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 G 1 =grade before the PVC G 2 =grade after the PVT L=length of the vertical curve

10. Vertical Curve Elevations Need G 1, G 2, and L A=G 2 -G 1 (%) r=A/100*L Elev x =(r/2)x 2 +g 1 x+Elev PVC g 1 is in decimal form X is distance measured from PVC

11. Calculating M and min/max elevations M=A*L/800 Max. or min. elev. Occurs @ x=-g 1 /r g 1 is in decimal form To get the actual elevation substitute x into the elevation equation: Elev x =(r/2)x 2 +g 1 x+Elev PVC

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/2x 2 + g 1 *x + Elev(PVC)

15. English-Crest XStar/2*X^2+G1*X+ PVC elev= Elev 0 33+00-.00003440+.010547.20 10034+00-.0000344100^2+.01100547.20547.86 20035+00-.0000344200^2+.01200547.20547.82 30036+00-.0000344300^2+.01300547.20547.11 40037+00-.0000344400^2+.01400547.20 545.70 (Check)

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 XSTAr/2X2G1XElev PVC Elev 01+00.00007810-.020450.00 1002+00.0000781100^2-.02100450.00448.78 1602+60.0000781160^2-.02160450.00448.80 2003+00.0000781200^2-.02200450.00449.12 3004+00.0000781300^2-.02300450.00451.03 3204+20.0000781320^2-.02320450.00451.60

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