The quarter method

3.5.3.3 Perpendicular tangent offset method

If the deflection angle is small (<20°), the curve can be set out by measuring perpendicular offsets from the tangents. If _Ox is the offset perpendicular to the tangent at _{Q, which is} at a distance of _{x from T}1, then drawing PP1 perpendicular to

the radial line _T1O. from the triangle PP1O,

or,

or,

The above relation has been used to create Table 3-3 based on selected curve radius, to determine the perpendicular offsets for setting out the curve.

Procedure for setting out curve: 1. Set out _{PI and find the deflection}

angle using Chart 3-1. If the deflection angle is less than _20°, proceed with step 2, otherwise use another method.

2. Set out _{BC and EC at distance equal} to the determined tangent length from _PI.

3. Set out temporary pegs along the tangents at 10m distance starting from _{BC and EC and as many as will} fit within the tangent length.

4. From Table 3-3 using the given radius, read off the offsets and fit as many offset points as possible.

Note that this method should only be used for gentle curves with deflection angle less than 20°. For curves with deflection angle greater than 20° for tangents longer than tabulated, another method as explained above should be used.

     PO

1 2



PO

2



PP

1 2





2 2 2 x

R O

R x

2 2 x

O R 

R x

Figure 3-22: Perpendicular Tangent Offset Method

Figure 3-23: Perpendicular Offset Method x

Q Ox

Perpendicular offsets from tangent (m) for deflection angle < 20° Tangential

Distance x (m) 10 20 30 40 50 60 70 80 90 100

Curve Radius

R (m) Offset distance Ox from beginning and end of curve (m)

150 0.33 1.34 160 0.31 1.25 170 0.29 1.18 180 0.28 1.11 2.52 190 0.26 1.06 2.38 200 0.25 1.00 2.26 210 0.24 0.95 2.15 220 0.23 0.91 2.06 230 0.22 0.87 1.96 3.50 240 0.21 0.83 1.88 3.36 250 0.20 0.80 1.81 3.22 260 0.19 0.77 1.74 3.10 270 0.19 0.74 1.67 2.98 280 0.18 0.72 1.61 2.87 290 0.17 0.69 1.56 2.77 4.34 300 0.17 0.67 1.50 2.68 4.20 350 0.14 0.57 1.29 2.29 3.59 5.181 400 0.13 0.50 1.13 2.01 3.14 4.526 6.17 450 0.11 0.44 1.00 1.78 2.79 4.018 5.48 500 0.10 0.40 0.90 1.60 2.51 3.613 4.92 6.44 550 0.09 0.36 0.82 1.46 2.28 3.283 4.47 5.85 600 0.08 0.33 0.75 1.33 2.09 3.008 4.10 5.36 6.79 8.39 650 0.08 0.31 0.69 1.23 1.93 2.775 3.78 4.94 6.26 7.74 700 0.07 0.29 0.64 1.14 1.79 2.576 3.51 4.59 5.81 7.18 750 0.07 0.27 0.60 1.07 1.67 2.404 3.27 4.28 5.42 6.70 800 0.06 0.25 0.56 1.00 1.56 2.253 3.07 4.01 5.08 6.27 850 0.06 0.24 0.53 0.94 1.47 2.120 2.89 3.77 4.78 5.90 900 0.06 0.22 0.50 0.89 1.39 2.002 2.73 3.56 4.51 5.57 950 0.05 0.21 0.47 0.84 1.32 1.897 2.58 3.37 4.27 5.28 1000 0.05 0.20 0.45 0.80 1.25 1.802 2.45 3.21 4.06 5.01

Other approximate methods which may also be used are explained below:

3.5.3.4 The string line method Procedure for setting out curve:

1. Find the deflection angle using Chart 3-1, establish _{BC and EC at equal distance T} from _{PI. T should be divisible by an integer} number, i.e. _{4, 5, 6 etc.}

2. Divide the tangents into equal number of parts and number them as shown in Figure 3-24.

3. The points on the curve lie on the intersection of the lines _{1-1 with 2-2, 2-2 with 3-3, 3-3 with} 4-4 and so on, established with sisal twine or using ranging rods. Note that the number of points on the curve will always be one less than the number of parts into which the tangents are divided.

