School of the Built Environment

Setting Out of Transition Curves

1. Introduction

Transition curves, as their name suggests, are designed to allow a smooth transition from a straight section to a (circular) curve and to allow the gradual introduction of super-elevation (also known as “banking” on a racing circuit and “cant” on a rail track). Setting out of transition curves is done using the same techniques as for circular curves (theodolite and tape or EDM/co-ordinates) and the purpose of these notes is to introduce the calculations required to produce the necessary setting out data.

2. Forces on vehicles moving round curves

A moving object will continue moving in a straight line at constant velocity unless acted upon by a force. If the force acts in the same direction as the motion, then the object will accelerate or decelerate; if the force acts perpendicular to the line of motion, then the object will change direction. In order to turn a vehicle round a curve, it is therefore necessary to apply a sideways force – this is done by turning the front wheels. There are two important considerations when designing roads (and railways):

Comfort of occupants: When the vehicle moves round the curve, the occupants feel the sideways force because their bodies wish to continue moving in a straight line. If this force is too great (curve radius too tight) or it is applied too rapidly (moving from a straight to a sharp circular curve) then the occupants will feel discomfort.

Safety of vehicle: The sideways force is transmitted to the vehicle via the tyres at road surface level. If the force is too great for the grip of the tyres, skidding may occur. If the centre of gravity of the vehicle is high, then overturning may occur.

The centrifugal force, acting outwards as a vehicle moves round a curve, is expressed as follows:

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Figure 1: Centrifugal force exerted on a vehicle travelling round a curve.

P = W V2 centrifugal forceg R

Or P = V2 centrifugal ratioW g R

= V2 with V in km/h127 R R in m

g = 9.807 m/s2

For practical purposes, a “comfort range” is adopted for values of the centrifugal ratio, P/W:

Roads: P/W between 0.21 and 0.25 (typically 0.22)Railways: P/W = 0.125

From this can be derived an expression for the minimum radius required for a particular design speed:

Minimum = V2

Radius (P/W) x 127

This should be regarded as a minimum value; the actual radius adopted (which may be larger) may depend upon other factors such as the overall alignment of the road.

3. Entry to a curve

Centrifugal force leads both to passenger discomfort and to slipping/overturning forces on vehicles. On high-speed roads, and on many modern medium-speed roads, this is counteracted by the application of super-elevation. This is the tilting of the

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Velocity V

Centrifugal force P

Radius R

Vehicle

Circular curve

R

W

P

road surface so that the resultant force of the weight of the vehicle and the centrifugal force is close to normal with the road surface. In railway engineering, this is known as “cant”; on a motor racing circuit it would be called “banking”.

Figure 2: Super-elevation

The super-elevation angle, , is usually expressed as a gradient (e.g. 1 in 14) and is related to the velocity of the vehicle and the curve radius.

Clearly, this cannot be applied suddenly, but must be introduced gradually. Furthermore, sudden entry from a straight into a circular curve may result in “acceleration shock” to the passengers.

For both of the above reasons, it is preferable to change the radius of curvature gradually from the straight to the circular curve steadily along a transition curve.

Design of transition curves is based upon the change of curvature (the inverse of radius) at a steady rate from the straight (curvature = 0) to the circular portion of the curve (curvature = 1/R). The rate of change of radial acceleration, q, must also be kept constant whilst the vehicle negotiates the transition curve at constant velocity over time t. This results in the following relationships:

time taken to travel t = L/Valong transition curve

radial acceleration aR = V2/R

rate of change of q = aR / tradial acceleration

= (V2/R)/(L/V)

= V3/RL

i.e. L = V3

46.7 q R

with V in km/h and R in m.

An acceptable value of q is found to be 0.33 m/s3. This gives an expression for the minimum length of transition curve required. The actual length of transition used may also depend upon previous experience.

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Road surface

W

P

Resultant force

The completed curve now consists of

an entry transition

a central circular portion

an exit transition

Both of the transitions will have identical geometry, but will be of opposite hand.

