Sags & Tensions
Determining the specifics of the wire and cable attachments on a pole or other structure, is the most important feature to the analysis of any pole or pole line. The tensions, along with the associated wind load, determine the loads to be applied. Knowing the sag of the wire or cable will determine ground clearances and mid-span clearances. It is well established and provable that tension and the amount of sag that results are inversely related. As the tension in the wire/cable increases, the amount of sag that is created is reduced. The reverse is also true. A very approximate relationship has been used for years for a quick estimate of sag or tension, if you know one of the values already:
W = weight per unit length
L = Span length
T = Tension
D = Sag
Failure to calculate the Sag and Tension accurately is the leading cause of differences in the Analysis Results of Pole Line Design software tools. If you have these values wrong or too far off from being accurate, the loading on the structure will not represent reality. These inaccuracies can lead to undetected overloaded conditions or upgrade recommendations which are simply not necessary. There are considerable differences in how different software tools approach this requirement. Some tools expose to users how they calculate Sags & Tensions, while others provide very little insight. Of the tools in use in the Marketplace today, many create inaccurate predictions through the use of the following practices:
- Ruling Spans. If a tool calculates an average or Ruling Span and applies it to one or more pole spans, it is not calculating exactly what is required. It is making assumptions about how the structures behave in response to unequal span lengths along a pole line under different loading conditions. These assumptions may not be valid. In the case of Joint Use Distribution Pole Lines, this assumption is almost certainly to be inaccurate. More on this later.
- Pole Elevations. If a tool does not take differences in attachment height elevations into account, the tensions and sags calculated will be inaccurate, period. Elevation differences change the effective span length for the calculations, which changes the tension and sag values that should be used. This will be shown in detail later.
- Constant Modulus of Elasticity. Several tools in the Marketplace use one value for the major parameter of wires or cables – Modulus of Elasticity. This parameter is similar to a spring constant in the way it relates the amount it stretches with the amount of tension. The reality is that no wire or cable that exists has a purely constant modulus. The modulus (slope of the stress-strain curve) changes as more tension is applied. Failure to recognize this will result in inaccurate results, or the correct result by pure luck. Slack Tension situations would be an exception to this as the tensions are very low on the Stress/Strain curve.
- Wire Creep ignored. All wires and self-supporting cables in the Marketplace today will permanently stretch when exposed to constant tension over a long period of time, or as a result of being exposed to extreme tension loads. In response to this permanent stretching, the tension in the wire or cable will reduce and the sags will increase. This phenomenon has been well-known for several decades and can result in very different results than if it was not considered at all. Failure to consider this effect can lead to significantly inaccurate results.
- Using constant tension values. Some tools require that tensions be pre-determined and entered into a data file for their use. Different tension values are specified for different span lengths and loading criteria for the Loading Analysis stage of their software tool. While there is usually some adjustment to better relate these values to the actual span lengths of the pole line, these can never be as accurate as calculating the Sags & Tensions directly, for the following reasons:
- The amount of permanent stretch (creep) experienced by the wire/cable is dependent on the actual span length. As mentioned above, this has a significant impact on sag & tension. It is unlikely that a simple interpolation of values between the provided span lengths will produce accurate results.
- The amount of wind and ice load that must be extracted from the provided span and tension values for proper application to the pole structure for support must be estimated. This can be difficult, especially when the wind is applied at angles different than 90 degrees to the wire or cable.
Tolerance for Inaccuracy
If you don't care about accuracy of results, stop reading now and use whatever tools or methods you want.
If you're still reading, then we have some information to tell you. Whether you are an experienced Pole Line Designer, a Professional Engineer that is taking responsibility for the work or someone new to this field, it is important to understand and become completely comfortable with the concepts. With this knowledge you can critique the value of any Sag & Tension result generated by any means.
Understand your Material
In order to have some assurance of accuracy of results, it is essential to understand the material you are using and how it behaves under various circumstances. For Sags and tensions of wires and self-supporting cables this is very important as they do not behave in a linear fashion. Regardless of the material used (steel, aluminum, copper, fiberglass, Kevlar yarns, etc…) all materials have nonlinear characteristics that significantly impact the final results. This importance is compounded further when more than one type of material is present in the wire or cable.
This generic curve shows some important principles that are common to any material you are using:
- There needs to be some amount of elongation of the material/wire/cable in order for it to be under tension; even if it is barely measurable. In the figure above for a sample of material of length L, this is shown as ΔL.
- The ratio of the change in length to the original length is called "Strain".
