Study of the dynamic response of a transmission-line ...

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14th International Conference on Wind Engineering – Porto Alegre, Brazil – June 21-26, 2015

Study of the dynamic response of a transmission-line system to simulated artificial

wind

Fábio Paiva, Jorge Henriques, Pedro Almeida, Rui C. Barros

Department of Civil Eng., Faculty of Engineering, University of Porto, Porto, Portugal

email: [email protected] , [email protected] , [email protected], [email protected]

ABSTRACT: The present paper describe the study of the dynamic response of transmission-line system in Portugal to simulated

artificial wind. In the study, two different methodologies are compared, a static equivalent method as defined in Eurocode 3-3-1,

and non-linear dynamic analysis. As a main conclusion, the study observes a strong approximation between the results of both

methods.

KEY WORDS: Transmission-line system; Lattice Tower; Artificial wind, Eurocode 1-4, Non-Linear Dynamic Analysis

1 INTRODUCTION

There are several characteristics of electrical transmission line structures that make them different from other civil engineering

structures. The most important is that a transmission line system is, for all practical purposes, continuous. The lines extend for

miles (a large number of spans), and the spans generally have different lengths. This makes the analysis of dynamic behavior

difficult and very site-specific because the conductor behavior is significantly affected by adjacent span characteristics [1].

In the design of transmission lines, most calculations are based on static load cases. The environmental load cases are based on

statistical data of wind and ice accretion. They provide a good estimate of the extreme forces that a transmission line is subjected

to during its service life. In certain circumstances, the dynamic effects also need to be examined [2]. The present study describes

analyses of buffeting of a transmission-line system, for that a non-linear dynamic analysis (geometric non-linearities) is conducted.

In the present paper, a simulation procedure for analyzing transmission-line system is described in which representative wind

velocities, forces, line response and lateral tower loads are computed on a step by step basis. The structural model of transmission

–line system is fully detailed, which includes the system geometry and its materials. The simulated wind records are generated

using a web-application developed by NatHaz Modeling Laboratory. A brief summary of the main parameters for generating the

wind records is given. Finally the non-dynamic response of transmission-line system is studied, and compared with static-

equivalent method defined in Eurocode 3-3-1 [3] and EN50341 [4].

2 TRANSMISSIO-LINE SYSTEM CHARACTERISTICS

Transmission-line system geometry

A four-span section of the line (located in Portimão, Portugal), as shown in Figure 1

Figure 1, was selected for this study because it is on relatively level terrain (all the towers are at the same level) and is made up

of "typical" tower geometries and span lengths. The towers are implanted more than 5 km from the coast, in a terrain category

with area with low vegetation (such as grass) and with isolated obstacles.

The transmission-line system consists of the conductors, the wire shields, the insulator strings and hardware and the suspension

and dead-end structures which support the wires at the necessary electrical clearances. Usually, the conductors are stranded cables

composed of aluminum, galvanized steel or a combination of the two. The wire shields are grounded steel wires placed above the

conductors for lightning protection. Conductors are attached to suspension structures via insulator strings that are vertical under

normal operation conditions and are free to swing along the line whenever there is a longitudinal unbalanced load. At dead-end

structures, the insulator strings are anchored and in-line with the conductors [2].

The three lattice towers that support the conductor suspension system are shown in Figure 2. The towers in this study belong to

the CWS type which are used in the Portuguese National Electrical Network (REN in Portuguese), for high-voltage distribution

lines in Portugal. The CWS towers are used for alignment or small angle disposition of the cables. The towers have rectangular

base occupying an area of 6.90*5.20 m at the foundations, and total height of 50 m, as shown in Figure 2.

mailto:[email protected]




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Figure 2, the base dimensions decrease with the height of the tower. The upper body of the tower has square geometry with

dimensions 1.80*1.80 m. The lower body has a height of 34.00 m, and the rest 16.00m, in the upper body of the tower the cross

arms are executed. The conductors are then fixed to the insulators strings, which in their case are fixed to the cross arms (lower,

middle and the upper level), the wire shield is directly fixed into the top of the tower, and with an effective span between adjacent

towers of 450 m.

Figure 1. Transmission line system (Lateral view)

Figure 2. Transmission-line system (left) and lattice tower (right-3d view and frontal view)

The lattice steel towers comprise angles members (L-section), with a steel grade S355 (yield strength 355 MPa, ultimate stress

490 MPa as defined in the Eurocode 3-1 [5]).

Some considerations related with the structural model

To support the analysis of this transmission-line system structure, a numerical model was created in the program SAP2000 [6].

Numerical methods based on finite element analysis have the advantage of being applicable to specific line sections with variable

degrees of details. The influence of different line parameters can also be studied without having to erect costly scale models.

However, the results obtained with an analytical method depend on the numerical parameters selected and therefore necessitate a

thorough understanding of the concepts of dynamics and of the physical problem at hand. It is emphasized that careful validation

of analytical models with full-scale test results is still lacking [2].

The objective of the modeling approach, is to capture the essential features of the response of the coupled tower-cable system.

In addition to the inherent geometric non-linearities of suspended cables, it’s expected that the high wind speeds creates a change

in the boundary conditions of the system and induces large, non-linear cable and insulator string (were not modeled in this study)

displacements. To a lesser degree, P-delta (second order) effects on the displaced towers are also present [2].

