Vertical Curve Profile Views. Vertical Alignment Fundamentals. Offsets. Offset Formulas 9/17/2009

Transcription

1 9/17/009 Vertical Alignment Fundamentals Vertical Curve Profile Views CE 3 Transportation Engineering Dr. Ahmed Abdel-Rahim, Ph.D., P.E. Fig. 3.3 Fig. 3.4 Offsets Offsets are vertical distances from initial tangent to the curve Offset Formulas For an equal tangent parabola, A Y 00 L x ( G G1 ) 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 1 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 3 4 1

2 9/17/009 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 1% change in gradient The quantity L/A is termed K Distance to Zero Grade (low or high point) Computing high/low points for curves (provided the high/low point is not at a curve end) by, G1L x hl = K G1 or x hl G G1 Where x hl = distance from the PVC to the high/low point in feet Careful of units 1 st equation: if G 1 is in % then x hl is in feet if G 1 is in ft/ft then x hl is in stations nd equation: L can be in feet or stations and you will have x hl in similar units. 5 6 Example Problem A 1600-ft-long sag vertical curve (equal tangent) has a PVI at station and elevation 147 ft. The initial grade is 3.5% and the final grade is +6.5%. Determine the elevation and stationing of the low point, PVC, and PVT. Vertical Curve Through a Point Illustration Solution steps Use Equation 3.1 Determine parameters a, b, and c Substitute parameters into Equation 3.1 Solve for L as a quadratic equation Review problem Culvert clearance 7 8

3 9/17/009 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 10 SSD and Curve Design SSD formulation was given in Chapter and 3 Eq..50 ds = d + dr (Eq..50) V1 Eq. 3.1 SSD V1 tr a g G g Fig. 3.6 SSD Factors Important for crest curves Required sight distance Curve length Initial and final grades (which grade??) Eye and object heights SSD given in Table 3.1 using AASHTO values of a = 11. ft/s and tr =.5 sec

4 9/17/009 Minimum Curve Length Minimum curve length, based on parabola Using the equations L m A SSD Eq H1 H Eq H1 H L m SSD A for SSD L for SSD L Minimum Curve Length For adequate SSD use the following specifications: H1 (driver s eye height) = 3.5 ft (1080 mm) H (object height) =.0 ft (600 mm) Minimum Curve Length Substituting these values into previous two equations yields: For SSD < L For SSD > L US Customary A SSD L m L m SSD A Metric A SSD L m (3.15) L m SSD (3.16) A Example Problem 3.5 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 +1.0% and.0%. Determine the minimum length of vertical curve necessary to meet SSD requirements

5 9/17/009 Example Problem 3.5 Curve Where SSD > L Eq. 3.15, SSD < L Eq. 3.16, SSD > L Speed = 70 mph SSD = 730 ft Just reduce the final grade G1 = 1% G = -1% So A = (See Mathcad worksheet) Both are very similar, but choose because > SSD (730) Table 3. 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., 001. Notes About Both K-values Tables Because K = L/A L = SSD /158 (US Customary) For SSD < L Table 3., SSD calculations use G = 0 For larger grades > 3% calculate SSD Assume SSD < L Effects of assumption Equation for L m with SSD > L equal to or larger than other equation

6 9/17/009 Check SSD < L Assumption How would you test this assumption? (see equations 3.15 and 3.16) (see Matchcad worksheet) 6