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)
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)
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
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%
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
Vertical Curves-Definitions
Vertical curves are parabolic curves instead of circular curves
Crest Curves (3 types) Sag Curves (3 types)
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
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
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
Calculating M and 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
Vertical Curve Examples
English (crest) ---see below Metric (sag) ---on board if needed Comprehensive Curve (sag)---see
below
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?
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)
English-Crest
X Sta r/2* X^2+ G1* X+PVC
elev=Elev
0 33+00
-.0000344
0 +.01 0547.2
0547.20
10034+0
0-.000034
4100^2 +.01 100
547.20
547.86
20035+0
0-.000034
4200^2 +.01 200
547.20
547.82
30036+0
0-.000034
4300^2 +.01 300
547.20
547.11
40037+0
0-.000034
4400^2 +.01 400
547.20
545.70(Check)
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
Comprehensive Curve Example
Finding a PVI Fitting a curve Finding elevations on the curve Finding M Finding low point
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=450.00Sta 7+00; Elev=460.00
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 L2L1=160 ftL2=440 ftPVI Sta=2+60 (Sta 1+00 + 160’)PVI Elev=446.80’
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=.0001563r/2=.0000781
Fitting a Curve Between Two Set Tangents (4/6)
Elev (x)=r/2*X2+g1X+El PVC
X STA r/2 X2 G1 X Elev PVC
Elev
0 1+00 .0000781 0 -.02 0 450.00
450.00
100
2+00 .0000781 100^2
-.02 100 450.00
448.78
160
2+60 .0000781 160^2
-.02 160 450.00
448.80
200
3+00 .0000781 200^2
-.02 200 450.00
449.12
300
4+00 .0000781 300^2
-.02 300 450.00
451.03
320
4+20 .0000781 320^2
-.02 320 450.00
451.60
Fitting a Curve Between Two Set Tangents (5/6)
Determine Curve Elevation @ PVIM=AL/800=2’
PVI STA 2+60; Elev 446.8’Curve Elev @ PVI=446.80+2’=448.80’
Fitting a Curve Between Two Set Tangents (6/6)
Determine Low PointX=-g1/r=.02/0001563=127.96Sta=2+27.96
Elev @ x=127.96 =448.72’
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)
Next lecture
Checking sight distances on horizontal curves
Checking sight distances on vertical curves Crest-Stopping sight distance Sag-Headlight sight distance