Research on the Effect of Geometric Parameters on the Stress Concentration Factor of Multi-Planar KK-Joints and Carbon Fiber-Reinforced Polymer Wrapping Rehabilitation with Numerical Simulation

Abstract

The focus of this paper is on estimating the stress concentration factor of circular hollow section KK-joints with different geometric parameters and subsequently assessing the effectiveness of carbon fiber-reinforced polymer (CFRP) wrapping for repairing joints with cracks. Different geometric parameters, such as θ (brace inclination angle), γ (the ratio of the outer diameter to the wall thickness of the chord), and τ (the thickness ratio of the brace to the chord), were studied to investigate changes in stress concentration using numerical simulation. The results indicated that the stress concentration factor was most sensitive to changes in θ, followed by γ. Subsequently, the effect of crack length and depth was analyzed to simulate cracks in joints subjected to reciprocating load. The results showed that changing D from T/16 to T/2 (where T is the thickness of the chord) can cause more stress concentration, with an average of 8.37%. Next, damaged joints were wrapped in carbon fiber-reinforced polymer as a repair. Analysis of the effects of different layers and directions of polymer wrap revealed that even six layers of wrapping effectively reduced the stress concentration compared to the initial model. Finally, based on the results of parametric analysis and nonlinear fitting, a calculation formula for the stress concentration factor suitable for KK-joints under axial loads is proposed.

1. Introduction

In recent years, circular hollow section (CHS) structures have been widely used in various structures [1] due to their inherent advantages, such as the absence of an obvious bending weak axis and good economic benefits. As a core component of load transfer, the mechanical properties of intersecting joints play a decisive role in the stiffness, strength, and stability of the entire structural system. In civil engineering and marine engineering, the welded intersecting joints, such as X-joints and KK-joints (shown in Figure 1), are often the weakest part of the whole structure [2]. Especially when subjected to the action of high stress and local long-term reciprocating loads, the interior stress state of the multi-planar tubular joint is more complex. A slight change can have a greater impact on the stress concentration. Usually, the stress concentration factor (SCF) is an important reference to describe joint fatigue and evaluate the magnitude of the stress concentration. An increase in the SCF can even ultimately cause damage to the joint.

Joint damage is now common in heavy structures. Meanwhile, there are new requirements, especially in extremely important engineering structures, caused by the action of increasing loads and a change in function, so an increase in the SCF occurs. The reinforcement of joints through the use of carbon fiber-reinforced polymer (CFRP) wrapping has emerged as an approach to increase the fatigue life of intersecting joints. CFRP wrapping has the advantages of a high strength-to-weight ratio and corrosion resistance, and a remarkable decrease in the SCF has been observed [4,5,6,7,8,9]. The SCF of many types of joints has been studied [10,11,12], but not that of CFRP-wrapped KK-joints. Based on the influence of geometric parameters, revealing the effect of CFRP reinforcement on the SCF of damaged multi-planar CHS KK-joints is of great significance.

To explain the SCF distribution and SCF changes, many studies have found dimensionless parameters useful. The research on the influence of geometric parameters in uni-plane intersecting joints is relatively clear, and some SCF estimation methods have been widely used, like the IIW-XV-E [13] method. Shao [14] studied different dimensionless parameters for tubular T- and K-joints subjected to brace axial loading, found that γ and β parameters had different effects on the stress distribution, and further indicated that geometrical parameters affect the location of the peak stress along the weld toe. Furthermore, in a different loading condition, Cao et al. [15] established a highly accurate model considering residual stress. They summarized that variations in α, γ, β, ζ, and θ will not lead to movement of the hot spot, and increases in α, γ, and θ generally result in an increase in the SCF of both the chord and the brace. Chen et al. [16] proposed that in the intersecting area, there exists a coupling action on the SCF caused by the dimensionless parameters of a K-joint. However, the majority of studies are limited and neglect the complexities of engineering structures in practice.

Over the past decade, the evaluation of the SCF of multi-planar intersecting joints has typically been based on a modification of those for uni-planar intersecting joints. Additionally, several correction formulas have been proposed, such as those by Woghiren et al. [3] and Ahmadi et al. [17] and another by Ahmadi et al. [18]. In practice, based on the multi-planar joint design procedures presented in the CIDECT manual and API RP 2A [19], these methods cannot fit the new requirements [20], as the difference between the uni-planar SCF and the multi-planar SCF increases as the considered position along the weld toe becomes closer to the saddle position [21]. In addition, due to the complexity of the stress state, the sensitivity of the hot spot area to different dimensionless parameters varies. There has been considerable effort [3,17,21,22,23] aimed at improving multi-planar joint fatigue design equations. In particular, Woghiren et al. [3] agreed that the magnitude of the multi-planar effects depends on the load patterns and the relative geometrical locations of the brace members. Lotfollahi-Yaghin et al. [24] reported that the LR equations [22] proposed for uni-planar KT-joints to compute the SCFs in multi-planar DKT-joints can lead to considerably subpar results, and even the actual SCF is 2.27 times bigger than the SCF value of the corresponding uni-planar KT-joint. And Ahmadi [17] proposed that the multi-planarity effect is important in the stress distribution at the brace-to-chord intersection areas. They summarized that the finite element (FE) model with weld can be underestimated by 20% compared to that disregarding weld [25,26,27,28]. Thus, it can be seen that accurately evaluating the fatigue of multi-planar joints is relatively difficult and can be studied further.

