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The behavior of power transmission tower-line system subjected to spatially varying base excitations is studied in this paper. The transmission towers are modeled by beam elements while the transmission lines are modeled by cable elements that account for the nonlinear geometry of the cables. The real multistation data from SMART-1 are used to analyze the system response subjected to spatially varying ground motions. The seismic input waves for vertical and horizontal ground motions are also generated based on the Code for Design of Seismic of Electrical Installations. Both the incoherency of seismic waves and wave travel effects are accounted for. The nonlinear time history analytical method is used in the analysis. The effects of boundary conditions, ground motion spatial variations, the incident angle of the seismic wave, coherency loss, and wave travel on the system are investigated. The results show that the uniform ground motion at all supports of system does not provide the most critical case for the response calculations.

In China, the west-to-east power transmission project will play an important role in changing the uneven distribution of our country's energy resources. Transmission projects extend thousands of kilometers and cost billions of dollars to construct and maintain, and most of them will cross high-intensity earthquake zones. However, most of research attentions on it have been paid on the actions of static load, impulsive load, equivalent static wind load and so forth. There are no code provisions for earthquake design of transmission tower-line system. It is unrealistic to assume that the transmission towers and lines are safe to go through earthquakes without adequate analysis. There are several recent cases of damage to power lines during earthquakes. In the 1999 CHI-CHI earthquake, transmission towers and lines was damaged most severely, and a lot of lines were broken and some towers collapsed [

Transmission tower-line system of Sichuan electric network damaged by the Wenchuan earthquake.

In the past one or two decades, researchers have done some earthquake dynamic analysis on the transmission tower-line system. Noteworthy contributions to the related study of transmission towers include some work that has developed effective approaches to deal with the actual problems. Li et al. [

A major problem that arises in the analysis of the long span structures such as transmission tower-line system is the difference among the ground motion components affecting various support points of the structure. The system response using uniform support excitation is compared with the response using multiple support excitations which is a more realistic assumption. In this paper, spatially varying ground motions of real data from the dense digital arrays of strong motion seismographs in SMART-1 are selected. The seismic input waves for vertical and horizontal ground motions are also generated based on Code for Design of Seismic of Electrical Installations [

Three-dimension finite element tower-line system according to practical project is established. A finite-element computer program SAP2000 is selected to establish the model. As shown in Figure ^{#}, 2^{#}, and 3^{#}) and four-span line. The tower is 45.5 m high, and its weight is approximately 9.2 t. The structural members of the tower are made of angle steel with the elastic modulus of 206 GPa. The tower is modeled by 1369 space beam members and 107 nodes, and the connections of members are rigid. The transmission line is modeled by 200 two-node isoparametric cable elements with three translational DOFs at each node. The upper 8 cables are ground lines and lower 24 cables are single bundled conductor. The spans to adjacent towers are all 200 m. The base points of the transmission tower are fixed on the ground, and the connections between transmission towers and lines are hinged, and the side spans of the lines are hinged at the same height of middle tower.

Finite element model of three towers and four-span lines coupled system.

The initial axial force and large deformation effect of cable are taken into consideration. Under self weight, the cable spatial configuration is a catenary. Based on the coordinate system illustrated in Figure

Coordinates of a single cable under self weight.

The fundamental equation for geometric stiffness for a cable is very simple to drive. Consider the horizontal cable shown in Figure

Force acting on a cable element.

Spatially varying ground motions can be available directly from the seismometer arrays data if the distance between the supports of the structure under investigation is equal to that between the stations of the array considered [

The SMART-1 array was the first large array of digital accelerometers specially designed to investigate the near-field properties of earthquake ground motion. It was located in the northeast corner of Taiwan near city of Lotung on the Lanyang plain. The array consists of 12 triaxial strong-motion accelerometers located in each of three concentric circular rings having radius of 200 m, 1000 m, and 2000 m and one triaxial accelerometer located at the center of the array. Figure

The location map of SMART-1, Taiwan.

This paper will use the data from the largest event recorded that had an epicenter within close proximity to the array, event 45, the earthquake of 14 November 1986 [

Event 45 of SMART-1, Taiwan.

Results are presented in the following for the application of the proposed approach to data recorded at the centre station C00, two inner stations (I06 and I12). Figure

Displacement time-history of each station at a distance of 200 meters under event 45.

N-S component

W-E component

Vertical component

The variation in seismic ground motion affecting different supports of a long-span structure is influenced by three main factors. The first factor is the wave travel effect that results from the finite speed of seismic waves. The second factor is the coherency effect that results from the reflection and refraction of seismic waves. The third factor is the site effect. The first two factors are accounted for in this simulation while the site specific effects are beyond the scope of this study.

To account for the variation in seismic ground motion, a seismological approach based on the seismic wave propagation from the epicenter to the supporting structure may be used. Alternatively, a stochastic approach based on random vibration analysis may be adopted. The ground motion cross-power spectral density function of spatial ground motions at point

According to Penzien and Watabe’s research [

In this study, the transmission tower-line system is assumed to locate in the Chinese Seismic Intensity Zone 8 with peak longitudinal ground acceleration 0.2 g and locate in the medium firm soil. The intensity of the transverse component and vertical component, as stated in the code, is 0.85 and 0.65 times of the longitudinal component. Figure

Code response spectrums with 2% damping for Chinese Seismic Intensity Zone 8.

