 

THESIS

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Michigan State
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This is to certify that the
thesis entitled

Dynamic Effect of Traffic Loading on Highway Bridges

presented by

Muhammad Uzair Khan

has been accepted towards fulfillment
of the requirements for

Master's degree in Civil Engineering

 

 

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Major professor

Date 1/06/97

 

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MSUI MAMmotlvo 0 Inﬂation
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DYNAMIC EFFECT OF TRAFFIC LOADING
ON
HIGHWAY BRIDGES

By

Muhammad Uzair Khan

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Civil and Environmental Engineering

1997

ABSTRACT

DYNAMIC EFFECT OF TRAFFIC LOADING
ON
HIGHWAY BRIDGES

By

Muhammad Uzair Khan

Bridges like other structures are designed to sustain a variety of loads and load
combinations including those due to their own weight, traffic, snow, wind, earthquake
and more. It is the purpose of the present study, however, to focus on highway bridge
traffic loads in a very fundamental manner. When a highway bridge is subjected to a
moving load, the deﬂection and the stresses caused may be signiﬁcantly higher than when
the bridge is subjected to stationary loads. The mass and damping properties of crossing
vehicle(s) in addition, become intrinsic to the vibrating system changing the fundamental
frequencies from that determined for the structure alone. In the design of bridges, the
effect of moving loads has been and continues to be represented by a ‘dynamic load
factor’(DLF), historically termed as “impact” factor. A review of theoretical and
analytical methods used to determine the DLF is herein presented and the results
compared with values obtained from current bridge design codes of the USA, Ontario
(Canada) and Pakistan. Correlation of the theoretical approach with code speciﬁed values

is also discussed.

Dedicated
to
my parents who continue praying for my success
and
late Brigadier Muhammad Yousaf

I have no words to express the respect and love I have for him in my heart.

iii

ACKNOWLEDGMENTS

I avail this opportunity to thank the Military Engineer Services (M.E.S) and
Government of Pakistan for sponsoring my studies at Michigan State University, USA
under Split MS program of National University of Sciences and Technology (NUST),
Pakistan.

I would like to express my deep thanks and sincere appreciation to all persons
who have assisted me in preparation and completion of this report. I am especially
thankful to my thesis advisor Dr. William E. Saul. His advice, and professional guidance
have been invaluable for the completion of this study. Thanks are also due to my Thesis
Committee members, Dr. Ronald Harichandran and Dr. Frank Hatﬁeld, for pointing out
mistakes in the draft report, and making invaluable suggestions for overall improvement
of the ﬁnal report. I would also like to extend my deepest thanks to my instructors at
Military College of Engineering (MCE), Pakistan. who guided me in the best possible
way, and enabled me for consideration and selection for Split MS program at Michigan
State University.

My deepest love and thanks to my parents, who brought me up in the best
possible manner, for their unconditional and untiring support from the day I was born and
for their endless prayers for my success in every ﬁeld.

Above all, I am especially thankful to Almighty Allah who has blessed me with
honor and success in all walks of life.

Muhammad Uzair Khan

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

CHAPTER 1

INTRODUCTION

1.1 General

1.2 Historical Review of Bridge Loads

1.3 Dynamic response of Highway Bridges
1.4 Importance of Dynamic Analysis

1.5 Measurement of Dynamic Load Effect

1.6 Objective

CHAPTER 2

CONCEPT OF DYNAMIC LOAD FACTOR

2.1 Deﬁnition

2.2 Code Specifications on Dynamic Load Allowance

2.2.1 AASHTO LRFD Bridge Design Speciﬁcations - 1994 (USA)

ix

ll

11

13

13

2.2.2 AASHTO Bridge Design Speciﬁcations 1992 (USA)
2.2.3 Ontario Highway Bridge Design Code (OHBDC) 1991 (Canada)

2.2.4 Code of Practice Highway Bridges (CPHB) 1967 (Pakistan)

2.3 Derivation of Dynamic Load Factor

2.3.1 Empirical: Using “Rule of Thumb” Based on Experience

2.3.2 Theoretical Analysis - Computer Simulation

2.3.3 Model Tests

2.3.4 Field Tests

2.4 Modeling of the Vehicle and the Bridge

2.4.1 Vehicle Models

2.4.2 Bridge Models

CHAPTER 3

LITERATURE REVIEW

3.1 General

3.2 Analytical and Model Studies

3.3 Field or Test Studies

3.4 Factors Affecting Dynamic response

3.4.1

3.4.2

3.4.3

3.4.4

3.4.5

Bridge Span

Weight of the Vehicle
Speed of the Vehicle
Number of Loaded Lanes

Lateral Position of the Vehicles on the bridge

vi

15

16

16

19

20

21

21

23

25

25

27

54

77

78

78

79

79

79

3.4.6 Approach Surface Roughness and Initial Conditions
3.4.7 Number of Vehicles on the Bridge

3.4.8 Damping

3.4.9 Springing of the Vehicle

3.4.10 Type of Span

3.4.11 Axle Spacing

3.4.12 Speed Parameter

3.4.13 Frequency Ratio

3.4.14 Mass Ratio

CHAPTER 4

BEAM DEFLECTIONS DUE TO DYNAMIC LOADS

4.1 Differential Equation of Motion

4.2 Orthogonality Conditions

4.3 Forced Response

4.4 General Procedure

4.5 Effect of Damping

4.6 Example-1: Deﬂection Due to a Constant Force

4.7 Example-2: Deﬂection Due to a Suddenly Applied Stationary Load
4.8 Example-3: Deﬂection Due to a Time-variant Stationary Load
4.9 Deflection Due to a Constant Moving Force

4.10 Vibration effects on Highway Bridges

vii

79

80

81

81

82

82

83

84

84

86

86

89

96

97

98

99

101

104

107

114

CHAPTER 5

SUMMARY AND DISCUSSION
5.1 Development of Dynamic Load Factor
5.2 Vehicle and Bridge Models
5.3 Observed Values of Dynamic Load Factor (DLF)
5.4 Correlation of Field Investigations and Code Speciﬁcations
5.4.1 AASHTO 1992
5.4.2 AASHTO LRFD 1994
5.4.3 OHBDC 1991
5.4.4 CPHB 1967
5.5 Review of a Bridge Design Example
CHAPTER 6
CONCLUSIONS
6.1 Findings
6.2 Recommendations
APPENDIX
LIST OF REFERENCES

viii

119

119

120

121

122

124

126

127

129

131

144

144

147

149

160

Table 2.1

Table 5.1

Table 5.2

Table 5.3

LIST OF TABLES

Dynamic Load Allowance, IM
Comparison of Observed DLF Values
Comparison of Dynamic Effect Using Design Example

Composite Bridge Design Example

ix

14

121

138

142

Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 4.1
Figure 5.1
Figure 5.2

Figure 5.3

LIST OF FIGURES

Load without Mass 17
Oscillating Load without Mass 17
Load with Mass 17
Sprung Load with Mass 18
Load as a Moving Mass 116
Typical Cross-Section of Composite Highway Bridge 133
Time Vs Dynamic Load Factor (DLF) (Mode one only) 143

Time Vs Dynamic Load Factor (DLF) (Different Modes

1.1 General

Bridges like other structures are designed to sustain a variety of loads and load
combinations including those due to their own weight, trafﬁc, snow, wind, earthquake
and more. It is the purpose of the present study, however, to focus on highway bridge

trafﬁc loads and to do that in a very fundamental manner.

12 II'I'IB' [B'lll

1. Prior to the Industrial Revolution, say 1800, bridges were built of stone and
timber. Traffic loads were relatively small in comparison with the mass of the bridge and
moved slowly. The designer’s primary concern was to design the bridge to support its
own weight.
2. Over the 19th century and until after World War I several events took place
successively to dramatically change bridge design:
a. New Materials: The discovery of new construction materials such as cast iron,
wrought iron, reinforced concrete, and steel lead to lighter bridges.
b. Railroad: Introduced signiﬁcantly heavier and faster traffic loads.
c. Bridge Failures: Large number of bridge failures took place, indicating a lack
of sufﬁcient bridge design standards and codes.
d. Analysis methods: Analysis for static loads developed rapidly. Analysis for

dynamic loads lagged very badly. Designers attempted to overcome their lack

2

of knowledge of dynamic effects with empirical methods; they assumed
loading due to dynamic effects could be handled as if they were static and as a
function of the span length only.
e. Bridges were mostly property of the railroad companies.
3. 20th century - post World War I:
a. Development of prestressing (lighter weight concrete bridges).
b. Development of highways, auto and truck loads.
c. In the latter part of period - development of dynamic loading analytical methods
- adoption is slow.
(1. Development and adoption of standards and codes.
e. Highways and bridges are publicly owned (state-owned), control of loads
becomes a policy matter.
f. Design methods, especially codes and standards, continue to be strongly
affected by the 19th century empiricism.
4. Trends for the let century:
a. Higher speed traﬁic.
b. Higher loads - possibly.
c. Use of lighter materials of construction.
d. Computer design tools that fully utilize dynamic loading and corresponding

response of bridge.

3

1-3 MmicResmnmeighmxBridges

The primary function of a bridge is to bear traffic, and since traffic moves, it is a
time-variant or dynamic form of loading. The displacement of a bridge subjected to
dynamic loading is quite different from its behavior subjected to static loading even due
to the same force, unless the duration over which the time-variant force is applied is quite
long in comparison with the fundamental period of the structure. The dynamic effects of
trafﬁc loads can be signiﬁcant. Their one special feature is that they are variant both in
time and space [Fryba 1972]. The behavior of a bridge structure under the action of a
moving load is referred to as its ‘dynamic response’ and speciﬁcally signiﬁes its
displacement or another stress resultant as it varies with time. These are important
considerations in the design of bridges because of the following:

a. The dynamic interaction between the vehicle and the bridge can result in
signiﬁcant deﬂections and stresses in the bridge structure. Accordingly, the bridge
has to be designed for these increased or ampliﬁed values. Resonance or near
resonance conditions, that is when the frequency of the loading function
approaches a fundamental frequency of the structure, can result in signiﬁcant
increases in these values.

b. Certain combinations of frequency of vibration and amplitude may cause
discomfort and adverse psychological effects to pedestrians as well as other road
users. [Kashif 1992]

c. The trend for faster and heavier vehicles and the use of ultimate-strength design

criteria by most of the bridge design codes permitting possibility of lighter weight

4

more slender structures, which are more susceptible to the dynamic effects of

moving loads. [Saadeghvaziri 1993; Humar and Kashif 1993]

The dynamic response of bridges to moving loads depends upon the interaction
of the vehicle and the bridge which is a complex dynamic phenomenon [Humar and
Kashif 1993] because of the large number of different parameters which inﬂuence the
dynamic response. The major parameters affecting the dynamic response of the bridges to
vehicular loads are [Wright and Green 1959; Nowak and Hong 1991; Humar and Kashif
1993, 1995; Huang et al. 1993. 1995; Saadeghvaziri 1993; Chang and Lee 1994; Yang et
a1. 1995]:

1. Bridge structure characteristics:

a) the bridge dimensions
b) the support conditions
c) mass distribution

d) stiffness distribution
e) structural damping

9

In the analysis, parameters ‘a’ to ‘d are generally represented in terms of the
mode shapes and the frequencies of the bridge structure.
2. Vehicle characteristics:
a) the number of axles,
b) the axle spacing and its relation to the span length,

0) the mass and springing of the vehicle and their distribution to axles

d) the natural frequency of the vehicle

1.4

for:

5

e) the degrees of freedom assigned in the model, and
e) the damping (provided by the shock absorbers as well as interleaf
friction in the springs).
3. Speed (velocity) of the vehicle,
4. Ratio of the weight(s) of the vehicle(s) to that of the bridge
5. Surface roughness of the bridge as well as the approach
6. The path followed by the vehicle, while passing over the bridge
7. Frequencies of the application of the vehicle load due to multiple axles,
8. The number of vehicles on the bridge and their relative positions

9. Forces generated by the braking (deceleration) and acceleration of the vehicle,

10. The motion induced in the bridge before application of the load under

consideration, particularly in the case of continuous bridges.
11. The speed parameter i.e., the ratio of driving frequency of the vehicle to the

fundamental frequency of the bridge.

