Finite Element Modeling and Stress Analysis of a Six-Splitting Mid-Phase Jumper

: In this study, a ﬁnite element, fully three-dimensional solid modeling method was used to study the mechanical response of a steel-cored aluminum strand (ACSR) with a mid-phase jumper under wind load. A whole model (simplifying an ACSR into a solid cylinder) and a local model (modeling according to the actual structure of an ACSR) of the mid-phase jumper were established. First, the movement of the mid-phase jumper of the tension tower under wind load was studied based on the whole ﬁnite element model, and the equivalent Young’s modulus of the whole model was adjusted based on the local model. The results of the whole model were then imported into the local model and the stress distribution of each strand of the ACSR was analyzed in detail to provide guidance for the treatment measures. Therefore, the whole model and the local model complemented each other, which could reduce the number of model operations and ensure the accuracy of the results. test to verify the results of the ﬁnite element simulation and the comparison of the simulation and fatigue test results, the causes of the broken strand of the ACSR were discussed. Although this modeling method was applied to the stress and deformation analysis of a mid-phase jumper in this study, it can be used to study the bending deformation of rope structures with a complex geometry and the main bending deformation. In addition, the e ﬀ ect of the friction coe ﬃ cient on the bending of the mid-phase jumper was studied.


Introduction
Fatigue of the wires in an overhead conductor occurs within the couplings that restrict the conductor vibration from incurring a vortex-induced oscillation [1][2][3]. At present, most of the overhead conductors are made of steel-cored aluminum strands (ACSRs), and ACSR is widely used in power delivery systems around the world [4,5]. An aeolian vibration is one of the causes of a fretting fatigue failure of a conductor. Such failures invariably occur in suspension clamps, dampers, and spacers [6,7].
The northwest part of China has several areas with a strong wind, which significantly impacts the operation of transmission and transformation lines. According to statistics, from July 2015 to September 2016, 750 kV ultra-high voltage (UHV) transmission lines undertaking the task of power transmission from the west to the east in a strong wind areas suffered three strand breaks of the mid-phase jumper in succession, resulting in significant economic losses. Therefore, it is very important to study the stress and deformation distribution of the mid-phase jumper in terms of the movement under strong winds to provide guidance for the treatment measures.
Rizzo et al. [8] described a method based on outlier analysis and the wavelet transform for structural damage detection based on guided ultrasonic waves. Furthermore, this method was applied to the detection of notch-like defects in a seven-wire strand by using built-in magnetostrictive devices for ultrasound transduction. Castellano et al. [9] presented an innovative non-destructive experimental technique based on ultrasonic waves for a quantitative determination of the damage level in structural components. Neslušan et al. [10] presented a systematic study using a magnetic non-destructive evaluation of corrosion attack in rope wires via Barkhausen noise emission. Because of the complexity of the scene, it is difficult to detect the damage of a six-splitting mid-phase jumper directly. Through the establishment of the finite element model six-splitting of a mid-phase jumper, the damage under wind load can be calculated. This paper presents an effective finite element modeling method for a large steel-cored aluminum strand when a main bending deformation occurs.
Because of the interaction between strands, multi-layer strands have a variable bending stiffness. With the bending of the strands, they begin to gradually slide with respect to each other, resulting in a decrease in the bending stiffness. However, the structural model of the conductor studied here is larger and has a spatial bending form. Therefore, it is extremely difficult to establish a finite element model of a mid-phase jumper with equal proportions.
Some scholars have proposed different modeling methods to study the mechanical response of steel strands during a bending deformation. Raoof and Hobbs [11] idealized a strand as a series of concentric orthotropic cylinders, each of which is associated with a specific layer and its corresponding mechanical properties. The bending process of the wire was also considered. In addition, Lanteigne [12] studied the mechanical behavior of cables under an arbitrary combination of tension, torsion, and bending. The force model of the cables is universal.
Leclair and Costello [13] applied the love crank balance equation to each conductor and deduced a mechanical conductor model that can predict the stress of single twisted strands under axial, bending, and torsion loads. Hong et al. [14] established a mechanical model of the conductor bending behavior under tension, although the model was significantly simplified. For example, the contact between adjacent cables of the same layer is neglected in the established model, and the contact line is used instead of the contact point of the superimposed layer. In addition, the cross-section of the twisted strand after bending is still planar.
Judge et al. [15] used LS-DYNA to establish a full three-dimensional, elastic-plastic finite element model of multi-layer helical cables under a quasi-static axial load. The model uses three-dimensional solid elements to discretize each strand and simulates the contact types between all strands. The model also uses the law of non-linear hardening to explain the plastic deformation. Zhang et al. [16] succeeded in analyzing the flexural stiffness of the strands using a solid three-dimensional finite element model, although their research was limited to single-pitch, single-layer cables. To summarize, the method of full three-dimensional solid modeling considers the contact between wires, among other factors, which results in a large number of computations, and thus the length of a fully three-dimensional solid element finite element model is usually very short. To reduce the computational complexity of the model, some scholars have used a beam element instead of solid element to establish a finite element model of a strand. Zhou et al. [17] used a beam element to model a single-layer conductor and managed the contact interaction between the lines using a coupling equation between the corresponding nodes, and thus the slip between the lines was not considered. The model was applied to the analysis of strands under bending loads, which is still limited to single-layer strands with a small deflection. Beleznai et al. [18] also modeled a conductor. Each line contact was simulated using a spring element. The stiffness of the spring element was based on the Hertzian contact theory. Although the correctness of this method has been proved, its application was still limited to single or double layers with a small displacement. Lalonde et al. [19,20] proposed a finite element modeling method for multi-layer steel strands under multi-axial loads. This method utilizes a beam element and a three-dimensional line contact, which greatly reduce the computational complexity of the model under the premise of ensuring a high level of accuracy.
