Surveying‐I (130601)CHAPTER ‐4CHAPTER  4

CURVES

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Question asked in GTU‐TheoryQuestion asked in GTU Theory

1) Describe the procedure of setting out of simple) esc be t e p ocedu e o sett g out o s p ecircular curve by (i) Perpendicular offset fromtangent, and (ii) Rankine’s method of tangential

langle.. Dec‐20092) Why transition curves are introduced on

h i t l f hi h il ? Dhorizontal curves of highways or railways? Dec‐2009

3) Describe the method of setting a circular curve3) Describe the method of setting a circular curveby the method of offsets from the long chord.Dec‐2010

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Question asked in GTU‐TheoryQuestion asked in GTU Theory

4) Discuss the method of setting out a circular curve) gwith two theodolite. What are its advantagesand disadvantages over Rankine’s method Dec‐20102010

5) What are the elements of simple circular curve?Define with figure and give their relationship.March‐2010

6)  Why are curves provided? State various types of         curves with sketchcurves with sketch.

7)  Draw the neat sketch of simple circular curve showing various elements of it. Dec‐2011g

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Question asked in GTU‐TheoryQuestion asked in GTU Theory

8) Enumerate the parts of a compound curve8) Enumerate the parts of a compound curveand describe the relationship between themJan‐2013Jan 2013.

9) What is vertical curve? Explain different typesof vertical curves Jan 2013of vertical curves. Jan‐2013.

10) Explain following terms (i) Compound curve(ii) P i f i i (iii) T Di(ii) Point of intersection (iii) Tangent Distance(iv) Mid OrdinateMay‐2012.

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Lecture outlineLecture outline

• IntroductionIntroduction• Theory and setting out methods of simple circular curvecircular curve

• Elements of compound and Reverse curve.• Transition curve• Transition curve• Types of Transition curveC bi d• Combined curve

• Types of vertical curve.

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What is curve?What is curve?

• Why Curve?Why Curve?

f C• Use of Curve.

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Components of Highway Design

Horizontal Alignment

Plan View

Horizontal Alignment

V ti l Ali t

Profile View

Vertical Alignment

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Horizontal AlignmentT d ’ ClToday’s Class:• Components of the horizontal alignment• Properties of a simple circular curve

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Types of CurveTypes of Curve

• CurvesCurves

• Horizontal Curve Vertical Curve

Circular Curve Transition Curve Summit Curve Valley Curve

1) Simple curve 1) Cubic parabola

2) Compound Curve 2 ) Spiral Curve2) Compound Curve 2 ) Spiral Curve

3) Reverse Curve 3) Lemniscate20 September 2013

Types of Circular Curve

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Types of Circular Curve

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Types of Circular Curve

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Definition and Notation of Simple Curve

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Definition and Notation of Simple Curve• 1) Back tangent or First Tangent ‐ AT₁

– Pervious to the curve

d d2) Forward Tangent or Second tangent‐ B T₂‐ Following the curve.3) Point of Intersection ( P.I.) or Vertex. (v)If the tangents AT₁ and BT₂ .are produced theyill t i i t ll d th i t fwill meet in a point called the point of

intersection4)Point of curve ( PC ) Beginning Point T of a4)Point of curve ( P.C.) –Beginning Point T₁ of acurve. Alignment changes from a tangent tocurve.

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Definition and Notation of Simple Curve

) f• 5) Point of Tangency ‐ PT– End point of curve ( T₂ ) is called..6) Intersection Angle (Ø )) g (Ø )‐ The Angle AVB between tangent AV and tangent VB iscalled...

7) Deflection Angle (∆ )7) Deflection Angle (∆ )The angle at P.I. between tangent AV and VB is called..

8)Tangent Distance –It is the distance between P.C. and P.I.

9) External Distance – CIThe distance from the mid point of the curve to P.I.The distance from the mid point of the curve to P.I.It is also called the apex distance.

10) Length of curve – lI i h l l h f f PC PTIt is the total length of curve from P.C. to P.T.

