I. INTRODUCTION
1.1. OBJECT AND SCOPE
Although the response of simplespan
highway bridges to the influence of moving
vehicles has been studied at some length (see,
for example, References 21 and 26), there has
been comparatively little attention given to
the behavior of continuous highway bridges.
A general theory for the analysis of
continuous bridges, together with an explor
atory study of the effects of the various
parameters, have been presented in Reference
18, two chapters of which have been reprinted
in the Appendix. In addition, a simplified
method of analysis and a limited number of
numerical solutions have been reported in
References 17 and 23. A detailed bibliography
of the contributions in this area can be found
in the Appendix, page 57. This bibliography
refers both to theoretical studies(3)(7)* and
to model tests and field tests. (1) (8) (10) (11)
(13) Although these studies have contributed
a good deal of useful information, they have
not been as comprehensive and conclusive as
would be desired.
The object of the investigation re
ported here was twofold:
(a) To obtain information which will
lead to a better understanding
of the dynamic behavior of three
span continuous bridges under
moving vehicles, and
Superscripts refer to items in the Refer
ence section.
(b) To develop concepts and simple
approximate relations that may
be used to estimate the magni
tude of the dynamic effects that
may be produced under prescribed
conditions.
This study which parallels the one re
ported recently for simplespan bridges,(26)
was limited to a consideration of threespan
continuous bridges composed of steel girders
and concrete deck. The study was based on the
method of analysis and computer program re
ported in Reference 18. In this method, the
bridge is idealized as a single continuous
beam, and the vehicle is represented as a
multiaxle sprung load which incorporates the
effect of the interleaf friction in the sus
pension springs. For the details of the
method, the reader is referred to the original
source, for convenience, two chapters of
Reference 18 describing the development of the
theory are included in the Appendix.
An attempt has been made to study in a
systematic manner the effect of the numerous
parameters that influence the response of the
bridge. Primary emphasis was placed on the
effect of initially oscillating twoaxle
vehicles. However, the singleaxle represen
tation was also included as an aid in inter
preting the solutions obtained for the more
realistic but more involved twoaxle represen
tation. The effects of a moving constant
force and two moving constant forces were also
investigated to define the conditions under
which these relatively simple solutions are
applicable to the highway bridge problem and
to provide a convenient frame of reference
for interpreting the solutions obtained for
the more realistic representations of the
problem.
The investigation included the follow
ing basic steps:
(a) A study of the characteristics
of existing threespan highway
bridges to establish the realis
tic range of the various para
meters that influence the
response, and
(b) The compilation of a large
number of numerical solutions
for a range of the parameters
and the interpretation of these
solutions.
A total of approximately 800 solutions
were obtained and analyzed for this purpose.
The principal effort in the interpretation of
these solutions was to assess the relative
importance of the various parameters and to
retain in the final analysis only the most
significant ones.
It is shown that the effect of such
uncontrollable parameters as the initial
phase of the vertical oscillation of the
vehicle, the phase difference between the
motions of the individual axles and the
initial values of the interleaf friction in
the suspension system of the vehicle are
quite significant. However, these are not
factors that can possibly be incorporated in
any design provision.
Chapter I I is devoted to a study of the
characteristics of existing bridges and
vehicles. Chapter III discusses the magnitude
of the maximum static effects. In Chapters IV
and V the dynamic effects induced by constant
moving forces and smoothly moving vehicles are
studied. In Chapter VI the results obtained
for vehicles with small amplitudes of initial
oscillation are reported, and in Chapter VII
are given the results for vehicles with large
amplitudes of initial oscillation. A brief
summary of some of the major findings of this
effort is given in Chapter VIII.
Throughout this study the words "bridge"
and "beam" are used interchangeably.
* 0
II. CHARACTERISTICS OF BRIDGES AND VEHICLES
2.1. GENERAL
This chapter presents information on
the characteristics of threespan continuous
bridges and of heavy commercial vehicles. The
parameters required to define the properties
of the bridgevehicle system (see Appendix,
page 42) can be conveniently classified into
the following three groups:
(a) Parameters related to the bridge
(b) Parameters related to the vehicle
(c) Dimensionless parameters expressing
the relationship between corres
ponding characteristics of the
bridge and the vehicle.
2.2. CHARACTERISTICS OF THREESPAN CONTINUOUS
BRIDGES
2.2.1. General
The sources of information used in this
study include the Standard Plans for Highway
Bridge Superstructures of the U. S. Bureau of
Public Roads (hereafter abbreviated as B.P.R.),
(9)
the Design Charts for IBeam Highway
Bridges of the State of Illinois, (28) and
assorted information on threespan continuous
bridges on which field tests have been per
formed and reported. The latter bridges were
designed by five different states and are
described in References 1, 8, 10, 11, and 13.
The study is restricted to bridges of
the Ibeam type composed of steel girders and
a reinforced concrete deck. Most of the
bridges studied had a prismatic crosssection
and equal side spans, and thus conformed to
the limitations of the computer program
available for this study.(18) However, the
results of this study may also be applied to
other bridge types for which the values of the
fundamental parameters are within the range
considered in this investigation.
The natural frequencies of the contin
uous beam used to approximate the actual
bridge can be expressed in the form
J 2
E l
(2.1)
where c is the circular natural frequency in
radians per second, El is the flexural rigidity
of the crosssection of the beam, m is its
mass per unit of length, L is the length of
the central span, A is a dimensionless para
meter which depends on the ratio of the side
span to the center span, and the subscript j
is an integer denoting the order of the
natural frequency under consideration. The
natural frequency in cycles per second, (f ) ,
is given by the equation
b j
(f j  2
(2.2)
The values of ?\. corresponding to the
first three natural frequencies of threespan
continuous beams of uniform crosssection and
equal side spans are plotted in Figure 2.1 as
a function of the ratio of side span length to
center span length, a, hereafter referred to
simply as span ratio. Also included in this
figure are sketches of the corresponding modes
of vibration. These data were evaluated by
application of the method described in
Reference 14.
2.2.2. Bureau of Public Roads Bridges
These bridges are of the Ibeam type
and are designed either for H1544 loading or
for H20S1644 loading. The lengths of the
individual spans are in the ratio of 4:5:4,
and the overall length ranges from 130 feet
to 260 feet. Thus the shortest bridge has
spans of 40 feet:50 feet:40 feet, and the
longest bridge has spans of 80 feet:100 feet:
80 feet. The bridges designed for the H1544
loading have a roadway width of 24 feet, a
concrete slab 61/4 inches thick, and curbs
2 feet 6 3/4 inches wide and 101/4 inches
high. The entire deck is supported on four
wideflange steel girders. For the bridges
designed for the H20S1644 loading, the road
way width is 28 feet, the slab thickness is
6 inches, the curbs are 3 feet 3/4 inch wide
and 101/4 inches high, and a total of five
wideflange steel girders are used to support
the deck. The crosssections of both bridge
types are shown in Figure 2.2. The single
beam used to represent the bridge in the
analysis is further idealized in the manner
shown in Figure 2.3. For a more detailed
description of the mathematical model used,
see the Appendix, page 44.
The weight per unit of length of the
bridges was determined from the value of the
dead load reaction specified in the Standard
Plans. The results are tabulated in column 2
of Table 2.1 and are also displayed graphical
ly in Figure 2.4. This quantity was also
evaluated by computing the weight per unit
length of an interior girder and of its
tributary slab and multiplying the result by
the number of girders. Because of the pres
ence of heavy overhanging curbs in these
bridges, the unit weight computed by the
second method was found to be approximately
83 per cent of that determined by the first
method.
The fundamental frequency of vibration
of each bridge was computed for both composite
and noncomposite action assuming the bridge
to behave as a single beam of uniform cross
section and uniform mass per unit length. The
roadway surface was considered to be horizon
tal and smooth. The results are presented in
columns 4 and 6 of Table 2.1, and displayed
graphically in Figure 2.5. For composite
action, the moment of inertia used was that of
the entire cross section of the bridge at mid
span. The modular ratio was taken as n = 10.
For noncomposite action, the moment of iner
tia used was the sum of the corresponding
quantities for the beams, slab, and curbs of
the bridge at midspan, each taken with respect
to its own centroidal axis. This approach
neglects the continuity between the slab and
the curbs.
The span ratio, a, (i.e. the ratio of
lengths of the side span and center span) is
0.8 in all B.P.R. Standard Bridges. For this
value of a, the value of N for the fundamental
mode of vibration is A = 3.53, and the funda
mental period is
2 F
= 2trL m
b =2 El
(3.53)
(2.3)
Several studies were made to assess the
sensitivity of Tb to the approximations and
assumptions made in the evaluation of the
various parameters entering into Equation 2.3.
It was found that if the weight of the hand
rail and curbs is disregarded, the natural
period would decrease roughly by 10 per cent.
If the slab thickness were decreased by
1/2 inch, the reduction in the period would be
of the order of 8 per cent. When considering
composite action between the slab and the
girders, if the modular ratio is taken as 7
instead of as 10, the natural period would
increase roughly by 8 per cent. If non
composite action is considered, the period
would increase significantly, especially for
bridges with short spans. For instance, for
a 40 feet:50 feet:40 feet bridge designed for
H20S1644 loading, the period increases from
0.15 second for composite action to 0.24
second for noncomposite action.
It should be noted that the effect of
the coverplates over the piers was not con
sidered in the results presented. However,
since the slab is likely to be partially
cracked over the piers, neglecting the cover
plates tends to compensate for the reduction
in stiffness resulting from cracking of the
slab.
2.2.3. Illinois Bridges
The bridges considered in this section
are threespan continuous Ibeam bridges
proportioned on the basis of the design charts
used by the State of Illinois. These charts
give the beam size required for a given slab
thickness as a function of the length of the
side span. They allow consideration of beam
spacings from 5 feet to 7 feet in increments
of 6 inches.
The beam size required in a given case
is considered to be the same for both the
Hl544 and the H20S1644 loadings. For
bridges with a central span length equal to or
greater than 65 feet, the slab thickness is
taken as 7 inches irrespective of the design
load involved. However, for values of L
smaller than 65 feet, the slab thickness is a
function of the design load. For the H20S16
44 loading, a slab thickness of 7 inches is
used, whereas for the H1544 loading, slab
thicknesses of 61/2 inches and 63/4 inches
are used depending on whether the beam spacing
is smaller or greater than 6 feet, respective
ly. These slab thickness include a 1/2inch
allowance for wearing surface.
The bridges considered in the design
charts range in length from 94.8 feet to 333.0
feet. The span ratio a equals 0.8547 for
bridges with a center span length L smaller
than 80 feet, and 0.7806 for bridges with a
value of L equal to or greater than 80 feet.
In the computations summarized here, these
ratios were taken as 0.85 and 0.78,
respectively.
Although for bridges with L > 80 feet
coverplates are used over the intermediate
supports, their effect was neglected in the
computation of stiffnesses but was considered
in the computation of the weight. This weight
was assumed to be uniformly distributed along
the bridge. The weight per linear foot of the
bridge was computed on the basis of the weight
of a beam plus its tributary slab times the
number of beams. The results are presented in
column 2 of Table 2.2, and plotted in Figure
2.4.
The fundamental frequency of vibration
was computed for both composite and non
composite action using a single beam and its
tributary slab. The results are given in
columns 4 and 6 of Table 2.2, and in Figure
2.5.
The computation of the properties of
the Illinois bridges was made for all the
allowable values of beam spacing mentioned
above; however, the results given in this
section are the average of the results obtain
ed for the five values of beam spacing
considered.
2.2.4. Other Bridges
These bridges were also of the Ibeam
type, designed either for an H1544 or
H20S1644 loading. The information on these
bridges was obtained from reports of field
tests performed by several state highway de
partments and universities (References 1, 8,
10, 11 and 13). Their lengths range from
85.6 feet to 402 feet, and the span ratios
range from 0.527, in the case of one Michigan
bridge, to 0.984 for a bridge in Ohio. In
most cases no information was available con
cerning the width of the roadway and the
dimensions of the curbs, but in general the
weight per unit length of bridge and the
moment of inertia of the cross section for
composite action were given. The measured or
computed values of the fundamental frequency
of vibration of the bridges were also reported
in most cases, and these data are given in
columns 4 and 5 of Table 2.3. Where suffi
cient information was available, the funda
mental natural frequency of these bridges was
also evaluated by the procedure used for the
B.P.R. Standard Bridges considering both com
posite action and noncomposite action. This
information is summarized in columns 6 and 7
of Table 2.3.
In some of the cases reported, infor
mation on the amount of bridge damping as a
percentage of critical was also given. This
information is presented in column 8 of
Table 2.3.
2.3. CHARACTERISTICS OF COMMERCIAL VEHICLES
2.3.1. General
Studies of the geometric and dynamic
characteristics of heavy commercial trucks
tractorsemitrailer combinations have been
ported previously in Reference 18, based
essentially on data obtained from truck manu
facturers and on a report of the Society of
Automotive Engineers.(27) Additional
information was obtained from tests performed
by the Texas Agricultural and Mechanical
(12)
College and in the course of the AASHO
Road Test Project in Ottawa, Illinois.(25)
The pertinent information has been summarized
in Reference 26.
The characteristics of the vehicles can
conveniently be classified as follows:
(a) Geometric characteristics, namely
the dimensions of the vehicle and
the distribution of the weight
among the axles.
(b) Dynamic characteristics, such as
natural frequencies of vibration
and coefficients of interleaf
friction for the axles.
(c) Initial conditions of each axle at
the time the vehicle enters the
bridge.
Throughout this study, the word
"initial," refers to the instant the front
axle of the vehicle crosses the first abutment
of the bridge.
2.3.2. Geometric Characterisitcs of Vehicles
The most important vehicle characteris
tics are the axle spacing and the distribution
of the total vehicle weight among the axles.
This study is limited to a consideration of
loads having one or two axles, as shown sche
matically in Part A of Figure 2.6. The
singleaxle loading is obviously used for
simplicity. The twoaxle loading simulates
either the two axles of a heavy truck, or the
drive and rear axles of a tractorsemitrailer
combination. The loads on the two axles were
considered to be equal. Since the weight of
the front axle of the tractor is usually less
than about 1/4 of the weight of each of the
other axles, it can generally be neglected.
Each axle is presented by two springs
in series and a frictional device in parallel
with the upper spring. The two springs simu
late the tire and the suspension springs of
the vehicle. The symbols kt and k denote the
stiffness of the tires and of the suspension
springs, respectively. The stiffness of the
combined tire suspension system, that is of
the two springs acting in series, is denoted
by kts and is related to kt and ks by
vehicle were taken as
1 1 1
kt k1 k
ts t s
(2.4)
ft = 3.5 cps
f = 2.1 cps
The frictional device simulates the
friction that exists between the leave of the
suspension springs.
The load deformation relationship for
a singleaxle loading under increasing static
load has been described in detail in
Reference 19, and will not be repeated here.
However, the diagrams given in Part B of
Figure 2.6 are reproduced from Reference 19
for concurrent reference. The quantity P de
notes the force exerted by an axle on its
surface of support, and F denotes the fric
tional force in the suspension springs.
For the purpose of this study a
"standard" vehicle was defined. It consists
of a twoaxle vehicle with two equal axle
loads which may attain a maximum value of 32
kips each. The spacing between the axles may
vary from 14 to 35 feet, except for the
special case of a singleaxle load, for which
the spacing is zero. It may be noticed that
these values of load and spacing resemble very
closely the last two axles of the design load
H20S1644 of the AASHO.(16) Additional
characteristics of the "standard" vehicle are
given in the following sections.
2.3.3. Dynamic Characteristics of the Vehicles
From the data referred to in the pre
ceding subsection, average values of natural
frequencies of oscillation were obtained for
each axle, assuming the effective mass on the
axle to vibrate either on the tire springs,
with the suspension springs blocked, or on the
combined tiresuspension system with the two
springs acting in series. These frequencies
are denoted by ft and f ts, respectively.
Throughout this study, the natural frequencies
of oscillation for each axle of the "standard"
These are average values for the available
data. It may be noted that
f t /k 1/2
ts t) 0.6
t kt /
For the majority of the problems con
sidered, the limiting value of the frictional
force F' for each axle of the "standard"
vehicle was taken as 15 per cent of the static
reaction on that axle. The ratio F'/Pst is
known as the coefficient of interleaf friction
and will be denoted by the symbol ip.
Another important parameter of the
vehicle is the dynamic index, i, which is a
measure of the rotary inertia of the vehicle.
In the absence of specific information on
this parameter, a value of i=l was used
throughout this study. For this value of i,
the bounce and pitching frequencies of
vibration of the twoaxle load unit are equal
to the frequency of each axle.
There is assumed to be no damping of
viscous type either in the tires or in the
suspension system.
2.3.4. Initial Conditions of Vehicle
It is seldom that a vehicle enters the
bridge with its suspended mass in equilibrium
for vertical motion. It is therefore neces
sary to investigate the effect of initially
oscillating vehicles. The intial oscillation
of the vehicle may arise either from the ever
present irregularities of the approach pavement
or from a sharp discontinuity between the
approach pavement and the entrance of the
bridge.
The method of analysis assumes that,
while the vehicle is on the approach pavement,
the vertical oscillation of each axle is of
Equation 2.4.
8
the simple harmonic type. The initial value
.th
of the interacting force P. between the i
axle and the pavement is expressed as
P. = (1 + Ci cose.) Pst
SI I st
hence the amplitude of the variation of the
(2.5) interacting force can be expressed as
where C. is the amplitude of the initial force
variation, 0. is the phase angle between the
time at which the force attains its maximum
value and the time at which the front axle
enters the bridge, and P . is the static
st,i
reaction on the axle. For a twoaxle loading,
the phase angles eI and 02 will in general
be different. The quantity 8O = 0I  02 will
be referred to as the phase difference of the
vehicle.
In order to specify the initial motion
of the vehicle it is also necessary to con
sider the initial value of the interleaf
frictional force for each axle and the
limiting value of this force.
The values of C. and 0. depend upon
such variables as the dimensions and location
of the irregularities and/or discontinuities,
and upon the speed of the vehicle. The
initial value of the frictional force depends,
in addition, on the past history of the de
formation, and may have any value between F'
and F'.
It can be shown that for the case of a
singleaxle loading with a linearly elastic
loaddeformation relationship the maximum
deformation experienced by the supporting
spring when the moving vehicle encounters a
sharp discontinuity on its path is equal to,
or greater than, the height of the discontinu
ity, regardless of the shape of the leading
front of the discontinuity. (15) If the de
formation of the spring is assumed to be equal
to the height of the discontinuity, the varia
tion in the interacting force between the
vehicle and the floor is
AP = k * h
Inasmuch as
C becomes
C = P = h
P A
st st
A =   , the expression for
S42f 2 '
h4x2f 2
C = v
g
(2.6)
from which the height h of a discontinuity
required to produce a given variation in
interacting force can be obtained from the
expression
h = 9.86 C/f 2
(2.7)
For example, for a vehicle with an axle
frequency of 3.5 cps, the height of discon
tinuity required to produce a variation of
interacting force of 0.15 Pst is only 0.12
inch. To obtain a value of C = 0.50P , it is
obvious that a larger height of discontinuity
is required. However, for this larger varia
tion of interacting force, the limiting
frictional force in the suspension springs,
will.be exceeded and the effective frequency
of oscillation of the vehicle will change from
3.5 cps to 2.1 cps. Using the latter value of
f in Equation 2.7 the value of h for C =
0.50Pst becomes 1.12 inches.
In this study, the following two values
of C. were adopted as "standard": 0.15 and
0.50. The value of 0.50 is considered to
define the effect of a fairly large disconti
nuity at the abutment, or of a large irregu
larity on the approach pavement located close
to the entrance of the bridge. The value of
Ci = 0.15, on the other hand, is considered to
st
k s
st
be representative of a vehicle for which the
initial oscillation is due to the roughness of
the approach pavement.
The phase angle, e., in the "standard"
vehicle was allowed to vary between the
limits of 0° and 3600. The initial value of
interleaf friction was assigned the values of
F', zero, and F'.
2.4 BRIDGEVEHICLE RELATIONSHIPS
2.4.1. Parameters of Problem
A listing of the dimensionless param
eters required to define completely the re
sponse of the bridgevehicle system is
presented in the Appendix, page 55. In this
section only the most important of these
parameters will be given. These include
(a) The speed parameter, a, defined as
a = v Tb/2L
where v is the speed of the vehicle, Tb is
the fundamental natural period of vibration
of the bridge, and L is the length of the
center span. The speed of the vehicle is
assumed to be constant while the vehicle
crosses the bridge.
(b) The weight ratio, R, defined as
R = W/Wb
where W is the total weight of the vehicle,
and Wb is the weight of the center span of
the bridge.
(c) The frequency ratios
cpt = ft/fb and Cts = f ts/fb
where ft is the natural frequency of the
vehicle vibrating on its tires, fts is the
natural frequency when the vehicle vibrates
on its tiresuspension system, and fb is the
fundamental natural frequency of the bridge.
(d) The axlespacing parameter is s/L
where s is the axle spacing of a twoaxle
loading, and L is the length of the center
span.
Throughout this study, the bridge sur
face is considered to be smooth and horizon
tal, and all the numerical solutions were
obtained using 600 steps of integration for a
singleaxle loading and a proportionally
greater number for a twoaxle loading.
2.4.2. Practical Range of Parameters
2.4.2.1a. Speed Parameter
The vertical and horizontal axes of the
diagram shown in Figure 2.7 show the relation
ship between the fundamental period of
vibration and the center span length for the
various bridges considered in this study. The
remaining axes form a nomogram for the
determination of the speed parameter C. The
use of this nomogram is illustrated in the
lower part of the figure. Note that the
vehicle speed is expressed in miles per hour,
whereas the length of the center span, L, is
expressed in feet.
For the bridges considered and a vehi
cle speed of 70 mph, the value of a can be
shown to vary from 0.13 to 0.20 for bridges
with composite action and from 0.21 to 0.27
for bridges with noncomposite action. In the
numerical solutions to be presented, a is
considered to vary from 0.06 to a maximum
value of 0.24.
2.4.2.2b. Weight and Frequency Ratios
These ratios were evaluated for the
"standard" vehicle, for which W = 64 kips and
f = 3.5 cps, and for the bridges considered
in the second section of this chapter. The
weight ratio, R, was found to range between
0.10 and 0.54, the extreme values being
somewhat rare. In the majority of the
numerical solutions presented, the value of R
was taken between 0.175 and 0.350.
For bridges with composite action, the
value of Vt was found to range between 0.3 and
1.8, and for noncomposite action, between
0.5 and 2.3.
The interrelationship between Tt and R
is shown in Figure 2.8. The following
equations were determined using the method of
least squares.
For composite action: cp = 0.149 R'06
(2.8)
For noncomposite action: tP = 0.265 R094
(2.9)
2.4.2.3c. Axle Spacing Parameter
Since s varies between 14 and 35 feet
for a twoaxle loading and L varies from 35 to
130 feet for the bridges studied, the possible
range of variation of the axle spacing param
eter s/L is
* 0.1 < s/L < 1.0
However, inasmuch as bridges with span lengths
of the order of 35 feet are not very common,
the larger ratios are not encountered fre
quently. In this study, s/L was varied from
0 to 0.5.
2.4.3. Summary
The graphs in Figure 2.9 summarize the
range of the four parameters referred to
above. The "standard" vehicle defined pre
viously was considered, and the bridge
properties were averaged for all threespan
continuous bridges having the same length of
center span and a span ratio, a, close to 0.8.
The speed of the vehicle was taken as 70 mph.
The graphs are plotted as a function
of the length of center span, and it can be
seen that, except for the extreme points in
each curve, the ranges of parameters adopted
throughout this study encompass all the
values encountered in practice. *
III. RESPONSE OF BEAMS UNDER STATIC CONDITIONS
3.1. GENERAL
As in previous studies (l8a, , it
was found to be convenient and desirable to
express the dynamic response of the beam in
two parts: (a) a crawl or static component
representing the effect of the vehicle moving
slowly across the span, and (b) a dynamic
increment component representing the differ
ence between the total effect and the
corresponding crawl effect. It becomes
necessary, therefore, to investigate first
the behavior of the beam under slowly moving
loads.
For a prismatic beam traversed by two
forces of equal magnitude, the parameters that
influence the static response of the beam are:
(a) The axle spacing parameter s/L and
(b) The ratio of the side span length
to the center span length of the
beam a.
The effect of these parameters was investiga
ted by determining influence lines for moment
and shear for several sections along the
length of the beam and for several combina
tions of s/L and a. This study was limited to
two loads of equal magnitude. The axle
spacing s was varied from zero (oneaxle load)
to O.5L in increments of O.IL. The values of
a considered were 0.6, 0.8 and 1.0. The
various effects were evaluated at sections
spaced at intervals of O.lL along the span.
A portion of these data was obtained from
(2)
existing tables , whereas the remainder was
evaluated on an IBM 650 computer.
3.2. PRESENTATION OF DATA
3.2.1. Effect of Axle Spacing
Figures 3.1 through 3.3 present in
fluence lines for moment at three sections of
a threespan continuous beam of prismatic
section and a sidespan ratio a = 0.8. The
sections considered are at the center of the
center span, over the second interior support,
and at a distance of 0.4225 aL from the right
hand abutment. The latter section defines the
position for which a single concentrated load
produces the absolute maximum moment in the
righthand span. The moments at these
sections are denoted by Mc, M3, and M4'
respectively, and the moments at a distance
of 0.4225 aL from the lefthand abutment and
over the first interior support are denoted
by MI and M2, respectively. (These sections
are shown on the sketch in the upper right
hand corner of Figure 3.6) Because of the
symmetry of the structure and the symmetry of
the load, the influence lines for MI and M2
are the reflected images of the influence
lines for M4 and M3, provided the reflected
curves are shifted horizontally so that they
start at the lefthand abutment.
It should be noted in these figures
that, as the axle spacing parameter increases,
the magnitude of the maximum moment decreases,
and the portions of the curves where the
ordinates are large become flatter. This
trend is more pronounced in Figures 3.1 and
3.3, where positive moments are considered.
Figures 3.4 and 3.5 represent
influence lines for shear at sections to the
right of the lefthand abutment and to the
right of the first interior support,
respectively. These shears are denoted by SI
and S 2, respectively. It can be seen that an
increase in the axle spacing parameter de
creases the magnitude of the maximum shear at
a given section.
In Figure 3.6 the maximum value of the
moments Mi, M2, Mc, M3, and M4 are plotted as
a function of the axle spacing parameter s/L.
Each of these moments has been normalized with
respect to the corresponding moment produced
by a singleaxle load of the same total
weight. The latter moments are identified by
the subscript o and their values are given at
the upper lefthand corner of the figure. It
can be seen that the negative moments over the
supports are considerably less sensitive to
variations in s/L than the positive moments.
The curves in Figure 3.6 can be approx
imated by the following empirical expressions:
For M1 (and M4),
= 1  2 (s/L) f (s/L)2
1 = 0.5
1 o
"I ="
For M2 (and M3),
H2 5 2
M = 1   (s/L)
2 0
0 < s/L < 0.5
s/L > 0.5
0 < s/L < 0.4
MH 3
T2 _ = 0.92  3 (0.7  s/L)2 0.4 < s/L < 0.8
M 2) 02
For Mc,
M
o = 0.42 + 0.9(0.8  s/L)2 0 < s/L < 0.8
c o
Corresponding curves for maximum value
of the four reactions and of the shear at the
extreme left end of each span are shown in
Figure 3.7 as a function of the axle spacing
parameter. The ordinates for each curve are
normalized with respect to the corresponding
shear produced by a singleaxle load of the
same total weight. The sketch at the upper
righthand corner of the figure identifies
the four reactions, and the sections at which
the shears are considered. It can be seen
that all curves descrease with increasing axle
spacing and that the end reactions are the
most sensitive to variations in s/L.
The horizontal bars in Figure 3.8
define the "zones of influence" for the
moments MI through M 4. This term denotes the
position of the twoaxle load for which the
maximum value of an effect at a given section
is not smaller than 75 per cent of the
absolute maximum value of that effect. The
"zones of influence" were determined from
Figures 3.1 through 3.3 and from corresponding
figures for other values of s/L. These zones
are not symmetric with respect to the section
under consideration because the position of
the twoaxle load in this figure is expressed
in terms of the position of the front axle
rather than the position of the center of
gravity of the two loads. It can be seen that
these zones of influence, the significance of
which will be discussed in subsequent sections,
are a function of the axle spacing parameter
and that they increase with increasing value
of this parameter.
The curves in Figure 3.9 give the
maximum positive and maximum negative values
of the bending moment at all sections along
the span for four different values of the axle
spacing parameter s/L. The span ratio of the
beam, a, is taken as 0.8. These curves are
essentially the envelopes of sets of curves
similar to those given in Figures 3.1 through 3.3.
M
(M I
It can be seen that the greater the
axle spacing parameter, the flatter the
envelope curves are in the region where their
ordinates are maximum. For example, for s/L =
0, the length of the portion of the center
span where the positive bending moment is
greater than 95 per cent, the maximum positive
moment is of the order of 1/8 L, whereas for
s/L = 0.5, the corresponding length is of the
order of 1/3 L. This trend, which is most
pronounced in the case of positive moments is
of important consequence in the interpreta
tion of the dynamic response of the system.
It has been shown 24) that the peak
value of the dynamic increment for an effect
at a given section of the beam, when normal
ized with respect to the corresponding
maximum static effect, may be considered to
be constant for some length of the bridge on
either side of that section. If, in addition,
the static effect within that region can be
considered to be constant, then the maximum
dynamic effect in that region will be equal
to the sum of the maximum static effect and
the maximum value of the dynamic increment
for any section within that region. This
approach, which leads to an upper bound
estimate of the dynamic effect at the section
under consideration, is discussed in greater
detail in subsequent chapters.
It should also be noted that the
flattening trend of the curves observed for
the maximum positive moments does not apply in
the case of negative moments. In the latter
case the curves remain fairly sharp, although
the maximum static value decreases somewhat
with increasing values of s/L.
In Figure 3.10, corresponding curves
for the maximum absolute value of the shear
at any section along the span are plotted for
several values of s/L. No distinction is made
here between positive and negative shears.
Although the maximum values of the shear
decrease with increasing value of s/L, the
curves remain essentially parallel to each
other.
It should be noted that all the results
presented in this subsection have been ob
tained for span ratio a = 0.8. The data
presented in Figures 3.6, 3.7, 3.9, and 3.10
are tabulated together with additional data
in Tables 3.1 through 3.4.
3.2.2. Effect of Span Ratio
The effect of the span ratio was
studied by obtaining plots similar to those
given in Figure 3.9 for two additional span
ratios a = 0.6 and 1.0. These results are
summarized in Figure 3.11 and tabulated in
Table 3.5.
The lefthand side of Figure 3.11 gives
the maximum positive and maximum negative
values of the bending moment at all sections
along the span for three different bridges
with the same overall length but different
values of span ratio a due to a singleaxle
load. Corresponding curves for a twoaxle
load with a value of s/L = 0.3 are given at
the righthand side of the figure. In each
case, the quantity L denotes the length of the
center span of the bridge under consideration.
The ordinates of the curves are expressed in
terms of WLo.8, where W is the total weight
of the twoaxle load, and LO.8 is the length
of the center span of the bridge with a span
ratio of 0.8. Thus the ordinates of the
curves can be compared directly. Similar
curves were obtained for other values of the
axle spacing parameter.
It should be noticed that for the case
of a = 0.6, the absolute maximum value of
the moment in the first span is much smaller
than the maximum moment in the center span,
whereas for a = 1.0 the inverse is true.
Both for positive and for negative moments,
and for singleaxle and twoaxle loads, the
curve for a = 0.8 seems to average the
curves obtained for the other two values of
the span ratio.
In the remainder of this report, major
emphasis will be placed on the behavior of
threespan continuous bridges with a side
span ratio a = 0.8. The latter value leads
to a nearly balanced design for positive mo
ments in all the spans, and gives the smallest
value of negative moments over the intermedi
ate supports. As noted in the previous
chapter, this ratio of sidetocenter span is
the most commonly used in practice.
* * *0
IV. RESPONSE TO CONSTANT MOVING FORCES
4.1. GENERAL
Previous studies(26) have shown that
under certain conditions the variations of
the forces exerted by the vehicle on the
bridge surface, the socalled interacting
forces, are quite small. Consequently, these
forces may, for all practical purposes, be
considered to be constant. It is desirable,
therefore, to investigate first the response
of the beam to the action of one or two moving
forces of constant magnitude.
The response of the beam in this case
depends on the sidespan ratio a, the axle
spacing parameter s/L, and the speed param
eter of the system a, defined by the equation
v Tb
2L
in which v denotes the speed of the vehicle in
feet per second, Tb denotes the fundamental
natural period of vibration of the beam in
seconds, and L denotes the length of the
center span in feet.