In this study, a finite element, fully three-dimensional solid modeling method was used to study the mechanical response of a steel-cored aluminum strand with a mid-phase jumper under a bending deformation. A whole model (simplifying an ACSR into a solid cylinder) and a local model (modeling according to the actual structure of an ACSR) of the mid-phase jumper were established. First, the movement of the mid-phase jumper of the tension tower under wind load was studied based on the whole finite element model, and the equivalent Young's modulus of the whole model was adjusted based on the local model. The results of the whole model were then imported into the local model, and the stress distribution of each strand of the ACSR was analyzed in detail under a bending deformation to provide guidance for the treatment measures. Therefore, the whole model and the local model complemented each other, which could reduce the number of model operations and ensure the accuracy of the results. The feasibility of this method was verified through a comparison with the results of subsequent experiments. Although this modeling method was applied to the stress and deformation analysis of a mid-phase jumper in this study, it can be used to study the bending deformation of rope structures with a complex geometry and the main bending deformation. The finite element modeling method of a mid-phase jumper presented in this paper can be implemented in any general finite element software. Abaqus (2018, Dassault Systemes Simulia Corp, Johnston, RI, USA) was also used in this study. Figure 1a shows the location of a mid-phase jumper on a tension tower and Figure 1b shows the lateral view of a mid-phase jumper on a tension tower. Because of the tension at both ends of the tower, large-span wires are subjected to a significant amount of tension. However, for a mid-phase jumper, owing to its different spatial structure, the force differs from that of a large-span wire. As shown in Figure 1a, b, the lower end of the mid-phase jumper is fixed by a tension clamp and the upper end is suspended by an insulator string, which shows a spatial tortuous distribution. Compared with a large-span wire, the initial tension applied to the mid-phase jumper is very small, and thus the initial tension applied to the mid-phase jumper can be neglected. Figure 1c shows an image of the motion of the lower end of a mid-phase jumper near a fixed position under level-7 wind. It is seen that the mid-phase jumper oscillates along the A-B direction under a strong wind, and its range of motion is slightly larger than the diameter of the six-splitting spacer (650 mm). In addition, the oscillation frequency is 5-8 times per minute. As shown in Figure 1d, the broken position of the mid-phase jumper is at the outlet of the tension clamp. The common motion forms of an overhead conductor under wind load are breeze vibration, subspan vibration, galloping, etc. [21][22][23]. The frequency of aeolian vibration is generally 3-150 Hz, and the amplitude is 0.01-1d (d is the diameter of the conductor), which occurs when the wind speed is small. The frequency of galloping is 0.08-3 Hz, and the amplitude is much larger than that of aeolian vibration, which often occurs on the iced conductor with a high wind speed. Subspan oscillation refers to the vibration of a conductor between adjacent spacer bars under wind load. The amplitude of subspan oscillation is generally 5-10 mm and the frequency of oscillation is 1-2 Hz. To sum up, the movement form of the mid-phase jumper is different from the breeze vibration, the subspan vibration, and the galloping, which is a large-scale cyclic swing under a strong wind.  Figure 2a shows the structure of the mid-phase jumper and the related data size (mm). It is seen that the mid-phase jumper includes a six-splitting, steel-cored aluminum strand and a six-splitting spacer. Figure 2b shows a side view of a mid-phase jumper on a tension tower. Figure 2c shows the solid model of the mid-phase jumper imported into Abaqus, which simplified the steel-core aluminum wire into a solid rod and was established in SolidWorks. (2016, Dassault Systemes Simulia Corp, Johnston, RI, USA) Because the lower end of the mid-phase jumper was rigidly connected through a tension clamp, all degrees of freedom of the lower end of the mid-phase jumper were fixed. As shown with the A arrows, the fixed ends of the lower of the mid-phase jumper were the outlet position of the tension clamp. In the actual structure of the mid-phase jumper, the top of the insulator string was fixed on the tension tower. According to the assembly of the insulator string and the midphase jumper, the maximum displacement of the middle section of the mid-phase jumper deviating from the east-west direction was limited to 30 mm, and other displacement degrees of freedom and all rotational degrees of freedom were limited, as shown with the B arrows. Because of the large deformation of the mid-phase jumper under a strong wind, geometric nonlinearity was opened during the analysis step of Abaqus. The quadratic tetrahedron stress element (C3D10) with a high accuracy was used in the whole model. The number of elements and nodes were approximately 3.5 × 10 5 and 6.6 × 10 5 , respectively.  Figure 2a shows the structure of the mid-phase jumper and the related data size (mm). It is seen that the mid-phase jumper includes a six-splitting, steel-cored aluminum strand and a six-splitting spacer. Figure 2b shows a side view of a mid-phase jumper on a tension tower. Figure 2c shows the solid model of the mid-phase jumper imported into Abaqus, which simplified the steel-core aluminum wire into a solid rod and was established in SolidWorks. (2016, Dassault Systemes Simulia Corp, Johnston, RI, USA) Because the lower end of the mid-phase jumper was rigidly connected through a tension clamp, all degrees of freedom of the lower end of the mid-phase jumper were fixed. As shown with the A arrows, the fixed ends of the lower of the mid-phase jumper were the outlet position of the tension clamp. In the actual structure of the mid-phase jumper, the top of the insulator string was fixed on the tension tower. According to the assembly of the insulator string and the mid-phase jumper, the maximum displacement of the middle section of the mid-phase jumper deviating from the east-west direction was limited to 30 mm, and other displacement degrees of freedom and all rotational degrees of freedom were limited, as shown with the B arrows. Because of the large deformation of the mid-phase jumper under a strong wind, geometric nonlinearity was opened during the analysis step of Abaqus. The quadratic tetrahedron stress element (C3D10) with a high accuracy was used in the whole model. The number of elements and nodes were approximately 3.5 × 10 5 and 6.6 × 10 5 , respectively.

