Solutions Manual to accompany
Principles of Highway Engineering and Traffic Analysis, Analysis, 4e By Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski
Chapter 2 Road Vehicle Performance U.S. Customary Units
Copyright © 2008, 2008, by John Wiley & Sons, Inc. All rights reserved.
Preface The solutions to the fourth edition of Principles of Highway Engineering and Traffic Analysis 1 were prepared with the Mathcad software program. program. You will notice several notation conventions that you may not be familiar with if you are not a Mathcad user. Most of these notation conventions are self-explanatory or easily understood. The most common Mathcad specific notations in these solutions relate to the equals sign. sign. You will notice the equals sign being used in three different contexts, and Mathcad uses three different notations to distinguish between each of these contexts. The differences between these equals sign sign notations are explained as follows. follows.
•
The ‘:=’ (colon-equals) is an assignment operator, that is, the value of the variable or expression on the left side of ‘:=’is set equal to the value of the expression on the right side. For example, in the statement, L := := 1234, the variable ‘L’ is assigned (i.e., set equal to) the value of 1234. Another example is x := y + z. In this case, x is assigned assigned the value of y + z.
•
The ‘=’ (bold equals) is used when the Mathcad function solver was used to find the value of a variable in in the equation. For example, in the the equation , the = is used to tell Mathcad that the value of the expression on the left side needs to equal the value of the expression on the right side. Thus, the Mathcad solver can be employed to find a value for the variable ‘t’ that satisfies this relationship. This particular example is from from a problem where the function function for arrivals at some time ‘t’ is set equal to the function for departures at some time ‘t’ to find the time to queue clearance.
•
The ‘=’ (standard equals) is used for a simple numeric evaluation. For example, referring to the x := y + z assignment used previously, if the value of y was 10 [either by assignment (with :=), or the result of an equation solution (through the use of =) and the value of z was 15, then the expression ‘x =’ would yield 25. Another example would be as follows: s := 1800/3600, with s = 0.5. That is, ‘s’ was was assigned the value of 1800 divided by 3600 (using :=), which equals 0.5 (as given by using =).
Another symbol you will see frequently is ‘→’. In these solutions, it is used to perform an evaluation of an assignment expression in a single statement. For example, in the following statement, , Q(t) is assigned the value of 2 Arrivals(t) – Departures(t), and this evaluates to 2.2t – 0.10t .
Finally, to assist in quickly identifying the final answer, or answers, for what is being asked in the problem statement, yellow highlighting has been used (which will print as light gray). 1
www.mathcad.com
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 2.11
Determine the m aximum g rade. 3500 ne := 60
i := 0.035
ε o := 3.2
r :=
14 12
W
:= 2500 lb
(given)
assume F=F e calculate velocity
V
:=
2 ⋅ π ⋅ r ⋅ ne ⋅ ( 1
− i) V
εo
= 128.9
(Eq. 2.18)
ft/s
calculate aerodynamic resistance
ρ := 0.002378
CD := 0.35
ρ
Ra := ⋅ CD ⋅ Af ⋅ V 2
2
Af := 25 (Eq. 2.3)
Ra
= 172.99
lb
calculate rolling resistance V ⎞ ⎛ f rl := 0.01 1 + ⎝ 147 ⎠ Rrl := f rl ⋅ W
(Eq. 2.5)
Rrl
= 46.93
lb
calculate engine-generated tractive effort Me := 200 Fe :=
nd := 0.90
Me ⋅ ε o ⋅ nd
Fe
r
= 493.71
lb
(Eq. 2.17)
calculate grade resistance Rg := Fe Rg
− Ra − Rrl
(Eq. 2.2)
= 273.79
solve for G G :=
G
Rg
(Eq. 2.9)
W
= 0.1095
therefore G = 11.0%
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Al ter nat iv e cal cu lat io n f or gr ade, u si ng tr ig rel ati on sh ip s
θg := asin
⎛ Rg ⎞ ⎝ W ⎠
θg = 0.1097 degθg := θg ⋅ degθg
radians 180
π
convert from radians to degrees
= 6.287
tan deg = opposite side/adjacent side
G := tan
(θg) ⋅ 100
G
= 11.02
%
Thus, error is minimal when assuming G = sin
θg for small to medium grades
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 2.12 Determine the torqu e the engine is produ cing and the engine speed.
− ΣR
Fe
γ m ⋅m ⋅a
at top speed, acceleration = 0; thus, F e V
:= 124⋅
5280
V
3600
− ΣR
0
= 181.867 ft/s
(given)
calculate aerodynamic resistance
ρ := 0.00206
CD := 0.28
ρ
Ra := ⋅ CD ⋅ Af ⋅ V 2
2
Af := 19.4 Ra
= 185.056
(Eq. 2.3)
calculate rolling resistance
⎛ + ⎝
⎞ 147 ⎠ V
f rl := 0.01 ⋅ 1 W
f rl
= 0.022
(Eq. 2.5)
:= 2700
(given)
Rrl := f rl ⋅ W
Rrl
(Eq. 2.6)
= 60.404
Rg := 0 sum of resistances is equal to engine-generated tractive effort, solve for Me Fe := Ra + Rrl i := 0.03
Fe Me
+ Rg
ηd := 0.90
Me ⋅ ε 0 ⋅ ηd
ε 0 := 2.5
r :=
Me :=
r
= 114.548
(Eq. 2.2)
Fe = 245.46 12.6 12
Fe ⋅ r
(Eq. 2.17)
ε 0 ⋅ η d
ft-lb
Knowing velocity, solve for ne
V
ne
2 ⋅ π ⋅ r ⋅ ne ⋅ ( 1
− i)
ε0 = 71.048
rev s
V ⋅ε 0
ne := 2 ⋅ π ⋅ r ⋅ ( 1 ne ⋅ 60 = 4263
(Eq. 2.18)
− i) rev min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Rear - wheel drive
μ := 0.2
f rl := 0.011
h := 20
L := 120
lf := 60
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
2
Sal
V2 :=
2
γ b ⋅ V1 − V2 2 ⋅ g ⋅ ( η b ⋅ μ m + f rl − 0.03) 2
V1
−
2 ⋅ Sal ⋅ g ⋅
(Eq. 2.43)
(η b ⋅ μ m + f rl − 0.03) γb
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 2.24 Determine if th e driver shoul d appeal the ticket.
μ := 0.6 (for good, wet pavement, and slide value because of skidding) γ b := 1.04 V2
:= 40 ⋅
g := 32.2
5280
V2
3600
ηb := 0.95
= 58.667
S := 200
(given)
f rl := 0.015
Solve for the initial velocity of the car using theoretical stopping distance
2
S
V1
2
γ b ⋅ V1 − V2 2 ⋅ g ⋅ ( ηb μ ⋅ + f rl − 0.04)
V1
:=
2⋅S ⋅g⋅
(ηb ⋅ μ + f rl − 0.04) γb
+ V2
2
(Eq. 2.43)
= 100.952
V1 ⋅
3600 5280
= 68.83
mi h
No, the driver should not appeal the ticket as the initial velocity was higher than the speed limit, in addition to the road being wet.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
G
= 0.059
G ⋅ 100 = 5.93
%
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Multiple Choice Problems Determine the minimum t ractive effort.
CD := 0.35
2
A f := 20 ft
ρ := 0.002045
Problem 2.31 slugs ft
V := 70⋅
⎛ 5280 ⎞ ⎝ 3600 ⎠
ft s
3
(given)
W := 2000 lb
G := 0.05
grade resistance R g := 2000G ⋅
R g
= 100
R a
= 75.44
(Eq. 2.9)
lb
aerodynamic resistance
ρ
2
R a := ⋅ CD⋅ A f ⋅ V 2
lb
(Eq. 2.3)
rolling resistance
⎛ ⎝
f rl := 0.01⋅ 1 +
V ⎞ 147 ⎠
R rl := f rl⋅ W
f rl = 0.02
R rl = 33.97
(Eq. 2.5)
lb
(Eq. 2.6)
summation of resistances
F := R a
+
R rl + R g
F = 209.41 lb
(Eq. 2.2)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Using mi/h instead of ft/s for velocity mi
V := 70
h
⎛ ⎝
f rl := 0.01⋅ 1 +
V ⎞ 147 ⎠
R rl := f rl⋅ W R a :=
ρ 2
F := R a
2
f rl = 0.01
R rl = 29.52
lb
= 35.07
lb
⋅ CD⋅ A f ⋅ V
R a
+
F = 164.6
R rl + R g
lb
2) not including aerodynamic resistance V := 70⋅
5280 3600
F := R rl + R g
F
= 129.52
lb
3) not including rolling resistance F := R a
+
R g
F
= 135.07
lb
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the acceleration.
Problem 2.32 V := 20⋅
h := 20
Cd := 0.3
5280 3600
V = 29.33
ft
A f := 20
s
ρ := 0.002045
μ := 1
M e := 95 r :=
ft
W
2
slugs ft
3
L := 110
in
lf := 50
in
lb
(given)
ηd := 0.90
ft
12
:= 2500
ε o := 4.5
ft-lb
14
in
aerodynamic resistnace
ρ
2
R a := ⋅ Cd ⋅ A f ⋅ V 2
R a
= 5.28
(Eq. 2.3)
lb
rolling resistance f rl := 0.01⎛ 1 + ⎝
(Eq. 2.5)
V ⎞ 147 ⎠
R rl := 0.01⋅ ⎛ 1 +
⎝
V ⎞ 147 ⎠
⋅ 2500
R rl = 29.99
(Eq. 2.6)
lb
engine-generated tractive effort
Fe :=
M e⋅ ε o ⋅ ηd
Fe
r
= 329.79
(Eq. 2.17)
lb
mass factor
γ m := 1.04 +
0.0025ε o
2
(Eq. 2.20)
γ m = 1.09
lr := 120 − lf
acceleration
μ⋅ W⋅ ( lr + f rl⋅ h) Fmax :=
a :=
Fe
L
1+
μ⋅h
lb
(Eq. 2.15)
L
− R a − R rl
γ m⋅ ⎛
Fmax = 1350.77
2500 ⎞
⎝ 32.2 ⎠
a
= 3.48
ft s
(Eq. 2.19)
2
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Use a mass factor of 1.04
2) Use Fmax instead of Fe
γ m := 1.04
a :=
γ m := 1.091
Fe
a :=
− R a − R rl
ft a
2500 ⎞ γ m⋅ ⎛ ⎝ 32.2 ⎠ Fmax − R a
− R rl
2500 ⎞ γ m⋅ ⎛ ⎝ 32.2 ⎠
= 3.65
s
2
ft a
= 15.53
a
= 15.9
s
2
3) Rear wheel instead of front wheel drive
μ⋅ W⋅ ( lf − f rl⋅ h) Fmax :=
L
1−
μ⋅ h L
Fmax = 1382.22
a :=
Fmax − R a
γ m⋅ ⎛
− R rl
2500 ⎞
⎝ 32.2 ⎠
ft s
2
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the percentage of braking force. V := 65⋅
5280
ft
3600
s
Problem 2.33
μ := 0.90
L := 120
in
lf := 50 in
h := 20
in
lr := L − lf
(given) in
determine the coefficient of rolling resistance
⎛ ⎝
f rl := 0.01⋅ 1 +
V ⎞ 147 ⎠
f rl = 0.02
(Eq. 2.5)
determine the brake force ratio lr + h ⋅ BFR frmax:= lf − h ⋅
(μ + f rl) (μ + f rl)
BFR frmax = 2.79
(Eq. 2.30)
calculate percentage of braking force allocated to rear axle PBFr :=
100 1 + BFRfrmax
PBFr = 26.39 %
(Eq. 2.32)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Use front axle equation 100 PBFf := 100 − 1 + BFRfrmax
PBFf = 73.61 %
(Eq 2.31)
2) Use incorrect brake force ratio equation
BFR frmax:=
PBFr :=
(μ + f rl) lf + h⋅ ( μ + f rl) lr − h ⋅
100 1 + BFRfrmax
BFR frmax = 0.76
PBFr = 56.94 %
3) Switch l f and l r in brake force ratio equation
BFR frmax:=
PBFr :=
(μ + f rl) lr − h ⋅ (μ + f rl)
lf + h⋅
100 1 + BFRfrmax
BFR frmax = 1.32
PBFr = 43.06 %
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the theoretical stoppi ng dis tance on level grade. CD := 0.59
V := 80⋅ 2
A f := 26 ft
5280
ft
3600
s
Problem 2.34 (given)
μ := 0.7
γ b := 1.04
η b := 0.75
(assumed values)
Coefficient of Rolling Resistance
⎛
⎞
V 2
f rl := 0.01⋅ 1 +
⎝
f rl = 0.014
147 ⎠
(Eq. 2.5)
Theoretical Stopping Distance
S :=
2
γ b ⋅
V1
2
− V2
(
2⋅ g ⋅ η b⋅ μ
S = 412.8
)
+
f rl
s
2
(Eq. 2.43)
ft
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Not dividing the velocity by 2 for the coeffeicent of rolling resistance
⎛ ⎝
f rl := 0.01⋅ 1 +
S :=
γ b⋅
2
V1
V ⎞ 147 ⎠ 2
− V2
(
2⋅ g⋅ η b⋅ μ
+
S
)
f rl
= 409.8
s
2
ft
2) Using mi/h instead of ft/s for the velocity V1 := 80
V2 := 0
⎛ ⎝
S :=
γ b⋅
2
V1
(
⎞
V
f rl := 0.01⋅ 1 +
2
147 ⎠ 2
− V2
2⋅ g⋅ η b⋅ μ
V := 80
+
)
f rl
S
= 192.4
s
2
ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
3) Using γ = 1.0 value
γ b := 1.0 S :=
2
γ b ⋅
V1
2
− V2
(
2⋅ g ⋅ η b⋅ μ
+
S = 397.9
)
f rl
s
2
ft
Problem 2.35
Determine the stopping si ght dis tance. V := 45⋅
5280
ft/s
(given)
3600 ft
a := 11.2
s
tr := 2.5
2
g := 32.2
s
ft s
2
( assumed )
Braking Distance 2
d :=
V 2⋅ g⋅
d
⎛ a ⎞ ⎝ g ⎠
= 194.46
ft
(Eq. 2.47)
Perception/Reaction Distance d r := V⋅ tr
d r = 165.00 ft
(Eq. 2.49)
Total Stopping Distance d s
:= d +
d r
d s
= 359.46
ft
(Eq. 2.50)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) just the braking distance value d
= 194.46
ft
2) just the perception/reaction distance value d r = 165.00
ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
3) use the yellow signal interval deceleration rate a := 10.0 2
d :=
d s
V
⎛ a ⎞ 2⋅ g⋅ ⎝ g ⎠
:= d +
d r
d
= 217.80
ft
d s
= 382.80
ft
Problem 2.36
Determine the vehicle sp eed.
CD := 0.35
G := 0.04 2
A f := 16 ft W
:= 2500
V1 := 88⋅
S := 150
γ b := 1.04 η b := 1
ft slugs
lb
ρ := 0.002378
5280
ft
3600
s
g := 32.2
ft
ρ 2
⋅ CD⋅ A f
K a
μ := 0.8
3
f rl := 0.017
ft s
K a :=
(given)
2
= 0.007
Given V2 := 0
S
⎡ η ⋅ μ ⋅ W + K ⋅ V 2 + f ⋅ W + W⋅ G ⎤ b a 1 rl ⋅ ln 2⋅ g⋅ K a ⎢ ⎥ 2 ⎣ η b⋅ μ ⋅ W + K a⋅ V2 + f rl⋅ W + W⋅ G⎦ γ b⋅ W
(Eq. 2.39)
( )
V2 := Find V2
V2 = 91.6
V2 1.467
= 62.43
mi h
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers:
1) Use 0% grade G := 0.0 Given V2 := 0
⎡ η ⋅ μ ⋅ W + K ⋅ V 2 + f ⋅ W + W⋅ G ⎤ b a 1 rl ⋅ ln 2⋅ g⋅ K a ⎢ ⎥ 2 W + K a⋅ V2 + f rl⋅ W + W ⋅ G η ⋅ μ ⋅ b ⎣ ⎦ γ b⋅ W
S
( )
V2 := Find V2
V2
V2 = 93.6
1.467
mi
= 63.78
h
2) Ignoring aerodynamic resistance G := 0.04
V2 :=
V2 1.467
2
V1
−
(
S⋅ 2⋅ g ⋅ η b⋅ μ
+
)
f rl + G
V2 = 93.3
γ b
(Eq 2.43) rearranged to solve for V2
mi
= 63.6
h
3) Ignoring aerodynamic resistance and using G = 0
G := 0
V2 :=
V2 1.467
2
V1
= 64.9
−
(
S⋅ 2⋅ g⋅ η b ⋅ μ
γ b
+
)
f rl + G
V2 = 95.2
mi h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany
Principles of Highway Engineering and Traffic Analysis, 4e By Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski
Chapter 3 Geometric Design of Highways U.S. Customary Units
Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Preface The solutions to the fourth edition of Principles of Highway Engineering and Traffic Analysis 1 were prepared with the Mathcad software program. You will notice several notation conventions that you may not be familiar with if you are not a Mathcad user. Most of these notation conventions are self-explanatory or easily understood. The most common Mathcad specific notations in these solutions relate to the equals sign. You will notice the equals sign being used in three different contexts, and Mathcad uses three different notations to distinguish between each of these contexts. The differences between these equals sign notations are explained as follows. •
The ‘:=’ (colon-equals) is an assignment operator, that is, the value of the variable or expression on the left side of ‘:=’is set equal to the value of the expression on the right side. For example, in the statement, L := 1234, the variable ‘L’ is assigned (i.e., set equal to) the value of 1234. Another example is x := y + z. In this case, x is assigned the value of y + z.
•
The ‘=’ (bold equals) is used when the Mathcad function solver was used to find the value of a variable in the equation. For example, in the equation , the = is used to tell Mathcad that the value of the expression on the left side needs to equal the value of the expression on the right side. Thus, the Mathcad solver can be employed to find a value for the variable ‘t’ that satisfies this relationship. This particular example is from a problem where the function for arrivals at some time ‘t’ is set equal to the function for departures at some time ‘t’ to find the time to queue clearance.
•
The ‘=’ (standard equals) is used for a simple numeric evaluation. For example, referring to the x := y + z assignment used previously, if the value of y was 10 [either by assignment (with :=), or the result of an equation solution (through the use of =) and the value of z was 15, then the expression ‘x =’ would yield 25. Another example would be as follows: s := 1800/3600, with s = 0.5. That is, ‘s’ was assigned the value of 1800 divided by 3600 (using :=), which equals 0.5 (as given by using =).
Another symbol you will see frequently is ‘→’. In these solutions, it is used to perform an evaluation of an assignment expression in a single statement. For example, in the following statement, , Q(t) is assigned the value of 2 Arrivals(t) – Departures(t), and this evaluates to 2.2t – 0.10t .
Finally, to assist in quickly identifying the final answer, or answers, for what is being asked in the problem statement, yellow highlighting has been used (which will print as light gray). 1
www.mathcad.com
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 3.6 Compute the difference in design curve lengths for 2005 and 2025 designs. G1 := 1
G2 := −2
A := G2 − G1
A = 3
(given)
find required L for 70 mi/h design speed K := 247
(Table 3.2)
L2005 := K ⋅ A
L2005 = 741
(Eq. 3.17)
SSD2005 := 730 V := 70⋅
(Table 3.1)
5280
V = 102.667
3600
H1 := 3
H2 := 1
g := 32.2
G := 0
(given)
For 2025 values, a increases by 25% and t r increases by 20% a2025 := 11.2⋅ 1.25
a2025 = 14
t r2025 := 2.5 ⋅ 1.2
tr2025 = 3
Calculate required stopping sight distance in 2025 V
S2025 := 2⋅g⋅
2
⎡⎛ a2025 ⎞ ⎣⎝
g
⎠
⎤
+ V ⋅ t r2025
S2025 = 684.444
(Eq. 3.12)
−G
⎦
Using this distance, calculate required minimum curve length in 2025 2
A ⋅ S2025
L2025 :=
200⋅
(
(Eq. 3.13)
)2
H1 +
H2
L2025 = 941.435
Diff := L2025 − L2005
Diff = 200.43
ft
Al ter nat iv e Sol ut io n 2
L2002 :=
A ⋅ SSD2005 2158
L2005 = 741
L2025 − L2005 = 200.435
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
)2
(
K c ⋅ 4 + Gcon 200
−
(
)
4.0 ⋅ Kc ⋅ 4 + Gcon 100
+
(
Gcon ⋅ Ltotal − Kc ⋅ 4 + Gcon 100
)
− Ks ⋅ Gcon
2
+
K s ⋅ Gcon 200
Δ elev
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
(
G con − G 1s 200
)
2
⋅K s
+
G con
(
)
⋅ L total − G con − G 1s ⋅ K s − G con − G 2c ⋅ K c + 100
(
G con − G 2c 200
)
2
⋅K c
elev diff +
(
⎡
G con − G 1s ⋅ K s ⋅ G 1s
⎣
100
)
+
G con − G 2c ⋅ K c ⋅ G 2c
)⎤
100
(
⎦
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 3.24 Determine the station of th e PT. sta PC := 12410 e := 0.06
sta PI := 13140
V := 70
(given)
g := 32.2
f s := 0.10
(Table 3.5)
calculate radius Rv :=
( V ⋅ 1.467)
(
g ⋅ e + f s
2
)
Rv = 2046.8
since road is single-lane,
(Eq. 3.34)
R := Rv
R = 2046.8 T := sta PI − sta PC
T = 730
knowing tangent length and radius, solve for central angle T
R ⋅ tan
⎛ Δ ⎞ ⎝ 2 ⎠
Δ := 2 ⋅ atan
⎛ T ⎞ ⎝ R ⎠
Δ = 39.258deg
Δ := 39
(Eq. 3.36)
calculate length L :=
π 180
⋅ R⋅ (Δ )
sta PT := sta PC + L
L = 1393.2 sta PT = 13803.229
(Eq. 3.39) sta PT = 138 + 03.23
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 3.28 Determine the radius and s tationing of the PC and PT. sta PI := 25050 e := 0.08
g := 32.2
V := 65
(given)
Δ := 35
f s := 0.11
(Table 3.5)
calculate radius
Rv :=
( V ⋅ 1.467)
(
2
Rv = 1486.2
)
g ⋅ f s + e
(Eq. 3.34)
ft
since the road is two-lane with 12-ft lanes R := Rv + 6
R = 1492.2
ft
calculate length and tangent length of curve L :=
π 180
⋅R⋅Δ
T := R ⋅ tan
L = 911.534
⎡⎛ Δ ⎞ ⋅ deg⎤ ⎣⎝ 2 ⎠ ⎦
(Eq. 3.39)
T = 470.489
(Eq. 3.36)
sta PC := sta PI − T
sta PC = 24579.511
sta PC = 245+79.51
sta PT := sta PC + L
sta PT = 25491.044
sta PT = 254+91.04
Problem 3.29 Give the radius, degree of curvature, and length of cur ve that you would recommend.
Δ := 40
2 10-ft lanes
(given)
for a 70 mph design speed with e restricted to 0.06, R := Rv + L :=
D :=
π 180
5 2
⋅ R⋅ Δ
18000
π ⋅R
R = 2052.5
L = 1432.92
D = 2.79
Rv := 2050 ft
(Table 3.5)
ft
ft
degrees
(Eq. 3.39)
(Eq. 3.35)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 3.32 Determine a maximum safe speed to the nearest 5 mi/h.
Δ := 34
(given)
e := 0.08
PT := 12934 L := PT − PC
PC := 12350 L = 584
12 since this is a two-lane road with 12-ft lanes, M := 20.3 + s 2 L
π 180
⋅ R⋅ Δ
Rv := R − 6
R :=
L ⋅ 180
π ⋅Δ
R = 984.139
Ms = 26.3 (Eq. 3.39)
Rv = 978.139
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 3.33 Determine the distance that mu st be cl eared from the inside edge of t he inside l ane to pro vide adequate SSD. V is 70 mi/h SSD := 730
(Table 3.1)
Rv := 2050
⎛
Ms := Rv⋅ 1 − cos
⎝
(Prob. 3.29)
⎛ 90⋅ SSD ⋅ deg ⎞ ⎞ ⎝
π ⋅ Rv
(Eq. 3.42)
⎠ ⎠
Ms = 32.41 To inside edge of inside lane (subtracting 1/2 of lane width) Ms − 5 = 27.41
ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 3.35 Determine the length of th e horizontal curve. G1 := 1
G2 := 3
Ls := 420
Δ := 37
As := G2 − G1 Ks :=
Ls As
(given) e := 0.06
As = 2 Ks = 210
(Eq. 3.10)
safe design speed is 75 mi/h (K = 206 for 75 mi/h) Rv1 := 2510
or
V := 75
(Table 3.5)
f s := 0.09 ( V ⋅ 1.467)
2
Rv2 := 32.2 ⋅ fs + e
(
)
Rv2 = 2506.31
since the road is two-lane with 12-ft lanes, L :=
π 180
⋅ R⋅ Δ
(Table 3.3)
L = 1624.76
ft
R := Rv1 +
(Eq. 3.34) 12 2
R = 2516 (Eq. 3.39)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 3.36 Determine Determine the station of th e PT. PT. G1 := −2.5
G2 := 1.5
Δ := 38
e := 0.08
L := K ⋅ A
L = 824
A := G2 − G1
A = 4
K := 206
(given)
PVT := 2510 (Eq. 3.10)
PVC := PVT − L
PVC = 1686
PC := PVC − 292
PC = 1394
Rv := 2215
(Table 3.5)
since the road is two-lane, 12-ft lanes R := Rv + L :=
12 2
π ⋅ R⋅ Δ 180
PT := PC + L
R = 2221
ft
L = 1473.02
ft
PT = 2867.02
(Eq. 3.39) sta PT = 28 + 67.02
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Multiple Choice Problems Determine Determine the elevation elevation of the lowest poin t of t he curve. G1 := −4.0 c := 500
G2 := 2.5
Probl em 3.38 3.38
L := 4 stations
(given)
ft
stationing and elevation for lowest point on the curve dy dx
:= ( 2a⋅ x + b)
(Eq. 3.1)
0
(Eq. 3.3)
b := −4.0
a :=
x :=
G2 − G1 2⋅ L
− b 2⋅ a
(Eq. 3.6)
a = 0.813
(Eq. 3.1)
x = 2.462 stations
Lowe Lowest st Poi Point nt sta stati tion onin ing: g:
(100 (100 + 00) 00) + (2 (2 + 46) 46) = 102 102 + 46 46
Lowest Point elevation:
y := a⋅ x
( 2) + b⋅ x + c
y = 495.077 ft
(Eq. 3.1)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Miscalculation
y := 492.043
2) Miscalculate "a"
a :=
G1 − G2 2⋅ L
ft
a = −0.813
( 2) + b⋅ x + c
y := a⋅ x
3) Assume lowest point at L/2
Station = 102 + 46
y = 485.231
ft
Station = 102 + 00 x := 2
( 2) + b⋅ x + c
y := a⋅ x
y = 495.25
ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Determine Determine the station of
T := 1200
PT.
Δ :=
ft
Problem 3.39 ⋅ 0.5211180
(given)
π
Calculate radius T
R :=
⎡⎛ Δ ⎞
tan [mc]
⎣⎝ 2 ⎠
⋅ deg⎤
R = 4500.95 ft
(Eq 3.36)
L = 2345.44 ft
(Eq 3.39)
⎦
Solve for length of curve L :=
π 180
⋅ R ⋅ Δ
Calculate stationing of PT stationing PC = 145 + 00 minus 12+00 = 133 + 00 stationing PT = stationing PC + L = 133 + 00 plus 23 + 45.43 = 156 + 45.43
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Add length of curve to stationing PI stationing PT = 145 + 000 plus 23 + 45.43 = 168 + 45.43 2) Use radians instead of degrees T
R := tan
L :=
⎛ 0.5211 ⎞ ⎝
π 180
2
R = 4500.95 ft
⎠
⋅ R ⋅ 0.5211
L = 40.94
ft
stationing PT = 133 + 00 plus 40 + 94 = 173 + 94 3) add half of length to stationing PI stationong PT = 145 + 00 plus 11 +72.72 = 156 + 72.72
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 3.40
Determine Determine the o ffset.
G1 := 5.5 %
G2 := 2.5 %
x := 750 ft
L := 1600 ft
(given)
determine the absolute value of the difference of grades A := G1 − G2
A=3
determine offset at 750 feet from the PVC
Y :=
A
2
200 200⋅ L
⋅x
Y = 5.273 ft
(Eq 3.7)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: Answers: 1) Use Ym equation. Ym :=
A⋅L 800
Ym = 6
ft
(Eq 3.8)
Yf = 24
ft
(Eq 3.9)
2) Use Yf equation. Yf :=
A⋅L 200
3) Use 0.055 and 0.025 for grades. G1 := 0.055
G2 := 0.025
A := G1 − G2 Y :=
A 200L ⋅
2
⋅x
A = 0.03 Y = 0.053 ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Determine the minimum length of curve.
V := 65⋅
⎛ 5280 ⎞ ⎝ 3600 ⎠
ft
Problem 3.41
G1 := 1.5
s
G2 := −2.0
(given)
ignoring the effect of grades using Table 3.1, SSD for 65 mi/h would be 645 ft (assuming L > SSD) SSD := 645
(Table 3.1)
ft
A := G1 − G2
A = 3.50
2
Lm :=
A ⋅ SSD 2158
Lm = 674.74
ft
(Eq. 3.15)
674.74 > 645
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers 1) assume L < SSD Lm := 2⋅ SSD −
2158
Lm = 673.43 ft
A
(Eq. 3.16)
2) Misinterpret chart for 70 mi/h 2
SSD := 730
Lm :=
A ⋅ SSD 2158
Lm = 864.30 ft
3) Assume SSD is equivalent to L m
SSD := 645
Lm := SSD
therefore
Lm = 645.00 ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the stopping si ght dis tance.
V1 := 35⋅
a := 11.2
5280
ft
3600
s
ft s
g := 32.2
2
Problem 3.42 G :=
ft s
2
3
(given)
100
tr := 2.5 s
(assumed)
Determine stopping sight distance
2
V1
SSD := 2⋅ g ⋅
⎛ a
− G ⎞
⎝ g
+ V1⋅ tr
SSD = 257.08 ft
(Eq 3.12)
⎠
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Assume grade is positive (uphill)
2
SSD :=
V1
⎛ a + G ⎞ 2⋅ g ⋅ ⎝ g ⎠
+ V1⋅ tr
SSD = 236.63 ft
2) Use g = 9.81 m/s 2 instead of g = 32.2 ft/s 2 g := 9.81
m s
2 2
SSD :=
V1
a 2⋅ g ⋅ ⎛ − G ⎞ ⎝ g ⎠
+ V1⋅ tr
SSD = 249.15 ft
3) Miscalculation SSD := 254.23 ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the minimum length o f the vertical curve. G1 := 4.0
G2 := −2.0
H1 := 6.0 ft
Problem 3.43
H2 := 4.0 ft
(given) S := 450 ft
V := 40⋅
5280
ft
3600
s
Calculate the minim length of vertical curve A := G1 − G2
200⋅
Lm := 2⋅ S −
(
)
H1 +
H2
A
2
Lm = 240.07 ft
(Eq 3.14)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Use equation 3.13 2
A⋅ S
Lm := 200⋅
(
H1 +
)
H2
Lm = 306.85 ft
2
(Eq 3.13)
2) Use AASHTO guidelines for heights and equation 3.13 H1 := 3.5 ft
H2 := 2.0 ft 2
A⋅ S
Lm := 200⋅
(
H1 +
)
H2
Lm = 562.94 ft
2
3) Solve for S and not Lm
Lm := 450 ft S :=
Lm + 200⋅
(
H1 + 2
)
H2
2
S = 1304.15 ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany
Principles of Highway Engineering and Traffic Analysis, 4e By Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski
Chapter 4 Pavement Design U.S. Customary Units
Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Preface The solutions to the fourth edition of Principles of Highway Engineering and Traffic Analysis 1 were prepared with the Mathcad software program. You will notice several notation conventions that you may not be familiar with if you are not a Mathcad user. Most of these notation conventions are self-explanatory or easily understood. The most common Mathcad specific notations in these solutions relate to the equals sign. You will notice the equals sign being used in three different contexts, and Mathcad uses three different notations to distinguish between each of these contexts. The differences between these equals sign notations are explained as follows.
•
The ‘:=’ (colon-equals) is an assignment operator, that is, the value of the variable or expression on the left side of ‘:=’is set equal to the value of the expression on the right side. For example, in the statement, L := 1234, the variable ‘L’ is assigned (i.e., set equal to) the value of 1234. Another example is x := y + z. In this case, x is assigned the value of y + z.
•
The ‘=’ (bold equals) is used when the Mathcad function solver was used to find the value of a variable in the equation. For example, in the equation , the = is used to tell Mathcad that the value of the expression on the left side needs to equal the value of the expression on the right side. Thus, the Mathcad solver can be employed to find a value for the variable ‘t’ that satisfies this relationship. This particular example is from a problem where the function for arrivals at some time ‘t’ is set equal to the function for departures at some time ‘t’ to find the time to queue clearance.
•
The ‘=’ (standard equals) is used for a simple numeric evaluation. For example, referring to the x := y + z assignment used previously, if the value of y was 10 [either by assignment (with :=), or the result of an equation solution (through the use of =) and the value of z was 15, then the expression ‘x =’ would yield 25. Another example would be as follows: s := 1800/3600, with s = 0.5. That is, ‘s’ was assigned the value of 1800 divided by 3600 (using :=), which equals 0.5 (as given by using =).
Another symbol you will see frequently is ‘→’. In these solutions, it is used to perform an evaluation of an assignment expression in a single statement. For example, in the following statement, , Q(t) is assigned the value of 2 Arrivals(t) – Departures(t), and this evaluates to 2.2t – 0.10t .
