1

Dynamic Load Factors for Transmission Towers

Due to Snapped Conductors

P. Jayachandran1, M.ASCE., James F. Hannigan

2, Mark S. Browne

2 and Brian M. Reynolds

2

1Worcester Polytechnic Institute, Worcester, MA, 01609

2National Grid, Westborough, MA, 01582

Introduction

The ASCE Manual on Electrical Transmission Line Structural Loading No.74 (ASCE,1991),

describes longitudinal loads on structures, due to snapped conductors in its section 3.3. It

essentially specifies that longitudinal loads resulting from unbalanced wind or ice on adjacent

spans should be sufficiently resisted to prevent a failure of the structure. Additionally, it says

longitudinal loading resulting from snapped wires, insulator failure and component failure

should be considered in the structural design to avoid a cascading failure of the transmission

line. The ASCE manual provides residual static load factors (RSL) as a function of the span

length to sag ratio and the span length to insulator length ratio. Wire tension multiplied by the

longitudinal load factors predicts the final residual static tension in the wire after all dynamic

effects from the wire break have vanished. This is a static load factor applied to the design of

towers.

The towers are assumed to have rigid supports, and 10 equal spans between the wire break

and the next dead load. The RSL longitudinal load factors given are the minimum required

static loads to be resisted by the structures to avoid failure, and not dynamic effects. These

RSL values range from 1.0 to 1.5 in practice. These longitudinal loads act on the support

structure in the direction away from the failure of cables and will be added to the effects of all

permanent loads.

In this paper, effects of dynamic loadings due to snapped conductors are examined based on

analytic approaches advanced by Thomas, et al (Thomas and Peyrot, 1972,1981,1982). Often,

it is not economical to design and maintain a transmission line system such that it provides

sufficient strength to withstand large dynamic loads at each tower structure. An economic

design of a line system requires that the failure of a limited number of towers is acceptable, if

the overall system is protected from a cascading type failure. The acceptable number of

structural failures should be assessed based on the utility company’s design philosophy and

required reliability levels.

ASCE Manual 74 gives the Longitudinal Load Factor which may be used to estimate the

unbalanced longitudinal load. It is shown here in Figure 1. The Load Factor is a function of

span length to sag ratio, and it varies from 1.0 to 1.5. Research by EPRI indicates that the

span length to sag ratios vary from 10 to 100 (ASCE,1991). The wire tensions multiplied by

the longitudinal load factors provide approximate design loads that include dynamic effects,

structural stiffness and insulator lengths. The load factors are given for rigid structures such as

guyed or lattice towers. They are also given for flexible structures such as single poles, which

may undergo large elastic displacements. These load factors are based on the assumption that

collapse of one or two structures in each direction from the initiating event is acceptable to

avoid a cascading failure. The dynamic load factors computed in this paper have a mean value

of 1.4.

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Dynamic Response - Snapped Conductors

Three methods are used for the study of loads due to snapped conductors and also insulators:

1) static analyses to obtain the equilibrium position and residual forces or residual static load

factors; 2) full- scale or small scale experimental programs to determine the loads and

resulting stress values; and 3) a dynamic analysis of cable structural systems due to snapped

conductors and wind loads. ASCE Manual uses RSL derived from these effects in the range

of 1.0-1.5. A time history of dynamic loads, by a forcing function F(t) will provide peak

forces and time of occurrence of peaks and also the energy content of forces.

Some measurements of these forces, F(t), due to snapped conductors have been made by

Peyrot (1972), Lee, Kluge, and Thomas (1978, 1980, 1981, 1982). A time history of F(t)

measured in these tests with typical values encountered in practice, is shown in Figure 4.

Relative magnitudes shown are derived from practice based on similar National Grid towers

in the northeast United States. Dynamic analysis is done for towers with various cable

lengths, ranging from 400 feet to 1000 feet, typical of towers used by National Grid. Typical

towers in Western New York area are shown in Figures 2 and 3.

Dynamic Response - Numerical Integration Techniques

Constant velocity and linear acceleration methods are used to find displacement, velocity, and

acceleration at discrete time intervals. In the constant velocity method, acceleration y” can be

written as follows:

y”(s) =

2

*

1**

dtcm

dt

sysycsyksF

(1)

Where: y(s) = displacement at time step s;

y(s-1) at step (s-1);

dt = time step, usually T/20

where: T = Period = 2* Pi / w;

w = m

k

Pi = 3.14159

m = mass,

c = damping coefficient;

k = stiffness;

c = zeta* 2 * sqrt[k*m]

zeta = damping ratio, usually 0.01 for steel and 0.02 for concrete.

Displacement at time t can be written as follows:

y(s+1) = 2*"1*2 dtsysysy (2)

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In these equations, s = current time step, s+1 = next time step and dt = time step.

