Seismic responses of transmission tower-line system under multi-component ground motions

Abstract

In this study, shaking table tests, theoretical research, and finite element modeling analysis were conducted on seismic responses of a transmission tower-line system under multi-component ground motions. The rocking component of ground motion was determined from the original seismic record by the wavelet analysis. The shaking table tests for scale models (single tower and tower-line system) that satisfy the similarity ratio of an actual tower-line system, were performed under multiple directions of motion. Dynamic equations under multi-component (horizontal, coupled horizontal, and tilting (CHT), and coupled vertical, horizontal, and tilting (CVHT)) ground motions were first derived in theoretical studies. The finite element model based on the actual transmission tower line system was built, and the response of the system under the identical working condition was calculated. It is found that the rotational component of the ground motion cannot be overlooked and significantly impacts the seismic response of the tower-line system, which is reflected especially as the vertical motion is coupled. The tilting ground motion, the additional second-order effect, and the foundation tilting jointly cause the increase in the response amplitude and significant asymmetric displacement effect. Since this effect of tilting ground motions, the weakening effect of the transmission line to reduce the responses for the displacement and acceleration of the tower-line system is also weakened. It is, therefore, essential to consider the adverse effects of multi-component ground motions on the designing of transmission tower systems in high-intensity areas.

Keywords

transmission tower-line system multi-component ground motions additional second-order effect shaking table test finite element analysis

Introduction

Translational ground motions refer to the primary objects in seismic observations and studies. However, when seismic waves travel along the ground, the velocity, period and phase vary with each point in waves, inducing rocking motions accompanied by horizontal movements. However, the rocking motion has been difficult to obtain, which makes the researchers ignore the effect of the rocking motion, or estimate it not so precisely. The rocking ground motions can be split into two types based on the direction. The rocking motion along the vertical direction is considered as torsion, and the along with X and Y directions are tilting effects, as presented in Figure 1, while the great impact on the buildings caused by tiling motions, especially on high-rise structures, such as transmission towers.

Figure 1.

Six directions of seismic movements.

As a typical high-rise flexible structure, it is generally believed that the transmission tower structure is mainly damaged by wind and ice cover loads, however, there are also collapses and damages to transmission tower lines from the records of some past earthquakes (Hall et al., 1996; NCREE 1999, Zhang and Zhao, 2009; Liu and Liu, 2013), which all proved that earthquake damages are another factor that cannot be ignored. The research on the seismic performance of transmission towers is mainly about the effects of translational ground motions on the structure, however, the rocking ground motions (especially the tilting effects) to the structure are barely researched before.

In our recent research, it is found that the impact of tilting ground motions on the seismic performance of transmission tower structures cannot be ignored, especially when it is coupled with horizontal and vertical ground motions, that is, it is necessary to investigate the seismic performance of multi-component ground motion, which refers to the effect of horizontal, vertical and rocking ground motions, to the transmission tower-lines system.

Research in rocking ground motions

In the 1960s, the rotational components in earthquakes were first recognized (Trifunac, 2009), while in the 1990s, the rocking or tilting motion has been increasingly studied. Graizer (1991), Nigbor (1994), Stedman et al. (1995) used their way to measure or observed the rocking motions. Takeo (1998) effectively recorded rocking components and also (Takeo and Ito, 1997) highlighted that the angular sensor can detect rotational motions in some special cases. In the last decade, with the development of observation instruments, increasing records of rotating components have been recorded. Liu and Pike (2016) proposed a rotation sensor based on micro-electro-mechanical system (MEMS) technology, which can measure the rotational component of ground motion in the region. In 2017, the multiple ring laser gyroscope ROMY was first used to observe ground rotation (Hand, 2017), and On January 23, 2018, the rotational component from the M7.9 earthquakes in the Gulf of Alaska was recorded on ROMY (Gebaur et al., 2020), a newly opened multi-component ring laser observatory in Fürstenfeldbruck, Germany, which is the first time to directly observe the ground motion of all 6° of freedom of a teleseism.

Over the recent decades, except for the recording and research with special purposes, the effect of rocking motions has been significantly identified in several earthquakes (e.g.the Northridge earthquake in the United States) mentioned by Graizer (2006) and Gomberg (1997), the Chi-chi earthquake in Taiwan cited by Huang (2003), as well as the Wenchuan earthquake in China from the article of Peng and Li (2012). The dip angle attributed to the earthquakes in tilting ground motion can range from 0.8° to 3.1°. which indicated extremely serious damage caused by the rocking ground motions.

Since there is little empirical data on observed rocking components in the seismic time-history record, the effect of the rotation component of ground motion cannot be ignored, and is very important to obtain the rocking components in seismic research. In the past decade, good progress has been made in measuring the rotational components of ground motions with specific instruments. However, the device that can measure the rotational components of ground motions including the large ring laser gyroscope does not have a direct aim for seismic engineering, and its main purpose is to detect the movement of the earth. The most significant drawback of these instruments and devices is their size and cost (Sollberger and Igel, 2020), and improvement for the device is now in progress. Nevertheless, it will be some time before this is fully applied to the actual measurement record, and it will be some time before seismological researchers realize that the rotational components of ground motions are necessary for analysis. Thus, the available rotational component measurement records are relatively rare, especially for engineers and technicians, when they choose the translational seismic wave to be analyzed, in most cases, they cannot directly obtain the appropriate corresponding rotational components. Accordingly, how to obtain rotational components from relatively rich translational seismic records is still very important for scientific research and technical analysis, and the major methods of scholars at present to extract rotational components from translation records are as follows:

One method to acquire the rotational component complies with the theory of elastic waves. As suggested by Trifunac (1982), the Fourier spectrum of amplitude in the rotational component was acquired from the horizontal component. Lee and Trifunac (1985) extended an extracting horizontal acceleration method to torsional acceleration records, in which wave passage effects and dispersion were considered as well. Castellani and Boffi (1986) delved into the effects of surface waves on the rocking component and built mathematical models to analyze the tilting spectrum; besides, the results were compared with the response spectrum. Li et al. (2004) applied the elastic wave theory to calculate tilting ground motion by complying with the rotation degree of freedom for P, SV waves, and Rayleigh waves. Based on the theory of elastic plane wave propagation, Wei and Luo (2010), and Zembaty (2009) calculated and analyzed the responses for the spectrum of ground motion rotational from the measured records of earthquakes. Basu et al. (2012) split the three translational ground motions into body waves, which are reassembled to generate rotational time series; subsequently, the rotation components were obtained. Singla and Gupta (2019a) employed the planar wavefront model to synthesize the Fourier spectrum of the rotating ground motion; besides, the spatial variation relationship of the body wave amplitude was exploited by relying on the translational data near the epicenter.

