CSE29357 Highway Engineering

Revision Handout Transition curve

Section Summary

Transition Curve
Length of transition curve is related to the rate of change of radial acceleration: 𝐿 = 𝑉 3 /𝑐𝑅
(Normally, 𝑐 ranges from 0.3 to 0.86 𝑚/𝑠 3 )

Spiral transition curve (𝑟𝑙 = 𝑅𝐿 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)

𝜃 = 𝑙 2 /2𝑅𝐿 (exterior deflection angle)
𝑦 = 𝑙 (distance along tangent)
𝑥 = 𝑙 3 /6𝑅𝐿 = 𝑦 3 /6𝑅𝐿 (offset from tangent)
𝛿 = 𝑙 2 /6𝑅𝐿 (polar deflection angle)

Shift, 𝑠 = 𝐿2 /24𝑅
(if shift <0.3m, transition curve is not necessary)

Tangent distance
𝐿 𝛼
𝑇𝐼 =+ (𝑅 + 𝑆) tan
2 2
Where 𝑆 is the shift, and 𝛼 is the angle subtended by the original circular curve

Apart from being shifted inward, the circular curve became shorter after introducing the transition
curve, the length of the circular portion of the curve becomes 𝑅(𝛼 − 2𝜃𝐿 ) where 𝜃𝐿 is the exterior
deflection angle at 𝑙 = 𝐿 (i.e. 𝜃𝐿 = ? ? ?)

Superelevation transition
Cross section: normal crown (for drainage) to full superelevation (circular curve)
Modify the roadway cross section:
- Tangent runoff
- Superelevation runoff

For appearance or comfort, we limit the difference in longitudinal slope between the centerline and
edge of travelled way (usually 1/200 or 0.5%)

Refer back to lecture notes on how the length of transition curve is related to the length of tangent/
superelevation runoff (and also how to develop from normal crown to full superelevation in absence
of transition curve).

CSE29357 Revision Handout – Transition curve

Reading:
Roess (2011), Section 3.3.4, 3.3.5
Banks (2002), section 4.3.3, 4.4

Banks, J. H. (2002). Introduction to transportation engineering (2nd ed.. ed.). Boston: Boston :
McGraw-Hill.
Roess, R. P. (2011). Traffic engineering (4th ed.. ed.). Upper Saddle River, NJ: Upper Saddle River,
NJ : Pearson.

CSE29357 Revision Handout – Transition curve

In-class Worked Example

You do not learn much from reading, pay attention to the common
misconceptions and ask questions!

Question 1 – Setting out of transition curve

Consider a 80m spiral transition curve that connects a tangent to a 500m-radius circular curve. Set
out the transition curve at 20m intervals using
(i) Offset from tangent
(ii) Deflection from tangent

Solution
Given 𝐿 = 80𝑚, 𝑅 = 500𝑚
𝐿2
Shift, 𝑠 = 24𝑅 = 0.533𝑚 > 0.3𝑚 Thus, a transition curve is necessary
𝑙3 𝑦3
Offset from tangent, 𝑥 = 6𝑅𝐿
= 6𝑅𝐿
𝑥 𝑦2
(Polar) Deflection angle from tangent, 𝛿 = 𝑦 = 6𝑅𝐿 (radians, need to convert to degrees)
Distance along curve Distance along tangent Offset Deflection Angle, 𝛿
𝑙 (𝑚) 𝑦 (𝑚) 𝑥 (𝑚) Radians Degrees
0 0 0.0000 0.0000 0.0000
20 20 0.0333 0.0017 0.0955
40 40 0.2667 0.0067 0.3820
60 60 0.9000 0.0150 0.8594
80 80 2.1333 0.0267 1.5279
The column ‘Deflection angle in radians’ is an intermediate step, need not to be shown. It is presented here
for students to check their calculations.

Question 2 - Circular + Transition + Superelevation

A two-lane highway (each lane is 3.6m wide) with a design speed of 100 km/h has a 400m radius
horizontal curve connecting tangents with bearings of 𝑁75𝑜 𝐸 and 𝑆78𝑜 𝐸.

