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; m)

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

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

Calculating M and min/max elevations M=A*L/800 Max. or min. elev. 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

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 %. 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?

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

English-Crest XStar/2*X^2+G1*X+ PVC elev= Elev ^ ^ ^ ^ (Check)

English-Crest Highest Elev x=-g1/r=145.35’ Plug x in elevation equation to get: Highest elevation is ’ at Sta

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= Sta 7+00; Elev=460.00

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’

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=

Fitting a Curve Between Two Set Tangents (4/6) Elev (x)=r/2*X2+g1X+El PVC XSTAr/2X2G1XElev PVC Elev ^ ^ ^ ^ ^

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’

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