1. 1
Highway and railway geometric
design
John Parkin
Professor of Transport Engineering
john.parkin@uwe.ac.uk

2. 1. Introduction to trunk roads and
motorways
2. Introduction to urban streets
3. Design speeds and visibility
4. Horizontal geometry
5. Vertical geometry
6. Railway geometry
2

3. 1 Introduction to trunk roads and
motorways

4. 4
Design Manual for Roads and Bridges:
Applies to trunk roads and motorways

5. Main landing page
https://www.standardsforhighways.co.uk/dmrb/
Road layout standards
https://www.standardsforhighways.co.uk/dmrb/search?discipli
ne=ROAD_LAYOUT
5

6. Some basic definitions
Strategic Road Network:
• nationally significant roads used for the
distribution of goods and services, and a
network for the travelling public.
• In legal terms, it can be defined as those roads
which are the responsibility of the Secretary of
State for Transport (managed by Highways
England).
Trunk Road:
• Any road on the Strategic Road Network
6

7. 7
https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachme
nt_data/file/826533/Network_management_13-8-19.pdf
Dark blue/red: smart motorway
Thick blue: motorway
Thin blue: trunk A roads

8. 8
Some basic definitions
Single and dual carriageway roads:
A dual carriageway has two carriageways
(one in either direction) that are
physically separated, usually by a central
reservation

9. 9
Basic definitions cont…
In the UK we make a distinction between urban and rural
roads
Urban roads
 Urban motorway: “A motorway with a speed limit of 60
mph or less within a built up area”
 Urban all-purpose road: “An all purpose road within a built
up area, either a single carriageway with a speed limit of
40 mph or less or a dual carriageway with a speed limit of
60mph or less.”
DMRB TA 79/99

10. 10
Basic definitions cont…
In the UK we make a distinction between urban and rural
roads
Rural roads
 “All purpose roads and motorways that are not generally
subject to a local speed limit”
DMRB TA 46/97

11. 11
How much carriageway is
required for a new road?
 This is dependent on anticipated traffic volumes

12. Traffic flow ranges: New rural
roads
 Flow ranges assist designers in determining which
carriageway standards are most likely to be
economically/operationally acceptable
 Guidelines were in DMRB 5.1 – TA 46/97, but are now
withdrawn (but given on next two slides as overall
guidance)
 Recommends Annual Average Daily Traffic (AADT) flow
levels for determining carriageway type/width in the case
of rural roads

13. Traffic flow ranges: New rural
roads
Road Standard ID AADT
Single 2-lane S2 Up to 13,000
Wide single 2-lane WS2 6,000 to 21,000
Dual 2-lane all purpose D2AP 11,000 to 39,000
Dual 3-lane all purpose D3AP 23,000 to 54,000
Dual 2-lane motorway D2M Up to 41,000
Dual 3-lane motorway D3M 25,000 to 67,000

14. Typical rural dual carriageway
14

15. 15

16. Typical rural single carriageway
16

17. 17

18. Traffic flow ranges – Urban roads
 For urban roads, flow range guidelines were given in
DMRB 5.1 – TA 79/99, now withdrawn
 This gave maximum HOURLY vehicle capacities for
determining carriageway type/width in the case of urban
roads
 Five urban road classes are defined: UM, UAP1-UAP4...

19. Urban road classes

20. Traffic flow ranges – Urban roads
 Determination of class dependent on speed limit, level of
frontage development, frequency of side-road accesses
and level of on-street parking/loading etc.
 Hourly capacities sub-divided in each class according to
overall carriageway width and the number of lanes
 Quoted ‘maximum’ capacities (vph) are the highest ‘one
way’ flows achievable in the busiest direction (60/40
directional split assumed)

21. Traffic flow ranges – Urban roads

22. Typical urban road (DMRB)
22

23. 1. Introduction to trunk roads and motorways
2. Introduction to urban streets
3. Design speeds and visibility
4. Horizontal geometry
5. Vertical geometry
6. Railway geometry
23

24. 2 Introduction to urban streets

25.  Manual for Streets and Manual for Streets
2
 Applies to non-trunk roads in populated
areas
 “The strict application of DMRB to non-
trunk routes is rarely appropriate for
highway design in built up areas,
regardless of traffic volume”.
 Department for Transport (2007) Manual for Streets. Thomas
Telford, London, UK.
https://assets.publishing.service.gov.uk/government/uploads/sys
tem/uploads/attachment_data/file/341513/pdfmanforstreets.pdf
 Chartered Institution of Highways and Transportation (2010)
Manual for Streets 2. Chartered Institution of Highways and
Transportation, London, UK.
https://www.gov.uk/government/publications/manual-for-
streets-2
25

26. Principles of inclusive design
 places people at the heart of the design process;
 acknowledges diversity and difference;
 offers choice where a single solution cannot
accommodate all users;
 provides for flexibility in use; and
 provides buildings and environments that are convenient
and enjoyable to use for everyone.

27. Principle functions of streets
 place;
 movement;
 access;
 parking; and
 drainage, utilities and street lighting.

37. 1. Introduction to trunk roads and motorways
2. Introduction to urban streets
3. Design speeds and visibility
4. Horizontal geometry
5. Vertical geometry
6. Railway geometry
37

38. Local and mode specific guidance
 Local authority guidance on layout of roads in residential and
industrial areas
– See, for example, Transport for London’s ‘Streets Toolkit’ design
guidance
 Local Transport Note 1/20 Cycle Infrastructure Design.
 Also, see Parkin, J. (2018) Designing for cycle traffic, ICE
Publishing, available from the ICE Virtual Library as an e-book
38

