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Unformatted text preview: The Pennsylvania State University Department of Civil and Environmental Engineering CE 321: Highway Engineering Class 19 – Vertical Curvature (Part 1) Spring 2008 Vertical Curves Crest Curve Design Sag Curve Design Types of Vertical Curves Vertical Curve Fundamentals y = ax 2 + bx + c assuming a constant rate of change of slope and equal tangent lengths. When x = 0, y = C = elevation on curve dy / dx = 2ax + b = slope (rise / run) at x = 0 => slope is dy/dx = b = G 1 (the initial slope) Vertical Curve Fundamentals d 2 y/dx 2 = 2a = rate of change of slope 2a = (G 2 G 1 ) / L a = (G 2 G 1 ) / 2L therefore: y x = a x 2 + G 1 x + Elevation at PVC Where: L (sta.), x (sta.) High / Low Point on Curve When dy/dx = 0 then a = (G 2 G 1 ) / 2L therefore, dy/dx = 2ax + b = 0 2ax + G 1 = 2ax =  G 1 2(G 2 G 1 )(x)/2L =  G 1 then: x =  G 1 L / (G 2 G 1 ) Vertical Curve Design Vertical Curves maximum grades depend on: – Design Speed – Type of Terrain – Type of Highway – Length of Grade Vertical Curve Design Grades also affect: – fuel consumption – speed – safety or crashes (speed differential) Vertical Curve Design Design Speed Flat / Rolling Mountainous 50 mi/hr 5% 7% 60 mi/hr 4% 6% 70 mi/hr 3% 5% (+ 2 % for secondary / local roads) Example Problem 1 A 600 ft equal tangent sag vertical curve has the PVC at station 170+00 and elevation 1000 ft. The initial grade is 3.5 % and the final grade is 0.5 %....
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This note was uploaded on 06/08/2009 for the course CE 321 taught by Professor Petrucha during the Spring '02 term at Penn State.
 Spring '02
 Petrucha
 Environmental Engineering

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