Finally, to assist in quickly identifying the final answer, or answers, for what is being asked in the problem statement, yellow highlighting has been used (which will print as light gray). 1
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Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.1 Determine radial-horizontal st ress. P := 5000
p := 100
(given)
E := 43500 a :=
P
a
p ⋅ π z := 0
at z a
and r
=0
A := 1.0
a
= 3.989
(Eq. 4.3)
r := 0.8
= 0.201
H := 1.97987
(Table 4.1)
Solving for Poisson's ratio
Δ z := 0.016
(given)
Find μ:
Δz
p⋅ ( 1 +
μ) ⋅ a
⋅ ⎡ ⋅ A + (1 − μ) ⋅ H⎤ z
⎣a
E
⎦
(Eq. 4.6)
μ = 0.345 function C := 0
σr := p⋅ 2μ ⋅ A + σr = 84.47
and function F := 0.5 C + ( 1 − 2μ) F
(Eq. 4.5)
lb 2
in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine Modulus of Elasticity. contactarea
P := 6700
in
lb
(given)
contactarea
a :=
a
2
:= 80
Problem 4.2
π
= 5.046
inches
P
p := a
(Eq. 4.3)
2
⋅π lb
p
= 83.75
2
in
z := 2 z a
r := 0
and
r
= 0.3963
=0
a
Using linear interpolation: A := 0.62861
H := 1.35407
and
Δ z := 0.035
Δz E :=
μ := 0.5
p ⋅ ( 1 +
μ) ⋅ a
z
⎣a
E p ⋅ ( 1 +
⋅⎡ ⋅A +
μ) ⋅ a
⋅⎡ ⋅A + z
⎣a
Δz 4
E = 1.678 × 10
(Table 4.1) (given)
⎤ ⎦
(1 − μ )⋅ H
(Eq. 4.6)
⎤ ⎦
( 1 − μ)⋅ H
lb 2
in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.3 Determine load applied to th e wheel.
z
:= 0 z
=
a A
r := 0
and
r
0
a
:= 1
=
a := 3.5
and
(given)
0
C := 0
F
:= 0.5
H := 2
(Table 4.1)
σ r := p [ 2μ ⋅ A + C + ( 1 − 2μ )F]
(Eq. 4.5)
lb
σ r := 87
2
in
Therefore,
p
:=
σ r 2μ
p
+ 0.5 − μ
:=
σ r μ + 0.5
Find μ by substituting p into Equation 4.6
Δ z := 0.0165
p ⋅ ( 1 + μ ) ⋅ a
Δz
z
(Eq. 4.6)
⎦
⎛ 87 ⎞ ⋅( 1 + μ ) ⋅ 3.5 ⎝ μ + 0.5 ⎠ ⋅[ 0 + ( 1 − μ ) ⋅2.0] 43500
μ =
:=
⋅⎡ ⋅A + ( 1 − μ ) ⋅H⎤
⎣a
E
Δz
p
(given)
0.281 87
p
μ + 0.5
:=
P
:= a2( p π ⋅ )
P
:= 12.25⋅( p ⋅ π )
P
=
lb 2
(Eq. 4.3)
p π ⋅
4285.20
111.349
in
P
a
=
lb
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.4
Determine the vertical stress, radial-horizontal stress, and deflection.
:= 25
z z a
=
r := 50
and
r
1.969
a
A
:= 0.0116
p
:= 101.5
=
and
a
(given)
:= 12.7
3.937
B := − 0.0041 C := 0.01527
F
:= −0.005465
H := 0.22418
(Table 4.1)
μ := 0.4
σ z := p ⋅( A + B)
(Eq. 4.4)
lb
σz =
0.761
2
in
σ r := p ⋅ 2μ ⋅A + C + (1 − 2μ )F σ r =
(Eq. 4.5)
lb
2.381
2
in E := 36250
Δ z :=
p ⋅ ( 1 +
Δz =
μ ) ⋅a
⋅⎡ ⋅A + (1 − μ ) ⋅ H⎤ z
⎣a
E
−3
7.833 × 10
⎦
(Eq. 4.6)
inches
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.5 Determine which truck will c ause more pavement damage. a1 := 0.44
a2 := 0.2
a3 := 0.11
D1 := 3
D2 := 6
D3 := 8
(given)
M 3 := 1.0
(given)
M2 := 1.0
SN := a1⋅ D1 + a2⋅ D2⋅ M 2 + a3⋅ D3⋅ M 3
(Table 4.6)
(Eq. 4.9)
SN = 3.4
Using linear interpolation for Truck A
(Table 4.2, 4.3, and 4.4)
single12kip := 0.2226 single23kip := 2.574 Total18kipESALA := single12kip + single23kip Total18kipESALA
= 2.7966
Using linear interpolation for Truck B single8kip := 0.0046 tandem43kip := 2.706 Total18kipESALB := single8kip + tandem43kip Total18kipESALB = 2.7106
Therefore, Truck A causes more damage.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.6 How many 25-kip singl e-axle loads can be c arried before the pavement reaches its TSI? a1 := 0.44
a2 := 0.18
a3 := 0.11
D1 := 4
D2 := 7
D3 := 10
(given)
M2 := 0.9
M3 := 0.8
(given)
(Table 4.6)
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M 3
(Eq. 4.9)
SN = 3.774 ZR := −1.282 So := 0.4
(Table 4.5)
ΔPSI := 2.0
MR := 5000
lb
(given)
2
in
Using Eq. 4.7:
⎡⎡ x := ⎢⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢⎢ ⎣⎣
⎤ ⎤ + 2.32⋅ log( MR )⎥ − 8.07⎥ 1094 ⎤ ⎥ ⎥ 0.40 + ⎡ 5.19 ⎣ (SN + 1) ⎦ ⎦ ⎦ log
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
x = 5.974 x
5
W 18 := 10
W 18 = 9.4204645× 10
Interpolating from Table 4.2 to determine equivalent 25-kip axle load equivalency factor: TableValue1 :=
TableValue2 :=
3.09 + 4.31
TableValue1 = 3.7
2 2.89 + 3.91
TableValue2 = 3.4
2
EquivFactor := 3.7 −
⎛ TableValue1 − TableValue2 ⎞ ⋅ ( SN − 3) 1 ⎝ ⎠
EquivFactor = 3.4678 W 18 EquivFactor
= 2.716554×
5
10
25-kip loads
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.7 Determine minimum acceptable soil resilient modu lus. a1 := 0.35
a2 := 0.20
a3 := 0.11
D1 := 4
D2 := 6
D3 := 7
(given)
M2 := 1
M3 := 1
(given)
(Table 4.6)
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M 3
(Eq. 4.9)
SN = 3.37
Axle Loads (interpolating):
(Table 4.2, 4.3, and 4.4)
single10kip := 0.1121300 ⋅ single18kip := 1⋅ 120 single23kip := 2.578100 ⋅ tandem32kip := 0.8886100 ⋅ single32kip := 9.87130 ⋅ triple40kip := 0.546100 ⋅ TotAxleEqv := single10kip + single18kip + single23kip + tandem32kip + single32kip + triple40kip W 18 := TotAxleEqv⋅ 10⋅ 365 6
W 18 = 3.106 × 10 ZR := −1.036
ΔPSI := 2.2
(Table 4.5)
So := 0.30
⎡ log ( W 18) ⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢ ⎣ 3
MR = 9.0098× 10
⎤ + 2.32⋅ log( MR )⎥ − 8.07 1094 ⎤ ⎥ 0.40 + ⎡ 5.19 ⎣ (SN + 1) ⎦ ⎦ log
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
lb 2
in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine minimum acceptable soil resilient modu lus. a1 := 0.35
a2 := 0.20
D1 := 4
D2 := 6
a3 := 0.11
M2 := 1.0
Problem 4.8 (Table 4.6)
D3 := 7
(given)
M3 := 1.0
(given)
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M 3
(Eq. 4.9)
SN = 3.37
Axle Loads (interpolating):
(Table 4.2, 4.3, and 4.4)
single10kip := 0.1121300 ⋅ single18kip := 1⋅ 20 single23kip := 2.682100 ⋅ tandem32kip := 0.8886100 ⋅ single32kip := 9.87190 ⋅ triple40kip := 0.546100 ⋅
TotAxleEqv := single10kip + single18kip + single23kip + tandem32kip + single32kip + triple40kip W 18 := TotAxleEqv⋅ 10⋅ 365 ZR := −1.036
ΔPSI := 2.2
(Table 4.5)
So := 0.30
(given)
Using Eq. 4.7:
⎡ log ( W 18) ⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢ ⎣ 4
M R = 1.1005× 10
⎤ + 2.32⋅ log( MR )⎥ − 8.07 1094 ⎤ ⎥ 0.40 + ⎡ 5.19 ⎣ (SN + 1) ⎦ ⎦ log
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
lb 2
in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the soil resilient modul us of th e soil used in design.
Problem 4.9
a1 := 0.44
a2 := 0.40
(Table 4.6)
D1 := 4
D2 := 4
D3 := 8
(given)
M2 := 1.0
M3 := 1.0
(given)
a3 := 0.11
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M 3
(Eq. 4.9)
SN = 4.24
Axle Loads (interpolating):
(Tables 4.2 and 4.3)
single8kip := 0.039251300 ⋅ = 51.025 tandem15kip := 0.0431900 ⋅ = 38.79 single40kip := 22.16420 ⋅ = 443.28 tandem40kip := 2.042200 ⋅ = 408.4 TotAxleEqv := single8kip + tandem15kip + single40kip + tandem40kip TotAxleEqv = 941.495 6
W 18 := TotAxleEqv⋅ 12⋅ 365 = 4.1237× 10 ZR := −0.524 So := 0.5
(Table 4.5)
ΔPSI := 2.0
(given)
Using Eq. 4.7:
⎡ log( W 18) ⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢ ⎣ MR = 5250.1
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
⎤ + 2.32⋅ log( MR )⎥ − 8.07 1094 ⎡ ⎤ ⎥ 0.40 + 5.19 ⎣ ( SN + 1) ⎦ ⎦ log
lb 2
in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.10 Determine the reduction i n pavement design life. SN := 3.8
So := 0.40
PSI := 4.7
W 18 := 1800⋅ 365⋅ N
TSI := 2.5
ZR := −1.645
(given)
(for 95% reliability)
(Table 4.5)
CBR := 9
Current Design Life:
ΔPSI := PSI − TSI
ΔPSI = 2.2
MR := 1500⋅ CBR
MR = 1.35 × 10
(given) 4
Using Eq. 4.7:
⎡ log ( W 18) ⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢ ⎣
⎤ + 2.32⋅ log( MR )⎥ − 8.07 1094 ⎤ ⎥ 0.40 + ⎡ 5.19 ⎣ (SN + 1) ⎦ ⎦ log
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
6
W 18 = 8.0730705× 10 N before :=
W 18 1800⋅ 365
N before
= 12.288
3
1800⋅ 1.3 = 2.34 × 10
Nafter :=
W 18 2340⋅ 365
Δyears := N before − Nafter
Nafter = 9.452
Δyears = 2.836
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.11 How many years would yo u be 95% sure the pavement will last? ZR := −1.645
(Table 4.5)
MR := 5000
ΔPSI := 1.9
So := 0.45
(given)
a1 := 0.35
a2 := 0.20
a3 := 0.11
D1 := 6
D2 := 9
D3 := 10
(given)
M2 := 1.0
M3 := 1.0
(given)
(Table 4.6)
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M3
(Eq. 4.9)
SN = 5
Axle Loads (interpolating):
(Table 4.2, 4.3, and 4.4)
single2kip := 0.000420000 ⋅ single10kip := 0.088200 ⋅ tandem22kip := 0.18⋅ 200 single12kip := 0.189410 ⋅ tandem18kip := 0.077410 ⋅ triple50kip := 1.22⋅ 410 TotAxleEqv := single2kip + single10kip + tandem22kip + single12kip + tandem18kip + triple50kip TotAxleEqv = 670.86
Using Eq. 4.7:
⎡ log ( W 18) ⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢ ⎣
⎤ + 2.32⋅ log( MR )⎥ − 8.07 1094 ⎡ ⎤ ⎥ 0.40 + 5.19 ⎣ (SN + 1) ⎦ ⎦ log
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
6
W 18 = 3.545637× 10 N :=
W 18 TotAxleEqv⋅ 365
N = 14.48
years
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.12 Determine the Struc tural Number. N := 15
(given)
Assume: SN := 4.0
Axle Loads: single10kip := 0.1025000 ⋅ single24kip := 2.89⋅ 400 tandem30kip := 0.6951000 ⋅ tandem50kip := 4.64⋅ 100 TotAxleEqv := single10kip + single24kip + tandem30kip + tandem50kip 3
TotAxleEqv = 2.825× 10
7
DirW18 := TotAxleEqv⋅ 365⋅ N
DirW18 = 1.5466875× 10
PDL := 0.8
(Table 4.11)
DesignLaneW 18 := 0.8⋅ DirW18 7
DesignLaneW 18 = 1.23735× 10 ZR := −1.282
(Table 4.5)
So := 0.4
ΔPSI := 1.8 MR := 13750
Using Eq. 4.7:
⎡ log( DesignLaneW 18) ⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢ ⎣ SN = 4
⎤ + 2.32⋅ log( MR )⎥ − 8.07 1094 ⎡ ⎤ ⎥ 0.40 + 5.19 ⎣ ( SN + 1) ⎦ ⎦ log
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
Therefore the assumed value of 4 is good.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.13 Determine the probablilit y (reliability) that this pavement will last 20 years before reaching it s terminal s erviceability . a1 := 0.44
a2 := 0.44
a3 := 0.11
D1 := 5
D2 := 6
D3 := 10
(given)
M2 := 1.0
M3 := 1.0
(given)
(Table 4.6)
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M 3
(Eq. 4.9)
SN = 5.94 N := 20
years
Axle Loads (interpolating):
(Tables 4.2, 4.3, 4.4)
single20kip := 1.538200 ⋅ tandem40kip := 2.122200 ⋅ single22kip := 2.26480 ⋅ TotAxleEqv := single20kip + tandem40kip + single22kip TotAxleEqv = 913.12 W 18 := TotAxleEqv⋅ N⋅ 365 So := 0.6
M R := 3000
6
W 18 = 6.665776× 10
ΔPSI := 2.0
(given)
Using Eq. 4.7
⎡ log ( W 18) ⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢ ⎣
⎤ + 2.32⋅ log( MR )⎥ − 8.07 1094 ⎡ ⎤ ⎥ 0.40 + 5.19 ⎣ (SN + 1) ⎦ ⎦ log
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
ZR = −0.92773 R := 82.3
%
(interpolating from Table 4.5)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.14 Determine the probabili ty that t he pavement wil l have a PSI greater th an 2.5 after 20 years. PSI := 4.5
(given)
TSI := 2.5
ΔPSI := PSI − TSI
ΔPSI = 2
MR := 12000 So := 0.40 a1 := 0.35 D1 := 4
a2 := 0.18
a3 := 0.11
D2 := 6
D3 := 8
(given)
M2 := 1.0
M3 := 1.0
(given)
(Table 4.6)
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M 3
(Eq. 4.9)
SN = 3.36 N := 20
years
W 18 := 1290⋅ N⋅ 365
6
W 18 = 9.417 × 10
Using Eq. 4.7:
⎡ log ( W 18) ⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢ ⎣
⎤ + 2.32⋅ log( MR )⎥ − 8.07 1094 ⎡ ⎤ ⎥ 0.40 + 5.19 ⎣ (SN + 1) ⎦ ⎦ log
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
ZR = −0.1612 R := 56.40
%
(interpolating from Table 4.5)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.15 Determine probabilit y of pavement lastin g 25 years. a1 := 0.35
a2 := 0.20
a3 := 0.11
D1 := 4
D2 := 10
D3 := 10
M2 := 1.0
M3 := 1.0
SN := a1⋅ D1 + a2⋅ D2 M 2 + a3⋅ D3⋅ M 3 N := 25
(Table 4.6) (given) (given)
SN = 4.5
(Eq. 4.9)
years
Axle load equivalency factors from Tables 4.2 and 4.3, interpolating for SN = 4.5 single20kip := 1.49⋅ 400 tandem35kip := 1.24⋅ 900 TotAxleEqv := single20kip + tandem35kip 3
TotAxleEqv = 1.712× 10
W 18 := TotAxleEqv⋅ 365⋅ 25
So := 0.45
CBR := 8
M R := 1500⋅ CBR
PSI := 4.2
TSI := 2.5
ΔPSI := PSI − TSI
4
MR = 1.2 × 10
(given)
Using Eq. 4.7: log
(
)
log W 18
ZR ⋅ So
+
9.36(log( SN + 1) )
− 0.20 + 0.40 +
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
⎡ ⎤ 5.19 ⎣ ( SN + 1) ⎦ 1094
+
(
) − 8.07
2.32⋅ log MR
ZR = −1.265
Probability is approximately 89.7% (interpolating from Table 4.5), that PSI will equal 2.5 after 25 years; thus, probabilities less than this will correspond to PSIs above 2.5 after 25 years
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.16
Determine the struc tural numb er to be used s o the PCC and flexible pavements have the same life expectancy. D := 10
So := 0.35
R = 90%
PSI := 4.6
Sc := 700
TSI := 2.5 6
Ec := 4.5⋅ 10
CBR := 2
J := 3.0
Cd := 1.0
ΔPSI := PSI − TSI
(given)
ΔPSI = 2.1
k := 100
(Table 4.10)
ZR := −1.282
(Table 4.5)
Rigid Pavement Design Using Eq. 4.19:
⎛ ΔPSI ⎞ ⎝ 3.0 ⎠
0.75 ⎡ ⎤ Sc Cd ⋅ ( D − 1.132) x := ZR ⋅ So + 7.35⋅ ( log ( D + 1) ) − 0.06 + + ( 4.22 − 0.32⋅T SI) ⋅ log ⎢ 7 ⎤ 0.75 ⎡ 18.42 ⎤ ⎤ ⎥ ⎡ 1.62410 ⎡ ⋅ 215.63 J D ⋅ − ⎢ ⎥ 1+ 0.25 ⎢ ⎢ ⎢ E ⎥ ⎥ ⎥ 8.46 ⎛ c ⎞ ⎣ ( D + 1) ⎦ ⎣ ⎣ ⎣ ⎝ k ⎠ ⎦ ⎦ ⎦
log
x = 7.156 x
W 18 := 10
7
W 18 = 1.433 × 10
PDL := 0.80
For 3 lanes and a conservative design, 6
(
RemainingW18 := W 18 − CurrentW 18
)
7
CurrentW 18 := 13⋅ 10 ⋅ PDL = 1.04 × 10
6
RemainingW18 = 3.928 × 10
Rigid Pavement Axle load equivalency factors from Tables 4.7 and 4.8, for D = 10 single12kip := 0.175 tandem24kip := 0.441
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
TotAxleEqvrigid := single12kip + tandem24kip RemainingTrucks :=
RemainingW18
TotAxleEqvrigid = 0.616 6
RemainingTrucks = 6.376× 10
TotAxleEqvrigid
Flexible Pavement Design Flexible Pavement Axle load equivalency factors from Tables 4.2 and 4.3 Assume SN = 5 single12kip := 0.189 tandem24kip := 0.260 TotAxleEqvflex := single12kip + tandem24kip RemainingW18 := RemainingTrucks⋅ TotAxleEqvflex
TotAxleEqvflex = 0.449 6
RemainingW18 = 2.863 × 10
3
MR := 1500⋅ CBR
MR = 3 × 10
W 18 := RemainingW18
Using Eq. 4.7:
log
(
)
log W 18
ZR ⋅ So
+
9.36(log( SN + 1) )
− 0.20 +
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
1094 ⎤ 0.40 + ⎡ 5.19 ⎣ (SN + 1) ⎦
+
(
) − 8.07
2.32⋅ log MR
SN = 5.07
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.17 Determine pavement slab thic kness. P := 10000 a :=
p := 90
P
= 5.947
a
p ⋅ π
E := 4200000
(given) (Eq. 4.3)
in
μ := 0.25
k := 150
(given)
σe := 218.5 3 ⎞ P ⎞ ⎛ ⎛ E⋅ h ⎞ ⎛ 0.529⋅ ( 1 + 0.54μ) ⋅ ⋅ log − 0.71 2 4 ⎝ h ⎠ ⎝ ⎝ k⋅ a ⎠ ⎠
σe
h
= 10
inches
Problem 4.18
Determine the interior st ress. E := 3500000
μ := 0.30
Δ i := 0.008195
⎡ l
E⋅ h
l := 30.106
⎤
3
h := 8
P := 12000
γ := 0.577215
(given)
Euler's constant
0.25
(Eq. 4.13)
⎣ 12⋅ (1 − μ )⋅ k ⎦ 2
lb
k = 199.758
3
in
Δi
a
2 1 ⎞ ⎛ ⎛ a ⎞ 5 ⎞ ⎛ a ⎞ ⎤ ⎛ ⋅ 1+ ⋅ ln +γ− ⋅ 4 ⎠ ⎝ l ⎠ ⎦ 2 ⎣ 2⋅ π ⎠ ⎝ ⎝ 2⋅ l ⎠ ⎝ 8⋅ k ⋅ l
P
= 4.32
σi :=
⎡
(Eq. 4.12)
inches
3⋅ P⋅ ( 1 + 2⋅ π⋅ h
σi = 297.87
2
μ)
2 2⋅ l ⎞ 3⋅ P⋅ ( 1 + μ ) ⎛ a ⎞ ⎛ ⎛ ⎞ ⋅ ln + 0.5 − γ + ⋅ 2 ⎝ ⎝ a ⎠ ⎠ ⎝ l ⎠ 64⋅ h
(Eq. 4.11)
lb 2
in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.19 Determine the modulus of elasticity o f the pavement. h := 10
P := 17000
μ := 0.36
al := 7
k := 250
Δ c := 0.05
⎡ ⎛ al ⎞ ⎤ 1.205 − 0.69⋅ 2 ⎣ ⎝ l ⎠ ⎦ k⋅ l P
Δc
⋅
(given)
(Eq. 4.18)
l = 38.306
⎡ l
E⋅ h
(
⎣ 12⋅ 1 −
⎤
3
0.25
(Eq. 4.13)
)
2
μ ⋅ k ⎦
6
E = 5.6219543× 10
lb 2
in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.20 Determine the interior and edge stresses, as well as th e interior and edge slab deflections. h := 12
k := 300
E := 4000000
P := 9000
μ := 0.40
a := 5
l :=
⎡
E⋅ h
⎤
3
l = 38.883
⎣ 12⋅ (1 − μ )⋅ k ⎦
σi :=
2⋅ π⋅ h
μ)
2
(given)
0.25
2
3⋅ P⋅ ( 1 +
γ := 0.577215 Euler's constant
2 2⋅ l ⎞ 3⋅ P⋅ ( 1 + μ ) ⎛ a ⎞ ⎛ ⎛ ⎞ ⋅ ln + 0.5 − γ + ⋅ 2 ⎝ ⎝ a ⎠ ⎠ ⎝ l ⎠ 64⋅ h
(Eq. 4.13)
(Eq. 4.11)
lb
σi = 111.492
2
in
2 1 ⎞ ⎛ ⎛ a ⎞ 5 ⎞ ⎛ a ⎞ ⎤ ⎛ Δ i := ⋅ 1+ ⋅ ln +γ− ⋅ 2 ⎣ 4 ⎠ ⎝ l ⎠ ⎦ 2⋅ π ⎠ ⎝ ⎝ 2⋅ l ⎠ ⎝ 8⋅ k ⋅ l
P
⎡
Δ i = 2.45809×
−3
10
(Eq. 4.12)
inches
3 ⎞ P ⎞ ⎛ ⎛ E⋅ h ⎞ ⎛ σe := 0.529⋅ ( 1 + 0.54⋅ μ ) ⋅ ⋅ log − 0.71 2 4 ⎝ h ⎠ ⎝ ⎝ k⋅ a ⎠ ⎠
(Eq. 4.15)
lb
σe = 155.051
2
in
Δ e := 0.408⋅ ( 1 +
Δ e = 9.391 ×
0.4μ) ⋅
−3
10
⎛ P ⎞ 2 ⎝ k⋅ l ⎠
(Eq. 4.16)
inches
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.21 Considering Ex. 4.5, determine which tr uck wi ll cause mor e pavement damage? Axle load equivalency factors for truck A: single12kip := 0.175 single23kip := 2.915 TotAxleEqvA := single12kip + single23kip TotAxleEqvA
= 3.09
18 kip ESAL
Axle load equivalency factors for truck B: single8kip := 0.032 tandem23kip := 5.245 TotAxleEqvB := single8kip + tandem23kip TotAxleEqvB = 5.277
18 kip ESAL
Therefore, Truck B causes more damage.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.22
Determine Estimated Daily Truck Traffic. D := 11
N := 20
ZR := −1.282
Sc := 600
So := 0.35
Cd := 0.8
PSI := 4.8
J := 2.8
TSI := 2.5
(given)
6
Ec := 4⋅ 10
ΔPSI := PSI − TSI
k := 150
ΔPSI = 2.3 Using Eq. 4.19:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd ⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ ⎡ 1.62410 ⎡ 0.75 − ⎡ 18.42 ⎤ ⎤ ⎥⎥ ⋅ 215.63J⋅ D ⎢ 1+ 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ 0.25 ⎥ ⎥ ⎥ ⎣ (D + 1) ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
7
W 18 = 1.162 × 10
Axle load equivalency factors from Tables 4.7, 4.8, and 4.9, for D = 11 single20kip := 1.58 tandem26kip := 0.619 triple34kip := 0.593 TotAxleEqv := single20kip + tandem26kip + triple34kip TotAxleEqv = 2.792
DailyDesignLaneTraffic :=
W 18 365⋅ N⋅ TotAxleEqv
DailyDesignLaneTraffic = 570.2
trucks/day (for design lane)
For 3 lanes and a conservative design, TotalTraffic :=
DailyDesignLaneTraffic PDL
PDL := 0.80 TotalTraffic = 712.8
trucks/day (total for 3 lanes)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.23 Determine how long p avement will last with new loading and the additional lane. D := 11
Cd1 := 1.0 6
Ec := 5⋅ 10
So := 0.3
Sc := 700
J := 3.0
PSI := 4.7
(given)
TSI := 2.5 N1 := 20 years
CBR := 25 k := 290
(Table 4.10)
ZR := −1.645
(Table 4.5)
ΔPSI := PSI − TSI
ΔPSI = 2.2
Before Using Eq. 4.19:
⎡
0.75 ⎡ ⎤⎤ Sc Cd1⋅ ( D − 1.132) x1 := ⎢ ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + + ( 4.22 − 0.32⋅T SI) ⋅ log⎢ ⎥ 7 ⎤ 0.75 ⎡ 18.42 ⎤ ⎤ ⎥ ⎡ ⎡ ⎢ 1.624⋅ 10 215.63 J D ⋅ − ⎢ ⎥⎥ 1+ 0.25 ⎢ ⎢ ⎢ ⎢ ⎛ Ec ⎞ ⎥ ⎥ ⎥ ⎥ 8.46 ( D 1 ) + ⎣ ⎦ ⎢ ⎥ ⎣ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦⎦
log
⎛ ΔPSI ⎝ 3.0 ⎠
x1 = 7.514 x1
W 18before := 10
7
W 18before = 3.265 × 10
Axle load equivalency factors from Tables 4.7 and 4.8, for D = 11 single18kip := 1.0 tandem28kip := 0.850 TotAxleEqv1 := single18kip + tandem28kip DailyDesignLaneTraffic :=
TotAxleEqv1 = 1.85
W 18before 365⋅ N1⋅ TotAxleEqv1 3
DailyDesignLaneTraffic = 2.418× 10
trucks/day
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
For 2 lanes designed conservatively: PDL1 := 1.00
(Table 4.11)
TotalDailyTraffic :=
DailyDesignLaneTraffic
3
TotalDailyTraffic = 2.418× 10
PDL1
trucks/day
After Cd2 := 0.8
(given)
Using Eq. 4.19:
⎡
0.75 ⎡ ⎤⎤ Sc Cd2⋅ ( D − 1.132) x2 := ⎢ ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + + ( 4.22 − 0.32⋅T SI) ⋅ log⎢ ⎥ 7 ⎤ 0.75 ⎡ 18.42 ⎤ ⎤ ⎥ ⎡ ⎡ ⎢ 1.624⋅ 10 215.63 J ⋅ D − ⎢ ⎥⎥ 1+ 0.25 ⎢ ⎢ ⎢ ⎢ ⎛ Ec ⎞ ⎥ ⎥ ⎥ ⎥ 8.46 ( D + 1 ) ⎣ ⎦ ⎢ ⎥ ⎣ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦⎦
log
⎛ ΔPSI ⎝ 3.0 ⎠
x2 = 7.182 x2
7
W 18after := 10
W 18after = 1.522 × 10
Axle load equivalency factors from Tables 4.7 and 4.8 for D = 11 single20kip := 1.58 tandem34kip := 1.96 TotAxleEqv2 := single20kip + tandem34kip 3
TotalDailyTraffic = 2.418× 10
With addition of third lane,
TotAxleEqv2 = 3.54
trucks/day
(from Before condition)
PDL2 := 0.80
DailyDesignLaneTraffic2 := TotalDailyTraffic⋅ PDL2 3
DailyDesignLaneTraffic2 = 1.934 × 10 N2 :=
W 18after 365⋅ TotAxleEqv2 DailyDesignLaneTraffic2
N2 = 6.1
years
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.24 Determine the required slab th ickness i f a 20-year design life is used. N := 20
(given)
Assume: D := 11
Axle loads:
(Tables 4.7, 4.8 and 4.9)
single22kip := 2.67580 ⋅ tandem25kip := 0.5295570 ⋅ tandem39kip := 3.55⋅ 50 triple48kip := 2.58⋅ 80 TotAxleEqv := single22kip + tandem25kip + tandem39kip + triple48kip TotAxleEqv = 899.715 ZR := −1.645
(Table 4.5)
W 18 := TotAxleEqv⋅ N⋅ 365 6
W 18 = 6.5679195× 10 Sc := 600
Cd := 0.9
ΔPSI := 1.7
Ec := 5⋅ 10
J := 3.2
TSI := 2.5
So := 0.4
k := 200
6
(given)
Using Eq. 4.19:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd ⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ ⎡ 1.62410 ⎡ 0.75 − ⎡ 18.42 ⎤ ⎤ ⎥⎥ ⋅ 215.63J⋅ D ⎢ 1+ 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ 0.25 ⎥ ⎥ ⎥ + ( D 1 ) ⎣ ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
D = 11.43
inches
So assumption was close.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.25 Determine the design modulus o f ruptu re. k := 300
Cd := 1.0
ΔPSI := 2.0
D := 8.5
Ec := 4⋅ 10
TSI := 2.5
J := 3.0
So := 0.5
N := 12
6
ZR := −0.524
(given)
(Table 4.5)
Axle loads:
(Tables 4.7, 4.8 and 4.9)
single8kip := 0.03251300 ⋅ tandem15kip := 0.06575900 ⋅ single40kip := 26⋅ 20 tandem40kip := 3.645200 ⋅ TotAxleEqv := single8kip + tandem15kip + single40kip + tandem40kip 3
TotAxleEqv = 1.3504× 10
W 18 := TotAxleEqv⋅ N⋅ 365 6
W 18 = 5.9148615× 10
Using Eq. 4.19
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd ⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ ⎡ 1.62410 ⎡ 0.75 − ⎡ 18.42 ⎤ ⎤ ⎥⎥ ⋅ 215.63J⋅ D ⎢ 1+ 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ 0.25 ⎥ ⎥ ⎥ + ( D 1 ) ⎣ ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
Sc
= 575.343
lb 2
in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.26 Determine the assumed soil resilient modulu s. D := 10
PSI := 4.7
So := 0.35
Ec := 6⋅ 10
TSI := 2.5
k := 190
Sc := 432
ΔPSI := PSI − TSI
Cd := 0.8
J := 3.0
ΔPSI = 2.2
6
ZR := −1.282
(given)
(Table 4.5)
Rigid Pavement: Axle loads from Tables 4.7, 4.8 and 4.9: single20kip1 := 1.58⋅ 100 tandem42kip1 := 4.74⋅ 100 TotAxleEqvrig := single20kip1 + tandem42kip1 TotAxleEqvrig = 632
Flexible Pavement: SN := 4
Axle loads from Tables 4.2, 4.3 and 4.4: single20kip2 := 1.47⋅ 100 tandem42kip2 := 2.43⋅ 100 TotAxleEqvflex := single20kip2 + tandem42kip2 TotAxleEqvflex = 390
Using Eq. 4.19:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd ⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ ⎡ 1.62410 ⎡ 0.75 − ⎡ 18.42 ⎤ ⎤ ⎥⎥ ⋅ 215.63J⋅ D ⎢ 1+ 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ 0.25 ⎥ ⎥ ⎥ ⎣ (D + 1) ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
6
W 18 = 1.4723512× 10
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
N :=
W 18 TotAxleEqvrig⋅ 365
N = 6.383
years
Now applying Eq. 4.7 with: W 18 := N⋅ TotAxleEqvflex⋅ 365
⎡ log ( W 18) ⎢ZR ⋅ So + 9.36(log( SN + 1) ) − 0.20 + ⎢ ⎣ 3
MR = 3.67 × 10
5
W 18 = 9.08571× 10
⎤ + 2.32⋅ log( MR )⎥ − 8.07 1094 ⎡ ⎤ ⎥ 0.40 + 5.19 ⎣ (SN + 1) ⎦ ⎦ log
⎛ ΔPSI ⎞ ⎝ 2.7 ⎠
lb 2
in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
What slab thickness sho uld have been used? D := 8
Sc := 700
R = 90%
Cd := 1.0
So := 0.3
J := 3.0
PSI := 4.6 TSI := 2.5 CBR := 25
Problem 4.27 (given)
6
Ec := 5⋅ 10 Nyears
ΔPSI := PSI − TSI
:= 20
ΔPSI = 2.1
ZR := −1.282
(Table 4.5)
k := 290
(Table 4.10)
PDL := 0.75
(Table 4.11, for # Lanes = 4)
Using Eq. 4.19:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd ⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ ⎡ 1.62410 ⎡ 0.75 − ⎡ 18.42 ⎤ ⎤ ⎥⎥ ⋅ 215.63J⋅ D ⎢ 1+ 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ 0.25 ⎥ ⎥ ⎥ + ( D 1 ) ⎣ ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
6
W 18 = 5.7402× 10
Ignored Traffic in Design Axle load equivalency factors from Tables 4.7, and 4.8. D = 11 is assumed as a result of an iterative process. single22kip := 2.40 tandem30kip := 1.14 TotAxleEqv := single22kip + tandem30kip TotAxleEqv := 3.54
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
AddW 18 := 1000⋅ TotAxleEqv⋅ PDL⋅ 365⋅ Nyears 7
AddW 18 = 1.938 × 10
NewW18 := W 18 + AddW 18
W18 not originally included in design
NewW18
= 2.512 ×
7
10
Using Eq. 4.19: log⎛
ΔPSI ⎞
⎡ S C ⋅ (D0.75 − 1.132) ⎤ c d log( NewW18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + + ( 4.22 − 0.32⋅ T SI) ⋅ log⎢ 7 ⎤ 0.75 ⎡ 18.42 ⎤⎤ ⎥ ⎡ 1.62410 ⋅ 215.63J⋅ ⎡D − ⎢ ⎥ 1+ 0.25 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ ⎥⎥ ⎥ ⎣ ( D + 1) ⎦ ⎣ ⎣ ⎣ ⎝ k ⎠ ⎦⎦ ⎦ ⎝
D = 10.216
3.0
⎠
inches
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.28 Determine the number o f years the pavement will l ast.