As complete line sections are modeled, cable elements are needed to simulate the conductors and the wire shields. Such elements

must accommodate initial strains and simulate the slackness of the cable. As a result, the software must offer large kinematics

formulations (large-stress and large-displacements) [2]. Although towers modeling can vary within the full spectrum of possible

idealizations (like rigid supports or soil-structure interaction), for the present case every structural member was represented with

sufficient detail. According to Al-Bermani and Kitipornchai [7] the behavior of the lattice structure requires a consideration of the

L=450 m L=450 m L=450 m L=450 m

Wind load zone and

wind direction studied

50 m

34 m

6

,4 m

6

,4 m

3

,2 m

6,9 m

11 m

5 m

12 m

10 m

3d-view Front-view TL-system

3


non-linearities of system, principally, geometric non-linearity, material non linearity and joint flexibility/slippage. It was also

noted that tower members are asymmetric, thin-walled angle sections and eccentrically connected so that the detailed behavior is

difficult to achieve. For the present study only the geometrical non-linearities were taken into consideration. Thus a frame-truss

model (with pin supports) was used for the analysis, because they approximate more to the real structural behavior. Generally the

use of frame elements are more realistic since main leg members are usually continuous with multi-bolted connections that can

transfer secondary bending moments [2]. The boundary conditions of the extreme cables are modelled as longitudinal linear

springs, and pinned in the other directions. The detailed tower models behave according to a linear elastic model, it is also assumed

in SAP2000 small strains, therefore material nonlinearity and geometric nonlinearity effects are independent.

3 ARTIFICAL WIND SIMULATION

Introduction

As transmission towers are usually quite high, while their stiffness quite soft, and their weight quite light, such structures are

very sensitive to wind load. Accurate simulation of wind load of transmission towers is the primary problem of anti-wind design.

Generally design loads are determined by wind tunnel tests, field measurements, or numerical simulation. Stochastic wind

simulation is often divided into two parts, including stationary and non-stationary wind simulation. Both the classical linear

filtration method and harmonic superposition method (wave superposition) belong to stationary stochastic method simulation

method. The non-stationary stochastic wind simulation can be realized by the autoregressive method (AR), empirical mode

decomposition method (EMD), spectral representation method (SR), spline-interpolation-based FFT approach (SFFT) (digital

filtering) [8]. These techniques vary in their applicability, complexity, computer storage requirements and computing time.

Wind General Characteristics

Time-history of wind velocity along the wind direction contains two parts, the average of wind velocity, and the turbulent wind

velocity. In a given period of time, the average wind velocity and its direction do not change along with time; while the turbulent

wind velocity change stochastically with the time and must be analyzed according to the stochastic vibration theory. The wind

simulation considers mainly the turbulent wind. The total wind velocity in any point of a structure (1) is the sum of the average

wind velocity and the turbulent wind velocity [8]:

𝑈(𝑧, 𝑡) = 𝑈(𝑧) + 𝑢(𝑡) (1)

𝑈(𝑧, 𝑡) is the total wind velocity in structures, �̅�(𝑧) average wind velocity, u(t) turbulent wind velocity.

3.2.1 Mean wind speed

A logarithmic law is used to describe the variation of mean wind speed with elevation above the Earth surface within the

atmospheric boundary layer. The logarithmic law is considered slightly more accurate for larger heights but is also more difficult

to use [9].

Taking into consideration Eurocode 1-4 [10], the mean wind velocity varies with height and is defined as the average value of

wind speed for a 10 minutes period at an appropriate height above ground. It depends on the terrain roughness and orography and

on the basic wind velocity 𝑣𝑏 and should be determined according to (2) and (3):

𝑣𝑏 = 𝑐𝑑𝑖𝑟𝑐season(𝑧)𝑣𝑏,0 (2)

𝑈(𝑧) = 𝑣𝑚(𝑧) = 𝑐𝑟(𝑧)𝑐0(𝑧)𝑣𝑏 (3)

where 𝑐𝑟(𝑧)) is the roughness factor, 𝑐0(𝑧) is the orography factor (usually taken as 1), 𝑐𝑑𝑖𝑟 directional factor (taken as 1),

𝑐season season factor also taken as 1 and 𝑣𝑏,0 fundamental value of the basic wind velocity taken as 27 m/s.

The wind action calculated according to the Eurocode 1-4 gives characteristic values which are obtained from base values of

wind speed and wind pressure, corresponding to a probability of annual exceedence of 2% (i.e., 0.02); that is equivalent to a return

period of 50 years. In some situations it is advantageous to consider a fundamental velocity with a probability of annual exceedence

(𝑝) different from 0,02 (or 2%).

3.2.2 Turbulent wind speed

To describe the turbulent wind velocity component, is necessary to recur to wind power spectrum. At all times, the observation

results showed that the turbulent wind power spectrum obeys approximately to a Gaussian distribution [11]. The fluctuating wind

can be regarded as a 3D turbulent flow composed by the along-wind, across-wind, and vertical-wind component. The capability

characteristics of fluctuating wind range can be described by the power spectrum in various directions [8]. According to the

characteristics of transmission tower-line system, only the along-wind dynamic response is considered in this study.