Besides the first-mentioned method, records from multi-station procedures (MPS) or dense arrays are feasible approaches to obtain tilting components as well. By complying with a wide array of records from EI-Centro seismic waves, Niazi (1986) assessed the rotational effect of a long, narrow, and rigid foundation. Castellani and G. Boffi (1986) indicated a correlation between vertical motion and the rocking movement spectrum, suggesting the different ratios of the rocking spectrum and translational spectrum. Laouami and Labbe (2002) acquired the rotational components with data from stations of large-scale seismic testing arrays in Lotuong, Taiwan. They calculated the rotational component for each station by separating differences between translational acceleration along the traveling line by the distance between the two stations.

Based on different responses induced by tilting ground motion between the horizontal and vertical spectrum from the single-pendulum, Graizer (2006, 2009; Graizer and Kallan (2008)) proposed the third method to obtain the rocking ground motions. This method ensures the cut-off frequency by comparing the horizontal and vertical seismic waves of the Fourier spectrum, as well as obtaining the rocking movement time history by flitting the horizontal component.

Except for the study on the obtaining rotational component of ground motion, some scholars also conducted studies in some other fields. Lee et al. (2009) revealed some relationships between the rotational component and the translational component. some other researchers Bokowski et al. (2018); Castellania and Guidotti (2012); Guidoti et al. (2018); Lan and Jia (2018) analyzed the seismic effect of high-rise structures under the rotational component. Singla and Gupta (2019b) delved into the evaluation of the rotational effect in the soil layer. Rodda and Basu (2019) studied the auto-spectral density of the rotational component with consideration of random vibration. Zhang et al. (2020) applied the novel surface fitting method (SFM) for extracting rotational components from the translational earthquake and found the rotational components contribute to the seismic responses with respect to the displacements and accelerations at nodes, as well as the element’s internal force distribution and base reactions.

The effect of rocking ground motions on the structure

In addition to this direct record from Northridge, Chichi, or Wenchuan, the effect of this tilting effect produced by the rotational component on the structure also cannot be ignored, and in the analysis of the impact of the rotational component for ground motion on the structure, the high-rise structure has always been a concern for researchers. Guidotti et al. (2018) reported that the inter-story drifted up to 15% given the rotational component of robust ground motion when studying the damages of the highest local building-Grand Chancellor Hotel in the 2011 Mw 6.2 Christchurch earthquake. Moreover, Lan and Jia (2018) identified that the rotational effect of ground motion on the high pier bridge would enhance the bending moment and shear force at the bottom of the pier, and the increasing height also causes an increase in the bending moment. Moreover, Castellani and Guidotti (2012) has already highlighted that high-rise structures with long-period rotational effects depend on the amplitude of the response spectrum of the vertical component. Ecem and Gökhan (2019) delved into the rotation effect of the bridge and highlighted that the rotation effect will produce tilt and rotation on the bridge deck as well, and the impact on the high-speed bridge should not be neglected. Bokowski et.al. (2018) investigated the effect of coupled horizontal and tilt ground motion on a 160m high reinforced concrete chimney; as demonstrated from the results, the bending moment increased to 18% on the top and 65% at the bottom, and he also reported that the rotation effect is directly proportional to the translational moment.

Study on seismic performance of transmission tower-line system

Great achievements have been made in the research of transmission tower line systems in the past 30 years, and many aspects of the response under translational ground motions have been deeply and carefully analyzed, however, still focuses on the effect of translational ground motions: Li and Wang (1991) started earlier the response of the system consisting of long-span transmission lines and their supporting towers to seismic motions. Tian and his team have made fruitful results on the seismic performance of the transmission tower system in the past 10 years. The progression of collapse for a transmission line system was discussed by Tian et al. (2017a, 2017b). Furthermore, he (Tian and Li 2013) and Li and Xie (2015) revealed the mass effect and nonlinear vibration effect of a transmission line to a tower by the shaking table test, demonstrating that both effects would significantly hinder the dynamic response of the tower. Tian et al. (2012, 2018a) also analyzed the seismic performance of the transmission tower line system under multiple directions of translational ground motions using finite element analysis and test, and the results indicate that the ground motions affect the dynamic response of the transmission tower line system, and the transmission line has a significant impact on the structural response. These analysis methods and results of the transmission tower line system also enlighten the analysis of the sea-crossing cable-stayed bridges (Li et al., 2018) which are also affected by multiple directions of translational ground motions.Tian also studied non-uniform seismic excitations (Tian et al., 2014), and the effect of spatially varying ground motions (Tian et al., 2016, 2017c) on the transmission lines system, and found both non-uniform excitations and spatially varying ground motions cause larger tower dynamic responses.

The existing modeling and analysis of transmission tower structure primarily involve theoretical calculation, shaking table test, as well as finite element modeling analysis. The theoretical calculation mainly uses the multiple particle model in stiffness and mass for theoretical analysis (e.g., the theoretical equivalence proposed by Qu et al. (2003), and Zhang et al. (2011)). At this stage, scholars generally used finite element modeling analysis alone, or combined shaking table test and finite element analysis to research the response of transmission tower structure in numerous aspects. For instance, Tian et al. (2018a, 2018b) employed both finite element software and shaking table test for dynamic response analysis of transmission tower-line systems considering the SSI effect and proposed seismic amplification factor to examine the effect of SSI effect. Yuan et al. (2020) also investigated the effect of different seismic inputs on the transmission tower-line system using finite element analysis. There are some uncertainties in the above analysis methods (e.g., the simulation of transmission tower components, the simulation mode of transmission lines, and the failure principle of components), which should be further analyzed to ensure reasonable results.

Although the above studies have made great achievements, the lack of analysis of the influence of rocking ground motions on the transmission towers makes the research insufficient. From the response analysis of these high-rise structures under rotational ground motions in the last section, it can be inferred that as a typical high-rise structure the transmission tower is also significantly affected by the rocking ground motions, and it is time to further strengthen the analysis for the response of rocking ground motions to the transmission tower and tower-line system.

