Curves MCQ Quiz - Objective Question with Answer for Curves - Download Free PDF

Explanation:

Deflection Angle

Definition: A deflection angle is the angle formed by a survey line with the prolongation (or extension) of the preceding survey line. It is measured either to the right or left of the prolongation of the previous line. This type of angle is crucial in surveying and is generally used in route surveying, such as for roads, railways, or pipelines, where changes in direction occur along the path.

Working Principle: In surveying, to determine the direction of a new line concerning the previously established line, the deflection angle is measured. This angle can be either clockwise (right) or counterclockwise (left), and it helps surveyors establish accurate alignments and ensure that the project layout is consistent with the intended design.

Measurement: A deflection angle is measured using a theodolite or a total station instrument. The prolongation of the preceding survey line is extended, and the angle between this prolongation and the new line is measured.

Types of Deflection Angles:

• Right Deflection Angle: If the new line deviates to the right of the prolongation of the previous line, the angle is termed a right deflection angle.
• Left Deflection Angle: If the new line deviates to the left of the prolongation of the previous line, the angle is termed a left deflection angle.

Importance in Surveying:

• Deflection angles are critical for setting out curves and ensuring the alignment of structures such as roads and railways.
• They help in avoiding obstacles and ensuring that the layout follows the design plan.
• Accurate measurement of deflection angles ensures that there are no errors in the alignment of successive survey lines.

Concept:

For the given curve:

Tangent length \(\left( \text{T} \right)=\text{R}\tan \frac{\text{ }\!\!\Delta\!\!\text{ }}{2}\)

Length of curve \(\left( \text{l} \right)=\frac{\text{ }\!\!\pi\!\!\text{ R }\!\!\Delta\!\!\text{ }}{180}\)

Long chord \(\left( \text{L} \right)=2\text{R}\sin \frac{\text{ }\!\!\Delta\!\!\text{ }}{2}\)

External distance \(\left( \text{E} \right)=\text{R}\left( \sec \frac{\text{ }\!\!\Delta\!\!\text{ }}{2}\text{ }\!\!~\!\!\text{ }-1 \right)\)

Mid-ordinate \(\left( \text{M} \right)=\text{R}\left( 1-\cos \frac{\text{ }\!\!\Delta\!\!\text{ }}{2}\text{ }\!\!~\!\!\text{ } \right)\)

The ratio of long chord to tangent length of a simple circular curve:

= \({2\text{R}\sin \frac{\text{ }\!\!\Delta\!\!\text{ }}{2}}\over{\text{R}\tan \frac{\text{ }\!\!\Delta\!\!\text{ }}{2}}\)= 2cos(Δ°/2)

The ratio of long chord to the length of simple circular curve:

= \({2\text{R}\sin \frac{\text{ }\!\!\Delta\!\!\text{ }}{2}}\over {\frac{\text{ }\!\!\pi\!\!\text{ R }\!\!\Delta\!\!\text{ }}{180}}\) = 360° sin (Δ°/2)/Δπ 

Explanation:

• A reverse curve is made up of two arcs having equal or different radii bending in opposite direction with a common tangent at their junction.
• Their centers lie on opposite sides of the curve.
• The reverse curve is used when the straights are parallel or intersect at a very small angle.
• Reverse curves are also called a serpentine curve or the S curve because of their shape.

Simple curve:

• A simple circular curve has the property that it connects two straight lines with a curve of the constant radius at all points on the curve and connects the two straight-line tangentially.

Compound curve:

• Many times it is not possible to provide a curve of constant radius to connect the two straight lines. In that case, we provide more than one curved of different radii to connect them.

Transition curve:

• While going through a straight road, if a curve is suddenly encountered, we feel a jerk. In order to avoid such a sudden jerk, a Transition curve provided.
• This is a curve of the varying radius is provided which takes off from the straight line, turns gradually, and finally meets the curve i.e. attains the same radius as that of the curve.

Explanation:

• The Linear Method involves direct distance measurements using a chain or tape along the tangent and curve without requiring angular measurements.

• This method eliminates the need for intermediate angular observations and is practical when a theodolite is unavailable or angular measurements are difficult to take.

Analysis of Other Options:

1. Rankine’s Method of Tangential Angles 

   • This method uses angular measurements from the tangent point and is dependent on theodolite observations.