I I [D '!l'

The dynamic analysis of highway bridges subjected to vehicular traﬁic is essential

1. Accurate understanding of bridge behavior subjected to trafﬁc loads
2. Providing criteria for the accurate design of bridges

3. Research to develop more accurate and simpliﬁed techniques for analysis and

design

6

4. Preparing for future developments i.e.,

a. higher speeds

b. possible heavier loads

c. lighter bridges

d. higher strength conventional materials
1) concrete; reinforced, plain or prestressed
2) steel
3) timber
4) combination of the above

6. new materials
1) metals (e.g., aluminum)
2) composites (ﬁberglass, ﬁber-reinforced concrete, ﬁber-

reinforced-plastics)

In order to have a better and more accurate understanding of bridge behavior
subjected to traffic loads and to provide criteria for the accurate design of bridges
subjected to moving loads, research has been underway from the beginning of the railway
era (cc 1820) searching for reasonable techniques to predict the dynamic response of
bridges under the action of moving loads. Accordingly, numerous (analytical, model test
and ﬁeld) studies have been carried out to develop a relationship between the static and
the dynamic responses of a bridge, and consequently to propose a simple and reliable

procedure to incorporate the dynamic effect of moving loads in the design of bridges.

7

This approach ignores many of the dynamic features listed above such as frequency
relationships.

Historically, a relationship misnamed in literature generally as an ‘Impact Factor’,
and more exactly as the “Dynamic Load Factor” (DLF) has been used to simplify, or
over-simplify, the design. It is one purpose of this study to ascertain the limits of this
procedure and to suggest alternatives for greater accuracy. This factor is numerically
equal to the ratio of the maximum dynamic response due to a vehicular load to the
maximum static response produced by the same load acting statically minus one. In
general this factor is represented as a percentage of the maximum static response.

Attempts were made to relate the “impact factor” with only one of the
inﬂuencing parameters (most probably either due to lack of knowledge during the initial
period of investigation about other inﬂuencing parameters, or for the ease of application
in design offices) most notably, the span length and the natural frequency of the bridge.
These factors were adopted by some highway bridge design code authorities as well, for
example, some codes speciﬁed ‘impact factors’ in relation to the natural frequency of the
bridge [OHBDC 1983], and others with reference to the span length [AASHTO 1992;
CPHB 1967] of the bridge. Later investigations [Boehning 1953; Eichmann 1954; Wright
and Green 1959,1964; Walker and Veletsos 1966; Walker 1968; Tan and Shore 1968a, b;
Nieto-Ramirez and Veletsos 1966; Veletsos and Huang 1970] revealed that the dynamic
response depends upon a number of other factors, given above, as well. Some
researchers investigated the ‘impact factors’ incorporated into codes [AASHTO], and

showed that the code speciﬁed values underestimate the dynamic response in certain

8

cases (for example, for the short span bridges), and overestimate in others [Fleming and
Romualdi 1961; O’Connor and Pritchard 1985; Ibanathan and Wieland 1987; Galdos et
al. 1993; Huang et a1. 1993; Yang et al. 1995]. The relevant code speciﬁcations regarding
the ‘impact factor’ that is, the dynamic response of bridges to trafﬁc loads were changed

later on [OHBDC 1991; AASHTO LRFD 1994].

1.5 MW

The dynamic effect of moving loads on a bridge, as represented by a
dimensionless factor, may be mathematically deﬁned as the increase in the maximum
response (deﬂection, moment or shear) of a bridge structure due to a moving load, over
the maximum response (deﬂection, moment or shear) of the structure caused by the same
load when acting statically. This is usually represented as a percentage of the maximum
static response, may be termed as dynamic increment, DI. Symbolically the dynamic

increment, D1, is given by the percentage

_ Rdm (xi) " Rsm (xi)
_ Rsm (xi)

DI X 100

 

where Rdm (xi): max[Rd (x,, t j)] is the maximum amplitude of the dynamic
response,

and R8'“ (xi ) = max[Rs (xi )] is the maximum static response of the bridge at section
xi under the action of moving vehicles. Bridge design codes generally express the

dynamic effects of moving vehicles in terms of D1. On the other hand, theoretically the

9

dynamic load effects are computed in terms of ‘Dynamic Load Factor’ (DLF), deﬁned as
the the ratio of maximum dynamic deﬂection to the maximum static deﬂection under the

same load.

1.6 mum

The objective of this study is to review the most basic approach for the analysis of
the dynamic response of highway bridges under the action of vehicular trafﬁc and to
compare it with current practice; this includes:

a. A review of theoretical basis to obtain the DLF

b. Current code design speciﬁcations for dynamic effect of vehicular loads, and

c. To look for the correlation between (a) and (b) above.

To reach the objective, basic concepts pertaining to the representation and
measurement of the dynamic effect of moving loads have been described in chapter 2.
These concepts include the description of different methods (empirical, theoretical, model
tests, and ﬁeld tests), as well as the speciﬁcations of the bridge design codes currently in
use in the USA, Ontario (Canada), and Pakistan. This is followed by a review of some
important analytical and ﬁeld test studies carried out on the problem of the bridge
dynamics, in chapter 3. Theoretical basis of measuring dynamic response for some simple
load models has been discussed in chapter 4. A summary of the important results
obtained in chapters 3 and 4 ,and discussion thereof has been made in chapter 5 that also

contains a discussion on correlation of code speciﬁed values with the results of research

10

studies reviewed in chapter 3. Chapter 5 also includes a comparison of the dynamic
increment values for a typical simple span composite bridge, obtained by using the four
different bridge design codes, as well as theoretical load models. The report concludes
with chapter 6, containing the ﬁndings as well as certain recommendations regarding the

dynamic effect of moving loads on highway bridges.

*t***********************

W

2.1 Deﬁnition

Dynamic response is deﬁned as the history of the variation of a stress resultant
such as deﬂection, bending moment or a stress plotted as ﬁmction of time. Applied to a
bridge, it is the response of the bridge when acted upon by a moving load or vehicle. One
effect is to cause vibration of the bridge, a response that is signiﬁcantly different from the
static response since it is multi-valued and changes with time, involves frequency, and
has an interactive relationship between the load and the structure. The practical outcome
of this response is generally to cause an increase in the stress resultant over the static
magnitude of response, that is the response under a stationary load. This increase is
generally different for different bridges, depending upon a number of factors such as the
mass, stiffness, and damping of the bridge; the mass, suspension, damping, and velocity
of the vehicle; as well as the condition of the approach and the bridge surface.

The ratio of the magnitude of a speciﬁc parameter such as deﬂection y(t)

compared to the deﬂection by the same load applied statically ysmic = y“ has various

names such as the Dynamic Load Function. The quantity represented by the ratio of the
maximum magnitude of the response caused by dynamic loading to that due to the
statically applied load minus one also has many names such as Dynamic Load Factor
(DLF), Dynamic Ampliﬁcation Factor (DAF) and others. Through misunderstanding of
the dynamic phenomenon this ratio has been called an “impact factor” in bridge design.

This error is being rectiﬁed but continues to persist. (Hitting a nail with a hammer is

11

12

impact - the true duration of the event is very short-lived). The term ‘impact’ or ‘impact
factor’ , has been and is still being applied incorrectly to represent this dynamic effect.
The reason being that ‘impact’ is a phenomenon related to impulse - an effect involving
action of a force acting over a very small period of time causing change in momentum of
the body on which it is acting. Thus when an object having a mass and moving at some
velocity ‘impacts’ upon another body the change in momentum in the two bodies is
equivalent to an ‘impulse’; a dramatic change in force over a very small time interval.

This fact is now being recognized in the literature and different bridge design
codes as well as by various researchers who are now using more appropriate terms like
dynamic load allowance [AASHTO LRFD 1994, OHBDC 1991], dynamic increment
(D1) [Heywood 1995], dynamic increment factor (DIF) [Galdos et al. 1993] ), dynamic
ampliﬁcation factor [Humar and Kashif 1993, 1995; Kashif 1992] and dynamic load
factor (DLF) [Hwang and Nowak 1991] to represent the dynamic effect of moving loads
quantitatively.

In the present study this effect will be generally referred to as the ‘dynamic load
factor’ (DLF). The dynamic load factor, may be deﬁned as the ratio the maximum
dynamic response to the maximum static response. Symbolically DLF is given by the

relation

where Rdm (x,)=max[Rd (x,,tj)] is the maximum amplitude of the dynamic

response, and Rm (xi) = max[Rs (xi )] is the maximum static response of the bridge at

13

section xi under the action of moving vehicles. Representing the maximum static
response by ys, m, and the maximum dynamic deﬂection response by y (M m“, the

dynamic load factor (DLF) is given by the expression

DLF = Ydyn max

ySI max

2.2 W
2.2.1 W

The American Association of State Highway and Transportation Officials
(AASHTO) Load and Resistance Factor Design (LRFD) Bridge Design Speciﬁcations -
1994 (AASHTO LRFD 1994) deﬁnes the “impact factor” as “Dynamic Load
Allowance”, denoted by IM. The dynamic load allowance (DLA) is the percentage by
which the static effects of the design truck or tandem are to be increased to account for
the dynamic effects of trafﬁc loads. The percentage speciﬁed for the DLA is not
applicable to pedestrian loads or the design lane load (Lane load is a uniformly
distributed force, usually attributable to automobiles). The factor to be applied to the
static load has been speciﬁed as: (l + IM/lOO). The DLA percentages are given in Table l
on the next page.

Relevant AASHTO LRF D code provisions regarding the dynamic load allowance

are available in Appendix.

14
Table 2.1 : Dynamic Load Allowance, 1M

 

Component IM

 

Deck joints - All Limit States 75%

 

 

All Other Components

 

‘ Fatigue and Fracture Limit State 15%

 

* All Other Limit States 33%

 

 

 

 

 

2.2.2 W
In the AASHTO Highway Bridge Design Speciﬁcations - 1992, the dynamic

effect of vehicle loads was accounted for through an “impact allowance” calculated using

the expression:

_ 50
L+125

 

in which

I = impact fraction (maximum 30 percent);

L = length in feet of the portion of the span that is loaded to produce the
maximum stress in the member.

To calculate the dynamic effect, live load was required to be multiplied by the
‘impact’ fraction calculated from the above expression. The dynamic effect thus obtained
when added to the live load gave the total effect of the live load on the structure. Further

details on AASHTO 1992 provisions on the “impact factor” are given in the Appendix.