6
k := 200
Ec := 5⋅ 10
PSI := 4.7
So := 0.30
D := 8
Sc := 600
TSI := 2.5
ZR := −1.036
J := 3.2
Cd := 1.0
(given) (Table 4.5)
ΔPSI := PSI − TSI ΔPSI = 2.2
Axle loads:
(Tables 4.7, 4.8 and 4.9)
single10kip := 0.084300 ⋅ single18kip := 1.00⋅ 200 single23kip := 2.75⋅ 100 tandem32kip := 1.47⋅ 100 single32kip := 10.1⋅ 30 tandem40kip := 1.16⋅ 100 TotAxleEqv := single10kip + single18kip + single23kip + tandem32kip + single32kip + tandem40kip 3
TotAxleEqv = 1.0662× 10
Using Eq. 4.19:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd ⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ ⎡ 1.62410 ⎡ 0.75 − ⎡ 18.42 ⎤ ⎤ ⎥⎥ ⋅ 215.6 3J⋅ D ⎢ 1+ 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ 0.25 ⎥ ⎥ ⎥ + ( D 1 ) ⎣ ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
6
W 18 = 2.8833862× 10 N :=
W 18 TotAxleEqv⋅ 365
N = 7.409
years
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.29 Determine the number o f years the pavement will l ast with a 95% reliability. 6
k := 200
Ec := 5⋅ 10
PSI := 4.7
So := 0.30
D := 8
Sc := 600
TSI := 2.5
ZR := −1.645
J := 3.2
Cd := 1.0
(given) (Table 4.5)
ΔPSI := PSI − TSI ΔPSI = 2.2
Axle loads:
(Tables 4.7, 4.8 and 4.9)
single10kip := 0.084300 ⋅ single18kip := 1.00⋅ 200 single23kip := 2.75⋅ 100 tandem32kip := 1.47⋅ 100 single32kip := 10.1⋅ 30 tandem40kip := 1.16⋅ 100 TotAxleEqv := single10kip + single18kip + single23kip + tandem32kip + single32kip + tandem40kip 3
TotAxleEqv = 1.0662× 10
Using Eq. 4.19:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd ⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ ⎡ 1.62410 ⎡ 0.75 − ⎡ 18.42 ⎤ ⎤ ⎥⎥ ⋅ 215.6 3J⋅ D ⎢ 1+ 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ 0.25 ⎥ ⎥ ⎥ ⎣ (D + 1) ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
6
W 18 = 1.893227× 10 N :=
W 18 TotAxleEqv⋅ 365
N = 4.865
years
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.30
Determine the design life. D := 10
MR := 5000
Ec := 4500000
Cd1 := 0.8
ΔPSI := 1.9
Sc := 900
Cd2 := 0.6
So := 0.45
J := 3.2
k := 300
(given)
TSI := 2.5
ZR := −1.645
(Table 4.5)
Axle Loads (interpolating):
(Tables 4.7, 4.8, and 4.9)
single2kip := 0.000220000 ⋅ = 4.0
single12kip := 0.175410 ⋅ = 71.75
single10kip := 0.081200 ⋅ = 16.2
tandem18kip := 0.132410 ⋅ = 54.12
tandem22kip := 0.305200 ⋅ = 61.0
triple50kip := 3.02410 ⋅ = 1238.2
TotAxleEqv := single2kip + single10kip + tandem22kip + single12kip + tandem18kip + triple50kip TotAxleEqv = 1445.27
Using Eq. 4.19 with Cd=0.8:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd1⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ ⎡ 1.62410 ⎡ 0.75 − ⎡ 18.42 ⎤ ⎤ ⎥⎥ ⋅ 215.6 3J⋅ D ⎢ 1+ 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ 0.25 ⎥ ⎥ ⎥ + ( D 1 ) ⎣ ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
6
W 18 = 8.1485079× 10 N :=
W 18
N
TotAxleEqv⋅ 365
= 15.45
years
Using Eq. 4.19 with Cd=0.6:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd2⋅ ( D log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ 7 ⎤ ⎡ 1.62410 ⎡ 0.75 − ⎡ 18.42 ⎤ ⎤ ⎥⎥ ⋅ 215.6 3J⋅ D ⎢ 1+ 8.46 ⎢ ⎢ ⎢ ⎛ Ec ⎞ 0.25 ⎥ ⎥ ⎥ ⎣ (D + 1) ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
6
W 18 = 3.0464048× 10
N :=
W 18 TotAxleEqv⋅ 365
N
= 5.77
years
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.31 Determine how the design life of the soi l changes.
k := 150
Sc := 800
So := 0.45
D := 12
J := 3.0
Cd := 1.0
6
Ec := 6⋅ 10
ΔPSI := 2.0
TSI := 2.5
(given)
N1 := 20
ZR := −1.645
(Table 4.5)
DailyTruckTraffic := 1227.76
trucks day
Axle loads:
(Tables 4.7, 4.8 and 4.9)
single16kip := 0.599⋅ DailyTruckTraffic single20kip := 1.590⋅ DailyTruckTraffic tandem35kip := 2.245⋅ DailyTruckTraffic TotAxleEqv := single16kip + single20kip + tandem35kip 3
TotAxleEqv = 5.444× 10
Using Eq. 4.19:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd ⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ 0.75 ⎡ 18.42 ⎤ ⎤ ⎥ ⎡ 1.62410 ⋅ 215.6 3J⋅ ⎡ D − ⎢ ⎥ 1+ 0.25 E ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ 8.46 ⎛ c ⎞ ⎣ (D + 1) ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
7
W 18 = 3.7684418× 10 W 18
N2 := TotAxleEqv⋅ 365
N2 = 18.965
Δ N := N1 − N2 Δ N = 1.03
year reduction
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine how the design life of the soi l changes. k := 190
Sc := 600
So := 0.45
D := 12
J := 3.0
Cd := 1.0
6
Ec := 6⋅ 10
ΔPSI := 2.0
TSI := 2.5
Problem 4.32 (given)
N1 := 20
ZR := −1.645
(Table 4.5)
DailyTruckTraffic := 1227.76
trucks day
Axle loads:
(Tables 4.7, 4.8 and 4.9)
single16kip := 0.599⋅ DailyTruckTraffic single20kip := 1.590⋅ DailyTruckTraffic tandem35kip := 2.245⋅ DailyTruckTraffic TotAxleEqv := single16kip + single20kip + tandem35kip 3
TotAxleEqv = 5.444× 10
Using Eq. 4.19:
⎛ ΔPSI ⎝ 3.0 ⎠
0.75 ⎡ ⎤ − 1.132) Sc Cd ⋅ ( D + ( 4.22 − 0.32⋅ TSI) ⋅ log⎢ log( W 18) ZR ⋅ So + 7.35⋅ ( log( D + 1) ) − 0.06 + 7 ⎤ 0.75 ⎡ 18.42 ⎤ ⎤ ⎥ ⎡ 1.62410 ⋅ 215.6 3J⋅ ⎡ D − ⎢ ⎥ 1+ 0.25 E 8.46 ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎛ c ⎞ ⎣ (D + 1) ⎦ ⎣ ⎣ ⎣ ⎝ k ⎦⎦ ⎦
log
7
W 18 = 1.4857318× 10 N2 :=
W 18 TotAxleEqv⋅ 365
N2 = 7.477
years
Δ N := N1 − N2 Δ N = 12.52
year reduction
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 4.33
Determine Determine the deflection at a point at a depth of 20 inches. lb
p := 90
μ := 0.45
2
a :=
in z := 20
r := 20
in
10 2
E := 45000
lb 2
in
(given)
in
use Table 4.1 z a
= 4.0000
r
and
= 4.0000
a
and C := 0.00492 and
therefore A := 0.01109 and
F := 0.00209 and
B := 0.00595
⎤ ⎦
Δ z = 0.0021
H := 0.17640
deflection
Δ z :=
p ⋅ ( 1 +
μ)⋅a
⋅⎡ ⋅A + z
⎣a
E
( 1 − μ) ⋅ H
inches
(Eq 4.6)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Solve for radial-horizontal stress
σr := p ⋅ [ 2⋅ μ ⋅ A +
C + ( 1 − 2⋅ μ ) ⋅ F]
σr = 1.3599
inches
(Eq 4.5)
2) Solve for vertical stress
σz := p⋅ ( A +
σz = 1.5336
B)
inches
(Eq 4.4)
3) Careless with rounding A := 0.01
Δ z :=
H := 0.18
and
p ⋅ ( 1 + E
μ) ⋅ a
⋅⎡ ⋅A + z
⎣a
⎤ ⎦
( 1 − μ) ⋅ H
Δ z = 0.0020
inches
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Determine Determine the d eflection. lb
p := 100
a :=
2
in z := 0 in
r := 0
Problem 4.34
12
μ := 0.50
in
2
E := 55000
lb
(given)
2
in
in
(assumed since on pavement surface)
Use Table 4.1 z a
r
=0
a
A := 1.0
=0
B := 0
C := 0
F := 0.5
H := 2.0
(Table 4.1)
deflection
Δ z :=
p ⋅ ( 1 +
μ)⋅a
E
⋅⎡ ⋅A + z
⎣a
⎤ ⎦
( 1 − μ) ⋅ H
Δ z = 0.016 in
(Eq 4.6)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers 1) Use stress equation
σr := p ⋅ [ 2⋅ μ ⋅ A +
C + ( 1 − 2μ ) ⋅ F]
σr = 100
(Eq. 4.5)
in
2) Use diameter instead of radius a := 12 in
Δ z :=
p ⋅ ( 1 +
μ) ⋅ a
⋅⎡ ⋅A + z
⎣a
E
( 1 − μ ) ⋅ H⎤
⎦
Δ z = 0.033
3) Use F instead of H in deflection equation
a := 6
Δ z :=
p ⋅ ( 1 + E
μ) ⋅ a
⋅⎡ ⋅A + z
⎣a
⎤ ⎦
( 1 − μ) ⋅ F
Δ z = 4.091 ×
−3
10
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 4.35
Deteremine Deteremine the structur al number of the pavement. pavement. a1 := 0.35
a2 := 0.18
a3 := 0.11
(Table 4.6)
D1 := 5 in
D2 := 9 in
D3 := 10 in
(given)
M2 := 0.90
M 3 := 1.0
(given)
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M3
(Eq. 4.9)
SN = 4.31
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers 1) Use M2 = 1.0 M2 := 1.0
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M3
SN = 4.47
2) Use Hot-mix asphaltic concrete Structural-Layer Coefficient M2 := 0.90
a1 := 0.44
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M3
SN = 4.76
3) Use crushed stone Structural-Layer Coefficient a1 := 0.35
a2 := 0.14
SN := a1⋅ D1 + a2⋅ D2⋅ M2 + a3⋅ D3⋅ M3
SN = 3.98
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 4.36
Calculate Calculate the edge stress (in lb/in 2 ). P := 25000 lb
h := 11.0 in
p := 120
lb
E := 4500000
2
in
μ := 0.20
k := 180
lb 2
in
(given)
lb 3
in
radius of relative stiffness
l :=
⎡
E⋅ h
⎤
3
0.25
l = 41.23
⎣ 12⋅ ( 1 − μ )⋅ k ⎦ 2
(Eq. 4.13)
radius of tire footprint
a :=
P p⋅ π
a
= 8.14
in
(Eq. 4.14)
stress 3 ⎞ P ⎞ ⎛ ⎛ E⋅ h ⎞ ⎛ σe := 0.529⋅ ( 1 + 0.54⋅ μ) ⋅ ⋅ log − 0.71 2 4 ⎝ h ⎠ ⎝ ⎝ k⋅ a ⎠ ⎠
(Eq. 4.15)
σe = 383.76 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Alternative Answers 1) Use al for radius of tire footprint
3 ⎞ P ⎞ ⎛ ⎛ E⋅ h ⎞ ⎛ σe := 0.529⋅ ( 1 + 0.54⋅ μ) ⋅ ⋅ log − 0.71 2 4 ⎝ h ⎠ ⎝ ⎝ k⋅ al ⎠ ⎠
al := a⋅ 2
σe = 310.85 2) Solve for deformation instead of stress P Δ e := 0.408⋅ ( 1 + 0.4μ) ⋅ ⎛ ⎞ Δ e = 0.036 2 ⎝ k⋅ l ⎠
(Eq. 4.16)
3) Solve for corner loading stress
⎡
⎛ al ⎞ σc := ⋅ 1− 2 ⎝ l ⎠ h ⎣ 3⋅ P
⎤
0.72
⎦
(Eq. 4.17)
σc = 372.37
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 4.37
Determine Determine the corner deflection fr om the load.
P := 10000 lb
p := 110
lb
h := 9.5 in
2
μ := 0.18
in
(given) lb
E := 4100000
k := 175
2
in
lb 3
in
radius of relative stiffness
l :=
⎡
E⋅ h
⎤
3
0.25
⎣ 12⋅ ( 1 − μ )⋅ k ⎦ 2
l = 36.267
(Eq 4.13)
= 5.379
(Eq 4.14)
radius of tire footprint
a :=
P
a
p⋅ π
al := a⋅ 2
al = 7.608
deformation
Δ c :=
⎡ ⎛ al ⎞⎤ 1.205 − 0.69⋅ 2 ⎣ ⎝ l ⎠⎦ k⋅ l P
⋅
Δ c = 0.046 in
(Eq 4.18)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Alternative Answers 1) Use a instead of al
Δ c :=
P
⋅ ⎡1.205 − 0.69⋅ ⎛
⎣ k⋅ l 2
a ⎞⎤
⎝ l ⎠⎦
Δ c = 0.048 in
2) Solve for stress instead of deformation
⎡
⎛ al ⎞ σc := ⋅ 1− 2 ⎣ ⎝ l ⎠ h 3⋅ P
⎤
0.72
⎦
σc = 224.435
in
(Eq 4.17)
3) Solve for total interior deflection 2 ⎡ ⎛ 1 ⎞ ⎛ ⎛ a ⎞ 5 ⎞ ⎛ a ⎞ ⎤ Δ i := ⋅ 1+ ⋅ ln +γ − ⋅ 2π ⎠ ⎝ ⎝ 2l ⎠ 4 ⎠ ⎝ l ⎠ ⎦ 2 ⎣ ⎝ 8⋅ k ⋅ l
P
Δ i = 0.005
in
(Eq 4.12)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the corner deflection fro m the load. P := 12000 lb
lb
p := 100
h := 12 in
2
(given)
in
μ := 0.16
k := 250
lb 3
Problem 4.38
E := 4650000
in
lb 2
in
Determine al P
a :=
a
p⋅ π
al := a⋅ 2
= 6.1804
al = 8.7404
(Eq 4.14)
(For corner of slab)
Determine radius of relative stiffness
l :=
⎡
E⋅ h
⎤
3
0.25
⎣ 12⋅ ( 1 − μ )⋅ k ⎦ 2
l = 40.7178
(Eq 4.13)
Δ c = 0.0306 in
(Eq 4.18)
Calculate the deflection
Δ c :=
⎡ ⎛ al ⎞⎤ 1.205 − 0.69⋅ 2 ⎣ ⎝ l ⎠⎦ k⋅ l P
⋅
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers: 1) Solve for corner stress
⎡
⎛ al ⎞ σc := ⋅ 1− 2 ⎣ ⎝ l ⎠ h 3⋅ P
⎤
0.72
lb
σc = 167.4348
⎦
(Eq 4.17)
2
in
2) Solve for edge loading deflection
Δ e := 0.408⋅ ( 1 +
0.4⋅ μ) ⋅
⎛ P ⎞ 2 ⎝ k⋅ l ⎠
Δ e = 0.0126 in
(Eq 4.16)
3) Use a in place of a l
Δ c :=
P
⋅ ⎡1.205 − 0.69⋅ ⎛
⎣ k⋅ l 2
a ⎞⎤
⎝ l ⎠⎦
Δ c = 0.0319 in
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany
Principles of Highway Engineering and Traffic Analysis, 4e By Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski
Chapter 5 Fundamentals of Traffic Flow and Queuing Theory U.S. Customary Units
Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Preface The solutions to the fourth edition of Principles of Highway Engineering and Traffic Analysis 1 were prepared with the Mathcad software program. You will notice several notation conventions that you may not be familiar with if you are not a Mathcad user. Most of these notation conventions are self-explanatory or easily understood. The most common Mathcad specific notations in these solutions relate to the equals sign. You will notice the equals sign being used in three different contexts, and Mathcad uses three different notations to distinguish between each of these contexts. The differences between these equals sign notations are explained as follows.
•
The ‘:=’ (colon-equals) is an assignment operator, that is, the value of the variable or expression on the left side of ‘:=’is set equal to the value of the expression on the right side. For example, in the statement, L := 1234, the variable ‘L’ is assigned (i.e., set equal to) the value of 1234. Another example is x := y + z. In this case, x is assigned the value of y + z.
•
The ‘=’ (bold equals) is used when the Mathcad function solver was used to find the value of a variable in the equation. For example, in the equation , the = is used to tell Mathcad that the value of the expression on the left side needs to equal the value of the expression on the right side. Thus, the Mathcad solver can be employed to find a value for the variable ‘t’ that satisfies this relationship. This particular example is from a problem where the function for arrivals at some time ‘t’ is set equal to the function for departures at some time ‘t’ to find the time to queue clearance.
•
The ‘=’ (standard equals) is used for a simple numeric evaluation. For example, referring to the x := y + z assignment used previously, if the value of y was 10 [either by assignment (with :=), or the result of an equation solution (through the use of =) and the value of z was 15, then the expression ‘x =’ would yield 25. Another example would be as follows: s := 1800/3600, with s = 0.5. That is, ‘s’ was assigned the value of 1800 divided by 3600 (using :=), which equals 0.5 (as given by using =).
Another symbol you will see frequently is ‘→’. In these solutions, it is used to perform an evaluation of an assignment expression in a single statement. For example, in the following statement, , Q(t) is assigned the value of 2 Arrivals(t) – Departures(t), and this evaluates to 2.2t – 0.10t .
Finally, to assist in quickly identifying the final answer, or answers, for what is being asked in the problem statement, yellow highlighting has been used (which will print as light gray). 1
www.mathcad.com
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine space-mean speed at capacity and free-flow speed.
⎡
u
q
3.5 k ⎞ ⎤ ⎛ uf ⋅ 1 − ⎣ ⎝ k j ⎠ ⎦
k ⋅u
k j := 225
qcap := 3800
(given)
⎡
3.5 k ⎞ ⎤ ⎛ uf ⋅ k − k ⋅ ⎣ ⎝ k j ⎠ ⎦
at capacity,
d
q
Problem 5.1
(Eq. 5.14)
0
dk
⎡ 0
1
⎢ ⎣
k cap qcap
− 4.5 ⋅
⎛ k
cap
⎜ ⎝
k j
⎞⎤
3.5 3.5
⎟⎥ ⎠⎦
= 146.4 k cap ⋅ ucap
(Eq. 5.14)
qcap ucap := k cap
ucap
qcap
uf := k cap
− kcap ⋅
⎛ kcap ⎞ ⎝
k j
3.5
= 25.96
mi h
uf = 33.37
mi h
⎠
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
a
= −3.222
b := −2 ⋅ a ⋅ u b
= 193.333
now, when
u2
= 51.58
q := 1400
mi/h
at q = 0, the free flow speed is as follows q := 0
u
= 60
mi/h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.3 Determine the capacity, speed at capacity, and densit y at o ne quarter capacity.
(
q ( k ) := 50 ⋅ k d
q (k)
− 0.156⋅ k
2
)
(given)
→ 50 − .312 ⋅ k
dk 50 − .312 ⋅ kcap
0
k cap := 160
(
)
qcap := q k cap
qcap
ucap := k cap
qcap
= 4006.4
ucap
= 25.04
veh h mi h
at 25% of capacity
q := q
qcap 4
= 1001.6
k1
= 21.5
k2
= 299
veh h
veh m veh m
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.4
Calculate the flo w, average speed, and d ensity of the traffic stream in this lane. hbar := 3 q :=
q
k
(Eq. 5.4)
hbar q ⋅ 3600 = 1200
veh/hr/ln
1 s bar
= 0.007
u :=
(given)
1
= 0.333
k :=
s bar := 150
q ⋅ 3600 k ⋅ 5280
(Eq. 5.13) k ⋅ 5280 = 35.2
u
= 34.09
veh/mi/ln
mi/hr
Also
q :=
k :=
u :=
3600 hbar 5280 sbar s bar hbar
q
= 1200
veh/hr/ln
k
= 35.2
veh/mi/ln
u
= 50
ft/s
u⋅
3600 5280
= 34.09
mi/hr
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.5
Determine the time-mean speed and space-mean speed. Lane 1 30 mi/h (44 ft/s)
Lane 2 45 mi/h 66 (ft/s)
Lane 3 60 mi/h (88 ft/s)
Spacing := 2640 ft 2640 Hdwy1 := 44
= 60
2640 Hdwy2 := 66
sec
2640 Hdwy3 := 88
= 40
= 30
For 30 minutes t := 30
⎛ 60 ⎞ Freq1 := t ⋅ ⎝ Hdwy1 ⎠
Freq1 = 30
⎛ 60 ⎞ Freq2 := t ⋅ ⎝ Hdwy2 ⎠
Freq2 = 45
⎛ 60 ⎞ Freq3 := t ⋅ ⎝ Hdwy3 ⎠
Freq3 = 60
Speed1 := 30 mi/h
ut_bar :=
Speed2 := 45
Speed1 ⋅ Freq1 + Speed2 ⋅ Freq2 Freq1
ut_bar = 48.33
vehicles in 30 minutes
+
Freq2
+
+
Speed3 := 60 Speed3 ⋅ Freq3
Freq3
mi/h
1
us_bar :=
1 Freq1 + Freq2
us_bar = 45.0
+
Freq3
⋅
⎡Freq ⋅ ⎛ 1 ⎞ + Freq ⋅ ⎛ 1 ⎞ + Freq ⋅ ⎛ 1 ⎞⎤ 1 Speed 2 Speed 3 Speed 1 ⎠ 2 ⎠ 3 ⎠⎦ ⎣ ⎝ ⎝ ⎝
mi/h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.6 Determine the time-mean speed and space-mean speed.
u1 := 195⋅
5280 3600
5280 u3 := 185⋅ 3600
= 271.333
2.5 ⋅ 5280 u2 2.5 ⋅ 5280
secperlap3 :=
u3 2.5 ⋅ 5280
secperlap4 :=
laps3 :=
u3
u1
secperlap2 :=
laps1 :=
= 286
2.5 ⋅ 5280
secperlap1 :=
total_sec
u1
u4
:= 30⋅ 60
secperlap1 total_sec secperlap3
TMS :=
laps1 ⋅ u1
ft/s
= 47.368
secperlap3
= 48.649
secperlap4
= 50
u4 = 264
= 1800
laps2 :=
laps3
= 37
laps4 :=
+
laps3
+
+
laps3 ⋅ u3 + laps4 ⋅ u4
laps4
total_sec secperlap2 total_sec secperlap4
total_laps
laps2
= 38
laps4
= 36
= 150
TMS⋅
3600 5280
= 187.67
mi/h
1
:=
1 total_laps
SMS
secperlap2
= 39
laps2 ⋅ u2
u2 = 278.667
3600
total_laps
TMS = 275.244
SMS
+
= 46.154
laps1
total_laps := laps1 + laps2
5280
5280 u4 := 180⋅ 3600
secperlap1
total_sec
total_sec
u2 := 190⋅
ft/s
= 275
ft/s
⋅
⎛ laps ⋅ 1 + laps ⋅ 1 + laps ⋅ 1 + laps ⋅ 1 ⎞ 1 u 2 u 3 u 4 u 1 2 3 4 ⎠ ⎝ SMS ⋅
3600 5280
= 187.5
mi/h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.7 Determine the space-mean speed. 195 + 190 + 185 + 180 4
= 187.5
mi/h
Problem 5.8 Estimate the probability of having 4 cars in an interval. t := 13
sec
− q⋅ ( t) P ( h ≥ 13)
(Eq. 5.26)
3600
− 0.0036 ⋅ q
0.6
q :=
e
e
ln ( 0.6)
−0.0036
qsec
:=
q 3600
q
= 141.9
qsec
= 0.039
veh/h
veh/s
during 30 sec intervals,
x := 30 ⋅ qsec
P ( n) :=
x
n
x
= 1.182
−x
⋅e n!
P ( 4)
= 0.025
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine how m any of t hese 120 intervals have 3 cars arriving. t := 20
Problem 5.9
sec (Given)
18
P ( 0)
120 ( λ ⋅ t)
P ( n)
⋅e n!
λ = 0.095
P ( n) := P ( 3)
− λ⋅ t
n
(Eq. 5.23)
veh/s
( λ ⋅ t)
n
− λ⋅ t
⋅e n!
= 0.1707
P ( 3) ⋅ 120 = 20.48
intervals
Extra
λ ⋅ 20 = 1.897
veh/20 s
λ ⋅ 60 = 5.691
veh/min
λ ⋅ 3600 = 341.5
veh/h
− qt P ( 0)
e
(Eq. 5.26)
3600
− q( 20 ) 18 20 0.15
− q( 0.00556 )
e
ln ( 0.15) q := −
3600
e
−q ( 0.00556)
ln ( 0.15) 0.00556
q
= 341.2
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.10 Determine percentages for headways. vehicles := 1.9
time := 20
vehicles
q :=
sec
(given)
q = 0.095
time
tgreat := 10
tless
:= 6
(given) (Eq. 5.26)
− q⋅ t
P ( t) := e
P ( h ≥ 10)
(
) = 0.387
P(h >= 10) = 0.387
P tgreat P (h
≥ 6)
(
P tless P (h 1
) = 0.566
< 6)
1 − P (h
≥ 6)
− P ( tless ) = 0.434
P(h < 6) = 0.434
Problem 5.11 Determine the probability of an accident. q := 280
t := 1.5
+
2.5 → 4.0
− q⋅ t P (h
≥ 4)
e
(given) (Eq. 5.26)
3600
− q⋅ t e
3600
P (h
= 0.733
< 4)
1 − P ( h ≥ 4)
− q⋅ t 1
−e
3600
= 0.267
Probability of an accident = 0.267
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.12 Determine the driver reaction times th at would make the probabili ty of an accident o ccurin g equal to 0.15. q := 280 we want P (h
t
≥ t)
= 2.09
(given) P (h
< 4)
0.15
so,
0.85
s
so the driver reaction time should be, t
− 1.5 = 0.59
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.14 Determine the t otal vehicl e delay.
λ1 := 8
veh/min
for t <= 20
λ2 := 0
veh/min
for t > 20 to t <= 30
λ3 := 2
veh/min
for t > 30
μ := 4
veh/min
for all t
Queue at t = 30 min Q30 := λ1 ⋅ 20 + λ2 ⋅ 10 − μ ⋅ 30 Q30 = 40 for queue dissipation after 30 minutes, note that:
t
= 20
min
so queue dissipation occurs at, t
+
30 = 50
min
using areas, the total delay can be calculated as,
Dt :=
20 ⋅ 80 2
+
( 80 + 40) ⋅ 10 2
+
40 ⋅ 20
Dt
2
= 1800
veh − min
Also t1 := 20 t2 := 10 t3 := 20 D t :=
Dt
1 2
⋅t 1⋅
( t 1 ⋅ λ 1) − ( t 1 μ⋅ )
+
1 2
⋅t 2⋅
( λ 1 ⋅t 1) − (μ ⋅ t 1)
+
(λ 1 ⋅t 1 + λ 2 ⋅ t 2) − μ ⋅( t 1 + t 2)
+
1 2
⋅ t 3⋅ (λ 1 ⋅ t 1 + λ 2 ⋅ t 2) − μ ⋅ (t 1 +
= 1800
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
)
t2
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.16 Determine when the queue will dissip ate and the to tal delay.
λ ( t) := 4.1 +
0.01t
Arrivals( t) :=
⌠ ⎮ λ ( t) dt → 4.10 ⋅ t + ⎮ ⌡
Departures( t) :=
= 15.34
veh/min
(given)
-3 2
5.00⋅ 10
⋅t
⌠ ⎮ μ dt → 12 ⋅ t ⎮ ⌡
Departures( t) := 12⋅ ( t
t
μ := 12
veh/min
− 10)
for service starting 10 min after arrivals
min
Check that total arrivals equals total departures after 15.34 min Arrivals( t)
= 64.065 = 64.065
Departures( t)
Calculate total delay (using triangular area below departures function)
⌠ t Dt := ⎮ ⌡0
4.1 ⋅ t
+
0.005 ⋅ t dt
⌠ t Dt := ⎮ ⌡0
4.1 ⋅ t
+
0.005 ⋅ t dt
Dt
= 317.32
2
− 0.5 ⋅ b ⋅ h
2
− [ ( 0.5) ⋅ ( t − 10) ⋅ ( Departures( t) ) ]
veh-min
Alternative total delay calculation (taking integrals for both arrival and departure functions Q ( t) := Arrivals( t)
⌠ ⎮ Dt := ⎮ ⌡ Dt
− Departures( t) ⌠ t ⌠ t Dt := ⎮ Arrivals( t ) dt − ⎮ ⌡0 ⌡10
Q ( t ) dt
= 317.32
Departures( t) dt
veh-min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.17 Determine time when qu eue clears.
λ ( t) := 5.2 − 0.01t
μ ( t ) := 3.3 +
2.4 ⋅ t
(given)
Integrate to obtain # of arrivals at specific time Arrivals( t) :=
⌠ ⎮ λ ( t) dt → 5.20 ⋅ t − 5.00⋅ 10-3 ⋅ t2 ⎮ ⌡
Integrate to obtain # of departures at specific time
⌠ ⎮ μ ( t) dt → 3.30 ⋅ t + Departures( t ) := ⎮ ⌡
1.20 ⋅ t
2
Since vehicle service begins (i.e., toll booth opens) 10 minutes after vehicles begin to arrive, Departures( t ) := 3.3( t
− 10) +
1.2 ⋅ ( t − 10)
2
Time to queue clearance: At what time does arrival rate equal constant 10 veh/min?
t
= 2.792
tcon := t
+
10
(
Departures tcon
t
tcon
= 12.792
time at which service becomes constant at 10 veh/min
) = 18.565
= 22.265
queue clears 22 minutes and 15.9 seconds after 7:50 AM
Check that total arrivals equals total departures Arrivals( 22.265)
= 113.3
Departures 10( 22.265 − 12.792)
= 94.73
94.73 + 18.568 = 113.3
arrivals after arrival rate becomes 10 veh/mi add arrivals before constant arrival rate
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.18 Determine the total delay and the lo ngest queue length .
λ := 6
μ ( t) := 2 +
veh/min
0.5 ⋅ t
veh/min
(given)
Integrate to obtain # of arrivals and departures at specific time
⌠ ⎮ λ dt → 6 ⋅ t Arrivals( t) := ⎮ ⌡ ⌠ ⎮ μ ( t) dt → 2. ⋅ t + Departures( t) := ⎮ ⌡
.250 ⋅ t
2
Time to queue clearance Q ( t) := Arrivals( t)
− Departures( t) → 4. ⋅ t − .250 ⋅ t
2
or Q ( t) :=
⌠ ⌠ ⎮ λ dt − ⎮ μ ( t) dt → 4. ⋅ t − .250 ⋅ t2 ⎮ ⎮ ⌡ ⌡
When q(t) = 0, queue is cleared
= 16
t
Total delay is area between arrival and departure curve
⌠ 16 Delay := ⎮ ⌡0
Q ( t) dt
→ 170.667
veh − min
Maximum queue length d
Q ( t)
4 − 0.5 ⋅ t
0
dt t :=
4
t
0.5
Q ( t) := 4 ⋅ t
=8
− 0.25 ⋅ t
2
Q ( 8)
= 16
cars
Check Arrivals( 8)
= 48
Departures( 8)
= 32
Q := Arrivals( 8)
− Departures( 8) → 16.000
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.19 Determine the delay and queue information .