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Because the transmission towers are usually high and slender structures, Simiu spectrum proposed by Simiu in 1974 was adopted

in this article [9]. Additionally, because the problem in study has a 2d relevance, 2d coherence function is used, for that the work

of Davenport [12] and Simiu and Scanlan is referred.

In order to facilitate the use of stochastic simulation, the NatHaz Modeling Laboratory has developed a web-based simulation

portal [13], which allow simulation of multivariate Gaussian random processes with prescribed spectral characteristics. The

NatHaz online wind simulator (NOWS), available at http://windsim.ce.nd.edu, enables users to simulate stationary random wind

fields beyond temporal and geographical boundaries using an intuitive and user-friendly interface [14].

NOWS also offers the flexibility of selecting one of four different simulation schemes, for the present study the discrete

frequency function with FFT was chosen [15]. To convert to 10 min averaging period wind speed of Eurocode (UT=600s) to the

three second gust speed defined in ASCE-7-1998 [17] (UT=3s), the method described in [18] was used. For that a relationship of

1.46 (UT=3s= 1.46*UT=600s) between those two velocities was determined, resulting in three gust speed equal to UT=3s=39.4 m/s. An

exposure category type C, similar to the terrain category II of Eurocode 1-4, was also used.

The wind histories were generated using the following data: a cut-off frequency (fc) equal to 2.5 Hz was defined in the web

application with a total frequency of N=1500, resulting in Δt=1/(2*fc) =0.2 s and T=2*N*Δt=600s. The parameters for the wind

simulation were chosen as compromise between the large computational effort due to the large number of wind fields generated

and the known consequences to the accuracy of the values. A summary of the wind loads points considered in this study is indicated

in Table 1.

Wind load models

The wind mean speed considered in this study 𝑣𝑏,0 the basic wind velocity is assumed to be 27 m/s, and terrain category type II

(z0=0.05 and zmin=3) by the Eurocode 1-4 (Portuguese National Annex). No orography effects were consider in this study, also

the directional and season factor were considered equal to 1.0. The variation of the mean wind with height (Figure 3), vm(z), follow

the procedure described in Eurocode 1-4. Only wind orthogonal to the transmission line axis (transverse direction) was studied,

with one tower and 2 spans under wind force (Figure 1).

The fluctuating wind speed simulation were conducted for 10 minutes with time intervals of 0.2 s. Because of the high number

of nodes of the transmission tower, it is difficult to simulate wind velocity time history for every node. Some simplification is

necessary, therefore for each panel of the transmission tower, the wind load was applied in the upper four nodes (see also Table

2). As shown in Figure 3, the tower is divided into a total of 14 panels.

Figure 3. Panels zones, wind simulation points (identified by the dots) and mean wind velocity-vertical profile

3

2

1

4

5

6

7

9

10

11

12

13

14

8

Longitudinal

Tra

nsv

erse

25,4

vm (z) {m/s]

Z [m]

22,6

28,5

30,5

31,8

33,5

34,3

35,5

35,1

Node 4

Node 3

Node 2

Node 1

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Each cable (conductor and wire shield) was divided into 18 elements, but the wind load was only applied in to 10 points per

span (450/50). Only the internal part of the transmission line system (central tower and the 2 consecutive spans) are studied in this

paper (Figure 1). Table 2 describes the wind load position, area of wind pressure, force coefficient, mean wind force and total

static-equivalent load as defined Eurocode 1-4.

Table 1. Summary of wind load point of the coupled system under study

Transmission line components No. of wind points

Conductor (each span) 10

Wire Shield (each span) 10

Lattice Tower 4*14 (panel)

Total of wind turbulent series in

the transmission-line system 208

Figure 4 shows the simulated fluctuating wind velocities for the panels 1 and 12.

Figure 4. Fluctuating wind speed at the panels 1 (left) and 12 (right)

The along wind-load loads are calculated by the following equation (4):

𝐹𝑑 (𝑡) =1

2𝜌𝑐𝑓𝐴[(𝑈(𝑧) + 𝑢(𝑡)]2

(4)

Where A is the area acted by wind in Table 2, 𝑐𝑓 is force coefficient, 𝜌 is the air density.

The wind force was divided into two components, the mean component (5) and turbulent component (6).

�̅�𝑚𝑑(𝑡) =

1

2𝜌𝑐𝑓𝐴[(𝑈(𝑧)]2

(5)

𝐹𝑢𝑑(𝑡) = 𝜌𝑐𝑓𝐴 ∗ 𝑈(𝑧) ∗ 𝑢(𝑡) +

1

2𝜌𝑐𝑓𝐴[(𝑢(𝑡)]2

(6)

For that two non-linear static analysis were conducted, first with just the dead load and then with the wind mean component,

subsequently a nonlinear dynamic analysis of the turbulent part was performed.