CFRP, used as repair material in uni-planar and multi-planar tubular joints, plays an important role in reducing fatigue damage and ovalization displacement around the weld. It can smooth the plug area, hinder the occurrence of yielding stresses, and allow yielding to occur at the crown point to decrease the SCF [4,5,29,30]. Others [31,32,33] have also investigated the effect of FRP on the SCF in intersecting joints. Lesani et al. [29,30] investigated T/Y-joints wrapped with glass fiber-reinforced polymer (GFRP) and concluded that GFRP could significantly improve fatigue strength. Pantelides et al. [4] and Fam et al. [5] successively reinforced cracked aluminum CHS K-joints using unidirectional FRP plates, which effectively restored their static ultimate bearing capacity. Fam et al. [5] provided evidence against Pantelides’ [4] conclusion that CFRP is more effective than GFRP. Different wrapping methods and different repair layers also have a significant impact on the SCF, as different layers of FRP provide different condensation effects, while fiber orientation affects the force transmission path [32,33,34,35,36]. Fu et al. [35] and Tong et al. [35] studied the effects of different FRP wrapping and reinforcement layers. Their results showed that the SCF range is 15% to 20%. Sadat et al. [35] investigated the influence of several FRP parameters on the SCF; they revealed that the SCF of a 1 mm thick carbon/epoxy layer decreased by about 50% with different parameters. On this basis, Sadat et al. [34] conducted a detailed numerical analysis of FRP thickness and fiber orientation. However, the majority of these studies are limited to KK-joints.

Overall, research on the effects of multi-planar geometric parameters on the SCF has predominantly focused on other joints, while comparatively less attention has been given to KK-joints. In light of the considerations above, this study offers the following advantages: (1) A SCF database was established based on 243 models, encompassing five geometric parameters of KK-joints and four parameters of reinforcement. (2) Parameter sensitivity and CFRP rehabilitation efficiency were analyzed and appraised. On this basis, this study examined CFRP wrapping for rehabilitating cracked KK-joints and investigated the effect of CFRP layers, CFRP orientation, crack length, and crack depth on the SCF. Furthermore, 48 models were selected for advanced studies with FRP to enhance the fatigue performance of cracked CHS-KK-joints. Finally, this study conducted an error analysis of the formula used to predict the SCF in different situations, which also analyzes the fitting relationship between different geometric parameters and the SCF of CHS KK-joints. In engineering, this formula can calculate the stress concentration factor for KK-joints under axial loads in various conditions, and it can predict SCF in different scenarios.

2. Verification of the FE Model

Because few tests were carried out on CHS KK-joints, this section uses FE software ABAQUS 2021 to verify the fatigue strength of CHS K-joints subjected to axial loads, as studied by Tong et al. [35]. Two specimens, including K1-13 and CK1-13 in Tong’s test, were chosen for the verification of the FE model in this section.

2.1. Establishment of the FE Model

2.1.1. Component Creation and Material Properties

The geometrical parameters and shapes of the chord and brace were consistent with those set by Tong et al. [35], as shown in

Table 1. Parameters of test specimens.

Specimen No.		Nominal Geometric Parameters						Number of CFRP Layers
Unstrengthened	Strengthened	Chord d0 × t0 (mm)	Brace d1 × t1 (mm)	θ	2γ	τ	β	Chord CFRP (ncf0) 1	Connecting CFRP (ncf1)
K1-13	CK1-13	219 × 8	127 × 6	45°	27.4	0.75	0.58	1	3

¹ n_cf0/1 means the wrapping layers.

. To ensure the accuracy of the verification results, FE verification models and CFRP parameters were fully consistent with Tong’s test, as exemplified in Figure 2. Furthermore, as the primary factor causing the change in structural stiffness and stress concentration [17], the weld is considered, as shown in Figure 3, to reflect the change in stiffness at the connection part.