3.1. Shape of the transition

To satisfy the criterion that rate of change of curvature must be constant along the transition curve, we must calculate the shape of the curve as follows.

At a distance l into the transition, measured from the transition tangent point T1, the radius of curvature will be r. At l = 0, i.e. at the transition point, r is infinite, and reduces to the circular curve radius, R, at the circular curve tangent point T1, where the transition curve joins the circular portion. This situation is shown in the diagram below:

Figure 3: Shape of the transition curve.

The radius is related to the length l by the equation derived above, i.e.

l = V3

46.7 q r

which can be re-written l = K since velocity is r constant

From figure 3, because angle d is small, we can write:

r d = dl

i.e. d = 1 dl = l dl r K

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d

dl

T1

T1

R

l

Entry tangent

Tangent to circular curve

This may now be integrated, inserting limits:

= l2 = l2

2 K 2 r l

= l2

2 R L

or l = (2 R L )½

This is the equation of the clothoid or Euler spiral. It is regarded as the ideal shape for the transition curve. The angle consumed by the transition curve, , is

= L (radians)2 R

= L x 180 (degrees)2 R

The clothoid is regarded as the “ideal” shape for a transition curve, as it satisfies our criteria for all values of . However, although it appears simple to write down, it is not easy to translate into a form that can be used readily for setting out in the field.

3.2. Setting out the transition curve

The curve that has now been formed is composite in nature, i.e. it contains a number of different curves, which are designed to fit together. There is a transition curve at each end and a central circular portion. As a whole, the composite curve is symmetrical, though it should be remembered when setting out that, unlike a circular curve, transition curves are not themselves symmetrical – one end has a greater radius of curvature than the other, so they are not reversible!

A transition curve can be set out in exactly the same way as a circular curve, i.e. by setting up a theodolite at the tangent point, aligning the telescope with the Intersection Point, swinging through the appropriate angles and taping chords between consecutive pegs. This is the simplest method, and the one on which the calculations are based. Other methods can be adopted, such as setting out using a Total Station set up off the curve, or from a point other than the tangent point for some practical purpose; however, in each case the same initial calculations are made and then adapted to suit as required.

In order to derive suitable equations relating chord length and deflection angle, we must examine the shape of the transition curve. First, look at it in terms of orthogonal co-ordinates, using the tangent as a reference axis:

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Figure4: Co-ordinates of the transition curve.

Imagine a point some distance, l, along the curve from the tangent point. It will have co-ordinates X and Y as shown in figure 4 and the angle between the tangent and the line joining T1 and p is . Direct calculation of for specific values of l (i.e. chainages) requires the use of tables of standard data, though such calculations would be made now by computer. For hand calculation, a series expansion of the clothoid is available:

X = l - l5 + l9 - l13 +…..40(RL)2 3456(RL)4 599040(RL)6

Y = l3 - l7 + l11 -….. 6RL 336(RL)3 42240(RL)5

tan = + 3 - 5 + 7 -….3 105 5997 198700

These are clearly impractical. However, for values of up to about 3, the higher order terms can be neglected, resulting in simpler formulae from which setting-out data can be readily calculated:

Cubic Spiral: Y = l3/6RL = /3

Cubic Parabola: Y = X3/6RL = /3

These are sufficiently accurate for most applications where hand calculation is likely to be employed, and the following procedure is based upon the cubic spiral, which is a better approximation to the clothoid.

3.2.1.Setting-out calculations for the cubic spiral

The equations in the above section allow us to calculate the deviation angle (i.e. the angle set on the theodolite when it is set up at the tangent point) directly. Note that, unlike the circular curve calculations, this is not cumulative.

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X

Y

p

T1 Entry tangent

T1

Deviation angle = /3 = l2/6RL (radians)

i.e. = 180 l2

6 R L

The tangent length of the composite curve (i.e. transition + circular + transition) will be needed for setting out the first tangent point, T1, and for calculation of the chainages of setting out points. It is obtained by first considering the properties of an equivalent circular curve, which has the same centre as the circular portion of the curve to be set out, but has a slightly larger radius. The relationship between the two curves is shown in the following diagram:

Figure 5: Equivalent circle and shift for a composite curve.

