Horizontal Curves (circular, spirals)

References:

• Wright and Ashford, sections 12-9 and 12-10,
• Surveying textbook (Moffitt and Bouchard, 9th ed.), esp. pp. 549-551
• CE 211 (111) Lab Manual, p65-68

Circular Curve Notation (see figure below):

• Point of Intersection (PI): the point at which the two tangents to the curve intersect
• Delta Angle: the angle between the tangents is also equal to the angle at the center of the curve
• Back Tangent: for a survey progressing to the right, it is the straight line that connects the PC to the PI
• Forward Tangent: for a survey progressing to the right, it is the straight line that connects the PI to the PT
• Point of Curvature (PC): the beginning point of the curve
• Point of Tangency (PT): the end point of the curve
• Tangent Distance (T): the distance from the PC to PI or from the PI to PT
• External Distance (E): the distance from the PI to the middle point of the curve
• Middle Ordinate (M): the distance from the middle point of the curve to the middle of the chord joining the PC and PT
• Long Chord (LC): the distance along the line joining the PC and the PT
• Length of Curve: the difference in stationing along the curve between the PC and the PT

Review: Circular Curve Stakeout

From CE 211 Lab Manual

Circular curves are laid out in the field by measuring off a series of chords along the arc. The following chord lengths may be substituted for arc lengths without undue loss of precision:

• 100 foot chords for curves up to 4 degrees
• 50 foot chords for curves up to 10 degrees
• 25 foot chords for curves up to 25 degrees
• 10 foot chords for curves up to 100 degrees

The normal way of staking out a circular curve is to lay out the delta angle and tangent length at and from the PI, then lay out the curve from the PC to the PT using a series of appropriate deflection angles and chords. The curve staking notes are prepared prior to the actual field layout. The normal field layout procedure may be described stepwise as follows:

1. Set instrument at PI, horizontal circle indexed to zero, telescope direct. Plunge telescope, sight along back tangent using lower motion and set PC.
2. Plunge telescope to direct position, lay off delta angle with upper motion and set PT.
3. Set instrument at PC, horizontal circle indexed to zero, telescope direct. Sight PI using lower motion.
4. Lay off consecutive appropriate deflection angles and chords, using upper motion, until PT is reached.
5. Sometimes it is necessary to "move up" on the curve because an obstruction blocks your vision.

Moving up on a curve

From CE 211 Lab Manual

The steps involved in moving the instrument "up on the curve" are as follows:

1. Set instrument to intermediate curve station, horizontal circle indexed to zero, telescope direct. Plunge telescope, sight PC using lower motion.
2. Plunge telescope to direct position, lay off deflection angle for instrument station toward curve using upper motion. This will place the line of sight tangent to the curve at the instrument station.
3. Continue laying off consecutive deflection angles and chords as before, using the upper motion, until the PT is reached.

Review:

• The angle from the PC to a point on a circular curve is one-half of the central angle subtended.

  

• To the first full station, the angle is d₁/2.

  

• The degree of curvature, D, subtends one station. (therefore, d₁/D = distance to first station/100).
• Between each station, add half of the degree of curve to the deflection angle.

  

• To lay out the curve, we also need the chord distance to the point. The chord length for one full station would be 2*R*sin(D/2), so for 15 degree curve = 99.71' (which differs from arc distance by .29')
• Since we usually layout the curve in half stations or less, the difference is smaller, and possibly negligible. For example, for the same 15 degree curve, the chord distance for 50' of arc would be 2(381.93)sin(7.5/2) = 49.96, a difference of only .04' (note that the angle subtended by 50' is D/2 or 7.5 degrees).

  

Spiral Curves in Design:

References:

Wright & Ashford, pp. 420-422
Moffitt & Bouchard, pp. 572
CE 211 Manual, pp. 69-75

General Information on Spirals

The introduction of the circular curve at the PC takes place at a point but drivers and vehicles do not make directional changes instantaneously.

It is also common practice in constructing curves on highways to tip or superelevate the pavement downward toward the inside of the curve to aid in the riding quality and safety for vehicles navigating the curve. Again it is not practicable or advisable to introduce the superelevation instantaneously. If introduced on the tangent where it is not needed, the driver must steer into it slightly with a negative steering angle. If introduced all on the curve some area of negative superelevation will generally result or the introduction will be done so quickly that both the riding quality and the visual attractiveness of the highway suffer.

A solution is to introduce both the curvature and superelevation at a gradual rate using an easement curve that gradually changes in radius from infinity to some finite value where the associated circular curve begins. In short, a spiral curve is required. There are a number of identifiable curves that spiral, but their mathematical differences do not affect their usefulness on highways.

The geometry of the spiral curve is more rigorous that that of the circular curve and handbook tables are the usual way of working out the deflection angles needed to lay out a spiral curve in the field. The discussion has been worked out with reference to Route Location and Design, 5th ed., Hickerson, Thomas F., New York: McGraw-Hill, 1964, for the appropriate tables.

The spiral curve element generally selected by the designer is the length of the spiral "l_s". The choice is usually made to introduce superelevation slowly enough so as not to exceed certain relative slopes between pavement edge and centerline grades. As a minimum, spiral curve lengths should not be shorter than the distance covered in two seconds at highway design speed.

Spiral Notation

• TS: point of change from tangent to spiral
• SC: point of change from spiral to circle
• CS: point of change from circle to spiral
• ST: point of change from spiral to tangent
• l: spiral arc length from TS to any point on the spiral
• l_s: total length of spiral from TS to SC
• Theta: central angle of spiral are l
• Theta_s: the spiral angle = central angle of spiral arc l_s
• Phi: spiral deflection angle at the TS from initial tangent to any point on the spiral
• D_s: degree of curve of the spiral at any point, and R = its radius
• D_c: degree of curve of the shifted circle to which the spiral becomes tangent at the SC, and R-c the radius fo the circle
• Delta: total central angle of the circular curve
• Delta_c: central angle of circular arc of L-c extending from the SC to the CS
• y_s: tangent offset of the SC with reference to the TS and the initial tangent
• x_s: tangent distance for the SC
• p: offset from the initial tangent to the PC of the shifted circular curve
• k: abscissa of the shifted PC referred to the TS
• T_s: total tangent distance = distance from PI to TS or ST
• E_s: = total external distance = distance from PI to midpoint of curve

Selected formulas for spiral curves:

Example Problem - Regular Deflection Angles:

Example Problem - Moving up on a Spiral:

What is delta backsight? It is the angle to turn from a backsight point (TS in this case) to get tangent to the curve (the red line is tangent to the curve). ...

What is delta forward? It is the angle to turn after you have become tangent to the curve, to get to the next stake point (SC in this case). ...