4. Place intermediate pegs if necessary to form smooth curve with string line.

5. Establish centreline - and chainage pegs. 3.5.3.5 The Offset Method

This is a trial and error approach for re-establishing existing alignments.

For any given radius and assumed tangential distance x the offset y is given by the formula:

From this formula, Table 3-4 has been generated for determining the offsets for setting out the curve.

Radius R (m) Distance x (m) Offset y (m) Distance x (m) Offset y (m) 500 10 0.20 20 0.80 450 10 0.22 20 0.89 400 10 0.25 20 1.00 350 10 0.29 20 1.14 300 10 0.33 20 1.33 250 10 0.40 20 1.60 200 10 0.50 20 2.00 150 10 0.67 20 2.67 100 10 1.00 20 4.00 50 5 0.50 10 2.00

Table 3-4: Data for setting out by offset method

2

x

y

_R

Procedure for setting out the curve:

1. Establish _{PI as the intersection of the straight} centre lines.

2. Choose the beginning of the curve _{BC on} one tangent.

3. Assume distance _{x (normally 10m) and use} Table 3-4 to determine the offset _y.

4. Set out point _{A on the tangent at distance x} from _BC.

5. Set out point _{B at perpendicular offset y/2} from A.

6. Set out point _{C at distance x from B on the} extension of _BC.

7. Set out point _{D at perpendicular offset y from} C.

8. Repeat until you reach the other tangent at point _H.

9. Set out point _{I on the extension of F-H.} 10. Set out point _{J at perpendicular offset y/2}

from _{I. This should be on the straight centre} line.

lf it is not possible to reach the other tangent satisfactorily, you will have to select another starting point _{BC and} repeat the setting out.

3.5.4 Problems in setting out curves

Due to certain site circumstances, the following cases of constraints may occur during setting out of curves: • Case 1. Point of intersection (_{V) not accessible.}

• Case 2. Point of curve (_T1) not accessible.

• Case 3. Point of tangency (_T2) not accessible.

• Case 4. Three Straights to be joined by a Curve. • Case 5. Curve must pass through a Fixed Point. The solutions of the above problems are discussed below.

Case 1. Point of intersection (V) not accessible.

Sometimes, the point of intersection (_{V) falls in a lake, river,} pond, thick forest or any other inaccessible place. In such a case, it is not possible to determine the deflection angle (f) by the methods described earlier. It is also not possible to locate the points _T1 and T2 by measuring the tangent distance T from

V. Therefore the following procedure may be employed to overcome the situation in setting out the curve.

Procedure:

1. Establish two points M and N on tangents which are inter- visible.

2. Find angles a and b (using chart 3-1) 3. Measure the length _{MN accurately} 4. Deflection angle f = a + b

5. Calculate the lengths _{MV and NV by solving the triangle MNV (using the Sine Rule)}

Figure 3-26: Point of intersection not accessible

Figure 3-25: The Offset Method

x

i.e.

sinMV

sinMN

sinNV

sin

sin

MV MN









sin

sin

NV MN









6. Calculate the tangent lengths _VT1 and VT2

 

1 2

tan 2

VT VT R





7. Determine the lengths _MT1 and NT2:

1 1

MT VT MV



2 2

NT VT NV



8. Locate the points _T1 by measuring a distance MT1 from M. Likewise locate point T2

9. Use the Quarter method to set out the curve.

Case 2. Point of curve (_T1) not accessible.

Procedure:

1. Select a point M on the line _VT1 near the point T1 but clear of the obstruction. Measure the distance MV.

2. Determine the distance _MT1

 

1

Tangent length

MT



T MV

3. Select another point _{N on the line VT}1 produced on the

other side of the obstruction.

4. Determine the chainage of _{N from the field records.} 5. Select another point _{Q on one side of the line MN,}

such that _{NQ and MQ are perpendicular, and measure} distances _{MQ and NQ. Determine the length MN using} Pythagoras theorem:



 

2



2

MN



MQ



NQ

6. Determine the chainage of T1:

1 1

Chainage of T

Chainage of N MN MT



7. Determine the chainage of _T2:

2 1

In document Good_Practice_Guide_to_Labour-Based_Methods_2013.pdf (Page 47-52)