Consideration will first be given to
the response of a uniform beam with three
equal spans, because it is possible to obtain
a relatively simple approximate solution to
this case. The results of the analysis of
this simple case provide considerable insight
into the behavior of the system and suggest
certain approximate relations for estimating
the magnitude of the maximum effects in the
beam.
4.2. RESPONSE OF BEAM WITH EQUAL SPANS
4.2.1. Approximate Analytical Solution
In Reference 3 an approximate series
solution is given for the dynamic response of
a uniform threespan continuous beam traversed
by a force of constant magnitude W moving at a
constant speed. In this solution only the
contributions of the sinusoidal modes of
oscillation, i.e. 1st, 4th, 7th, etc. are
considered.
In the following discussion this
solution will further be approximated by re
taining only the first term of the series.
Then the expression for the deflections for
the time that the force is on the beam becomes
2 WL3 _ t
y(x,t) = s in 2t T
3 4 El 1  a 2 Tb
a  sin 21 t  sin 
1  a b L
(4.1)
where y(x,t) denotes the deflection at time t
for a section located at a distance x from the
lefthand abutment, and El is the flexural
rigidity of the cross section of the beam.
Time is measured from the instant the force
crosses the left abutment.
In Equation 4.1 the quantity
2 WL3 t . x
  sin 2tn  sin 
3 4ElI b
represents the first term approximation to
the static deflection at a time t for an
arbitrary section x. The deflection at mid
span is
2 WL3sin 2n t
3 4El T b
The exact value
be shown to be
(yst )ct = 1.12
of the latter deflection can
WL t
 sin 21tU  (4.
n El b
2)
where the symbol c denotes the center, or mid
span, of the beam. The sine term may also be
vt
written in the form sin t .
For deflections at midspan, we shall
now modify Equation 4.1 as follows:
3
WL2 1 t
Yc t = 1.12 E 2 sin 20t b
, A4El 1_2 b
S  4 sin 2x  (4.3)
3 4El 12 Tb
In effect, we are replacing the first term
approximation to the static deflection by the
exact value given in Equation 4.2. Note,
however, that this change is made only for
the first term of Equation 4.1. The second
term representing the oscillatory component
of the response is left unchanged.
If Equation 4.2 is subtracted from the
two sides of Equation 4.3 and the resulting
expression is normalized with respect to the,
exact value of the maximum static deflection
at midspan, as given by Equation 4.2, the
following expression for the dimensionless
dynamic increment is obtained.
D 2 T
(D.I.)D = a2 sin 2 t5 
c 1  a b
 0.595 a sin 2x L (4.4)
 2 Tb
The subscript D identifies the deflection at
the center of the center span.
The dynamic increment consists of two
components. The first component, correspond
ing to the first term on the righthand side
of Equation 4.4, represents a series of three
halfsine waves with an amplitude a 2/(1  a 2)
and a wave length L. The second component,
corresponding to the second term in
Equation 4.4, represents a sinusoidal oscilla
tion with a period equal to the fundamental
period of vibration of the beam and an
amplitude of 0.595ca/(l  a ). In other words,
the curve for (D.l.)Dc oscillates about the
sinusoidal curve represented by the first term
in Equation 4.4, instead of about the zero
base line. It should be noted that, unlike
the corresponding expression for a simply
supported beam, 26) Equation 4.4 is not
independent of the section under consideration.
Specifically, the coefficient of the second
term in the equation will change when the
section under consideration is changed.
Proceeding in a similar manner, the
modified firstterm approximation to the
dynamic increment for moment at midspan can
be shown to be given by the equation
a2 t
(D.I.)M = 2 sin 2t T
c 1  a b
5 t
 0.386 a sin 2x 
1  a2 Tb
(4.5)
The corresponding expression for
dynamic increments of deflection at a point
located at a distance 0.42L from the lefthand
abutment is
(D.I.) f(x)
DI I  a2
(4.6)
 0.455 a sin 2n t
1  a2 Tb
where f(x) represents the crawl curve for
deflection at D1.
4.2.2. Comparison of the Approximate and
Exact Solutions
In Figure 4.1 the time histories of
dynamic increments for deflection and moment
at the center of the center span, as deter
mined from Equations 4.4 and 4.5 are compared
with the "exact" solutions evaluated on the
ILLIAC by use of the program referred to
previously. These solutions are for a value
of the speed parameter a = 0.15. Also shown
in dashed lines are the curves about which
the dynamic increment curves oscillate. It
can be seen that the agreement between the
approximate and exact solutions is quite good,
particularly for deflections, for which the
effect of the higher modes is less signifi
cant than for moments.
A similar comparison is shown in
Figure 4.2 for the dynamic increments for
deflection D in the first span. As might be
expected, the agreement between the exact and
the approximate solutions is not as good in
this case as before, because of the more
significant contribution of the higher modes
of vibration, which are neglected in the
approximate solution. The beatlike effect of
the exact curve is attributed to the fact that
the first and second natural frequencies of
vibration have very nearly the same value. A
better approximation to the dynamic increments
at sections away from midspan could be ob
tained by incorporating the contribution of
the second mode of oscillation; however, inas
much as this mode is not sinusoidal, the
resulting expressions are quite involved and
will not be presented.
Referring back to Figure 4.1, it can be
seen that the shape of the dynamic increment
curves for moment and deflection at midspan
are very similar, the ordinates of the moment
curve being smaller than those for deflection.
This relationship is in agreement with theore
tical predictions discussed in Reference 18c.
In fact, for the system under consideration
the predicted ratio of
(D. I.)M
cDI. = 0.649
(D.I.)
c
is in good agreement with the value obtained
from Figure 4.1 by comparing corresponding
points in the curves for moments and deflec
tions.
In Figure 4.3 the time histories of
dynamic increments for moment and deflection
are compared for sections away from midspan.
The upper half of the figure refers to a
section located at a distance of 0.42L from
the lefthand abutment, and the lower half of
the figure refers to the corresponding section
in the righthand span. For these sections
the theoretical relationship between moments
and deflections is
(D.l.)M (D.I.)M
7 =(D.I.) T= 0.703 (1 + e)
( DI D4
where e is a timedependent factor which
expresses the contribution of the higher modes
of vibration. For the numerical solutions
presented, the average value of this ratio is
of the order of 0.6.
4.3. EFFECTS OF A SINGLE CONSTANT FORCE ON A
BEAM WITH A SIDESPAN RATIO a = 0.8
4.3.1. Representative History Curves
In Figure 4.4 comparison is made of the
dymamic increment curves for moment and de
flection at midspan of a uniform beam with a
sidespan ratio a = 0.8. It can be seen that,
in spite of the fact that the lengths of the
side spans are different from that of the
center span, the curves are still very similar
to each other. If the quantity e referred to
above is assumed negligible, the following
ratio of dynamic increments for moment and
deflection may be predicted:
(D. I.) M
(D..) = 0.786
c
This ratio is in good agreement with the one
determined on the basis of the numerical data
that have been presented.
Figures 4.5 a and 4.5 b give time
histories of dynamic increments for moment at
sections of maximum positive moment and of
maximum negative moment, respectively. The
quantities MI and M4 represent moments at
sections 0.42aL from the lefthand and right
hand abutment, respectively; and M2 and M3
represent moments over the left and the right
intermediate supports. In these curves the
influence of the higher modes of oscillation
is particularly noticeable. It can also be
seen that the ordinates of the dynamic
increments for moment over the intermediate
supports are considerably greater than those
for moment in the side spans.
4.3.2. Effect of the Speed Parameter
In Figure 4.6 the amplification factors
for maximum moment and maximum deflection at
the center of the center span are plotted as a
function of the speed parameter, a, for values
of a as high as 0.60. As previously noted the
maximum possible value of a for highway
bridges is about 0.24. It can be seen that.
the curves are undulating and that the peaks
of the undulations increase with increasing
values of the speed parameter. The reason for
the undulating nature of these curves has been
discussed in Reference 18b.
In Figures 4.7a through 4.8 similar
plots are given for the moments and deflec
tions at sections 1 and 4 in the first and
third spans, and for moments M2 and M3 over
the intermediate supports.
Figures 4.9a and 4.9b give enlarged
views of the diagrams presented in the pre
ceding four figures for values of a smaller
than 0.24, which as previously noted represent
the realistic range of a for highway bridges.
The dashed lines in Figure 4.9a give the
amplification factors obtained from the
expression
(A.F.)z = 1 + (D.I.)
z z
(4.7)
where (D.I.) is the maximum value of the
dynamic increment under consideration for the
"region of influence" as defined in Section
3.2.1. The subscript Z denotes the location
and type of response considered, i.e., Mc, Dc,
M2, etc. These dashedline curves provide
good upper bounds to the amplification
factors. (24) The dotted lines represent
empirical relations obtained by approximating
the upper envelopes of the amplification
curves by straight lines. The equations for
these lines are
(A.F.)M = 1 + 0.5a
c
(A.F.)D = 1 + 0.72a
(4.8)
(4.9)
The terms of 0.7a in Equation 4.9 is slightly
greater than the value obtained by dividing
the term 0.50 of Equation 4.8 by the ratio
given in Section 4.3.1. Note that within the
realistic range of avalues, the maximum
amplification factor for M is about 1.12, and
for D it is about 1.16.
c
4.4 RESPONSE TO TWO MOVING CONSTANT FORCES
4.4.1. Representative Time Histories
Figure 4.10 presents time histories of
dynamic increments for moment at the center
of a uniform threespan continuous beam
traversed by two constant forces of equal
magnitude spaced at a distance s. These
forces may be considered to represent the
effects of the drive and rear axles of a
tractorsemitrailer combination. The speed
parameter of the system is taken as a = 0.10.
Results are presented for values of the axle
spacing parameter s/L in the range between
zero and 0.5. The ordinates of these curves
are normalized with respect to the maximum
static value of the moment for the particular
axle spacing considered. There are three
important eras in each history curve, corres
ponding to the cases where: (a) only the
first force, corresponding to the front axle
of the vehicle, is on the bridge, (b) both
forces are on the bridge, and (c) only the
second force is on the bridge.
With one exception, a striking simi
larity exists between these curves and the
corresponding curves reported previously for
the case of simplespan bridges.(26) The
exception refers to the fact that in the
present case the contribution of the higher
modes of vibration of the bridge is more
significant than for simplespan bridges. It
can be seen that for values of s/L equal to
0, 0.2, and 0.4, the component of the response
produced by the second force adds to that
produced by the first force, producing dynamic
increments which are comparatively large. The
period of oscillation of these dynamic incre
ments corresponds to the fundamental natural
period of the bridge. On the other hand, for
values of s/L equal to 0.1, 0.3 and 0.5, the
component produced by the second force cancels
that due to the first force, leaving only a
component of small amplitude, with a period
of oscillation corresponding to the third
natural mode of the bridge. It should also
be noted that the dynamic increments for
values of s/L = 0.2 and 0.4 are larger than
those for s/L = 0. This result is due to the
fact that the maximum static effect, with
respect to which the dynamic increments have
been normalized, decreases with increasing
axlespacing.
As might be expected from the material
already presented, the dynamic increment
curves oscillate, not with respect to a
horizontal base line passing through the
bridge supports, but with respect to a base
line resembling the crawl deflection curve of
the bridge. These base lines are shown as
dashed lines in the figure. It is of some
interest to note that for s/L = 0.5, the
response curve exhibits sharp discontinuities
at the instants each force reaches midspan.
This is probably associated with the fact that
at these instants one of the axles passes over
an interior support.
Similar results were obtained for other
values of a and for effects at sections away
from midspan. Figures 4.11a and 4.11b give
the results for moment at a section 0.42aL
from the lefthand abutment, and at a section
over the lefthand interior support, respec
tively. The same value of speed parameter
a = 0.10 is considered. It can be seen that
the addition and cancellation of effects are
not as well defined as in the case of moments
at midspan due to the more pronounced influ
ence of the higher modes of oscillation.
When certain amount of cancellation does occur,
the remaining oscillation of the dynamic
increment histories shows a well defined beat
like phenomenon, with a beating period corres
ponding to that between the second and third
modes of vibration of the bridge. This
beating effect is more pronounced in the time
histories of dynamic increment for M2. The
results presented indicate that, for moments
at sections away from midspan, the influences
of the second and even the third modes of
vibration of the bridge are of significance
in the response.
4.4.2. Response Spectra
In Figure 4.12 spectrum curves of
maximum dynamic increments for moment at mid
span are plotted versus the axlespacing
parameter s/L for three different values of ta.
In this figure, the dynamic increments are
given not as a function of the corresponding
maximum static effect, but in terms of WL,
where W is the total weight of the load.
When expressed in this manner, the effect of
the axlespacing parameter can more readily
be evaluated.
The curves for all values of a are un
dulating, with the peaks corresponding to
those values of s/L for which the effects of
the individual forces are additive, and the
valleys corresponding to the values of s/L for
which the component effects cancel. The peak
values of response increase with increasing
value of the speed parameter, and, in all
cases, the response due to a single constant
force may be considered to represent an upper
bound to the effects produced by two constant
forces.
Similar results, for sections away from
midspan, are given in Figures 4.13a through
4.13d. The addition and cancellation of
effects is not as marked in this case as
before, because of the more significant con
tribution of the higher modes of vibration;
and the resulting curves, although undulatory
in nature, do not follow a welldefined
pattern.
For simplespan bridges acted upon by
constant forces, it was shown in Reference 26
that response is due almost entirely to the
contribution of the fundamental mode of
oscillation, and that the magnitude of the
response is governed by the ratio of s/L and
a. Specifically, when (s/L)/a is zero or an
even integer, the effects of the two forces
are additive, whereas when this ratio is an
odd integer, the component effects cancel.
This result also applies to a lesser degree to
the continuous bridges considered in this
study. This is illustrated in Figure 4.14,
where the maximum dynamic increments for
moment and for deflection at midspan are
plotted as a function of (s/L)/a. The ordi
nates of the figure have been normalized with
respect to the maximum dynamic increments for
the case of s/L = 0. In spite of the scatter,
the solid line indicates a definite trend,
which agrees with the results obtained for
simplespan bridges. Similar results, however,
do not apply to effects away from midspan.
4.4.3. Summary
In Figure 4.15 the maximum dynamic
moment at the center of the center span is
plotted as a function of the speed parameter
for several values of s/L. The scale on the
left expresses the moment in absolute terms,
as a multiple of WL, whereas the ordinates on
the right express the moment in the form of an
amplification factor. That is, the dynamic
moment is normalized with respect to the maxi
mum static moment corresponding to the partic
ular value of s/L under consideration. This
plot clearly illustrates that the level of the
response decreases significantly with increas
ing s/L, but that the value of the amplifica
tion factor generally increases with increasing
s/L.
Figure 4.9a shows that for a single
force, the dynamic increment at midspan may be
approximated by the expression
0.5a(Mc )
where (Mc ) is the value of the maximum static
moment at midspan in terms of WL. The results
presented in Figures 4.12 and 4.13 also indi
cate that the maximum values of the dynamic
increment for two forces, when expressed in
absolute terms, are no greater than those
obtained for a single force. Accordingly, the
maximum dynamic increment for moment at mid
span due to two equal forces can be taken as
Mc + 0.5a(Mc )
c co
where M is the maximum static moment pro
duced by the forces. The amplification factor
for maximum moment is therefore
(A.F.) = + 0.5a (4.10)
Mc = +Mc/(Mc )o
where the ratio M c/(M ) is a function of s/L
and is given approximately by Equation 3.3.
In Figure 4.15 the results obtained by
this equation are shown in dashed lines. It
can be seen that this equation leads in gen
eral to conservative results.
The amplification factors for maximum
moments in the side spans MI and M4 and for
maximum moments over the interior supports,
M2 and M3, may be approximated in a similar
manner by the following expressions:
For MI and M4,
(A.F.)M = (A.F.)M 1 + 0. (4.11)
I M4 o/(M
* * *
For M2 and M3'
(A.F.)M = (A.F.)M = 1 + 0.8a
(4.12)
where MI/(M)o is given by Equation 3.1.
In Equation 4.12 the term involving the
parameter a is not divided by M2/(M ) because
the dynamic increments for twoaxle loads, as
shown in Figures 4.13c and 4.13d, are appre
ciably smaller than those for a singleaxle
load, and because the range of variation of
M2/(M2 o is small.
In Figures 4.16 and 4.17 results ob
tained by Equations 4.11 and 4.12 are plotted
as dashed lines and compared with correspond
ing curves for four different values of s/L
obtained with the computer program described
previously. In these curves, only the larger
of the two effects at symmetrical sections of
the bridge are presented. The data used to
plot the curves presented in this chapter to
gether with some additional data are summariz
ed in Table 4.2. *
V. RESPONSE TO SMOOTHLY MOVING SPRUNG VEHICLES
5.1. GENERAL
In this chapter the study of the re
sponse of threespan continuous bridges to
singleaxle and twoaxle smoothly moving
vehicles is presented. The vehicle is assumed
to be moving smoothly at the time it enters
the bridge; that is, its mass is assumed to be
in a state of static equilibrium. Further
more, the bridge deck is considered to be
horizontal and smooth. It is obvious that
this case is still not realistic, because in
practice the vehicle is generally in a state
of vertical oscillation at the time it enters
the bridge. This oscillation may be due to
the unevenness of the approach pavement or to
a grade discontinuity at the first abutment,
or to both factors.
Consideration of a smoothly moving,
sprung load introduces two new parameters into
the problem:
(a) The ratio of the total weight of
the vehicle or load to the weight
of the center span of the bridge,
referred to as the weight ratio R.
(b) The ratio of the natural frequency
for each axle of the vehicle when
vibrating on its tires to the
fundamental natural frequency of
the bridge, hereafter called the
frequency ratio ft
Throughout this chapter the limiting
value of the friction between the leaves of
the suspension springs is assumed to be so
large that the vehicle vibrates only on its
tires. The subsequent sections show that this
is indeed a reasonable assumption for the
values of interleaf friction encountered in
practice.
In the following sections, first the
effects of the parameters Pt, R, and a are
studied for a singleaxle loading, and then
the response of the bridge under a twoaxle
vehicle is investigated. The axle spacing
parameter s/L is introduced in combination
with the speed parameter in a manner similar
to that followed in the previous chapter for
the case of two constant moving forces.
5.2. RESPONSE TO SINGLEAXLE, SMOOTHLY MOVING
SPRUNG VEHICLES
5.2.1. Representative History Curves
Figure 5.1 presents the time histories
of dynamic response of the vehicle and of the
bridge for five different values of the weight
ratio, R. The response of the vehicle is
expressed by the history of the interacting
force, which is the force exerted by the tires
of the vehicle on the surface of the bridge.
The response of the bridge is expressed in
terms of the histories of dynamic increments
for moment at the center of the bridge. It
should be recalled here that the dynamic
increment of a given effect has been defined
as the difference between the dynamic value of
that effect at a given time and the corres
ponding static value. It can be seen that the
peak value of the interacting force seems to
increase almost linearly with R, whereas the
r'7
A
0 n\
I .
6.5
6.o
5.0
4.5
4.0
3.5
3.0
0.
5
1st
mode
2nd
mode
3rd
mode
0.7
Span Ratio, a
a L Am  A&  &
aL
3
"2
FIGURE 2.1 FREQUENCY COEFFICIENTS FOR THE FIRST THREE NATURAL MODES OF
VIBRATION OF A THREESPAN CONTINUOUS PRISMATIC BEAM
L
aL
L
1
1
L.
2'6 3A:" 24' 2'6 /I4
10 ' A
(a) HIL54 Design Load
V'0 3/1A: 28' 3'0 31Ar
10 1/"
6q
(b) mo08644 Design Load
FIGURE 2.2 SCHEMATIC CROSS SECTION OF B.P.R. "STANDARD" BRIDGES
I GaL L aL
FIGURE 2.3 BRIDGE MODEL USED IN ANALYSIS
b.V I I i
(a) Bridges Designed for B1514 Loading
5.0o 
0
+
0
0
00
3.0 
+ B.P.R. "Standard" Bridges
o Illinois Bridges
2.0
6.
(b) Bridges Designed for H2081644 Loading
+
+ + 0
+ 0
+
4.0 0 0. 0 0 0
o0
0 0
.( 0
40 60 80 100 120
Length of Center Span, L, in ft.
FIGURE 2.4 WEIGHTSPAN RELATIONSHIP FOR THREESPAN CONTINUOUS IBEAM BRIDGES
S
i
I
6
U
PS
0
U
.54
~.4
50
.54
I
I
Center Span Length, L, in feet
FIGURE 2.5 FREQUENCYSPAN RELATIONSHIP FOR THREESPAN CONTINUOUS IBEAM BRIDGES
I
y
ictional
vice
(a) SingleAxle and TwoAxle Loadings
(b) Interacting Force, P, and Frietional Force, F,
Versus Axle Displacement, y
FIGURE 2.6 CHARACTERISTICS OF VEHICLE
I
0
Vehicle Speed, v, in m.p.h.
I
0
Directions:
FIGURE 2.7
Center Span Length, L, in feet
1. Identify bridge type and pertinent values of L and Tb
2. Draw line 1 connecting values of L and v
3. Draw line 2 through value of T and intersection of line
1 and auxiliary line, and extnd until it intersects the ascale
NOMOGRAM FOR SPEED PARAMETER (ThreeSpan Continuous IBeam Bridges)
S.=0.265R0 .94
+
\t
'.0
\ 0
2.C
1.5
1.0
0.9
0.8
0.7
uomposite non(oM
Illinois Bridges Actio Ac0ion
B.P.R. "Standard" 4 +
Br Ages
0.15
0.2
Weight
)oeitO
10
10 O
\o
\ 0
s0.149R1.06
t;^
0.25 0.3
Ratio, R  W/Wb
FIGURE 2.8 RELATIONSHIP BETWEEN FREQUENCY RATIO AND WEIGHT RATIO
(ThreeSpan Continuous IBeam Bridges and "Standard" Vehicle,
W = 64 kips, ft = 3.5 cps)
4
0
\ o
o\
0
0 \
0.3
7
'C,
N'
I
.09 0.1
p
0.4 0.5 0.6
tc
v«u
0\
0\
\
0
S.. ~
1
0
Center Span Length, L, in feet
FIGURE 2.9 REALISTIC RANGE OF BRIDGEVEHICLE PARAMETERS
(ThreeSpan Continuous IBeam Bridges, "Standard" Vehicle)
0
I
0
0
I
11f
I
I
I7
Hi
m
0
It
<0
SU
Jo
I '
0
S E
F (
0 0
< 3
i o
I
E
S
S0
a) (
D s<
 I
4
C
'T jo sun U I N '4uemuN
I
0 co
0_
II
U,
+ 2
C
C
1
 I 
L)
u
:
i
F
cl EH
S^
o
I 5
uJ_
'IAj JOU U UTI a; "O
co
0
0
J 11
u
0 o
0n0
U.
Lz
L W o
64 F 8
Hd Lu
a . z
O L
<. 1 1
 z
LLJ
 V U a
Z '
(a ==
LLI u
F
L o
4
im Jo Ouwn UT N 'wIae=N
I
A JO m8=8, UT Ig '.TSuq
 v!
I
0
o_
O
w, co
0
LU II
I
Z O
z 
0
S 
I 0
..J 0
d W
I
Cu
,+ LO
* U
z s
AA < Q)
0 >
4) 1
S t/)
L o
S0
, . 4<
0
4.
U D
I I
(n 01
0)
Q S
o
Ln
UJ E
Z S
 4
U c
Zj~
U.
z
in
 LAI
A JO em.Ies u; *g <.iwq8g
LU
U
eo
af
0
a
o o5I
SII
O 0 0
 U «X
o x o 0 0 0 0
;;TSA T»wL ems Jo Pwoi GTxVa2UtTS o01, rP q u 1,ueam *XW 0
pWo'I OEXvO.IZ o0. 7np 9 4I* 1U. MU *XWVN H
Id
0m
U')
LUl
LL
I J_
< 00
D
IU
0
go 0 E
LIJ
< o
43I D
0
il
< CL
Lo
*. U) 0
§ S
D (
t.2
020
4
C:
0
4
D
U)^
i c
y U (1
1 2
1 S2
00
040
4P43
im0~
'I
0
Vo
FIGURE 3.7
MAXIMUM STATIC SHEARS AND REACTIONS DUE TO TWO EQUAL FORCES
(Uniform, ThreeSpan Continuous Beam; a = 0.8)
Axle Spacing Parameter, s/L
s/L = 0.5
 M
I ~ 
0.4
0.3
0.2
0.1
0
M
A2
e/L

0
= 0.5
0.4
0.3
0.2
0.1
0.1
0
I
/L = 0.5
0.4
0.3
0.2
0.1
0
s/L =
0.4
0.3
0.2
0.1
M3
0.5
0.2 0.4 0.6 0.8
Position of Front Axle, t = x/(l + 2a)L
FIGURE 3.8 "ZONES OF INFLUENCE" FOR STATIC MOMENTS MI
THROUGH M4 IN A THREESPAN CONTINUOUS BEAM SUBJECTED
TO TWO EQUAL LOADS (a = 0.8)
s/L = 0.51
0.4
0.2
A
I I
0
g
a
U
V
U
4
4)
U
4)
0
4)
I
Distance from Left Abutment to Section under Consideration, in Tera of (1 + 2a)L
FIGURE 3.9 CURVES FOR MAXIMUM POSITIVE AND MAXIMUM NEGATIVE STATIC
MOMENTS ALONG THE SPAN DUE TO TWO EQUAL FORCES (a = 0.8)
Distance from Left Abutment to Section under Consideration, in terms of (1 + 2a)L
FIGURE 3.10 CURVES FOR ABSOLUTE MAXIMUM STATIC SHEARS ALONG THE
SPAN DUE TO TWO EQUAL FORCES (a = 0.8)
,,~~~ . I °a o.8
, W ...... . A a . o.6
L .o.8
* s ~ .
Distance from Left Abutment to Section under Consideration, in Term. of (1 + 2a)L
FIGURE 3.11 COMPARISON OF MAXIMUM POSITIVE AND MAXIMUM NEGATIVE STATIC MOMENTS
ALONG THREESPAN CONTINUOUS BEAMS WITH SAME OVERALL LENGTH BUT
DIFFERENT SPAN RATIO (Two Equal Loads)
B
p
.4
.4
S
a
.4
a
i
S
I
Solution
A AA ,
I I I I I I
0 0.2 o.4 0.6 0.8
Position of Load, S = x/(l + 2a)L
FIGURE 4.1
TIMEHISTORIES OF DYNAMIC INCREMENTS FOR EFFECTS AT MIDSPAN
(a = 1.0; Single Constant Moving Force, a = 0.15)
0
O1
I
o
14
L0
Exact Solution
Solution
I
0.4
Position of Load,
I
0.6
I  x/(1 + 2a)L
FIGURE 4.2
TIMEHISTORY OF DYNAMIC INCREMENT FOR DEFLECTION
(a = 1.0; Single Constant Moving Force, a = 0.15)
AT x = 0.42 aL
I I
0
,

0
'4
I4
'.4
S
'4
0.2 0.4 0.b
Position of Load, ( = x/(1 + 2a)L
FIGURE 4.3 COMPARISON OF HISTORIES OF DYNAMIC INCREMENT FOR MOMENT AND
DEFLECTION AT SECTIONS 0.42 aL FROM EITHER ABUTMENT
(a = 1.0; Single Constant Moving Force, a = 0.15)
I
AA
0 0.2
FIGURE
0.4 0.6
Position of Load, e  x/(1 + 2a)L
4.4 COMPARISON OF HISTORIES OF DYNAMIC INCREME
MOMENT AND DEFLECTION AT MIDSPAN
(a = 0.8; Single Constant Moving Force, a = 0.15)
NT FOR
I I I I I
I I
m
...... $ I
AA
I I I I I I I I
0 0.2 0.4 0.6 0.8
Position of Load, S = x/(1 + 2a)L
FIGURE 4.5a TIMEHISTORIES OF DYNAMIC INCREMENTS FOR MOMENTS AT
SECTIONS 0.42 aL FROM EITHER ABUTMENT
(a = 0.8; Single Constant Moving Force, a = 0.15)
o.
3°.
H
M
0
L .
03.4
i
0
L .
I
A I
/N
Position of Loas, t  x/(1 + 2a)L
FIGURE 4.5b TIMEHISTORIES OF DYNAMIC INCREMENTS FOR
MOMENTS OVER INTERIOR SUPPORTS
(a = 0.8; Single Constant Moving Force, a = 0.15)
0
z
I
z
LL.
0
U z 0
0 U
u z >
< o
4 iz <
4 )
e  0
<
. r
LL
L o
0
II
Li
Ut
a a. oH oN oj ol04a UOTvTojTTldV mTplcw
I
.T jo 7 .zo .Jo4oa uoToTyjTEudov mxzwm
Z 0
z
X U
Ll
20
0< 
LIL 0
I L
0
<0
U 0
+) z o
0 <c
  e
< .
00
<f !
C O
>( 
_j
0
. .
la 0o w zoj .xo^ova uoTl.OTT[dmv T umprc
LL. a
0 LU
Z
Z r
Li 0
+4
In C
w 0
CL
F *
Z 0
II
0
SI .2o .xoj . joi o uonuouTlV wapM
0.05 0.10 0.15 0.20 0.25
Speed Parameter, a
FIGURE 4.9a MAXIMUM AMPLIFICATION FACTORS FOR MOMENT
AND DEFLECTION AT MIDSPAN AS A FUNCTION OF a
(a = 0.8; Single Constant Moving Force)
U°
'4
4)
I
C)
I)
I
42
S
j
Speed Parameter, a
FIGURE 4.9b MAXIMUM AMPLIFICATION FACTORS FOR MOMENTS AT
SECTIONS AWAY FROM MIDSPAN AS A FUNCTION OF a
(a = 0.8; Single Constant Moving Force)
A ^4
.v W
0.06
s/L = 0
0o.o6
0o.1
0.00
U
o0.2
o.o6r
/4AR ANA .AAA M\R
S second Force Enters Bridge
A ftA A A AAAA A A A Aft AAl
R[7VVvV VVv v v RUVI A1vu T
W~I (UUA A /
V VVVVVVU
L1
0.5 0 0.4 0.8 1.3 1.8 2.2
Position of Second Force, (2  vt2/L
FIGURE 4.10
TIMEHISTORIES OF DYNAMIC INCREMENT FOR MOMENT AT MIDSPAN
(a = 0.8; Two Constant Moving Forces, a = 0.10)
0.o06
II
L
I'
L
0.0oL
=
.Second Force Enters Bridge
A. AA\ A A^ A A A n Al
I I I I
0.5 0 0.4 0.8 1.3 1.8 2.2 2.6
Position of Second Force, (2 = vt2/L
FIGURE 4.11a TIMEHISTORIES OF DYNAMIC INCREMENT FOR MOMENT AT x = 0.42 aL
(a = 0.8; Two Constant Moving Forces, a = 0.10)
6,m
'4
0
{O
0
O
E
4"
0
C)
0
0
'
4,t
0
I
0
'
0.06
0.06
I
0.3
A .
0
tt
s/L  0.5
AA
rv
ill
 UI
flAt nR/In~
¶JV71TA
'I
'F
IL
I I I I I I
0.5 0 0.4 0.8 1.3 1.8 2.2 2.6
Position of Second Force, (2 = vt2/L
FIGURE 4.11a (CONTINUED)
O
Do
;
U
0.06
4ý
0
4)
e 0.06
0.12
0.0
S0.06
0.06
 °o.4
AA
1
i
ý v
ýýA
v
v
A A A A
V v vv T" vvvv
Force Enters Bridge
I I I I I I 1 I
0.5 0 0.4 0.8 1.3 1.8 2.2 2.6
Pogition of Second Force. (2  vt2/L
FIGURE 4.11b TIMEHISTORIES OF DYNAMIC INCREMENT
FOR MOMENT OVER LEFTHAND INTERIOR SUPPORT
(a = 0.8; Two Constant Moving Forces, a = 0.10)
S
0
V
'r4
0
0
S'
0
I
C
'4
I
0
'
11
I
4
±71
Second Force Enters Bridge
fluA AflR. UAhflhF
lI V \ uvlIU\llI V
V LI" 1.J U
I I l l I I I
0.5
0 0.4 0.8 1.3 1.8
Position of Second Force, 12  vt2/L
FIGURE 4.11b (CONTINUED)
0.0
6
WVJFTO
_ A
I I i i i I I I
Z
0
(0
Lii
J
x
u
0
I
C.
*.