Establishment of Local Model
As shown in Figure 1d, the actual fracture position of the mid-phase jumper is the outlet position of the tension clamp. In order to accurately analyze the stress distribution of the actual fracture position, a local model was established. The local model established in this study consisted of four helical layers wound around a straight cylinder at the center. In addition, adjacent spiral layers were wound in the opposite directions to reduce the internal torque caused by the winding. The formula for calculating the radius of each helical layer is shown in Equation (1). The formula for calculating the helix angle is shown in Equation (2). In Equations (1) and (2), R i , d i , d core , m i , and a i indicate the layer radius (mm), wire diameter (mm), diameter of the central steel wire (mm), pitch diameter ratio, and helical angles of the helical layers, respectively. 1 core i i a r c t a n ( ) m a π = (2) Table 1 shows the mechanical properties of the materials and modeling parameters of the local finite element model. Figure 3a shows the sectional diagram of the local finite element model. As shown in Figure 3b, a section of all strands was fixed at one end of the local model. The fixed end was located at the outlet of the tension clamp. By setting the motion coupling constraints in Abaqus, the other end was coupled with the central point, all six degrees of freedom of the constrained area were selected to be coupled with the coupling points, and the constrained end face then became a rigid plane. There was no relative displacement between the nodes in the constrained area, only a rigid motion with the selected coupling point. In this case, the displacement boundary condition (Ya) could be applied to the coupling point.

Establishment of Local Model
As shown in Figure 1d, the actual fracture position of the mid-phase jumper is the outlet position of the tension clamp. In order to accurately analyze the stress distribution of the actual fracture position, a local model was established. The local model established in this study consisted of four helical layers wound around a straight cylinder at the center. In addition, adjacent spiral layers were wound in the opposite directions to reduce the internal torque caused by the winding. The formula for calculating the radius of each helical layer is shown in Equation (1). The formula for calculating the helix angle is shown in Equation (2). In Equations (1) and (2), R i , d i , d core , m i , and ai indicate the layer radius (mm), wire diameter (mm), diameter of the central steel wire (mm), pitch diameter ratio, and helical angles of the helical layers, respectively. Table 1 shows the mechanical properties of the materials and modeling parameters of the local finite element model. Figure 3a shows the sectional diagram of the local finite element model. As shown in Figure 3b, a section of all strands was fixed at one end of the local model. The fixed end was located at the outlet of the tension clamp. By setting the motion coupling constraints in Abaqus, the other end was coupled with the central point, all six degrees of freedom of the constrained area were selected to be coupled with the coupling points, and the constrained end face then became a rigid plane. There was no relative displacement between the nodes in the constrained area, only a rigid motion with the selected coupling point. In this case, the displacement boundary condition (Y a ) could be applied to the coupling point.   The hexahedral incompatible element (C3D8I) was used in the local model. The number of elements and nodes were approximately 1.3 × 10 5 and 1.8 × 10 5 , respectively. The default contact behavior in the Abaqus/Standard was the "hard contact." The surface contact was set between each strand. Although numerous contact algorithms can be used to establish contact pairs, the contact is extremely complex owing to the large number of surface-to-surface contacts involved. Therefore, the contact algorithm was set as the augmented Lagrange algorithm, which improved the convergence efficiency of the model operation under the premise of ensuring the accuracy. The default value of the contact stiffness was adopted, and the expansion coefficient of the contact stiffness was 1.
Although Utting and Jones [24] explored the friction effect between steel wires, they were not specified regarding the friction coefficient. Judge et al. [15] and Lalonde et al. [19] applied a friction coefficient μ of 0.115 to all contact points. The present simulation used the same coefficient value. When the mid-phase jumper sways significantly under wind load, the main deformation of the midphase jumper is bending. Therefore, the bending stiffness of the whole model was made the same as that of the local model by adjusting the Young's modulus parameters, and the adjusted Young's modulus was then input into the whole model for the calculations. Finally, the motion behavior of the whole model under wind load was obtained. The adjustment process of the Young's modulus of the whole model is described in detail in Section 3.

Application of Wind Load on the Whole Model
A mid-phase jumper oscillates under wind load. According to Bernoulli's equation, the wind load on the conductor is as shown in Equation (3): ( In Equation (3), P, γ, g, v, and θ indicate the wind load (N), the air density (kg/m 3 ), acceleration of gravity (m/s 2 ), wind speed (m/s), and angle between the wind direction and conductor axis, respectively. For a large-span conductor, an aerodynamic coefficient and a nonuniformity coefficient were introduced. According to Equation (3), the main factors influencing the wind load on the conductor are the wind speed and angle θ between the wind direction and conductor. When θ is 90°, the formula for calculating the wind load is shown in Equation (4): The hexahedral incompatible element (C3D8I) was used in the local model. The number of elements and nodes were approximately 1.3 × 10 5 and 1.8 × 10 5 , respectively. The default contact behavior in the Abaqus/Standard was the "hard contact." The surface contact was set between each strand. Although numerous contact algorithms can be used to establish contact pairs, the contact is extremely complex owing to the large number of surface-to-surface contacts involved. Therefore, the contact algorithm was set as the augmented Lagrange algorithm, which improved the convergence efficiency of the model operation under the premise of ensuring the accuracy. The default value of the contact stiffness was adopted, and the expansion coefficient of the contact stiffness was 1.
Although Utting and Jones [24] explored the friction effect between steel wires, they were not specified regarding the friction coefficient. Judge et al. [15] and Lalonde et al. [19] applied a friction coefficient µ of 0.115 to all contact points. The present simulation used the same coefficient value. When the mid-phase jumper sways significantly under wind load, the main deformation of the mid-phase jumper is bending. Therefore, the bending stiffness of the whole model was made the same as that of the local model by adjusting the Young's modulus parameters, and the adjusted Young's modulus was then input into the whole model for the calculations. Finally, the motion behavior of the whole model under wind load was obtained. The adjustment process of the Young's modulus of the whole model is described in detail in Section 3.

