In this paper, a large-scale commercial finite element software ANSYS is used to simulate a K-joint of a transmission tower, and the FEM of the K-joint was established by 3D 8-node solid element SOLID185, as shown in Figure 1. There are five types of contact pairs in the model, which are the contact between the main angle steel and the gusset plate, the contact between the branch angle steel and the gusset plate, the contact between the nut and the steel member, the contact between the bolt head and the steel member, and the contact between the bolt shank and the bolt hole. The 3D surface-to-surface contact element CONTA173 and the 3D target element TARGE170 are used as the contact element and the target element, respectively. And the bolt pretension force is applied by the preload element PRETS179. Hexahedral mesh is adopted in the model, and refined division operation is carried out at the contact part of steel members. The elastoplastic model with a strengthened section is adopted for the constitutive relationship of members and bolts, as shown in Figure 2. In Figure 2, E is the elastic modulus, ε _y is the yield strain, ε _u is the ultimate strain, f _y is the yield stress; f _u is the ultimate stress.

In this paper, simulations of twelve loading scenarios are carried out. The thickness of each component in this model is 10 mm, in which the main material is L160 × 10 angle steel and the branch member is L80 × 10 angle steel. The effects of preload force, friction coefficient of the component, eccentricity, bolt hole diameter, strength of connector and bolt grade on the hysteretic performance of K-joints are studied, as listed in Table 1. In order to study the effect of bolt slip of K-joint, different loading modes were applied to Group A and Group B, respectively, in which tension compression cyclic load is applied to single branch angle steel of Group A, and tension compression cyclic load is applied to double branch angle steels of Group B, as shown in Figure 3. According to the Chinese code, the load is first loaded according to 25%, 50%, and 75% of the yield load, and each stage is cycled once. After the yield is reached, it is gradually increased according to 10% of the ultimate load, and each stage is cycled three times (Zhang and Li, 2017).

When the single branch angle steel of the K-joint is subjected to cyclic loading, the stress state of the K-joint is equivalent to the bolt lap joint. However, when double branch angle steels are subjected to cyclic loading because the resultant force of branch angle members bearing the load does not pass through the centroid of the bolt group on the main angle steel, the moment will be generated at the centroid of the bolt group, resulting in the sliding of the branch angle steel. Meanwhile, the gusset plate will rotate by eccentric force. In this state, the stress state of the K-joint is more complicated and it is also more consistent with the actual stress state of the K-joint. The load-displacement hysteresis curves under two different loading conditions are shown in Figure 4. The curve takes the displacement of the loaded branch angle steel as the horizontal axis and the load as the vertical axis. It can be seen that the ultimate load and failure position of Group B are different from those of Group A.

The ultimate load of Case A1 is 371.25 kN, the ultimate load of Case B5 is 300 kN, and the ultimate load is reduced by 23.75%. The reason for this phenomenon is that when the single branch angle steel of the K-joint bears the load F0, the resultant force at the centroid is only F 0 / 2 . At this time, the shear force of the bolt of the branch angle steel is greater than that of the bolt of the main member. Figure 5 shows the strain nephogram of the bolt. In Figure 5, the maximum plastic strain of the bolt of the main member is 0.153, while the maximum plastic strain of the bolt of the branch angle steel is 0.64, which indicates that the bolt of the branch angle steel is damaged before the bolt of the main member.

When branch angle steels on both sides of the K-joint are subjected to a load F ₀ of one tension and one compression at the same time, the resultant force at the centroid is 2 F 0 , the moment is larger, the rotation angle of the gusset plate is larger. And the bolt of the main member is subjected to a greater load, which will be failure before the bolt of the branch member, as shown in Figure 6, that is, the most unfavorable load condition for the K-joint is one side of the branch member is tensioned and one side of the branch member is compressed. At this time, the maximum plastic strain of the bolt of the main member is 0.216, while the maximum plastic strain of the bolt of the branch angle steel is 0.122.