ILl
u
0
* a
a *Z
0

I >
0 04
tf s
C)
4i
ULJ
U3
IL
Ia jo me UT u q4ueDN .zoj 4nuotouw opeUna UmwpNi
co in
R
a
I
aI
'4
I
J
Axle Spacing Parameter, s/L
FIGURE 4.13
MAXIMUM DYNAMIC INCREMENTS FOR MOMENT AT SECTIONS AWAY
FROM MIDSPAN AS A FUNCTION OF AXLE SPACING
(a = 0.8; Two Constant Moving Forces)
0 0.1 0.2 0.3 0.4
Axle Spacing ParamOter, s/L
FIGURE 4.13 (CONTINUED)
0
I
8
I
I
b
U
U
I
I
j
(,/L)/=
FIGURE 4.14 MAXIMUM DYNAMIC INCREMENTS FOR MOMENT AND
DEFLECTION AT MIDSPAN AS A FUNCTION OF (s/L)/a
(a = 0.8; Two Constant Moving Forces)
I
I
I<
I»
V
E
C,
4
Psi
4
C)
.5.4
~.4
.5.4
I
Speed Paramter, a
FIGURE 4.15 MAXIMUM DYNAMIC MOMENT AT MIDSPAN AS A FUNCTION OF a
(a = 0.8; Two Constant Moving Forces)
a
r4
4j
§
i
I>
$4
'.4
I
0.05 0.10 0.15 0.20 0.25
Speed Paramter, a
FIGURE 4.16 MAXIMUM DYNAMIC MOMENT AT SECTIONS 1 OR 4 AS A FUNCTION OF a
(a = 0.8; Two Constant Moving Forces)
.10J .ZO *zj* . m m
rIA JO MUU.S UT C .tO 3
OOoa$es .o ummoN opnmk anI pmx
0IT
o1.05 v vV "
1.900 cV
S0.95
0
$4
0
H
(a) R = 0.175
1 'O r' A AV A /\
o.io vV vv vv
(b) R  0.3
H
0 0.2 0.4 0.6 0.8 1.0
Position of Load, = x/(1 + 2a)L
FIGURE 5.1 TIME HISTORIES OF DYNAMIC RESPONSE
(a = 0.8; SingleAxle Load, 4t = 1.0, a = 0.15)
A A
AfAA
V\yvv
A AVM
vV
AA \AnA" A\
V v VJ V
(d) R = 0.5
SAA A
V\h
A/
r'I V va
Vi \
fM
K A A A
(e) R 1.oo0
0.2 0.4 0.6 0.8
Position of Load, t  x/(l + 2a)L
FIGURE 5.1 (CONTINUED)
1.10
1.00
0.90
V
0.10
A
0.10o[
1.20
1.10
1.00
NI
p
II I
1.

\
4 1.05
S0.10
0.10  V Constant Force Solution
(a) 't = 0.5
4 1.05
0 1.95
0.95
0.
ii
E
H
0.2 0.4
Position of Load,
I I
0.6 (
I  x/(1 + 2a)L
FIGURE 5.2 TIME HISTORIES OF DYNAMIC RESPONSE
(a = 0.8; SingleAxle Load, R = 0.175, a = 0.15)
1.0
1.0
O.
r.'_A 
v 4ý \ý)
I
.8
*1
'<4
11
«i
0
Weight Ratio, R
FIGURE 5.3 EFFECT OF WEIGHT RATIO ON MAXIMUM VALUES OF RESPONSE
(a = 0.8; SingleAxle Load, (t = 1.0, a = 0.15)
4)
C,
§
I
5
Frequency Ratio, *t
FIGURE 5.4 EFFECT OF FREQUENCY RATIO ON MAXIMUM VALUES OF RESPONSE
(a = 0.8; SingleAxle Load, a = 0.15)
R  0.50 R 0.175
1 . 1 0 C *
(c) Monent M4
R = 0.50
1 .20  J  
R = 0.175
\,,. / "\/\,
1.10op,/ 
(d) Mcmnent M3
1.00 _ 1 11 1I
Frequency Ratio, *t
FIGURE 5.4 (CONTINUED)
Speed Parameter, a
FIGURE 5.5 EFFECT OF SPEED PARAMETER ON INTERACTING FORCE
(a = 0.8; SingleAxle Load)
42
a4
I
*r4
Speed Parameter, a
FIGURE 5.6 EFFECT OF SPEED PARAMETER ON MAXIMUM MOMENT AT MIDSPAN
(a = 0.8; SingleAxle Load)
R  0.175
I I
I I
I I
I I
'II
Speed Parameter, a
FIGURE 5.7 EFFECT OF SPEED PARAMETER ON MOMENT M4
(a = 0.8; SingleAxle Load)
'I
'.4
8
*5
,I\
K\ /
/
/
/
A ..
I
/
Speed Parameter, a
FIGURE 5.8 EFFECT OF SPEED PARAMETER ON MOMENT M3
(a = 0.8; SingleAxle Load)
3r
0
d
3
.4
4
I
C
43
C)
4
i
I
Speed Parameter, a
FIGURE 5.9 COMPARISON OF MOMENTS AT SYMMETRIC SECTIONS AS A FUNCTION OF a
(a = 0.8; SingleAxle Load, R = 0.30, <t = 1.0)
I
04
0
'g
.4
.41
5
Frequency Ratio, 0t
FIGURE 5.10 COMPARISON OF MOMENTS AT SYMMETRIC SECTIONS AS A FUNCTION OF t
(a = 0.8; SingleAxle Load, R = 0.30, a = 0.18)
I
0
I4
4)
I1
i
f4
0
0
Position of Rear Axle, (2 = vt2/L
FIGURE 5.11 TIMEHISTORIES OF DYNAMIC INCREMENTS FOR MOMENT AT MIDSPAN
(a = 0.8; TwoAxle Load, R = 0.175, 4t = 1.0, a = 0.10)
Axle Spacing Paramter, s/L
FIGURE 5.12 EFFECT OF AXLE SPACING PARAMETER ON INTERACTING FORCE
(a = 0.8; TwoAxle Load, R = 0.175)
Axle Spacing Parameter, s/L
FIGURE 5.13 EFFECT OF AXLE SPACING PARAMETER ON MAXIMUM
DYNAMIC INCREMENT FOR MOMENT AT MIDSPAN
(a = 0.8; TwoAxle Load, R = 0.175)
i
0
I
I
I
5
~j5
02o00 I I I
Axle Spacing Parameter, s/L
FIGURE 5.14 EFFECT OF AXLE SPACING PARAMETER ON
MAXIMUM DYNAMIC INCREMENT FOR M4
(a = 0.8; TwoAxle Load, R = 0.175)
U
0
I
U
H
e)
I
j
0
a
q.
ai
I
a
M
I
Axle Spacing Parameter, s/L
FIGURE 5.15 EFFECT OF AXLE SPACING PARAMETER ON
MAXIMUM DYNAMIC INCREMENT FOR M3
(a = 0.8; TwoAxle Load, R = 0.175)
U
.4
.4a
i
V4
:°
I
I
&
Frequency Ratio, *t
FIGURE 5.16 COMPARISON OF RESPONSE OF SINGLEAXLE AND
TWOAXLE LOAD AS A FUNCTION OF <t
(a = 0.8; R = 0.175, a = 0.15)
Frequency Ratio, 0t
FIGURE 5.16 (CONTINUED)
Static _j
Eq. 4.10 l 
0.3 .
1.10
1.00
1.10
1.00
1.30
L.2U
1.10
1 mt
     .         _   ~
Static7 __
 .~ . /
J.L»C\
1.10
1.00
I I I I I I I I I I I I
0.10
0.15
I I I I I
0.20
0.25
Speed Parameter, a
FIGURE 5.17 MAXIMUM DYNAMIC MOMENT AT MIDSPAN AS A FUNCTION OF a
(a = 0.8; TwoAxle Load, R = 0.175, pt = 1.0)
V.2u
0.18
0.16
0.12
0.08
X
1+
0
V4
T4
g
C,
J
i 7i
_ " Static
0.05
Eq. 4.10 l ,

,
_e»z
9
0
*H
'4
I
~f4
0
~.4
43
C
V
I
Speed Parameter, a
FIGURE 5.18 MAXIMUM DYNAMIC MOMENT AT SECTIONS 1 OR 4 AS A FUNCTION OF a
(a = 0.8; TwoAxle Vehicle; R = 0.175, ýt = 1.0)
0
0
U
0.05 0.10 0.15 0.20 0.25
Speed Parater, a
FIGURE 5.19
MAXIMUM DYNAMIC MOMENT AT SECTIONS 2 OR 3 AS A FUNCTION OF a
(a = 0.8; TwoAxle Vehicle, R = 0.175, 4t = 1.0)
riF~
[I
Sa S 0.15, 53 sols.
Ia > 0.15, 38 sols.
For M or M
SI I
1.1 1.2 1.5
40
20
I
9
I v
w
I IFor M
1.1 1.2
0 r
A0
I I
For M
C
U I I
0 1 1 1 1 1 I.
1.4 1.5
I I
v y 1.m a. . .J. .3 . J .. 5
Amplification Factor for Moment
FIGURE 5.20 PERCENTAGE DISTRIBUTION OF AMPLIFICATION FACTORS
(a = 0.8; SingleAxle, Smoothly Moving Vehicle)
U
.
 . = i i J

.
rLo 1
=
=
 , =
I.Ajk
0
I
L)
U
<
IL
s o
0.. 
< 
4 O
0
LU 0
LL o
O > E
I
4 z 4
UJ 0
U <
S i
4 z X
0
L.
0
i
u_
5 ujl
C1
LA
ui
OL
r4
vgososqy usq. .zo.eaz *.*vy q.TA uoTnTos jo qnuoos.
wsaegqy uwq.4 .v.azS **vy qTA suootnlog J O Uoojoed
(n
0
I
(C)
U
z
0
I
U
 D
v
 0
z .
 v
E °
C)
LLI .
0 0
0  CD
4) ~ () .1
IL
S " oj
o ' o
0
I
V4)
D
LU.
F 0
0
4)
 ,0
0
E E
V) c
0
C
LU
U °
LL
LAI
ui^
C4
C4
0
*L
\ \ 198 solutions
80
K ~ NiorN
BO _______ ____________________
\l^
60v 1
so K2 or K3
40,
20 _ i I i T~ ~  . i
1.0 1.1 1.2 1.3
Amplification Factor for Monent
FIGURE 5.23 EFFECT OF SECTION UNDER CONSIDERATION ON
CUMULATIVE PERCENTAGES OF AMPLIFICATION FACTORS
(a = 0.8; Smoothly Moving Vehicle, 0.06<a<0.24, 0<s/0<0.5)
U
43
5T
T v T.
DI
I I I I I I I I I I I
0 0.2 0.4 0.6 0.8 1.0
Position of the Load, t = x/(1 + 2a)L
(a) Timehistories of Interacting Force, P
FIGURE 6.1 TIMEHISTORIES OF DYNAMIC RESPONSE
(a = 0.8; SingleAxle Load with 15 Per Cent Initial Oscillation,
R = 0.175, a = 0.15, e = ', )
T
T
*t  0.6
It  1.0
I ftrt
I I I I I I i I I I I
0 0.2 0.4 0.6 0.8 1.0
Position of the Load,   x/(l + 2a)L
(b) Timehistories of Dynamic Increments for Moment at Midspan
FIGURE 6.1 (CONTINUED)
15
I
0
14
Phase Angle, 0, In degrees
FIGURE 6.2 COMPARISON OF MAXIMUM INTERACTING FORCE
FOR DIFFERENT ERAS OF RESPONSE
(a = 0.8; SingleAxle Load with 15 Per Cent
Initial Oscillation, a = 0.15, u = )
U4J
a4
Phase Angle, e, in degrees
FIGURE 6.3 DEPENDENCE OF ABSOLUTE MAXIMUM INTERACTING FORCE ON R AND <t
(a = 0.8; SingleAxle Load with 15 Per Cent
Initial Oscillation, a = 0.15, v = )
.K=mOJ 14uamojoai OTUS uzuMnMPcu
0
8
z 0 4 ii
, Z
0 n L
< < C
Sr
. UL r ..
tL 0 C;
 U r a
S LL. Z
0 0
S 0 " c
C) L 0
< .
LLu
Z
WuMnoo .OJ .zoW.v uoTZaDTJTd1UTaV tuMnprw
o R = 0.175
o R = 0.35
a R = 0.5
i
1.2
1.1
1.C
90 1. 0
Phase Angle, 0,
270
in degrees
FIGURE 6.5a UPPER ENVELOPES OF SPECTRA OF MAXIMUM AMPLIFICATION
FACTORS FOR MOMENT AT MIDSPAN AS A FUNCTION OF e
(a = 0.8; SingleAxle Load with 15 Per Cent
Initial Oscillation, a = 0.15, p =  )
For 4t = 1.0
o t1.0
o t = 0.6
t = 1.5
For R = 0.175
I I I I I I I
1.2
1.1
1.0
0.0
0
'A
v.  .
  ,.. o
»
1 _
0U
'4
I
j
peoU Ioj wol.oW UoTru;pOtJday amw=PM
o R = 0.175
1.2 o R = 0.55
AR = 0.5
1.
1.0 F  = o.15
90 180 270
Phase Angle, 0, in degrees
FIGURE 6.6a EFFECT OF INTERLEAF FRICTION ON RESPONSE AT MIDSPAN
(a = 0.8; SingleAxle Load with 15 Per Cent Initial
Oscillation, 1t = 1.0, a = 0.15, y = 0.36, F. = 0)
r 1 I
V .y v  I I II     I  i 
,. 4 .4 ,4 .4 .4 i r 6
z
<
0_
0
Us
Z 0
o c
S
Z
0 C
I e
LU 4j
U')
Z 1
0 L)
0
LU
Z .I)
0 
H '_ 0
i 0 I
_.I
LU 
iU
z , "'
LL .,.
LI
UI 0
LU
U
LU II
Y
10
LU
Ul
9,usmow xoj jvOBe uoflpoTJTTd1IV ninmTXVW
(b) Moments over Interior Supports
Points Refer
with 15%
S=w 4
o
0
A
to Solutions for Vehicle
Initial Oscillation
= 0.15
* R = 0.175
* R = 0.35
A R = 0.5
0.7 0.9 1.1 1.3
Frequency Ratio, 4t
IF GURE 6 . 7 COMPARIS LEVELS OF RESPONSE
SMOOTHLY MOVING AND INITIALLY OSCILLATING VEHICLE
(a = 0.8; SingleAxle Load, a = 0.15, y = 0.36, Fi = 0)
0.5
AA A
43
V..
0.
0.
0.
S0.
0.
I I I I I I II
0 0.4 0.8 1.3 1.8 2.2 2.6 2.9
Position of Front Axle, (1 1 vt1/L
(a) Axles Initially in Phase
FIGURE 6.8 RESPONSE CURVES FOR TWOAXLE VEHICLE WITH
15 PERCENT INITIAL OSCILLATION
(a = 0.8; s/L = 0.3, a = 0.15, R = 0.175, 4t = 1.0, e1 = 1800, u =
AA A ,
4
*4
'4^
U
E
I
I
FIGURE 6.8 (CONTINUED)
I I I I I I I I
0 0.4 0.8 1.3 1.8 2.2 2.6 2.9
Position of Front Axle, t1 1 vt,/L
(b) Axles Initially 180° Out of Phase
1.2
1.(
1.2
1.2
1.1
1.0
1.2
1.1
' 1.2
1.1
1.0
0.9
Phase Angle of Front Axle, e1, in degrees
FIGURE 6.9 SPECTRA OF RESPONSE VERSUS PHASE ANGLE OF
FRONT AXLE FOR SEVERAL PHASE DIFFERENCES
(a = 0.8; TwoAxle Vehicle with 15 Per Cent Initial
Oscillation, s/L = 0.3, a = 0.15, R = 0.175, ¢t = 1.0, p = )
x
4
0
4.
141
0
r4
l4
Phase Angle of Front Axle, 01, in degrees
(a) Effects at Midspan
FIGURE 6.10 COMPARISON OF UPPER ENVELOPES OF RESPONSE FOR
SINGLEAXLE AND TWOAXLE VEHICLES WITH 15 PER CENT INITIAL OSCILLATION
(a = 0.8; a = 0.15, Ae = 00, w = )
^i
4a
0
.d
o
.'Q
,U
a .
.4 4 4
nMOH =J O .zoqi uoTJ'4.vOTThfTiY mumPX
.3
0
4,
Eq 6 i
     
Speed Parameter, a
FIGURE 6.11 MAXIMUM DYNAMIC MOMENT AT MIDSPAN AS A FUNCTION OF a
(a = 0.8; TwoAxle Vehicle with 15 Per Cent Initial Oscillation,
R = 0.175, t = 1.0, Ae = 00, v = 0.15, y = 0.36, Fi = 0)
54
0
11
UH
4)
0
k
0
4)
S
'4
'4
4
I
Speed Parameter, a
FIGURE 6.12 UPPER ENVELOPES OF SPECTRA OF RESPONSE AS A FUNCTION OF a
(a = 0.8; TwoAxle Vehicle with 15 Per Cent Initial Oscillation,
R = 0.175, <t = 1.0, Ae = 00, v = 0.15, y = 0.36, Fi. = 0)
u.J
SI
<
<
_1 z
D
z
0
o 
z
IE
So
L 0
()
U z
LL
fI U
00
II
LU Z
SLL
I.
LL
LL C
w
0, ~
.4 W'
U H13
vgsToaqcy wuqo .z..waa *j*vy q.A suoonTog jo qunaaJa
o
l)
OLl
J
x Z
D
E U
54 0
3 0 r
8 ..
0
* < 1 0
0< _
LL 0
II
L Zc
_ <
Li
uj 
I1
5
U3 0
UJ =
waposoqy uq. ae«eaj y q*4* " a soqT~on S Jo qnUJ.zaZ

Auplification Factors for Moment
FIGURE 6.15 EFFECT OF SECTION UNDER CONSIDERATION ON
CUMULATIVE PERCENTAGES OF AMPLIFICATION FACTORS
(a = 0.8; Single and TwoAxle Vehicles with 15 Per Cent
Initial Oscillation, y = 0.36)
g
S
S
0
*
:4
0
I)
S
C,
:4
*
0
'I
54
04
I
I
44~
c o
CO
0)
E II
<u
> C
0
"0
r II
o 11
Q LJ
fO
0
0 0
0
CD
II
0
r* *
.__1
II
I "{:
Q.) ¢'y
cn 0
t
U
0
0
LL.
0
z
I
U (
z > c
LL
0
" 0 co
. X 1
u C
LL
 uI
 :2
« a
c1.01,
P
p
rst
SteadyState
0
roiLTon or loaa, c m x/' i. * ea)u
FIGURE 7.3 COMPARISON OF TIMEHISTORIES ON INTERACTING FORCE FOR
VEHICLE MOVING ON A RIGID PAVEMENT AND ON A BRIDGE
(SingleAxle Load; y = 0.36, v = 0.15, Fi = 0, e = 0°)
P
at
P
t
SteadyState
~ A
Position of Load, I = x/(1 + 2a)L
FIGURE 7.4 EFFECT OF INITIAL FRICTION ON TIMEHISTORIES OF
INTERACTING FORCE FOR A LOAD MOVING OVER A BRIDGE
(a = 0.8; SingleAxle Load; y = 0.36. u = 0.15,
R = 0.175, *t = 1.0, a = 0.15, e = 0°)
FIGURE 7.5
a Time to Attain SteadyState Vibration on Tires
Tt " Natural Period of Load Vibrating on Tires
EFFECT OF STIFFNESS RATIO, y, ON TIME REQUIRED
TO ATTAIN STEADYSTATE VIBRATION
(SingleAxle Load Moving on Rigid Pavement, Fi = 0, e = 0°)
8
0
*Sl
U^
N3
:ii
0
s Time to Attain SteadyState Vibration on Tires
T Natural Period of Load Vibrating on Tires
FIGURE 7.6 EFFECT OF INITIAL FRICTIONAL FORCE, F., ON TIME
REQUIRED TO ATTAIN STEADYSTATE VIBRATION
(SingleAxle Load Moving on Rigid Pavement, y = 0.36, e = 00°)
54
.3
*1
I

Phase Angle, 9, in degrees
FIGURE 7.7a SPECTRA OF RESPONSE VERSUS PHASE ANGLE FOR SINGLEAXLE LOAD
(a = 0.8; C = 0.50, a = 0.15, R = 0.175, qt = 1.0, y = 0.36, v = 0.15)
Moment M1
Phase Angle of Front Axle, 01, in degrees
FIGURE 7.7b SPECTRA OF RESPONSE VERSUS PHASE ANGLE OF
FRONT AXLE FOR TWOAXLE VEHICLE
(a = 0.8; s/L = 0.3, C = 0.50, a = 0.15, R = 0.175,
=t = 1.0, y = 0.36, v = 0.15, Ae = 0°)
1.2
U
1.1
1.0
*i 1.2
1.2
1.1
1.0
0.9
Phase Angle, 6, in degrees
FIGURE 7.8a COMPARISON OF UPPER ENVELOPES OF SPECTRA OF MAXIMUM
RESPONSE AT MIDSPAN FOR DIFFERENT AMPLITUDES OF INITIAL OSCILLATION
(a = 0.8; SingleAxle Load, a = 0.15, y = 0.36, F. = 0)
4,
4)
'
I)
WuuOMO JoJ 20o.'oeJ uoTIv'TJTT(IV immnIpva
d
'4
e
p4
j
4
Phase Angle of Front Axle, 01, in degrees
FIGURE 7.9 SPECTRA OF RESPONSE VERSUS PHASE ANGLE OF
FRONT AXLE FOR SEVERAL PHASE DIFFERENCES
(a = 0.8; TwoAxle Vehicle with 50 Per Cent Initial
Oscillation, s/4 = 0.3, a = 0.15, R = 0.175, ýt = 1.0, u = )
5
2
h
I
I
j»
I
I'
Phase Angle of Front Axle, 01, in degrees
FIGURE 7.10 COMPARISON OF UPPER ENVELOPES OF SPECTRA
OF RESPONSE FOR VARIOUS AMPLITUDES OF INITIAL OSCILLATION
(a = 0.8; TwoAxle Vehicle, s/L = 0.3, a = 0.15, R = 0.175, it = 1.0, p = )
I
0
I
Frequency Ratio, 0*
FIGURE 7.11 COMPARISON OF MAXIMUM LEVELS OF RESPONSE FOR
SMOOTHLY MOVING AND INITIALLY OSCILLATING VEHICLES
(a = 0.8; SingleAxle Load, a = 0.15, y = 0.36, F. = 0)
0°
0
I
'5
Frequency Ratio, 0t
FIGURE 7.12 SPECTRA OF RESPONSE VERSUS FREQUENCY RATIO
(a = 0.8; SingleAxle Load with 50 Per Cent Initial
Oscillation, a = 0.15, R = 0.175, ot = 1.0, y = 0.36, e = 00, F. = 0)
1.2
1.0 Ment_ _
 (b) Moment M

1 11
1.6
1.0
/ 'I .p0.15 /
1\ / \ /
(c) Moment M4
0.8 I I I I I I I
0.5
Frequency Ratio, *t
FIGURE 7.12 (CONTINUED)
Frequency Ratio, *t
FIGURE 7.13 SPECTRA OF RESPONSE VERSUS FREQUENCY RATIO
FOR UNDAMPED VEHICLE (a = 0.8; SingleAxle Load with 50
Per Cent Initial Oscillation, a = 0.15, e = 00)
X,
I
5
Frequency Ratio, 4t
FIGURE 7.13 (CONTINUED)
j
x
<L
.a
I
IJ
Lj,
z
I ** n
0
< 0
0 .1
 
J E o
* 0
0
UJ LU to
0 ) 0
LU
a L_ Q1
X > t
0 1
F
U
U z
 i
LU
LL
wSspasqy uwq .z 4.as *j"*y qq.T suoT.nTcos jo q.uao.aa
es:pasqy uq .roea. *I'V q! l:A sooTynTog jo q.uao.zd
TABLE 2.1
CHARACTERISTICS OF B.P.R. IBEAM BRIDGES
Side Span Ratio a = 0.8
(1) (2) (3) (4) (5) (6) (7)
L w Wb Composite Action Noncomposite Action
ft kip/ft kips f1, cps Tl, sec fl, cps T1, sec
For H1544 Loading
3.74
3.82
3.91
4.08
4.17
4.34
4.43
4.55.
4.66
4.79
5.00
5.24
186.8
229.2
274.0
326.3
374.9
433.5
5.51
4.43
3.68
2.97
2.67
2.37
0.182
0.226
0.272
0.336
0.375
0.422
For H20S1644 Loading
221.4
272.8
326.3
383.4
449.7
524.4
6.67
5.33
4.46
3.80
3.14
2.73
0.150
0.188
0.224
0.263
0.318
0.366
3.32
2.72
2.29
1.91
1.76
1.59
4.17
3.41
2.95
2.55
2.17
1.96
0.302
0.367
0.436
0.524
0.568
0.630
0.240
0.293
0.340
0.392
0.461
0.510
TABLE 2.2
CHARACTERISTICS OF ILLINOIS IBEAM BRIDGES
Side Span Ratio a is 0.85 for L < 80 ft, and
0.78 for L > 80 ft.
(1) (2) (3) (4) (5) (6) (7)
L w Wb Composite Action Noncomposite Action
ft kip/ft kips f, cps Tl, sec fl, cps T1, sec
For H1544 Loading
3.40
3.44
3.51
3.57
3.64
3.71
3.57
3.62
3.69
3.75
3.81
3.88
3.95
4.02
4.11
4.03
4.10
4.16
4.25
4.37
4.47
4.57
4.69
4.79
4.88
4.95
118.9
137.7
158. 1
178.7
200.0
222.3
11.46
9.48
8.34
7.39
6.65
6.15
0.087
0.105
0.120
0.135
0.150
0.163
For H20S1644 Loading
125.0
144.7
165.9
187.4
209.6
232.8
256.9
281.2
308.5
322.4
348.2
374.7
404.0
437.4
469.8
503.1
539.3
574.5
610.0
643.8
11.37
9.41
8.30
7.33
6.59
6.20
5.61
5.19
4.79
4.12
3.88
3.54
3.25
2.95
2.69
2.53
2.39
2.26
2.11
1.97
0.088
0.106
0.120
0.136
0.152
0.161
0.178
0.193
0.209
0.243
0.258
0.283
0.308
0.339
0.372
0.395
0.418
0.442
0.474
0.508
7.10
5.94
5.33
4.78
4.36
4.15
6.99
5.84
5.24
4.69
4.27
4.07
3.75
3.51
3.28
2.77
2.64
2.43
2.26
2.07
1.92
1.83
1.76
1 .68
1.58
1.49
0.141
0.168
0.188
0.290
0.230
0.241
0.143
0.171
0.191
0.213
0.234
0.246
0.267
0.285
0.305
0.362
0.379
0.412
0.442
0.482
0.520
0.546
0.569
0.597
0.634
0.671
*For L > 60 ft, the characteristics of bridges designed for HI1544
loading are identical to those of bridges designed for H20SI644.
TABLE 2.3
CHARACTERISTICS OF OTHER BRIDGES
L a w Fund. Natural Frequency, fl, cps Damping
Ref. Percent
Ref. Bridge ft kip/ft Observed Computed Values Percent
No. Designation Critical
(1) (2) (3) (4) (5) (6)t (7) (8)
For H1544 Loading
Doyles Branch
Kansas 1
Kansas 2
Range Creek
Laws Creek
Big Creek
Maine 2
CL4137
Nevada 2
Texas 4
Johnson Co.
Missouri 2
Little Muddy
Pecatonica
Kansas 3
Kansas 4
Kansas 5
Kansas 6
Indiana
Arkansas 3
B2 of 38114
Kansas 7
Jackson Byp.
Kansas 8
33.0
43.0
49.0
49.3
60.7
62.5
65.0
70.0
71.8
75.0
86.0
90.0
104.2
113.0
57.5
62.5
67.5
72.5
75.0
80.0
80.6
82.8
92.0
93.0
0.80
0.74
0.82
0.78
0.78
0.78
0.71
0.80
0.80
0.79
0.78
0.67
0.78
0.78
4.17
2.81
2.92
2.81
2.95
2.94
3.30
6.17
2.62
3.24
3.48
3.63
4.75
5.20
For H20S1644 Loading
0.83
0.84
0.85
0.86
0.83
0.81
0.53
0.82
0.80
0.84
3.22
3.26
3.31
3.36
3.88
3.39
3.53
5.27
5.20
(29) Kansas 9
(1) HA5021
*From original reference
t and tt Computed in this
respectively.
103.0 0.85 3.58
  2.56 1.76
153.5 0.98  2.75 2.48
study for composite action and noncomposite action,
12.40
6.50
5.20
4.80
2.91
3.40
2.80
2.40
3.56
5.28
4.86
12.40
8.75
7.02
6.50
5.20
4.80
4.71
4.56
3.85
3.40
3.59
2.80
2.40
5.59
4.78
4.60
4.45
4.10
3.68
3.16
5.35
4.41
3.02
2.89
2.46
2.32
3.53
3.06
2.98
2.95
2.67
2.48
2.15
TABLE 3.1
MAXIMUM STATIC MOMENTS AT FIVE DIFFERENT
SECTIONS DUE TO TWO EQUAL FORCES
Side Span Ratio a = 0.8; Each Force of Magnitude W/2
s/L MI or M4 M2 or M3 M
I r2 3 c
(a) In terms of WL
0.1660
0.0877
(b) In terms of Corresponding Moment for s/L = 0
0
0.1
0.169
0.2
0.3
0.338
0.4
0.5
0.6
0.675
0.7
0.8
1.000
0.869
0.783
0.745
0.637
0.601
0.549
0.500
0.500
0.500
0.500
0.500
1.000
0.986
0.962
0.948
0.885
0.855
0.800
0.852
0.903
0.920
0.922
0.913
0.1685
1.000
0.861
0.776
0.741
0.642
0.609
0.559
0.498
0.456
0.436
0.431
0.418
TABLE 3.2
MAXIMUM STATIC SHEARS AND REACTIONS AT DIFFERENT
SECTIONS DUE TO TWO EQUAL FORCES
Side Span Ratio a = 0.8; Each Force of
Magnitude W/2 Results given in terms of W
s/L SI, RI or R4 S2 S3 R2 or R3
1.000
0.923
0.847
0.774
0.705
0.642
1.000
0.963
0.919
0.866
0.810
0.750
1.000
0.962
0.915
0.858
0.795
0.727
1.000
0.995
0.978
0.953
0.918
0.875
0T) i^ t 10 tj o Lr
^o I CM o~ ocD
to vJ " 0D 0) OD
o oo m r ao(
CD N00)0 a

m0 r CM 0 00 0C
l N 0) 0) Ca Co
L co co * CO
in 00co U') LO Lo
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TABLE 4.1
MAXIMUM STATIC EFFECTS FOR A THREE SPAN BEAM SUBJECTED
TO TWO CONSTANT FORCES EACH OF MAGNITUDE W/2
Prismatic Beam with Side Span Ratio a = 0.8
Ds I or D4 D MI or M4 M2 or M3 M R2
L WL3/EI WL /EI WL WL WL W
0
0.1
0.169
0.2
0.3
0.338
0.4
0.5
0.6
0.675
0.7
0.8
0.0077
0.0075
0.0072
0.0070
0.0061
0.0058
0.0051
0.0040
0.0039
0.0039
0.0039
0.0039
0.0106
0.0104
0.0101
0.0099
0.0090
0.0086
0.0079
0.0067
0.0053
0.0043
0.0041
0.0037
0.1660
0.1442
0.1300
0.1237
0.1057
0.0998
0.0912
0.0830
0.0853
0.0830
0.0830
0.0831
0.0877
0.0865
0.0844
0.0831
0.0776
0.0750
0.0702
0.0747
0.0792
0.0807
0.0809
0.0801
0.1685
0.1450
0.1307
0.1248
0.1081
0.1026
0.0942
0.0839
0.0768
0.0735
0.0726
0.0705
1.000
0.995
0.985
0.978
0.953
0.940
0.918
0.875
0.825
0.784
0.769
0.708
TABLE 4.2
AMPLIFICATION FACTORS FOR MAXIMUM EFFECTS IN A THREESPAN BEAM
SUBJECTED TO TWO MOVING CONSTANT FORCES EACH OF MAGNITUDE W/2
Uniform, Undamped Beam with Side Span Ratio a = 0.8
Results for Deflection are Expressed in terms of Maximum Static Effect at Midspan.
