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LEC: Solve for the elements of horizontal or vertical curves in highway or railway.
LAB: Conduct fieldwork exercises in laying out horizontal curves and report results in an organized manner.
MODULE 5: HORIZONTAL AND VERTICAL CURVES
During the survey of the alignment of a project involving roads or railways, the direction of the line may change due to some unavoidable circumstances. The angle at which the direction changed is known as the deflection angle. For it to be possible for a vehicle to run easily along the road or railway track, the two straight lines (the original line and the deflected line) are connected by an arc which is known as the curve of the road or track (lceted.com). Those curves that change the alignment or direction are known as horizontal curves, and those that change the slope are vertical curves.
TYPES OF HORIZONTAL CURVES A curve may be simple, compound, reverse, or spiral. Compound and reverse curves are treated as a combination of two or more simple curves, whereas the spiral curve is based on a varying radius.
1. Simple Curve is an arc of a circle. It is the most commonly used curve. The "sharpness" or "flatness" of the curve depends on the radius of the circle. The curve becomes "flatter" as the radius increases.2. Compound Curve is frequently used by surveyors due to the terrain. Typically, this curve is made up of two simple curves that are connected together and curving in the same direction.3. Reverse Curve is created by joining two simple curves that are curving in the opposite directions. The surveyor must only utilize this curve when absolutely necessary for safety concerns.4. Spiral Curve is a curve with varying radius used on railroads and somemodern highways. It provides a transition from the tangent to a simple curve or between simple curves in a compound curve.
1. Simple Curve is an arc of a circle. It is the most commonly used curve. The "sharpness" or "flatness" of the curve depends on the radius of the circle. The curve becomes "flatter" as the radius increases.2. Compound Curve is frequently used by surveyors due to the terrain. Typically, this curve is made up of two simple curves that are connected together and curving in the same direction.3. Reverse Curve is created by joining two simple curves that are curving in the opposite directions. The surveyor must only utilize this curve when absolutely necessary for safety concerns.4. Spiral Curve is a curve with varying radius used on railroads and somemodern highways. It provides a transition from the tangent to a simple curve or between simple curves in a compound curve.
ELEMENTS OF A SIMPLE CURVE
1. Point of Curvature (PC)The point of curvature is the point on the back tangent where the circular curve begins.2. Point of Tangency (PT)The point of tangency is the point on the forward tangent where the curve ends.3. Point of Intersection (PI)The point of intersection is the point where the back and forward tangents intersect. Sometimes the point of intersection is designated as V (vertex).4. Radius (R)The radius is the distance from the center of a circle or curve represented as an arc or segment. The radius is always perpendicular to the back and forward tangents.5. Degree of Curve (D)The degree of curve defines the sharpness or flatness of the curve.6. Tangent Distance (T)The tangent distance is the distance along the tangents from the PI to the PC or to the PT. These distances are equal on a simple curve. T = R tan(I/2)
1. Point of Curvature (PC)The point of curvature is the point on the back tangent where the circular curve begins.2. Point of Tangency (PT)The point of tangency is the point on the forward tangent where the curve ends.3. Point of Intersection (PI)The point of intersection is the point where the back and forward tangents intersect. Sometimes the point of intersection is designated as V (vertex).4. Radius (R)The radius is the distance from the center of a circle or curve represented as an arc or segment. The radius is always perpendicular to the back and forward tangents.5. Degree of Curve (D)The degree of curve defines the sharpness or flatness of the curve.6. Tangent Distance (T)The tangent distance is the distance along the tangents from the PI to the PC or to the PT. These distances are equal on a simple curve. T = R tan(I/2)
7. Angle of Intersection (I)The angle of intersection is the deflection angle at PI. Its value is either computed from the preliminary traverse angles or measured in the field.8. External Distance (E)The external distance (also called the external secant) is the distance from the PI to the midpoint of the curve. The external distance bisects the interior angle at PI. E = R sec(I/2) - R9. Middle Ordinate (m)The middle ordinate is the distance from the midpoint of the curve to the midpoint of the long chord. The extension of the middle ordinate bisects the central angle. m = R - R cos(I/2)10. Length of Curve (LC)The length of curve is the length of curve from the PC to PT, measured along the curve. LC = (π R I)/180° 11. Long Chord (L)The long chord is the straight-line distance from the PC to the PT. L = 2R sin (I/2)12. First Deflection Angled1 = 2sin-1 (C1/2R)13. Last Deflection Angled2 = 2sin-1 (C2/2R)
SAMPLE PROBLEM #1
In the first sample problem, the given quantities are the radius, the angle of intersection, and some distance from the PC. The first thing I did is find the angle that the distance made with the center of circle. Then, using the formula to find the length of curve, I derived a formula to find the theta or the angle. For instance, the angle that I got is 10° 34' 34.21''. Lastly, using the formula for the middle ordinate, I was able to find the x which is 3.367m.
For the second problem, the given quantities are the angle of intersection and the external distance. In this problem, I simply used the external distance formula to derive the formula for radius. A lot of algebraic manipulations happened before I arrive to the final answer of 226.942m radius.
Just like the computation for missing data, the problems involving vertical curves also need sketch or drawing to visualize the missing quantities in the curve. Further, one should familiarize the different parts of the curve and their formulas to easily solve a problem involving curves. Though there are three more kinds of horizontal curve namely, compound, reverse, and spiral, our instructor was able to discuss up to the simple curve because of the limited time.
SAMPLE PROBLEM #2
REFLECTION
I became aware of the significance of civil engineers in the construction of roads and railways throughout this module. Before a curve can be made, numerous computations should be made to ensure the safety of the motorists and the public. Road curves avoid obstructions of a straight road and make gradients more comfortable. Curve also helps avoid construction over expensive land and makes road travel stable with steep grades. Also, it brings a graduation change to direction which gives comfort to the passengers. When it comes to the topic, I learned that there are curves may be horizontal or vertical. Those curves that change the alignment or direction are known as horizontal curves, and those that change the slope are vertical curves. Further, there are four kinds of horizontal curve, namely, simple curve, compound curve, reverse curve, and spiral curve. In computing for the missing quantities in a simple curve, it is imperative to first sketch the given curve to have a full understanding of the problem and to eventually establish the strategy on how to solve it.
I became aware of the significance of civil engineers in the construction of roads and railways throughout this module. Before a curve can be made, numerous computations should be made to ensure the safety of the motorists and the public. Road curves avoid obstructions of a straight road and make gradients more comfortable. Curve also helps avoid construction over expensive land and makes road travel stable with steep grades. Also, it brings a graduation change to direction which gives comfort to the passengers. When it comes to the topic, I learned that there are curves may be horizontal or vertical. Those curves that change the alignment or direction are known as horizontal curves, and those that change the slope are vertical curves. Further, there are four kinds of horizontal curve, namely, simple curve, compound curve, reverse curve, and spiral curve. In computing for the missing quantities in a simple curve, it is imperative to first sketch the given curve to have a full understanding of the problem and to eventually establish the strategy on how to solve it.
REFERENCES:
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