39. 1. Local Transport Note 1/20 Cycle infrastructure design.
https://www.gov.uk/government/publications/cycle-infrastructure-design-ltn-120
2. Parkin, J. (2018) Designing for cycle traffic: international principles and practice. ICE
Publishing, London.
3. CIHT (2019) Streets and transport in the urban environment. (Source documents on
planning and designing for walking, cycling and public transport)
https://www.ciht.org.uk/knowledge-resource-centre/resources/streets-and-transport-in-the-
urban-environment/
4. Butler, R., Salter, J., Stevens, D., Deegan, B. (2019) CYCLOPS – Creating Protected
Junctions. Greater Manchester Combined Authority, Transport for Greater Manchester.
http://www.jctconsultancy.co.uk/Symposium/Symposium2018/PapersForDownload/CYCLOPS
%20Creating%20Protected%20Junctions%20-
%20Richard%20Butler%20Jonathan%20Salter%20Dave%20Stevens%20TFGM.pdf

40. Accessing Designing for cycle traffic
1. Go to the UWE library home page
https://www.uwe.ac.uk/study/library
2. At the bottom you will see ‘Databases
A-Z at the bottom. Click the link.
3. Then click the link for ‘I’ to take you to
a page with the
4. Institution of Civil Engineers….
5. Once you have entered the virtual
library put ‘Designing for cycle traffic’
in the search and you should find it.

41. • Design guidance is extensive and requires
interpretation and evaluation.
• Design requires creativity and an
understanding of user needs.
• Language is important because it affects
thinking.

42. ‘vulnerable road user’: admitting
the system is failing
‘Non-motorised road user’: a
persuasive definition.
‘cycle path’: a path is for
walking
False comparisons: ‘motor traffic
and cyclists’.
‘Active travel’: Cf ‘Inactive
travel’
‘cycle traffic’: Cf ‘fietsverkeer’,
‘radverkehr’, ‘cykeltrafik’.
‘cycleway’: Cf ‘carriageway’ and
‘footway’
‘Rider’? Cf driver
 

43. Principles for designing for
cycle traffic

44. • Cycles are vehicles capable of
speed.
• Cycles are heterogeneous with
many different dimensions and
characteristics.
• The design speed is the principle
characteristic of relevance for
design.

45. Design guidance 40 km/hr 36
km/hr
30
km/hr
20
km/hr
Highways England (2016) Gradients of
3%
All routes Absolute
minimum
(see
note)
Cycling Embassy for
Denmark (2012)
Gradients of
5%
Gradients
of 3%
Not
explicitly

46. • In addition to the design speed, it is
important within the envelope of provision to
cater for cyclists of all abilities and disabilities.
• Cycling is not the same as walking and
separate provision is required.
• Cyclists have a kinematic envelope that needs
to be taken into account in design.
• The environment for cycling needs to be
attractive and comfortable.

47. 47
Where should cycle traffic be?
Local Transport Note 1/20 Cycle Infrastructure Design

48. Speed limit
(mph)
Motor traffic volume
(Annual Average
Daily Traffic, two-
way) Route type
Peak hour
cycle flow
One-way
width
(metres)
Two-way
width
(metres)
Buffer
distance
from
carriageway
(metres)
70 All volumes Cycle track
< 150 2.50 3.00
3.50
150 to 750 3.00
4.00
> 750 4.00
60 All volumes Cycle track
< 150 2.50 3.00
2.50
150 to 750 3.00
4.00
> 750 4.00
50 All volumes Cycle track
< 150 2.50 3.00
2.00
150 to 750 3.00
4.00
> 750 4.00
40 All volumes Cycle track
< 150 2.50 3.00
1.00
150 to 750 3.00
4.00
> 750 4.00
30
< 5,000
Cycle Lanes
All
volumes
2.00
Light segregation All volumes 2.50
> 5,000 Stepped Cycle tracks
< 150 2.50
150 to 750 3.00
> 750 4.00
> 5,000 Cycle track
< 150 2.50 3.00
0.500
150 to 750 3.00
4.00
> 750 4.00
20
< 2,500
Mixed traffic < 75 N/A
Mixed traffic or cycle
street
50 to 250 N/A
Cycle street > 200 N/A
2,500 to 5,000
Mixed traffic < 75 N/A
Mixed traffic or cycle
lane
50 to 250 2.00

49. Networks
• Comfortable, attractive and safe networks required
• Full capabilities realized through filtered permeability (new build
and retro-fit)
• Start building the network from major attractors
• Direction signing essential
• Cycle dismount signs are discriminatory, replace with warning
signs

50. • Two-way cycle traffic in one-way
streets
• Opening up closures
• Exemptions from banned turns
• Bridges

51. Signing

52. 1. Introduction to trunk roads and motorways
2. Introduction to urban streets
3. Design speeds and visibility
4. Horizontal geometry
5. Vertical geometry
6. Railway geometry
52

53. 3 Design speeds and visibility

54.  Highway Link Design concerns the geometry of the
road alignment, both horizontally and vertically
 Alignment design must ensure that standards of
curvature, visibility and super-elevation are provided for a
DESIGN SPEED consistent with anticipated vehicle
speeds
 Design process seeks to achieve value for money and
acceptable standard of safety

55. Design Speed
 Road alignment should be consistent with
design speed
 The DMRB standard provides an alignment
appropriate for 85th percentile driver (in
wet conditions)

56. Design factors
Design speed dictates:
 Stopping sight distance (SSD)
 Full overtaking sight distance (FOSD)
 Horizontal alignment
 Vertical alignment
 Co-ordination of horizontal and vertical
alignments

57. 57
Design speed related parameters

58. 58
Urban design speed determination
• Design speed is linked to speed limit
• Permits a small margin for speeds
greater than limit

59. Cycle traffic design speed
59
Design guidance 40 km/hr 36
km/hr
30
km/hr
20
km/hr
Highways England
(2016)
Gradients of
3%
All routes Absolute
minimum
Cycling Embassy for
Denmark (2012)
Gradients of
5%
Gradients
of 3%
Not
explicitly
stated
CROW (2017) Outside
built-up
areas
‘Normal
situations
’