All Other Data are in terms of Maximum Static Effect at Section under Consideration
For Maximum Static Effects see Table 4.1
s Amplification Factors For

L D1 D D M M M M M R
1 c 4 1 2 c 3 4 2
0 0.06 0.740 1.019 0.758 0.989 1.014 0.974 1.038 1.010 1.021
0.09 0.764 1.055 0.771 1.042 1.083 1.043 1.056 1.019 1.024
0.10 0.778 1.055 0.723 1.027 1.066 0.998 1.089 1.004 1.023
0.12 0.758 1.077 0.755 0.976 1.108 1.027 1.065 1.043 1.073
0.14 0.724 1.018 0.781 0.993 1.067 0.993 1.085 1.027 1.088
0.15 0.741 1.105 0.768 1.020 1.051 1.060 1.089 1.011 1.078
0.16 0.756 1.110 0.820 1.041 1.061 1.054 1.152 0.907 1.073
0.18 0.772 1.030 0.739 1.050 1.186 1.008 1.183 0.986 1.072
0.20 0.774 1.140 0.778 1.024 1.213 1.095 1.143 1.052 1.052
0.21 0.773 1.147 0.758 1.016 1.167 1.109 1.126 1.050 1.026
0.24 0.781 1.023 0.906 1.047 1.215 0.933 1.078 1.182 1.021
0.30 0.884 1.274 0.855 1.165 1.155 1.215 1.273 1.008 1.050
0.40 1.002 1.183 1.024 1.245 1.311 1.027 1.322 1.237 1.063
0.50 1.051 1.733 1.040 1.176 1.572 1.550 1.818 1.213 1.348
0.60 1.068 2.013 1.351 1.096 1.978 1.726 1.499 1.584 1.594
0.1 0.06 0.729 1.029 0.737 0.997 1.013 1.019 1.040 1.008 1.010
0.09 0.750 1.011 0.756 1.024 1.053 1.017 1.059 1.040 1.011
0.10 0.724 1.019 0.736 1.001 1.065 1.019 1.066 1.007 1.024
0.12 0.737 1.031 0.729 1.010 1.039 1.028 1.030 1.011 1.045
0.15 0.721 1.066 0.738 0.998 1.022 1.056 1.026 1.028 1.003
0.18 0.744 1.032 0.726 1.022 1.075 1.036 1.030 0.998 1.000
0.20 0.761 1.119 0.769 1.036 1.080 1.102 1.067 1.044 0.987
0.21 0.767 1.113 0.738 1.043 1.050 1.104 1.087 1.038 0.990
0.24 0.785 1.009 0.851 1.083 1.156 1.004 1.029 1.147 0.997
0.169 0.20 0.762 1.084 0.754 1.055 1.110 1.087 1.064 1.029 1.059
0.2 0.10 0.723 1.058 0.748 1.029 1.072 1.057 1.063 1.020 1.033
0.15 0.747 1.010 0.735 1.058 1.060 1.021 1.096 1.048 1.093
0.20 0.761 1.064 0.743 1.053 1.149 1.061 1.074 1.015 1.056
0.3 0.06 0.693 1.005 0.697 1.006 1.033 0.992 1.030 1.001 1.020
0.09 0.691 1.038 0.690 1.002 1.025 1.010 1.033 1.019 1.030
0.10 0.712 1.019 0.708 1.031 1.029 1.010 1.031 1.016 1.039
0.12 0.699 1.044 0.743 1.013 1.047 1.057 1.012 1.001 1.039
0.15 0.738 1.064 0.761 1.028 1.054 1.112 1.134 1.055 1.062
0.18 0.710 1.130 0.764 1.073 1.071 1.078 1.124 1.028 1.010
0.20 0.715 0.990 0.690 1.010 1.154 1.101 1.074 1.058 1.036
0.21 0.727 1.068 0.778 0.991 1.228 1.136 1.041 1.146 1.031
0.24 0.770 1.147 0.798 1.061 1.187 1.109 1.055 1.206 1.038
TABLE 4.2 (Continued)
Amplication Factors For
L D D D M M2 M M3 M4 R2
1 c 4 1 2 c 3 4 2
0.338 0.20 0.700 1.014 0.707
0.4 0.10
0.15
0.20
0.5 0.06
0.09
0.10
0.12
0.15
0.18
0.20
0.21
0.24
0.671
0.726
0.706
0.634
0.647
0.623
0.656
0.638
0.721
0.733
0.727
0.669
1.081
1.022
1.065
1.043
1.048
1.019
1.118
1.093
1.075
1.060
1.103
1.255
0.689
0.739
0.731
0.616
0.629
0.632
0.673
0.628
0.704
0.730
0.733
0.767
0.6 0.20 0.776 1.067 0.792
0.675 0.20 0.968 1.212 1.025
0.7 0.20 1.017 1.213 1.097
0.8 0.20 1.121 1.358 1.170
1.057 1.055
1.036
1.030
1.077
0.994
1.048
1.027
0.980
1.027
1.053
1.026
1.020
1.052
1.073
1.081
1.117
1.005
1.062
1.035
1.064
1.053
1.041
1.086
1.122
1.062
1.021 1.055
1.023 1 .103
1.023 1.117
1.023 1.162
1.142 1.078 1.135
1.044
1.051
1. 167
0.959
1.040
1.004
1.058
1.002
1.034
1.090
1.138
1.001
1.065
1.104
1.125
1.007
1.049
1.053
1.040
1.071
1.079
1.112
1.122
1.175
0.998
0.980
1.138
0.992
1.054
1.007
1.048
1.011
1.005
1.130
1.109
1.215
1.002 1.044 0.961
1.066 1.140 1.081
1.124 1.147 1.131
1.214 1.084 0.998 1.096
1.039
1.067
1.062
1.074
1.028
1.050
1.040
1.054
1.051
1.059
1.078
1.089
1.098
1.103
1.117
1.129
TABLE 5.1
AMPLIFICATION FACTORS FOR MAXIMUM EFFECTS IN A THREESPAN
BEAM SUBJECTED TO A TWOAXLE, SMOOTHLY MOVING, SPRUNG LOAD
Uniform Undamped Beam with Side Span Ratio a = 0.8
Identical Axles with a Static Reaction per Axle of W/2,
Dynamic Index = 1.0 and p = o
Results for Deflection are Expressed in Terms of Maximum Static Effect at Midspan.
All Other Data are in terms of Maximum Static Effect at Section Under Consideration.
For Maximum Static Effects See Table 4.1
s Amplification Factors For
L D, D D 4
S0.15c 4
0 0.15 0.732 1.078 0.733
1.062
1.152
1.117
1.233
1.022
1.098
1.221
1.051
1.043
1.171
1.040
1.085
1.098
1.082
1.058
1.142
1.033
1.104
1.092
0.731
0.807
0.799
0.945
0.726
0.782
0.925
0.745
0.762
0.877
0.708
0.749
0.788
0.675
0.725
0.741
0.637
0.631
0.740
0 0.15 0.750 1.082 0.801
0.18 0.785 1.109 0.761
M M2 Mc
1 2 c
R = 0.175, (t = 0.3
1.010 1.023 1.037
R = 0.175, et = 0.5
1.046 1.062 0.981
1.023 1.114 1.075
1.042 1.172 1.127
1.004 1.251 1.183
1.016 1.082 1.021
1.012 1.085 1.089
1.020 1.107 1.202
1.046 1.074 1.050
1.075 1.113 1.063
1.032 1.081 1.176
1.046 1.024 1.003
1.030 1.076 1.118
1.015 1.102 1.181
1.047 1.071 1.012
1.046 1.081 1.064
1.071 1.109 1.241
1.039 1.012 0.983
1.028 1.060 1.013
1.015 1.070 1.127
R = 0.175, et = 0.6
1.040 1.053 1.026
1.056 1.259 1.058
M M4 R2
3 4 2
1.116 0.972
1.072
1.093
1.247
1.302
1.061
1.034
1.202
1.059
1.122
1.187
1.042
1.128
1.158
1.068
1.125
1.200
1.066
1.069
1.087
1.012
1.056
1.065
1.265
0.998
1.084
1.243
1 .023
1.060
1.191
1.026
1.046
1.076
1.031
0.993
1.192
1.046
0.966
1.205
1.122 1.005
1.149 1.061
1.080
1.025
1.038
1.038
1.029
1.031
1.000
0.997
1.028
1.047
1.042
1.040
1.046
1.018
1.063
1.052
1.071
1.053
1.054
1.062
1.088
1.028
0.3 0.15 0.736 1.083 0.739 1.037 1.066 1.111
0 0.10
0.15
0.18
0.20
0.1 0.10
0.15
0.20
0.2 0.10
0.15
0.20
0.3 0.10
0.15
0.20
0.4 0.10
0.15
0.20
0.5 0.10
0.15
0.20
0.790
0.738
0.771
0.774
0.737
0.731
0.749
0.735
0.761
0.746
0.714
0.738
0.710
0.670
0.725
0.706
0.628
0.635
0.727
1.129 1.035 1.038
TABLE 5.1 (Continued)
Amplification Factors For
L D1 Dc D4 M I M2 Mc M3 M4 R2
0 0.15 0.764 1.147
0.18 0.805 1.019
0.844
0.805
R = 0.175, tt = 0.7
1.058 1.106 1.056
1.078 1.239 0.981
0.3 0.15 0.737 1.132 0.719 1.047 1.035 1.137
R = 0.175, (t = 0.8
0 0.15 0.775 1.079 0.797 1.071 1.088 0.960
0.18 0.825 1.104 0.815 1.104 1.162 1.101
0.3 0.15 0.745 1.115 0.699 1.063 1.117 1.052
0 0.15 0.785 1.094
0.18 0.842 1.160
0.824
0.3 0.15 0.759 1.100 0.745
R = 0.175, 4t = 0.9
1.073 1.092 0.999
1.129 1.175 1.145
1.079 1.123 1.093
R = 0.175, et
0 0.06
0.075
0.09
0.10
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.20
0.21
0.24
0.1 0.06
0.09
0.10
0.12
0.15
0.18
0.20
0.21
0.24
0.756
0.744
0.755
0.759
0.726
0.728
0.755
0.788
0.810
0.835
0.850
0.876
0.883
0.889
0.720
0.749
0.730
0.724
0.767
0.809
0.833
0.843
0.854
1.015
1.035
1.060
1.059
1.080
1.064
1.043
1.102
1.110
1.040
1.089
1.152
1. 161
1.120
1.014
1.014
1.036
1.060
1.092
1.063
1.154
1.146
1.081
0.743
0.743
0.732
0.763
0.731
0.764
0.802
0.763
0.787
0.827
0.748
0.837
0.756
0.959
0.719
0.729
0.733
0.736
0.731
0.736
0.796
0.761
0.897
0.989
0.999
1.036
1.004
0.982
0.994
1.017
1.062
1.107
1.134
1.147
1.115
1.092
1.057
1.003
1.022
0.991
1.015
1.058
1.108
1.133
.142
1.139
1.029
1.051
1.066
1.087
1.077
1.068
1.136
1.127
1.114
1.132
1.218
1.281
1.257
1.301
1.013
1.054
1.076
1.040
1.032
1.084
1.157
1.141
1.203
= 1.0
0.984
1.009
1.030
1.030
1.050
0.990
1.029
1.066
1.064
0.919
1.049
1.105
1.150
0.969
1.009
1.022
1.031
1.053
1.081
1.063
1.134
1. 140
1.057
1.107 1.020
1.235 1.073
1.156 1.009
1.129 0.998
1.182 1.079
1.083 1.017
1.188 
1.118 1.048
1.026 1.019
1.022
1.033
1.051
1.085
1.017
1.101
1.095
1.168
1.172
1.154
1.136
1.246
1.250
1.199
1.020
1.035
1.077
1.024
1.031
1.013
1.095
1.117
.147
1.019
1.019
1.011
0.976
1.002
1.036
0.986
0.966
0.984
1.120
1.025
1.119
1.041
1.177
1.004
1.012
1.001
1.017
1.013
1.027
1.078
1.060
1.185
1.078
1.043
1 .037
1.055
1.075
1.063
1.081
1.048
1.094
1.019
1.027
1.031
1.036
1.102
1.107
1.115
1.123
1.102
1.087
1.110
1.102
1.090
1.057
1.006
1.014
1.026
1.026
0.991
1.014
1.041
1.040
1.047
TABLE 5.1 (Continued)
Amplification Factors For
L Da D D4 M M2 M M M R2
1 c 4 I 2 c 3 4 2
0.2 0.10 0.718
0.15 0.779
0.20 0.756
0.3 0.06
0.09
0.10
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.20
0.21
0.24
0.4 0.10
0.15
0.20
0.5 0.06
0.09
0.10
0.12
0.15
0.18
0.20
0.21
0.24
0.705
0.697
0.727
0.718
0.740
0.763
0.774
0.772
0.766
0.752
0.716
0.707
0.765
0.705
0.699
0.697
0.607
0.643
0.626
0.664
0.659
0.687
0.692
0.682
0.658
1.041
1.050
1.131
1.018
1.031
1.003
1.045
1.099
1.135
1.081
1.082
1 .152
1.164
1 .065
1.061
1 .144
1.073
1.071
1.184
1.027
1 .032
1.036
1.115
1.038
1.199
1.207
1.136
1.218
0.728
0.764
0.734
0.695
0.694
0.706
0.738
0.701
0.734
0.751
0.732
0.723
0.796
0.686
0.763
0.776
0.692
0.703
0.692
0.603
0.631
0.634
0.666
0.647
0.661
0.723
0.727
0.714
1.018
1.074
1.088
0.973
1.020
1.014
0.987
1.007
1.047
1.090
1.124
1.154
1 .148
1 .100
1.085
1 .039
1.010
1.065
1.141
0.991
1.044
1.018
0.979
1.051
1.108
1.075
1.062
1.064
1.075
1.116
1.149
1.035
1.037
1.038
1.053
1.074
1.066
1.045
1.068
1.074
1.083
1.106
1.214
1.215
1.054
1.045
1.091
1.010
1.044
1.043
1.021
1.087
1.080
0.986
1.024
1.061
1.040
1.041
1.133
1.012
1.012
1.024
1.081
1 .078
1 .080
1.109
1.106
1.095
1 . 100
1.102
1.146
1.104
1.013
1 .072
1.223
0.973
1.037
1.030
1.033
1.003
0.965
1.120
1.221
0.983
R = 0.175, Ot = 1.1
0 0.15 0.784 1.103 0.798 1.042 1.123 0.997
0.18 0.853 1.011 0.832 1.156 1.194 1.033
0.3 0.15 0.782 1.107 0.739 1.099 1.069 1.077
R = 0.175, <t
0 0.15 0.773 1.030 0.763 1.018 1.109
0.3 0.15 0.785 1.112
0 0.18 0.854 1.108
0.767
= 1.2
0.958
1.097 1.112 1.082
1.054
1.116
1.141
1.020
1.048
1.058
1.083
1.025
1.094
1.105
1.041
1.031
1.090
1.084
1.112
1.128
1.052
1.153
1.217
1.012
1.050
1.063
1.057
1.084
1.105
1.150
1.076
1. 100
1.020
1.044
1.050
1.018
1.026
1.022
1.005
1.009
1.017
1.060
0.999
1.136
1.079
1.005
1.166
1.118
0.988
0.967
1.137
0.983
1.050
0.997
1.018
0.984
1.110
0.988
1 .169
1.162
1.164 0.974
1.179 1.074
1.127 1.029
1.139 1.019
1.073 1.147
R = 0.175, 0t = 1.25
0.777 1.148 1.181 1.112 1.230 1.006
1 .035
1.067
1 .052
1.014
1.012
1.033
1.065
1.060
1.070
1.099
1.095
1.057
0.998
1.040
1.041
1.027
1.081
1.058
1.079
1.017
1.036
1.045
1.059
1.074
1.072
1. 104
1.092
1.112
1.112
1 .102
1.068
1.064
1.058
1.087
TABLE 5.1 (Continued)
Amplification Factors For
L D D c D4 M M Mc M3 M4 R2
I c 4 1 2 c 3 42
R = 0.175, 4t = 1.3
0 0.15 0.768 1.079 0.797 0.997 1.119 0.982
0.3 0.15 0.787 1.118 0.701 1.087 1.121 1.108
R = 0.175, (tb
0 0.15 0.757 1.102 0.813 0.983 1.101
0.18 0.836 1.153 0.805 1.121 1.163
= 1.4
1.012
1.094
0.3 0.15 0.785 1.101 0.729 1.075 1.112 1.115
1.094 1.074
1.032 1.107
1.080 1.099
1.214 1.121
1.052 1.034
R = 0.175, 0t = 1.5
0 0.10
0.15
0.18
0.20
0.1 0. 10
0.15
0.20
0.2 0.10
0.15
0.20
0.3 0.10
0.15
0.20
0.4 0.10
0.15
0.20
0.778
0.743
0.827
0.873
0.742
0.735
0.838
0.716
0.764
0.789
0.712
0.783
0.766
0.706
0.730
0.733
0.5 0.10 0.617
0.15 0.631
0.20 0.728
0 0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.739
0.779
0.793
0.738
0.770
0.770
0.748
1.055
1.109
1 .153
1.173
1.025
1.060
1.104
1.026
1.032
1.054
1.019
1.085
1.104
1.085
1.067
1.157
1.038
1.081
1.148
1.029
1.034
1.064
1.196
1.185
1.335
1.287
0.779
0.826
0.831
0.792
0.732
0.766
0.772
0.759
0.770
0.804
0.720
0.723
0.831
0.682
0.659
0.769
0.621
0.667
0.701
0.761
0.786
0.787
0.812
0.870
1.052
0.806
1.007
0.979
1 .098
1 .157
1 .024
1.028
1.136
1.015
1.038
1.114
1 .003
1.062
1 .102
0.994
0.954
1 .162
1 .007
0.989
1.107
1 .063
1 .061
1. 188
1.059
1.041
1.003
1 .034
1 .090
1.115
1.095
1 .065
1 .107
1 .205
1 .080
1.139
1 .199
1 .031
1 .067
1 .087
1.033
1.025
1.067
1.075
1 .028
1.066
1.110
1.066
1.036
1.064
1.027
1.118
1.112
1.042
1.076
1.185
1 .020
1.000
1.082
R = 0.3, t = 0.5
t
0.985
1 .063
1 .011
1 .025
1 .036
0.975
0.979
1 .020
1 .086
1 .086
1.160
1.205
1 .269
1.147
0.994
1.024
1.006
1.093
1.201
1 .220
0.927
1.019
1.140
1.184
1.234
1.018
1.051
1.133
1.033
1.054
1.086
1.031
1.060
1.058
1 .068
1.031
1.051
1.019
1.003
1 .030
1.046
1.089
1.094
1.124
1.312
1.403
1.351
1.066
1.046
1.122
0.979
1.018
1.043
1.010
1.075
1.063
1.095
0.992
0.996
1.125
1.059
1.052
0.993
1.025
1.035
0.896
0.977
1.065
1.070
1.059
1.155
1 .367
1.020
1.044
1.059
1 .025
1 .11 1
1 .039
1 .057
1 .038
1.078
1 .143
1.027
0.999
1 .024
1 .064
1 .042
1 .060
1.048
1 .039
1.031
1.080
1 .060
1 .097
1.046
1.058
1.090
1.036
1.035
1.033
1.007
1.055
1.060
1.081
TABLE 5.1 (Continued)
Amplification Factors For
L D D D M M M M M R
I c 4 1 2 c 3 4 2
R = 0.3, Ot = 0.6
0 0.18 0.793 1.167 0.764 1.059 1.295 1.118
R = 0.3, Ot = 0.7
0 0.18 0.822 1.070 0.888 1.093 1.284 0.986
R = 0.3, t = 0.8
0 0.18 0.851 1.138 0.785 1.131 1.162 1.109
R = 0.3, (t = 0.9
0 0.18 0.872 1.227 0.867 1.165 1.233 1.192
R = 0.3, (t =
0 0.06
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.20
0.21
0.24
0.748
0.745
0.748
0.750
0.737
0.724
0.739
0.773
0.808
0.832
0.863
0.881
0.917
0.930
0.951
1.020
1.033
1.048
1.060
1.031
1.081
1.032
1.105
1.123
1.062
1.059
1.170
1.224
1 .158
1.196
0.747
0.732
0.733
0.760
0.762
0.735
0.785
0.801
0.751
0.817
0.830
0.769
0.883
0.768
0.964
0.989
1.024
1.029
0.989
0.922
1.001
1.018
1.040
1.085
1.136
1.170
1.189
1.159
1.130
1.065
1.011
1.038
1.068
1.085
1.076
1.071
1.067
1.133
1.116
1.155
1.169
1.191
1.260
1.248
1.354
1.0
1.006
1.011
1 .023
1.037
0.978
1 .049
0.996
1.065
1.060
1.052
0.936
1.097
1.115
1.138
0.937
R = 0.3, (t = 1.1
0 0.18 0.888 1.068 0.832 1.198 1.202 1.023
R = 0.3, (t = 1.2
0 0.18 0.888 1.062 0.868 1.192 1.128 1.031
R = 0.3, Ot = 1.3
0 0.18 0.877 1.098 0.793 1.175 1.120 1.058
R = 0.3, (t = 1.4
0 0.18 0.867 1.119 0.783 1.152 1.149 1.075
1.135 1.061
1.231 1.144
1.268 1.035
1.199 1.118
1.039
1.062
1.086
1.052
1.137
1.057
1.081
1.058
1.218
1.265
1.179
1.118
1.281
1.307
1.432
1 .028
0.985
1.006
0.984
1 .035
0.981
1.055
1.049
0.957
1 .081
1.137
1.000
1.151
1.055
1.137
1.092 1.051
1.134 1.026
1.157 1.040
1.163 1.077 1.075
1.056
1.064
1 .082
1.069
1.045
1.032
1.013
1.029
1.033
1.087
1.107
1.105
1.127
1.116
1.101
1.120
1.122
1.119
1.106
1.139
1.138
1.124
TABLE 5.1 (Continued)
s Amplification Factors For
L D1 Dc D4 Ml M2 Mc M3 M4 R2
R = 0.3, 4t = 1.5
1.017
1.026
1.033
1.061
1.143
1.037
1.337
0.750
0.736
0.745
0.819
0.782
0.841
0.964
0.3 0.15 0.738 1.112 0.727
0.3 0.15 0.739 1.076 0.730
0.3 0.15 0.744 1.153 0.758
0.3 0.15 0.756 1.174 0.777
0.3 0.15 0.776 1.166 0.742
0 0.10 0.749
0.15 0.813
0.20 0.927
0.1 0.10 0.731
0.15 0.801
0.20 0.882
0.2 0.10 0.730
0.15 0.796
0.20 0.823
0.3 0.10 0.743
0.15 0.796
0.20 0.754
1.064
1.139
1.262
1.026
1.101
1.226
1.026
1.048
1.179
1.050
1.041
1.106
1.051
1.037
1.197
1.073
1.040
1.214
0.752
0.731
0.882
0.741
0.735
0.836
0.747
0.794
0.810
0.743
0.749
0.736
0.690
0.715
0.678
0.635
0.663
0.657
1.035
1 .051
1 .001
1.004
1.128
1.222
1.236
1.022
1.069
1 .146
1.057
1.137
1.082
1.291
0.982
1.018
1.002
0.978
1.088
1.005
1.283
R = 0.35, 0t = 0.5
1.037 1.112 1.121
R = 0.35, *t = 0.6
1.051 1.071 1.079
R = 0.35, Ot = 0.7
1.070 1.038 1.135
R = 0.35, At = 0.8
1.092 1.130 1.100
R = 0.35, 4t = 0.9
1.111 1.141 1.153
R = 0.35, ot =
0.985
1.094
1.172
0.998
1.102
1.199
1.036
1.091
1.186
0.998
1.123
1.166
0.991
1 .102
1.195
1.006
1.063
1.115
1.051
1.099
1.254
1.078
1.044
1.149
1 .083
1.116
1 .205
1.056
1.076
1 .139
1.052
1 .059
1.059
1.084
1.126
1.007
1.0
1 .039
1 .057
1.117
1.028
1.083
1.207
1.036
1.039
1.174
1.018
1.099
1.177
0.999
1.140
1.208
1.053
1.059
1.178
1.042
1.048
1.100
1.182
1.153
1 .145
1.327
0.990
1.010
1.000
1.114
1.072
1.132
1.118
1.136 1.024
1.113 1.050
1.170 1.055
1.133 1.096
1.052 1.038
1.035
1.213
1.268
1.069
1.110
1.113
1.062
1.164
1.235
1.020
1.041
1.157
1.017
1.090
1.236
1.068
1.057
1.079
0.988
0.965
1.140
1.022
1.002
1.135
1.056
1.086
1.134
1.024
1.111
1.054
0.953
1.042
1.080
1.019
1.011
0.997
0 0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.753
0.769
0.768
0.750
0.857
0.930
0.994
1.035
1.048
1.081
1.061
1.044
1.086
1.143
1 .032
1.015
1.045
1.057
1.082
1.019
1 .118
1.111
1 .002
0.988
1.021
1.010
1.069
1.089
1 .029
1.101
1.038
1.074
1.045
1.142
1.075
1.107
1.126
0.4 0.10
0.15
0.20
0.5 0.10
0.15
0.20
0.719
0.714
0.719
0.655
0.686
0.698
TABLE 5.1 (Continued)
Amplification Factors For
L DI Dc D4 M M2 Mc M3 M4 2
R = 0.35, 4t = 1.1
0.3 0.15 0.809 1.102 0.708 1.136 1.005 1.083 1.140 1.037 1.079
R = 0.35, Pt = 1.2
0.3 0.15 0.813 1.153 0.793 1.136 1.096 1.111 1.174 1.085 1.053
R = 0.35, Ot = 1.3
0.3 0.15 0.821 1.174 0.832 1.122 1.129 1.111 1.150 1.157 1.024
R = 0.35, t = 1.4
0.3 0.15 0.821 1.175 0.790 1.113 1.118 1.124 1.134 1.187 1.042
R = 0.35, (t = 1.5
0.3 0.15 0.811 1.156 0.762 1.097 1.083 1.127 1.063 1.157 1.092
R = 0.5, Ot = 0.3
0 0.15 0.718 1.027 0.700 0.992 0.972 0.992 1.159 0.892 1.124
R = 0.5, 0t 0.5
0 0.15 0.756 1.260 0.797 1.025 1.231 1.131 1.172 1.026 1.045
0.3 0.15 0.739 1.141 0.708 1.044 1.144 1.145 1.151 1.032 1.020
R = 0.5, Dt = 0.6
0 0.15 0.786 1.044 0.892 1.062 1.152 0.954 1.111 1.026 1.080
R = 0.5, <t = 0.7
0 0.15 0.802 1.097 0.786 1.099 1.091 1.039 1.241 1.012 1.128
0.3 0.15 0.750 1.138 0.773 1.087 1.037 1.114 1.173 1.077 1.049
R = 0.5, (t = 0.8
0 0.15 0.811 1.142 0.807 1.122 1.134 1.038 1.192 1.089 1.078
R = 0.5, 4t = 0.9
0 0.15 0.821 1.110 0.817 1.128 1.093 1.009 1.096 1.074 1.060
0.3 0.15 0.786 1.183 0.708 1.126 1.137 1.165 1.104 1.026 1.054
R = 0.5, Ot = 1.0
0 0.15 0.823 1.152 0.762 1.118 1.073 1.059 1.124 1.008 1.064
0.3 0.15 0.806 1.093 0.779 1.138 1.112 1.148 1.060 1.065 1.068
TABLE 5.1 (Continued)
D 4
0.78
0.73i
0.79
0.79£
0.81
0.78c
0.77
0.811
0.781
Amplification Factors For
MI M M c
R = 0.5, et = 1.1
1 1.102 1.066 1.112
8 1.148 1.058 1.118
R = 0.5, 4t = 1.2
1 1.086 1.144 1.105
R = 0.5, 1t = 1.3
9 1.074 1.199 1.044
3 1.130 1.124 1.155
R = 0.5, Ot = 1.4
9 1.068 1.192 1.013
R = 0.5, ct = 1.5
9 1.063 1.188 1.019
8 1.135 1.235 1.144
R = 1.0, (t = 1.0
8 1.192 1.294 1.146
D
L
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
D
0.792
0.818
0.811
0.798
0.832
0.787
0.784
0.815
0.925
C
1 .154
1 .019
1 .127
1.103
1.151
1 .078
1.064
1 .178
1 .142
R2
1 .093
1.054
1 .098
1.064
1 .058
1.048
1.044
1.119
1.070
M 3
1.149
1 .147
1.115
1 .142
1 .136
1.169
1 .124
1 .142
1 .216
M 4
1.092
1 .138
1 .094
1 .055
1 .100
1 .003
1 .005
1 .105
1.055
S.
TABLE 6.1
AMPLIFICATION FACTORS FOR MAXIMUM EFFECTS IN A THREESPAN BEAM SUBJECTED
TO A TWOAXLE, UNDAMPED VEHICLE WITH 15% INITIAL OSCILLATION
Uniform Undamped Beam with SideSpan Ratio a = 0.8
Identical Axles with a Static Reaction per Axle of W/2,
Dynamic Index = 1.0 and 1 = o
Results for Deflection are Expressed in Terms of Maximum Static Effect at Midspan.
All Other Data are in Terms of Maximum Static Effect at Section Under Consideration.