λ ( t) := 5.2 − 0.20t
μ := 3
veh/min
veh/min
(given)
Integrate to obtain # of arrivals and departures at specific time
⌠ ⎮ λ ( t) dt → 5.20 ⋅ t − .100 ⋅ t2 Arrivals( t) := ⎮ ⌡ Departures( t) :=
⌠ ⎮ μ dt → 3 ⋅ t ⎮ ⌡
Time to queue clearance Q ( t) := Arrivals( t)
− Departures( t) → 2.20 ⋅ t − .100 ⋅ t
2
or
⌠ ⌠ 2 ⎮ λ ( t ) dt − ⎮ μ dt → 2.20 ⋅ t − .100 ⋅ t Q ( t) := ⎮ ⎮ ⌡ ⌡ When Q(t)=0, queue is cleared
t
= 22
Total delay is area between arrival and departure curve
⌠ 22 Delay := ⎮ ⌡0
Q ( t) dt
→ 177.467
veh − min
Maximum queue length d
Q ( t)
2.2
− .2 ⋅ t
t :=
0
dt Q ( 11)
= 12.1
2.2 0.2
t
= 11
veh
Wait time for 20th vehicle
− 20 0 t = 6.667
3t
time when 20th vehicle departs
wait := 6.667 − 4.183
wait
= 2.48
min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.21 Determine delay and maximum queue length.
λ ( t) := 1.2 +
μ := 12
veh/min
0.3t
veh/min
(given)
Integrate to obtain # of arrivals at specific time
Arrivals( t) :=
⌠ ⎮ λ ( t) dt → 1.20 ⋅ t + ⎮ ⌡
Departures( t) :=
.150 ⋅ t
2
⌠ ⎮ μ dt → 12 ⋅ t ⎮ ⌡
Departures( t) := 12⋅ ( t
− 10)
for service starting 10 min after arrivals
For time to queue clearance, set arrival rate = to departure rate
1.2 ⋅ t
t
+
0.15 ⋅ t
= 13.729
2
12 ⋅ ( t
− 10)
min
⌠ t ⌠ t Dt := ⎮ Arrivals( t) dt − ⎮ ⌡0 ⌡10 Dt
= 159.04
Departures( t) dt
veh-min
By inspection, maximum queue is at t = 10 (point at which service begins)
Qmax ( t ) := Arrivals( t ) Qmax ( 10)
= 27
at t=10
veh
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the average length of queue, average time spent in the system, and average waiting ti me spent in the queue.
Problem 5.23
M/M/1
μ := 5 ρ :=
λ := 4
λ μ
(given)
ρ = 0.8
(Eq. 5.27)
2
ρ Qbar := 1−ρ tbar := wbar :=
1
μ−λ λ μ ⋅ ( μ − λ)
Qbar = 3.2
veh
(Eq. 5.31)
tbar = 1
min/veh
(Eq. 5.33)
wbar = 0.8
min/veh
(Eq. 5.32)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.24
Determine the average length of queue, average time spent in the system, and average waiting ti me spent in the queue. M/D/1
μ := 3 λ ρ := μ
λ := 2 ρ = 0.667 ρ
Qbar := 2⋅ (1
− ρ) ρ
− ρ)
−ρ 2 ⋅ μ ⋅ ( 1 − ρ ) 2
(Eq. 5.27)
2
wbar := ⋅ ⋅ (1 2 μ tbar :=
(given)
Qbar = 0.667
veh
(Eq. 5.28)
wbar = 0.333
min/veh
(Eq. 5.30)
tbar = 0.667
min/veh
(Eq. 5.29)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.26 Determine the new distrib ution r ate required to c lear the queue.
λ ( t) := 120 + 8 ⋅ ( t − 30)
(from Problem 5.25)
μ ( t)
(from Problem 5.25, with an unknown service rate)
μ ⋅ ( t − 30)
for dissipation at t = 60 (8:45 - 9:45): 120 + 8 ⋅ ( t
− 30)
μ ⋅ ( t − 30)
μ = 12
veh/min
Problem 5.27 Determine the arrival rate.
μ := 4
veh/min
tc := 30
min
lane_cap := 30
(time to queue clearance) vehicles
at queue clearance
λ ⋅ tc
μ ⋅ ( tc − tp)
Where t p is the time until processing begins (i.e., time at which queuing lane becomes full) tp
lane_cap
λ
substituting expression for t p and solving for arrival rate gives,
λ ⋅ tc λ=2
μ⋅
⎛ tc ⎞ t − ⎝ c λ ⎠ veh/min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.28 Develop an expression for determining p rocessing rates in terms of x.
λ ⋅t
x
λ
and since x
t
2
at queue dissipation
2
λ ⋅ t μ ⋅ ( t − 13) substituting gives x
μ
x 2
− 13
Problem 5.29 Determine the time until queue dissipation. 1 2
⋅b⋅h
Delay is calculated by finding area of triangle formed by uniform arrival and departure rates
3600
b := 30
h :=
3600⋅ 2
μ := 4 h
μ
time at which processing begins
= 60
h
b
= 240
veh/min
min
(given)
time until queue to clearance after arrival of first vehicle
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.30 Determine when th e queue dissipates, the total delay, and t he length of the long est queue.
λ ( t) := 4.3 − 0.22t μ := 2
veh/min
(given)
veh/min
Arrivals( t) :=
⌠ ⎮ λ ( t) dt → 4.30 ⋅ t − .110 ⋅ t2 ⎮ ⌡
⌠ ⎮ μ dt → 2 ⋅ t Departures( t) := ⎮ ⌡ At time of queue clearance, Arrivals( t)
4.3 ⋅ t
t
Departures( t)
− 0.11 ⋅ t
= 20.91
2
2⋅ t
min
for total delay,
⌠ t Dt := ⎮ Arrivals( t) dt − ⌡0 Dt
= 167.59
1 2
⋅ t ⋅ ( μ ⋅ t)
second term is triangular area formed by uniform departures
veh − min
length of queue at time t is, Q ( t) := Arrivals( t)
− Departures( t) → 2.30 ⋅ t − .110 ⋅ t
2
for maximum length of queue, d
Q ( t)
0
0.22t
− 2.3
dt t :=
t
2.3 0.22
= 10.455
Q ( t)
= 12.02
veh
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.31 Determine how many veh/min sho uld be pro cessed. length of queue at time t is
Q ( t)
⌠ t ⎮ ⌡0
Q (t )
3.3 ⋅ t
⌠ ⎮ μ dt 3.3 − 0.1 ⋅ t dt − ⎮ ⌡ 2
− 0.05 ⋅ t − μ ⋅ t
for maximum d
Q ( t)
0
3.3
−μ
3.3
− 0.1 ⋅ t − μ
dt
t
0.1
first substitute for t, then find departure rate, 2
Q(t)
⎛ 3.3 − μ ⎞ − 0.05 ⋅ ⎛ 3.3 − μ ⎞ − μ ⋅ ⎛ 3.3 − μ ⎞ 3.3 ⋅ ⎝ 0.1 ⎠ ⎝ 0.1 ⎠ ⎝ 0.1 ⎠
μ = 2.41
4
veh/min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.32 Determine the probability that the number of trucks wil l exceed 5.
λ := 1.5 ρ :=
μ := 2
λ μ
veh/min
(given)
ρ = 0.75
(Eq. 5.27)
N := 1 1
P0 :=
+
1
P0
ρ 1−ρ
(Eq. 5.34)
= 0.25
n
Pn
ρ ⋅ P0 n− N
N
(Eq. 5.36)
⋅ N!
1
P1 :=
ρ ⋅ P0 1− N
N
⋅ N!
P1
= 0.1875
P2
= 0.1406
P3
= 0.1055
P4
= 0.0791
P5
= 0.0593
2
P2 :=
ρ ⋅ P0 2− N
N
⋅ N!
3
P3 :=
ρ ⋅ P0 3− N
N
⋅ N!
4
P4 :=
ρ ⋅ P0 4− N
N
⋅ N!
5
P5 :=
ρ ⋅ P0 5− N
N
⋅ N!
Psum := P0 + P1 + P2 Pgreater_than_5 := 1
+
P3
+
− Psum
P4
+
P5
Psum
= 0.822
Pgreater_than_5 = 0.178
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.33 Determine the number o f spaces.
1
P0 :=
1
1+
P0
2
1!
2
+
2
2!
3
+
2
3!
(Eq. 5.34) 4
+
2
4!
5
2
+
5!
6
+
2
( 6!) ⋅ 0.5
= 0.134
Pn_greater_than_N
P0 ⋅ ρ
N+ 1
⎛ − ⎝ N ⎠
N! ⋅ N ⋅ 1
ρ := 2
ρ ⎞
(Eq. 5.37)
(from Example 5.12)
N := 6
P6 :=
P0 ⋅ ρ
N+ 1
⎛ − ρ ⎞ ⎝ N ⎠
N! ⋅ N ⋅ 1
P6
= 0.006
(this is less than 1%, but also check 5 spaces)
N := 5
P5 :=
P0 ⋅ ρ
N+ 1
⎛ − ρ ⎞ ⎝ N ⎠
N! ⋅ N ⋅ 1
P5
= 0.024
so 6 spaces must be provided, because with only 5 spaces, the probability of 2.4% exceeds 1%
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.34 Determine the total time spent in system by all vehicles in a one hour period.
λ := 430⋅
μ :=
1 10
1 60
⋅ 60
λ μ
ρ :=
λ = 7.167
veh/min
μ=6
veh/min
(given)
ρ = 1.194
(Eq. 5.27)
N := 2
ρ N
= 0.597 1
P0 := 1+
Qbar :=
tbar :=
ρ + 1!
P0 ⋅ ρ
ρ
P0
= 0.252
(Eq. 5.34)
⎛ − ρ ⎞ ⎝ N ⎠
2! ⋅ 1
N+ 1
N! ⋅ N
ρ+
2
⋅
⎡ 1 ⎤ ⎢ ⎛ ρ ⎞ 2 ⎥ 1− ⎣ ⎝ N ⎠ ⎦
Qbar
λ
Qbar = 0.662
veh
(Eq. 5.38)
t bar = 0.259
min/veh
(Eq. 5.40)
so total time spent in one hour: 430⋅ tbar = 111.4
min/h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.35 Determine the minimum number of boo ths needed.
λ := 500⋅ μ := ρ :=
1 15
1 60
⋅ 60
λ μ
λ = 8.333
veh/min
μ=4
veh/min
(given)
ρ = 2.083
(Eq. 5.27)
N must be greater than 2 for the equations to apply, try 3 booths N := 3 1
P0 :=
2
1
Qbar :=
wbar :=
+
ρ ρ + + 1! 2!
P 0 ⋅ ρ
N+ 1
⋅
N! ⋅ N
ρ+
Qbar
λ
⎛ ⎝
3! ⋅ 1 −
1
(Eq. 5.34)
N ⎠
Qbar = 1.101
wbar = 0.132
μ
= 0.098
ρ ⎞
⎡ 1 ⎤ ⎢ ⎛ ρ ⎞ 2 ⎥ 1− ⎣ ⎝ N ⎠ ⎦
−
wbar ⋅ 60 = 7.924
ρ
P0
3
veh
(Eq. 5.38)
(Eq. 5.39)
min/veh
sec/veh
try 4 booths N := 4 1
P0 :=
2
1
Qbar :=
wbar :=
+
3
ρ ρ ρ + + + 1! 2! 3!
P 0 ⋅ ρ
N+ 1
N! ⋅ N
ρ+
Qbar
λ
wbar ⋅ 60 = 1.526
⋅
ρ
1
μ
sec/veh
= 0.119
(Eq. 5.34)
veh
(Eq. 5.38)
⎛ − ρ ⎞ ⎝ N ⎠
4! ⋅ 1
⎡ 1 ⎤ ⎢ ⎛ ρ ⎞ 2 ⎥ 1− ⎣ ⎝ N ⎠ ⎦
−
P0
4
Qbar = 0.212
wbar = 0.025
min/veh
(Eq. 5.39)
so a minimum of 4 booths must be open
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the tim e-mean speed of the miniv ans.
Problem 5.36
t1 := 98 s l := 3
t2 := 108 s
t3 := 113 s
t4 := 108 s
t5 := 102 s
(given)
n := 5 minivans
miles
velocities l
V1 :=
t1 l
V3 :=
t3 l
V5 :=
t5
⋅ 3600
mi
⋅ 3600
mi
⋅ 3600
mi
h
h
V2 :=
V4 :=
l t2 l t4
⋅ 3600
mi
⋅ 3600
mi
h
h
h
time-mean speed n
∑ ut
u t :=
i
(Eq 5.5)
ui
=1 n
V1 + V2 + V3 + V4 + V5 n
ut
= 102.332
mi h
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers 1) Use space-mean speed equation us
:=
1 1 n
⋅ ⎛
1
⎝ V1
+
1 V2
+
1 V3
+
1 V4
+
us
1 ⎞
= 102.079
mi
(Eq 5.9)
h
V5 ⎠
2) Use all vehicles and not just minivans V6 :=
u t :=
3 101
⋅ 3600
mi
V7 :=
h
3 85
⋅ 3600
mi
V8 :=
h
V1 + V2 + V3 + V4 + V5 + V6 + V7 + V8
ut
8
3 95
= 107.417
⋅ 3600
mi h
mi h
3) Inattention to rounding V1 := 110
u t :=
mi
V2 := 100
h
V1 + V2 + V3 + V4 + V5 n
mi h
V3 := 96
mi h
V4 := 100
ut
mi h
= 102.4
V5 := 106
mi h
mi h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the probability that more than five vehicles wil l arrive in a one-minute interval.
λ :=
400
veh
3600
s
n 0 := 0
n1 := 1
t := 60 s
n2 := 2
Problem 5.37
(given)
n3 := 3
n 4 := 4
n 5 := 5
solve for the probabilities of 5 and fewer vehicles
P0 :=
P1 :=
P2 :=
P3 :=
P4 :=
P5 :=
( λ ⋅ t)
( λ ⋅ t)
( λ ⋅ t)
( λ ⋅ t)
( λ ⋅ t)
( λ ⋅ t)
n0
⋅e n0!
n1
⋅e n1!
n2
⋅e n2!
n3
⋅e n3!
n4
⋅e n4!
n5
⋅e n5!
− λ⋅ t P0 = 0.0013
(Eq 5.23)
− λ⋅ t P1 = 0.0085
− λ⋅ t P2 = 0.0283
− λ⋅ t P3 = 0.0628
− λ⋅ t P4 = 0.1047
− λ⋅ t P5 = 0.1397
solve for 6 or more vehicles P := 1 − P0 − P1 − P2 − P3 − P4 − P5
P
= 0.6547
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers 1) Do not account for P(0) P := 1 − P1 − P2 − P3 − P4 − P5
P
= 0.656
P
= 0.7944
2) Do not account for P(5) P := 1 − P0 − P1 − P2 − P3 − P4
3) Solve just for P(6)
n6 := 6
P6 :=
( λ ⋅ t)
n6
⋅e n6!
− λ⋅ t P6 = 0.1552
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 5.38
At wh at t im e do es t he m axi mu m q ueu e len gt h o cc ur .
λ ( t) := 1.8 +
0.25⋅ t
− 0.003t ⋅
2
μ ( t) := 1.4 +
0.11⋅ t
t := 60 min
(given)
set the integral arrival and departure rates equal to each other
⌠ t ⎮ 1.8 + ⌡0
0.25⋅ t
1.8⋅ t
0.125t
+
− 0.003t⋅
2
2
− 0.001t⋅
Q( t ) := −0.001⋅ t
3
+
⌠ t ⎮ 1.4 + ⌡0
d t
3
0.07⋅ t
1.4⋅ t 2
+
+
0.11⋅ t d t
0.055t ⋅
2
0.4⋅ t
solve for time at which maximum queue length occurs d d t
Q( t)
2
−.003⋅ t +
0.14⋅ t
+
0.4
0
use quadratic equation
t1 :=
t2 :=
−0.14 +
(0.142) − 4⋅ −0.003⋅ 0.4 2⋅ −0.003
−0.14 −
(0.142) − 4⋅ −0.003⋅ 0.4 2⋅ −0.003
t1 = −2.7
min
t2 = 49.4 min
(this is the reasonable answer of the two provided by the quadratic equation)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers 1) Solve for total vehicle delay
⌠ 60 ⎮ 1.8 + ⌡0
⌠ 60 0.25⋅ t − 0.003t ⋅ d t − ⎮ 1.4 + ⌡0 2
0.11⋅ t d t
= 60
min
2) Take the t1 value as positive t := 2.701 min
3) Set Q(t) = 0 and assume t is in hours Q( t ) := −0.001⋅ t 3
−0.001⋅ t +
t :=
−0.07 +
3
+
0.07⋅ t
2
+
0.07⋅ t
2
0.4⋅ t
+
0.4⋅ t 0
(0.072) − 4⋅ −0.001⋅ 0.4 2 − 0.001
t
=0
t⋅ 3660 = 19.4
min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the average waiting time for t his qu euing system.
λ :=
200
veh
60
min
μ :=
60
veh
15
min
Problem 5.39 (given)
calculate traffic intensity
λ μ
ρ :=
ρ = 0.833
(Eq 5.27)
calculate average waiting time in queue (M/M/1)
w :=
λ μ⋅ ( μ − λ )
w
= 1.25
min
(Eq 5.32)
veh
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 1) Assume M/D/1 w :=
ρ 2⋅ μ ⋅ ( 1 − ρ)
w
= 0.625
min
(Eq 5.29)
veh
2) Find average time spent in the system t :=
1
t
μ−λ
= 1.5
min
(Eq 5.30)
veh
3) Use average queue length equation 2
ρ Q := ( 1 − ρ)
Q = 4.167
min
(Eq 5.28)
veh
(note: the units for the average queue length equation are actually in "veh" so that should be an obvious sign this is the inocrrect equation to use.)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the average time spent in the syst em.
λ :=
60
veh
60
min
μ :=
60
veh
30
min
Problem 5.40 (given)
compute the average time spent in the system t :=
1
t
μ−λ
=1
min
(Eq 5.33)
veh
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers 1) Assume M/D/1 queuing
ρ :=
λ μ
t :=
2−
ρ 2⋅ μ ⋅ ( 1 − ρ)
t
= 0.75
min
(Eq 5.30)
veh
2) Use average waiting time equation w :=
λ μ⋅ (μ − λ )
w
= 0.5
(Eq 5.32)
min veh
3) Keep units in veh/hr
λ := 60
μ := 120
t :=
1
μ−λ
t
= 0.017
min veh
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
How would the probability of waiting i n a queue change if a fourth tol l both w ere opened? 850
λ :=
μ :=
60
Problem 5.41
60 12
(given)
N := 3
determine probability of having no vehicles in system
ρ :=
λ μ
P0 :=
ρ = 2.833
1 2
ρ ρ 1+ + + 1! 2!
P0 = 0.013
3
ρ
⎛ ⎝
3!⋅ 1 −
(Eq 5.34)
ρ ⎞
N ⎠
determine probability of having to wait in a queue N + 1
P3 :=
P0⋅ ρ
⎛ ⎝
N !⋅ N ⋅ 1 −
P3 = 0.847
ρ ⎞
(Eq 5.37)
N ⎠
determine probability of having to wait in queue for four toll booths N := 4 1
P0 :=
2
1+
3
ρ ρ ρ + + + 1! 2! 3!
4
ρ
⎛ ⎝
4!⋅ 1 −
P0 = 0.048
(Eq 5.34)
P4 = 0.313
(Eq 5.37)
ρ ⎞
N ⎠
N + 1
P4 :=
P0⋅ ρ
⎛ ρ ⎞ N !⋅ N ⋅ 1 − ⎝ N ⎠
calculate difference
P3 − P4 = 0.534
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers 1) Solve for probability for just 3 toll booths P3 = 0.847
2) Solve for probability for just 4 toll booths P4 = 0.313
3) Use Equation 5.36 for 4 toll booths n := 15.247
(approximate value for average length of queue
N := 3 n
ρ ⋅ P0
Pn3 := N
n− N
⋅ N !
Pn3
= 0.09
Pn4
= 0.003
N := 4
n
ρ ⋅ P0
Pn4 := N Pn3
n− N
⋅ N !
− Pn4 = 0.088
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany
Principles of Highway Engineering and Traffic Analysis, 4e By Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski
Chapter 6 Highway Capacity and Level of Service Analysis U.S. Customary Units
Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Preface The solutions to the fourth edition of Principles of Highway Engineering and Traffic Analysis 1 were prepared with the Mathcad software program. You will notice several notation conventions that you may not be familiar with if you are not a Mathcad user. Most of these notation conventions are self-explanatory or easily understood. The most common Mathcad specific notations in these solutions relate to the equals sign. You will notice the equals sign being used in three different contexts, and Mathcad uses three different notations to distinguish between each of these contexts. The differences between these equals sign notations are explained as follows. •
The ‘:=’ (colon-equals) is an assignment operator, that is, the value of the variable or expression on the left side of ‘:=’is set equal to the value of the expression on the right side. For example, in the statement, L := 1234, the variable ‘L’ is assigned (i.e., set equal to) the value of 1234. Another example is x := y + z. In this case, x is assigned the value of y + z.
•
The ‘=’ (bold equals) is used when the Mathcad function solver was used to find the value of a variable in the equation. For example, in the equation , the = is used to tell Mathcad that the value of the expression on the left side needs to equal the value of the expression on the right side. Thus, the Mathcad solver can be employed to find a value for the variable ‘t’ that satisfies this relationship. This particular example is from a problem where the function for arrivals at some time ‘t’ is set equal to the function for departures at some time ‘t’ to find the time to queue clearance.
•
The ‘=’ (standard equals) is used for a simple numeric evaluation. For example, referring to the x := y + z assignment used previously, if the value of y was 10 [either by assignment (with :=), or the result of an equation solution (through the use of =) and the value of z was 15, then the expression ‘x =’ would yield 25. Another example would be as follows: s := 1800/3600, with s = 0.5. That is, ‘s’ was assigned the value of 1800 divided by 3600 (using :=), which equals 0.5 (as given by using =).
Another symbol you will see frequently is ‘→’. In these solutions, it is used to perform an evaluation of an assignment expression in a single statement. For example, in the following statement, , Q(t) is assigned the value of 2 Arrivals(t) – Departures(t), and this evaluates to 2.2t – 0.10t .
Finally, to assist in quickly identifying the final answer, or answers, for what is being asked in the problem statement, yellow highlighting has be en used (which will print as light gray). 1
www.mathcad.com
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.1 Determine the hourly volume. BFFS := 65
(given)
f LW := 1.9
(Table 6.3)
f LC := 0.80
(Table 6.4)
f N := 0.0
(Table 6.5)
f ID := 0.0
(Table 6.6)
FFS := BFFS − fLW − f LC − f N − f ID
FFS = 62.3
vp := 1616
(Eq. 6.2)
(Table 6.1, by interpolation)
PT := 0.10
(given)
ET := 2.5
(Table 6.8)
1 f HV := 1 + PT⋅ ET − 1
(
PHF := 0.90
V := vp ⋅ PHF ⋅ N ⋅ f HV ⋅ f p
) f p := 1.0
f HV = 0.87
(Eq. 6.5)
N := 3
(given)
V = 3794
veh/h
(Eq. 6.3)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.2 Determine Determine the g rade length. V := 5435
(given)
FFS := 62.3
(From Problem 6.1)
vp := 2323
(Table 6.1)
PHF := 0.90
vp
f p := 1.0
N := 3
(given)
V (Eq. 6.3)
PHF ⋅ N ⋅ f HV ⋅ f p
f HV = 0.867 PT := 0.10
f HV
(given)
1
(
)
1 + PT⋅ ET − 1
(Eq. 6.5)
ET = 2.54 Using Table 6.8, the length is found to be between 0.75-1.0 mi
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 6.3 Determine Determine the maximum numb er of large trucks and buses. first, determine the heavy vehicle factor assume urban freeway BFFS := 70
(given)
f LW := 0.0
(Table 6.3)
f LC := 0.0
(Table 6.4)
f N := 4.5
(Table 6.5)
f ID := 0.0
(Table 6.6)
FFS := BFFS − fLW − f LC − f N − f ID PHF :=
1800
PHF = 0.6429
700⋅ 4
f p := 1.0 vp
FFS = 65.5
(Eq. 6.2)
(Eq. 6.4)
N := 2
(given)
1800
1400
PHF ⋅ N ⋅ f HV ⋅ f p
f HV
(Eq. 6.3)
interpolate from Table 6.1 to find v p
vp := 1680 + ( FFS − 65) ⋅
f HV :=
1400
⎛ 1770 − 1680 ⎞ ⎝ 70 − 65 ⎠
vp = 1689
f HV = 0.8289
vp
now, determine the number of trucks ET := 2.5 f HV
1
(
)
1 + PT⋅ ET − 1
1 + 1.5 ⋅ PT
(Table 6.7)
(rolling terrain)
1
1
1 + PT⋅ ( 2.5 − 1)
1 + 1.5PT
(Eq. 6.5)
1 f HV
PT = 0.1376 1800⋅ PT = 247.7143
therefore 247 trucks and buses
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 6.4 Determine Determine the level of servic e V := 1800 PT :=
(given)
180
PT = 0.1
V
ET := 4.0
(Table 6.9)
1 f HV := 1 + PT⋅ ET − 1
(
PHF := 0.643 vp :=
V PHF ⋅ N ⋅ f HV
f HV = 0.769
)
(given)
N := 2 vp = 1819.6
(Eq. 6.5)
pc/h/ln
(Eq. 6.3)
max service flows at 65 mi/h FFS from Table 6.1 LOS C = 1680 LOS D = 2090 therefore the LOS is D Alternative solution: by interpolation, D :=
V 64
26.0 < 28.1 < 35.0
D = 28.125
pc/mi/ln
(Eq. 6.6)
therefore the LOS is D
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Probl em 6.5 6.5 Determine Determine the driver population factor . PHF := 0.80
PT := 0.08
PR := 0.06
N := 3
V := 3900
(given)
Solve for f HV ET := 2.0
ER := 3.0
Table 6.7
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
)
(
)
f HV = 0.833
Freeway is operating at capacity, so for FFS = 55 mi/h
vp
(Eq. 6.5)
vp := 2250
(Table 6.1)
V PHF ⋅ fHV ⋅ N ⋅ f p
f p = 0.867
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Probl em 6.6 6.6 Determine Determine the n ew level of servic e.
BFFS := 60 f LW := 1.9
(Table 6.3)
f N := 0.0
(Table 6.5)
f LC := 0.6
(Table 6.4)
f ID := 0.0
(Table 6.6)
FFS := BFFS − fLW − f LC − f N − f ID
FFS = 57.5
mi/h
(Eq. 6.2)
calculate heavy vehicle adjustment PT := 0.12
PR := 0.06
ET := 4.5
ER := 4.0
(Table 6.7)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
)
(
)
f HV = 0.625
(Eq. 6.5)
calculate LOS before and after lane addition N1 := 2
N2 := 3
2300 + 2250 2 vp
f p := 0.90
= 2275
V PHF ⋅ fHV ⋅ N1 ⋅ f p
PHF := 0.88
vp := 2275
(Table 6.1 by interpolation) interpolation)
V := vp ⋅ PHF ⋅ f HV ⋅ N1 ⋅ f p
(Eq. 6.3)
V = 2252.25 V vp2 := PHF ⋅ fHV ⋅ N2 ⋅ f p vp2 = 1516.667
(Eq. 6.3)
pc/h/ln
LOS D (Table 6.1, by interpolation)
max service flow: 1495 pc/h/ln for LOS C 1965 pc/h/ln for LOS D Calculate FFS after lane addition f LC := 0.4 For vp of 1517,
D :=
vp2 S
FFS := BFFS − fLW − f LC − f N − f ID
FFS = 57.7
S := FFS
D = 26.285
pc/mi/ln
LOS D - 26 < 26.3 < 35
(Eq. 6.6)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 6.7 Determine the LOS. find the analysis flow rate ( 1500 1500⋅ 0.03 0.03) + ( 1000⋅ 0.04) 2500 2500 5280
= 0.473
= 0.034
(average grade)
mi
PT := 0.05
(given)
ET := 2.0
(Table 6.8)
1 f HV := 1 + PT⋅ ET − 1
(
N := 2.0
f HV = 0.952
)
PHF := 0.90
V vp := PHF ⋅ N ⋅ f HV ⋅ f p
f p := 1.0 vp = 1166.667
V := 2000
pc/h/ln
(Eq. 6.5)
(given)
(Eq. 6.3)
determine FFS BFFS := 65
(given)
f LW := 0.0
(Table 6.3)
f LC := 1.8
(Table 6.4)
f N := 4.5
(Table 6.5)
f ID := 0.0
(Table 6.6)
FFS := BFFS − fLW − f LC − f N − f ID
FFS = 58.7
(Eq. 6.2)
pc/mi/ln therefore LOS is C
(Eq. 6.6)
find level of service for vp = 1167, S = FFS
D :=
vp FFS
D = 19.88
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine density and level of service before and after the ban.
Problem 6.9
Before: PT := 0.06
PB := 0.05
PTB := PT + PB
(given)
PTB = 0.11
ETB := 2.5
(Table 6.7)
1 f HVTB := 1 + PTB⋅ ETB − 1
(
PHF := 0.95
f HVTB = 0.858
)
f p := 1.0
V vp := ⋅ f p ⋅ N PHF ⋅ fHVTB
N := 4
V := 5400
vp = 1655.526
(Eq. 6.5)
(given)
(Eq. 6.3)
BFFS := 70
(given)
f LW := 1.9
(Table 6.3)
f LC := 0.4
(Table 6.4)
f N := 1.5
(Table 6.5)
f ID := 3.7
(Table 6.6)
FFS := BFFS − fLW − f LC − f N − f ID
FFS = 62.5
S := 62.5
D :=
(Eq. 6.2) (Figure 6.2)
vp
D = 26.5
S
pc/mi/ln
LOS D
(Eq. 6.6)
After:
(
)
Vnew := V ⋅ 1 − PT
Vnew = 5076
NumBuses := V ⋅ PB
NumBuses = 270
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
PBnew :=
NumBuses PBnew = 0.053
Vnew
EB := 2.5
(given)
1 f HVB := 1 + PBnew⋅ EB − 1
(
)
f HVB = 0.926
(Table 6.7) (Eq. 6.5)
vp :=
Vnew PHF ⋅ fHVB ⋅ f p ⋅ N
vp = 1442.368 (Eq. 6.3)
S := 62.5
D :=
vp S
D = 23.1
pc/mi/ln
LOS C
(Figure 6.2) (Eq. 6.6)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.10 Determine the density, v/c ratio, and LOS before and after the strike. calculate free-flow speed BFFS := 65
(given)
f LW := 0.0
(Table 6.3)
f LC := 0.0
(Table 6.4)
f N := 3.0
(Table 6.5)
f ID := 5.0
(Table 6.6)
FFS := BFFS − fLW − f LC − f N − f ID
FFS = 57 mi/h
(Eq. 6.7)
calculate volume after bus strike V 1 := 3800
PT := 0.02
PB := 0.04
(
)
V2 := V1 − PB ⋅ V1 + 6 ⋅ P B ⋅ V1
PHF := 0.90
PTB := PT + PB
(given)
V2 = 4560
calculate heavy vehicle adjustments before and after bus strike ETB := 4.0
ET := 5.0
(Table 6.8)
1 f HVTB := 1 + P TB⋅ E TB − 1
(
)
PT :=
Calculate new P T 1 f HVT := 1 + P T⋅ E T − 1
(
N := 3
)
f HVTB = 0.847 V1 ⋅ PT V2
PT = 0.017
f HVT = 0.938
f p := 1.0 V1
vp1 := PHF ⋅ N⋅ f HVTB⋅ f p
(Eq. 6.5)
(given)
vp1 = 1660.741
(Eq. 6.3)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
vp1 c
= 0.732
D = 29.14
D :=
vp1 FFS
pc/mi/ln
LOS D
V2 vp2 := PHF ⋅ N ⋅ f HVT⋅ f p vp2 c
= 0.794
D :=
(Eq. 6.6)
(Eq. 6.3)
vp2 = 1801.481
vp2 FFS
D = 31.6
pc/mi/ln
LOS D
(Eq. 6.6)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.11 Determine level of service. PT := 0.06
PR := 0.02
(given)
ET := 1.5
ER := 1.2
(Table 6.7)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
V := 1300
)
(
PHF := 0.85
V vp := PHF ⋅ N ⋅ f HV ⋅ f p
)
f p := 0.95 vp = 832.322
f HV = 0.967
(Eq. 6.5)
N := 2
(given)
pc/mi/ln
(Eq. 6.3)
FFS := 45
(given)
LOS C, from Figure 6.2
Problem 6.12 Determine the number o f vehicles added. FFS := 45
(given)
vp := 1900
(from Table 6.11)
PHF := 0.85
f p := 0.95
PT := 0.06
PR := 0.02
ET := 1.5
ER := 1.2
N := 2
(given)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
)
(
)
f HV := 0.967 (from Problem 6.11)
Vprev := 1300 V := vp ⋅ PHF ⋅ N ⋅ f HV ⋅ f p
V = 2967
V − Vprev = 1667.239
1667 vehicles can be added to the traffic flow
(Eq. 6.3)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.13 Determine Determine the level of service. Estimate FFS BFFS := 55 + 5
(posted speed limit + 5 mi/h)
f LW := 1.9
(Table 6.3)
f LC := 0.4
(Table 6.13)
f M := 0
(Table 6.14)
f := 3.75 A
(Table 6.15, by interpolation)
FFS := BFFS − fLW − f LC − f M − A f
FFS = 53.95
(Eq. 6.7)
calculate analysis flow rate PT := 0.08
PR := 0.02
(given)
ET := 3.0
(Table 6.8)
ER := 3.0
(Table 6.9)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
V := 4000
)
veh/h
(
PHF := 0.90
V vp := PHF ⋅ N ⋅ f HV ⋅ f p S := 51.5 D :=
f HV = 0.833
) N := 3
vp = 1871.345
f p := 0.95 pc/h/ln
(Eq. 6.5)
(given) (Eq. 6.3)
(Figure 6.4)
vp
(Eq. 6.6)
S
D = 36.34 LOS E
veh/mi/ln (Table 6.11 or Figure 6.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Probl em 6.14 6.14 Determin Determin e LOS before before and after the developm ent. Before: PT := 0.10
PR := 0.03
(given)
ET := 2.5
ER := 2.0
(Table 6.7)
1 f HV := 1 + P T⋅ ET − 1 + P R ⋅ ER − 1
f HV = 0.847
(Eq. 6.5)
PHF := 0.95
V := 2300
(given)
(
)
(
N := 2
V vp := N ⋅ PHF ⋅ f HV ⋅ f p BFFS := 50 + 5
)
f p := 0.90 vp = 1587.135
(Eq. 6.3)
BFFS = 55
(given)
f LW := 6.6
(Table 6.3)
TLC := 6 + 3
TLC = 9
(Eq. 6.8)
f LC := 0.65
(Table 6.13, by interpolation)
f M := 0.0
(Table 6.14)
f := 1.0 A
(Table 6.15)
FFS := BFFS − fLW − f LC − f M − A f
FFS = 46.75
S := 46 D :=
vp S
(Eq. 6.7) (from Figure 6.4)
D = 34.5
which is LOS D
(Eq. 6.6)
After: (Table 6.15)
f := 3.0 A FFS := BFFS − fLW − f LC − f M − A f
FFS = 44.75
(Eq. 6.7)
Vnew := 2700 Vnew vp := N ⋅ PHF ⋅ f HV ⋅ f p
vp = 1863.158
(from Figure 6.4)
S := 42 D :=
vp S
(Eq. 6.3)
D = 44.36
which is LOS E
(Eq. 6.6)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 6.15 Determine Determine the level of servic e before and after after tru cks are allowed. Before: PB := 0.02
(given)
EB := 3.5
(Table 6.8)
1 f HVB := 1 + PB ⋅ EB − 1
(
V := 1900
f HVB = 0.952
)
PHF := 0.80
V vp := PHF ⋅ N ⋅ f HVB ⋅ f p1 LOS LOS C
N := 2
(Eq. 6.5)
f p1 := 1.0
vp = 1246.875
pc/h/ln
(given)
(Eq. 6.3)
(fro (from m Tab Table le 6.1 6.11, 1, wit with h FFS FFS = 55 55 mi/h mi/h))
After: Vnew := V + 150 Buses := PB ⋅ V PBnew :=
PT :=
Buses = 38
Buses Vnew
150
PBnew = 0.019
PT = 0.073
Vnew
PTB := PBnew + PT
PTB = 0.092
ETB := 2.5
(Table 6.8)
1 f HV := 1 + PTB⋅ ETB − 1.
