Experimental Study of Hot Spot Stress for Spatial CHS KK-Joints

Abstract

In offshore structures such as offshore wind turbine jacket foundations, prolonged exposure to wind and wave loads can lead to fatigue failure, especially at the joint connections. Currently, international codes primarily evaluate the fatigue performance of tubular joints using the hot spot stress (HSS) method, with stress concentration factor (SCF) being the most crucial parameter. Numerous studies have investigated the SCF of tubular joints and proposed corresponding calculation formulas. However, most of these studies have focused on simple basic loading cases. To comprehensively understand the distribution pattern of the HSS method at spatial tubular joints, this study selects the spatial CHS KK-joints as the research subject and performs the HSS testing on spatial KK-joints. The research indicates that, in CHS joints, the distribution of the HSS on the chord side remains consistent under tensile or compressive loading conditions along the brace axis. Additionally, within spatial joints, the loading conditions of neighboring braces near the reference brace significantly affect the distribution of HSS on the chord side and exhibit varying spatial effects on the SCF. Furthermore, this study applies recommended calculation formulas from codes to analyze the experimental results. The findings reveal significant discrepancies between the SCF calculation formulas recommended by CIDECT and the test results, rendering it inadequate for accurately calculating the SCF of spatial KK-joints. Therefore, further research is required to develop suitable calculation formulas for SCFs in spatial KK-joints.

1. Introduction

To address increasingly dire environmental and energy challenges, the development of clean energy has emerged as a pivotal direction for sustainable progress worldwide. Wind power, as a form of clean energy, has garnered escalating attention. Due to the abundant wind resources available at sea and the minimal utilization of land, the installed capacity of offshore wind power has been steadily rising [1].

The jacket structure (Figure 1) serves as a commonly employed fixed foundation configuration for offshore wind turbine systems, comprising a steel tower section, a jacket section, and pile foundations. The jacket section typically consists of multiple vertically oriented chord members interconnected by bracing members. The bracing members intersect at a single point on the chord member, forming a joint. Due to the prolonged exposure of offshore wind structures to the cyclic loads of wind and waves, fatigue effects become apparent [2], particularly at the joint where the unique structural configuration tends to induce stress concentration, thereby further amplifying the influence of fatigue loads.

Currently, the commonly used fatigue design method for joints in international codes is the S-N curve method based on hot spot stress (HSS) [3]. Research has revealed that HSS mainly occurs at the weld between the braces and the chords. Therefore, a thorough understanding of stress concentration phenomena at intersecting joints is crucial for the fatigue design of structures.

In numerous previous studies and codes, the stress concentration factor (SCF) is defined as the ratio of HSS at the weld toe to the nominal stress of the brace. Conversely, if the SCFs of the joints are known, multiplying them by the corresponding fatigue load allows for the convenient determination of the HSS at the weld toe of the joints. This method is known as the HSS method. Evidently, it is crucial to identify the corresponding SCFs for the HSS method. Researchers have conducted extensive investigations in this regard. Mohamed et al. [4] conducted SCF tests and fatigue experiments on three T-joints, subjecting three identical T-joints to heat treatment. The results indicate that fatigue cracks initiate at the saddle point of the joint, and the higher the specimen temperature, the longer the fatigue life. Furthermore, a comparison of the experimental results with CIDECT code [5] and API code [6] reveals that the calculated values of CIDECT tend to be unsafe, while those of API tend to be conservative. Lie et al. [7] and N’ Diaye et al. [8] performed HSS analysis on the tubular T-joints using the finite element (FE) method. Gho et al. [9] established an FE model consisting of 192 fully overlapped tubular K-joints and conducted parameter analysis. The results demonstrate that the SCF extreme point occurs at the saddle point of the overlapping joints, leading to the derivation of the SCF calculation formula for overlapped tubular K-joints based on the FE results. In addition, researchers have separately investigated the fatigue performance of intersecting joints in rectangular steel tubes [10,11,12]. They have conducted fatigue and HSS tests on T-, X-, and K-joints, combined with FE analysis to perform parameter analysis, and proposed corresponding SCF calculation formulas for tubular joints. In addition, Nassiraei [13] conducted a parameter analysis on the local joint flexibility of CHS T/Y-connections strengthened with a collar plate and proposed a recommended design method.

