1. CTC 440 Vertical Alignment

2. Objectives
What is a profile?
What are 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; m)

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: 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 M and min/max elevations
M=A*L/800 (A is in %)
Max. or min. elev. 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 ’.
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= =-2.75 (in %)
Find r=A/(100*L)=
Find r/2=
Find Elevations at even stations:
Elev(x)=r/2x2 + g1*x + Elev(PVC)

15. English-Crest X Sta r/2* X^2+ G1* X+ PVC elev= Elev 33+00 -.0000344
33+00
+.01
547.20
100
34+00
100^2
547.86
200
35+00
200^2
547.82
300
36+00
300^2
547.11
400
37+00
400^2
545.70
(Check)

16. English-Crest (highest elev)
Highest Elev x=-g1/r=145.35’
Plug x in elevation equation to get:
Highest elevation is ’ at Sta

17. English-Crest (using M to calculate curve elev at PVI Station
M=AL/800=-2.75*400/800=-1.375’
At PVI STA 35 the elevation = ’
The elevation on the curve at STA 35 = ’-1.375’=547.82’ (checks w/ Table value)

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

19. 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

20. Fitting a Curve Between Two Set Tangents (2/6)
Find PVI Station and Elevation:
*L1= *L2
L1+L2=600’
Solve for L1 and L2
L1=160 ft
L2=440 ft
PVI Sta=2+60 (Sta ’)
PVI Elev=446.80’

21. 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= r/2=

22. Fitting a Curve Between Two Set Tangents (4/6)
Elev (x)=r/2*X2+g1X+El PVC X
STA
r/2 X2 G1
Elev PVC
Elev
1+00
-.02
450.00
100
2+00
100^2
448.78
160
2+60
160^2
448.80
200
3+00
200^2
449.12
300
4+00
300^2
451.03
320
4+20
320^2
451.60

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

24. Fitting a Curve Between Two Set Tangents (6/6)
Determine Low Point X=-g1/r=.02/ = Sta= x= =448.72’

25. 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)

26. Review Questions With respect to vertical alignments: What is a PVC?
What is a PVI?
What is a PVT?
What is M?
What is G1 and G2?
What is A?
What kind of curve is used?
What is used instead of tangents?
How do you determine elevations at any station on a tangent?
How do you determine elevations at any station on a curve?
How do you determine M?
How to you find a low or high point?
Why is it important to find high and low point stations and elevations?

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