60. Rural design speed
determination
1. Select trial design speed
2. Draw up initial alignment
3. Measure alignment constraint over
sections >= 2km
4. Re-calculate design speed and
compare against trial design speed
5. Relax alignment to achieve cost /
environmental savings
6. Re-check speeds following changes
60

61. Design Speed
Speeds vary according to the impression of constraint that
the road alignment and layout impart to the driver (CD 109)
 Alignment constraint (Ac)
– Bendiness and forward visibility
 Layout constraint (Lc)
– Road width, verge width, no. of access junctions
V50wet = 110 – Lc - Ac

62. 62
Alignment constraint
Dual
carriageways
θ1
θ2
10
6
.
6
B
Ac 

45
2
60
12
B
VISI
Ac 


L
B



4
3
2
1
1
...
1
1
1
V
V
V
V
n
VISI




Single
carriageways
B, Bendiness
VISI, harmonic
mean visibility
Harmonic mean visibility: Forward visibility
measured every 50m or 100m where visibility is
consistent

63. 63
Layout constraint

64. 64
Original TD 9/93 chart
c
c A
L
Speed 

110

65. CD109 chart
65

66. 66
Speed band determination

67. Relaxations
• The standard provides an
alignment appropriate for 85th
percentile driver
• A relaxation provides an
alignment one or more design
speed bands below this
• Allowable to reduce construction
cost / environmental impact
Salter and Hounsell (1996)
67

68. Relaxations
Relaxation to a number of design steps below desirable minimum
may be made where cost and/or environmental savings are
significant, and safety not compromised
• Relaxations should only be applied to SSD, horizontal
curvature, vertical curvature and superelevation.
• Values for SSD, horizontal curvature and vertical curvature
shall not be less than the value for 50 kph design speed
• Relaxations for SSD and vertical curvature for crest and sag
curves are not permitted on the immediate approaches to
junctions.
68

69. Stopping sight distance
The distance required by a driver to stop a
vehicle when faced with an obstruction
Includes
• Perception – reaction distance:
• 2 secs
• Braking distance:
• comfortable braking deceleration: 0.25g
Salter and Hounsell (1996)
70

70. 71

71. 72
Stopping sight distance
Measurement of SSD:
• Drivers eye height between 1.05 – 2.0m
• Object height between 0.26m – 2.0m
above road surface
• Checked in vertical and horizontal plane

72. 𝑣 = 𝑢 + 𝑎𝑡
0 = 𝑢 + 𝑎𝑡
𝑢 = −𝑎𝑡
𝑡 =
𝑢
−𝑎
73

73. 74
Stopping sight distance
Desirable
minimum (V in
km/hr)
One step below
desirable
minimum (V in
km/hr)
𝑆𝑆𝐷 = 2𝑢 +
𝑢
2
. 𝑡𝑏 = 2𝑢 +
𝑢
2
.
𝑢
0.25𝑔
= 𝑢 2 +
𝑢
0.5𝑔
0.555𝑉 + 0.01573𝑉2
0.555𝑉 + 0.01049𝑉2

74. MfS2 stopping sight distance
For light vehicles
• 1.5 seconds perception and reaction time for and
• 0.45g deceleration.
• It is recommended that the volume and proportion of
buses and heavy goods vehicles is considered when
considering deceleration rates.
• Account is also taken of the longitudinal gradient of the
street, resulting in the formula, as follows
• t is the perception and reaction time,
• 𝑑 is the deceleration and
• 𝑎 is the longitudinal gradient (positive for upgrades)
75
𝑆𝑆𝐷 = 𝑢𝑡 +
𝑢2
2(𝑑 + 0.1𝑎

75. Design
Speed
(km/hr
)
Minimum
stopping sight
distance
(metres)
Highways
England /
(CROW)
Sight
distance
in motion
at 10 and
5 seconds
(metres)
CROW
40 47 / (-) 111 / 56
30 31 / (40) 83 / 42
20 17 / (21) 56 / 28_

76. Full overtaking sight distance
For single carriageway roads:
• Sections provided with adequate forward
visibility to enable safe overtaking using
opposing traffic lane
• FOSD should be available between 1.05
and 2m above centre of carriageway
• FOSD is greater than SSD
77

77. 78
Full overtaking sight distance

78. 79
Components of FOSD
D1 is the overtaking distance
D2 is the closing distance
D3 is a safety margin
A A
B B
C C
D1 D3 D2

79. 80
Components of FOSD
D1 is 10 seconds. (Assumed to starts 2 steps below design speed
and accelerate to design speed, hence approximately equal to a
constant speed one step below design speed)
D2 is 10 seconds at design speed
D3 is a safety margin equal to one fifth of D2.
A A
B B
C C
D1 D3 D2

80. 81
Manual for Streets visibility
• It has been shown that increased
forward visibility is linked to higher
speeds
• For urban areas MfS recommends
limiting forward visibility (to manage
speed) i.e.
• Forward visibility = SSD

81. 82
Manual for Streets visibility

82. 20mph in urban areas
DfT Circular 1/2006 ‘Setting local speed
limits’ (para 89 et seq.)
• mean speeds (not 85%ile speeds) now
considered
• mean speed of 24mph or less for a
20mph limit to be set without other self-
enforcing measures.
• The key factor in setting speed limits is
what the road looks like to the road
user.
83

83. • 20 mph speed limits are just signed limits
• 20 mph speed zones have traffic calming and
entry features
• We demonstrated a statistically significant
reduction in fatalities for Bristol (Bornioli, A.,
Bray, I., Pilkington, P., Parkin, J., 2019)
• Wales has a task group now to implement
20mph national default for residential roads.
• For more information see 20’s Plenty website
84