In Tong’s test, all steel tubes were made from low-carbon structural steel of grade Q235B, conforming to Chinese standard GB/T700-2006 [37], with a yield stress of 235 MPa. In this section, a simplified bilinear stress–strain model, comprising the elastic and hardening stages, was used in this FE model. The density is 7850 kg/m³, elastic modulus is 201 GPa, Poisson’s ratio is 0.28, and plastic strain is 0.31. However, the nominal stress and strain above should be converted to the true stress and strain. To accurately analyze the mechanical properties, it is also necessary to consider the damage evolution of steel. Therefore, the Bao–Wierzbickis constitutive model [38] was used, and its constitutive equation is expressed as:

$${\epsilon }^{-pl}=\left\{\begin{array}{ll}\infty \hfill & \eta \le {\eta }_0\hfill \\ {\displaystyle \frac{C_1}{1+3\eta }}\hfill & -{\eta }_0\le \eta \le 0\hfill \\ C_1+(C_2-C_1){\left({\displaystyle \raisebox{1ex}{$\eta $}\!\left/ \!\raisebox{-1ex}{${\eta }_0$}\right.}\right)}^2\hfill & 0\le \eta \le {\eta }_0\hfill \\ {\displaystyle \frac{C_2{\eta }_2}{\eta }}\hfill & {\eta }_0\le \eta \hfill \end{array}\right.$$

(1)

where η₀ is taken as 1/3, and C₁ and C₂ are taken as 0.8 and 1.33275 [39], respectively.

In this paper, the material properties of CFRP were consistent with those used by Tong [35]. However, the failure of material strength is a gradual process, meaning that after reaching ultimate strength, CFRP bears a decreasing load until it fails at the maximum strain [40]. Therefore, the Hashin constitutive model was used, and its damage degradation is expressed as:

$$\begin{array}{c}\left\{\begin{array}{cc}{\left({\displaystyle \frac{{\sigma }_{22}}{Y_T}}\right)}^2+{\left({\displaystyle \frac{{\sigma }_{12}}{S_{12}}}\right)}^2=1\hfill & {\sigma }_{22}>0\hfill \\ {\sigma }_{11}=X_c\hfill & {\sigma }_{11}<0\hfill \\ {\left({\displaystyle \frac{{\sigma }_{11}}{X_T}}\right)}^2+{\left({\displaystyle \frac{{\sigma }_{12}}{S_{12}}}\right)}^2=1\hfill & {\sigma }_{11}>0\hfill \\ {\left({\displaystyle \frac{{\sigma }_{22}}{2Y_T}}\right)}^2+\left[{\left({\displaystyle \frac{Y_C}{S_{12}}}\right)}^2-1\right]{\displaystyle \frac{{\sigma }_{22}}{Y_C}}+{\left({\displaystyle \frac{{\tau }_{12}}{S_{12}}}\right)}^2\hfill & {\sigma }_{22}<0\hfill \end{array}\right.\\ d_{\mathrm{c}}={\displaystyle \frac{{\delta }_{eq}^f\left({\delta }_{eq}-{\delta }_{eq}^0\right)}{{\delta }_{eq}\left({\delta }_{eq}^f-{\delta }_{eq}^0\right)}},M=diag\left(\begin{array}{ccc}{\displaystyle \frac{1}{1-d_f}}& {\displaystyle \frac{1}{1-d_m}}& {\displaystyle \frac{1}{1-d_s}}\end{array}\right)\\ \left\{\begin{array}{c}{d}_f=\left\{\begin{array}{cc}{d}_f^t& {\sigma }_1>0\\ d_f^c& {\sigma }_1<0\end{array}\right.\hfill \\ d_s=1-\left(1-d_f^t\right)\left(1-d_m^t\right)\left(1-d_m^c\right)\hfill \\ d_m=\left\{\begin{array}{cc}{d}_m^t& {\sigma }_2>0\\ d_m^c& {\sigma }_2<0\end{array}\right.\hfill \end{array}\right.\end{array}$$

(2)

where d_c was 0, CFRP became damaged, CFRP was completely damaged, and d_c reached 1.

2.1.2. Mesh, Load, and Boundary Conditions

The model was a symmetrical loaded multi-planar K-joint. By establishing an 8-node linear-reduced integral element C3D8R with mesh adaptation and adjustment, the model possessed reasonable hourglass stiffness and mesh density. Then, it considered welding, putty, and FRP wrapping under cyclic loading. CFRP materials were simulated using a two-dimensional (8-node), double-bending, thin-shell element (S8R5). The S8R5 element is suitable for modeling small deformation materials within the elastic range, compared to the 4-node hyperbola shell (S4R). Since the SCF only involves calculations in the elastic stage before yielding, the S8R5 element is more suitable for simulating CFRP materials. And the CFRP reinforcement length was recommended by Tong et al. [35].

In this section, load and boundary conditions were set according to Tong et al. [35]. Specifically, axial pressure was applied to the top end of the chord. Ideally, two braces would be subject to equivalent axial tension and compression according to the force balance condition.

2.1.3. Interaction Setting and CFRP Direction

Because of the reinforcement effect of composite materials, the surface of the weld is uneven, affecting the performance of the fibers. To improve the accuracy of the FE model, a small amount of repair material (putty) is typically applied to the surface to create a concave fillet transition. The material parameters of the putty were tested, revealing that its ingredients include putty, glass aerosol, and resin, as well as promoters and catalysts.