2.2.3 W

The Ontario Highway Bridge Design Code-1991 (OHBDC 1991) speciﬁes the

dynamic load allowance (DLA) with reference to the number of axles of the vehicle, the

15

maximum value being 0.40 for a single axle load and the minimum 0.25 for vehicles with
three or more axles. For a 2-axle vehicle, the dynamic load allowance speciﬁed is 0.30.
These values are for normal loading condition with a smooth approach road. In case of
unusual approach surface or bump at expansion joint and single axle loads, the DLA is
increased to 0.5 from 0.4 within a length of 3 m or one-tenth of the span, whichever is
greater from the joint.

For further details about the DLA refer to the Appendix.

2.2.4 WWW“)

The existing Code of Practice Highway Bridges (CPHB) 1967 of Pakistan
provides the following formula to account for the dynamic effect of trafﬁc loads in the
design of highway bridges:

_ 15
L+20

 

where

I = impact fraction (maximum 30 percent)

L = length of span in feet

In this case, the dynamic effect of live load is obtained by multiplying the
standard Truck-train loading by the “impact” fraction 1 obtained from the above formula

More details about CPHB 1967 code requirements on the calculation of dynamic
effect of vehicle loads are in the Appendix.

23 D'I' [D .1 1E1 [DIE]
Numerically, the dynamic load factor (DLF) is obtained by subtracting one from

the ratio of the response (deﬂection, moment etc.) of a bridge structure under the action

16

of a moving load or vehicle to that obtained under the same load when applied statically
to the same structure. In the actual design of a bridge, the static live load is multiplied by
this fraction to obtain the dynamic effect of the live load, which is then added to other
loads (dead, static live, snow, earthquake etc.) to check the resistance of various
components of the bridge being designed.

The dynamic load factor, is generally derived in following ways:

2.3.1 We;

The performance of existing bridges provides data which can and is used for
“rule-of-thumb” or empirical rules for design without any detailed calculations or as a
result of intuitive thought perhaps with some logical justiﬁcation. In this manner a
constant value or a formula may be adopted to calculate the dynamic effect of moving
loads for application to particular types of bridges. These relationships may apply in
limited cases or within narrow bounds only, they nevertheless constitute rules for design

such as the DLF. Waddell (1918) reports one example of thistype of DLF.

2.3.2 IhmtieaLAnalxsiL-annuteLSimulatinn

For a theoretical analysis the moving load and the bridge are generally modeled
by making some reasonable assumptions and the bridge response under the moving load
is calculated by employing well recognized mathematical and analytical techniques. For
example, the load, moving at a constant speed across the bridge, may be modeled as:

1. a single force

2. a single unsprung mass

l7

3. a multi-axle unsprung load

4. a single sprung load

5. a combination of sprung and unsprung loads,
6. a multi-axle sprung load

A few load models are shown below:

 

i—«—l" e
l MI I

FIG. 2.1: Load without Mass

 

 

 

 

I‘_Vt__iPsith
l mu l

FIG. 2.2: Oscillating Load without Mass

 

 

 

 

 

FIG. 2.3: Load with Mass

 

 

 

18

 

 

_1T
.3

['11
—s

FIG. 2.4: Sprung Load with Mass

 

 

 

Whereas the bridge may be idealized as:

1. an elastic beam (simple, continuous or cantilever),

2. a plate (isotropic or orthotropic) structure, or

3. a combination of beam and plate structures.

The analytical expression for a force or mass moving with constant velocity over
a simply supported span is an exercise designed to obtain a “complete” or “exact”
solution for this unique problem.

It is necessary to clarify here that “complete” or “exact” are relative terms
meaning that the solution completely or exactly satisﬁes the differential equation.
However, the differential equation is based on certain assumptions such as elastic
behavior , a simply supported span, negligible defamation due to shear and rotary inertia,
and other constraints placed on the behavior of the load. The solution of the continuum
equation that is, the partial differential equation, thus is a specialized format, an idealized
situation. The solution of this idealized problem provides a basis for understanding the

effects of trafﬁc loads. A study of this basic situation and a review of its variation should

19

provide signiﬁcant information regarding the parameters, their interaction, and details of
the dynamic response due to moving loads. However, although this idealized model and
its solutions provide knowledge, in all but a few cases it does not exactly model the
speciﬁcs of most bridges. Real bridges conform to a large array of types, conﬁgurations,
materials , and loads.

There is an array of methods for determining the dynamic properties of structures,
including any shape of bridge. Most involve modeling the structure as a system of
discrete multiple degree of freedom set of masses and structural elements. These methods
allow for computation of natural frequencies, numerical integration for determination of
response, mode superposition, discrete models, etc. Determining the dynamic properties
of a bridge is a well understood process. Knowledge gained from the solutions of the
continuum problem described above can provide help for modeling loads for discrete
analysis and for interpretation of the expected response details.

This is a more reliable method of ﬁnding the DLF, especially if supplemented
with model tests. A number of researchers have used this method. [Inglis 1934; Scheffy
1956; Fleming and Romualdi 1961; Wen and Veletsos 1962; Sundara Raja Iyengar and
Jagadish 1968, 1970; Veletsos and Huang 1970; Ibanathan and Wieland 1987; Chang and

Lee 1994; Humar and Kashif 1993, 1995; Huang et al. 1995; Yang and Lin 1995]

2.3.3 Modellests
A small sample bridge, the model, similar in dynamic characteristics to an actual

one, the prototype, may be subjected to simulated representative models of actual

20
vehicles in the laboratory, to study its dynamic response. The results obtained from such

a laboratory test may also be used to an'ive at suitable values or an empirical relation to
reﬂect the dynamic effect of moving loads in actual bridge design [Boehning 1953;
Eichmann 1954; Tung et al. 1956; Walker 1968; Tiedeman et al. 1993].

This is a more accurate method, especially if used in conjunction with theoretical
analysis, and the results of both are in good agreement with each other. However, it is
expensive as well as time consuming, and since late 1950’s has declined in use in favor of

computerized simulation methods.

2.3.4 W

In some cases, tests are carried out on existing bridge structures, instrumented
suitably to measure dynamic response under real trafﬁc loads [Biggs and Suer 1956;
Edgerton and Beecroft 1956; Hayes and Sbarounis 1956; Foster and Oehler 1956; Fenves
et al. 1962; Shepherd and Aves 1973; Billing 1984; Cantieni 1984; O’Connor and
Pritchard 1985; Ibanathan and Wieland 1987; Chan and O’Connor 1990; Huang et al.
1993; Chang and Lee 1994; Heywood 1995]. Based on statistical results of these ﬁeld

tests, the dynamic load factor may be calculated from the relation

DLF = Ydyn max

yst max

where ydyn m" is the maximum dynamic deﬂection, and

y," m, is the maximum static deﬂection for the same load.

21

Since these tests are can'ied out on existing full-sized bridges using actual
vehicles, the results and deductions made therefrom are generally more reliable and
realistic, especially if these tests are sufficient in number and cover the effects of all the
factors and practical conditions that inﬂuence the dynamic response of a bridge structure
the most adversely. The drawback of this method , in addition to its cost, is that the data
obtained is for that particular bridge only. extrapolation or interpolation of results must
rely heavily on an understanding of dynamic response characteristics obtained
analytically. Further, dynamic analysis of the subject bridge, the one instrumented for
testing , is a necessity if the results of the test are to be fully understood and correctly

interpolated.

2.4 MW
2.4.1 W

In the analytical studies of the dynamic response of bridges under the action of
moving loads or vehicles, the mass of the vehicle has been modeled in the following

ways:

2.4.1.1 AuMnxinaLuaiﬂaxingMncmuMam

This assumption holds good for the cases where the inertia (mass) of the vehicle is
negligible in comparison with that of the bridge. Although less realistic than a sprung
load model, a moving force model [Inglis 1934; Sundara Raja Iyengar and Jagadish

1968]:

22
1. helps in understanding the effect of the speed parameter ( ratio of the load

velocity to one-half of the bridge period of vibration).
2. indicates the relative inﬂuence of various modes of vibration to the dynamic
response of the structure.
Several researchers have carried out studies based on this vehicle model [Inglis
1934; Tan and Shore 1968a, b; Timoshenko et al. 1974; Warburton 1976; Sridharan and

Mallik 1979; Yang and Lin 1995].

2.4.1.2 AuMnxinaMass

In cases where the inertia of the vehicle is comparable with that of the bridge, it
cannot be neglected. In such cases, the load can be modeled as a moving mass. Several
studies have been carried out using this vehicle model as well [Blejwas et al. 1979;

Ibanathan and Wieland 1987; Akin and Moﬁd 1989; Yang and Lin 1995].

2.4.1.3 mm

For more accurate vehicle models, the dynamic characteristics of the various
vehicle components like body (mass), suspension system and tires etc. are considered
separately as lumped masses, to obtain more realistic response of the bridge, as a
combined effect due to interaction of the various components. This sort of vehicle models
have also been used by researchers. [Hwang and Nowak 1991; Wang and Huang 1992 ;

Yang and Lin 1995]

23
2.4.2 BridELModels

For dynamic analyses bridges may be modeled as a continuum requiring the
solution partial differential equations or as a discrete solution with a ﬁnite number of
kinematic degrees of freedom which results in a set of ordinary differential equations. In
the analytical studies of vehicle-bridge interaction, a bridge may further be modeled in

the following ways:

2.4-2.1 W

The simplest way in which a bridge can be modeled is as a beam (simply
supported, continuous or cantilever) of length equal to the span or spans of the bridge. A
number of studies have been carried out using this model.[Biggs et al. 1959; Fleming and

Romualdi 1961; Yang and Lin 1995]

2.4.2-2 W

In some cases, research has been carried out by modeling the bridge as a two-
dimensional or plate structure. In this case, the bridge deﬂection is considered along the
length as well as width of the bridge [Yoshida and Weaver 1971; Sundara Raja Iyengar
and Jagadish 1968, 1970; Humar and Kashif 1995].

2.4.2.3 W

In some studies, research on dynamic behavior of bridges has been carried by
modeling the deck slab as a plate, and the girder as a beam element. [Yamada and

Veletsos 1958; Oran and Veletsos 1961]

2.4.2.4 Studies Using Einithlement and EinittStn'n Methods

Some researchers have carried out studies on the dynamic response of bridges,

using ﬁnite-strip methods [Cheung and Cheung 1972] or ﬁnite element methods

24
[Rabizadeh and Shore 1975; Saadeghvaziri 1993]. These inherently result in a discrete

system for mathematical analyses.

