IOWA STATE COLLEGE
OF AGRICULTURE AND MECHANIC ARTS
OFFICIAL PUBLICATION
Vol. XXI February 28, 1923 No. 39
THE ECONOMICS OF HIGHWAY GRADES
By
T. R. AGG
Highway Engineer
IOWA ENGINEERING EXPERIMENT STATION
and
Professor of Highway Engineering
IOWA STATE COLLEGE. AMES, IOWA
hi
BULLETIN 65
E 153
ablShed »we
ENGINEERING EXPERIMENT STATION
AMES, IOWA
Publmhed •weekly by Iowa State College of Agriculture and Mechanic Arts, Ames, Iowa.
Entered as Second-class matter, and accepted for mailing at special rate of postage provided
for in Section 429, P. L. & R., Act August 24, 1912, authorized April 12, 1920.
IOWA STATE COLLEGE
OF AGRICULTURE AND MECHANIC ARTS
OFFICIAL PUBLICATION
Vol. XXI February 28, 1928 No. 39
THE ECONOMICS OF HIGHWAY GRADES
By
T. R. AGG
Highway Engineer
IOWA ENGINEERING EXPERIMENT STATION
and
Professor of Highway Engineering
IOWA STATE COLLEGE, AMES, IOWA
BULLETIN 65
ENGINEERING EXPERIMENT STATION
AMES, IOWA
Published weekly by Iowa State College of Agriculture and Mechanic Arts, Ames, Iowa.
Entered as Second-class matter, and accepted for mailing at special rate of postage provide!
for in Section 429, P. L. & It., Act August 24, 1912, authorized April 12, 1920.
STATE BOARD OF EDUCATION
Members
Hon. D. D. Murphy, President Elkader
Hon. George T. Baker Davenport
Hon. Ohas. R. Brenton Dallas Center
Hon. P. K. Holbrook Onawa
Hon. Edw. P. Schoentgen Council Bluffs
Hon. W. C. Stuckslager Lisbon
Hon. Anna B. Lawther '....Dubuque
Hon. Pauline Lewelling Devitt Oskaloosa
Hon. C. H. Thomas Creston
Finance Committee
Hon. W. R. Boyd, Chairman Cedar Rapids
Hon. Thomas Lambert Sabula
Hon. W. H. Gemmill, Secretary Des Moines
ENGINEERING EXPERIMENT STATION
Station Council
(Appointed by the State Board of Education)
Raymond A. Pearson, LL. D President
Anson Marston, C. E Professor
Warren H. Meeker, M. E Professor
Fred Alan Fish, M. E. E. E Professor
Allen Holmes Kimball, M. S Professor
O. R. Sweeney, M. S., Ph. D Professor
Louis Bevier Spinney, B. M. E Professor
Fred R. White, B. C. E Chief Engineer, Iowa Highway Commission
Station Staff
Raymond A. Pearson, LL. D President, Ex-officio
Anson Marston, C. E Director and Civil Engineer
H. E. Pride, B. S. in C. E Bulletin Editor
Roblev Winfrey, B. S. in C. E Assistant Bulletin Editor
J. B. Davidson, B. S. M. E Agricultural Engineer
Allen Holmes Kimball, M. S Architectural Engineer
Paul E. Cox, B. S. Cer. Eng Ceramic Engineer
O. R. Sweeney, M. S., Ph. D Chemical Engineer
Almon H. Fuller, C. E Civil Engineer
Fred Alan Fish, M. E. E. E Electrical Engineer
Warren H. Meeker, M. E Mechanical Engineer
A. K. Friedrich, E. M Mining Engineer
Max Levine, S. B Bacteriologist
Lulu Soppeland, M. S Assistant Bacteriologist
D. A. Moulton, B. S. in Cer. Eng Ceramic Engineer
G. W. Burke, Ch. E Chemist
C. H. Geister, B. S Assistant Chemist
William J. Sclilick, C. E Drainage Engineer
John Edwin Brindley, A. M., Ph. D Engineering Economist
Clyde Mason, B. S. in E. E., B. S. in C. E Engineer
S. L. Galpin, Ph. D Geologist
T. R. Agg, C. E Highway Engineer
Louis Bevier Spinney, B. M. E Illuminating Engineer and Physicist
C. J. Myers, B. S. in M. E Mechanical and Electrical Engineer
Charles S. Nichols, C. E Sanitary Engineer
J. H. Griffith, M. S Structural Materials Engineer
F. W. Hallgren Mechanician
INVESTIGATION OF THE ECONOMICS OF HIGHWAY GRADES.
Introduction. The grades for rural highways are generally estab¬
lished in accordance with empirical rules that have been developed
as a result of experience. There is usually an agreed maximum grade
and an attempt is made to keep beloAv it so far as topographical
conditions permit and below that maximum grade there is little uni¬
formity in procedure. The rules followed are for the most part those
originally developed for highways used by horse-drawn traffic, modi¬
fied for motor traffic as experience and judgment dictate.
In 1919 the Iowa Engineering Experiment Station began an inves¬
tigation of the economics of grades for highways, based on motor-
drawn rather than horse-drawn traffic. The work has been con¬
tinued until the present time and it will require several years more
to evaluate all of the factors that enter into the problem. This report
indicates the progress that has been made up to the present time,
and presents some conclusions that seem to be established by the
data already obtained.
An analysis of the problem from the purely theoretical standpoint,
considering only the fuel consumption and speed of motor vehicles,
indicated that the three factors controlling the economical maximum
grades for motor vehicles were length of grade, resistance to trans¬
lation on the particular road surface and maximum tractive effort
of the vehicle. In order to study the relation between these factors
and to determine with absolute certainty that others did not enter
into the problem, it was found desirable to obtain records of fuel
consumption and other pertinent data for typical vehicles operating
under normal conditions on hilly roads.