Determine
(i) The superelevation rate, the length of spiral if the difference in grade between the
centerline and the edge of the travelled way is limited to 1/200 and the allowable rate
of change of centripetal acceleration is 0.4𝑚/𝑠 3
(ii) The stations of TS, SC, CS, and ST, given that the temporary station of the PI is 150 + 00
The length of the spiral should be rounded up to the next highest 1m interval. Normal crown is
2.5%

Determine superelevation of circular curve

According to TPDM table 3.3.3.1 Appropriate radii and superelevation, for design speed V =
100km/h, we cannot find a R-value with radius = 400m (R = 400m is between R3 and R4).
Thus, determine superelevation using the equation for minimum curve radius (45% of centrifugal
force balanced by superelevation)
𝑉2
45% ( )=𝑒
127𝑅
1002
𝑒 = 0.45 × = 0.089 ≅ 0.09
127(400)

CSE29357 Revision Handout – Transition curve

Determine length of transition curve
Criteria 1: rate of change of superelevation
𝐷𝑒 1
≤ ⇒ 𝐿 ≥ 200 × 3.6 × 0.09 = 64.8𝑚
𝐿 200
[What is D? why D is 3.6 instead of 7.2 (for 2 lanes)?]

Criteria 2: rate of change of centripetal acceleration

𝑉3 (100/3.6)3
≤𝑐 ⇒ ≤ 0.4 ⇒ 𝐿 ≥ 133.96𝑚
𝑅𝐿 400𝐿
Thus, the length fo transitive curve should be 134m

Remarks:
The required superelevation runoff is 64.8m (if we use 1/200 instead of assuming other values),
𝑒 2.5
and the corresponding tangent runoff is 𝑒𝑁𝐶 𝐿 = 9 × 64.8 = 18𝑚
𝑑
Thus, the proposed length of transition curve is sufficient to accommodate both the tangent runoff
and superelevation runoff (recall lecture slide p.127).

Check shift
𝐿2
𝑠 = 24𝑅 = 1.87𝑚 > 0.3𝑚. A transition curve is necessary.

Critical points

Given: incoming tangent 𝑁75𝑜 𝐸, outgoing

tangent 𝑆78𝑜 𝐸

Central angle of circular curve [use a sketch to

help if necessary]
𝛼 = 180 − (75 + 18) = 27𝑜 = 0.47 𝑟𝑎𝑑

Tangent distance
𝛼
𝑇𝐼 = 𝐿 = 2 + (𝑅 + 𝑆) tan
2
134 0.47
= + (400 + 1.87) tan = 163.48𝑚
2 2

Angle subtended by the circular portion of

𝐿 137
curve, 𝛼 − 2𝜃𝐿 = 0.47 − 2 (2𝑅) = 0.47 − 2 ( 2
×
400) = 0.135 𝑟𝑎𝑑

Length of circular portion of the curve , 𝑅(𝛼 −

2𝜃𝐿 ) = 400 × 0.435 = 54𝑚

Given station of PI is 150 + 00

TS station = PI station – tangent distance = (150 + 00) – 163.48 = 148 + 36.52

SC station = TS station + L = (148 + 36.52) + 134 = 149 + 70.52
CS station = SC station + length of circular portion = (149 + 70.52) + 54 = 150 + 24.52
ST station = CS station + L = (150 + 24.52) + 134 = 151 + 58.52

CSE29357 Revision Handout – Transition curve

Question 3 – Superelevation transition

Compute the tangent runoff, superelevation runoff and draw the superelevation transition diagram
based on Q2. The length of tangent runoff should be in multiples of 20m, and superelevation
transition ends exactly at SC.

Based on Q2,
Superelevation runoff 𝐿𝑟 ≥ 64.8𝑚, round to 80m
𝑒 0.025
Tangent runoff, 𝐿𝑡 = 𝑁𝐶 = × 80 = 22.22m
𝑒𝐿𝑟 0.09
De
[If we use Lr = 80m, the corresponding rate of change of superelevation is Δg = = 0.00405. If we
Lr
use tangent runoff of 22.22m, this ratio 0.00405 can be maintained. But if we round the tangent runoff
to another convenient value, say 25m, the rate of change of superelevation of that segment is NOT
0.00405, an inconsistent rate of change is very “odd” for drivers (violate aesthetic/ comfort criteria).
Thus, we keep the 22.2m in here.]

Total runoff = 80 + 22.2 = 102.22m

Station of SC = 149 + 70.52
Start of tangent runoff = (149 + 70.52) – 102.22 = 148 + 68.32
End of tangent runoff = (148 + 68.32) + 22.22 = 148 + 90.52

Cross slope
Location station
Inner Edge Outer Edge
Tangent
(This point is arbitrary, just to show the cross slopes 148 + 60.00 -0.025 -0.025
before to the changes take place
Begin tangent runoff 148 + 68.30 -0.025 -0.025
End tangent runoff/ being superelevation runoff 148 + 90.52 -0.025 0
all lanes have same slope 149 + 12.74 -0.025 0.025
end of superelevation runoff/ SC 149 + 70.52 -0.090 0.090

(Try to produce this sketch yourself at home without looking)

Superelevation Transition
0.1
0.08
0.06 148+12.74
0.04
Cross slope

0.02
148+68.32
0
-0.0214850 14870 14890 14910 14930 14950 14970 14990
-0.04
148+60.00 148+90.52
-0.06
-0.08
-0.1
148+70.52
Inner Edge Outer Edge

CSE29357 Revision Handout – Transition curve

Revision Exercises

Please attempt these yourself before checking the answers.

