---
title: "Length of transition curve (criteria)"
subject: "Transportation Engineering I"
chapter: 3
key: "semester_6/transportation_engineering_i:tc-length-methods"
source: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/tc-length-methods
site: BCT Engineering
---

# Length of transition curve (criteria)

Transportation Engineering I · Chapter 3: Geometric Design of Highway

- Asked in: 2078 Poush, 5 marks; 2073 Magh; 2072 Ashwin; 2071 Magh; 2070 Bhadra; 2066 Magh (old course), 8 marks; 2065 Kartik (old course), 8 marks; 2063 Kartik (old course), 5 marks (asked 8 times)
- Video: [Length of Transition Curve in Highways – Formula, Calculation & IRC Standards #highway #rrbje #sscje](https://www.youtube.com/watch?v=_vkIevHnWyw) — Civil Site Plus
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/tc-length-methods
- Whole chapter: https://www.bctengineer.com/md/semester_6/transportation_engineering_i/chapter-3

## Question

Derive an expression for the length of the transition curve to be introduced between the straight and the circular path on a horizontal curve from two criteria (rate of change of centrifugal acceleration and rate of introduction of superelevation); describe the methods of designing the length of the transition curve.

## Short answer

Transition curve length is determined by three criteria: rate of change of centrifugal acceleration (Ls = 0.0215V³/(C*R)), rate of superelevation introduction (Ls = N*e*(W+We)), and IRC empirical formulas (Ls = 2.7V²/R for plain/rolling, Ls = V²/R for mountainous). The largest value is adopted, rounded to 5m, with minimum ~30m. Shape is typically a spiral (clothoid).

## Answer

A transition curve is introduced between a straight and a circular curve so that the centrifugal acceleration and the superelevation are introduced gradually. Its length is found from the following criteria, and the largest length is adopted.

#### 1. Rate of change of centrifugal acceleration
When a vehicle moves at speed $v$ on a circular curve of radius $R$, the centrifugal acceleration is $v^2/R$. On the transition curve it grows from 0 to $v^2/R$ in time $t = L_s/v$. If the allowable rate of change of acceleration is $C$ (m/s³):

$$
C = \frac{v^2/R}{L_s/v} = \frac{v^3}{L_s R}
\quad\Rightarrow\quad
L_s = \frac{v^3}{C R}
$$

With $v = V/3.6$ (V in km/h): $L_s = 0.0215\,\dfrac{V^3}{C R}$.

IRC: $C = \dfrac{80}{75+V}$, limited to $0.5 \le C \le 0.8$ m/s³.

#### 2. Rate of introduction of superelevation
The pavement is raised from the normal camber to the full superelevation $e$ over the transition length at a rate of 1 in $N$.
Let $E$ = total rise of the outer edge relative to the inner edge = $e\,(W + W_e)$ when the pavement is rotated about the inner edge. Then

$$
L_s = N\,e\,(W+W_e)
$$

(For rotation about the centre line, the rise of the outer edge is $\dfrac{(e+n)(W+W_e)}{2}$ and $L_s = N\dfrac{(e+n)(W+W_e)}{2}$, with $n$ = camber.)

Typical $N$ (IRC): 1 in 150 for plain and rolling terrain, 1 in 60 for mountainous and steep terrain (NRS 2070 uses similar values).

#### 3. Empirical formula (IRC)

$$
L_s = \frac{2.7\,V^2}{R}\ \text{(plain and rolling)},\qquad
L_s = \frac{V^2}{R}\ \text{(mountainous and steep)}
$$

#### Design procedure
1. Find the length by each of the three methods.
2. Adopt the **largest value,** rounded up to a multiple of 5 m.
3. Check the minimum (not less than about 30 m, or the distance travelled in 2-3 s) and make sure the length is not so large that no circular arc is left between the transitions (shift check).

#### Shape of transition
The most common type is the **spiral (clothoid),** in which the radius $r$ at a distance $l$ along the curve satisfies $r\,l = R\,L_s$ (the radius varies inversely with the length). It is set out by the cubic parabola $y = \dfrac{x^3}{6RL_s}$. The shift of the circular curve is $s = \dfrac{L_s^2}{24R}$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Length from rate of change of centrifugal acceleration is:
   A. Ls = v^3 / (C R)
   B. Ls = v^2 / (C R)
   C. Ls = C R / v^3
   D. Ls = v R / C
   Answer: A. Acceleration v^2/R is reached in time Ls/v, so C = v^3/(Ls R) and Ls = v^3/(CR). (Key point: Centrifugal acceleration)

2. For rate of introduction of superelevation of 1 in N, rotating about the inner edge, Ls =
   A. e (W+We)/N
   B. N e (W+We)
   C. N (W+We)
   D. e N / (W+We)
   Answer: B. Rise of outer edge e(W+We) is spread over a slope 1 in N: Ls = N e (W+We). (Key point: Superelevation criterion)

3. Which length is adopted for design?
   A. The smallest of the three
   B. The average
   C. The largest of the three, rounded up to a multiple of 5 m
   D. The first computed
   Answer: C. The largest value satisfies all criteria. (Key point: Adopted length)

4. In the spiral transition, how does radius r vary with distance l?
   A. r is constant
   B. r l = R Ls, so radius varies inversely with length
   C. r increases with l
   D. r = l
   Answer: B. For the clothoid r l = R Ls, so radius falls from infinity to R. (Key point: Spiral shape)

## Related in Transportation Engineering I

- [Transition curve length and shift, 7.2 m pavement](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-62j-tc-shift) — Transportation Engineering I, chapter 3
- [Composite curve, 65 kmph, centrifugal ratio 1/5](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-71bh-composite-65) — Transportation Engineering I, chapter 3
- [Transition curve for 310 m radius](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-80c-tc-310) — Transportation Engineering I, chapter 3
- [Transition curve 400 m radius, 85 kmph](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-tc-400-85) — Transportation Engineering I, chapter 3, asked 2×
- [Transition curve 625 m radius, 90 kmph](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-75b-tc-625) — Transportation Engineering I, chapter 3
- [Transition curve 220 m, pavement 7 m](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-77c-tc-220) — Transportation Engineering I, chapter 3

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