Personnel. The investigation was carried out by the Iowa Engi¬
neering Experiment Station at Ames, of which Dean Anson Marston
is director. In addition to the author, Prof. W. L. Foster of the civil
engineering department at Ames, was engaged on the investigation
for a part of the time. C. P. Lewellen, Robley Winfrey, II. II. Bowen
and B. H. Piatt assisted in the preparation of the tables and diagrams
and in the field work. The Iowa Highway Commission cooperated
throughout by loaning trucks and small equipment.
Vehicles employed. A Buick five-passenger automobile, a pneu¬
matic tired truck of 1 ton capacity and a solid tired truck of 31/2
tons capacity were employed in the study of highway grades. The
trucks were equipped by the Iowa Highway Commission and loaned
for the work. The passenger car was purchased by the writer for
use in the investigation. Table I gives the weight and equipment of
each of the vehicles.
— 4 —
Purpose of fuel consumption measurements. The purpose of the
fuel consumption determinations was to establish the relation be¬
tween the amount of gasoline used when vehicles are traveling over
hilly roads and when traveling on level roads. A number of sec¬
tions of highway were selected in the vicinity of Ames, upon which
various rates and lengths of grade were represented and a number
of preliminary runs were made over these roads. After the general
trend of the results of the preliminary runs became apparent, a sec¬
ond series of runs was made under close control and the data so ob¬
tained are the basis for the conclusions presented herein relative to
the effect of highway grades on fuel consumption and speed. It
should be pointed out that fuel consumption is not a direct measure
of the power developed by a gasoline motor but it is believed that
the results so obtained are comparative and sufficiently accurate for
the purposes of this investigation. Moreover it is necessary to inter¬
pret the effect of highway grades in terms of fuel in order to prop¬
erly evaluate the possible saving in fuel costs resulting from grade
reduction.
TABLE I.—EQUIPMENT OF VEHICLES.
Vehicle
Gross Weight
Size and Type of Tires
Remarks
Buick
1.85 tons
33x4 U. S. Royal cord, inflation 65 to 70
lb. per sq. in.
Weight includes flow-meter ap¬
paratus, driver and observer.
Car and tires nearly new at
beginning of the series of runs.
Light
aviation
army truck
4.14 tons
Front, 35x5 Goodyear cord all weather
tread; rear, twin 35x5 Goodyear cord all
weather tread: inflation 85-90 lb. per
sq. in.
Truck and all equipment nearly
new at beginning of the series
of runs.
Heavy
aviation
army truck
7.00 tons
Front, 36x6 solid; rear, 36x10 dual solid.
Truck and all equipment new at
beginning of the series of runs
Sections of road utilized. The sections of highway selected for the
study of grades and the kinds of road surfaces are indicated in Table
II, but reference should be made to the profiles that appear in Figs.
8 to 37 for information relative to the grades.
Gasoline measurement. In the earlier experiments measurement
was made of the total quantity of gasoline consumed during each
trip up a hill and each trip down, but considerable difficulty was
experienced in measuring accurately the small quantities of gasoline
used. It was also found that on hilly roads total fuel consumption
of itself had little significance. It became apparent that to properly
analyze the effect of grades it was necessary to secure a continuous
record of the rate of fuel consumption so that a gasoline profile
could be platted for comparison with a road profile. This involved
the development of an accurate recording flowmeter for measuring
the rate of gasoline consumption. Many devices were tried, in which
— 5 —
the changing level of the gasoline in a tube of uniform diameter was
measured by a float of some sort. None of these proved to be adapted
to the service required. With the heavier trucks there is always con¬
siderable vibration which causes trouble with instruments of the float
type. H. W. Dietert, who was then a senior student in mechanical
engineering at Ames, devised a method of measuring the loss of head
in a column of gasoline, due to the flow to the motor, by means of an
ingenious float system and constructed an instrument that gave ac¬
curate results. But the float arrangement in the instrument was too
delicate for truck service and the instrument had to be discarded.
It was finally discovered that a calibrated diaphragm, such as is
used in gages designed for low pressures, could be used to measure
the velocity head due to the flow of gasoline to the carbureter.
TABLE II.—LOCATION OF ROADS FOR GRADE INVESTIGATIONS.
Flowmeter Test
Section
Location
Type of Surface
1
Lincoln Highway. 1 mile east of Ames, Iowa.
Gravel surface in good condition.
2
Jefferson Highway 6 miles south of Ames, Iowa.
Gravel surface in good condition.
3
Lincoln Highway approach to Des Moines River from east.
Gravel surface, fair.
4
Lincoln Highway approach to Des Moines River from west.
Gravel surface, fair.
5
South end of Merle Hay Road near Des Moines, Iowa.
Monolithic brick in good condition.
6 | North end of Merle Hay Road near Des Moines, Iowa.
Monolithic brick in good condition.
7
Primary Road No. 2 west from city limits of Des Moines,
Iowa.
New Portland cement concrete.
8 j West Grand Ave., near city limits of Des Moines, Iowa.
Asphalt filled brick in fair condition.
9 West 28th Street, Des Moines, Iowa, between West Grand
| Ave., and Kingman Boulevard.
Sheet asphalt in good condition.
10
Primary Road No. 15 about 2]A miles north of Ames, Iowa.
Good gravel roads
11
41st Street, Des Moines, Iowa, Franklin Ave., to point
near University Ave.
Sheet asphalt in good condition.
12
Jefferson Highway 2 miles south of Ames, Iowa.
Gravel surface in good condition.
The Ames flowmeter. Fig. 1 shows the flowmeter in its final form.
In principle it is a piezometer ring connected up in such a way that
the pressure head is constant. The velocity head at the piezometer
ring, which varies with the quantity of gasoline flowing, is measured
by means of the calibrated diaphragm. Since the maximum velocity
head is about 4 ounces the movement of the diaphragm is small and
there is insufficient power to operate a pencil for recording purposes,
but a device was worked out for punching a record by means of an
electric spark.
