- Gain a better understanding of the equations used in horizontal bends
- Results from the full and simplified equations are compared under varying WR to Tin ratios
- How the full equation produces most accurate results across multiple scenarios
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.
|Conduit Bend Equations|
|Tout = Tin * cosh(wμθ) + (sinh(wμθ) * Sqrt(Tin2 )+ (WR)2)||Equation 1 (full)|
|Tout = Tin * ewμϴ||Equation 2 (simplified)|
|Tout = Tension Coming Out of the Bend (lbf, kg, kN)|
|Tin = Tension Coming into the Bend (lbf, kg, kN)|
|w = Weight Correction Factor (dimensionless)|
|μ = Coefficient of Friction (COF) (dimensionless)|
|ϴ = Angle of Bend (radians)|
|W = weight of the cable (lbs,kg)|
|R = radius of the bend (ft, m)|
|e = Natural Log Base (constant)|
Equation 2 relates to equation 1 as follows:
- When Tin >> WR, the radical in equation 1 approaches Tin, that is Sqrt(Tin2 )+ (WR)2) → Tin
- Equation 1 then simplifies to Tout = Tin * (cosh(wϴμ) + sinh(wϴμ))
- Which, by definition, simplifies to Tout= Tin* ewμϴ (equation 2)
So, the accuracy of equation 2 depends on the WR to Tin ratio. The accuracy of equation 2 deceases as the WR/Tin ration increases.
Comparison of the equations based on WR/Tin 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/Tin 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/Tin ratio of 0.30. AEIC suggests the validity of equation 2 at WR/Tin ratios < 0.5, which is about a 2.6% difference. We see that when Tin = WR (ratio is 1) the calculational difference is ~8.8%. After that, the results quickly diverge.
The implications of these differences
Higher WR/Tin 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.
|Inputs for Figure 2 Graph|
|Tin = 1000 (incoming tension)||W = 5 (cable weight per length)|
|μ = 0.2 (coefficient of friction factor)||w = 1 (weight correction factor)|
|ϴ = bend angle that is varied from 90 to 0 degrees (1.57 to 0 radians)|
|R = bend radius starts at 63.66 and increases as the angle decreases to maintain a constant cable length of 100 in the bend arc. Note that this also means a constant cable weight of 500 in the bend.|
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: Tout = Tin + 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 Graph1). 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.
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/Tin ratio.
This analysis applies to horizontal bends. A future post will focus any similar issues with the vertical component bend equations.