Question 4
A transition curve is to be inserted to connect a straight to a circular arc of radius 600m. If the design
speed of the road is 100 km/h and the permitted rate of change of radial acceleration is 0.3 m/sec3,
tabulate the data required to set out the transition curve using 20m intervals from the tangent point
by
(a) Offset from the tangent; and
(b) Deflection angle (in degrees)
The point TS is at station 1 + 230

Question 5
A circular curve with a radius of 350m is connected to tangents with Δ = 0.349 𝑟𝑎𝑑 by 60-m spiral
transition curves. If the station of the TS is 1+054, determine the station of ST and the tangent
distance.

Question 6
Set out a transition curve that connects a tangent to a circular curve of 450m radius. The exterior
deflection angle  at SC is 17 degrees, station of SC is 22 + 400. Compute the necessary data at 50-m
intervals beginning at the first full station after TS. Round the length of transition curve to the nearest
meter.
Hints:
- In this question, either setting out methods are not very accurate (what are the conditions?)
- The setting out table is long, you are recommended to do it with Excel.

Question 7
On a single-2-lane carriageway (carriageway width = 10.0m), a tangent section connecting directly
to a circular curve with 4% superelevation (i.e. without transition curve). The normal crown is 0.5%
and the allowable rate of change of superelevation is 1 in 200. Since there’s no transition curve,
assume 80% of the total runoff is achieved on the tangent section. Determine the length of the
superelevation runoff (round up to 20m) and draw the superelevation transition diagram. Given the
station of PC is 1 + 054.

CSE29357 Revision Handout – Transition curve

Revision Exercises
Do NOT read the solution without trying to work on it yourself!
Please consult Dr Siu if you have questions after going through the solution.

Question 4
A transition curve is to be inserted to connect a straight to a circular arc of radius 600m. If the design
speed of the road is 100 km/h and the permitted rate of change of radial acceleration is 0.3 m/sec 3,
tabulate the data required to set out the transition curve at 20m intervals from the tangent point by
(a) Offset from the tangent; and
(b) Deflection angle (in degrees)
The point TS is at station 1 + 230.

Solution
1000 3
𝑉3 (100× )
Length of transition curve, 𝐿 = 𝑐𝑅
= 3600
0.3×600
= 119.07𝑚
Check shift
𝐿2
Shift, 𝑠 = 24𝑅 = 0.985𝑚 >0.3m, a transition curve is required

Setting out of transition curve

𝑦3 𝑙2
Offset from tangent, 𝑥 = 6𝑅𝐿; Deflection from tangent, 𝛿 = 6𝑅𝐿 (in radians)
Station Distance along Offset from
tangent, y (m) tangent, 𝑥 (m)𝑥 = Deflection angle,
(= distance along 𝑦3 𝛿 (degrees)
curve, 𝑙 (m) 6𝑅𝐿
1 + 230 (TS) 0 0.000 0.0000
1 + 240 10 0.002 0.0115
1 + 260 30 0.063 0.1203
1 + 280 50 0.292 0.3323
1 + 300 70 0.800 0.6532
1 + 320 90 1.701 1.0829
1 + 340 110 3.105 1.6157
1 + 349.07 (ST) 119.07 3.938 1.8965
[Note: the STATIONS are in multiples of 20, not 𝑙 in multiples of 20]
A sketch of the transition curve looks like this, try to figure out how x and 𝛿 are measured
120

100

80
y (m)

60

40

20

0
0 1 2 3 4
x (m)

CSE29357 Revision Handout – Transition curve

Question 5
A circular curve with a radius of 350m is connected to tangents with Δ = 0.349 𝑟𝑎𝑑 by 60-m spiral
transition curves. If the station of the TS is 1+054, determine the station of ST and the tangent
distance.