Based on the analysis of the existing literature, it can be found that there is still some insufficient in obtaining and verifying the rotational component of the ground motion. Moreover, in recent seismic analysis for the transmission tower line system, either the rocking ground motion is not considered, or the tilting angle or inclination of the foundation caused by rocking ground motions are not regarded in the research of seismic responses from a transmission tower. In addition, the shaking table test for the transmission towers under rocking ground motion, which can intuitively show the rocking effect is still missing. Therefore, an introduction to the improved method of obtaining the rotational component is included in the article first, after acquiring the rocking component from a seismic record, the dynamic response of the transmission tower-line system under the multi-component ground motion (including the rocking component) was further analyzed and studies. The research on the response of transmission tower structures under multi-component ground motions can also be used as a reference for the seismic response of other similar high-rise flexible structures.

Method of obtaining the rocking ground motion in the article

The method of obtaining the rocking ground motions is based on Multi-station and dense array records, which is not extensively adopted since it is only applicable to areas with relevant instruments set. Thus, normally two main methods can be adopted to gain the rocking component by extracting the general ground motion, i.e., the method based on the theory of elastic wave, as well as the method in comparison with the Fourier spectrum.

Since the seismic wave is a kind of elastic wave in the rock formation, obtaining the rotational component of ground motion by using elastic wave theory was feasible. However, this method assumes that an elastic wave spreads in an ideal isotropic homogeneous horizontal layered medium, and no plastic deformation in the earthquake occurs. In reality, nonlinear deformation appears in the near-field of the seismic epicenter and can increase the ground tilt in this region. Therefore, the rocking component recovered using elastic wave theory may underestimate the rocking ground motions (Basu et al., 2012), which makes the method cannot estimate the rocking component of the ground motion precisely. On the other hand, the method used in comparison with the Fourier spectrum, proposed by Graizer (2006), is convenient to calculate the rocking ground motion, but determining the characteristic frequency is difficult, and then obtaining the rocking component correctly, which makes the method also need to be improved.

Like the real and artificial earthquake sequences can be used for the fragility curves of the structure from Yaghmaei-Sabegh and Mahdipour-Moghanni (2019), the wavelet analysis is an also important method for seismic research of artificial earthquake sequences, for the method, is capable of resolving the different frequencies and exhibiting the capability in comparison to the frequency in the overall range, can be adopted to remedy the drawbacks of the Fourier spectrum method. Yaghmaei-Sabegh, Jorge Ruiz-García (2016), for instance, have used wavelet transform analysis to analyze the energy distribution and frequency characteristics of the Varzaghan–Ahar earthquake in 2012. The present method is based on the principle proposed by Graizer (2006) and improved with the Wavelet analysis method to obtain tilting components from the horizontal component. The procedure using wavelet analysis to obtain the rocking ground motion can be summarized as follows:

i. Select a suitable wavelet basis and several decomposition layers to decompose the horizontal and vertical components of the ground motion;

ii. Wavelet coefficients of each layer of the horizontal and vertical components were compared, and similar and different parts are obtained;

iii. Select a reasonable threshold of the wavelet coefficients for the horizontal component, and wavelet coefficients are chosen or filtered based on this, to reconstruct the rotation displacement time history, and the filtered out is the horizontal component of the ground motion.

iv. Divide the obtained rocking component by g to obtain the rocking angular displacement time history of the earthquake, and then the acceleration time history can also be obtained.

As mentioned above, due to the setting requirements, ensuring that suitable dense array records can be found for each typical translational ground motion to directly obtain the rocking component is difficult, but the correctness of the rotation component obtained by the wavelet analysis method can be verified by the limited records of the existing dense array. Select the records from dense array stations SMART, which can directly calculate the rocking motions by a surface distribution method. Based on the definition for the angular displacement of rocking ground motion, the way to calculate it was established as equation (1):

θ_{i} (t) = \frac{u_{i B} (t) - u_{i A} (t)}{Δ_{A B}}

(1)

u_iA and u_iB are the vertical displacement component recorded by two adjacent stations and is the distance between two stations. The angular displacement of rocking ground motion calculated based on equation (1) is set as angular displacement 1, while the displacement obtained by wavelet analysis is set as angular displacement 2, and the two displacements are compared in Figure 2.

Figure 2.

Comparison between displacement.

Comparing with two curves, it can be concluded that the coincidence degree is good at the beginning and middle of the period when the earthquake amplitude is large, while in the later period of the earthquake, the amplitude difference is not big and the changing trend is close. the rocking component obtained by wavelet analysis is in good agreement with the component calculated from recorded waves from multiple stations, which proves the correctness of the wavelet analysis method to obtain rocking ground motions.

However, the amplitude of the rocking ground motion recorded this time is small, and the effect on the actual structure is not ideal. For this reason, we selected seismic waves with obvious rocking effects in the actual seismic records for calculation. The most typical one refers to seismic data recorded at the Sylmar-Pacoima Dam’s upper left abutment during the Northridge earthquake provided. They measure a ground surface residual tilt of 3.1° due to the rocking ground motions, and Figure 3 presents the horizontal and the vertical acceleration time history. The data had 1001 points, the interval is 0.02 s, and the duration is 20s. The displacement time history of rocking ground motion using wavelet analysis from the translational ground motions is indicated in Figure 4(a) and the rotation displacement reaches nearly 3.1°(0.054 rad), complying with actual observation values for the effect of tilting motions. The rotation acceleration time history is also obtained from displacement and indicated in Figure 4(b), which can be applied in theoretical analysis and tests.

Figure 3.

Time history of translational acceleration from the Northridge earthquake. (a) Time history of horizontal acceleration. (b) Time history of vertical acceleration.

Figure 4.

Time history of tilt displacement/acceleration of ground motion. Shaking table Test of Transmission System under Multi-component Ground motion. (a) Time history of tilt displacement. (b) Time history of tilt displacement.

This article focuses on the dynamic response of the transmission tower-line system under the rocking ground motions. The key factor of the study is the effect of the rotational component of ground motion. The interaction of soil and structure(SSI) and multiple support excitation was ignored to ensure the result was only affected because of rotation movement. Thus, the test and analysis only consider the response of the tower-line system on a rigid foundation and with the seismic wave in a single-point input way, which makes the difference in dynamic response only caused by rocking seismic action.

To determine the effect of multi-component ground motions (e.g. Rocking and horizontal motion) on the transmission tower line system, shaking table tests of the scale model for a single transmission tower and three-tower two-line system, from the prototype of an actual structure, under multiple directions of ground motion were performed.

The practical parameters of the tower were presented as follows: the transmission system consisted of four transmission lines arranged at an identical height; with the tower height of 81.80 m, the space between tower feet at the bottom reached 17.18 m, and the distance of the system between towers was 500 m. The tests were performed in the structural dynamics laboratory at Chongqing Traffic Research and Design Institute. The institute was equipped with an earthquake simulation shaking table with 6 freedoms on 3 different axes. The most significant acceleration of the shaking table is 1.0 g and the size of the table is 3m $\times$ 6 m.

