FUNDAMENTALS OF SURVEYING : Curves

Curves

> Curves are used on highways or railways when there is change in direction or alignment of road. > These are circular, parabolic and spiral in shape.

> Circular curves are further divided as simple, compound and reverse.

1) Simple curve;

> Simple curve is the one which consists of single arc of a circle.

> It is tangential to both straight lines.

> A sharpness of curve is designated by radius (in Q UK) or degree of curve (in USA, Canada

and India)

Degree of curve:

> The angle subtended by a chord of fixed length

> In India a chord of 20m is generally taken to calculate the degree of curve,

360⁰ o 2πR

D0 o 20m

1146

R =_ D for 20 m chain

1719

R =_ D , for 30m chain

Nomen cloture of curve:/ Definitions and Notations: Backward tangent / Bank tangent / First tangent (AT₁)

Tangent pervious to the curve is called back tangent. Forward tangent or second tangent (T₂B)

The tangent following the curve.

Point ofintersection or vertex(V)

If tangent AT₁ and BT₂ are produced they will meet at point V called as point of

intersection.

Point of curve (PC) (T₁)

The point from where the alignment changes from tangent to curve.

Point of tangency (T₂) (PT)

End point ofthe curve where the alignment changes from curve to tangent.

Deflection angle / Deviation angle:

The difference between the slopes oftwo tangents.

Tangent distance/tangent length:

It is the distance between P.C. and P.I. i.e. T₁V

Also the distance between PI and PI i.e. VT₂

T = T₁V =VT₂= R.Tan

External distance or Apex distance (E)

Distance between midpoint of curve and PI.

Distance VC

 

E = OV – OC = R. sec

       -1

   

2

Mid ordinate / versed sine (M) (distance CD)

it is the ordinate from mid point of long chord to mid – point of the curve.

M = C D = O C – OD = R – R cos ( /2)

                      
M=R 1-cos

          = R.versin   

        

2                         

2

Length of curve (L):

Total length of curve from P.C. to P.T. l = R.D, when  in radian π

l =R.D, ____ when in degrees.



180

Long chord: (T₁T₂)

Chord joining PC to PT.

Δ

T T = 2R sin = lenght of long chord

1 2                                                                             2
Right hand curve:

If the curve deflects to the right ofthe direction ofprogress of survey, it is called right hand

curve Left hand curve:

The curve which deflects to the left of the direction ofprogress of survey.

Chainages:

Chainage of T₁ = chainage of V – tangent length

Chainage of T₂ = chainage of T₁ + length of curve.

Setting out simple curves:

1) Linear methods: Only chain or tape is used. Used when high degree of accuracy is not required and when curve is short.

2) Angular methods:

An instrument such as theodolite is used with or without a chain or tape.

1) Linear methods of setting out:

a)      By ordinates or offsets from the long chord.

b)      By successive bisection of arc

c)      By offsets from the tangents

d)     By offsets from chord produced (or by deflection distances)

2) By ordinates or offsets from long chord

Let, L = length of long chord, Mid ordinate O₀ = OC – OD

2

2 L

 

R - R -              

 

2

= R-Rcos (A/2)

O_x = ordinate at a distance x from the mid-point of the chord. O_x = EF = E₁ D = E₁O – OD

O = R - x - (R -O ₀ )

2 2

x

Approximate method:

EG × 2R = T₁F ×FT₂, let T₁F = x O. 2R = x (L - x)

x

O = x	x L- x (           )
	2R

x’—> dirt from tangent pt T₁

2)      By successive bisection of arcs or chords (using versed sine) CD = R - R cos (A/2)

C1D1= C2D2= R -Rcos (A/4) C₃ D₃= R -R cos (A/8) And so on.

