Compound Curves A compound curve curve consists of of two (or more) circular curves between curves between two main tangents joined at point of compound compound curve ( PCC PCC ). ). Curve at PC at PC is is designated as 1 ( R1, L1, T 1, etc) and curve at PT at PT is is designated as 2 ( R2, L2, T 2, etc).
Elements of compound curve
PC = = point of curvature
PT = = point of tangency
PI = = point of intersec intersection tion
PCC = = point of compound curve
T 1 = length of tangent of the first curve
T 2 = length of tangent of the second curve
V 1 = vertex of the first curve
V 2 = vertex of the second curve
I 1 = central angle of the first curve
I 2 = central angle of the second curve
I = angle of intersection = I 1 + I 2
Lc1 = length of first curve
Lc2 = length of second curve
L1 = length of first chord
L2 = length of second chord
L = length of long chord from PC to PT
T 1 + T 2 = length of common tangent measured from V 1 to V 2
θ = 180° - I
x and y can be found from triangle V 1-V 2- PI .
L can be found from triangle PC-PCC-PT
Finding the stationing of PT Given the stationing of PC Sta PT=Sta PC+Lc1+Lc2Sta PT=Sta PC+Lc1+Lc2 Given the stationing of PI Sta PT=Sta PI−x−T1+Lc1+Lc2Sta PT=Sta PI−x−T1+Lc1+Lc2
Reversed Curve Reversed curve, though pleasing to the eye, would bring discomfort to motorist running at design speed. The instant change in direction at the PRC brought some safety problems. Despite this fact, reversed curves are being used with great success on park roads, formal paths, waterway channels, and the like.
Elements of Reversed Curve
PC = point of curvature
PT = point of tangency
PRC = point of reversed curvature
T 1 = length of tangent of the first curve
T 2 = length of tangent of the second curve
V 1 = vertex of the first curve
V 2 = vertex of the second curve
I 1 = central angle of the first curve
I 2 = central angle of the second curve
Lc1 = length of first curve
Lc2 = length of second curve
L1 = length of first chord
L2 = length of second chord
T 1 + T 2 = length of common tangent measured from V 1 to V 2
Finding the stationing of PT Given the stationing of PC Sta PT=Sta PC+Lc1+Lc2Sta PT=Sta PC+Lc1+Lc2 Given the stationing of V 1 Sta PT=Sta V1−T1+Lc1+Lc2Sta PT=Sta V1−T1+Lc1+Lc2
Reversed Curve for Nonparallel Tangents The following figure is an example reversed curves of unequal radii connecting non-parallel tangents.
Reversed Curve for Parallel Tangents The figure below is an example of reversed curves of unequal radii connecting two parallel roads.
Terminologies in Simple Curve
PC = Point of curvature. It is the beginning of curve.
PT = Point of tangency. It is the end of curve.
PI = Point of intersection of the tangents. Also called vertex
T = Length of tangent from PC to PI and from PI to PT . It is known as subtangent.
R = Radius of simple curve, or simply radius.
L = Length of chord from PC to PT . Point Q as shown below is the midpoint of L.
Lc = Length of curve from PC to PT . Point M in the the figure is the midpoint of Lc.
E = External distance, the nearest distance from PI to the curve.
m = Middle ordinate, the distance from midpoint of curve to midpoint of chord. I = Deflection angle (also called angle of intersection and central angle). It is the angle of intersection of the tangents. The angle subtended by PC and PT at O is also equal to I , where O is the center of the circular curve from the above figure.
x = offset distance from tangent to the curve. Note: x is perpendicular to T . θ = offset angle subtended at PC between PI and any point in the curve
D = Degree of curve. It is the central angle subtended by a length of curve equal to one station. In English system, one station is equal to 100 ft and in SI, one station is equal to 20 m.
Sub chord = chord distance between two adjacent full stations. Sharpness of circular curve The smaller is the degree of curve, the flatter is the curve and vice versa. The sharpness
of simple curve is also determined by radius R. Large radius are flat whereas small radius are sharp.
Formulas for Circular Curves The formulas we are about to present need not be memorized. All we need is geometry plus names of all elements in simple curve. Note that we are only dealing with circular arc, it is in our great advantage if we deal it at geometry level rather than memorize these formulas. For easy reference, the figure shown in the previous page will be shown again here.