84. Other speed limit considerations
• collision and casualty savings;
• traffic flow and emissions;
• journey times for motorised traffic and journey-
time reliability;
• the environmental impact;
• the level of public anxiety;
• the level of severance caused by fast-moving
traffic;
• conditions and facilities for vulnerable users;
• the costs of measures and their maintenance;
• the cost and impact of signing;
• the cost of enforcement. 85

85. 1. Introduction to trunk roads and
motorways
2. Introduction to urban streets
3. Design speeds and visibility
4. Horizontal geometry
5. Vertical geometry
6. Railway geometry
86

86. 4 Horizontal geometry

87. 89
Horizontal curves
R
vehicle at rest travelling around a curve of radius R

88. 90
Horizontal curves: Forces
w=m.g
N1
P
N2
• N1 and N2: normal contact forces between tyres and road surface
• w: weight of the object of mass m in the Earth’s gravitational field, acting from the centre of mass
• P: frictional force between the tyres and the surface, acting towards the centre of the curve

89. 91
Slipping
m.v2
R
m.v2
R
μ.N
Force required to keep the body moving in a circle:
Force provided by friction between tyres and surface:
At point of slipping the speed has increased to
the point where :
m.v2
R
μ.m.g
=
= μ.m.g
Inertial resistance to
the centripetal
acceleration
w=m.g
N1 N2
N = N1 + N2

90. Overturning
92
P

91. Overturning: Forces
93
N1 N2
mg
a a
h
Geometry:
• h – vertical distance between road
surface and centre of mass
• a - horizontal distance between
centre of either tyre and centre of
wheel base
Forces:
• N1 and N2: Normal contact forces
at tyres
• mg: weight of vehicle in Earth’s
gravitational field
• F: Horizontal force required to keep
the vehicle moving around a curve
F

92. Overturning: Forces
94
N1 N2
mg
a a
h
There is no vertical acceleration:
N1 + N2 = mg (eq 1)
The minimum horizontal force required
to keep the vehicle moving around the
curve of radius R is given by:
F = mv2 / R F

93. Overturning: Moments
95
N1 N2
mg
a a
h
When the vehicle is not overturning, it
is in ‘rotational equilibrium’.
Moments of forces, acting on either
side of the vehicle’s centre of mass
(clockwise and anticlockwise), will be
equal:
F.h + N1.a = N2.a F

94. Overturning: Moments
96
N1 N2
mg
a a
h
Solving for N1:
F.h + N1.a = N2.a
N1.a – N2.a = - F.h
N1.a – N2.a = (-m.v2/R) . h
N1 – N2 = -m.v2.h / R.a (eq 2)
Adding (eq 1) to (eq 2)
N1 + N2 + N1 – N2 = mg - m.v2.h / R.a
2N1 = mg - m.v2.h / R.a
2N1 = m (g - v2.h / R.a)
N1 = m /2 . (g - v2.h / R.a)
F

95. Overturning: Moments
97
N1 N2
mg
a a
h
Solving for N1:
N1 = m /2 . (g - v2.h / R.a)
Solving for N2:
Similarly, it can be shown that
N2 = m /2 . (g + v2.h / R.a)
F

96. Overturning: Toppling
98
N1 N2
mg
a a
h
N1 = m /2 . (g - v2.h / R.a)
As the speed of the vehicle increases,
N1 tends to zero.
The point of toppling around the outer
wheel occurs when N=0. This is when
the inner tyre is no longer in contact
with the road surface i.e. when:
g = v2.h / R.a
Solving for the toppling speed v gives:
v = √g.R.a / h
F

97. Super-elevation
• With super-elevation, a vehicle’s
weight contributes to the force
required to keep the body moving
around a horizontal curve, in
addition to the frictional force
99

98. 100
N
w=m.g
P
Super-elevation
α
To move around a curve, we need to generate a horizontal force
equivalent to mv2/R
With super-elevation, this force is generated by:
i) the horizontal component of the Normal force (N), plus
ii) the horizontal component of friction (P)

99. 101
N
w=m.g
P
Super-elevation
α
Horizontal component of Normal: N sin
α
α

100. 102
N
w=m.g
P
Super-elevation
α
Horizontal component of Friction: P cos α
α
Horizontal component of Normal: N sinα

101. 103
N
w=m.g
P
Super-elevation
α
α
N sin + P cos = mv2/R
α α
N sin + 𝜇 N cos = mv2/R
α α
The point of slipping occurs when:

102. 104
N
w=m.g
P
Super-elevation
α
There is no vertical acceleration
Therefore, resolving vertically( & ignoring negligible vertical component of P):
N cos - mg = 0
Hence:
N=mg / cos α
α
α

103. 105
Super-elevation
N sin + 𝜇 N cos = mv2/R
α α
(mg . sin / cos ) + (𝜇 mg cos / cos ) = mv2/R
α α α α
𝑣2
𝑔. 𝑅
= 𝑇𝑎𝑛𝛼 + 𝜇
Substituting for N and simplifying:

104. Converting v(m/s) to V(kph)and setting g = 9.81 gives
Where “e” is the superelevation. In the UK superelevation, e,
is limited to generating 0.45 of the centripetal force, F. Hence
superelevation e, is given by
As a percentage this is:
106
𝑉2
3.62 × 9.81 × 𝑅
= 𝑒 + 𝜇 =
𝑉2
127𝑅
𝑒 =
0.45. 𝑉2
127𝑅
=
𝑉2
283𝑅
𝑒% =
100. 𝑉2
283𝑅
=
𝑉2
2.83𝑅
=
0.353. 𝑉2
𝑅

105. 107
Design speed related parameters

106. • There are is minimum horizontal and vertical geometry for cycle traffic that
should be adhered to.
Design
speed
(km/hr)
Minimum
horizontal
curve radius
(metres)
Highways
England /
(CROW)
Minimum
taper
40 57 / (-) 1:11
30 32 / (20) 1:8
20 14 / (10) 1:6
Lower limit
for stability
(12)
5 / (5) N/A