- The more Strain that exists, the higher is the tension and stress in the material (stress = Force/Area)
- The slope of the Stress/Strain curve at any point (Stress/Strain) is known as the Material's Modulus of Elasticity. It can be evaluated as an average value over a certain range, or at a specific point on the curve.
The shape of the Stress/Strain curve for different materials can be completely different. They can useful in providing insight into how the material will behave in your application. Here are a few material examples:
In all types of material there is a region of stress (force/cross sectional area) where the material is almost perfectly elastic. That is, it returns to its original length when the load is removed. Beyond this point the wire/cable will be longer when the load is removed than before it was applied. The material is now permanently stretched. The region of stress in the curve where this permanent stretching happens is called the "plastic" region. Within this region the material can undergo what is called "Strain Hardening", which may allow the material to handle additional stress up to what is called the Ultimate Tensile Strength (UTS), where it breaks.
Manufacturers of Wire and Self-supporting Cables for Power or Communication purposes are typically asked to provide fourth-order polynomial equations that describe the product's Stress/Strain behavior. These polynomials are expected to be curve-fitted to actual Stress/Strain test results for validation purposes. This mathematical approximation of the Stress/Strain curves is essential for accurate computer modeling of Sags and Tensions. They are considered valid up to a strain of 0.5%. After that point a straight-line projection (tangent) to the curve is extended, if required. This curve is called the "Initial Curve" as it represents the material's response to initial stresses.
Most materials used in Power Conductors, Wires and Self-Supporting Cables will stretch plastically (permanently) if the stress applied exceeds a certain Yield Point. For simplicity purposes, many materials would define this point as the point where the Strain reaches 0.2%; which is cited in many metal testing criteria and ASTM tests. In reality though, this point can vary with material and is only useful for conceptual purposes. Some materials will plastically stretch a small amount under small loads and then much more significantly under heavy loads. Some plastically stretch throughout their Stress/Strain curve, such as most aluminum alloys. After the initial amount of plastic stretch occurs, no further plastic stretch will occur in this same material sample up to the same maximum extent that stress was initially applied. The straight line that occurs after maximum load on any material is called the "Final with Load" curve.
Final with Load Curve
The Final Load Curve is a large contributor to the Sag and Tension results. It represents how the material behaves after it has seen the worst tension and loading condition expected. More on this later.
Let's look at an example of a 3M 1272 ACCR Conductor:
The violet curve shown represents the steel core of the wire. Its upper curve shows the Initial Curve as stress is gradually applied to the core. The lower violet line shows the core's behavior as the stress is removed. It follows the Final Load Curve for the Core. The light blue curve represents the Aluminum outer wires of the conductor and follows the same logic; except it is easier to see the difference between the Initial and Final Load Curves. The dark blue curves show how the core and outer wires, in their proper proportions, may come together to respond as a unit.
Another significant behavior of most wire/cable materials is that even modest, everyday tension loads on it will cause some permanent plastic stretch. This behavior is called Creep. Some wires/cables may have been installed for years, but have never experienced the maximum design loads that would have created permanent stretch that way. We must account for this to ensure our design considers this possibility. If a constant tension was exerted onto a material sample for a long period of time, the amount of creep that would develop would be highest at the beginning and then reduce over time to negligible amounts.
Wire and Self-Supporting Cable manufacturers place their materials under tension for 1,000 hours to determine the amount of creep that will occur based on the loads applied. Separate test results are normally generated at 15%, 20%, 25% and 30% of the Ultimate or Rated Tensile Strength of the material. These results are extrapolated on log paper to estimate the amount of creep expected in 100,000 hours (~ 10 years). After this point the additional creep that could develop is considered by the Industry to be so negligible that it is not worth considering any further. The 10 year projection is expressed in terms of a polynomial, similar to the Initial Curves, for ease of use within software tools. This polynomial represents what is called the "Final with Creep" curve for the material.
The above discussion on curve creation (Initial, Final with Load, Final with creep) assumes a constant known temperature for the material sample. All materials that are currently used in pole line construction expand with an increase in temperature. Said another way, these materials would have a positive coefficient of thermal expansion. This means that the physical length of wires and cables will increase on hot days or under heavy electrical loads. This is fairly intuitive to most that wires/cables would sag most on hot days than in the winter. Therefore it is important to consider what temperature to use for the sag and tension result you are seeking.
