Comparison of the Influence of Double-Limb Double-Plate Joint on the Stability Bearing Capacity of Triangular and Quadrilateral Transmission Tower Structures

Abstract

The axial stiffness of the connection joints in a transmission tower will affect the stability bearing capacity of the tower. The axial stiffness of different forms of connection joints has different effects on the stability bearing capacities of triangular and quadrilateral lattice towers. This paper takes triangular and quadrilateral lattice towers as the comparative research objects and considers the influence of the stiffness of the single-limb, single-plate joint (SLSPJ) and double-limb, double-plate joint (DLDPJ) of the tower. Under vertical load, the vertical stability bearing capacities of triangular and quadrilateral transmission towers are studied from hte three aspects of theoretical analysis, numerical simulation and test result analysis. The influence rules of the SLSPJ and DLDPJ on the vertical stability bearing capacities of triangular and quadrilateral transmission towers are clarified. Through the energy method, considering the influence of the axial stiffness of connection joints, the calculation expressions of the vertical stability bearing capacities of triangular and quadrilateral lattice towers are derived. Through quantitative analysis, it is found that the axial stiffness of the connection joints has a more significant influence on the vertical stability bearing capacities of triangular lattice towers. The finite element models of the triangular and quadrilateral lattice towers including the SLSPJ and DLDPJ are further established. Through nonlinear finite element analysis, it is found that the DLDPJ can improve the vertical stability bearing capacity of the triangular lattice tower by 22.7% and the quadrilateral lattice tower by 14.9%. Through theoretical calculation, the expressions of the vertical stability bearing capacities of the triangular and quadrilateral lattice towers including the SLSPJ and DLDPJ are obtained. Combined with the test results of the SLSPJ and DLDPJ, it is found that the DLDPJ can improve the vertical stability bearing capacity of the triangular lattice tower by 23.4% and the quadrilateral lattice tower by 15.6%. The research results show that the DLDPJ can improve the vertical stability bearing capacities of triangular and quadrilateral lattice towers. The improvement effect of the vertical stability bearing capacity of the triangular lattice tower is 1.50∼1.52 times that of the quadrilateral lattice tower. The research results can provide a reference for the engineering popularization, application, and design of the DLDPJ in transmission towers.

1. Introduction

In recent years, with the continuous improvement in people’s living standards and the increasing demand for power consumption [1,2,3], the power infrastructure industry has shown a trend of rapid development. The transmission system is an important guarantee for the development of the power industry. As the main part of the power transmission system, transmission towers play a very important role in the transmission system [4,5]. A transmission tower is mainly composed of rod members and connection joints, which act together to resist the applied external load [6]. The bearing capacity of the transmission tower will directly affect the safety of the transmission system. The shape, rod members, and connection joints of transmission towers are closely related to structural safety and economic feasibility [7,8,9]. In the power infrastructure systems of countries all over the world, transmission towers mostly use steel lattice towers [10], while triangular and quadrilateral lattice towers are the most used structural forms in transmission system. The connection members of towers mostly use steel pipes and angled steel, and the rod members form the whole structure of the tower through connection joints. Transmission towers have many connection forms between the rod members [11,12,13,14,15], and the form of the tower connection joints has different effects on the tower with different shapes.

