9.1 - Two straights AC and CB
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Chainage and Basic Definitions
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Today, we will learn about chainage, which is essentially a system for measuring distance along a road or rail line. Chainage can help us understand positions related to curves.
What exactly is the difference between chainage and regular distance measurements?
Good question! Chainage is a continuous measure along the alignment, starting from a fixed point, which allows us to easily reference locations on a curve or straight. For example, if we begin measuring at zero, every meter represents an increment of chainage.
So if I measure 1 kilometer from our starting point, the chainage would be 1000 meters?
Exactly! Chainage simplifies communication in engineering projects. Now, let’s see how we determine the length of tangents next.
Calculating Tangent Lengths
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To find tangent lengths, we use the formula: Length of tangent = R tan(Δ / 2). R is the radius of the curve and Δ is the deflection angle.
What is the deflection angle again? Can you explain how it works?
The deflection angle is the angle between the two straight lines. It's crucial as it affects the radius and, thus, the tangent length and overall curve layout. If we have an angle of 36° and a radius of 300m, how would we calculate the tangent length?
I think it would be… Length = 300 * tan(18°)?
Well done! The calculated length will indeed inform the proper placement of the curve.
Length of Curve Calculation
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Moving on, let’s calculate the length of the curve using the formula: Length of curve = RΔ(π/180). Here, we need to convert the angle Δ from degrees to radians for this computation.
So, if we have R as 300m and Δ as 36°, it translates to…
Length will be 300 * 36 * π / 180!
Precisely! Just ensure your calculator is set to radians when calculating.
Can you give us an example with actual numbers?
Certainly! If R is 300m, the length of the curve would be 188.5m.
Real-World Application Examples
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Let’s apply these calculations to a real-world example. If our chainage of the point of intersection is identified at 2140m with a normal chord length of 20m, what would our tangent length and chainage of the P.C. be?
We could calculate the tangent with R tan(Δ / 2) again, right?
Yeah! And then we use that to find the PC?
Exactly! By always using our formulas together, we can derive meaningful results in our projects.
Introduction & Overview
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Quick Overview
Standard
The section outlines the methodology for calculating geometric parameters for circular curves connecting two straight lines. Through various examples, it demonstrates how to derive the lengths of tangents, chainages at critical points, and the necessary data for practical application in civil engineering.
Detailed
Detailed Summary
This section elaborates on the procedures necessary to set out a circular curve between two straight lines AC and CB. It emphasizes geometric concepts that govern circular alignment in road and railway engineering, with specific details regarding chainages, deflection angles, radius calculation, and tangent lengths. The primary calculations include the following key points:
- Chainage Calculation: Determining the position of points on the road using a defined chainage system (measured in meters or chains).
- Tangent Length Calculation: Utilizing the formula
Length of tangent = R tan(Δ / 2)where R is the curve radius and Δ is the deflection angle, to compute the necessary tangent lengths required for setting out curves. - Curve Length Calculation: The length of the circular curve is given by the formula
Length of curve = RΔ(π/180), indicating how length changes with radius and deflection. - Example Applications: The section provides multiple numerical examples, vividly portraying how the theoretical aspects translate into practical solutions, reinforcing the importance of accuracy in geospatial design in engineering fields.
Understanding these concepts is essential for engineers to accurately lay out road and railway curves, ensuring safety and efficiency in transport systems.
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Apex and Deflection Angle
Chapter 1 of 4
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Chapter Content
Chainage of apex V = 1190 m, Deflection angle D = 36°, Radius R = 300 m, Peg interval = 30 m.
Detailed Explanation
This section begins by introducing the key parameters needed for setting out a curve based on two straight paths or tangents, AC and CB. The apex, or point of intersection of the tangents, is positioned at a chainage of 1190 meters. The deflection angle, which is the angle change at this point, is specified as 36 degrees. Additionally, the radius of the intended curve to connect the two straights is established as 300 meters. Understanding these parameters is crucial for accurately establishing the curve for successful road alignment.