However, when the bolt slip is not considered, the ultimate load of K-joints under the two loading methods is 375 kN and 373.8 kN, respectively, with a difference of 0.32%. At this time, the loading method has little effect on the ultimate load. Therefore, the main difference between Case A1 and Case B5 is due to the influence of bolt slip on the K-joint. Figure 7 shows the load-displacement hysteresis curve without considering the bolt slip. It can be seen that when the bolt gap is not considered, the hysteresis curve is fusiform and full in shape, which also shows that it is necessary to consider the bolt slip of the K-joint.

The trend of structural stiffness variation under cyclic loading can be obtained from the hysteretic curve. It can be seen from Section 3.1 that the bearing capacity of double branch angle steel loading is lower than that of single branch angle steel loading, and the K-joint is in the most unfavorable stress state. Therefore, this section mainly analyzes the hysteresis curve of double branch angle steel loading.

The hysteresis curve of each working condition is shown in Figure 8. It can be seen that the fullness of the hysteresis curve is affected by the parameters such as bolt preload, friction coefficient, eccentricity and other parameters. Compared with Figures 8D–F, it can be seen that with the increase of bolt preload, the shape of the hysteresis curve is plumper. This trend is the same as Figures 8A, B, D. The reason for this phenomenon is that increasing the preload of the bolt can make the components of the bolt connection area more closely fit, to a certain extent, the effect of increasing the friction force is achieved, so more energy is dissipated during the cycle. Figures 8C, D show the hysteretic curves of the joints with different eccentricities. When the eccentricity increases from 50 mm to 100 mm, the ultimate load decreases from 300 kN to 275.4 kN. This is because the same load can produce greater moment when the eccentricity increases, and the rotation angle of the gusset plate will also increase, resulting in the failure of the joint. However, it has little effect on the shape of the hysteresis curve, because the load-displacement curve analysis focuses on the deformation of the branch angle steel rather than the rotation of the gusset plate. Compared with Figures 8G–I, the bolt grade is increased from 4.8 to 8.8, and the ultimate load is 239.25, 350 and 425 kN, respectively, which shows that increasing the bolt grade can effectively improve the bearing capacity of the joint, and meanwhile, the curve becomes more full.

As can be seen from Figure 8, the slope of the load-displacement curve decreases gradually during each cycle, indicating that the stiffness of the joint is degraded and the residual deformation is accumulated. The reason for this phenomenon is the yielding of members and bolts and the slipping of bolts. In addition, in the process of repeated loading, due to the existence of bolt clearance, both the bolt of the main angle steel and the bolt of the branch angle steel will slip a certain distance, which leads to the occurrence of the pinching phenomenon, making the hysteresis curve a pinched anti-S shape. If the clearance between the bolt and the screw hole is eliminated, a welding-like effect will be produced, and the hysteresis curve will also be in a full spindle shape, as shown in Figure 7. This shows that reducing the clearance and avoiding bolt slippage can improve the seismic performance and energy dissipation capacity of the K-joint.

According to the Chinese code (China Academy of Building Research, 2015), the secant stiffness K _i under each cycle is calculated by Eq. 1. K i = + F i + − F i + X i + − X i (1) where + F _i and -F _i are the load values of the ith forward and reverse peak points; +X _i and -X _i are the displacement values of the ith forward and reverse peak points.

The secant stiffness K _i of the load-displacement curve of each cycle is shown in Figure 9. It can be seen from the figure that the secant stiffness of Case B4 and Case B8 suddenly decreases. As grade 4.8 ordinary bolts are used in Case B8, the shear strength is very low, and the bolts yield earlier, resulting in the early stage of the cycle joint stiffness decreased. Case B4 is similar to welding because there is no clearance. It can be seen from the shape of the hysteresis curve that there is an obvious yield point. When the component reaches yield, the stiffness of the joint will suddenly decrease first and then decrease slowly, while the stiffness degradation of other cases is generally gentle.

The stiffness of K-joints with larger eccentricity for the same material is generally lower. At the same time, the overall stiffness of the joint can be improved by increasing the strength of the component and upgrading the bolt grade.

In addition, Table 2 shows the initial stiffness K ₀ of K-joints under different conditions. It can be seen from the table that the initial stiffness of the K-joints in the elastic state has little different.