In the past decade, the research on transmission towers has mainly focused on the nonlinearity of towers [16,17,18], the mechanical properties of connection joints [9,14,15,19], and the local mechanical properties of towers [7,20,21]. The research of transmission towers mainly focuses on the nonlinear mechanical properties of the whole tower. Wu et al. [16] conducted a nonlinear stability analysis of a high-steel hyperbolic cooling tower of five structural systems considering various distributions and the amplitude of the defects. Gan et al. [22] carried out the full-scale tower test of unequal leg transmission towers. Considering the influence of the slip effect of connection joints, they studied it through test and numerical simulation. The predicted deformation, ultimate bearing capacity, and failure mode are in good agreement with the test results. The results showed that the ultimate bearing capacity of the whole tower changed little with the increase in the length of the transition section, and the maximum change is about 0.8%. Therefore, the transition section has little effect on the ultimate bearing capacity of the whole tower. Tian et al. [17] carried out full-scale experimental research on lattice steel pipe transmission towers and proposed a buckling softening failure model, which can more accurately predict the initial failure members, failure process and ultimate bearing capacity of the transmission tower. Fu et al. [8,23] analyzed the stress and displacement distribution of tension towers under eight different load conditions through the full-scale tower test. Then, the finite element model of the test tower is established, and the buckling instability of the tower is simulated by using the initial defects. Wang et al. [24] proposed a suitable defect simulation method to accurately evaluate the ultimate bearing capacity of the transmission tower. Bezas et al. [10] considered relevant defects, geometric and material nonlinearity, simulated the tower using fully nonlinear finite element software, and proposed two analytical models to predict the critical load of this buckling mode. The research on the structural connection joints of transmission towers mainly focuses on the theoretical derivation, testing and numerical simulation of the innovation and mechanical properties of various connection joints. Taha et al. [6] studied the mechanical behavior and failure mode of the connection bolt joints of three lattice transmission towers and obtained the axial stiffness of the lap bolt joints from the load displacement curve of the test. Ma et al. [11] carried out experimental and numerical simulation research on the new single double-transformation connection joint and modified the design theory of the gusset plate. Li et al. [12,25] studied the ultimate bearing capacities of plane K-type and multi plane DK type transmission tower connection joints through tests and numerical simulations and found that the equivalent moment ultimate bearing capacity of the plane DK joint is about 0.89∼1.0 times that of the equivalent moment of the K joint. Balagopal et al. [26] proposed a simplified calculation model of bolted joints considering the axial and rotational stiffnesses of tower connection joints. Abdelrahman et al. [27] proposed an improved joint slip model for the second-order nonlinear analysis of transmission towers. The local analysis of transmission towers is mainly conducted to study the mechanical properties of the tower legs, cross arms and the constituent components of the towers. Xie et al. [4] studied the influence of tower diaphragms on the bearing capacity of the tower. By adding diaphragm beams, the structural performance can be rationalized, and the bearing capacity can be greatly improved. Under vertical load, a single plate can increase the ultimate strength by 18.3% and a double plate can increase the ultimate strength by 17.6%. Through finite element analysis and small-scale tests, Kim [7] studied the influence of internal and external frame shape changes on the ultimate strength and deformation of triangular and quadrilateral lattice tower legs and proposed a square frame tower leg structure with reduced secondary support members. Chen et al. [18] found that the greater the length difference between the tower legs, the greater the strength ratio of the main column members and that the buckling failure of the primary and secondary members will directly affect the overall stability of the tower. Sharaf et al. [20] conducted experimental and numerical studies on the mechanical properties of the wooden cross arm of the transmission tower. Prasadrao et al. [21] studied the importance of “K” and “X” support forms and connection details in the whole performance of the tower. Gunathilaka et al. [28] developed the design formula based on the yield line theory to find the appropriate minimum thickness requirement of the base plate fixed by four bolts. Selvaraj et al. [29] evaluated the mechanical properties of composite cross arms through experimental test, analytical solution and finite element simulation. The components of the transmission tower and the form of the connection joints will affect the mechanical properties of the whole structure. Buckling can reduce the bearing capacity of the lattice tower by 30%, and the sliding of connection joints will reduce the bearing capacity of the tower by 6% [30]. In theory, the tower connection joint will affect the bearing capacity of the structure, but the research mainly focuses on the influence of bolt connection, and the research on the influence of joint connection form on the tower structure is relatively lacking. Yan et al. [31] carried out the experimental study of the DLDPJ and found that the DLDPJ can greatly reduce the stress of connecting plates and components and improve the bearing capacity of joints, but its effect in the whole tower has not been studied. Zhao et al. [32], combined with the experimental study of the DLDPJ, further explored their role in the quadrilateral tower structure and obtained the corresponding influence law. Although people have studied the mechanical characteristics and failure mechanisms of the transmission tower for a long time, it is difficult to establish a simple and unified theory due to the different shapes, sizes, connection joints, rod members and external load conditions of transmission towers [33].

The connection joint plays the role of connecting the components of the tower. The form of the connection joint will directly affect the stability bearing capacity of the whole transmission tower. In the analysis of the transmission tower, it is usually assumed that the connection joints are hinged. However, due to manufacturing difficulties, this assumption is difficult to realize [1]. Therefore, considering the influence of the form of connection joints on the mechanical properties of the whole tower, taking triangular and quadrilateral lattice towers as the research objects, this paper studies the influence of the new DLDPJ on the stability bearing capacities of triangular and quadrilateral lattice towers from three aspects of theoretical derivation, numerical simulation and test analysis. The research results will provide a reference basis for the application of the DLDPJ in transmission towers.

2. Calculation Method of Stability Bearing Capacity of the Transmission Tower

2.1. Stability Bearing Capacity of the Triangular Lattice Tower

The triangular transmission tower is a truss structure with a batten combination. According to the constraint state of the tower in practice, the calculation of the transmission line tower can be regarded as a cantilever structure with a fixed bottom. The calculation model is shown in Figure 1.

The steel pipes in the transmission tower are mostly thin-walled rod members. When calculating the moment of inertia of the tower section, the distance between the steel pipes in the lattice section is relatively large. Relative to the overall moment of inertia of the lattice section, the moment of inertia of the pipe to its own axis can be ignored. It is assumed that the section size of the leg member pipe is $\varphi D_c\times t_c$, the cross-sectional dimension of angle steel is $\mathrm{L}{b}_a\times t_a$, the cross-sectional area of the steel pipe can be approximately taken as $A_c=\pi D_c{t}_c$; through calculation, the moment of inertia of the lattice section around the x and y axes is:

$$I_x=I_y=\frac{\pi D_c{t}_c{b}^2}{2}$$

(1)

In Equation (1), $D_c$ and $t_c$ are the outer diameter and wall thickness of the steel pipe section, respectively; b is the root opening of the tower.

According to Equation (1), the moment of inertia of the triangular lattice tower around the x and y axes is basically equal. Therefore, the bending capacity of the triangular lattice tower in the x and y directions is equivalent. In the calculation model, the leg member of the tower is a steel pipe, and the cross-sectional area is $A_1$; angle steel is adopted for the diagonal member, with a cross-sectional area of $A_2$, and the included angle between the diagonal members and the horizontal direction is $\theta$. It is assumed that the curve of the transmission tower instability is sinusoidal, i.e.,

$$y=asin{\displaystyle \frac{\pi x}{2l}}$$

(2)

In Equation (2), l is the total height of the tower, and x is the distance from the top of the tower.