The transmission line system model was built at a scale of 1:30, and the similarity parameters of the test model are listed in Table 1. For the size of the shaking table is 6m, and the horizontal spacing of the scale model was set to 2 m in the actual installation.

Table 1.

Similarity parameters of the test model.

Quantity	Relation	Similarity coefficient
Elastic modulus (E)	S _E	1.0
Length(l)	S _l	1/30
Mass(m)	$S_{m} = S_{l}^{2}$	1/900
Stiffness (k)	$S_{K} = S_{E} S_{l}$	1/30
Time (t)	$S_{t} = S^{1 / 2}$	$1 / \sqrt{30}$
Acceleration $\ddot{x}$	$S_{x} = S_{l} / S_{t}^{2}$	1.0
Rotation displacement $α$	$S_{α} = 1.0$	1.0
Rotation acceleration $\ddot{α}$	$S_{\ddot{α}} = S_{α} / {S_{t}}^{2}$	30

To offset the difference between the length of the transmission line for the actual model and the size of the prototype caused by the size of the shaking table. The method of adding additional mass to the steel-stranded wire to simulate the transmission lines in actual length was adopted. Given the consistent relationship between the basic period and the theoretical scale model, the mass of the additional chain on a transmission line reached 2.2 kg, while the simulated transmission line is steel-stranded wire with a nominal diameter of 3 mm. The original steel-stranded wire and the one used in the test with additional chains are indicated in Figure 5. The first three frequencies for vibration modes based on the scaled theoretical model (span 16.67 m) were compared with the practical model (span 2 m) as indicated in Table 2.

Figure 5.

The original steel stand and the one with the additional chain used in the test. (a) Original strand cable. (b) Steel strand with additional chain in test.

Table 2.

Companion between theoretical and piratical model.

Item	Theoretical model	Practical model
Spacing (m)	16.67	2
Frequency of 1^st order	0.71	0.75
Frequency of 2^nd order	1.37	1.41
Frequency of 3^rd order	1.42	1.51

As compared with the frequency of the practical test model, it was found that the first three order frequencies of the two models were well consistent with each other. Using the steel-stranded wire with chain as additional mass can eliminate the error due to the insufficient length of the actual model in the shaking table test. Since the transmission tower model is made strictly according to the similarity ratio, and the transmission line model (the original steel stranded wire with additional mass) also satisfies the similar ratio to the actual tower line model, Therefore the test model of the whole tower-line system is satisfying the similarity ratio of the original structure, and the results of test model can represent the response from the actual tower line system.

The main tower in the test exhibited a height of 2.73 m, the spaces between towers’ feet were 0.3 m, and the horizontal bars were 1.2 m above the tower. The equilateral roof steel applied as the main and web members in the test model was L30 × 2.5, and the diagonal braces were round bars exhibiting a diameter of 3 mm welded with each member bar together. The material used in the model is Q345 steel, with the yield strength of the steel being 345 N/mm². Counterweight boxes weighing 10.5 kg were set at the top and middle layers of the tower, which could be exploited to regulate the fundamental frequency of the tower by altering the weight of each box. Figure 6 illustrates the design drawings of the front elevation and side facade for the transmission system model.

Figure 6.

Evaluation view of the model. (a) Front evaluation view. (b) Side evaluation view.

The tower feet were firmly fixed to the shaking table by four bolts; the main structures were connected by welding. The final design of the transmission tower model is illustrated in Figure 7(a). The transmission lines and insulators were simulated by steel-stranded wire with chains, which were connected to towers. The three-tower two-line models are illustrated in Figure 7(b).

Figure 7.

Test models. (a) Single-tower model. (b) Three-tower two-line model.

The arrangement of the vibration pickup is presented in Figure 8. The accelerometer set on the top of the model for the transmission system was KISTLER 8310A10 with a measurement range of 5g, while the largest frequency response was 2100 Hz. The accelerometer set on the top of the test table was KISTLER 8310A2 exhibiting a measurement range of 2g, while the maximum frequency response was 1400 Hz. The laser displacement meter was set on the top of the tower; its type was ILD1401-200(000) with a range of 200 mm.

Figure 8.

Arrangement of the vibration pickup. (a) Arrangement of accelerometer. (b) Arrangement of displacement meter.

The acceleration time history of seismic waves employed in the test was from the 1994 Northridge earthquake as we mentioned in part 3 which is indicated in Figures 3 and 4, and the duration of the earthquake was compressed with the similarity parameters of the test model (Table 1), and the time of acceleration time history reached 3.65s.

As required by the specification of the code from the seismic design of the building (GB 50011-2010), the summit acceleration of horizontal ground motion adopted for rare earthquakes of 9° was 6.2 m/s². The same adjustment coefficient of horizontal ground motion is then used in multiplying the rocking ground motions, and the summit value of rocking ground motion is 0.68 rad/s². Since the effect of the transmission line, the calculation and analysis of the two directions of the transmission tower system were different. The transverse and longitudinal directions of the transmission tower-line system and their relationships with the single tower and transmission lines are given in Figure 9, in which the longitudinal direction or X-axis represents the direction of parallel transmission lines, and the transverse direction or Y-axis denotes the direction perpendicular to the transmission lines.

Figure 9.

Diagrammatic sketch of different directions of the transmission tower system.

In the test, a vibration pickup was arranged on the shaking table to collect the actual acceleration time history of the shaking table. Three different conditions of input in the test were implemented, i.e., horizontal; coupled horizontal and tilting (CHT); coupled vertical, horizontal, and tilting (CVHT) ground motion, as listed in Table 3. In the table, the tower-line system 1 represents three loading conditions, i.e., under horizontal direction UX2, in both horizontal direction UX2 and rotational direction RY2, as well as in horizontal direction UX2, vertical direction UZ2, and rotational direction RY2; besides, the tower-line system 2 represents the loading condition in horizontal direction UY2, in both horizontal direction UY2 and rotational direction RX2, as well as in horizontal direction UY2, vertical direction UZ2 and rotational direction RX2.

Table 3.

Test conditions.