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Definition and Notation of Simple Curve11) Long Chord11) Long Chord– It is the chord joining P.C. to P.T., T₁ T₂ is a long chord.12) Normal Chord:

A chord between two successive regular station on a curve isA chord between two successive regular station on a curve iscalled normal chord. Normally , the length of normal chord is 1chain ( 2o mt).

13) Sub chordThe chord shorter than normal chord ( shorter than 20 mt) iscalled sub chord)

14) Versed sine – Distance CDThe distance between mid point of long chord ( D ) and the apexpoint C, is called versed sine. It is also called mid‐ ordinate ( M).

15) Right hand curve:If the curve deflects to the right of the direction of the progressof survey.

16) Left hand curveIf th d fl t t th l ft f th di ti f thIf the curve deflects to the left of the direction of the progressof survey.

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Designation of curveThe sharpness of the curve is designated by two ways.ways.

( 1 ) By radius ( R)

( 2) B D f C t ( D )( 2) By Degree of Curvature ( D ) 

( 1 ) By radius ( R)

Curve is known by the length of its radius‐ RCurve is known by the length of its radius R

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Designation of curve( 2) By Degree of Curvature ( D )

Chord Definition Arc Definition     

The Angle subtended atthe centre of curve by ah d f 30 20 i

The Angle subtended atthe centre of curve by an

chord of 30 or 20 mt. iscalled degree ofcurvature.

arc of 30 or 20 mt.length is called degree ofcurve.

If an angle subtended atthe centre of curve by a

Used in America, canada,India etc

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the centre of curve by achord of 20 mt is 5° , thecurve is called 5° curve

India etc.

Relation between Radius and degree of curve.( ) B h d d fi iti(a) By chord definition

The angle subtended at the centre of curveby a chord of 20 mt. is called degree of curve.

R = radius of curve.D = degree of curve.PQ = 20 mt. = Length of chord.g

From Triangle PCO 

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Relation between Radius and degree of curve.

When D is small,           may taken equal to

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Relation between Radius and degree of curve.

(b) By Arc Definition :

The angle subtended at the centre of curve byan arc of 20 mt. length is called degree ofcurve.

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Elements of Simple circular curve

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Elements of simple circular curveElements of simple circular curve

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Elements of Simple circular curve

• T₁ = P.C.= Point of tangency=Point of curve.

• T₂ = P.T.= Second point of tangency.T₂   P.T.  Second point of tangency.

• V or I  = P.I. = Point of intersection.

∆ D fl ti l• ∆ = Deflection angle.

• Ø = Intersection angle.

• R = Radius of curve.

• CD= Mid ordinate (M)CD  Mid ordinate (M)

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Example 1• A circular curve has a radius of 150 mt and 60⁰deflection angle. What is its degree(i) By arcdeflection angle. What is its degree(i) By arcdefinition and 9ii) by chord definition.

• Solution:• Solution:

(i) By arc definition Assuming chord length 30mt

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Elements of Simple circular curve

• 1) Length of curve ( l)• * If curve is designated by Radius:g y

l = Length of arc T₁ C T₂ = R * ∆          ‐When ∆ is in Radian

‐ When ∆ is in degree.

* If curve is designated by degree:• Length of arc =20 mt.• Length of curve  

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Elements of Simple circular curve

2) Tangent length ( T):

• VT and VT are the tangents length Elements• VT₁ and VT₂ are the tangents length Elements of simple circular curve

T VT VT t t l th• T = VT₁ = VT₂ = tangent length

• From ∆ VT₁O 

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Elements of Simple circular curve

3) Length of chord ( L ):Elements of simple circular curvecircular curve

• In the figure T₁ ,T₂ is a long chord.

L th f l h d L T T 2 * (T D)• Length of long chord =L =T₁T₂ =  2 * (T₁ D).

• From triangle T₁DO,

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Elements of Simple circular curve

4) External Distance ( E ): or Apex distanceElements of simple circular curve

• In the figure VC  is an external distance.• External distance = E= VC= OV –OC Length of• From triangle VT₁O.

E= OV‐OC   ( OC = R)

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Elements of Simple circular curve

• 5) Mid ordinate ( M ): = Distance – CD

• Also known as versed sine of the curve.