The bending moment at any position on the central axis of the triangular lattice tower structure is:

$$M=F_{p}y=F_{p}asin\frac{\pi x}{2l}$$

(3)

The shear force is:

$$F_Q=\frac{dM}{dx}=F_{p}a\frac{\pi }{2l}cos\frac{\pi x}{2l}$$

(4)

Under the action of the bending moment, the calculation expression of the leg member axial force of a triangular lattice tower is:

$$F_N=\frac{M{y}_j}{\sum y_i^2}$$

(5)

In Equation (5), $y_j$ is the distance from the centroid of the leg member to the x axis of the inertia axis; $\sum y_i^2$ is the sum of the squares of the distances from the centroid of all leg members to the x axis of the inertia axis.

Calculated by truss, the axial forces of leg members of the triangular lattice tower are $F_{Nt}^{11}$ and $F_{Nt}^{12}$. The axial force of the diagonal member is $F_{Nt}^2$:

$$F_{Nt}^{11}=\pm \frac{2\sqrt{3}M}{3b}=\pm F_p\frac{2\sqrt{3}a}{3b}sin\frac{\pi x}{2l}$$

(6)

$$F_{Nt}^{12}=\pm \frac{\sqrt{3}M}{3b}=\pm F_p\frac{\sqrt{3}a}{3b}sin\frac{\pi x}{2l}$$

(7)

$$F_{Nt}^2=\pm \frac{F_Q}{4cos{30}^{\circ }cos\theta }=\pm F_p\frac{2a\pi }{8\sqrt{3}lcos\theta }cos\frac{\pi x}{2l}$$

(8)

In Equations (6)–(8), $F_{Nt}^{11}$ and $F_{Nt}^{12}$ are the axial forces of the leg members of the triangular tower; $F_{Nt}^2$ is the axial force of the diagonal member of the triangular tower; b is the root opening of the tower; and $\theta$ is the angle between the diagonal member and the horizontal axis, as shown in Figure 1.

The strain energy of the triangular lattice tower is:

$$U_t=\sum \frac{F_N^2{s}_i^t}{2E{A}_i^t}$$

(9)

In Equation (9), $s_i^t$ and $A_i^t$ are the rod length and cross-sectional area of each rod member of the triangular tower, respectively; E is the elastic modulus of the member material.

The axial force is substituted as follows:

$$U_t=\frac{1}{2E}\left(\sum _1^n\frac{{\left(F_P\frac{2\sqrt{3}a}{3b}sin\frac{\pi x}{2l}\right)}^{2}d}{A_1^t}+\sum _1^{2n}\frac{{\left(F_P\frac{\sqrt{3}a}{3b}sin\frac{\pi x}{2l}\right)}^{2}d}{A_1^t}+\sum _1^{4n}\frac{{\left(F_P\frac{2a\pi }{8\sqrt{3}lcos\theta }cos\frac{\pi x}{2l}\right)}^2\frac{b}{cos\theta }}{A_2^t}\right)$$

(10)

In Equation (10), $A_1^t$ is the cross-sectional area of the leg member of the triangular tower; $A_2^t$ is the cross-sectional area of the diagonal member of the triangular tower; d is the height of the tower internode; b is the root opening of the tower; $\theta$ is the angle between the diagonal member and the x-axis; n is the number of internodes of the tower; the total number of diagonal members is the sum of $4n$ rods; and the total number of leg members is the sum of $3n$ rods.

According to the characteristics of the transmission tower, we can obtain:

$$d=\Delta x\approx dx$$

(11)

Using a variation of Equation (12):

$$\begin{array}{cc}\hfill & \sum _1^n{\left(sin\frac{\pi x}{2l}\right)}^{2}d\approx {\int }_0^l{\left(sin\frac{\pi x}{2l}\right)}^{2}dx=\frac{l}{2}\hfill \\ \hfill & \sum _1^{2n}{\left(sin\frac{\pi x}{2l}\right)}^{2}d\approx 2{\int }_0^l{\left(sin\frac{\pi x}{2l}\right)}^{2}dx=l\hfill \\ \hfill & \sum _1^{4n}{\left(cos\frac{\pi x}{2l}\right)}^{2}d\approx 4{\int }_0^l{\left(cos\frac{\pi x}{2l}\right)}^{2}dx=2l\hfill \end{array}$$

(12)

Using Equation (13):

$$d=2btan\theta$$

(13)

The strain energy calculation expression of the triangular lattice tower becomes:

$$U_t=\frac{F_p^{2}l{a}^2}{2E}\left[\frac{1}{A_1^t{b}^2}+\frac{{\pi }^2}{48l^{2}tan\theta }·\frac{1}{A_2^t{cos}^3\theta }\right]$$

(14)

Considering the effect of bending deformation of the triangular lattice tower, the potential external load energy is:

$$U_P^t=-F_p{\int }_0^l\frac{1}{2}{\left(y^{\prime }\right)}^{2}dx=-F_p\frac{a^2{\pi }^2}{16l}$$

(15)

Using the energy method [34,35], and according to the stationary value condition of the external load potential energy $\frac{d{E}_p^t}{da}$, the critical load expression of the triangular lattice tower is obtained as:

$$F_{P_{cr}}^t=\frac{{\pi }^{2}E{I}_t}{l^2}\frac{1}{4+\frac{{\pi }^2}{12}{\left(\frac{b}{l}\right)}^2\frac{A_1^t}{A_2^{t}sin\theta {cos}^2\theta }}$$

(16)

In Equation (16), $I_t=\frac{A_1^t{b}^2}{2}$ is the moment of inertia of the triangular lattice tower section to the centroid axis, as shown in Figure 1.