Application of Wind Load on the Whole Model
A mid-phase jumper oscillates under wind load. According to Bernoulli's equation, the wind load on the conductor is as shown in Equation (3): In Equation (3), P, γ, g, v, and θ indicate the wind load (N), the air density (kg/m 3 ), acceleration of gravity (m/s 2 ), wind speed (m/s), and angle between the wind direction and conductor axis, respectively. For a large-span conductor, an aerodynamic coefficient and a nonuniformity coefficient were introduced. According to Equation (3), the main factors influencing the wind load on the conductor are the wind speed and angle θ between the wind direction and conductor. When θ is 90 • , the formula for calculating the wind load is shown in Equation (4): In Equation (4), a f indicates the asymmetrical coefficient of the wind pressure. When the wind speed is less than 20 m/s, a f is taken to be 1, and when the wind speed is between 20 and 30 m/s, it is taken to be 0.85. In addition, K a indicates the aerodynamic coefficient and is taken to be 1, A f represents the projected area of the conductor in the direction of the wind load, and v f indicates the speed (m/s). Finally, P f indicates the wind load (N). The amplitude change of the wind load applied on the mid-phase jumper is shown in Equation (5): In Equation (5), A mp indicates the amplitude of wind load applied, and t indicates the time (s). The wind load is converted from the kinetic energy of the wind, and the kinetic energy consumption of the wind perpendicular to the conductor is the largest; therefore, the wind speed component perpendicular to the strand determines the actual wind load, and that parallel to the conductor has little effect. Table 2 shows the applied wind speed level to the whole model, the corresponding wind speed, and the magnitude of the applied wind load when θ was 90 • . Westerly winds dominate the northwest part of China throughout the year. In this study, the mechanical response of the mid-phase jumper to a westerly wind was mainly considered. As shown in Figure 4, the wind load was applied to the whole model, and the arrow in the figure represents the wind load and wind direction.
In Equation (4), a f indicates the asymmetrical coefficient of the wind pressure. When the wind speed is less than 20 m/s, a f is taken to be 1, and when the wind speed is between 20 and 30 m/s, it is taken to be 0.85. In addition, K a indicates the aerodynamic coefficient and is taken to be 1, A f represents the projected area of the conductor in the direction of the wind load, and v f indicates the speed (m/s). Finally, P f indicates the wind load (N). The amplitude change of the wind load applied on the mid-phase jumper is shown in Equation (5): In Equation (5), A m p indicates the amplitude of wind load applied, and t indicates the time (s).
The wind load is converted from the kinetic energy of the wind, and the kinetic energy consumption of the wind perpendicular to the conductor is the largest; therefore, the wind speed component perpendicular to the strand determines the actual wind load, and that parallel to the conductor has little effect. Table 2 shows the applied wind speed level to the whole model, the corresponding wind speed, and the magnitude of the applied wind load when θ was 90°. Westerly winds dominate the northwest part of China throughout the year. In this study, the mechanical response of the mid-phase jumper to a westerly wind was mainly considered. As shown in Figure 4, the wind load was applied to the whole model, and the arrow in the figure represents the wind load and wind direction.

Model Solution
The whole model used the Newton-Raphson algorithm to solve non-linear problems with a large displacement. The local model used the Newton-Raphson method to solve the contact and material nonlinearity problems. All simulations conducted in this study were implemented on a 2.3 GHz, 24-core CPU and a computer with 64 GB of memory.

Model Solution
The whole model used the Newton-Raphson algorithm to solve non-linear problems with a large displacement. The local model used the Newton-Raphson method to solve the contact and material nonlinearity problems. All simulations conducted in this study were implemented on a 2.3 GHz, 24-core CPU and a computer with 64 GB of memory.

