Horizontal Bend Analysis in Directionally Bored Ducts

Horizontal bend pulling equations

There are two forms of the bend pulling equations used to estimate the pulling tension through a horizontal bend.  Equation 1 is the “full” form equation.  Equation 2 is a  simplification that is accurate when the tension entering the bend is much greater than the cable weight in the bend.

Equation 2 relates to equation 1 as follows:

• When T_in  >> WR, the radical in equation 1 approaches T_in, that is Sqrt(T_in² )+ (WR)²)  → T_in
• Equation 1 then simplifies to T_out = T_in * (cosh(wϴμ) + sinh(wϴμ))
• Which, by definition, simplifies to T_out= T_in* e^wμϴ  (equation 2)

So, the accuracy of equation 2 depends on the WR to T_in ratio. The accuracy of equation 2 deceases as the WR/Tin ratio increases.

Comparison of the equations based on WR/T_in ratio

We see that equation 1 will always calculate a greater tension than equation 2 because there is no cable weight in equation 2.  The calculational difference between the two equations depends on the specific w, μ, ϴ, W, and R.  Graph1 below plots the percent difference (w = 1, μ = 0.2 and ϴ = π/2, 90°) against WR/T_in ratios.

Graph 1.  Equation Divergence vs WR/Tin Ratio

The callouts show some key points in the comparison.  Equation 2 has diverged from equation 1 by ~1% at a WR/T_in ratio of 0.30.  AEIC suggests the validity of equation 2 at WR/T_in ratios < 0.5, which is about a 2.6% difference.  We see that when T_in = WR (ratio is 1) the calculational difference is ~8.8%. After that, the results quickly diverge.

The implications of these differences

Higher WR/T_in ratios (above 0.3) are the result of either a low tension entering the bend or a large bend radius. The latter is common in directionally bored duct. Directional bored bends can have large radii with low displacement angles. What happens to the equations in this situation?

To simplify, the analysis below does not specify units. It will hold for any appropriate force unit (typically lbf, kgf, or N), as long as the weight (typically lb./ft, kg/m, or N/m) and radius (ft or m) use the equivalent unit.

Graph 2 compares the calculated tension from the two equations. As the degree of bend is decreased the radius is increased to maintain a constant weight of cable in the bend. The specific inputs for the graph are below.

Graph 2.  Tension Calculations Comparing Equations 1 and 2

Equation 1 results are graphed in blue and equation 2 (simplified) in red.  Additional perspective is provided by the tension calculated as if the length of cable in the bend arc were a straight run (green line).  Because this example was set up with a constant length of cable in the arc, there is a constant calculated straight section add-on of 100 (µWL) to total 1100 when added to the incoming tension of 1000 (see equation 3 below).

Equation 3:      T_out = T_in + µWL (straight section equation)

The WR/Tin ratio in the 90° calculations in Graph 2 is 0.32 which produces an initial 1+% difference (see Graph 1). The results diverge as the angle of bend decreases.

We see the calculated tensions from equation 1 approach the straight section tension as the bend angle decreases toward zero. This is what we expect since the decreasing bend approaches a straight section, with a zero (0) bend angle being a straight section.

But equation 2 approaches a tension lower than an equivalent straight section. As the bend angle approaches 0, the multiplier approaches 1. The result is no “add-on” at all to incoming tension. We know the cable in the bend must add some tension. But we also know that the results are not in a ratio area where the approximation is valid, so it’s use is not appropriate.

Pull-Planner software

Besides the large radius bends and low bend angle effects shown above, there can also be an issue with lightweight cable, such as fiber optic. The simplest solution is to use the  equations in all horizontal bend calculations, which is what our Pull-Planner™ software does. There is no downside since the software does the calculation work, and the results are valid regardless of the WR/T_in ratio.

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Reference

Underground Extruded Power Cable Pulling Guide, AIEC Publication CG-5-15, 3rd Edition, Association of Edison Illuminating Companies (AEIC), January 2015.