Length of tangent, T Length of tangent (also referred to as subtangent) is the distance from PC to PI . It is the same distance from PI to PT . From the right triangle PI-PT-O, tanI2=TR tan I2=TR
T=RtanI2T=Rtan I2
External distance, E External distance is the distance from PI to the midpoint of the curve. From the same right triangle PI-PT-O, cosI2=RR+Ecos I2=RR+E
R+E=RcosI2R+E=Rcos I2 E=RsecI2−R E=Rsec I2−R
Middle ordinate, m Middle ordinate is the distance from the midpoint of the curve to the midpoint of the chord. From right triangle O-Q-PT , cosI2=R −mR cos I2=R −mR
RcosI2=R −mRcos I2=R −m m=R −RcosI2m=R−Rcos I2
Length of long chord, L Length of long chord or simply length of chord is the distance from PC to PT . Again, from right triangle O-Q-PT , sinI2=L/2R sin I2=L/2R
RsinI2=L/2Rsin I2=L/2 L=2RsinI2L=2Rsin I2
Length of curve, L
c
Length of curve from PC to PT is the road distance between ends of the simple curve. By ratio and proportion, LcI=2πR 360∘LcI=2πR360 ∘
Lc=πRI180∘Lc=πRI180 ∘
An alternate formula for the length of curve is by ratio and proportion with its degree of curve.
LcI=1stationDLcI=1stationD Lc=1station×IDLc=1station×ID SI units: 1 station = 20 m
Lc=20IDLc=20ID
English system: 1 station = 100 ft
Lc=100IDLc=100ID
If given the stationing of PC and PT Lc=Stationing of PT− Stationing of PC Lc=Stationing of PT− Stationing of PC
Degree of curve, D The degree of curve is the central angle subtended by an arc (arc basis) or chord (chord basis) of one station. It will define the sharpness of the curve. In English system, 1 station is equal to 100 ft. In SI, 1 station is equal to 20 m. It is important to note that 100 ft is equal to 30.48 m not 20 m. Arc Basis In arc definition, the degree of curve is the central angle angle subtended by one station of circular arc. This definition is used in highways. Using ratio and proportion,
1stationD=2πR 360∘1stationD=2πR360 ∘ SI units (1 station = 20 m): 20D=2πR 360∘20D=2πR360 ∘
English system (1 station = 100 ft):
100D=2πR 360∘100D=2πR360 ∘
Chord Basis Chord definition is used in railway design. The degree of curve is the central angle subtended by one station length of chord. From the right triangle shaded in green color,
sinD2=halfstationR sin D2=halfstationR SI units (half station = 10 m): sinD2=10R sin D2=10R
English system (half station = 50 ft):
sinD2=50R sin D2=50R
Minimum Radius of Curvature Vehicle traveling on a horizontal curve may either skid or overturn off the road due to centrifugal force. Side friction f and superelevation e are the factors that will stabilize this force. The superelevation e = tan θ and the friction factor f = tan ( phi ). The minimum radius of curve so that the vehicle can round the curve without skidding is determined as follows.
From the force polygon shown in the right tan(θ+ϕ)=CFWtan (θ+ϕ)=CFW
tan(θ+ϕ)=Wv2gRWtan (θ+ϕ)=Wv2gRW tan(θ+ϕ)=Wv2WgR tan (θ+ϕ)=Wv2WgR tan(θ+ϕ)=v2gR tan (θ+ϕ)=v2gR The quantity v2/gR is called impact factor. Impact factor
if =v2gR if=v2gR
Back to the equation tan (θ + phi ) = v2/gR tan(θ+ϕ)=v2gR tan (θ+ϕ)=v2gR
tanθ+tanϕ1−tanθtanϕ=v2gR tan θ+tanϕ 1−tan θtanϕ =v2gR Recall that tanθ=etan θ=e and tanϕ=f tanϕ =f e+f 1−ef=v2gR e+f1−ef=v2gR But ef ≡0ef ≡0, thus e+f=v2gR e+f=v2gR Radius of curvature with R in meter and v in meter per second
R=v2g(e+f)R=v2g(e+f) For the above formula, v must be in meter per second (m/s) and R in meter (m). For v in kilometer per hour (kph) and R in meter, the following convenient formula is being used. R=(vkmhr)2(1000mkm×1 hr3600 sec)2g(e+f)R=(vkmhr)2(1000mkm×1 hr3600 s ec)2g(e+f) R=v2(13.6)2g(e+f)R=v2(13.6)2g(e+f)
R=v2(3.62)g(e+f)R=v2(3.62)g(e+f) R=v2(3.62)(9.80)(e+f)R=v2(3.62)(9.80)(e+f) Radius of curvature with R in meter and v in kilometer per hour R=v2127(e+f)R=v2127(e+f) Using the above formula, R must be in meter (m) and v in kilometer per hour (kph).
