1.3.2 - By offsets perpendicular to the tangents
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Interactive Audio Lesson
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Introduction to Offsets and Setting Out Curves
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Today, we will discuss the technique of using offsets perpendicular to the tangents to set out curves. Can anyone tell me what a tangent is in this context?
Isn't it a straight line that touches the curve at just one point?
Exactly! And when we offset from the tangents, we create a series of points that help us plot the curve accurately along a defined path. The formula we'll use to calculate these offsets is \( O = R - \sqrt{R^2 - x^2} \). Can anyone break down this formula?
So, R is the radius, and x is the distance from the tangent point to the point where we're measuring the offset?
Correct! Remember that the offset O will tell us how far we need to go from the tangent to reach the curve. Let's continue to the next step.
Exact vs. Approximate Calculations
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Now, let’s compare the exact calculations with the approximate ones. Why do you think we would use an approximate formula like \( O \approx \frac{x^2}{2R} \)?
I think it’s probably because it’s simpler and easier to calculate for larger radii.
Exactly! When the radius is large, the offsets become much smaller in comparison to the main circle. This approximation saves time while maintaining accuracy. Can anyone think of situations where this simplification is beneficial?
Maybe when laying out long roads or highways where we use large radius curves?
Well said! Efficiency in calculations is vital in large-scale projects. Let’s move on to the practical steps in setting out the curve.
Step-by-Step Procedure for Setting Out Curves
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To set out the curve precisely, we must follow a specific procedure. Can anyone list the first step?
We need to locate the tangent points and measure distances along the tangent.
Great! After that, we calculate the offsets for each measured distance using the formulas we covered. What do we do next?
We mark these offsets on the curve?
Yes! And remember, the curve is symmetrical; therefore, we can use the same values on both sides. Why is this symmetry important?
It helps to ensure that the road or path is uniform and safe for travel!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section outlines the techniques for setting out simple circular curves by using offsets perpendicular to tangents, including variations in calculations for large radius curves, and provides a step-by-step procedure for the method, emphasizing its applicability for sharp curves with favorable external ground conditions.
Detailed
Detailed Summary
In this section, we examine the method of setting out circular curves by offsets that are perpendicular to the tangents. This technique involves using intersecting lines, geometrical relationships, and specific formulas to determine the positions along the curve. The approach is particularly useful in scenarios where sharp curves are needed, especially when the ground outside the curve is amenable to measuring distances accurately.
Key Points Covered:
- Exact Calculations: The exact offset is calculated based on the radius and the distance from the point of tangency. The formula provided is:
- For perpendicular offsets: \( O = R - \sqrt{R^2 - x^2} \)
This equation relates the radius of the curve, the distance to the point, and the required offset.
- Approximate Calculations: For larger radius curves, a simplified formula is used:
- \( O \approx \frac{x^2}{2R} \)
This approximation is valid due to the very small size of \( O^2 \) when compared to \( 2R \).
- Setting Out the Curve: The procedural steps include:
- Locating tangent points and measuring equal distances along the tangents,
- Calculating offsets using the formulas provided,
- Continuing the process until the apex is reached and then replicating for the other half.
The method is widely accepted in civil engineering, particularly for road works and railways, where precision in curvature layout is critical.
Audio Book
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Overview of Offsets Perpendicular to the Tangents
Chapter 1 of 3
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Chapter Content
In Figure 2.8, O = PP which is the perpendicular offset at P at a distance of x from T along the tangent AB. Draw PP line parallel to BT.
PP = PT = x, and TP = PP = O.
Now TP = OT – OP, where OT = R, and OP = R² – x².
Detailed Explanation
This chunk introduces the concept of setting out curves by taking perpendicular offsets from the tangents. When we have a point P along a tangent, a perpendicular line is drawn to the circle at this point to establish the offset. Essentially, this helps determine the distance from the tangent to the actual curve.
