--- title: "Composite curve by deflection angle method" subject: "Surveying II" chapter: 6 key: "semester_4/surveying_ii:composite-curve-deflection-angle-method" source: https://www.bctengineer.com/q/semester_4/surveying_ii/composite-curve-deflection-angle-method site: BCT Engineering --- # Composite curve by deflection angle method Surveying II · Chapter 6: Curves - Asked in: 2078 Poush; 2078 Baisakh; 2079 Chaitra, 4 marks; 2076 Baisakh, 4 marks; 2075 Bhadra, 4 marks (asked 5 times) - Video: [Elements and Procedure to Setout Combined Curve - Horizontal Curve - Surveying-II](https://www.youtube.com/watch?v=-_yAtlMXi5s) — Ekeeda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/composite-curve-deflection-angle-method - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-6 ## Question Explain about the laying out method of composite curve (two end transition and central circular) by deflection angle method by supporting sketch. ## Short answer A composite curve is set out by deflection angles: first transition from T1 using δ = 573l²/(RL) minutes; at S1, shift instrument, set common tangent, then Rankine deflections for the circular curve; finally, set the second transition from T2 backwards to S2 with the same transition formula. Junction checks ensure correctness. ## Answer A **composite curve** is a circular curve with a transition (spiral) curve at each end. The deflection angle method sets out the transition curves from T1 and T2 and the central circular curve from the junction points. ```text IP / \ / \ T1 ~~~S1 ______ S2~~~ T2 (spiral) (arc) (spiral) |<-L->|<-Lc->|<-L->| ``` #### Elements to compute $R$ = radius of the circular arc, $L$ = length of transition (from $L = V^3/(\alpha R)$), $\Delta$ = deflection angle. - Shift: $s = \dfrac{L^2}{24R}$ - Spiral angle: $\phi_s = \dfrac{L}{2R}$ rad - Tangent length: $T_s = (R + s)\tan\dfrac{\Delta}{2} + \dfrac{L}{2}$ - Central angle of the circular arc: $\Delta_c = \Delta - 2\phi_s$; length $L_c = R\Delta_c$ (in radians) - Chainages: T1 = IP − $T_s$; S1 = T1 + L; S2 = S1 + $L_c$; T2 = S2 + L #### Setting out **1. First transition (from T1).** Set the theodolite at T1 with zero reading on the IP. For a peg at distance $l$ along the curve (cubic spiral $y = x^3/6RL$): $$ \delta = \frac{l^2}{6RL}\ \text{rad} = \frac{573\,l^2}{RL}\ \text{minutes} $$ The pegs are set out by turning angle $\delta$ and measuring the chord (taken equal to the arc). The last peg at $l = L$ is S1 with $\delta_L = \phi_s/3$. Check the position of S1 from the data. **2. Circular curve (from S1).** Shift the instrument to S1. Take a back-sight on T1. The angle between the chord S1T1 and the common tangent at S1 is $\phi_s - \phi_s/3 = \dfrac{2\phi_s}{3}$. So, after the back-sight, plunge the telescope and turn it through $\dfrac{2\phi_s}{3}$ to get the direction of the tangent at S1, and set the horizontal circle to zero there. Then set out the arc by Rankine's deflection angles from this tangent: $$ \delta = \frac{1718.9\,c}{R}\ \text{minutes} $$ for a chord $c$ (first sub-chord, full chords, last sub-chord). The cumulative angle at S2 must equal $\Delta_c/2$, which checks the work. **3. Second transition (from T2).** Set up at T2, back-sight the IP (the tangent), and set out pegs backwards from T2 towards S2 using the same deflection angles $\delta = 573\,l^2/(RL)$ minutes, where $l$ is the distance of the peg from T2. The pegs should meet the circular curve at S2. **Check:** The pegs of the circular curve from S1 and the pegs of the second transition from T2 must meet at S2. The method needs only a theodolite and a tape, but it is slow because of the shifting of the instrument at the junction points. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For a composite curve of deflection angle Δ and spiral angle φs, what is the central angle of the circular arc? A. Δ + 2φs B. Δ - 2φs C. Δ - φs D. Δ/2 - φs Answer: B. Each transition uses up a spiral angle φs, so Δc = Δ - 2φs and Lc = R·Δc. (Key point: Central angle of circular arc) 2. Which deflection angle sets out a peg at distance l along the first transition from T1? A. δ = 1718.9 l/R minutes B. δ = l/(2R) radians C. δ = 573 l^2/(R L) minutes D. δ = 573 l/(R L) minutes Answer: C. For the cubic spiral y = x^3/6RL, δ = l^2/(6RL) rad = 573 l^2/(RL) minutes. (Key point: Deflection angle on the transition) 3. At S1, after back-sighting T1, how is the common tangent found? A. Plunge the telescope and turn through 2φs/3 B. Turn through φs/3 without plunging C. Turn through φs D. Turn through Δc/2 Answer: A. The chord S1T1 makes φs - φs/3 = 2φs/3 with the common tangent, so after plunging the telescope is turned through 2φs/3. (Key point: Setting the tangent at S1) 4. What check applies to the circular curve set out from S1? A. Cumulative deflection at S2 equals Δ/2 B. Cumulative deflection at S2 equals φs C. Cumulative deflection at S2 equals Δc/2 D. Cumulative deflection at S2 equals 2φs/3 Answer: C. The arc subtends Δc at the centre, so the last cumulative Rankine deflection must be Δc/2, and the pegs must meet the second transition at S2. (Key point: Check at S2) ## Related in Surveying II - [Transition deflection angle derivation and laying out](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atransition-deflection-angle-derive-laying-out) — Surveying II, chapter 6 - [Simple circular curve by deflection angle method](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asimple-curve-deflection-angle-method-sketch) — Surveying II, chapter 6 - [Derive deflection angle of transition curve](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atransition-deflection-angle-derivation) — Surveying II, chapter 6, asked 3× - [Function of transition curve, deflection angle](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atransition-curve-function-deflection-angle) — Surveying II, chapter 6 - [Transition curve R=300 m, deflection 40°](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atransition-curve-80kmph-radius-300) — Surveying II, chapter 6 - [Transition curve R=2000 m, chainages, deflection table](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atransition-curve-2000m-160kmh-chainage-table) — Surveying II, chapter 6 ## Related in other subjects - [Composite curve with 48 degree deflection](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-72m-composite-60) — Transportation Engineering I, chapter 3 - [Transition curve length from spiral angle 10.32](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-73bh-tc-spiral) — Transportation Engineering I, chapter 3, asked 2× - [Transition curve from spiral angle 8.35](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-69bh-tc-spiral) — Transportation Engineering I, chapter 3, asked 2× - [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