Dynamic behaviour of pre-stressed concrete transmission poles under synoptic wind loading

Abstract

A numerical model capable of predicting the dynamic response of pre-stressed concrete transmission poles under both the mean and fluctuating components of synoptic wind loads is developed in this study. A full non-linear dynamic analysis is conducted under a time history variation of wind velocity. The peak total responses, such as conductors’ reactions and poles’ base moments, are determined from this analysis. The same analysis is repeated in a quasi-static manner. Dynamic amplification factors (DAF), defined as the ratio between the maximum response based on a non-linear dynamic analysis and the corresponding value based on a quasi-static analysis, are calculated for the poles and the conductors to quantify the dynamic impact of synoptic wind loads. This factor is used to assess the importance of including the resonant component while estimating the response of the transmission poles. In addition, gust response factors (GRF) defined as the ratio between the peak and mean responses are evaluated and compared to GRF recommended by ASCE-74 (2010). A parametric study is conducted on three pre-stressed concrete transmission line systems. The mean value of the incoming wind speed is the main variable included in the parametric study. It is found that the resonant effect is dominant in the conductors at low wind speeds and the poles exhibit high dynamic response at higher wind speeds.

Keywords

pre-stressed concrete poles synoptic wind turbulent wind dynamic analysis quasi-static analysis

Introduction

Transmission line structures are critical elements of the modern infrastructure system as they carry electricity from the generation stations to the distribution systems. Those structures should be designed to resist different types of environmental loads including wind effects. For synoptic wind, the wind velocity field consists of a mean component, which is stationary over a period typically taken between 10 min and 1 h, and a fluctuating component (turbulent), which varies with time. The turbulent component can trigger a dynamic response of the structure, which can lead to failure from the gust effect. Many literature studies have reported failures of transmission line structures under dynamic wind loads such as the studies conducted by Li (2000), Savory et al. (2001) and Li and Bai (2006).

The main components of a transmission line system are the conductors, insulators, ground wires, and the supporting towers. The supporting towers can be in the form of lattice steel towers or pole-type structures. Most of the previous studies which investigated the dynamic behavior of transmission line systems under wind loads focused on conductors and on lattice steel towers as the main supporting system (Momomura et al., 1997; Horr et al., 2004). Loredo-Souza and Davenport (1998) concluded that the aerodynamic damping of the conductors has a major effect on the dynamic behavior of transmission line structures. The contribution of the resonant component to the response of self-supported and guyed transmission lattice structures was found relatively low in the study conducted by Aboshosha et al. (2016). That was attributed to the difference between the loading and the tower frequencies and the decoupling of the mode shapes of both the lattice steel transmission towers and the conductors. Hamada et al. (2017) investigated the structural response of a multi-span transmission line system using wind tunnel testing of four aero-elastic guyed towers and conductors. The study concluded that the transmission line system behaves in a quasi-static manner under various wind speeds and the resonant component is less significant especially under higher wind speeds. Ibrahim et al. (2019) concluded that the resonant component of the longitudinal force applied to transmission towers due to downburst loading is not significant. In addition, Li et al. (2018) studied the dynamic behavior of a transmission line system due to the insulator breakage caused by excessive ice loads. It was concluded that the vibrations of the system increase with the increase of ice loads leading to the cascading collapse of the transmission line system. Fu et al. (2020) conducted non-linear dynamic analysis to assess the fragility of a transmission tower under combined wind and ice loads. The results showed that the wind angle of attack and the length of the conductor spans have a major effect on the fragility of the tower.

Regarding pole structures, few studies were performed to assess their vulnerability to dynamic loads. Chen et al. (2006), Lantrip (1995) and Polyzois et al. (1998) conducted a number of studies to identify the free-vibration modes of transmission poles. Dai and Chen (2008) studied the effect of the pre-stress level on the modal behavior of pre-stressed concrete poles. Chen and Dai (2010) concluded that strong coupling exists between the poles and the conductors’ vibrations. The reduced redundancy of pole-type structures makes them more vulnerable to dynamic excitations (Chen and Dai 2010).

The studies conducted on the pole-type structures were mainly focusing on the free-vibration dynamic properties of the poles. However, there is a lack in the literature in studying the forced vibrations of pole-type structures. Dai (2009) performed a time history analysis to assess transmission pole structures’ response under blast loading. To the best of the authors’ knowledge, the current study is the first study to assess the dynamic performance of transmission poles under both mean and fluctuating synoptic wind loads. Among different types of pole transmission line structures, pre-stressed concrete poles are wide spread compared to other types such as wooden and steel poles. This is due to their low installation and maintenance costs in addition to the corrosion resistivity. Hence, pre-stressed concrete pole systems are considered in this study.

The complexity of performing dynamic analysis of transmission line structures arises from the fact that the stiffness properties of those structures are frequency-dependent. This is attributed to the fact that the conductor’s stiffness depends on the tension forces arising from the wind loading. This might lead to coupling between the response of conductors and supporting towers (Simiu and Scanlan 1996; Madugula 2002; IEC 2003; Chen and Dai 2010).