2.2. Stability Bearing Capacity of the Quadrilateral Lattice Tower

Similarly, the calculation of the quadrilateral lattice tower is also simplified to a cantilever structure with a fixed bottom and a free upper part. The calculation model is shown in Figure 2.

According to Equations (2)–(4), in the quadrilateral lattice tower, the axial force $F_{Nq}^1$ of the leg member and the axial force $F_{Nq}^2$ of the diagonal member can be obtained by the approximate calculation of the truss:

$$F_{Nq}^1=\pm \frac{M}{2b}=\pm F_p\frac{a}{2b}sin\frac{\pi x}{2l}$$

(17)

$$F_{Nq}^2=\pm \frac{F_Q}{4cos\theta }=\pm F_p\frac{a\pi }{8lcos\theta }cos\frac{\pi x}{2l}$$

(18)

In Equations (17) and (18), $F_{Nq}^1$ is the axial force of the leg member of the quadrilateral tower; $F_{Nq}^2$ is the axial force of the diagonal member of the quadrilateral tower; b is the root opening of the tower; $\theta$ is the angle between the diagonal member and the x-axis, as shown in Figure 2.

The strain energy of the quadrilateral lattice tower is:

$$U_q=\sum \frac{F_N^2{s}_i^q}{2E{A}_i^q}$$

(19)

In Equation (19), $s_i^q$ and $A_i^q$ are the length and cross-sectional area of each rod member of the quadrilateral tower, and E is the elastic modulus of the member material.

The axial force can be substituted as follows:

$$U_q=\frac{1}{2E}\left(\sum _1^{4n}\frac{{\left(F_p\frac{a}{2b}sin\frac{\pi x}{2l}\right)}^{2}d}{A_1^q}+\sum _1^{4n}\frac{{\left(F_p\frac{a\pi }{8lcos\theta }cos\frac{\pi x}{2l}\right)}^2\frac{b}{cos\theta }}{A_2^q}\right)$$

(20)

In Equation (20), $A_1^q$ is the cross-sectional area of the leg member of the quadrilateral tower; $A_2^q$ is the cross-sectional area of the diagonal member of the quadrilateral tower; d is the height of the tower internode; b is the root opening of the tower; $\theta$ is the angle between the diagonal member and the x-axis; n is the number of internodes of the tower; the total number of diagonal members is the sum of $4n$ rods; and the total number of leg members is the sum of $4n$ rods.

Using the variation in Equations (11) and (21):

$$\begin{array}{cc}\hfill & \sum _1^{4n}{\left(sin\frac{\pi x}{2l}\right)}^{2}d\approx 4{\int }_0^l{\left(sin\frac{\pi x}{2l}\right)}^{2}dx=2l\hfill \\ \hfill & \sum _1^{4n}{\left(cos\frac{\pi x}{2l}\right)}^{2}d\approx 4{\int }_0^l{\left(cos\frac{\pi x}{2l}\right)}^{2}dx=2l\hfill \end{array}$$

(21)

Combining Equation (13), the strain energy of the quadrilateral lattice tower can be written as:

$$U_q=\frac{F_p^{2}l{a}^2}{4E}\left[\frac{1}{A_1^q{b}^2}+\frac{{\pi }^2}{32l^{2}tan\theta }·\frac{1}{A_2^q{cos}^3\theta }\right]$$

(22)

The external load potential energy is:

$$U_P^q=-F_p{\int }_0^l\frac{1}{2}{\left(y^{{}^{\prime }}\right)}^{2}dx=-F_p\frac{a^2{\pi }^2}{16l}$$

(23)

Using the energy method [34,35], and according to the stationary value condition of the external load potential energy $\frac{d{E}_P^q}{da}=0$, the critical load expression of the quadrilateral lattice tower is obtained as:

$$F_{Pcr}^q=\frac{{\pi }^{2}E{I}_q}{l^2}\frac{1}{4+\frac{{\pi }^2}{8}{\left(\frac{b}{l}\right)}^2\frac{A_1^q}{A_2^{q}sin\theta {cos}^2\theta }}$$

(24)

In Equation (24), $I_q=4A_1^q{\left(\frac{b}{2}\right)}^2=A_1^q{b}^2$ is the moment of inertia of the quadrilateral lattice tower section to the centroid axis, as shown in Figure 2.

3. Influence of the Joint Axis Stiffness on Triangular and Quadrilateral Lattice Towers

In the tower structure, the axial stiffness of the connection joint will affect the stability bearing capacity of the tower, and the strain energy of the connection joint of the tower can be expressed as:

$$U=\sum \frac{F_N^{2}kb/cos\theta }{2E{A}_j}$$

(25)

In Equation (25), k is the ratio of the length of the connection joint to the total length of the rod member, referred to as the joint length coefficient; $A_j$ is the cross-sectional area of the equivalent spring joint.

