# Supplementary Information

## K Value

Vertical curves (sag or summit) can also be designed using the value K as an input parameter instead of the minimum Radius/Length defined at Design Standard.

The input parameter K can be employed for both circular (default) and parabolic vertical curves.

## Activating K - Value as input

Run the function Options and activate the tab Vertical. Mark the check box K - Value as input.

The table Input V of the user interfaces will be having an additional column for K - Value.

Define the K - Value in the cells of curve elements of the column K.

The values Radius/Kv, Horizontal Length, and K are interrelated and if one of these values is edited, other values will be updated automatically.

Figure 1: User Interface for Circular Curves

Figure 2: User Interface for Parabolic Curves

## K - Value concept

For a simple parabola, the rate of change of grade per unit length (horizontal) of the curve is a constant Q, and is equal to the algebraic difference between intersecting tangent grades (%) divided by the length of the curve in meters; expressed as: Q = A/L in percent per meter.

The reciprocal K is defined as the horizontal distance in meters which results in a 1% change in grade: K = L/A.

The reciprocal L/A, termed K, is the horizontal distance in meters required to make a 1% change in gradient and is, therefore, a measure of curvature.

The quantity K is useful in determining the horizontal distance from the vertical point of curvature (VPC) to the high point (in case of the summit) curves or the low point (in case of sag curves). This point, where the slope is zero, occurs at a distance from the VPC equal to K times the approach gradient.

Figure 3: K - Value concept and details

For simplicity, a parabolic curve with an equivalent vertical axis centered on the vertical point of intersection (VPI) is usually used in roadway profile design.

The vertical offset from the tangent varies as the square of the horizontal distance from the curve end (point of tangency).

The vertical offset from the tangent grade at any point along the curve is calculated as a proportion of the vertical offset at the VPI, which is AL/800, where the symbols are shown in the above figure.

For each design speed and sight distance configuration, a single value of K defines the length of the curve L for all values of A.

Also, for design purposes; a vertical curve can be plotted using a circular curve of radius R, which is approximately very close to a parabolic curve by the relationship: R = 100*K.

## Design Criteria - Vertical Curves

Vertical curves to effect gradual changes between tangent grades may be any one of the summit or sag types shown in figure Types of Vertical Curves.

Vertical curves should be simple in application and should result in a design that is;

Sight distance

Comfort

Pleasing in appearance

Adequate for drainage

The major control for safe operation on crest vertical curves is the provision of ample sight distances (at least the stopping sight distances) for the design speed.

For driver comfort, the rate of change of grade should be kept within tolerable limits. This consideration is most important in sag vertical curves where gravitational and vertical centripetal forces act in opposite directions.

Appearance also should be considered in designing vertical curves.

A long curve has a more pleasing appearance than a short one; short vertical curves may give the appearance of a sudden break in the profile (vertical alignment) due to the effect of foreshortening.

Drainage of curbed roadway on sag vertical curves needs careful profile (vertical alignment) design.

### Sight Distance Requirements

The horizontal length of the vertical curve (L) for a given sight distance (Ls) is calculated by the following expressions:

If the horizontal length of a curve (L) is less than the sight distance (Ls):

L = (2*Ls)-(L/A) — — — — — EQ(1)If the horizontal length of a curve (L) is greater than the sight distance (Ls):

L = (Ls2)*A/C — — — — — EQ(2)

Where,

L is the horizontal length of the vertical curve (meter)

Ls is the sight distance (meter)

A is the algebraic difference of vertical grading (%)

C is the sightline constant

Substituting the vertical curve parameter K for L/A in EQ(2) gives:

K = (Ls2)/C — — — — — EQ(3)

Sight distance formula:

Ls = (0.284*V*tr)+(V2)/(254.3*(f+s)) — — — — — EQ(4)

Where,

V is design speed (kmph)

tr is the reaction time (sec)

f is friction coefficient

s is grade (ratio)

Sightline constant (C) for summit curves:

C = 200*[√(h1)+√(h2)]2 — — — — — EQ(5)

Where,

h1 is the driver's eye height above the roadway surface (meter)

h2 is the height of the object on the roadway surface (meter)

Sightline constant (C) for sag curves:

C = 200*[h+Ls*Tan(q)] — — — — — EQ(6)

Where,

Ls is stopping sight distance

h is headlight mounting height

q is the elevation angle of the headlight beam (+ is upwards)

Sightline constant (C) for sag curves at overhead obstructions:

C = 200*[√(H-h1)+√(H-h2)]2 — — — — — EQ(7)

Where,

H is headroom for the vehicle from the road surface (height of overhead obstruction)

Using the above equations, the value of K can be obtained for the following conditions:

K value for summit curves and stopping sight distance when L < Ls

K value for summit curves and stopping sight distance when L > Ls

K value for summit curves and passing sight distance when L < Ls

K value for summit curves and passing sight distance when L > Ls

K value for sag curves and stopping sight distance when L < Ls

K value for sag curves and stopping sight distance when L > Ls

K value for sag curves at overhead obstruction when L < Ls

K value for sag curves at overhead obstruction when L > Ls

### Comfort Requirements

Human beings subjected to rapid changes in vertical acceleration feel discomfort. However, vertical acceleration only becomes critical in the design of sharp sag curves.

Refer road authority standards for maximum values of vertical acceleration generated when passing from one grade to another.

The vertical component of acceleration normal to the curve, when traversing the path of a parabolic vertical curve at uniform speed is a function of K.

The equation for the vertical component of acceleration, a = V2/(12960*K)

Where V is design speed (kmph)

### Appearance Requirements

For very small changes of grade, the vertical curve has little effect on the appearance of the road's profile (vertical geometry) and may usually be omitted.

Short vertical curves can, however, have a significant effect on the appearance of a road's profile. Therefore, vertical curves for small changes of the grade should have K values significantly greater than those needed for minimum sight distance reasons. This is particularly important on high standard roads, especially for sag curves.

Refer to the road authority standards for a maximum grade change, where vertical curves may be omitted and the minimum length of vertical curve necessary to give a satisfactory appearance.

Longer curves are preferred where they can be achieved without conflict with other design requirements, such as drainage.

### Parabolic Curves

Enabling 3rd-degree (parabolic curves) for the summit and sag curves. By default 2nd-degree curves will be used for the curve elements (sag/summit) of vertical alignment. Parabolic curves can also be used instead of 2nd-degree curves. To activate parabolic curves, use the following procedure:

Step 1: Open the dialog box System Properties and select the tab Advanced:

Start > Control Panels > System

Or activate the shortcut menu of the icon My Computer and select Properties.

Step 2: Click the button Environment Variables. The dialog box Environment Variables will pop up.

Step 3: System variables: Click the button New. The dialog box New System Variable will pop up.

Step 4: Define the value VIPS at the field Variable Name: and value P at the field Variable Value:.

The change made will come in the act once the computer is rebooted.

The newly created system variable will be listed.

Note: To switch back to simple curves for vertical alignment, delete the system variable VIPS.

### Scale Symbols

The construction objects are drawn like symbols or print, e.g. example fixed points, radius, clothoid, etc. The drawing scale determines the size of the symbol. If the scale is changed from, for example, 1:1000 to 1:4000 the symbol size will increase.

If the drawing scale is changed while the Alignment design is running, the CAD command Redraw or Regen is recommended to update the symbol size to the new scale.

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