The main objectives of the current study are to assess the non-linear dynamic response of pre-stressed concrete transmission pole structures under mean and turbulent synoptic wind components using a numerical tool developed and validated in this study. The developed tool is then used to identify the range of velocities at which the resonant component has to be considered for a number of pre-stressed concrete transmission pole systems having different conductors’ spans. In addition, the gust response factors of the pole systems are evaluated and compared to the gust response factors obtained using the expressions incorporated in the ASCE74 (2010).

The study is divided into six sections. In section Numerical model, a description of the numerical model developed and validated is outlined. Section Description of the considered pre-stressed concrete pole systems provides details about the three different pre-stressed concrete transmission pole systems considered in the study. A sample of the results of the dynamic and quasi-static analyses of the three different systems is presented in section Results of analyses of the considered TL systems under turbulent wind. The variation of dynamic amplification factors and gust response factors with mean wind speeds is obtained from the analyses. The gust response factors are then compared to the ones evaluated using ASCE74 (2010) provisions in section DAF and GRF variations with wind speeds. The findings and conclusions obtained from the study are presented in section Conclusion.

Numerical model

Aboshosha and El Damatty (2015) conducted nonlinear dynamic analysis of multi-spanned conductors using an in-house numerical model. This model was extended to be capable of performing both quasi-static and dynamic analyses for steel lattice transmission line structures under turbulent wind loading by Aboshosha et al. (2016). This model was validated using the results of the multi-span guyed transmission line aero-elastic wind tunnel test conducted at the Boundary Layer Wind Tunnel Laboratory (BLWTL) at University of Western Ontario by Hamada et al. (2017). A comparison between the numerical model and the test results showed a very good agreement as reported by Aboshosha et al. (2016). In the current study, Aboshosha et al. (2016) model is extended further to account for the dynamic behavior of pre-stressed concrete poles under turbulent wind loading. This is done by incorporating the following together: (1) a procedure to generate turbulent wind field developed by Chen and Letchford (2004a), Chen and Letchford (2004b) and Chay et al. (2006), (2) the non-linear model for the conductors previously developed and validated by Aboshosha and El Damatty (2015) and (3) a non-linear finite element model for self-supported and guyed pre-stressed concrete poles previously developed and validated by Ibrahim et al. (2017) and Ibrahim and El Damatty (2019).

The incorporation of the three developed and validated tools forms together a unique package capable of predicting the dynamic and quasi-static responses of pre-stressed concrete transmission pole structures under the mean and fluctuating components of a wind field.

A brief description of the procedure used to generate the turbulence is provided in the next subsection. This is followed by a description of the various steps involved in the dynamic analysis. In each step, the related numerical details are explained so the reader can gain an understanding about the features and capabilities of the entire numerical model.

Turbulent wind field generation

Synoptic winds are decomposed into mean and fluctuating velocity components. Chen and Letchford (2004a), Chen and Letchford (2004b) and Chay et al. (2006) developed a numerical technique to generate the fluctuating wind velocities. In this technique, the Power Spectral Density (PSD), which describes the energy of the wind fluctuations in the frequency domain, developed by Von Karman (1948), was used to evaluate turbulent velocities. This technique is adopted in the current study. The length scale is considered equal to 67 m according to the ASCE 74 (2010) assuming an open terrain exposure. A sample of the variation of the produced velocity with time at a height of 20 m for a mean wind speed of 15 m/sec is shown in Figure 1.

Figure 1.

Generated turbulent velocity for open terrain exposure at 20 m height.

Steps for performing dynamic and quasi-static analyses

The response of pre-stressed concrete transmission poles under mean and fluctuating synoptic wind is highly non-linear. This is attributed to the non-linearity of the conductors due to sagging, pre-tensioning, and insulators’ stiffness. In addition to that, the pre-stressed concrete poles’ behavior is non-linear due to stress–strain non-linear relationship of concrete and pre-stressing strands, cracking and long-term effects such as creep, shrinkage, and strands’ relaxation.

The wind forces on the pre-stressed concrete transmission poles can be calculated using ASCE 74 (2010) based on the following equation

F_{w i} = \frac{1}{2} ρ_{a} G R F C_{f} A {(Z_{v} V_{i})}^{2}

(1)

where F_wi is the force developing in the i direction, ρ_a is the density of air = 1.225 (Kg/m³), GRF is the gust factor, C_f is the drag force coefficient, A is the nodal projected area perpendicular to i direction, and Z_v is the terrain factor and V_i is the mean or turbulent wind velocity in the i direction (units m/sec). For conductors and circular concrete poles, the value of the drag coefficient is taken equal to 1.0 according to ASCE 74 (2010) guidelines, and the same value is recommended for gust and terrain factors.

The dynamic behavior of transmission line structures subjected to fluctuating synoptic wind can be evaluated using full non-linear dynamic analysis under the instantaneous value of the wind velocity which includes both the mean and the fluctuating components. This method is very time consuming and hence not practical. Sparling and Wegner (2007) developed a technique which significantly reduces the computational time without compromising the accuracy of the solution.

