2.7.3. Characteristics of a vertical curve

The characteristic of a parabolic curve is that the gradient changes from point to point but the rate of change in grade remains constant. Hence, for finding the length of the vertical curve, the rate of change of grade should be an essential consideration as this factor remains constant throughout the length of vertical curve.

Vertical curves are used to create a smooth transition between two different gradients. The special characteristic of parabolic curves is that while the gradient may change continuously at different points, the rate at which these changes occur is consistent. This means that if you were to measure the gradient at the start of the curve, it would smoothly change to another gradient type by the end without sudden dips or rises. Understanding the rate of change of gradient is crucial for determining how long the curve should be.

Imagine driving on a road that gently rises and falls in a parabolic manner. At first, you might be ascending at a gentle slope, and as you continue, the road gradually levels out or may even descend smoothly. This makes it comfortable for you as a driver, allowing you to maintain speed and not feel jolted by sudden changes.

Generally, the recommended rate of change of grade is 0.1% per 30 m at summits and 0.05% per 30 m at sags.

The design of vertical curves incorporates standard recommendations that provide guidelines on how steep the gradient should change along the curve. For summit curves (where the gradient rises), a change of 0.1% for every 30 meters is suggested, while for sag curves (where the gradient drops), a smaller change of 0.05% per 30 meters is preferred. This ensures the curve is gradual enough to allow drivers to maintain control and visibility.

Think of it like a roller coaster. If the inclination to ascend is too steep without a gradual slope beforehand, riders might feel uncomfortable. By controlling the rate of change gently, the roller coaster ride can be thrilling yet safe, similar to the principles of designing vehicle roadways.

In Figure 2.22, AB and BC are two gradient lines intersecting at point B. A vertical curve (T FT) is to be introduced between these two gradients. If g1% is the gradient of line AB (In this case, it is +ve), g2% is the gradient of line BC (In this case, it is -ve), r is the rate of change of grade, T1 and T2 are the tangent points at the beginning of the curve and at the end of the curve.

When two differently sloped roadways intersect, there needs to be a vertical curve connecting them smoothly. The curve transition between the gradient of line AB (which is positive) and line BC (which is negative) allows for a continuous flow. This design helps avoid sudden changes that could lead to discomfort or danger for vehicles traversing the switch between upwards and downwards slopes.

Picture walking from a hill down to a valley. If there is a staircase with just steps, your movement would be jerky, but instead, if there was a gentle slope or ramp connecting the two elevations, your transition would be smooth and comfortable, much like how vertical curves function on roads.

It is assumed that length of the vertical curve is equal to the length of the two tangents i.e., T1 T2 = BT1 + BT2. Using the algebraic difference of the two grades, the length of the vertical curve, L, can be calculated as: $L = \frac{g_1 - g_2}{r}$

The process of calculating the length of the vertical curve involves understanding the geometry of the intersecting gradients. By estimating the distance between the tangent points and relating that to the algebraic difference of the gradients, the length of the vertical curve can be calculated. This balance is essential to maintain both the safety and comfort of vehicle passengers.

Think of it like designing a smooth driveway that connects a steep road to a gentle slope. If you measure how much change in incline there is, you can plan how long your driveway should be to let cars transition smoothly from the steep road to the gentle slope.

The chainage of beginning of the curve (T1) = Chainage of B – L/2, Chainage of end of the curve (T2) = Chainage of the intersection point + half the length of the vertical curve.

Chainage refers to the distance measured along the road. When determining where to begin and end the vertical curve, you take the chainage at the intersection point B and adjust it based on the length of your calculated curve (L). This helps define where drivers will first notice the curve and where they will exit it.

Imagine a race track where the start and end points of a curve are marked. To calculate where drivers will enter the curve, you'd start accounting from the point where the straight segment meets the curve. This ensures that every driver knows exactly where to begin turning.

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