Chen et al. [14], Musa et al. [15], and Kim et al. [16] have conducted SCF experiments on circular concrete-filled steel tubular (CFST) joints and comparative specimens of circular hollow steel (CHS) joints. The forms of the joints include Y-, K-, and KT-joints. These research results all indicate that filling the chord of the steel joint with concrete can achieve a more uniform distribution of stiffness at the intersection, alleviate stress concentration phenomena, and lead to varying degrees of decrease in the SCFs. Nassiraei et al. [17] investigated HSS distribution in tubular T/Y-joints reinforced with FRP and proposed the corresponding formula for calculating SCF.

From the previous studies, it can be observed that there has been extensive research on the HSS testing of joints under basic loads. However, there is still a lack of experimental studies on the HSS testing of tubular joints with spatial joints. Marine structures, during their operational phase, confront a highly complex load environment [18]. The force conditions of tubular joints are inevitably subjected to multiple operating conditions, necessitating research on the spatial interactions of these joints.

In this study, a spatial loading test setup for static and fatigue loading of spatial tubular joints was designed. By conducting HSS tests on spatial CHS KK-joints under various operating conditions, the influence of each condition on the distribution of the HSS at the joints was investigated. Furthermore, by comparing test results with existing codes, recommendations are proposed for subsequent design work.

2. Hot Spot Stress Test

2.1. Specimen Design

To investigate the distribution of the HSS in KK-joints in different combinations of loads, the creators of this study designed and fabricated a spatial CHS KK-joints. The dimensions of the specimen were obtained by scaling down the jacket foundation structure for offshore wind turbines, with a scaling ratio of 1:10. The notations used are given in

Table 1. Notation list.

Parameters and Sets
D	Chord diameter
tc	Chord wall thickness
L	Chord length
d	Brace diameter
tb	Brace wall thickness
l	Brace length
θ	Angle between brace and chord
Φ	Angle between two planes formed by the braces
β	d/D
γ	D/(2T)
τ	t/T
α	2L/D
g	Gap between two braces
ζ	g/D
σy	Steel yield stress
E	Steel Young modulus
rc	Chord radius
rb	Brace radius

.

The diameter of the chord member D is 219.1 mm, with a wall thickness t_c of 6.4 mm and a length L of 2000 mm, approximately nine times the diameter of the chord member, while excluding the influence of end constraints on the joint region [19]. The diameter of the brace member d is 114.0 mm, with a wall thickness t_b of 6.0 mm, and the bracing member length l is set at 700 mm (Figure 2). Reinforcing ribs were installed at the ends of both the chord and brace members to enhance the end sections and prevent premature failure due to stress concentration.

The specimen is composed of four braces and one chord, with the brace ends cut into intersecting lines and directly welded to the surface of the chord. The axis of each brace forms a 45-degree angle with the axis of the chord, and the two planes formed by the braces have a 90-degree angle between them. The four braces axes intersect at a single point on the chord axis. In offshore wind turbine jacket structures, the quality requirements for welded joints are high due to long-term exposure to combined wind, wave, and current loads. The specimens in this study meet the welding requirements specified by the AWS code [20].

As marine structures continue to evolve, the demands placed on structural steel are increasing. However, for the purposes of this study, commonly used low-carbon steel still suffices to meet the testing requirements for the HSS tests. The selected steel material in this study is Q355B steel. Additionally, to ensure the overall integrity of the structure and prevent the influence of unnecessary welds on the test results, seamless steel tubes are used for all the members. Material property tests were conducted on the steel samples, and the obtained characteristics (yield stress σ_y and Young modulus E of the steel) of the specimens are presented in

Table 2. Material properties of specimen.

Members	Thickness (mm)	σy (MPa)	E (MPa)
Chord	6.4	365	219.59
Brace	6.0	397	222.82

.

