Tension Methods

Wire tensions play a considerable role in creating loads on a structure. How utilities have incorporated wire tensions in their design practices varies from utility to utility. To accommodate this, SPIDAcalc gives users two methods to set wire tensions that are used during a load analysis – static and dynamic. Using the Client Editor, users can select which tension method is used for each wire stored in the client file library.

Dynamic Tensions

When the tension method is set to dynamic for any wire, SPIDAcalc will perform a tension calculation at the beginning of the load analysis process. In setting up the client file, the user should enter an initial stringing tension (unloaded) or a final place tension (unloaded) at a specified temperature.

During the analysis, SPIDAcalc will apply the temperature, wind, and ice specified by the given load case to dynamically calculate the wire’s horizontal tension. After the tension is determined, the appropriate load factors (multipliers) are applied. The resulting forces are used to determine how the structure reacts and the final stress load. The values displayed in Detailed Analysis Results and on reports reflect the factored wire tensions. For co-rotational analysis, the option “Update Tensions Based on Displacement” may affect dynamic wire tensions.

Static Tensions

When the tension method is set to static for any wire, SPIDAcalc will not perform a horizontal tension calculation during the load analysis. Instead, the tension value set in the client file is used. The static tension (loaded) entered by the user typically represents a design, or worst case, tension (inclusive of temperature, wind, and ice) for the given wire. It implies that a calculation has already been completed prior to entry in the client file to determine this tension.

During the analysis, SPIDAcalc will use the static value as supplied in the client file for analysis. Wind and ice will still be applied to the wire to determine the overall effect on supporting elements (e.g. pole, guy wire, etc.), but no additional horizontal tension calculations will be performed. The appropriate load factors (multipliers) will still be applied to the static tension indicated in the client file. The resulting forces are then used to determine how the structure reacts to these loads and the final stresses. The values displayed in Detailed Analysis Results and on reports reflect the factored wire tensions.

Average Length on Main Span

“Average Length on Main Span” is an option that can be enabled in any load case. When enabled, the distance of the Previous and Next wire endpoints (WEPs) are averaged when determining which tension group from the client file is used for each wire during a load analysis. For example, if the span lengths of the Next and Previous WEPs are 125’ and 175’, SPIDAcalc will use the tension associated with a span of 150’ for both spans ((125+175)/2).

When applying wind and ice to the conductors in the design, the actual design span length will be used, not the average. For those conductors set to the dynamic tension method, a tension imbalance may occur longitudinally due to the weight of ice applied to the longer span of the design.

The “Average Length on Main Span” option will only apply to the WEPs designated as Previous and Next. Wires placed from the pole to other types of WEPs will utilize the entered span length to determine the appropriate tension group for the wire.

Tension Adjustment Factor

The Tension Adjustment Factor (TAF) in SPIDAcalc is used to customize the generic tensioning data in the client file to more accurately model unique or non-standard designs and structures. The most common scenario is for unique short or slack span structures that do not use standard tensions on the conductors.

This additional factor is applied to the tension pulled from the client file for the wire. Therefore, a TAF less than 1 will reduce the tension, while a TAF greater than 1 will increase the tension. The TAF can be found in the Component Properties Panel for any wire.

 

 

Tension Calculations - General Setup

This is a vastly condensed and summarized version of the tension calculations inside SPIDAcalc. Please see other documentation for a complete derivation of equations.

We start with the general catenary equation that describes the geometry of a wire hanging between any two supports:

(1)

 

where:

x_c = Horizontal coordinate of the lowest point.
y_c = Vertical coordinate of the lowest point.
a = Catenary constant H/w, where H is the horizontal tension, and w is the linear weight of the wire.

 

We then define where our two supports should be placed and the distances between them. Two known points (0,0) and (s,h) where point (0,0) is the current pole, and (s,h) is the wire endpoint. s is horizontal distance and h is height to the wire endpoint.

Using point (0,0) in the general equation, and after some algebra, leads to:

(2)

 

Using point (s,h) and some very similar algebra leads to:

(3)

 

These equations are used to solve for the lowest point along the span. This is important for both calculating the change in horizontal tension due to changes in environmental loading as well as for calculating the weight-span for inclined lines.

Arc Length

We also need to know the arc length of the wire, and how the wire length changes to accurately calculate the tension. This can be done by applying the general equation for the arc length of any curve between two points.

Solving for the arc length L between the two endpoints:

(4)

 

Applying the catenary formula and simplifying:

(5)

 

Substituting in for x_c from Eq 3 and several steps of algebra leads us to:

(6)

Change in State - Solve for Design Tension

A general equation for change in arc length is:

(InitialArcLength) + [ChangeInArcLength] = (FinalArcLength)

(7)

 

or:

(InitialArcLength) + [(ΔfromTemp) + (ΔfromStress/Strain)] = (FinalArcLength)

(8)

 

which becomes:

(9)

 

where:

L₁ = Initial arc length
L₂ = Final arc length
α = Thermal expansion coefficient
t₁ = Temperature in initial state
t₂ = Temperature in final state
T₁ = Horizontal tension in initial state
T₂ = Horizontal tension in final state
ρ = Cross-sectional area of wire
E = Modulus of elasticity of wire

Weight-Span and Uplift

Unlike the horizontal tension, the vertical component of wire tensions is significantly affected by changes in wire geometry due to wind forces. To accurately calculate the vertical force each WEP supports, this new wire geometry must be determined.