2. Offset from Chord Method 

   • This method involves offset distances from the chords of the curve, which still requires reference points and angular checks.

3. Angular Method using a Theodolite 

   • This method relies completely on angular measurements, making it unsuitable for a technique that predominantly uses a chain or tape.

Thus, the Linear Method is the correct choice since it is based on distance measurements along tangents and the curve, without requiring angular observations.

Concept:

Versine of curve: The versine is the perpendicular distance of the midpoint of a chord from the arc of a circle.

where, M = Versine of curve, Δ  = Deflection angle and R = Radius of curve

\(M = R \times (1 - Cos{\Delta\over 2})\)

Calculation:

Given: R = 600 m,  Δ  = 120⁰

\(M = 600 \times (1 - cos{120\over 2})=300m\)

Concept:

For the given curve:

Tangent length \(\left( {\rm{T}} \right) = {\rm{R}}\tan \frac{{{\rm{\;}}{\rm{Δ }}{\rm{\;}}}}{2}\)

Length of curve \(\left( {\rm{l}} \right) = \frac{{{\rm{\;}}\pi {\rm{\;R\;}}{\rm{Δ }}{\rm{\;}}}}{{180}}\)  

Long chord \(\left( {\rm{L}} \right) = 2{\rm{R}}\sin \frac{{{\rm{\;}}{\rm{Δ }}{\rm{\;}}}}{2}\)  

External distance \(\left( {\rm{E}} \right) = {\rm{R}}\left( {\sec \frac{{{\rm{\;}}{\rm{Δ }}{\rm{\;}}}}{2}{\rm{\;}}{\rm{\;}}{\rm{\;}} - 1} \right)\)  

Mid-ordinate \(\left( {\rm{M}} \right) = {\rm{R}}\left( {1 - \cos \frac{{{\rm{\;}}{\rm{Δ }}{\rm{\;}}}}{2}{\rm}} \right)\)  

Calculation:

 Given angle of intersection = 120 ° 

But we need to find deflection Angle Δ 

Angle of intersection + Deflection angle = 180°

∴ Deflection angle, Δ = 60° 

Length of long chord = 2 × 600 × sin (60°/2) = 600 m.

Explanation:

Length of the curve of radius R and deflection angle Δ is given by:

\({\rm{L}} = \frac{{2{\rm{\pi R}}}}{{360}} \times {\rm{\Delta }}\)

Where,

R = Radius of curve in “km”, Δ = Deflection angle, and L = Length of the curve,

It can be concluded from the above expression that:

∴ Length of the curve is directly proportional to the radius of curve as well as deflection angle of curve.

Concept:

For 20 m chain length,

Degree of curve (D) is given by:

\(D = \frac{{1146}}{R}\)

For 30 m chain length,

Degree of curve (D) is given by:

\(D = \frac{{1719}}{R}\)

Calculation:

Given,

L = 20 m

R = 573 m

\(D = \frac{{1146}}{{573}}\)

\(D = {2^o}\)

Explanation:

Rankine Method of tangential angles:

• Rankine method is based on the principle that the deflection angle to any point on a circular curve is measured by one half the angle subtended by the arc from Point of curve to that point.
• The method is most accurate since each point is fixed independently.

Two theodolite method: 

• This method is used when the ground is unsuitable for chaining and is based on the principle that the angle between the tangent and the chord is equal to the angle which that chord subtends in the opposite segment.
• It is use if the topography is rough or field condition is difficult.

Tacheometric method: 

• this method is less accurate than “Rankine Method of tangential angles”.
• It is used when the ground is rough and chaining cannot be done accurately.

Concept:

Versine of curve

The versine is the perpendicular distance of the midpoint of a chord from the arc of a circle.

The relationship between radius and versine of a curve in various units is shown below:

\(V = \frac{{{C^2}}}{{8R}}\) .............(V, C, R are in same units, say, m or cm)

\(V = \frac{{125{C^2}}}{R}\) ......... (V in mm, C in m, and R in m)

\(V = \frac{{1.5{C^2}}}{R}\)...........(C and R in feet and V in inches)

where, 

V = Versine of curve

C = Length of chord

R = Radius of curve

Concept:

(a) if the length of the chord is 20 m

\(D^{∘}={1146\over R}\)

Where R = Radius of the curve in m

D∘ = Degree of curve

(b) If the chord length is 30 m

\(D^{∘}={1719\over R}\)

Calculation:

Given data:

Curve radius (R) = 688 m

Chord length (L) =?

