09 - Chapter-9 - Geometric Design Manual-2002 Chapter 9...

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Chapter 9 Geometric Design Manual-2002 Vertical Alignment Ethiopian Roads Authority Page 9-1 9V ERTICAL A LIGNMENT 9.1 Introduction The two major aspects of vertical alignment are vertical curvature, which is governed by sight distance criteria, and gradient, which is related to vehicle performance and level of service. The following text gives the formula and features of the vertical curve; gives values for maximum and minimum gradients; indicates gradient requirements through villages; develops the criteria for incorporation of a climbing lane; and provides vertical clearance standards. 9.2 Vertical Curve Formula Vertical curves are required to provide smooth transitions between consecutive gradients. The simple parabola is specified for these. The parabola provides a constant rate of change of curvature, and hence acceleration and visibility, along its length and has the form: elevation of BVC Where r = rate of change of grade per section (%) g1 = starting grade (%) g2 = ending grade (%) L = length of curve (horizontal distance m) y = elevation of a point on the curve x = distance in stations from the BVC (meters/100) BVC = beginning of the vertical curve EVC = end of the vertical curve A related formula is: Where y = vertical distance from the tangent to the curve (meters) x = horizontal distance from the start of the vertical curve (meters) G = algebraic difference in gradients (%) L = length of vertical curve (meters). Examples of crest and sag vertical curves are shown in Figures 9-1 and 9-2, respectively. L g g r 1 2 = + + = x g 2 rx y 1 2 2 L x 200 L * G y » ¼ º « ¬ ª =
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Chapter 9 Vertical Alignment Geometric Design Manual- 2002 Page 9-2 Ethiopian Roads Authority Figure 9-1: Crest Curve Figure 9-2: Sag Curve Example: two grade lines intersect at Station 2+200 where the point of vertical intersection (PVI) elevation is 239.5 m. The starting grade is –6 percent and the ending grade is +2 percent. The length of curve is 400 m. Compute the elevation at station 2+200. Elev (B.V.C.) = 239.5 + 0.06(200) = 251.5 % 00 . 2 400 ) 6 ( 2 L g g r 1 2 + = = = 000 2 2 400 200 2 BVC + = + = 200 000 2 200 2 x = + + = 5 . 243 5 . 251 ) 2 )( 6 ( 2 ) 2 ( 02 . 0 y 2 = + + =
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Chapter 9 Geometric Design Manual-2002 Vertical Alignment Ethiopian Roads Authority Page 9-3 9.3 Crest and Sag Curves The formulae for design of crest and sag vertical curves can be rather complex to apply, and thus the design is best accomplished through the application of a computer program, or by use of design charts. Figures 9-4 and 9-5 show the minimum length requirements for crest and sag curves, respectively, for differing design speeds and algebraic differences in grade. Example: Starting grade = -6%, ending grade = +2%, design speed = 100 km/hr. Algebraic difference in grade = 8%, sag curve. From Figure 9-2: 400-meter length The minimum lengths of crest and sag curves have been designed to provide sufficient stopping sight distance. The design is based on minimum allowable "K" values, as defined by the formula: K=L/A Where K = limiting value, horizontal distance required to achieve a 1% change in grade L = length of vertical curve (m) A = Algebraic difference in approach and exit grades (%)
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09 - Chapter-9 - Geometric Design Manual-2002 Chapter 9...

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