For Maximum Static Effects See Table 4.1
8 Amplification Factors For
SD D D M M M M M R
(deg.) 1 c 4 1 2 c 3 4 2
s/L = 0, R = 0.175, ot = 0.6
0.820
0.785
0.740
0.686
0.748
0.804
0.852
0.872
0.879
0.862
0.850
0.842
0.910
0.885
0.847
0.803
0.772
0.746
0.713
0.750
0.811
0.866
0.895
0.916
0.954
0.988
1.084
1.172
1.241
1.267
1.267
1.225
1.164
1.101
1.048
0.986
1.180
1.183
1.170
1 .142
1.096
1.104
1.129
.143
.166
1.172
1.180
1.182
0.777
0.844
0.905
0.938
0.935
0.913
0.864
0.796
0.773
0.728
0.696
0.722
s/L
0.825
0.786
0.811
0.826
0.838
0.827
0.815
0.846
0.885
0.898
0.906
0.873
0.941
0.888
0.875
0.907
0.974
1.060
1.140
.193
1.206
1.174
1.106
1.021
= 0, R =
1 .209
1.209
1.169
1.101
1 .023
0.955
0.916
0.916
0.983
1.063
1.134
1.175
1.190
1.223
1.230
1.200
1.191
1.214
1.195
1.137
1.666
0.989
1.029
1.119
0.847
0.865
0.939
1.021
1.098
1.169
1.204
1.192
1.135
1.048
0.956
0.882
.167
1.142
1.089
1.109
1.154
1.166
1.141
1.204
1.261
1.280
1.282
1.240
0.175, <t = 1.0
1.213 1.096 1.231
1.191 1.087 1.197
1.152 1.074 1.170
1.106 1.058 1.137
1.066 1.044 1.105
1.043 1.036 1.114
1.042 1.037 1.142
1.064 1.045 1.177
1.103 1.059 1.209
1.149 1.075 1.231
1.189 1.088 1.235
1.212 1.096 1.222
0.917
0.997
1.076
1.138
1.173
1.184
1.147
1 .072
0.979
0.893
0.837
0.848
0.932
0.910
0.955
0.994
1.046
1.102
1.121
1.099
1.075
1.058
1.037
0.999
1.168
1.089
1.082
1.077
1.124
1.136
1.132
1.110
1.195
1.231
1.247
1.226
1.238
1.223
1.197
1.166
1.116
1.059
1.011
1.023
1.064
1.122
1.179
1 .222
0.15 0
30
60
90
120
150
180
210
240
270
300
330
0.15 0
30
60
90
120
150
180
210
240
270
300
330
TABLE 6.1 (Continued)
e Amplification Factors For
(deg.) D1 Dc D4 Ml M2 Mc M3 M4 R2
s/L = 0, R = 0.175, 4t = 1.5
0.893
0.893
0.895
0.883
0.852
0.812
0.775
0.758
0.781
0.808
0.844
0.875
0.946
0.977
1.018
1.029
1.048
1.063
1.071
1.054
1.018
0.970
0.926
0.895
1.174
1.171
1.180
1.156
1.187
1.222
1.203
1.201
1.193
1.153
1.134
1.144
1.086
1.057
1.031
1.009
0.983
0.963
0.965
0.993
1.031
1.066
1.091
1.098
s/L = 0, R = 0.35, t = 1.0
0.778
0.792
0.799
0.808
0.799
0.776
0.783
0.805
0.807
0.804
0.781
0.759
1.199
1.185
1.146
1.094
1.041
1.002
0.988
1.003
1.042
1. 100
1.147
1.185
1.143
1 .148
1.153
1 .144
1.122
1.095
1.069
1 .051
1.058
1.085
1.114
1.135
1.039
1.032
1.032
1.039
1.050
1.063
1.075
1.081
1.081
1.075
1.063
1.050
0.15 0
30
60
90
120
150
180
210
240
270
300
330
0.15 0
30
60
90
120
150
180
210
240
270
300
330
0.15 0
30
60
90
120
150
180
210
240
270
300
330
1.172
1.170
1.146
1.156
1.152
1.164
1.169
1.166
1.156
1.142
1.139
1.148
0.821
0.838
0.842
0.847
0.823
0.805
0.775
0.757
0.749
0.746
0.744
0.779
0.896
0.864
0.827
0.806
0.770
0.729
0.733
0.782
0.834
0.876
0.895
0.907
0.881
0.875
0.869
0.834
0.794
0.751
0.766
0.804
0.846
0.877
0.891
0.896
1 .159
1.150
1.134
1.121
1.106
1.091
1.081
1.079
1.091
1 .1 11
1.137
1.151
1.226
1.209
1 .191
1.162
1.117
1.095
1.108
1 .126
1.149
1.182
1.197
1.221
1.195
1.191
1.177
1.156
1.134
1.117
1.109
1.128
1.149
1.164
1.170
1.187
s/L = 0, R = 0.5, (t = 1.0
0.771 1.201 1.125 1.037
0.782 1.178 1.140 1.044
0.789 1.139 1.136 1.054
0.788 1.094 1.117 1.065
0.780 1.056 1.105 1.075
0.768 1.035 1.079 1.081
0.775 1.036 1.045 1.081
0.778 1.059 1.028 1.075
0.769 1.098 1.035 1.064
0.750 1.142 1.041 1.053
0.744 1.180 1.087 1.043
0.757 1.202 1.108 1.037
1.083
1.112
1.120
1.131
.128
1.103
1.063
1.018
0.980
0.963
0.997
1.041
0.882
0.866
0.884
0.940
1.002
1.055
1.083
1.080
1.058
1.029
0.982
0.928
0.947
0.944
0.958
0.986
1.020
1.050
1.070
1.073
1.058
1.031
0.997
0.966
1.158
1.175
1.214
1.212
1.189
1.176
1.205
1.179
1.148
.158
1.185
1.191
1.157
1.153
1.139
1.120
1.100
1.086
1.079
1.084
1.097
1.117
1.136
1.151
1.168
1.132
1.078
1.092
1.098
1.096
1.085
1.068
1.058
1.103
1.143
1.171
1.193
1.155
1.132
1.131
.152
1.189
1.232
1.270
1.293
1.294
1.273
1.236
1.217
.195
1.181
1.135
1.135
1.186
1.231
1.247
1.231
1.250
1.263
1.238
TABLE 6.1 (Continued)
Amplification Factors For
D D4 M1 M2 M M3 M4 R2
c 4 1 2 c 3 4 2
s/L = 0.3, R = 0.175, Ot = 0.6, L8 = 00
0.710
0.702
0.799
0.882
0.876
0.775
1.170
1.227
1.182
1.013
1.005
1.101
1.131
1 .073
1.187
1.226
1.216
1.172
1.292
1.269
1 .128
1.257
1.197
.169
1.291
1.250
1.089
0.986
1.114
1.202
/L = 0.3, R = 0.175, et = 1.0, Aý = 0°
0.795
0.796
0.785
0.765
0.744
0.720
0.706
0.713
0.734
0.757
0.775
0.783
1.192
1 .140
1.063
1.011
1.044
1.066
1 .11 1
1.167
1.218
1.254
1.249
1.239
1.116
1 .127
1.085
1.080
1.088
1 .085
1.071
1 .050
1.028
1.011
1.037
1.068
1.236
1.164
1.091
1.045
1.037
1.048
1.119
1.191
1.254
1.275
1 .294
1.267
1.142
1 . 125
1.104
1 .083
1 .067
1.060
1.069
1.093
1.121
1 .144
1.157
1.156
1.095
1.051
1.222
1.243
1.213
1.055
1.112
1.102
1.079
1.047
1.015
1.047
1.078
1.092
1.110
1.116
1.123
1.121
/L = 0.3, R = 0.175, (t = 1.0, LO = 600
0.15 0
60
120
180
240
300
0.15 0
30
60
90
120
150
180
210
240
270
300
330
0.15 0
30
60
90
120
150
180
210
240
270
300
330
0.869
0.872
0.787
0.651
0.723
0.817
0.801
0.739
0.687
0.732
0.770
0.801
0.838
0.870
0.889
0.898
0.883
0.847
0.822
0.785
0.750
0.713
0.730
0.759
0.780
0.810
0.834
0.850
0.853
0.847
1.278
1.218
1.072
1 .144
1.146
1.188
s
1.178
1.107
1.079
1.074
1.076
1.104
1 .156
1.215
1.251
1.273
1.263
1 .236
s
1.208
1.169
1.118
1.085
1.068
1.058
1.074
1.117
1. 167
1.195
1.224
1.214
1.222
1. 192
1.161
1.110
1.053
1.004
1.016
1.077
1.141
1.191
1.213
1.228
1.100
1.104
1.109
1.113
1.114
1.114
1 .111
1.111
1.111
1.108
1.104
1.099
1.132
1.133
1.130
1 .111
1 .075
1.032
1.001
1.021
1.043
1.068
1.096
1 .111
(deg.)
0.788
0.840
0.818
0.814
0.794
0.766
0.733
0.715
0.709
0.697
0.727
0.760
1.184
1.171
1.135
1.094
1.053
1.031
1.042
1.069
1.112
.150
1.187
1.197
1.101
1. 127
1.141
1.098
1.072
1.074
1.068
1.056
1.041
1 .027
1 .027
1.047
1.141
1.136
1.090
1.096
1.154
1.192
1.177
1.190
1.179
1.146
1.101
1.056
1.021
1.008
1.019
1.060
1.104
1.142
1.115
1.156
1.181
1.184
1.165
1.128
1.083
1.042
1.016
1.018
1.033
1.074
TABLE 6.1 (Continued)
Amplification Factors For
D D D0 M M M M M R
(deg.) 1 c 4 1 2 c 3 4 2
s/L = 0.3, R = 0.175, 4t = 1.0, 68 = 120°
0.740
0.780
0.809
0.822
0.827
0.811
0.779
0.742
0.729
0.721
0.711
0.700
1.119
1 .137
1 .142
1.146
1.146
1.130
1.110
1.083
1.047
1.046
1.066
1.093
1 .046
1 .058
1 .084
1 .097
1 .065
1 .058
1 .053
1 .063
1 .082
1 .086
1 .075
1 .051
1 .154
1 .150
1.159
1 .155
1.143
1 .132
1.108
1 .079
1 .059
1.076
1.114
1.147
1.070
1.049
1.067
1.099
1.134
1.160
1.172
1. 166
1 .155
1.141
1.116
1 .088
1.103
1.127
1.143
1.141
1.126
1.096
1.053
1.021
1.037
1.058
1.071
1.067
s/L = 0.3, R = 0.175, 4t = 1.0, £8 = 1800
0.15 0
30
60
90
120
150
180
210
240
270
300
330
0.15 0
30
60
90
120
150
180
210
240
270
300
330
0.15 0
60
120
180
240
300
0.15 0
60
120
180
240
300
0.702
0.720
0.761
0.796
0.816
0.819
0.812
0.789
0.756
0.721
0.708
0.698
0.999
1.033
1.082
1 .134
1 .173
1.191
1.211
1.199
1.179
1 .129
1 .058
0.989
1.069
1.071
1.077
1.070
1.063
1.082
1.073
1.048
1 .092
1.125
1 .133
1.112
1 .148
1.161
1.145
1.132
1.168
1 .202
1.210
1.207
1.183
1 .133
1 .070
1 .109
1.1 11
1.080
1 .028
1.040
1.092
1.147
1.191
1.212
1.205
1.185
1 .160
1.125
0.792
0.792
0.788
0.780
0.770
0.762
0.756
0.755
0.767
0.779
0.788
0.789
0.730
0.749
0.773
0.794
0.816
0.831
0.832
0.816
0.792
0.759
0.732
0.719
0.733
0.774
0.818
0.834
0.811
0.772
0.788
0.707
0.747
0.839
0.873
0.866
0.712
0.723
0.738
0.746
0.750
0.743
1.039
1.035
1.095
1.134
1.136
1.095
1.152
1.255
1.270
1.220
1.168
1.162
1 .105
1.113
1 .125
1.131
1 .126
1.113
1.057
1.113
1. 152
1 .137
1.099
1.024
/L = 0.3, R = 0.35, 4t = 1.0, 6z = 0°
0.772
0.793
0.782
0.751
0.725
0.728
1.169
1.064
1.017
1 .125
S.228
1.233
1.152
1.100
1.041
1.090
1.148
1 .168
1.204
1.174
1 .100
1.102
1.139
1.231
1.049
1.101
1.131
1.148
1.146
1.138
1.109
1.073
1.035
1.018
1.033
1 .039
1.092
1.045
0.998
1.023
1.052
1.077
1.168
1.135
1.085
1.053
1.098
1.152
1.149
1. 144
1.146
1.128
1.094
1.053
1.015
1.027
1.059
1.087
1.108
1.136
1.059
1.067
1. 103
1.133
1.150
1. 156
1. 150
1.121
1.078
1.031
1.012
1.040
S
1.099
1.097
1 .080
1.072
1.087
1.100
s
1.158
1.112
1.034
1.070
1.088
1.149
/L = 0.3, R = 0.175, ®t = 1.5, A = 0°
1.051
1.083
1.119
1.150
1.167
1.166
1.147
1.115
1.078
1.047
1.033
1.044
1.049
1.045
1.055
1.078
1.105
.132
1.149
1.153
1.142
1.120
1.092
1.066
1 .170
1.148
1.112
1.143
1.198
1.181
1 .105
1.122
1.118
1.097
1.079
1.084
TABLE 6.2
AMPLIFICATION FACTORS FOR MAXIMUM EFFECTS IN A THREESPAN BEAM SUBJECTED TO A
TWOAXLE, DAMPED VEHICLE WITH 15% INITIAL OSCILLATION AND AXLES INITIALLY IN PHASE
Uniform Undamped Beam with SideSpan Ratio a = 0.8
Identical Axles with a Static Reaction per Axle of W/2,
Dynamic Index = 1.0, p = 0.15 and 7 = 0.36
Results for Deflection are Expressed in Terms of Maximum Static Effect at Midspan.
All Other Data are in Terms of Maximum Static Effect at Section Under Consideration.
For Maximum Static Effects See Table 4.1
0 1 Amplification Factors For
, 1 D D D M M M M M R
(deg.) 1 c 4 1 2 c 3 4 2
s/L = 0, R = 0.175, st = 0.6, F. = 0
0.15 270
300
330
360
0.06 0
0.09
0.12
0.15
0.18
0.21
0.24
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.06 240
0.09
0.12
0.15
0.18
0.21
0.24
0.15 0
60
120
180
240
270
300
330
0.850
0.798
0.755
0.717
0.779
0.799
0.793
0.830
0.826
0.815
0.783
0.781
0.759
0.798
0.754
0.730
0.745
0.824
0.801
0.822
0.794
0.796
0.883
0.944
0.968
0.830
0.811
0.754
0.762
0.796
0.858
0.862
0.846
1.073
0.984
1.031
1.100
0.703
0.729
0.820
0.886
1 .158
1.030
0.932
0.916
0.972
1.062
1.157
1.174
1.034
0.952
0.938
0.971
s/L = 0, R = 0.175, It = 1.0, F.
1.073
1 .098
1.118
1.147
1.135
1.066
1.173
1.052
1.044
1.061
1.0&3
1.057
1.116
1.112
1.031
1.066
1.142
.164
1.136
1.225
1.155
1.147
1.150
1.083
1.152
1.164
1.167
1.144
1.148
0.752
0.782
0.829
0.780
0.814
0.840
0.848
0.794
0.771
0.831
0.816
0.800
0.852
0.913
0.777
0.792
0.775
0.827
0.719
0.782
0.942
0.780
0.812
0.816
0.778
0.827
0.886
0.811
0.761
0.980
0.942
1.076
1.133
1.046
0.932
0.914
1.036
1.000
1.071
1.006
0.985
0.965
1.035
1.064
1.108
0.912
1.013
1.192
1.244
1.223
1.133
1.116
1.006
1.007
1.013
1.063
1.136
1 .161
1.048
1.046
1.060
1.084
1.218
1.172
1.376
1.027
1.090
1.068
1.068
1.212
1.147
1.375
1.073
1.074
1.060
1.078
1.116
1.111
1.187
1.084
1 .118
1.068
1.114
1.078
1 .148
1.163
1.124
0.965
1.024
1.056
1.070
1.100
1.086
1.057
1.009
1.027
1.021
1.040
1.082
1.092
1.048
1.024
1.028
1.059
1.092
0.999
1.131
1.031
1.070
1.050
1.040
1.072
1.092
1.076
1.093
1.091
1.250
1.158
1.089
1.071
= 0
1.064
1.041
1.043
1.101
1.125
1.217
1.173
1.085
1.039
1.042
1.098
1.153
1.220
1.166
1.050
1.044
1.046
1.143
1.034
1.195
1.287
1.101
1.138
1.098
1 .156
1 .143
1.223
1.174
1.135
0.914
0.911
0.983
1.055
0.982
0.910
0.973
0.943
1.083
1.119
1.105
0.948
1.008
1.085
1.053
1.070
1.136
1.179
1.004
1.083
1.041
1.026
0.963
1.035
1.212
0.943
0.948
1.053
1.023
1.026
1.042
0.938
0.905
1.194
1.144
1.054
1.048
1.072
1.036
1.086
1.111
1.131
1.081
1.093
1.064
1.064
1.086
1.095
1.078
1.064
1.028
1.059
1.075
1.067
1.053
1.043
1.083
1.015
1.111
1.153
1.095
1.088
1.053
1.123
1 .160
1.136
TABLE 6.2 (Continued)
l Amplification Factors For
a 1 D D D M M M M R
(deg.) 1 c 4 1 2 c 3 4 2
s/L = 0, R = 0.175, st = 1.5, F. = 0
0.810
0.853
0.861
0.862
0.970
0.932
0.957
1.023
1.146
1.084
1.131
1.131
s/L = 0, R = 0.35, 4t =
0.754
0.794
0.851
0.803
0.831
0.876
0.831
0.803
0.803
0.796
0.756
0.770
0.760
1.014
0.948
1.113
1.135
1.075
0.972
0.912
1.135
1.111
1.038
1.056
1.074
1.156
1.038
1.070
1.088
1.108
1.178
1.220
1.398
1.108
1.120
1.114
1.088
1.037
1.108
1.064
1.064
1.027
0.993
1.0, F.
1.036
1.023
1.054
1.043
1.152
1.106
1.201
1.043
1.038
1.035
1.067
1.083
1.053
1.107
1.125
1.125
1.109
1.071
1.095
1.091
1.167
1.194
1.106
1.279
1.167
1.165
1.076
1.084
1.142
1.175
1.133
1.133
1.081
1.087
1.131
1.148
0.15 270
300
330
360
0.06 0
0.09
0.12
0.15
0.18
0.21
0.24
0.15 0
60
120
180
240
300
0
60
120
180
240
300
1.131
1.102
1.055
1.081
1.111
1.163
1.132
1.110
1.126
1.095
1.016
1.095
1.050
1.067
1.052
1.067
1.063
1.039
s/L = 0.3, R = 0.175, O = 1.0, An = 0, F =
\ i
0.769
0.766
0.813
0.837
0.839
0.841
0.837
0.837
0.820
0.761
0.777
0.815
0.859
0.854
0.844
0.773
0.785
0.823
0.853
0.775
0.740
0.786
0.697
0.759
0.796
0.803
0.749
0.799
0.747
0.758
0.810
0.784
0.711
1.041
1.109
1.093
1.042
1.046
1.117
1.180
1.029
1.112
1.089
1.021
1. 163
1.203
1.110
1.062
1.037
1.067
1.073
1.044
1.209
1.155
1.082
1.034
1.058
1.059
1.083
1.165
1.137
1.064
0.988
1.098
1.046
1.073
1.145
1.055
1.049
1.048
1.042
1 .043
1.100
1.187
1.140
/L = 0, R = 0.5, Ot = 1.0, F. = 0
t I
1.140
1.120
1.125
1.136
= 0
1.098
1.077
.140
1.097
1.262
1.130
1.383
1.097
1.103
1.134
1.199
1.215
1.190
0.814
0.787
0.798
0.772
0.750
0.777
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.744
0.757
0.797
0.825
0.697
0.737
0.736
0.760
0.766
0.757
0.794
0.691
0.757
0.743
0.723
0.786
0.805
0.746
0.968
1.040
1 .084
1.121
0.969
0.962
0.964
0.919
1 .107
1.124
1.086
0.919
0.938
0.791
1.012
1 .008
0.913
0.968
1 .003
1.043
1.037
1.027
0.984
0
1.070
1.126
1.005
1.049
1.044
1.207
1.134
1.076
1.098
0.999
1.030
1.113
1.243
1.087
1.147
1.115
1.097
1.146
1.018
1.097
1.065
1.056
1.069
1.100
1.118
1.056
1.042
1.098
1.085
1.059
1.112
1.028
1.046
1.105
1.066
1.056
1.072
1.108
1.060
1.051
1.127
0.993
1.039
1.025
1.094
1.062
1.063
1.082
0.998
1.068
1.054
1.143
1.146
1.146
1.180
1.187
1.150
1.087
1 .038
1.049
1.056
1.055
1.131
1.101
1.064
1.071
1.056
1.067
1.165
1.105
1.187
1.066
1.027
1.092
1.097
1.089
1 .108
1 .148
1.060
1 .068
1.039
1.067
1.072
1.150
1.053
TABLE 6.2 (Continued)
l Amplification Factors For
(deg.) D D D4 Ml M M M3 M4 R2
(deg.) 1 c 4 1 2 c 3 4 2
1. 120
1.036
1.096
1.144
1.227
1.100
.188
1.056
1.055
1.067
1.135
1. 144
1 .147
0.695
0.759
0.714
0.751
0.832
0.788
0.805
0.760
0.774
0.723
0.754
0.751
0.774
1.060
1.028
1.041
1.201
1.254
1.045
1.033
1.042
1.008
1.021
1.164
1.201
1.188
1.048
1.083
1.031
1 .033
1.166
1.189
1.216
1.073
1.082
1.059
1.026
1.033
1.032
1.062
1.072
1.125
1. 167
1.169
1.141
1.142
1.046
1 .048
1.043
1 .156
1.167
1.182
s/L = 0.3, R = 0.175, t = 1.0, n
1.117
1.039
1.158
1. 170
1.109
1.127
0.798
0.766
0.741
0.764
0.770
0.787
1.187
1.025
1.151
1.245
1.198
1.207
1.057
1.064
1.036
1.029
1.086
1.081
1.175
1 .036
1.163
1.201
1.159
1 .152
0.746
0.771
0.776
0.862
0.818
0.688
0.768
0.697
0.709
0.758
0.840
0.862
0.821
0.798
0.695
0.850
0.884
0.818
0.839
0.786
0.802
0.767
0.838
0.854
0.715
1.098
1.100
1.050
1 . 159
1.171
1.064
0.742
0.744
0.739
0.698
0.760
0.762
1.090
1.068
1.031
1.111
1.192
1.089
s/L = 0.3, R = 0.35,
1.071
1.088
1.056
1.089
1.175
1.242
0.748
0.727
0.734
0.768
0.752
0.840
0.994
1.060
1.087
1.129
1.209
1.196
1.095
1.058
1.069
1.050
1.050
1.062
1.032
1.068
1 .045
1 .122
1 .184
1 .093
1.084
1.081
1.076
1.060
1.094
1.111
1 .068
1.097
1.105
1.067
1 .100
1 .060
1.117
0.988
1.024
1.011
1.112
1.121
1.164
1.264
1.049
1 .067
1.030
1.083
1.112
1.085
= 00, F. = F'
1.147 1.112
1.097 1.056
1.112 1.086
1.054 1.098
1.094 1.096
1.138 1.130
= 00, F. = F'
1.091 1.024
1.117 1.035
1.086 1.023
1.084 1.082
1.118 1.097
1.089 1.052
Ct = 1.0, zA = 00, F.
1.055 1.065 1.064
1.059 1.102 1.074
1.068 1.046 1.046
1.084 1.045 1.110
1.170 1.205 1.159
1.250 1.197 1.160
= 0
1.049
1.043
1.047
1.057
1.091
1.158
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.15 0
60
120
180
240
300
0.15 0
60
120
180
240
300
0.15 0
60
120
180
240
300
s/L = 0.3, R = 0.175, t = 1.0, A0
1.095
1.051
1.058
1.040
1.040
1.051
1.044
1.127
1.149
1 .082
1.065
1.040
1.110
1.173
1.145
1 .053
1.082
1.106
1.104
1.103
1.079
1.092
1.021
1.060
1 . 140
1.041
1.069
1.035
1.012
1.058
1.021
0.06
0.09
0.12
0.18
0.21
0.24
0 0.726
0.744
0.783
0.754
0.790
0.775
TABLE 7.1
AMPLIFICATION FACTORS FOR MAXIMUM EFFECTS IN A THREESPAN BEAM SUBJECTED
TO A SINGLEAXLE, UNDAMPED VEHICLE WITH 50% INITIAL OSCILLATION
Uniform Undamped Beam with SideSpan Ratio a = 0.8
Identical Axles with a Static Reaction per Axle of W/2,
Dynamic Index = 1.0 and u = t
Results for Deflection are Expressed in Terms of Maximum Static Effect at Midspan.
All Other Data are in Terms of Maximum Static Effect at Section Under Consideration.
For Maximum Static Effects See Table 4.1
Amplification Factors For
t D Dc D4 M M2 Mc M3 M R
s/L = 0, R = 0.175, 0 = 0°
0.05 0.6 1.088 1.567 1.178
1.0 1.096 1.674 0.964
0.15 0.3
0.5
0.6
0.7
0.8
0.9
1 .0
1.1
1.2
1.3
1 .4
1.5
0.692
0.956
1. 126
1 164
1.150
1.161
1.193
1.197
1.197
1 .129
1.117
1.040
1 .175
1.841
1.289
1.783
2.178
1.761
1.462
1.473
1 .363
1.308
1.349
1.310
0.895
0.937
1 .021
1.422
0.975
1.243
1.131
1.046
1 .178
1 .228
1 . 120
1.050
0.18 0.6 0.906 1.879 1.1
1.0 1.229 1.352 1.1
0.15 0.6 1.107 1.699 1.1
1.0 0.946 1.407 1.1
0.15 1.0 1.117 1.465 1.069
0.15 0.3
0.5
0.7
0.9
1.0
1 .1
1.2
1.3
1.4
1.5
0.679
0.913
1 .136
1.103
1.055
1.019
1 .013
0.992
0.966
0.916
1.141
1.843
1.326
1.516
1.294
1.328
1 .262
1.184
1.151
1.141
0.844
0.663
1.099
0.881
0.877
0.915
0.952
0.849
0.813
0.854
1.492 1.495 1.275 1.564 1.468
1.364 1.382 1.324 1.593 1.223
0.807
0.866
1.070
1.214
1.391
1.571
1.551
1 .500
1.439
1.329
1.284
1.178
1.449
1.710
1.521
1.465
1.635
1.467
1.479
1.550
1.676
1 .681
1.589
1.638
1.022
1 .578
0.849
1 .649
1.697
1.332
1 . 164
1.113
0.977
1 .167
1 .252
1.226
1.214
1.702
1.533
1.583
1.713
1.479
1.782
1.819
1.677
1.822
1.739
1.436
1.081
1.062
0.925
1.658
1.043
1 .451
1.142
1.213
1.497
1.612
1.369
1.215
49 0.783 1.722 1.575 1.686 1.177
20 1.543 1.669 1.132 1.768 1.033
s/L = 0, R = 0.175, , = 1800
47 1.397 1.541 1.621 1.471 1.478
37 0.943 1.272 1.003 1.250 1.483
s/L = 0, R = 0.3, e1 = 0°
9 1.470 1.410 1.152 1.505
s/L = 0, R = 0.5, 01 = 0°
0.788
0.829
1.179
1.381
1.393
1.335
1.258
1.192
1.141
1.096
1.371
1.751
1.238
1.402
1.292
1.255
1.259
1.194
1.205
1.329
0.991
1.584
1.090
1.154
1.018
1.112
1.118
1.051
1.066
1.013
1.145
1.696
1.699
1.537
1.644
1.607
1.743
1.686
1.610
1.360
0.999
1.026
0.771
1.259
1.072
0.934
1 .272
1.224
1.001
0.919
0.939
1.471
1.436
1.558
1.449
1.256
1.637
1.317
1.245
1.505
1.461
1.444
1.732
1.735
1.664
1.351
1 .488
1.443
1 .074
1 .330
1.534
1.520
1.635
1.381
1.441
1.321
1.189
1.080
1.207
1.428
TABLE 7.1 (Continued)
Amplification Factors For
a t D Dc D4 M M M M3 M R
s/L = 0, R = 1.0, 01 = 0°
0.15 0.38
0.42
0.46
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0.613
0.684
0.792
1.010
1.097
1.144
1.148
1.155
1.126
1.103
1.058
1.037
0.989
1.352
1.626
1.708
1.509
1.338
1.449
1.374
1.242
1.117
1.178
1.279
1.282
1.286
1.181
1.468
1.270
1.170
0.974
1.092
1.195
0.857
1 .050
1.071
0.902
0.720
0.870
0.639
0.609
0.704
0.987
1.149
1.263
1.343
1.367
1.380
1.384
1.356
1.308
1.261
1.105
1.336
1.518
1.795
1.487
1.332
1.493
1.632
1.546
1.432
1.480
1.363
1.306
0.885
1.176
1.522
1.036
0.910
1.224
1.451
1.224
0.988
1.003
1.125
1.140
1.186
1.640
1.832
1.912
1.462
1.571
1.874
1.798
1.636
1.494
1.630
1.597
1.547
1.580
1 .408
1.851
1.395
1.600
1.024
1.367
1.373
0.904
1.250
1.226
1 .000
0.957
1 .078
1.556
1.607
1.546
1.601
1.525
1.715
1.605
1.497
1.622
1 .572
1.544
1.407
1.396
TABLE 7.2
AMPLIFICATION FACTORS FOR MAXIMUM EFFECTS IN A THREESPAN BEAM SUBJECTED
TO A TWOAXLE, DAMPED VEHICLE WITH 50% INITIAL OSCILLATION
Uniform Undamped Beam with SideSpan Ratio a = 0.8
Identical Axles with a Static Reaction per Axle of W/2
Dynamic Index = 1.0, a = 0.15, p = 0.15 and 7 = 0.36
Results for Deflection are Expressed in Terms of Maximum Static Effect at Midspan.
All Other Data are in Terms of Maximum Static Effect at Section Under Consideration.
For Maximum Static Effects See Table 4.1
0 Amplification Factors For
t D D D M M M M M R
(deg.) 1 c 4 1 2 c 3 4 2
0.4 0
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0.6 0
60
120
180
240
300
1.0 0
60
120
180
240
300
1.5 0
60
120
180
240
300
1.062
0.644
0.676
0.820
0.853
0.890
0.928
0.956
0.920
0.950
0.967
0.995
0.676
0.702
0.977
1.041
0.919
0.800
0.928
0.879
0.788
1.039
0.966
0.902
0.995
0.921
0.819
0.848
0.811
0.927
2.527
0.949
1.293
1.366
1.032
1.231
1.315
1.171
.172
1.269
1.245
.169
0.293
1.350
1.238
1.165
1.016
1.381
1.315
1.312
1.250
1.193
1.228
1.207
1.169
1.113
1.192
1.233
1.219
. 142
s/L
1.279
0.899
0.772
0.895
0.876
0.770
0.920
0.871
0.914
0.844
0.857
0.864
0.772
0.948
0.724
0.902
0.861
0.933
0.920
0.871
0.912
0.824
0.841
0.924
0.864
0.859
0.712
0.916
0.843
0.900
= 0, R =
0.961
0.795
0.672
0.755
0.842
0.914
1.033
1.123
1.241
1.298
1.318
1 .327
0.672
0.883
1.318
1.391
1.155
0.804
1.033
0.942
0.856
1.261
1.277
1.227
1 .327
1.216
1.066
0.915
0.902
1.131
0.175, F. = 0
I
2.197
1 .060
1.270
1.320
1.263
1.246
1.196
1.288
1.359
1.388
1.336
1 .324
1.270
1.287
1.166
1.262
1.189
1.379
1.196
1.221
1.107
1.349
1.186
1.307
1.324
1.291
1.322
1.265
1.186
1.270
2.136
0.901
1.222
1.184
0.876
1.152
1.107
1.063
1.084
1.137
1.057
0.961
1.222
1.241
1.122
1.045
0.935
1.186
1.107
1.056
1.130
0.993
1.128
0.942
0.961
0.938
1.045
1.223
1.168
0.969
2.270
1.200
1.261
1.241
1.277
1.261
1.159
1.275
1.486
1.478
1.496
1.309
1.261
1.322
1.250
1.102
1.128
1.225
1.159
1.205
1.101
1.392
1.288
1.267
1.309
1.330
1.289
1.361
1.221
1.170
1.425
1.120
1.024
1.037
1.077
0.887
1.219
1.019
1.022
0.919
1.095
1.169
1.024
1.188
0.976
1.101
0.992
1.167
1.219
1.174
1.173
0.922
0.902
1.027
1.169
1.160
0.971
0.963
1.010
1.195
1.642
1 .357
1.420
1 .272
1 .326
1.250
1 .143
1.213
1 .334
1.394
1.262
1 .322
1.420
1 .294
1.132
1 .077
1 .097
1 .228
1 .143
1.126
1 .006
1.126
1 .124
1.180
1.322
1.381
1.289
1.282
1.198
1.226
TABLE 7.2 (Continued)
0 Amplification Factors For
t (de1.) D D c D M M M M M R
S(deg.) = c 4 1 2 c 3 4 2
s/L = 0, R = 0.175, F. = F'
1.068
0.929
0.947
1.337
1.398
1 .225
1.207
1 .183
1.145
1.414
1.283
1.249
1.022
1.031
1.131
0.911
0.996
1 .006
s/L = 0, R = 0.175, F. = F'
836 0.992 1.357 1.251
914 0.915 1.262 1.114
899 0.842 1.078 1.104
683 1.164 1.255 1.088
805 1.234 1.080 1.134
939 1.229 1.313 0.983
s/L = 0, R = 0.35, F. = 0
1 .003
0.940
0.851
1 .276
1.318
1.192
1.252
1.165
1.012
1 .300
1 .169
1.275
1.134
1 .022
1 .056
0.966
1.037
0.980
1.0 0
60
120
180
240
300
1.0 0
60
120
180
240
300
1.0 0
60
120
180
240
300
0.5 0
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.0 0
60
120
180
240
300
0.878
0.879
0.829
1.069
1.052
0.907
1.005
0.884
0.813
0.964
0.918
0.900
0.942
0.875
0.759
1.018
0.976
0.909
0.635
0.672
0.786
0.820
0.886
0.942
0.981
1 .006
0.990
1.002
0.976
0.942
0.869
0.744
1.006
0.972
0.941
1.257
1.263
1.241
1 .154
1.137
1.243
1.318
1.345
1.229
1.192
1.244
1.171
1 .112
1 .199
1.094
1.032
1.105
1.134
1.006
1.238
1.390
1.220
1.061
1.143
1.195
1.225
1.210
1.196
1.157
1.143
1.105
1.037
1.008
1.076
1.150
0.948
1 .196
1.214
1.018
0.899
1.090
.120
1.060
1.046
1.082
1.096
1.208 1.090
1.119 1.048
1.040 1.016
1.252 0.992
1.172 1.023
1.249 1.007
1.147
1.166
1.190
1 .303
1.381
1 .257
1 .106
1.224
1 .048
1 .375
1 .172
1 .249
1.191
1.204
1.239
1.346
1.327
1.152
1 .233
1.291
1.263
1.183
1.258
1.257
1.249
1.315
1.275
1.311
1.3 11
1.257
1.223
1.271
1.316
1.296
1.154
0.932
0.886
0.922
0.783
0.772
0.848
0.
0.c
0.8
0.(
0.
