18. Example 2.26:

Length of vertical curve L = (g − g ) r * 30 = (0.5 – (−0.7)) 0.1 * 30 = 360 m

The length of the vertical curve is calculated using the change in grades (g1 and g2). Here, g1 is +0.5% and g2 is -0.7%. We first convert these percentages into decimals (0.005 and -0.007) and then multiply by the rate of change of grade (0.1% per 30m). Thus, we find that the length L of the vertical curve is 360 meters.

Think of a roller coaster that needs to smoothly connect two slopes. The length of the gentle slope (vertical curve) is crucial for a smooth ride. Just like we calculated the curve's length to ensure safety and comfort, engineers do so for roads to ensure vehicles can transition from one slope to another without sudden drops or rises.

L / 2 = 180 m Chainage of O = chainage of A – L/2 = 500 – 180 = 320 m Chainage of B = chainage of A + L/ 2 = 500 + 180 = 680 m

The chainage, or the location along the road's alignment, is calculated based on the length of the vertical curve. We divide the total length by 2 to find the distance to the start (O) and the end (B) of the curve from the point of intersection (A), which is at chainage 500m. Thus, we find that O is 320m and B is 680m.

Imagine you're walking a path that curves up and down; your position is marked with markers (chainages). The midpoint of the curve indicates where the path is smoothest, just like in our calculation, where we mark the beginning and end of the curve where the grade changes. Knowing these positions helps us determine where to plant trees or build fences along the path.

RL of point of intersection, A = 330.75 m RL of O = 330.75 - g L / 200 = 330.75 - 0.5 (360)/ 200 = 329.85 m RL of B = 330.75 - g L/ 200 = 330.75 - 0.7 (360)/ 200 = 329.49 m

The Reduced Levels (RL) at points O and B are calculated from the RL of point A (the intersection). We use the change in grade multiplied by the length of the vertical curve divided by 200 to find the elevation at points O and B. Here, the calculations yield RLs of 329.85m for O and 329.49m for B.

Think of finding how high you are at different points along a hill by measuring from sea level. If you're at point A, you can use the slopes (the grades) to see how high you will be at points O and B. Just like we calculate the heights using the changes in slope, hikers might track elevation to ensure they don't backtrack or encounter steep drops unexpectedly.

RL of mid-point E of chord OB = ½ (RL of O + RL of B) = ½ (329.85 + 329.49) = 329.67 m RL of F (vertex of curve) = ½ ( RL of B + RL of A) = ½ (330.75 + 329.67) = 330.21 m

To determine the elevation at the midpoint of the curve (E), we average the elevations at points O and B. This simple formula accounts for the smooth gradient of the curve. The vertex (F) is found by averaging the elevation at point A and the mid-point E, showing the highest point of the curve.

Imagine a children’s play slide that rises gently before dropping. The highest point on the slide corresponds to the vertex of our curve, while the average height at the halfway mark represents the midpoint. Knowing where the highest points are helps ensure that the slide is safe and enjoyable for kids.

Check: AF = (g – g ) L/ 800 = [0.5 −(− 0.7)]* 360/ 800 = 0.54 m

To verify our calculations for the change in elevation (AF) between A and F, we apply a check using the grades and the length of the curve. This confirms that the difference calculated aligns with our expectations based on grade changes.

Think of double-checking a recipe to ensure the flavors balance after adding ingredients. Just like you might taste and adjust a dish, engineers confirm their measurements to provide safe and smooth transitions in roads.

The differences in RLs at various chainages are summarized in a table to visualize the data clearly.

The elevations at different chainages are organized in a table format for easy reference. Each value showcases how the elevation changes along the curve from O to B, indicating how the vertical curve transitions smoothly.

This is like creating a chart of your savings over a year's time. By visually plotting this data, you can easily see growth or dips across different months. Similarly, the table of RLs helps us understand grade changes visually over the course of a vertical curve.

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