Sags and Tensions
Now that we understand how a material will respond to changes in loads and temperature, we need to use this to determine the Sags and Tension values we are looking for. The first question then should be "What are you looking for"? Code requirements of the NESC and CSA will identify some design requirements that you need to meet, but can be less specific on how to calculate them. Each code set will specify certain load cases and require that the design requirements of the standard be tested against the worst case conditions (for that requirement). The end result is that just one calculation for sag and tension of a wire or cable is inadequate. Several values must be calculated for evaluation against the various code requirements. For instance, maximum sag never occurs under the same conditions as maximum tension.
One thing you should note from the Material discussion is that finding the worst load condition (as seen from the material's perspective) is highly important as that helps define the "Final after Load" conditions for the wire or cable. The extent that the material experiences the plastic region of the Initial Curve has a significant impact on Final after Load values. The higher it goes on the curve, the more permanent stretch there will be. The temperature and load experienced under every-day conditions (assumed to be 10°C for most cases), defines the "Final after Creep" values.
So we have three different curves (Initial, Final after Load and Final after Creep) for how this material behaves. Which one should we use? The answer is all three, as the design requirement is to meet the code requirements under all expected conditions. Load conditions plus material conditions. Let's review each of these expected conditions in the material:
- Initial. The day after a wire or cable is strung; it could experience any of the expected load conditions, from no loading to maximum loading.
- Final after Load. The wire or cable may have experienced a maximum loading condition already. Some permanent stretching has occurred.
- Final after Creep. The wire or cable may have never experienced a significant loading event in its lifetime, but has existed for several years undergoing some amount of creep. The actual amount of creep could be anywhere from none (which follows the Initial curves) to a maximum predicted value, dependent on the length of time since it was first installed.
As you can see, you must calculate sag and tension values using all three curves and then test the worst value against the code requirements for various load conditions. This is because the wire or cable could be in any one of these states or some variation of the three. To meet code requirements you must test against all expected conditions in the material and these three curves are the major variations recognized in the Industry all over the world. Determining values using each curve individually establishes the maximum values for that scenario. Since actual wires could be in a state that is a mixture of these scenarios, using the three curves ensures that the worst case condition is always used for testing against code requirements. It also ensures that your design will be compliant with today's code not just for today, but for the expected lifetime of the structure . Since the actual occurrence of specific load conditions are not predictable (since weather is involved), this approach ensures that all realistic scenarios are included when testing against code requirements.
To determine the Creep curve, you must know the stress value expected on the Initial curve under everyday loads (no wind or ice, 10°C). Find this spot on the Initial curve. Then use the creep polynomials to estimate the amount of strain after creep. This will be either equal to (low probability) or greater than the strain value from the Initial curve. Now draw a line through this point that parallels the Final Load curve. The slope of both of these lines is equal to the Final Modulus value for that material, which is provided by the manufacturer. For any load condition expected on the Creep curve that is higher than the point where it intersects the Initial curve, the Initial curve is expected to be followed.
After now establishing the three curves (Initial, Final after Load, Final after Creep), we can now proceed to answer any of the code requirements that we have. Remember, for each requirement we must use the worst value expected of all real load and material conditions. For an example, what is the maximum sag for the wire we evaluated? This would typically be for ground clearance requirements. In essence you need to test all possible load conditions required for code compliance, plus any other you feel are warranted. For this requirement there are a few conditions that are typically the most significant:
- Thermal conditions – the maximum sag at the thermal temperature for the wire of any of the three curves. In this case the thermal temperature is normally 100C with no wind or ice. The sag value required is the maximum value obtained from any of the three curves.
- Maximum Vertical Load at the highest temperature – this is most often a condition where ice is formed on the wire, the wind is low or zero and the temperature is as high as can be expected with the ice still attached. The circulating air can be higher than the wire as it is the wire's temperature, not the air, which determines the thermal elongation of the wire. In many locations this is 0° C, no wind and maximum ice. The sag value required is the maximum value obtained from any of the three curves.
The codes will specify exact temperatures to evaluate some of the requirements, such as mid-span separations of attachments on the same line or crossing over/under others. These may vary based on the type of wire or cable. The same process would be applied for those temperatures and load conditions, looking for the maximum/worst value from all three curves. Although in these examples we were interested in the resulting sag values, the same process and rationale is used when looking for maximum tension.
Get Your Starting Point
The three curves discussed (Initial, Final after Load, Final after Creep) are an integral component of determining the actual Sag and Tension values, but don't provide the values we need easily by themselves. To get started, we need a starting point. That is a known, measurable and verifiable condition along with either a tension or sag measurement. Something you could send a survey crew and possibly a Line Crew with the appropriate tools to measure in the field. This is critical to any calculation as it provides the linkage between reality and the calculations. As you already know, you can take any piece or wire or cable and get different sag values by changing its stringing tension. If you change this starting condition, all other load and material condition tests will yield different results for sags and tensions. So it is critical to know your starting point.