107. Transition curves
• Transition curves are provided to enable vehicles moving
at high speeds to make the change from a tangent
(straight) section to the curved section in a safe and
comfortable way.
110

108. 111

109. Transition curves
• Transitions are a spiral shape.
• There are many types of spiral, but the one used for
transitions in alignment design is the Clothoid Spiral
(also known as Euler’s spiral or the Cornu Spiral)
because its curvature increases linearly with the distance
along the spiral.
112

110. 113
The radius of curve rises linearly with
distance along transition, hence:
Where R is the circular curve radius in
metres and L is the length of the
Clothoid Spiral in metres.
A is the Clothoid Parameter in metres
squared
Many characteristics of the spiral can
be derived from this parameter
Source:
https://pwayblog.com/2016/07/03/the-
clothoid/
L
R
A .
2


111. 114
Transition curves
Rate of gain (or loss) of radial acceleration is given by
Re-arranging to make L the subject
Hence, from the Clothoid Spiral equation 𝐴2
= 𝑣3
𝑞. Converting
so speed is in 𝑘𝑚 ℎ𝑟 (𝑣 = 𝑉 3.6) gives the following equation
as used in the UK:
Where q is the rate of gain of radial acceleration and is usually
limited to 0.3m/s3 in the UK. In difficult circumstances it may
be increased to 0.45 or 0.6 m/s3.
R
q
V
L
.
.
7
.
46
3

𝑞 =
𝑣2
𝑅
𝑡
=
𝑣2
𝑅
𝐿 𝑣
𝐿 =
𝑣3
𝑞. 𝑅

112. Superelevation and crossfall
• Crossfall is applied to allow for drainage
• Super-elevation is achieved by incrementally
‘rolling-over’ the drainage crossfall:
115
Rollover
Drainage crossfall on straight Super-elevation on curve

113. 116

114. Application of superelevation
117
Superelevation
applied within the
transition

115. 1. Introduction to trunk roads and
motorways
2. Introduction to urban streets
3. Design speeds and visibility
4. Horizontal geometry
5. Vertical geometry
6. Railway geometry
118

116. 5 Vertical geometry

117. Vertical curves
Vertical curves are parabolic, although very gentle ones
approximate to a circular arc.
The formula:
gives the length of the curve (metres) connecting an entry
grade of g1% to an exit grade of g2%.
120
𝐿 = 𝐾(𝑔1 − 𝑔2
g1 g2
K
L

118. Vertical curves
• ‘K’ values are given in Table 3 of TD9/93 for crest and
sag curves.
• The ‘K’ values refer to the radius of the vertical curve
• All K values satisfy comfort requirement of 0.3m/s2
vertical radial acceleration
121
g1 g2
K
L

119. Vertical curves
The curvature equation is:
where y is a vertical offset distance from the projected entry
gradient and x is a horizontal distance.
122
𝑦 =
𝑔2 − 𝑔1 . 𝑥2
200. 𝐿

120. Vertical curves
The curvature equation is:
where y is a vertical offset distance from the projected entry
gradient and x is a horizontal distance.
Hence to calculate centreline levels along the curve, where A
is the level at the start of the curve, use the equation:
123
𝑦 =
𝑔2 − 𝑔1 . 𝑥2
200. 𝐿
A +
𝑔1.𝑥
100
+
𝑔2−𝑔1 .𝑥2
200.𝐿
level along line of
projected entry gradient
the y offset (-ve for a crest
curve, +ve for a sag curve)

121. 124
Design speed related parameters

122. 125
Design speed related parameters

123. Vertical curves
The curvature equation is:
where y is a vertical offset distance from the projected entry
gradient and x is a horizontal distance.
Gradients are normally limited in the following ways:
• 3% for motorways (max. to 4%)
• 4% for all-purpose dual carriageways (max. 8%)
• 6% for all purpose single carriageways (max. 8%) 126
𝑦 =
𝑔2 − 𝑔1 . 𝑥2
200. 𝐿

124. Vertical curves
• The FOSD value allows for overtaking on a crest and the
vertical curvature values allow for stopping.
• For sag curves, headlight distance governs the K values
up to 70km/hr and above this comfort is the ruling
criterion.
127
g1 g2
K
L

125. CROW (2017) Celis Consult (2014)
Target severity
(0.075)
Upper limit of
severity
(0.200)
Gradi
ent
Height
differe
nce
Maxim
um
length
of
gradie
nt
(metre
s)
Height
differe
nce
Maxim
um
length
of
gradie
nt
(metre
s)
Height
differe
nce
Maxim
um
length
of
gradie
nt
(metre
s)
Sever
ity
2.00
% 3.8 188 10.0 500
2.50
% 3.0 120 8.0 320
3.00
% 2.5 83 6.7 222 15 500 0.45
3.50
% 2.1 61 5.7 163 10.5 300
0.367
5
4.00
% 1.9 47 5.0 125 8 200 0.32
4.50
% 1.7 37 4.4 99 4.5 100
0.202
5
5.00
% 1.5 30 4.0 80 2.5 50 0.125
5.50
% 1.4 25 3.6 66
6.00
% 1.3 21 3.3 56

126. Plan
Part of layout of
A30 improvement
scheme, Cornwall.
Drawings by
CORMAC
129
https://infrastructure.planninginspectorate.gov.uk/wp-
content/ipc/uploads/projects/TR010014/TR010014-000129-
2.05%20Long%20Sections%20and%20Cross%20Sections.pdf

127. Long section
130

128. Cross-
Section
131

129. 6 Railway geometry

130. Introduction
133
• A train driver cannot vary the
horizontal movement of the train,
so speed and alignment have to
be more closely aligned than for
highways
• Aim is to achieve constant running
speed
• Gradients are limited to minimise
tractive and braking effort
required.
• Train traffic is of different types
(passenger, freight, light rail etc.),
and the mix of traffic and its speed
determines the alignment.

