Vertical curve design on rail alignment: A 0.70% upgrade meets a 0.65% downgrade at a summit. If the permitted rate of change of grade is 0.10% per chain, what is the required length of the vertical curve?

Introduction / Context:
Vertical curves provide smooth transitions between grades for ride comfort, safety, and sight distance. Designers limit the rate of change of grade to a maximum per unit length (often per chain in railway practice).

Given Data / Assumptions:

• Upgrade = +0.70%.
• Downgrade = −0.65% (summit curve).
• Allowable rate of change = 0.10% per chain.
• 1 chain is the reference length unit used for the rule.

Concept / Approach:
The total algebraic change in grade across the curve equals the sum of magnitudes at a summit (since signs are opposite): Δg = 0.70% + 0.65% = 1.35%. Minimum curve length L (chains) is Δg divided by the permitted rate per chain.

Step-by-Step Solution:

Verification / Alternative check:
Check limiting rate using 14 chains: 1.35/14 ≈ 0.096%/chain ≤ 0.10%/chain, so the requirement is satisfied.

Why Other Options Are Wrong:

• 10, 12 chains: Too short; exceed the permitted rate.
• 16, 18 chains: Acceptable but longer than necessary in typical exam context seeking the minimum satisfying design.

Common Pitfalls:

• Subtracting grades (appropriate for valley combinations) instead of adding for a summit.
• Failing to round up to meet the limit.

Final Answer:
14 chains