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Vertical Alignment

Vertical Alignment. CTC 440. Objectives. Understand the basics of profiles Understand the basics of vertical curves. Grades and Elevations. Grade-A change in elevation per unit horizontal length (+ or - % in direction of stationing)

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Vertical Alignment

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  1. Vertical Alignment CTC 440

  2. Objectives • Understand the basics of profiles • Understand the basics of vertical curves

  3. Grades and Elevations • Grade-A change in elevation per unit horizontal length (+ or - % in direction of stationing) • Elevations are usually computed to the nearest one-hundredth of a foot or to the nearest one-thousandth of a meter (43.32 ft; 21.262m)

  4. Examples • Computing a grade between two stations with known elevations • Computing an unknown elevation at a station given the known elevation at another station and given the grade between the stations

  5. Compute a grade between two pts w/ known elevations • Sta 1+00; Elev.=198.30’ • Sta 2+00; Elev.=203.80’ • Grade=change in elev/change in length • Answer: Grade=+5.50%

  6. Compute an unknown elevation at a station • Sta 5+30; Elev.=270.00’ • Grade between 5+30 & 7+70=-2.5% • Calculate elev at 7+70 • Answer: 264.00 feet

  7. Vertical Curves-Definitions • Vertical curves are parabolic curves instead of circular curves • Crest Curves (3 types) • Sag Curves (3 types)

  8. Other Vertical Curve Parameters • PVI-Point of Vertical Intersection-intersection of the two grades • PVC-Point of Vertical Curvature-beginning of the vertical curve • PVT-Point of Vertical Tangency-end of the vertical curve

  9. Other Vertical Curve Parameters • M,middle ordinate-A computed correction, which must be applied to the PVI to determine the elevation at the midpoint of the vertical curve • G1=grade before the PVC • G2=grade after the PVT • L=length of the vertical curve

  10. Vertical Curve Elevations • Need G1, G2, and L • A=G2-G1 (%) • r=A/100*L • Elevx=(r/2)x2+g1x+ElevPVC • g1 is in decimal form • X is distance measured from PVC

  11. Calculating Mand min/max elevations • M=A*L/800 • Max. or min. elev. Occurs @ x=-g1/r • g1 is in decimal form • To get the actual elevation substitute x into the elevation equation: • Elevx=(r/2)x2+g1x+ElevPVC

  12. Vertical Curve Examples • English (crest) ---see below • Metric (sag) ---on board if needed • Comprehensive Curve (sag)---see below

  13. English-Crest • A vertical crest curve with a length of 400’ is to connect grades of +1% and -1.75%. The PVI is located at station 35 and has an elevation of 549.20’. • What are the elevations of the PVC, PVT and at all full stations on the curve?

  14. English-Crest • Elev at PVC=549.2-(1%*200’)=547.20’ • Elev at PVT=549.2-(1.75%*200’)=545.70’ • Find A=-1.75-1=-2.75 (in %) • Find r=A/(100*L)=-.0000688 • Find r/2=-.0000344 • Find Elevations at even stations: • Elev(x)=r/2x2 + g1*x + Elev(PVC)

  15. English-Crest

  16. English-Crest • Highest Elev occurs @ x=-g1/r=145.35’ • Plug x in elevation equation to get: • Highest elevation is 547.93’ at Sta 34+45.35

  17. Comprehensive Curve Example • Finding a PVI • Fitting a curve • Finding elevations on the curve • Finding M • Finding low point

  18. Fitting a Curve Between Two Set Tangents (1/6) Given: • G1=-2%; G2=+3% Two Fixed Points (sta/elev is set): • Sta 1+00; Elev=450.00 • Sta 7+00; Elev=460.00

  19. Fitting a Curve Between Two Set Tangents (2/6) Find PVI Station and Elevation: • 450-.02*L1=460-.03*L2 • L1+L2=600’ Solve for L1 and L2 • L1=160 ft • L2=440 ft • PVI Sta=2+60 (Sta 1+00 + 160’) • PVI Elev=446.80’

  20. Fitting a Curve Between Two Set Tangents (3/6) If we’re constrained at Sta 1 and 7 then the maximum vertical curve length we can fit is 160’+160’=320’ L=320’ A=5% r=A/100L=.0001563 r/2=.0000781

  21. Fitting a Curve Between Two Set Tangents (4/6) Elev (x)=r/2*X2+g1X+El PVC

  22. Fitting a Curve Between Two Set Tangents (5/6) Determine Curve Elevation @ PVI M=AL/800=2’ PVI STA 2+60; Elev 446.8’ Curve Elev @ PVI=446.80+2’=448.80’

  23. Fitting a Curve Between Two Set Tangents (6/6) Determine Low Point X=-g1/r=.02/0001563=127.96 Sta=2+27.96 Elev @ x=127.96 =448.72’

  24. General Rules for Establishing Vertical Alignment • Goal-Provide a uniform, comfortable ride and safe vehicle operation • Balance cut/fills • Grades>=0.5% to prevent drainage problems • Check SSD/HSD • Check driveway and intersecting road tie-ins • Keep simple (few curves, flat curves, gradual grades) • Check clearances (over/under bridge, over culverts)

  25. Next lecture • Checking sight distances on horizontal curves • Checking sight distances on vertical curves • Crest-Stopping sight distance • Sag-Headlight sight distance

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