Electricity plays a significant role in social development and various industries. After years of development, China has built a large-scale power transmission network that inevitably passes through regions with complex terrain (Cai and Wan, 2021). As the conQA23ng nodes between high-voltage transmission lines and the ground, transmission towers are often built on steep slopes of mountains to reduce transmission distance and power loss (Xu et al., 2022). This terrain condition increases the possibility of geological hazards. Transmission towers are susceptible to catastrophic damage caused by rockfall impacts, which can further lead to local instability (Cai et al., 2019; Xu et al., 2023) and collapse of the entire tower (Xu et al., 2021; Xue et al., 2023). Rockfall is a major disaster that seriously threatens the structural safety of transmission towers in mountainous areas.

Researchers have carried out a large number of studies on transmission towers to ensure the stable operation of the power grid (Li et al., 2022; Li, et al., 2023). In the field of finite element analysis. The study conducted by Fu and Li. (2016) investigated the dynamic response of tower systems under wind and rain loads. It was found that the average displacement amplification of the structure was 22% higher under combined wind and rain loads than when under wind loads alone. He S et al. (2019) developed a model for a three-tower, two-span tower-line system to investigate the response of the towers to wind-induced vibrations. The results indicated that the dynamic response was higher than the static response, with a stress ratio of 1.43. Morgan and Swift. (1964) identified that the behavior of the tower-line system was affected by the height of ice shedding to varying degrees. The study results revealed that ground clearance of the lower conductor of the loaded span drops below the safety limit for ice loads beyond 1 pound per foot. The study conducted by Yang and Hong. (2016) examined the nonlinear and non-elastic response of tower-line systems exposed to downburst loading. The outcomes of the investigation suggested that the load-carrying capacity curve of an individual tower was able to represent the curvature for the tower-line system. Zhou et al. (2022) performed physical model testing and numerical analysis on transmission tower landslides caused by rainfall. The study identifies the failure process and modes of these landslides. Gong and Zhi (2020) performed an analysis on the failure mode of transmission towers impacted by earthquakes. The outcomes of their study demonstrated that the failure mode of the towers, under severe seismic impacts, was in-plane global collapse as a result of local failures. According to Tian et al. (2019) a collapse analysis of transmission tower was conducted to evaluate their structural durability when exposed to varying wind angles. The results show that the bending deformation in the mid-section of the tower body was significantly better than the shear deformation, ultimately leading to the failure of its main components. The analysis of transmission tower line systems using tower and cable models is complicated and time-consuming due to the wide range of tower types. As a result, the numerical calculations required for such models can be very complex.

Buildings in mountainous areas are prone to the impact of rockfall disasters caused by earthquakes, landslides, and debris flows (Happ and Noble, 1993). In terms of the conditions for rockfall occurrence, the collapsed rock masses are mostly concentrated on slopes with a gradient greater than 32°, and after detaching from the bedrock, they fall and collide continuously with trees, buildings, and other objects, with impact velocities mostly below 20 m/s. Although the rockfall impact belongs to low to medium-speed impact, the large mass of falling rocks can still acquire significant energy during the falling process (Wyllie, 2014). In terms of the analytical solutions for displacement response under impact, Zhang et al. (2023) derived a formula to measure the largest displacement caused by kinetic energy during impact. The study found that doubling of the impactor’s mass results in a 150% increase in the maximum displacement. Wang et al. (2014) proposed two enhanced analytical techniques rooted in forced vibration theory and variable separation method to determine the natural frequency and mode shape of a uniformly cantilevered beam. Prasada Rao et al. (2004) derived a formula for predicting the natural frequency of tower-like structures based on theoretical deformation and geometric parameters. In addition, Yan et al. (2018) investigated the influence of rockfall impact angle on the impact response of reinforced concrete slabs. Their findings revealed that both the sphericity of the impacting object and the angle exert impact on the RC slabs. Prakash et al. (2021) conducted a study on the dynamic prediction of beam behavior subject to rockfall impact. The study found that the safety of the structure is primarily influenced by the impact mass. According to analysis of existing literature, there are few relevant studies on the analytical solutions of displacement response of transmission towers under rockfall impact, which is contrary to the increasing scale and potential protection needs of mountainous transmission networks.

Rockfall disasters seriously threaten the safety of mountainous transmission networks. In order to carry out impact-resistant design of structures and understand the impact of rockfall loads on transmission tower structures effectively, it is necessary to conduct in-depth research and take protective measures. This paper proposes an approximate algorithm for the fundamental frequency of tower structures based on the energy method and combination synthesis method. It derives the displacement response of iron towers under falling rock impact, verifies the proposed approximate algorithm through ABAQUS finite element numerical simulation, and parametric analysis was also carried out. This paper was organized as follows. Section 2 is devoted to theoretical derivation for the fundamental frequency and lateral dynamic displacements of the tower structure; Case study of a transmission tower was carried out in section 3; And conclusions are drawn in section 4.

