plate shape factors corresponding to σ_M and σ_B, respectively.

3) The size of the crack-tip propagation step is calculated.

The appropriate crack depth propagation step size Δa_i is selected, which is generally advisable for Δa_i = 0.1a_i–1. The smaller the propagation step size, the longer the calculation time. The propagation step if too large, results in a large error in the calculation of crack propagation.

According to Paris’ law, the load cycles corresponding to all points on the crack surface of the same propagation step, are equal. Δc is be calculated as follows:

(17)\mathrm{\Delta }c=\mathrm{\Delta }a{\left(\frac{\mathrm{\Delta }{K}_{\text{c}}}{\mathrm{\Delta }{K}_{\text{a}}}\right)}^m.

The crack depth a_i and crack length c_i is calculated as follows:

(18)a_i=a_0+\mathrm{\Delta }{a}_i,

(19)c_i=c_0+\mathrm{\Delta }{c}_i.

4) Determine whether the crack expands according to the decision conditions. When the decision conditions are satisfied, the crack continues to expand. Otherwise, crack propagation terminates.

The decision conditions are

(20)\mathrm{\Delta }K>K_{\text{th}},

where K_th is the threshold value of fatigue crack propagation, and its value is related to the component material.

The crack shape should also be satisfied as follows:

(21)a_i⩽a_{\text{c}}\text{or}{c}_i⩽c_{\text{c}},

5) Remaining life predictions. According to Paris’ law [37], the fatigue life is calculated as follows:

(22)N={\int }_{a_0}^{a_{\text{f}}}\frac{1}{C{\left(\mathrm{\Delta }K\right)}^m}\text{d}a,

where a_f is final crack depth.

Fig.21 shows a summary of the fatigue life prediction process of weld joints based on the fracture mechanics method.

5.3 Remaining fatigue life prediction of CFRHS K-joints

The remaining fatigue life of the CFRHS K-joints was predicted according to Fig.21. A three-dimensional stress field near the brace-chord intersection was obtained by the combination of the SCF equations along the tube outer wall, given by Jiang et al. [10], and the DoB equation through the tube thickness presented in this paper. The remaining life prediction of CFRHS K-joints under a balanced-axial force theory put the data in Tab.7 into practice. The remaining fatigue life can be calculated as follows:

(23)N=\sum _{i=1}^n\mathrm{\Delta }{N}_i=\frac{1}{C}\sum _{i=1}^n\left(\frac{\mathrm{\Delta }{a}_i}{{\left(\mathrm{\Delta }K\right)}_i^m}\right),

where the number of sub steps for crack propagation analysis represented symbolically by i, Δa_i is the increment of crack propagation, and ΔN_i is the crack propagation life corresponding to Δa_i..

Tab.8 shows the remaining fatigue life prediction of the CFRHS K-joints with initial cracks under a balanced axial force. The SCFs of joint 2 and joint 1 are similar, but the DoB difference is 0.29, and the residual fatigue life difference is 1.583 million cycles. Additionally, a reduction of 37.17% was found in the DoB, resulting in a decrease of 66.85% in the fatigue life. The CFRHS K-joints with same HSSs, may have completely different fatigue lives due to the different DoBs.

6 Conclusions

The objective of this study was to improve the accuracy of fatigue assessment methods including the S-N curve method and fracture mechanics method for CFRHS K-joints, by considering the through-the-thickness stress characterized as DoBs. The FE model was developed and validated using the test result of a CFRHS truss. A database of DoBs was created and 190 numerical models were established under the balanced axial force, and four non-dimensional geometric parameters were evaluated. The parametric equation to predict DoBs of CFRHS K-joints was proposed through multiple regression analysis.

The main conclusions are as follows.

1) Compared with RHS joints, the σ_O, σ_I, and σ_M at CFRHS joints decreased, however, the value of the DoB increased, which indicated that the fatigue performance of RHS joints is improved after filling the chord with concrete.

2) The DoB values were positively associated with β and τ, and negatively related with 2γ. θ had little effect on the DoB.

3) An accurate evaluation of the proposed parametric equation showed that they matched well with the numerical results with R² being equal to 0.955, and the relative errors of DoB_P and DoB_R are less than 5%. In addition, the proposed parametric equation also satisfied the judgment criteria of DoE.

4) A reduction of 6.4% in DoB and 101.1% increase of SCF in CFRHS K-joints, was compared to CHS K-joints. Utilizing only SCF to evaluate the fatigue behavior without considering the influence of the DoB would cause errors.

5) To consider the influence of DoB, a correction factor was introduced to the HSS S-N curve, and the remaining life prediction process based on fracture mechanics for weld joints was presented.

6) A reduction of 37.17% was found in the DoB, resulting in a decrease of 66.85% in the fatigue life. The CFRHS K-joints with the same HSSs, may have completely different fatigue lives due to the different DoBs.

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