Sparling and Wegner (2007) technique was followed by Aboshosha and El Damatty (2015) in analyzing various transmission line conductors subjected to synoptic and non-synoptic wind. A very good matching was obtained for the conductor responses with the corresponding values obtained from fully non-linear dynamic analyses. As such, Sparling and Wegner (2007) technique is incorporated in the current study. The steps and the details of the technique are explained below:

Step 1: Non-linear static analysis under the mean loads

Non-linear static analysis is conducted to obtain the transmission pole systems’ response under the mean wind component (M). The non-linear behavior of a pre-stressed concrete pole system is evaluated using two separate analyses for the conductors and the pre-stressed concrete poles which are outlined below:

Modeling of conductors

Non-linear static analysis of the conductors is conducted under the synoptic wind mean component using the semi-analytical method developed by Aboshosha and El Damatty (2014), Aboshosha and El Damatty (2015). This technique accounts for conductors’ sagging, pre-tensioning forces, and insulator’s stiffness. The conductors’ reactions and displacements are calculated using this technique.

Modeling of pre-stressed concrete poles

The non-linear static behavior of the poles is investigated using the non-linear finite element model developed and validated by Ibrahim et al. (2017). Frame elements are used to model the pre-stressed concrete poles which are subjected to axial forces and bending moments. In this technique, the effects of non-linear stress–strain relationship of concrete and pre-stressing strands as well as creep, shrinkage, and relaxation of strands are taken into consideration. The concrete poles are analyzed under mean wind component in addition to the conductor reactions previously calculated in Modeling of conductors.

Step 2: linear dynamic analysis under the fluctuating loads

Linear dynamic analysis of the system is conducted under the fluctuating wind component to obtain the fluctuating response (F). The pole system properties used in the linear dynamic analysis such as stiffness (K), aerodynamic, and structural damping (C) are calculated based on the non-linear analysis conducted in step 1. The fluctuating response (F) obtained in this step is equal to the summation of the resonant (R) and the background (B) responses. It should be noted that (B) represents the response of the pole system to the fluctuating wind component using quasi-static analysis (i.e., without including the dynamic effect) as shown later in Equation 6.

The linear dynamic analysis of a pre-stressed concrete pole system is conducted using step by step Newmark’s integration method (Bathe, 1996). The equation of motion of the transmission pole system is expressed by Equation 2

[M] {\ddot{u}} + [C] {\dot{u}} + [K] {u} = {F_{w} (t)}

(2)

where

[M]

is the mass matrix,

[C]

is the damping matrix, [K]is the stiffness matrix, and

{F_{w} (t)}

is the fluctuating dynamic wind load vector at time t,{u},

{\dot{u}}

, and

{\ddot{u}}

are the displacement, velocity and acceleration responses of the system, respectively.

The stiffness matrix of a transmission pole system [K] is formed by linearly adding the stiffness matrices of the pole and the conductors’ elements. The Poles are discretized into 20 frame elements. Each element has two nodes with 6 degrees of freedom per node (i.e., 3 displacements and 3 rotations), while each conductor span is modeled using 10 two nodded-cable elements.

The damping matrix $[C]$ results from the aerodynamic damping as well as the structural damping of the transmission pole systems. Bachmann et al. (1995) stated that the structural damping of the conductors can be ignored compared to the aerodynamic damping. As such, the damping matrix[C] can be expressed by Equation 3

[C] = [\begin{matrix} C_{c a} & 0 \\ 0 & C_{p a} + C_{p s} \end{matrix}]

(3)

where C_ca and C_pa are the aerodynamic damping matrices for the conductors and the poles and C_ps is the structural damping of the pole.

The aerodynamic damping matrices of the conductors and the transmission poles, C_ca and, C_pa are diagonal matrices that have damping coefficients, C_ai, at locations of the degrees of freedom corresponding to the wind direction. The damping coefficients at the poles and the conductors’ nodes are expressed by Equation 4

C_{a i} = ρ C_{d i} A_{i} V_{i}

(4)

where ρ is the air density which is taken equal to 1.225 kg/m³, C_di is the drag coefficient at node i, A_i is the projected area of the conductor or the pole elements around node i, and V_i is the applied mean wind speed.

The structural damping of the transmission poles C_ps is modeled using Raleigh damping as shown in Equation 5

[C_{p s}] = α [M_{p}] + β [K_{p}]

(5)

where M_p and K_p are the mass and stiffness matrices of the pole,

α

and

β

are constants controlling the structural damping. In this study,

α

and

β

are chosen to obtain an un-cracked pole damping that is equal to 2% and a cracked pole damping of 5% for the first two pole vibration modes according to Loredo-Souza and Davenport (2003) and Newmark and Hall (1982).

It should be mentioned that the pre-stressed concrete pole systems considered in this study are designed to remain un-cracked under a synoptic wind speed of 40 m/sec according to ASCE 123 (2012) and ASCE 74 (2010). Hence, the structural damping ratio used for the poles under wind speeds less than or equal to the cracking wind speed (i.e., 40 m/sec) is assumed to be 2%. However, the damping ratio is assumed to be 5% under higher wind speeds.

