Catenary CurvesThe catenary is the shape that a free hanging flexible cable or wire assumes. The following gives equations for the case where the wire is of uniform mass per unit length, and the supporting points are at the same height. Cases where the ends are not at the same height, or where there are point loads (i.e. dipole antennas supported at ends with a feed line hanging in the middle), can also be calculated, but these equations won't do it. The picture below defines some terminology and the reference points.

We assume that the origin is at the center of the span. Total span = L

Sag in the cable = hSo, the coordinates of the endpoints are (+/- L/2,h).

The weight per unit length = w Total length of wire/cable = S Length along the cable from the origin = s Fh is the horizontal force component everywhere, and is equal to half the tension at

the center.Maximum TensionSince the National Electrical Safety Code (NESC) minimum vertical clearance between conductors at supports and between the conductors and ground are a function of the sag of the conductors, some utilities might choose to reduce the sag by installing their aerial conductors at the highest tension for which they are allowed without exceeding the NESC limits. By limiting the conductor sags, the structure heights can be reduced and/or the spans can be increased, both of which reduce the cost of construction. What is the maximum tension? The NESC in Rule 261H, page 179, limits the tension of open supply conductors and overhead shield wires to 60 percent of the rated breaking strength (RBS) under the loading conditions specifi ed in Rule 250B, Table 250-1. For example, in the Heavy Loading District, the 60 percent limit is with the conductor at 0°F, with one half inch of radial ice and a 4 Lb./Sq. Ft. wind (about 40-mph wind). The same rule also limits the initial unloaded tension

at 60°F to 35 percent of the RBS and the fi nal unloaded tension at 60°F to 25 percent of the RBS. Safety Factor

Table 1Since the NESC requirements are maximums, it is important to use some safety factor in the design to insure that the design tensions do not violate the NESC requirements. To do this, most utilities install their conductors at less tension than the NESC maximums. One approach might be to limit the tensions to 90 percent of that allowed by the NESC. The resulting sag/tension computer program printout might then look like the following for a 200 foot span of 477 AAC “COSMOS” conductor in the NESC heavy loading district (see table 1).Note that the maximum tension is 4514 lb. The stringing sags and tensions are the initial sags and tensions at 30°, 60°, 90°, and 120°F. The crew installing the conductor usually extrapolates between the four temperatures to determine the stringing sag/tension corresponding to the actual conductor temperature at time of tensioning. The maximum sag is 5.31 feet, the final sag at the conductor’s maximum operating temperature. In this example, the utility has chosen 212°F as the maximum operating temperature. For further information on conductor maximum operating temperatures, see my article on “Conductor Ampacity” in the March/April 2004 IAEI News and my article “Conductor Hyperthermia” in the July/August 2002 IAEI News. Depending upon the construction standard that a utility uses for dead-ending conductors, there might be further limitations on the conductor tension based upon the rated strength of the dead-end hardware and anchoring system. In areas where the soil strength is low for example due to moisture or sand, keeping the conductor tensions lower improves reliability because it makes the anchoring system strength less critical. Is a 10 percent safety factor enough to insure that the NESC maximum tensions are

not exceeded under all conditions? To answer this question, we have to look at what conditions effect conductor tension.Ruling SpanLimiting the tensions to 90 percent of that allowed by the NESC might be fine if all the span lengths between dead-ends of a proposed line are the same. For transmission lines being constructed across open land for great distances, keeping the span lengths the same is usually achievable. However, for most distribution line construction along highways and rear lot lines, keeping the span lengths the same can be difficult. In states where utilities have to get permission from property owners to install poles, the property owners prefer the poles be installed at the side property lines rather than in front of their houses. Since the property widths along the highway may vary greatly, the utility is forced to install poles with diverse span lengths. To better understand how to deal with spans of different length, let’s consider a simple example. A developer has constructed a new road extending 1,184 feet off of an existing road and is selling off property on both sides of the new road for the construction of houses. The lot sizes vary greatly because of the terrain and natural features like streams. To serve the new houses, the utility decides to install the poles at the side property lines along the new road. Not every property corner has a pole. Some spans cross two lots. The resulting design consists of eight spans; 140′, 135′, 124′, 50′, 200′, 275′, 150′, and 110′ in length. If we run a sag/tension program for each of the span lengths and limit tension to 90 percent of the NESC limits, we will find that the final tensions vary greatly span to span even though the tension limits are the same. The final tensions are different because the span lengths are different. To reduce construction time and cost and to reduce the differences in tension from one span to the next, the industry accepted method for stringing the conductors over the eight spans is to string them all at the same time at the same tension. Since we want to install the conductors at the same tension for all the spans, what tension do we use? The industry accepted method to determine the tension to be used for all the spans is to calculate a weighted average of the span lengths. The weighted average is called the “ruling span.” The ruling span is the square root of the sum of each span length cubed divided by the sum of the spans.

