2A_Fundamentals of Traffic Flow

KAAF UNIVERSITY COLLEGE

Civil Engineering Department College of Engineering

__________________________________

Transportation Engineering I

CIV 367

Lecture 2A_ Fundamentals of Traffic Flow

Kwasi Agyeman – Boakye ( [email protected])


Traffic Flow

Traffic Flow Theory

Traffic flow theories deal with the mathematical relationship between the primary elements of the traffic stream which include flow, speed and density

Traffic Flow Facilities

For the analyses of traffic streams the movement is categorized as interrupted flows or uninterrupted flows. Volume or traffic flow is a parameter common to both uninterrupted- and interrupted-flow facilities, but speed and density apply primarily to uninterrupted flow.

Uninterrupted Flow

A category of facilities that have no fixed causes of delay or interruption external to the traffic stream; examples include freeways and unsignalized sections of multilane and two-lane rural highways.

Interrupted Flow

A category of traffic facilities characterized by traffic signals, stop signs, or other fixed causes of periodic delay or interruption to the traffic stream.


Primary Elements of Traffic Volume and Flow

Volume (v)- The total number of vehicles that pass over a given point or section

of a lane or roadway during a given time interval; volumes can be expressed in

terms of annual, daily, hourly, or sub hourly periods.

Flow (q)- The equivalent hourly rate at which vehicles pass over a given point or

section of a lane or roadway during a given time interval of less than 1 h, usually

15 min.

Qn: For four continuous 15 minutes interval

counts that were made on a section of highway

between 5 and 6 pm the following volumes were

observed; 1000 veh, 1200 veh, 1100 veh and

1000 veh. Determine the Volume per hour and

the Flow.

Volume(V)=1000+1200+1100+1000=4,300 veh/hr

Flow (q) = 1200/0.25= 4800 veh/hr

Time Interval Volume for Time Intervals (veh)

Rate of Flow For Time Interval

(Veh/hr)

5:00 – 5:15 PM 1000 1000/0.25 = 4,000

5:15 – 5:30 PM 1200 1200/0.25 = 4,800

5:30 – 5:45 PM 1100 1100/0.25 = 4,400

5:45 – 6:00 PM 1000 1000/0.25 = 4,000

5:00 – 6:00 PM V = 4,300 Q = 4,800


Primary Elements of Traffic Stream Speed

Speed - It is the distance travelled by a vehicle during a unit time, often expressed in kilometres per hour (km/hr). There are two main types of speed known as time mean speed and space mean speed. The other speeds include average running speed, average travel speed and free flow speed.

Average Running Speed- It is the length of the segment divided by the average running time of vehicles to traverse the segment. Running time includes only time that vehicles are in motion.

Average Travel Speed - It is the length of the segment divided by the average travel time of vehicles traversing the segment, including all stopped delay times. It is also a space mean speed.

Free Flow Spee - The average speed of vehicles on a given facility, measured under low-volume conditions, when drivers tend to drive at their desired speed and are not constrained by control delay

From the time – space diagram it can be observed that the speed of a vehicle at any time t is the slope of the time – space diagram for that vehicle at time t. It is observed that Vehicles 1 and 2 are moving at constant speeds because the slopes of the associated graphs are constant. Vehicle 3 moves at a constant speed between time zero and time t3 , then stops for the period t3 to t”3 and then accelerates and eventually moves at a constant speed.


Time Space Diagram

Time

1

2

3 4

5

6

d1

d3

d2

d4

d3-4

h3-4

Slope = U1

0 t3 t3 ‘’ t5 t6

Dis

tan

ce



Time Mean Speed ( TMS)

Is the average speed of all vehicles occupying a given section of highway or lane over some specified period

Space Mean Speed ( SMS)

Is the average speed of all vehicles passing a point on a highway or lane over some specified time period.

time (s)

Dis

tance (

m)

5 10 15

100

v2

40

50

v3 v1



Three vehicles are recorded with speeds of 20, 10, and 8 m/s. The time to

traverse 100 m is 5sec, 10 sec, and 15 sec, respectively. Determine the SMS

and the TMS. Ans TMS = 12.67sec, SMS=10sec

Relationship between TMS and SMS

Space mean speed is always less than time mean speed, but the difference

decreases as the absolute value of speed increases.

SMS = 1.026x TMS – 3.042

Qn. Four vehicles are recorded with

speeds 20, 15, 10 and 5 m/s over a

section 200m. Determine the SMS.


Primary Elements of Traffic Stream Density and Headway

Density/Concentration (k) is the number of vehicles (or pedestrians) occupying a given length of a lane or roadway at a particular instant. It is averaged over time and is usually expressed as vehicles per kilometer (veh/km) or passenger cars per kilometer (pc/km).