— 6 —
Iii Fig. 1 the diaphragm is shown at A, and the needle valve and
piezometer ring at B. The vacuum tank at C is used to give nearly
constant head and the spare tank at D was used to permit checking
the records but may be dispensed with in most cases and connection
be made to the supply tank on the vehicle. E is the cable of a taxi¬
meter drive which is used to operate the paper rolls and F is a clock
which is arranged to momentarily close a circuit and punch a hole
in the paper every 7 seconds. The distance between the time marks
on the record paper is proportional to the vehicle speed and furnishes
the data for a speed curve of the run. G is a spark coil for giving
the high tension current for punching the records on the paper.
Fig. 1—The Ames gasoline flowmeter apparatus attached to Buick passenger car.
The pitot tube shown over the head light on the car is connected
to an air speed indicator on the instrument board of the car and
enables the observers to eliminate records which may have been
influenced by wind effects.
The record paper is drawn through the recording device by means
of rollers geared to the front wheel of the vehicle, so that the paper
travel is in a fixed ratio to the vehicle travel. For passenger vehicles
the paper moves approximately ^2 inch I)er 100 feet of vehicle travel.
For trucks the paper travel is somewhat less.
In using the meter, the paper travel is first determined by cali¬
bration runs over a measured course. The needle valve is then set
— 7 —
so as to give a record about 1 y<% inches high when the vehicle is
operating so as to use the maximum amount of gasoline. The meter
is then calibrated and is ready for use.
There are two sources of error in the instrument: (a) errors due
to the flow of gasoline due to the operation of the float valve in the
carbureter and (b) variations of the pressure head due to the fluc¬
tuations of the gasoline level in the vacuum tank. The accuracy of
the meter has been tested in various ways and for ordinary operat¬
ing conditions the maximum error is about 5 per cent.
In Fig. 2, there is shown the calibration curve obtained with the
instrument adjusted for a Buick touring car. It is convenient in
L5
aa
O
<
6
£6 £6 30 X 34 3b 38 40 4£
Calibration Piagram
for
Ames Gasoline Flowmeter
Batck Model
Ames bwa Ajg. IF, I9FI
/
t
O £ 4 6 6 iO /£ 14 /6 id £0 ££ £4
Height of Pecord m Thirtieths of an Inch
Fig. 2—Typical calibration curve for the Ames flowmeter wnen attached to Buick
passenger car. The flowmeter must be adjusted to give the desired flow and
calibrated for each type of vehicle.
computing the records to measure the height of the gasoline curve
with the "30" scale and for that reason the curve is platted with
thirtieths of an inch an abscissae.
Fig. 3 is from a photograph of a part of one of the records. The
gasoline consumption curve on the record was a series of small holes
punched through the paper by a high voltage spark. It will be noted
that the time spark punctures a group of three or four holes in the
paper. The measurements for speed are made from the left hand
hole in a group to the corresponding hole in the next group. The
zero line is constant and is drawn on the record after it is removed
from the instrument.
Using the flowmeter. Having established the constants for vehicle
and instrument, the regular series of runs were completed as rapidly
as the weather permitted. During these runs care was taken to
have the motor warmed up before starting and to avoid windy days.
The rate of gasoline consumption per ton-mile for a 0.0 grade on the
particular type of surface was first established and then the records
were taken on the selected grades. Five round trips were usually
7Seconds
Zero Line-j
Fig. 3—A portion of a record slip showing the variation in gasoline now and the
small holes punched by electric sparks
made, but three only, were averaged for most of the diagrams that
accompany this report. There is shown in Fig. 4, the gasoline profiles
for each of five runs over one of the routes and these indicate the
consistency secured in successive runs.
Results of fuel consumption measurements. It was noted that the
effect of grade upon fuel consumption was first studied by measuring
the quantity of gasoline used in ascending grades of various per¬
centages. Table III shows data which are typical of those obtained
in that way. It should be pointed out that while the fuel required
for ascending these grades is in excess of that required on 0.0
grades with the same type of surface, the totals do not truly repre¬
sent the effect of the grade upon traffic moving in both directions.
If it be assumed that no fuel is required for the vehicle on the trip
down the hill (none would be required on these particular grades)
the ton-miles per gallon for the round trip would be double that
shown in Table III. The ton-miles per gallon for a round trip esti¬
mated in that manner are less than is obtained in traveling on 0.0
grades, as will be apparent from the data in Table III.
If a vehicle is allowed to accelerate in descending a hill the mo-
— 9 —
mentum attained will carry it some distance beyond the bottom
of the hill before the speed is reduced to that at the beginning of the
descent. This momentum effect must be taken into account in com¬
puting the effect of grade on fuel consumption. It is very difficult
to make this correction if the fuel consumption is measured, without
supplementary data relative to speed. This was one reason for
adopting the flowmeter method of studying the effect of grades on
fuel consumption.
The results shown in Table III indicate that there is an increase
in fuel consumption when the vehicle is operated on any gear other
than high, but such a statement must be properly construed. The
variation in motor efficiency with the speed and with throttle open¬
ing enter in to complicate the problem. If in ascending a hill the
TABLE III.—QUANTITY OF GASOLINE USED IN ASCENDING HILLS.
Fourth gear is "high," first gear is "low."
Per Cent Grade
Gear
Vertical Distance Traveled
Ton-Miles Obtained
Road Surface
in Feet
Per Gallon
5.8
2d
290
7.50
Gravel
5.1
2d
602
11.401
5.1
2d
536
11.25 \
11.22
Monolithic brick.
5.1
2d
418
11.00 J
3.7
3d
561
14.201
13.20
3.7
3d
561
12.20 /
Monolithic brick.
0.0
High
21.20
Gravel.