131. Horizontal Alignment Elements
Three elements:
1. Straights
2. Curves
3. Transition curves
Rule of thumb: Practical minimum element length:
• Length in metres is half design speed in km/hr
• E.g. 40m for a design speed of 80kph
• Absolute minimum is 20 metres
135

132. Horizontal Alignment Elements
Left and Right Hand Curves
• Maximum radius: 10,000m
• Not possible to construct a more gentle curve to line and level
• Minimum radius:
• Determined by design speed and cant (super-elevation); and
• Curving ability of rolling stock
136

133. Cant (E): Super-elevation
137
The difference in height
between the inside rail head
and outside rail head - E

134. UK Track Gauge
138
1502mm between centre line
1435mm

135. Equilibrium Cant
The cant that would need to be applied
such that:
The effect of the inertial resistance
to the centripetal force is balanced
by the component of the vehicle’s
weight acting in the opposite direction
139

136. Equilibrium Cant
140
𝐸𝑞 = 11.82
𝑉2
𝑅
B

137. θ
mv2 / R
mg
Equilibrium Cant - Derivation
Resolve the two forces horizontal to the banked track and
equate the horizontal components

138. θ
mv2 / R
mg
Equilibrium Cant - Derivation
Resolve the two forces horizontal to the banked track and
equate the horizontal components
𝑚. 𝑔. 𝑠𝑖𝑛𝜃 =
𝑚. 𝑣2
𝑅
. 𝑐𝑜𝑠𝜃

139. θ
mv2 / R
mg
Equilibrium Cant - Derivation
Now rearrange in terms of θ and simplify
𝑚. 𝑔. 𝑠𝑖𝑛𝜃 =
𝑚. 𝑣2
𝑅
. 𝑐𝑜𝑠𝜃

140. θ
mv2 / R
mg
Equilibrium Cant - Derivation
Now rearrange in terms of θ and simplify
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
=
𝑣2
𝑔. 𝑅

141. θ
mv2 / R
mg
Equilibrium Cant - Derivation
Now rearrange in terms of θ and simplify
𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
=
𝑣2
𝑔. 𝑅
tan 𝜃 =
𝑣2
𝑔. 𝑅

142. θ
mv2 / R
mg
Equilibrium Cant - Derivation
What is tan approximately equal to in relation to the track
geometry (the equilibrium cant E, and the distance between the
rail head centre line B)?
tan 𝜃 ≈?
~B (=1.502m)
E
θ

143. θ
mv2 / R
mg
Equilibrium Cant - Derivation
What is tan approximately equal to in relation to the track
geometry (the equilibrium cant E, and the distance between the
rail head centre line B)?
tan 𝜃 ≈
𝐸
𝐵
~B (=1.502m)
E
θ

144. θ
mv2 / R
mg
Equilibrium Cant - Derivation
Substitute for tan in
tan 𝜃 ≈
𝐸
𝐵
~B (=1.502m)
E
θ
tan 𝜃 =
𝑣2
𝑔. 𝑅
Using:
And express in terms of E
Then substitute for B=1.502m , g=9.80655 m/s2

145. Equilibrium Cant - Derivation
UK standard gauge, G, is 1435 mm between rail heads. Dimension
between centre lines of the rail heads is 1502 mm.
Resolving parallel to banking:
𝑚. 𝑔. 𝑠𝑖𝑛𝜃 =
𝑚. 𝑣2
𝑅
. 𝑐𝑜𝑠𝜃
As
𝐸𝑞
𝐵
≈ 𝑡𝑎𝑛𝜃 =
𝑣2
𝑅. 𝑔
where 𝐸𝑞 is the equilibrium cant
𝐸𝑞 =
𝐵. 𝑣2
𝑅𝑔
= 0.1532.
𝑣2
𝑅
Where 𝐵 = 1.502𝑚 and 𝑣 is in m/s and 𝑅 is in metres.
149

146. Equilibrium Cant - Derivation
Now convert for 𝑣 (m/s) in km/hr:
[Hint what factor do you multiply km/hr by to convert to m/s?]
𝐸𝑞 = 0.1532.
𝑣2
𝑅
=?
150

147. Equilibrium Cant - Derivation
Now convert for 𝑣 (m/s) in km/hr:
𝐸𝑞 = 0.1532.
𝑣2
𝑅
= 0.1532.
𝑉2
𝑅
.
10002
36002
= 0.01182
𝑉2
𝑅
151

148. Equilibrium Cant - Derivation
Converting for 𝑣 (m/s) in km/hr, the equation becomes:
𝐸𝑞 = 0.1532.
𝑣2
𝑅
= 0.1532.
𝑉2
𝑅
.
10002
36002
= 0.01182
𝑉2
𝑅
Finally, convert for 𝐸𝑞 (m) in mm:
𝐸𝑞 =?
152

149. Equilibrium Cant - Derivation
Converting for 𝑣 (m/s) in km/hr, the equation becomes:
𝐸𝑞 = 0.1532.
𝑣2
𝑅
= 0.1532.
𝑉2
𝑅
.
10002
36002
= 0.01182
𝑉2
𝑅
Converting for 𝐸𝑞 (m) in mm, the equation becomes:
𝐸𝑞 = 1000 × 0.01182
𝑉2
𝑅
= 11.82
𝑉2
𝑅
153

150. Cant deficiency
On single speed lines, cant can be designed so that it precisely
counteracts the inertial resistance to the centripetal force
On mixed traffic lines with different train speeds, a ‘deficiency’
of cant can be allowed
But ‘cant deficiency’ has to be limited to limit the maximum
force (acting outwards) created at the rail.
The applied cant plus the deficiency equals Equilibrium Cant,
as follows
𝐸𝑞 = 𝐸𝑎 + 𝐷 = 11.82.
𝑉2
𝑅
Where 𝐸𝑎 is the applied cant and 𝐷 is the deficiency.
154