Because force was applied in stages and carefully controlled to ensure that the specimens worked within a linear elastic range, CFRP would not undergo debonding or peeling. Consequently, the layup of a CFRP–adhesive composite section was modeled in ABAQUS. And a “Tie” constraint was employed between CFRP and steel tubes to streamline the calculation process.

In this paper, the wrapping direction of the circumferential fibers of the chord and axial fibers of the brace is defined as 90°, while the corresponding other direction of FRP is defined as 0°. To achieve optimum effect considering practical convenience, the FE model added two directions of 45° and −45°.

2.2. Analysis and Extraction of SCFs

The fatigue life of joints is generally assessed using the S-N curve method, which determines the number of cyclic loads that the structure can endure before fatigue failure by calculating hot spot stress at the weld. This study calculated the hot spot stress based on the linear extrapolation concluded by Ahmadi to provide sufficient information regarding the extraction of SCFs from an FE model, as shown in Figure 4.

2.3. Results Between Tong’s Test and Verification Models

The conclusion is shown in

Table 2. The comparison of SNCF between the test data and verification models.

Test [37] Result	Chord Tensile Side			Compressive Side			Brace Tensile Side			Compressive Side
	C0	C90	C180	C0	C90	C180	C0	C90	C180	C0	C90	C180
K1-13	0.37	2.7	3.1	1.32	2.44	4.13	1.59	1.38	1.91	1.37	1.54	2.80
CK1-13	0.42	2.31	2.52	1.29	2.00	3.07	1.51	1.24	1.63	1.32	1.36	2.19
Verification Models	Chord Tensile Side			Compressive Side			Brace Tensile Side			Compressive Side
	B0	B90	B180	B0	B90	B180	B0	B90	B180	B0	B90	B180
K1-13	0.42	2.6	3.3	1.3	2.44	3.3	1.57	1.37	2.32	1.37	1.56	2.15
CK1-13	0.40	2.28	2.0	1.20	2.04	2.82	1.58	1.22	1.75	1.52	1.43	1.88

, where SNCF means the strain concentration factors in Tong’s test (the ratio of hot spot strain to the corresponding nominal strain, i.e., SNCF = ε_HSS/ε_n). It can be seen that the results of the verification models are acceptable.

3. Design Scheme for the Numerical Analysis Model

The FE tool was used to simulate the fatigue performance of KK-joints subjected to axial loads. A total of 243 finite element models were generated, and the influence of dimensionless geometric parameters on the SCF was explored. The outer diameter R of the chord was taken as 406 mm, and the length L was taken as 2460 mm, while the length l of the brace was 2200 mm. The thickness T of the chord and the thickness t of the brace were defined as variables. Then, 48 cracked finite element models were selected from the initial 243 models for advanced study. The influence of the length and depth of the joint cracks was studied. Therefore, the most sensitive parameters were identified and selected, and then the cracked joints were externally bonded with a CFRP sheet. Finally, the effect of different CFRP layers and directions on the SCF were studied. The material properties and interactions of the KK-joint model followed the relevant settings of the verification model detailed in Section 2.

Specific details of KK-joints are shown in Figure 5. In KK-joint models, the entire structure is subdivided into three areas: the refined zones (intersection area of the chord and brace), coarse zones (the part of the chord and brace far from the intersecting area), and extrapolation zones. Lie et al. [41] suggested that doubling the mesh density may lead the fatigue strength to converge rapidly. As a result, the mesh size of the refined zones of this model was 10 mm, and that of the coarse zones was 20 mm. The FE model is shown in Figure 6. An axial force was applied to the reference point at the end plate of the brace. A simplified FE model is shown in Figure 7.

4. Effect of Geometrical Parameters on SCFs

4.1. Details of Parametric Investigation

This study referred to the parameter range provided by the CIDCT specification. As a result, a database was established containing various geometric parameters of CHS KK-joints (from

Table 3. Database of CHS KK-joints.

Groups	Specimen NO.	Parameter	Range 1	Values	R = 406 mm L = 2460 mm l = 2200 mm
R12 R18 R24	O30/45/60-B0.4/0.5/0.6-T0.4/0.7/1-Gi 2	β	0.2–0.95	0.4, 0.5, 0.6
		τ	0.2–1.0	0.4, 0.7, 1.0
		θ	30–90°	30°, 45°, 60°
		γ	7.5–32	12, 18, 24

¹ The parameter range is provided by the CIDCT. ² G_i (i = 20, 60, 100) means that g is 20 mm, 60 mm, or 100 mm, respectively; g means the longitudinal clearance of the brace.

).

The 243 models were divided into three groups, i.e., R₁₂, R₁₈, and R₂₄, in which the values of γ are 12, 18, and 24, respectively. For example, the meaning of specimen R₁₈-O₃₀-B_0.5-T₁-G₂₀ in Table 3 is that the value of θ, γ, β, τ, and g are 30° (O₃₀), 18 (R₁₈), 0.5 (B_0.5), 1.0 (T₁), and 20 mm (G₂₀), respectively.

