Design Elements Vertical Milos N. Mladenovic Assistant Professor Department of Built Environment

Transcription

1 Design Elements Vertical Milos N. Mladenovic Assistant Professor Department of Built Environment

2 Outline Basic elements of roadway vertical profile design Basic parameters of a vertical curve Elevation of different points along a vertical curve Stopping sight distance for crest and sag curves Passing sight distance for crest curves 2

3 Highway Alignment in 2D 3

4 Design Steps/Data Examples Topographic (contour) maps Photogrammetric reconnaissance survey Identification of alternative alignments Preliminary selection of preferred alignment Design of final alignment Environmental and social impact assessment Detailed design and construction 4

5 Cut and Fill 5

6 Types of Vertical Curves 6

7 Vertical Curve G1 = initial roadway grade in percent or m/m (this grade is also referred to as the initial tangent grade, viewing previous figure from left to right) G2 = final roadway (tangent) grade in percent or m/m A = absolute value of the difference in grades (initial minus final, usually expressed in percent), A = G2 G1 PVC = point of the vertical curve (the initial point of the curve) PVI = point of vertical intersection (intersection of initial and final grades) PVT = point of vertical tangent, which is the final point of the vertical curve (the point where the curve returns to the final grade or, equivalently, the final tangent) L = length of the curve in stations or ft measured in a constant-elevation horizontal plane 7

8 Vertical Curve 8

9 Crest vs. Sag curve Crest curve can be formed in a transition point between: An ascent in a descent An ascent that goes from steep to shallow A descent that goes from a gradual to steep A sag curve can be formed at the transition point between: A descent to an ascent An ascent that goes from gradual to steep A descent that goes from steep to gradual 9

10 Quadratic Parabola Radius The choice of radius depends on: safety issues e.g., sight distance driving factors e.g., centrifugal acceleration aesthetic aspects e.g., sharp bends or uneven stretches 10

11 Parts of Vertical Projection Longitudinal profile of the terrain: height of the ground above the road axis Gradient (inclination): height changes along the road axis Incline graph: lengths of the road gradients and quadratic parabolas with determining factors Altitude data: altitude of the terrain and gradients at characteristic points from the horizontal projection Distance points: taking over distance points along the axis from the horizontal projection 11

12 Vertical Alignment Basics Grade tangents connected with parabolic vertical curves The length of a vertical curve is measured along the horizontal alignment The desirable maximum grade ranges 2% for freeways 6% for local streets Higher grades may be unavoidable combined effect of gradient and lengths (composite) length of grade at locations used by heavy vehicles 12

13 Percent Time Spent Following PTSF is the average percentage of travel time that vehicles must travel behind slower vehicles due to the lack of passing opportunities (because of geometry and/or opposing traffic). 13

14 Max Gradient according to RAS-L VE [km/h] sa [%] sbi, Bii [%]

15 Curve Equation Parabolic curves are generally used for design Parabolic function y = ax 2 + bx + c y = roadway elevation x = distance from PVC c = elevation of PVC Also usually design for equallength tangents i.e., half of curve length is before PVI and half after 15

16 First Derivative of Curve Equation First derivative gives slope dy dx 2ax b At PVC, x = 0, so b dy dx G 1 G1 is initial slope (in m/m) as previously defined 16

17 Second Derivative of Equation Second derivative gives rate of change of slope 2 d y 2a 2 dx However, the average rate of change of slope, by observation, can also be written as 2 d y G2 G1 2 dx L a G 2 G 2L 1 17

18 Offsets Offsets are vertical distances from initial tangent to the curve 18

19 Offset Formulas Examples For an equal tangent parabola, Y A 2 200L x Y = offset (in m or ft) at any distance, x, from the PVC A and L are as previously defined Y m AL 800 offset at the curve midpoint AL Y f offset at the end the curve

20 K Values The rate of change of grade at successive points on the curve is a constant amount for equal increments of horizontal distance K= L/A the horizontal distance required to effect a 1% change in gradient and is, therefore, a measure of curvature 20

21 K Values The K-value can be used directly to compute the high/low points for crest/sag vertical curves (provided the high/low point is not at a curve end) x hl = K G 1 Where x = distance from the PVC to the high/low point 21

22 Example A 520-ft long equal tangent crest vertical curve connects tangents that intersect at station and elevation 1325 ft. The initial grade is +4.0% and the final grade is 2.5%. Determine the elevation and stationing of the high point, PVC and PVT. 22

23 L = 520 ft PVI is at stapvi = elevpvi = 1325 G1 = 4% G2 = -2.5% A = = 6.5 Elevation and stationing at PVC, high point, and PVT? 23