Condition	Tests model	Tests model	Wave components			Description
Condition	Tests model	Tests model	Horizontal (m/s²)	Vertical (m/s²)	Tilting (rad/s²)	Description
1	Single tower	W1	0.5	—	—	White noise
2		UX1	6.2	—	—	Horizontal
3		UX1+RY1	6.2	—	0.68	CHT
4		UX1+UZ1+RY1	6.2	5.77	0.68	CVHT
5	Tower line system	W2	0.5	—	—	White noise
6		UX2	6.2	—	—	Horizontal
7		UX2+RY2	6.2	—	0.68	CHT in X direction
8		UX2+UZ2+RY2	6.2	5.77	0.68	CVHT in X direction
9		UY2	6.2	—	—	Horizontal
10		UY2+RX2	6.2	—	0.68	CHT in Y direction
11		UY2+UZ2+RY2	6.2	5.77	0.68	CVHT in Y direction

By numerical calculation and white noise sweep, the frequency of the original actual transmission tower and the theoretical calculation value of the scale model could be determined. the basic natural vibration period of the original tower-line system was similar to the calculated results of the tower-line structure, which can prove the integral tower-line structure model satisfied the requirement of dynamic characteristics for shaking table tests.

Figure 10 plots the displacement time history curves on top of the model under horizontal, CHT, and CVHT ground motions for the single tower, while the displacement time history curve in longitudinal and transverse directions on the top layer of the tower-line system are presented in Figure 11.

Figure 10.

Displacement time history curves on top for the single tower.

Figure 11.

Displacement time history curves on top of the tower-line system in two directions. (a) In the longitudinal direction. (b) In the transverse direction.

The acceleration time history curves on the top of the model under horizontal, CHT, and CVHT ground motions for the single tower are plotted in Figure 12, while those in longitudinal and transverse directions on the top layer of the tower-line system are plotted in Figure 13.

Figure 12.

Acceleration time history curves on top for the single tower.

Figure 13.

Acceleration time history curves on top of the tower-line system in two directions. (a) In the longitudinal direction. (b) In the transverse direction.

The responses of amplitude for displacement and acceleration on the top of the tower under horizontal, CHT, and CVHT ground motions in the longitudinal (Tower lines system1) and transverse direction (Tower lines system2) for a single tower and the tower-line system are listed in Tables 4 and 5. Amplification factor 1 represents the amplified range of increase between horizontal and coupled horizontal, and tilting (CHT) ground motion, and Amplification factor 2 refers to the amplified range between horizontal and CVHT ground motion.

Table 4.

Horizontal displacement amplitude of top of the transmission tower.

Model	Single tower	Tower lines system1	Tower lines system2
Horizontal	5.27	4.85	4.92
CHT	6.41	6.01	6.27
CVHT	6.98	6.65	6.88
Amplification1	21.6%	23.9%	27.4%
Amplification2	32.2%	37.1%	39.8%

Table 5.

Horizontal acceleration amplitude of top of the transmission tower.

Model	Single tower	Tower lines system1	Tower lines System2
Horizontal	11.71	10.65	10.56
CHT	13.99	12.84	13.57
CVHT	15.19	14.31	15.01
Amplification1	19.5%	20.5%	28.5%
Amplification2	29.7%	34.3%	42.3%

Tables 6 and 7 list the reduction in response amplitude between the single tower and tower-line system under horizontal and CHT, horizontal and CVHT ground motions.

Table 6.

Reduction in displacement amplitude.

Model	Direction	Single tower	Tower lines system	Decreasing amplitude (%)
Horizontal	Longitudinal	5.27	4.85	7.9
Horizontal	Transverse	—	4.92	8.2
CHT	Longitudinal	6.41	6.01	6.2
CHT	Transverse	—	6.27	6.1
CVHT	Longitudinal	6.98	6.65	4.7
CVHT	Transverse	—	6.88	5.8

Table 7.

Reduction in acceleration amplitude.

Model	Direction	Single tower	Tower lines system	Decreasing amplitude (%)
Horizontal	Longitudinal	11.71	10.65	9.1
Horizontal	Transverse	—	10.21	8.8
CHT	Longitudinal	13.99	12.84	8.9
CHT	Transverse	—	13.57	6.9
CVHT	Longitudinal	15.19	14.31	5.8
CVHT	Transverse	—	15.01	6.6

Given the data presented in Figures 10–13 and Tables 4–7, the conclusions can be drawn as follows:

I. Compared with the horizontal situation, the displacement and acceleration of CHT movement on the top of the respective model increased to a certain degree. The displacement amplitude of the single tower and tower transmission system increased by more than 20% (except for a special case in Table 5). It is therefore indicated that the horizontal displacement and acceleration response increased significantly on top of a transmission tower as a typical high-rise structure. which means the effects of rocking motion should not be overlooked. Moreover, compared with CHT ground motion, the growth of displacement and acceleration under CVHT ground motion was significant as well: 30% or more. It is suggested that the vertical seismic motion and the rocking ground motions significantly impacted the response of the structure.

ii. Under the CHT and CVHT conditions, each model on top revealed the curves of displacement time history shift from baseline 0 as indicated in Figures 10 and 11, which induced the asymmetric effect in displacement. Certain differences between horizontal displacement in positive and negative directions appeared. However, horizontal acceleration time history curves fluctuate along baseline 0, exhibiting no asymmetry of the curves.

iii. Under identical ground motions, the horizontal displacement and acceleration responses of the tower-line system were lower than those of the single-tower model. The reason for this decreasing trend is a comprehensive influence effect, including the influence of tension from the transmission line, as well as the damping and mass effects, among which the main influence is the mass effect of the line and the nonlinear vibration effect of the structure.

The decrease in displacement and acceleration of the whole system was largely attributed to the mass effect of the transmission line and the nonlinear vibration effect on the structure. The decreasing effect in the transverse direction was more obvious than that in the longitudinal direction.

iv. Compared with the horizontal condition, the weakening effect produced by the line in acceleration and displacement between the tower transmission line and the single tower, decreased under CHT and CVHT ground motions, suggesting that the weakening effect of transmission lines on the main structure of the system was affected by the rocking ground motions, demonstrating that the rocking movement should be examined in the design of a transmission-line system, particularly when coupled with the vertical direction.

Theoretical analyses for tower-line system under multi-component ground motions

From the data obtained from the test, it can be concluded that the dynamic response of the test model under the action of CHT and CVHT ground motions has been increased to a certain extent, and a shift of displacement is also generated. With the research of the dynamic response of the analysis model for transmission tower structure, the dynamic equation is used to investigate the causes of the above phenomena.

According to the concentrate quality method, the mass of each layer model structure is concentrated to a mass point on the elevation, and the rigidity of each member in the model from the test is equivalent to the rigidity in the multiple particles model, which is assigned to the mass-less rigid rod and connecting each mass point to ensure that the test model is equivalent to the multiple particle model in stiffness and mass, as indicated in Figure 14.