4.2. The Effect of Different Geometric Parameters on Stress Contours

For brevity, twelve representative specimen graphs in Figure 8a–l are presented and summarized here to study the effect. With different stress variations, the results showed that the stress concentration occurred in the area near the intersection of the brace and the chord. A trend can be seen where, as the geometric parameter values increase, the high-stress area in the stress contours spreads from the weld to the surrounding areas. In all models, the maximum stress generally occurs between the saddle and the crown heel. The stress concentration mainly occurs at the crown heel, while stress is low in the saddle of the intersecting area.

The results show that the stress contour undergoes significant changes with the increase in θ. From Figure 8b,e,g,i, it can be seen that with the increase in γ, the maximum stress area continues to move towards the saddle, and a low-stress area begins to appear at the crown toe. The stress in the cross-region at the saddle points of each brace is relatively low, and the stress distribution with different γ values shows approximately the same variation characteristics. For the four groups shown in Figure 8a–c,d–f,g–i,j–l, the results showed that the increase in g has little effect on the high-stress area and even alleviates the degree of the stress peak. In Figure 8b,g,i, the distribution of τ affected the stress contours, indicating that the stress concentration spreads around the weld. Additionally, the low-stress zone is compressed by the intersecting area as the brace radius increases with the increasing value of β, as seen in Figure 8a,e,g.

4.3. Effects of Geometrical Parameters on the SCF

4.3.1. Results of β

Figure 9 shows the changes in the value of β on the SCF for definite values of γ (where γ = 12, 18, and 24) to eliminate the impact of γ. An increase in β with a constant chord diameter results in an increase in the brace diameter, and the changes in the SCF are consistent with the trend of changes in the brace diameter. Figure 9a,b show the SCF increase range is from 1.86% to 5.21%. Because of the influence of multi-plane interactions, the axial force applied to a brace can generate additional stress on adjacent braces [42]. Meanwhile, the increase in the diameter of the brace caused by an increase in β leads to a larger multi-planar interaction factor (MIF) [23], and the unloaded brace will bear more loads. Figure 9c shows a larger but smoother SCF.

Further analysis shows that when the parameter γ = 24, the growth in SCFs corresponds to γ = 12 or γ = 18. However, when controlling parameter θ at 45° and 60°, specimens T_0.7-G₂₀ and T_0.7-G₁₀₀ showed a trend of first increasing and then decreasing, with the SCF of the former specimen increasing by 7.1% and decreasing by 4.3%, while the latter increased by 5.7% and decreased by 2.1%. When θ is 60°, the SCF of specimens T_0.7-G₁₀₀ and T_0.7-G₆₀ was measured while β < 0.5, and a downward trend appeared, but it was not significant. The SCF decreased by 2.9%, 5.2%, and 1.6%, respectively, due to the large gap parameter g.

4.3.2. Results of τ

Figure 10 shows the changes in the value of τ on the SCF for definite values of γ (where γ = 12, 18, and 24) to eliminate the impact of γ. A significant increase in the SCF can be concluded. As τ increases, the stiffness difference between the chord and brace decreases, which weakens the influence of the other parameters. For example, in Figure 10b,c, the ratio maintaining a lower stiffness for the chord reduced the influence of θ = 45 and θ = 45.

Figure 10a shows an increase in the SCF from 27.4% to 29.2%, with γ remaining at 12. Particularly, for many specimens, such as B_i-G_i in Figure 10b, when τ < 0.7, the increase in the SCF is about 32.5%, but there is smoothing when τ > 0.7. The reason for the above changing pattern can be that the thickness t increases with the growth in τ, leading to a larger axial stiffness, and the intersection of the brace and chord would have a larger SCF when unloaded.

4.3.3. Results of γ

Figure 11 shows the changes in the value of γ on the SCF for definite values of τ (where τ = 0.4, 0.7, and 1.0). It is concluded, based on Figure 11a,b, that increasing the diameter of the chord while simultaneously decreasing its thickness leads to a decrease in radial stiffness, which results in transmitting less load. An excessive diameter of the tubular joint will lead to a greater SCF when γ > 18, such as B₂-G_i and B₃-G_i. Similarly, the parameter γ has a greater impact on the SCF, with an average increase of 26.98% in Figure 11c. Regardless of whether the values of both γ and θ are low or high, the effect of the multi-plane interaction is weak. Therefore, a thinner chord thickness significantly increases the weld impact and exacerbates the changes in the SCF.