Table 2We then run a sag/tension computer program on the ruling span to determine the stringing tensions to be used for all the spans. The resulting sag/tension printout might look similar to the information in table 2.Note that the stringing tensions that would be used for all the span lengths are the initial tensions at 30°, 60°, 90°, and 120°F from the above printout. Again, the crew installing the conductors will have to extrapolate between the four temperatures to determine the stringing tension corresponding to the actual conductor temperature at time of tensioning.Effect on Tension

Table 3If we run a sag/tension program for each of the diff erent span lengths using the initial tensions from the above ruling span printout, we would fi nd that spans shorter than the ruling span have less sag and higher tension than the ruling span and spans longer than the ruling have greater sag and less tension than the ruling span. Since the shortest span will have the highest tension, to insure that the design tensions do not exceed the NESC maximums, we have to run the sag/tension computer program on the shortest span length (50′) limit-ing the initial tension at 120°F to 1102 lb. Note that I have chosen the conductor stringing temperature to be 120°F. I did this because the final sag and tension is also a function of the temperature of the conductor when it is tensioned. Stringing the conductors at 120°F is the worst case scenario. Spans shorter than the ruling span will have higher tensions if tensioned at 120°F. Spans longer than the ruling span will have greater sag if tensioned at 120°F. The resulting sag/tension program printout might look like table 3.Note that the initial tension of 2977 lb at 60°F exceeds the NESC maximum tension of 35 percent of RBS (2926 lb). This means that the 10 percent safety factor we assumed at the beginning is not enough for this application. Some utilities limit the initial conductor tension to 50 percent, instead of 60 percent, of the rated breaking strength under the loading conditions specified in Rule 250B, the initial conductor tension at 0°F, instead of at 60°F, to 35 percent of the RBS, and the final conductor unloaded tension at 0°F, instead of at 60°F, to 25 percent of the RBS.Effect on SagSince most NESC clearances are a function of the sag of the conductor and spans longer than the ruling span have greater sag, it is important to determine the actual sag of the longer spans for calculation of the clearances for the longer spans. When the tension limits

used to design a line are close to the NESC tension limits, ruling span has very little eff ect on the sag of the spans that are longer than the ruling span.For example, consider the eight span example discussed above. If the tension of each span were limited to 90 percent of the NESC limits, the 212°F final sag of the 275′ span would be 7.75 feet. If we install the 275′ span at the same initial tension as for the 185.5 foot ruling span, the 212°F final sag would be 7.79 feet. When the tension limits used to design the line are considerably lower than the NESC limits, ruling span does have a significant effect on the sag of spans that are longer than the ruling span. For example, if the tension limits of the same eight span examples discussed above were limited to 50 percent of the NESC limits, the effect of ruling span on the 275-foot span would increase the 212°F final sag by almost 11 percent.

Ruling Span: The term “ruling span” is one of the most frequently used yet misunderstoodand misused term in the design, staking, and construction of overhead lines. “Ruling span” is loosely used with several different meanings. The term should be preceded by a descriptive adjective to identify its specific meaning.

3.2 The Ruling Span Theory: During stringing and sagging, the conductors are placed on travelers at the supports and are dead-ended at the ends of the stringing section of the line.While the conductor is on travelers and frees to move between spans, the conductor tension and length in any span is a function of the combined average tension of all the spans and the total conductor length of the dead-ended stringing section.

3.2.1 When all the spans have equal lengths and the supports are of equal elevation, the behavior of the conductor in each span will be the same and can be determined by the equations for the dead-end span. When the spans are of unequal length and the supports are of varying elevations, the mathematics become too complicated to be easily calculated. Therefore, it is necessary to simplify the problem.