Figure 14.

Equivalent from single tower model for the multiple-particle model.

The model under horizontal and rocking ground motions based on the multiple particle model is presented in Figure 15, and description of the symbols is listed in Table 8.

Figure 15.

Multiple particle model of transmission tower under CHT motion.

Table 8.

Description of the symbols.

Symbols	Descriptions
M _i	The mass of the structural particle
$β_{i}$	The secant angle of particle i after the structural deformation, and $β_{i} = X_{i} / H_{i}$
${\ddot{X}}_{g}$	The horizontal acceleration of ground motion
${\ddot{Z}}_{g}$	The horizontal acceleration
(M)	Mass matrix of the structure
(K_G)	Geometric stiffness matrix of the structure
H _i	The relative height of particle i to the ground,
x _i	The relative displacement of structural particle i to the ground
$α_{g} / {\ddot{α}}_{g}$	The rocking displacement and acceleration of ground motion
[I]	The identity matrix
[K₀]	Stiffness matrix of the structure
(C)	Damping matrix of the structure

Besides the effect of horizontal and rocking ground motion on the structure, the effect of the inclined deformation on the foundation should be considered as well. The additional rotation increases the arm of bending moment in the $P - Δ$ effect, which amplifies the second-order effect of the structure, and this effect increases with the rise of the structure. At this time, the $P - Δ$ effect for high-rise structures as transmission towers should consist of two parts: the conventional $P - Δ$ effect which is expressed as $[M] {β} g$ in Figure 15 is one part; while the foundation inclination caused by the rotation component which leads to the additional $P - Δ$ effect is the other part, and should be used as another seismic excitation for the dynamic response of the structure. The dynamic equation of the transmission tower under coupled horizontal and tilting motion (CHT) can be obtained in equation (2),

\begin{array}{l} [M] {\ddot{X}} + [C] {\dot{X}} + [K_{0}] {X} = - [M] {\ddot{X}}_{g} \\ + [M] {β} g - [M] [H] {\ddot{α}}_{g} + {[M]}_{h} {I} g α_{g} \end{array}

(2)

the second part on the right of the equation

[M] {β} g

denotes conventional

P - Δ

effect; the third part

- [M] [H] \overset{. .}{α_{g}}

represents the effect of rocking motion; the last part

[M] [I] g α_{g}

refers to the additional

P - Δ

effect caused by the rotation of the foundation.

As presented in the literature review, existing studies mentioned that the vertical seismic effect increased to some extent from the rotation effect. To conduct more in-depth studies, the motion in the vertical direction should be considered as well, and the $P - Δ$ effect is also modified for the vertical effect. Based on the analysis under CHT ground motions, the $P - Δ$ effect under CVHT ground motions can also be split into two parts: the former part is replaced $[M] {β} g$ in equation (2) to $[M] {β} (g - {\overset{. .}{Z}}_{g})$ , and the latter can also be modified $[M] {I} g α_{g}$ to $[M] {I} (g - {\overset{. .}{Z}}_{g}) α_{g}$ The model under CVHT motion is illustrated in Figure 16. The dynamic equation of the transmission tower under CVHT ground motions with the inclined foundation deformation can be obtained in equation (3).

\begin{array}{l} [M] {\ddot{X}} + [C] {\dot{X}} + [K_{0}] {X} = - [M] {\ddot{X}}_{g} - [M] [H] {\ddot{α}}_{g} \\ + [M] {I} (g - {\ddot{Z}}_{g}) α_{g} + [M] {β} (g - {\ddot{Z}}_{g}) \end{array}

(3)

Figure 16.

Multiple particle model of transmission tower under CVHT motion.

The tower-line system was explored based on the conclusion from the analysis of a single tower. Each transmission line fell into five-part, corresponding to four turning points with lumped mass. For the tower-line system with all transmission lines located at the identical height as presented in the test, lines could be equivalent to one. Simplification of the transmission tower-line system is shown in Figure 17. Accordingly, based on single tower calculation, given the equivalence and simplification of the transmission line, the dynamic equation of the tower-line system under CHT ground motion is written in equation (4)

\begin{array}{l} {[M]}_{s} {\ddot{X}} + [C] {\dot{X}} + ({[K_{0}]}_{s} - {[K_{G}]}_{s}) {X} = - {[M]}_{s} {{\ddot{X}}_{g}} \\ - {[M]}_{s} {[H]}_{s} {{\ddot{α}}_{g}}_{h} + {[M]}_{s} g {α_{g}}_{s} \end{array}

(4)

Figure 17.

Simplification of transmission tower-line system.

The dynamic equation of the tower-line system under CVHT ground motion is expressed as equation (5)

\begin{array}{l} {[M]}_{s} {\ddot{X}} + [C] {\dot{X}} + {[K_{0}]}_{s} {X} = - {[M]}_{s} {\ddot{X}}_{g} - {[M]}_{s} {[H]}_{s} {\ddot{α}}_{g} \\ + {[M]}_{s} [I] (g - {\ddot{Z}}_{g}) α_{g} + {[M]}_{s} {β}_{s} (g - {\ddot{Z}}_{g}) \end{array}

(5)

The main matrix and vector in the mentioned equations are expressed in the longitudinal direction (used both for CHT and CVHT). For other specific parameters of the listed equations, please reference the research by Li and Wang (1997):

\begin{array}{l} {[M]}_{s} = {[\begin{array}{c} {[M]}_{l i n e} & {[M]}_{c o u p l e d} \\ {[M]}_{c o u p l e d} & {[M]}_{t o w e r} \end{array}]}_{(3 + N) \times (3 + N)} {[K_{0}]}_{s} = {[\begin{array}{c} {[K]}_{l i n e} & [O] \\ {[O]}^{T} & {[M]}_{t o w e r} \end{array}]}_{(3 + N) \times (3 + N)} {[H]}_{(s)} = d i a g {[0,0,0, h_{1}, h_{2 \dots} . . . h_{N}]}_{(3 + N) \times (3 + N)} \\ {{\ddot{X}}_{g}}_{s} = {[0,0,0, \underset{N}{\underset{⏟}{{\ddot{X}}_{g} . . . {\ddot{X}}_{g}}}]}_{(3 + N) \times 1}^{T} {α_{g}}_{s} = {[0,0,0, \underset{N}{\underset{⏟}{α_{g} . . . α_{g}}}]}_{(3 + N) \times 1}^{T} \end{array}

From the dynamic equations, it can be explained that the increase in dynamic response in the test, is partly due to the impact of rocking dynamic response. When the seismic intensity is large, the influence of this effect on the structure cannot be ignored. For high-rise structures with large structural heights, such as transmission towers in the test, the residual displacement and additional $P - Δ$ effects caused by rocking ground motions should also be considered, when under the CHT and CVHT ground motions. Such phenomenon can be explained: the tiling movement from rocking ground motion leads to the foundation residual inclination, which results in additional displacement, and when added to the ordinary displacement, the asymmetric additional deviation in displacement the structure will then appear. Except for the increase in acceleration and displacement caused by rocking ground motions, the inclination of the structure and offset in displacement will not only increase the displacement amplitude of the structure under the action of rocking ground motion, but also aggravate the deformation of the motion, and even cause its collapse and damage.