3.1. Calculation Method of the Triangular Lattice Tower Considering Joints Axial Stiffness

Considering the axial stiffness of the connection joints, the strain energy equation of the triangular lattice tower becomes:

$$\begin{array}{cc}\hfill & U_t=\frac{1}{2E}\left(\sum _1^n\frac{{\left(F_p\frac{2\sqrt{3}a}{3b}sin\frac{\pi x}{2l}\right)}^{2}d}{A_1^t}+\sum _1^{2n}\frac{{\left(F_p\frac{\sqrt{3}a}{3b}sin\frac{\pi x}{2l}\right)}^{2}d}{A_1^t}+\right.\hfill \\ \hfill & \left.\sum _1^{4n}\frac{{\left(F_p\frac{2a\pi }{8\sqrt{3}lcos\theta }cos\frac{\pi x}{2l}\right)}^2\left(1-k\right)\frac{b}{cos\theta }}{A_2^t}+\sum _1^{8n}\frac{{\left(F_p\frac{2a\pi }{8\sqrt{3}lcos\theta }cos\frac{\pi x}{2l}\right)}^{2}k\frac{b}{cos\theta }}{A_j}\right)\hfill \end{array}$$

(26)

Combining transformation Equation (27), we obtain:

$$\sum _1^{8n}{\left(cos\frac{\pi x}{2l}\right)}^{2}d\approx 8{\int }_0^l{\left(cos\frac{\pi x}{2l}\right)}^{2}dx=4l$$

(27)

Equations (12), (13) and (27) are solved simultaneously to obtain:

$$U_t=\frac{F_p^{2}l{a}^2}{2E}\left[\frac{1}{A_1^t{b}^2}+\frac{{\pi }^2}{48l^{2}tan\theta }·\frac{1-k}{A_2^t{cos}^3\theta }+\frac{{\pi }^2}{24l^{2}tan\theta }·\frac{k}{A_j{cos}^3\theta }\right]$$

(28)

From $\frac{d{E}_P^t}{da}=0$, the expression of the critical load can be obtained:

$$F_{Pcr}^t=\frac{{\pi }^{2}EI}{l^2}\frac{1}{4+\frac{{\pi }^2}{12}{\left(\frac{b}{l}\right)}^2\frac{\left(1-k\right)A_1^t}{A_2^{t}sin\theta {cos}^2\theta }+\frac{{\pi }^{2}k}{6}{\left(\frac{b}{l}\right)}^2\frac{A_1^t}{A_{j}sin\theta {cos}^2\theta }}$$

(29)

3.2. Calculation Method of the Quadrilateral Lattice Tower Considering Joints Axial Stiffness

Considering the axial stiffness of the connection joints, the strain energy equation of the quadrilateral lattice tower becomes:

$$\begin{array}{cc}\hfill & U_q=\frac{1}{2E}\left(\sum _1^{4n}\frac{{\left(F_p\frac{a}{2b}sin\frac{\pi x}{2l}\right)}^{2}d}{A_1^q}+\sum _1^{4n}\frac{{\left(F_p\frac{a\pi }{8lcos\theta }cos\frac{\pi x}{2l}\right)}^2\left(1-k\right)\frac{b}{cos\theta }}{A_2^q}\right.\hfill \\ \hfill & \left.+\sum _1^{8n}\frac{{\left(F_p\frac{a\pi }{8lcos\theta }cos\frac{\pi x}{2l}\right)}^{2}k\frac{b}{cos\theta }}{A_j}\right)\hfill \end{array}$$

(30)

Combining Equations (21), (22), and (27), we can obtain:

$$U_q=\frac{F_p^{2}l{a}^2}{4E}\left[\frac{1}{A_1^q{b}^2}+\frac{{\pi }^2}{32l^{2}tan\theta }·\frac{1-k}{A_2^q{cos}^3\theta }+\frac{{\pi }^2}{32l^{2}tan\theta }·\frac{k}{A_j{cos}^3\theta }\right]$$

(31)

From $\frac{d{E}_P^q}{da}=0$, the expression of the critical load can be obtained:

$$F_{Pcr}^q=\frac{{\pi }^{2}EI}{l^2}\frac{1}{4+\frac{{\pi }^2}{8}{\left(\frac{b}{l}\right)}^2\frac{\left(1-k\right)A_1^q}{A_2^{q}sin\theta {cos}^2\theta }+\frac{{\pi }^{2}k}{4}{\left(\frac{b}{l}\right)}^2\frac{A_1^q}{A_{j}sin\theta {cos}^2\theta }}$$

(32)

3.3. Influence of the Joint Axial Stiffness on Stability Bearing Capacity of Triangular and Quadrilateral Lattice Towers

In order to obtain the influence of the axial stiffness of the connection joints on the stability bearing capacities of the triangular and quadrilateral lattice towers, a mathematical model of the two-story tower cell was established for analysis, namely, $l=nh$, $n=2$, as shown in Figure 3. The tower model adopts the same root opening and internode height, i.e., $b=h$. The cross-sectional dimensions of rod members and equivalent joints of the triangular and quadrilateral lattice towers are $A_1=A$, $A_2=A/5$, $A_j=\zeta A$, the length coefficient of the connection joint of the tower is $k=0.3$.

