17.3.3.2. Case b. Length of summit curve less than sight distance

The second case is illustrated in figure 17:4. From the basic geometry, one can write:

L = h1 + h2 + S =

(17.2)

Therefore for a given L, h and h to get minimum S, differentiate the above equation with respect to h1 and equate it to zero.

In Case b, we are looking at the scenario where the length of the summit curve (L) is less than the stopping sight distance (S). The relationship between these variables can be described using basic geometry. By rearranging the components related to hills and sight distances, we can differentiate the equation to find the optimal height at which an obstruction should be positioned. This is significant because if the road length does not allow for adequate sight distance, it affects safety drastically.

Imagine you are driving up a hill. If the hill is too steep and extends beyond your line of sight, you might not see a car coming from the opposite direction until it's too late. This is similar to how we must ensure that curves in the road are designed in a way that maintains visibility, making sure drivers can react in time to avoid accidents.

Therefore,

dS/dh1 = -n2 + n1 = 0 (17.3)

Solving for n1, n1 = N√(h1 * h2)/(h1 - h2) (17.4)

Here, we differentiate the sight distance function (S) with respect to h1 to find out where the relationship reaches an optimum point. By setting this derivative to zero, we can solve for n1 (the gradient at point 1) in a way that accounts for the heights of both the driver and the obstruction. This calculation is crucial since it determines how much height adjustment you might need to make for effective visibility.

Think of setting up a camera to take a picture of something from a distance. If you're too low or too high, you won't get the shot right. Similarly, adjusting the height (h1 and h2) allows for a clearer picture of the road conditions—a metaphor for ensuring drivers have the right sight distance.

Now we can substitute n back to get the value of minimum value of L for a given n1, n2, h1 and h2.

L = (h1 + h2)/2 (17.5)

After calculating n1, we can substitute this value back into our equation to find the minimum length (L) of the summit curve. This relationship between the heights and gradients helps ensure that we design roads that meet safety standards without compromising visibility. The formula indicates that the total length needs to encompass the heights to optimize sight distance.

Imagine you’re planning a picnic on a hill. You want to make sure the picnic blanket is spread out in a way that everyone can see the view without obstructions. Similarly, when designing a road, we want to ensure that heights of elements aren’t limiting sightlines and also adhere to safety standards.

Therefore, L = 2√(2h1) + √(2h2)/(N) (17.6) When stopping sight distance is considered, the height of driver’s eye above the road surface (h1) is taken as 1.2 metres, and height of object above the pavement surface (h2) is taken as 0.15 metres.

The final formula gives us a consistent way to calculate the minimum length of the summit curve based on various factors including the driver's eye level and the height of potential obstructions. This ensures that as we design roads, we build in enough length for safe stopping distances. Knowing these specifics helps city planners and engineers create roads that are safer and more efficient.

Visualize a driver approaching a crest in the road. If the driver can see over the obstacle (the summit), they can react in time. Just like ensuring there’s enough clear space in front of you while driving to stop in time, the road must provide that same amount of reaction space right before the summit curve.

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