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Quiz 1 Equation Sheet

163CIVIL ENGINEERING

Transportation Models

See INDUSTRIAL ENGINEERING for optimization models and methods, including queueing theory.

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DENSITY k (veh/mi)

SPE

ED

v (

mph

)

DENSITY k (veh/mi)

VO

LU

ME

q (

veh/

hr)

CAPACITY

VOLUME q (veh/hr)

SPE

ED

v (

mph

)

CA

PAC

ITY

Vertical Curves: Sight Distance Related to Curve Length

S ≤ L S > L

Crest Vertical Curve

General equation:

h1 = 3.50 ft and h2 = 2.0 ft:

L =2

21 2100( 2 2 )

ASh h+

L =2

2,158

AS

L = 2S −( )2

1 2200 h h

A

+

L = 2S −2,158

A

Sag Vertical Curve

(based on standard headlight criteria)

L =2

400 3.5

ASS+

L = 2S −400 3.5S

A+

Sag Vertical Curve

(based on riding comfort) L =2

46.5

AV

L =2

1 28002

ASh hC +

−L = 2S − 1 2800

2

h hC

A

+−

Sag Vertical Curve

(based on adequate sight distance under an overhead structure to see an object beyond a sag vertical curve)

C = vertical clearance for overhead structure (overpass) located within 200 feet of the midpoint of the curve

( ) ( )

( )

Standard Criteria:

Horizontal Curves

Side friction factor (based on superelevation)

2

0.0115

Ve fR

+ =

Spiral Transition Length Ls =33.15V

RC

C = rate of increase of lateral acceleration [use 1 ft/sec3 unless otherwise stated]

Sight Distance (to see around obstruction) HSO = R 28.65

1 cosS

R−

HSO = Horizontal sight line offset

( )[ ]

164 CIVIL ENGINEERING

Horizontal Curve Formulas

D = Degree of Curve, Arc ���������PC = Point of Curve (also called BC)PT = Point of Tangent (also called EC)PI = Point of IntersectionI = Intersection Angle (also called Δ) Angle Between Two TangentsL = Length of Curve, from PC to PTT = Tangent DistanceE = External DistanceR = RadiusLC = Length of Long ChordM = Length of Middle Ordinatec = Length of Sub-Chordd = Angle of Sub-Chordl = Curve Length for Sub-Chord

.R D5729 58

=

sinR

ILC

2 2=

_ i

tancos

T R II

LC22 2

= =__

ii

L RI DI

180 100= =r

cosM R I1 2= - _ i8 BcosE R

R I 2+

= _ i

cosRR M I 2- = _ i

sinc R d2 2= _ i

l Rd 180=rb l

cosE R

I 21 1= -_ i= G

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+ Latitude

– Latitude

– Departure + Departure

LATITUDES AND DEPARTURES

165CIVIL ENGINEERING

Vertical Curve Formulas

TANGENTOFFSET

BACK TANGENT

VERTICAL CURVE FORMULASNOT TO SCALE

DATUM

FORWARDTANGENT

L

x

y Eg2

g 1

YPVC

PVC

PVT

PVI

L = Length of Curve (horizontal) g2 = Grade of Forward Tangent

PVC = Point of Vertical Curvature a = Parabola Constant

PVI = Point of Vertical Intersection y = Tangent Offset

PVT = Point of Vertical Tangency E = Tangent Offset at PVIg1 = Grade of Back Tangent r = Rate of Change of Grade

x = Horizontal Distance from PVC to Point on Curve

xm = Horizontal Distance to Min/Max Elevation on Curve = ag

g gg L

21

1 2

1- =-

Tangent Elevation = YPVC + g1x and = YPVI + g2 (x – L/2)

Curve Elevation = YPVC + g1x + ax2 = YPVC + g1x + [(g2 – g1)/(2L)]x2

y ax a Lg g E a L r L

g g2 2

2 2 12

2 1= =-

= =-b l

EARTHWORK FORMULAS

Average End Area Formula, V = L(A1 + A2)/2Prismoidal Formula, V = L (A1 + 4Am + A2)/6,

where Am = area of mid-section, and

L = distance between A1 and A2

Pyramid or Cone, V = h (Area of Base)/3

AREA FORMULAS

Area by Coordinates: Area = [XA (YB – YN) + XB (YC – YA) + XC (YD – YB) + ... + XN (YA – YN – 1)] / 2

Trapezoidal Rule: Area = w h h h h h h2n

n1

2 3 4 1f+

+ + + + + -c m w = common interval

Simpson’s 1/3 Rule: Area = w h h h h2 4 3, , , ,

kk

nk

k

nn1

3 5

2

2 4

1= + + +

f f=

-

=

-

e eo o> H! ! n must be odd number of measurements

w = common interval

Vertical Curve Offsets Parabolic Equations

2

200x

LAY = y = ax2 + bx +c

A = ∣ G1 – G2∣ *A is in percent form. Where y = roadway elevation at distance x from the PVC.

KAL= 2 1

2G GaL−= ; b = G1; c = ELEVPVC

1GKxhl ×= *keep in mind that you must use

either station/% or ft/decimal for x/Gi.

1

4. Regression Equation?

2. Size within Data Extremes?

3. Number of Data Points?

1. Compatible with ITE Land Use Code?

A B

Col

lect

Loc

al D

ata

Source: ITE Trip Generation Handbook, 2nd Edition

Yes

Yes

Yes

No

No

1 or 2

3-5 3-5

6 +

No

Selection of ITE Rates/Equations, or Collection of

Local Data

16

If number of data points between 3 and 5, analysts are encouraged to collect local data, but can proceed to Step 4.

5. Standard Deviation 110 percent?

6. Data Cluster Okay?

Use Weighted Average Rate

Col

lect

Loc

al D

ata

A

Yes

No

No

Yes

17

Figure 3.1

Source: ITE Trip Generation Handbook, 2nd Edition

Selection of ITE Rates/Equations, or Collection of Local

Data (cont.)

2

7. 20 or More Data Points?

Use Regression Equation

8a. R2 0.75? And Within Cluster?

8B. Std Dev 110%? And Within Cluster?

If 8A is yes & 8B is yes

If 8A is yes & 8B is no

If 8A is no & 8B is yes

If 8A is no & 8B is no

Choose Line at Cluster

Use Weighted Average Rate

Use Regression Equation

Collect Local Data

B

Yes

No

18

Figure 3.1

Source: ITE Trip Generation Handbook, 2nd Edition

Selection of ITE Rates/Equations, or Collection

of Local Data (cont.)