-15

-10

-5

0

5

10

15

20

0 100 200 300 400 500 600

u(t

) (m

/s)

Time (s)

-15

-10

-5

0

5

10

15

0 100 200 300 400 500 600

u(t

) (m

/s)

Time (s)

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Table 2. Wind load position, area of wind pressure, force coefficient and mean wind force

Panel Height of

panel (m)

Level of application

of the load (m)

Area

[m2]

cf F,mean

(kN)*

F,T (kN)

EC1-4

1 4.10 4.10 2,818 3,22 2,90 6,78

2 2.99 7.09 1,898 3,23 2,47 5,41

3 5.98 13.07 3,316 3,22 5,45 11,19

4 5.98 19.05 3,128 3,15 5,73 11,43

5 5.98 25.03 3,006 3,03 5,80 11,40

6 4.62 29.65 2,247 2,91 4,38 8,58

7 3.02 32.67 1,441 2,80 2,79 5,45

8 1.33 34.00 0,723 2,61 1,32 2,58

9 1.60 35.60 0,735 2,77 1,44 2,82

10 4.80 40.40 1,972 2,87 4,17 8,17

11 1.60 42.00 0,703 2,81 1,47 2,89

12 4.80 46.80 1,876 2,91 4,20 8,29

13 1.60 48.40 0,703 2,81 1,54 3,04

14 1.60 50 0,685 2,83 1,52 3,02

F,mean *mean component for the non-linear dynamic analysis, F,T .static-equivalent wind load EC1-4

The structural factor for the tower according to the Eurocode 1-4 gives a value of 0,91 (with cd=1,01 and cs=0,90). The logarithmic

decrement of structural damping in the fundamental mode was taken as 0.05 (lattice steel tower with ordinary bolts). These values

only affects the determination of total wind load F,T (mean and turbulent) which will be compare to the results of the non-linear

dynamic analysis.

The static equivalent load in the cables were determined according with EN50341-1. The conductor and wire shield force

coefficients were taken as 1.2.

4 STRUCTURAL RESPONSE OF THE TRANSMISSION-LINE SYSTEM TO DYNAMIC LOADING

It’s important to guarantee a dynamic interaction between turbulent wind forces (buffeting) and the conductors for the coupled

structural system (supporting structure, conductors and wire shields), to ensure that a correct assessment of the dynamic response

of the structure is accomplished. For high wind speeds, and for the situation where the wind acts perpendicular to the transmission

line, the wind action can induce high angular displacements to the conductors [18].

Modal analysis

A modal analysis is used to determine the vibration characteristics of a structure while it is being designed. Hence, the goal of

a modal analysis is determining the natural frequencies and mode shapes. The right hand side of the equation of motion (7) is

considered to be zero, i.e. f(t) = 0.

𝑀�̈� + 𝐶�̇� + 𝐾𝑋 = 0 (7)

Where M, C and K are mass, damping and stiffness matrices, and �̈� is the response acceleration vector, �̇� is the response

velocity vector and 𝑋 is the response displacement vector.

A modal analysis can also be taken as a basis for other more detailed dynamic analyses such as a transient dynamic analysis, a

harmonic analysis or even a spectrum analysis based on the modal superposition technique. Mode shapes describe the

configurations into which a structure will naturally displace. Typically, lateral displacement patterns are of primary concern. Mode

shapes of low-order mathematical expression tend to provide the greatest contribution to structural response. The modal analysis

is a linear analysis. The determination of the natural frequencies in SAP2000 was based in Eigen vectors.

A total of three modal analysis will be conducted, first, just the lattice tower, then only the cables and finally the coupled-system

(under initial stresses), this will allow a better understating of the dynamic behavior of each part in coupled system [19]. To obtain

meaningful results when performing modal analysis of a structure with cables, load cases in which dead and other loads are applied

should use the option for stiffness at the end of nonlinear load case. Modes will then be based on the stiffness of the cables under

applied loading.

The transmission line is constituted by 6 conductors and 2 wire shields. Both of the cables are aluminum-steel in all it extension.

The mechanical properties of the cables are listed in Table 3.

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Table 3. Conductor line and wire shield mechanical proprieties

Type Outside

diameter [mm]

E-Modulus

[GPa]

Transversal cross-

section[mm2]

Mass per unit

length [kg/m]

Ultimate cable

Strength (kN)

Line expansion

coefficient [1/C]

Conductor line 28.60 70 484.5 1.618 128.49 19.4E-06

Wire shield 16.00 104 152.8 0.706 77.08 15.3E-06

4.1.1 Tower modal analysis

Table 4 shows the first five frequencies of the lattice tower without the conductor and wire shield, also the direction involved in

the model shape is defined, as well the percentage of mobilized modal masses. The first two modes (longitudinal and transverse

bending) and the forth mode (torsion) are illustrated in Figure 5.

Table 4. Natural vibration frequency and mobilized modal mass for the lattice tower

Mode Frequency

[Hz] Direction

Modal mass (%)

Total=12746,45 kg

mlongitudinal mtransverse mvertical

1

2

3

4

5

1,33

1,61

4,63

4,80

5,20

Longitudinal

Transverse

Local mode (lower part)

Torsional

Longitudinal

48%

0%

0%

0%

20%

0%

45%

0%

0%

0%

0%

0%

0%

0%

0%

Figure 5. Mode shapes of the tower for mode 1, 2 and 4 (from left to right)

In Table 5 a comparison of the natural frequency obtained through SAP 2000 with some empirical expression available in the

literature (Glanville and Kwok [19] and Holmes [20]), derived from full-scale testes is made.

