Circular Curves in Geometry

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Questions and Answers

What does the intersection angle (I) represent in a simple circular curve?

The angle subtended at the center of the circle.

The angle formed by the radius and the tangent.

The angle by which the forward tangent deflects from the back tangent. (correct)

The angle between two tangent lines.

Which of the following correctly describes the term 'middle ordinate' in the context of a circular curve?

The distance from the mid-point of the long chord to the mid-point of the circular curve. (correct)

The distance from the point of intersection to the tangent distance.

The length of the arc between the point of curvature and the point of tangency.

The distance from the vertex to the mid-point of the circular curve.

In circular curves, what is the relationship between the radius (R) and the degree of curve (D0)?

Degree of curve is independent of radius.

Sharpness can be described by either radius or degree of curve. (correct)

A smaller radius indicates a smaller degree of curve.

A larger radius means a sharper curve.

What does 'long chord (C)' refer to in a circular curve?

The straight-line distance from point of curvature to point of tangency. Signup and view all the answers

Which statement accurately defines the term 'tangent distance (T)' in the context of simple circular curves?

The distance from the vertex to either the point of curvature or the point of tangency. Signup and view all the answers

What is the formula for calculating offset Y in the context of the given procedures?

Y = R - (R^2 - X^2) Signup and view all the answers

How is the offset at 10m calculated based on the provided information?

0.84m Signup and view all the answers

What is the significance of the deviation angle Δ in the calculations?

It influences the calculation of offsets. Signup and view all the answers

In the context of offsets from the long chord, what characteristic is primarily utilized?

It uses two tapes for accurate measurement. Signup and view all the answers

What does the length T represent in the calculation procedures?

The measured distance along the tangent. Signup and view all the answers

What is the angle $ heta_i$ between the tangent to the transition curve at point i and the entry straight calculated from?

The distance along the transition curve Signup and view all the answers

In vertical curves, what does a positive algebraic difference between gradients indicate?

Sag Curve Signup and view all the answers

What is the main purpose of vertical curves in road design?

To ensure adequate visibility and passenger comfort Signup and view all the answers

Which type of vertical curve is characterized by equal lengths from PVC to PVI and from PVI to PVT?

Symmetrical Parabola Curve Signup and view all the answers

What gradient is represented by '1 in 25'?

4 percent Signup and view all the answers

Which of the following is true regarding unsymmetrical parabolic curves?

The lengths from PVC to PVI and PVI to PVT are not equal Signup and view all the answers

What results from the radial force acting on a vehicle in a vertical curve?

Increased risk of skidding Signup and view all the answers

What determines the radius of curvature at the junction of the transition curve and the simple curve?

The radius $R$ Signup and view all the answers

What is the significance of the Vertex (V) in a parabolic vertical curve?

It is the point of intersection of the grade line. Signup and view all the answers

Which equation represents the elevation of points on the curve based on the rate of change of grade?

EPC = EPVC + g1X + (r/2)X^2 Signup and view all the answers

Which point is defined as the point of tangency where the parabolic vertical curve meets the forward grade?

PVT Signup and view all the answers

What does the symbol 'e' represent in the context of a vertical curve?

Offset from the vertex to the curve Signup and view all the answers

In the equation for determining offsets, which variable represents the length of the curve?

L Signup and view all the answers

What gradient does the back tangent have in the example provided?

-4 percent Signup and view all the answers

What is the correct expression for the elevation on the tangent?

EPT = EPVC + g1x Signup and view all the answers

How is the length of the vertical curve (L) defined?

The horizontal distance from PVC to PVT. Signup and view all the answers

What is the elevation at the PVC station (4+920)?

503.20m Signup and view all the answers

At a distance of 20m from the EPVC, what is the calculated EPC?

501.90m Signup and view all the answers

What is the value of y when calculating the elevation on a curve?

1.2 Signup and view all the answers

What is the elevation at station 5+000?

501.20m Signup and view all the answers

For an upward gradient of 3% meeting a downward gradient of 2%, what is the difference in elevation at station 1000.00?

3.50m Signup and view all the answers

What factor is used to modify the elevation on a curve based on the length of the curve?

4e/L Signup and view all the answers

At an offset of +1.20 at station 5+000, how does the elevation change compared to the previous station?

-0.675m Signup and view all the answers

What does g1 represent in this context?

Gradient of curve Signup and view all the answers

How is the elevation on a tangent (EPT) calculated from EPVC?

EPT = EPVC + g1x Signup and view all the answers

What happens to the elevation as one moves further along the tangent from EPVC?

It increases dependent on g1 Signup and view all the answers

What is the calculation used to determine EPC at the full station of 4+940?