4.3.4. Results of θ

From Figure 12, it can be seen that as τ changes to 0.4 or 0.7, the variation in each histogram is basically the same, showing a significant upward trend, with an increase of 32.21–36.93%. And according to Figure 12a, the percentage change in the SCF can reach 32.95–36.36% in specimen B_i-G₃ as the force transmitted from the brace to the chord increases in a sinusoidal proportion with the increase in θ. However, the histograms show that the SCF increases for most specimens with the increase in θ, and the range is from 22.13% to 24.56% in Figure 12b,c. This is because when chord stiffness is lower, a large MIF bears more loads and simultaneously reduces the sensitivity to angle changes. But specimens B₂-G₁ and B₁-G₂ exhibit the opposite trend when θ > 45°, with the SCF decreasing by about 1% because the chord bears more axial load and less radial load. Therefore, in practice, the θ value should be considered as an important parameter in fatigue performance design. At the same time, more attention should be paid to the fatigue phenomenon of large-angle tubular joints to avoid structural damage.

4.3.5. Results of g

In the same conditions, the effect of changes in the longitudinal clearance of the brace tends to be consistent. As shown in Figure 13a,b, in this condition, the tendency of the SCF in specimens R_i-B_0.4 and R_i-B_0.5 increases by an average of 4.41%. However, when the longitudinal clearance of the brace increases, the multi-plane interaction diminishes. Moreover, the wall thickness values of the chord and the radius of the brace are not excessively large or small, and the load transmitted by the joints and brace is borne by the chord, resulting in a small stress concentration or less SCF increases in specimens R₂-B_i, R₃-B_0.4, and R₃-B_0.5. In Figure 13c, the tendency of the SCF is not well in accordance with the change pattern described above. However, when the values of g become 100 mm, each change in γ and β is significant as the influence of the multi-plane interaction diminishes. And due to the lower sensitivity of g, the change in the SCF remains significant with different angles.

5. Analysis of the SCF of Cracked KK-Joints Strengthened with FRP

5.1. The Establishment of Reinforced Models

It is well known that defects in a structure can enlarge the stress concentration, and an increased load can also enlarge the stress concentration. As a result, cracks will inevitably occur due to defects, increased load, or function extension as the service life of a tubular joint increases. However, cracks with sharp notches and a large length–width ratio tend to induce a higher stress concentration compared to other defects. In this paper, cracks were simulated with ABAQUS XEFM (Extended Finite Element Method).

Therefore, to obtain conservative results, this study artificially defined parameters as unfavorable values after considering the sensitivity to the SCF. Meanwhile, the crack parameters were described as crack length (C) and crack depth (D). The detailed naming of specimens is shown in

Table 4. Numberings of KK-joints with cracks and the FRP-reinforced grouping with one layer or two.

Groups	Specimen NO.	One-Layer-Reinforced			Two-Layer-Reinforced
		Specimen NO.	Direction 1		Specimen NO.	Direction
O30,crack O45,crack O60,crack	C4/6/8/10-D16	FRPa/b/c-a FRPa/b/c-b FRPa/b/c-c	Chord	Brace	FRPd-d FRPd-e FRPe-d	Chord	Brace
	C4/6/8/10-D8		[0°]/ [45°]/ [90°]	[0°]		[90°/0°]	[90°/0°]
	C4/6/8/10-D4			[45°]		[90°/0°]	[0°/90°]
	C4/6/8/10-D2			[90°]		[0°/90°]	[90°/0°]

¹ [0°], [45°], [90°], [90°/0°], and [0°/90°] are replaced with a, b, c, d, and e for short, respectively. The alphabet before the dash in the FRP subscript represents the FRP wrapping direction on the chord, and that after the dash means the FRP direction on the brace.

, where the meaning of specimen O_45,crack-C₆-D₈-FRP_c-a is that the values of θ, C, D, and the wrapping direction are 45° (O_45,crack), 6 mm (C₆), 1/8T (D₈), and chord wrapping direction 90° and the brace 0° (FRP_c-a), respectively.

5.2. Effect of Crack Depth and Length on the SCF

The tendency of SCF changes is shown in Figure 14a–c, where each graph selects θ, the most sensitive geometrical parameter, as a definite value to eliminate the impact of this parameter. As shown in Figure 14, when changing D from T/16 to T/2, the SCF variation pattern tends towards consistency, with an increased range of 7.82% to 8.93%. Simultaneously, the change in C leads to an increase in the SCF of 2.64% to 2.97%. Therefore, the crack depth has a greater impact on the SCF. Because deeper cracks make the material more discontinuous and even cause discontinuity in most tube sections, this results in greater mechanical stress concentration. The material concentration stress caused by defects or damage further exacerbates the mechanical stress concentration. However, when θ is 60°, changes in D can cause more stress concentration than those in C, and the overall impact of cracks is higher than other patterns.

In order to explore the impact of changes in D and C in a visual way, we normalized the SCF of the cracked CHS KK-joints obtained from different crack depths and lengths with the SCF of uncracked KK-joints to provide a reference for future practical engineering. The results are shown in Figure 14d.

