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

Transcription

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

2 Outline Highway alignment Vehicle cornering forces Minimum radius Circular curve elements Transition curve Design horizontal curve for stopping sight distance

3 Highway Alignment in 2D 3

4 Highway Alignment in 3D 4

5 Highway Positioning The length of the facility is measured along the horizontal alignment of a control line usually the center line of a highway and usually expressed in terms of 100-m stations from a reference point. 5

6 Straight Segments + Shortest connection between two points + Traffic aspects (overtaking space, arrangement of intersections) + Grouping of different transport modes (e.g., road and rail) + Landscape areas (flat valleys) - Incorrect assessment of oncoming and following traffic speed - Risk of drivers being blinded at night by oncoming vehicles - Fitting the straight line into the hilly landscape - Risk of drivers being tired because of monotonous driving style 6

7 Horizontal Curve Horizontal curve transitions the roadway between two straight (tangent) sections Main concern is safety, but designers are also concerned with drainage Other aspects: costs, comfort, environmental protection Vehicle cornering capability is a key concern in horizontal curve design 7

8 Vehicle Cornering 8

9 Forces acting on a vehicle during cornering 9

10 Vehicle Cornering Forces R v = radius defined to the vehicle s traveled path in m = angle of incline in degrees e = number of vertical m of rise per 100 m of horizontal distance W = weight of the vehicle in N W n = vehicle weight normal to the roadway surface in N W p = vehicle weight parallel to the roadway surface in N F f = side frictional force (centripetal, in N) F c = centripetal force (lateral acceleration x mass, in N) F cp = centripetal force acting parallel to the roadway surface in N F cn = centripetal force acting normal to the roadway surface in N K = curvature = C (scale factor) * 1 / R 10

11 Vehicle Cornering Some basic horizontal curve relationships can be derived by noting that: W p + F f = F cp From basic physics this equation can be written as [with Ff = fs(wn + Fcn)]: W sin f s W WV cos gr v 2 sin WV gr v 2 cos f s is the coefficient of side friction 11

12 Vehicle Cornering Dividing both sides of the previous equation by W cos yields: 2 V tan f s 1 f s tan gr v The term tan is referred to as the superelevation of the curve and is denoted e (e = 100 tan ). Being conservative and ignoring the normal component of centripetal force (f s tan ), and with e = 100 tan, the above equation can be rearranged as follows: 2 V Rv e g( fs )

13 Example 1 A roadway is designed for a speed of 110 km/h. At one horizontal curve, it is known that superelevation is 4% and that coefficient of side friction is Determine the minimum radius curve that will provide for the safe vehicle operation.! Minimum radius does not represent the desired design radius. From the perspective of the driver, the longer the radius the better. 13

14 Minimum Radius Selected value of e is critical as high rates of superelevation can cause vehicle steering problems on the horizontal curve On the contrary, in cold climates, ice on the roadway can reduce f s such that vehicles traveling less than the design speed on an excessively superelevated curve could slide inward off the curve by gravitational forces Usually, e is 10% but in cold climates 8% is recommended Selecting a superelevation, e, a design speed, V, and using maximum side friction, f s, a minimum radius is obtained Following table gives AASHTO guidelines for selecting values of e and f s 14

15 AASHTO Guidelines Example 15

16 Curve Options There a few options available for curve types to connect tangent sections: Simple circular curve Reverse curves Compound curve Spiral curve The circular curve has a single, constant radius. 16

17 Simple Circular Horizontal Curve R = radius, usually measured to the centerline of the road, in m = central angle of the curve in degrees PC = point of curve (the beginning point of the horizontal curve) PI = point of tangent intersection PT = point of tangent (the ending point of the horizontal curve) T = tangent length in m M = middle ordinate in m E = external distance in m L = length of curve in ft (m) 17

18 Circular Curve Formulas D T R R R tan 2 Degree of curve: Angle subtended by a 100-m arc along the horizontal curve. Measure of sharpness of the curve. Tangent length 1 E R cos( / 2) 1 M R1 cos 2 L R 180 External distance Middle ordinate Curve length * Measured from the centerline of the road 18

19 Min Radii vs. Min Curve Length Example RAS-L V [km/h] Min R [m] Min L [m]

20 Coordinating Sequence of Radii Example Radii of consecutive curves must be in balanced ratio Radius after straight segment: If L >= 300 m Then min R > 400 m If L < 300 m Then min R > L 20

21 Driving Line vs. Curvature Graph Smoot movement of the steering wheel? Practical vs. theoretical driving line Aesthetics sharp bends Transition to greater camber going towards the inner side of the bend 21

22 Euler Spiral and Transition Curve Has constantly changing curvature Used for developing transition curve Transition curve connects straight-line segments (tangents) with circular curves Transition curves are usually introduces on highspeed sections Clothoid parameter A = sqrt ( L * R) 22

23 Horizontal Projection vs. Clothoid Curvature The curvature alters in a linear fashion with the arc length The driver can turn steering wheel with a constant angular velocity 23

24 From the Driver s Seat 24

25 Development of Superelevation Banking of the cross section is needed on the curved portion of the facility but it is not necessary along the tangent segments of the horizontal alignment AB tangent runout BE superelevation runoff 25

26 Stopping Sight Distance (SSD) Sight distance restrictions on horizontal curves occur from obstructions (e.g., buildings, rock outcroppings) When such an obstruction exists, the stopping-sight distance is measured along the horizontal curve from the center of the traveled lane (the assumed location of the driver s eyes). For a specified stopping distance, some distance, M s (the middle ordinate of a curve that has an arc length equal to the stopping sight distance), must be visually cleared so that the line of sight is such that sufficient stopping-sight distance is available. 26

27 SSD and Clearance 27

28 SSD Assumptions Equations for computing stopping-sight distance relationships for horizontal curves can be derived by first determining the central angle, s, for an arc equal to the required stopping-sight distance (this is not the same as the central angle). Assumption is that the length of the horizontal curve exceeds the required SSD. 28

29 SSD Equations SSD R 180 v s s 180SSD R v Substituting into the general equation for the middle ordinate of a simple horizontal curve gives: M s R v 1 cos 90SSD R v Solving the above equation for SSD gives: SSD R 90 v cos 1 R v M R v s 29

30 Example 2 A horizontal curve is being designed for a new twolane highway (12-ft lanes). The PI is at station , design speed is 65 mi/h, and a maximum superelevation of 0.07 ft/ft is to be used. If the central angle of the curve is 38 degrees, design a curve for the highway by computing the radius and stationing of the PC and PT? 30

31 Example 2 statpi = g=32.2 V=65 e = 0.07 Δ = 38 fs = 0.11 Rv = (V 1.467)2 g(e+fs) = Since the road is two-lane with 12 ft lanes R = Rv + 6 = L = R tan[(δ/2)deg] = statpc = statpi T = statpt = statpc + L =