# Horizontal Curves - Minnesota State University Moorhead

Horizontal Curves Circular Curves Degree of Curvature Terminology Calculations Staking Transition Spirals

Calculations Staking Circular Curves I Intersection angle Portion of a circle R - Radius I Defines rate of change R

Degree of Curvature D defines Radius Chord Method Arc Method R = 50/sin(D/2) (360/D)=100/(2R)

R = 5729.578/D D used to describe curves Terminology PC: Point of Curvature PC = PI T PI = Point of

Intersection T = Tangent PT: Point of Tangency PT = PC + L L = Length Curve Calculations L = 100I/D T = Rtan(I/2) L.C. = 2Rsin(I/2)

E = R(1/cos(I/2)-1) M = R(1-cos(I/2)) Curve Calcs - Example Given: D = 230 5729.578 R 2291.83' 2.5 22.5 T 2291.38 tan 455.87' 2 PC (175 50) ( 4 55.87) 170 94.13 22.5 L 100

900.00' 2.5 PT (170 94.13) (9 00) 179 94.13 Curve Calcs - Example Given: D = 230 R 2291.83' 22.5 L.C. 2(2291.83) sin 894.23' 2 22.5 M 2291.831 cos 44.04'

2 1 E 2291.83 1 44.90' cos 22.5 2 Curve Design Select D based on:

Highway design limitations Minimum values for E or M Determine stationing for PC and PT R = 5729.58/D T = R tan(I/2) PC = PI T L = 100(I/D)

PT = PC + L Curve Design Example Given: I = 7430 PI at Sta 256+32.00 Design requires D < 5 E must be > 315 Curve Staking

Deflection Angles Transit at PC, sight PI Turn angle to sight on Pt along curve Angle enclosed = Length from PC to Pt = l Chord from PC to point = c l l D

D, , 100 2 200 c 2 R sin 2 R sin( ) 2 Curve Staking Example D 2 30' , PC 170 94.13 l17100 5.87' , 5.87 2.5 0 04'24" 200 R 2291.83, c 2(2291.83) sin(0 4'24" ) 5.87' 105.87( 2.5 ) 17200

1 19'24" 200 c17200 2(2291.83) sin(1 19'24" ) 105.86' Curve Staking If chaining along the curve, each station has the same 100(c: 2.5 ) 100 1 15' 200 c100 2(2291.83) sin(1 15' ) 99.99' With the total station, find and c, use stake-out 405.87(2.5 ) 17500 5 04'24"

200 c17500 2(2291.83) sin(5 04'24" ) 405.34' Computer Example Moving Up on the Curve Say you cant see past Sta 177+00. Move transit to that Sta, sight back on PC. Plunge scope, turn 7 34 24 to sight on a tangent line. Turn 115 to sight on Sta 178+00.

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