Matching Adjustment of Whole Model and Local Model
On the one hand, in order to accurately simulate the deformation of a six-splitting mid-phase jumper under wind load, it is necessary to adjust the equivalent Young's modulus of the whole model to make it have the same bending resistance as the local model. On the other hand, in order to more accurately simulate the stress of each strand of a conductor at the outlet of the tension clamp, it is necessary to ensure the accuracy of the deformation of the whole model introduced into the local model, that is, to ensure that the whole model has the same bending resistance as the local model. Therefore, it is very important for the accuracy of the simulation results to adjust the Young's modulus of the whole model and obtain the equivalent Young's modulus of the whole model based on the local model.
The Young's modulus of the whole model is really the key of this paper, which was determined using the following method. In order to determine the Young's modulus of the whole model, as shown in Figure 3c, the whole microsegment model with a length of 120 mm was established, which was consistent with the cross section of the whole model (27.63 mm). By adjusting the Young's modulus of the whole microsegment model, the whole microsegment model and the local model had the same bending stiffness, and then the adjusted Young's modulus was input into the whole model such that the whole model had the same bending stiffness as the local model. A hexahedral incompatible element (C3D8I) was used in the whole microsegment model. The number of elements and nodes were approximately 2.4 × 10 4 and 2.6 × 10 4 , respectively. One end of the whole microsegment model was completely fixed, while the other end was coupled with the central point, and the displacement boundary condition (Y a ) was applied to the coupling point. The boundary conditions of the microsegment whole model were in good agreement with those of the local model.
The Y a of the coupling point and the force (F) needed to achieve the Y a were obtained by changing Y a . As shown in Figure 5a, for the local model, the relationship of F − Y a was approximately linear. By continuously adjusting the Young's modulus of the whole microsegment model, the relationship of F − Y a under different Young's moduli can be obtained, and the relationship of F − Y a was approximately linear. As shown in Figure 5a, when the Young's modulus of the whole microsegment model was 6.25 GPa, the F − Y a lines of the whole microsegment model and the local model almost coincided. Therefore, when the Young's modulus of the whole microsegment model was 6.25 GPa, the whole microsegment model and the local model had a similar bending stiffness. Then, 6.25 GPa was input into the whole model, which had the same bending stiffness as the local model.
As shown in Figure 5b, for the same Y a , the difference in F between the whole microsegment model and the local model was defined as ∆F. Based on an F corresponding to a Y a of the local model, the ratio of ∆F to F corresponding to Y a of this model was the error when applying this method. As shown in Figure 5b, Y a had a larger error of 0-0.25 mm and a smaller error of 0.25-0.6 mm. According to the simulation results in the fourth section, Y a corresponding to a strong wind was between 0.25-0.60; therefore, this study mainly considered the part of Y a between 0.25 and 0.60 mm. The absolute error of this part was less than 10%, which is within a reasonable range.
model and the local model was defined as ΔF. Based on an F corresponding to a Ya of the local model, the ratio of ΔF to F corresponding to Ya of this model was the error when applying this method. As shown in Figure 5b, Ya had a larger error of 0-0.25 mm and a smaller error of 0.25-0.6 mm. According to the simulation results in the fourth section, Ya corresponding to a strong wind was between 0.25-0.60; therefore, this study mainly considered the part of Ya between 0.25 and 0.60 mm. The absolute error of this part was less than 10%, which is within a reasonable range.  Figure 6a shows a displacement nephogram of the whole model under a westerly, level-6 wind. Figure 6b shows the displacement nephogram of the whole model with a length of 1000 mm at the outlet of the tension clamp. The middle section of the mid-phase jumper swung back and forth along the east-west direction. The swing of the mid-phase jumper in this direction caused a greater bending moment at the lower area of the mid-phase jumper.

Results and Discussion
It was assumed that the deflection of the conductor with the maximum displacement at the lower area of the whole model under wind load was Y b at 89 mm away from the outlet of the tension clamp. When Y b corresponding to different wind loads was imported into the local model, the maximum equivalent stress (σ a ) corresponding to the outlet position of the tension clamp of the local model under different wind loads could be obtained. As shown in Figure 6c, Y b changed with the swing of the whole model. When the displacement of the middle part of the mid-phase jumper along the negative X direction reached the maximum, Y b also reached the maximum (t = 3 s). This maximum value of Y b (0.258 mm) was imported into the local model to obtain the stress distribution at the outlet of the tension clamp under the level-6 wind, as shown in Figure 6d. It is seen that the stress concentration appeared near the fixed end of the local model, that is, the stress concentration appeared near the outlet of the tension clamp and the maximum stress appeared here. This explains why the actual mid-phase jumper breakage occurred at the outlet of the tension clamp.
As shown in Figure 6e, with an increase in the wind load, Y b increased, accelerating rapidly after a level-4 wind, which was due to the rapid increase in the wind load applied to the whole model. As shown in Figure 6f, with an increase in the wind load, the value of σ a of the outermost, third, and second layers of the aluminum conductor increased, where the σ a of the third layer was the largest. The σ a of the outermost and third layers were higher than the yield strength of the conductor under the level-6 wind. The difference between σ a of the outermost layer and σ a of the third layer increased gradually with the wind load increases, and the difference was approximately 10 MPa under the level-6 wind load. As shown in Figure 6g, the steel layers had not reached their yield strength (275 MPa) when all three aluminum layers yielded under the level-7 wind load, although σ a on the first layer was the largest at this time. This was because the Young's modulus of the first layer was much larger than that of the aluminum because of its steel material, which resulted in a larger stress on the first layer of the steel core.

Verification of the Fatigue Test
Combined with the finite element simulation, the fatigue test was carried out on the fatigue test platform of an ACSR. Figure 7 shows the three-dimensional design of the fatigue test platform of an ACSR. The initial adjustment device was fixed to the tension clamp at the end of the ACSR, and the

