Question

Traffic flow on a one-lane road section (no passing is allowed) has the capacity of 1,000 vehicles/hour and free-flow speed of 40 km/hour. There exists a railway grade crossing at the end of the road section. On one day, a train is passing through the crossing and the gate is closed. Consequently the vehicles are required to stop behind the gate while the train is passing. Three minutes later, the gate is opened and the vehicles immediately start moving again. (a) Calculate the jpm density and the density at capacity (critical density) using the Greenshields' model. Sketch the flow-density graph and show the values of capacity, jam density, and critical density in the graph. (b) Traffic flow in normal traffic condition is characterized by a volume of 640 vehicles/hour and a density of 20 vehicles/km. Using the shock wave theory, determine the length of the platoon when the gate is opened. (c) Determine how long it would take for the platoon to dissipate after the gate is opened. Assume that there is no congestion on the road downstream of the crossing.

Answer 1

Answer;

Given;

free flow speed(v)= 40km/hr

capacity(q)= 1000veh/hr

we know from the fundamental equation of traffic flow,

q=k×v

1000=k×40

k =1000/40   =25veh/km

Now,

(a)Jam Density by Greenshield’s model

q =k×v / 4

q=25×40/ 4    =250veh/hr

So, Jam density = 250 veh/hr

Now, Maximum density= Half of Jam density, i.e

K_m (Maximum Density or critical Density) =k_j(Jam Density) /2

So,Critical Density k=250/2 =125 veh/km

q _max=k_j v_f /4

        =250×40/4

       q_max=2500 veh/hr

qm (2500) E- Flow(9) - - - - ok isterik) kj (250)

(b)Length of platoon when the gate was opened

Lets first characterize the approach and platoon conditions

APPROACH CONDITIONS PLATOON CONDITIONS

q_a= 640 veh/hr q_p= 0 veh/hr

u =40 km/hr u=0

k= 20 veh/km     k=k_j =250 veh/km

u=(0-640) / (250-20) = -2.782 km/hr

Length of the que after 3 minutes = u * t

=2.782 * (3/60)

Length of que   =0.1391 km

(c) Time for the platoon to dissipate after opening of the gate

Again lets characterize the platoon and release conditions

PLATOON CONDITIONS RELEASE CONDITIONS

q=0 veh/hr q=q_m=2500 veh/hr

v=0 km/hr v=40 km/hr

k=250 veh/km k=k_m(no congestion) = 250/2 =125 veh/km

u=(0-2500) / (250-125) =-20 km/hr

U=-20-(-2.782)= 17.218 km/hr

t=L/U

t=0.1391/17.218 = 0.008 hr = 0.417 min