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4.2 Track Geometry 4.2 1. 2. 3. 4. Learning Outcomes Identify elements of a simple road curve Compute design values related to horizontal alignment of simple curves Identify elements of a modern road curve Compute design values related to the horizontal alignment of modern road curves 5. Define and explain the purpose of road superelevation 6. Compute design values related to road superelevation Railway curve • Modern road curves are composed of a simple curve and transition curves at the ends (also called spiral curve) • The purpose of a transition curve is to to gradually increase the curvature as the vehicle enters the curve • Why? An abrupt change in curvature will also cause abrupt application of centrifugal force on the wheels. • A railway curve has to be provided also with superelevation to provide the necessary centripetal force for the curvature Banking / leaning / superelevation / railway cant Photo by: Science Photo Library Simple curve (Circular curve) Simple curve (Circular curve) Sample problems – simple curve 1. A railway track needs to change direction from 𝑁 5°22′13" 𝐸 to S 78°06′ 52" 𝐸. According to the design speed, the radius of curvature should be 125.53 m. If a simple curve is to be provided a) What will be its length? b) How far from the point of intersection should the curve start? Problem #1 Solution a) What will be its length? (𝐿𝑐 ) 𝐼 𝑃𝑇 𝑃𝐼 S 78°06′ 52" 𝐸 𝐿𝑐 = 𝑅𝜃 𝐼 = (90 − 5°22′13") + (90 − 78°06′ 52") 𝐿𝑐 𝑅 = 125.53 𝑚 𝑃𝐶 𝑳𝒄 = ___________𝒎 𝑂 𝐼 𝑇 𝑃𝐼 𝐼 = __________________rad 𝐿𝑐 = 125.53 𝐼 𝐼 𝑁 5°22′13" 𝐸 𝑃𝑇 b) How far from the point of intersection should the curve start? (Length of Tangent) 𝑇 𝑅 = 125.53 𝑚 𝑃𝐶 (rad) 𝐼 2 𝑇 = 𝑅 tan 𝐼 2 𝑻 = _____________𝒎 𝑂 Sample problems – simple curve 2. An engineer needs to design a simple curve for a change in road direction from 𝑁 42°10′ 39" 𝑊 to S 38°19′ 46" 𝑊 and has to stake on the ground points along the curve using a total station (for the construction). According to the design speed, the radius of curvature should be 100.0 m. a) Starting from the PI, how far should he move to get to station PC? b) He needs to stake out 10 equidistant points along the curve from station PC (excluding PC, last point is PT). What are the offset angles and chord distances he needs to stake out? c) To check his work, he transfers back to PI, and bisects the angle between PC and PT to locate the midpoint of the curve. What distance should M be from PI? Problem #2 Solution 𝑃𝐼 𝐼 a) Starting from the PI, how far should he move to get to station PC? (Length of Tangent) 𝑇 𝑃𝐶 𝑃𝑇 𝑁 42°10′ 39" 𝑊 𝐼 2 ′ S 38°19 46" 𝑊 𝑅 = 100.0 𝑚 𝐼 = 90 − 42°10′ 39" + 90 − 38°19′ 46" 𝐼 = _________________rad 𝐼 𝑇 = 𝑅 tan 2 𝑻 = _____________𝒎 b) He needs to stake out 10 equidistant points along the curve from station PC (excluding PC, last point is PT). What are the offset angles and chord distances he needs to stake out? For S1: 𝐼 𝜃= = __________rad 10 𝐼 𝛼= 𝑆1 𝐿𝑆1 𝜃 𝜃 2 𝑃𝐶 𝑳𝑺𝟏 𝜃 𝛼 = = _____________rad 2 𝜃 = 2𝑅 sin = _________________𝒎 2 Problem #2 Solution c) To check his work, he transfers back to PI, and bisects the angle between PC and PT to locate the midpoint of the curve. What distance should M be from PI? 𝑃𝐼 𝐼 𝐸 𝑀 𝑃𝑇 ′ S 38°19 46" 𝑊 𝑇 𝑃𝐶 𝐸= 𝑁 42°10′ 39" 𝑊 𝐼 2 𝑂 𝑅 = 100.0 𝑚 𝑇 𝐼 sin 2 − 𝑅 = _______________𝒎 Sample problems – simple curve 3. A simple curve has a radius of 125 m. What is its equivalent degree of curvature in English and SI systems? Solution: a) English system: (1 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 = 100 𝑓𝑡) 𝑆 = 1 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 𝑅 = 125 𝑚 = 410.1 𝑓𝑡 𝑆 = 𝑅𝜃, 𝜃= 𝑆 𝑅 = 100 410.1 𝑅 𝑫 = 𝜃 = 0.243843 rad = 𝟏𝟑. 𝟗𝟕𝟏° 𝐷 b) SI system: (1 𝑠𝑡𝑎𝑡𝑖𝑜𝑛 = 20 𝑚) 𝑅 = 125 𝑚 𝑆 = 𝑅𝜃, 𝜃= 𝑆 𝑅 = 20 100 𝑫 = 𝜃 = 0.2 rad = 𝟏𝟏. 𝟒𝟓𝟗° Compound curve Reverse curve