3.2.2 By making certain reasonable assumptions concerning the behavior of the conductor in a series of spans supported on travelers, the mathematics can be simplified to a manageable equation. The assumptions for the “Ruling Span Theory” are:• The supports are at equal elevations since the span lengths are large compared to the difference in elevation of supports. (Resultant errors will be negligible.)• The horizontal tension is constant throughout the stringing section since variations inspan lengths will not be great enough to cause a measurable difference in the horizontaltension between any two adjacent spans.

• The uneven spans are replaced by a series of equal spans such that the total length of theconductor and the horizontal tension of the section is unchanged. Thus the sag andtension characteristics of a single dead-end span can could be used to predict the sagbehavior in any of the spans in the section.

3.3.1 The theoretical ruling span equation is not exact because of the assumptions made. Since its accuracy is sufficient for most line designs, it is the equation used most often to calculate the ruling span for new overhead distribution lines.

3.3.2 Since the horizontal tension and the unit conductor weight are assumed constant throughout the stringing section, the sag of any span in the section can be calculated by the parabolic sag equations given earlier in Section 2.3 of this bulletin.

3.3.3 The above ruling span formula is the ruling span in its true sense. It has been called ruling span, theoretical ruling span, actual ruling span, true ruling span, and equivalent span. It is an equivalent span length based on the total length and average tension of the conductor in a series of spans that is being pulled up and sagged in one operation. It is, therefore, a function of all the spans included in the stringing section.

3.3.4 After being tied in, each span virtually becomes a dead-end span with approximately the same tension as the theoretical ruling span. When the tied spans in the section are of different lengths, changes in temperature, loading and elongation due to creep will cause differences in tension between the spans. These differences in tension will cause a flexing or bending of poles and arms.

3.3.5 This ruling span “rules” the behavior of the sagged section of line. The sag characteristics of the ruling span set the sag characteristics of every span in the section. If conductors are installed using a sag-tension table with the wrong ruling span, actual final sags and tensions will not be the same as predicted. The greater the difference, the greater the error!

3.4 Design Ruling Span: One or more assumed ruling spans, based on experience, has to be used for the field design of a new line because the theoretical ruling span of a line section cannot be determined until after the line is staked. If the land is reasonably flat, it is appropriate to use a ruling span that approximates the level ground span. The required ground clearance may be subtracted from the attachment height of the lowest conductor to

determine the sag limited by ground clearance. This sag value can then be used to determine a ruling span length whose sagis approximately equal to the sag allowed by the basic structure height. For rugged terrain, a ruling span that is longer than the level ground span is usually more effective.After staking, the theoretical ruling span should be calculated and compared with the design ruling span.

Using a design ruling span appreciably different from the theoretical ruling span of the section will produce unpredictable sags and tensions. Slack sags may cause clearance problems while tightly drawn spans may cause uplift problems. Higher than predicted tensions may exceed the permitted load on support assemblies or may cause aeolian vibration problems.

3.5 Estimated Ruling Span: Knowledge gained from a reconnaissance of the proposed line route may make it possible to estimate a ruling span. A traditional “rule of thumb” equation that may be helpful in the estimation of a ruling span is:Se = Average Span + 2/3 (Maximum Span – Average Span)

3.5.1 Use this rule for estimating the ruling span with caution! This “rule of thumb,” used indiscriminately, has significantly different sags and tensions than the true ruling span equation. Even one span much longer than the average span may cause the estimated ruling span to be much greater than the actual theoretical ruling span. This formula should only be used for estimating the ruling span when the actual spans are not yet known. When the spans are known, the theoretical equation should be used

3.6 Effects of the Wrong Ruling Span: The greater the difference between the theoretical ruling span and the design ruling span, the greater the variation will be between the actual and predicted sag and tension values. The magnitude by which actual sags and tensions will differ from the predicted values is a function of conductor temperature and loading.• If the design sag is greater than the theoretical sag, then the actual sag of the installed conductors will be less than the predicted sag. This condition will lead to increased conductor tensions, which may exceed the permitted loads of support structures and guying assemblies.• If the design sag is less than the theoretical sag, then the actual sag of the installed conductors will be greater than the predicted sag. This condition may result in inadequate ground clearances.