By conducting linear dynamic analysis on the transmission pole system, the fluctuating response of the system (F) can be obtained. This response includes the background component (B) and the resonant component (R).

Step 3: linear quasi-static analysis under the fluctuating loads

A quasi-static analysis of the transmission pole system under the fluctuating wind component is conducted. Similar to step 2, the stiffness used in the quasi-static analysis is calculated based on the non-linear analysis performed in step 1. The main purpose of conducting the quasi-static analysis is to obtain the background response of the system (B). The background response is obtained throughout solving Equation 6

[K] {u} = {F_{w} (t)}

(6)

Step 4: Obtaining R, T, and QS

The background response (B) obtained from step 3 is subtracted from the fluctuating response (F) obtained from step 2 so that the resonant response (R) can be evaluated. As such, the total dynamic response (T) (i.e., including the dynamic effect) can be determined by adding the mean (M), background (B), and resonant (R) responses. However, the quasi-static response (QS) (i.e., neglecting the dynamic effect) is obtained by adding the mean (M) response to the background one (B). The steps are summarized and illustrated in the flowchart in Figure 2.

Figure 2.

Flow chart for the dynamic analysis of pre-stressed concrete pole systems.

Description of the considered pre-stressed concrete pole systems

The numerical model is employed to evaluate the dynamic behavior of three different pre-stressed concrete poles. The total dynamic responses are computed as well as the quasi-static responses so that the resonant component effect can be assessed. Dynamic amplification factors (DAF) and gust response factors (GRF) are then evaluated for various wind speeds. The DAF is defined as the ratio between the peak total dynamic and the peak quasi-static responses while the GRF is defined as the ratio between the peak and the mean response. The calculated GRFs are compared to the corresponding values evaluated using Davenport’s expressions incorporated in the ASCE74 (2010).

Three transmission pole systems are considered in this study. The common geometric and material properties of the three systems are shown in Table 1.

Table 1.

Common properties of the considered pole systems.

Pre-stressed concrete pole properties
Unsupported height (m)	Concrete compressive strength		Reinforcement
25.4	75.8		M10 low relaxation pre-stressing strands
Conductors properties
Number of conductors at each pole	Projected area (m²)	Weight per unit length (N/m)	Insulator length (m)	Sag value relative to span (%)	Cross arm length (m)
2	0.04	30	2.1	2	2.4

The three systems differ in terms of conductor spans, number of strands and the dimensions of the bottom and top cross sections of the poles. The purpose of considering those three different systems is to investigate the effect of varying the conductor spans on the values of DAF and GRF. The specific properties of the three systems are given in Table 2, where, D_outtop, D_outbottom, D_intop, and D_inbottom are the outside top diameter, outside bottom diameter, inside top diameter, and inside bottom diameter of the concrete poles, respectively. N_strands are the number of low relaxation pre-stressing strands assumed in the poles. The cross-section properties of the poles are selected such that the poles of the three systems remain un-cracked under a synoptic wind speed of 40 m/sec based on the ASCE 123 (2012) and ASCE 74 (2010) guidelines.

Table 2.

Properties of the considered pole systems.

Conductors		Pre-stressed concrete pole
Span(m)	Sag(m)	D_outtop (mm)	D_intop (mm)	D_outbottom (mm)	D_inbottom (mm)	N_strands
100	2	280	140	740	547	20
200	4	596	196	1034	586	20
300	6	1096	396	1558	807	24

Results of analyses of the considered TL systems under turbulent wind

A parametric study is conducted on three pre-stressed concrete pole systems by varying the applied mean wind speed from 10 to 60 m/sec with an increment of 5 m/sec. As mentioned before, the stiffness of a transmission pole system is frequency dependent. By changing the value of the mean wind load applied on the pole system, the frequency of the whole system changes leading to a different response. The numerical model described earlier is employed to analyze the three transmission pre-stressed concrete pole systems.

Samples of the results of the parametric study are provided in this section. Firstly, the numerical model is employed to quantify the variation of mean, background, resonant, total dynamic, and quasi-static responses with time for a sample mean wind speed of 40 m/sec applied to the 100 m pole system. In subsection Identifying the 200 m pole system frequencies, the power spectral density (PSD) responses of the conductors’ reactions and the pole base moments are evaluated for the 200 m pole system under two mean wind speeds of 15 and 50 m/sec, respectively. As such, the frequencies of the transmission pole system under those two different mean wind speeds can be identified. In subsection Variation of the peak responses with wind speed for the 300 m pole, the variation of two different peak responses (conductor reaction and base moment) with the applied mean wind speed is presented for the 300 m pole system.

Variation of the mean, resonant, and background responses of the 100 m pole system with time

According to NRC (2005) and AIJ (2004), the turbulence statistics are typically stable over the range of 600–3600 s. As such, the duration time used in the dynamic analysis is taken equal to 600 s. Figures 3(a and b) show a sample of the predicted conductor and pole base moment responses, respectively. The shown responses are for a transmission pole system with a conductor span of 100 m subjected to a mean wind speed of 40 m/s. As shown in the figure, mean, background, and resonant responses are obtained. By summing all of those responses the total response which includes the dynamic effect is evaluated. Summation of the mean and background responses leads to obtaining the quasi-static response (i.e., neglecting the dynamic effect) of the pole system.