Table 5. Natural vibration frequency (Hz) for SAP 2000, empirical expressions

SAP 2000 Glanville and Kwok

75/h

Holmes

750 (Wb+Wt)/h2

1,33 1,5 2,6 2,1

Where Wb and Wt are the width of the lattice tower at the bottom and top and h is tower height.

For the tower in study Wb varies form 5.2 m to 6.9 m at the base, and Wt=1.8 m. The expression defined by Glanville and Kwok

agrees reasonable well with the frequency calculated by SAP 2000.

4.1.2 Cable modal analysis

Table 6 shows the primary mode frequencies for the conductor and the wire shield under dead load without towers. According

to Yasui [19] the influence of the mean wind load increases slightly the frequencies, because increases the tensile forces (cable

tauter). Therefore in the following modal analysis only cable with dead load is studied.

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Table 6. Natural vibration frequency and mobilized modal mass for the conductors and wire shield

Mode

Conductor Wire shield

Frequency

[Hz]

Modal mass (%)

Total=801,50 kg

Frequency

[Hz]

Modal mass (%)

Total=350,78 kg

m

out-of plane

m

in-plane

m

out-of plane

m

in-plane

1

2

3

4

5

0,18

0,32

0,35

0,35

0,52

85%

0

0

0

9%

0

76%

0

0

0

0,19

0,31

0,37

0,37

0,55

86%

0%

0%

0%

9%

0%

79%

0%

0%

0%

6 0,53 0 17% 0,56 0% 15%

Figure 6 illustrates the first 6 mode shapes for cables of the transmission-line system under study.

Figure 6. Mode shapes of the cables for mode 1 to 6

4.1.3 Coupled system (lattice tower + cables)

The coupled system was formed by the three towers and the corresponding cables. Remote spans are modelled as linear static

springs, in the longitudinal direction. According to Desai [22] KL is computed from (8):

1

𝑘𝐿

=𝐿

𝐴𝐸+

𝑃𝑧2𝐿𝑥

3

12𝐻3

(8)

Where L and Lx are the total and horizontal span lengths between adjacent towers,

Pz is the total vertical load intensity;

H horizontal component of static tension;

AE axial stiffness of the cable;

Table 7 shows the main natural frequencies, involved components and modal mass mobilized. The vibration modes of the

coupled system in the longitudinal and transverse direction are shown in Figure 7 and Figure 8.

Table 7. Natural vibration frequency, modal shapes description and mobilized modal mass for the coupled system

Mode Frequency

[Hz]

Mode Shape Modal mass (%)

Involved component Tower displacement

direction

mlong mtrasv mvert

17

51

385

0,16

0,27

0,95

Cables-out-of-plane

Cables-in-plane

Tower + cables

-

-

Longitudinal

5%

0%

0%

0%

0%

33%

0%

22%

0%

846 1,59 Tower + cables Transverse 14% 0% 0%

The free vibration analysis results indicated a strong coupling phenomena between the tower and the cables. As expected cable

dominates the lower vibration modes, which means that their dynamic effects is transmitted to the tower structure. These cable

low modes play an important role in the dynamic response of transmission tower under wind loads. The main structure (tower)

vibration at lower modes is not easily identified because of the cable large displacements. At higher modes, local vibration of the

tower dominates the response of the system. These higher modes are of interest in study of the dynamic response of the coupled

system under seismic excitation.

Out-plane-

mode 1

In-plane

mode 2

In-plane-

mode 3

Out-of-plane mode 4

Out-of-plane mode 5

In-plane mode 6

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Another conclusion of the previous analysis, is the significant reduction of the frequency in the longitudinal direction (1,33-0,95

Hz) and the small reduction in the transverse direction (1,61-1,59 Hz).

Figure 7. Mode shape of the coupled system for mode 385

Figure 8. Mode shape of the coupled system for mode 846

These results enable an understanding of one of the most important aspects of the dynamic system behavior (tower and cable),

the conductor out-of-plane oscillation under the action of wind excites the tower dominant vibration modes. As noted by [23] the

low frequency means that, when the structures is exposed to turbulent component of the wind load, the fluctuating response of the

low-damped tower-cables coupled system in the along-wind and across-wind directions can be significant.

Time domain analysis

A transient dynamic analysis is a technique which is used to determine the time history dynamic response of a structure to

arbitrary forces varying in time. This type of analysis yields the displacement, strain, stress and force time history response of a

structure to any combination of transient or harmonic loads.

To obtain a solution for the equation of motion (9) a time integration has to be performed.

𝑀�̈�𝑡 + 𝐶�̇�𝑡 + 𝐾𝑋𝑡 = 𝐹𝑡𝑑 (9)

Where 𝐹𝑡𝑑 is the fluctuating wind force and the suffix and aerodynamic damping force and the suffix 𝑡 is used to indicate time.

The stiffness matrix K, under dead load and mean wind force, is obtained through the followings non-linear static analysis: in

the first step, initial coordinates and tensile forces of the cables, insulators and towers under a dead load are computed from a

catenary curve; in the second step, a non-linear static analysis is performed under the wind mean force is applied at each point,

thus both deformations and tensile forces of the members are computed, taking into account their geometrical stiffness and the

change in relative angles between wind direction and members. After this two steps the equilibrium equation is obtained and the

time history response analysis is carried out as in (9).

