Comparison of observation of actual span with calculated model
No wind loading
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 Inverted-L 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.
Table 1: Observations of actual span
| 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.
Table 2: Estimate of sag using parabolic approximation
| 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=(WS2)/(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.
Table 3: Estimate of sag using catenary
| 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.
Table 4: Comparison of sag using different methods
| Method |
Value |
| Measured |
0.324m |
| Parabolic approximation |
0.319m |
| Catenary |
0.316m |
Wind loading
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.
Table 2: Estimate of sag using parabolic approximation
| 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=(WS2)/(8T) |
Table 3 shows a catenary of the span loaded to 100% of GBS with safety
factor 3.5.
Table 3: Estimate of sag using catenary
| 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.
Changes
| Version |
Date |
Description |
| 1.01 |
24/09/2006 |
Initial. |
| 1.02 |
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| 1.03 |
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| 1.04 |
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V1.01 24/09/06 02:08:46 -0600.
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