0.0
High
29.70
Monolithic brick.
motor is operated in high gear at very slow speed the fuel consumed
will be greater than when the motor is operated at higher speed in
a lower gear, all other conditions being equal. This is an additional
reason for adopting the flowmeter method of studying the effect of
grade on fuel consumption.
The flowmeter records were therefore adopted as a basis for
establishing the effect of grades on fuel consumption and the dia¬
grams shown herein are based on records made under normal operat¬
ing conditions by drivers of ordinary skill. A study of the diagrams
in Figs. 8 to 37 will show how clearly each change in grade or operat¬
ing condition is reflected in the rate of fuel consumption.
Rules for grades. A further study of the diagrams in Figs. 8 to 37,
indicates that if fuel consumption only is considered, the economical
highway grade is determined by the following conditions:
1. The most economical rate of grade for descending traffic is that
rate of grade that will permit the vehicle to descend without attain¬
ing an unsafe speed and without the use of the brake.
— 10 —
It will be apparent that for a specific vehicle at any certain time
there will be a rate of grade that will comply with the above, but
when consideration is given to all types of vehicles and their varied
resistance to translation it is apparent that a general average must
be accepted in the rate of grade adopted for descending traffic. It
is also necessary to establish the allowable limits in speed for the
several classes of vehicles before computing the grades.
The usual driving practice is to descend a hill with the motor
throttled down so that to some extent the motor serves as a brake.
If the driver were to coast down hill with the motor declutched an
entirely different condition would arise. The motor would idle at
slow speed and little gasoline would be required to keep it turning
over. The acceleration from the grade would be greater than with
the clutch in and could be almost wholly utilized in ascending the
next hill or in coasting on the level. By operating the vehicle in
that way, highways that are a succession of grades can be traveled
with lower gasoline consumption than is required for level roads.
Likewise the most economical minus grades for this operating condi¬
tion are lower than for the usual method of driving.
2. The economical maximum rate of grade for ascending traffic
is that which will permit the vehicle to ascend without changing
gears and at a speed that permits the motor to operate most efficiently.
The power required when a vehicle is operating on low gear is
greater than when operating on high gear because of the increase of
internal friction in the vehicle. With some motors this is partly
compensated for by the increased motor efficiency near full throttle,
but in general the motor uses less fuel on high gear than on any
other, for identical road conditions.
3. The economical maximum rate of grade will depend upon the
resistance to translation of the vehicle, the length of the grade, the
allowable limits' of speed and the tractive power of the vehicle.
To establish values for all of the factors that have a bearing on
highway grades much information will be required that is now
available only in fragmentary form. The following discussion is
based on the best data now available and is intended to be illustra¬
tive only, although the curves shown are believed to be reasonably
close to the correct values.
— 11 —
Theoretical Considerations.
On the basis of the three principles mentioned in the preceding
paragraphs, certain relations can be established between per cent
of grade, length of grade, characteristics of the road surface and
the vehicle.
Case I. Vehicle descending the grade. Let the section from 1 to 2
in Fig. 5 represent a portion of the road profile, which may be preceded
and followed by a plus grade of any per cent, and
R = resistance to translation of vehicle in pounds per ton.
W = weight of vehicle in tons.
L = length of grade in feet, measured along the surface
of the road.
P = per cent of grade.
Vx = velocity of vehicle at (1), in feet per second.
V2 = velocity of vehicle at (2), in feet per second.
Si == speed of vehicle at (1), in miles per hour.
S. = speed of vehicle at (2), in miles per hour.
2 A I to2 = summation of kinetic energy of all rotating parts of
vehicle,
g = 32.2.
2000W
g
total resistance to translation of vehicle in pounds,
difference in elevation of points 1 and 2, in feet.
At position 1, the kinetic energy, Ek, of the vehicle is
2000W v2 , yfjyj 2,
Ek — A g * i+E(^4lto )i
At position 2, the kinetic energy, E'k, of the vehicle is
J/2000W » Wl/T 2.
E k = A—g—2)2
The gain (or loss) in kinetic energy during the time the vehicle
moves from position 1 to position 2 is equal to the algebraic differ¬
ence between Ek and E'k The loss in potential energy is 2000Wh
and the work done against rolling resistance is rL. Whence,
2000Wh—rL= J# 20^Wy2_^^ + |~s (1)
— 12 —
The term S^Ito2 can be evaluated experimentally for any type
of vehicle. For the vehicles employed in these investigations
S^Iw2 was determined exclusively of the fly wheel and other parts
ahead of the neutral gear, in terms of per cent of the kinetic energy
of translation as follows:
Buick passenger car 5 per cent
Pneumatic tire truck 6 per cent
Solid tire truck 10 per cent
It should be noted that this quantity will vary considerably for
the various types of vehicles and may be considerably larger than
10 per cent for trucks of large capacity which have very heavy
wheels. It will also be considerably larger if account is taken of the
rotating parts ahead of the neutral gear.
In the following analysis the kinetic energy of rotation for a pas¬
senger automobile will be assumed at 5 per cent of that due to trans¬
lation and for trucks, 10 per cent of that due to translation.
For passenger automobiles equation (1) may therefore be written.
2000Wh—rL = 1.05 (^mV/)-1.05 (^mVi2) (2)
For motor trucks equation (1) may be written:
2000Wh—rL = 1.10 (^mV22)-1.10 (^mVi2) (3)
Since resistance to translation is usually expressed in terms of
pounds per ton and it is more convenient to use speed in miles per
hour than velocity in feet per second, equation (2) may be rewritten
in the form:
Equation (3) may be written in the form:
Equations (4) and (5) signify that if a vehicle arrives at position
(1) at a speed of Sx miles per hour and proceeds to position (2) with¬
out the application of power, the speed at position (2) will be S2, Pm
being the per cent of grade, L the length of grade in feet and R the
resistance to translation in pound per ton.
— 13 —
APPLICATION.