1. The point P is where the curve meets the tangent.
2. The distance OP represents a radius adjusted by the perpendicular distance from the tangent to the circle's edge.
3. The total offset O can then be computed using the relationship between the circle's radius and the lengths derived from the offsets.
Examples & Analogies
Imagine if you're standing on a straight road (the tangent) and need to measure how far a tree (the curve) is from that road. By walking directly away from the road (the perpendicular direction from the tangent), you can measure the tree’s distance accurately rather than trying to measure at an angle, which can be tricky.
Exact and Approximate Offset Formulas
Chapter 2 of 3
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Chapter Content
So O = R – R² – x² (exact) (2.14).
Approximately, the formula may be obtained similarly as equation 2.13; O = (approximate) (2.15).
Detailed Explanation
In this section, we look at the formulas used to calculate offsets. The exact formula provides a precise calculation for the offset from the tangent based on the radius and the distance x:
1. The exact formula reflects the geometric properties of the circle, ensuring accurate calculations based on the fundamental definitions of radius and distance.
2. The approximate formula is useful in practical scenarios where quick estimations are necessary and higher precision isn't critical. This makes it more convenient in everyday applications.
Examples & Analogies
Think of using a ruler to measure a piece of paper. If you want a precise length, you'd measure carefully (exact method). However, if you're just estimating how long it is for a quick decision (approximate method), you might not worry if it's exactly right, as long as you're close enough.
Procedure of Setting Out the Curve
Chapter 3 of 3
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Chapter Content
(i) Locate the tangent points T and T.
(ii) Measure equal distances, say 15 or 30 m along the tangent from T.
(iii) Set out the offsets calculated by any of the above methods at each distance (say x), thus obtaining the required points on the curve.
(iv) Continue the process until the apex of the curve is reached.
(v) Set out the other half of the curve from the second tangent; being symmetrical in nature.
Detailed Explanation
This chunk details the step-by-step method for setting out the curve using the perpendicular offsets:
1. Locate Tangent Points: Identify the points where the tangents meet the curve.
2. Distance Measurement: Measure predetermined intervals along the tangent line, which serve as reference points for offsets.
3. Calculate Offsets: Calculate the offsets at each of these measured points using the formulas established earlier.
4. Construct the Curve: Mark the curve points based on these offsets. Once you reach the curve's peak, reflect this process for the other side, ensuring both halves mirror each other to maintain symmetry.
Examples & Analogies
Imagine baking a cake. First, you determine how wide the cake will be (locating tangent points), then measure equal rounds of batter to fill the pan (measuring distances), pour in the batter with some spread from the center (set offsets), and finally you ensure both sides are evenly done as the batter rises in the oven (ensuring symmetry).
Key Concepts
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Offsets Perpendicular to Tangents: A method used in setting out curves, involving calculations for offsets based on radius and distance.
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Exact vs. Approximate Formulas: Distinctions between precise and simplified formulas for calculating offsets in curves.
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Practical Steps: Procedures involved in setting out curves, including locating tangents, measuring distances, and marking offsets.
Examples & Applications
When laying out roads, if the radius of the curve is 50 meters and the distance from the tangent to a point is 10 meters, the offset can be calculated using the formula.
In a project with large curves, an approximate calculation simplifies the process and speeds up work without sacrificing much accuracy.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
To find the offset far and wide, subtract the root to find the guide.
Stories
Imagine a builder laying a road. He measures carefully from straight edges to create a smooth curve, using simple formulas to guide him.
Memory Tools
R.O.D.: Radius, Offset, Distance – key measurements for setting curves.
Acronyms
S.O.C.
'Setting Out Curves' – remember the main activity we are discussing!
Flash Cards
Glossary
- Offset
The distance measured from a reference line (e.g., tangent) to a point on a curve.
- Tangent
A straight line that touches a curve at a single point.
- Radius (R)
The distance from the center of a circle to any point on its circumference.
- Exact Calculation
The precise mathematical calculation based on established formulas.
- Approximate Calculation
An estimation that simplifies the calculation, usually for larger circles.
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