Horizontal and Vertical Curve Design

Transcription

1 Horizontal and Vertical Curve Design CE 576 Highway Design and Traffic Safety Dr. Ahmed Abdel-Rahim Horizontal Alignment Horizontal curve is critical. Vehicle cornering capability is thus a key concern in horizontal curve design Vehicle Cornering Example A roadway is being designed for a speed of 70 mi/h. At one horizontal curve, it is known that the superelevation is 8.0% and the coefficient of side friction is 0.0. Determine the minimum radius of curve (measured to the traveled path) that will provide for safe vehicle operation. forces acting on a vehicle during cornering. V (70*.467) Rv ft e g( f ) 3.( ) s 00

2 Insights on Minimum Radius Selected value of e is critical Because high rates of superelevation vehicle steering problems in cold climates, ice on the roadway can reduce f s. R min obtained with Maximum e Assuming design speed Maximum side friction (f s ) V Rv e g( fs ) 00 AASHTO guidelines for selecting values of e and f s. Curve Options There are 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. The following figure shows the basic elements of a simple circular curve. Simple Circular Horizontal Curve Simple Circular Horizontal Curve Fig. 3.3 R = radius, usually measured to the centerline of the road, in ft (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 ft (m), M = middle ordinate in ft (m), E = external distance in ft (m), and L = length of curve in ft (m).

3 Circular Curve Formulas D 8000 R R T R tan E R cos( / ) M R cos L R 80 Degree of curve: Angle subtended by a 00-ft arc along the horizontal curve. Fig. 3.3 Vertical Curve Profile Views 0 Notation Notation (cont.) Curve point PVC: Point of Vertical Curvature PVI: Point of Vertical Intersection PVT: Point of Vertical Tangency Curve positioning and length usually referenced in stations Stations represent 000 m or 00 ft e.g., 58.5 ft (i.e., stations & 58.5 ft) G = initial tangent grade G = final tangent grade A = the absolute value of the difference in grades (generally expressed in percent) A = G G L = the length of the vertical curve measured in a horizontal plane 3

4 Fundamentals Parabolic curves are generally used for design Parabolic function y = ax + bx + c y = roadway elevation x = distance from PVC c = elevation of PVC Also usually design for equal-length tangents i.e., half of curve length is before PVI and half after Quiz Question Given the following, determine the distance from the PVC to the lowest point on the curve (given equation for y(x) L = 00 G = -7% G = % PVI elev = 00 ft PVI sta = First Derivative of Function y = ax + bx + c dy/dx = ax + b low or high point x = -b/(a) = x hl Because dy/dx is the curve grade At PVC, x = 0 and curve grade is G G = a(0) + b b = G Second Derivative of Function Second derivative of y slope rate of change (note that it is constant) Alternative form for slope rate of Gchange, G a L Thus, and d y a dx d y G G dx L x hl G L G G 5 Substituting for a and b 6 4

5 Grade Variation w.r.t. x Offsets Say: a = -0.45; b = 3%; c = 500 ft; PVC PVT G Offsets are vertical distances from initial tangent to the curve Set: G(x) = dy/dx dy/dx = ax + b G Input values for a and b: G(x) = -0.9(x) + 3 (x is in stations) Variation of G(x) Slope of G(x) Grade at high pt. 7 8 Offset Formulas For an equal tangent parabola, A Y 00L x ( G G ) OR Y ax x L Y = offset (ft) at any distance, x, from the PVC x, A, and L are as previously defined careful with units st equation: if A is in %...x and L should be in feet If A is in ft/ft L should be in stations and x in feet nd equation: If grade is in %...x and L should be in station If grade is in ft/ft x and L should be in feet Vertical Curve Offsets Offsets are vertical distances from initial tangent to the curve 9 0 5