Degree of the curve (\(D^{∘}\)) =?

Let the length of the chord is 20

\(Degree\, of\, curve(D^{∘})={1146\over 688}\)

\(Degree\, of\, curve(D^{∘})=1.665^{∘}\)

Degree of curve 1.665^∘ not given an option

Let the length of the chord is 30

\(Degree\, of\, curve(D^{∘})={1719\over 688}\)

\(Degree\, of\, curve(D^{∘})=2.4985\approx 2.5^{∘}\)

\(Degree\, of\, curve(D^{\circ})=2.5^{\circ}\)

Concept:

Angle of intersection:

It is the angle between the back tangent and forward tangent. The surveyor either computes its value from the preliminary traverse station angles or measures it in the field. it can be denoted by ϕ. 

Curve:

Curves are used in highways and railways where there is a need to change the direction of motion. In general, Curves are provided at the intersection of the straight lines. Circular Curves are of two types:

1). Horizontal curve: A horizontal curve is provided at the point where the two straight lines intersect in a horizontal plane.

2). Vertical Curve: A vertical curve is provided at the point where the two straight lines at different gradients intersect in the vertical plane.

For the given curve:

Tangent length \(\left( \text{T} \right)=\text{R}\tan \frac{\text{ }\!\!Δ\!\!\text{ }}{2}\)

Length of curve \(\left( \text{l} \right)=\frac{\text{ }\!\!\pi \;\!\!\text{ R }\;\!\!Δ\!\!\text{ }}{180}\)

Long chord \(\left( \text{L} \right)=2\text{R}\sin \frac{\text{ }\!\!Δ\!\!\text{ }}{2}\)

External distance \(\left( \text{E} \right)=\text{R}\left( \sec \frac{\text{ }\!\!Δ\!\!\text{ }}{2}\text{ }\!\!~\!\!\text{ }-1 \right)\)

Mid-ordinate \(\left( \text{M} \right)=\text{R}\left( 1-\cos \frac{\text{ }\!\!Δ\!\!\text{ }}{2}\text{ }\!\!~\!\!\text{ } \right)\)

Concept:

The offset for the transition curve are found from-

\({\bf{offset}}\;\left( {\bf{y}} \right) = \frac{{{\ell ^3}}}{{6{\bf{RL}}}}\)

Where,

ℓ = measured along the curve (m),

R = radius of the curve (m), and

L = length of the curve (m)

∴ For offset at junction point, ℓ = L

\({\bf{y}} = \frac{{{{\bf{l}}^3}}}{{6{\bf{RL}}}} = \frac{{{{\bf{L}}^3}}}{{6{\bf{RL}}}} = \frac{{{{\bf{L}}^2}}}{{6{\bf{R}}}}\)         ---(1)

Also, we know Shift of the curve,

\({\bf{S}} = \frac{{{{\bf{L}}^2}}}{{24{\bf{R}}}}\)         ----(2)

From (1) and (2)

\(\frac{{\bf{y}}}{{\bf{S}}} = 4\;\)

Concept:

Given: Radius of the curve = 300 m

Formula:

Degree of curve is given by :

\({\rm{D}} = \frac{{1720}}{{\rm{R}}}{\rm{\;degree}}\),

Where,

D = Degree of the curve (degrees) and R = radius of the curve (m)

Calculation:

Degree of curve,

\({\rm{D}} = \frac{{1720}}{{\rm{R}}} = \frac{{1720}}{{300}} = 5.733^\circ \)

Concepts:

Reverse Curve: When two simple circular curves of equal or different radius,s bending or curve in opposite directions with a common tangent at their junction, their centers being on opposite sides of the curve. 

Characteristics of Reverse Curve:

• They are also called S curves as they appear like the English letter ‘S’.
• They are provided between two parallel lines or the angle between them is very small.
• They are suitable for railway yards but not suitable for highways.
• No superelevation is required.
• Wherever reverse curves are necessary to use, long lengths of straight portion (minimum 36 m length) should be provided in between two reverse curves.