Meanwhile, with the angle θ increase, the crack size will slightly exacerbate the impact of the stress concentration. It can be seen that the SCF is more sensitive to the existence and development of cracks, as cracks exacerbate the material stress concentration at intersecting joints. In practice, attention should be paid to the growth of cracks in the hot spot area, especially when monitoring the degree of crack penetration. Therefore, it is crucial to reinforce cracked KK-joints and analyze the effectiveness of FRP.

5.3. Results of Cracked Joints Strengthened with CFRP

5.3.1. Model Scheme of FRP-Reinforced Joints with Initial Cracks

The crack length and crack depth mentioned above were selected as the conservative adverse parameters to ensure reliability. The chord and brace were reinforced in directions of 0°, 45°, and 90° to explore the influence and effectiveness of the CFRP direction on the SCF. Additionally, 0°, 45°, and 90° were replaced with a, b, and c for brevity, as shown in Table 4.

5.3.2. Reinforced with One-Layer CFRP

The normalized results are shown in Figure 15, where each group takes the average values and eliminates the impact of θ. The conclusion indicates that the effect trends are consistent, with the optimal being 90° wrapping for the chord due to the strongest binding effect on the cracks.

For the brace, with the winding direction controlled consistently in Figure 15a, and only the winding direction of the chord changed, the optimization was from 1.42% to 1.87%. Similarly, for the chord, the optimization result was 1.42%. And the SCF reduction rate was from 1.33% to 2.01% when controlling the 45° chord winding direction.

When the winding direction of the FRP on the brace was controlled consistently in Figure 15b, and only that of the chord was changed, the effect of FRP reinforcement was best at 90° and then at 0°. The best SCF value was 12.963, with the best optimization rate being 1.42%. Meanwhile, the optimization range was between 0.83% and 1.31% when the brace FRP wrapping was 45°.

Similarly, as shown in Figure 15c, the SCF optimization range was between 0.6% and 0.89%. And the SCF improvement ranged from 0.96% to 1.20% when controlling the 45° winding direction of the chord, while the best optimization rate was 1.45% when maintaining the 90° direction.

5.3.3. Reinforced with Two-Layer CFRP

It can be concluded that with one-layer FRP reinforcement, when the winding direction is 45° for both the chord and brace, the impact on the SCF is not significant compared to the other two directions. Therefore, when studying reinforcement with two-layer FRP, two stacking winding angles of 0° and 90° were considered for FRP reinforcement. And it can be seen that the exchange in the winding directions between the upper and lower layers will cause changes, and the optimal direction for the chord is to choose 0° for the first layer.

When the winding direction of two stacking FRP layers on the brace was controlled at 30° in Figure 16a, the reinforcement effect was best in the first layer with a 0° direction. The values of d-e and e-e changed from 13.21 without reinforcement to 12.18 and 12.12, respectively, with the SCF decreasing by 7.79% and 8.25%.

In Figure 16b, d-e and e-e changed from 15.571 without reinforcement to 14.487 and 14.398, with decreases of 6.96% and 7.53%, respectively. Similarly, when only the winding direction of the brace FRP in the upper and lower layers changed, the FRP reinforcement effect is best when the winding direction of the first layer is 0°, achieving an average optimization of 6.93%. However, the conclusion still holds when θ = 60°, as shown in Figure 16c.

5.3.4. Reinforced with Multi-Layer CFRP

Overall, the best winding directions are 0° and 90°. On this basis, it can be seen that with multi-layer wrapping, the SCF grows steadily with increasing θ, as shown in Figure 17. And six-layer reinforcement indicates a good reinforcement effect.

The results showed that two-layer reinforcement can significantly reduce the SCF, with an optimization of 8.25%. When using four-layer CFRP reinforcement, the SCF roughly recovered to the same level as before without cracks, changing from 13.21 to 11.321, with an SCF decrease of 14.3%. And the SCF was reduced from 13.21 to 10.563, with an optimization of 20.04%. The SCF of KK-joints roughly recovered to the same level as before without cracks. When using six-layer CFRP reinforcement, the SCF is even better than the initial model, especially when θ is higher, with the optimization rate reaching 18.8%.

5.4. Analysis of CFRP-Strengthened KK-Joints with Cracks

In summary, due to the strongest binding effect on cracks, a larger CFRP winding angle can reduce the SCF, especially when winding at the chord. The reason may be that CFRP bears unidirectional force, and a larger angle can bear more loads. Therefore, controlling the 90° winding direction at the chord results in a better optimal SCF value. And vertical CFRP on the chord and brace can achieve an optimal SCF value. The conclusion of two-layer reinforcement is also consistent with that of one-layer reinforcement. It can be concluded that the optimal angle for the first layer is 0°.

When θ = 30°, the KK-joints recovered to approximately the same SCF as that of four-layer reinforcement without cracks. When θ = 45° and 60°, the KK-joints recovered to a state that was generally similar to the one without cracks after six layers of reinforcement. Through the analysis, it can be concluded that a good reinforcement effect of FRP on the SCF requires a smaller θ between the brace and chord.