2.2. Loading and Measurement System

To investigate the spatial interaction mechanism of spatial KK-joints, it is necessary to employ a loading method that combines multiple working conditions, specifically requiring the individual loading of the four braces. Therefore, a spatial loading test setup suitable for statically and dynamically loading spatial tube joints is developed in this study, as illustrated in Figure 3. This test setup primarily consists of four independently controlled MTS loading actuators, supporting columns, oblique support devices, actuator supports, and specimen supports, among other components. The stability of the connections between the various parts is ensured using high-strength friction-type bolts. Additionally, all components placed on the reaction floor are connected to the floor with prestressed steel bars, preventing any relative slippage between the parts. During the installation of the specimen, it is necessary to first utilize a laser calibration instrument to position the specimen at the center of the loading device and align the axes of the braces and actuators, ensuring that the axes of the actuators and the braces coincide, thus preventing any additional loads during the loading process. Furthermore, prior to applying the load to the specimen, it is advisable to conduct a preliminary loading test, which serves to assess the alignment between the specimen and the test setup, as well as the stability of the measurement system.

Due to the effect of notching in the weld joint, direct measurement of stress at the weld toe is not feasible during testing. The prevailing approach commonly employed is the indirect method, which is currently known as the extrapolation method. In accordance with the recommendations provided by CIDECT [5] and IIW codes [21], the extrapolation method for tubular joints primarily encompasses linear extrapolation and quadratic extrapolation. Linear extrapolation involves linear calculations based on two measurement points in the extrapolation region to determine the stress at the weld toe, whereas quadratic extrapolation entails secondary calculations utilizing multiple measurement points. The codes suggest that the linear extrapolation method should be used for circular steel tubular joints. Consequently, the linear extrapolation method for measuring HSS is utilized in the tests for this study.

The arrangement of measurement points in the extrapolation region for the linear extrapolation method is illustrated in Figure 4. As depicted in the diagram, the linear extrapolation method requires the placement of only two strain gauges in the extrapolation region of the weld toe. CIDECT [5] has provided recommendations regarding the positioning of strain gauges, as shown in

Table 3. Arrangement of strain gauges in linear extrapolation method.

Distance from Weld Toe	Chord		Brace
	Saddle	Crown	Saddle	Crown
Lr,min *	0.4·tc		0.4·tb
Lr,max **	0.09rc	$0.4\times \sqrt[4]{r_c{t}_c{r}_b{t}_b}$	$0.65\times \sqrt{r_b{t}_b}$

* Minimum value for L_r,min is 4 mm. ** Minimum value for L_r,max is L_r,min + 0.6 t_b.

. In the table, r_c represents the radius of the chord, while r_b denotes the radius of the brace. Finally, in welded tubular joints, the crown point refers to the highest point in the joints, while the saddle point refers to the lowest point in the joints.

In various international codes, there are some discrepancies regarding the stress (strain) component of SCFs. The question arises whether the principal stress (strain) or the stress (strain) perpendicular to the weld toe should be utilized. IIW [21] argues in favor of adopting the maximum principal stress (strain), while AWS [20] and API [6] advocate for using the stress (strain) perpendicular to the weld toe. Although in theory, the fatigue crack propagation direction in tubular joints tends to expand towards the direction perpendicular to the maximum principal stress, the limitations of current testing techniques prevent the precise measurement of the maximum principal stress. Additionally, the direct superposition of maximum principal stress is not feasible under different combinations of loads. Moreover, it is more convenient to arrange the strain gauges perpendicular to the weld toe in experimental measurements. Furthermore, CIDECT [5] suggests that the differences between these two types of stresses near the weld toe are not significant. Therefore, CIDECT [5] recommends measuring the strain solely using strain gauges positioned perpendicular to the weld toe, without the need for employing strain gauges that measure the principal strain. In this study, the measurement of strain perpendicular to the weld is chosen.

Hence, the formula for calculating the hot spot strain at the weld toe based on linear extrapolation is as follows:

$${\epsilon }_{\perp W}=\frac{L_{r,max}{\epsilon }_{\perp E1}-L_{r,min}{\epsilon }_{\perp E2}}{L_{r,max}-L_{r,min}}$$

(1)

where ${\epsilon }_{\perp W}$ is the strain at the weld toe obtained by linear calculation; ${\epsilon }_{\perp E1}$ and ${\epsilon }_{\perp E2}$ represent the perpendicular strain measured at the first and second measurement points, respectively, in the extrapolation region, perpendicular to the weld toe; L_r,min and L_r,max denote the distances from the weld toe to the first and second extrapolation points in the extrapolation region, as specified in Table 3. In the study, L_r,min is set to 4 mm on both the chord side and the brace side. L_r,max is determined to be 9.9 mm at the saddle point and 8.9 mm at the crown point on the chord side, while it is uniformly 12 mm on the brace side. Interpolation is employed for the points between the crown point and the brace point.