151. Why allow ‘cant deficiency’?
For mixed traffic lines
• Equilibrium cant will apply to the maximum
running speed (to counter the centrifugal force)
• How will the banked track ‘feel’ at lower
running speeds [consider the extreme case of a
stationary train on a banked track]?
155

152. Why allow ‘cant deficiency’?
• Allows higher running speeds and
hence greater line capacity
• Too much banking is
uncomfortable, particularly at lower
speeds
156

153. Description All locations Continuously
Welded Rail
Jointed track
Maximum cant:
 On curved track
 At station platforms
150mm
110mm
Deficiency on plain line 110mm 90mm
Deficiency on switches and
crossings:
 On through line
 On turnout
 With negative cant on turnout
At switch toes
110mm
90mm
90mm
125mm
90mm
90mm
90mm
125mm
Maximum cant gradient not
steeper than:
 Desirable
 Absolute
1 in 600
1 in 400
Cant gradient not flatter than:
 Desirable
 Absolute 1 in 1500
1 in 2500
Rate of change of cant and cant
deficiency on plain line and
switches and crossings
 Desirable
 Absolute
35mm/s
55mm/s
157
Source: Network Rail (2010) Track Design Handbook

154. Applied Cant
The amount of cant applied in practice is usually the lesser of:
𝐸𝑎 =
2
3
. 𝐸𝑞 and maximum allowable cant (150 mm).
158

155. Line speed and cant
To ensure that safety limits and comfort criteria are not
exceeded, limits are placed on the line speed based on the
cant. Re-arranging the equation for cant, we get:
𝑉
𝑚𝑎𝑥 = 0.29 𝑅 𝐸𝑎 + 𝐷𝑚𝑎𝑥
As an example, with a maximum applied cant of 150mm, and
a maximum cant deficiency of 110mm (on continuously
welded rail), the maximum speed on a curve of 2km radius
would be 209 km/hr (130 mph).
159

156. Horizontal Alignment Elements
Curves
• Maximum radius: 10,000m
• Minimum determined by design speed and cant and
curving ability of rolling stock
On tight bends gauge widening to overcome screech (for trains
without bogies)
• 140 – 200 m Radius: 6 mm widening
• 110 – 140 m Radius: 12 mm widening
• <100 m Radius: 18 mm widening
160

157. 161
Corner wear
Level track Curved track
Outer rail head wears on the inside top face and the inner rail
wears across the whole rail head. Contributing to wear are:
• Wheels of different size
• Tyres in different stages of wear
• Wagons of different wheelbases
• Varying amounts of play in the axle boxes

158. Cant Gradient
The rate at which applied cant is ‘built up’
over a transition curve
162

159. Cant Gradient
Cant is applied in 5 mm increments.
The cant gradient depends on:
• The line speed limit;
• The proximity of permanent speed restrictions e.g. at
junctions and tunnels);
• Track longitudinal gradients (that may cause reduction
below line speed limit);
• Relative importance of types of traffic.
163

160. Cant Gradient
Cant is applied over the transition curve length, where a
transition exists, and can be expressed as:
𝐶𝑎𝑛𝑡 𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 1 𝑖𝑛
𝑇𝐿
𝐶
Where 𝑇𝐿 is the transition length and 𝐶 is the cant.
Generalising and taking account of units, a cant gradient of
1 to N may have N calculated as follows:
𝑁 =
1000. 𝐿
𝐸𝑎
Where 𝐿 is the length over which the cant is applied in metres
and 𝐸𝑎 is in millimetres.
Maximum and minimum cant gradients are specified
164

161. Description All locations Continuously
Welded Rail
Jointed track
Maximum cant:
 On curved track
 At station platforms
150mm
110mm
Deficiency on plain line 110mm 90mm
Deficiency on switches and
crossings:
 On through line
 On turnout
 With negative cant on turnout
At switch toes
110mm
90mm
90mm
125mm
90mm
90mm
90mm
125mm
Maximum cant gradient not
steeper than:
 Desirable
 Absolute
1 in 600
1 in 400
Cant gradient not flatter than:
 Desirable
 Absolute 1 in 1500
1 in 2500
Rate of change of cant and cant
deficiency on plain line and
switches and crossings
 Desirable
 Absolute
35mm/s
55mm/s
165
Source: Network Rail (2010) Track Design Handbook

162. Rate of change of cant
• Is limited to 35 – 55 mm/s
• Why does the ‘rate of change of cant’ &
‘rate of change of cant deficiency’ need
to be limited?
166

163. Rate of change of cant
• Is limited to 35 – 55 mm/s
• Why does the rate of change of cant
need to be limited?
i. To reduce ‘roller coaster’ effect if cant is applied to
quickly
ii. If ‘cant deficiency’ builds up too quickly passengers
will feel a sharp outwards force
• The rate at which cant can be
comfortably built up may determine the
transition curve length
167

164. Can be expressed in terms of transition length:
𝑑𝐸
𝑑𝑡
=
∆𝐸.𝑉
3.6𝐿
𝑑𝐷
𝑑𝑡
=
∆𝐷.𝑉
3.6𝐿
Where
Δ𝐸 is the change in cant over the transition length (mm)
Δ𝐷 is the change in cant deficiency over the transition length
(mm)
𝑉 is the design speed (km/hr)
𝐿 is the length of the transition curve (m)
168
Rate of change of cant

165. The transition length may be given by the following equation
𝑇𝐿 =
𝐶
𝑟
. 𝑣
Where 𝑇𝐿 is the transition length in m
𝐶 is the cant or cant deficiency whichever is the greater (mm),
𝑟 is the rate of change of cant (mm/s)
𝑣 is the line speed in m/s.
169
Transition length

166. Transition setting out
Cubic parabola if 𝛼 is less than 4°.
170

167. Transition setting out
𝜙 =
𝑙2
2𝑅𝐿
𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
90𝐿
𝜋𝑅
𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑋 = 𝑙 −
𝑙5
40 𝑅𝐿 2
𝑆 =
𝐿2
24𝑅
−
𝐿4
2668 𝑅3
𝑌 =
𝑙3
6𝑅𝐿
−
𝑙7
336 𝑅𝐿 3
𝐶 =
𝐿
2
−
𝐿3
240𝑅2
Source: Network Rail track design
handbook, Section C2.5
171
Clothoid spiral if 𝛼 is more than 4°.