In applying equations (4) and (5) to the establishment of highway
grades, Si and S. will be assumed in accordance with driving prac¬
tice. R will be a value determined by experiment and since L is
known for any hill, P can be computed. The per cent grade computed
in this way will be one which will insure that the vehicle to which the
value of R is applicable will descend the hill without the use of
power and without attaining an unsafe speed. If the computation
results in a value for P that is greater than that of the existing hill,
then no change should be made in the grade because the vehicle will
use power in descending the hill in order to maintain its speed.
Values for Sx and S2 can be adopted for any specific location with¬
out great controversy since safe driving practice is fairly well estab¬
lished.
The critical factor in equations (4) and (5) is R, the resistance to
translation. Much confusion exists as to what is meant by "trac¬
tive resistance" and how it is to be used. This is discussed at some
length in Bulletin 64, "Investigation of the Resistance to the Trans¬
lation of Motor Vehicles," of the Iowa Engineering Experiment
Station. It is necessary, however, to consider certain aspects of the
subject herein. The factor R is used in equations (4) and (5) to
determine the grade upon which the vehicle will coast at uniform
speed without the application of power.
If it is desired to find the coasting grade for the vehicle in high
gear with the ignition off, then R must include all resistance of the
road surface, all internal resistance of the vehicle and all air resistance.
A car is never operated in that way except when the motor is used to
assist in braking.
After long consideration and study of the performance of many ve¬
hicles on all sorts of grades it has been concluded that grades should
be based on the value of R that is obtained when the vehicle is running
declutched. This value of R includes rolling and air resistances and
the internal resistance up to the clutch. The diagrams showing max¬
imum grades that are included herein, are based on values of R deter¬
mined in that way. It is apparent that from the standpoint of conserv¬
ing energy it is immaterial whether, when a vehicle is coasting, the
motor is turned over by fuel or by the effect of gravity. It will make
some difference in the grade adopted however.
Ca-se II—Vehicle ascending the hill. The symbols used here have the
same significance as when used in connection with Case I and there is
added:
T=tractive effort of the motor at the road surface in pounds per ton
of weight of vehicle.
Following the discussion of descending traffic the following equa¬
tions may be written for ascending traffic :
-14 —
The kinetic energy at position 1 is:
-n _t/2000wtr2 , v/t/t
ek = //2 g \ )l
The kinetic energy at position 2 is:
E'k =^^^V!+2(^Iu!)2
The loss or gain in potential energy from position 2 to position 1 is
equal to the algebraic difference between Ek and E'k- The gain in
potential energy is 2000Wh, the energy gained from the motor is WTL
and the energy required to overcome rolling resistance is rL.
Whence,
WTL = 2000Wh+rL+[^mV12— ^mV22]+[2^Ico2)i—S(#Iu2)2] .. (6)
The kinetic energy of rotating parts may be provided for by multi¬
plying the quantity representing the kinetic energy of translation by a
suitable factor. As before 1.05 is the factor adopted herein for pas¬
senger automobiles and 1.10 for trucks, which does not include 2^Iw2
for rotating parts ahead of the clutch.
Equation (6) may be written in the following form for passenger
vehicles:
p _ T R . 3.50 .g2 o2\ (n\
20 20 L ' 2 ^
For motor trucks the form is:
T R 3.65 fo2 cA ,
20 " "20 L~ ( 2 (8'
By means of formulas (7) and (8) the maximum grade for ascending
vehicles can be computed if T and R are known. The variations to be
expected in the value of R have already been discussed. T is a func¬
tion of the size of motor and the gear ratio of the vehicle. Obviously it
is considerably larger for low gear than for high gear.
Applications of Theory of Grades.
The theory of highway grades must be based on the performance of
groups of vehicles rather than upon that of a single vehicle. An at¬
tempt is therefore made to establish the characteristics of two com-
TABLE IV.—POWER OF CHARACTERISTICS OF TYPICAL COMMERCIAL
VEHICLES.*
No. in Group
Rated Capacity in Tons
Average Tractive
Studied
Effort in Lb. Per
Ton in High Gear
3
KandM
160
3
1
180
3
IK
150
3
2
130
4
2K
120
3
3
125
4
3K
130
2
4
110
4
5
115
Mean for the composite vehicle assumed at 130 lb. per ton when loaded.
•Based on data furnished by Dr. H. C. Dickinson, Research Director Society of
Automotive Engineers.
TABLE V.—CHARACTERISTICS OF THE COMPOSITE COMMERCIAL VEHICLE.
Speed Range in Miles
Per Hour, High Gear
Tractive Effort on the Several Gears
In Lb. Per Ton
High
3d
2d
Low
Rolling Resistance in Lb. Per Ton
Pavements
Gravel, Macadam and
Good Earth Surfaces
15 to 25
130
160
250
380
TABLE VI.-
-COMPARATIVE FUEL CONSUMPTION ON VARIOUS GEARS.
(Commercial Vehicles.)
Type of Vehicle
Ton Miles!
'er Gallon of Gasoline on Paved Surfaces
Remarks
Low
Gear
Second
Gear
Third
Gear
High
Gear
Truck with solid tires
Truck with pneumatic tires..
Averages
9
11
10
16
15
15.5
25
22
23.5
30
30
30
Averages of five series of runs.
TABLE VII.—RELATIVE SPEEDS WITH THE SEVERAL GEARS.
(Commercial Vehicles.)
Gear
Speed in m. p. h.
Per Cent of High
Gear Speed
High
25.00
100
3d
15.00
60
2d
8.75
35
Low
5.00
20
— 16 —
posite vehicles, the one being typical of automobiles and the other of
commercial vehicles. These composite vehicles represent an average
vehicle as determined by a study of all pertinent data now available
and especially the information developed during the Ames investiga¬
tions. Undoubtedly future investigations will indicate certain desir¬
able modifications of the characteristics given herein, but it is unlikely
that such modifications as develop will have any considerable effect on
the limiting grades suggested herein.