In the literature, several time integration algorithms are discussed in detail ([24]). They can be broadly classified into implicit

and explicit methods. Considering the stability of these two types of integration methods we notice that implicit methods are

usually unconditionally stable which means that different time step sizes can be chosen without any limitations originating from

the method itself. Explicit methods on the other hand are only stable if the time step size is smaller than a critical one which

typically depends on the largest natural frequency of the structure. Due to the small time step necessary for stability reasons

explicit methods are typically used for short-duration transient problems in structural dynamics.

By taking into account the earlier consideration an implicit method will be chosen, for that SAP2000 has a variety of common

methods available to perform direct-integration time-history analysis (Newmark, Wilson, Collocation, Hilber-Highes-Taylor and

Chung and Hulbert). In the non-linear dynamic analysis the Newmark method, was the preferred method. The Newmark method

with parameters γ= 0.5 and β = 0.25, as used to solve the dynamic equation, which is the same as the average acceleration method

(also called the trapezoidal rule). Using the parameters indicated in the Newmark method, offers the highest accuracy of the

available methods, but may permit excessive vibrations in the higher frequency modes, i.e., those modes with periods of the same

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order as or less than the time-step size. For best results, the smallest time step practical should be used. Different values of γ and

β and time-step should be used, to be sure that the solution is not too dependent upon these parameters [6].

In the time history analyses of the structure, turbulent wind forces are applied directly into to the defined nodes (see section

3.3.). Modelling the structure allowed the inherent stiffness of each structure to dictate how the loading was distributed among its

supporting members. The mass of the structure is lumped at the structure nodal points, the masses are assumed to have only

translational degree of freedom.

One final aspect, of the analytical model development, that requires addressing, is the topic of damping. Damping is a very

important component of most dynamic models for three main reasons. First, the level of damping within a dynamic system dictates

how long the system will oscillate significantly once it has been excited. The second reason is that relatively high levels of damping

(i.e. ≥ 20% of critical) can have a significant effect on the system’s natural frequencies and mode shapes. Finally, increasing

damping reduces the maximum amplitude of the oscillations [25]. There are many mechanisms of damping in a structure the most

identified of which are material damping and interfacial damping. The material damping contribution comes from a complex

molecular interaction within the material, thus the damping is dependent on type of material, methods of manufacturing and

finishing processes. Since the material microscopic properties may differ from one sample to the other one, estimation of material

damping me be complicated. The interfacial damping mechanism is Coulomb friction between members and connections of a

structural system and non-structural components. Experiments have shown that most of the energy dissipation mechanisms

through the structure are dependent on displacement amplitude rather than frequency of the structure. Therefore the most

appropriate form of formulating damping in structures may be friction damping [26]. The problem of a formulation based on

friction damping is that it would need a great amount of time and computer power to analyze a structure. That’s why in typical

engineering practice viscous damping models are used for the sake of simplicity as they lead to the linear analysis of equation of

motion. It’s important to emphasize that the viscous damping is frequency dependent and not displacement dependent. Since

viscous damping is used to linearly model the structural behavior, any source of nonlinearity is relatively unknown and

consequences of using this model are ignored [26].

As referred, the damping of structure is assumed to be viscous and frequency dependent for the sake of convenience in analysis.

Rayleigh damping known as proportional damping or classical damping model expresses damping as a linear combination of the

mass and stiffness matrices:

𝐶 = 𝛼𝑀 + 𝛽𝐾 (10)

Whrere α and β are real scalars,

Knowing all matrices of the structure, M,C,K the response of the system can be calculated with a time integration method. It’s

important to have in consideration that the accuracy of response may be questionable, due to the fact that this approach is based

on two main assumptions, a) The model used for damping of a structure is viscous and b) the approach is formulated for the linear

response of the structure which may not be the available situation for all cases (i.e. structures with nonlinearities) [26].

In some research studies the damping measured was “total” damping, which includes both inherent structural damping and any

aerodynamic damping. SAP2000 includes damping in its dynamic time history analysis application, through the classic

formulation of Rayleigh damping with user-defined quantities, as being proportional to the mass and stiffness matrix. SAP2000

also allows users to either specify coefficients α and β directly, or in terms of the critical-damping ratio either at two different

frequencies, f (Hz).

A target damping ratio equal to ξ=2 % was used for every mode of vibration, the conditions associated with the proportionality

factors simplify as follows:

𝜉𝑖 = 𝜉𝑗 = 𝜉 ; 𝛽 =2𝜉

𝑤𝑖+𝑤𝑗 ; 𝛼 = 𝑤𝑖𝑤𝑗𝜉 (11)

𝜔 is the natural frequency

The 3D-FEM model was analyzed in the time domain (total time of 650s), with time step equal to 0.1s. The wind forces were

considered in to act transverse to the transmission line axis. Due to random nature of the dynamic loading, it is important to assess

the reliability of the response obtained. As one of the aim of the analysis is to compare EC1-4 methodology with the non-linear

dynamic analysis, which is made on the 10 min wind, the transient analysis as also evaluated for maximum response for 10 min

interval, plus an additional of 50 seconds without turbulent wind load. Due to the length of the calculations the number of samples

as limited to a manageable size. For that a total of 6 samples, were considered to be sufficient to show the overall trend of structural

response.