(
)
f HV = 0.879
f p2 := 0.97
(given)
Vnew vpnew := PHF ⋅ N ⋅ f HV ⋅ f p2 LOS LOS D
(Eq. 6.5)
vpnew = 1502.577
(Eq. 6.3)
(fro (from m Tab Table le 6.11 6.11,, wit with h FFS FFS = 55 55 mi/ mi/h) h)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 6.16 Determine Determine the directional hourly v olume. calculate FFS BFFS := 55 + 5
BFFS = 60
(posted speed limit + 5 mi/h)
f LW := 1.9 f LC := 0.4
(Table 6.3) (4 ft right shoulder + 6 ft left (undivided))
(Table 6.13)
f M := 1.6
(Table 6.14)
f := 2.5 A
(Table 6.15)
FFS := BFFS − fLW − f LC − f M − A f
FFS = 53.6
mi/h
(Eq. 6.7 )
calculate heavy vehicle adjustment PT := 0.08
(given)
ET := 1.5
(Table 6.7)
1 f HV := 1 + PT ⋅ E T − 1
(
f HV = 0.962
)
(Eq. 6.5)
calculate directional hourly volume f p := 0.90
PHF := 0.80
Roadway at capacity, so V := vp ⋅ PHF ⋅ N ⋅ f HV ⋅ f p
N := 2
vp := 2062 veh/h V = 2855
(given) (Table 6.11, by interpolation 2000 - 2100)
veh/h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 6.17 Determine Determine the level of service. calculate the heavy vehicle factor V := 1990 PT :=
trucks := 140
buses := 40
trucks + buses
PR :=
rvs := 10
(given)
PT = 0.09
V rvs
PR = 0.005
V
ET := 1.5
(Table 6.10)
ER := 1.2
(Table 6.7)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
)
(
)
f HV = 0.956
(Eq. 6.5)
determine the level of service N := 2
PHF := 0.85
(given)
f p := 1.0
(Assumed)
V vp := N ⋅ PHF ⋅ f HV ⋅ f p BFFS := 5 + 50
vp = 1225
pc/h/ln
(Eq. 6.3)
BFFS = 55
(given)
f LW := 0.0
(Table 6.3)
f LC := 0.0
(Table 6.13, 6 ft shoulders, divided
f M := 0.0
(Table 6.14)
f := 1 A
(Table 6.15)
FFS := BFFS − fM − f LW − f LC − A f
FFS = 54
(Eq. 6.7)
S := FFS D :=
vp S
D = 22.68
pc/mi/ln
therefore LOS is C
(Eq. 6.6)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 6.18 Determine Determine the number o f vehicles th at can be added. find initial volume FFS := 50 PT := 0.06
PR := 0.02
(given)
ET := 4.5
(Table 6.8)
ER := 4.0
(Table 6.9)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
N := 3
)
PHF1 := 0.90
vp1 := 1300
(
f HV = 0.787
)
f p := 0.92
(Eq. 6.5)
(given)
(from Table 6.11, for FFS = 50 mi/h, LOS C)
V1 := vp1 ⋅ PHF1 ⋅ N ⋅ f HV ⋅ f p
V1 = 2543
(Eq. 6.3)
find final volume vp2 := 2000
(capacity conditions)
PHF2 := 0.95
(given)
other variables are same as above V2 := vp2 ⋅ PHF2 ⋅ N ⋅ f HV ⋅ f p Vadded := V2 − V1
V2 = 4129 Vadded = 1586
(Eq. 6.3)
vehicles
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley Wiley & Sons, Inc. All rights reserved. reserved.
Problem 6.19 Determine how many access points mu st be blocked. calculate analysis flow rate V := 2500 + 200 PT :=
trucks := 200
trucks
(given)
PT = 0.074
V
(Table 6.8) (Table 6.8)
ET := 2.0 1 f HV := 1 + PT⋅ ET − 1
(
)
f HV = 0.931
V15 := 720 PHF :=
(given)
V
PHF = 0.938
V15 ⋅ 4
f p := 1.0
(Eq. 6.4)
N := 2
V vp := PHF ⋅ N ⋅ f HV ⋅ f p
(given)
vp = 1547
(Eq. 6.3)
find BFFS FFS := 55
(given)
f LW := 1.9
(Table 6.3)
f LC := 0.2
(Table 6.13)
f M := 1.6
(Table 6.14)
for A f , first convert the 15 access points in 0.62 mile to the number of access points per mile 15 0.62 f := 6.25 A
= 24.2 (using a rounded value of 25 access points/mile)
BFFS := FFS + fLW + f LC + f M + A f
BFFS = 64.95 mi/h
(Table 6.15) (Eq. 6.7)
find density and LOS
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
S := 54.5
D :=
vp S
(interpolated from Fig. 6.4, for FFS = 55) (Eq. 6.6)
D = 28.4
LOS D
(Table 6.11)
Determine the number of access points that need to be blocked D := 26
S :=
vp D
(to achieve LOS C)
S = 59.5
(Eq. 6.6)
FFS := 60
(interpolated from Fig 6.4)
f := BFFS − FFS − f LW − f LC − f M A
f = 1.25 A
(Eq. 6.7)
for A f = 1.25 mi/h (reduction in FFS), access points per mile = 5 (from Table 6.15) f ⋅ 4) 0.62 = 3.1 ( A 15 − 3 = 12
access points over 0.62 mile (4 to 1 relationship between f A and access points/mile) access points need to be blocked
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.20 Determine the level of service. ATS: V := 180 V PHF
PHF := 0.90
(given)
= 200
PT := 0.15
(given)
ET := 1.7
(Table 6.18)
1 f HV := 1 + PT⋅ ET − 1
(
f HV = 0.905
)
(Eq. 6.5)
f G := 1.00
(Table 6.15)
V vp := PHF ⋅ fG ⋅ f HV
vp = 221
(Eq. 6.11)
FFS := 65
(given)
f np := 1.9
(Table 6.19, by interpolation)
ATS := FFS − 0.00776⋅ vp − f np
(Eq. 6.12)
ATS = 61.385
(Table 6.21)
mi/h
LOS is A
PTSF: PT := 0.15
(given)
ET := 1.1
(Table 6.18)
f HV :=
1
(
f HV = 0.985
)
1 + PT⋅ ET − 1
V := 180
(Eq. 6.5)
PHF := 0.90
(given)
f G := 1.00
(Table 6.17)
V vp := PHF ⋅ fG ⋅ f HV
vp = 203
− 0.000879 ⋅ vp BPTSF := 100⋅ 1 − e
f dnp := 22.0 PTSF := BPTSF + fdnp
(Eq. 6.11)
BPTSF = 16.342
(Eq. 6.14) (Table 6.20, by interpolation)
PTSF = 38.342 %
LOS is B (Table 6.21)
(Eq. 6.13)
lower LOS (PTSF) governs, LOS is B
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.21 Determine th e level of servic e. ATS: V := 540 V15 :=
PHF := 0.87 V
(given)
V15 = 620.69
PHF
PT := 0.05
PR := 0.10
(given)
ET := 1.9
ER := 1.1
(Table 6.18)
1 f HV := 1 + P T⋅ ET − 1 + PR ⋅ ER − 1
(
)
(
f HV = 0.948
)
f G := 0.93
(Eq. 6.5) (Table 6.17)
V vp := PHF ⋅ fG ⋅ f HV Vf := 275
vp = 704.116
(Eq. 6.11)
SFM := 57
FFS := SFM + 0.00776 ⋅
(given)
Vf
FFS = 59.251
f HV
(Eq. 6.9)
f np := 3.0
(Table 6.19)
ATS := FFS − 0.00776⋅ vp − f np
ATS = 50.787
LOS B
(Eq. 6.12)
PTSF: PT := 0.05
PR := 0.10
ET := 1.5
ER := 1.0
(given) (Table 6.18)
1 f HV := 1 + P T⋅ ET − 1 + PR ⋅ ER − 1
(
)
(
f HV = 0.976
)
(Table 6.17)
f G := 0.94 V vp := PHF ⋅ fG ⋅ f HV
vp = 676.816
− 0.000879 ⋅ vp BPTSF := 100⋅ 1 − e
(Eq. 6.11)
BPTSF = 44.839
f dnp := 17.5
PTSF := BPTSF + fdnp
(Eq. 6.5)
(Eq. 6.14) (Table 6.20)
PTSF = 62.339
LOS C
(Eq. 6.13)
lower LOS (PTSF) governs, LOS is C
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.22 Determine the level of service. ATS: PT := 0.04
PB := 0.03
PR := 0.01
(given)
PT := PT + PB ET := 1.2
ER := 1.0
(Table 6.18)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
VNB := 440 V PHF
)
(
VSB := 360
f HV = 0.986
)
V := VNB + VSB
V = 800
(Eq. 6.5)
PHF := 0.87
= 919.54
f G := 1.00 vp :=
(given)
(Table 6.17) V
vp = 932.414
PHF ⋅ fG ⋅ f HV
(Eq. 6.11)
BFFS := 60
(given)
f LS := 1.7
(Table 6.16)
f := 4 A
(Table 6.15)
FFS := BFFS − fLS − A f
FFS = 54.3
(Eq. 6.10)
f np := 0
(Table 6.19)
ATS := FFS − 0.00776⋅ vp − f np
ATS = 47.064
mi
LOS C
(Eq. 6.12)
h
PTSF: ET := 1.1
ER := 1.0
(Table 6.18)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
f G := 1.00
)
(
)
f HV = 0.993
(Eq. 6.5) (Table 6.17)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
V vp := PHF ⋅ fG ⋅ f HV
vp = 925.977
− 0.000879 ⋅ vp BPTSF := 100⋅ 1 − e
VNB V
= 0.55
(Eq. 6.11)
BPTSF = 55.689
so a 55/45 split
f dnp := 0.0
(Table 6.20)
PTSF := BPTSF − fdnp PTSF = 55.689
(Eq. 6.14)
(Eq. 6.13) LOS C, from both PTSF and ATS
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.23 Determine the percentage of trucks . V15 := 720
(given)
V := V15 ⋅ 4
V = 2880
vp := 3200 Using ATS adjustment values f G := 0.99
(Table 6.17)
PHF := 1 V f HV := PHF ⋅ fG ⋅ vp
f HV = 0.909
(Eq. 6.11)
ET := 1.5
f HV
(Table 6.18)
1
(
(Eq. 6.5)
)
1 + PT⋅ ET − 1
PT = 0.2
PT⋅ 100 = 20
%
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.24
Determine the maximum percentage of no-passing zones. Determine the flow rate for PTSF PT := 0.08
PB := 0.02
PR := 0.05
(given)
PTB := PT + PB ET := 1.8
ER := 1.0
(Table 6.18)
1 f HV := 1 + PTB ⋅ ET − 1 + PR ⋅ ER − 1
(
V := 500
)
(
)
f HV = 0.926
(Eq. 6.5)
PHF := 0.85
(given)
f G := 0.77
(Table 6.17)
V vp := PHF ⋅ fG ⋅ f HV
(Eq. 6.11)
vp = 825.057 must find new values, since 825 > 600 f G := 0.94
(Table 6.17)
ET := 1.5
ER := 1.0
(Table 6.18)
1 f HV := 1 + PTB⋅ ET − 1 + PR ⋅ ER − 1
(
)
(
)
f HV = 0.952
(Eq. 6.5)
V vp := PHF ⋅ fG ⋅ f HV vp = 657.071
(Eq. 6.11)
is between 600 and 1200 pc/h, so ok
Determine the percent no-passing zones − 0.000879 ⋅ vp BPTSF := 100⋅ 1 − e
BPTSF = 43.874
(Eq. 6.14)
to maintain maximum passing zones and still LOS B, from Table 6.22, PTSF = 55% PTSF := 55 f dnp := PTSF − BPTSF percentnp := 24.32
f dnp = 11.126 %
(Eq. 6.13, rearranged to solve for f d/np ) (from Table 6.20, with triple interpolation)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.26
Determine the freeway's AADT. PT := 0.08
(given)
ET := 2.5
(Table 6.7)
1 f HV := 1 + PT⋅ ET − 1
(
)
PHF := 0.85
f HV = 0.893
(Eq. 6.5)
N := 2
(given)
BFFS := 70
(assumed)
f LW := 1.9
(Table 6.3)
f LC := 1.2
(Table 6.4)
f N := 4.5
(Table 6.5)
f ID := 0.0
(Table 6.6)
FFS := BFFS − fLW − f LC − f N − f ID S := 51.5
( interpolated)
D := 45.0
(for capacity)
D
vp
FFS = 62.4
(Eq. 6.2)
(Eq. 6.6)
S
vp = 2317.5
vp
V PHF ⋅ N ⋅ f HV ⋅ 1.0
(Eq. 6.3)
V = 3517.634 DDHV := V K := 0.12
D := 0.6
AADT⋅ K ⋅ D
DDHV
AADT = 48856
veh day
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternate Solution: vp := 2325
vp
(capacity at FFS = 62.4, interpolated from Table 6.1 or 6.2)
V PHF ⋅ N ⋅ f HV ⋅ 1.0
V = 3529.018 AADT :=
V K ⋅D
AADT = 49014
veh day
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.27 Determine the total directional tr affic volum e. f p := 1.0 PT := 0.10
N := 3
vp := 1430
PR := 0.05
(Table 6.1)
PHF := 0.94
(given)
K := 0.20 ET := 2.5
ER := 2.0
(Table 6.7)
1 f HV := 1 + PT⋅ E T − 1 + P R ⋅ E R − 1
(
)
V := vp ⋅ PHF ⋅ N ⋅ f p ⋅ f HV VD :=
V K
(
)
V = 3360
VD = 16802
f HV = 0.833
(Eq. 6.5)
(Eq. 6.3)
veh/day
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 6.28
What is the density o f the traffic stream?
V := 2000 veh
V15 := 600
BFFS := 70
f lw := 6.6
PT := 0.12
mi
f lc := 0.8
h
ET := 2.5
mi h
f N := 3.0
)
(
FFS = 52.1
)
f HV = 0.806
v p = 992
PHF⋅ N⋅ f HV⋅ f p
S
h
PHF = 0.833
600⋅ 4
V
v p
f id := 7.5
mi h
(Tables: 6.3, 6.4, 6.5, 6.6, respectively)
(Eq 6.2)
(Table 6.7)
V
for vp = 992,
D :=
mi
ER := 2.0
(
(given)
(assumed)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
v p :=
N := 3
f p := 1.0
FFS := BFFS − flw − f lc − f N − f id
PHF :=
PR := 0.06
(Eq 6.5)
(Eq 6.4)
(Eq 6.3)
S := FFS
D = 19.04
pc/mi/ln
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers 1) using BFFS instead of FFS (with S = BFFS) S := BFFS v p
D :=
D = 14.171
S
2) not adjusting for PHF PHF := 1.0 V v p := PHF⋅ N⋅ f HV⋅ f p
v p = 826.667
v p
D :=
D = 15.867
52.1
3) use level terrain PCE values from Table 6.7 instead of rolling terrain ET := 1.5 f HV :=
1
(
V 600⋅ 4
(
)
f HV = 0.933
PHF = 0.833
V PHF⋅ N⋅ f HV⋅ f p
S := FFS
D :=
)
1 + PT⋅ ET − 1 + PR ⋅ ER − 1
PHF :=
v p :=
ER := 1.2
v p S
v p = 857.6 S = 52.1
D = 16.461
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine is th e level of service. PHF := 0.84
BFFS := 70
V := 2500 veh
mi
Problem 6.29 PR := 0.04
N := 3 lanes
f p := 1
h
(given)
(assumed)
f lw := 6.6
(Table 6.3)
f n := 3.0 (Table 6.5)
f id := 2.5
(Table 6.6)
f lc := 0.4 (Table 6.4)
ER := 4.0
(Table 6.7)
Determine free flow speed FFS := BFFS − flw − f n − f id − f lc
FFS = 57.5
mi
(Eq 6.2)
h
Determine heavy vehicle factor f HV :=
1
(
)
1 + PR ⋅ ER − 1
f HV = 0.89
(Eq 6.5)
v p = 1111.11 pc/h/ln
(Eq 6.3)
Determine the flow rate V v p := PHF⋅ N⋅ f HV⋅ f p
Interpolation of Table 6.1 Interpolation of Table 6.1 with a vp of 1111.11 pc/h/ln yields a level of service of C
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Alternative Answers: 1) LOS D 2) LOS A 3) LOS B
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the p eak-hour volume. PT := .06 + .04
Problem 6.30
PR := .03
PHF := 0.81
f p := 0.94
N := 2 lanes
v p := 1250 pc/h/ln
BFFS := 60.0
ER := 2.0
mi h
ET := 2.5
(Table 6.7)
f lc := 1.3
(Table 6.13, with TLC = 4 + 2 = 6 from Eq 6.8)
(Table 6.7)
(given)
f lw := 1.9
(Table 6.3) f M := 0.0
(Table 6.14)
f A := 1.25 (Table 6.15, by interpolation)
Determine free flow speed FFS := BFFS − flc − f lw − f M − f A
FFS = 56
mi
(Eq 6.7)
h
Determine heavy-vehicle factor f HV :=
1
(
)
(
)
1 + PT⋅ ET − 1 + PR ⋅ ER − 1
f HV = 0.847
(Eq 6.5)
Determine peak-hour volume V := v p ⋅ PHF⋅ N ⋅ f HV⋅ f p
V = 1613.14
veh
(Eq 6.3, rearranged)
h
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers: 1) Reverse V and vp V := 1250
veh
v p :=
h
V PHF⋅ N⋅ f HV⋅ f p
v p = 969
veh
V = 1329.79
veh
(different units, supposed to be pc/h/ln)
h
2) Use f p of 1.0 f p := 1.0
V := v p⋅ PHF⋅ N ⋅ f HV⋅ f p
h
3) switch bus/truck and recreational vehicle percentages PR := 0.10 f HV :=
PT := 0.03
f p := 0.94
1
(
)
(
)
1 + PT⋅ ET − 1 + PR ⋅ ER − 1
V := v p⋅ PHF⋅ N ⋅ f HV⋅ f p
f HV = 1
V = 1288.21
veh h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Estimate the free flow speed (in mi/h).
N := 3 lanes
PostedSpeed := 50
BFFS := PostedSpeed + 5
Problem 6.31
mi
(given)
h
BFFS = 55
look up adjustments TLC := 6 + 2
(LCL = 6 ft since undivided highway)
mi f LW := 0.0 h f M := 1.6
mi
(Table 6.3)
(Table 6.14)
h
f LC := 0.9
mi
(Eq 6.8)
(Table 6.13)
h
f A := 0.5 (Table 6.15, with interpolation)
calculate free-flow speed FFS := BFFS − fLW − f LC − f M − f A
FFS = 52
mi
(Eq 6.7)
h
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
A six-lane undivided multilane highway (three lanes in each direction) has 12-ft lanes with 2-ft shoulders on the right side. Estimate the free flow speed (in mi/h).
N := 3 lanes
PostedSpeed := 50
BFFS := PostedSpeed + 5
mi
(given)
h
BFFS = 55
look up adjustments TLC := 6 + 2 f LW := 0.0 f M := 1.6
mi
(Table 6.3)
h mi
(Eq 6.8)
(LCL = 6 ft since undivided highway)
(Table 6.14)
h
f LC := 0.9
mi
(Table 6.13)
h
f A := 0.5 (Table 6.15, with interpolation)
calculate free-flow speed FFS := BFFS − fLW − f LC − f M − f A
FFS = 52
mi
(Eq 6.7)
h
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers 1) Use freeway free-flow speed equation f LC := 1.6
mi h
f N := 3.0
mi h
FFS := BFFS − fLW − f LC − f N − f ID
f ID := 7.5
mi h
FFS = 43
mi
(Eq 6.2)
h
2) Use TLC = 2-ft f LC := 2.8
mi h
FFS := BFFS − fLW − f LC − f M − f A
FFS = 50
3) Misinterpolation of Table 6.15 mi f A := 2.5 h FFS := BFFS − fLW − f LC − f M − f A
FFS = 48
mi h
mi h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the analysis flow rate, based on average travel speed. PHF := 0.92
V := 550
mi
PT := .07 + .03
h
PR := .07
Problem 6.32 (given)
determine heavy vehicle adjustment factor f G := 0.93
(Table 6.17; >600-1200)
ET := 1.9
(Table 6.18; >600-1200)
ER := 1.1
(Table 6.18; >600-1200)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
)
(
)
f HV = 0.912
(Eq 6.5)
determine 15-min passenger car equivalent flow rate v p :=
V PHF⋅ f G⋅ f HV
v p = 705
pc/h
(Eq 6.11)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers 1) Misuse Tables 6.17 and 6.18 f G := 0.71
(Table 6.17; 0-600)
ET := 2.5
(Table 6.18; 0-600)
ER := 1.1
(Table 6.18; 0-600)
f HV :=
1
(
)
(
f HV = 0.864
)
1 + PT⋅ ET − 1 + PR ⋅ ER − 1
V v p := PHF⋅ f G⋅ f HV
v p = 974
pc/h/ln
2) Don't include percentage of buses PT := 0.07 f G := 0.93
(Table 6.17; >600-1200)
ET := 1.9
(Table 6.18; >600-1200)
ER := 1.1
(Table 6.18; >600-1200)
1 f HV := 1 + PT⋅ ET − 1 + PR ⋅ ER − 1
(
)
V v p := PHF⋅ f G⋅ f HV
(
f HV = 0.935
)
v p = 688
pc/h
3) Use Equation 6.3 for flow rate N := 2
f p := 1
v p :=
V PHF⋅ N ⋅ f HV⋅ f p
f HV := 0.912 v p = 328
pc/h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the number of lanes required to pr ovide at least LOS D. PHF := 0.88
D := 0.65
f p := 1.0
f HV := 1.0
FFS := 70
mi h
AADT := 75000
Problem 6.33 (given)
(assumed) (Figure 6.7)
K 60 := 0.11
Calculate directional design-hour volume DDHV:= K60 ⋅ D⋅ AADT
(Eq 6.16)
From Table 6.1, the maximum service flow rate for LOS D at 70 mi/h is 2150 pc/h/ln.
Assume a four-lane freeway. Use Eq 6.3 N := 3 lanes v p :=
DDHV PHF⋅ N⋅ f p ⋅ f HV
v p = 2031.25
pc/h/ln
(Eq 6.3)
This value of vp is less than 2150 pc/h/ln, so six lanes (both directions) is sufficient.
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 4 lanes, 8 lanes, 10 lanes
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany
Principles of Highway Engineering and Traffic Analysis, 4e By Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski
Chapter 7 Traffic Control and Analysis at Signalized Intersections U.S. Customary Units
Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Preface The solutions to the fourth edition of Principles of Highway Engineering and Traffic Analysis 1 were prepared with the Mathcad software program. You will notice several notation conventions that you may not be familiar with if you are not a Mathcad user. Most of these notation conventions are self-explanatory or easily understood. The most common Mathcad specific notations in these solutions relate to the equals sign. You will notice the equals sign being used in three different contexts, and Mathcad uses three different notations to distinguish between each of these contexts. The differences between these equals sign notations are explained as follows.
•
The ‘:=’ (colon-equals) is an assignment operator, that is, the value of the variable or expression on the left side of ‘:=’is set equal to the value of the expression on the right side. For example, in the statement, L := 1234, the variable ‘L’ is assigned (i.e., set equal to) the value of 1234. Another example is x := y + z. In this case, x is assigned the value of y + z.
•
The ‘=’ (bold equals) is used when the Mathcad function solver was used to find the value of a variable in the equation. For example, in the equation , the = is used to tell Mathcad that the value of the expression on the left side needs to equal the value of the expression on the right side. Thus, the Mathcad solver can be employed to find a value for the variable ‘t’ that satisfies this relationship. This particular example is from a problem where the function for arrivals at some time ‘t’ is set equal to the function for departures at some time ‘t’ to find the time to queue clearance.
•
The ‘=’ (standard equals) is used for a simple numeric evaluation. For example, referring to the x := y + z assignment used previously, if the value of y was 10 [either by assignment (with :=), or the result of an equation solution (through the use of =) and the value of z was 15, then the expression ‘x =’ would yield 25. Another example would be as follows: s := 1800/3600, with s = 0.5. That is, ‘s’ was assigned the value of 1800 divided by 3600 (using :=), which equals 0.5 (as given by using =).
Another symbol you will see frequently is ‘→’. In these solutions, it is used to perform an evaluation of an assignment expression in a single statement. For example, in the following statement, , Q(t) is assigned the value of 2 Arrivals(t) – Departures(t), and this evaluates to 2.2t – 0.10t .
Finally, to assist in quickly identifying the final answer, or answers, for what is being asked in the problem statement, yellow highlighting has be en used (which will print as light gray). 1
www.mathcad.com
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problems 7.1 Determine the maximum length of the effective red.
λ := ρ :=
800
λ = 0.222
3600
λ μ
ρ = 0.533
C := 60
μ :=
1500 3600
μ = 0.417
(given) (Eq. 5.27)
(given)
at queue clearing r + t0
+
10
C
t0
C − r − 10
t0
ρ ⋅ r (1 − ρ )
C − r − 10
r
= 23.3
so
(Eq. 7.7)
ρ ⋅ r (1 − ρ ) sec
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.3 Determine the average vehicle delay. C := 60
g := 25
(given)
r := C − g
r = 35
(Eq. 7.5)
s :=
λ :=
1400
s
3600 500
λ = 0.139
3600
μ := s ρ :=
= 0.389
(given)
μ = 0.389
λ μ
ρ = 0.357
(Eq. 5.27)
2
d :=
r
2 ⋅ C⋅ (1
d
− ρ)
= 15.88
s/veh
(Eq. 7.12)
Problem 7.4 Determine queue dissipation time. r := 40
λ := μ :=
Q r 1440 3600
ρ :=
λ μ
tc :=
ρ ⋅ r (1 − ρ )
Q := 8
(Given)
λ = 0.2
given vehicles over a given time period
μ = 0.4
(Given)
ρ = 0.5
tc
= 40
(Eq. 5.27)
sec
(Eq. 7.7)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.5
Determine the arrival rate. calculate the effective red time
μ :=
1500
μ = 0.416667
3600
C := 60
(Given)
D := 5.78 ⋅ 60
D = 346.8
for delay, a graph of the queue gives the area between the arrival and departure curves as C⋅ (9
Area1
− 4 + λ ⋅ g)
triangle above rectangle for full cycle
2 rectangle for full cycle
C⋅ 4
Area2
g⋅(λ ⋅C + 4
Area3
− 2)
triangle under departure function for green time
2
Area1 + Area2 + Area3
D
the λs will cancel out, giving C⋅ ( 9
D
−4+
g)
2
+ C⋅ 4 −
g ⋅ ( C + 4 − 2) 2
plug in variables to solve for g
g
= 43.2
r := C − g
r
= 16.8
sec
(Eq 7.5)
calculate the arrival rate by inspection of the queue diagram
λ ⋅ r
9
−4
λ = 0.298
veh/s
λ ⋅ 3600 = 1071.4
veh/h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.7 Determine the arrival rate.
μ :=
3600
(given)
μ=1
3600
C := 60 at queue dissipation
λ ⋅ ( C − 8) r
μ ⋅ ( C − 8 − r )
13
λ
substituting and solving for λ
⎛ C − 8 − 13 ⎞ λ ⎠ ⎝
λ ⋅ ( C − 8)
μ⋅
λ = 0.5
λ ⋅ 3600 = 1800
r :=
13
λ
g := C − r
veh/h
r = 26
g = 34
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.8 Determine the t otal vehicl e delay. find the arrival rate
μ :=
1800
μ = 0.5
3600
C := 80
veh/s (given)
s
arrival rate during eff. green = 2λ (arrival rate during effective red)
Δ q := 7.9 − 2
difference between queue at beginning and end of effective red
Δ q = 5.9 by inspection of a queuing diagram, we find that equating vertical distances gives 2
+ λ ⋅ ( C − g) + 2λ ⋅ g μg
(1)
(note that r = C - g )
and
λ ⋅ ( C − g)
Δq
or g
C−
Δq λ
(2)
with two equations and two unknowns (λ, g), substituting (2) into (1) gives 2
+ λ⋅
⎡ ⎛ Δ q ⎞⎤ ⎛ Δ q ⎞ ⎛ Δ q ⎞ C− C− + 2 λ ⋅ ⋅ C− μ⋅ C − ⎣ ⎝ λ ⎠⎦ ⎝ λ ⎠ ⎝ λ ⎠
solving for λ yields two solutions
λ1 = 0.118
veh/s
λ2 = 0.157
veh/s
both λ's are feasible, so we must find two delays
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
determine total delay based on first arrival rate g1 := C −
Δq
g1
λ1
= 29.8
r 1 := C − g1 r 1
(Eq 7.5)
= 50.2
D1 :=
(2
+
7.9) ⋅ r 1 2
+
7.9 ⋅ g1 2
D1
= 366.2
veh-sec
determine total delay based on second arrival rate
g2 := C −
Δq
g2
λ2
= 42.4
r 2 := C − g2 r 2
(Eq 7.5)
= 37.6
D2 :=
(2
+
7.9) ⋅ r 2 2
+
7.9 ⋅ g2 2
D2
= 353.6
veh-sec
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.8 Alternate Solution with 8.0 vehicles in the queue. Determine the t otal vehicl e delay. find the arrival rate
μ :=
1800
μ = 0.5
3600
C := 80
veh/s (given)
s
arrival rate during eff. green = 2λ (arrival rate during effective red)
Δ q := 8 − 2
difference between queue at beginning and end of effective red
Δq = 6 by inspection of a queuing diagram, we find that equating vertical distances gives 2
+ λ ⋅ ( C − g) + 2λ ⋅ g μg
(1)
(note that r = C - g )
and
λ ⋅ ( C − g)
Δq
or g
C−
Δq λ
(2)
with two equations and two unknowns (λ, g), substituting (2) into (1) gives
⎡ ⎛ Δ q ⎞⎤ ⎛ Δ q ⎞ ⎛ Δ q ⎞ + 2 λ ⋅ ⋅ C− μ⋅ C − 2 + λ⋅ C − C − ⎣ ⎝ λ ⎠⎦ ⎝ λ ⎠ ⎝ λ ⎠ solving for λ yields two solutions
λ1 = 0.125
veh/s
λ2 = 0.15
veh/s
both λ's are feasible, so we must find two delays
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
determine total delay based on first arrival rate g1 := C −
Δq
g1
λ1
= 32
r 1 := C − g1 r 1
(Eq 7.5)
= 48
D1 :=
(2
+
8) ⋅ r 1 2
+
8 ⋅ g1
D1
2
= 368
veh-sec
determine total delay based on second arrival rate
g2 := C −
Δq
g2
λ2
= 40
r 2 := C − g2 r 2
(Eq 7.5)
= 40
D2 :=
(2
+
8) ⋅ r 2 2
+
8 ⋅ g2 2
D2
= 360
veh-sec
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.9 Determine the t otal vehicl e delay. s
s := 1800
μ :=
C := 60
r := 25
g := C − r 1 Dt := ⋅ [ 6 2
g
+
(6
3600
= 35
μ = 0.5
λ :=
900 3600
λ = 0.25
(given)
(Eq. 7.5)
sec
+ λ ⋅ 60) ] ⋅ 60 −
1 2
⋅ ( g ⋅ μ ) ⋅ g
Dt
= 503.75
veh-sec
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.10 Determine the total delay. 1064
λ :=
λ = 0.296
3600
s := 2640
veh/s
μ :=
veh/h
s
μ = 0.733
3600
veh/s
(given)
solving for t 3
+ λ ⋅t
t :=
10
7
t
λ
= 23.7
sec
solving for effective red r := t
+
r = 31.7
8
sec
g := 15
(given)
C := r + g
C = 46.7
(Eq. 7.5) b2 := g
b1 := C h1 := [ 3
+
(3
+ λ ⋅ C) ]
1 Dt := ⋅ b1 ⋅ h1 2
1 Dt := ⋅ [ 3 2
+
−
(3
1 2
h2 := g ⋅ μ
⋅ b2 ⋅ h2
+ λ ⋅ C) ] ⋅ C −
Dt
1 2
= 379.6
⋅ ( g ⋅ μ ) ⋅ g
veh-sec
Dt
= 379.6
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.11 Determine the t otal vehicl e delay. find the arrival rate 1800
μ :=
μ = 0.5
3600
C := 80
veh/s
(given)
s
6
λ
r C − r
g
(Eq. 7.5)
substituting and solving for r ( 10 +
λ ⋅ g) − μ ⋅ g
[ 10 +
λ ⋅ ( C − r ) ] − μ ⋅ ( C − r )
2
⎡ 10 + 6 ⋅ ( C − r )⎤ − μ ⋅ (C − r ) r ⎣ ⎦
8
+
480
480 r
− 6 − 40 +
+
0.5 ⋅ r 38
0.5 ⋅ r
2
− 38⋅ r +
r = 16
λ :=
r
2
2
0.5 ⋅ r 0
480
0
6
λ = 0.375
r
find total delay 1 Dt := ⋅ ( 4 2 Dt
+
10) ⋅ r +
= 496
1 2
⋅ [ 10 +
[ 4 + ( 0.375⋅ C) ] ] ⋅ ( C − r )
−
1 2
⋅ [ μ ⋅ ( C − r ) ] ⋅ ( C − r )
veh − sec
Alternative Solution
⌠ C 4⋅ C + ⎮ ⌡0
⌠ C−r 0.375⋅ t dt − ⎮ 0.5 ⋅ t dt = 496.006 ⌡0
veh − sec
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.12 Determine effective green and red ti mes and the t otal vehicle delay. C := 60
s (given)
λ ( t) := 0.22 + Arrivals( t) :=
Arrivals( C)
.012 ⋅ t
⌠ ⎮ λ ( t) dt → .220 ⋅ t + ⎮ ⌡
= 34.8
g :=
⋅t
veh
λC= μg
to clear at the end of the cycle
μ :=
-3 2
6.00⋅ 10
3600
μ=1
3600 Arrivals( C)
μ
r := C − g
veh/s
(given)
g
= 34.8
sec
r
= 25.2
sec
(Eq. 7.5)
determine total delay
Delay :=
⎛ ⌠ 60 ⎮ ⌡0
⎝
⎞ Arrivals( t) dt
⎠
−
1 2
g ⋅ ( μ ⋅ g)
Delay = 222.5
veh − sec
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.13 Determine the t otal vehicl e delay.
