Theatre Design & Technology - Fall 2019 - 48

assumes the cable is loaded by a constant
weight per foot in the horizontal direction,
rather than a constant weight per foot
along the cable length.
In most cases, the sag of the bridle
leg is so slight that the difference in the
lengths of the straight-line cord and the
curve of the sagging bridle leg is not significant. But, when the bridle legs are
long, and the weight of legs are relatively
high compared to the load, the difference
can be significant. When this occurs, the
rigger must know the length of this curve
in order to make the actual bridle leg the
correct length.
The amount of sag in the bridle leg,
and therefore the length of the bridle leg,
is determined by four factors: the length
of the leg, the weight per foot of the material (usually wire rope) that the leg is
made from, the tension on the leg, and the
angle of the cord line. As a result,
* The longer the leg, the greater the sag
* The less tension on the leg, the greater
the sag
* The greater the leg weights per foot, the
greater the sag
* The closer the angle of the cord is to
horizontal, the greater the sag
Because we assume that the bridle
legs are weightless, that leads us to another estimation-that the tension on the
bridle legs is the same at the apex (the
lower end) and at the anchor (the upper
end). It is not. The Upper End Tension of
the bridle legs is greater than the Lower
End Tension because it must include part
of the weight of the wire rope that makes
up the bridle leg. This may seem trivial,
but it is necessary for accurate modeling
of the rigging condition being analyzed.
This additional tension is calculated using the equations:
Upper End Tension on Leg 1 = Lower End
Tension on Leg 1 + (V1 x Weight per foot of
Leg 1)
Upper End Tension on Leg 2 = Lower End
Tension on Leg 2 + (V2 x Weight per foot of
Leg 2)
The diagram that follows illustrates
the general relationship between the
parts of the bridle. Lines L1 and L2 represent the theoretical lengths of the bridle
legs, and lines S1 and S2 represent the
actual bridle legs with their sagging catenary curved shapes.

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THEATRE DESIGN & TECHNOLOGY

|

FALL 2019

To find the length and the tension of
the bridle legs (S1 and S2), we use a twosegment method that requires the equations of equilibrium be solved iteratively
(the process of repeating a set of instructions a specified number of times or until
a specific result is achieved). Creating a
"loop" to perform these tasks can be done
in Microsoft Excel by creating a VBA (Visual Basic for Applications) macro.
The authors created a spreadsheet
to demonstrate how such a macro works.
Download it here: www.springknollpress.
com/RiggingMath/downloads/SagCalc.
xlsm Note: spreadsheets that contain
macros end in .XLSM instead of .XLSX.
Also, when you open this spreadsheet,
Excel will warn you that it contains a
macro and ask you if you want to enable
or disable the macro. Of course, you want
it enabled.
To calculate the length of the bridle
legs, we model the two bridles as catenary curves and identify the line tensions to
produce horizontal and vertical equilibrium for the specified load position. We
start with the weightless line estimate for
the lower end tension on the bridle legs.
This is not the true lower end tension
under the catenary model. Because of
the line weight of the cable, the angle of
the LINE TENSION is LOWER than the
cord slope at the load. Thus, to produce
adequate vertical force, the line tension
must be higher than that of the weightless
line estimate, so that vertical force on Leg

Fig. 1

1 plus the vertical force on Leg 2 is equal
to the Load. Because vertical force on Leg
1 must be equal to the true lower end tension of Leg 1 times the sine of the true angle, our algorithm methodically increases
the line tension until these equilibrium
conditions are satisfied.
When the algorithm has determined
that the horizontal forces on the apex are
equal and the sum of the vertical forces
at the apex equal the load, the process
is terminated and the calculated lengths
and tensions, along with the message
"Equilibrium Conditions Satisfied" are
displayed in the spreadsheet. The macro
also checks to ensure that there are no
calculation errors that would cause the
program to freeze. If so, the message "Solution could not be found" is displayed.
Our macro also calculates and displays the weight of each leg, the angle of
the cord for each leg, and the mid-span
perpendicular deflection distance for
each leg in order to help the user understand more about each leg. Let's look at
the four sections of the spreadsheet.
The upper left section is the diagram
of the bridle setup displayed above. This
diagram relates to the upper right section of the spreadsheet, the part where
the user enters the data for the bridle.
In the Tool (see figure 1), the usermust enter seven data items, three related to Leg 1 (Blue), three related to Leg
2 (Orange), and the Load on the apex
(Green). We have given you the weight
per foot of three sizes of wire rope. Each
leg can be made of a single size of wire
rope, but the two legs do not have to be
from the same size wire rope. After the
user has filled-in these cells, the user
needs to click on the button that is labeled "Click to Calculate Bridle Length."
Doing this runs the macro and updates
http://www.springknollpress.com/RiggingMath/downloads/SagCalc http://www.springknollpress.com/RiggingMath/downloads/SagCalc
Theatre Design & Technology - Fall 2019

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