Cubic parabolic transition curve for highways: select the standard relation (offset from tangent) commonly used for setting out.
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Ay = x^2 / (2R)
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By = x^3 / (6 R L)
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Cy = x^3 / (3 R)
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Dy = (R x^3) / (6 L)
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Enone of these.
Answer
Correct Answer: y = x^3 / (6 R L)
Explanation
Introduction / Context:Transition curves provide a gradual change of curvature from tangent to circular arc. A widely used highway transition is the cubic parabola, chosen for its simple field equations and smooth curvature variation.
Given Data / Assumptions:
- x = distance measured along the tangent from the tangent point.
- y = offset from the tangent to the transition curve.
- R = radius of the circular curve that the transition leads into.
- L = length of the transition curve.
Concept / Approach:For a cubic parabola, curvature varies linearly with arc length. The standard highway setting-out relation between the tangent offset y and station distance x is y = x^3 / (6 R L) (for small angles), enabling quick computation of offsets.
Step-by-Step Solution:
Start from linear curvature assumption: k = s / (R L).Relate small-angle geometry to derive offset y proportional to x^3.Obtain y = x^3 / (6 R L) as the practical setting-out formula.Verification / Alternative check:Compare with spiral geometry tables; the cubic-parabola approximation yields close agreement for typical transition lengths, validating field use.
Why Other Options Are Wrong:Option (a) is a parabolic form unrelated to cubic transition; (c) and (d) have incorrect dimensions and omit L or misplace R and L; thus they do not represent the standard highway relation.
Common Pitfalls:Using large-angle geometry beyond the range where the approximation holds; mixing up coordinate systems (offset from chord vs tangent).
Final Answer:y = x^3 / (6 R L)