3)             By Offsets from the tangent:

When A and R both are small then curves can be set out by offsets from tangent

a) Perpendicular offsets: offset at a distance x from T₁

O_x = AB = T₁C

O_x = OT₁ – OC

O = R - R²-x²

O_x=

x

2R

Approximate formula. b) Radial offsets:

O_x= AB = OA - OB

O = R²+ x²-R

x

O_X =___ approximate formula
^x 2R

4) By deflection distances (OR offsets from the chord produced)

C1 _> c _  C1

2S1= R          '31 2R

O1= S1C1

O1=

C2 1 2R

O₂= BB₂ = BB₁ +B₁ B₂ C₂ S₁ + C₂ S₂

C ^C1  + C C2  ² 2R ² 2R

O2= ^C2 (C₁+ C₂) 2R

C

3

O3= _2R (C₂+ C₃)

On= 2 R Cn (Cn- 1 + C_n )

Angular methods:

Rankines method of define angles

δ = 1

C1 2R

C₁ 180

δ = 1  ×              ×60min

¹ 2R π

1718.09

δ1=  _R x C₁ min

1718.09 R

δ₂=_________ x C₂

1718.09

δ =________ ×C

n                                              n

R

Total deflection angle,

A 1 = δ₁

A 2 = δ₁ + δ₂

A 3 = δ₂ + δ₃

A n = δ_n-1 + δ_n

A n = A /2, check

Compound curve:

T₁ D T₂ is a two centered compound curve having two circular arcs T₁D and DT₂ meeting at a

common po0int D known as the point of compound curvature (PCC).

T₁ – pt. of curve, T₂ – pt. of tangency,

O₁ & O₂ – center of the two arcs,

R_s – smaller radius (T₁O₁)

R_L – longer radius (T₂O₂)

D1D2 – common tangent

When it is not possible to connect tow tangents by one circular curve, it becomes necessary to

take a suitable common tangent, and set out two curves ofdifferent radii to connect the rear and

forward tangents. This curve is known as compound curve.

A 1 = deflection angle between rear and common tangent

A 2 = deflection angle between the common and forward tangent

 = total deflection angle =  1+ 2

t_s = length of the tangent to the arc (T₁D) having smaller radius.

_ 

A 1

t = R tan           = T D = D D

s          s                       1       1            1

 

2

t_L = length of tangent to the arc DT₂ having longer radius.

__ 

A 2

t = R tan            = DD = D T

L              L                    2               2 2

 

2

From  VD₁D₂, we have

D D                     D V D V

1 2                 1______ 2

=            =

sin (180-A) sin A              sin A

2                           1

D1D2 = t_s + t_L

t + t

___ s                         L

D V =                      = sin A

1                                                                             2

sin (180-A) t + t

___ s                         L

D V =                       =sin A

2                                                                             1

sin(180-A)

   1   2

T_s = tangent distance T₁V corresponding to shorter radius,

t +t s_______________________________________________________________________________________________________________________________________________________________________________________________________________________ L T = T D +D V= t                                                                                =sin A S                                                                                    1 1                                        1                               s sin 180-A                                                                                                                                                                                                                                                 

2

T_L = Tangent distance T₂V corresponding to longer radius t + t

s L

T =T +                        =sin A

L L                                                                             1

sin 180-A

           

Note: centrifugal ratio,

Allowable value ofCR,

In roads = 1/4

In railways = 1/8

Reverse curve:

A reverse curve consists of two simple curves of opposite direction that join at a common

tangent point called as the point of reverse curvature (PRC).

Necessity:

1) These are provided when two straights run parallel to each other or include a very small angle of

intersection.

2) Provided generally in mountainous countries, in cities, and in the layout ofrailway spur tracks and cross- over.

Reverse curve should be avoided on highways and main railway lines where speeds are high due to following reasons

1)        There is sudden change in CF at point of reverse curve (PRC). From one side to other which may cause jerk.

2)        No opportunity to elevate outer edge at PRC.

3)        There is sudden change in directions which is uncomfortable for passengers and is objectionable.

Steering along reverse curve is dangerous in case of highways and the driver has to be very cautions.