PARABOLIC CURVES SYMMETRICAL (Recall) HORIZONTAL CURVES TYPES VERTICAL CURVES TYPES Simple Crest/Summit those which change the alignment of the road from downhill to uphill simple arc provided in the road to impose a curve between the two straight lines Compound combination of two simple curves combined together to curve in the same direction In designing crest vertical curves it is important that the grades be not] too high which makes it difficult for the motorists to travel upon it Reverse combination of two simple curves combined together to curve in opposite directions Sag change the alignment of the road from uphill to downhill Transition/Spiral curve that with varying radius SYMMETRICAL PARABOLIC CURVES VERTICAL CURVE are used to provide gradual change between two adjacent vertical grade lines PARABOLA provides a gradual change in direction along the curve because its second derivative is constant ELEMENTS OF A PARABOLIC CURVE PC (Point of Curvature) also known as Beginning of Vertical Curve (BVC) PT (Point of Tangency) also known as End of Vertical Curve (EVC) PI (Point of Intersection of Tangents) also known as Point of Vertical Intersection (PVI) L (Length of the Parabolic Curve) NOTE: +ππ Υ −π2 (downward) −ππ Υ +π2 (upward) projection of the curve onto a horizontal surface which corresponds to the plan distance A – grade change from PC to PT) π – (vertical distance between PC and PI) π – (vertical distance between PT and PI) H – (vertical distance between PI and the curve) πΊπ – horizontal distance from PC to the highest (lowest) point of the summit (sag) πΊπ – horizontal distance from PT to the highest (lowest) point of the summit (sag) π‘π – vertical distance between PC and the highest (lowest) point of the summit (sag) π‘π – vertical distance between PT and the highest (lowest) point of the summit (sag) ππ − grade (in percent) of back tangent (tangent through PC) ππ − grade (in percent) of forward tangent (tangent through PT) PROPERTIES OF A SYMMETRICAL PARABOLIC CURVE AND ITS GRADE DIAGRAM 1. The length of parabolic curve L is the horizontal distance between PC and PT. 2. PI is midway between PC and PT. πΈ 3. The curve is midway between PI and the midpoint of the chord from PC to PT. 4. The vertical distance between any two points on the curve is equal to area under the grade diagram. The vertical distance c = Area. ππΈ 5. The grade of the curve at a specific point is equal to the offset distance in the grade diagram under that point. The grade at point Q is equal to gQ. Considering the isolated figure below, πΏ π 2 1 π³ (π − ππ ) π π V π1 (%) π― ππ ππ π2 (%) π― PC π³/π ππ πΏ π 2 2 PT π³ π³ (π − ππ ) π π π― PC π³/π π³ PT π³/π ππ Applying the square property of parabola, π³ (ππ − ππ ) π― π = π³ π π³π ( π) π³ ππ― π (ππ − ππ ) = π³π π³π π³ π― = (ππ − ππ ) π Considering the isolated figure below, πΏ π 2 1 π³ (π − ππ ) π π V π1 (%) π― ππ ππ π2 (%) πΏ π 2 2 π― PC π³/π ππ ππ π― PC π³/π ππ π³/π ππ Solving for the vertical offset at any point on the curve using squared property of parabola, ππ π― π = π³ ππ ( )π π PT π³ PT π³ π― ππ ππ π― π = π³ ππ ( )π π π³/π ππ Considering the isolated figure below, Locating the highest(summit) or lowest point of the curve (for this case, the highest), ππ π³ πΊπ = ππ − ππ ππ π³ πΊπ = ππ − ππ measured from PC measured from PT PROB NO. 1 A parabolic curve has a descending grade of -0.8% which meets an ascending grade of 0.4% at sta 10 + 020. The max allowable change of grade per 20m station is 0.15. Elevation at station 10 +020 is 240.60m. a. What is the length of the curve? b. Compute the elevation of the lowest point of the curve c. Compute the elevation at Sta 10 + 000. PROB NO. 2 A symmetrical vertical summit curve has tangents of +4% and -2%. The allowable rate of change of grade is 0.3%per 20 meter station. Stationing and elevation of Pt is at 10 +020 and 142.63m respectively. a. Compute the length of curve. b. Compute the distance of the highest point of curve from pc c. Compute the elevation of the highest point of curve. PROB NO. 1 A parabolic curve has a descending grade of -0.8% which meets an ascending grade of 0.4% at sta 10 + 020. The max allowable change of grade per 20m station is 0.15. Elevation at station 10 +020 is 240.60m. a. What is the length of the curve? b. Compute the elevation of the lowest point of the curve c. Compute the elevation at Sta 10 + 000. PROB NO. 2 A symmetrical vertical summit curve has tangents of +4% and -2%. The allowable rate of change of grade is 0.3%per 20 meter station. Stationing and elevation of Pt is at 10 +020 and 142.63m respectively. a. Compute the length of curve. b. Compute the distance of the highest point of curve from pc c. Compute the elevation of the highest point of curve. PROB NO. 3 A vertical parabolic sag curve of Lapulapu underpass has a grade of -4% followed by a grade of 2% intersecting at station 12 + 150.60 at elevation 124.80m. above sea level. The change of grade of the sag curve is restricted to 0.6% a. Compute the length of curve b. Compute the elevation of the lowest point of the curve c. Compute the elevation at station 12 + 125.60. PROB NO. 4 A vertical summit parabolic curve has its PI at Sta 14 +750 with elevation 76.30m. The grade of the back tangent is 3.4% and the forward tangent of -4.8%. If the length of curve is 300m a. Compute the location of the vertical curve turning point from the PI b. Compute the elevation of the vertical curve turning point in meters c. Compute the stationing of the vertical curve turning point. PROB NO. 5 A horizontally laid circular pipe culvert having an elevation of its top to 26.0m crosses at right angles under a proposed 120m highway parabolic curve. The point of intersection of the grade lines is at station 5 + 216 and its elevation is 27.0m while the culvert is located at station 5 + 228. The backward tangent has a grade of 3% and the grade of the forward tangent is -1.6%. a. Compute the stationing of the highest point of curve. b. Compute the elevation of the highest point of curve. c. Under this conditions, what will be the depth cover over the pipe?