During the 10 min periods used in the analysis, several high peaks in the structural response occur. It is shown in the following

figures (Figure 9 to Figure 12) the displacement time history of the nodes 1 to 4 (Figure 3) in the along-wind (x), cross-wind (y)

and in the vertical direction (z) for the first sample.

11


Figure 9. Time histories- displacements at node 1




0,00

0,05

0,10

0,15

0,20

0,25

0 200 400 600

x (m

)

Time (s)

Along-Wind Direction- N1

-0,15

-0,10

-0,05

0,00

0,05

0,10

0,15

0,20

0 200 400 600

y (m

)Time (s)

Cross-Wind Direction-N1

-0,10

-0,09

-0,08

-0,07

-0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0,00

0 200 400 600

z (m

)

Time (s)

Vertical Direction-N1

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0 200 400 600

x (m

)

Time (s)

Along-Wind Direction-N2

-0,20

-0,10

0,00

0,10

0,20

0,30

0 200 400 600

y (m

)

Time (s)


-0,14

-0,12

-0,10

-0,08

-0,06

-0,04

-0,02

0,00

0 200 400 600

z (m

)

Time (s)


0,00

0,10

0,20

0,30

0,40

0,50

0 200 400 600

x (m

)

Time (s)


-0,30

-0,20

-0,10

0,00

0,10

0,20

0,30

0,40

0 200 400 600

y (m

)

Time (s)


-0,12

-0,10

-0,08

-0,06

-0,04

-0,02

0,00

0 200 400 600

z (m

)

Time (s)


0,00

0,10

0,20

0,30

0,40

0,50

0,60

0 200 400 600

x (m

)

Time (s)


-0,30

-0,20

-0,10

0,00

0,10

0,20

0,30

0,40

0 200 400 600

y (m

)

Time (s)


-0,10

-0,08

-0,06

-0,04

-0,02

0,00

0 200 400 600

z (m

)

Time (s)


12


In Table 8 the maximum displacement for nodes 1 to 4 in the along-wind, cross-wind and vertical directions, are determined,

with the correspondent mean value (�̅�𝑒), standard deviation (𝜎𝑒), coefficient of variation (COVe) for the 6 samples analyzed.

Table 8. Maximum displacement for the 6 samples, for nodes N1, N2, N3 and N4

N1 N2 N3 N4

Sample x y z x y z x y z x y z

1 0,202 0,149 0,095 0,317 0,223 0,127 0,449 0,304 0,113 0,517 0,344 0,077

2 0,228 0,108 0,107 0,358 0,161 0,144 0,509 0,217 0,129 0,587 0,245 0,088

3 0,205 0,156 0,097 0,323 0,230 0,130 0,459 0,312 0,116 0,529 0,353 0,079

4 0,212 0,181 0,100 0,333 0,266 0,133 0,473 0,357 0,120 0,546 0,402 0,082

5 0,220 0,144 0,103 0,345 0,213 0,138 0,489 0,287 0,124 0,564 0,324 0,085

6 0,203 0,130 0,096 0,319 0,195 0,128 0,452 0,265 0,115 0,521 0,301 0,078

�̅�𝑒 0,208 0,146 0,099 0,328 0,218 0,132 0,466 0,295 0,118 0,537 0,334 0,080

𝜎𝑒 0,010 0,025 0,005 0,016 0,035 0,006 0,024 0,047 0,006 0,027 0,053 0,004

C0Ve (%) 5% 17% 5% 5% 16% 5% 5% 16% 5% 5% 16% 5%

Note: x-along-wind vertical, y-across-wind direction and z-vertical direction

To illustrate the variability of the maximum displacements values from the 6 samples, Table 8 summarizes this study. The table

shows that the values obtained have low standard deviation, with COVe smaller than 5%, except for across-wind direction with

COVe around 17%. From the present study of the 6 samples, it was observed that the values could range from 11-40%. Therefore

it is concluded, that the 6 samples are sufficient to represent the expected range of dynamic response.

Power spectrum density of transmission line system

Figure 13 shows the power spectrum densities (PSD) at Node 3 of the along-wind, across and vertical direction.

Figure 13. Power spectral density (PSD) at node 3 in the along-wind (left), across-wind (center) and vertical direction (right)

Although the frequency-dependent stiffness, as mentioned earlier in this work, the cables and tower contribute to the dynamic

response of the structure in different ranges, below 0.9 Hz for the cables and between 0.90 to 3.5 Hz for the tower. From the

analysis of PSD at node 3 the fundamental frequencies of the tower in the longitudinal and transverse can be easily identified.

Even the PSD of the vertical direction, the transverse frequency of the lattice tower is clearly presented. The peaks in the lower

range of frequency of the PSD, are not clearly defined, but rather many small peaks occur. This condition demonstrate a distributed

dynamic effect induced by the cables on the tower over the range 0.2 to 0.9 Hz.

Comparison between static equivalent and non-linear dynamic methods

For the comparison of results of the non-linear dynamic analysis with the static equivalent method of Eurocode 3-3-1, the mean

of the maximum from the 6 samples and the corresponding standard deviation were used.