Figure 3.

(a and b) Conductor reaction and pole base moment for the 100 m pole system under.

It should be noted that the peak total dynamic response is higher than the peak quasi-static response due to the inclusion of the dynamic effect. The DAF for the conductor reaction and the pole base moment is found to be equal to 1.07 and 1.14, respectively. This indicates that the resonant component is more significant in the pole’s base moment than that in the conductor’s reaction under a mean wind speed of 40 m/sec.

Identifying the 200 m pole system frequencies

As mentioned earlier, the transmission pole systems’ frequencies (f) are dependent on the value of the mean wind load. In this subsection, the frequencies of the 200 m pole system are identified under two different wind speeds of 15 m/sec and 50 m/sec, respectively. The transmission pole system frequency and the conductor frequency are evaluated using the power spectral density (PSD) analysis. PSD curves are obtained for both the dynamic and quasi-static pole base moment and the conductor reaction responses.

Figure 4 and Figure 5 show PSD curves of the above quantities for wind speeds of 15 m/sec and 50 m/sec, respectively.

Figure 4.

(a and b) PSD curves for the 200 m pre-stressed concrete pole under V_mean=15 m/s.

Figure 5.

(a and b) PSD curves for the 200 m pre-stressed concrete pole under V_mean=50 m/s.

It should be mentioned that the frequencies of the conductors and transmission pole systems can be identified at the values corresponding to the peaks of the PSD curves using the following steps:

- Peaks that appear in the dynamic response and do not appear in the quasi-steady response reflect natural frequencies. Those natural frequencies can be for the conductor or the poles.

- Peaks associated with conductor natural frequencies appear in both conductor and pole responses; however, it has stronger and more pronounced contribution in the case of conductor response (Figure 4a).

- Peaks associated with pole response appear mainly in the pole response and not (or minimally) in the conductor response (Figure 4b and Figure (5b))

As such, the conductors’ and the pole system’s frequencies under a mean wind speed of 15 m/sec are 0.22 Hz and 0.95 Hz, respectively. Although the conductors’ and the transmission pole system’s frequencies under a mean wind speed of 50 m/sec are 0.42 Hz and 0.32 Hz, respectively.

By analyzing the PSD curve peaks under the mean wind speed of 15 m/sec, the dynamic effect is found to be not significant (i.e., area under the peaks in the PSD curve is less than 10% of the total area) due to the large difference between the conductor and pole system frequencies (i.e., 0.22 Hz and 0.95 Hz).

It should be noted that a clear reduction in the pole frequency occurs by increasing the mean wind speed (from 0.95 under 15 m/sec to 0.32 Hz under 50 m/sec). This is attributed to the fact that the considered pre-stressed concrete poles are designed to remain un-cracked under a mean wind speed of 40 m/sec. As such, any increase beyond the design mean wind speed will lead to the cracking of the pole and consequently a decrease in the pole system stiffness and its frequency. By experiencing more dynamic vibrations, the pole system’s frequency becomes closer to the conductor’s frequency (0.32 Hz and 0.42 Hz) and both fall in the range of strong wind excitation (typically less than 1 Hz), which magnifies the resonant component. Hence, the pole structure becomes more vulnerable to wind excitations.

Variation of the peak responses with wind speed for the 300 m pole

Peak total and quasi-static responses are evaluated for the 300 m pole system. Figure 6 shows the variation of the peak responses with the change in the mean velocity. Two responses are selected for comparison purpose which represent: (i) conductor reaction and (ii) pole base moment. Figure 6 indicates that the peak values of the responses increase with the increase of the velocity. The figure also indicates that the pole generally exhibits larger differences in the ratios between the peak total and quasi-static responses than the conductors for wind speeds greater than 40 m/sec. This is due to the cracking of the pole under mean wind speeds higher than 40 m/sec which leads to more dynamic excitations.

Figure 6.

(a and b) Peak responses for the conductors and the pre-stressed concrete pole.

Comparison is made for the peak conductor reaction and pole base moment between the 300 m span system and the 100 m span system at 40 m/sec. It is found that the dynamic analysis predicts a peak conductor reaction and a pole base moment for the 300 m span system that are approximately 3 times the corresponding values for the 100 m span system.

DAF and GRF variations with wind speeds

In this section, the variation of the dynamic amplification factor (DAF) and gust response factor (GRF) with different mean wind speeds for the three different pole systems is presented. GRF values are then compared with the values recommended by ASCE 74 (2010).

DAF variation for different responses with wind speed

The dynamic amplification factor (DAF) is defined as the ratio between the peak total response and the peak quasi-static response as expressed in Equation 7

D A F = \frac{{\hat{R}}_{T}}{{\hat{R}}_{Q S}}

(7)

where

{\hat{R}}_{T}

is the peak total response and R_QS is the peak quasi-static response.