Furthermore, the strain concentration factor (SNCF) of the joints could be obtained by dividing the hot spot strain by the nominal strain.

$$\mathrm{SNCF}=\frac{{\epsilon }_{\perp W}}{{\epsilon }_N}$$

(2)

where ε_N is the nominal strain of the braces, which is mesured in this study by strain gauges.

Furthermore, according to Hooke’s Theorem, there exists a relationship between the SCF and the SNCF as expressed by the following equation:

$$\mathrm{SCF}=\mathrm{SNCF}\times \left(1+v{\epsilon }_{\parallel W}/{\epsilon }_{\perp W}\right)/\left(1-v^2\right)$$

(3)

where ${\epsilon }_{\parallel W}$ represents the strain parallel to the weld toe, calculated using the same method as ${\epsilon }_{\perp W}$, and v denotes the Poisson’s ratio of the steel material.

$${\epsilon }_{\parallel W}=\frac{L_{r,max}{\epsilon }_{\parallel E1}-L_{r,min}{\epsilon }_{\parallel E2}}{L_{r,max}-L_{r,min}}$$

(4)

where ${\epsilon }_{\parallel E1}$ and ${\epsilon }_{\parallel E2}$ represent the perpendicular strain measured at the first and second measurement points, respectively, in the extrapolation region, parallel to the weld toe.

Therefore, in order to obtain the SCF more accurately, this study employed a strain gauge parallel to the weld toe beside each strain gauge perpendicular to the weld toe. Additionally, due to the impracticality of measuring all positions along the weld toe during the experimental process, strategically positioned strain measurement points were selected in this study. In particular, the strain distribution around the weld toe of the joints under spatial effects needs to be considered. Hence, eight measurement points are placed for every brace region. The first measurement point is located at the 0° position, which is at the crown point near the side of the chord. Eight measurement points are distributed at 45° intervals along the weld toe, with the coordinate origin being the intersection of the brace axis and the surface of the chord. The positions of the four braces corresponded consistently to the strain gauges. The arrangement diagram and numbering of the strain gauges are illustrated in Figure 5. It should be noted that the points at 0°/360° and 180° are referred to as the crown points of the joints, whereas the points at 90° and 270° are referred to as the saddle points of the joints. The actual arrangement of the gauges on the specimen is shown in Figure 6.

In order to obtain the coefficient of hot spot stress, it is necessary to further measure the axial strain of the brace as a nominal strain. In this study, four uniformly distributed circumferential axial strain gauges are placed at the midpoint of the brace to avoid the influence of stress concentration at the node and end regions on the test results.

2.3. Load Design Rules

Four actuators on the test setup enable independent loading, providing various load combinations for the joints. Within the IIW code [18], three reference combinations are provided for spatial KK-type joints: the first is force-symmetric loading, the second is unilateral loading, and the third is force anti-symmetric loading, as illustrated in Figure 7.

To fully understand the stress conditions of spatial tubular joints, five loading conditions were proposed for HSS testing, as shown in

Table 4. Loading conditions for specimen.

Load Case	B1		B2		B3		B4
	T *	C	T	C	T	C	T	C
L1	1 **	−1	0	0	0	0	0	0
L2	1	−1	0	0	0	0	−1	1
L3	1	−1	−1	1	0	0	0	0
L4	1	−1	−1	1	1	−1	−1	1
L5	1	−1	−1	1	−1	1	1	−1

* T and C respectively represent the tensile or compressive conditions of the brace in the axial direction. ** One unit of load.

. Furthermore, to present a clearer depiction of the loading rules for each loading condition, the loading rules from the table are depicted in Figure 8. It should be noted that the listed loading conditions in Table 4 are implemented by first selecting a reference brace and applying the corresponding load condition to that reference brace. Given the symmetrical construction of the specimen, any brace can serve as the reference brace. Additionally, since the steel material remains in an elastic working state throughout the HSS tests, each brace is repeatedly subjected to the load conditions listed in Table 4. It is important to note that in general, marine structures subjected to long-term loading rarely experience fatigue loads that cause the material to enter the plastic working stage. Therefore, it is necessary to ensure that the stress on the steel material during the HSS tests does not exceed the yield strength of the steel.