The analysis of the finite element model of the tower-line system under multi-component ground motions

With the simplified model in the theoretical calculation, the response of the transmission tower line system under multi-component ground motions can be studied. However, the simplified model may not be so accurate to study the effect of rocking motions on the transmission tower-line system precisely, while occasional errors and other errors in the scale model in the test may also lead to inaccurate results. To delve into the effect of the transmission line of the tower-line system under multi-component motions more precisely, and to compare with the result of the test, finite element models were built using the software SAP2000.

The transmission tower refers to a typical spatial structure, which can be simulated by a truss model, truss-beam model, and beam model. Given the details of the transmission tower structure, the moment could be transferred between the mainframe member and the web member, which could be simulated with the beam model; while the diagonal brace and chord were connected by a single bolt, and similar to a hinge joint, which could be more appropriately simulated with the spatial bar system. In the simulation of the truss-beam model, the rotation was generally considered at the joint plate, and other nodes were regarded as hinge joints; other members of the transmission tower could be simulated by the space bar and beam elements. Thus, the truss-beam model complies with the practical situation and applies to the simulation of the whole transmission tower structure. After the establishment of the finite element model, the modal analysis shows that the first two modes of the single tower are bending, the third mode is the overall torsion as indicated in Figure 18, and the higher mode is mainly about local vibration of the lower part of the tower.

Figure 18.

Deformation diagram of the first three modes for the single tower. (a) Bending in first and second modes. (b) Torsion in third modes.

As a type of high-rise and flexible structure, the transmission line only bears tensile force, instead of pressure and bending moment. Based on the nonlinearity of the transmission line, it could be considered a single cable structure in analysis and simulated by the catenary curve cable element. The mentioned finding can be exploited to confirm the assumption that the self-weight of the cable exhibits an even distribution along the length of the curve and in accordance with the hypothetical of incrementally distributed along the horizontal axis and uniformly distributed along the horizontal axis. The geometric stiffness matrix of the cable element generally varied with the random swing of the cable, and the analysis diagram for the cable is indicated in Figure 19.

Figure 19.

Stress diagram of cable element.

The length of the horizontal cable is L, assuming that the cable element is subject to the initial tension F at time t, while at the time $t + Δ t$ , the node forces of i and j at both ends of the cable element under the seismic motion are expressed as, $F_{i} (t + Δ t) F_{j} (t + Δ t)$ and $T (t + Δ t)$ respectively, while the lateral displacement of both ends of the cable is v_i and v_j, i.e., an equilibrium state, and $F_{i} (t + Δ t)$ $F_{j} (t + Δ t)$ is expressed by the following matrix equation to satisfy the balance equation of force:

[\begin{array}{c} F_{i} (t + Δ t) \\ F_{j} (t + Δ t) \end{array}] = \frac{T (t + Δ t)}{L} [\begin{array}{c} 1 & - 1 \\ - 1 & 1 \end{array}] [\begin{array}{c} v_{i} \\ v_{j} \end{array}] = K^{G} [\begin{array}{c} v_{i} \\ v_{j} \end{array}]

(6)

The finite element model of three towers and two lines system was then built according to the actual transmission tower size and span as indicated in Figure 20.

Figure 20.

Finite element model of transmission tower-line system.

After completing the modeling of the transmission tower and transmission line, the multi-component ground motion input should also be considered. As concluded from the theoretical analysis in the last part, the effect of rocking ground motion can be split into three parts: the rocking motion itself, the conventional and additional $P - Δ$ effect, and in the finite element calculation, the conventional $P - Δ$ effect $[M] {β} g$ can be equated with $[M] {β} g = [K_{G}] {X}$ , while $[K_{G}]$

Expressed as $[K_{G}] = d i a g [M_{1} g / H_{1}, M_{2} g / H_{2}, . . . M_{N} g / H_{N}]$ , and dynamic equation is rewritten as equation (7)

\begin{array}{l} [M] {\ddot{X}} + [C] {\dot{X}} + ([K_{0}] - [K_{G}]) {X} = - [M] [I] {\ddot{X}}_{g} \\ - [M] [H] {\ddot{α}}_{g} + {[M]}_{h} [I] g α_{g} \end{array}

(7)

By modifying the geometric stiffness to consider the $P - Δ$ effect and turning on the option of geometric nonlinearity, the calculation of finite element analysis is then based on the dynamic equation from theory analysis for CHT ground motions, and in the same way for CVHT ground motions. Vibration and deformation for the transmission lines appeared in the low-order model of the transmission tower-line system, as indicated in Figure 21. The overall bending and torsion of the tower in the tower-line system appear in the higher order, and typical deformation of the tower were in Figure 22.

Figure 21.

Vibration and deformation of transmission lines in the tower-line system.

Figure 22.

Deformation of a transmission tower in the tower-line system. (a) Bending deformation of transmission tower. (b) Torsion deformation of transmission tower.

The nonlinear time history analysis of single tower and tower-line systems with finite element models under the CHT and CVHT ground motions were calculated respectively, and the results of the calculation were compared with the test results to deepen the study on the effect of multi-component ground motions with titling component to the transmission tower-line system. A comparison of displacement and acceleration on top for a single tower between finite element and test under CHT ground motion is presented in Figure 23, while the comparison for the tower-line system between finite element and test under CHT ground motion is indicated in Figure 24 (with the longitudinal direction as an instance). The results of the finite element analysis were adjusted according to the ratio of the scale model and actual structure size to compare the result.

Figure 23.

Comparison of displacement and acceleration on top for a single tower. (a) Comparison of displacement. (b) Comparison of acceleration.

Figure 24.

Comparison of displacement and acceleration on top of the tower-line system. (a) Comparison of displacement. (b) Comparison of acceleration.

