Which is the following situation created for valley curve ?

Question

Examoogle · Accepted Answer

The correct answer is Option D: All of the above. A valley curve (or sag curve) is a vertical curve provided at the dip or intersection of two gradients where the convexity is downward. It is formed when the deviation angle is negative, which occurs when a downgrade is followed by an upgrade, a downgrade is followed by another milder downgrade, or a level/plane surface is followed by an upgrade.

Step-by-Step Solution & Concepts:
A valley curve (or sag curve) is a vertical curve provided at the dip or intersection of two gradients where the convexity is downward. It is formed when the deviation angle is negative, which occurs when a downgrade is followed by an upgrade, a downgrade is followed by another milder downgrade, or a level/plane surface is followed by an upgrade.

Working Principle: Valley curves are designed based on headlight sight distance (HSD) requirements during night driving and rider comfort. When a vehicle transitions through a sag curve, the downward acceleration creates an additional centrifugal force acting downward in the same direction as gravity, requiring careful design of the curve length to limit discomfort and satisfy sight distance.

Key Formulas: $N = n_1 - (-n_2) = n_1 + n_2$ — Total deviation angle for downgrade followed by upgrade; $L = \frac{N S^2}{1.5 + 0.035 S}$ — Length of valley curve for headlight sight distance ($S; $L = 2S - \frac{1.5 + 0.035 S}{N}$ — Length of valley curve for headlight sight distance ($S > L$); $L = 2 \left( \frac{N v^3}{C} \right)^{1/2}$ — Length of valley curve based on comfort criterion

Key Points: A valley curve exhibits downward convexity, which causes the center of curvature to lie above the road profile.; Unlike summit curves where stopping sight distance (SSD) during the day governs design, valley curves are governed by Headlight Sight Distance (HSD) at night.; A cubic parabola ($y = a x^3$) is generally preferred for valley curves as it provides a smooth transition.

Advantages: Provides smooth transition between intersecting gradients.; Ensures safe Headlight Sight Distance (HSD) during night driving.; Limits discomfort caused by the impact of centrifugal forces.

Disadvantages: Requires proper drainage management at the lowest point (sag point) to prevent water ponding.; Involves higher construction costs due to earthwork excavation in deep valleys.

Applications: Used in highway geometric design at dips, river crossings, and underpasses.; Applied wherever a descending gradient transitions into an ascending gradient or flatter grade.

Additional Info: Standard IRC specifications recommend designing valley curves for Headlight Sight Distance equal to Stopping Sight Distance (SSD).. Option A: Downgrade followed by upgrade creates a sag (valley) curve.. Option B: Steeper downgrade followed by a milder downgrade also creates a sag (valley) curve as deviation angle $N = -n_1 - (-n_2) = -(n_1 - n_2)$ is negative.. Option C: A level plane surface ($0\%$) followed by an upgrade ($+n_2\%$) yields deviation angle $N = 0 - n_2 = -n_2$, forming a valley curve.

Answer: D is correct — All listed gradient combinations result in downward convexity with a negative deviation angle, forming a valley curve.

Exam Tip: Remember: Summit curves are governed by daytime SSD/Overtaking Sight Distance, while Valley curves are governed strictly by Headlight Sight Distance (HSD) at night and comfort criteria.

Examoogle · Answer

Option A: the downgrade followed by an upgrade

Examoogle · Answer

Option B: the downgrade followed by another downgrade

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Option C: A plane surface followed by upgrade

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