According to Equations (16), (24), (29), and (32), the calculation expressions of the stability bearing capacities of triangular and quadrilateral lattice towers with and without the joint axial stiffness are obtained. If the ratio coefficient of the calculation results with and without the joint axial stiffness is $\chi$, then the expression is:

$${\chi }_t=\frac{\left(48\sqrt{2}+5{\pi }^2\right)\zeta }{48\sqrt{2}\zeta +3.5{\pi }^2\zeta +0.6{\pi }^2}$$

(33)

$${\chi }_q=\frac{\left(32\sqrt{2}+5{\pi }^2\right)\zeta }{32\sqrt{2}\zeta +3.5{\pi }^2\zeta +0.3{\pi }^2}$$

(34)

In Equations (33) and (34), ${\chi }_t$ and ${\chi }_q$ are the ratio coefficients of the load-bearing capacities of triangular and quadrilateral lattice towers with and without the axial stiffness of the joints, respectively, and $\zeta$ is the axial stiffness variation coefficient of the equivalent spring connection joints.

According to Equations (29) and (32), take $\zeta =1/5$ to calculate the stability bearing capacities of triangular and quadrilateral lattice towers as the reference bearing capacities, take $\zeta =1/5\sim 3$ to calculate the stability bearing capacities of triangular and quadrilateral lattice towers, and record their ratio coefficients to the reference bearing capacities as ${\chi }_t^r$ and ${\chi }_q^r$, respectively. The relationship curves between the ratio coefficients $\zeta$ and the axial stiffness variation coefficients ${\chi }_t^r$, ${\chi }_q^r$ of the triangular and quadrilateral lattice towers are obtained as shown in Figure 4.

4. Influence of the DLDPJ on the Bearing Capacity of the Tower

4.1. Basic Dimensions of the Tower Model

The DLDPJ is a new type of steel pipe and angle steel connection joint proposed by Zhao et al. [31,32]. As shown in Figure 5b, the DLDPJ can reduce the eccentric load of the traditional SLSPJ and improve the stability bearing capacity of the tower. Due to the different axial stiffnesses between the traditional connection joint and DLDPJ, the influence on the stability bearing capacities of the triangular and quadrilateral lattice tower is also different. By establishing the triangular and quadrilateral lattice tower models, including the traditional joint and DLDPJ, and further combining the test, theory and finite element analysis, the influence law of the DLDPJ on the stability bearing capacities of triangular and quadrilateral lattice towers is obtained.

In order to study the influence of the DLDPJ on the stability bearing capacities of triangular and quadrilateral lattice towers, the cell numerical models of triangular and quadrilateral two-layer transmission towers including the SLSPJ and DLDPJ are established by ANSYS. The root opening of the tower model is 4 m ($b=4$ m), and the height is 8m ($h=4$ m), as shown in Figure 6.

In the tower, the size of the connection members of the SLDPJ and DLDPJ shall be set according to the test model, the leg member shall be steel pipe $\varphi 140\times 8\mathrm{mm}$, angle steel $\mathrm{L}50\times 5\mathrm{mm}$ is used for the diagonal members, the thickness of the SLSPJ connection plate is $t=8$ mm, and the thickness of the DLDPJ connection plate is $t=4$ mm. The shape and component model of the connection joint are shown in Figure 7.

4.2. Finite Element Analysis of the Tower Model

In order to accurately predict the behavior of the structure, an accurate finite element model is essential and very important [17]. The bolted joints of the angle steel members of the lattice transmission tower will have joint eccentricity [36]. In the traditional analysis of the transmission tower, the rod element is mostly used for calculation, and the tower connection joints are mostly simplified as hinge joints, and the calculation model is quite different from the actual model. The multi-scale finite element model can accurately analyze the complex tower joint problems, such as stress distribution, plastic development, failure mode, and ultimate bearing capacity [1]. Therefore, according to the model joint size of the SLSPJ and DLDPJ, the more practical triangular and quadrangular lattice transmission tower models are established for numerical simulation analysis.

Firstly, according to the triangular and quadrilateral lattice tower models and the size of connection joints, the geometric models of triangular and quadrilateral lattice towers including the SLSPJ and DLDPJ are established, as shown in Figure 8 and Figure 9.

Secondly, the finite element models of triangular and quadrilateral lattice tower cells are formed. The members and connection joints of triangular and quadrilateral lattice tower models are made of steel, with elastic modulus $E=210\mathrm{GPa}$, Poisson’s ratio $\nu =0.3$, and density $\rho =7850\mathrm{kg}/{\mathrm{m}}^3$. The shell63 shell element with four modes in the ANSYS finite element is used to simulate the tower model, and the thickness of the shell element is set according to the thickness of the member profile and gusset plate. The grid size of the shell element will affect the calculation results of the stability bearing capacity of the tower. Therefore, on the premise of meeting the calculation accuracy, the grid density at the connection joint of the tower will appropriately increase. According to the previous numerical simulation and test research, the bolt sliding in the tower connection joints has a great impact on the displacement but has little impact on the stability bearing capacity of the whole tower [37,38,39,40]. Since this paper mainly studies the stability bearing capacity of the transmission tower, the influence of bolt sliding can not be considered, so the connection gusset plate at the bolt hole connecting the small angle steel and the angle steel diagonal member can adopt the coupling treatment method. The effect of the bolt connection is simulated through the coupling connection joint method, and the triangular and quadrilateral lattice tower finite element models including the SLSPJ and DLDPJ are further established, as shown in Figure 10 and Figure 11.