Starting with the general catenary geometry for wires:

(10)

 

where:

x_c = Horizontal coordinate of the lowest point
y_c = Vertical coordinate of the lowest point
a = Catenary constant H/w, where H is the horizontal tension and w is the linear weight of the wire

 

Taking the derivative:

(11)

 

The line tangent to the arc of the wire at our pole (0,0) is therefore:

(12)

 

This is, in turn, the direction of the force from the wire.

 

From geometry, the vertical component of the force is:

(13)

(14)

 

where m₁ and m₂ are the slopes of two lines, and θ is the angle between them. The horizontal portion of our situation is clearly parallel to the x-axis and has a slope of zero. Substituting this into the equation, we get:

(15)

 

(16)

 

Using this information, we now can set y′(0) = m₁ = tan(θ) and apply Equation 13.

 

(17)

 

Or:

(18)

 

Looking back at the equation for arc length, we see that the right side of the equation is precisely the arc length from the current pole at (0,0) to the lowest point, (x_c,y_c) multiplied by -w. The negative would imply the force is downward. Equation 18 is the precise equation SPIDAcalc uses to calculate the vertical force each wire exerts on the poles to which it is attached.

 

Test Scenario

As a sanity check, if we now expand sin h  we get:

(19)

 

(20)

 

When H ≫ x_cw, the second term is negligible, and the first term dominates.

 

(21)

 

That is to say, the vertical force is approximately equal to the distance to the lowest point, multiplied by the linear weight of the wire.

Sag Calculations

Sag is defined as the maximum separation between the catenary curve and the inclined straight line (line of sight) between supports.

In the figure above, the values reported in SPIDAcalc are labeled including what the reported sag is measuring, and where it occurs in the span. SPIDAcalc also reports the lowest point in the catenary curve, which, for typical scenarios, occurs near the point of maximum sag.

The inclined straight line, or span line, is defined as such:

(22)

 

The vertical separation between Y (x) and the catenary curve y(x) is simply:

(23)

 

(24)

 

Solving for the maximum of D(x), x_D, we get:

(25)

 

Solving for x_D results in:

(26)

 

The vertical coordinate y_D is then calculated from the catenary curve:

(27)

 

Canceling terms:

(28)

(29)

 

Simplifying the equation and combining terms:

(30)

 

Using these results in Equation 23 gives us a final equation for sag:

(31)

 

The reported sag, as well as the arc length that the wind forces act upon, are calculated before load factors are applied. This is to ensure that worst-case sag and wind forces are used, as increasing the tension with a load factor would decrease the sag and arc length of the wire.This also avoids the rare edge case in which load factors would be set to zero to avoid a particular load affecting the structure. In the scenario of a zero-tension load factor, sag would be infinite and the wind forces would act on an infinite area, causing nonsensical results.

Uplift

Wire sag calculations become particularly confusing for uplift scenarios. For this reason, it is worth taking a moment to discuss the differences in wire geometry when uplift occurs, even if there is not a significant change in the mathematics.When uplift occurs, the theoretical lowest point in the catenary curve is not between the two supports, as shown in the figure below. When this happens, the calculated distance to the lowest point becomes negative, while the point of maximum sag remains near the center of the span. In SPIDAcalc, an uplift scenario can be confirmed by looking at the reported vertical force for each wire. If the wire has a positive vertical force, the wire is pulling upwards on its attachment point.

 

Tension from Sag and Temperature

Using Equation 24, it is straightforward to derive the tension of the wire if given the separation, D, between the line of sight and the wire.

The derivation begins by using the approximation for cosh, in the equation for D(x):

(32)

 

Using the Taylor series for cos h:

(33)

 

Simplifying and canceling terms:

(34)

 

Using the approximation for y_c from Equation 3:

(35)

 

From here, the derivation is simply algebra and canceling terms:

(36)

 

(37)

 

(38)

 

(39)

 

(40)

 

 

 

Finally, we can solve for the catenary constant, a, of the wire:

(41)

 

which leads us to the final equation solved for the horizontal tension of the wire as dependent on the input separation between line of sight and the wire geometry D, the horizontal distance from the current wire attachment that the separation was measured x, the horizontal distance to the far attachment s, and the linear weight of the wire w:

(42)

 

Once the tension of the wire is solved for, along with the temperature to which the sag and tension refer, the same calculation as before can be used to determine the wire tension at any other temperature or state.