501098357-Examples for Transportation engineering problems.pptx

Traffic Flow Analysis

1. Examples 1. Fundamentals of traffic Flow 3/28/2021 Mohammed A. 1 1

2. Examples 1. The figure shown below that the vehicles traveling at constant speed on two lane highway between section X and Y with their positions and speeds obtained at an instant of time by photogrammetry. An observer located at point X observes four the vehicles passing point X during the period of 2minutes . The velocities of the vehicles measured as 45, 45, 40 and 30 km/ hr respectively. Therefore, calculate the flow, density, space mean speed and time mean speed. 3/28/2021 Mohammed A. 2 2

3. Examples Given data number of vehicles n = 4 The velocities of the vehicles vehicle A = 45 km/ hr vehicle B = 45 km/ hr vehicle C = 40 km/ hr vehicle D = 30 km/ hr The time period T = 2minutes Required calculate: a) flow, b) density, c) time mean speed and d) space mean speed 3/28/2021 Mohammed A. 3 3

4. Examples Solution Flow The flow can be expressed by the formula: = 120 veh / hr b) Density It can be given by the formula: = 13.3 veh /km   c) time mean speed It can be found by the equations: = 40 km/ hr d) space mean speed It can be also found by: L is the distance between X and Y. Now, the time t i takes the i th vehicle to travel from X toY at speed u i can be found by:   3/28/2021 Mohammed A. 4 4

5. Examples = 24 sec = 24 sec = 27 sec = 36 sec   Therefore, the space mean speed becomes: = 10.81 m/sec = 38.92km/ hr Note: the time mean speed is always higher than the space mean speed . i.e 40 km/ hr ……ok!!!   3/28/2021 Mohammed A. 5 5

6. Examples 2. Two set of vehicles are timed over a kilometer, and the flows are also recorded. In the first sets, four vehicles are take 52,56,63 and 69 seconds when the flow is 1500 veh /hr. In the second set, four vehicles are take 70,77,74 and 79 seconds when the flow is 1920 veh /hr. These vehicles have been timed over space, then calculate the density, space mean speed, free flow speed, jam density and capacity for the given two sets of vehicles? Given: Data for the first set of vehicles: Data for the 2nd set of vehicles: Number of vehicles, n = 4 Number of vehicles, n = 4 Timed over a kilometer, ti Timed over a kilometer, ti 52,56,63 and 69 seconds 70,77,74 and 79 seconds The flow, q = 1500 veh /hr. The flow, q = 1920 veh /hr. 3/28/2021 Mohammed A. 6 6

7. Examples Required: the density, space mean speed, free flow speed, jam density and capacity Solution: the density, It can be found by: = 60 sec. for the 1 st set of vehicles = = 75 sec. for 2 nd set of vehicles Now, the density for both sets of vehicles can be: = 25 veh /km for 1 st set of vehicles = 40 veh /km for 2 nd set of vehicles   3/28/2021 Mohammed A. 7 7

8. Examples 2. space mean speed It can be given by the relations: = = 60 km/ hr 1 st set = = 48 km/ hr 2 nd set of vehicle 3. free flow speed In order to determine the free flow speed we Need to develop the equation using Green shield model:   Now, substituting the values of speed and density into the equation and get the required values. 60K j = U f * K j - 25U f ……..1 48K j = U f * K j - 40U f ……..2 Multiplying equation 1 by 8 and equation 2 by -5 and we get: 480Kj = 8UfKj – 200Uf….1 -240Kj = -5UfKj + 200Uf….2 Add the two equations and we get: 240Kj = 3UfKj Uf = 80 km/hr. 3/28/2021 Mohammed A. 8 8

9. Examples 4. jam density To determine jam density substitute the value of free flow speed into equation 1 and we get: 60Kj = UfKj - 25Uf…1 Kj = 100 veh /km 5. Capacity It can be calculated by: q cap = 2000 veh /hr.   3/28/2021 Mohammed A. 9 9

10. Examples 3. The speeds of five vehicles were measured (with radar) at the midpoint of a 0.5-mile section of roadway. The speeds for vehicles 1, 2, 3, 4, and 5 were 44, 42, 51, 49, and 46 mi/h, respectively. Assuming all vehicles were traveling at constant speed over this roadway section, calculate the time-mean and space-mean speeds. Given data 3/28/2021 Mohammed A. 10 Vehicles Speeds (mi/ hr ) 1 44 2 42 3 51 4 49 5 46 10

11. Examples Required: Time mean speed Space mean speed Solution: The time mean speed is given by: = 46.4 mi/ hr the space mean speed is given by: ok!!! = 46.17 mi/ hr   3/28/2021 Mohammed A. 11 11