λ1 := λ2 := μ :=
800 3600 500 3600
1200 3600
λ1 = 0.222
λ1 ⋅ 40 = 8.89
λ2 = 0.139
λ2 ⋅ ( 120 − 40) = 11.11
μ = 0.333
μ ⋅ 20 = 6.667
(given)
By inspection of the queuing diagram, (with 8.89 vehicles arriving and 6.67 departing at the end of the first cycle and 14.45 and 13.34 arriving and departing at the end of the second cycle) we find that the first cycle delay is (using triangles)
D1 :=
⎛ 40⋅ 8.89 − ⎝ 2
20⋅ 6.67 ⎞ 2
D1
⎠
= 111.1
veh − sec
similarly, second cycle delay is:
D2 := 40⋅ ( 2.23)
+
( 14.45 − 8.89) ⋅ 40 2
−
( 13.34 − 6.67) ⋅ 20 2
D2 = 133.7
veh − sec
Therefore, total delay 2 cycles later is 111.1 + 133.7 = 244.8
veh − sec
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.16 Determine the sum of crit ical flow ratios. NBL := SBL :=
330 365
SBL
1750 1125
NBTR :=
3400 1075
SBTR :=
EBL :=
NBL = 0.194
1700
3300
110
WBL :=
80
EBTR :=
WBTR :=
250 1750 285 1800
Σvs := SBL +
phase 2 critical
SBTR = 0.326
= 0.169
WBL
600
phase 1 critical
NBTR = 0.331
EBL
650
= 0.209
phase 3 critical
= 0.133
EBTR = 0.143
WBTR = 0.158
NBTR + EBL
Σvs = 0.709
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.18 Determine X c. C := 95
(given)
L := 5 ⋅ 4
L
Σvs := 0.235 +
= 20 0.250 + 0.170 + 0.125
(given)
Σvs = 0.78
C
Xc
L ⋅ Xc Xc
− Σvs
(Eq. 7.20)
= 0.988
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.20 Determine the minim um cy cle length and p hase effective green times. calculate cycle length flow ratios - 0.225, 0.175, 0.200, 0.150 L := 5 ⋅ 4
lost time per phase - 5 sec
(given) L
= 20
sec
Xc := 0.85
(given)
Σvs := 0.225 +
Cmin :=
0.175 + 0.200 + 0.150
L ⋅ Xc Xc
Cmin
− Σvs
= 170
Σvs = 0.75
(Eq. 7.20)
sec
calculate green times for each phase
g1 := ( 0.225) ⋅
g2 := ( 0.175) ⋅
g3 := ( 0.200) ⋅
g4 := ( 0.150) ⋅ g1
+
⎛ Cmin ⎞ Xc
⎝ ⎠ ⎛ Cmin ⎞ Xc
⎝ ⎠ ⎛ Cmin ⎞ Xc
⎝ ⎠ ⎛ Cmin ⎞ ⎝
Xc
⎠
g2 + g3 + g4 + L
= 170
g1
= 45
g2
= 35
g3
= 40
sec
(Eq. 7.22)
g4
= 30
sec
(Eq. 7.22)
sec
sec
sec
(Eq. 7.22)
(Eq. 7.22)
checks
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.21 Determine the minimum cycle length and effective green times. calculate the minimum cycle length SBL :=
365
NBTR := EBL :=
SBL
1750 1125
= 0.209
NBTR = 0.331
3400
110
EBL
650
= 0.169
Σvs := SBL +
NBTR + EBL
L := 4 ⋅ 3
= 12
L
(given)
Σvs = 0.709 (given)
L ⋅ 0.9 Cmin := 0.9 − Σvs Cmin
= 56.451
C := 60
(Eq. 7.20)
rounds to 60 sec
sec
calculate the effective green times
Σvs⋅ C Xc := C−L
Xc
= 0.886
⎛ C ⎞ g1 := SBL ⋅ ⎝ Xc ⎠
g1
= 14.1
sec
⎛ C ⎞ g2 := NBTR⋅ ⎝ Xc ⎠
g2
= 22.4
sec
⎛ C ⎞ g3 := EBL ⋅ ⎝ Xc ⎠
g3
= 11.5
sec
g1
+
g2 + g3 + L
= 60
sec
(Eq. 7.20)
checks
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.22 Determine the northbo und approach delay and level of service. calculate northbound left delay C := 60
gL := 14.127 NBL
XL :=
sL ⋅
s L := 1700
XL = 0.824
⎛ gL ⎞
(given)
(Eq. 7.22)
⎝ C ⎠
⎛ gL ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ d1L := ⎛ gL ⎞ 1 − XL ⋅ ⎝ C ⎠ T := 0.25
c L := sL ⋅
NBL := 330
2
k := 0.5 gL
(Eq. 7.15)
I := 1.0
(assumed)
c L = 400.265
C
d2L := 900⋅ T⋅
PF := 1.0
d1L = 21.76
⎡ ⎣
(Eq. 7.6)
(XL − 1) + (XL − 1)
2
+
8 ⋅ k ⋅ I⋅ XL ⎤ cL ⋅ T
⎦
d3L := 0
dL := d1L ⋅ PF
+
d2L = 17.322
(Eq. 7.16)
(assumed)
d2L + d3L
= 39.082
dL
(Eq. 7.14)
calculate northbound through delay gTR := 22.411 XTR :=
NBTR := 1125
NBTR
⎛ gTR ⎞ s TR⋅ ⎝ C ⎠
⎛ gTR ⎞ 0.5 ⋅ C⋅ 1 − C ⎠ ⎝ d1TR := ⎛ gTR ⎞ 1 − XTR⋅ C ⎝
s TR := 3400
(given)
XTR = 0.886
2
d1TR = 17.597
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
c TR := s TR⋅
gTR
T := 0.25
k := 0.5
d2TR := 900⋅ T⋅ PF := 1.0
cTR = 1269.957
C
⎡ ⎣
I := 1.0
(XTR − 1) + (XTR − 1)
2
+
8 ⋅ k ⋅ I⋅ XTR ⎤ c TR⋅ T
⎦
d2TR = 9.312
d3TR := 0
dTR := d1TR⋅ PF
+
d2TR + d3TR
dTR = 26.909
calculate northbound total delay
d :=
NBL ⋅ dL
+
NBTR⋅ dTR
NBL + NBTR
d = 29.7
sec/veh
LOS is C
(Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.23 Determine the southbo und approach delay and level of service. calculate southbound left delay C := 60
gL := 14.127 SBL
XL :=
sL ⋅
XL
⎛ gL ⎞
s L := 1750
(given)
= 0.886
(Eq. 7.22)
⎝ C ⎠
⎛ gL ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ d1L := ⎛ gL ⎞ 1 − XL ⋅ ⎝ C ⎠ c L := sL ⋅
SBL := 365
gL
T := 0.25
cL
C
d1L = 22.158
⎡ ⎣
(Eq. 7.15)
= 412.038
k := 0.5
d2L := 900⋅ T⋅ PF := 1.0
2
(Eq. 7.6)
I := 1.0
(assumed)
(XL − 1) + (XL − 1)
2
+
8 ⋅ k ⋅ I⋅ XL ⎤ cL ⋅ T
⎦
d3L := 0
dL := d1L ⋅ PF
+
d2L = 23.316
(Eq. 7.16)
(assumed)
d2L + d3L
dL
= 45.474
(Eq. 7.14)
calculate southbound through and right delay gTR := 22.411 XTR :=
SBTR := 1075
s TR := 3300
(given)
SBTR XTR = 0.872
⎛ gTR ⎞ s TR⋅ ⎝ C ⎠
⎛ gTR ⎞ 0.5 ⋅ C⋅ 1 − C ⎠ ⎝ d1TR := ⎛ gTR ⎞ 1 − XTR⋅ C ⎝
2
d1TR = 17.463
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
c TR := s TR⋅
gTR
T := 0.25
k := 0.5
d2TR := 900⋅ T⋅ PF := 1.0
cTR = 1232.605
C
⎡ ⎣
I := 1.0
(XTR − 1) + (XTR − 1)
2
+
8 ⋅ k ⋅ I⋅ XTR ⎤ c TR⋅ T
⎦
d2TR = 8.658
d3TR := 0
dTR := d1TR⋅ PF
+
d2TR + d3TR
dTR = 26.121
calculate southbound total approach delay
d :=
SBL ⋅ dL SBL
+ +
SBTR⋅ dTR SBTR
d
= 31
sec/veh
LOS = C
(Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.24 Determine the westbound approach delay and level of service. calculate westbound left delay C := 60
XL :=
gL := 11.462
WBL := 80
WBL
XL
⎛ gL ⎞ sL ⋅ ⎝ C ⎠
⎛ gL ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ d1L := ⎛ gL ⎞ 1 − XL ⋅ ⎝ C ⎠ c L := s L ⋅
gL
T := 0.25
⎡
(
⎣
PF := 1.0
= 0.698
(Eq. 7.22)
d1L = 22.653
(Eq. 7.6)
I := 1.0
(assumed)
− 1) +
(
XL
2
− 1) +
8 ⋅ k ⋅ I⋅ XL⎤ cL ⋅ T
⎦
d3L := 0
dL := d1L ⋅ PF
+
(Eq. 7.15)
= 114.62
k := 0.5
d2L := 900⋅ T⋅ XL
(given)
2
cL
C
sL := 600
d2L = 29.769
(Eq. 7.16)
(assumed)
d2L + d3L
dL
= 52.422
(Eq. 7.14)
calculate westbound through and right delay gTR := 11.462 XTR :=
WBTR := 285
sTR := 1800
(given)
WBTR XTR = 0.829
⎛ gTR ⎞ s TR⋅ ⎝ C ⎠
⎛ gTR ⎞ 0.5 ⋅ C⋅ 1 − C ⎠ ⎝ d1TR := ⎛ gTR ⎞ 1 − XTR⋅ C ⎠ ⎝
2
d1TR = 23.326
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
c TR := s TR⋅ T := 0.25
gTR
c TR = 343.86
C k := 0.5
⎡
(
I := 1.0
) + (XTR − 1)
d2TR := 900⋅ T⋅ XTR − 1
⎣
PF := 1.0
2
+
8 ⋅ k ⋅ I⋅ XTR⎤ c TR⋅ T
⎦
d2TR = 20.101
d3TR := 0
dTR := d1TR⋅ PF
+
d2TR + d3TR
dTR = 43.427
calculate westbound total approach delay
d :=
WBL ⋅ dL WBL
+ +
WBTR ⋅ dTR WBTR
d
= 45.4
sec/veh
LOS is D
(Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.25 Determine the eastbound approach delay and level of service. calculate eastbound left delay C := 60
gL := 11.462 EBL
XL :=
sL ⋅
XL
⎛ gL ⎞
sL := 650
(given)
= 0.886
(Eq. 7.22)
⎝ C ⎠
⎛ gL ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ d1L := ⎛ gL ⎞ 1 − XL ⋅ ⎝ C ⎠ c L := s L ⋅
EBL := 110
gL
T := 0.25
cL
C
(
d2L := 900⋅ T⋅ XL
⎣
d1L = 23.632
(Eq. 7.15)
= 124.172
k := 0.5
⎡
PF := 1.0
2
− 1) +
(Eq. 7.6)
I := 1.0
(assumed)
(XL − 1)
2
+
8 ⋅ k ⋅ I⋅ XL⎤ cL ⋅ T
⎦
d3L := 0
dL := d1L ⋅ PF
+
d2L = 54.559
(Eq. 7.16)
(assumed)
d2L + d3L
dL
= 78.191
(Eq. 7.14)
calculate eastbound through and right delay gTR := 11.462 XTR :=
EBTR := 250
sTR := 1750
(given)
EBTR XTR = 0.748
⎛ gTR ⎞ s TR⋅ ⎝ C ⎠
⎛ gTR ⎞ 0.5 ⋅ C⋅ 1 − C ⎠ ⎝ d1TR := ⎛ gTR ⎞ 1 − XTR⋅ C ⎠ ⎝
2
d1TR = 22.905
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
c TR := s TR⋅
gTR
T := 0.25
cTR = 334.308
C k := 0.5
⎡
(
I := 1.0
) + (XTR − 1)
d2TR := 900⋅ T⋅ XTR − 1
⎣
PF := 1.0
2
+
8 ⋅ k ⋅ I⋅ XTR⎤ c TR⋅ T
⎦
d2TR = 14.191
d3TR := 0
dTR := d1TR⋅ PF
+
d2TR + d3TR
dTR = 37.096
calculate eastbound total approach delay
d :=
EBL ⋅ dL EBL
+ +
EBTR⋅ dTR EBTR
d
= 49.65
sec/veh
LOS is D
(Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.26 Determine the overall i ntersection d elay and LOS. Phase 1
Phase 2
Phase 3
NBL := 330
NBTR := 1125
EBL := 110
SBL := 365
SBTR := 1075
WBL := 80 EBTR := 250 WBTR := 285
dNB := 29.7
dSB := 31.0
dWB
:= 45.4
dEB := 49.65
(from previous questions)
Using Eq. 7.28:
dtotal :=
dtotal
( NBL + NBTR) ⋅ dNB
= 33.8
+
( SBL
( NBL + NBTR) sec/veh
+
+
SBTR) ⋅ dSB
( SBL
+
LOS is C
SBTR)
+
( WBL
+
WBTR) ⋅ dWB
+
( WBL
+
WBTR)
+
+
( EBL
( EBL
+
+
EBTR) ⋅ dEB
EBTR)
(Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.28 Determine the minim um cy cle length and effective green times fo r each phase. find minimum cycle length EBL :=
245
WBTR :=
SBL :=
NBL :=
EBL
1750 1030
= 0.14
WBTR = 0.303
3400
255
SBL
1725 225
= 0.148
NBL = 0.132
1700
Σvs := EBL +
WBTR + SBL
L := 4 ⋅ 4
L
Cmin :=
Cmin
= 16
+
Σvs = 0.723
NBL
Xc := 0.95
L ⋅ Xc Xc
(Eq. 7.20)
− Σvs
= 66.996
rounds to 70 seconds
C := 70
sec
find effective green times
Σvs⋅ C Xc := C−L
Xc
C g1 := EBL ⋅ Xc
g1 = 10.5
sec
C g2 := WBTR⋅ Xc
g2 = 22.6
sec
C g3 := SBL ⋅ Xc
g3 = 11
sec
C g4 := NBL ⋅ Xc
g4 = 9.9
sec
g1
+
g2 + g3 + g4 + L
= 0.937
= 70
checks
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.29 Determine the northbo und approach delay and level of service. calculate northbound left delay C := 70
gL := 9.884 NBL
XL :=
sL ⋅
NBL := 225
(given)
XL = 0.937
⎛ gL ⎞
(Eq. 7.22)
⎝ C ⎠
⎛ gL ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ d1L := ⎛ gL ⎞ 1 − XL ⋅ ⎝ C ⎠ c L := sL ⋅
s L := 1700
gL
cL
C
T := 0.25
2
d1L = 29.752
= 240.04
k := 0.5
d2L := 900⋅ T⋅ PF := 1.0
⎡ ⎣
(Eq. 7.15)
(Eq. 7.6)
I := 1.0
(given)
(XL − 1) + (XL − 1)
2
+
8 ⋅ k ⋅ I⋅ XL ⎤ cL ⋅ T
⎦
d3L := 0
dL := d1L ⋅ PF
+
d2L = 43.883
(Eq. 7.16)
(given)
d2L + d3L
dL = 73.634
calculate northbound through and right delay gTR := 9.884 XTR :=
NBTR := 215
s TR := 1750
(given)
NBTR XTR = 0.87
⎛ gTR ⎞ s TR⋅ ⎝ C ⎠
⎛ gTR ⎞ 0.5 ⋅ C⋅ 1 − C ⎠ ⎝ d1TR := ⎛ gTR ⎞ 1 − XTR⋅ C ⎠ ⎝ c TR := s TR⋅
gTR C
2
d1TR = 29.429
c TR = 247.1
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
T := 0.25
k := 0.5
d2TR := 900⋅ T⋅
PF := 1.0
⎡ ⎣
(
I := 1.0
)+
XTR − 1
(
)
XTR − 1
2
+
8 ⋅ k ⋅ I⋅ XTR ⎤ c TR⋅ T
⎦
d2TR = 31.652
d3TR := 0
dTR := d1TR⋅ PF
+
d2TR + d3TR
dTR = 61.082
calculate northbound total approach delay
d :=
NBL ⋅ dL
+
NBTR⋅ dTR
NBL + NBTR
d
= 67.5
sec/veh
LOS is E
(Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.30 Determine the southbo und approach delay and level of service. calculate eastbound left delay C := 70
gL := 11.0
SBL := 255
SBL
XL :=
sL ⋅
XL
⎛ gL ⎞
= 0.9
(given)
(Eq. 7.22)
⎝ C ⎠
⎛ gL ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ d1L := ⎛ gL ⎞ 1 − XL ⋅ ⎝ C ⎠ c L := s L ⋅
s L := 1725
gL
cL
C
T := 0.25
2
(
d2L := 900⋅ T⋅ XL
⎣
PF := 1.0
(Eq. 7.15)
= 271.1
k := 0.5
⎡
d1L = 29.2
− 1) +
(Eq. 7.6)
I := 1.0
(assumed)
(XL − 1)
2
+
8 ⋅ k ⋅ I⋅ XL⎤ cL ⋅ T
⎦
d3L := 0
dL := d1L ⋅ PF
+
d2L = 41.3
(Eq. 7.16)
(assumed)
d2L + d3L
dL
= 70.5
(Eq. 7.14)
calculate southbound through/right delay gTR := 11.0
XTR :=
SBTR := 235
SBTR
⎛ gTR ⎞ s TR⋅ ⎝ C ⎠
⎛ gTR ⎞ 0.5 ⋅ C⋅ 1 − C ⎠ ⎝ d1TR := ⎛ gTR ⎞ 1 − XTR⋅ C ⎠ ⎝
s TR := 1750
XTR = 0.9
(given)
(Eq. 7.22)
2
d1TR = 28.7
(Eq. 7.15)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
c TR := s TR⋅
gTR
T := 0.25
C k := 0.5
⎡
(
(assumed)
I := 1.0
) + (XTR − 1)
d2TR := 900⋅ T⋅ XTR − 1
⎣
PF := 1.0
(Eq. 7.6)
cTR = 275
2
+
8 ⋅ k ⋅ I⋅ XTR⎤ c TR⋅ T
⎦
d2TR = 27.2
(assumed)
d3TR := 0
dTR := d1TR⋅ PF
+
(Eq. 7.16)
d2TR + d3TR
dTR = 55.9
(Eq. 7.14)
calculate southbound total approach delay
d :=
SBL ⋅ dL SBL
+ +
SBTR⋅ dTR SBTR
d
= 63.5
sec/veh
LOS is E
(Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.31 Determine the westbound approach delay and level of service. calculate westbound left delay C := 70 XL :=
gL := 10.455
WBL := 230
s L := 1725
(given)
WBL XL = 0.893
⎛ gL ⎞ sL ⋅ ⎝ C ⎠
⎛ gL ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ d1L := ⎛ gL ⎞ 1 − XL ⋅ ⎝ C ⎠ c L := sL ⋅
gL
T := 0.25
2
d1L = 29.222
(Eq. 7.15)
c L = 257.641
C k := 0.5
d2L := 900⋅ T⋅ PF := 1.0
(Eq. 7.22)
⎡ ⎣
(
XL
− 1) +
(Eq. 7.6)
I := 1.0
(
XL
(assumed) 2
− 1) +
8 ⋅ k ⋅ I⋅ XL ⎤ cL ⋅ T
⎦
d3L := 0
dL := d1L ⋅ PF
+
d2L = 34.079
(Eq. 7.16)
(assumed)
d2L + d3L
dL
= 63.301
(Eq. 7.14)
calculate westbound through and right delay gTR := 22.623 XTR :=
WBTR := 1030
WBTR
(given)
XTR = 0.937
⎛ gTR ⎞ s TR⋅ ⎝ C ⎠
⎛ gTR ⎞ 0.5 ⋅ C⋅ 1 − C ⎠ ⎝ d1TR := ⎛ gTR ⎞ 1 − XTR⋅ C ⎝
s TR := 3400
2
d1TR = 23.001
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
c TR := s TR⋅ T := 0.25
gTR
k := 0.5
d2TR := 900⋅ T⋅ PF := 1.0
c TR = 1098.831
C
⎡ ⎣
I := 1.0
(XTR − 1) + (XTR − 1)
2
+
8 ⋅ k ⋅ I⋅ XTR ⎤ c TR⋅ T
⎦
d2TR = 15.732
d3TR := 0
dTR := d1TR⋅ PF
+
d2TR + d3TR
dTR = 38.733
calculate westbound total approach delay
d :=
WBL ⋅ dL WBL
+ +
WBTR ⋅ dTR WBTR
d
= 43.2
sec/veh
LOS is D
(Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.33 Determine the overall intersection d elay and level of service. calculate overall intersection delay by a weighted average of approach delays EBL := 245
WBL := 230
SBL := 255
NBL := 225
EBTR := 975
WBTR := 1030
SBTR := 235
NBTR := 215
dNB := 67.50
dSB := 62.86
dWB
:= 43.22
dEB := 41.85
Using Eq. 7.28:
dtotal :=
dtotal
( NBL + NBTR) ⋅ dNB
= 48.7
+
( SBL
( NBL + NBTR) sec/veh
+
+
SBTR) ⋅ dSB
( SBL
LOS is D
+
SBTR)
+
( WBL
+
WBTR) ⋅ dWB
+
( WBL
+
WBTR)
+
+
( EBL
( EBL
+
+
EBTR) ⋅ dEB
EBTR) (Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.34
Determine the optimal cycle length and correspond ing effective green times.
EBL :=
245
1030
WBTR :=
SBL :=
NBL :=
EBL
1750
WBTR = 0.303
3400
255
SBL
1725 225
(from Prob. 7.17)
= 0.148
NBL = 0.132
1700
Σvs := EBL +
= 0.14
WBTR + SBL
+
Σvs = 0.723
NBL
calculate total lost time L := 4 ⋅ 4
L = 16
calculate optimal cycle length 1.5 ⋅ L + 5 Copt := 1.0 − Σ vs
Copt
= 104.7
(Eq. 7.21)
sec
round up to
Copt := 105
sec
calculate phase effective green times
g1 := EBL ⋅
⎛ Copt − L ⎞ ⎝ Σvs ⎠
g1
= 17.2
sec
(Eq. 7.22)
⎛ Copt − L ⎞ g2 := WBTR ⋅ ⎝ Σvs ⎠
g2
= 37.3
sec
(Eq. 7.22)
⎛ Copt − L ⎞ g3 := SBL ⋅ ⎝ Σvs ⎠
g3
= 18.2
sec
(Eq. 7.22)
⎛ Copt − L ⎞ ⎝ Σvs ⎠
g4
= 16.3
sec
(Eq. 7.22)
g4 := NBL ⋅ g1
+
g2
+
g3
+
g4
+
L
= 105
checks
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.36 Determine the new level of service for the nor thbound approach. Volumes increase 10% from previous values, calculate new volumes NBL := 90⋅ 1.10
NBL = 99
NBT := 340⋅ 1.10
NBT = 374
NBR := 50 ⋅ 1.10
NBR = 55
calculate northbound left delay C := 65
gL := 15.8 NBL
XL :=
sL ⋅
s L := 475
(previously calculated)
XL = 0.857
⎛ gL ⎞
(Eq. 7.22)
⎝ C ⎠
⎛ gL ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ d1L := ⎛ gL ⎞ 1 − XL ⋅ ⎝ C ⎠ T := 0.25
2
k := 0.5
d1L = 23.523
(Eq. 7.15)
I := 1.0
(assumed)
c L := 337
(previously calculated)
d2L := 900⋅ T⋅
⎡ ⎣
⎡
(XL − 1) + (XL − 1)
PF := 1.0
⎣
2
+
8 ⋅ k ⋅ I⋅ XL⎤ ⎤ cL ⋅ T
⎦⎦
d2L = 23.508
d3L := 0
dL := d1L ⋅ PF
+
d2L + d3L
(Eq. 7.16)
(assumed)
dL
= 47.031
LOS for left turn is D
(Eq. 7.14)
calculate northbound through and right delay C := 65
XTR :=
gTR := 15.8 NBT + NBR
⎛ gTR ⎞ s TR⋅ ⎝ C
sTR := 1800
(previously calculated)
XTR = 0.98
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
⎛ gTR ⎞ 0.5 ⋅ C⋅ 1 − C ⎠ ⎝ d1TR := ⎛ gTR ⎞ 1 − XTR⋅ C ⎠ ⎝ T := 0.25
k := 0.5
2
d1TR = 24.447
(Eq. 7.15)
I := 1.0
(assumed)
c TR := 1292
d2TR := 900⋅ T⋅
(previously calculated)
⎡ ⎣
⎡
(XTR − 1) + (XTR − 1) ⎣
2
+
8 ⋅ k ⋅ I⋅ XTR⎤ ⎤ cTR⋅ T
(Eq. 7.16)
⎦⎦
d2TR = 20.788 PF := 1.0
(assumed)
d3TR := 0
dTR := d1TR⋅ PF
+
d2TR + d3TR
dTR = 45.235
(Eq. 7.14)
LOS for through/right turn is D calculate northbound total approach delay
d
:=
NBL ⋅ dL + ( NBT + NBR) ⋅ dTR NBL + NBR + NBT
d
= 45.6
sec/veh
LOS is D
(Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.37 Determine the yellow and all-red times. V
:= 30⋅
5280
V
3600
tr := 1.0
= 44
a := 10.0
g := 32.2
(assumed)
w := 60
l := 20
(given)
G := 0.08
calculate yellow time Y := tr +
V 2⋅a
Y := 3.0
+ 2⋅ g⋅G
Y
= 2.749
rounds to 3.0 seconds
(Eq. 7.23)
sec
calculate all-red time AR :=
w+ l
AR
V
AR := 2.0
= 1.818
rounds to 2.0 seconds
(Eq. 7.24)
sec
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
d1
⎛ g ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ :=
c := s ⋅
c
C
= 18.481
(Eq. 7.15)
C
= 1292
⎡ ⎣
d2 := 900⋅ T⋅ ( X − 1) PF := 1
d1
g
1 − X⋅ g
2
⎡( X − 1) 2 + 8 ⋅ k ⋅ I⋅ X⎤ ⎤ c ⋅T ⎦⎦ ⎣
+
d2 = 7.265
d3 := 0
dEBTR := d1 ⋅ PF
+
(Eq. 7.16)
(assumed) d2
+
d3
dEBTR = 25.746
Calculate volume-weighted delay for eastbound approach VEBL := 300
VEBTR := 1100
VEBL ⋅ dEBL
dEB :=
+
VEBL
VEBTR⋅ dEBTR
+
VEBTR
dEB
= 31.628
(Eq. 7.27)
Calculate delay for westbound left-turn lane group C := 65
g := 12.5
X := WBL ⋅
d1
C
s := 1750
X = 0.744
g
⎛ g ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ := 1
c := s ⋅
g C
− X⋅ c
d1
g
= 24.74
(Eq. 7.15)
C
= 336.538
⎡ ⎣
d2 := 900⋅ T⋅ ( X − 1) PF := 1
2
+
⎡( X − 1) 2 + 8 ⋅ k ⋅ I⋅ X⎤ ⎤ c ⋅T ⎦⎦ ⎣
d3 := 0
dWBL := d1 ⋅ PF
+
d2 = 13.849
(Eq. 7.16)
(assumed) d2
+
d3
dWBL
= 38.589
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Calculate delay for westbound through/right-turn lane group g := 24.7
s := 3400
X := WBTR⋅
d1
C
X = 0.889
g
⎛ g ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ := 1
c := s ⋅
− X⋅
d1
g
= 18.872
(Eq. 7.15)
C
g
c
C
⎡ ⎣
d2 := 900⋅ T⋅ ( X − 1) PF := 1
2
= 1292
⎡(X − 1)2 + 8 ⋅ k ⋅ I⋅ X⎤ ⎤ c ⋅T ⎦⎦ ⎣
+
d2
= 9.426
d3 := 0
dWBTR := d1 ⋅ PF
+
(Eq. 7.16) (assumed)
d2
+
dWBTR = 28.297
d3
Calculate volume-weighted delay for westbound approach VWBL := 250
dWB
:=
VWBTR := 1150
VWBL ⋅ dWBL VWBL
+ +
VWBTR ⋅ dWBTR VWBTR
dWB
= 30.135
(Eq. 7.27)
Calculate delay for northbound left-turn lane group g := C 65 := 15.8 X := NBL ⋅
d1
s := 575 C
X = 0.644
g
⎛ g ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ := 1
c := s ⋅
g C
− X⋅
2
g
d1
= 22.076
(Eq. 7.15)
C
c
= 139.769
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
⎡ ⎣
d2 := 900⋅ T⋅ ( X − 1) PF := 1
⎡( X − 1) 2 + 8 ⋅ k ⋅ I⋅ X⎤ ⎤ c ⋅T ⎦⎦ ⎣
+
d2
= 20.632
d3 := 0
dNBL := d1 ⋅ PF
+
(Eq. 7.16) (assumed)
d2
+
dNBL = 42.708
d3
Calculate delay for northbound through/right-turn lane group s := 1800 g 15.8 C
X := NBTR⋅
d1
X = 0.891
g
⎛ g ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ := 1 − X⋅
c := s ⋅
g
c
C
d1
g
(Eq. 7.15)
= 23.771
C
= 437.538
⎡ ⎣
d2 := 900⋅ T⋅ ( X − 1) PF := 1
2
⎡( X − 1) 2 + 8 ⋅ k ⋅ I⋅ X⎤ ⎤ c ⋅T ⎦⎦ ⎣
+
d2
= 22.964
d3 := 0
dNBTR := d1 ⋅ PF
(assumed)
+
d2
+
d3
dNBTR = 46.735
Calculate volume-weighted delay for northbound approach VNBL := 90
dNB :=
(Eq. 7.16)
(Eq. 7.27)
VNBTR := 390
VNBL ⋅ dNBL + VNBTR⋅ dNBTR VNBL + VNBTR
dNB
= 45.98
Calculate delay for southbound left-turn lane group C := 65
X := SBL ⋅
g := 15.8 C g
s := 550
X = 0.524
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
d1
⎛ g ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ := 1 − X⋅
c := s ⋅
g
(Eq. 7.15)
C
= 133.692
⎡ ⎣
d2 := 900⋅ T⋅ ( X − 1) PF := 1.0
d1 = 21.336
g
c
C
2
⎡( X − 1) 2 + 8 ⋅ k ⋅ I⋅ X⎤ ⎤ c ⋅T ⎦⎦ ⎣
+
d2
= 13.896
d3 := 0
dSBL := d1 ⋅ PF
+
d2
(Eq. 7.16)
(assumed)
+
d3
dSBL
= 35.232
Calculate delay for southbound through/right-turn lane group g := 15.8
s := 1800
X := SBTR⋅
d1
C
X = 0.846
g
⎛ g ⎞ 0.5 ⋅ C⋅ 1 − ⎝ C ⎠ := 1 − X⋅
c := s ⋅
g
c
C
d1
g
= 23.438
(Eq. 7.15)
C
= 437.538
⎡ ⎣
d2 := 900⋅ T⋅ ( X − 1) PF := 1.0
2
⎡( X − 1) 2 + 8 ⋅ k ⋅ I⋅ X⎤ ⎤ c ⋅T ⎦⎦ ⎣
+
d2
= 17.916
d3 := 0
dSBTR := d1 ⋅ PF
+
(Eq. 7.16)
(assumed)
d2
+
d3
dSBTR = 41.355
Calculate volume-weighted delay for southbound approach VSBL := 70
dSB :=
VSBTR := 370
VSBL ⋅ dSBL VSBL
+ +
VSBTR⋅ dSBTR VSBTR
dSB
= 40.381
(Eq. 7.27)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Calculate volume-weighted delay for entire intersection
( VEBL + VEBTR) ⋅ dEB + ( VWBL + VWBTR) ⋅ dWB + ( VNBL + VNBTR) ⋅dNB + ( VSBL + VSBTR) ⋅ dS d := ( VEBL + VEBTR) + ( VWBL + VWBTR) + ( VNBL + VNBTR) + ( VSBL + VSBTR) d
= 34
LOS of the intersection is C
(Eq. 7.28) (Table 7.4)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.40 Determine the new mini mum pedestrian gr een time. Nped := 20
W E := 6
L := 60
(given)
SP := 4.0 GP := 3.2
(assumed)
+
L SP
+
0.27 ⋅ Nped
GP
= 23.6
sec
(Eq. 7.25)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
⎛ second ⎞ ⎜⎜ ⎟⎟ term ⎝ ⎠
second term (continuation of equation above)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
⎛ second ⎞ ⎜⎜ ⎟⎟ term ⎝ ⎠
second term (continuation of equation above)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
What is the g/C ratio fo r this approach? PF := 0.641
Problem 7.43 (given)
The arrival type is AT-5 for a highly favorable progression quality.
f p := 1.00
(Table 7.6 for AT-5)
R p := 1.667
(Table 7.7 for AT-5)
Solve for g/C using Equations 7.31 and 7.32 PF = [1 - R p*(g/C)]*(f p) / [1 - (g/C)]
1− gC := R p
−
PF f p PF
gC
= 0.35
f p
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.44
Determine the arrival type for thi s approach.
g := 35 s arrivalgreen
C := 100
:= 66
veh
(given)
arrivaltotalcycle
:= arrivalgreen +
105
(given)
Calculate g/C ratio gC :=
g
gC
C
= 0.35
Calculate PVG
PVG:=
arrivalgreen
PVG = 0.386
arrivaltotalcycle
Solve for R p usi ng Eq. 7.32
R p :=
PVG gC
R p
= 1.103
(Eq. 7.32)
Use Table 7.7 to determine arrival type
An Rp of 1.103 falls in the range for arrival type 3 (AT-3).