3. Test Results

3.1. Data Processing

Before processing the experimental results, it is necessary to make certain assumptions:

1.Nominal strain gauges were installed at the midpoint positions of each brace in the test. During the preloading process prior to the start of the test, it is necessary to ensure that the values of the four nominal strain gauges under the same brace do not differ significantly. Ultimately, the nominal strain of that brace is calculated according to Equation (5) and compared with the theoretically calculated results (calculated by Equation (6)) to ensure the accuracy of the measured outcome.

$${\epsilon }_N=\frac{{\epsilon }_1+{\epsilon }_2+{\epsilon }_3+{\epsilon }_4}{4}$$

(5)

$${\epsilon }_N=\frac{{\sigma }_N}{E}=\frac{4F_A}{\pi E\left[d^2-{\left(d-2t_b\right)}^2\right]}$$

(6)

where ε₁, ε₂, ε₃ and ε₄ are nominal strains of the four measuring points along the brace.

2. During the testing process, each brace was used as a reference brace for identical load conditions. Therefore, it can be postulated that the specimen underwent four repeated tests for each loading condition. In actuality, some errors are inevitably encountered during the manufacturing and welding processes of the specimen. Furthermore, in the context of large-scale structural testing, it is unfeasible to guarantee perfect alignment of the loading apparatus. Consequently, the mean value of the results from the four repeated tests was adopted as the ultimate outcome.

3. The existing research generally holds the view that the HSS of the chord in the CHS joint is significantly higher than that of the brace. In the HSS test, the focal point is primarily on the distribution of the HSS on the side of the chord. Consequently, this study mainly focuses on the HSS distribution on the chord side of the spatial KK-joints.

3.2. Discussion

The SCF distribution curves obtained from the test are summarized as shown in the Figure 9. The following distribution patterns can be derived from the figures.

• From the overall situation, it can be observed that for the spatial CHS KK-joints, the HSS on the chord side are almost the same under both tension and compression conditions of the brace subjected to various loading conditions. This indicates that for the CHS KK-joint, a single set of calculation formulas is sufficient to determine the distribution of HSS in the chord under axial loading of the brace.
• From Figure 9a, it can be observed that when a single brace is subjected to axial loading, the SCF is approximately symmetric about the crown point. However, due to the presence of other braces at the 90° position, the stiffness at that location is greater than at the 270° position. Therefore, the SCF value at the 270° position of the chord will be larger than the SCF value at the 90° position. When a single brace is subjected to axial loading, the SCF values at all measurement points on the chord are greater than 1, with the maximum value occurring at the outer side of the chord at the 270° position, which is 9.26.
• Upon comparing Figure 9a,b, it can be observed that the stress patterns in these two scenarios resemble those of Y-joints and spatial YY-joints. It is evident that the axial load on the reference brace, as indicated by the SCF values, is significantly influenced by the axial load on the other brace. Particularly noteworthy is the substantial increase in the SCF values at the 90° and 270° positions, which correspond to the saddle points on both sides of the chord. At the 90° position, the SCF is 13.93, representing an improvement of 84.99%, while at the 270° position, the SCF is 12.80, reflecting a 38.98% enhancement. Through force analysis, it can be deduced that in spatial tubular joints, when one side of the reference brace is subjected to load, a lateral force is exerted on the chord, thereby altering the strain distribution on one side of the chord. Additionally, the CHS tube undergoes localized deformation under the load applied by the brace. Hence, under the influence of the other brace, the strain and SCF at the 90° and 270° positions of the chord side increase.
• When comparing Figure 9a,c, it can be observed that the stress patterns in these two scenarios resemble those of Y-joints and K-joints. Similar to the previous finding, the inclusion of load on other brace alters the distribution of SCF on the reference brace. However, unlike before, in the case of K-joints under loading conditions, the SCF of the brace experiences a significant reduction. Through force analysis, it can be inferred that when the other side brace is subjected to axial force, it acts in the opposite direction to the axial load on the reference brace. In the axial load mode of Y-joints, the chord experiences a lateral force, resulting in lateral deformation. However, in the axial action mode of the K-joints brace, the perpendicular components of the axial forces of the two braces in relation to the chord’s direction will cancel each other out, thereby reducing the lateral deformation of the chord at the reference brace and consequently decreasing the SCFs.
• The SCF distribution of spatial KK-joints under the axial forces of braces is depicted in Figure 9d,e. The loading conditions in Figure 9d,e correspond to the spatial loading scheme of Figure 9c, representing the three force modes specified in the IIW codes [21] for spatial KK-joints. The two braces loaded in Figure 9c are designated as the reference plane. It can be observed that the loading conditions illustrated in Figure 9d,e involve the application of spatial loads on one side of the reference plane. Although the SCF in Figure 9c is not perfectly symmetrical about the crown point, the SCF values on either side of the crown point are similar. Similar to Y-joints, the SCF at 90° is smaller than that at 270° due to structural issues. Through force analysis, it can be deduced that when axial forces are applied to the brace adjacent to the reference brace of Y-joints, the resulting axial force components cause greater deformation in the chord compared to the Y-joints, thereby increasing the SCF of the chord. However, in spatial KK-joints, the axial force components of the two braces on the opposite plane balance each other, reducing the lateral deformation of the chord and consequently decreasing the chord’s SCF. Nevertheless, the local deformation of the chord under the force state of the braces on the opposite plane persists. When the axial load direction on the reference brace aligns with that of the brace on its side, it will reduce the SCF of the chord at the 90° location. Conversely, when the axial load direction on the reference brace opposes that of the brace on its side, it will increase the SCF of the chord member at the 90° location.