168. Virtual transition length
• On larger radius curves and lower designs speeds,
transitions may be omitted.
• If so, two thirds of the cant is developed on the straight
with the remainder developed on the curve.
• As the first bogie enters the curve, the rail vehicle will
experience increasing inertial resistance to the centripetal
force up to the maximum centrifugal force when the rear
bogies has joined the curve.
• As a result of the above, the wheelbase between the bogies
is in effect a virtual transition length.
• Bogie wheelbase for design is assumed to be 12.2m on the
mainline and 10.06m on London Underground.
172

169. 173

170. Reverse curves: buffer locking
Value of 𝐵 should be less than 300 mm calculated as follows.
𝐵 =
𝐿2−𝑊2−𝐷2
8𝑅𝑒
for the condition where 𝐷 < 𝐿 − 𝑊
𝐵 =
𝐿−𝑊 𝐿+𝑊−𝐷 2
16𝑊𝑅𝑒
for the condition where 𝐷 > 𝐿 − 𝑊 <
𝐿2−𝑊2
2
Where, 𝐵 , 𝐿 , 𝑊 and 𝐷 are defined in Figure 5.6 and the
equivalent radius of the two curves is given as 𝑅𝑒 =
𝑅1𝑅2
𝑅1+𝑅2
. 174

171. Gradients
The gradient: percentage or ‘rise over the run’, 1 in N.
A rise of 1.5 metres over 750 metres gives a percentage of
0.2%, or a rise over run of 1 in 500. Note that
𝑁 =
100
𝐺%
Gradients: as flat as possible
• Usually: 0.2% and 1.5% (1 in 500 to 1 in 66)
• Absolute maximum: 2.5% (1 in 40)
• At station platforms and in sidings: less than 0.2% (1 in
500)
• In tunnels: greater than 0.5% (1 in 200) for drainage
reasons.
175

172. Ruling and compensated
gradients
Ruling grade: maximum gradient on straight track at which
the heaviest train can maintain speed using its maximum
tractive effort.
On a curve, there will be more resistance and so maximum a
‘compensated grade’ calculated:
𝐺𝑐 = 𝐺 −
70
𝑅
Where 𝐺𝑐 is the compensated grade, 𝐺 is the grade and 𝑅 is
the curve radius in metres.
176

173. Vertical alignment
Instantaneous grade change of less than 0.15% are
acceptable.
Otherwise, vertical curves are provided to connect grades of
different gradient:
• Minimum radius of a vertical curve is normally 1,000m
• Practical maximum is 40,000m.
177

174. General form of a parabolic curve is a quadratic of the form:
𝑦 = 𝑎. 𝑥2 + 𝑏. 𝑥 + 𝑐
Differentiating with respect to 𝑥, where 𝑥 is the (horizontal)
length of the curve gives the gradient at points on the curve:
𝑑𝑦
𝑑𝑥
= 2. 𝑎𝑥 + 𝑏
At the origin (0,0) (and assuming no vertical shift, i.e. c=0)
and the gradient,
𝑑𝑦
𝑑𝑥
, is = 𝑏
We can express the gradient as a 1 in N, so
𝑏 =
1
𝑁1
And at some distance 𝑙 along L, the gradient is
1
𝑁2
= 2𝑎𝑙 +
1
𝑁1
Differentiating again with respect to 𝑥 gives the rate of change
of gradient, and this is the inverse of the radius of the curve
because curvature is the inverse of the radius of curvature.
𝑑2𝑦
𝑑𝑥2
= 2𝑎 = −
1
𝑅𝑣
178

175. As vertical acceleration, assuming a design limit of 3% of g:
𝑣2
𝑅𝑣
≤ 0.03𝑔
So:
𝑅𝑣 ≥
𝑣2
0.03𝑔
As, from above
1
𝑁1
−
1
𝑁2
= −2𝑎𝑙 =
𝑙
𝑅𝑣
So:
𝐿 ≥
1
𝑁1
−
1
𝑁2
.
𝑣2
0.03𝑔
179

176. 181
The levels along the vertical curve at
points 𝑙 along the curve are calculated
as follows:
𝐴 +
𝐺1. 𝑙1
100
+
𝐺1 − 𝐺2 . 𝑙2
200. 𝐿
Where 𝐴 is the reduced level at the
origin of the vertical curve.
The length, 𝐿, of the vertical curve is calculated based on a
limit on the maximum vertical acceleration as follows
• Desirable 0.0225g
• Maximum for sag 0.0325g; for a crest 0.0425g
• Exceptionally, 0.06g

177. High speed lines
• French TGV: PAX only;
3.5% maximum gradient;
300 km/hr; minimum
curve radius is 6000m
(TGV Atlantique has a
4500 metre minimum
curve and maximum
gradient of 2.5%.)
• Japanese Shinkansen
(bullet train): PAX only
• German Deutsche Bahn
Inter-City Express: PAX
and freight; 317 km/hr;
third of route in tunnel;
mixed traffic means
gentler gradients. 182

178. High Speed 2
Is 360 km/hr too fast?
• Very gentle curves mean more
tunnel needed
• Longer braking distances
mean fewer trains per hour
• Acceleration and deceleration
energy intensive
• Speed limited at points
• Maximum tractive effort on
gradients reduces maximum
speed
• Ballast blown by high speeds
may mean solid track bed
183