There are generally two types of headway, time headway and space headway. Generally when it is used in the generic as headway the time headway is the one being referred to.

Headway(h) – It is the time between successive vehicles as they pass a point along the lane, also measured between common reference points. Going back to the space time diagram identify the headway between vehicles 3 and 4 at d1 . Space headway/Spacing (d) – The distance between successive vehicles in a traffic lane, measured from some common reference point on the vehicle such as the front bumper or front wheel. Identify the space headway between vehicles 3 and 4 at time t5 in the space time diagram.


Headway, Flow and Density

Travel Time ( T) or Distance (X)


Speed, Flow and Density


Speed, Flow, Density and Headway

Traffic in a congested multilane highway

lane is observed to have an average

spacing of 80m and an average headway of

3.8s. Estimate the rate of flow, density and

speed of traffic in this lane.

Ans. All working is per lane.

Average Spacing = 80m/veh

Average Headway = 3.8s/veh

Flow =3600 = 947.37veh/h/lane

3.8

Density = 1000 = 12.5veh/km/lane

80

Speed=Flow rate=947.37 =75.79km/hr/lane

Density 12.5


Assignment

Qn 1. At a given location, the space mean speed is measured as 70km/h and the rate of flow as 1,600 pc/h/lane. What is the density at this location for the analysis period?

Qn 2. A traffic stream displays average vehicle headways of 2.2sec at 80km/hr. Compute the density and rate of flow for this traffic stream.


Relationship Between Speed and

Density

Greenshield Relationship

Greenshields (1934) proposed the simplest representation between the two variables, assuming

a linear relationship. U = a+bk

Where densities are near to zero and the speed at which the car can be driven is determined

solely by the geometric design and layout of the road; such a speed is termed free-flow speed

As more vehicles use the section of highway, the density of the flow will increase and their

speed will decrease from their maximum free-flow value (uf) as they are increasingly more

inhibited by the driving manoeuvres of others.

If traffic volumes continue to increase, a point is reached

where traffic will be brought to a stop, thus speeds will

equal zero (u = 0), with the density at its maximum

point as cars are jammed bumper to bumper (termed)

jam density, kj). Thus, the limiting values of the

relationship between speed and density are as follows:

When k = 0, u = uf, When u = 0, k = kj.



Density

Greenberg Logarithm Relationship

Greenberg notes that the relationship is not quite linear Greenberg Relationship

but slightly concave. This is given by an equation;

u = a loge bk

However in this formula uf has a value of infinity

Underwood Exponential Relationship

Underwood also expresses another relationship in

the exponential form. This is given by an equation;

u =ae-bk

Here kj has a value of infinity. Underwood Relationship


Relationship Between Flow and

Density

Combining Equation 1 with the fundamental flow equation;

Establishing the density at which maximum

flow occurs;

Fundamental Diagram of Traffic Flow



Flow

In order to derive this relationship, Equation 1 is turned upside down

Combining this equation with the fundamental flow equation;

In order to determine the speed at maximum flow;


Maximum Flows I

Combining equations 3 and 5 the following is

obtained for maximum flow;

Qn 1.Two platoons of cars are timed over a distance of

0.5km. Their flows are recorded. The first group is timed

at 40 seconds, with the flow at 1350 vehicles per hour.

The second group take 45 seconds, with a flow of 1800

vehicles per hour. Determine the maximum flow of the

traffic stream.

Qn 2. Two sets of vehicles are timed over a kilometre,

and the flows are also recorded. In the first set, four

vehicles take 52,56,63 and 69 seconds when the flow is

1500 veh/h. In the second, four vehicles take 70,74,77

and 79 seconds when the flow is 1920 veh/h. Determine

the maximum flow of the traffic stream.


Maximum Flows II

Where k1= 30 veh/km and u1=45km/h equation 2

becomes;

45 = a -30b………….3

Where k2 = 45veh/km and u2=40km/h equation 2

becomes;

40 = a – 45b………..4

Solving the equations simultaneously ;

a = 55 and b = 1/3

Therefore uf = 55 and

Uf = 1/3 , hence kj = 165

Kj

q max = UfxKj = 55x165 = 2269veh/h

4 4

Question 1

Distance = 0.5km

First Group, t1 = 40s q1 = 1350veh/h

2nd Group , t2 = 45s q2 = 1800veh/h

First Group, u1= q1/t1 = 0.5x3600/40 = 45km/h

2nd Group , u2= q2/t2 = 0.5x3600/45 = 40km/h

First Group, k1= q1/u1=1350/45 = 30 veh/km

2nd Group , k2= q2/u2=1800/40 = 45veh/km

Using the normal linear relationship

2A_Fundamentals of Traffic Flow

Documents

Transcript of 2A_Fundamentals of Traffic Flow