2a_fundamentals of traffic flow
TRANSCRIPT
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])
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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.
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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
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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.
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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
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Primary Elements of Traffic Stream Speed
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
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Primary Elements of Traffic Stream Speed
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.
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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.
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Headway, Flow and Density
Travel Time ( T) or Distance (X)
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Speed, Flow and Density
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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
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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.
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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.
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Relationship Between Speed and
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
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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
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Relationship Between Speed and
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;
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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.
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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