0.
s/L = 0, R = 0.5, F. = 0
I
0.785
0.663
0.729
0.829
0.910
1 .002
1 .090
1.150
1 .159
1.179
1 .195
1 .002
0.955
0.878
1.283
1.319
1.161
0.882
0.867
0.855
0.804
0.770
0.872
1.003
1.225
1.368
1.286
1.211
1.208
1.307
1.364
1.295
1.191
1 .152
1 . 180
1.116
1.141
0.881
0.906
0.949
1.137
1 .172
1 .208
1.056
0.903
1.087
1.141
1 .174
1 .154
0.895
0.955
0.977
1 .127
0.969
1.040
0.951
1 .162
1.108
0.976
1 .067
1 .165
1.155
1.115
1.108
1.129
1.147
1.016
1.021
1.022
1.088
1.149
1.079
1. 146
1.133
1. 188
1.288
1.201
1.074
1 .1 1 1
1.072
1. 192
1.235
1.165
1.041
1.115
1.143
1.143
1.330
1.427
1.295
1.393
1.330
1.242
1.227
1.201
1.257
1.309
1.270
1.242
1.170
1.106
1.136
1.133
1.175
0.913
0.811
0.921
0.809
0.894
0.867
0.841
0.817
0.865
0.866
0.812
0.867
0.861
0.862
0.802
0.777
0.844
TABLE 7.2 (Continued)
S lAmplification Factors For
t deg.) DI D D M M. M M3 M R2
(deg.) 1 c 4 1 2 c 3 4 2
1 .266
1.112
1.221
1.282
1 .304
1.132
1 .269
1 .269
1.172
1 .290
1 .168
1.191
1.188
1 . 103
1.041
1.160
1.125
1 . 159
1 .255
1.206
1. 140
1.337
1 .243
1. 179
1 .200
1 .277
1.201
1.193
1.143
1.219
s/L = 0.3, R = 0.175, Az = 00, F. = 0
0.841 1.151 1.364 1.279 1.176
0.848 1.075 1.204 1.210 1.086
0.820 1.060 1.292 1.211 1.100
0.722 1.336 1.107 1.336 1.371
0.689 1.229 1.193 1.347 1.341
0.764 1.203 1.255 1.190 1.220
0.695
0.746
0.756
0.783
0.803
0.793
0.739
0.721
0.758
0.843
0.784
0.753
1.313
1.157
1.329
1.371
1.264
1.095
.164
1.192
1.158
1 .144
1 .069
1.150
1.203
1.115
1.099
1.354
1.081
1 .159
1.275
1.201
1.138
1.303
1.227
1.252
1.244
1.275
1. 199
1.288
1.201
1.163
1.128
1.074
1.088
1.239
1.157
1.141
1.121
1.253
1.108
1.223
1.055
1.237
1.184
1.047
1.076
1.190
1.111
1.066
0.6 0
60
120
180
240
300
1.0 0
60
120
180
240
300
1.5 0
60
120
180
240
300
1.0 0
60
120
180
240
300
1.0 0
60
120
180
240
300
1.0 0
60
120
180
240
300
0.808
0.684
0.645
0.969
0.909
0.850
0.987
0.874
0.958
0.901
0.825
0.789
0.802
0.834
0.809
0.863
0.799
0.848
0.901
0.861
0.975
0.887
0.813
0.729
1.062
0.943
0.954
0.913
0.825
0.851
0.852
0.915
0.917
0.849
0.830
0.726
1.186
1 .158
1 .364
1.355
1.242
1 .103
1 .1 11
1.092
1.215
1.440
1.143
1 .152
1.203
1.177
1 .191
1.316
1.185
1.174
1.070
1.187
1.198
1.246
1.045
1.217
s/L = 0.3, R = 0.175, LA = 00, F. = F'
0.725 1.412 1.288 1.231 1.214
0.783 1.270 1.155 1.312 1.278
0.721 1.287 1.066 1.203 1.129
0.812 1.373 1.243 1.197 1.180
0.783 1.259 1.063 1.209 1.047
0.741 1.159 1.148 1.172 1.235
s/L = 0.3, R = 0.175, AG = 600, F. = 0
I
1.110 0.732
1.251 0.765
1.160 0.739
1.247 0.840
1.192 0.735
1.194 0.831
1.141
1.208
1.231
1.238
1.271
1.096
1.071
1.108
1.146
1.224
1.174
1 .124
1.100
1.274
1.188
1 .237
1.163
1.159
1 . 164
1.228
1.059
1.089
1.074
1.223
1 .146
1.173
1.098
1.161
1.051
1.159
1. 103
1.087
1.147
1 .11 1
1.084
1.100
1.109
1.027
1.035
1.150
1.070
1.059
s/L = 0.3, R = 0.175, zAG = 0°, F. = F'
0.703
0.726
0.766
0.816
0.787
0.818
1.293
1 .166
1.135
1.174
1.199
1.091
1.125
1.114
1 .086
1 .172
1 .102
1 .166
1.199
1.221
1.214
S.148
1.151
1 .168
1.025
1 .123
S.123
1 .247
1.103
1 .182
1.189
1.188
1 .031
1 .166
1.054
1.155
1.212
1. 123
1.051
1.113
1.076
1.189
1.087
1.040
1.135
1.151
1.063
1.135
1.100
1.131
.135
1.140
1.071
1.084
0.985
1.108
1.167
1.184
1 .001
1.145
)
)
I
i
)
)
)
i
)
TABLE 7.2 (Continued)
t Amplification Factors For
(deg.) D1 Dc D4 M1 M2 Mc M3 M4 2
s/L = 0.3, R = 0.175, nO = 1200, F. = 0
I
1.132
1 .097
1.201
1.249
1.167
1.257
0.703
0.746
0.785
0.784
0.775
0.776
1.090
1 .142
1.268
1.084
1 .146
1 .095
1.075
1.146
1 .123
1.321
1 .088
1 .227
1.137
1.107
1 .246
1 .252
1 .185
1 .233
1.152
1.295
1 .049
1.202
1.071
1.192
s/L = 0.3, R = 0.175, 6z = 1800, F. =
I
1.10f
1.091
1 .055
1.208
1.143
1.130
1.118
1.11
1 .08£
1 .22C
1.175
1. 1 6(
0.743
0.744
0.697
0.799
0.720
0.837
1.082
1.116
1.109
1.057
1.015
1.012
1.120
1 .143
1.046
1.228
1.107
1 .068
1 .091
1.127
1.074
1.201
1 .194
1.114
1 .144
1 .305
1.117
1 .164
1 .056
1 .143
s/L = 0.3, R = 0.35, zA = 00, F. = 0
3 0.752 1.288 1.189 1.096 1.150
1 0.796 1.151 1.081 1.125 1.235
9. 0.775 1.319 1.184 1.147 1.178
9  1.331 1.309 1.220 1.220
5 0.752 1.235 1.103 1.132 1.041
0 0.803 1.092 1.105 1.119 1.272
1.0 0
60
120
180
240
300
1.0 0
60
120
180
240
300
1.0 0
60
120
180
240
300
0.818
0.825
0.945
0.764
0.783
0.706
0.781
0.773
0.805
0.744
0.722
0.733
0.968
0.854
0.948
0.872
0.804
0.787
0.989
1.026
1.206
1.127
1.080
1.062
0
1 .025
1.040
1.055
1.137
1.092
1 .167
1.049
1 .131
1 .071
1 .036
1 .127
1 .193
1.237
1.046
1 .188
1.022
1.159
1.154
1.213
1.085
1 .157
1.055
1.123
1.170
1.226
1.101
1.203
1.096
1.152
6
5
0
TABLE 7.3
AMPLIFICATION FACTORS FOR MAXIMUM EFFECTS IN A THREESPAN BEAM SUBJECTED
TO A SINGLEAXLE, DAMPED VEHICLE WITH 50% INITIAL OSCILLATION
Uniform Undamped Beam with SideSpan Ratio a = 0.8
Identical Axles with a Static Reaction per Axle of W/2,
Dynamic Index = 1.0, R = 0.175, 7 = 0.36 and F. = 0
Results for Deflection are Expressed in Terms of Maximum Static Effect at Midspan.
All Other Data are in Terms of Maximum Static Effect at Section Under Consideration.
For Maximum Static Effects See Table 4.1
Amplification Factors For
C1 D Dc M M M M M R
<t = 1.0
0.05 0.25 0.971 1.215 0.913 1.018 1.170 1.112 1.152 1.150 1.175
0.15 0.05 1.071 1.064 0.780 1.019 1.245 0.773 1.267 0.892 1.349
0.10 1.000 1.205 0.800 0.952 1.259 1.166 1.233 0.991 1.310
0.15 0.928 1.315 0.920 1.033 1.196 1.107 1.159 1.219 1.143
0.25 0.989 1.235 0.957 1.172 1.242 0.935 1.258 1.173 1.067
0.18 0.05 0.853 1.616 0.874 0.733 1.691 1.268 1.422 1.050 1.318
0.10 0.792 1.469 0.811 0.751 1.646 1.074 1.298 0.934 1.271
0.15 0.803 1.303 0.804 0.836 1.460 0.935 1.462 0.968 1.120
0.25 0.920 1.277 0.979 0.948 1.323 1.084 1.254 1.130 1.174
ot = 1.5
0.15 0.05 1.092 1.275 0.912 1.478 1.320 0.936 1.310 1.018 1.288
0.10 1.036 1.270 0.836 1.402 1.273 1.045 1.365 1.017 1.353
0.15 0.995 1.169 0.864 1.327 1.324 0.961 1.309 1.169 1.322
0.25 1.010 1.078 0.912 1.281 1.374 0.907 1.310 1.235 1.477
Flexible
(a) Simply
beam with lumped masses
Supported Bridge
Flexible beam with lumped masses
(b) ThreeSpan Continuous Bridge
FIGURE Al BRIDGE MODELS
FIGURE A2 REPRESENTATION OF A TRACTORTRAILER TYPE VEHICLE
(a) Model for ThreeAxle Vehicle
(b) Model for TwoAxle Vehicle
(c) Model for SingleAxle
Load Unit
FIGURE A3 VEHICLE MODELS
id,
Z
d
u
E (
m E
S C0 0 0
 : U
 r . O 
uC 4u o
L
0 U a 4 VL
COu a M
0  . 0 
apou z ,
r qf
Q)
.0
0
0
C
N
N
Ll
LL
0
LLI
a
I
z
(n

0
LJ
o
0
ILu
.
1r
z
fn
0
I
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UJ
tv
37
 .
C
0
n
C
 U
O
L.
a. I
Z
D
0
C4 !
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LL
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I
LLn
L<
10
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LU
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 0)
4. 0
U
0
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.i.
(a) Pu Diagram
a e
FIGURE A6
VERSUS
INTERACTING
SHORTENING
FORCE, P, AND FRICTIONAL FORCE, F,
OF SUSPENSIONTIRE SYSTEM, u
LOCATIONS FOR WHICH DYNAMIC RESPONSE WAS CALCULATED
b f
U
FIGURE A7
peak value of the dynamic increment for
moment at midspan, although increasing with R,
does not increase significantly even when the
value of R is doubled. On the other hand,
the detailed features of the response curves
do change with changes in R.
Figure 5.2 presents similar time
histories for a fixed value of R and three
different values of tP. The values of t
considered are 0.5, 1.0, and 1.5, which
represent a lower bound, an average value, and
an upper bound, respectively, to the values
encountered in practice. As a basis of
comparison, the time history of dynamic incre
ment for moment at midspan due to a constant
moving force with the same value of the speed
parameter is plotted as a dashed line. It
should be noted that for the values of Pt
considered, the maximum variation in the
interacting force does not exceed 10 per cent
of its static value. Based on this observa
tion, one may conclude that the response of
the bridge should not differ very much from
that observed under the action of a moving
constant force. This fact can be seen in the
timehistories of dynamic increment for moment
at midspan, for which the order of magnitude
of the dynamic increments is the same as that
predicted by the constant force solution
irrespective of the value of Pt involved; how
ever, the details of the waves change with
changes in the frequency ratio. For it = 0.5,
the agreement between the curves is excellent
both with respect to phase and amplitude, but
for large values of Pt a phase shift is
apparent.
5.2.2. Effect of Weight Ratio
Figure 5.3 shows the effect of weight
ratio on the response of the vehicle and the
bridge. The upper half of the figure gives a
spectrum curve of the maximum value of the
interacting force as a function of R, whereas
the lower half shows spectra of maximum ampli
fication factors for moment at five sections
along the bridge, also as a function of R.
The results presented in this figure are for
values of a = 0.15 and Pt = 1.0; however,
similar results were also obtained for other
realistic combinations of these parameters.
It can be seen that within the realistic range
of R values, the maximum variation in the
interacting force does not exceed 15 per cent
of its static value. Previous studies(25)
have justified the use of 15 per cent of the
static weight of each axle as a very common
limiting value of the interleaf friction in
the suspension springs; therefore, the assump
tion that the vehicle oscillates only on its
tires is fully justified, since the interleaf
friction will not be mobilized in most cases
of smoothly moving vehicles.
The lower part of Figure 5.3 shows that
within the practical range of R the variation
in the amplification factor of any effect due
to a change in R is of the order of 10 per
cent of its static value, and that, in general,
the amplification factors increase with in
creasing R.
5.2.3. Effect of Frequency Ratio
The effect of the frequency ratio on
the response of the vehicle is shown in
Figure 5.4a. In this figure, spectrum curves
for the maximum value of the interacting force
are given as a function of the frequency ratio
Pt, for the two limiting values of R = 0.175
and R = 0.50 considered in this study. It is
noteworthy that the peak value of the inter
acting force remains fairly constant through
out the range of values of Ct considered. The
effect of the frequency ratio on the response
of the bridge is shown in Figures 5.4b to
5.4d, where spectra of peak values of the
amplification factors for moments at various
sections are plotted as a function of Pt for
the same two values of R. The response of the
bridge to variations in Pt is erratic, and no
general trend can be observed; however, it is
interesting to note that the amplification
factors for moment over the second interior
support, M3, are consistently larger than
those obtained for M or M4. These curves
c e
also show that the response is insensitive to
the weight ratio, although there is a slight
trend for the absolute maximum value of the
amplification factors to increase with
increasing R.
The results presented in Figures 5.3
and 5.4 are for a value of the speed parameter
a = 0.15. It is interesting to find out how
the responses of the vehicle and the bridge
are influenced by the weight ratio and the
frequency ratio at other values of a. This is
shown in Figures 5.5 through 5.8, where
spectra of maximum interacting force and of
peak amplification factors for Mc, M4, and M3
are shown as a function of C. In the upper
half of each figure, the results are presented
for an average value of frequency ratio,
Pt = 1.0, and two values of R, whereas in the
lower half an average Rvalue of 0.30 and
three values of ct are considered. The peak
value of the interacting force is nearly
proportional to R, whereas the effect of R is
unimportant insofar as the response of the
bridge is concerned. Within the range of a
investigated, the larger values of response
were observed for the smaller values of cpt
It is very important to observe again the
general trend of increasing values of response
with increasing values of the speed parameter,
and that the largest amplification factors
observed, of the order of 1.4, correspond to
moments over the second interior support.
5.2.4. Comparison of Effects at Symmetric
Sections
In the preceding figures, the response
of the bridge was studied only for sections at
midspan and to the right of midspan. In
Figures 5.9 and 5.10 a comparison is made of
the response of the bridge at corresponding
sections on either side of midspan. It can be
observed that the peak value of the spectrum
curve for MI is slightly greater than the
corresponding value for M4, and that the in
fluence of the parameters a and Pt is more
erratic for M4 than that for MI. The latter
result is attributed to the fact that the
maximum value of M4 occurs at a later time
than MI and, consequently, influence of the
higher modes of oscillation is more marked.
We can observe, finally, that the influence of
a and tP on the effects over both interior
supports is erratic and of the same order of
magnitude in both cases, but the amplification
factors observed are significantly larger than
those observed at sections in the side spans.
5.3. RESPONSE TO TWOAXLE, SMOOTHLY MOVING
SPRUNG VEHICLES
To make possible a direct comparison
with the results obtained for two constant
forces, the static weight of each axle was as
sumed to be equal to onehalf of the total
weight of the vehicle. This assumption agrees
with the H20S16 representation of the drive
and rear axles of a tractorsemitrailer combi
nation given by AASHO.(16) The natural fre
quencies of vibration on the axles were consid
ered to be the same, and the dynamic index of
the vehicle, which is a measure of the rotary
inertia of the twoaxle unit, see Appendix,
page 42, was taken equal to one. For this
value of dynamic index, the pitch and bounce
modes of vibration are uncoupled; that is, if
the system is forced to start oscillating in
its bounce mode and then released, it will vi
brate freely in this mode only. The opposite
is true if it is forced to start oscillating
in its pitch mode. In our case, however, the
vibration of a twoaxle vehicle is influenced
by the movement of the bridge. It will always
have a combination of both pitch and bounce
motions, due mainly to the change of phase it
undergoes when passing from one span to the
next. Inasmuch as both modes of oscillation of
the vehicle will be excited, it was felt that
the effect of the dynamic index was of minor
importance, and it was arbitrarily set equal
to 1.0 throughout this study.
5.3.1. Representative History Curves
Figure 5.11 shows the possibility of
cancellation or superposition of effects
produced by the individual axles for systems
with a fixed value of a and several values of
s/L. In this figure, history curves of
dynamic increment for moment at the center of
the center span are plotted in terms of their
corresponding maximum static values. The
weight ratio was taken as R = 0.175, and the
frequency ratio Tl = 1.0. For s/L = 0.1,
there is a definite reduction in the level of
the response at the time the rear axle enters
the bridge; however, this reduction is not as
large as the one observed in Figure 4.10 for
the case of two constant forces. In the time
histories shown, in the region where cancel
lation of the primary components of response
does occur, the influence of the second and
third modes of oscillation of the bridge can
be seen. On the other hand, for the values
of s/L equal to 0.2 and 0.4, we notice that
the superposition of effects is not as com
plete as the one observed in the case of
constant forces. Thus the resulting dynamic
increments are smaller than those obtained
with two constant forces. Since the addition
of effects is not complete, it is concluded
that the dynamic increment corresponding to a
singleaxle solution represents a conservative
estimate of the dynamic effect produced by a
twoaxle vehicle of the same weight, frequency,
and speed.
5.3.2. Effect of Axle Spacing Parameter
Figures 5.12 through 5.15 show the
effect of the axle spacing parameter s/L on
the presponse of the vehicle and the bridge
for several combinations of the parameters a
and Pt . The weight ratio R has been taken
equal to 0.175 in the remainder of this
chapter; however, very similar results were
also obtained for other values of R.
In Figure 5.12 the peak value of the
interacting forces in the two axles is plotted
versus s/L. The upper half of the figure gives
the results for Tt = 1.0 and two values of a,
and the lower half gives corresponding results
for a = 0.15 and two values of P. It is seen
that larger values of a give larger values of
peak interacting force for both axles and the
entire range of s/L values. A somewhat
similar trend can be observed for the larger
values of Ct considered in the lower half of
the figure. It is very interesting to notice
that in most cases the peak value of the inter
acting force for the rear axle is greater than
the corresponding value for the front axle.
This can be explained by the fact that, where
as the front axle encounters the bridge in a
condition of static equilibrium, the response
of the rear axle is amplified by the motion of
the bridge induced by the front axle.
The upper parts of Figures 5.13 through
5.15 give spectra of maximum dynamic incre
ments for moments M , M4, and M3 for a value
of Pt = 1.0 and three values of a, whereas the
lower parts give corresponding spectra for
a = 0.15 and three values of t . The ordi
nates are expressed in terms of WL to make
possible a direct comparison. It can be
observed that throughout the range of s/L the
larger dynamic increments for all the effects
considered correspond to the larger values of
the speed parameter. It can also be seen that
the peaks of the spectra correspond to the
combinations of a and s/L for which the ratio
(s/L)/a is zero or an even integer, whereas
the valleys correspond to those combinations
for which (s/L)/a is an odd integer. This
result is consistent with that found in the
case of two constant moving forces. The
influence of Tt is somewhat more difficult to
predict, although in general, the larger
values of dynamic increment correspond to the
smaller values of Pt when s/L = 0. However,
this trend seems to reverse itself for other
values of s/L. Also notice that the value of
the peak dynamic increment decreases with
increasing values of s/L, since the super
position of the effects of the individual
axles is not as good as in the case of two
constant forces.
In Figures 5.16a through 5.16d the
response for a twoaxle vehicle with s/L = 0.3
is plotted as a function of Pt and compared
with corresponding response for a singleaxle
vehicle. The value of a is 0.15. The combi
nation of s/L and a used is such as to produce
addition of effects. It can be seen that
although the peak dynamic increments for
moments at the sections considered are depen
dent on Pt, their dependence is somewhat
erratic for both the singleaxle and the two
axle vehicles. For moments over the second
intermediate support, the singleaxle solution
always induces larger dynamic increments than
the twoaxle solution. For moments at midspan
and at a section 0.42 aL from the right abut
ment (M4), the singleaxle solution induces
larger dynamic increments than the twoaxle
solution throughout the entire range of Pt,
except in a region very close to Pt = 1.2,
where the twoaxle solution goes slightly
above the curve corresponding to the single
axle case. Therefore, perfect addition of the
effects of both axles in the twoaxle vehicle
occurs only at this particular value of Vt'
and in general, the singleaxle solution pro
vides a conservative upper bound to the ef
fects of a twoaxle vehicle.
It should be remembered that the static
effects for a twoaxle loading are smaller
than for a single axle. Consequently, even
if the dynamic increment for a twoaxle vehicle
is not quite as large as that for the corres
ponding singleaxle loading, the amplification
factor for the twoaxle loading will be the
greater. This fact is illustrated in Figure
5.17 where the maximum moments at midspan are
plotted as a function of the speed parameter
for four different values of s/L. The ordi
nates on the left express the total dynamic
moment in terms of WL; the scales on the right
give the amplification factor in terms of the
corresponding maximum static effects. By com
paring this figure with corresponding curves
obtained for the case of constant forces shown
in Figure 4.15, one can see that results are
very similar, although the amplification
factors are slightly larger for smoothly moving
sprung vehicles. The undulating nature of the
curves has been explained in Section 4.4.2.
The approximation to the amplification factor
for M given by Equation 4.10 for the four
values of s/L considered is shown by dashed
lines. It can be seen that this approximation
is still conservative in most cases.
The results presented in Figure 5.17
were obtained for cPt = 1.0 and R = 0.175.
Table 5.1 presents results for combination of
these parameters and for effects at sections
away from midspan
Figures 5.18 and 5.19 show similar
curves for effects at sections away from mid
span, both for the results obtained with the
computer program and with Equations 4.11 and
4.12. It is seen that the approximate ex
pressions are less conservative than for the
case of moving constant forces.
* .
VI. RESPONSE TO VEHICLES WITH SMALL AMPLITUDES OF INITIAL OSCILLATION
6.1. GENERAL
In this chapter, the effect of an
initially oscillating vehicle is studied on
the assumption that the initial amplitude of
variation of the interacting force for each
axle of the vehicle is 15 per cent of the
static weight of the axle. This initial
oscillation may be due to irregularities of
the approach pavement and/or to a small dis
continuity in grade at the first abutment.
As mentioned in Section 2.3.4, the
interacting force between the ith axle and
the pavement at the time the front axle enters
the bridge is expressed by the equation
P. = (1 + C. cos e.) P
i I I s t ,i
in which C., denotes the amplitude of varia
tion of the interacting force from its static
value, and 0. denotes the initial phase
angle. Since the pavement irregularities
which excite the vertical oscillations of a
twoaxle vehicle affect both axles in a some
what similar manner, the value of C. will be
assumed to be the same for both axles. The
phase angle, however, will be considered to
be different. The phase difference between
the two axles will be denoted as
A0 = 01  02
The parameters C, el, and ,A are the three
additional parameters which are considered
for an initially oscillating vehicle. For
singleaxle vehicles, 01 is simply denoted as
0. Throughout the present chapter the
limiting value of the interleaf friction for
each axle is considered to be 15 per cent of
the axle load (i.e. the coefficient of inter
leaf friction ýi for each axle is taken as
0.15) and the initial value of the frictional
force for each axle is taken equal to zero.
Finally, the ratio of the stiffness of the
tiresuspension system to the stiffness of the
tires, 7, is taken as 0.36.
6.2. RESPONSE TO A SINGLEAXLE VEHICLE
6.2.1. Representative History Curves
Figure 6.1a presents time histories of
interacting force for a singleaxle load with
C = 0.15, R = 0.175, a = 0.15, 0 = 00, and
three different values of c . The coefficient
of interleaf friction, p, is taken as infinity
so that the suspension spring cannot engage.
It can be seen that the detailed fea
tures of the curves vary considerably with a
change in t . When the frequency ratio is
low, the frequency of variation of the inter
acting force correspond to the natural fre
quency of the vehicle indicating that the
response of the vehicle is virtually unaffect
ed by the oscillation of the bridge. For the
higher frequency ratios, however, the effect
of the bridge vibration is considerably more
pronounced. Note, in particular, that for
ct = 1.5 the interacting force has oscilla
tions the frequency of which corresponds to
the second natural frequency of the bridge,
f2 = 1.52 fb. This is probably due to the
fact that the frequency of the vehicle is
this case is very close to the second natural
frequency of the bridge. The period of the
vehicle is indicated in the figure by T and
the first two natural periods of the bridge
by Tb and T2, respectively. It can be further
seen that the peak value of the interacting
force is the same in all three cases, and
that the limiting value of the interleaf
friction (I.15 Pst) is exceeded only occasion
ally, and then only by a small amount. It
should be recalled, however, that the histo
ries presented are for a specific value of
0 = 00. Considerably larger values of P were
observed for other values of 0. This is
discussed further in the next section.
Figure 6.1b gives timehistories of
dynamic increment for moment at midspan cor
responding to the histories of interacting
force just presented. As might be expected,
the peak values of the dynamic increments are
consistently larger than those presented in
Figure 5.2 for smoothly moving vehicles.
6.2.2. Study of the Interacting Force
Figure 6.2 shows a plot of the inter
acting force in each span of the bridge for
a vehicle with 15 per cent initial oscillation
as a function of the phase angle 0 for two
combinations of weight ratio and frequency
ratio. The coefficient of interleaf friction
is I = o and the speed parameter is a = 0.15.
It can be seen that for the lighter vehicle
the peak value of the interacting force is of
the order of 1.18Pst and occurs while the
vehicle is in the first or in the third span.
For the heavier vehicle this peak interacting
force is of the order of 1.24Ps and occurs
when the vehicle is in the center span. It
should be noted that the absolute maximum
value of P is highly dependent on 0, a
parameter that cannot be controlled in
practice. Even for a given span, the value
of 0 which corresponds to the maximum inter
acting force cannot be predicted, and the
value of 0 corresponding to the peak value of
P is different for each span.
Figure 6.3 is a plot of the peak value
of the interacting force, regardless of the
span in which it occurs, as a function of the
phase angle 0. In the upper half of the
figure results are presented for Pt = 1.0 and
three different values of R; whereas in the
lower half, R is equal to 0.175 and three
different values of Pt are considered. It can
be seen that there is not a definite trend in
the dependence of the peak value of P on the
phase angle, although a value of 0 = 1800
always gives the lowest value of P. It can
also be noted that the larges values of P
max
correspond to the largest value of R, and/or
the smallest value of t"
6.2.3. Response of the Bridge
Figure 6.4 is a comparison plot of the
peak amplification factors and the peak
dynamic increments for moments M and M as a
c
function of the phase angle 0. The upper half
of the figure corresponds to moments at a
section 0.42 aL from the first abutment, and
the bottom half corresponds to moments at
midspan. The results presented are for a
singleaxle vehicle with the parameters indi
cated in the figure. Two values of frequency
ratio are considered, 0.6 and 1.5, which
correspond to the lower and upper limiting
values of practical significance, respective
ly. It can be observed that the dynamic
increments are not as sensitive to changes in
the phase angle as the amplification factors,
consequently, it would be possible to obtain
the peak value of the dynamic increments with
fewer solutions than required for the peak
amplification factors. As would be expected
the expression 1.0 + (D.I.) represents an
upper bound to the maximum value of the
amplification factor; however, this upper
bound may be quite conservative in certain
cases. The spectrum curves of peak values
of dynamic increments provide useful informa
tion for purposes of interpretation of solu
tions; however, to gain a better insight into
the sensitivity of the true amplification
factors to variations of the various param
eters, the remainder of this discussion will
be concerned with amplification factors
rather than with dynamic increments. It
should be noticed that the influence of the
frequency ratio is more noticeable when
studying amplification factors than when
studying dynamic increments.
Inasmuch as the phase angle 0 is un
predictable, our main interest will center on
the general level of the maximum response
throughout the entire range of this parameter
rather than on the detailed features of the
response curves as a function of 0. In
Figures 6.5a and 6.5b, instead of plotting
each of the spectrum curves in its entirety,
only the portion of each that gives the
maximum amplification factor for a given value
of 0 is presented as a function of 0. That
is, the curves in Figures 6.5 are actually
envelopes of families of curves similar to
those shown in the left half of Figure 6.4.
In the upper half of Figures 6.5a and
6.5b the envelopes of peak amplification
factors for moment at the five critical
sections studied throughout this report are
plotted for a fixed value of Pt and three
different values of R; in the bottom half of
each figure, similar envelopes are given for
a fixed value of R and three different values
of P . It should be noted that the largest
amplification factors correspond to moments
over the interior supports, with the higher
level of response obtained for moments over
the right hand interior support. As might be
expected, the amplification factors for moment
in the first side span generally are larger
than those for moment in the third side span.
Figures 6.6a and 6.6b illustrate the
manner in which the maximum response at each
of the five critical sections considered is
affected by the friction in the suspension
springs of the vehicle. The curves represent
upper envelopes of spectrum curves of peak
amplification factors for moment for three
different values of R. The solid curves are
for p = o, that is, a vehicle for which there
is no energy dissipation due to interleaf
friction, and the dashed curves are for a
value of t = 0.15. It can be seen that the
effect of interleaf friction is to reduce the
peak values of the response and to decrease
the dependence of the response on the phase
angle 9. In the solutions for ( = 0.15, the
initial value of the interleaf friction F. was
assumed to be zero and the stiffness ratio 7
was taken as 0.36. The effects of these
parameters and of t will be studied in greater
detail in the following chapter.
Since the phase angle 9 is one of the
important parameters of the problem and since
this parameter is difficult to control or to
determine in practice, one must consider the
entire possible range of 0 in order to evalu
ate the maximum possible response of the
bridge.
Figure 6.7 compares the maximum values
of response computed for vehicles with 15
per cent initial oscillation when considering
the entire range of 0 with those obtained for
smoothly moving vehicles having the same
value of the speed parameter. The results
are presented as amplification factors for
moment as a function of the frequency ratio
t. The curves for smoothly moving vehicles
were obtained for a value of R = 0.175, and
are given as envelope curves in the figure.
The uppermost part of the figure gives the
maximum effects at section I and 4 of the side
spans, the solid portion of the line corres
ponding to the region of 1 where MI is larger
than M4, and the dashed portion, to the region
where M4 governs. Similarly, in the middle of
the figure, the solid portion represents the
region of Ct where M2 governs, and the dashed
portion represents the region where M3 is
larger than M2. The response corresponding
to initially oscillating vehicles is presented
only for the three values of Pt for which the
entire range of 9 was investigated. The open
points correspond to undamped vehicles, and
the solid points to vehicles with k = 0.15.
As might be expected, the amplification
factors for initially oscillating vehicles
are significantly larger than those for
smoothly moving vehicles. Also, the amplifi
cation factors for damped vehicles (p = 0.15),
even for the small initial oscillations con
sidered in this chapter, are generally smaller
than those obtained for undamped vehicles
(P = w). It is also significant to note that,
for the entire range of Pt values considered
in this study, the results for the initially
oscillating vehicle may be considered to be
independent of Pt, when the complete range of
0 is covered.
6.3. RESPONSE TO A TWOAXLE VEHICLE
6.3.1. Representative Time Histories
In Figures 6.8a and 6.8b, representa
tive time histories are given for twoaxle
vehicles with 15 per cent initial oscillation
and two different values of the phase differ
ence A 0 between axles. The coefficient of
interleaf friction is considered to be in
finite. The upper half of each figure shows
the time histories of the interacting force
of each axle, and the lower half shows the
corresponding time history of dynamic incre
ments for moment at midspan. In Figure 6.8a
both axles are initially in phase, whereas in
Figure 6.8b they are initially 1800 out of
phase.
It can be seen in these figures that in
both cases the peak amplitudes of variation of
the interacting forces are somewhat greater
than those observed in Figure 6.1 for single
axle vehicles, and the interacting forces
exceed fairly frequently the value of 0.15P t'
which represents the limiting value of the
interleaf friction for most vehicles. It
follows, then, that the effects of ki would be
more important in this case than for the
singleaxle vehicles considered in the pre
ceding section. It is of some interest to
note also that when initially in phase with
one another the axles become 1800 out of
phase at the end of the run, and when initial
ly 1800 out of phase they tend to be almost
in phase at the end of the run. In Figure
6.8a a change in phase is apparent in the rear
axle when the front axle passes from the first
to the second span, and in Figure 6.8b a
similar change occurs when the front axle
passes from the second span to the third
span. The larger dynamic increments for M
c
occur in both figures when the axles are in
phase. These dynamic increments are larger
than those obtained for the corresponding
singleaxle vehicle considered in Figure 6.1.