Ill*It************************

3.1 General

Determination of the response of bridges to moving loads i.e., vehicles, is very
important for bridge users and bridge design engineers because dynamic response creates
important considerations. Vibration may cause unpleasant physiological and
psychological effects on bridge users, and stress resultants, e.g., deﬂections and bending
moments in the bridge may increase considerably from that obtained under the action of
static loads. The magnitude of these increases will vary in amount depending upon a
number of characteristics of the bridge as well as the moving load. Thus the
determination of reasonable estimates of structural loads and the response to these loads
becomes increasingly more complex. One possible approach for estimating the maximum
effect due to trafﬁc loads is to increase the static load by a certain percentage or ratio
based on assumptions or statistical data to account for the dynamic effect of the load, and
use this increased load to design the bridge. A similar approach is used in many of the
bridge design codes in the world. This approach may be sufﬁcient for many short span
stiff bridges; however, it yields insufﬁcient information concerning bridges that because
of span length or ﬂexibility are more sensitive to vibration. Dynamic response is a
complex phenomenon, affected by a number of factors which require greater
computational effort. The maximum values of key stress resultants, such as bending
moment at a critical section or mid-span deﬂection, due to dynamic loading is usually

expressed in terms of the ratio of the maximum value due to dynamic loading to the value

25

26

if the structure or structural element were loaded with the same force applied statically.
This ratio in some form may be called an amplitude factor, a dynamic load factor, a
dynamic increment, or in bridge design, incorrectly labeled as an impact factor.
Accordingly, it is necessary to investigate the response of bridges to traveling loads, and
to arrive at suitable methodologies for use in the design and analysis process. Highway
bridges that are very stiff, usually short span - such as a masonry arch, that is, bridges
with a relatively short period, may be designed using static or quasi-static methods.
Bridges with a longer natural period, usually longer span and more ﬂexible are more
sensitive to dynamics and may have to be analyzed for its dynamic behavior either in part
or requiring a full dynamic simulation. The large majority of highway bridges are short to
medium span and built of masonry, concrete, steel, or a concrete steel composite for
which static trafﬁc loads are dominant and design utilizes the quasi-static approach. For
this purpose the static load is increased by the dynamic load factor referred to earlier,
labeled an “impact factor” in some codes (striking a nail with a hammer is an example of
impact - the duration over which the force is applied is quite short! hardly what occurs
for normal traffic loads). The study and investigation of the response of bridges to
moving loads or vehicles can be traced back to 1849. Since then a number of
investigations have been carried out to understand the behavior of bridges subjected to
moving loads, and to explore the factors or parameters which are major contributors to
this phenomenon. An overview of important research on the dynamic response of bridges

subjected to moving loads and relevant to this study, is given below:

27

For the sake of clarity, these studies are discussed in the following two main
categories:
a. Analytical and Model Studies

b. Field or Test Studies

3.2 W

3.2.1 The history of the study of the behavior of bridges under moving loads can be
traced back to 1847 when a British Royal CommissionI was appointed to investigate the
possibility of application of iron for use in construction of railway structures. The
Commission was the ﬁrst to investigate the effect of moving loads on bridges. After
carrying out experiments on the effect of a rolling load, the Commission reached the
conclusion that in the case of a light and ﬂexible simple beam, the increase in deﬂection
was dependent on the velocity of the load. It was also found that in certain cases, the
deﬂection caused by the moving load was two or three times the static deﬂection.
Professor R. Willis, a member of the Commission, tried to formulate the problem through
analytical techniques and consequently developed a differential equation to calculate the
deﬂection of the beam under the point of the load. In developing the deﬂection equation,
Willis assumed that:
a. the mass of the beam is negligible,

b. the beam is undamped, and

 

' The original reference was not available. The information quoted in this section has been mainly taken
from the report of Wright and Green (1959).

28

c. the load is a concentrated mass
According to Willisz, the deﬂection of the bearnyp under the point of application
of load P(t) is proportional to the contact pressure of the moving load expressed as a mass

m u and can be computed from the equation for the static deﬂection of the beam i.e.,

_ P(t)X§ (Ir-x1»)2
P" 3LEI

 

--------- (I)

where xP is the distance from left support, the origin, to the point of application of the

load. For ﬁnding P(t), the inertia force, using the relationship

dt d x (It a
is added to the rolling load of the unsprung mass mu

2 2
mug+mu-d—y-mug+muv2 d—1

dt‘ ’ dx‘

assuming the velocity v, of the load to be constant. Eq(1) changes to

 

v2 dzy] xf. (L — xp )2

3LEI """"'(2)

This equation was solved by G. G. Stokes3 and the solution was in tabular form.
This solution, however, holds good only for the cases where the ratio of the weight of the
moving load to that of the bridge is large. In the case of present day medium span
bridges, this ratio is generally not more than 0.2. Accordingly, Stoke’s solution is not

applicable [Wright and Green 1959].

 

2 See footnote on page 26.
3 See footnote on page 26.

29
3.2.2 In 1905, A. N. Krylov4 attempted a theoretical solution of the problem of dynamic

behavior of a bridge, by assuming that:
a— the mass of the load is negligible,
b— the moving load P(x, t) can be replaced by a moving force rnu g.
Krylov represents the state of vibration of a simple beam under the action of a

moving force with the differential equation

 

4 M 2 °°
ZXZ... L8 Zt¥=P(x,t)=3-t?—gzsm in X Sin ZIWfbt --------- (3)
180

El
Based on Krylov’s solution, for most load cases, the approximate amplitude of the

vibration can be found from the expression:

ZmuL’ﬁ

y =-——
P MgEIfb

sin 21tfb t sin? ......... (4)

where ii is equal to —V—
2 L

M8 is the mass of the beam or girder

fl, is the natural frequency of unloaded beam or bridge

3.2.3 H. H. Jeffcotts (1929) was the ﬁrst researcher who included the mass of both the
moving load and the bridge in his attempt to get an analytical solution to the bridge

vibration problem. He neglected the effect of damping and used an iterative procedure to

 

‘ See footnote on page 26.
5 See footnote on page 26.

30

obtain two solutions of the problem; one after neglecting the inertia term, and the other

including the inertia terms. The method he used does not converge in certain cases.

3.2.4 For the analytical and experimental research carried out in Great Britain by the
Bridge Stress Committee, C. E. Inglis (1934) formulated the expressions for the natural

frequency of loaded and unloaded simply supported spans, respectively, given below:

n2 EIL
2nfbl=ff x n """""" (5)
Mg +2m sin2 PT

1:2 I E I L
7‘ b L2 Mg ( )

where fbl is natural frequency of the loaded beam or bridge.

 

 

Inglis showed that for beams with non-uniform rigidity, the natural frequency can
be obtained quite accurately, using the value of the rigidity at mid-span. Using a
harmonic analysis, he also developed the expressions for the following cases of loading:
a. A massless oscillating force ( P sin 1: Q t, where Q is the frequency of
the force) with damping of the bridge
b. An oscillating force with damping of the bridge:
The magnitude of vibration was obtained using a series solution. The

approximate maximum amplitude is given by the relation

P N2
_ 2 --------- 7
Kb \/(w/21I:) +(Cbl)2 ( )

 

 

where K, is the stiffness of the span,

31
and Cbl is the damping coefﬁcient of the loaded span.

c. Vibrations considering locomotive springing: According to Inglis, to determine
the vibration of medium and long span railway bridges accurately, the

damping factors (of these bridges) must be known.

3.2.5 The research work carried out by C. F. Scheffey (1951)‘5 was most probably the
ﬁrst study especially related to the dynamic behavior of highway bridges (the earlier
studies concentrated on railroad bridges). The main objective of the study was to predict
a rational impact allowance for the design of highway bridges. Scheffey developed the
solution of the differential equation of motion of a simple prismatic beam traversed by a
rolling mass with a constant velocity. His solution is similar to that obtained by Krylov
(1905). Scheffey showed that in the case of a single load rolling smoothly over a simple
span with a velocity of 60 mph, the maximum dynamic increment (i.e., addition to the
static deﬂection due to dynamic effect of rolling load) is nearly 8 percent for bridge spans
in the range of 20 to 100 feet, and nearly 4 percent in the case of a span of 300 feet
[Wright and Green 1959].

Scheffey also showed that in the case of a sprung vehicle, with a spring friction
about 10 percent of the vehicle weight, and entering a span with its springs in an
equilibrium position, no spring action (that is, the oscillation) would take place in bridge
spans longer than 40 feet. His results also lead to the conclusion that the phenomenon of

resonance may take place for simply supported loaded spans of about 300 feet or more.

 

6 See footnote on page 26.

32

He commented that while considering maximum dynamic effects, representative vehicles

should be used rather than the equivalent design loads.

3.2.6 The analytical research carried out by Dr. A. Hillerborg7 (1951) is important in
that it was one of the most complete analyses performed on the general problem of
dynamics of bridges considering the effects of the following parameters [Wright and
Green 1959]:

“a. the mass of the load,

b. the mass of the beam,

c. the velocity of the load,

d. the spring mounting of the load,

e. viscous damping of the beam ( internal and external), and

f. spring action of the load springing” ( must be the spring action of the vehicle’s

suspension )

The analysis is an improved and modiﬁed version of the theory presented by
Inglis (1934). Hillerborg also obtained the expressions for the dynamic bending moment
and the shearing force as well as a general equation for the motion of a simply supported
girder and several moving loads. In addition, he provided numerical solutions for the
following cases [Wright and Green 1959]:

“a. a non-sprrmg load without damping of the beam

b. a non-sprung load with damping of the beam, and

 

7 See footnote on page 26.

33

c. a sprung load.”

The results of the analyses were compared with model tests, and both the
numerical and the test results were in a good agreement. In the case of the bending
moment, however, the analyses gave the dynamic increment only for a range of variables

that are generally not applicable to the dynamic behavior of highway bridges.

3.2.7 The main objective of the research reported under the title “Highway Bridge
Impact Investigations” and others [Boehning 1953; Eichmann 1954] carried out from
19951 to 1955 at the Engineering Experiment Station, University of Illinois8 [Wright and
Green 1959], was the development of a rational method for the prediction of the dynamic
increments of the bending moment and the deﬂection, for a simple span under the action
of normal commercial vehicles. Many of the important factors capable of affecting the
response (deﬂections and stresses) of loaded highway bridges were considered in the
analysis. Model tests, corresponding to the conditions adopted in the analysis were
carried out to check the validity of the results of the analysis. Equations of motion, in this
case, were derived using the energy-principles and considering the energies stored in the
bridge and the vehicle as a result of the vehicle-bridge interaction.

The test models were designed to be dynamically similar to a simple span I-beam
highway bridge of span length 60 it, weight 3000 lb./ﬁ., stiffness 20x109 1h.-ii2 and
fundamental frequency 6.4 Hz, and traversed by a 30,000 lb. single axle load. The

suspension system of the loading carriage was designed in accordance with

 

' Some of these references were not available. Accordingly, the report of Wright and Green (1959) is the
major source of information provided in this section.

34
manufacturer’s data about loaded trucks i.e., the natural frequency of the suspension

system in the range of 1.5 to 1.84 Hz and the ratio between the sprung and unsprung
weights from 7.0 to 8.0.
The following parameters represent the range of an average vehicle-bridge-
system:
R=0.167, R,/R2 =7.0, 0L=0.04 to 0.15, F= 3.85

Weight of sprung part of vehicle __ ms
Weight of bridge m

 

where RI =

 

9

__ Weight of unsprung part of vehicle _ mu
Weight of bridge m

 

 

R2

9

_ Weight of vehicle -R _ m

Weight of bridge m8

 

_ One - half the period of bridge _ V
Time required for vehicle to cross span 2 F L ’

 

0i and

F- Natural frequency of bridge

 

- Natural frequency of vehicle

The following are the parameter ranges used for both the model tests and the
analytical solutions:

R = 0.20 to 0.60, Rl /R2 = 7.0, a = 0.06 to 0.20, and F = 2.0 to 5.0

The model was handled carefully to have control over the state of motion of the
carriage, as it entered the beam,

The following are among the assumptions that were made in the analysis [Wright
and Green 1959]:

“a. that simple slab or beam and slab bridges behave as simple ﬂexible beams,

35
b. that the structures are linearly elastic,

c. that the vehicle is considered to be a single, or tandem, axle load, consisting of
a sprung part (chassis and payload) and an unsprung part (wheels and frame)
connected by a linear spring,

(1. that the ﬂexibility of the vehicle tires can be neglected,

e. that the vehicle speed is constant over the span,

f. that the vehicle wheels are in contact with the span throughout the crossing,

g. that the span is stationary before loading,

h. that the damping of the vehicle-bridge system, camber and deck roughness can

be neglected, and

j. that the vehicle path is along the center-line of the span.