I. The composite commercial vehicle. The data in Table IV show the
characteristics of a number of commercial vehicles now regularly
manufactured in the United States.
15
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$ 7
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For s4utomob>iies .-
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For Commercial Vehic/es
P- ^_j_3.63
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Length of Grade in Hundreds of Feet
Fig. 6—Curve showing the limits for economical plus grades.
The rolling resistance of the composite commercial vehicle, based on
data obtained in the Ames investigations of tractive resistance, at a
road speed of 25 miles per hour is 40 pounds per ton, when operating
on good pavements. This will also apply to earth and gravel roads in
excellent condition. For gravel roads in average condition it is 50
pounds per ton, and this will also apply to earth roads in good condi¬
tion.
With the data from Tables V, VI and VII, maximum grades for
ascending traffic ("plus" grades) may be established which are ap-
— 17 —
plicable to commercial vehicles. These are shown in Fig. 6 and are
based on equation (8).
In order to evaluate the saving to traffic that results from the reduc¬
tion of plus grades, two factors must be considered.
a. The relative fuel economy on high gear and the lower gears.
b. The value of the time lost due to operating on gears other than
high.
From the data given in Table Y it appears that, with good pave¬
ments, the composite vehicle can ascend, on low gear, a continuous
grade of 17 per cent; on second gear one of 10.5 per cent and on third
gear one of 6 per cent. In high gear it can ascend the grades shown
in the diagram in Fig. 6 In high gear, momentum may be utilized in
ascending the grade and should be considered within the limits of
speed adopted. When the vehicle is operating in any gear other than
high, momentum effects should be disregarded.
Few data are available relative to the rate of fuel consumption on the
several gears, but Table VI indicates the average of a number of trials
of vehicles used in the Ames investigatons.
Allowable Expenditure for Reducing Plus Grades.
If grades do not exceed the limit for economical plus grades shown
in Fig. 6, no saving in time or fuel will result from reducing them, (as¬
suming that characteristics of the composite vehicle are correct.) For
grades in excess of those shown in Fig. 6, it is first necessary to de¬
termine which gear must be used before the value of the fuel saved by
reducing the grade can be estimated. When it has been determined
which gear will be used for the particular hill, the justifiable expendi¬
ture for reducing the grade can be determined from equation (9), tak¬
ing into account only fuel saving.
Let Sc=the sum in dollars that can be expended for reducing excessive
plus grades based on fuel saving to commercial vehicles.
hx=height of the hill that would result from making the grade
equal the economical plus grade.
h=actual height of an existing hill, which must be not greater
than 17 per cent if vehicles are to operate at all.
Vc=annual commercial vehicle traffic ascending the hill in millions
of tons.
C=cost of gasoline in dollars per gallon.
N=the ratio of the fuel consumption on the gear used to high
gear.
(Note : For the vehicles used at Ames, N—3 for low gear, 1.9 for 2d
gear, and 1.3 for 3d gear.)
Then Sc=5200 CVc (Nh—h,)
(9)
— 18 —
It still remains to evaluate loss in time on excessive grades, which is
a significant factor for commercial vehicles, but one that is difficult to
estimate. The speed that may be obtained on any gear is Indicated by
the data in Table VII, which is based on many trials with the various
vehicles used in the Ames investigations. The loss in time in traveling
up a hill, due to the necessity for traveling for a part of the distance on
the lower gears, may be estimated by the following formula :
H=n[-^- ( e-1 )] (10)
Where H=the annual loss in time in hours.
n=the number of commercial vehicles ascending the hill per an¬
num.
A=the average speed on level roads.
d=the distance traveled on the lower gears in miles.
e=the ratio of the speed on the particular gear considered to the
normal speed in high gear.
For commercial vehicles about 20 per cent of the operating cost is
fuel cost and about 60 per cent is made up of time factors. The com¬
posite commercial vehicle will require about % gallon of gasoline an
hour per ton of weight. The sum that may be expended in grade re¬
duction to eliminate lost time may be estimated from the following
equation.
D=56HC (11)
Where H=the time lost on a particular hill as determined by equa¬
tion (10).
D=the sum in dollars that may be expended to prevent loss
of time.
C=price of gasoline in dollars per gallon.
Commercial vehicles with trailers. The principles already set forth
apply to trucks with trailers but the curves showing economical grades
do not apply because the values of R and of tractive effort upon which
they are based are not correct for combinations of trucks and trailers.
No data covering these combinations are as yet available.
II. The composite automobile. The composite automobile is assumed
to have the characteristics as shown in Table VIII. With these data and
equation (7) maximum economical grades for ascending traffic ("plus"
grades), which are applicable to automobiles, can be established. These
are shown in Fig. 6.
Allowable Expenditure for Reducing Plus Grades.
The justifiable expenditure for reducing plus grades to the maximum
shown by the diagram in Fig. 6 can be computed from equation (12),
taking into account only fuel saving.
— 19 —
Sa=the sum in dollars that can be expended for reducing ex¬
cessive plus grades, based on fuel saving to automobiles.
h1=height of hill corresponding to the theoretical maximum
plus grade.
h=actual height of hill which must not exceed 22.5 per cent
if the composite automobile is to climb it.
Va=annual automobile traffic ascending the hill in millions of
tons.
C=Cost of gasoline in dollars per gallon.
N=the ratio of the fuel consumption on the gear used to high
gear.
(Note: For the vehicles used at Ames N=1.9 for low gear and 1.3 for
intermediate.)
Sa=5200 CVa [ 1.9 (Nh—hj) ] (12)
TABLE VIII.—CHARACTERISTICS OF THE COMPOSITE AUTOMOBILE.
Speed Range
in
Miles Per Hour
Tractive E
in Pou
fort on the Se
nds Per Ton
veral Gears
R
in
oiling Resistance
Pounds Per Ton
25 to 40
High
2d
Low
Pavements
Gravel, macadam, good
earth.
