A valley curve is formed by a descending grade of 1 in 25 meeting an ascending grade of 1 in 30. What should be the length of the curve from the viewpoint of headlight sight distance for a stopping sight distance of 127.3 m?
- A173.4 m
- B181.7 m
- C189.3 m
- D199.5 mCorrect
Solution
Concept
The design of a valley curve based on headlight sight distance ensures that the headlight beam distance at night is at least equal to the stopping sight distance (SSD).
Formula
N = n1 - n2 L = (N × S²) / (1.5 + 0.035 × S) for L > S
Step-by-step solution
1. Calculate the total deviation angle N: N = |(-1/25) - (1/30)| = 1/25 + 1/30 = 11/150 = 0.07333 2. Calculate the required length of valley curve L (assuming L > S = 127.3 m): L = (N × S²) / (1.5 + 0.035 × S) L = (0.07333 × 127.3²) / (1.5 + 0.035 × 127.3) L = 1188.39 / 5.9555 = 199.54 m 3. Verify the assumption L > S: Since 199.5 m > 127.3 m, the assumption holds true.
Explanation
The deviation angle N is the algebraic difference between the two gradients. Assuming the length of the valley curve L is greater than the stopping sight distance S, the standard IRC formula based on headlight sight distance gives L = 199.5 m.
Why the other options are wrong
A: Incorrect calculated value resulting from arithmetic error. B: Result obtained if wrong grade difference or formula denominator is used. C: Result obtained if improper rounding of N is used.
Shortcut
Multiply (11/150) by S² and divide by (1.5 + 0.035 S) directly in one calculator step.
Common mistake
Adding grades incorrectly or using the summit curve formula instead of the valley curve formula.
Correct answer: D) 199.5 m