3. A section of highway has vertical and horizontal curves with the same design speed. A vertical curve on this highway connects a 1% and a 3% grade and is 420ft long. If a horizontal curve on this highway is on a two-lane section with 12ft lanes and has a central angle of 37 degrees and a superelevation of 6%, what is the length of the horizontal curve

Answer / Explanation

To be able to answer the question above, we need to understand that as related to the question, the transition curves used between tangents and circular curves or between two circular  curves have significantly different radii.

Hence,

To calculate the minimum length of spiral curve necessary:

We must recall the formula: Ls = 3.15V³ / RC

Where:

Ls = length of spiral (ft)

V = design speed (mph)

R = curve radius (ft)

C = rate of increase of lateral acceleration (ft/s³) *design value = 1 ft/s³

But before we ca continue with the above formula, we need to calculate the absolute difference in the grade of the sag curve as shown below:

Utilizing the formula:

G = (I(G₂) - (G₁)I)

G = (I+3.0) - (+1.0)I)

G = +2.0%

Now, moving forward to determine the value of K by using the formula below or relation as follows:

K = L / G

That is:

K = 420 / 2.0

K = 210

Now that we understand that K = 210, we now refer to the table of interpolation to calculate the value of the design speed:

Therefore:

V = ( 75 + (80 - 75 / 231 - 206) x ( 210 - 206)

Solving the above further, we arrive at:

V = 75.8 m/h

Now, also referring back to the question which states that for e = 6%

Also, referencing the interpolation table to check for the value of e, we have:

Rv = 2587.616 feet

Therefore, R = 2587.616 feet  + 12 / 2

R = 2593.616 feet

Now, to calculate the length of the horizontal curve as follows,

we have:

Ls = 3.15V³ / RC = (π / 180 ) RC

Therefore,

Ls =  (π / 180 ) x 2593.616 x 37°

Ls = 1674.88 feet

Hence, we can conclude that the length of horizontal curve is 1674.88 ft