Nonlinear static analysis is very important for understanding the performance, possible bearing capacity, design defects and instability of structures [21]. During the manufacturing, transportation and assembly of transmission tower, geometric defects will inevitably appear, including joint deviation, initial buckling and eccentricity of components, residual stress, etc. [24]. The buckling of the lattice tower is mostly caused by additional eccentric force caused by geometric defects [41]. Therefore, considering the nonlinear influence of geometric defects of the tower, the arc length method, which has a significant effect on structural jump buckling analysis, is used to study the nonlinear stability bearing capacities of triangular and quadrilateral lattice towers under vertical load. In the tower model, the upper and lower loading plates are used to apply loads and constraints. As shown in Figure 12, the tower bottom plate is constrained in the X, y and Z directions in the Cartesian coordinate system, and the top plate is uniformly loaded. The nonlinear stability bearing capacities of triangular and quadrilateral lattice towers can be obtained by the arc length method.

Considering the influence of geometric defects of the tower, the whole load displacement process of triangular and quadrilateral lattice towers is analyzed. It is assumed that the materials of the members and joints are in the stage of elastic deformation during the loading process. The geometric defects of the tower mainly consider the influence of the installation deviation of the initial member position and construction defects. The first-order buckling mode of the tower is taken as the distribution mode of the geometric defects of the tower in the initial state, and the maximum defect value of the geometric defects can be 1/300 [16,42] of the tower height h. The arc length method is used to analyze the whole process of geometric nonlinear buckling of the tower. The geometrically nonlinear ultimate stability values of triangular and quadrilateral lattice towers with the SLSPJ and DLDPJ are obtained, respectively.

Through the analysis, the vertical displacement nephograms of triangular and quadrilateral lattice towers with the SLSPJ and DLDPJ are obtained, as shown in Figure 13 and Figure 14.

The load–displacement relationship curves of the stability bearing capacities of triangular and quadrilateral lattice towers under the influence of the SLSPJ and DLDPJ are shown in Figure 15.

The results of the load displacement curves of triangular and quadrilateral lattice towers in Figure 15 show that the DLDPJ can improve the nonlinear stability bearing capacity of the tower. For the triangular lattice tower, the DLDPJ can improve the nonlinear stability bearing capacity of the transmission tower by 22.7%; for quadrilateral towers, the DLDPJ can improve the nonlinear stability bearing capacity of the transmission tower by 14.9%. Through numerical analysis, it is found that the DLDPJ has a greater impact on the nonlinear stability bearing capacity of the triangular lattice tower, and the impact on the triangular lattice tower is 1.52 times that of the quadrilateral lattice tower.

4.3. Joint Test Analysis and Verification

In Equations (29) and (32), the calculation expression of the cross-sectional area $A_j$ of the tower connection joint equivalent to the spring joint can be expressed as:

$$A_j=\frac{Fl}{E\Delta l}$$

(35)

According to Equation (35), the ratio of equivalent area of the SLSPJ and DLDPJ is:

$$\frac{A_{js}}{A_{jd}}=\frac{{\kappa }_s{l}_s}{{\kappa }_d{l}_d}$$

(36)

In Equation (36), $A_{js}$ and $A_{jd}$ are the equivalent cross-sectional areas of the SLSPJ and DLDPJ, respectively; ${\kappa }_s$ and ${\kappa }_d$ are the equivalent elastic stiffness of the SLSPJ and DLDPJ, respectively; $l_s$ and $l_d$ represent the connection lengths of the SLSPJ and DLDPJ, respectively.

According to the test results of the bearing capacities of the SLSPJ and DLDPJ [32], the test models of SLSPJ and DLDPJ are shown in Figure 16, the loading system of the SLSPJ and DLDPJ models is shown in

Table 1. Load classifications of the specimens of SLSPJ and DLDPJ.

Loading Percentage		1	2	3	4
Load size (kN)	SLSPJ	60	105	150	destruction
	DLDPJ	60	100	destruction	-

, and the load–displacement curve of the SLSPJ and DLDPJ test is shown in Figure 17. Combined with the results of model test, taking ${\kappa }_s=11.4\mathrm{kN}/\mathrm{mm}$, ${\kappa }_d=35.6\mathrm{kN}/\mathrm{mm}$ and $l_s=l_d$, we can obtain $A_{jd}/A_{js}=3.12$.

Take the cross-sectional dimensions of the tower rod members as $A_1=A$, $A_2=A/5$, $A_{js}={\zeta }_{s}A$, $A_{jd}={\zeta }_{d}A$. According to Equation (29), the stability bearing capacities of the triangular lattice tower including the SLSPJ and DLDPJ are:

$$F_{Pcr}^{st}=\frac{3{\pi }^{2}E{\zeta }_{s}A}{96{\zeta }_s+3.5\sqrt{2}{\pi }^2{\zeta }_s+0.6\sqrt{2}{\pi }^2}$$

(37)

$$F_{Pcr}^{dt}=\frac{3{\pi }^{2}E{\zeta }_{d}A}{96{\zeta }_d+3.5\sqrt{2}{\pi }^2{\zeta }_d+0.6\sqrt{2}{\pi }^2}$$

(38)

In Equations (37) and (38), $F_{Pcr}^{st}$ and $F_{Pcr}^{dt}$ are the stability bearing capacities of triangular lattice towers including the SLSPJ and DLDPJ, respectively; ${\zeta }_s$ and ${\zeta }_d$ are the axial stiffness variation coefficients of the SLSPJ and DLDPJ, respectively, which are equivalent to spring joints.