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.45
Determine the progression adjustm ent factor.
g := 35 s arrivalgreen
C := 100 s
:= 66
veh
arrivalcycle := arrivalgreen
(given)
+
105
veh
(given)
Calculate g/C ratio
gC :=
g
gC = 0.35
C
Calculate PVG PVG:=
arrivalgreen arrivalcycle
PVG = 0.386
Determine R p R p :=
PVG
R p
gC
= 1.103
(Eq 7.32)
Determine f p For an Rp of 1.103, the arrival rate is AT-3 by use of Table 7.7 f p := 1.00
(Table 7.6 using AT-4)
Calculate PF
PF :=
( 1 − PVG) ⋅ f p 1−
⎛ g ⎞
PF = 0.945
(Eq 7.31)
⎝ C ⎠
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the displayed green time for the traffic movement. r := 37 s
C := 70 s
Y := 5 s
AR
:= 2
s
t L := 4 s
Problem 7.46 (given)
Determine the effective green time for a traffic movement g := C − r
g
= 33
s
(Eq 7.5)
Calculate the displayed green time for a traffic movement G := g
− Y − AR +
G = 30 s
tL
(Eq 7.3)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Solve for effective green time g
= 33
s
2) Use all-red time in Eq 7.5 r := 2 G := g
g := C − r
− Y − AR +
tL
g
= 68
s
G = 65 s
3) Add AR time to effective red, and then subtract from cycle length r := 37 s
G := C − ( AR
AR
+
r)
:= 2 G = 31
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.47
Determine the average delay per vehicle.
μ :=
1900
veh
3600
s
R := 58 s
λ :=
tL := 4 s
550
veh
3600
s
g := 28 s
Y := 3 s
(given) AR
:= 2
s
Determine effective red time and cycle length r := R
+
tL
r
C := r + g
= 62.0
(Eq 7.4)
s
C = 90.0 s
(Eq 7.5)
Calculate traffic intensity
ρ :=
λ μ
(Eq 5.27)
ρ = 0.3
Determine average delay per vehicle 2
λ ⋅ r 1 d avg := ⋅ 2⋅ ( 1 − ρ) λ ⋅ C
d avg
= 30.1
s
(Eq 7.12)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answer: 1) Use displayed red time in Eq 7.12 2
λ ⋅ R 1 d avg := ⋅ 2⋅ ( 1 − ρ) λ ⋅ C
d avg
= 26.3
s
2) Solve for effective green time using 28 seconds as displayed green time G := 28 s
g := G + Y + AR − tL
C := r + g
g
= 29.0
(Eq 7.3)
s
C = 91.0 s 2
d avg
:=
λ ⋅ r 1 ⋅ 2⋅ ( 1 − ρ) λ ⋅ C
d avg
= 29.7
s
3) Solve for total vehicle delay g := 28 s
C := r
+
g
C = 90.0 s
2
λ ⋅ r Dt := 2⋅ ( 1 − ρ)
Dt
= 413.3
s
(Eq 7.11)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the average approach delay (in seconds). v := 750
veh
s
h
:= 3200
Problem 7.48
veh h
(given) C := 100 s
g := 32 s
T := 0.25
d 3 := 0 s
PF := 1.0
k := 0.5 (pretimed control)
(assumed)
I := 1.0 (isolated mode)
Calculate v/c (X) ratio c := s ⋅ X :=
g
= 1024.000
c
C
v
X
c
veh
(Eq 7.6)
h
= 0.732
Calculate uniform delay
⎛ g ⎞ 0.5⋅ C⋅ 1 − ⎝ C ⎠ d 1 := ⎛ g ⎞ 1 − X⋅ ⎝ C ⎠
2
d 1 = 30.198 s
(Eq 7.15)
Calculate random delay
⎡ − 1) + ⎣
d 2 := 900⋅ T⋅ ( X
(X
2
− 1) +
8⋅ k ⋅ I⋅ X ⎤ c⋅ T
⎦
= 34.8
s
d 2 = 4.633 s
(Eq 7.16)
Calculate the total delay d := d 1⋅ PF + d 2 + d 3
d
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers: 1) Use d1 as final answer.
d 1 = 30.198 s
2) Use d2 as final answer
d 2 = 4.633 s
3) Use T of 15 minutes rather than 0.25 hrs T := 15
⎡ − 1) + ⎣
d 2 := 900⋅ T⋅ ( X
d := d 1⋅ PF + d 2 + d 3
(X d
2
− 1) +
= 35.0
8⋅ k ⋅ I⋅ X⎤ c⋅ T
⎦
d 2 = 4.808
s
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 7.49
Calculate the sum of flow ratios for th e critical lane groups. Calculate flow ratios Phase 1 EBL :=
250 1800
WBL :=
300 1800
Phase 2
EBL = 0.139
WBL = 0.167
EBTR :=
1200 3600
WBTR :=
1350 3600
EBTR = 0.333
WBTR = 0.375
Phase 3 SBL :=
75 500
100 NBL := 525 SBTR := NBTR :=
420 1950 425 1950
SBL = 0.15 NBL
= 0.19
SBTR = 0.215 NBTR = 0.218
Calculate sum of flow ratios for critical lane groups Yc := WBL + WBTR + NBTR
Yc
= 0.76
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Use incorrect Phase 1 value Yc := EBL + EBTR + NBTR
Yc
= 0.69
2) Use split phasing for NB & SB Yc := WBL + WBTR + NBL
+ NBTR
Yc
= 0.95
3) Choose the smallest flow ratio for each phase. Yc := EBL + EBTR + SBL
Yc
= 0.622
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Calculate the minimum cy cle length.
L := 12 s
Xc
:= 0.9
Problem 7.50
SumCritFlowRatios:= 0.72
(given)
Calculate minimum cycle length
Cmin :=
L⋅ Xc Xc
Cmin = 60.0 s
− SumCritFlowRatios
(Eq 7.20)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Solve for optimum cycle length Copt
:=
1.5⋅ L + 5
Copt
1.0 − SumCritFlowRatios
= 82.1
s
(Eq 7.21)
2) Switch sum and Xc in Eq 7.21 SumCritFlowRatios:= 0.9
Cmin :=
L⋅ Xc Xc
− SumCritFlowRatios
Xc
:= 0.72
Cmin = −48.0s
use absolute value
3) Use Xc = 1.0 Xc
:= 1.0
Cmin :=
SumCritFlowRatios:= 0.72 L⋅ Xc
Xc
− SumCritFlowRatios
Cmin = 42.9 s
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the sum of th e yellow and all-red times.
G :=
4
w := 60 ft
100
V := 40⋅
⎛ 5280 ⎞ ⎝ 3600 ⎠
tr := 1.0
l := 16 ft
ft
g := 32.2
s
(given)
ft s
a := 10.0
s
Problem 7.51
2
(assumed)
ft s
2
Determine the yellow time V
Y := tr +
2⋅ a
Y = 3.599s
+ 2⋅ g⋅ G
(Eq 7.23)
Determine the all red time AR :=
+
w
l
AR
V
= 1.295
s
Sum Y & AR AR
+
Y = 4.894 s
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Use assumed value of 20.0 ft for vehicle length l := 20.0 ft AR
+
AR :=
+
w
l
AR
V
= 1.364
s
Y = 4.962 s Y = 3.599s
2) Use Y value only
3) The percent grade is not divided by 100 G := 4
l := 16.0 ft
Y := tr + AR := AR
+
w
V 2⋅ a
+
+ 2⋅ g⋅ G
l
V
AR
Y = 1.211
= 1.295
s
Y = 2.507 s
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany
Principles of Highway Engineering and Traffic Analysis, 4e By Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski
Chapter 8 Travel Demand and Traffic Forecasting U.S. Customary Units
Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Preface The solutions to the fourth edition of Principles of Highway Engineering and Traffic Analysis 1 were prepared with the Mathcad software program. You will notice several notation conventions that you may not be familiar with if you are not a Mathcad user. Most of these notation conventions are self-explanatory or easily understood. The most common Mathcad specific notations in these solutions relate to the equals sign. You will notice the equals sign being used in three different contexts, and Mathcad uses three different notations to distinguish between each of these contexts. The differences between these equals sign notations are explained as follows.
•
The ‘:=’ (colon-equals) is an assignment operator, that is, the value of the variable or expression on the left side of ‘:=’is set equal to the value of the expression on the right side. For example, in the statement, L := 1234, the variable ‘L’ is assigned (i.e., set equal to) the value of 1234. Another example is x := y + z. In this case, x is assigned the value of y + z.
•
The ‘=’ (bold equals) is used when the Mathcad function solver was used to find the value of a variable in the equation. For example, in the equation , the = is used to tell Mathcad that the value of the expression on the left side needs to equal the value of the expression on the right side. Thus, the Mathcad solver can be employed to find a value for the variable ‘t’ that satisfies this relationship. This particular example is from a problem where the function for arrivals at some time ‘t’ is set equal to the function for departures at some time ‘t’ to find the time to queue clearance.
•
The ‘=’ (standard equals) is used for a simple numeric evaluation. For example, referring to the x := y + z assignment used previously, if the value of y was 10 [either by assignment (with :=), or the result of an equation solution (through the use of =) and the value of z was 15, then the expression ‘x =’ would yield 25. Another example would be as follows: s := 1800/3600, with s = 0.5. That is, ‘s’ was assigned the value of 1800 divided by 3600 (using :=), which equals 0.5 (as given by using =).
Another symbol you will see frequently is ‘→’. In these solutions, it is used to perform an evaluation of an assignment expression in a single statement. For example, in the following statement, , Q(t) is assigned the value of 2 Arrivals(t) – Departures(t), and this evaluates to 2.2t – 0.10t .
Finally, to assist in quickly identifying the final answer, or answers, for what is being asked in the problem statement, yellow highlighting has be en used (which will print as light gray). 1
www.mathcad.com
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.1 Determine the number o f peak hour trips generated. income := 20
household_size := 2
neighborhood_employment:= 1.0
nonworking := 2
(given)
From Example 1: trips ex1 := 0.12 + 0.09 ⋅ ( household_size) trips ex1
= 0.37
+ 0.011⋅ ( income) − 0.15⋅ ( neighborhood_employment)
trips per household
trips ex1 ⋅ 1700 = 629
total shopping trips
From Example 2: trips ex2 := 0.04 + 0.018⋅ ( household_size) trips ex2
= 0.576
trips per household
trips ex2 ⋅ 1700 = 979
(
total := 1700⋅ tripsex1 total
+ 0.009⋅ ( income) + 0.16⋅ ( nonworking)
total social/recreational trips
+ tripsex2)
= 1608.2 trips
629 + 979 = 1608 total shopping and social/recreational trips
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.2 Determine the amount o f addition al retail employment needed. income := 20 trips := trips
size := 2
retail
retail := 100
(given)
trips per household x 1700 = 100
1700
= 0.059
trips
0.12 + 0.09⋅ size
0.520 − 0.059 0.15
+ 0.011⋅ income − 0.15⋅ employment
= 3.073
= 3.075
employment
employment ⋅ 100 − retail
= 207.451
so,
208 more retail employees
Problem 8.3 Determine the change in the number of peak hour social/recreational tri ps. income := 15
size := 5.2
nonworking := size
− working
workingnew := 1.2 ⋅ working nonworkingnew := size
working := 1.2 nonworking = 4 workingnew = 1.44
− workingnew
incomenew := 1.1income trips := 0.04 + 0.018⋅ size
(given)
nonworkingnew = 3.76
incomenew = 16.5
+ 0.009⋅ income + 0.16 ⋅ nonworking
trips new := 0.04 + 0.018⋅ size
trips
+ 0.009⋅ incomenew + 0.16 ⋅ nonworkingnew
trips new⋅ 1500 − trips ⋅ 1500 = −37.35
= 0.909 tripsnew = 0.884
trips
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.4 Determine the number of peak-hour sh opping t rips and the probabili ty that the household wil l make more than 1 peak-hour shop ping tri p. calculate the number of peak hour trips house := 4
income := 150
retail := 3
(given)
BZi := −0.35 + 0.03⋅ house + 0.004⋅ income − 0.10⋅ retail BZi
λi := e
λi = 1.07
BZi
= 0.07
(Example 8.4)
vehicle trips
(Eq. 8.3)
find the probability of making more than 1 trip
− λi
( ) :=
e
P Ti
Ti
⋅ λi
(Eq. 8.2)
Ti!
= 0.342
P ( 0)
P ( 1)
= 0.367
Pmore := 1
− P ( 0) − P ( 1)
Pmore = 0.291
Problem 8.5 Determine the number of peak-hour social/recreational tr ips and t he probability that the household wil l not make a peak-hour social/rec trip. calculate the number of peak-hour trips coeff size
:= 0.025
size := 5
coeff income := 0.008
coeff nonwork := 0.10
(given)
income := 100
nonwork := 5 − 3
(given)
BZi := −0.75 + coeff size ⋅ size BZi
+ coeff income ⋅ income + coeff nonwork⋅ nonwork
(given)
= 0.375 BZi
λ i := e
λ i = 1.455
vehicle-trips
(Eq. 8.3)
calculate the probability of zero peak-hour trips Ti := 0
(given)
− λi
( ) :=
P Ti
P ( 0)
e
Ti
⋅λ i
Ti!
(Eq. 8.2)
= 0.233
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.6 Determine the number of w ork tri p vehicles that leave during th e peak hour. usersB := 750
userseach_B := 15
usersB
buses :=
buses
userseach_B
cars DL := 2380
= 50
cars SR := 870 carsSR
cars := carsDL + cars
(given)
cars
2
(given)
= 2815
+ buses = 2865 total trip vehicles
Problem 8.7 Determine how m any workers w ill t ake each mode. cost DL := 8.00
cost SR := 4.00
cost B := 0.50
traveltimeDL := 20
traveltimeSR := 20
traveltimeB := 25
(given)
From Example 8.5: UDL := 2.2
− 0.2 ⋅ cost DL − 0.03 ⋅ traveltimeDL
UDL = 0
USR := 0.8
− 0.2 ⋅ cost SR − 0.03 ⋅ traveltimeSR
USR = −0.6
UB := −0.2 ⋅ cost B
− 0.01 ⋅ traveltimeB
UB
= −0.35
Using Eq. 8.7: UDL
e
PDL :=
UDL
e
USR
+e
UB
PDL = 0.444
PDL⋅ 4000 = 1775
drive alone
PSR = 0.244
PSR ⋅ 4000 = 974
shared ride
+e
USR
e
PSR :=
UDL
e
USR
+e
UB
+e
UB
PB :=
e UDL
e
USR
+e
UB
PB
= 0.313
PB ⋅ 4000 = 1251
bus
+e
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.8 Determine how many shopp ing trip s will b e made to each of the four shopping centers. coeff dist := −0.455
coeff space := 0.0172
U A1 := coeff dist ⋅ ( 2.5)
+ coeff space ⋅ ( 200)
U A1
= 2.303
U A2 := coeff dist ⋅ ( 5.5)
+ coeff space ⋅ ( 150)
U A2
= 0.078
U A3 := coeff dist ⋅ ( 5.0)
+ coeff space ⋅ ( 300)
U A3
= 2.885
U A4 := coeff dist ⋅ ( 8.7)
+ coeff space ⋅ ( 600)
U A4
= 6.362
P A1
= 0.016
P A2
= 0.002
P A3
= 0.029
P A4
= 0.952
(given)
Using Eq. 8.7: U A1
e
P A1 := U A1 U A2 U A3 U A4 +e +e +e e U A2
e
P A2 := U A1 U A2 U A3 U A4 e +e +e +e U A3
e
P A3 := U A1 U A2 U A3 U A4 e +e +e +e U A4
e
P A4 := U A1 U A2 U A3 U A4 e +e +e +e trips := 4000 Trips to each shopping center: trips 1 := trips ⋅ P A1
trips1
= 66
trips 2 := trips ⋅ P A2
trips2
=7
trips 3 := trips ⋅ P A3
trips3
= 118
trips 4 := trips ⋅ P A4
trips4
= 3809
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.9 Determine the new distributi on of the 4000 shopping tr ips. coeff dist := −0.455
coeff space := 0.0172
U A1 := coeff dist ⋅ ( 2.5)
+ coeff space ⋅ ( 200)
U A1
= 2.303
U A2 := coeff dist ⋅ ( 5.5)
+ coeff space ⋅ ( 500)
U A2
= 6.098
U A4 := coeff dist ⋅ ( 8.7)
+ coeff space ⋅ ( 600)
U A4
= 6.362
(given)
Using Eq. 8.7: U A1
e
P A1 := U A1 U A2 U A4 +e +e e
P A1
= 0.01
P A2
= 0.43
P A4
= 0.56
U A2
e
P A2 := U A1 U A2 U A4 e +e +e U A4
e
P A4 := U A1 U A2 U A4 e +e +e trips := 4000 Trips to each shopping center: trips 1 := trips ⋅ P A1
trips1
= 39
trips 2 := trips ⋅ P A2
trips2
= 1721
trips 4 := trips ⋅ P A4
trips4
= 2241
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.10 Determine how muc h comm ercial floor sp ace is needed. coeff dist := −0.455 trips := trips 4000
4000 3
coeff space := 0.0172 trips
(given)
= 1333
= 0.333
therefore P A := 0.333 as before U A4 := 6.362 which means that U A := 6.362 so for shopping center 1
U A
coeff dist ⋅ ( 2.5)
+ coeff space ⋅ ( space1)
space1 = 436.017
space1 ⋅ 1000 = 436017
2
ft
and for shopping center 2
U A
coeff dist ⋅ ( 5.5)
space2 = 515.378
+ coeff space ⋅ ( space2) space2 ⋅ 1000 = 515378
2
ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.11 Determine the new distri bution of s hoppin g trips by destin ation and mode. time2auto := 15 ⋅ 0.8
time2auto = 12
time2bus := 22 ⋅ 0.8
time2bus
(given)
= 17.6
coeff := 0.012 U A1 := 0.6
− 0.3 ⋅ 8 + coeff ⋅ 250
U A1
= 1.2
UB1 := −0.3 ⋅ 14 + coeff ⋅ 250
UB1
= −1.2
U A2 := 0.6
U A2
= 1.8
UB2
= −0.48
P A1
= 0.323
PB1
= 0.029
P A2
= 0.588
PB2
= 0.06
− 0.3 ⋅ time2auto + coeff ⋅ 400
UB2 := −0.3 ⋅ time2bus
+ coeff ⋅ 400
Using Eq. 8.7: U A1
e
P A1 := U A1 UB1 e +e
U A2
+e
UB2
+e
UB1
PB1 :=
e U A1
e
UB1
U A2
+e
+e
UB2
+e
U A2
e
P A2 := U A1 UB1 e +e
U A2
+e
UB2
+e
UB2
PB2 :=
e U A1
e
UB1
+e
U A2
+e
UB2
+e
total := 900 trips A1 := total ⋅ P A1
trips A1
= 290
trips B1 := total ⋅ PB1
tripsB1
= 26
trips A2 := total ⋅ P A2
trips A2
= 529
trips B2 := total ⋅ PB2
tripsB2
= 54
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.12 Determine how much c ommercial floor s pace must be added to shoppi ng center 2. total := 900
trips A2
P A2 :=
total tripsB2
PB2 :=
P1 :=
trips A2 := 355
total
total
tripsB2 := 23
P A2
= 0.394
PB2
= 0.026
− ( trips A2 + trips B2)
P1
total
U A1 := 1.2
(given)
= 0.58
UB1 := −1.2
(given)
let x = e UA2 + e UB2
U A1
e
P1
U A1
e x
UB1
+e
UB1
+e
(Eq. 8.7)
+x
= 2.622
time2auto := 15 + 4
time2bus := 22 + 4
time2auto = 19
time2bus
x
0.6−0.3⋅ time2 auto + 0.012 ⋅ space
e
space
= 26
− 0.3⋅ time2 bus + 0.012 ⋅ space
+e
= 499.918
added_space := space − 400 added_space = 99.918
added_space⋅ 1000 = 99918
2
ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.13 Determine the distribution of trips among possible destinations. coeff pop := 0.2
coeff dist := −0.24
U A1 := coeff pop⋅ ( 15.5)
coeff space := 0.09
(given)
+ coeff dist ⋅ ( 7.5) + coeff space ⋅ ( 5)
U A1
= 1.75
U A2 := coeff pop⋅ ( 6.0)
+ coeff dist ⋅ ( 5.0) + coeff space ⋅ ( 10)
U A2
= 0.9
U A3 := coeff pop⋅ ( 0.8)
+ coeff dist ⋅ ( 2.0) + coeff space ⋅ ( 8)
U A3
= 0.4
U A4 := coeff pop⋅ ( 5.0)
+ coeff dist ⋅ ( 7.0) + coeff space ⋅ ( 15)
U A4
= 0.67
Using Eq. 8.7: U A1
e
P A1 := U A1 U A2 U A3 U A4 e +e +e +e
P A1
= 0.494
P A2
= 0.211
P A3
= 0.128
P A4
= 0.168
U A2
e
P A2 := U A1 U A2 U A3 U A4 e +e +e +e U A3
e
P A3 := U A1 U A2 U A3 U A4 +e +e +e e
U A4
e
P A4 := U A1 U A2 U A3 U A4 e +e +e +e
trips := 700 trips 1 := trips ⋅ P A1
trips1
= 345
trips 2 := trips ⋅ P A2
trips2
= 148
trips 3 := trips ⋅ P A3
trips3
= 90
trips 4 := trips ⋅ P A4
trips4
= 117
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problems 8.14 Determine the distr ibutio n of the 700 peak-hour s ocial/recreational tri ps. coeff pop := 0.2
coeff dist := −0.24
U A1 := coeff pop⋅ ( 15.5)
coeff space := 0.09
(given)
+ coeff dist ⋅ ( 7.5) + coeff space ⋅ ( 5)
U A1
= 1.75
U A2 := coeff pop⋅ ( 6.0)
+ coeff dist ⋅ ( 5.0) + coeff space ⋅ ( 10)
U A2
= 0.9
U A3 := coeff pop⋅ ( 0.8)
+ coeff dist ⋅ ( 2.0) + coeff space ⋅ ( 8 + 15)
U A3
= 1.75
U A4 := coeff pop⋅ ( 5.0)
+ coeff dist ⋅ ( 7.0) + coeff space ⋅ ( 15)
U A4
= 0.67
Using Eq. 8.7: U A1
e
P A1 := U A1 U A2 U A3 U A4 e +e +e +e
P A1
= 0.361
P A2
= 0.154
P A3
= 0.361
P A4
= 0.123
U A2
e
P A2 := U A1 U A2 U A3 U A4 e +e +e +e U A3
e
P A3 := U A1 U A2 U A3 U A4 +e +e +e e U A4
e
P A4 := U A1 U A2 U A3 U A4 +e +e +e e trips := 700 trips 1 := trips ⋅ P A1
trips1
= 253
trips 2 := trips ⋅ P A2
trips2
= 108
trips 3 := trips ⋅ P A3
trips3
= 253
trips 4 := trips ⋅ P A4
trips4
= 86
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.15 Determine how muc h additional amusement floor space must be added. trips1total := 500⋅ 0.40
trips1total
= 200
P1 := 0.40 P2and3 := 1.0 − P1 U A2 := 2.124
P2and3 = 0.6
UB2 := 0.564
U A3 := 0.178
UB3 := −2.042
(From Ex. 8.8)
let x = e UA1 + e UB1
U A2
P2and3
e
U A2
e x
x
UB2
+e
UB2
+e
U A3
+e
U A3
+e
UB3
+e
UB3
+e
+x
= 7.631 ( 0.9− 0.22 ⋅ 14 + 0.16⋅ 12.4+ .11⋅ space )
e
space
( − 0.22 ⋅ 17 + 0.16⋅ 12.4+ .11⋅ space )
+e
= 18.5
added_space := space added_space = 5.5
− 13
added_space⋅ 1000 = 5523
2
ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.16 Determine how muc h additional amusement floor space must be added. trips := 700
trips2 := 250
trips 2 P A2 := trips
P A2
U A1 := 1.75
= 0.357
U A3 := 0.4
U A4 := 0.67
(previously calculated)
U A2
P A2
e U A1
e U A2
U A2
+e
+e
U A4
+e
= 1.631
coeff pop := 0.2
U A2
(Eq. 8.7) U A3
coeff dist := −0.24
coeff pop⋅ ( 6.0)
coeff space := 0.09
(given)
+ coeff dist ⋅ ( 5.0) + coeff space ⋅ ( space2)
space2 = 18.128 space2 − 10.0 = 8.128
8.128⋅ 1000 = 8128
2
ft
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.18 Determine the user equilibrium and system opti mal route travel times, total travel time, and rout e flows. User equilibrium: t1 x1
8
t2
1
+ 2 ⋅ x2
q := 4
(given)
+ x2 q
t1 8
+ x1
t2
+ x1 1 + 2 ⋅ ( q − x1)
Route flows: x1
= 0.333
x 1 ⋅ 1000 = 333
x 2 := q − x 1
veh/h
x 2 = 3.667
x 2 ⋅ 1000 = 3667
veh/h
Route travel times: t1 := 8
+ x1
t2 := 1
+ 2 ⋅ x2
t1
= 8.33 t2
= 8.33
min min
x 1 ⋅ t 1 ⋅ 1000 + x 2 ⋅ t2 ⋅ 1000 = 33333
veh − min
total travel time
System Optimal: S (x)
S (x)
dx 1
+ x 1) + x 2 ⋅ (1 + 2 ⋅ x2)
q − x1
x2
d
(
x1 ⋅ 8
(
x1 ⋅ 8
S ( x)
+ x 1) + (q − x 1) ⋅ 1 + 2 ⋅ ( q − x1)
−9 + 6 ⋅ x1
setting the derivative = 0
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Route flows: x1
= 1.5
x 2 := q
− x1
x1 ⋅ 1000 = 1500 x2
= 2.5
veh/h x 2 ⋅ 1000 = 2500
veh/h
Route travel times: t1 := 8 + x 1 t2 := 1 + 2 ⋅ x 2
t1 = 9.5 t2
=6
min min
x 1 ⋅ t1 ⋅ 1000 + x2 ⋅ t2 ⋅ 1000 = 29250
veh − min
total travel time
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.19 Determine how many vehicl e hours wil l be saved. hourstotal := 875.97
veh − h
(from Example 8.12)
For System Optimal:
(4 + 1.136⋅ x1) ⋅ x1 + (3 + 3.182⋅ x2) ⋅ x2
z x1
6 − x2
z
4 + 1.136⋅ 6 − x2
(
2
4.318⋅ x2
z d
z
dx x 2 :=
) ⋅ (6 − x2) + (3 + 3.182⋅ x2) ⋅ x2
− 14.632⋅ x2 + 51.264
8.636⋅ x2
− 14.632 0
14.632 8.636
x 2 = 1.694
x 1 := 6 − x 2
x 1 = 4.306
t1 :=
t2 :=
(4 + 1.136⋅x 1) ⋅ x1 ⋅ 1000 60
(3 + 3.182⋅ x2) ⋅ x2 ⋅ 1000
hourstotal
60
− t1 − t2 = 0.962
t1
= 638.052
veh − h
t2
= 236.956
veh − h
veh − h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.20 Determine the reduction in peak-hour traffic demand needed. Before reconstruction c 1 := 2.5
c 2 := 4
q := 3.5 S (x )
t1
+ 3⋅
c1
⎝ ⎠
t2
4
+ 2⋅
⎛ x2 ⎞
(given)
c2
⎝ ⎠
thousand vehicles
( 2 + 1.2 ⋅ x1) ⋅ x1 + (4 + 0.5⋅ x2) ⋅ x2 q − x1
x2 S (x )
( 2 + 1.2 ⋅ x1) ⋅ x1 +
S (x)
1.7x1
d
2
⎛ x1 ⎞
2
S ( x)
x 1 :=
+ 0.5 ⋅ ( 3.5 − x1) ⋅ ( 3.5 − x1)
− 5.5 ⋅ x1 + 20.125
3.4x1
dx 1
4
− 5.5
5.5 3.4
x 2 := q − x 1
0
x1
= 1.618
veh/hr
x2
= 1.882
veh/hr
( ) tot_trav2 := ( 4 + 0.5 ⋅ x2) ⋅ x2 ⋅ 1000 tot_trav1 := 2 + 1.2 ⋅ x1 ⋅ x1 ⋅ 1000
tot_trav1
+ tot_trav2 = 15676.5 veh-min
tot_trav1
= 6375.433 veh-min
tot_trav2
= 9301.038 veh-min
total vehicle travel time (before construction)
During reconstruction minimize
S
( 2 + 1.2 ⋅ x1) ⋅ x1 + (4 + 1 ⋅ x2) ⋅ x2 with ( 2 + 1.2 ⋅ x1) ⋅ x1 + 4 + 1( q − x1) (q − x1)
S
2.2 ⋅ x1
S
d dx
2
S
q
− x1
2
− 2 ⋅ x1 + q − 4 ⋅ q − 2 ⋅ q ⋅ x 1
4.4 ⋅ x1
q := 2.2 ⋅ x1
x2
− 2 − 2q 0
−1
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Also, from travel times: 15.676
( 2 + 1.2⋅x 1) ⋅ x1 + (4 + 1⋅ x2) ⋅ x2
15.676
2 ⋅ x1
+ 1.2 ⋅ x1 + 4 ⋅ ( q − x1) + (q − x 1)
15.676
2 ⋅ x1
+ 1.2 ⋅ x1 + 4 ⋅ q − 4x1 + q − 2 ⋅ q ⋅ x 1 + x1
15.676
2.2x1
2 2
2
x2
q
− x1
2
2
2
2
− 2 ⋅ x1 + 4q + q − 2 ⋅ q ⋅ x 1
2.2 ⋅ x1 − 1
Substituting q 15.676
with
2
2.2 ⋅ x1
2
− 2 ⋅ x 1 + 4 ⋅ ( 2.2 ⋅ x1 − 1) + (2.2 ⋅ x1 − 1) − 2 ⋅ ( 2.2 ⋅ x1 − 1) ⋅ x 1
Solving for x1 yields: x1
= 1.954
x1 ⋅ 1000 = 1954 veh/h
q := 2.2 ⋅ x1 − 1 x 2 := q
− x1
Δq := 3.5 − q
q = 3.299 x2
= 1.345
thousand vehicles x 2 ⋅ 1000 = 1345
Δq ⋅ 1000 = 201
veh
veh/h
reduction in peak-hour traffic demand needed
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.22 Determine the user equilibrium t raffic flows .