The model tests carried out were for the cases of:
a. an unsprung mass with both single and tandem axles,
b. a sprung mass with no initial vertical velocity or displacement, and

c. a sprung mass with initial velocity and displacement.”

For the case of high weight ratios (i.e., the ratio of the vehicle weight to that of the
bridge), the results of analytical and experimental studies for the mid-span deﬂection
ampliﬁcation factors were in good agreement. When plotted against or, the speed

parameter, the ampliﬁcation factors for both the mid-span deﬂections and the bending

36

moment, showed a similar trend. The studies also reported an increase in the dynamic
response of the loaded beam as a result of an increase in the speed parameter on.

From the test results, it was observed that unsprung single axles invariably
produced larger though similar, dynamic effects than the tandem axles.

' In the case of weight ratios greater than 0.20, the maximum dynamic effects for
the completely unsprung vehicle were almost independent of the weight ratio. For the
higher weight ratios the maximum ampliﬁcation factors were observed at lower speeds,
there was no change in the magnitude of the peak values.

In the case of a sprung vehicle with zero vertical displacement and velocity, the
dynamic effects were similar to those produced by an ‘equivalent’ unsprung vehicle. In
this case, the equivalent weight was the sum of the weight of the unsprung portion of the
sprung vehicle plus some fraction of the sprung weight.

For normal heavy vehicles, the equivalent weight is nearly the same as the sprung
portion of the vehicle carriage. The sprung vehicle with vertical motion or displacement
may be responsible for greater dynamic effects in the bridge than those produced by the
‘equivalent’ unsprung vehicle because of the energy stored in the vehicle springs when

the vehicle enters the span.

3.2.8 The studies carried out at Massachusetts Institute of Technology9 (M.I.T) in early

19505 were aimed at developing an analytical procedure for the prediction of the motion

 

9 See footnote on page 32.

37
of simply supported bridges under the action of a moving sprung axle load. [Biggs and

Suer 1956; Wright and Green 1959]:

Results of the ﬁeld tests lead to the main conclusion that the vertical oscillation of
the vehicle itself was the main inﬂuence on the motion of the bridge and that this was
caused by the roughness of the approach .

The assumptions made in the analysis, were:

“a. that only ﬁrst mode vibration need be considered,

b. that the vehicle system has one degree of freedom, and that its entire weight is

located at the center of gravity of the vehicle mass,

c. that only viscous damping is present in the bridge vehicle system, and

d. that the deﬂected shape of the structure is at all times sinusoidal.”

The sprung-vehicle-bridge system was represented by two single-degree of
freedom systems.

The results of the laboratory studies, lead to the conclusion that a simple span
beam could be represented as a single degree of freedom system , and that a single
concentrated sprung load could reasonably approximate a model two axle truck.

The following ﬁrrther assumptions were made in the study [Wright and Green
l959k

“a. that approach roughness induces an initial oscillation of the vehicle, the

bridge deck being perfectly level,
b. that damping may be neglected,

c. that the load travels along the center-line of the structure,

38
d. the static and dynamic rigidities of the structure have the same

characteristics,
e. that the vehicle is completely sprung, and
f. that the bridge is at rest when a vehicle enters”
The parameters considered in the study were:

00 - the frequency of the vehicle crossing,

R = 31 i.e., the ratio of the total mass of the load to that of the total

mg

mass of the beam or girder,

fl where f" is the natural frequency of the vehicle system in cps (Hz),

V

and F = fb f i.e., the frequency ratio.

V

The results of this study, obtained for the case of a mass ratio of 0.10 and an

irregularity parameter I of 0.30, (given by the relation I = K, Z," =F R SEA where K, is

u
the spring constant of the vehicle system, Z," is the amplitude of the sprung mass motion
of the vehicle, and 8 5., is the maximum static mid-span deﬂection), are as follows:
a. W:
The maximum dynamic deﬂection takes place for values of the phase
angle of the vehicle motion (1) between it and 21:. Using an approximation the

maximum dynamic deﬂection in terms of frequency is given by

39

g1 =0.50F+1.15 --------- (8)

b. W

In the case of frequency ratios less than 2.5, the dynamic deﬂection
increases to 1.74 for a ratio just greater than unity. The dynamic effect decreases
rapidly as the frequency ratio approaches zero. For frequency ratios F greater
than 2.5, variation in the dynamic deﬂection values was small. It was observed
that the dynamic deﬂection was dependent only upon the ratio of the frequencies
rather than the actual values of vehicle and bridge frequency. It was further noted,
from the model tests, that a beating effect occurred for low frequency ratios when
most of the energy transferred from the bridge system to the vehicle system and
vice-versa. The latter was responsible for large dynamic deﬂections of the
structure over a short time interval.

The frequency ratios range selected was based on bridge and vehicle

frequencies of 2 to 8 Hz and 1 to 3 Hz respectively. The natural frequency of the

bridge spans, f , was taken as 3—'LS—O Hz where L is the span length in feet.

c. Raughncss:

The effect of initial vehicle amplitude upon the dynamic deﬂection was
observed to be linear for a given frequency ratio and increased with increasing
irregularity parameter I. It was further reported that the dynamic deﬂection was
also a function of the frequency ratio, and that for frequency ratios near unity, the

effect of the roughness was greater.

40
d. Massﬁanc:

For mass ratios greater than 0.15, the dynamic deﬂection was almost
independent of frequency ratio. The dynamic deﬂection increased for mass ratios
less than 0.15 and the rate of the increase was dependent upon the frequency ratio.
The results were presented as design curves [Biggs et al. 1956]"), which might be

used to predict the maximum dynamic deﬂection of the mid-span of a medium span
girder or stringer bridge, resulting from a two axle vehicle with‘a heavily loaded rear axle,
traveling down the center line of the bridge. It was commented that it was necessary to

know the dynamic characteristics of both the bridge and vehicle to apply the curves to

design [Wright and Green 1959].

3.2.9 Wright and Green (1959) presented a collection and review of the analytical and
ﬁeld studies carried out up to that time on the problem of vibration in bridges. Their
report discussed the merits and demerits of these studies besides mentioning the

assumptions made and the results obtained, as mainly reported herein up to this point.

3.2.10 Fleming and Romualdi (1961) provided analytical study of single span and three
span continuous highway bridges modeled as discrete multiple degree of freedom
systems subjected to moving loads. The regular differential equations of motion were
derived for both cases, and numerically solved with a digital computer using integration

over ﬁnite intervals of time. To study the effects of vehicle characteristics (mass,

 

'0 As reported by Wright and Green (1959).

41
springing, velocity, damping) and the bridge geometry, the dynamic response of the

bridge was related to the following dimension-less parameters:

a _ time of crossing
fundamental period of bridge

 

_ mass of load

 

mass of bridge

_ Fundamental period of load
p Fundamental period of bridge

 

More than one thousand results were obtained by varying values of the various
variables affecting the dynamic response and analyzed in terms of above mentioned
dimension-less parameters. The spans used in the model study ranged from 20 to 140 ft
for the case of single span, and in the case of continuous spans, the outer spans ranged
from 40 to 80 fl and the central spans from 50 to 100 ft.

On the basis of their results, it was concluded that for unsprung loads, the most
important factor was the velocity of the load. It was commented that the main effect of a
change in the mass of the load was to change the velocity at which the maximum
dynamic response would occur. It was observed that due to change in mass of the load,
the change in the maximum produced dynamic effect was not large, however, it was
expected that the change in dynamic load effect could be more pronounced in the case of
continuous bridges. It was reported that the springing of the load, and the surface
condition of the approach of the bridge were the two most important factors for the
dynamic response. It was also concluded that initial vibration of the load, caused by

settlement or depression in the bridge approach could result in very large DLFs.

42
3.2.11 Another notable and detailed analytical study on the problem of dynamic response

of highway bridges was due to Wen and Veletsos(l962). The study was carried out for
simple span highway bridges. The bridges analyzed were I-beam type steel-girder
concrete deck slab bridges, and the span lengths analyzed were 20, 45 and 70 feet. The
bridge parameters considered were span length, total weight, frmdamental natural
frequency of vibration, deviation of bridge surface from a straight line, and the initial
dynamic condition of the bridge. The parameters considered in the case of the vehicle
were its speed, total weight, distribution of weight among sprung and unsprung
components, axle spacing, effective spring constant for the axle, and the initial dynamic
condition of the vehicle. The analysis was carried out using a computer program in terms
of the following dimension-less parameters:

1. Speed parameter, given by

V Tb
a = —-— ------ 9
2 L ( )
where or is the speed parameter,

V is the speed of the vehicle,
Tb is the ﬁmdamental natural period of the bridge, and
L is the span length of the bridge
In the case of short span bridges, for all practical situations, Tb is
proportional to the span length L. Thus, the speed parameter or is in fact a
function of the vehicle speed only.

2. Weight and Weight-distribution parameters:

43
a. The weight parameter was deﬁned as the ratio of the total weight of the

vehicle to that of the bridge .
b. Weight distribution parameters consisted of :
I. the ratio of the static reactions on two axles, and
II. the ratios of the unsprung weight for each of two axles to total
weight of the vehicle .
3. Frequency parameters:

For each axle ‘j’ (j = 1 or 2), a separate frequency ratio was deﬁned as

 

 

(f,)j _ Natural Frequency of j‘h axle (10)
f b Fundamental Natural Frequency of Bridge
where the axle frequency (fV ) j was deﬁned as
l K j
= — — --------- 11
v 2“ M, ( )
where K j is the estimated stiffness of ‘j’th axle, and

M j is the sprung mass of ‘j’th axle.

4. Rotary Inertia Parameter was stated to be a measure of the resistance of the
sprung mass of the vehicle against pitching motion, and was deﬁned by the ratio

.1

_2_— """"" (12)
jZMjai

where J is the polar moment of inertia of sprung mass about its

centroidal axis,

44

M j is the sprung mass of ‘j’th axle, and

a- is the horizontal distance between the ‘j’th axle and the

centroid of the sprung mass.
General conclusions drawn from the study, were:

a. The magnitude of the maximum dynamic effects were proportional to
the speed of the vehicle.

b. In the case of axle spacing greater than the bridge length, dynamic
effects under 2-axle vehicle loading were greater than that for the single
axle loading.

c. For the variable axle spacing, keeping all other factors constant, a quasi-

resonance condition was observed when the time interval of application
of two successive axles over a point was equal to the fundamental natural
period of the bridge.