The increase for the twoaxle case is due to
two factors: First, the interacting forces
are somewhat greater, and second (and more
important) the maximum static effects used to
normalize the ordinates of the curves are
smaller for the twoaxle vehicle. In general,
it can be seen that the detailed features of
the response curves are highly sensitive to
AG, a parameter which cannot be controlled in
practice.
6.3.2. Response of the Bridge
In Figure 6.9 the effect of the
parameter !A is studied in greater detail by
means of spectrum curves for maximum amplifi
cation factors for moment at the five critical
sections along the bridge. These curves,
plotted as a function of the phase angle of
the front axle, l0, refer to a twoaxle,
undamped vehicle with s/L = 0.3, a = 0.15,
R = 0.175, Pt = 1.0, and four different
values of AG. Similar data are available for
other combinations of parameters and are
summarized in Table 6.1. These data indicate
that the absolute maximum response at a
section may be considered to be independent
of the phase angle AG, provided the entire
possible range of 0l is investigated. In the
remainder of this chapter only the inphase
condition zA = 00 will be considered.
Figures 6.10a and 6.10b give a compari
son of spectra of amplification factors for
moments at several sections along the bridge
for a singleaxle and a twoaxle undamped
vehicle with 15 per cent initial oscillation
and a value of a = 0.15. The axle spacing
parameter of the twoaxle vehicle is taken as
s/L = 0.3 to give the combination of s/L and
a which prodces addition of effects. The
upper half of each figure gives the envelope
of spectrum curves for Tt = 1.0 and several
values of R, and the lower half gives a
similar envelope for R = 0.175 and several
values of P . The results for the single
axle vehicle are reproduced from Figure 6.5.
It can be seen that the amplification factors
for the twoaxle load are in general greater
than those for the single axle load. This is
due mainly to the fact that the maximum static
effects are smaller for the twoaxle loads.
The results presented in Figures 6.10a
and 6.10b are limited to a value of a = 0.15.
In Figures 6.11 and 6.12 spectrum curves are
plotted as a function of a for both a single
axle vehicle and a twoaxle vehicle with
s/L = 0.3. In Figure 6.11 the results for
moment at midspan are given for three
different values of 8, and in Figure 6.12
only the absolute maximum values of the
response for these values of 0 are presented.
In all cases i = 0.15, R = 0.175, and Pt = 1.0.
6.4. APPROXIMATE EMPIRICAL RELATIONS
From a study of the data presented in
this and the preceding chapters it is felt
that the following equations represent with
reasonable accuracy the absolute maximum
values of dynamic effects that may be induced
in threespan continuous bridges of the type
considered in this study. The equations are
for vehicles with 15 per cent initial oscil
lation and are obtained by modifying the
corresponding expressions presented earlier
for constant moving forces.
The maximum amplification factor for
moment at midspan may be taken as
(A.F.)M = 1.05 + 0.a
c Mc (Mc ) o
(6.1)
where the ratio Mc/(M ) is a function of s/L
and is given approximately by Equation 3.3.
The maximum amplification factors for
moment at sections 1 and 2 are approximated
by the equations
(A.F.) = 1.10 + 0.8
M M1 / (M )
(A.F.)M =
")2
1.10 + 0.8a
(6.2)
(6.3)
where M1/(M ) is given approximately by
Equation 3.1. The effects at sections M4 and
M3 will in general be smaller than those at
sections MI and M2, respectively.
In Figure 6.11 the results predicted
by Equation 6.1 are compared with the actual
data. The results of Equation 6.1 are greater
than those obtained with the computer program,
but it should be remembered that the data are
applicable to specific values of Pt, R, and p.
If one were to consider a range of these pa
rameters, it is believed that the difference
between the results predicted by Equation 6.1
and those obtained with the computer program
would be reduced significantly. *
VII. RESPONSE TO VEHICLES WITH LARGE AMPLITUDES OF INITIAL OSCILLATION
7.1. GENERAL
This chapter is devoted to a study of
the dynamic effects produced by twoaxle
vehicles for which the initial amplitude of
variation of the interacting force for each
axle is 50 per cent of the static reaction of
the axle. Such large amplitudes of initial
oscillation may be induced by a large dis
continuity at the bridge entrance.
The parameters to be considered in
this case are the same as those mentioned in
the preceding chapter. However, since the
initial amplitude of variation of the inter
acting force is several times greater than
the limiting value of interleaf friction in
most vehicles, the energy dissipation due to
interleaf friction will be considerably more
important than before, and the assumption of
i =  can no longer be used. Unless otherwise
noted, in the solutions to be presented the
limiting value of interleaf friction will be
assumed to be 0.15 Pst (i.e. p = 0.15).
Furthermore, the two axles will be considered
to be identical; the static reaction per axle
will be taken as W/2, and the dynamic index
of the vehicle will be taken equal to 1.0.
7.2. EFFECT OF INTERLEAF FRICTION ON VARIATION
OF INTERACTING FORCES
To obtain an idea of the manner in
which the variation of the interacting force
may be affected by interleaf friction, the
response of a singleaxle loading running
over a rigid pavement was first studied.
Figure 7.1 obtained by means of a
phaseplane diagram (15) gives the time
history of the interacting force for such a
load on the assumptions that the initial
value of P = 1.5 Pst, the initial value of the
interleaf friction F. = 0, the limiting value
of the interleaf friction is 0.15 Pst, and the
ratio of slopes of the two segments of the
loaddeformation diagram 7 = 0.36. The time
scale has been normalized with respect to the
natural period Tt of the system when vibrating
on its tires. The scale of interacting force
is normalized with respect to the static
reaction of the axle. The solid portions of
the curve represent the portions of the run in
which the vehicle oscillates only on its tires,
whereas the dashed portions correspond to the
vehicle oscillating on its combined tire
suspension system acting in series. The
changes in the slope of the curve at the
junction of solid and dashed portions are due
to the changes in the stiffness of the system
when the limiting frictional force is over
come, and the suspension spring acts in series
with the tire spring.
It is important to note that after each
cycle of oscillation the amplitude of varia
tion of the interacting force decreases in
magnitude due to the energy dissipated by the
interleaf friction. After a few cycles, a
steadystate condition is attained in which
the double amplitude of variation is equal to
2i P st For the particular example considered,
this steadystate condition is reached at a
time equal to 2.7 Tt.
The curve presented in the preceding
figure is compared in Figure 7.2 with the
corresponding curves obtained for values of
F. equal to F' and F', where F' = p P is
st
the limiting value of the interleaf friction.
All other parameters are the same as before.
It is seen that the steadystate amplitude of
variation of the interacting force is the
same in all three cases; however, this steady
state condition is reached in the shortest
time when F. = F' and in the longest time
when F. = 0.
In Figure 7.3 the curve presented in
Figure 7.1 is replotted so that it can be
compared with that obtained for the same load
moving over a flexible bridge. The bridge
parameters are indicated in the figure. The
relationship between the abscissas used in
this plot and the plot of Figure 7.1 is
obtained by multiplying the numerator and
denominator of the quantity t/Tt by
2( + 2a) LvTb and making use of the identities
vt = x and a = vTb/2L. Proceeding in this
manner one obtains
t _ (1 + 2a) Tb x 1 + 2a .
T 20 T (1 + 2a)L 20a t
t t
For the particular case considered, a = 0.15
and ýt = 1.0; accordingly,
t
T = 8.66
t
It can be seen that the two curves, especially
in their initial stages, are in good agreement
between each other, indicating that the
flexibility of the bridge does not signifi
cantly affect the initial portion of the
curves when the interacting forces are large.
Note in particular that the time of the
"steadystate" condition is predicted quite
accurately by the approximate solution.
Figure 7.4 gives the time histories of
interacting force for a load moving over the
bridge for two different values of F.. Al
though the general appearance of the curves is
quite similar, the detailed features differ
and clearly reflect the influence of the
bridge motion.
The data presented above and some
additional data reported previously 19) show
that the "steadystate" amplitude of variation
of the interacting force is always equal to or
less than P ,st and that the time required to
reach this "steadystate" condition depends
on:
(a) the initial amplitude of variation
of the interacting force C,
(b) the coefficient of interleaf
friction p,
(c) the initial value of the friction
al force F., and
(d) the ratio of the slopes of the two
segments of the loaddeformation
relation y.
It turns out that for a load moving over a
rigid pavement the time required for a steady
state condition t can be expressed in a
s
dimensionless form by plotting t /Tt versus
P/ý where
= C  Fi/Pst
(7.1)
In Figure 7.5 the ratio t s/Tt is plot
ted in this form for several values of 7 and a
value of F. = 0. It can be seen that the
value of t s/Tt increases with decreasing ratio
of /V. Decreasing coefficients of interleaf
friction obviously lead to smaller residual
amplitudes of variation. However, these
smaller amplitudes are obtained at the expense
of time. It can also be seen that the values
of t /Tt generally increase with increasing
values of 7. For instance, for k = 0.15,
C = 0.50, and Fi = 0, that is, p/P = 0.3,
t /Tt will be 2.5 for a stiffness ratio of
0.36 and will be of the order of 5.0 for a
stiffness ratio of 0.5.
Figure 7.6 presents similar curves for
a fixed value of 7 = 0.36 and three different
values of the initial friction. It is seen
that the shape of the curves is approximately
the same for all values of F., although for a
given value of p/5 the steadystate condition
is reached earlier when the initial frictional
force is F' than when F. = F'. This result
is in apparent disagreement with the discus
sion of Figure 7.2, but it should be noted
that Figure 7.2 was obtained for a constant
value of C. Consequently, the solutions in
that case correspond to different values of
i/P. It is apparent that the influence of F.
on t is small. For example, for i/P = 0.30,
a change in F. from F' to F' produced a
change in ts of the order of 1.0 T t. The
resulting change is still smaller when one
considers fixed values of ui and C. For
example, for p = 0.15 and C = 0.50 the values
of p/B corresponding to F. = F', F. = 0, and
F. = F' are 0.23, 0.30, and 0.43, respective
ly, and the maximum change in t is only
0.2 T .
7.3. RESPONSE OF THE BRIDGE
7.3.1. Effect of Initial Friction
Figures 7.7a and 7.7b show spectra of
maximum amplification factors for moments
MI, M2, and Mc for singleaxle and twoaxle
damped vehicles. The spectra are given as a
function of 01 for three different values of
the initial frictional force. The remaining
parameters are identified on the figure. For
the twoaxle vehicle, both axles are consider
ed to be initially in phase.
It can be seen from these figures that
the maximum difference in amplification
factors due to a variation in F. between its
limiting values of F' and F' is of the order
of 0.20 and that the difference decreases as
one considers effects at sections closer to
midspan. In this connection it should be
noted that values of F. varying from F' to F'
are not realistic for all possible values of
01. In fact, the limiting values of F. can
only occur for 01 equal to 00 or to 1800. For
other values of 0l, the range of variation
of F. will be smaller. Specifically, for
values of 61 between 0° and 1800, F. will be
bounded between F' and a value which is
smaller than F' and decreases with increasing
aI. For values of 01 between 180° and 360°,
F. will be bounded between F° and a value
which is greater than F' and increases with
increasing 0 l In the remainder of this study
the initial frictional force will be taken as
zero.
7.3.2. Effect of Amplitude of Initial
Oscillation
In Figure 7.8a a comparison is made
between the amplification factors for moment
at midspan produced by a singleaxle vehicle
with 15 per cent initial oscillation, both
damped and undamped, and the effects produced
by a damped singleaxle vehicle with 50 per
cent initial oscillation. The curves repre
sent upper envelopes of spectra for several
values of R or Pt. The upper part of the
figure refers to a value of Tt = 1.0 and
values of R equal to 0.175, 0.35, and 0.5,
whereas the lower part refers to corresponding
results for R = 0.175 and values of Pt equal
to 0.6, 1.0, and 1.5. It can be seen that the
peak values obtained for damped vehicles with
50 per cent initial oscillation are about the
same as those obtained for undamped vehicles
with 15 per cent initial oscillation. This
is due to the fact that, since the coefficient
of interleaf friction in these solutions is
V = 0.15, the "steadystate" amplitude of
variation of the interacting force is 0.15
P st. Therefore, once the vehicle attains its
"steadystate" condition, it acts essentially
as an undamped vehicle with 15 per cent
initial oscillation.
In Section 7.2 it was shown that for a
load with y = 0.36, F. = 0, and i/P = 0.3 the
steadystate condition is reached at a time
of 2.7 T t. On the other hand, the time re
quired for the load to reach the center of the
center span equals 1.3 t/2a. Now, for Tt =
1.0 and a = 0.15, the latter expression gives
a value of 4.33 T . In other words, the
"steadystate" condition will be obtained
before the load reaches midspan. In fact,
this condition will be attained when the
vehicle is at a distance of 0.81L from the
first abutment, that is, slightly after it
enters the center span.
The results for a damped vehicle with
50 per cent initial oscillation are in general
somewhat greater than those for a damped
vehicle with 15 per cent initial oscillation.
However, this agreement is limited only to
effects at midspan. For sections closer to
the bridge entrance the difference in the
magnitude of the effects induced by vehicles
with 15 per cent and 50 per cent initial
oscillation is quite appreciable. This can be
seen in Figure 7.8b, which gives information
similar to that given in Figure 7.8a but for
sections away from midspan. In this figure,
only the larger of the two values computed for
symmetric sections of the bridge is reported.
Considering that the peak values of the
upper and lower parts of Figures 7.8a and 7.8b
are about the same, and further considering
that most of these curves exhibit several
maxima of the same order of magnitude, it is
concluded that the absolute maximum value of
the response may be assumed to be independent
of both the weight ratio and the frequency
ratio, provided one covers the complete range
of values for the phase angle 0e1
Figure 7.9 gives spectra of amplifica
tion factors for moment at the five critical
sections of the bridge for a twoaxle vehicle
and four different values of the phase
difference between axles, A6. These solutions
refer to a damped vehicle with 50 per cent
initial oscillation. These curves show that
the response is quite sensitive to the value
of z0, but that the maximum peak of the
response for A9 = 0° and the entire range of
01 values is, in general, a good measure of
the absolute maximum value of the response
obtained.
The upper envelopes of these curves are
compared in Figure 7.10 with the upper enve
lopes of corresponding curves for an undamped
vehicle with 15 per cent initial oscillation
presented previously in Figure 6.9. Again,
the amplification factors for moment at mid
span for damped vehicles with large initial
oscillations are only slightly greater than
those for undamped vehicles with small
initial oscillations.
In Figure 7.11 the general level of the
response throughout the range of 0 for a
singleaxle damped vehicle with 50 per cent
initial oscillation is compared with corres
ponding results for damped and undamped
vehicles with 15 per cent initial oscillation
and with results obtained for smoothly moving
vehicles. The results are given in the form
of amplification factors for several effects
and plotted as a function of the frequency
ratio Tt . It can be seen that, within the
range of parameters considered, the maximum
level of response may be considered to be
independent of the frequency ratio, and that
the amplification factors due to vehicles with
50 per cent initial oscillation are of the
order of 0.10 to 0.15 greater than the corres
ponding values due to vehicles with 15 per
cent initial oscillation.
7.3.3. Effect of Coefficient of Interleaf
Friction
Figures 7.12a through 7.13 give spectra
of amplification factors for moment at dif
ferent sections of the bridge, both for damped
and undamped singleaxle vehicles with 50 per
cent initial oscillation. The spectra are
plotted as a function of Ct and refer to solu
tions with a = 0.15 and 0 = 0°. These data
are included to illustrate that the dynamic
effects due to undamped vehicles are signifi
cantly greater than those induced by damped
vehicles.
000
VIII. SUMMARY
8.1. GENERAL
The numerical data presented in this
report are derived from a theory in which the
bridge is idealized as a single beam of uni
form cross section and mass per unit length.
Throughout this study, the bridge has been
considered to be horizontal and smooth.
However, the effects of irregularities on the
approach pavement and of discontinuities at
the bridge entrance have been taken into
account by considering the effects of initial
ly oscillating vehicles.
No effort has been made in this study
to relate the results of the theory with the
results of field tests reported elsewhere in
the literature, (1,8,10,11,13) because the test
programs generally provided insufficient
information about the characteristics of the
test vehicles.
8.2. SUMMARY OF RESULTS
The following is a brief summary of
some of the principal results of this
investigation.
(a) Under static conditions, a side
span ratio of a = 0.8 leads to nearly equal
maximum positive moments in all the spans and
to the smallest value of negative moments over
the intermediate supports. All studies of
dynamic effects considered in this report were
made for threespan continuous bridges with a
side span ratio of 0.8.
(b) The amplification factors for the
various effects generally increase with
increasing value of the speed parameter a
which is defined in Section 2.4.1. Therefore,
for design purposes the value of a corres
ponding to the maximum expected vehicle speed
should be used. Note that a depends also on
the fundamental natural period of the bridge
and on the length of the center span, L.
However, for the type of bridges considered,
the natural period may be considered to
increase linearly with L.
(c) The dynamic effects in a system
for which the ratio (s/L)/a is zero or an even
integer are generally larger than those when
this ratio is an odd integer.
(d) The response of the bridge is
insensitive to variations in the weight ratio
R. Although, for smoothly moving vehicles
the maximum dynamic effects tend to increase
with R, for initially oscillating vehicles the
reverse is true. However, the total dynamic
effects obviously are greater for the heavier
vehicles.
(e) The influence of the frequency
ratio ~t is erratic and not easy to define.
The results presented indicate that, within
the complete possible range of the phase
angle of the initial motion of the vehicle
0, the maximum amplification factors for fixed
values of a and R may be considered to be
independent of the frequency ratio t . It
follows that, for design purposes, one may
consider an average value of Pt and an average
value of R, provided the phase angle 0 is varied
between the limiting values of 0° and 3600.
(f) The effect of the phase difference
between axles LA and of the initial value of
the frictional force F. may be considered to
be of secondary importance, provided the phase
angle 0 is varied over its complete possible
range. For convenience both A0 and F. may be
taken equal to zero.
(g) The interleaf friction in the sus
pension system of the vehicle is an important
source of energy dissipation and should be
considered in studies of bridge response. The
amplification factors computed on the assump
tion that the vehicle is a linearly elastic,
undamped system may be extremely overconser
vative, particularly for vehicles with large
amplitudes of initial oscillation.
(h) The large the initial amplitudes of
variation of the interacting forces, the
greater will be the dynamic effects on the
bridge. In this study the maximum value of
initial oscillation considered was 50 per cent
of the static reaction of the axles. Larger
amplitudes, while possible, are believed to be
unlikely, and, if they do occur, they will be
associated with small values of the speed
parameter a.
(i) In general, the dynamic moments due
to initially oscillating vehicles are larger
in the first span than corresponding moments
in the third span. Similarly, the dynamic
moments over the first interior support are
generally greater than those over the second
interior support. However, since vehicles may
enter the bridge from either direction, the
larger of the two sets of values must be used
for design purposes.
0() For bridges without a sharp dis
continuity between the approach pavement and
the first abutment, the amplification factor
for maximum moments may be estimated from the
expressions presented in Equations 6.1 through
6.3. The effect of discontinuities at the
bridge entrance may be estimated from the data
presented in Chapter VII. These equations and
data refer to threespan continuous bridges of
the Ibeam type, but may also be applied for
other bridge types for which the fundamental
parameters are within the range considered
in this investigation.
For bridges of unusual design not cov
ered by the range of parameters investigated,
the following procedure may be used if an
"exact" analysis of the problem is warranted.
This approach requires the use of a highspeed
computer.
(1) Design the bridge for static loads
making some "reasonable" allowance for dynamic
effects.
(2) Compute the maximum possible value
of a and representative values of cP and R for
the vehicles that may travel over the bridge.
(3) Considering the parameters deter
mined in the preceding item and values of
p = 0.15, 7 = 0.36, F. = 0, and some reason
able value of C based on the expected condi
tion of the approach pavement, investigate the
response of the bridge for the complete range
of the phase angle 0 of a singleaxle vehicle.
(4) Use the absolute maximum amplifica
tion factor for M1, M2, and Mc to check the
design and to redesign the structure if
necessary. *
REFERENCES
1. L. E. Vandegrift, Vibration Studies of
Continuous Span Bridges (Engineering
Experiment Station Bulletin No. 119),
Columbus, Ohio: Ohio State University,
1944.
2. Otto Ernst Bollinger, Tabellen fur
Durchlaufende Trager, Zurich: Schweizer
Druck und Verlagshaus Ag., 1949.
3. R. S. Ayre, G. Ford, and L. S. Jacobsen,
"Transverse Vibration of a TwoSpan Beam
Under the Action of a Moving Constant
Force," Journal of Applied Mechanics,
Vol. .17 (1950), pp. 112.
4. L. S. Jacobsen and R. S. Ayre, Transverse
Vibration of One and of TwoSpan Beams
Under the Combined Action of a Moving
Mass Load and a Moving Alternating Force
(Structural Dynamics Technical Report 10),
Stanford, Calif.: Stanford University,
1950.
5. R. S. Ayre, L. S. Jacobsen, and C. S. Hsu,
"Transverse Vibration of One and of Two
Span Beams Under the Action of a Moving
Mass Load," Proceedings, First U. S.
National Congress on Applied Mechanics
(American Society of Mechanical Engi
neers), 1951, pp. 8190.
6. L. S. Jacobsen and R. S. Ayre, Transverse
Vibration of One and of TwoSpan Beams
Under the Action of a Moving Combined
Load Consisting of a SpringBorne Mass
and an Alternating Force (Structural
Dynamics Technical Report 12), Stanford,
Calif.: Stanford University, 1951.
7. L. S. Jacobsen and R. S. Ayre, Transverse
Vibration of MultiSpan Continuous Beams
Under the Action of a Moving Alternating
Force (Structural Dynamics Technical
Report 15), Stanford, Calif.: Stanford
University, 1951.
8. First Progress Report, Highway Bridge
Impact Investigation, Urbana, Ill.:
University of Illinois Civil Engineering
Department, 1951, pp. 911.
9. Standard Plans for Highway Bridge Super
structures, Washington, D.C.: U. S.
Department of Commerce, Bureau of
Public Roads, 1956.
10. G. M. Foster and L. T. Oehler, "Vibration
and Deflection of RolledBeam and Plate
Girder Bridges," Vibration and Stresses
in Girder Bridges (Highway Research Board
Bulletin 124), Washington, D.C.:
National Research Council Highway
Research Board, 1956, pp. 79110.
11. J. M. Hayes and J. A. Sbarounis,
"Vibration Study of ThreeSpan Continuous
IBeam Bridge, Vibration and Stresses in
Girder Bridges (Highway Research Board
Bulletin 124), Washington, D.C.:
National Research Council Highway
Research Board, 1956, pp. 4778.
12. S. Mitchell and G. F. Borrmann, "Vehicle
Loads and Highway Bridge Design,"
Proceedings, American Society of Civil
Engineers, Vol. 83, ST4 (July, 1957),
Paper 1302.
13. L. T. Oehler, Vibration Susceptibilities
of Various Highway Bridge Types, Michigan
State Highway Department, Research
Laboratory Testing and Research Division
Report No. 272, January, 1957.
14. A. S. Veletsos and N. M. Newmark,
"Natural Frequencies of Continuous
Flexural Members," Transactions, American
Society of Civil Engineers, Vol. 122
(1957), pp. 249278.
15. L. S. Jacobsen and R. S. Ayre, Engineer
ing Vibrations, New York: McGrawHill
Book Co., 1958, pp. 160187.
16. Standard Specifications for Highway
Bridges, Washington, D.C.: American
Association of State Highway Officials,
1960, p. 9.
17. J. F. Fleming, "The Effect of Load
Characteristics and the Bridge Geometry
upon Highway Bridge Impact," Ph.D.
thesis, Carnegie Institute of
Technology, 1960.
18. T. Huang and A. S. Veletsos, Dynamic
Response of ThreeSpan Continuous High
way Bridges (Civil Engineering Studies,
Structural Research Series No. 190),
Urbana, Ill.: University of Illinois
Civil Engineering Department, 1960.
18.a. Ibid, p. 47.
18.b. Ibid, p. 54.
18.c. Ibid, p. 66.
19. J. A. Nieto, A Study of Effect of Inter
leaf Friction on the Dynamic Response of
SimpleSpan Highway Bridges  Part C,
Tenth Progress Report, Highway Bridge
Impact Investigation, Urbana, Ill.:
University of Illinois Civil Engineering
Department, October, 1960.
20. W. H. Walker, Studies of Dynamic Behavior
of SimpleSpan Highway Bridges  Part B,
Tenth Progress Report, Highway Bridge
Impact Investigation, Urbana, Ill.:
University of Illinois Civil Engineering
Department, October, 1960.
21. C. Oran and A. S. Veletsos, Analysis of
Static and Dynamic Response of Simple
Span, Multigirder Highway Bridges (Civil
Engineering Studies, Structural Research
Series No. 221), Urbana, Ill.: Univer
sity of Illinois Civil Engineering
Department, 1961.
22. W. H. Walker, J. A. Nieto, and A. S.
Veletsos, Summary of Activities  PartA,
Eleventh Progress Report, Highway Bridge
Impact Investigation, Urbana, Ill.:
University of Illinois Civil Engineering
Department, October, 1961, Section 4.
23. J. F. Fleming and J. P. Romauldi,
"Dynamic Response of Highway Bridges,"
Proceedings, American Society of Civil
Engineers, Vol. 87, ST7, Part I
(October, 1961), Paper 2655, pp. 3161.
24. A. S. Veletsos and W. H. Walker,
"Dynamic Behavior of Highway Bridges,"
paper presented at the Annual Meeting
of the American Society of Civil
Engineers, Detroit, October, 1962.
25. S. J. Fenves, A. S. Veletsos, and C. P.
Siess, Dynamic Studies of Bridges on
the AASHO Road (Civil Engineering
Studies, Structural Research Series
No. 227), Urbana, Ill.: University of
Illinois Civil Engineering Department,
1962.
26. W. H. Walker and A. S. Veletsos, Response
of SimpleSpan Highway Bridges to Moving
Vehicles (Engineering Experiment Station
Bulletin No. 486), Urbana, Ill.:
University of Illinois, 1966.
27. TractorTrailer Ride, New York:
Technical Board of Society of Automo
tive Engineers, Inc. (485 Lexington
Ave.)
28. Standard IBeam Bridges, Taken from
design charts of the State of Illinois,
Private Communication.
29. "Survey of Characteristics of Continuous
Bridges," Unpublished report of the
University of Illinois Civil Engineer
ing Department.
APPENDIX
This Appendix is a reprint of Chap
ters 2 and 3 from a report titled Dynamic
Response of ThreeSpan Continuous Highway
Bridges by Tseng Huang and A. S. Veletsos,
Civil Engineering Studies, Structural Research
Series No. 190, University of Illinois,
September, 1960. This report is Reference 20
listed on page 58.
METHOD OF ANALYSIS
4. IDEALIZATION OF BRIDGE AND VEHICLE
4.1. IDEALIZATION OF BRIDGE
It is assumed that during vibration the
deflection configuration of the bridge in the
transverse direction remains the same at all
times. Accordingly, the bridge may be repre
sented by a beam. In the analysis of the
beam, the actual distributed mass is lumped
into a series of point masses, spaced at equal
intervals within each span. However, the
flexibility of the beam is considered to be
distributed. Thus the actual system which
has an infinite number of degrees of freedom
is replaced by a system for which the number
of degrees of freedom is equal to the number
of mass concentrations used. Figure I shows
the replacement system for a simplespan
bridge and a threespan continuous bridge.
Damping in the bridge is assumed to be
viscous. In the actual system the damping
resistance is distributed along the length of
the bridge. In the replacement system this
resistance is assumed to be concentrated at
the points of mass concentration, as shown by
the dashpots in Figure 1.
4.2. IDEALIZATION OF VEHICLE
Since the bridge has been idealized as
a beam, the width of the vehicle and conse
quently, the rolling effect cannot be con
sidered in the analysis. Even when treated as
a plane system, a vehicle is a very complex
mechanical system. However, insofar as its
effect on a bridge is concerned it may be
represented by one or two rigid bodies
supported on a series of springs and dashpots.
Figure 2 shows diagrammatically the
detailed features of what is believed to be a
complete representation of a tractortrailer
combination. All shaded areas in this figure
are considered to be rigid bodies. The
quantity WI represents the weight of the
tractor mounted on its suspension system. The
quantity i I is the dynamic index(28) of the
tractor. This is a measure of the rotary
moment of inertia of the weight W, and it is
defined as the ratio of the radius of gyration
squared to the product of the horizontal dis
tances between the two supports and the center
of gravity of the weight. The dashpots at the
center of gravity of W1 represent damping
resistances against vertical motion and rotary
motion. The rigid bar represents the chassis
of the tractor and its weight is designated
as w4. The point masses, with weight wI and
w2, represent the mass of the axles, springs,
and tires for the two axles. The quantities
W2, i2, and w3 refer to the trailer and have
the same meaning as that of the corresponding
quantities for the tractor. For convenience
in presentation, the weights WI and W2 are
referred to as "sprung" weights and the re
maining weights are referred to as "unsprung"
weights.
The dynamic characterisitcs of the tire
tires for each axle of the vehicle are
represented by a spring and a dashpot. The
suspension system for each axle is represented
by a massless spring, a dashpot, and a
frictional device. The dashpot accounts for
the effects of shock absorbers or air sus
pension, and the frictional device accounts
for any frictional force that may develop in
the suspension system, particularly in the
leaf springs. The value of the frictional
force developed at any time is designated by
F and the limiting or maximum possible value
is designated by F'. As long as F' < F < F'
for a particular axle, the suspension spring
for that axle is inactice (i.e. only the tire
spring deflects), and the effective stiffness
of that axle is equal to the stiffness of the
tires. On the other hand, if f = + F', both
springs are active and the effective stiffness
is that of the two springs acting in series.
The characteristics of the suspensiontire
system for a simplified case will be explained
further in Article 6.2.
In the present analysis the above
system is further simplified by (a) neglecting
all sources of viscous damping and (b) re
placing the "unsprung" weights by concentrated
"sprung" weights as shown in Figure 3a. In
this replacement the weight of the chassis,
designated as w4 in Figure 2, is incorporated
into the weights wI and w2. This replacement
is justified by the fact that the "unsprung"
weights are quite small in comparison to the
"sprung" weights. For a representative
tractor the ratio of the total "unsprung"
weights to the "sprung" weight is about 1/7,
and for a trailer it is for all practical
purposes negligible. In addition to the
threeaxle load unit, in Figure 3 are shown
specialized models for a twoaxle and a single
axle load unit.
With its velocity specified, the three
axle load unit shown in Figure 3a has three
degrees of freedom. The parameters which
define its characteristics are:
(a) the weight distribution parameters
which include the weights W,, W2, wl, w2 and
w3, and the dynamic indices il and i2;
(b) the geometrical parameters which
include the axle spacings H£ and 2', and the
ratios a through a5, as defined in Figure 3a;
(c) the stiffness parameters for the
tires and the suspension springs; for the ith
axle (i=1,2,3), the stiffness of the tires is
denoted by kt,i and the stiffness of the tires
and the suspension springs when acting in
series is denoted by k ts,i;
(d) the friction parameters, for the
suspension systems of the vehicle. For the
ith axle this is the limiting frictional
force Fi.
5. METHOD OF ANALYSIS
5.1. ASSUMPTIONS
The analysis is based on the ordinary
beam theory, which neglects the effects of
shearing deformation and axial forces. In
addition, since the mass is treated as a
series of point masses, the effect of rotary
moment of inertia does not enter in the solu
tion. The vehicle is assumed to remain in
contact with the bridge at all times, and its
angular displacements are considered to be
small. It is further assumed that no longitu
dinal force can develop at the junction of the
tractor and trailer. This junction is known
as the "fifth wheel pivot." Finally, all
springs of the vehicle are considered to be
elastic.
5.2. COORDINATES
The motion of the vehiclebridge system
is expressed in terms of the coordinates z.
and yr shown in Figure 4. The coordinate z.
denotes the vertical displacement, measured
from a fixed horizontal plane, of the point
of support of the vehicle mass for the ith
axle. The coordinate y denotes the deflec
th
tion of the r node point of the beam. This
deflection is measured from the static
equilibrium position when the bridge is
subject to its own weight alone. Both coor
dinates z and y are considered to be positive
when downward.