Verification of the Fatigue Test
Combined with the finite element simulation, the fatigue test was carried out on the fatigue test platform of an ACSR. Figure 7 shows the three-dimensional design of the fatigue test platform of an ACSR. The initial adjustment device was fixed to the tension clamp at the end of the ACSR, and the pretension and other parameters of the ACSR could be adjusted according to the working conditions on site. The axial adjustment device can exert axial tension on the ACSR. The device for transverse motion acted directly on the ACSR, causing it to move transversely, and the amplitude of the conductor swing can be adjusted. A steel-cored aluminum strand of LGJ-400/50 (Dazheng electric wire Corp, cangzhou, Hebei, China) was pressed in the tension clamp and assembled on the fatigue test platform of the ACSR. A displacement of 89 mm (Y b ) at the outlet of the tension clamp was measured using a dial indicator. The Y b simulated by the finite element method under the level-7 wind load was set to the maximum displacement (Y b ) of 89 mm at the outlet of the tension clamp during the fatigue test, and a cyclic fatigue test was then conducted. Three groups of experiments were conducted. Table 3 shows that the cyclic fatigue life of the conductor was between 150,000 and 300,000 times without or with very little tension.  Table 3 shows that the cyclic fatigue life of the conductor was between 150,000 and 300,000 times without or with very little tension.  As shown in Figure 8a, there was an obvious stress concentration at the outlet of the tension clamp for the outermost conductor and the third layer of conductor. As shown in Figure 8b,c, σa of the third layer of the conductor (79.7 MPa) was greater than σa of the outermost conductor (71.4 MPa), which was very interesting. Under the guidance of the finite element results, when one strand of the outermost conductor broke during the fatigue test, the outermost conductor was cut off completely, as shown by arrow A in Figure 9. Interestingly, at this time, the third layer of conductor had almost broken, as shown by arrow B in Figure 9. This indicates that at the outlet of the tension clamp, the crack of the third layer of conductor occurred earlier than the outmost conductor, which shows the accuracy of local model simulation results. As shown in Figure 8a, there was an obvious stress concentration at the outlet of the tension clamp for the outermost conductor and the third layer of conductor. As shown in Figure 8b,c, σ a of the third layer of the conductor (79.7 MPa) was greater than σ a of the outermost conductor (71.4 MPa), which was very interesting. Under the guidance of the finite element results, when one strand of the outermost conductor broke during the fatigue test, the outermost conductor was cut off completely, as shown by arrow A in Figure 9. Interestingly, at this time, the third layer of conductor had almost broken, as shown by arrow B in Figure 9. This indicates that at the outlet of the tension clamp, the crack of the third layer of conductor occurred earlier than the outmost conductor, which shows the accuracy of local model simulation results.
which was very interesting. Under the guidance of the finite element results, when one strand of the outermost conductor broke during the fatigue test, the outermost conductor was cut off completely, as shown by arrow A in Figure 9. Interestingly, at this time, the third layer of conductor had almost broken, as shown by arrow B in Figure 9. This indicates that at the outlet of the tension clamp, the crack of the third layer of conductor occurred earlier than the outmost conductor, which shows the accuracy of local model simulation results.   Figure 10a shows the macroscopic morphology of the fatigue fracture. As shown in Figure 10b, the fatigue band with the characteristics of bainite lines was observed. These fatigue bands were thick, short, deep, wide, and discontinuous, which obviously accorded with the characteristics of low cycle fatigue fracture bands. Therefore, under the level-7 wind load, the stress concentration of the third layer at the outlet of the tension clamp had yielded. To summarize, the results of the fatigue tests were in good agreement with the simulation results, which proved the accuracy of the local model simulation results. As shown in Figure 11a, after cutting the outermost conductor, it was found that the third layer of the conductor has obvious wear on the surface at the outlet of the tension clamp. As shown in Figure 11b, after the experiment, several wear defects could be seen on the surface of the third layer of conductor at the outlet of the tension clamp after cutting. The outer surface of the third layer led to wear defects due to the sliding of the conductor in the process of moving. These defects became the source of cracks in the third layer of conductor, and the cracks eventually continued to expand in  Figure 10a shows the macroscopic morphology of the fatigue fracture. As shown in Figure 10b, the fatigue band with the characteristics of bainite lines was observed. These fatigue bands were thick, short, deep, wide, and discontinuous, which obviously accorded with the characteristics of low cycle fatigue fracture bands. Therefore, under the level-7 wind load, the stress concentration of the third layer at the outlet of the tension clamp had yielded. To summarize, the results of the fatigue tests were in good agreement with the simulation results, which proved the accuracy of the local model simulation results.   Figure 10a shows the macroscopic morphology of the fatigue fracture. As shown in Figure 10b, the fatigue band with the characteristics of bainite lines was observed. These fatigue bands were thick, short, deep, wide, and discontinuous, which obviously accorded with the characteristics of low cycle fatigue fracture bands. Therefore, under the level-7 wind load, the stress concentration of the third layer at the outlet of the tension clamp had yielded. To summarize, the results of the fatigue tests were in good agreement with the simulation results, which proved the accuracy of the local model simulation results. As shown in Figure 11a, after cutting the outermost conductor, it was found that the third layer of the conductor has obvious wear on the surface at the outlet of the tension clamp. As shown in Figure 11b, after the experiment, several wear defects could be seen on the surface of the third layer of conductor at the outlet of the tension clamp after cutting. The outer surface of the third layer led to wear defects due to the sliding of the conductor in the process of moving. These defects became As shown in Figure 11a, after cutting the outermost conductor, it was found that the third layer of the conductor has obvious wear on the surface at the outlet of the tension clamp. As shown in Figure 11b, after the experiment, several wear defects could be seen on the surface of the third layer of conductor at the outlet of the tension clamp after cutting. The outer surface of the third layer led to wear defects due to the sliding of the conductor in the process of moving. These defects became the source of cracks in the third layer of conductor, and the cracks eventually continued to expand in the process of repeated cyclic movement of the conductor, resulting in the fracture of the third layer of conductor.

Verification of the Strain Gauges Test
In order to verify the accuracy of the finite element model, strain gauges were pasted at the outlet of the clamp. When Yb changed, σa was measured under different wind loads and compared with the simulated stress. Because the strain gauge needed to be pasted on a smooth surface, the surface of the single strand of aluminum wire to be pasted with the strain gauge was lightly polished with a file. The thickness of the polished part accounted for about one fifth of the diameter of the single strand. The model of static strain instrument used was YE2538A (Yutian Technology Company, Wuxi, China). The strain gauge used was a conventional small resistance strain gauge with the size of 0.5 mm × 0.5 mm and a sensitivity of 2.00 kV. Figure 12 shows the installation position of strain gauges. As shown in Figure 13, with the increase of the wind load, the σa obtained using the simulation and σa obtained using the test both increased, and the maximum error between σa obtained using the simulation and σa obtained using the test was about 20 MPa. On one hand, the error was the error of experimental measurement, on the other hand, it was the error of the finite element numerical calculation itself. In conclusion, although there was an error between σa obtained using the simulation and σa obtained using the test, it also proved that the accuracy of the finite element model results were within a reasonable range.