1S5
400
600
50
65
Relative fuel consumption in ton—miles per gallon 1, high gear, 28; 2d gear, 20; low gear, 14
: i —;
Relative average spppfls rn thp spvrri'£T""rs J hirh ge°r 100; 2d gear. 85; low gear 25
The value of time saved is even less readily estimated for automo¬
biles than for commercial vehicles and no attempt is made herein to do
so. It is true that a very large amount of automobile traffic is occa¬
sioned by business necessity and time is an important factor. Likewise
the automobile is used extensively for pleasure. Until traffic census
data are available to show the relative amounts of the two classes of
automobile traffic it is futile to attempt to estimate the value of time
saving to automobile traffic. The analysis would follow the method
outlined for commercial vehicles.
Allowable expenditure for reducing minus grades for both commer¬
cial vehicles and automobiles. The maximum economical minus grades
for the composite commercial vehicles and for the composite automo¬
bile are shown in Fig. 7. These are based on equations (4) and (5).
If a vehicle must be controlled by the brake when coasting down a
grade, the energy absorbed by the brake is equal to that required to
raise the vehicle a vertical distance equal to the excess rise of the hill.
If the average total efficiency of the vehicle be assumed at 10 per cent
and the fuel loss due to excess grade be capitalized at 4 per cent, the
— 20 —
justifiable expenditure for reducing any grade to the economical max¬
imum can be computed in the following manner:
Let P^the average rate of grade of the existing hill in per cent.
Pm=the maximum economical minus grade in per cent.
Sm=the sum of money in dollars that may be expended for reduc¬
ing the grade.
Vt =total annual traffic descending the grade in millions of tons.
d1=the length of the existing hill in 100 foot stations.
C=the price of gasoline in dollars per gallon.
S„,=5200 CVt (Pi—Pm) dx,
(13)
If
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P„= 3f(Sf-S?)+ §0
e=40, S, =15, Sz=£5
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Length of Ornde in Hundreds of feet
70
Fig-. 7—Curve showing the limits for economy minus grades.
Summary
In the foregoing discussion there is presented a tentative economic
theory of highway grades. The value of the deductions depends upon
the sufficiency of the experimental data and the accuracy of the as¬
sumptions relative to the characteristics of the composite vehicles.
There has been analyzed a great mass of data bearing on the subject
and extensive field investigations have been in progress for 3 years,
but the subject involves so many variables that the results presented
herein can hardly be considered more than the first step in the estab-
— 21 —
Iishment of an adequate theory of highway grades. Amplifications and
refinements to the theory will be made by those interested as rapidly
as pertinent experimental data are available.
If the principles presented herein are considered in connection with
trunk-line highways, where the annual traffic may reach several million
tons, it will be apparent that the actual value of the fuel saved by
grade reduction may reach very significant sums.
The value of lost time due to excessive grades is an even larger sum,
where the volume of commercial vehicle traffic is large, and undoubt¬
edly a similar loss accrues to a considerable percentage of the automo¬
bile traffic. It is highly important to establish a basis for evaluating
lost time, but it is one of those elusive quantities which are difficult to
analyze.
It seems to have been established that momentum grades on rural
highways are economical both from the standpoint of fuel and time
and that, under certain circumstances, less fuel will be required on a
road with an undulating grade line than on one with very flat grades.
It is also shown that no economy results from the reduction of long
— 22 —
grades of less than about 3 per cent and that short grades may reach
8 per cent without adversely affecting either fuel consumption or av¬
erage speed.
Fuel cost is an item amounting to only 15 or 20 per cent of the cost
of operation of a motor vehicle, but a saving of one-tenth of the fuel
annually consumed in a state will amount to a large sum.
Almost invariably those things that lower fuel consumption also
lower maintenance costs for the vehicle and thus indirectly effect a
fuel saving.
One interesting fact brought out in the fuel consumption runs was
the marked saving in fuel that results from coasting down hill with the
motor declutched. This is a perfectly feasible way to drive an auto¬
mobile but may be dangerous with a commercial vehicle. It should not
be practiced with any kind of vehicle on long grades or where safety
considerations require the vehicle to be kept under control.
Safety, aesthetics and drainage considerations in connection with
grade reduction are outside the scope of this investigation, although
they are factors that must always be considered in connection with
highway improvement.
p?ori
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TLE OF MERLE. NAY PCM?
NORTH HILL. -SEPZS.50
BUCK TOUPING CAP
160
JUL
120
100
80
60
40
0.52L.
SUMMARY Of
CLUTCH—DIPTl
IN NORTH
IN SOUTH
OUT SOUTH
IN NORTH
"NXSAL,
150 STATIONS
160
<§0_
OAS CONSUMPTION
PER/00070N MILES
Jgj| A\A4ZOS
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SPEED
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HIPH PEAR BRAI
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Q4SOUNE
RUN NORTH
CLUTCH IN
AV&ALPEP TON MILE-00484
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PUN SOUTH
CEUTCH IN
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PER 70NMtLE-OOaSF
Fig. 11
OT MERLE HAY ROAp
x3ltth hill
series si
touring cab
SUMHABY Or OAS
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CONSUMPTION
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PUN SOUTH
CLUTCH UN
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AUGAL PCP TONMtLE - 0.Q36S
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AVGAL PCR TON MILE'0.0£4I
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— LZ —
— 82 —
6asounc consumption in gallons pc* ton mil c
cuwnoNs in rtcr
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— 31 —
— 28 —
CO
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Fig. ZI
— SS —
— 39 —
Fig. 27
— 41 —
— 43 —
— 9* —
Fig. .35
BULLETINS OF THE ENGINEERING EXPERIMENT STATION
*No. 1. The Iowa State College Sewage Disposal Plant Investigations.
♦No. 2. Bacteriological Investigations of the Iowa State College Sewage.