It can be shown that the line OQ, from the centre of the circle to the tangent point of the equivalent circle, bisects the transition curve T1 T1 and that the shift, S, is

S = L2 / 24R

From the dimensions of the equivalent circular curve, we can obtain its tangent length, distance QI:

QI = (R + S) tan (/2)

And, since length T1Q is half the length of the transition curve, we can write

T1I = L / 2 + (R + S) tan (/2)

This allows us to establish the chainage of the first tangent point T1 and hence the distance from the tangent point to the chainage pegs along the transition curve as well as the chainage of the tangent point to the circular curve, T1.

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Q

S

I

T2T1T1

T2

O

R

3.2.2.The circular portion of the curve

The central circular arc can be set out in exactly the same way as a purely circular curve, i.e. from the tangent point using theodolite and tape. The difference in the case of a composite curve is that the tangents to the circular arc are not the same as the entry and exit tangents for the composite curve – the intersection angle will be lessened and the point at which they intersect will be closer to the curve than the point I, as shown in the diagram below.

Figure 6: Geometry of the circular portion of a composite curve.

It can be seen that the angle of the circular arc is

C = 2

Hence the length of the circular arc is

LC = R C

180

This allows the chainages of the two remaining tangent points, T2 and T2, to be calculated. After calculating the positions along the circular arc of the chainage pegs, (cumulative) deflection angles for the entry and exit sub-chords and the standard chord are calculated in the usual way:

d = 28.648 lC / R

The curve can now be set out by setting up the theodolite at T1 and taping between pegs. In order to do this we must establish the orientation of the tangent, since we can no longer sight onto the Intersection Point, I. This can be done by looking at the geometry of the transition curve.

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I

T2T1T1

T2

O

-2

R

Figure 7: Transition curve and Back Angle.

From the properties of the cubic spiral, we know that

= / 3

and the back angle can therefore be calculated as

back = 2 / 3angle

The theodolite can therefore be oriented at T1 by sighting back to T1 and turning the alidade through 180 + 2/3 (clockwise). The telescope will then be facing along the tangent to the circular arc, ready to continue setting out.

Note that, for a left-hand curve, the alidade must be rotated through 180 - 2/3 (clockwise). This is because all of the curve angles will have been reversed, but the theodolite is still graduated in a clockwise direction.

3.2.3.The exit transition

The entire circular curve has now been set out, assuming that there have been no obstructions, from one tangent point, usually the entry tangent point T1. However, because the transition curve is not symmetrical, the formulae used above to calculate deviation angle are only valid for angles from the start of the transition, i.e. T1 or T2. We must therefore move directly to T2 to set out the second transition curve. The position of T2 can be found by linear measurement along the exit straight from the intersection point, and checked from T1 by measuring the angle between either straight and the line T1-T2, which should be equal to /2.

Figure 8: Checking the position of the two tangent points.

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T2T1

I

required orientation

Tangent to circular arc

back angle

T1

Entry tangent

T1

/2

/2

It is important to remember that chainage is still running from T2 to T2, even though we are calculating deviation angles from T2 to T2 and, in all probability, setting out from T2. Also worth checking in the field with the theodolite set up at T2, before the transition is set out, is the position of T2. The formula used to calculate deviation angles in the second transition is exactly the same as before, except that it will be of the opposite hand.

The procedure should become clear when the following example is followed.

4. Example

A right-hand composite curve is to be set out at 25m intervals from the following data:

Intersection angle 09 32 46Forward chainage of IP 3738.621mRadius of circular curve 850mLength of transition 70m

Calculations:

Shift: S = L2 / 24 R = 702 / (24 x 850)

= 0.240m

Tangent length:T1I = L/2 + (R + S) tan(/2)

= 35 + 850.24 tan (09 32 46/2)

