^{By David C. Gelder, P.E. }

^{Based on “The Catenary Exposed – Understanding Theoretical Conductor Behavior in Transmission Lines” presented at TSDOS 2017. Reproduced with permission.}

**Generational Challenge**

The Eiffel Tower, Golden Gate Bridge, Brooklyn Bridge, Empire State Building, the list of modern engineering marvels goes on. For Millennials—with technology just a touchscreen away—it is astounding to consider all these iconic structures were engineered not with fancy apps, but with old-fashioned pencil and paper. Clearly, feats of great engineering were possible long before powerful computers and sophisticated software.

A huge advantage of these modern tools is obviously, speed. Even relatively new transmission engineers can generate complex designs much quicker than their predecessors. Long alignments complete with structures, wires, strength, loads, staking coordinates, and plan and profiles are comparatively simple to assemble.

These tools have become available at a critical time. Increased energy demand, aging infrastructure, and retiring Baby Boomers have all led to a significant influx of young engineers into power delivery. Fortunately, these bright individuals have quickly learned the trade and become expert users of design software. However, the rapid turnover raises concerns about how effectively the knowledgebase of fundamental hand-calculations – used for over a century – are being transferred.

For all the benefits of technology, there is risk of creating a generation of transmission engineers too reliant on and too accepting of software outputs. As a result, Millennials may be unable to identify mistakes readily apparent to engineers of an earlier era who did their own calculations. So, how will young engineers overcome this potential computer crutch?

**Knowledge Sharing**

One way to combat this generational challenge is by creating a culture of knowledge sharing. Webinars, training sessions, lunch-and-learns, and other events are ways for engineers of all ages to share learning. A main focus should be on combining theoretical and practical knowledge.

To illustrate, it seems the ‘sweet spot’ of engineering design and analysis is achieved when experimental and/or field data are combined with hand calculations and computer analysis using finite element software (Figure 1). Troublingly, a design lacking any one of these components may appear sound, but could be off mark due to complexities of actual structural behavior. Most experienced engineers are familiar with these types of blunders and have seen examples during their career.

**Complex, Not Complicated**

Knowledge sharing may begin with the most important shape in power lines—the ‘Catenary.’ The remainder of this article introduces the catenary, or hanging conductor, and basic formulas to demystify theoretical conductor behavior to young engineers. These hand-calculations can be used to reproduce software results, and thus demonstrate true mastery of fundamentals.

Although conductor sag is of paramount importance in transmission engineering, few contemporary engineers have taken the time to fully understand the catenary, probably because of perceived complexity. While the shape of the hanging conductor under uniform load is complex, it’s not complicated to understand.

So transmission engineers, get ready to sharpen…your…pencils!

**Statics of Catenary**

Recall Statics 101 where the world was idealized as rigid bodies and point loads. Similarly, the discussion of conductor sag begins with a simple static catenary, or hanging cable, that is:

- Simply supported at the ends,
- Subjected to a uniform force per unit length,
- Perfectly flexible, and
- Inextensible (i.e. unable to be stretched).

As such, the catenary deforms so that internal forces result in no shear or bending moment, only tension. Figure 2 depicts a typical asymmetric catenary for reference.

**Catenary Calculations**

As shown in Figure 2, the catenary has simple supports at the ends and a uniform force per unit length, which may be due to either internal body forces (self-weight) or externally applied forces of surrounding fluid (wind, water, etc.). The catenary constant, *a*, is both simple and incredibly useful. Equation 1 defines this value equal to horizontal tension, *F _{H}* (force) divided by cable unit weight,

*w*(force/length). Thus,

*a*, has corresponding units of length.

*a = F _{H} / w *(1)

The value simply denotes a local axes directly below the low point or “belly” of the curve at a distance equal to the catenary constant. Hanging conductors in transmission lines have relatively low curvature, meaning a very high catenary constant—perhaps between 1,000 and 10,000 ft. This means that the theoretical datum for a typical span lies thousands of feet below the surface of the earth! Remember, the catenary constant should not be confused with distance of cable to ground.

Now, Equation 2 gives the relationship of position in terms of the catenary constant, *a*:

** y (x) = a / 2 (e ^{x/a} + e ^{-x/a}) = a * cosh(x/a) (2)**

This equation can be used to plot basic catenaries in Excel, understanding that the shape is based on horizontal tension and conductor unit weight. From here, several other useful relationships can be determined. For instance, slope is the derivative of position. Wire length and tension can also be derived. These equations are the basis of commercial transmission line software, such as PLS-CADD^{®}.

While most transmission engineers are familiar with this basic catenary equation of position, there are three reasons they may find it intimidating:

- Unfamiliarity with hyperbolic functions, which are more intuitively understood using the definition with Euler’s number,
*e*= 2.718; - Accepting the catenary constant must be solved iteratively, not directly; and,
- Confusion regarding transformation from local coordinates in the catenary equation to global coordinates in engineering drawings such as station and elevation.

Just remember this: the catenary constant is very important to the computer. Typically the catenary constant is calculated first based on a known or desired catenary shape. Then the computer can easily calculate and output tensions, sags, and wire lengths for a variety of conductor temperatures.

In summary, Statics is the first lesson of the catenary. This is simply a basis for understanding forces within the member. Additional topics including stress-strain, thermal expansion, and creep are built upon this initial understanding, similar to how Mechanics introduces these topics to the study of a beam.

**Conclusio**

Millennials surely will make valuable contributions to the practice of engineering. The trick is simply to think outside the software box and master the fundamentals. Managers should support creating a culture of knowledge sharing. Project deadlines and budgets are real pressures, so fundamentals cannot be learned in one sitting. Instead, every project should be treated as an opportunity to learn and, hopefully, to share this knowledge with others.

**Content provided by TRC. For more information contact:**

David Gelder • Taylorsville, UT • Sr. Transmission Engineer • 385.355.3328 • [email protected]

Stephen Persutti • Rocky Hill, CT • Vice President of Utility Development • 860.202.4244 • [email protected]

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