24 stapvc = stapvi L/2 = = elevpvc = elevpvi (G1 * L/2) = High point is where 2ax + b = 0 a G 2 G 2L b = G1 = 0.04 xhigh = -b/2a = 320 = stahigh = stapvc + xhigh = = Y 1 A 2 200L x elevhigh = elevpvc + (G1 * disthigh) Yhigh =

25 stapvt = stapvi + L/2 = = Y f AL 200 elevpvt = elevpvc + (G1 * L) Yfinal = ft 25

26 Stopping Sight Distance When designing vertical curves we need to provide adequate stopping-sight distance (SSD), removing objects, embankments, or other restrictions Because curve construction is expensive, we want to minimize curve length, subject to adequate SSD SSD a = 3.4 m/s 2, t r =2.5s 2 V1 V1 a 2g G g t r 26

27 SSD Factors Length of the curve is a critical element for providing SSD Two different factors are important for crest curves The driver s eye height in vehicle, H 1 = 1.0 m Height of a roadway obstruction object, H 2 = 0.6 m 27

28 Minimum Curve Length for Crest Curve By using the properties of a parabola for an equal tangent curve, it can be shown that the minimum length of curve, L m, for a required SSD is L m A SSD 200 H 1 H H1 L m 2 SSD 2 2 A for SSD L H 2 2 for SSD L * Grade often not accounted for unless >3% 28

29 Design Control Table for Crest Curve Assuming L > SSD is a safe, conservative, assumption Knowing K and A obtain Lm = K*A 29

30 Design Factors for Sag Curve Since SSD is unrestricted on sag curves during daylight hours, nighttime conditions govern design The critical concern for sag curves is the headlight sight distance (area of headlight illumination) This length is a function of the height of the headlight above the roadway, H, and the inclined upward angle of the headlight beam, β 30

31 Minimum Curve Length for Sag Curve SSD < L L m 200 AS H S 2 tan β SSD > L L m 2S 200 H S A tan β H = 0.6 m β = 1ᵒ 31

32 Minimum Sag Curve Length and Design Control 32

33 Sag Curve Design at Specific Locations Obstacle h 1 = 3,0m H h 1 = 0.35m 33

34 Passing Sight Distance the initial maneuver distance (which includes drivers' perception/reaction time and the time it takes to bring the vehicle from its trailing speed to the point of encroachment on the left lane), the distance that the passing vehicle traverses while occupying the left lane, the clearance length between the passing and opposing vehicles at the end of the passing maneuver, and the distance traversed by an opposing vehicle during two-thirds of the time the passing vehicle occupies the left lane. 34

35 Design Control for Passing Sight Distance - AASHTO *Assumption L > PSD Design speed (mi/h) Passing sight distance (ft) Rate of vertical curvature, K*

36 Passing Sight Distance - RAS 36

37 Passing Sight Distance - RAS s1 passing sight distance [m] v2 speed of vehicle being passed [km/h] a1 acceleration of the passing vehicle [m/s^2] a2 deceleration of the passing vehicle [m/s^2] l1 length of the passing vehicle [m] l2 length of the vehicle being passed [m] d = sa + sb + l1 + l2 [m] sa = sb = v1 * ta ta action time, 1 second 37

38 SSD vs. PSD 38

39 Example 1 An equal tangent vertical curve has an initial grade of -2.5%. It is known that the final grade is positive, and that the low point is at the elevation 270 ft, and stationing The PVT of the curve is at the elevation 274 ft, and the design speed for the curve is 35 mi/h. Determine the station and elevation of the PVC and PVI. 39

40 Xlow from PVC = K * abs(g1) = 49 * 2.5 = ft PVCsta = Xlow sta = = y = ax 2 + bx + c b = G1 = a G G 2L A/100 2KA K

41 y = 270 = *(122.5) 2 + (-0.025)(122.5) + c c = elevation of PVC At PVT, added x is L y = al 2 + bl + c 274 = *L 2 + (-0.025)L L = ft PVIsta = PVCsta + L/2 = ( /2) = PVIelev = PVCelev + G1*(L/2) =

42 Example 2 A tangent section of highway has a 1.0% grade and ends at station and elevation 82 ft. It must be connected to another section of highway that has a 1.0% grade, and begins at station and elevation ft. The connecting alignment should consist of a sag curve, constant-grade section, and crest curve, and be designed for a speed of 50 mi/h. What is the lowest grade possible for the constantgrade section that will still complete this alignment? 42

43 Y fc Δy c Δy con Δelev Y fs Δy s Δy c + Δelev + Δy s = Y fs + Δy con + Y fc

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