1,00E-11

1,00E-09

1,00E-07

1,00E-05

1,00E-03

1,00E-01

1,00E+01

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

1,00E-11

1,00E-09

1,00E-07

1,00E-05

1,00E-03

1,00E-01

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

0.94 Hz

1,00E-11

1,00E-09

1,00E-07

1,00E-05

1,00E-03

1,00E-01

1,00E+01

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

≈1.56 Hz ≈1.56 Hz

13


4.4.1 Cable response

The cable responses, cable tension and transverse displacement of the conductor and the wire shield, from the static equivalent

method is compared to the non-linear dynamic analysis. Only the top conductor (which connects to node N3) and the wire shield

at node N4 are analyzed. To show the range of results from the dynamic analysis, the standard deviation of mean maximum values

for each parameter is also presented. Table 9 summarizes the results for the cables under study. The tension values represent the

average tension for the whole span, where in the case of transverse displacements they are determined at the mid span section. For

the conductor, it was observed that tensions calculated by the static equivalent were slighter inferior than the dynamic analysis,

by about than 10%. The same trend was observed for the transverse displacement. For the wire shield, the same behavior was

observed for tension, as for transverse displacements, and the static equivalent values were almost within the limits �̅�𝑒 ±-𝜎𝑒 from

the dynamic analysis.

Table 9. Cable results at Node 3 (top conductor) and Node 4 (wire shield)

EQ-S

Non-linear Dynamic

Parameter �̅�𝑒 𝜎𝑒

Conductor at Node 3

Tension (kN) 60,11 66,305 2,626

dT (m) 15,89 18,033 0,710

Wire shield at Node 4

Tension (kN) 32,24 34,826 1,032

dT (m) 16,85 19,102 0,729

Note: EQ-S, equivalent static method, dT -transverse displacement

4.4.2 Tower tip displacement (Node N3)

A summary of the tower tip displacement at nodes 1,2,3 and 4 is given in Table 10. In the x,z direction the results from the static

equivalent method agree quite well with the non-linear dynamics analysis, and almost all value are within the limits �̅�𝑒 ±-𝜎𝑒. For

y direction (tower longitudinal direction), the existence of displacements occur as consequence of spatial variability of turbulent

wind load, causing torsion in the lattice tower.

Table 10. Maximum displacement of the 6 samples, for nodes N1, N2, N3 and N4

EQ-S

Non-linear Dynamic

Node Direction �̅�𝑒 𝜎𝑒

N1 (m)

x 0,20 0,208 0,010

y 0,00 0,146 0,025

z 0,094 0,099 0,005

N2 (m)

x 0,313 0,328 0,016

y 0,00 0,218 0,035

z 0,125 0,132 0,006

N3 (m)

x 0,444 0,466 0,024

y 0,00 0,295 0,047

z 0,112 0,118 0,006

N4 (m)

x 0,512 0,537 0,027

y 0,00 0,334 0,053

z 0,076 0,080 0,004

4.4.3 Tower reaction

The tower reactions, for the non-linear dynamic analysis, gives values clearly superior to the static-equivalent method, Table 11.

The bending moments exhibits a significant difference between both methods, with the standard deviation values revealing a great

variation. For the longitudinal and transverse reaction, the differences between both methods are small. The longitudinal response

that only exist for the non-linear dynamic analysis, is a consequence of the turbulent wind time-series, which contain spatial

variability along the transmission line system.

14


Table 11. Tower reactions for the EQ-S and the non-linear dynamic

EQ-S

Non-linear Dynamic

Reaction �̅�𝑒 𝜎𝑒

RL (kN) 0,00 9,261 8,317

RT (kN) 206,33 257,187 10,985

ML (kN.m) 0,00 364,569 351,086

MT (kN.m) 7370,03 10122,840 462,820

Note: L-Longitudinal and T-Transverse direction, bending moments at the center of the tower base

5 CONCLUSION

The present study focus on the comparison of the structural response of a transmission-line system between the equivalent static

method as defined in Eurocode 3-3-1 and a non-linear dynamic analysis.

The difference between the static equivalent method and non-linear dynamic for the cable response, specifically for the cable

tension, are relatively small. This highlights the fact the overall wind loading magnitude along the cable spans for the static and

dynamic analysis are almost the identical. The same trend as encountered for the tower tip displacements, with similar results for

the static-equivalent method and non-linear dynamic analysis. In the case of the tower reactions, it can be observed that the static

method result in lower responses that the non-linear dynamic case. Therefore, for the case under study concluded that the dynamic

effects are not entirely taken into account by the static equivalent method defined in the Eurocode 1-4, the main difference between

both methods occur for the tower reactions.

ACKNOWLEDGEMENTS

This work was co-participated by funds from the project “VHSSPOLES-Very High Strength Steel Poles” (Faculty of

Engineering of the University of Porto, reference 21518) sponsored by the European Fund for Regional Development (FEDER)

through COMPETE (Operational Program Competitiveness Factors - POFC). The Authors acknowledge the financial support and

the opportunity to contribute to the development of the transmission towers testing site of Metalogalva (Trofa, Portugal).

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