RELATED RESOUCES:
HORIZONTAL ALIGNMENT
Curves used in horizontal planes to connect two straight tangent sections are called horizontal curves. Two types are used: circular arcs and spirals. Both are readily laid out in the field with surveying equipment. A simple curve (Fig. 1a) is a circular arc connecting two tangents. It is the type most often used. A compound curve (Fig. 1b) is composed of two or more circular arcs of different radii tangent to each other, with their centers on the same side of the alignment. The combination of a short length of tangent (less than 100 ft) connecting two circular arcs that have centers on the same side (Fig. 1c) is called a broken-back curve. A reverse curve (Fig. 1d) consists of two circular arcs tangent to each other, with their centers on opposite sides of the alignment. Compound, broken-back, and reverse curves are unsuitable for modern high-speed highway, rapid transit, and railroad traffic and should be avoided, if possible. However, they are sometimes necessary in mountainous terrain to avoid excessive grades or very deep cuts and fills. Compound curves are often used on exit and entrance ramps of interstate highways and expressways, although easement curves are generally a better choice for these situations. Easement curves are desirable, especially for railroads and rapid transit systems, to lessen the sudden change in curvature at the junction of a tangent and a circular curve. A spiral makes an excellent easement curve because its radius decreases uniformly from infinity at the tangent to that of the curve it meets. Spirals are used to connect a tangent with a circular curve, a tangent with a tangent (double spiral), and a circular curve with a circular curve. Figure 2 illustrates these arrangements.
The effect of centrifugal force on a vehicle passing around a curve can be balanced by superelevation, which raises the outer rail of a track or outer edge of a highway pavement. Correct transition into superelevation on a spiral increases uniformly with the distance from the beginning of the spiral and is in inverse proportion to the radius at any point. Properly superelevated spirals ensure smooth and safe riding with less wear on equipment. As noted, spirals are used for railroads and rapid-transit systems. This is because trains are constrained to follow the tracks, and thus a smooth, safe, and comfortable ride can only be assured with properly constructed alignments that include easement curves. On highways, spirals are less frequently used because drivers are able to overcome abrupt directional changes at circular curves by steering a spiraled path as they enter and exit the curves.
Degree of Circular Curve
The rate of curvature of circular curves can be designated either by their radius (e.g., a 1000-ft curve) or by their degree of curve. There are two different designations for degree of curve, the arc definition and the chord definition. By the arc definition, degree of curve is the central angle subtended by a circular arc of 100 ft (see Fig. 3a). This definition is preferred for highway work. By the chord definition, degree of curve is the angle at the center of a circular arc subtended by a chord of 100 ft (see Fig. 3b). This definition is convenient for very gentle curves and hence is preferred for railroads.
Circular Curve Equations
Circular Curve Stationing
Normally, an initial route survey consists of establishing the PIs according to plan, laying out the tangents, and establishing continuous stationing along them from the start of the project, through each PI, to the end of the job. After the tangents have been staked and stationed, the intersection angle (I) is observed at each PI and curves computed and staked. The station locations of points on any curve are based upon the stationing of the curve’s PI. To compute the PC station, tangent distance T is subtracted from the PI station, and to calculate the PT station, curve length L is added to the PC station.
Circular Curve Layout Methods
Except for unusual cases, the radii of curves on route surveys are too large to permit swinging an arc from the curve center. Circular curves are therefore laid out by more practical methods, including (1) deflection angles, (2) coordinates, (3) tangent offsets, (4) chord offsets, (5) middle ordinates, and (6) ordinates from the long chord.
Computing Deflection Angles and Chords
To stake the first station, which is normally an odd distance from the PC (shorter than a full-station increment), subdeflection angle δ_a and subchord c_a are needed.
Task 1: Circular Curve Layout with a Total Station
Assume that I=4°12’, the station of the PI is 64 + 27.46, and terrain conditions require the minimum radius permitted by the specifications to be 2864.79 ft (arc definition). Layout the circular curve using a total station and total chord method.
Steps
Regardless of the method used to stake intermediate curve points, the first steps in curve layout are:
(1) establishing the PC and PT, normally by measuring tangent distance T from the PI along both the back and forward tangents and
(2) measuring the total deflection angle at the PC from PI to PT.
With the instrument set up and leveled over the PC, it is oriented by backsighting on the PI, or on a point along the back tangent, with 0°00’ on the circle. The subdeflection angle of 0°46’33" is then turned. Meanwhile, the 22-ft mark of the tape is held on the PC. The zero end of the (add) tape is swung. This is station 63 + 00. Follow the same procedure to stake other stations.
Field Book Sample:
VERTICAL ALIGNMENT
The vertical alignment contributes significantly to a highway’s safety, aesthetics, operations and costs. Long, gentle vertical curves provide greater sight distances and a more pleasing appearance for the driver. The design of vertical alignment involves, to a large extent, complying with specific limiting criteria. These criteria include maximum and minimum grades, sight distance at vertical curves and vertical clearances. In addition, the designer should adhere to certain general design principles and controls that will determine the overall safety and operation of the facility and will enhance the aesthetic appearance of the highway. Curves are needed to provide smooth transitions between straight segments (tangents) of grade lines for highways and railroads. Because these curves exist in vertical planes, they are called vertical curves. An example is illustrated in Figure 1, which shows the profile view of a proposed section of highway to be constructed from A to B. A grade line consisting of three tangent sections has been designed to fit the ground profile. Two vertical curves are needed: curve a to join tangents 1 and 2, and curve b to connect tangents 2 and 3. The function of each curve is to provide a gradual change in grade from the initial (back) tangent to the grade of the second (forward) tangent. Because parabolas provide a constant rate of change of grade, they are ideal and almost always applied for vertical alignments used by vehicular traffic. Two basic types of vertical curves exist, crest and sag. These are illustrated in Figure 1. Curve a is a crest type, which by definition undergoes a negative change in grade; that is, the curve turns downward. Curve b is a sag type, in which the change in grade is positive and the curve turns upward. There are several factors that must be taken into account when designing a grade line of tangents and curves on any highway or railroad project. They include (1) providing a good fit with the existing ground profile, thereby minimizing the depths of cuts and fills, (2) balancing the volume of cut material against fill, (3) maintaining adequate drainage, (4) not exceeding maximum specified grades, and (5) meeting fixed elevations such as intersections with other roads. In addition, the curves must be designed to (a) fit the grade lines they connect, (b) have lengths sufficient to meet specifications covering a maximum rate of change of grade (which affects the comfort of vehicle occupants), and (c) provide sufficient sight distance for safe vehicle operation.
Vertical Curve Equations
Task 2: Crest Vertical Curve Layout with a Total Station
A grade g_1 of +5.00% intersects grade g_2 of -4.40% at a vertex whose station and elevation are 45 + 00 and 855.50 ft, respectively. An equal-tangent parabolic curve 200 ft long has been selected to join two tangents. Layout the crest vertical curve using a total station.