The generated ground displacements considering both incoherency and wave travel effects are shown in Figure ^{#} and tower 2^{#} is also calculated and compared to the model coherence loss function in Figure

Generated ground displacement considering both wave travel and incoherency effects.

Longitudinal component

Transverse component

Vertical component

Coherency function of the generated ground motion comparison with model.

Code response spectrum and the response spectrum of typical simulated ground motions.

Most of the commonly available computer programs for the seismic analysis of structures do not allow the introduction of multiple acceleration time histories. The use of the displacement time history of the ground motion instead of the acceleration time history may be appropriate for long-span structures [

The equation defining the response degrees of freedom “

Assuming that the mass matrix is diagonal and

Model sketch massless rigid element method.

The current versions of SAP2000 can accommodate multiple excitation analysis only if excitations are defined as displacements and not accelerations. Furthermore, if displacement is applied to a node which is part of an integrated system such as a set of interconnected plate elements representing the mat foundation, only the node excited would move and the other nodes connected to it are not displaced. The two horizontal and one vertical ground displacement histories are applied to the bottom end of these rigid pedestals and the supports of every tower are subjected to different displacements.

To analyze the response of transmission lines to spatial ground motion, the model shown in Figure ^{#} are subjected to the generated ground displacement at 0 m, the supports of tower 2^{#} are subjected to the generated ground displacement at 200 m, and the supports of tower 3^{#} are subjected to the generated ground displacement at 400 m.

The dynamic performance is analyzed in terms of axial force, shear force, and moment at the tower bases as well as the displacement of tower and cables. The first, second, third, and forth layers of cables are called cable 1, cable 2, cable 3, and cable 4, respectively. The numbers of cables and top node of tower are shown in Figure

The numbers of cables and top node of tower.

Currently, most researchers established three towers and two-span model and focused on the response of middle tower. Here, comparisons for the model of three towers and two spans (Model I) with the model of three towers and four spans (Model II) are done. Figure

Finite element model of transmission tower-line coupled system.

In order to study the effect of the boundary condition, I06-C00-I12 of event 45 is selected. Table

Maximum response of Model I and Model II.

Tower | Model I | Model II | Cable | Model I | Model II |
---|---|---|---|---|---|

Axial force (N) | 85776 | 66643 | Cable 1 (N) | 4938 | 5238 |

Moment (N·m) | 1661 | 1134 | Cable 2 (N) | 9342 | 9646 |

Shear force (N) | 948 | 668 | Cable 3 (N) | 9499 | 9613 |

Displacement (cm) | 6.33 | 4.63 | Cable 4 (N) | 9612 | 9849 |

Figure

Longitudinal displacement of top node of transmission tower.

In order to study the effect of the spatially varying ground motions, I06-C00-I12 and I07-C00-I01 of event 45 are selected. Figure

I06-C00-I12 and I07-C00-I01 of Event 45.

The results in Table

Maximum response of tower under I06-C00-I12 and I07-C00-I01 excitations.

Tower | I06-C00-I12 | I07-C00-I01 | ||
---|---|---|---|---|

Uniform | Multiple | Uniform | Multiple | |

Axial force (N) | 48073 | 66643 | 40580 | 70020 |

Moment ( | 858 | 1134 | 786 | 1091 |

Shear force (N) | 491 | 668 | 465 | 628 |

Displacement (cm) | 3.14 | 4.63 | 3.14 | 4.59 |

Maximum response of cable under I06-C00-I12 and I07-C00-I01 excitations.

Cable | I06-C00-I12 | I07-C00-I01 | ||
---|---|---|---|---|

Uniform | Multiple | Uniform | Multiple | |

Cable 1 (N) | 4679 | 5238 | 4598 | 5162 |

Cable 2 (N) | 8888 | 9646 | 8745 | 9642 |

Cable 3 (N) | 8692 | 9613 | 8661 | 9579 |

Cable 4 (N) | 8704 | 9849 | 8651 | 9861 |

Figure

Longitudinal displacement of top node of transmission tower.

The angle of incidence of the seismic wave is investigated by varying the direction of wave propagation with respect to the longitudinal direction of the system as shown in Figure

Various angle earthquake wave of Event 45.

Table

Maximum response of tower under various angle excitations.

Number | Degree | Axial force ( | Moment ( | Shear force (N) | Displacement (cm) |
---|---|---|---|---|---|

I | 66643 | 1134 | 668 | 4.63 | |

II | 70020 | 1091 | 628 | 4.59 | |

III | 58945 | 1100 | 641 | 4.28 | |

IV | 66532 | 1159 | 666 | 4.65 | |

V | 58065 | 966 | 552 | 3.64 |

Maximum response of cable under various angle excitations.