Solution
Given: R = 350m, L = 60m, 𝛼 = 0.349rad

𝐿2 602
Shift, 𝑠 = = = 0.4286𝑚 (>0.3m, transition curve is necessary)
24𝑅 24×350

𝐿 60
𝜃𝐿 = = = 0.0857 𝑟𝑎𝑑
2𝑅 2(350)

Angle subtended circular curve between SC and CS = 𝛼 − 2𝜃𝐿 = 0.349 − 2(0.0857) = 0.1776 𝑟𝑎𝑑
Length of arc SC – CS = 𝑅(𝛼 − 2𝜃𝐿 ) = 62.15𝑚

Station of ST = (1 + 054) + 𝐿 + 62.15 + 𝐿 = (1 + 236.15)

𝐿 𝛼
Tangent distance, 𝑇𝐼 = 2 + (𝑅 + 𝑆) tan 2 = 91.78𝑚

Question 6
Set out a transition curve that connects a tangent to a circular curve of 450m radius. The exterior
deflection angle  at SC is 17 degrees, station of SC is 22 + 400. Compute the necessary data at 20-m
intervals beginning at the first full station after TS. Round the length of transition curve to the nearest
meter.

Solution
𝐿 𝜋 L
⇒ 17𝑜 ×
𝜃𝐿 = =
2𝑅 180 2(450)
𝐿 = 267𝑚
Station of TS = (22 + 400) – L = 22 + 133

𝐿2
Shift, 𝑠 = 24𝑅 = 6.60𝑚 Shift > 0.3m, transition curve is necessary

Note that in this question, the 𝐿 > 0.4𝑅 and 𝜃𝐿 > 12𝑜 the setting out methods are not very accurate
(p.116) (since the higher order terms are truncated in deriving the expressions)

Let 𝑦 denote distance along tangent, 𝑥 is the offset from tangent and 𝛿 is the deflection from tangent
𝑦3 𝑙2
𝑥= ,𝛿 =
6𝑅𝐿 6𝑅𝐿

[Try the first two stations and then complete the rest with Excel
Distance from Tangent Offset from Tangent Deflection from Tangent
Station
y (m) x (m) 𝛿, (degrees)
0 22 + 133 0.0000 0.0000
7 22 + 140 0.0005 0.0039
27 22 + 160 0.0273 0.0579
47 22 + 180 0.1440 0.1756
67 22 + 200 0.4172 0.3568
87 22 + 220 0.9134 0.6016
107 22 + 240 1.6993 0.9099

CSE29357 Revision Handout – Transition curve

 127 22 + 260 2.8414 1.2819
147 22 + 280 4.4063 1.7174
167 22 + 300 6.4606 2.2166
187 22 + 320 9.0709 2.7793
207 22 + 340 12.3037 3.4056
227 22 + 360 16.2257 4.0954
247 22 + 380 20.9033 4.8489
267 22 + 400 26.4033 5.6659
(note that we are not using 0, 20, 40… in column 1)

Question 7 – Superelevation transition

On a single-2-lane carriageway (carriageway width = 10.0m), a tangent section connecting directly

to a circular curve with 4% superelevation (i.e. without transition curve). The normal crown is 0.5%
and the allowable rate of change of superelevation is 1 in 200. Since there’s no transition curve,
assume 80% of the total runoff is achieved on the tangent section. Determine the length of the
superelevation runoff (round up to 20m) and draw the superelevation transition diagram. Given the
station of PC is 1 + 054.

Length of superelevation runoff,

𝐷𝑒 1
Δ𝑔 = ≤
𝐿 200
10.0
⇒ 𝐿 ≥ 200𝐷𝑒 = 200 ( ) (4%) = 40𝑚
2
𝑒𝑁𝐶 0.5
Length of tangent runoff = 𝐿 = × 40 = 5𝑚
𝑒 4

Total runoff = tangent runoff + superelevation runoff = 45m

80% × 45 = 36𝑚 is achieved on the tangent section.

Thus, tangent runoff starts at (1 + 054) – 36 = 1 + 018

and the total runoff ends at (1 + 018) + 45 = 1 + 063

Cross slope
station
Location Inner Edge Outer Edge
Tangent
(This point is arbitrary, just to show the cross slopes 1 + 000 -0.005 -0.005
before to the changes take place
Begin tangent runoff 1 + 018 -0.005 -0.005
End tangent runoff/ being superelevation runoff 1 + 023 -0.005 0
all lanes have same slope 1 + 028 -0.005 0.005
PC
(cross slope still changing inside circular curve since there 1 + 054 -0.031 0.031
is no transition curve)
end of superelevation runoff 1 + 063 -0.040 0.040

CSE29357 Revision Handout – Transition curve

 Superelevation Transition
0.05
0.04
0.03
Cross slope 0.02
0.01 1 + 023
0
-0.01
1 + 018
-0.02
1 + 028
-0.03
-0.04 1 + 054
-0.05 1 + 063
1000 1010 1020 1030 1040 1050 1060 1070

Inner Edge Outer Edge

CSE29357 Revision Handout – Transition curve