Similar equations are available for the linear acceleration method.

The stiffness, k, is obtained by applying a lateral force P and computing the lateral

displacement of tower yst; then, k = sty

P. This analysis can be done using software such as

Mastan2, SAPIV or PLS-CADD. This study is mainly for lattice type of transmission line

structures; wood pole structures are not included herein.

Dynamic Load Factors

Once displacements have been determined, the dynamic load factor (DLF) can be computed.

DLF = st

max

y

y (3)

Here, ymax is the maximum dynamic displacement of the tower by constant velocity method as

in Eq.2. Design loads for the tower simply follow by multiplying the lateral loads by the

dynamic load factor or impact factor. This factor is in the range of 1 to 1.5. In the analysis of

the towers computed here, DLF is about 1.08 to 1.63 with a mean value of 1.4. The constant

velocity and linear acceleration techniques were both used. The advantage of using dynamic

analysis is that it allows a computer simulation of forces and consequent response parameters

such as displacement, velocity and acceleration, which will be used in the design of towers.

Shear and overturning moments are computed, to be used in the design of foundations of

towers.

Response of Towers - Methodology

Impact factors were determined by experimental studies by Govers, Mozer, Ferry-Borges and

Peyrot (1972). The impact factors were determined by conducting tests in which the

conductors and insulators were allowed to snap and impact loadings measured from their

dynamic effects. The impact factor depended on span length of cable, initial tension, insulator

length and flexibility and tower stiffness itself. Peyrot (1972) developed a semi-analytical

formula for the impact factor, based on the recoil of the conductor away from the break in the

cable and the dropping of the conductor. The time history of the F(t) measured had two peaks

and a decaying period, lasting a few seconds after the break. The widths of the peaks and the

area under the peaks have a significant effect on the response of the tower. The period of the

cable determines the time duration of the peaks. Figure 4 shows a typical forcing function

F(t).

Thomas and Peyrot (1972) have developed a dynamic analysis model with cable elements and

lumped masses and damping to compute the tower response by numerical integration.

Flexible tower and a rigid tower were connected by two cables, with an intermediate flexible

tower in between. Damping was used in the analysis. Linear acceleration technique was used,

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with 20 elements per cable. Time increments of 0.002 seconds were used. This is included in

the program Cable7 (16). Experiments conducted by Peyrot (1972), Lee and Kluge at

Wisconsin Power and Light Company (1980, 1981), also developed forcing functions with

cable tensions and tower forces. These forcing functions have two peaks with an initial period

equal to the cable frequency itself, and the second period about half of the initial period.

The typical forcing function shown in Figure 4 has two major peaks, followed by smaller

ones, decreasing until the equilibrium is achieved in the cable system after the cable-snapping

energy is dissipated. At the instant of cable rupture, when t = 0 second, the conductor tension

begins to drop. It remains about 5-10% of the initial tension, Ti for a period of time called the

slack time, and begins to rise to form the first peak. Peyrot (1972) has determined that this

first peak occurs when the insulator swings towards the horizontal due to the recoil of the

attached conductor away from the break.

The second peak occurs, when sufficient time has elapsed after the break for the conductor to

fall freely and bottom down (Thomas and Peyrot [1972]). Either the first or second peak,

which occur within about 0.5 seconds of the break, and can overlap under certain conditions,

is the critical peak for the maximum tension Tmax, in the conductor. The peak values used in

this paper come from experience in cable forces estimated in similar cases in National Grid.

Figure 5 shows the typical curve used in this study, with maximum values determined from

cable frequencies of National Grid towers summarized in Table 1.

Careful observations of the insulator displacement following the breaking of conductors show

that the peak tension occurs when the insulator is nearly horizontal. Consequently, the force

on the tower is essentially a horizontal force, instead of a vertical load, as is the case for the

static equilibrium position just prior to the break itself (Thomas and Peyrot [1972]).

Response of Towers - Analysis and Design

The dynamic analysis of towers was accomplished by using constant velocity and linear

acceleration methods (19, 20). Damping was specified. The time interval for numerical

integration was 0.002 seconds. The natural frequency of the modeled structure was 1.8 Hertz.

Typical time history values of forces and displacements are shown in Figures 4 and 5.

Maximum values of displacements are summarized in Table 1.

The response was computed for cable lengths in the range of 400 ft to 1000 ft. The DLF was

also computed as the ratio between ymax and yst values. These are shown in Table 1. These

values have a range of 1.08 to 1.6265. The mean value is 1.4. The forcing function F(t) was

determined from maximum values based on measurements in previous research and also from

past design experience in National Grid towers in the northeast United States. This is mainly

used to verify design values of displacements and lateral shear and moments at the

foundation.

