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The following compares measurements of a span with two calculated models, a parabolic approximation and a true catenary model.
The span is part of an InvertedL antenna, the wire is 2mm HDC.
The span was measured using a theodolite and level staff. A very slight breeze caused the wire to sway a little adding error to measurement of the vertical angle at the centre of the span. The measurements and data reduction are reported in Table 1.
Serial  Item  Value  Notes 
1  Vertical height of collimation plane  1.40m  measured using theodolite and staff 
2  Horizontal distance to LH end  16.1m  measured using 10:1 stadia marks on theodolite cross hair and staff 
3  Horizontal angle to LH end  0°00'00"  measured using theodolite 
4  Vertical angle to LH end  68°57'45"  measured using theodolite, zenith=0 
5  Horizontal distance to RH end  16.6m  measured using 10:1 stadia marks on theodolite cross hair and staff 
6  Horizontal angle to RH end  71°40'40"  measured using theodolite 
7  Vertical angle to RH end  73°14'48"  measured using theodolite, zenith=0 
8  Horizontal angle to "middle"  35°50'20"  measured using theodolite 
9  Vertical angle to "middle"  68°00'30"  measured using theodolite, zenith=0 
10  Horizontal distance to "middle"  13.05m  calculated from 2 & 5 
11  Height of LH end  7.59m  calculated using 1, 2 & 4 
12  Height of RH end  6.40m  calculated using 1, 5 & 7 
13  Height of "middle"  6.67m  calculated using 1, 9 & 10 
14  Run of span  19.15m  calculated using cosine rule from 2, 5, 3 & 6 
15  Rise of span  1.20m  calculated using 11 & 12 
16  Sag  0.324m  calculated using 11, 12 & 13 
Table 2 shows a parabolic approximation of the span. The greatest source of error is the measured tension due to friction in the halyard pulley and accuracy of the spring scale.
Serial  Item  Value  Notes 
1  Tension  40N  measured using spring scale 
2  Force per unit length (weight)  0.28N/m  calculated from material properties 
3  Span  19.15m  measured 
4  Sag  0.319m  calculated from 1, 2 & 3, Sag=(WS^{2})/(8T) 
Table 3 shows a catenary of the span. The greatest source of error is again the measured tension due to friction in the halyard pulley and accuracy of the spring scale.
Serial  Item  Value  Notes 
1  Tension  40N  measured using spring scale 
2  Force per unit length (weight)  0.28N/m  calculated from material properties 
3  Span  19.15m  measured 
4  Rise  1.20m  
5  Sag  0.316m  calculated from 1, 2, 3 & 4 
Table 4 compares the sag derived from the three methods. The three results are within reasonable tolerance given the accuracy of the measurements and known data.
Method  Value 
Measured  0.324m 
Parabolic approximation  0.319m 
Catenary  0.316m 
Measurements were not made under wind loading owing in part at least to the difficulty in proving a calibrated wind for observation. Nevertheless this section shows the sag required to fully load the wire, and therefore the minimum sag to assure survival of the design wind speed.
Wind loading at 40m/s (144km/h) would increase the force per unit length to 2.32N/m.
Table 2 shows a parabolic approximation of the span loaded to 100% of GBS with safety factor 3.5.
Serial  Item  Value  Notes 
1  Tension  386N  measured using spring scale 
2  Force per unit length (weight)  2.32N/m  calculated from material properties 
3  Span  19.15m  measured 
4  Sag  0.276m  calculated from 1, 2 & 3, Sag=(WS^{2})/(8T) 
Table 3 shows a catenary of the span loaded to 100% of GBS with safety factor 3.5.
Serial  Item  Value  Notes 
1  Tension  386N  measured using spring scale 
2  Force per unit length (weight)  2.32N/m  calculated from material properties 
3  Span  19.15m  measured 
4  Rise  1.20m  
5  Sag  0.278m  calculated from 1, 2, 3 & 4 
The two methods produce almost identical sags, the error in using the parabolic approximation for shallow spans is very low.
Version  Date  Description 
1.01  24/09/2006  Initial. 
1.02  
1.03  
1.04  
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