= 105.994m

Angle consumed by transition curve: = L / 2R x 180 / degrees

= 02 21 33

Chainages: first transitionCh T1 = 3738.621 - 105.994 = 3632.627m

Ch T1 = 3632.627 + 70 = 3702.627m

Setting-out pegs for the first transition are therefore required at the following chainages: 3650, 3675, 3700 and 3702.627 (T1). In each case, the following formula is used:

= 9.5493 l2 / RL

and the data tabulated as below:

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Chainage l (angle set on theodolite)

3632.627 (T1) 0 -3650 17.373 00 02 543675 42.373 00 17 173700 67.373 00 43 433702.627 (T1) 70 00 47 11

Check: 00 47 11 x 3 = 02 21 33 =

Circular curve

Back angle = 2 / 3 = (02 21 33) x 2/3

= 01 34 22

Length of curve:C = - 2 = 09 32 46 - 2(02 21 33)

= 04 49 40

LC = R C / 180 = x 850 x (04 49 40) / 180

= 71.622m

Chainages:Ch T2 = 3702.627m + 71.622 = 3774.249m

Ch T2 = 3774.249 + 70 = 3844.249m

Setting-out pegs for the circular arc are therefore required at the following chainages: 3725, 3750 and 3774.249 (T2).

Deflection angles:Entry sub-chord = 3725 - 3702.627 = 22.373mStandard chord = 25mExit sub-chord = 3774.249 - 3750 = 24.249m

From the equation d = 28.648 lC / R, the following data can be tabulated:

Chord length Deflection angle Angle set on theodolite22.373 00 45 14.6 00 45 1525 00 50 33.3 01 35 4824.249 00 49 02.2 02 24 50

Check: 02 24 50 x 2 = 04 49 40 = C

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Note: the above table of angles assumes that the theodolite is set up at T2, sighted back to T2, swung through 180 + back angle (= 181 34 22) and then set to zero. An alternative approach, if the theodolite is not an electronic type where the reading can be set to precisely zero at the touch of a button, is to compute the angles to be set on the theodolite assuming it has been set to zero on the back sight only:

Chainage Chord length Deflection angle

Angle set on theodolite

T2 back sight) - 00 00 00tangent - 181 34 22

3725 22.373 00 45 14.6 182 19 373750 25 00 50 33.3 183 10 103774.249 (T2) 24.249 00 49 02.2 183 59 12

Check: 183 59 12 - 181 34 22 = 02 24 50= / 2

Second transition curve:

Setting-out pegs for the second transition are required at the following chainages: 3774.249 (T2, as check), 3775, 3800 and 3825. In each case, the same formula as before is used. However, because the second transition is left-handed, the deviation angle must be subtracted from 360 to obtain the angle to be set on the theodolite (assuming it is set to zero when sighted onto the intersection point).

Chainage l angle set on theodolite

3774.249 (T2 check) 70 00 47 11 359 12 493775 69.249 00 46 11 359 13 493800 44.249 00 18 51 359 41 093825 19.249 00 03 34 359 56 26

Notice that the length of curve from T2 is decreasing as chainage increases.

The complete setting-out data can be summarised on the table on the next page.

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Setting-out data – summary

Chainage Chord, m (measured from previous peg)

Angle set on theodolite

Theodolite set up over T1, set to zero sighting intersection point. Taping starts from T1.3650 17.373 (from T1) 00 02 543675 25 00 17 173700 25 00 43 433702.627 (T1) 2.627 00 47 11

Theodolite set up over T1, set to zero sighting T1. Taping starts from T1.

3725 22.373 (from T1) 182 19 373750 25 183 10 103774.249 (T2) 24.249 183 59 12

Theodolite set up over T2, set to zero sighting intersection point. Taping starts from T2.3825 19.249 (from T2) 359 56 263800 25 359 41 093775 25 359 13 493774.249 (T2 check) 0.751 359 12 49

Further examples and exercises can be found in the following texts, or in any standard textbook on Surveying.

John Muskett “Site Surveying” 2nd Ed. Blackwell 1995, chapter 10

Arthur Bannister et al “Surveying” 7th Ed Longman 1998, chapter 11

W. Schofield “Engineering Surveying” 5th Ed Butterworth 2001, chapter 8

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