To facilitate viewing, the SCF values obtained from the calculations of various loading conditions are consolidated in

Table 5. Value of SCFs on the chord side under different loading conditions.

Location	L1-T	L1-C	L2-T	L2-C	L3-T	L3-C	L4-T	L4-C	L5-T	L5-C
0	3.15	3.16	4.04	4.04	1.78	1.81	3.48	3.51	0.66	0.53
45	4.10	4.15	6.11	6.06	2.93	2.86	4.42	4.37	1.79	1.67
90	7.45	7.62	14.00	13.85	4.15	4.22	6.59	6.58	2.30	2.38
135	7.72	7.79	9.16	9.12	4.29	4.36	6.60	6.62	2.13	2.11
180	6.69	6.73	7.58	7.48	4.72	4.80	6.00	6.03	3.55	3.35
225	7.48	7.62	9.40	9.33	4.85	4.91	4.63	4.62	5.35	5.30
270	9.16	9.26	12.79	12.81	5.03	5.03	5.21	5.15	4.83	4.93
315	4.64	4.65	6.02	5.98	3.23	3.12	4.09	4.08	2.29	2.23
360	3.15	3.16	4.04	4.04	1.78	1.81	3.48	3.51	0.66	0.53

. It can be observed that among all the conditions, the highest SCF value occurs at the 90° position of the chord under the L2 loading condition, with tensile and compressive SCF values of 14.00 and 13.85, respectively.

4. Comparison with Empirical Formulas

Currently, various international codes such as CIDECT [5], API [6], IIW [21], and DNV [22] typically employ the Efthymiou formula. The application range of this formula is as follows: 0.2 ≤ β ≤ 1.0, 0.2 ≤ τ ≤ 1.0, 8.0 ≤ γ ≤ 32, and 20° ≤ θ ≤ 90°.

Under the influence of axial force in the brace, the calculation formula of SCF for the T/Y-joints is as follows:

The saddle point of the chord:

$${\mathrm{SCF}}_{T/Y}=F_1\gamma {\tau }^{1.1}\left[1.11-3{(\beta -0.52)}^2\right]{\sin }^{1.6}\theta$$

(7)

The crown point of the chord:

$${\mathrm{SCF}}_{T/Y}={\gamma }^{0.2}\tau \left[2.65+5{(\beta -0.65)}^2\right]+\tau \beta \left(0.25\alpha -3\right)\sin \theta$$

(8)

When α is greater than or equal to 12, F₁ = 1; when α is less than 12, F₁ is calculated using Equation (9).