5.3. EQUATIONS OF MOTION FOR BRIDGE MODEL
Let P. be the interacting force between
th
the bridge surface and the i axle of the
vehicle. Then the equation of motion for the
concentrated mass at the rth node of the beam,
m , may be expressed as follows:
mr r + cmr r  Ry  Q Pi = 0 (1)
where yr is the deflection of the r node, as
previously defined, and a dot superscript
denotes one differentiation with respect to
time. The quantity Rj is defined as the
r
reactiondeflection coefficient and represents
the static reaction at the rth node point
induced by a unit deflection of the jth node
point, when all other nodes are supported
against deflection. A reaction is considered
as positive when directed upward. In an
analogous manner, Q' is defined as the reac
r
tionload coefficient and represents the
th
reaction at the r node point induced by a
concentrated unit load at the point of appli
cation of P. when all nodes are supported
against deflection. Obviously, when the unit
load is off the bridge, Qr 0.
In Equation I the first term represents
th
the inertia force for the r mass, the second
term represents the concentrated damping
force, and the last two terms represent the
total resisting force provided by the beam.
In particular, the third term denotes the
resisting force produced by the displacements
of the node points. The summation for this
term is extended over all node points. The
last term denotes the resisting force due to
the interactive forces P. when the nodes are
held against deflection. The summation for
this term is extended over all the axles
considered. It should be noted that the
interacting forces P. are not known at this
stage. The procedure used to evaluate these
forces is described in Article 5.5.
By application of Equation I to each
mass, one obtains as many equations as there
are degrees of freedom for the bridge model.
The quantities R depend only on the character
istics of the bridge model, whereas the
quantities Q depend both on the characteris
tics of the bridge and the position of the
load; hence the latter are timedependent
quantities. Both quantities can be evaluated
in a number of different ways. The procedure
used in this work will be described in
Articles 7 and 8.
Equation I is applicable to bridges
having any boundary conditions and any number
of spans, and is independent of whether the
cross section of the bridge is uniform or not.
It can, therefore, be applied to simple span,
continuous or cantilever bridges. The reac
tion coefficients R and Q will, of course, be
different in each case. It may be noted also
that the speed of the vehicle may vary
arbitrarily.
5.4. EQUATIONS OF MOTION FOR A VEHICLE
Let P . be the reaction at the ith
st,i
axle when the vehicle is in a position of
static equilibrium. With P. denoting the
dynamic reaction at any time t, the disturbing
force for the i axle is P.  P . and the equation of motion for a threeaxle vehicle can be
stated in the form:
131 f~p1
11 al2 a13 2l P  Pst,1
1 a a 2  P
12 a22 a23 2 w=  P2 Pst,2
13 a23 a33 3 3 Pst,
where g is the gravitational acceleration, W is the total weight of the veh
a33 are dimensionless coefficients given by the following expressions:
2 1W 2 +2 W1
all = (a1 + aa2i1)  + a5 (a3 + a3a4i2) + 
al2 = ala2 (il + a (1a5)(a3 + a3a4i2) 
al3 = a3a4a5 (1i2) 
2 1 2 2 W2 2
a22 = (a2 + ala2il )  + (1a5) (a3 + a3a4i2)  + 
a23 = a3a4 (1a5) (li2)
2 2 3
a33 = (a4 + a3a4i2) 2+ 
icle, and all through
(3)
The symbols in these expressions have already been defined. The details of derivation are
presented in Appendix A. In the following the matrix of the coefficients a is denoted as
matrix A.
Premultiplication of Equation 2 by the inverse of matrix A yields,
b11 b12 b13
bl12 b22 b23
b13 b23 b33
1 st, l
2 st,2
3 st,3
Since the matrix A is symmetric, its inverse,
matrix B, is also symmetric. For a case in
volving more than one vehicle, an equation of
the above form must be written for each
vehicle.
It can be shown that a sprung mass with
a value of dynamic index i=l is dynamically
equivalent to two separate point masses
attached directly to the supporting springs of
the distributed mass. The weight of the two
I
masses must be equal to the static reactions
produced by the distributed mass. By making
use of this fact, it is possible to consider
certain special cases of a threeaxle load
unit. The following cases are of special
interest.
(a) When i2 = 1 and a3 = 0, one
obtains a singleaxle load with a weight
W + w3 preceded by a twoaxle load unit. In
this case, the coefficients a 13 and a23 in
Equation 2 are equal to zero, and, conse
quently, in Equation 4, bl3 = b23 = 0. The
twoaxle load unit shown in Figure 3b can be
obtained from the above case by taking, in
addition, W2 + w3 = 0. In this case, the
coefficient a33 = 0, and the matrices A and B
are of the second order.
(b) When iI = 1, a5 = 0 and aI = 1,
one has a singleaxle load of weight W1 + wl
followed by a twoaxle load. In this case,
a12 = al3 = bl2 = bl3 = 0. The singleaxle
load unit shown in Figure 3c can be obtained
by taking, in addition, W = w = w = 0.
Then the matrices A and B reduce to all and
bl1, respectively.
(c) By taking il = i2 = 1, a5 = 0,
and a, = a3 = 1, one obtains three single
axle load units of weights W + wl, W2 + w
and w3. In this case, A and B are diagonal
matrices.
It should be noted that these special
ized load units can be obtained also by a
different combination of the parameters in
volyed.
5.5. EVALUATION OF INTERACTING FORCES
Equationsl and 4 are coupled through
the interacting forces P., which remain to be
evaluated. Let time t be measured from the
instant the first axle enters the bridge.
Then the interacting force at time t is given
be the equation
P P =PI t =
where P.i I
' t = o
t du.
+ k.  dr
i dr
is the initial value of P.,
k. is the instantaneous effective stiffness of
Ith
the suspensiontire system for the i axle at
any time T, and u. is the corresponding
shortening of the suspensiontire system.
If at the instant it enters the bridge,
the vehicle is at the position of static
equilibrium, the initial force P =
I t = o
P .. The instantaneous stiffness k. depends
on the magnitude of the frictional force F.
which, in turn, depends on the history of the
shortening u.. As previously noted, when the
frictional force F. for the i axle is less
than its limiting value F!, the quantity k. is
equal to the stiffness of the tires only,
whereas when F. = F!, k. is equal to the com
bined stiffness of the suspension springs and
the tires acting in series. The procedure
used to evaluate k. for a simplified case is
described in detail in Article 9.2.
The shortening u. can be expressed in
the form,
u. = zi + dpi  yp + constant (6)
th
where z. is the coordinate for the i axle,
as previously defined. The quantity dPi
represents the deviation of the bridge profile
from a horizontal line passing through the
first abutment for the point of application of
P., as shown in Figure 4, and it is positive
when upward. This deviation may be due to
dead load deflection, initial camber, grade,
vertical curve or roadway unevenness. The
quantity yp. represents the deflection of the
bridge at the point where P. acts. The
deflection yp. is measured from the static
equilibrium position of the beam, when acted
upon by its own weight, and it is positive
downward. This quantity is a function of the
coordinates y (i.e. of the deflections of
all node points) and of the magnitude and
position of the interacting forces P.. The
constant term, while irrelevant in subsequent
computation, is required in the above expres
sion since z. is measured from an arbitrary
reference line.
5.6. SUMMARY
Application of Equations I and 4 to
each concentrated mass of the bridge model
and to each axle of the vehicle yield a set of
simultaneous, second order differential
equations, equal in number to the number of
degrees of freedom of the bridgevehicle
system. In these equations the independent
variable is t and the dependent coordinates
are yr and z.
In the solution of equations of this
type, it is usual to express all timedepen
dent quantities, other than the coordinates
themselves, in terms of the coordinates and
the independent variable. In the present
case, the additional timedependent variables
in Equations I and 4 are the reactionload
coefficients, Q , and the interacting forces,
P.. With the vehicle speed specified, the
quantities Q can be expressed explicitly in
r
terms of the position of the load, which is a
function of t, and the characteristics of the
bridge model. However, the quantities P.
cannot be expressed explicitly, as can be
appreciated b/ an examination of Equation 5.
It can be seen that the right side of this
integral equation includes the quantities u.
and k., both of which are functions not only
of the coordinates y and z. and of other
physically determinable quantities, but also
of all interacting forces P.. Furthermore,
as explained in the preceding article, the
value of the instantaneous stiffness k. de
l
pends upon the past history of motion of the
entire system.
These equations can be solved conven
iently by a numerical method of integration in
which the evaluation of the interacting forces
P. is a major intermediate step.
As the integration of the differential
equations is carried out, the values of all
the coordinates and of the interacting forces
are determined. From these quantities the
values of the corresponding deflections,
moments and reactions at any desired section
may then be evaluated by statics.
It is to be emphasized that the equa
tions of motion presented in this chapter can
be applied also to the cases for which the
bridge material is nonlinear or even plastic.
For nonlinear elastic material, the reaction
deflection coefficients Rj and the reaction
r
load coefficients Qr in Equation I depend on
the value of the deflection at each node
point and on the magnitudes and locations of
the interacting forces P.. For the plastic
case, these two coefficients depend not only
on the quantities mentioned above, but also
on the deflection history of the node points.
APPLICATION OF METHOD TO ANALYSIS OF THREE
SPAN CONTINUOUS BRIDGES
This chapter is concerned with the
detailed application of the method presented
in the preceding chapter to the special case
of a threespan continuous bridge traversed by
a single vehicle.
6. SYSTEM CONSIDERED
The system considered is shown in
Figure 5; its characteristics are as follows:
6.1. BRIDGE
The bridge model is a threespan con
tinuous beam of equal side spans and uniform
flexural rigidity, El. The length of the
center span is denoted by L and the length of
a side span by aL. The center span is divid
ed into n equal panels of length  ah. The
n
nodes are numbered consecutively starting
with zero at the left abutment and termina
ting with (2n+m) at the right abutment. The
panel between nodes r and rI is designated
as the r panel. As before, the mass is
considered to be concentrated at the node
points.
6.2. VEHICLE
The vehicle is idealized by any one of
the systems shown in Figure 3. The following
additional assumptions are made: (a) both
the suspension springs and the tire springs
are linearly elastic, (b) the maximum fric
tional force which can be mobilized in the
suspension system of an axle is constant, and
(c) the speed of the vehicle is constant.
Available test data on trucks(28),(29)
show that the stiffness of the suspension
springs is fairly constant but that the stiff
ness of the tires is dependent on the inten
sity of the applied load. These tests show
further that the maximum frictional force
which can be mobilized in the suspension of
an axle is in general a complicated function
of the load transmitted through the axle and
depends on such factors as the condition and
the age of the springs. However, when the
variation in the magnitude of the interacting
force is small, the assumption of linear
elasticity for both springs and the assumption
of constant maximum frictional forces are
quite reasonable. These assumptions appear
to be acceptable even for large variations of
the interacting force. In selecting the
stiffness of the suspension spring and of the
tires of an axle, one should use the values
corresponding to a load equal to the static
reaction on that axle. Similarly, the value
of the limiting frictional force for an axle
should be determined for a mean load equal to
the static load on that axle.
The relationship between the inter
acting force, P, and the shortening, u, of
the combined suspensiontire system is shown
in Figure 6. Included in this figure, is
also a diagram showing the relationship be
tween u and the frictional force, F. As an
example, assume that a singleaxle load unit
is displaced from its position of equilibrium
(i.e. when P = Pst), and that the initial val
value of the frictional force is equal to zero.
As the displacement is increased, the fric
tional force first increases at the same rate
as the interacting force. Accordingly, the
initial paths of the Pu and Fu diagrams are
parallel. The suspension spring remains in
active and the stiffness of the system,
represented by the slope of line oa, is equal
to the stiffness of the tires, k . As the
displacement is increased further, the
frictional force will eventually attain its
limiting value F'. From that point on the
frictional force will remain constant and the
suspension spring will come into play. Ac
cordingly, the slope of the Pu curve becomes
equal to the stiffness, kts, of the suspension
and tire springs acting in series. If at
some instant, say the instant represented by
point b on the diagrams, the displacement is
decreased, the tire spring will rebound and
the suspension spring will remain idle. The
frictional force will then decrease at the
same rate as the interacting force, and the
unloading paths on the Pu and Fu diagrams
will be parallel to the initial paths. If the
displacement is decreased further, at an
instant represented by points c on the dia
grams the frictional force will become equal
On a loaddisplacement diagram, the mean
load is represented by a curve midway be
tween the loading and unloading curves.
to F1. Then both springs will act in series.
A possible path beyond this instant is re
presented by the lines cddeeffg.
It is clear that the values of P and F
depend not only on the value of u, but also
on the past history of u. To determine
whether the effective stiffness of the sus
pensiontire system is equal to k or k , it
is only necessary to know whether the locus
of Fu follows a horizontal or an inclined
line.
7. CHARACTERISTIC COEFFICIENTS OF BRIDGE
MODEL
The reactiondeflection coefficients,
Rj in Equation I are constants for a given
r
bridge model. These coefficients may be
evaluated in a number of different ways. The
method used in this study is based on the
modified moment distribution procedure intro
duced by T. Y. Lin(30).
The essential feature of Lin's proce
dure is that an unbalanced moment at a joint
is balanced and carried over to the other
joints just once to obtain the final moments.
The procedure makes use of the concept of the
effective stiffness and effective carryover
factors which are defined as follows: Con
sider a bar ab resting on nondeflecting
supports and elastically restrained against
rotation at end b by a coil spring having a
stiffness R. The moment at end a necessary
to produce a unit rotation at that end is
defined as the effective stiffness of that end
of the bar. Denoted by K', this stiffness is
a'
given by the equation,
K' = 1  ka,b kba b K (7)
SKb +R J a
where Ka and Kb are the Hardy Cross stiff
nesses of the bar for the ends a and b
respectively. Similarly ka,b and kb,a are
*a,b b,a
the Hardy Cross carryover factors from ends
a and b and from b to a, respectively. The
ratio of the moment produced at end b to the
applied moment at a is defined as the effec
tive carryover factor, k'a,b, and is given
by the equation;
k' = R k (8)
ab (1  ka,b kb,a )Kb + R ab
For a prismatic bar, Ka = Kb = K,
ka,b = k, =  1/2, and the above equations
become
K' = [1 4K  K
4 K + R
kiR
a,b K+ 2 R
2
For a continuous beam the coil spring symbol
izes the continuity of a particular span with
the adjacent spans.
In the course of calculating the
coefficients Rr by this procedure, one calcu
r
lates also the moments at the nodes due to a
unit displacement at the j node. These
moments are termed as momentdeflection
coefficients and are designated by J . In
r.
evaluating the coefficients Rj and J , the
r r
following quantities are used. In all cases,
it is assumed that the bridge model is sup
ported against deflection at the node points.
(a) Effective Stiffness Coefficients
Consider the portion of the bridge
model between the left hand abutment and the
th
r node as a beam continuous over nondeflec
tive supports at the nodes. Then the effec
tive stiffness of the beam at end r may be
stated as the product of a dimensionless
stiffness coefficient C and the quantity
4EI/h, where h refers to the length of a
panel in the center span of the bridge model.
By application of Equation 9 it can be shown
that the coefficient C is given by the
r
following recurrence formula:
h
C = hh [ r 01)
r r 4 h + (C)
h r1
r
th
where h is the length of the r panel. For
a panel on the center span, h = h; and for a
r
panel on a side span, h =  ah.
r n
It should be noted that, because of
symmetry. the dimensionless coefficient for
the stiffness at node r for the portion of
the beam between the r node and the right
hand support is equal to C2n+mr.
(b) Effective Distribution Factors
The effective distribution factor for
th
the right hand side of the r node, designa
ted as d , is given by the expression,
r
C2n+mr
r C +C (12)
r 2n+mr
The distribution factor for the left hand side
of the r node is 1  d .
r
(c) Effective CarryOver Factors
The effective carryover factor from
node r to node rl is designated as k'
r,rI
By application of Equation 10, one finds that
k' = Cr (13a)
r,rl 3 h + 2C
2 h r1
r
Since the beam is symmetrical about the center
line, it follows that
k' = k.
r,r+l 2n+mr,2n+mr1
(13b)
For the sake of brevity, in the following dis
cussion the quantity k' is designated as
r,r l
k'.
r
To determine the momentdeflection
coefficients J and the reactiondeflection
r. th
coefficients R , the j node of the model is
r
first displaced by a unit amount, and by keep
ing all nodes fixed against rotation the fixed
fixedend moments produced at the nodes (j 1),
j and (j + 1) are evaluated. The resulting
unbalanced moments (if h = hhj +1 there is no
unbalanced moment at the j node) are then
distributed and carried over by use of the
quantities given in Equations 12 and 13. The
final moments at the nodes yield the coeffi
cients J . The reactiondeflection coeffi
r.
cients R are next evaluated from the
r
equation
j  JJ jJ _ JJ
Rj r1 r r r+1
r h hr+l
r r+l
The quantities C and d are used only
r r.
to evaluate the coefficients R and J ,
r r
whereas the carryover factors k' and the
quantities Rj and J are used repeatedly in
r r
later stages of the solution.
Another quantity needed in subsequent
computation is the total angle change produced
th
at the r node when the beam is cut at the
th
r node and a unit bending moment is applied
on the two sides of that node. As before, all
nodes are assumed to be held against deflec
tion. This angle change is denoted by 0r and
is given by the expression,
r = C + C, 4E I
r 2n+mr
By use of Equation 13, the above expression
may be written as
h r 2 1 k1)
°r =El h 3 3 r(15)
h 2 1 1 ()
+rl  +  k r]
3r 3 2n+m
h
It should be emphasized that the
quantities defined in this article depend only
on the characteristics of the bridge model.
8. BASIC OPERATIONS
Certain operations are used repeatedly
in the numerical solution of the equations of
motion and in the computation of deflections,
bending moments and reactions. A systematic
treatment of these operations is desirable.
The operations involved are as follows:
Operation 1: Evaluate the moment and
deflection at any point of a simply supported
beam due to moments applied at the ends of the
beam.
Operation 2: Evaluate the moment and
deflection produced at any point of a simply
supported beam due to a concentrated load on
the beam. The governing expressions for
Operations I and 2 are quite simple.
Operation 3: Evaluate the moment at
the r node produced by the i axle load
P., when all nodes are held against deflection.
This moment is equal to the product of P. and
the momentload coefficient M . By Maxwell's
r
law of reciprocity, the latter quantity is
numerically equal to the deflection at the
i axle produced by a unit moment applied at
the rth node (with the continuity there cut)
divided by the coefficient 0 . The latter
r
coefficient is given by Equation 15. To
evaluate this deflection at the ith axle, the
moments at the ends of the panel supporting
the i axle are first calculated. These are
determined by multiplying successively the
effective carryover factors for the panels
between the r node and the nodes where the
moments are computed. The deflection at the
i axle is then computed by application of
Operation 1.
Operation 4: Calculate the reaction at
th th
the r node produced by the i axle load P.,
when all nodes are held against deflection.
This reaction is equal to the product of P.
and the reactionload coefficient Q . The
r
latter coefficient is also equal to the
deflection at the point of application of P.
th
due to a unit displacement at the r node.
To evaluate this deflection, first the moment
deflection coefficients, J, for the nodes on
either side of the panel supporting the ith
axle are selected, and then the deflection
produced by these moments are determined by
application of Operation 1. If the axle is on
a panel connected to the r node, this
deflection represents only one component of
the desired deflection. The additional compo
nent is obtained by considering the deflection
corresponding to a rigid body rotation for
that panel.
Operation 5: Evaluate the moment at
the r node due to known deflections of all
node points. This moment is equal to Z y. J .
J r
Operation 6: Compute the reaction at
th
the r node due to known deflections of all
node points. This is equal to Z y. R .
J r
The last two operations may involve
small differences of large numbers; therefore,
the individual products must be evaluated to
a large number of significant figures.
9. NUMERICAL INTEGRATION PROCEDURE
9.1. GENERAL
The equations of motion of the system
have been solved numerically by means of a
stepbystep method of integration. The time
required for the vehicle to cross the bridge
has been divided into a number of short
intervals and the equations of motion have
been "satisfied" only at these discrete
instants. Let it be assumed that the values
of the acceleration, velocity and displacement
of each coordinate of the system are know at
a time t s, and it is desired to find the
corresponding values at time ts+1, which
differs from t by a short interval At. The
s
method used to accomplish this consists of the
following basic steps. First, an assumption
is made regarding the manner in which the
acceleration of each coordinate varies within
the time interval. Second, the velocity and
displacement for each coordinate are deter
mined in terms of known accelerations, veloc
ities and displacements for the beginning of
where p is a dimensionless parameter speci
fying the variation of the acceleration within
the time interval; the quantity xk represents
the displacement of a coordinate (either y or
z.); and xk and xk represent, respectively,
the corresponding velocity and acceleration.
The subscripts s and s+l following a comma
identify quantities corresponding to t and
s
t s+1, respectively. For the numerical results
presented in this report P was taken equal to
1/6; this value corresponds to a linear varia
tion of acceleration.
The following iterative procedure was
used to evaluate the accelerations, velocities
and displacements of the coordinates at the
end of a time interval.
(1) Define the position of each axle on
the bridge for time ts+I.
(2) Assume that the accelerations y r,s+l
and is+l for the end of the time interval
are the same as those at the beginning of the
interval, and from Equations 16 and 17 evalu
ate the velocities r,s+l and zi,s+l and the
displacements y r,s+l and z. is+.
(3) Evaluate improved accelerations for
the yr coordinates proceeding as follows:
the interval and in terms of unknown accel
erations for the end of the interval. Next,
these unknown accelerations are evaluated by
"satisfying" the equation of motion at the end
of the time interval. The velocities and dis
placements for this time are finally deter
mined from the expressions established in the
second step.
In the present study, the following
generalized equations due to N. M. Newmark(31)
have been used.
ks + k  (At) k + R
xk,s+l k,s 2 k,s k,s+l
Xk,s+l = xk,s
+ (At)k,s + (~ P) (At)2 k,s
+ O(At)2 k,s+l
(a) By application of Equation I to
the first node (r=l), obtain an improved value
for yl,s+ . The major operation in this step
concerns the computation of the quantities
SR y s+l and Q P7 . The former
J Jj+1
quantity is obtained by Operation 6 and the
latter b/ repeating Operation 4 as many times
as there are axles. The values of P. used in
this computation are those applicable to the
beginning of the time interval (i.e. P is),
and the values of yj,s+l are those evaluated
in step 2.
(b) By application of Equations 16 and
17 calculate the values of y,s+! and yl,s+
corresponding to the accelerations determined
in step 3(a).
(c) Repeat steps 3(a) and 3(b) for the
remaining y coordinates (r = 2, 3, ...),
considering one coordinate at a time. For
each computation, use the latest available
values of yjs+l and yj,s+l.
(4) Evaluate improved accelerations for
the z. coordinates as follows:
(a) Compute the interacting force
P ls+ at the end of the time interval. The
various steps involved in this computation
are described in detail in the following
subarticle.
(b) From Equation 4 evaluate 2,s+l'
using the latest available value of P.i,s+l
For the first axle, the value of Pl,s+l used
is that evaluated in step 4(a), and the values
of P 2,s+1 and P3,s+1 are those obtained from
the preceding cycle.
(c) From the acceleration obtained in
step 4(b), determined improved values of
z l,s+ and zl,s+1 by use of Equations 16 and
17.
(d) Repeat steps 4(a), 4(b) and 4(c)
for the remaining axles (if any), considering
one axle at a time, always using the latest
available values of P i,s+ and zi,+1.
,S+1 i ,s+1l
(5) For each coordinate, compare the
newly derived value of acceleration with the
previously available value. If the difference
between the two values for any coordinate
exceeds a prescribed tolerance, repeat steps 3
through 5, always using the latest available
values of P i,s+ and y.j,s+. When all differ
ences are less than the prescribed tolerance,
the integration for this time interval is
considered to be completed. One then proceeds
to the next interval. If desired, the values
of reactions, bending moment and deflection
at any selected point may be evaluated before
proceeding to the next interval. Steps 3 and
4 constitute one cycle of iteration. To
illustrate the details of the procedure, a
numerical example is presented in Appendix B
for one step of integration.
9.2 EVALUATION OF P.
In the computation of P. it is assumed
that the effective stiffness of the suspension
tire system remains constant within a time
interval of integration. In other words, the
suspension spring is assumed to engage or
disengage at the end of a time interval. Under
this assumption, Equation 5 may be written in
the form:
P s+l = P is + (Aui)ki
or
PIs+ = Ps + (uis+  ui,s)ki (18)
where the subscripts s and s+l denote, as be
fore, quantitites corresponding to time t and
t s+, respectively. The quantities ui,s+1 and
k. are determined as follows:
(a) Computation of u.,s+1
The value of u i,s+ is determined by
application of Equation 6. The value of dPi
in this equation is specified, and the value
of the deflection z. is furnished by step 2 or
4(c) of the iterative procedure described in
Article 9.1. The deflection under the load,
ypi, is evaluated by superimposing the follow
ing three components: (i) deflection due to
the moments acting at the two ends of the
panel; (ii) deflection due to the force or
forces P. acting on the panel; and (iii)
deflection due to a rigid body displacement of
the panel.
The moments at the ends of the panel
are obtained with the aids of Operations 5 and
3. Then component (i) of the deflection is
obtained by Operation 1. The component (ii)
is obtained by application of Operation 2 for
each axle on the panel. The rigid body dis
placement is determined from the known deflec
tions of the end points of the panel and the
component (iii) is evaluated by simple
proportion.
Strictly speaking the deflection yPi
must be evaluated for each cycle of iteration
in the integration process, since the values
of y and P involved in the computation vary
from one cycle to the next. Inasmuch as this
computation is rather time consuming, an
approximation was used. This consists in
evaluating the first two components of yPi
only for the first iterative cycle of an
integration step. The third component was
evaluated for each cycle of iteration. The
results obtained by this approximation were
found to be in good agreement with the "exact"
values. A comparison is provided in Table 1
for a case for which the difference between
the two sets of solutions is likely to be
large. The response of the system at a few
selected sections was compared for a few
selected instants.
(b) Determination of k.
The effective stiffness of the suspen
siontire system for an axle is determined by
making use of the Fu diagram for that axle,
as shown in Figure 6. Let the frictional
force corresponding to u. be denoted by
Fis. In the Fu diagram, imagine a straight
line which passes through the point (ui,s,
Fis) and is parallel to the initial line oa.
Let u. be the abscissa of the point of
intersection of this inclined line and a
horizontal line corresponding to the positive
value of F'. Similarly, let u. represent
the point of intersection of this inclined
line with a horizontal line corresponding to
the .negative value of F'. Then
k. to be used in Equation 18 is
from the following criteria:
the value of
determined
Case Condition Consequence
I u. + Au. < u. k. = k t
,s  is t, i
Au. > o
2 u. + Zu. > uu k. = k .
IS I is I ts ,i
3 u. +u. >u.>u. k. = k
I,s I i,s I t,
Au. < o
4 u. + Au. < u. k. = k
I,s i  I,S i ts,i
It follows that the selection of k. depends
u 2
only on the value of Au, u and u . The value
of F need not be computed. For cases 1 and 3
the values of uu and u at time ts+1 are the
same as those at time ts, whereas for cases 2
and 4 they differ by the amount Zu.
9.3 INITIAL CONDITIONS
In order to start the integration
procedure, it is necessary to specify the
initial values of the deflection and velocity
of each node point, the velocity of each z
coordinate, the interacting force for each
axle, and the frictional force for the suspen
sion system of each axle. These values refer
to the time the front axle enters the bridge.
9.4. CHOICE OF TIME INTERVAL
In the application of the numerical
procedure described in Article 9.1, the time
interval used should be small enough so that
successive cycles of iteration converge and
the solution is stable. The criteria for
convergence and stability of this procedure
have been established by N. M. Newmark(31)
For p = 1/6, convergence and stability are
issued if
At < 0.389T
where T is the shortest natural period of
vibration of the system; in this case, the
system is the beamvehicle combination.
Strictly speaking, this period depends on the
position of the vehicle on the span and also
on whether the limiting frictional force of
the suspension system of the vehicle has been
overcome or not.
The total time between the instant the
front axle enters the bridge and the instant
the last axle leaves the bridge is (1 + 2a +
sI + s2) L/V. Let N be the number of steps
used for a complete solution, then
N > (1 + 2a)L + (s + s2)L
0.389 VT 0.389 VT
The right side of this inequality represents
the minimum number of steps required for a
complete solution, on the assumption that At
is constant and that the criteria for conver
gence and stability are independent of the
position of the vehicle on the bridge and the
condition of the vehicle. For the bridge
model considered with a=0.8, n=3 and m=4, the
shortest natural period T = 0.208Tb, where Tb
is the fundamental period of vibration of the
bridge model. In this case, Equation 19
reduces to
sI + s2
N > 16.1 + s1 2
a 0.1615 a
where
2L
a 2L (21)
For a singleaxle loading, the minimum
value of N given by Equation 20 is 215 when
a = 0.075, and 135 when a = 0.12. For a
multiple axle vehicle, the corresponding
minimum values of N are of course larger.
In Table 2 solutions are presented for
the maximum dynamic effects in a threespan
continuous bridge model considering different
values of N. The characteristics of the
system are defined in the table heading.
Solutions are given for a value of a = 0.075,
with three different values of N, and for a
value of a = 0.15 with two values of N. It can
be seen that differences between corresponding
solutions are generally small. For the
numerical results presented in the remaining
part of this report the value of a ranges
between 0.12 and 0.18. For these solutions a
constant value N = 600 was used.
10. COMPUTATION OF DEFLECTIONS, MOMENTS AND
REACTIONS
10.1. STATIC EFFECTS
The static effects are determined by
application of the basic operations described
in Article 8. It is only necessary to consider
n = m = 1 and P = P . In particular, the
deflection and moment at a prescribed point of
a span are determined in two steps. First, by
considering the span to be simply supported
the effects of the force or forces Pst acting
on that span are determined. To these effects
are added the effects produced by the moments
at the ends of the span considered. The
reaction at a support is obtained by applica
tion of Operation 4 for each axle on the b
bridge.
10.2. DYNAMIC EFFECTS
At the end of an integration step, the
deflections of the node points and the inter
acting forces are known. From this informa
tion the deflections of other points and the
magnitude of moments and reactions can be
evaluated as follows: The deflection of a
point within a panel is determined by the
addition of three deflection components in a
manner similar to that described in Article 9.2
in connection with the computation of Ypi.
Moments are evaluated in a similar way, with
the exception that only the effects of the end
moments and of the interacting forces need be
considered. The reaction at a support is
determined in two steps. First, the effect of
the interacting forces is calculated by con
sidering the beam to be held against deflec
tions at all node points. To this is added
the effect of the known deflections of the
nodes. The first component is determined by
Operation 4, and the second component by
Operation 6.
11. SUMMARY OF PARAMETERS
The parameters of the problem are
expressed in dimensionless form and include
the following:
Bridge Parameters
(1) The span ratio, a. This is the
ratio of the side span to the center span.
(2) The damping factor, c/c cr, where c
is the damping force per unit mass per unit
velocity, and ccr is the critical damping
coefficient corresponding to the fundamental
mode of vibration of the bridge.
Vehicle Parameters
(3) The distance parameters, a,, a3
and a5. As shown in Figure 3a, these param
eters define the locations of the centers of
gravity of the tractor and trailer and the
location of the "fifth wheel pivot".
(4) The weight distribution parameters
Wl/W W /W, w /W, w2/W and w3/W. (See
Figure 3a).
(5) The dynamic indices i and i2 for
the tractor and trailer, as defined in
Section 4.2.
(6) The coefficient of friction, pi,
for the suspension system of each axle. For
the ith axle; pi = F!/Psti.
BridgeVehicle Parameters
(7) The speed parameter a, defined
by Equation 21.
(8) The weight ratio W/Wb, where W is
the total weight of the vehicle and Wb is the
weight of the center span of the bridge.
(9) The frequency ratios ft/fb and
f ts/fb for each axle. The quantity fb is the
fundamental natural frequency of the bridge,
and ft and fts are pseudofrequencies which
are measures of the stiffnesses of the tire
and of the suspension springs for an axle.
The quantity ft represents the frequency of a
mass with a weight Pst vibrating on the tire
spring, whereas fts represents the correspond
ing frequency of the same mass vibrating on
the tire and suspension springs acting in
series. For the i axle,
f / 9g
f =  titi
ts,i 2 st,i/kts,i
(22a)
(22b)
When the limiting frictional force F! is so
large that the effective stiffness of the
suspensiontire system for the ith axle is
always equal to kti, or when F! is so small
that the effective stiffness may be considered
to be always equal to kts,i, then it is
necessary to specify a single frequency. This
frequency is denoted by f ..
(10) The profile variation parameter,
dpik ti/P st,i. The numerator of this expres
sion represents the change in the interacting
th
force for the i axle when the tire spring is
shortened by an amount equal to dPi.
(11) The axle spacing parameters, sl
and s2, defined by the equations
sI = 1I/L, and s2 = /L
in which t1 and 12 are the axle spacings, as
shown in Figure 3a.
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