Clark, et.al. (2006), have measured dynamic loads in the 4 chord members due to snapped

conductor loads at Southampton. The characteristic load response following the conductor

release was evident in all their tests, with an initial peak load associated with the impulse

nature of applied loading, followed by a residual load change. The release load is offset from

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the vertical axis of the tower (i.e., there is a transverse lever arm component due to cross-arm

geometry,) which causes a torsional response in addition to longitudinal sway response. This

is evidenced by the difference in tensile load recorded in legs A and B respectively and the

compressive loads recorded in legs C and D respectively. The global longitudinal response,

with a frequency of about 3 Hz, dominates over the torsional response (there are 17 peaks

between 10 and 15 second marks in the load-time curve.) The magnitudes of the tensile loads

are about 75 to 150 kN (legs A and B,) and the compressive loads are about 20 to 60 kN (legs

C and D). The Southampton tests also measured the loads on the footings of the 4 legs of the

main chord members.

Dynamic Response - Analysis

Displacements and forces are shown in Figure 5 for one tower with a typical cable length.

This shows the response of a tower which follows the force. The dynamic load factors are

determined based on the ratio of ymax and ystatic. The forcing function essentially follows the

snapped cable peaks 1 and 2 and a subsequent smaller peak, which diminishes after a few

seconds. The response, y(t), which is displacement, follows the forcing functions for different

cable lengths, ranging from 400 feet to 1000 feet. The displacements and accelerations were

calculated using constant velocity and linear acceleration methods, Biggs (1965), Irvine

(1972). The time increment delta used was 0.0.002 seconds. Element forces can be then

calculated using element stiffness matrices. The element forces vector {F} can be written as

follows :

{F} = [S] * [A]T * {y} (4)

Where displacements {x} in element axes are {x} = [A]T * {y}. Damping ratios in the range

of 0.01 to 0.02 were used. In Eq. 4, [S] is the Element Stiffness matrix and [A] is the Statics

Matrix of the structure. See McGuire and Gallagher (2000) and Wang (1972) for matrices

[A], [S] and {F}.

Results and Conclusions

Computer simulation of dynamic load factors for transmission line structures due to snapped

conductors is illustrated in this paper. The forcing functions F(t) due to snapped conductors

used here are based on experiments conducted at Wisconsin Power and Light Company by

Peyrot (1972), Lee, Kluge, and Ferry-Borges. Mathematical modeling developed by Thomas

and Peyrot (1972) was used to assess dynamic load factors on transmission line towers. This

is more realistic than procedures suggested in ASCE Manual on Transmission Line Structures

Loading, (1991), which uses residual static loads, RSL, for longitudinal loads on these towers

due to broken conductors.

ASCE philosophy is based on accepting the collapse of some towers, based on broken

conductors, so that a cascading type of failure is avoided. The dynamic load factor approach

used in this paper had load factors in the range of 1.08 to 1.6265, yielding a mean value of 1.4

for cable lengths in the range of 400 feet to 1000 feet. The dynamic load factor approach

permits computer simulation of response for different classes of towers, with relative ease

based on numerical integration of equations of motion, with damping specified.

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Consulting firms in the design of transmission towers could now use a load factor of 1.4 for

dynamic loads due to snapped conductors. The other loading conditions will be due to dead,

live, snow, ice and extreme winds. Load factors are used based on ASCE, Canadian and

NESC standards are used for suitable combinations. Gust effect factors are also used from

Canadian and ASCE standards. This is also outlined in the ASCE Manual on Transmission

Line Structural Loadings - (1991).

Dynamic Load Factor computed herein for the longitudinal loads due to snapped conductors

are used in the design of lattice type transmission towers. They form part of a rational method

of analysis of such towers, in the design. The snapped conductors essentially introduce a

dynamic load factor, in the design.

The design of tower elements-chord, diagonals and strut elements are carried out now, due to

loading combinations of dead and live loads, amplified by the dynamic load factors. Methods

introduced in this paper offer an alternate way of amplification factor for dynamic loads due

to snapped conductors. The ASCE Manual-27 offers a static load factor approach.

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References

[1] A. Angelos and S. Cluts (1977), “Unbalanced Forces on Tangent Transmission

Structures,” Paper A 77 220-7, IEEE PES Winter Meeting, New York, Jan.

[2] D. B. Campbell (1970), “Unbalanced Tensions in Transmission Lines,” Journal of the

Structural Division, ASCE, Vol. 96, ST 10, Oct.

[3] J. Ferry-Borges (1968), “Experimental Study of the Stresses Created by the Breakage

of Conductors in High-Voltage Lines,” Department of Public Works, National Civil

Engineering Laboratory, Lisbon, Portugal, November.

[4] A. Govers (1970), “On the Impact of Uni-Directional Forces on High-Voltage Towers

Following Conductor Breakage,” Paper No. 22-03 International Conference on Large

Electric Systems (CIGRE), Paris.

