Traverse Quality and Curve Setting-Out for a Two-Pier Bridge
A total station traverse of five stations has been run to establish control for a bridge with two piers 60 m apart. The sum of the measured interior angles is 540°00′42″; the perimeter of the traverse is 640 m; the unadjusted coordinate differences close with ΣΔE = +0.058 m and ΣΔN = −0.052 m. Departmental specification for control traverses is 1:5000 for linear misclosure and 20″√n for angular misclosure. The bridge design coordinates are on a national grid, computed with a projection scale factor of 0.9996 at the site. The mean site elevation is 300 m above the reference ellipsoid; earth radius can be taken as 6371 km. Separately, a simple circular curve of radius 150 m with a deflection angle of 48° has to be set out from a point of intersection whose chainage is 1250.000 m. The pier positions must be set to ±10 mm.
1. Evaluate the traverse against the specification and describe the adjustment procedure, with a numerical example.
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Angular misclosure: theoretical sum for a 5-sided closed traverse = (2n − 4) × 90° = 540°00′00″; measured 540°00′42″, so misclosure = +42″. Allowable = 20″√5 = 20 × 2.236 = 44.7″. Within tolerance, so the angles can be adjusted.
Correction per angle = −42″/5 = −8.4″ (equal distribution), applied before computing bearings.
Linear misclosure: W = √(0.058² + 0.052²) = √(0.003364 + 0.002704) = √0.006068 = 0.0779 m. Accuracy = 0.0779/640 = 1 in 8,215, better than the 1:5000 requirement.
Adjustment (Bowditch, suitable for a compass/total station traverse of similar leg lengths): correction to each ΔE = −0.058 × (leg length/640), and to each ΔN = +0.052 × (leg length/640). For a 128 m leg: ΔE correction = −0.058 × 0.20 = −0.0116 m; ΔN correction = +0.052 × 0.20 = +0.0104 m.
Judgement for the bridge: the traverse meets the departmental specification, but ±10 mm pier positions over 60 m require about 1:6000 relative accuracy between piers plus an independent check. Set out the piers from the adjusted coordinates, then verify the 60 m pier-to-pier distance by a direct calibrated baseline or GNSS static observation before drilling. Do not rely on a single traverse adjustment for setting out permanent bridge foundations.
2. Compute the elements of the simple circular curve and the chainages of tangent points, and comment on whether the curve can be accommodated.
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Elements (Δ = 48°, R = 150 m):
- Tangent length T = R tan(Δ/2) = 150 × tan 24° = 150 × 0.44523 = 66.78 m
- Curve length L = πRΔ/180 = π × 150 × 48/180 = 125.66 m
- Chainage of PI = 1250.000 m
- Chainage of first tangent point T1 = 1250.000 − 66.78 = 1183.22 m
- Chainage of second tangent point T2 = 1183.22 + 125.66 = 1308.88 m
- Apex distance E = R[sec(Δ/2) − 1] = 150(1.0946 − 1) = 14.19 m
- Mid-ordinate M = R[1 − cos(Δ/2)] = 150(1 − 0.91355) = 12.97 m
Setting out: from T1 and T2, peg at 10 m chainages using deflection angles δ = 1718.87 × c / R minutes (for c = 10 m, δ ≈ 114.6′ = 1°54′36″), or use the long-chord method with the tangent points already established.
Accommodation: the curve needs 66.78 m of tangent on each side and 12.97 m of shift at mid-curve. Check that at chainage 1308.88 the alignment has cleared any structure, and that for the bridge the pier positions are on the straight between tangents, not on the curve. Where the curve falls within the bridge length, the deck must be curved or widened - confirm with the structural designer before pegging.
3. Explain the national grid versus ground-distance discrepancy and quantify the error for a 1 km baseline.
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Cause: design coordinates are grid coordinates. Total station distances, however, are measured on the ground, and the local survey is at a mean elevation above the reference ellipsoid. Two reductions separate them:
- Projection scale factor on the grid = 0.9996 (Transverse Mercator, as used in Indian national grids).
- Elevation (sea-level) factor = R/(R + H) = 6371/(6371 + 0.300) = 0.999953.
Combined scale factor = 0.9996 × 0.999953 = 0.99955.
Error: ground distance must be multiplied by 0.99955 to get the grid distance. For 1 km: 1000 × (1 − 0.99955) = 0.45 m. Over the 60 m pier spacing, 60 × 0.00045 = 0.027 m = 27 mm, which is nearly three times the ±10 mm tolerance - enough to put the pier in the wrong place if the mixing of systems is not detected.
Procedure: set the total station to output grid coordinates and grid bearings (apply the combined factor to all measured distances and use the grid convergence / rotation), or work entirely in ground coordinates and transform once at the end. Never mix measured ground distances with grid coordinates. Record the combined factor, convergence and the transformation parameters in the setting-out report, and confirm with an independent check between the two pier positions before fixing the foundations.