13 civl235 route curves

7/29/2019 13 Civl235 Route Curves

1/9

Plane Surveying Route CurvesCivil Engineering 235 Department of Civil Engineering, UBC

Page 1 of 9

HORIZONTAL CIRCULAR CURVES:

a) Definitions

Route curves used to connect two tangents in a horizontal plane are defined in several ways. We can refer to a

curve by the degree of the curve or by the radius of the curve. The degree of the curve is defined as the anglesubtended by a sector of a circle with a given arc or chord length. In Imperial units the degree of curve, Dc, is

the angle subtended by (a) 100 ft chord (railway definition), or (b) 100 ft arc (highway definition). The metric

degree of curve, Dm, is the central angle subtended by an arc or chord of 10 metres. Thus the functions of the

1 metric curve are one-tenth of the corresponding functions of a 1 imperial curve (in feet) and the ratio ofmetric curves to foot curves is (10 m/100 ft) = (32.808/100) = .32808. However, in the metric system a curve is

more usually expressed by its radius, Rm. The figures illustrate the various definitions and the relationship

between degree of curve and radius of curve.


2/9


Page 2 of 9

b) Properties of a Circular Curve

Rc = radius

Lc = length of curve (arc)

Tc = sub-tangent

C = long chord

Ec = External distance

Mc = middle ordinate

P.I. = point of intersection

? = intersection or central angle

B.C. = beginning of curve (P.C)

E.C. = end of curve (PT)


3/9


Page 3 of 9

Basic relationships are:

)2

-1(R=Mor

)2cos-1(R=Mor

2cosR=M-R:M:OrdinateMiddle

)1-2Sec(R=Eor)1-

2Sec(R=Eor

2cos

1=

R

E+R:Ece,sExternalDi

2R2=Cor

2R2=C:ChordLong

D10=Lor

D100=L:LCurve,ofLength

2

R=Tor2=

R

T:TTangent,

R

L180=Angle,Central

D10=Lor

180R=LLength,

D

573=Ror

L180=RRadius,

mm

ccccc

mmcc

c

ccc

mmcc

m

cm

c

cc

mm

c

cc

c

c

c

m

cm

ccc

m

m

c

cc

cos

tan

sinsin

tantan


4/9


Page 4 of 9

c) Deflection Angles

For a curve of 100 ft arc

Deflection angle from tangent for a 100 ft arc is Dc/2.

Deflection angle from tangent for less than 100 ft arc, is proportionately less; i.e. for 27 ft arc deflection

is 27/100 x Dc/2.

We can use this property by running in any curve in 100 ft arcs and calculating each successive

deflection angle as 1/2 Dc. However, this means that we have to make a chain correction because the

chord length for a 100 ft arc is something less than 100 ft. It is given by C = 2 Rc sin Dc/2. [Corrections

to chord length less than 100 ft arc proportional.] [In practice we adjust for corrections to deflectionangles and chainage at the end of curve E.C.]

For a curve of 10 m arc:

Deflection angle, , from tangent is Dm/2.

Deflection angle from tangent for less than 10 m is proportionately less; i.e. for 2.7 m is =(2.7/10)(Dc/2).

Therefore, the deflection angles for each station on a metric curve (in which stations are 10 m apart)are shown below with the associated assumptions:

i) The metric degree of curvature (Dm) is the central angle to an arc 10 m long.ii) The bearing of the first 10 m of the curve deviates from the tangent by Dm/2; subsequent

chords vary from the preceding chord by Dm.


5/9


Page 5 of 9

iii) At the end of the curve (consisting of full 10 m stations) the tangent deviates from the last

chord by Dm/2 again.

d) Laying in a Circular Curve

Curves are normally run in the field by means of deflection angles or tangent offsets. For the deflection

angle method, the points (stations) on the curve are established by turning angles from the tangent

with a transit set-up on the BC; measuring along the chord the required distance between stations; and

establishing the point on the curve. An alternate method is to lay in the points on the curve by

measuring along the tangent a distance y = R sin [(57.3 x )/R] and "throwing" the curve point anoffset distance x = R - (R2 - y2)1/2. We use the deflection angle method in this case.