6. Fitting Relationships for the SCF of KK-Joints with Different Parameters

The SCF at each intersecting area is affected by parameters β, γ, τ, θ, and g, and the sensitivity level of the SCF to them can be ranked in a decreasing sequence as θ > γ > τ > β > g.

Hence, each parameter has a varying influence on the SCF. It is necessary to find an optimal formula that reflects the impact of changes in all parameters on the SCF and provides a prediction closest to the observed values. According to the FE database and parametric study, multiple nonlinear fittings were performed using MATLAB R2018a and ORIGIN 2021:

$$SCF=(0.031\beta +0.0835)(0.052\tau +0.0627)(0.188\theta +7.525)(\ln \gamma +0.376)(0.021g+21.5)$$

(3)

where the R² of this formula is 0.93 and the range of these parameters described are above. And the applicable range of Equation (3) is a 90° angle between braces.

To verify whether the accuracy of the fitted formula meets the requirement, the following formula is proposed:

$$SC{F}_{error}={\displaystyle \frac{SC{F}_{fit}-SC{F}_{FE}}{{SCF}_{FE}}}$$

(4)

where SCF_fit is the fitting formula SCF value and SCF_FE is the values of the SCF simulation in FE software.

To verify the proposed formulas, 92 groups of geometric parameters (from

Table 5. Database of CHS KK-joints in Gao et al. [43].

Geometric Parameters	Range	Value Interval
θ	30–60°	2.00°
γ	12–24	0.5
τ	0.25–0.75	0.05
β	0.3–0.5	0.04
g (mm)	50–150	10

) carried out by Gao et al. [43], along with the SCF values of 243 models in this study, were utilized. The SCF values of the fitting formula are visually compared in Figure 18a. The observation results showed that the error values between models and the formula results were within 10%, while a small portion of error values exceeded 10%. Three groups of data were selected for detailed analysis due to the large amount of data, as shown in Figure 18b–d. In summary, there are 324 group parameters with calculated error values less than 10% and 11 with values that exceeded 10%. The accuracy of the formula established in this paper has been demonstrated. As a result, this formula can be used for the theoretical analysis of the parameters of the models in this paper and for studying the change pattern of the SCF.

7. Conclusions

This paper used numerical simulation to analyze the SCF of CHS KK-joints subjected to axial loads. A comparative analysis was conducted on the sensitivity of the SCF to different geometric parameters. On this basis, the efficiency of carbon fiber-reinforced polymer (CFRP) wrapping on joints with damaged cracks was appraised. Firstly, the effect of crack length and depth were analyzed, and then the parameters of CFRP material, including the number of layers and winding direction, were studied. Finally, a fitting relationship for the SCF of KK-joints with different parameters was established. The conclusions are as follows:

(1) The sequence according to the sensitivity level is as follows: θ has the greatest impact when g is at its smallest. The influence levels that closely followed θ are γ, τ, β, and g. The brace inclination angle should be considered an important parameter in fatigue performance design, as it results in a larger SCF increase rate of 36.93%. In addition, the difference in the wall thickness between the chord and brace should be kept relatively small. And the distance between braces should be relatively large.

(2) It is shown that with the axial load, the SCF of intersecting joints is more sensitive to D. The results showed that when θ was 60°, changes in D caused more stress concentration than those in C, and the overall impact of cracks was higher than that of other patterns. Meanwhile, with the increase in the angle θ, increasingly larger crack sizes will slightly exacerbate the impact of the stress concentration. As a result, during the repair process, special attention should be paid to deeper cracks. Therefore, CFRP reinforcement should be used preferentially for deeper and wider cracks.

(3) The results showed that when one layer of reinforcement is used, CFRP has the strongest binding effect on cracks. It is more effective to wrap CFRP on the chord with a 90° winding. The following direction is 45°. And the conclusion for the brace is exactly the opposite. However, two-layer reinforcement only considers stacking at 0° and 90°. Research has found that both a chord and brace reinforced with [0/90] and [0/90] will have the best reinforcement effect. On this basis, this research indicated that when the number of reinforcement layers exceeds four, KK-joints recover to approximately the same fatigue strength as joints without cracks. And even when θ is 45° and 60°, six-layer CFRP has higher fatigue strength. As a result, the smaller the θ and the more layers there are, the larger the advantage of the reinforcement effect of CFRP.

(4) Based on the results of parametric analysis and nonlinear fitting, an SCF calculation formula suitable for KK-joints subjected to axial loads is proposed. The error range between the calculated and simulated values of the formula is within 10%, and the R² is 0.93, indicating that the proposed calculation formula is accurate. However, the applicable range of this formula is for a 90° angle between braces, and it is suitable for non-overlapping KK-joints with small spacing between braces.