12. Examples 4. Vehicle time headways and spacing were measured at a point along a highway, from a single lane, over the course of an hour. The average values were calculated as 2.5 s/ veh for headway and 200 ft / veh (61 m/ veh ) for spacing. Calculate the, flow, density and speed of the traffic. Given Data: Average headway = 2.5 sec/ veh Average spacing = 200 ft / veh or 61m/ veh Required: Flow Density a nd speed 3/28/2021 Mohammed A. 12 12

13. Examples Solution: Flow It can be found by the relation: = 0.4 veh /sec because the data were collected for an hour: sec/h = 1440veh/h II. Density It also calculated by using the formula: = 0.0164 veh /m   applying this spacing over the course of one kilometer: K = 16.4 veh /km III. Speed It is given by the formula: = 87.8 km/h It can be also computed by: = 24.4 m/s u = 24.4 m/s*3.6 = 87.8km/h   3/28/2021 Mohammed A. 13 13

14. Examples 5. A section of highway is known to have a free-flow speed of 55 mi/h and a capacity of 3300 veh /h. In a given hour, 2100 vehicles were counted at a specified point along this highway section. If the linear speed-density relationship applies, what would you estimate the space-mean speed of these 2100 vehicles to be? Given data: U f = 55 mi/ hr q cap = 3300 veh /h q = 2100 veh in a given hour 3/28/2021 Mohammed A. 14 14

15. Examples Required: the space-mean speed Solution: linear speed-density relationship is given by: Rearranging the above equation to solve for u, Now, determine the jam density using: Substitute the values of the given data in to the equation and we get: By applying quadratic equation we get the required space mean speed as: u = 44.08 mi/h or 10.92 mi/h From the results both of these speeds are feasible. 3/28/2021 Mohammed A. 15 15

16. Examples 6. Consider Greenshield’s linear speed-density model U= 55 – 0.45K , then drive flow-density relationships, and calculate Free flow speed, Jam density and Capacity.(where speed is in mi/ hr , density veh /mi and flow veh / hr ). Given data Linear speed- density model, U = 55 – 0.45K Required Drive flow- density relationships Free flow speed Jam density capacity 3/28/2021 Mohammed A. 16 16

17. Examples Solution Derivation of flow- density relationships U= 55 – 0.45K but q = u*k, u = q/k substitute (q/k) = 55 – 0.45K finally we get, q = 55k – 0.45k^2 it is parabolic relationship Free flow speed Free flow speed occurs when the density is zero. U= 55 – 0.45K, k=0 U f = 55 – 0.45 *0 U f = 55 mi/ hr Jam density Jam density occurs when speed is zero. U= 55 – 0.45K, u = 0 0 = 55 – 0.45* K j K j = 55/0.45 = 122.22 veh /mi capacity It is computed by: = 1680.53 veh / hr /ln   3/28/2021 Mohammed A. 17 17

18. Examples 8 .A section of highway has the following speed-flow relationship: q+100u = 300u -4u^2, then calculate the following : a) free flow speed b) density at capacity c) capacity d) the flow rate corresponding to a density of 80vpm. 3/28/2021 Mohammed A. 18 18

19. Examples 3/28/2021 Mohammed A. 19 19

20. 9 . The data shown below were obtained on a highway. Use regression analysis to develop Greenshields model/equation based on the given data shown below in the table. 3/28/2021 Mohammed A. 20 20 Examples

21. Solution: Based on Greenshield model there is a linear relation ship between traffic stream parameters. Therefore, in order to determine the equation we use linear regression with the general form shown below. Where, a and b are regression constant which is given by the formula: The computation is shown below in the table 3/28/2021 Mohammed A. 21 Examples 21

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23. we see that the speed in the Greenshields expression is represented by y in the estimated regression function, the mean free speed u f is represented by a , and the value of the mean free speed u f divided by the jam density k j is represented by - b. We therefore obtain: Finally the required regression equation becomes: y = 62.68 – 0.53x Where, y represents space mean speed and x represents density and we get 3/28/2021 Mohammed A. 23 Examples 23

24. 10. Redo example 9 using Greenberg model. Solution: Based on Greenberg there is logarithmic relationship between traffic stream parameters which can be given by the expression e see that in the Greenberg expression is represented by y in the estimated regression function, c ln kj is represented by a, c is represented by -b , and ln k is represented by x. We therefore obtain The computation is shown below in the table 3/28/2021 Mohammed A. 24 Examples 24

25. 3/28/2021 Mohammed A. 25 Examples 25 Since a = 145.06 and b = -28.68 , the speed for maximum flow is c = 28.68 mi/h. Finally , Then we get the equation

26. 3/28/2021 Mohammed A. 26 Examples 26 11. A driver takes 3.5 s to react a complex situation while traveling at a speed of 60 km/hr. how far does the vehicle travel before the driver initiates a physical response to the situation? Solution: The vehicle will travel : dr = 0.278vt, where v = 60km/h t = 3.5s Therefore, the distance traveled by the vehicle cab be: dr = 0.278*60*3.5 = 58.4 m

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29. 3/28/2021 Mohammed A. 29 Thank you 29