Design of Highway Vertical Alignment Chapter 16 Dr. TALEB M. AL-ROUSAN

Introduction The vertical and the horizontal layout of the highway make up the alignment. The design of alignment depends primarily on the design speed selected for the highway. The least cost alignment is one that generally takes the form of the natural topography. Often this is not possible, because designer has to adhere to certain standards that may not exist on the natural topography. Its important that the alignment has consistent standards to avoid sudden changes in the vertical and horizontal layout of the highway.

Introduction Cont. Horizontal and vertical alignment should be designed to complement each other for safe and more attractive highways. Proper balancing of the grades of tangents with curvatures of horizontal curves and the location of horizontal and vertical curves with respect to each other. Example on poor design: sharp horizontal curves placed at or near the top of crest curve or bottom of sag curve will create hazardous section of the highway.

Vertical Alignment Consists of straight sections of the highway known as grades, or tangents connected by vertical curves. The design of vertical alignment involves selection of suitable grades for the tangent sections and the design of the vertical curves. Topography of the area which the road traverses has a significant impact on the design of the vertical alignment.

Vertical Alignment/ Grades Grades affect the performance of vehicles (reduce speed) specially for trucks. Max grades on highways should be selected to limit grade effect on vehicle performance. Selection of max grade depends on Design speed. Design vehicle 4-5% grades have little or no effect on PC. Studies showed that speeds may increase up to 5% on downgrades and decrease by 7% on upgrades depending on percent and length of grade.

Vertical Alignment/ Grades Maximum grades have been established, based on operating characteristics of design vehicle. Max grade vary from 5% for DS of 70 mi/h to 12% for DS of 30 mi/h. See Table 16.4 for recommended values of max grades. Max grades in the Table should not be used frequently, particularly when grades are long and traffic include high % of trucks. on low volume rural highways, maximum grades may be increase by 2% When : grade length (< 500 ft) and roads are one-way in the downgrade direction,

Vertical Alignment/ Grades Minimum grades depend on the drainage condition of the highway. 0% grades may be used on uncurbed pavements with adequate cross-slopes to laterally drain the surface water. When pavements are curbed, longitudinal flow should be provided to facilitate the longitudinal flow of surface water. Min of (0.5%).

Vertical Curves Used to provide gradual change from one tangent grade to another so that vehicle may run smoothly as they traverse the highway. Usually parabolic in shape Classes or types (see Figure 16.11) Crest vertical curves Sag vertical curves Main criteria for design are: Provision of min stopping sight distance (associated with crest vertical curves only) Adequate drainage Comfortable in operation Pleasant appearance All four criteria are associated with sag vertical curves.

Crest Vertical Curves Conditions for min length: When sight distance > length of curve. When sight distance < length of curve. For the first condition (see Figure 16.12) For second condition (see Figure 16.13)

Crest Vertical Curves When (S > L) L min = 2S –[(200 (SQR(H1) + SQR(H2))2/ A] L: length of vertical curve (ft) S: sight distance (ft) = SSD = 1.47 u t + [u2 / (30 [(a/g) ± G)])] H1: height of eye above road surface = 3.5 ft H2: height of object above road = 2.0 ft G1: Slope of first tangent G2: Slope of second tangent A : grades algebraic difference = Abs [G1 – G2] PVC: Point of vertical curve PVI :Point of vertical intersection PVT: Point of vertical tangent L min = 2S – [2158/A] For (S>L)

Crest Vertical Curves When (S < L) L min = (A S2)/[(200 ((SQR(H1) + SQR(H2))2] L: length of vertical curve (ft) S: sight distance (ft) = SSD = 1.47 u t + [u2 / (30 [(a/g) ± G)])] H1: height of eye above road surface = 3.5 ft H2: height of object above road = 2.0 ft G1: Slope of first tangent G2: Slope of second tangent A : grades algebraic difference = Abs [G1 – G2] PVC: Point of vertical curve PVI :Point of vertical intersection PVT: Point of vertical tangent L min = (A S2)/2158 For (S< L)

Sag Vertical Curves Selection of min. length of sag vertical curve is controlled by: Sight distance provided by the headlight. Rider comfort Control of drainage General appearance see Figure 16.14) for headlight sight distance on sag vertical curve with (S>L).