According to Equation (32), similarly, the stability bearing capacities of quadrilateral lattice towers including the SLSPJ and DLDPJ are:

$$F_{Pcr}^{sq}=\frac{4{\pi }^{2}E{\zeta }_{s}A}{64{\zeta }_s+3.5\sqrt{2}{\pi }^2{\zeta }_s+0.3\sqrt{2}{\pi }^2}$$

(39)

$$F_{Pcr}^{dq}=\frac{4{\pi }^{2}E{\zeta }_{d}A}{64{\zeta }_d+3.5\sqrt{2}{\pi }^2{\zeta }_d+0.3\sqrt{2}{\pi }^2}$$

(40)

In Equations (39) and (40), $F_{Pcr}^{sq}$ and $F_{Pcr}^{dq}$ are the stability bearing capacities of quadrilateral lattice towers including the SLSPJ and DLDPJ, respectively.

According to the joint equivalent area relationship obtained from the test, take ${\zeta }_d=0.15$ and ${\zeta }_s=0.47$; the tower root opening and internode height are both h; the number of tower layers is $n=2$; the tower height is $l=nh$; and take the length coefficient $k=0.3$. According to Equations (37)–(40), it is calculated that the DLDPJ can improve the stability bearing capacity of the triangular lattice tower by 23.4%, and the stability bearing capacity of the quadrilateral lattice tower is increased by 15.6%. Through the test and theoretical analysis, it is found that the DLDPJ has a greater impact on the nonlinear stability bearing capacity of the triangular lattice tower, and the impact on the triangular lattice tower is 1.5 times that of the quadrilateral lattice tower.

4.4. Result Analysis

Through numerical, experimental and theoretical analyses, the effects of the SLSPJ and DLDPJ on the stability bearing capacities of triangular and quadrilateral lattice towers are studied. It is found that the DLDPJ can improve the stability bearing capacities of triangular and quadrilateral lattice towers. The effect of the DLDPJ on improving the stability bearing capacity of the tower is greater than that of the SLSPJ. The influence results of the DLDPJ compared with the SLSPJ are shown in

Table 2. Influence of the DLDPJ on stability bearing capacity.

	Improvement of Bearing Capacity(%)	Numerical Analysis	Experiment and Theory
Type of the Tower
The triangular lattice tower		22.7	23.4
The quadrilateral lattice tower		14.9	15.6
Influence degree (ratio) *		1.52	1.50

* The influence degree (ratio) refers to the ratio of the influence degree of the DLDPJ on the stability bearing capacities of triangular and quadrilateral lattice towers.

.

The results of numerical analysis, test and theoretical analysis in Table 2 show that the DLDPJ can improve the stability bearing capacity of the triangular lattice tower by 22.7∼23.4% and can improve the stability bearing capacity of the quadrilateral lattice tower by 14.9∼15.6%. The influence of the DLDPJ on the stability bearing capacity of the triangular lattice tower is 1.50∼1.52 times that of the quadrilateral lattice tower.

5. Conclusions

In this paper, considering the influence of the axial stiffnesses of connection joints, the theoretical calculation expression of the stability bearing capacities of triangular and quadrilateral lattice towers is derived by the energy method. Through numerical simulations, tests and theoretical analysis, the influence law of the DLDPJ on the stability bearing capacities of triangular and quadrilateral lattice towers is further studied, and the following conclusions are obtained:

• Considering the axial stiffness of the connection joints, the theoretical calculation expressions of the stability bearing capacities of triangular and quadrilateral lattice towers are obtained by the energy method. The influence of the change in the axial stiffness on the stability bearing capacities of triangular and quadrilateral lattice towers is studied. With the increase in the axial stiffness, the ratio coefficient of the tower stability bearing capacity increases in the early stage and then tends to be stable. Through quantitative analysis, it is found that the influence of the axial stiffness on the ratio coefficient of the triangular lattice tower is greater than that of the quadrilateral lattice tower.
• Furthermore, the finite element models of triangular and quadrilateral lattice towers including connection joints are established. Considering the influence of geometric nonlinearity, the influence of the laws of the SLSPJ and DLDPJ on the stability bearing capacities of triangular and quadrilateral lattice towers are obtained by using the arc length method. The results show that the DLDPJ can greatly improve the stability bearing capacity of the triangular lattice tower by 22.7% and can improve the stability bearing capacity of the quadrilateral lattice tower by 14.9%; the ratio of influence degree is 1.52.
• Through theoretical analysis, considering the axial stiffness of the SLSPJ and DLDPJ, the theoretical calculation expression of the stability bearing capacities of triangular and quadrilateral lattice towers is obtained. According to the test results of the SLSPJ and DLDPJ, the influence of the DLDPJ on the stability bearing capacities of triangular and quadrilateral lattice towers is further analyzed. The DLDPJ has a greater impact on the stability bearing capacity of the triangular lattice tower, increasing the stability bearing capacity of the triangular lattice tower by 23.4%, and it can increase the stability bearing capacity of the quadrilateral lattice tower by 15.6%; the ratio of influence degree is 1.50.

In this paper, considering the axial stiffness of the SLSPJ and DLDPJ, the theoretical calculation expressions of the stability bearing capacities of triangular and quadrilateral lattice towers are derived. The influence of the DLDPJ on the stability bearing capacities of triangular and quadrilateral lattice towers is studied from the three aspects of theoretical analysis, numerical simulation and test result analysis, and some research results are obtained. these results can provide reference for the popularization, application and design of the DLDPJ.