( ) := 8 + 0.5 ⋅ x1
q := 3
t1 x1
( ) := 1 + 2 ⋅ x2
( ) := 3 + 0.75⋅x 3
t 2 x2
t1 ( q)
= 9.5
t1 ( 0)
=8
t2 ( 0)
=1
t2 ( q)
=7
t3 ( 0)
=3
t3 x 3 t3 ( q)
(given)
= 5.25
So, Route 1 will never be used since t1 (0) > t 2 (q) > t 3 (q) therefore, t2 1
+ 2 ⋅ x2 3 + 0.75 ⋅x 3 x2 + x3
q 1
t3
+ 2 ⋅ x2 3 + 0.75 ⋅ ( q − x2)
x2
= 1.545
x 2 ⋅ 1000 = 1545
veh/h
x 3 ⋅ 1000 = 1455
veh/h
x 3 := q − x 2 x3
= 1.455
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.23 Determine equilibrium f lows and travel times before and after reconstruct ion begins. Let subscript 11 denote route 1 before construction Let subscript 12 denote route 1 during construction Let subscript 21 denote route 2 before construction Let subscript 22 denote route 2 during construction c 11 := 4
c 21 := 2
c 12 := 3
t12 - t 11 = 35.28 s or 0.588 min
c 22 := 2 (given)
Also, since for UE, t 1 = t 2 ; t 22 - t 21 = 0.588 min (1) t 22
10 + 1.5 ⋅x 22
t 21
10 + 1.5 ⋅x 21
Substituting the above two equations into (1):
(10 + 1.5 ⋅x 22) − (10 + 1.5 ⋅x 21)
0.588
and solving for x21 in terms of x 22 yields: x 21
x 22 − 0.392
( 2)
Also, for the traffic increase: x 22
1.685⋅ x21
or x 21
0.5935⋅ x22
Substituting into (2):
0.5935⋅ x22
x 22 = 0.964
x 22 − 0.392
x 22 ⋅ 1000 = 964
t22 := 10 + 1.5 ⋅x 22
t22 = 11.45
veh/h min
flow on route 2 after construction travel time after construction
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
t12 := t22
t12
6
⎛ x12 ⎞
+8
⎝ c12 ⎠
x 12 = 2.042
x 12 ⋅ 1000 = 2042
veh/h
flow on route 1 after construction
from User Equilibrium: t11 := t22 − 0.588
t11 = 10.86
min
travel time before construction
t21 := t11
t11
6
+ 8⋅
⎛ x11 ⎞ ⎝ c11 ⎠
x 11 = 2.429
t21
x 11 ⋅ 1000 = 2429
veh/h
flow on route 1 before construction
⎛ x21 ⎞
10 + 3
x 21 = 0.572
⎝ c21 ⎠ x 21 ⋅ 1000 = 572
veh/h
flow on route 2 before construction
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.24 Determine the user equilib rium flo ws. Determine for q = 10000
( ) := 2 + 0.5 ⋅ x1
q := 10
t1 x 1
t1 ( q)
=7
t1 ( 0)
=2
t2 ( 0)
=1
t2 ( q)
= 11
t3 ( 0)
=4
( ) := 1 + x2
t2 x 2
t3 ( q)
( ) := 4 + 0.2 ⋅ x3
t3 x 3
(given)
=6
so all routes might be used t1
t2
t3
x1 + x2 + x3
q x2
q − x1
t1
t2
2
+ 0.5 ⋅ x1 1 + x 2 1 + 0.5 ⋅ x1
x2 also 2
− x3
t1
t3
+ 0.5 ⋅ x1 4 + 0.2 ⋅ x3
(−2 + 0.5 ⋅ x1)
x3
0.2
−10 + 2.5 ⋅ x1
x3 so
(
) + (−10 + 2.5 ⋅ x1)
x1
= 4.75
x 1 ⋅ 1000 = 4750
t1
= 4.375
q
x 1 + 1 + 0.5 ⋅ x1
veh/h
t2 := t1 x 2 := t2
−1
x 2 = 3.375
x 2 ⋅ 1000 = 3375
veh/h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
t3 := t1 t3
x 3 :=
−4 x3
0.2
= 1.875
x3 ⋅ 1000 = 1875
veh/h
Determine for q = 5000
( ) := 2 + 0.5 ⋅ x1
q := 5
t1 x 1
t1 ( q)
= 4.5
t1 ( 0)
=2
t2 ( 0)
=1
t2 ( q)
=6
t3 ( 0)
=4
( ) := 1 + x2
t2 x 2
t3 ( q)
( ) := 4 + 0.2 ⋅ x3
t3 x3
(given
=5
so all routes might be used Routes 1 and 2 have the lowest free-flow travel time, so assume route 3 is not used t1
t2
+ 0.5 ⋅ x1 1 + x2 x 2 1 + 0.5 ⋅ x1 2
x2 1
q
− x1
+ 0.5 ⋅ x1 q − x 1
x1
= 2.667
x 2 := q
x 1 ⋅ 1000 = 2667
− x1
x2
= 2.333
veh/h x2 ⋅ 1000 = 2333
veh/h
Check to see if route 3 is used:
( ) = 3.333
t1 x 1
3.333 < 4
minutes (route 3 free-flow time)
Therefore the assumption that route 3 is not used is valid.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.25 Determine the minimum demand so that all routes are used. Flow on route 3 will begin when t1 = t 2 = 4 (which is route 3's free-flow time). so t1
2
x 1 :=
+ 0.5 ⋅ x1 4
4−2
x1
=4
x 2 := 4 − 1
x2
=3
q := x 1 + x 2
q
t2
1
0.5
+ x2 4
=7
thousand vehicles
If q > 7, flow on all three routes If q
≤ 7 flow on routes 1 and 2 only.
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.26 Determine the total person hours of travel for each situation. SOV veh := 2500
HOV 2veh := 500
HOV3veh := 300
Buses := 20
(given)
vehtotal := SOV veh + HOV2veh + HOV 3veh + Buses vehtotal = 3320 peopletotal := SOV veh⋅ 1 + HOV 2veh⋅ 2 + HOV 3veh⋅ 3 + Buses ⋅ 50 peopletotal = 5400 t 0 := 15
(given)
c 3 := 3600 a) open to all
⎡ t := t 0 ⋅ 1 + 1.15 ⋅
⎣
personhours :=
⎛ vehtotal ⎞ ⎝
c3
6.87
⎠
⎤ t = 24.89
⎦
t ⋅ people total 60
personhours = 2240
person − hours
b) 2+ lane c 1 := 1200
c 2 := 2400
⎡ t SOV := t 0 ⋅ 1 + 1.15 ⋅
⎛ SOV veh ⎞ c2
⎤
⎝
⎡
⎛ HOV2veh + HOV3veh + Buses ⎞
⎣
⎝
⎦
t SOV = 37.834
⎣
t HOV := t 0 ⋅ 1 + 1.15 ⋅
⎠
6.87
c1
6.87
⎤ ⎦
t HOV = 16.261
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
personhours :=
(
tSOV ⋅ SOVveh + tHOV ⋅ 2 ⋅ HOV2veh + 3 ⋅ HOV3veh + 50⋅ Buses
)
60
personhours = 2362
person − hours
c) 3+ lane
⎡ t2 := t0 ⋅ 1
⎣
+ 1.15 ⋅
⎡ t3 := t0 ⋅ 1
⎣
+ 1.15 ⋅
personhours :=
⎛ SOVveh + HOV2veh ⎞ ⎝
c2
⎠
⎛ HOV3veh + Buses ⎞ ⎝
(
c1
6.87
⎦
6.87
⎠
⎤ t2
= 94.903
t3
= 15.002
⎤ ⎦
) + t3 ⋅ (3 ⋅ HOV3veh + 50⋅ Buses)
t2 ⋅ SOVveh + 2 ⋅ HOV2veh
personhours = 6011
60
person − hours
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.27 Determine the total person-hours and the minimum m ode shift. t0 := 15
c2 := 2400
SOV
⎡
⎛ SOV ⎞ tSOV := t0 ⋅ 1 + 1.15 ⋅ ⎣ ⎝ c2 ⎠
:= 2500 − 500
⎤
6.87
⎦
= 19.93
tSOV
c 1 := 1200
HOV2 := 500
⎡ tHOV := t0 ⋅ 1
⎣
tHOV
+ 1.15 ⋅
HOV3 := 300
⎛ HOV2 + HOV3 + Buses ⎞ c1
⎝
⎠
Buses := 20 + 10
⎤
6.87
⎦
= 16.371
personhours :=
tSOV ⋅ SOV
+ tHOV ⋅ ( 2 ⋅ HOV2 + 3 ⋅ HOV3 + 50⋅ Buses ) 60
personhours = 1592
person − hours
determine mode shift for all 3 lanes open to all traffic
personhours := 2240.127
(From Problem 8.26)
personmin := personhours ⋅ 60 personmin SOV
= 134407.62
:= 2500
HOV := 500 + 300 + 20
HOVp := 500⋅ 2 + 300⋅ 3 + 20 ⋅ 50 c 1 := 1200
(given)
HOVp = 2900
c2 := 2400
⎡ ⎢
⎛ HOV + 6.87⎤ ⎡ ⎛ SOV − x ⎞ ⋅ ( SOV − x) + t0 ⋅ 1 + 1.15 ⋅ ⎜ t0 ⋅ 1 + 1.15 ⋅ ⎣ ⎝ c2 ⎠ ⎦ ⎣ ⎝ c1 x
= 43.826
⎞ 50 ⎟ ⎠ x
⎤ ⎥
6.87
⎦
⋅ ( HOVp + x )
personmin
so, 100 people must shift from SOV to Bus
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.28 Determine the user equilib rium and syst em optimal route flows and total travel times. t1
5
+ 3 ⋅ x1
t2
7
+ x2
q := 7
(given)
At System Optimal:
(
x1 ⋅ 5
S (x)
+ 3 ⋅ x1) + x 2 ⋅ ( 7 + x2)
q − x1
x2
(
x1 ⋅ 5
S (x) d
S ( x)
dx 1
+ 3 ⋅ x1) + ( q − x 1) ⋅ 7 + ( q − x1)
8 ⋅ x1
− 16 0
Route flows:
x1
=2
x 1 ⋅ 1000 = 2000
x 2 := q − x 1
x2
=5
veh/h
x 2 ⋅ 1000 = 5000
veh/h
Route travel times:
t1 := 5 t1
+ 3 ⋅ x1
t2 := 7
= 11
t2
(
TravelTimeSO := t1 ⋅ x 1 TravelTimeSO
= 82
+ x2
= 12
+ t2 ⋅ x 2) veh − min
At User Equilibrium: t1
5
t2
+ 3 ⋅ x1 7 + ( q − x1)
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Route flows:
x1
= 2.25
x 2 := q
x1 ⋅ 1000 = 2250
− x1
x2
= 4.75
veh/h
x 2 ⋅ 1000 = 4750
veh/h
Route travel times:
t1 := 5 + 3 ⋅ x 1 t1
= 11.75
TravelTimeUE := t1 ⋅ q TravelTimeUE
= 82.25
veh − min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.29 Determine the value of the derivative of the user equilibri um math program evaluated at the system optimal solut ion wit h respect to x 1 . q := 7 S (x)
S (x)
2
5 ⋅ x 1 + 1.5 ⋅ x1
S ( x)
dx 1 at d
⌠ ⎮ ( 7 + 3 ⋅ w) dw ⎮ ⌡
q − x1
x2
d
⌠ ⎮ ( 5 + 3 ⋅ w) dw + ⎮ ⌡
4 ⋅ x1
2
+ 49 − 7 ⋅ x1 + 24.5 − 7 ⋅ x 1 + 0.5 ⋅ x1
−9
x 1 := 2 (the SO solution) z
−1
dx 1
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.30 Determine the user equilibrium f lows and total hourly o rigindestination demand after the improvement.
⎛ x ⎞ t1 ( x , c ) := 3 + 1.5 ⋅ ⎝ c ⎠ c 1 := 2
2
t2 ( x , c ) := 5
c 2 := 1.5
+ 4⋅
⎛ x ⎞ ⎝ c ⎠
(given)
q := 6
(
) = 16.5
t1 0 , c 1
(
)=5
t2 q , c 2
t1 q , c 1 t2 0 , c 2
(
)=3
(
) = 21
so both routes may be used t1
t2
x2
q
− x1
Existing Condtions
3
+ 1.5 ⋅
x1
⎛ x1 ⎞
2
⎝ c1 ⎠
5
+ 4⋅
⎛ q − x1 ⎞ ⎝
c2
⎠
= 4.232
(
)
t1_exist := t1 x 1 , c 1
t1_exist
= 9.715
t2_exist := t1_exist x 2 := q − x 1 x2
5
+
4 ⋅ x2 2.5
5
+ 1.6 ⋅ x2
= 1.768
After capacity expansion, c 2_new := 2.5 t2_new
5
+ 4⋅
x 2_new c 2_new
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
− (t2_new)
q_new
q + 0.5 ⋅ t2_exist
q_new
q + 0.5 ⋅ t2_exist
q_new
6 + 0.5 ⋅ 9.715 −
q_new
8.3595 − 0.8⋅ x2_new
⎡ ⎣ ⎡ ⎣
⎛
x 2_new ⎞⎤
⎝
c 2_new
− 5 + 4⋅
⎠⎦
x 2_new ⎞⎤ ⎛ 5 + 4⋅ 2.5 ⎠⎦ ⎝
also, q_new
x1_new + x 2_new 8.3595 − x1_new
x 2_new t1
3
1.8
t2
+ 1.5 ⋅
⎛ x1_new ⎞ ⎝
c1
⎛ 8.3595 −x1_new ⎞ 1.8 ⎜ ⎟ 5 + 4⋅ ⎝ c2_new ⎠
2
⎠
x 1_new = 3.968
t1_new := 3 + 1.5 ⋅
x1_new⋅ 1000 = 3968
⎛ x1_new ⎞ ⎝
c1
veh/h
2
⎠
t1_new = 8.9
t2_new := t1_new x 2_new :=
t2_new − 5 4
⋅ c 2_new
q := x1_new + x 2_new
x2_new = 2.44
q
= 6.41
veh/h
x2_new⋅ 1000 = 2440
veh/h
total hourly origin-destination demand
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.31 Determine if the user equilibrium and sy stem optimal solu tions can be equal. t1
5
+ 4 ⋅ x1
t2
7
+ 2 ⋅ x2
(given)
At System Optimal:
(
x1 ⋅ 5
S (x)
q − x1
x2
2
5 ⋅ x1 + 4 ⋅ x1
S (x)
d
S
dx 1 x1
+ 4 ⋅ x1) + x 2 ⋅ ( 7 + 2 ⋅ x2)
+ 7 ⋅ ( q − x1) + 2 ⋅ ( q − x1)
2
−7 − 4 ⋅ q + 4 ⋅ x 1 + 5 + 8 ⋅ x 1 0
0.333q + 0.1667
At User Equilibrium: t1 5
t2
+ 4 ⋅ x1 7 + 2 ⋅ ( q − x1)
6x1
− 2q − 2 0
substituting, 2q + 1 − 2q − 2
0
This statement is false, therefore the solution is not possible
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.32 Determine the user equilib rium flo ws. t1 ( x ) := 5
+ 1.5 ⋅ x
t2 ( x) := 12 + 3 ⋅ x
t 3 ( x)
:= 2 + 0.2x
2
(given)
q := 4 t1 ( 4)
= 11
t1 ( 0)
=5
t2 ( 0)
= 12
t2 ( 4)
= 24
t3 ( 0)
=2
t3 ( 4)
= 5.2 Determine Which Routes Are Used
so only routes 1 and 3 will be used t1
5
t3 and x 1
+ 1.5( 4 − x3) 2
0.2 ⋅ x3 x3
4 − x3
2
2
+ 0.2 ⋅ x3
+ 1.5 ⋅ x3 − 9 0
= 3.935
x 3 ⋅ 1000 = 3935
x 1 := q − x 3
x1
( ) = 5.1
min
( ) = 5.1
min
t1 x 1 t3 x 3
= 0.065
veh/h x1 ⋅ 1000 = 65
veh/h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.33 Determine user equilibriu m route flo ws and total vehicl e travel time. qo := 4
t1
6
+ 4x1
t2
2
2
+ 0.5x2
(given)
For each minute of travel time greater than 2 minutes, 100 fewer vehicles depart qo
− 0.1( t2 − 2)
x1
+ x2
substitute for t 2 qo
2
− 0.1 2 + 0.5 ⋅x 2 − 2
x1
+ x2
solving for x1 x1
4 − x2
2
− 0.05x2
user equilibrium route flows means t1 = t 2 t1
6
2
+ 4 ⋅ x1 t2 2 + 0.5 ⋅ x2
plugging in x1 2
2
+ 0.5 ⋅ x2 2
0.7x 2
x2
6
2
+ 4 4 − x2 − 0.05 ⋅ x2
+ 4 ⋅ x2 − 20 0
= 3.204
t2 := 2
x 2 ⋅ 1000 = 3204
+ 0.5 ⋅ ( x2)
qnew := qo
2
t2
− 0.1( t2 − 2)
x 1 := qnew − x 2
veh/h
= 7.132 qnew = 3.487
x 1 = 0.283
Total travel time = q new⋅ t2 = 24.868
x 1 ⋅ 1000 = 283
veh/h
or 24,868 vehicle-min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.34 Determine the distribut ion of tr affic between restricted and unrestric ted lanes so that total person-hours are minimized. (subscripting: r = restricted, u2 = 2 person using unrestricted, u1 = 1 person using unrestricted)
( xr ⋅ tr ) ⋅ 2 + (xu2⋅ tu) ⋅ 2 + (xu1⋅ tu) ⋅ 1
z ( x) where
x u2 + x u1 and
xu
x u1
3
rewriting
(
)
2 ⋅ x r ⋅ 12 + xr
z ( x)
and x + x r u2
+ 2 ⋅ x u2 ⋅ 12 + 0.5 ⋅ ( 3.0 + xu2) + 3 12 + 0.5 ⋅ ( 3.0 + xu2)
4.0 2
2
z ( x)
24 ⋅ x r + 2 ⋅ xr
z ( x)
24 ⋅ 4 − xu2
z ( x)
96 − 24xu2 + 32 − 16xu2 + 2 ⋅ xu2
z ( x)
3 ⋅ xu2
d
+ 24⋅ xu2 + 3xu2 + x u2 + 36 + 4.5 + 1.5xu2
) + 2 ⋅ (4 − xu2) 2 + 24⋅ xu2 + 3xu2 + xu22 + 36 + 4.5 + 1.5xu2
(
2
2
− 11.5 ⋅ xu2 + 168.5
6 ⋅ xu2 − 11.5
z
dxu2
2
+ 24xu2 + 3xu2 + x u2 + 36 + 4.5 + 1.5xu2
0
x u2 := 1.916
so,
x r := 4.0 − 1.916
x r = 2.084
x r ⋅ 1000 = 2084
veh/h
x u := 3
xu
= 4.916
x u ⋅ 1000 = 4916
veh/h
+ xu2
Total person hours: tr := 12 + 2.083
t r = 14.083
min
tu := 12 + 0.5 ⋅ 4.916
t u = 14.458
min
2 ⋅ 2083⋅
14.083 60
+ 2 ⋅ 1916⋅
2624.1⋅ 60 = 157446
14.458 60
+ 3000⋅
14.458 60
= 2624.1
person − hr
person − min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.35 Determine what percentage would take rout e 1 and ho w muc h travel time would b e saved. At User Equilibrium: t1
t2
x1
⎛ 3x2 ⎞ 5+ ⎝ 2 ⎠ x2
2
⎛ x1 ⎞ t1 ( x1) := 5 + ⎝ 2 ⎠
3x2
⎛ x2 ⎞ 7+ ⎝ 4 ⎠
2
⎛ x2 ⎞ t2 ( x2) := 7 + ⎝ 4 ⎠
2
(given)
2
= 0.956
x 1 := 3 ⋅ x 2
x 1 = 2.869
q := x 1 + x 2
q
( )⋅
TravelTime:= t1 x1
= 3.825 q ⋅ 1000 60
TravelTime= 449.861
At System Optimal:
⎡
S (x)
2 ⎡ ⎛ x ⎞ 2⎤ ⎛ x1 ⎞ ⎤ 2 x1 ⋅ 5 + + x2 ⋅ 7 + ⎣ ⎝ 2 ⎠ ⎦ ⎣ ⎝ 4 ⎠ ⎦
q − x2
x1
⎡
S (x)
S (x)
d
2 ⎡ ⎛ x ⎞ 2⎤ ⎛ q − x2 ⎞ ⎤ 2 ( q − x 2) ⋅ ⎣ 5 + ⎝ 2 ⎠ ⎦ + x2 ⋅ ⎣ 7 + ⎝ 4 ⎠ ⎦
S ( x)
dx 2
x2
3
2
−0.1875⋅ x 2 + 2.86875⋅ x2 − 8.973⋅ x2 + 33.1155 2
−0.5625⋅ x 2 + 5.7375⋅ x2 − 8.973 0
= 1.929
x 1 := q − x 2
x1
= 1.896
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
percentage :=
x1 q
⋅ 100
percentage = 49.6
%
Total travel time:
( )
t1_SO = 5.899
( )
t2_SO = 7.232
t1_SO := t1 x 1 t2_SO := t2 x 2
TravelTimeSO :=
x 1 ⋅ t1_SO + x 2 ⋅ t2_SO 60
⋅ 1000
TravelTimeSO
= 418.892
Savings := TravelTime− TravelTimeSO Savings = 30.97
veh-min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.36 Determine the dif ference in total vehicle travel times between user equilibrium and system optimal solutions. At User Equilibrium: t1
5
+ 3.5 ⋅ x1 q−x2
S ( x)
5q − 5x2
dx 2
2
+ 0.5 ⋅ x2
x1
q − x2
(given)
x
S (x)
S ( x)
1
⌠ 2 2 ( 5 + 3.5 ⋅ w) dw + ⎮ ( 1 + 0.5w ) dw ⌡0
⌠ ⎮ ⌡0
d
t2
2
2
3
+ 1.75q − 3.5q ⋅ x2 + 1.75x2 + x 2 + 0.167x 2 2
−5 − 3.5q + 3.5x2 + 1 + 0.5x2
At System Optimal:
S (x)
5x1
2
5 ⋅ q − 5 ⋅ x 2 + 3.5 ⋅ q
S (x)
S ( x)
dx 2
d dx 2
2
+ x2 + 0.5x2
q − x2
x1
d
2
+ 3.5x1
2
1.5x2
SSO ( x )
2
1.5x 2
−
d dx 2
2
3
− 7 ⋅ q ⋅ x 2 + 3.5 ⋅ x2 + x 2 + 0.5 ⋅ x2
+ 7x2 − 4 − 7q 0
SUE ( x)
7
2
+ 7x2 − 4 − 7q − 0.5x2 + 3.5x2 − 4 − 3.5q
7
q := 3.57
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Finding total travel time at user equilibrium: 2
0.5x2
+ 3.5x2 − 4 − 3.5( 3.57)
0
x 2u := 3.226 by quadratic, so
gives
Total Travel Time
tt u := [ 5
x 1u := 3.57 − 3.226
+ 3.5( 0.344) ] ⋅ 3.57⋅ 1000
x1u = 0.344
tt u
22314
veh-min
Finding total travel time at system optimal: 2
1.5x2
+ 7 ⋅ x 2 − 4 − 3.5( 3.57)
gives
x 2s := 2.64 by quadratic, so
0
tt 1s := [ 5
+ 3.5( 0.927) ] ⋅ 0.927⋅ 1000
tt 2s := 1
+ 0.5 ( 2.64)
Total := tt 1s
+ tt 2s
2
⋅ 2.64 ⋅ 1000 Total = 19483
x 1s := 3.57 − 2.64 tt 1s
= 7642.651
tt 2s
= 11839.872
x1s
= 0.93
veh-min veh-min
veh − min
Difference := 22314 − 19482 Difference = 2832
veh-min
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the probability of t he household making th ree or more peak-hour trip s?
Problem 8.37
Solve Poisson regression BZi := −0.30 + 0.04⋅ ( 4)
e
BZ i
+ 0.005⋅ ( 85) − 0.12⋅ ( 2)
BZi = 0.045
= 1.046 vehicle trips
Calculate probability
P0 :=
e
P1 :=
e
P2 :=
e
⋅ (1.046 ) 0!
− 1.046
0
P0 = 0.351
⋅ (1.046 ) 1!
P1 = 0.368
⋅ (1.046 ) 2!
P2 = 0.192
− 1.046
(
(Eq 8.2)
1
− 1.046
(Eq 8.2)
2
(Eq 8.2)
) = 0.089
1 − P0 + P1 + P2
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Find probability of exatly three trips
P3 :=
e
⋅ (1.046 ) 3!
− 1.046
3
P3 = 0.067
2) Do not subtract probabilities from 1 P0 + P1 + P2 = 0.911
3) Find probability of greater than 3 trips
(
) = 0.022
1 − P0 + P1 + P2 + P3
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.38
How many workers wi ll use the shared-ride mode?
Cost DL
Cost DL := 5.50
Cost SR :=
TravTimeDL := 21
TravTimeSR := 21
3
Cost B := 1.25 TravTimeB := 27
Calculate utility expressions
(
) − 0.02⋅ ( TravTimeDL)
(
) − 0.04⋅ ( TravTimeSR )
UDL := 2.6 − 0.3 Cost DL
USR := 0.7 − 0.3⋅ Cost SR
(
) − 0.01( TravTimeB)
UB := −0.3⋅ Cost B
UDL = 0.530 USR = −0.690 UB = −0.645
Calculate probability of using carpool mode e
PSR := e
UDL
USR
+e
PSR ⋅ 5000 = 920
USR
+e
UB
PSR = 0.184
(Eq 8.7)
individuals ride in carpools
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers: 1) Solve for individuals riding in drive-alone autos
e
PDL := e
UDL
UDL
+e
USR
+e
UB
PDL = 0.623
PDL⋅ 5000 = 3117 individuals
2) Solve for individuals riding in buses e
PB := e
UDL
UB
+e
USR
+e
UB
PB = 0.193
PB⋅ 5000 = 963
individuals
3) Do not divide cost by three
(
) − 0.04⋅ ( TravTimeSR )
USR := 0.7 − 0.3⋅ Cost DL
e
PSR := e
UDL
USR = −1.790
USR
+e
USR
+e
UB
PSR = 0.070
PSR ⋅ 5000 = 349 individuals
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Problem 8.39
Determine the number of bus tri ps to sh opping p laza 2. UA1 := 0.25 − ( 0.4⋅ 15) UB1 := 0.0 − ( 0.5⋅ 18)
e
UA1
+e
UB2 := 0.0 − ( 0.5⋅ 19)
+ ( 0.013325 ⋅ )
+ ( 0.013325 ⋅ )
UB2
UB1
PB2⋅ 1100 = 21
UA2 := 0.25 − ( 0.4⋅ 16)
+ ( 0.013275 ⋅ )
e
PB2 :=
+ ( 0.013275 ⋅ )
+e
UA2
+e
PB2 = 0.019
UB2
trips
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Alternative Answers: 1) Solve for bus trips to plaza 1 e
PB1 := e
UA1
+e
UB1
UB1
PB1⋅ 1100 = 18
+e
UA2
+e
PB1 = 0.016
UB2
trips
2) Mix up commercial floor spacing UA1 := 0.25 − ( 0.4⋅ 15) UB1 := 0.0 − ( 0.5⋅ 18)
e
UA1
+e
PB2⋅ 1100 = 10
UA2 := 0.25 − ( 0.4⋅ 16)
+ ( 0.013325 ⋅ )
e
PB2 :=
+ ( 0.013325 ⋅ )
UB2 := 0.0 − ( 0.5⋅ 19)
+ ( 0.013275 ⋅ )
+ ( 0.013275 ⋅ )
UB2
UB1
+e
UA2
+e
PB2 = 0.009
UB2
trips
3) Solve for auto trips to plaza 2 UA1 := 0.25 − ( 0.4⋅ 15)
+ ( 0.013275 ⋅ )
UB1 := 0.0 − ( 0.5⋅ 18)
+ ( 0.013275 ⋅ )
PA2 :=
e e
UA1
+e
PA2⋅ 1100 = 597
UA2 := 0.25 − ( 0.4⋅ 16) UB2 := 0.0 − ( 0.5⋅ 19)
+ ( 0.013325 ⋅ )
+ ( 0.013325 ⋅ )
UA2
UB1
+e
UA2
+e
UB2
PA2
= 0.542
trips
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
How many additional vehicle-hours of travel time will be added to the system assuming user-equilibrium c onditions hol d?
Problem 8.40
Check to see if both routes are used t1(0) = 5 min
t 2(0) = 4 min
t1(4) = 9.57 min
t 2(4) = 8.35 min
Both routes are used since t 2(4) > t 1(0) and t1(4) > t 2 (0). Apply user equilibrium and flow conservation to solve for x1 and x 2 5 + 4/3.5(x 1) = 4 + 5/4.6(x 2) Flow conservation: x2 = 4 - x 1 5 + 1.143(x 1) = 4 +1.087(4 - x 1)
x1 := 1.501
x2 := 4 − x1
x2 = 2.5
Find total travel time in hours
⎡ ⎛ x1 ⎞⎤ ⎡ ⎛ x2 ⎞⎤ ⋅ 1501 + 4 + 5⋅ ⋅ 2499 5 + 4⋅ ⎣ ⎝ 3.5 ⎠⎦ ⎣ ⎝ 4.6 ⎠⎦ = 447.7
veh-h
60
Check route usage for reduced-capacity case t1(0) = 5 min
t 2(0) = 4 min
t1(4) = 9.57 min
t 2(4) = 12.70 min
Both routes are used since t 2(6) > t 1(0) and t1(6) > t 2 (0) Apply user equilibrium and flow conservation to solve for x1 and x 2 5 + 4/3.5(x 1) = 4 + 5/2.5(x 2) Flow conservation: x2 = 4 - x 1 5 + 1.143(x 1) = 4 +2(4 - x 1)
x1 := 2.227
x2 := 4 − x1
x2 = 1.8
Find total travel time in hours
⎡ ⎛ x1 ⎞⎤ ⎡ ⎛ x2 ⎞⎤ ⋅ 2227 + 4 + 5⋅ ⋅ 1773 5 + 4⋅ ⎣ ⎝ 3.5 ⎠⎦ ⎣ ⎝ 2.5 ⎠⎦ = 503.0
veh-h
60
Find additional vehicle hours 503.0 − 447.7 = 55.3
veh-h
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers: 1) Solve only for reduced-capacity case Additional vehicle hours = 503.0
veh-h
2) Use 2100 veh-h for route 2 capacity 5 + 4/3.5(x 1 ) = 4 + 5/2.1(x 2 ) Flow conservation: x2 = 4 - x 1 5 + 1.143(x1 ) = 4 +2.381(4 - x 1 )
x1 := 2.145
x2 := 4 − x1
x2 = 1.9
Find total travel time in hours
⎡ ⎛ x1 ⎞⎤ ⎡ ⎛ x2 ⎞⎤ 5 + 4⋅ ⋅ 2227 + 4 + 5⋅ ⋅ 1773 ⎣ ⎝ 3.5 ⎠⎦ ⎣ ⎝ 2.1 ⎠⎦ = 525.3
veh-h
60
Find additional vehicle hours 525.3 − 447.7 = 77.6
veh-h
3) Solve only for reduced-capacity case using 2100 veh-h route 2 capacity Additional vehicle hours = 525.3
veh-h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Determine the sys tem-optimal total t ravel time (in veh-h).
7
Route 1:
65 4
Route 2:
50
Problem 8.41
⋅ 60 = 6.46 min ⋅ 60 = 4.80 min
Performance functions: t1
6.46 + 4⋅ x1
t2
4.80 + x2
2
The basic flow conservation identity is: q = x 1 + x 2 = 5.5 Substitute the performance functions for routes 1 and 2 into Eq 8.9
(
) + x2⋅
S( x)
x1⋅ 6.46 + 4⋅ x1
S( x)
6.46⋅ x1 + 4x1
2
2
4.80 + x2 2
+ 4⋅ x2 + x2
From flow conseration, x1 = 5.5 - x 2 ; therefore,
) + 4⋅ ( 5.5 − x2) 2 + 4⋅ x2 + x23
(
S( x)
6.46⋅ 5.5 − x2
S( x)
x2
3
2
+ 4⋅ x2 − 46.46x2 + 156.33
Set the first derivative to zero to find minimum dS( x) dx2
2
3⋅ x2
+ 8⋅ x2 − 46.46 0
which gives
x2 := 2.822
and x := 5.5 − x 1 2
Find system-optimal travel times and system-optimal total travel time
( )
t1 := 6.46 + 4 x1
min
2
t2 := 4.80 + x2 t1 = 17.17
min t2 = 12.76
( 2680t⋅ 1) + (2822t⋅ 2) 60
x1 = 2.68
x2 = 2.82
= 1367.33 veh-h
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
Alternative Answers: 1) Simply add up individual travel times t1 and t2 t1 + t2 = 29.94 veh-h
2) Solve for total user-equilibrium travel time using t1 5500t ⋅1 60
= 1574.10
veh-h
3) Solve for total user-equilibirum travel time using t2 5500t ⋅2 60
= 1170.00
veh-h
Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.
How many vehicle-hours could be saved?
Problem 8.42
Solve for total number of vehicle hours using given distributions x1 := 2500
x2 := 2000
t1 := 12 + 2.5
t1 = 14.50 x1⋅ t1
Route 1:
t2 = 11.00
= 604.17
60 x2⋅ t2
Route 2:
t2 := 7 + 2⋅ 2.0
= 366.67
60
total = 970.84 vehicle-hours Solve with system-optimal traffic distribution S( x)
:= ( 12 + x1) ⋅ x1 + (7 + 2⋅ x2) ⋅ x2
With flow conservation, x1 = 4.5 - x 2 so that S( x)
2
:= 5⋅ x2 − 14⋅ x2 + 63
Set the first derivative to zero 10x2 − 14
x2 :=
0
t1 := 12 + x1
Route 1:
10
t1 = 15.10
1400t ⋅1 60
Route 2:
14
t2 := 7 + 2⋅ x2
t2 = 9.80
= 352.33
3100t ⋅2 60
x1 := 4.5 − x2
= 506.33
total = 858.66 vehicle-hours Hours saved = 970.84 - 858.66 = 112.18 vehicle-hours saved
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Solutions Manual to accompany Principles of Highway Engineering and Traffic Analysis, 4e, by Fred L. Mannering, Scott S. Washburn, and Walter P. Kilareski. Copyright © 2008, by John Wiley & Sons, Inc. All rights reserved.