(I. When the bridge was considered to be initially under vibratory motion,
the magnitude of the resulting dynamic effect was observed to almost
linearly increase with the amplitude of the initial oscillation considered.

e. The results of the analysis lead to the conclusion that the riding surface
roughness might be a source of large dynamic effects in bridges,

f. Since damping in both the vehicle and the bridge was ignored in the
analysis, the actual dynamic response would be less than that obtained in

the analysis.

45
3.2.12 Sundara Raja Iyengar and Jagadish (1968,1970) performed numerical studies on

dynamic response characteristics of two bridges, consisting of one each of slab- and slab-
and-beam type bridges. The bridges were modeled as two-dimensional orthotropic plate
structures, and the vehicle was modeled as a moving force in the ﬁrst study, and as a
moving mass in the second study. The general conclusions were as follows:

1. The mid-span response of the highway bridge was inﬂuenced mostly by the
ﬁrst three modes of vibration. Although this conclusion had been drawn with reference to
two bridges, it might be extended to other bridges where the frequencies of the ﬁrst three
modes were clustered together. It has been quoted with reference to Sundara Raja Iyengar
et al. (1967) that the frequencies of most of the highway bridges follow the above
mentioned pattern.

2. When the bridge was subjected to a concentric (acting along center line of the
bridge) moving force, the ampliﬁcation factors for the points below the moving force
followed the trend found using beam theory. The values of the ampliﬁcations found by
the orthotropic plate theory were smaller than the values found by the beam theory. When
the bridge, modeled as a plate structure, was under an eccentric moving force (acting
along a line parallel to the center line of the loading lane), the ampliﬁcations for points
below the moving force did not closely follow the trend observed in the case of beam
theory.

3. In the case of a moving force model, the maximum ampliﬁcation on the
transverse section was higher at points away from the line of loading, than that on the line

of loading. The largest ampliﬁcations were realized at the unloaded edge of a bridge

46

under an eccentric loading. This effect was more pronounced in the beam-and-slab bridge
than in the slab bridge. If the ﬁrst two frequencies of a bridge were very close, the
unloaded edge executed a beating type motion, when the bridge was subjected to an
eccentric moving force.

When the vehicle was modeled as a moving mass, the following conclusions were
drawn:

a. The maximum ampliﬁcation factors on the transverse section were larger for
points on the bridge away from the line of loading, than on the line of loading. It was
observed that in the case of an eccentrically loaded bridge, the effect was more
pronounced, and the unloaded edge experienced quite large ampliﬁcations irrespective of
the frequency and mass ratios. The ampliﬁcations in the beam and slab bridge were
reported to be much larger than in the slab bridge.

b. The ampliﬁcations given by the beam theory were slightly on the conservative
side when compared to the point of maximum ampliﬁcation in the orthotropic plate
theory.

c. The moment and deﬂection ampliﬁcations were about equal for points away
from the line of loading . Whereas for points on the line of loading, the maximum
moment ampliﬁcation factors were observed generally smaller than the deﬂection
ampliﬁcation factors. This effect was more pronounced for larger values of the speed
parameter.

It was observed that the maximum value of the moment at the mid-span point on

the line of loading always occurred when the load was at or very close to mid-span even

47

if the interaction force was not large at this instant. It was also concluded that for the
larger values of the speed parameter, the load would act near the third quarter point and
the maximum deﬂection ampliﬁcation occurred when the load had traveled beyond the
midspan. Thus, the variation of interaction force between the bridge and the moving
vehicle had a stronger inﬂuence on the maximum deﬂection rather than on the maximum
moment, for points on the line of loading. The above might be responsible for the
considerable differences between maximum ampliﬁcation factors for the deﬂection and
the moment for such points.

The frequency and mass ratios had deﬁnite, although secondary, inﬂuences on the
response characteristics. When the frequency ratio was between 0.6 and 1.0, large
dynamic effects were observed. The effect of mass ratio was found to be less pronounced,
although the ampliﬁcations generally increased with the mass ratio.

It was concluded that the initial oscillation generally lead to higher amplitude
oscillations all over the bridge. It was remarked that the maximum moment, at the mid-
span point on the line of loading, would be increased signiﬁcantly by initial oscillation
only if the load would be acting at or near the mid-span. It was commented that for a
ﬂexible bridge, an eccentric loading with initial oscillation might produce pronounced

oscillations at the unloaded edge.

3.2.13 The report of Veletsos and Huang (1970) provides a summary and a list of
references regarding studies conducted for predicting the dynamic response of highway

bridges at the University of Illinois since 1950. These studies were of following types:

48

a. Theoretical and numerical studies,

b. Laboratory studies of highway bridge models, and

c. Field tests on a number of specially designed bridges.

In the analytical studies, the bridge was modeled in two different ways. In the ﬁrst
case, it was assumed to behave like a single beam (used in the analysis of simple-span,
three span continuous, and three span cantilever bridges), and in the second (used to
analyze simple-span bridges only), it was treated as a plate continuous over a series of
ﬂexible beams. The main objective of the study was to present the most general method
of analysis developed in University of Illinois studies, using single-beam modeling of
bridges. The secondary objective of the study was to analyze other types of bridges
speciﬁcally with respect to susceptibility of the bridges to different kinds to vibrations.

In the method of analysis presented for the prediction of dynamic response of
bridges to moving loads, the bridge was modeled as a single linearly elastic beam with
distributed ﬂexibility and concentrated point masses . The vehicle was idealized as a
three-axle sprung load, taking into account the effect of ﬁiction present in the suspension
system.

On the basis of the results of the analysis and other relevant data[Nieto-Ramirez
and Veletsos 1966], it was concluded that cantilever type bridges are most susceptible to
vibrations caused by moving loads, simple-span and continuous-type bridges being next

in the same order.

49
3.2.14 Ibanathan and Wieland (1987) conducted a stochastic analysis to predict the

dynamic response of a simply supported box girder bridge subjected to a vehicle moving
at a constant speed. The bridge was modeled as a one-dimensional elastic beam, and the
vehicle as a single-point mass moving at constant speed.
From the results of analysis, it was concluded that:
a. at high speeds and at high ratios of the vehicle mass to the bridge mass, the
the mass of the vehicle had signiﬁcant effect on the response of the bridge,
b. the maximum response was almost unaffected by the presence of damping (in
either the bridge or the vehicle), and
c. mainly trucks traveling at high speeds were responsible for the maximrun
dynamic response, and the dynamic effect due to passenger cars could be

ignored.

3.2.15 The objective of the detailed analytical study performed by Hwang and Nowak
(1991) were:
a. to develop a procedure for the calculation of the dynamic load, and
b. to determine the statistical parameters of the dynamic load to be used in the
development of a reliability-based bridge design code.
The analysis was carried out for cases of :
a. composite steel girder bridges, and

b. prestressed concrete girder bridges

50

with simply supported spans of 12 to 30 m. Strains were considered to be in the elastic
range only.

For the dynamic analysis, the vehicle was considered to be composed of:

a. the vehicle body, represented by a distributed mass, and

b. tires, represented by linear elastic springs.
A three-axle single truck and a ﬁve-axle tractor-trailer were used in the analysis. The
bridge was modeled as a prismatic beam. The damping of the bridge was assumed to be
2% of the critical damping for each mode. A computer program was used to solve
differential equations of the motion for the vehicle-bridge system, using Newmark 13
method. For the three-axle single single truck, 20% of the total weight was assumed to be
acting on the front axle and the rest 80% on the rear axle. In the case of ﬁve-axle tractor-
trailor 60% of the total weight was assumed to be acting on the front axle and 40% on the
rear one.

A parametric study was carried out by calculating the dynamic load factors (as the
ratio of the maximum dynamic deﬂection to the maximum static deﬂection), as a
function of the gross weight, the speed, and the axle distance of the vehicle. Dynamic

load effects caused by one and two vehicles were also compared.

3.2.16 In the study carried out by Chang and Lee (1994), the vehicle was modeled as a 2
degree-of-freedom vibrating system, and the bridge was represented by a simply

supported beam. The study was carried out to ﬁnd the effects of the surface roughness,

51

vehicle speed, and the bridge span on the ‘impact factor’. The results were compared with
‘impact factor’ given by the Korean code(l982) i.e.,

20
50+L

 

--------- (13)

where L is the span length in meters.

Using 4 different vehicle models it was shown that the response was different
when the road roughness was taken into account than when the effect of road roughness
was neglected. It was also observed that ‘impact factors’ increased with surface
roughness, and vehicle speed. It was pointed out that the Korean code underestimated the
dynamic load effect, especially for vehicles traveling with high speeds over a long rough

bridge.

3.2.17 Humar and Kashif (1995) also treated the problem of bridge vibration
analytically. The bridge deck was idealized by rectangular thin plate elements. The plate
bending element was based on a triangular element derived by Hsiah (Clough and
Tocher 1966) for ensuring monotonic convergence. The vehicle was modeled as a single
sprung mass and a dashpot (representing viscous damping). Damping was ignored in the
bridge. The vehicle conﬁguration and the road roughness were also ignored.

From the study of dynamic response of a slab type structure under the action of moving

vehicles, the following conclusions were made:

52

a. The characterizing parameters of a bridge modeled as an isotropic plate
structure were the aspect ratio, the speed parameter or, the frequency ratio (p,
and the mass ratio K.

b. Two different bridges with the same four parameters had identical responses.

c. For an orthotropic plate model of the bridge, the governing parameters were a,
d), K , and factors (x) and (0) related to the geometry and stiffness of the plate.

d. For an isotropic plate model of the bridge, the maximum response of two
bridges with different aspect ratio were not the same but still quite close.

e. In general, dynamic ampliﬁcation factors for the presence of multiple vehicles
were lower than those for the single vehicle.

f. The response of two vehicles each of mass m, traveling along parallel paths but

one lagging behind the other was less than that produced by the two when
traveling parallel to each other.

g. For the isotropic plate case, the moment ampliﬁcation factors were generally

less than the deﬂection ampliﬁcation factors.

3.2.18 Huang et al.(l995) analyzed a thin walled box-girder bridge, using beam
elements. The vehicle used in the analysis was an AASHTO HSZO-44 truck, modeled as
an 11 degrees-of-freedom system. Four classes of road roughness (very good, good,

average, and poor) were considered in the analysis.

53
The results of the analysis were stated to be in good agreement with that obtained

by using a folded plate model [Jonesll 1972].

From the results of the study, it was concluded that:

a. A decrease in torsional rigidity of the end diaphragm, resulted in a
decrease of the ‘impact factors’ for simply supported box-girder bridges.

b. ‘Impact factors’ varied with change in transverse positions and number of
vehicles.

c. For ‘very good’ and ‘good’ road surfaces, the ‘impact factors’ were less than
0.10 and varied slightly with increased speed of the vehicle.

d. For the ‘average’ and ‘poor’ road surfaces, the ‘impact factors’ increased

signiﬁcantly.

3.2.19 Using ﬁnite element method approach for modeling the bridge and a modiﬁed
dynamic condensation method, Yang and Lin (1995) presented an analytical model for
the dynamic analysis of a vehicle-bridge interaction system. In the model, the vehicle was
represented by sprung lumped masses and dashpots, and the bridge was idealized with

beam elements and a rough pavement.