$$F_1=1-\left(0.83\beta -0.56{\beta }^2-0.02\right){\gamma }^{0.23}\exp \left(-0.21{\gamma }^{-1.16}{\alpha }^{2.5}\right)$$

(9)

Under the influence of axial force in the brace, the calculation formula of SCF for K-joints is as follows:

$${\mathrm{SCF}}_K=\left[{\tau }^{0.9}{\gamma }^{0.5}\left(0.67-{\beta }^2+1.16\beta \right)\sin \theta \right]{\left(\frac{\sin {\theta }_{max}}{\sin {\theta }_{min}}\right)}^{0.30}\times {\left(\frac{{\beta }_{max}}{{\beta }_{min}}\right)}^{0.30}\times \left[1.64+0.29{\beta }^{-0.38}\arctan \left(8\zeta \right)\right]$$

(10)

Furthermore, within the CIDECT code [5], a modified formula based on Equation (10) for calculating the SCF of spatial CHS KK-joints is proposed. The calculation formula is presented as follows.

$${\mathrm{SCF}}_{KK}=f_{\mathrm{geom} }{f}_{\mathrm{load} }{\mathrm{SCF}}_K$$

(11)

where the values of f_geom and f_load are determined by the geometric parameters and the load conditions. In the code, a singular parameter MCF has been employed to enhance Equation (11).

$${\mathrm{SCF}}_{KK}=MCF\cdot {\mathrm{SCF}}_K$$

(12)

where the value of MCF is typically associated with the angle Φ between the plane where the braces are located. Based on reference to the code, MCF is assigned a value of 1.25 for the L4 loading case and a value of 1 for the L5 loading case in this study.

The calculated values from the formulas in Equations (7)–(12) are summarized and compared to the test results, and the results are consolidated in

Table 6. Comparison between experimental maximum values of SCF and recommendation formula values.

Load Case		Test	Position	Recommendation Formula	Position	Error (%)
L1	T	9.19	270°	10.16	90°/270°	10.6
	C	9.26	270°			9.7
L2	T	14.00	90°			27.4
	C	13.85	90°			26.6
L3	T	5.03	270°	5.70	—	13.3
	C	5.03	270°			13.3
L4	T	6.60	135°	7.13	—	8.0
	C	6.62	135°			7.6
L5	T	5.35	225°	5.70	—	6.5
	C	5.30	225°			7.5

.

From the comparison results, it can be observed that the existing codes have relatively low precision in calculating the recommended SCFs for spatial KK-joints. Particularly in the L2 loading case, the calculation formula for this loading configuration is not proposed by the code. If the calculation formula for uniplanar Y-joints is used to calculate the SCF under the L2 condition, the maximum error will reach 27.4%. In the code, the calculated results of Equation (12) (obtained by correcting Equation (10)) have an error of between 6.5% and 8% from the test results.

Through the HSS testing for spatial KK-joints, it is evident that the distribution of SCFs for the joints varies significantly under different loading conditions. Additionally, the existing codes could not adequately account for the spatial effects of spatial joints. If the SCF calculation formula for joints under the single force mode is employed to calculate the SCF of spatial joints in actual structures, precise results are often unattainable. Therefore, it is necessary in subsequent work to propose corresponding SCF calculation formulas for spatial joints under different conditions.

5. Conclusions

An experimental investigation of HSS for spatial CHS KK-joints was proposed in this study. Initially, the test setup was proposed for loading spatial joints, followed by a detailed description of the HSS loading conditions and measurement device. Lastly, the HSS of the joints was tested under both individual and combined loading conditions, and the results are compared with the outcomes derived from existing recommended calculation formulas. The main conclusions are as follows:

• Under axial loading conditions in the brace direction, the distribution of the SCFs on the chord side of the CHS KK-joints is essentially similar to that under axial tensile forces in the brace direction.
• When the joints are subjected to spatial loads, the distribution of SCFs on the chord side, with reference to the brace, is greatly influenced by other force-bearing braces.
• The existing codes for calculating the SCFs of spatial joints primarily rely on the application of corrective parameters to the basic formulas. However, this approach leads to significant discrepancies in the calculated results and fails to accurately predict the SCFs of spatial joints. Therefore, it is necessary to propose suitable SCF calculation formulas specifically designed for spatial joints in order to enhance calculation accuracy.