The variations in the DAF with the mean wind velocity for the conductor reaction and the pole base moment responses is shown in Figures 7(a–c) for the 100, 200, and 300 m pole systems, respectively.

Figure 7.

(a–c) DAF for the three pre-stressed concrete pole systems.

By investigating the values and the trend of the DAF, the following findings are obtained:

• The DAF of the conductor reactions has a decreasing trend with the increase of the mean wind speed. This trend results from the increase of the aerodynamic damping with velocity which attenuates the background and resonant components as presented in Equation 2 and Equation 4.

• The DAF of the pole base moments is almost constant for mean wind speeds between 10 to 40 m/sec. The poles are considered un-cracked under those mean wind speeds. In this speed range, the maximum DAF is 1.13, 1.12, and 1.14 for the 100, 200, and 300 m pole systems, respectively.

• The DAF of the pole base moments increases at mean velocities above 40 m/sec as the poles crack and become more vulnerable to wind turbulent excitations. This results in coupled pole-conductor mode shapes, which is the main reason for the increase of the resonant component. The damping ratio for the cracked pole systems is assumed to be 5% according to Newmark and Hall (1982). This damping ratio limits the maximum values of the DAF to be 1.24, 1.25, and 1.27 for the 100, 200, and 300 m pole systems, respectively. After reaching the peak value, the frequencies of the pole system and conductors become different and the DAF decreases. For the three systems, the DAF variation with mean wind speed is found to be similar. As such, the dynamic effect has to be considered while analyzing the pre-stressed concrete pole systems under a normal wind speed higher than the design wind speed (i.e., 40 m/sec) regardless of the conductor spans.

GRF variation for different responses with wind speed

The gust response factors are evaluated for the poles’ base moment as the ratio between the peak total response, R_T and the mean response, M. This gust response factor is referred to as GRF_T and is plotted in Figures 8(a–c) for the 100, 200, and 300 m pole systems, respectively. Another gust response factor is also evaluated which represents the ratio between the peak quasi-static response, R_QS and the mean response, M. It is referred to as GRF_QS and is plotted in the same figure.

Figure 8.

(a–c) GRF for the base moment for the three pre-stressed concrete pole systems.

It can be noted from Figure 8 that the values of GRF_T are usually higher than the values of GRF_QS due to the inclusion of the resonant component.

Gust response factors are also evaluated using Davenport’s (1979) equations incorporated in the ASCE 74 (2010). These equations account for the mean, background, and resonant components. Gust response factors resulting from Davenport’s procedure is referred to as GRF_T-ASCE. In addition, the ASCE 74 (2010) uses another method by incorporating Davenport’s expressions in a simplified way by neglecting the contribution of the resonant component, which means it includes only the quasi-static response. The resulting gust response factor from this method is referred to as GRF_QS-ASCE. Both GRF_T-ASCE and GRF_Qs-ASCE are evaluated and are also plotted in Figure 8 for comparison purpose with GRF_T and GRF_Qs, respectively. Pole base moments are chosen for comparing the gust response factors for the three considered pole systems.

By comparing the gust response factors obtained from the ASCE 74 (2010) with the gust response factors obtained from the dynamic analyses, the following findings can be stated:

• In most of the cases, the gust response factor obtained from the ASCE that includes the dynamic effect (Davenport 1979), GRF_T-ASCE, is higher than the gust response factor obtained from the dynamic analyses, GRF_T. This leads to the conclusion that the ASCE 74 (2010) overestimates the dynamic response.

• In some of the cases, GRF_QS-ASCE is found to be higher than the corresponding GRF_QS values (over conservative). Although in other cases, GRF_QS are found close to or slightly higher than the obtained GRF_QS-ASCE values.

• By comparing GRF_T to GRF_QS-ASCE, it is found that GRF_T is consistently higher than GRF_QS-ASCE. This comparison can be used to assess the importance of including the dynamic effects. The larger the discrepancy between GRF_T and GRF_QS-ASCE, the more important is the dynamic effect. This discrepancy is found to be larger at the higher wind speeds when the poles crack and coupled pole-conductor modes exist. This implies that it is important to consider the dynamic effect when analyzing the poles under wind speeds that exceed the cracking wind speed (40 m/sec).

Conclusion

A numerical model capable of conducting dynamic analysis of pre-stressed concrete transmission lines is developed. The numerical model is employed to evaluate the dynamic behavior of three pre-stressed concrete pole systems having different spans. Peak total responses (i.e., including the dynamic effect) and peak quasi-static responses (i.e., neglecting the dynamic effect) are evaluated. Dynamic amplification factor (DAF), defined as the ratio between peak total responses to the peak quasi-static responses, is evaluated. Total gust response factor, GRF_T, defined as the ratio between peak total response to the mean response is evaluated using the results from the dynamic analyses and is compared with the quasi-static gust response factor, GRF_QS, which is defined as the ratio between peak quasi-static response to the mean response. Total and quasi-static gust response factors, GRF_T-ASCE and GRF_QS-ASCE, based on Davenport’s equations included in the ASCE 74 (2010) are also evaluated. The following conclusions are obtained:

• Conductor reactions exhibit larger dynamic amplification factor (DAF) than the pole base moments at wind speeds approximately below 40 m/sec. This trend results from the low conductor aerodynamic damping at lower wind speeds.