3.2.20 In the analytical study of Yang et al.(1995), the vehicle was modeled with
suspended lumped masses i.e., lumped masses supported by springs and dashpots

(viscous damping), whereas the bridge was modeled as a series of beam elements. The

 

” The original reference not available.

54

analysis was carried out only for “beam like” bridges consisting of simple and continuous
span bridges for all ( short, medium and long) spans. Separate formulas were proposed
for determining ‘impact factors’ in case of deﬂections, bending moments, and shear
forces for the simple as well as continuous bridges.

From the results of the analysis, it was concluded that:

a. AASHTO (1989) ‘impact forrnula’ underestimates the dynamic load effect of

moving loads in certain cases, especially in case of mid-span deﬂection, and
mid-span moment of simple span beams.

b. The suspension stiffness of the vehicle had negligible effect on the DLF.

c. The pavement roughness had little effect on the dynamic response, such that it
could be neglected while using the proposed formulas. (Although the effect of
pavement is generally signiﬁcantly pronounced, especially if the approach road
or the riding surface of the bridge contains a settlement or a small bump
[Fleming and Romualdi 1961; Heywood 1995]. )

d. When heavier bridge cross-sections were used, the ‘impact factor’ might be

slightly less than that predicted by the proposed formulas.

3.3 W
In case of ﬁeld or test studies the majority of the types of the highway bridges
considered consisted of the simple and continuous span slab-and-beam or slab-and-girder

bridges. Although in these tests generally the bridges were well-instrumented,

55

nevertheless, the test vehicles were not always true representatives of normal commercial

vehicles.

3.3.1 Some of the early test studies carried out in various parts of world up to about
1918, to determine the dynamic effect of moving loads on bridges, have been reported by
Waddell (1918). Although the original references of most of these studies are not
available, nevertheless these studies are important in that these highlight general trends
followed in the early studies on determination of dynamic load effect of moving loads on
bridges. Accordingly, the following information is mainly based on the report of Waddell
(1918)

Recognition of the existence of the dynamic effect caused by moving vehicles had
taken place at least in case of railway bridges as long ago as the middle of the 19th
century. Experimental determination of dynamic effects due to moving loads had
commenced as early as 1849 when an English engineer named Willisl2 performed some
very generalized experiments on a pair of 9 feet long bars, using a carriage running at
different speeds with variable loads. It was concluded that the deﬂections increased with
the speed of the carriage up to a certain point. The deﬂections recorded were in some
cases two or three times those obtained under static load. However, the bars used in the
tests were too ﬂexible to be compared with real bridges. Accordingly reasonable

prediction of load factors for real bridges was not possible [Waddell 1918].

 

'2 The original reference is not available. The information presented has been chieﬂy taken from Waddell
(1918).

56
In 1881, the deﬂections measured at the center and the quarter points of a 60-feet

continuous Howe truss span, by H. Sabinel3 and Professor S.W. Robinson resulted in
DLF values ranging from 6 to 12 percent for velocities at or under 30 mph. In a paper,
based on the results of the deﬂection tests conducted between 1881 and 1887 on Ohio
railroad bridges [Robinson 1887], Professor Robinson showed that the dynamic load
effects may be as high as 50 percent with an average of about 26.5 percent.

In France, a notable experimental work for determination of dynamic load effect
was started by Col. Charles Rabut”. The investigation yielded the DLF values from 10 to
50 percent. A formula for ‘impact factors’ of short spans was proposed as follows

[Waddell 1918]:

 

I= --------- (14)

where I is the impact factor, and

L is the span in meters

In 1893, J. Melanl5 of Austria carried out a mathematical study of dynamic effects
for different span lengths. His results indicated impact values of 0.77 to 0.15 for the span
lengths from 5 to 120 m respectively [Waddell 1918].

In 1887, CC. Schneider”, proposed the following formula:

_ 300
L +300

 

--------- (15)

 

'3 Same as footnote No. 12.

" See the footnote on page 54.
'5 See the footnote on page 54.
'6 See the footnote on page 54.

57

where I is the coefﬁcient of ‘impact’ and
L is the length of that portion of the span which is covered by the moving
load when the member under consideration receives its greatest live load
stress.

This formula remained in use until about 1912-13 [Waddell 1918].

Another important step towards the experimental determination of the dynamic
effect of moving loads on bridges was the formation of an American Railway
Engineering Association (AREA) committee for a thorough investigation of the impact
effect. The report of the committee17 issued in 1911, based on the experimental work
carried out on :

a. 21 plate-girder spans of lengths up to 100 feet, and

b. 24 truss spans of length ranging from 100 to 250 feet
for loaded cars with speeds ranging from 10 to greater than 60 miles an hour, proposed

following two formulas for calculation of impact [Waddell 1918]:

 

I=_—L2—— --------- (16)
1+
20,000
60
d I=— --------- 17
an L ( )

where I and L are deﬁned similar to that for the Schneider formula given above.

 

'7 See the footnote on page 54.

58
In a later report of the committee18 [AREA Proceedings 1916], a revised formula

has been given as [Waddell 1918]:

1
I=_—I}— --------- (18)
1+

30,000

 

In a paper (published in Volume CC of the Proceedings of the Institution of Civil
Engineers, Charles William Anderson)'9 , based on some experiments conducted in India
by the government bridge engineer H. S. Sales, proposed the following ‘impact’ formula
[Waddell 1918]:

_50
L+50

 

--------- (19)

The author of the paper admitted that the formula had no direct scientiﬁc basis [Waddell
1918]

After indicating the inadequacy of various impact formulas, Waddell (1918),
based on the analysis of these formulas, recommended the following formulas, for

various kinds of bridges:

 

 

 

Stnrcturepre EmmssdlmnacLEcnnula
1. Steam Railway Structures 1- 165A 20
_nL+lﬂ) """"* )
2. Highway and Electric-Railway 1- 100A {21
’ nL+200 ‘ )
Structures
" See the footnote on page 54.

'9 See the footnote on page 54.

59

where n is the number of tracks on the bridge,
L is the span length in feet, and
A = 1.00 for steel bridges with open or light ﬂoors,
= 0.75 for steel bridges with heavy, solid ﬂoors,
= 0.50 for reinforced concrete structures, except spandrel-ﬁlled arch bridges

= 0.25 for spandrel-ﬁlled arch bridges

3.3.2 The real effort on the problem of the highway bridge dynamic response started in
the early 19305 when a special committee of the American Society of Civil Engineers
published an important report on the subject of the ‘impact’ in highway bridges
[Committee 1931]. The recommendations made in this report, based on data obtained
from a series of ﬁeld tests carried out using 3-1/2 ton Liberty trucks traveling at a
maximum speed of 15 mph, became the basis of most of the speciﬁcations (on the
computation of dynamic effects of moving loads on highway bridges ) being used in the
19608 [AASHTO] and some of which are still being used [CPHB 1967]. In this report the
committee recommended, in the case of girders and trusses of highway bridges that:

a. in the case of span lengths less than 40 ft, the ‘impact increment’ (that is, the
dynamic load factor) for the stress should be assumed as 25 percent of the live
load stress, and

b. in the case of span lengths of 40 feet or more, the impact increment of the stress

should be assumed as a fraction of the live load stress determined by the

formula:

60

I- 50
L+160

 

--------- (22)

where I is the impact fraction, and

L is the span length in feet.

3.3.3 In 1944, Vandegrift20 performed tests on several types of bridges in Ohio, to
determine their dynamic properties and to check if some sort of damper could be ﬁxed to
the bridge to reduce vibration. The major conclusions drawn from the study were:

1. Rigidity: Composite action was present in simply supported and continuous
deck and slab bridges although not considered in the design. Accordingly, the stiffness
and natural frequency of the spans were increased. However, the stiffening provided by
the deck system did not increase the rigidity of through plate-girder bridges and trusses.

2. W: “First mode natural frequencies were usual for both
loaded and unloaded spans.” [Wright and Green 1959] (Might be similar in both loaded
and unloaded cases).

3. W: In the case of 3-span continuous bridge, the dampers

used were effective in reducing the amplitude of vibration.

3.3.4 Dynamic load effect was the main concern of the bridge tests reported by G. R.
Mitchell“ (1943-1952 D.S.I.R. - Great Britain ) [Wright and Green 1959]. The tested

bridges were stated to have relatively good approaches and the surfaces free of pot holes

 

2° See footnote on page 26.
2' See footnote on page 26.

61

or irregularities that were thought to be responsible for the initiation of vertical motion of
the vehicle. The spans of the bridges were in the range of 28 to 35 feet. The test vehicle
consisted of a tractor-trailer combination and a sprung suspension system with solid tires.
The vehicle axle spacing was about 24 feet and axle loads were 14 kips for the front and
50 kips for the rear axle. The results and conclusions of the study were as follows:

a. In one of the tests, the observed dynamic response was similar regardless of
whether the vehicle traveled along the center line, or in a traffic lane of the
bridge.”

b. It was noticed that the test vehicles had a vertical motion as they crossed the
bridges and, moreover, their wheels did not leave the deck of the bridge. In the
case of loaded bridges, the frequency of vibration was different than the natural
frequency of the unloaded bridge, and thus it was the case of a forced vibration.

c. For each axle of the trailer, values of the forced frequency of vibration were
found to be constant and did not change signiﬁcantly even from bridge to
bridge.

(I. In the small range (Original reference is not available. So the speed range is not
known) of vehicle speeds considered during the tests the amplitude of the bridge
vibration increased considerably with the speed of the test vehicle.

3.3.5 The tests reported by Norman22 (1949) [Wright and Green 1959] were carried out
to investigate the causes of signiﬁcant vibration of a bridge under normal loading

conditions. The bridge was a two lane, two span composite structure of total length 160

 

22 See footnote on page 26.

62
feet, with a 20 degree. skew angle and a slight grade (1/60). The bridge was designed for

H-20 S16 loading. The bridge approaches were reported to be good. The results and the
conclusions drawn were:

a. The maximum amplitude of motion occurred when the test vehicle had left one
span and had reached the middle of the adjacent span. Vibrations caused by the two-axle
test vehicle continued for one and a half minutes in some cases. The structure was so
ﬂexible that it started vibrating due to pedestrians walking over it.

b. For calculating the natural frequency the values of stiffness were computed from
measured static deﬂections. The bridge had a ﬁrst vibration mode equivalent to the
second mode of a simple span, with a 2% difference between the measured and computed
frequencies. The test vehicle did not affect the frequency of the bridge vibration.

c. The magnitude of the vibration of a bridge was dependent upon the vehicle speed;
the motion of the bridge was not much signiﬁcant at a speed of 37.2 mph whereas at 41.2
mph it became very pronounced. The latter speed did not correspond to the resonance
condition given by the product of vehicle axle spacing and natural frequency of the bridge
(32 mph).

d. The maximum recorded value of the amplitude was 1.7x10'3 inches, and the
natural frequency of the bridge 3.58 Hz. “Assuming the motion of the bridge to be
sinusoidal over this maximum cycle, the resulting velocity and acceleration were 38,000
micro-inches per second, and 0.002g. feet per (second)2 respectively. This motion was

very signiﬁcant for an observer on the bridge.” [Wright and Green 1959]

63
3.3.6 In the ﬁeld tests carried out at M.I.T.23 (Massachusetts Institute of Technology)

