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Geometric Design Horizontal Alignment • Curves (circular) • Tangents • Transitions Vertical Alignment • Curves (parabolic) • Tangents IT222 Plan and Profile Drawing Types of Horizontal Curves • • • • Simple Curve Compound Curves Reverse Curves Transition (Spiral) Curves Curve Parameter Use 1. 2. 3. What parameter is needed to find the beginning of a curve relative to the PI? What other parameters affect the value of the parameter in question 1? To prevent a subdivision street curve from being too close to a lot line, the centerline cannot be less than 65 ft from the PI. Parameter(s) to find R? Degree of Curvature Arc Definition Most frequently used in highway design It is the central angle formed by two radii that extend from the center of a circle to the ends of an arc measuring 100 feet long Degree of Curvature Arc Definition Ratio between the degree of curvature (D) and 360° is the same as the ratio between 100 feet of arc and the circumference (C) of a circle having the same radius D__ = 360° 100__ C Degree of Curvature Arc Definition Since the circumference of a circle = 2πR D__ = 360° R = 5729.578 / D 100__ 2πR Degree of Curvature Arc Definition Smaller D means larger R; Larger D means smaller R For a 100’ arc: D = 32 degrees, R = 179.05’ D = 8 degrees, R = 716.20’ Degree of Curvature – Chord Definition Most frequently used in railway practice Degree of curve is the central angle formed by two radii drawn from the center of the circle to the ends of a chord 100 feet long Degree of Curvature – Chord Definition The following trigonometric relationship exists: sin (D/2) = 50/R Solving for R: R = 50 / (sin(D/2)) Horizontal Alignment Principle consideration Length of radius or degree of curvature, based on Design speed Sight distance As limited by headlights or obstructions Horizontal Alignment The smaller the radius, the sharper the curve For high speed highways, curves must be flat and have large radius Typical radii 12,000 feet on interstate highways 1,000 feet on major arterials 500 feet on industrial access roads 150 feet on residential roads Geometry: Intersection or Delta Angle Angle PI-PC-PT = Delta / 2 because: Sum of the angles for the isosceles triangle PC-PI-PT = 180 The angle at the PI = 180 - Delta Therefore, the other two angles = (180 – (180 - Delta))/2 = Delta/2 Simple Curves PI Sta.4+10.23 PI Sta = PC Sta + T PT St a 3 .2 60 1+ a. St St a. = 5 PC +5 St 2.93 a + L PT PC Begin Project Sta 0+00 Laying Out a Horizontal Curve Surveyors stake out curves for construction Stakes placed at PC, PT and full stations (every 100 feet) For sharp curves, may need stakes at 25, 50 and 75-feet stations Two Methods Deflection Angle Method Tangent Offset Method Laying Out a Horizontal Curve Deflection Angle Method Curve distances assumed to start at PC Deflection angle: angle between tangent and chord Laying Out a Horizontal Curve PI (V) Deflection angle between the tangent and chord is half the PC subtended arc (A) δ/2 q PT (B) δ Laying Out a Horizontal Curve PI (V) Angle between 2 chords is half the subtended arc r PC (A) PT (B) D/2 D Laying Out a Horizontal Curve Staking out points on the curve Need: Deflection angles measured from the PC (cumulative) Lengths of the chords joining consecutive stations First deflection angle (δ1) and last deflection angle (δ2) are usually < 1 station C1 = 2R sin (δ1/2) C2 = 2R sin (δ2/2) PI (V) ) I or Delta δ1/2 r q TC/PC (A) s t p δ1 D D D D δ2 CT/PT (B) Laying Out a HorizontalPICurve (V) l1 δ1 = Δ L or I = l2 δ2 δ1/2 D/2 D/2 Deflection angles from the PC VAp = δ1/2 p PC VAq = δ1/2 + D/2 (A) VAr = δ1/2 + D/2 + D/2 VAs = δ1/2 + D/2 + D/2 + D/2 VAt = δ1/2 + D/2 + D/2 + D/2 + D/2 δ1 VAB = δ1/2 + D/2 + D/2 + D/2 + D/2 + δ2/2 q r D/2 s D/2 t δ2/2 D D D D δ2 PT (B) Laying Out a Horizontal Curve Entire horizontal curve can be laid out from PC Sighting deflection angle PI-PC-A = δ/2 Taping chord distance PC-A PC and PT can be found by solving for T and starting at PI ? ?? ? ? Degree of Curve vs Chord Length Degree of Curve Arc Length (ft) Chord (ft) 2o 100 99.995 3o 100 99.989 6o 100 99.954 10o 100 99.873 Nominal Chord Lengths – Guidelines for curve layout Da Nominal Chord Length to Use < 3o Full Stations 3o – 7o Half Stations 7o – 14o Quarter Stations HC Layout example HC Layout example - detail Field Notes Layout Table Profile from Surveying Text