• DAF of the poles has a constant trend under mean wind speeds up to the cracking wind speed (40 m/sec). Up to this wind speed, the maximum DAF do not exceed 1.13, 1.12, and 1.14 for the 100, 200, and 300 m pole systems, respectively. This is mainly due to the significant difference between the conductors’ and poles’ frequencies under mean wind speeds that are less than the cracking wind speed. This difference in frequencies exists for all the considered conductors’ spans.

• DAF of the pole base moments exhibits higher values when the mean wind speeds exceed 40 m/sec which is the design wind speed. This is attributed to the fact that the poles crack when the mean wind speed is higher than 40 m/sec. The poles’ cracking increases the vibrations of the whole transmission line system. This results in increasing the resonant component and hence increasing the dynamic effect. For mean wind speeds greater than the cracking wind speed, the maximum values of the DAF are 1.24, 1.25, and 1.27 for the 100, 200, and 300 m pole systems, respectively. As such, regardless of the conductor’s span, the dynamic effect becomes more important after cracking happens in the poles.

• GRF_T-ASCE, is over conservative and is usually higher than the gust response factor obtained from the dynamic analyses, GRF_T.

• GRF_QS-ASCE is found to be considerably less than GRF_T for the three pole systems especially for the wind speeds above the cracking wind speed.

Based on the above findings, it is concluded that dynamic effect has to be accounted while evaluating conductor peak responses especially for low velocity magnitudes. It is also concluded that dynamic effect has to be considered for the cracked pole systems that exhibit coupled conductor-pole modes as the DAF reached 1.24, 1.25, and 1.27 for the 100, 200, and 300 m pole systems, respectively.

Footnotes

Acknowledgments

The authors would like to acknowledge CEATI International Inc., Montreal, Quebec, Canada for funding this research.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ahmed M Ibrahim

References

Aboshosha

El Damatty

A.A.

, (2014), “Effective technique for the reactions of transmission line conductors under high intensity winds”, Wind and Structures. 18(3), 235–252. 10.12989/was.2014.18.3.235

Aboshosha

El Damatty

(2015) Dynamic Effect for Transmission Line Conductors under Downburst and Synoptic Winds. Wind and Structures 21(2): 241–272. 10.12989/was.2015.21.2.241

Aboshosha

Ibrahim

A.M.

El Damatty

A.A.

, et al. (2016) Dynamic Behaviour of Transmission Line Structures under Synoptic Wind Loads”. Montreal, QC, Canada: Proceeding of CIGRE-IEC colloquim.

AIJ (2004) RLB Recommendations for Loads on Buildings. Tokyo: Structural Standards Committee.

American Society of Civil Engineers (ASCE) (2010) Guidelines for Electrical Transmission Line Structural Loading. In: ASCE manuals and reports on engineering practice. New York, NY, USA, No. 74.

American Society of Civil Engineers (ASCE) (2012) “Prestressed Concrete Transmission Pole Structures”, ASCE manuals and reports on engineering practice. Reston, VA, USA, No. 123.

Bachmann

Ammann

W.J.

Deischl

, et al. (1995) Vibration Problems in Structures. Practical guidelines. Berlin: Birkhauser.

Bathe

K.J.

(1996) Finite Element Procedures in Engineering Analysis. Englewood Cliffs, New Jersey: Prentice-Hall.

Chay

M. T.

Albermani

Wilson

(2006) Numerical and analytical simulation of downburst wind loads. [Eye Science Electronic Resource] 28(2): 240–254. 10.1016/j.engstruct.2005.07.007

10.

Chen

Letchford

C.W.

(2004a) A deterministic–stochastic hybrid model of downbursts and its impact on a cantilevered structure. [Eye Science Electronic Resource] 26(5): 619–629. 10.1016/j.engstruct.2003.12.009

11.

Chen

Letchford

C.W.

(2004b) Parametric study on the along-wind response of the CAARC building to downbursts in the time domain. Journal of Wind Engineering and Industrial Aerodynamics 92(9): 703–724. 10.1016/j.jweia.2004.03.001

12.

Chen

S.E.

Ong

C. K.

Antonsson

(2006) Modal Behaviors of Spun-Cast Pre-Stressed Concrete Pole Structures. 24th Conference and Exposition on Structural Dynamics IMAC-XXIV.

13.

Chen

S.E.

Dai

(2010) Modal Characteristics of Two Operating Power Transmission Poles. Shock and Vibration 17(4–5): 551–561. 10.3233/SAV-2010-0547

14.

Dai

(2009) Dynamic Performance of Transmission Pole Stuructures under Blasting Induced Ground Vibration” A Dissertation Submitted to the. USA: faculty of The University of North Carolina at Charlotte, Charlotte.

15.

Dai

Chen

S.E.

(2008) Vibration of Spun-Cast Prestressed Concrete Poles. Orlando, Florida, USA: Proceedings IMAC – XXV.