The most desirable starting position is the known stringing tension of the wire or cable at a known temperature; assuming new material that was never placed before. You may know these values if they are recorded, or you may have organizational standards that call for stringing tensions at different temperatures for installation crews to use. It is generally accepted that existing wires or cables would have been strung to organizational standards, if no other data is available. This is based on the assumption that installation crews were trained to use these standards and were somehow tested for how close they were able to reach these values both during their training and possibly on different projects over time. Desired tensions can be achieved by several means such as direct measurement using a dynometer, striking the wire and counting the amount of time for reflections to return from the far end, or by using sighting boards which help to achieve the desired amount of sag between poles. Experienced linemen that have used a particular wire for many years are expected to get the desired tensions quickly and consistently; although checking their tension results periodically is also highly recommended.
When applying standard tensions, Installation Crews may have Organizational Standards that vary by the Ruling Span length. Since in any line there will be poles with shorter spans than the average, the Installation Crews are expected to pick which Ruling Span Length to use from the charts they have. This is expected to be a judgment call, mainly based on the area and existing average span lengths. For Communication wires/cables there are usually no differences in the proposed tension except for special applications. For Power Conductors, many utilities have different categories of Ruling Spans such as Service, Urban, Rural and Long-Span rural. Designers should use the same rationale for choosing an installation tension as what they can reasonably expect the Installation Crews to use.
Another interesting point about tensions is that while a conductor/cable may not be completely up to standard installation tensions, it should be expected to be brought up to proper tensions at any time as part of a maintenance activity. The key point here is that designers may not be involved in any re-tensioning activities. Therefore designers should expect that this is a possibility and ensure the line is designed appropriately to allow normal maintenance activities without their involvement.
Over-tensioning is something that may also happen for a variety of reasons. It could occur from excessive tension during stringing operations, or due to excessive re-tensioning of guy wires after normal stringing activities. A good Industry Test Objective to follow is to achieve the tension values within 10% accuracy to the desired values. It is impractical to mandate exact values from installation and maintenance forces. For Design purposes, a designer must assume the plant was strung to normal tensions unless they have knowledge or evidence to the contrary.
Another scenario you could run across is a wire or cable that is obviously strung to far less tension than normal. There can be specific design requirements for the pole line that require this, such as poles with no room for proper anchoring. In other cases you may visually notice some obviously slack wires or cables that should be re-tensioned. They could have been subjected to loads that exceeded what they were designed to expect (weather events, traffic accidents…). The pole line may also have some corner and deadend anchors that have pulled somewhat and may in fact be under-sized for the type of soil. In any event, under-tensioned wires and cables that could impact clearances must be addressed through re-tensioning and consideration of the existing anchors.
Slack Span Construction is a technique that can prove useful in dealing with difficult situations, as a last resort. It is prudent for the designer to consider all other options first as pole lines under full tension provide more reliability and are easier to maintain and build.
The good thing about Slack Spans is that you can always assume that the wire or cable is in a "new material" state, regardless of when it was initially constructed. This is because the everyday tension is very low on the Initial Curve and even heavy loads will not create very high stresses on this curve. If everyday tensions are low, long term creep will not be significant. If the wire or cable does not see high stresses, it will fundamentally only operate within the purely elastic area of the Initial Curve. This means that so little permanent stretch from either creep or plastic stretch will have occurred, that the material can be observed in its current state (sag/tension) and assumed to be "new material" for calculation purposes.
To capture the "starting condition" for slack spans, the best way is to estimate the existing sag of the wire or cable. Knowing this value, the span length, the temperature and the relative attachment heights will accurately define the tension (which defines the stress). It can also be advantageous if you have pole survey crews capturing similar data in the area as no other special tools are necessary. You can also directly measure the tension in the wire or cable, however this is usually more time consuming to coordinate and costly. You can also directly measure the tension on existing head-guys, in some situations, and iteratively adjust the tension in a whole-pole loading analysis until you reach values that appear reasonable for the same temperature. Some organizations also having been using some "rules of thumb" which could be very close, or close enough, to the appropriate values to use. For instance, for spans up to 20 meters, an everyday tension value of 100 Newtons (20 lbs.) might be used.