Verification of the Strain Gauges Test
In order to verify the accuracy of the finite element model, strain gauges were pasted at the outlet of the clamp. When Y b changed, σ a was measured under different wind loads and compared with the simulated stress. Because the strain gauge needed to be pasted on a smooth surface, the surface of the single strand of aluminum wire to be pasted with the strain gauge was lightly polished with a file. The thickness of the polished part accounted for about one fifth of the diameter of the single strand. The model of static strain instrument used was YE2538A (Yutian Technology Company, Wuxi, China). The strain gauge used was a conventional small resistance strain gauge with the size of 0.5 mm × 0.5 mm and a sensitivity of 2.00 kV. Figure 12 shows the installation position of strain gauges.

Verification of the Strain Gauges Test
In order to verify the accuracy of the finite element model, strain gauges were pasted at the outlet of the clamp. When Yb changed, σa was measured under different wind loads and compared with the simulated stress. Because the strain gauge needed to be pasted on a smooth surface, the surface of the single strand of aluminum wire to be pasted with the strain gauge was lightly polished with a file. The thickness of the polished part accounted for about one fifth of the diameter of the single strand. The model of static strain instrument used was YE2538A (Yutian Technology Company, Wuxi, China). The strain gauge used was a conventional small resistance strain gauge with the size of 0.5 mm × 0.5 mm and a sensitivity of 2.00 kV. Figure 12 shows the installation position of strain gauges. As shown in Figure 13, with the increase of the wind load, the σa obtained using the simulation and σa obtained using the test both increased, and the maximum error between σa obtained using the simulation and σa obtained using the test was about 20 MPa. On one hand, the error was the error of experimental measurement, on the other hand, it was the error of the finite element numerical calculation itself. In conclusion, although there was an error between σa obtained using the simulation and σa obtained using the test, it also proved that the accuracy of the finite element model results were within a reasonable range. As shown in Figure 13, with the increase of the wind load, the σ a obtained using the simulation and σ a obtained using the test both increased, and the maximum error between σ a obtained using the simulation and σ a obtained using the test was about 20 MPa. On one hand, the error was the error of experimental measurement, on the other hand, it was the error of the finite element numerical calculation itself. In conclusion, although there was an error between σ a obtained using the simulation and σ a obtained using the test, it also proved that the accuracy of the finite element model results were within a reasonable range.

Effect of the Friction Coefficient
Because the friction coefficient is affected by the metal type, chemical composition, surface state, deformation temperature, and contact pressure, the friction coefficient between aluminum strands is not easy to determine. This paper mainly studied the bending deformation of a conductor, and the contact between each strand of a conductor is extremely complex during the process of the bending deformation. Coulomb's law was introduced to study this problem. The tangential contact of each strand in the aluminum strand follows this law. The Coulomb friction model was used in Abaqus.
Papailiou et al. [25] proposed a conductor model that can withstand both the tension and bending loads, and considered the interlayer friction and slip during the process of wire bending. It was found that the bending analysis of the conductor was extremely sensitive to the selection of the friction coefficients between two adjacent layers of the conductors. The range of friction coefficients between cables was found to be between 0.55 and 0.9. Wharton et al. [26] measured the fretting fatigue strength of A1-4Mg-0.7Mn in contact with aluminum and copper. It was found that the friction coefficient of A1-4Mg-0.7Mn in contact with aluminum and copper was extremely low, but with an increase in the number of fatigue cycles, the friction coefficient increased to approximately 1.0 and remained unchanged at this value. Clearly, a conductor bending analysis is extremely sensitive to the choice of friction coefficient between two adjacent conductors. In the process of long-term service, the friction coefficient between strands of a conductor will change due to the wear between strands. In order to accurately evaluate the change of the friction coefficient in the service process of the midphase jumper, this section discusses the influence of the friction coefficient on the stress of the midphase jumper when other conditions remain unchanged. Thus, the friction coefficients between the steel-cored aluminum strands were set to 0.3, 0.5, and 0.7. At this time, the equivalent Young's modulus of the whole finite element model was 6.78, 6.68, and 7.67 GPa, respectively, and the other conditions remained unchanged. The influence of the friction coefficient is discussed next.
As shown in Figure 14, between level-1 and level-3 wind loads, the wind load was small, and thus σa and Yb were extremely small; therefore, the friction coefficient had little effect on the corresponding σa and Yb. Between level-4 and level-7 wind loads, when the friction coefficient was 0.115 to 0.50, σa and Yb first decreased and then increased with an increase of the friction coefficient under the same wind load; when the friction coefficient was 0.50 to 0.70, σa and Yb decreased again with an increase of the friction coefficient under the same wind load. Therefore, the friction coefficient had a great effect on the corresponding σa and Yb between level-4 and level-7 wind loads. The difference in σa between the outermost layer and the third layer reached the maximum under a level-7 wind load when the friction coefficient took the above four parameters, and the maximum difference did not exceed 20 MPa. In conclusion, the friction coefficient influenced the bending of the mid-phase jumper, but the effect was not great.