Examples & Analogies
Imagine two roads meeting at a junction: one road is for drivers going straight, and the other is for those turning. The point where they meet (the apex) is like a traffic light changing direction. The deflection angle represents how much you need to turn your steering wheel to remain on the new path (the curvature of the road). The radius of the curve is like the size of the turn; a larger radius means a gentler turn.
Calculating the Length of Tangent
Chapter 2 of 4
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Chapter Content
Length of tangent = R tan Δ/2 = 300 tan 36/2 = 97.48 m.
Detailed Explanation
To calculate the length of the tangent, we use the formula: Length of tangent = Radius (R) multiplied by the tangent of half the deflection angle (D). Here, we first compute the tangent of half of 36 degrees, which is half of the total angle of deflection. By plugging in the values, the computation yields a tangent length of approximately 97.48 meters. This length represents how far out the reference line (tangent) extends from the apex before the curve begins.
Examples & Analogies
Consider setting up a slide on a playground. The length of the slide from the top (the apex) to where it meets the ground (the tangent) is important to understand how steep it will be. The longer the slide, the gentler the slope, making it safer and more fun to use!
Finding Chainage of Tangent Point (T)
Chapter 3 of 4
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Chapter Content
Chainage of T = 1190 – 97.48 = 1092.52 m.
Detailed Explanation
After determining the length of the tangent, we can find the chainage for the tangent point (T) where the curve starts. This is calculated by subtracting the tangent length (97.48 m) from the chainage at the apex (1190 m). This results in a new chainage of 1092.52 m for the tangent point. Knowing the chainage of (T) is essential for setting out the curve accurately along the road.
Examples & Analogies
Imagine you are measuring a distance while walking on a trail. If you start at a specific point (the apex), step back a certain distance (the tangent length), you can determine your new location on the trail. This helps you know where to continue your hike, just like knowing where the road begins to curve.
Calculating Length of Curve
Chapter 4 of 4
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Chapter Content
Length of curve = RΔ(π/180) = 300 * 36 (π/180) = 188.50 m.
Detailed Explanation
The length of the curve is calculated using the formula: Length of curve = Radius (R) multiplied by the deflection angle (Δ) in radians, scaled by π/180 to convert degrees to radians. Substituting the known values gives us a curve length of approximately 188.50 meters. This length allows planners to know how much road space will be curved versus straight.
Examples & Analogies
Think of drawing a circle. The radius helps determine how wide the circle will be. Just like when you measure a piece of string to build a circular garden bed, knowing the right radius keeps everything neat and proportional. The curve length is essentially how long that string would need to be!
Key Concepts
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Chainage: A measurement indicating distance along a path, important for positioning curves.
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Tangent Length: A straight length connecting a point of intersection to a curve, central for layout.
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Deflection Angle: Indicating how much a line deviates from straight alignment, essential for curve calculations.
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Radius of Curve: The fixed distance from the curve's central point impacting the dynamics of layout.
Examples & Applications
Example 1: Given a deflection angle of 36° and radius of 300m, calculate the tangent length and resulting chainage.
Example 2: For a point of intersection chainage of 2140m, determine the tangent distance using a normal chord length of 20m.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Chainage measures distance - it’s key, Tangent lengths guide curves perfectly!
Stories
Imagine a road builder mapping a highway. He measures the distance from the starting point in chainages, ensuring every angle and curve is accurate as he goes along.
Memory Tools
RLT: Remember Lengths of Tangents to measure accurately.
Acronyms
CRD
Curve
Radius
Deflection - the essentials for laying out curves.
Flash Cards
Glossary
- Chainage
A linear measurement along a line or path, used in transportation engineering to denote distances from a starting point.
- Tangent Length
The distance along a straight line from the point of intersection to the point of curve.
- Deflection Angle
An angle between two intersecting lines or tangents representing the divergence from straight alignment.
- Radius of Curve
A fixed distance from the center of a curve to any point on its perimeter, crucial in determining curvature characteristics.
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