Service Wire Tensions
Service wires to residential and commercial buildings deserve special mention. Most wires or cables used for servicing customers can be run either pole to pole or pole to customer. When run pole to pole they are always tensioned to higher values than pole to customer, as the clearance and separation requirements are harder to achieve over these longer distances. Many organizations have different installation tensions defined for service connections than for mainline attachments. That is part of the answer as those tensions were established to meet the maximum service span length anticipated; which is usually 38 meters.
There is a requirement in the Electrical Codes of Canada (CEC Part 1) and USA (NESC), and perhaps other countries as well, for the power service connection point to withstand up to 600 pounds of combined load, under worst case conditions, from all attachments on the smallest of services from an amperage perspective. This covers residential and many commercial buildings. Larger commercial buildings would have higher tension load capacities with corresponding larger conduit and mast pipe sizes. In many cases, communication drops will attach to this same service mast as power. In other cases, especially commercial buildings, communication attachments have their own support at the building. This is important context as Power Utilities do not want to field complaints from customers if their power mast is pulled off their building. This has driven behavior in installation tensions for spans shorter than the maximum which should be considered.
Like mainline wire and cable attachments, it is convenient and desirable to assume with confidence that specific tensions were utilized at installation time. In the case of power service wires, often only tensions for the maximum span length are provided. This leaves designers wondering what values to use as the maximum values are obviously inappropriate for many typical installations. Also, it is impractical to perform survey measurements on all services when assessing a pole line. There are just too many of them, creating too much of a burden on the pole line owner or attacher to do a detailed accurate assessment. A simpler, but reasonable way is needed to estimate the tensions of power services.
One company in Canada surveyed and measured tensions on many different power service wire installations of various lengths in two different cities. From the results, they analyzed the data collected along with feedback they received from several Canadian Power Utilities on their installation practices. The outcome, which was suspected but no proven before, is that Power Utility technicians do not pull their service wires to full maximum tension unless it is needed for clearance purposes. In almost all cases, vertical sag below the attachment point at the customer's location of about 0.5m was being achieved at installation time. This is consistent with the desire of the power utility to stay well below the 600 pound limitation of most service masts; especially those of older buildings built before this load requirement was well understood. If the designer has no other information to use, the results of this study may prove useful to them.
Calculating Sags & Tensions
The above guidance for a starting point in calculations was necessary, but it doesn't set our starting point on the Initial Curve yet. For that you need both Stress (Tension/Area) and strain. A little more work is required. There are three sets of equations that need to be solved simultaneously in order to get this point. One set of equations is the Initial Curve itself, relating Stress (or tension) to Strain (or stretch), defined in polynomial form by the manufacturer:
The second equation is recognition of the thermal expansion coefficient for the material:
The third equation ensures that the wire or cable solution fits the physical restrictions placed upon it such as the specific span length, relative attachment heights, temperature and either a measured tension or a measured sag.
S= wire or cable length between attachment points
H = horizontal tension at maximum sag point
W = load per unit length
L = effective span length
It can be shown that this formula must be true for any wire or cable that has a constant weight per unit length along the span.
These three equations that define the behavior of the material and its physical restrictions are then solved, usually in an iterative manner, to land on a solution that satisfies all three. There are several methods available to software providers, some better than others.
A complication to this calculation exists when there are two types of materials in a single wire or cable. A common example is a 1/0 ACSR wire (aluminum covered steel reinforced). The three curves and thermal coefficients of both materials are all different. In these cases, the restrictions on both the core and outer materials must be satisfied at the same time. There is a manual graphical way of calculating these simultaneous equations that was developed by ALCOA many years ago. It is seldom used these days, but is still valid for any interested reader. What software tools can do for the user today is amazing compared to the manual processes of the past.
After you have your starting point on the Initial Curve you must then apply all reasonable load conditions to the wire or cable to find the one that stretches it the highest on that curve. This helps to define the Final after Load curve. You also need to determine where on the Initial Curve the average everyday "No Load" at 10C condition would lie. You need to specifically calculate the stress of this condition. From this you can solve for the applicable strain value after Creep from the Creep polynomial. Following this stress and new strain value down to zero strain using the slope of the Final Modulus of Elasticity will determine the amount of permanent stretch due to creep and the Final after Creep Curve.
This process will need to be done for every different material type within the wire or cable. Once these three curves are well defined, all other load conditions required by code (plus other reasonable or desired ones) should be tested and made available. Then for each code requirement, as mentioned before, the results from all three curves must be considered under all reasonable load conditions.