6 Offset Formulas For an equal tangent parabola, A Y 00L x ( G G ) OR Y ax x L Y = offset (ft) at any distance, x, from the PVC x, A, and L are as previously defined careful with units st equation: if A is in %...x and L should be in feet If A is in ft/ft L should be in stations and x in feet nd equation: If grade is in %...x and L should be in station If grade is in ft/ft x and L should be in feet K Values Rate of change in grade at successive points on the curve is Constant = L/A in percent per ft L/A distance required per % change in gradient The quantity L/A is termed K SSD and Crest Vertical Curve Design SSD and Curve Design Design vertical curves, to provide adequate stopping-sight distance (SSD) Minimize costs by minimizing curve length 4 6

7 SSD and Curve Design SSD V V a g G g t r SSD Factors Important for crest curves Required sight distance Curve length Initial and final grades (which grade??) Eye and object heights SSD using AASHTO values of a =. ft/s and tr =.5 sec 5 6 Minimum Curve Length Minimum curve length, based on parabola Using the equations L m L m A SSD 00 H H 00 SSD H H A for SSD L for SSD L Minimum Curve Length For adequate SSD use the following specifications: H (driver s eye height) = 3.5 ft (080 mm) H (object height) =.0 ft (600 mm) 7 8 7

8 Minimum Curve Length Substituting these values into previous two equations yields: Example Problem A highway is being designed to AASHTO guidelines with a 70-mph design speed and, at one section, an equal tangent vertical curve must be designed to connect grades of +.0% and.0%. Determine the minimum length of vertical curve necessary to meet SSD requirements Example Problem SSD < L SSD > L K Values for Adequate SSD Design Controls for Crest Vertical Curves Based on SSD US Customary Metric Stopping Rate of vertical Stopping Rate of vertical Design Design sight curvature, K a sight curvature, K a speed speed distance distance (mi/h) Calculated Design (km/h) Calculated Design (ft) (m) a Rate of vertical curvature, K, is the length of curve per percent algebraic difference in intersecting grades (A). K = L/A Source: American Association of State Highway and Transportation Officials, A Policy on Geometric Design of Highways and Streets, Washington, D.C.,

9 SSD and Sag Vertical Curve Design Sag Curve Design Factors Because SSD is unrestricted on sag curves during daylight hours, nighttime conditions govern design Thus, the critical concern for sag curves is the length of road illuminated by the vehicle s headlights, Function of the height of the headlight above the roadway, H, and upward angle of the headlight beam, b 35 Sag Curve (Profile View) Minimum Curve Length Like crest curves, equations for L m A SSD L m for SSD L 00H SSD tan b 00H SSD tan b L m SSD for SSD L A

10 Minimum Curve Length Current AASHTO design standards use the following specifications: H (headlight height) =.0 ft (600 mm) b (headlight angle) = Minimum Curve Length Substituting these values into the previous two equations yields: 38 If not sure which equation to use, assume SSD < L first (for either sag or crest curves) 39 K Values for Adequate SSD Passing Sight Distance Design Controls for Sag Vertical Curves Based on SSD US Customary Metric Stopping Rate of vertical Stopping Rate of vertical Design Design sight curvature, K a sight curvature, K a speed speed distance distance (mi/h) Calculated Design (km/h) Calculated Design (ft) (m) a Rate of vertical curvature, K, is the length of curve per percent algebraic difference in intersecting grades (A). K = L/A Source: American Association of State Highway and Transportation Officials, A Policy on Geometric Design of Highways and Streets, Washington, D.C., 00. Same assumptions as crest curve Set S = PSD, instead of SSD

11 Underpass Sight Distance A structure passing over a sag curve may block a driver s line-ofsight over the full length of the curve. Example Problem 3 An existing tunnel needs to be connected to a newly constructed bridge with sag and crest vertical curves. The profile view of the tunnel and bridge is shown in the Figure. Develop a vertical alignment to connect the tunnel and bridge by determining the highest possible common design speed for the sag and crest (equal tangent) vertical curves needed. Compute the stationing and elevations of PVC, PVI, and PVT curve points Example Problem 3 L L 00 or K A K A 00 s c ALS ALC Y s Y c 40 or A ( L S L C ) 40 A K S K C 80 s c Find the highest possible design speed that would satisfy this condition 47