CTC 440 Vertical Alignment

Objectives What is a profile? What are 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 (A is in %) 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 33+00 -.0000344 33+00 -.0000344 +.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)

English-Crest (highest elev) 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

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 = 549.20’ The elevation on the curve at STA 35 = 549.20’-1.375’=547.82’ (checks w/ Table value)

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=450.00 Sta 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 L2 L1=160 ft L2=440 ft PVI 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=.0001563 r/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 Elev PVC Elev 1+00 .0000781 -.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

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’

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’

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)

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?

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