EPC40 = EPVC + g1 * 20 Signup and view all the answers

What is the slope of the tangent calculated at the point of intersection of gradients?

3.0% Signup and view all the answers

Which elevation represents the peak height at 5+020?

501.075m Signup and view all the answers

What is the formula used to calculate the offset Yn when given a radius R and a distance Xn?

Yn = (R^2 - Xn^2) - a Signup and view all the answers

If the deviation angle is 60 degrees, what is the value of C when R = 50m?

43.3m Signup and view all the answers

In the example provided, what is the offset at 20m based on the calculated values?

1.14m Signup and view all the answers

Calculate the ordinates for points at intervals of 20m on a circular curve with a long chord of 120m. What is the versed sine when the offset is given as 5m?

5m Signup and view all the answers

What is the correct radius R when the offset Yn at 10m is calculated using Yn = (R^2 - Xn^2) - a?

60m Signup and view all the answers

For the given offset Yn = (R^2 - Xn^2) - [R^2 - (C^2)], what does C represent in this scenario?

The versed sine Signup and view all the answers

When calculating the offsets at various intervals from the tangents, what is the maximum value of Yn calculated in the example given?

4.57m Signup and view all the answers

Which of the following is NOT involved in the calculation of offsets in the given examples?

Cosine function Signup and view all the answers

When determining the ordinates for a circular curve having a long chord of 120m and a versed sine of 5m, what is the method of measurement?

Calculation at specified intervals along the chord. Signup and view all the answers

For a deviation angle of Δ = 45 degrees, what would be the value of C if R = 60m?

45.92m Signup and view all the answers

Study Notes

Unit Three Curve

Route Surveying encompasses surveying and mapping activities for planning, designing, and constructing transportation facilities like roads, railways, pipelines, and power transmission lines.
Alignment refers to the shape or geometry of a transportation route, including horizontal and vertical components (plan and profile views).
Horizontal alignment comprises a series of straight lines (tangents) connected by curves.
Vertical alignment consists of straight segments (gradients) connected by vertical curves.
Plan view elements include: bearing of tangents, angle of intersection, stationing, geometric data for horizontal curves, topography near the centerline, and details of existing structures affected by the project.
Profile view elements include: existing ground surface, proposed route grade line, grades of tangents, and vertical curve data.

Types of Curves and Their Uses

Curves are essential to gradually change direction in communication routes (roads, railways, canals).
Two main curve types exist: horizontal and vertical curves.
Horizontal curves connect tangents, usually circular but spirals may be used for transitions.
Sharpness of a circular curve is defined by its radius (R) or degree of curve (D°).
Simple circular curve is the most common type, joining two intersecting tangents.

Elements of Simple Circular Curve

Vertex (V): point of intersection of the two tangents.
Point of Curvature(PC): tangent point where curve begins.
Point of Tangency (PT): tangent point where curve ends.
Tangent Distance (T): distance from vertex to PC/PT.
Intersection Angle (I): the angle between the tangents.
Radius (R): radius of the circle.
External Distance (E): distance from vertex to the midpoint of the curve.
Long Chord (C): distance from PC to PT.
Middle Ordinate (M): distance from the midpoint of long chord to the curve midpoint.
Degree of Curve (D°): a way of describing the curve's sharpness, defined by the central angle subtended by a 20m arc or chord length.

Compound Circular Curve

Consists of two or more simple curves with varying radii. Used to provide smoother transitions when changing curvature.

Reverse Circular Curve

Two consecutive circular curves with the same or different radii. They might share a common tangent.

Spiral Curve or Transition Curve

Used to transition from tangent to circular curve and from circular curve back to tangent.
Gradually changes radius of curvature from infinity to a fixed value, to smoothly introduce radial force on vehicles.
Properties: radial force change is gradual and constant.
Another important function: smoothly introducing superelevation.

Vertical Curve

Used to connect gradients (slope changes) in the vertical plane.
Crest or summit curves have negatively calculated algebraic difference of gradients.
Sag or valley curves have positively calculated algebraic difference of gradients.
Generally parabolic, providing consistent rate of change of grade.

Equations of Vertical Curves

Symmetrical parabola equations are used.
Equations determine elevation(vertical offset) at any point along the curve given horizontal distances and other parameters.

Methods of Setting Out

Detail center line survey is often used before construction.
Deflection angle method uses theodolite measures along the curve using the chord lengths.
Linear methods uses offsets from tangents to define points on a curve.

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Description

This quiz covers essential concepts related to circular curves in geometry, including terms such as intersection angle, middle ordinate, and tangent distance. Test your understanding of radius, degree of curve, and the calculations involved in circular curve geometry.