6/9


Page 6 of 9

i) Procedure for laying in the curve using Imperial measurements:

1. Locate P.I.

2. Determine length of Tc from curve data and set B.C. in field.

3. Locate E.C. in field by measuring Tc forward from P.I.

4. Set transit up on B.C. with plate reading 000' sighting on the P.I.

5. Deflection angles: 85 + 17.20 1314'85 + 00 1243'84 + 00 943'83 + 00 643'82 + 00 343'81 + 00 043'80 + 76.00 00'

6. Chain in curve and check whether located E.C. checks in chainage and alignment.

If an intermediate set-up is required follow this procedure:

a) Set transit at intermediate station (say 83 + 00)


7/9


Page 7 of 9

b) Back sight on B.C. with plate and vernier's reading 000'.

c) Plunge telescope and turn vernier through 643' to get on "local tangent" and then afurther 300' to set station 84 + 00. Note that in practice telescope is swung through

943' for station 84 + 00, exactly the same as calculated above from the set-up onB.C.

d) If a set-up on 84 + 00 with a sight on 83 + 00 with vernier reading 643' and for 85 +00 deflection angle is 1243'.

e) In other words, on an intermediate set-up set on vernier the deflection angle for that

station you sight on and move forward in the usual manner.

ii) Procedure for laying in a curve using metric measurements:

The procedure is the same except,

a) the design standard for the curve will likely be the radius Rm of the curve instead of Dm, and

b) distance measures are in metres.

Example:

A curve of 100 m radius is to be run in between two tangents intersecting at = 45. The P.I.

chainage for #12 P.I. is 1 + 423.73 [or 1000 metres + 423.73 metres from some fixed reference point].We have also calculated the tangent bearings to be S8500'E and S4000'E. The stations on thecurve are to be at 10 m intervals.

1) Locate P.I.

2) Determine length of T from T = R tan /2 = 100 x tan 22.5= 41.42 m

3) Set B.C. and E.C. in the field

4) Set transit on B.C. and swing the deflection angle to the first full station [i.e. 1 + 390.00]

2R

573.769=

2

D

10.00

7.69=

2

D

382.31+1-

390.00+1=

m

mm

10


8/9


Page 8 of 9

The curve data is:

? = 45R = 100 m

T = 41.42 m

L = 78.54 m

P.I. #12 = 1+423.73

-T = 41.42

B.C. = 1+382.31

+L = 78.54

E.C. = 1+460.85

Dm = 543'48"Dm/2 = 251'54"

Calculation of deflection angles:

Station or Curve Deflection Angle Adjusted Deflection Angle

BC 1+382.31

1+390.00

1+400.00

1+410.00

1+420.00

1+430.00

1+440.00

1+450.00

1+460.00

0 = 01 = 212'12" [.769x251'54"]2 = 504'06"3 = 756'00"4 = 1047'54"5 = 1339'48"6 = 1631'42"

7 = 1923'36"8 = 2215'30"

0 = 01 = 212'12"2 = 504'05"3 = 755'58"4 = 1047'51"5 = 1339'44"6 = 1631'37"

7 = 1923'30"8 = 2215'23"

EC 1+460.85

Check9 = 2230'07" [.085x251'54"]9 = /2 = 2230'00"

9 = 2230'00"


9/9


Page 9 of 9

ROUTE CURVE EXERCISE:

The objective of the route curve exercise is to gain familiarity with the theory, design and layout of highwaycurves. In highway (and railway) location, the horizontal curves used at points of change in direction are arcs of

circles. The straight line connecting these circular curves are tangent to them and are called tangents. The field

exercise is for each party to design and lay out two circular curves connecting two tangents. The party as awhole will establish control points, measure the intersecting angle between tangents and assist with curve

design and layout.

Each party is required to layout a circular curve between two tangents on an imaginary road. Two stakes areset in the field, defining the beginning of the curve (Point B.C.) and an arbitrary point along the direction of thefirst tangent. The parties are provided with the chainage of the intersection (P.I.), radius of the curve, R, and the

central angle, ?. Follow the steps below to set up the station on the curve:

1. Calculate the chainage of the beginning (B.C.) and end (E.C.) of the curve.

2. Find the location of P.I. along the tangent line assigned to your group.

3. Set up the transit at P.I. and find the location of E.C.

4. Calculate the deflection angles for all the intermediate stations.

5. Lay the stations along the curve, including the point E.C.

Note that every individual should enter all the details of calculations and the field procedure in his/her fieldbook. After laying all the stations along the curve, the TAs will examine your calculations and the accuracy of

the stations, and assign the accuracy mark for all the members of the party.

Equipment per party:

Transit Chaining PinsSteel Tape Crayon

Plumb Bobs (3) Range PoleStakes (6)

13 civl235 route curves

Documents