16.

Davenport

A.G.

(1979) Proceedings, 5th International Conference on Wind Engineering. Fort Collins, Colorado: Pergamon Press, pp. 899–909.Gust response factors for transmission line loading.

17.

H.-N.

, et al. (2020) Fragility analysis of a transmission tower under combined wind and rain loads. Journal of Wind Engineering and Industrial Aerodynamics 199: 104098. 10.1016/j.jweia.2020.104098

18.

Hamada

King

J.P.C.

El Damatty

A.A.

, et al. (2017) The response of a guyed transmission line system to boundary layer wind. (Journal of Environmental Sciences China) 139: 135–152. 10.1016/j.engstruct.2017.01.047

19.

Horr

A.M.

Yibulayin

Disney

(2004) Nonlinear spectral dynamic analysis of guyed towers. Part II: Manitoba towers case study. Canada Journal of Civil Engineering 31(6): 1061–1076. 10.1139/l04-084

20.

Ibrahim

A.M.

El Damatty

A. A.

Elansary

A.M.

(2017) Finite element modelling of pre-stressed concrete poles under downbursts and tornadoes. (Journal of Environmental Sciences China) 153: 370–382. 10.1016/j.engstruct.2017.10.047

21.

Ibrahim

A.M.

El Damatty

A.A.

(2019) Behaviour and design of Guyed Pre-stressed Concrete Poles under Downbursts. Wind and Structures Journal 29(5): 339–359. 10.12989/was.2019.29.5.339

22.

Ibrahim

El Damatty

A. A.

Elawady

(2019) The Dynamic Effect of Downburst Winds on the Longitudinal Forces Applied to Transmission Towers. Frontiers in Built Environment Journal, ISSN: 2297–3362. 10.3389/fbuil.2019.00059

23.

International Electrotechnical Commission (IEC) (2003) Design Criteria of Overhead Transmission Lines. Geneva, Switzerland: International Standard IEC-60826”.

24.

Lantrip

T.B.

(1995) Identification of Structural Characteristics of Spun Prestressed Concrete Poles Using Modal Testing Methods. Department of Civil and Environmental Engineering, University of Alabama at Birmingham.

25.

C.Q.

(2000) A stochastic model of severe thunderstorms for transmission line design. Probab. Eng. Mech 15: 359–364. 10.1016/S0266-8920(99)00037-5

26.

Bai

(2006) High-voltage transmission tower-line system subjected to disaster loads. Progress in Natural Science 16(9): 899–911. 10.1080/10020070612330087

27.

Jia-Xiang

Hong-Nan

Xing

(2018) Dynamic Behaviour of Transmission- Line Systems Subject to Insulator Breakage. International Journal of Structural Stability and Dynamics 18(3): 1–24. 10.1142/S0219455418500360

28.

Loredo-Souza

Davenport

A.G.

(1998) The effects of high winds on transmission lines. J. Wind Eng. Ind. Aerodyn 74-76: 987–994. 10.1016/S0167-6105(98)00090-7

29.

Loredo-Souza

A. M.

Davenport

A. G.

(2003) The influence of the design methodology in the response of transmission towers to wind loading. Journal of Wind Engineering and Industrial Aerodynamics 91(8): 995–1005. 10.1016/S0167-6105(03)00048-5

30.

Madugula

M. K. S.

(2002) Dynamic Response of Lattice Towers and Guyed Masts”. New York, NY, USA: American Society of Civil Engineers (ASCE).

31.

Momomura

Marukawa

Okamura

, et al. (1997) Full-scale measurements of wind induced vibration of a transmission line system in a mountainous area. Journal of Wind Engineering and Industrial Aerodynamics 72: 241–252. 10.1016/S0167-6105(97)00240-7

32.

Newmark

N. M.

Hall

W. J.

(1982) Earthquake Spectra and Design. Berekley, CA, USA: Earth-quake Engineering Research Institute.

33.

NRC National Building Code of Canada (2005) Associate Committee on the National Building Code. Ottawa: National Research Council.

34.

Polyzois

Raftoyiannis

I.G.

Ibrahim

(1998) Finite elements method for the dynamic analysis of tapered composite poles. Composite Structures(43): 25–34. 10.1016/S0263-8223(98)00088-9

35.

Savory

Parke

Zeinoddini

, et al. (2001) Modeling of tornado and microburst induced wind loading and failure of a lattice transmission tower. [Eye Science Electronic Resource] 23: 365–375. 10.1016/S0141-0296(00)00045-6

36.

Simiu

Scanlan

(1996) Wind Effects on Structures: Fundamentals and Applications to Design. Third Edition. New York, NY: John Wiley & Sons.

37.

Sparling

Wegner

(2007) Estimating peak wind load effects in guyed masts. Wind and Structures 10(4): 347–366. 10.12989/was.2007.10.4.347

38.

Von Karman

(1948) Progress in the Statistical Theory of Turbulence. Proceedings of the National Academy of Sciences of the United States of America 34(11): 530–539. 10.1073/pnas.34.11.530