The lateral dynamic displacements of a lattice transmission tower structure can be expressed as a partial differential equation m ∂ 2 Y z , t ∂ t 2 + C ∂ Y z , t ∂ t + ∂ 2 ∂ z 2 ∂ 2 Y z , t ∂ z 2 E I = Q z , t (1) where Y(z,t) = transverse dynamic displacement, E = Young’s modulus of material, C = tower damping intensity, m = Tower mass per unit length, Q(z,t) = impact load function. The transmission tower is a complex structure that can be reduced as a cantilever beam with a varying cross-section (Jiang et al., 2011; Wang et al., 2015). As a result, the inertia of cross Section 1 can be simply represented by I z = I 0 1 + β z L   0 ≤ z ≤ L (2)

Where β = the change rate of tower section.

The continuous lateral displacement Y(z,t) is represented by the finite sum the production of shape functions and generalized coordinates (Erturk and Inman, 2011) Y z , t = ∑ i = 1 n φ i z y i t (3) where φ_i(z) = Characteristic function of linear uniform cantilever transfer tower, yi(t) = generalized coordinates over time.

The impact of rockfall is a transient effect. And the impact of rock falling on the tower can be approximated by the impact function F (z,t) in terms of piece-wise function (Wang et al., 2014) F z , t = P δ z − l 0 sin π t t 0   0 ≤ t ≤ t 0 0 t 0 ≤ t (4)

Where P and t ₀ = the half sine pulse force magnitude and duration, respectively; l ₀ = the height of the bottom end of the transmission tower from the impact point, δ = unit pulse function.

After substituting Eq. 3 into Eq. 1 and multiplying φ_j(z) on both sides of Eq. 1, then integrate each term of Eq. 1 over span L and divide by m, it yields (Wang et al., 2017): y ¨ i + 2 ξ i ω i y ˙ i + ω i 2 y i = ∫ 0 L F z , t φ i z m d x (5) in which ξ i = c 2 m ω i

The right side of Eq. 5 can be formulated as ∫ 0 L φ i z F z , t m d x = ∫ 0 L δ z − l 0 φ i z q t m d x = φ i l 0 q t m (6) in which q t = P ⁡ sin π t t 0   0 ≤ t ≤ t 0 0   t 0 ≤ t (7)

After substituting Eq. 6 into Eq. 5, the generalized time-dependent coordinates y_i can be determined y i = − φ l 0 P π e − ξ i ω i t ω i 1 − ξ i t 0 sin ω i 1 − ξ i 2 t − ϕ 1 − sin π t 0 t − ϕ 2 1 − π ω i t 0 2 2 + 2 π ξ i ω i t 0 2   t 0 ≥ t ≥ 0 (8) y i = φ l 0 P π 1 − ξ i t 0 ω i e t 0 − t ξ i ω i ⁡ cos 1 − ξ i 2 t ω i − ϕ 1 × sin 1 − ξ i 2 t 0 ω i 1 − π ω i t 0 2 2 + 2 π ξ i ω i t 0 2 − φ l 0 P π 1 − ξ i t 0 ω i e − ξ i ω i t ⁡ sin 1 − ξ i 2 t ω i − ϕ 1 × 1 + e ξ i ω i t 0 ⁡ cos 1 − ξ i 2 t 0 ω i 1 − π ω i t 0 2 2 + 2 π ξ i ω i t 0 2 t 0 ≤ t (9) in which tan ⁡ ϕ 1 = 2 ξ i 1 − ξ i 2 1 − π ω i t 0 2 − 2 ξ i 2 , tan ⁡ ϕ 2 = 2 ξ i π ω i t 0 1 − π ω i t 0 2

The eigenfunction of a transmission tower with cross arms is given by (Erturk and Inman, 2011) φ i z = A i B i sin z λ i L − sinh z λ i L + cos z λ i L − cosh z λ i L (10) where A_i = the modal constant, L = the height of the tower, λ_i = the eigenvalue, and B_i is given by B i = sin ⁡ λ i + M λ i m L cos ⁡ λ i − cosh ⁡ λ i − sinh ⁡ λ i cos ⁡ λ i − M λ i m L sin ⁡ λ i − sinh ⁡ λ i − cosh ⁡ λ i (11)

The eigenvalue of a system can be determined by solving the characteristic equations (Erturk and Inman, 2011) 1 + cos ⁡ λ i ⁡ cosh ⁡ λ i = M λ i m L sin ⁡ λ i ⁡ cosh ⁡ λ i − cos ⁡ λ i ⁡ sinh ⁡ λ i (12)

Once the eigenvalues solved in Eq. 12 and then substitute Eqs 8–10 into Eq. 3, the lateral dynamic displacements of a transmission tower equipped with cross arms subjected to rockfall impact can be determined.