Effect of the Friction Coefficient
Because the friction coefficient is affected by the metal type, chemical composition, surface state, deformation temperature, and contact pressure, the friction coefficient between aluminum strands is not easy to determine. This paper mainly studied the bending deformation of a conductor, and the contact between each strand of a conductor is extremely complex during the process of the bending deformation. Coulomb's law was introduced to study this problem. The tangential contact of each strand in the aluminum strand follows this law. The Coulomb friction model was used in Abaqus.
Papailiou et al. [25] proposed a conductor model that can withstand both the tension and bending loads, and considered the interlayer friction and slip during the process of wire bending. It was found that the bending analysis of the conductor was extremely sensitive to the selection of the friction coefficients between two adjacent layers of the conductors. The range of friction coefficients between cables was found to be between 0.55 and 0.9. Wharton et al. [26] measured the fretting fatigue strength of A1-4Mg-0.7Mn in contact with aluminum and copper. It was found that the friction coefficient of A1-4Mg-0.7Mn in contact with aluminum and copper was extremely low, but with an increase in the number of fatigue cycles, the friction coefficient increased to approximately 1.0 and remained unchanged at this value. Clearly, a conductor bending analysis is extremely sensitive to the choice of friction coefficient between two adjacent conductors. In the process of long-term service, the friction coefficient between strands of a conductor will change due to the wear between strands. In order to accurately evaluate the change of the friction coefficient in the service process of the mid-phase jumper, this section discusses the influence of the friction coefficient on the stress of the mid-phase jumper when other conditions remain unchanged. Thus, the friction coefficients between the steel-cored aluminum strands were set to 0.3, 0.5, and 0.7. At this time, the equivalent Young's modulus of the whole finite element model was 6.78, 6.68, and 7.67 GPa, respectively, and the other conditions remained unchanged. The influence of the friction coefficient is discussed next.
As shown in Figure 14, between level-1 and level-3 wind loads, the wind load was small, and thus σ a and Y b were extremely small; therefore, the friction coefficient had little effect on the corresponding σ a and Y b . Between level-4 and level-7 wind loads, when the friction coefficient was 0.115 to 0.50, σ a and Y b first decreased and then increased with an increase of the friction coefficient under the same wind load; when the friction coefficient was 0.50 to 0.70, σ a and Y b decreased again with an increase of the friction coefficient under the same wind load. Therefore, the friction coefficient had a great effect on the corresponding σ a and Y b between level-4 and level-7 wind loads. The difference in σ a between the outermost layer and the third layer reached the maximum under a level-7 wind load when the friction coefficient took the above four parameters, and the maximum difference did not exceed 20 MPa. In conclusion, the friction coefficient influenced the bending of the mid-phase jumper, but the effect was not great.

Conclusions
In this paper, a finite element, fully three-dimensional solid modeling method was used to study the mechanical response of a steel-cored aluminum strand (ACSR) with a mid-phase jumper under a wind load. The whole model (simplifying an ACSR into a solid cylinder) and a local model (modeling according to the actual structure of an ACSR) of the mid-phase jumper were established. First, the movement of the mid-phase jumper of the tension tower under a wind load was studied based on a whole finite element model, and the equivalent Young's modulus of the whole model was adjusted based on the local model. The results of the whole model were then imported into the local model, and the stress distribution of each strand of the ACSR was analyzed in detail to provide guidance for the treatment measures. Therefore, the whole model and the local model complemented each other, which could reduce the number of model operations and ensure the accuracy of the results. Although this method was applied to the stress and deformation analysis of a mid-phase jumper in this paper, it can be used to study the bending deformation of rope structures with a complex geometry and a main bending deformation.
The analysis showed that the swing of the mid-phase jumper in the east-west direction caused a greater bending moment at the lower area of the mid-phase jumper, which led to the stress concentration appearing near the outlet of the tension clamp. This explained why the actual midphase jumper breakage occurred at the outlet of the tension clamp. The maximum stresses of the outermost and third layers were higher than the yield strength under a level-6 wind. The difference in σa between the outermost and third layers increased gradually when the wind load increased, and the difference was approximately 10 MPa under a level-6 wind load.
Interestingly, σa of the third layer of the conductor (79.7 MPa) was greater than σa of the outermost conductor (71.4 MPa), which was verified by the results of the fatigue tests. Based on the analysis of the surface wear of the third layer of the conductor, the reason for the fracture of the third layer of conductor was explained. The feasibility of the element modeling method was verified through a comparison with the results of a subsequent series of experiments.

Conclusions
In this paper, a finite element, fully three-dimensional solid modeling method was used to study the mechanical response of a steel-cored aluminum strand (ACSR) with a mid-phase jumper under a wind load. The whole model (simplifying an ACSR into a solid cylinder) and a local model (modeling according to the actual structure of an ACSR) of the mid-phase jumper were established. First, the movement of the mid-phase jumper of the tension tower under a wind load was studied based on a whole finite element model, and the equivalent Young's modulus of the whole model was adjusted based on the local model. The results of the whole model were then imported into the local model, and the stress distribution of each strand of the ACSR was analyzed in detail to provide guidance for the treatment measures. Therefore, the whole model and the local model complemented each other, which could reduce the number of model operations and ensure the accuracy of the results. Although this method was applied to the stress and deformation analysis of a mid-phase jumper in this paper, it can be used to study the bending deformation of rope structures with a complex geometry and a main bending deformation.
The analysis showed that the swing of the mid-phase jumper in the east-west direction caused a greater bending moment at the lower area of the mid-phase jumper, which led to the stress concentration appearing near the outlet of the tension clamp. This explained why the actual mid-phase jumper breakage occurred at the outlet of the tension clamp. The maximum stresses of the outermost and third layers were higher than the yield strength under a level-6 wind. The difference in σ a between the outermost and third layers increased gradually when the wind load increased, and the difference was approximately 10 MPa under a level-6 wind load.
Interestingly, σ a of the third layer of the conductor (79.7 MPa) was greater than σ a of the outermost conductor (71.4 MPa), which was verified by the results of the fatigue tests. Based on the analysis of the surface wear of the third layer of the conductor, the reason for the fracture of the third layer of conductor was explained. The feasibility of the element modeling method was verified through a comparison with the results of a subsequent series of experiments.
In addition, the effects of the friction coefficient on the bending of a mid-phase jumper were studied. It was found that the friction coefficient influenced the bending of the mid-phase jumper, but the effect was not great.

Conflicts of Interest:
The authors declare no conflict of interest.