[5] L. Haro, B. Magnusson and K. Ponni (1956), “Investigations on Forces Acting on a

Support After Conductor Breakage,” Paper No. 210, International Conference on

Large Electric Systems (CIGRE), Paris.

[6] J. Lummis and K. E. Lindsey (1974), “Computer-Aided Design of Longitudinal Loads

on Flexibility Supported Transmission Lines,” Paper C74 388-5, IEEE PES Summer

Meeting, Anaheim, California, July.

[7] J. Lummis and J. C. Pohlman (1974), “Flexible Transmission Structures Operate to

Suppress Cascading Failures,” Paper C74 055-0, IEE PES Winter Meeting, New York,

N.Y., Jan.

[8] J. D. Mozer, et al (1978), “Longitudinal Unbalanced Loads on Transmission Line

Structures,” Final Report Project 561, EPRI, August.

[9] J. D. Mozer, J. C. Polhman, and J. F. Fleming (1977), “Longitudinal Load Analysis of

Transmission Line Systems,” Paper F 77 221-5, IEEE PES Winter Meeting, New

York, Jan.

[10] A. H. Peyrot, et al (1978), “Longitudinal Loading Tests on A Transmission Line,”

Final Report Project 1096-1, EPRI, September.

[11] A. H. Peyrot, and A. M. Goulois (1978), “Analysis of Flexible Transmission Lines,”

Journal of the Structural Division, ASCE, Vol. 105, ST 5, May.

[12] A. H. Peyrot (1980), “Marine Cable Structures,” Journal of the Structural Division,

ASCE, Vol. 106, No. ST 12, December.

[13] A. H. Peyrot, R. O. Kluge, and J. W. Lee (1980), “Longitudinal Loads From Broken

Conductors and Broken Insulators and Their Effects on Transmission Lines,” IEE

Transactions on Power Apparatus and Systems, Vol. PAS-99, No. 1, Jan.-Feb.

[14] A. H. Peyrot, J. W. Lee, H. G. Jensen, and J. D. Osteraas (1981), “Application of

Cable Element Concept to a Transmission Line with Cross Rope Suspension

Structures,” IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No.

7, July.

[15] M. B. Thomas (1981), “Broken Conductor Loads on Transmission Line Structures,”

Ph.D. Thesis, the University of Wisconsin, Madison, June.

[16] M. B. Thomas and A. H. Peyrot (1981), “Cable7 – A Broken Conductor Analysis

Program – User’s Manual,” EPRI Project RP-1096-3, July.

[17] M. B. Thomas and A .H. Peyrot (1982), “Dynamic Response of Ruptured Conductors

in Transmission Lines”, Proceedings, IEEE – PES Winter Meeting, New York, NY,

Feb. 2, 82-WM-038-8, pp.1-6.

[18] Biggs, J. M. (1965), Structural Dynamics, McGraw Hill.

[19] Irvine, M.(1972), Structural Dynamics, Allyn and Bacon.

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[20] McGuire, W., Gallagher, R.H. and Ziemien, R. (2000), Matrix Structural Analysis,

John Wiley and Sons.

[21] Clark, M., Richards, D.J. and Clutterbuck (2006), “Measured Dynamic Performance

of Electricity Transmission Towers Following Controlled Broken Wire Events “,

CIGRE Paper, Paris, paper B2-313, pp.1-8.

[22] Wang, C.K. (1972), Computer Methods of Matrix Structural Analysis, International

Text Book Co.

[23] Guidelines for Electrical Transmission Line Structural Loading (1991), ASCE Manual

and Reports on Engineering Practice No.74, ISBN 0-87262-825-6.

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Figure 1

10

Figure 2

11

Figure 3

12

Figure 4

13

Deflection v. Time

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80

Time (s)

Dis

pla

ce

me

nt

(in

)

Displacement Force (Transposed)

Figure 5

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Dynamic Load Factors for Towers

Lc

(Length)

Te

(Sec)

We

(rad/s)

fe

(Hertz)

Ymax

(Const. vel)

Ymax

(Lin. Accl) DLF

Te =

We

2

We = m

k

fe = Te

1

DLF =

st

max

y

y

400’ 0.1413 44.47 7.08 4.072 4.071 1.4995

500’ 0.177 35.55 5.65 3.878 3.877 1.4280

600’ 0.212 29.622 4.72 3.162 3.161 1.1644

700’ 0.247 25.44 4.04 2.929 2.929 1.0786

800’ 0.279 22.563 3.59 3.539 3.540 1.3032

900’ 0.3314 18.963 3.02 4.405 4.405 1.6221

1000’ 0.365 17.24 2.743 4.417 4.417 1.6265

yst = k

P = 2.7156 in.

Table 1