A comparison of displacement and acceleration on top for a single tower between finite element and test under CVHT ground motion is indicated in Figure 25, while the comparison for the tower-line system between finite element and test under CVHT ground motion is drawn in Figure 26 (with the longitudinal direction as the instance).

Figure 25.

Comparison of displacement and acceleration on top for a single tower. (a) Comparison of displacement. (b) Comparison of acceleration.

Figure 26.

Comparison of displacement and acceleration on top of tower-line system. (a) Comparison of displacement. (b) Comparison of acceleration.

Under the CHT ground motions of the finite element analysis, whether to input the additional $P - Δ$ effect is compared with the test results for the single tower, which are listed in Table 8, and the amplitude of the finite element considering the additional effect is named as finite element result 1, and with not considered is named as finite element result 2. Compared with the analysis of finite elements and test results, the following conclusions can be drawn:

i. On the whole, the results of finite element calculation are similar to test results, whether under the CHT or CVHT ground motions. It is reasonable to use the attached chain as additional mass to simulate the original wire in the scale model. The result from finite element analysis, based on a dynamic equation considering additional $P - Δ$ effects caused by rocking ground motions, which complied with that of the test result proves the correctness of the theory.

ii. The time history curves of finite element models and tests on the top layer both indicated movement from baseline, which demonstrates the rocking component take a great influence on the transmission tower. With the impact of the tilting and additional $P - Δ$ effect, an obvious influence on the dynamic response appears on the transmission tower line system, which proves the rocking component and the concomitant effect should not be ignored in the actual calculation and analysis.

iii. Compared with a single tower, the trend of results finite element analysis in the tower-line system is smaller than that of a single tower, which indicated a certain damping effect appeared for the transmission lines. The differences between the test and the finite element calculation are slightly large after time 2.0s, associated with the convergence criterion used in the finite element calculation. The result of the influence from the transmission tower line system under the rotational component in the actual situation should be between the test and the finite element analysis.

Conclusion

In the present study, shaking table tests, theoretical analysis, and finite element modeling calculations are conducted to study the response of the transmission tower line system under multi-component ground motions. the response of a single tower and tower-line system under horizontal, CHT, and CVHT ground motion is investigated in the shaking table test, and the dynamic equations of a single tower and tower-line system under CHT and CVHT ground motion are derived with the simplified model. The differences in responses between finite element analysis and test results under CHT and CVHT ground motion are compared. Based on the above research results, the following conclusions are drawn:

i. It is feasible to use the wavelet analysis method to extract the rocking component from the horizontal seismic ground motions. The accuracy of using the method to obtain the rotation component from the translational ground motion has also been verified. The ground motion rotation component obtained by this analysis method can be used to analyze the dynamic response of actual structures under rocking ground motions.

ii. Significant impact could be observed on the tower-line system because of rocking ground motions, which cannot be neglected. Compared with horizontal movement, there were noticeable increases in displacement and acceleration responses on the top of the tower model under CHT and CVHT ground motions: the displacement amplitude of the single tower and tower transmission system increased by more than 20% under CHT ground, and increased about 30% or more under CVHT ground motion. Moreover, the increases under CVHT are greater than that in CHT ground motions, demonstrating that the effects of tilting and vertical ground motion should be considered together for the transmission line system.

iii. The amplitude of displacement and acceleration for the tower-line system was smaller than that of the single tower under horizontal, CHT, and CVHT movements. an observable weakening effect appears in horizontal displacement and acceleration, which is attributed to the action of the transmission line to the tower structure. The weakening effect, however, from transmission lines to towers will decrease when considering tilting movement compared with only horizontal movement. Therefore, in actual analysis and research, the influence of transmission lines on the seismic response of transmission towers should not be overestimated, preventing the possible rotation component to weaken this effect, and resulting in the lack of structural bearing capacity.

vi. Besides horizontal ground motions, and conventional $P - Δ$ effects, the rocking ground motion and the additional $P - Δ$ effect caused by the rocking ground motion jointly together, and impact the movement of displacement and amplitude enhancement under multi-component ground motions, which was found in the test result and confirmed in the finite element analysis, and the last two items which always ignored in the research should also be taken seriously on for the high rise structures like transmission tower.

The effect of rocking ground motion will increase the dynamic response of the transmission tower, and coupled with the vertical motions the dynamic response will be more obvious. Moreover, the weakening effect of the transmission lines to the tower-line system on the seismic response to the structure will be reduced by the rocking ground motions. The high-rise flexible transmission tower is not only directly affected by translational ground motions and traditional $P - Δ$ effects but also affected by rotational ground motion and the additional second-order effect caused by the rotation motions. This additional $P - Δ$

effect is found for the first time, which acts as an additional term in the dynamic equation, and is confirmed by comparison between the result from finite element analysis and test, the influence of which on the transmission tower-line system cannot be ignored. Therefore, for the seismic response analysis of high-rise flexible structures such as transmission towers, special attention should be paid to the impact of rocking ground motions and the additional $P - Δ$ effect produced by them on the structures.

Lastly, this study is an explanation and summary of the preliminary analysis of the response of the transmission tower-line system under multi-component ground motions, and there are still some contents that need to be in-depth researched and continued: The analysis considering soil-structure interaction (SSI effect) to the transmission tower line system under CHT and CVHT ground motions is still insufficient; for structures including transmission tower-line systems and long-span bridge structures where non-uniform seismic input should be considered on the structure, the effect of rotational components should be considered as well; although the analysis and test on the dynamic response of the structure under the multi-component ground motions have been complete, and the damage analysis of the structure is preliminarily conducted, the collapse analysis of the structure under multi-component ground motions is still at the beginning stage, whether simulation analysis or the test. There are still many difficulties and problems in how to conduct effective research collapse analysis under CHT and CVHT ground motions.

Besides the above problems, there are still some aspects to be analyzed: how to simulate and analyze the rotation ground motion of the high-frequency part; how to analyze the rotation ground motion in the far field, and how to introduce the effect of the tilting ground motion on the structure into the actual design. Hopefully, this article can give more researchers some new thoughts and inspiration, to promote the effect of rotational components on structures.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work has been supported by the National Natural Science Foundation of China under Grant No. 51678462, and Sanya Yazhou Bay Science and Technology City Administration Scientific research project (Grant No. SKJC-KJ-2019KY02), and the Hainan Special PhD Scientific Research Foundation of Sanya Yazhou Bay Science and Technology City (Grant No. HSPHDSRF-2022-03-008).

ORCID iDs

Ying Hu

Yong Lin Pi

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