--- title: "Extra widening: reasons and derivation" subject: "Transportation Engineering I" chapter: 3 key: "semester_6/transportation_engineering_i:extra-widening-derive" source: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/extra-widening-derive site: BCT Engineering --- # Extra widening: reasons and derivation Transportation Engineering I ยท Chapter 3: Geometric Design of Highway - Asked in: 2081 Chaitra, 4 marks; 2078 Chaitra, 3+5 marks; 2078 Poush, 4+4 marks; 2076 Bhadra; 2073 Magh, 8 marks; 2067 Mangsir (old course), 6 marks; 2065 Chaitra (old course), 1+2+5 marks; 2064 Poush (old course), 1+2+2+3 marks (asked 8 times) - Video: [Extra Widening on Curves | Theory & Derivation Explained | PU | PoU | CTEVT Civil Engineering](https://www.youtube.com/watch?v=SzjKekAmhec) โ€” Civil Gyan Kendra ๐Ÿ–‹๏ธ - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/extra-widening-derive - Whole chapter: https://www.bctengineer.com/md/semester_6/transportation_engineering_i/chapter-3 ## Question Define extra widening. State the necessity/reasons for extra widening of pavement at horizontal curves and derive an expression for the extra widening required (mechanical and psychological widening); mention the methods of providing it. ## Short answer Extra widening is additional carriageway width on horizontal curves due to off-tracking (rear wheels track inside), steering difficulty, front overhang swing, and clearance needs. Total widening We = Wm + Wps, where mechanical widening Wm = n*lยฒ/(2R) and psychological widening Wps = V/(9.5โˆšR). Methods: provide on inner edge only, equally on both edges, or gradually along transition curve. ## Answer **Extra widening** is the additional width of carriageway provided on a horizontal curve, over and above the normal width of the straight road. #### Need (reasons) 1. **Off-tracking:** the rear wheels of a long vehicle do not follow the front wheels; they track inside, so the vehicle occupies more width. 2. **Difficulty of steering:** on a curve the driver tends to keep a greater clearance from the pavement edge (psychological effect). 3. **Wider swing path of the front overhang,** so the vehicle may encroach on the adjacent lane. 4. **Greater clearance** is needed between two vehicles that pass or overtake on a curve, because their lateral position varies with speed. 5. Therefore, extra widening $W_e = W_m + W_{ps}$ (mechanical + psychological). #### Mechanical widening ($W_m$) Let $l$ = length of wheelbase (rear axle to front axle), $R_1$ = radius of the path of the outer front wheel, $R_2$ = radius of the path of the outer rear wheel, $R$ = mean radius of the curve, and $n$ = number of traffic lanes. ```text O (centre of curve) |\ | \ R2 | \ R1 | \ | \ rear axle +-----+ front wheel <- l -> ``` The rear axle points towards the centre $O$, so the centre, the outer rear wheel and the outer front wheel form a right triangle with $R_1$ as the hypotenuse: $$ \begin{aligned} R_2^2 &= R_1^2 - l^2 \\ W_m = R_1 - R_2 &= R_1 - \sqrt{R_1^2 - l^2} \approx \frac{l^2}{2R} \end{aligned} $$ For $n$ lanes, each vehicle gives the same off-tracking, so $$ W_m = \frac{n\, l^2}{2R} $$ #### Psychological widening ($W_{ps}$) Empirical formula (IRC): $$ W_{ps} = \frac{V}{9.5\sqrt{R}} $$ where $V$ is the design speed in km/h and $R$ is the radius of the curve in m. #### Total extra widening $$ W_e = \frac{n\, l^2}{2R} + \frac{V}{9.5\sqrt{R}} $$ IRC takes $l = 6.1$ m for the design vehicle. Widening is generally provided for $R < 300$ m, since for larger radii it is negligible (NRS 2070/IRC practice). #### Methods of providing 1. **On the inner edge only** - preferred on hill roads, where the widening is given on the inner (hill) side. 2. **Equally on both edges (inner and outer)** - common on plain roads, so the centre line stays central. 3. **Introduce the widening gradually:** along the transition curve, so that the full width is reached at the start of the circular curve; on a simple circular curve without a transition, about two-thirds of the widening is attained on the tangent and one-third on the curve (the same as the superelevation run-off), giving a smooth edge. 4. Add the **widening to the pavement** and keep the shoulders as usual. Example: for $V = 60$ km/h, $R = 100$ m, 2 lanes, $l = 6.1$ m: $W_e \approx 2(6.1)^2/200 + 60/(9.5\times10) = 0.37 + 0.63 \approx 1.0$ m. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Total extra widening is the sum of: A. Mechanical and psychological widening B. Camber and superelevation C. Lag and braking widening D. Inner and outer widening only Answer: A. We = Wm + Wps. (Key point: Components) 2. What causes mechanical widening? A. Driver fear B. Superelevation C. Rear wheels tracking inside the path of front wheels (off-tracking) D. Road camber Answer: C. Off-tracking of long vehicles needs extra width. (Key point: Off-tracking) 3. What is the mechanical widening for n lanes, wheelbase l, radius R? A. n l / (2R) B. l^2 / (2R) C. n l^2 / R D. n l^2 / (2R) Answer: D. Each lane gives l^2/(2R) off-tracking, so Wm = n l^2/(2R). (Key point: Mechanical formula) 4. Psychological widening is given by: A. V^2 / (127R) B. V / (9.5 sqrt(R)) C. n l^2 / 2R D. V^2 / (254 f) Answer: B. IRC empirical formula is Wps = V/(9.5 sqrt(R)). 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