• Mid ordinate =M= CD= OC‐OD

• From ∆T₁DOFrom ∆T₁DO 

• M = OC‐ OD

•

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Setting out of single Circular curve

• First step‐ Locate tangent point

• ‐ By tape measurements.

‐Intersection of both tangents point V‐ Point   of intersection.

‐ Set theodolite at V and measure angle Ø

Ø‐ Ø ( Measure by theodolite)

‐ Calculate tangent length

‐ Fix point  T₁T₂

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Setting out of single Circular curve

• Chainage of tangents:

• ‐ Point A is the starting point of chain line

• Chainage of point V, B, D are measured from        point A.p

• ‐ Chainage of T₁ = Chainage of V‐ T ( Tangent length)

• T Chainage of T + Length of curve (l)• T₂ =Chainage of T₁ + Length of curve (l)

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Setting out of single Circular curveN l h d d S b h d• Normal chord and Sub chord:

• ‐For alignment pegs are driven.h di b i ll 20• The distance between two pegs is normally 20m

• Peg station are called main stations.• The chord joining the tangents point T₁ and the firstmain peg station is called First sub chord.All th h d j i i dj t t ti• All the chord joining adjacent peg stations arecalled full chord or normal chord.

• The length of normal chord is 20 mt• The length of normal chord is 20 mt.• The point joining last main peg station and tangentT₂ is called last sub chordT₂ is called last sub chord.

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Methods of Setting out of single Circular curve

• Two Methods

• 1) Linear Methods

• 2) Angular Methods.

• 1) Linear Methods

( ) ff d f h l h d• ‐ (i) By offsets or ordinate from the long chord.

• (ii) By successive bisection of arcs or chords.

• (iii) By offsets from the tangents.

• (iv) By offsets from the chord produced(iv) By offsets from the chord produced.

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(i) By offsets or ordinate from the long chord.

R = Radius of curveR = Radius of curveO0  = Mid ordinateOx = Ordinate at distance xT1, T2 = tangents pointL = Length of long chord.

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(ii) By successive bisection of arcs or chords.

• T1 T2= L• T1‐T2= L

• T1‐C = L

• T2‐C = L

• C‐C1, C‐C2=L,

• C1‐T1, C2‐T2=L

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(iii) By offsets from the tangents.• Two types

• Radial offset Perpendicular offsetp

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Angular Method• Used when length of curve is large• More accurate than the linear methods• More accurate than the linear methods.• Theodolite is used

• The angular methods are:1) Rankine method of tangential angles.

OR One theodolite method

2) Two theodolite method.

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Obstacles in setting out simple curves

• Case –I -When P.I. is inaccessibleC II Wh P C i i ibl• Case –II -When P.C. is inaccessible

• Case –III -When P.T. is inaccessible• Case –IV - When both P.C. and P.T. is

inaccessible.• Case –V - When obstacles to chaining.

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TRANSITION CURVE

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Requirement of transition curve• Tangential to straight

M t i l t ti ll• Meet circular curve tangentially

• At origin curvature should zero.

• Curvature should same at junction of circular curve.

• Rate of increase of curvature = rate increase of super elevation.p

• Length of transition curve = full super elevation attainedelevation attained.

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Purpose of transition curvep• Curvature is increase gradually.• Medium for gradual introduction of• Medium for gradual introduction of superelevation

• Provide Extra widening gradually• Provide Extra widening gradually• Advantages• Increase comfort to passenger on curve• Increase comfort to passenger on curve• Reduce overturning• Allow higher speed• Less wear on gear, tyre

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Types of transition curve• Cubic parabola

• For railway• ‐ For railway

• Spiral or Clothoid

‐ Ideal transition

‐ Radius α Distance

• Lemniscates

‐ Used for road‐ Used for road

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Vertical curve

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Length of vertical curveLength of vertical curve

• Total change of gradeTotal change of grade• Length of vertical curve =‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐• Rate of change of grade• Rate of change of grade

• g2 – g1• =     ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐• r• g1, g2  = Grades in %• r = Rate of change of grade

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