Q14. A lateral shift in the transition curve is given by:
📚 Detailed Explanation: The Transition Curve Shift Formula
This question isolates the standalone formula for lateral shift already applied numerically in Q6 — reinforcing the exact relationship between transition curve length, circular curve radius, and the resulting inward shift of the circular arc.
Fig: The shift S is the small inward displacement of the circular arc needed to accommodate the transition curve smoothly.
The lateral shift formula is derived from the geometry of a cubic spiral/clothoid transition curve meeting a circular arc, and is given by:
S = L² / (24R)
where L is the length of the transition curve and R is the radius of the circular curve it connects to. This same formula was applied numerically in Q6, where L = 48 m and R = 200 m gave S = 0.48 m.
Why the Other Options Are Wrong
A — L/24R: Uses L to the first power instead of squared — dimensionally and numerically incorrect, since shift must scale with the square of the transition length (a longer transition curve causes a disproportionately larger shift).
C — L²/4R: Uses the wrong denominator constant (4 instead of 24) — this would overstate the shift by a factor of 6 compared to the correct formula.
D — L/240R: Also uses L to the first power (incorrect) and an incorrect denominator constant, compounding two errors at once.
Key Concepts for Students
- Memorise S = L²/24R exactly — the squared L and the 24 are both essential: This is one of the most frequently tested formulas in transition curve design, appearing both as a standalone formula question (here) and embedded in applied numerical problems (Q6).
- Shift exists because of the changing radius of the transition curve: Since a transition curve’s radius decreases continuously from infinity (at the straight) to R (at the circular curve), connecting it smoothly requires the circular arc’s centre to be offset inward by S from its theoretical (untransitioned) position.
- Always double-check whether L needs to be the governing (maximum) design value first: As shown in Q6, when multiple candidate transition lengths exist, this shift formula must be applied using the correctly selected governing length, not any of the individual (possibly shorter) candidate values.