Sag Vertical Curves Min. Length When S > L L min = 2S –[(200 (H1 + S tan β) / A] H= height of headlight = 2 ft (β) is angle of upward inclination of the headlight beam = 1o L min = 2S –[(400 + 3.5 S) / A] for (S>L) When S < L L min = (A S2)/[(200 (H + S tan β] L min = (A S2)/(400 + 3.5 S] for (S<L)

Sag Vertical Curves Cont. To provide safe conditions on a sag curve, the length of curve must be such that the light beam sight distance (S) be at least equal to the SSD. The comfort criterion for the design of sag curves require that both gravitational and centrifugal forces act in combination . Comfortable ride will be provided if the radial acceleration is not greater than 1 ft/sec2. L min/ comfort = (A U2) / 46.5 U: design speed in mi/h

Sag Vertical Curves Cont. The drainage criterion for sag curves is essential when the road is curbed. Maximum length is given here rather than min. To satisfy drainage criterion a min. grade of (0.35%) be provided within (50 ft) of the level point of the curve. Max length > min. lengths for other criteria for speeds up to 60 mph and = min. length for a 70 mph. The criterion of general appearance is satisfied by: L min /appearance = 100 A

Elevation of Crest Vertical Curves L min = 2S – [2158/A] For (S>L) L min = (A S2)/2158 For (S< L) when S< L, L min can be rewritten as: L min = KA For (S< L) K = length of vertical curve per percent change in A. K= rate of vertical curvature See Table 16.5 for design controls of crest vertical curves. It has been found that when S> L, L min can are not practical design values and generally not used. General practice for the S>L case is to: set min. limits ranging from (100 – 325 ft) or min length can be set at (3 times) design speed.

Elevation of Crest Vertical Curves Cont. After determining the length of the crest vertical curve. The elevation of the curve at regular intervals can then be determined. Done by considering the properties of the parabola (Y = a X2) See Fig. 16.15 Note: in the figure, the length of the vertical curve is the its horizontal projection not the length along the curve (L = T1 + T2 = 2T)

Elevation of Crest Vertical Curves Cont. (L = T1 + T2 = 2T) From properties of a parabola with (Y = a X2), the rate of change of slope (second derivative) = 2a…………. a is constant If the total change in slope is A, then 2a = A/(100L)… a = [A/ (200 L)] Curve equation: Y = [A/(200 L)] X2 When x=L/2 , the external distance (E) from PVI to the curve is found by: E = [A/(200L)](L/2)2 = [AL /800] Since stations are given in 100 ft intervals E = AN/8…… N= length of the curve in stations. Any point on the curve can be given in terms of E, by substituting A=800E/L in the curve equation Y = E [X/( L/2)]2

Location of high Point (Turning Point) of Crest Vertical Curves High point (turning point) is of interest to the designer because of drainage requirements. Xhigh = distance from BVC to the turning point Xhigh = [(100L)/(G1-G2)](G1/100) Xhigh = [LG1/ (G1-G2)] Yhigh = difference in elevation between BVC and the turning point. Yhigh = (LG12)/(200(G1-G2))

Procedure for the Design of Crest Vertical Curve Determine min. length of curve to satisfy sight distance (S>L or S<L). Determine from layout plans the station and elevation of the PVI. Compute elevations of the BVC and end of vertical curve EVC. Compute offsets (Y) from the tangent to the curve at equal distance (100-ft) beginning with the first whole station after the BVC. Compute elevations on the curve. See Example 16.4 for design of crest vertical curve.

Elevation of Sag Vertical Curves Same format as crest vertical curves. The headlight sight distance requirement is used for design purposes since SSDs based on it are generally within limits of accepted values. L min = 2S –[(400 + 3.5 S) / A] for (S>L) L min = [(A S2)/(400 + 3.5 S)] for (S<L) See Table 16.7 for design controls of sag curves which are expressed in terms of K (rate of vertical curvature). Computation of the elevation at different points on the curve takes the same form as for the crest vertical curve. But, offset (Y) is added to tangent elevation since the curve elevation is higher than the tangent. See Example 16.5 for design of sag vertical curve.