The fundamental frequency of the vibration of a transmission tower structure using energy method is given by (Gu and Yin, 1983) ω 2 = E ∫ 0 L I z d 2 x f z d z 2 2 d z + k ′ G ∫ 0 L A z d x s z d z 2 d z γ g ∫ 0 L A z x f z + x s z 2 d z + ∑ i = 1 n W i g ∫ 0 L δ z − l i x f z + x s z 2 d z (13) where x_f (z), x_s (z) = displacement due to bending and shear deformation of the structure, respectively; G, E = shear modulus and modulus of elasticity of the structure, respectively; W_i = concentrated mass load, k' = 5/6 (rectangular section), G = 3E/8, and cross-sectional area A = A₀(1+βz/L).

Since the variation of the cross-section and additional cross arms of the transmission tower, it is difficult to calculate the fundamental frequency. Herein the combined synthesis method is used to calculate the simplified calculation of multiple subsystems. According to the combination synthesis method, its fundamental frequency can be approximated by a combination of the fundamental frequencies of subsystem (Gu and Yin, 1983). 1 ω 2 ≤ 1 ω 1 2 + 1 ω 2 2 + ∑ i = 1 n 1 ω 3 2 + 1 ω 4 2 i (14)

Where ω 1 = the subsystem frequency composed of tower mass distribution and bending stiffness; ω 2 = the subsystem frequency composed of tower mass distribution and shear stiffness; ω 3 = the subsystem frequency composed of cross arm mass and bending stiffness near the top tower; ω 3 = the subsystem frequency composed of cross arm mass and shear stiffness near the top tower; and n = number of additional masses. ω 1 2 = E ∫ 0 L I z d 2 x f z d z 2 2 d z γ g ∫ 0 L A z x f 2 z d z (15) ω 2 2 = k ′ G ∫ 0 L A z d x s z d z 2 d z γ g ∫ 0 L A z x s 2 z d z (16) ω 3 2 i = E ∫ 0 L I z d 2 x f z d z 2 2 d z W i g ∫ 0 L δ z − l i x f 2 z d z (17) ω 4 2 i = k ′ G ∫ 0 L A z d x s z d z 2 d z W i g ∫ 0 L δ z − l i x s 2 z d z (18)

The selected deflection function must satisfy the geometric boundary conditions. Since a transmission tower structure can be idealized as a cantilever beam with a stationary end and a free end, x_f represents the deflection due to bending and it could be chosen as follows (Erturk and Inman, 2011) x f z = C f 3 z L 2 − z L 3 (19)

C_f is unknown coefficient. And x_s represents the displacements due to shear deformation and it could be taken the first two Fourier steps as follows C_f is unknown coefficient. And x_s represents the displacements due to shear deformation and it could be taken the first two Fourier steps as follows x s z = C s ⁡ sin π z 2 L = C s π z 2 L − π 3 z 3 24 L 3 (20)

C_s is unknown coefficient.

By substituting Eq. 19 into Eqs 15, 17 and substituting Eq. 20 into Eqs 16, 18, we obtain: ω 1 2 = 36 E ∫ 0 L I z L − z 2 d z γ g ∫ 0 L A z 3 z 2 L − z 3 2 d z (21) ω 2 2 = k ′ G ∫ 0 L A z π 2 4 l 2 − π 4 z 2 8 l 4 + π 6 z 4 64 l 6 d z γ g ∫ 0 L A z π z 2 l − π 3 z 3 24 l 3 2 d z (22) ω 3 2 i = 36 E ∫ 0 L I z L − z 2 d z W i g x f 2 l i (23) ω 4 2 i = k ′ G ∫ 0 L A z π 2 4 l 2 − π 4 z 2 8 l 4 + π 6 z 4 64 l 6 d z W i g x s 2 l i (24)

And finally, the fundamental frequency of a transmission tower structure can be determined by substituting Eqs 21–24 into Eq. 14

In this paper, we present an algorithm that estimates the fundamental frequency of tower structures. The algorithm combines the energy method and the combination synthesis method to provide an accurate approximation. The bending and shear deformations of transmission towers, as well as the effects of the distribution and variable cross-section characteristics of cross-arms near the tower top are also under consideration. The displacement response of the tower under rockfall impact is derived. The parametric analysis was also carried out. The following conclusions are drawn.

(1) The peak impact force has a larger impact on the displacement response of transmission towers when compared to the impact action time, impact height, and number of cross-arms.

The algorithm proposed in this article is not only applicable for the calculation of the fundamental frequency and displacement response of transmission towers, but also suitable for other tower-like structures. However, the article did not take into account the cable effect on transmission towers. The dynamic displacement responses of tower-like structures under cables constraint conditions needs further research.