Number | Degree | Cable 1 (N) | Cable 2 (N) | Cable 3 (N) | Cable 4 (N) |
---|---|---|---|---|---|

I | 5238 | 9646 | 9613 | 9849 | |

II | 5162 | 9642 | 9579 | 9861 | |

III | 5830 | 10416 | 9915 | 9816 | |

IV | 5644 | 10667 | 9603 | 10105 | |

V | 5219 | 9665 | 9326 | 10218 |

The above analyses demonstrated the importance of boundary conditions, ground motion spatial variation, and the incident angle of seismic wave on the transmission tower-line system responses. As discussed above, ground motion spatial variation is induced by wave passage and coherency loss. In the following, these two effects on ground motion spatial variations are investigated separately in detail to examine their influence on the transmission tower-line system.

To investigate the influence of spatially varying ground motions on the middle tower, highly, intermediately, weakly correlated, and uncorrelated ground motions are considered. It should be noted that the correlation as low as uncorrelated does not usually occur at short distances, unless there are considerable changes in the local geology from one support to the other. The parameters are given in Table

Parameters for coherency loss functions.

Coherency loss | ||||
---|---|---|---|---|

Highly | ||||

Intermediately | ||||

Weakly |

Coherency functions of the simulated spatially varying ground motions.

The maximum response of tower under various degrees of coherency is shown in Table

Maximum response of tower under various degrees of coherency.

Coherency | Axial force (N) | Moment ( | Shear force (N) | Displacement (cm) |
---|---|---|---|---|

Uniform | 23704 | 562 | 318 | 2.44 |

Highly | 34578 | 929 | 515 | 3.75 |

Intermediately | 46670 | 1170 | 656 | 5.18 |

Weakly | 69875 | 1415 | 813 | 6.86 |

Uncorrelated | 83342 | 2037 | 1157 | 9.41 |

Cable’s displacement ratio.

In order to obtain a representative analysis, various degrees of coherency of spatial ground motions should be considered. Neglecting loss of coherency between spatial ground motions may result in substantial underestimations of system responses.

Wave propagation will cause a phase delay between spatial ground motions. The phase delay depends on the separation distance and the wave propagation apparent velocity. Previous study revealed that wave propagation apparent velocity is quite irregular [

Maximum response of tower under different wave travel excitations is shown in Table

Maximum response of tower under different wave travel excitations.

Apparent velocity | Axial force (N) | Moment ( | Shear force (N) | Displacement (cm) |
---|---|---|---|---|

Uniform | 23704 | 562 | 318 | 2.44 |

200 m/s | 90550 | 2969 | 1716 | 12.3 |

400 m/s | 68679 | 2185 | 1262 | 8.39 |

800 m/s | 34578 | 929 | 515 | 3.75 |

1600 m/s | 27173 | 667 | 381 | 2.46 |

Cable’s axial force ratio.

The above results demonstrate that the spatial ground motion phase difference has a significant effect on the structural responses. Neglecting spatial ground motion phase difference may lead to erroneous estimation of system responses. As shown in Table

The effect of the spatial variation of earthquake ground motion on the response of the transmission tower-line system has been investigated in this paper. The members of transmission tower are modeled by beam elements and the nonlinear dynamic behavior of cables is taken into account. The input of ground motion is taken as displacement time histories. The real data from the close digital arrays of strong motion seismographs in SMART-1 are selected. Artificial ground displacement records are also developed and used in the analysis. The nonlinear time history analytical method is used in the analysis. The influence of the boundary condition, spatially varying ground excitations, incident angle of the seismic wave, coherency, and wave travel on the system are considered. Following conclusions can be obtained based on the above studies.

The boundary condition has an obvious effect on the response of the system. In order to obtain accurate results, three towers and four-spans model must be taken in the analysis.

The case of uniform support excitation does not produce the maximum response in the system. The multiple support excitations, which is a more realistic assumption, can result in larger response. The effect of spatially varying ground motions cannot be neglected.

The incident angle of the seismic wave has a slight effect on the responses of system. Assuming that the longitudinal of the ground motion and the direction of the wave propagation coincide with the longitudinal direction of the system could not obtain the maximum responses of the system.

The coherency loss has a significant effect on the response of the system. The uncorrelated ground motion gives bigger responses than other cases. In order to obtain a representative analysis, the various degrees of coherency should be considered.

The assumed velocity of propagation of seismic waves has a significant effect on the response of system to seismic ground motion. In order to obtain a representative analysis of the system, an accurate estimation of the wave velocity is required.

Based on the obtained results, uncorrelated ground motion and the apparent velocity of 200 m/s provide the most critical case for the response calculations. It should be noted that many studies have been reported on the ground motion spatial variation effect on bridges, viaducts, pipelines, and dams; very limited study on transmission tower-line system can be found in the literature. This study demonstrates that the ground motion spatial variation effect is very important to transmission tower-line system. As many cat head type towers, cup towers, and guyed towers are of transmission systems, more studies are deemed necessary to further investigate the ground motion spatial variation effects on responses of these systems.

This work is supported by the National Natural Science Foundation of China under Grant no. 50638010 and the Foundation of Ministry of Education for Innovation Group under Grant no. IRT0518. This support is greatly appreciated.