♦No. 3. Data of Iowa Sewage and Sewage Disposal.
♦No. 4. Bacteriological Investigations of the Iowa State College Sewage Disposal Plant.
♦No. 5. Chemical Composition of Sewage of the Iowa State College Sewage Disposal Plant.
♦No. 6. Tests of Iowa Common Brick.
♦No. 7. Sewage Disposal in Iowa.
♦No. 8. Tests of Dry Pressed Brick Used in Iowa.
♦No. 9. Notes on Steam Generation with Iowa Coal.
♦No. 10. Dredging by the Hydraulic Method.
♦No. 11. An Investigation of Some Iowa Sewage Disposal Systems.
♦Vol. II, No. 6. The Good Roads Problem in Iowa.
♦Vol. Ill, No. 1. Tests of Cement.
♦Vol. Ill, No. 2. State Railroad Taxation.
♦Vol. Ill, No. 3. Steam Generation with Iowa Coal.
♦Vol. Ill, No. 4. Incandescent Lamp Testing.
♦Vol. Ill, No. 5. Steam Pipe Covering Tests.
♦Vol. Ill, No. 6. The Assessment of Drainage Districts.
♦Vol. IV, No. 1. Tests of Iowa Limes.
♦Vol. IV. No. 2. Holding Power of Nails in Single Shear.
♦Vol. IV, No. 3. Miracle Contest Papers for 1908. (Theses on Cement and Concrete.)
♦Vol. IV, No. 4. Miracle Prize Papers for 1909. (Theses on Cement and Concrete.)
♦Vol. IV, No. 5. Sanitary Examination of Water Supplies.
♦Vol. IV, No. 6. Sewage Disposal Plants for Private Houses.
No. 25. Electric Power on the Farm.
No. 26. The Production of Excessive Hydrogen Sulfid in Sewage Disposal Plants and Conse¬
quent Disintegration of the Concrete.
No. 27. A Study of Iowa Population as Related to Industrial Conditions.
No. 28. History of Road Legislation in Iowa.
No. 29. Cost of Producing Power with Iowa Coals.
No. 30. The Determination of Internal Temperature Range in Concrete Arch Bridges.
♦No. 31. The Theory of Loads on Pipes in Ditches, and Tests of Cement and Clay Drain Tile
and Sewer Pipe.
No. 32. A Topographical Survey of the Spirit and Okoboji Lakes Region.
No. 33. House Heating Fuel Tests.
No. 84. The Use of Iowa Gravel for Concrete.
No. 85. The Iowa Engineering Experiment Station and its Service to Industries of the State.
No. 86. Report of the Investigations on Drain Tile of Committee C-6, American Society for
Testing Materials.
No. 87. Illuminating Power of Kerosenes.
No. 38. Electric Central Station Operation in Iowa.
No. 89. Good Roads and Community Life.
♦No. 40. An Investigation of Iowa Fire Clays.
No. 41. Sewage Disposal for Village and Rural Homes.
No. 42. A Study of Oil Engines in Iowa Power Plants.
No. 43. Practical Handling of Iowa Clays.
No. 44. Locomotive Tests with Iowa and Illinois Coals.
No. 45. Investigations of Gravel for Road Surfacing.
No. 46. Electric Pumping, with Results of Tests and Operating Records.
No. 47. The Supporting Strength of Sewer Pipe in Ditches, and Methods of Testing Sewer
Pipe in Laboratories to Determine their Ordinary Supporting Strength.
No. 48. The Early Purchase and Storage of Iowa Coal.
No. 49. An Investigation of Tests of Iowa Shale Drain Tile.
No. 50. The Theory of Underdrainage.
No. 51. Recommendations for Farm Drainage.
No. 52. The Spacing and Depths of Laterals in Iowa Underdrainage Systems and the Rate of
Runoff from Them.
No. 53. Load Concentrations on Steel Floor-Joists of Wood-Floor Highway Bridges.
No. 54. An Investigation of the Protective Values of Structural Steel Paints.
No. 55. Lighting for Country Homes and Village Communities.
No. 56. Traffic on Iowa Highways.
No. 57. Supporting Strength of Drain Tile and Sewer Pipe Under Different Pipe-Laying Con¬
ditions.
No. 58. Possibilities of Pottery Manufacture from Iowa Clays.
No. 59. Effects on Concrete of Immersion in Boiling Water and Oven Drying.
No. 60. Method of Proportioning Concrete Materials—Screened and Unscreened Gravel.
No. 61. Estimation of the Constituents of Portland Cement Concrete.
No. 62. Bacteria Fermenting Lactose and Their Significance in Water Analysis.
No. 63. Preliminary Impact Studios—Skunk River Bridge on the Lincoln Highway - near
Ames, Iowa. _ •
No. 64. Resistances to the Translation of Motor Vehicles.
No. 65. The Economics of Highway Grades.
♦Out of print.
Bulletins not out of print may be obtained free of charge upon request addressed to The
Director, Engineering Experiment Station, Sta. A, Ames, Iowa.
THE COLLEGE
The Iowa State College of Agriculture and Mechanic
Arts conducts work along five major lines:
AGRICULTURE
ENGINEERING
HOME ECONOMICS
INDUSTRIAL SCIENCE
VETERINARY MEDICINE
The Graduate College conducts advanced research and
instruction in all these five lines.
Four, five and six-year collegiate courses are offered in
different divisions of the College. Non-collegiate courses
are offered in agriculture, home economics and trades and
industries. Summer sessions include graduate, collegiate
and non-collegiate work. Short courses are offered in
the winter.
Extension courses are conducted at various points
throughout the state.
Research work is conducted in the Agricultural and
Engineering Experiment Stations and in the Veterinary
Research Laboratory.
Special announcements of the different branches of the
work are supplied, free of charge, on application. The
general bulletins will be sent on request.
Address, The Registrar,
IOWA STATE COLLEGE,
Ames, Iowa