[Wright and Green 1959], ﬁve simply supported bridges with spans from 69 to 114 feet
were subjected to a two axle moving vehicle, to observe the mid span deﬂections. Two of
these bridges were later on retested using an instrumented two axle vehicle as well.
The study lead to the following major conclusions:
1. Rigidity: The values of simple static deﬂections and natural frequencies of vibration
revealed that the bridges generally had a greater rigidity than that computed from the
design data.
2. W: The loaded bridges exhibited the forcing frequency of the test
vehicles instead of the natural frequency of the bridge. Originally the bridge vibrated at
its natural frequency, later on the forcing frequency dominated the bridge motion. The
irregularity of the deck of one bridge modiﬁed the vehicle motion, and correspondingly,
resulting in a change in the frequency of the bridge vibration from a forced to a natural
frequency. In the unloaded condition the bridges vibrated with their ﬁrst mode frequency.
The natural frequency of a simple beam bridge was found to be approximated by

the empirical equation:

_1.21r EIL

_.— ......... 23
21.2 M ( )

s
where fl, is the natural frequency of the bridge, Hz

L is the span length,

E1 is the ﬂexural rigidity, and

 

23 See footnote on page 26.

64

Mg is the mass of the beam or girder

3. Amplitude: No apparent relationship could be found between the amplitude of
vibration under load and vehicle speed. In case of a smoothly rolling load, the observed
amplitudes were larger than that obtained by theory. The largest amplitudes were
observed either in the case of an irregular bridge approach or when the natural frequency
of the span was low compared to the other spans. The surface roughness of the bridge
deck was not thought to be a major cause in producing large amplitudes[Wright and
Green 1959]. Assuming viscous damping, the mean values of damping coefficient were

found to be 0.007 for girder bridges and 0.022 for stringer bridges.

4. Vehieleﬁﬁeets: The rear axle load was varied up to 82 per cent of the total vehicle
load. The results lead to the conclusion that variation of the forcing frequency was
proportional to the frequency of load variation. The magnitude of this variation was a
function of pavement irregularity and thought to be a primary factor affecting the
amplitude of vibration. The expansion joints were found to have a signiﬁcant effect on
the axle load variation. The observed frequency of (dynamic) load variation was in
agreement with the computed natural frequency of the loaded tires. It was observed that
in the case of the test vehicle, the actual vehicle springing was due to the tires, rather than
the suspension system. It was pointed out that a vehicle with a blocked suspension system
(totally unsprung vehicle load) might result in much larger bridge deﬂections than from a

vehicle with a free system.

65
5. Analysis: All the results obtained through analytical studies, mentioned earlier (section

3.2.8 above), were in good agreement with the corresponding ﬁeld test results.

6. Design: It was warned that the present tendency toward long span slender bridges
should be regarded with caution because combination of low natural frequency
(approaching that of heavy vehicles) and approach road surface irregularities might result

in excessive vibration for these structures.

3.3.7 Hayes and Sbarounis (1954) reported observations made on the dynamic and
static characteristics of a three-span continuous beam and slab bridge. The approaches
and deck of the test structure were described as smooth.

Composite action was found to be present in various amounts between the deck
and the beams, and was responsible for increased structural rigidity. However, a
comparison of static and dynamic strain measurements showed that the composite action
was reduced under dynamic loading.

During and as a result of the motion of the test vehicle, the natural frequency of
the structure was changing continuously and was found to be a function of the
percentage of the composite action present between beam and slab for varying positions
of the test vehicle.

Whenever the period of load application coincided with the natural period of the
structure, the amplitudes were a maximum. At low speeds, the tandem axles of the test

vehicle lead to a resonant condition; whereas at higher speeds the tandem axles acted

66

dynamically as a single axle, the axle spacing of the test vehicle being a major factor.
Reportedly, the natural frequency of the test vehicle had no coincidence with the

frequencies of the bridge motion resulting from dynamic load tests.

3.3.8 Tests carried out by the Michigan State Highway Department24 (1952-1957) to
determine the static and dynamic characteristics of several types of highway
bridges,[Foster 1952; Foster and Oehler 1956; Wright and Green 1959] were of two
types:
a. extensive tests on two bridges, using a wide variety of test vehicles, and
b. tests on a variety of bridge types, using only one standard instrumented test
vehicle, to compare the vibration susceptibility of the bridge types:
a. WWW:
These tests were carried out on the following two bridges:
I. An eight span bridge, consisting of a three-span continuous (steel) plate
girder and slab structure and ﬁve simple spans, and
II. a bridge consisting of six simple spans.
Following were the major ﬁndings from these tests:
1. Rigidity: The ratio of the actual to computed deﬂections of the spans under load, was
observed to be less than 40 per cent, the deﬂections being computed in accordance with
AASHO speciﬁcations25 (1953)[Wright and Green 1959]. It was concluded that the

increase in rigidity was due to composite action of the deck and girders.

 

2’ See footnote on page 32.
25 See footnote on page 32.

67
2. Eregneneyefﬂibmtien: In general, the bridge vibrations exhibited a ﬁrst mode natural

frequency. The natural frequencies of the spans, not designed for composite action, were
calculated considering the presence of the composite action. The difference between the
computed and the ﬁeld values of natural frequency was found to be less than 4 per cent.
The decay of the residual vibrations was observed to have a recurring pattern, although it

was not regular.

3. Amplmideandﬂuratieneﬂibtetien: The effect of the period of axle load application

(i.e., axle spacing/vehicle speed) was very signiﬁcant. It was observed that when this
period was a simple fraction of the natural period of the span, the amplitude and duration
of vibration of the bridge were maximum, except in the case of two axle vehicles -
probably because of unequal load distribution between axles. The effect of tandem axles
was found to be smaller than that of an equivalent single axle. Considering viscous
damping, the damping coefficients obtained were 0.025 for beam and slab bridges, 0.011
for plate girder and slab bridges, and 0.004 for the continuous bridge. The bridges were
generally of composite concrete-deck over steel-girder construction.

b. ﬁenemljests: In this part of study, thirty four spans were tested using a standard
instrumented 2S-1 vehicle. The bridges tested were beam-and- slab type simple,
continuous or cantilever-type structures. The data recorded consisted of the mid-span
deﬂections and axle load variation of the test vehicle.

The main conclusions reached were [Wright and Green 1959]:

68
“l. Rigidity: Using AASHO speciﬁcations (1953), the ratio of observed to theoretical ,

deﬂection for the bridges tested was found to be less than 0.50. The average ratio was:
a) for simple spans, with composite design, 0.32,
b) for simple spans, without composite design, 0.16,
c) for continuous spans, either steel beam and slab without composite design or
reinforced concrete, less than 0.35, and
d) for cantilever-type bridges with composite design, for both anchor arm span
and suspended span, 0.44 and without composite design 0.22.” [Wright and

Green 1959]

2. mm: The spans exhibited a ﬁrst mode vibration signiﬁcantly. The
computed frequencies were in a very good agreement with the ﬁeld results. The
numerical method of A. S. Veletsos and N. M. Newmark (1955)”, was used for
calculating the frequency values of the continuous beam or girder and slab bridges.
Whereas, for the continuous reinforced concrete and cantilever-type spans the
cumbersome inﬂuence coefﬁcient method (Thomson 1953)27 was used. It was observed
that the suspended spans of the cantilever-type bridges did not vibrate as simple spans but

with a frequency dependent upon the structure as a whole.

3. Amplimdeandﬂuratieneﬂyrhratrdn: It was observed that the loaded simple spans

with natural frequencies of less than 6.5 Hz generally developed greater amplitudes than

 

‘6 Reported by Wright and Green (1959).
27 Same as footnote No. 26.

69

spans with frequencies of less than 6.5 Hz. The cantilever bridges were observed to be the
most ﬂexible. “The duration of vibration of the various types of spans was approximately
proportional to their observed rigidity. The mean values of damping factor obtained for
the various bridge types were:

“a. for simple spans, composite and non-composite design, 0.012,

b. for continuous beam or girder and slab spans, non composite design, 0.010,

c. for continuous reinforced concrete spans, 0.020, and

d. for cantilever-type spans, both composite and non-composite design, 0.010.”

[Wright and Green 1959]

4. Mekn'eie: The test vehicle consisted of two heavily loaded axles; one with a
conventional suspension system and the other connected directly to the vehicle frame of
the vehicle. The DLFs for the various loaded spans were proportional to the variation of
the maximum axle load except in cases of the cantilever spans and the extremely rigid
non-composite simple spans. “The maximum load variation was found to be proportional

to vehicle speed, and varied with the irregularity of the bridge approaches.”

“5. Mihratienﬁuseeptihility: An attempt was made to compare the various type of bridges

tested with respect to human perception of motion, by using Janeway’s28 perceptibility
limits [Wright and Green 1959]. The results suggested simple spans and cantilever type

structures give the same degree of unpleasantness under the action of the test vehicle.

 

2’ See footnote on page 26.

70

However, the ﬁeld observers considered the cantilever structures the more unpleasant of

the two types.”

3.3.9 “Two similar three-span continuous and plate-girder bridges were tested by
Edgerton and Beecroft (1953) of the Oregon State Highway Department. One Bridge was
studied because of reports of perceptible vibration and ﬂoor panel damage, while the
second study was made for comparison. The test vehicle approximated an H-20 S-16
design load.

The conclusions reached from the tests were:
1. Rigidity: Composite action between the girders and the ﬂoor slab, though not

considered in design, increased the stiffness of the spans.

2. W: Two modes of vibration were evident (ﬁrst symmetric mode
and ﬁrst asymmetric mode of vibration). No forcing frequencies were observed.
Computed frequencies for the bridges were about 10 per cent lower than observed. (The
structures had varying moment of inertia and dead weight; the frequencies for the bridges
were computed in terms of an equivalent prismatic structure).

3. Amplitude: Values of amplitude for both bridges were dependent upon the speed of the
test vehicle. The values of amplitude obtained could be associated with deviations in the

deck proﬁles of the two bridges.”

71
3.3.10 “Observations were carried out by the California Highways Department29 [Wright

and Green 1959] on a beam and slab bridge to determine a) the degree of composite
action present (in design such action was not considered), and b) the load distribution
characteristics of the structure.”

“The bridge was composed of a concrete deck supported by three longitudinal
steel beams. the general layout of the 23-span structure was such that the girders were
continuous over two spans, with the third span a suspended (might be cantilever or
overhanging) span. The vehicle used during the tests was similar to a standard two-axle
H-truck but weighed nearly 34 tons”

“The records of deﬂection indicated that dynamic deﬂection was a ﬁrnction of

vehicle speed.” [Wright and Green 1959]

3.3.11 The report of Fenves et al. (1962) gave a brief review of the program of the
AASHO Road Test bridges, and presented some of the major results of investigation. The
dynamic tests were performed on 15 test bridges, using 14 test vehicles. More than 1800
test runs were carried out with different combinations of vehicles and bridges. Vehicle
speeds ranged from 10 to 60 mph. The bridges consisted of composite and non-composite
steel, and reinforced, and prestressed concrete. Two and three axle vehicles were used in
the tests. The dynamic response was studied with respect to four dimensionless
parameters, namely the speed parameter, the weight ratio, the frequency ratio and the axle

spacing ratio. Spans of test bridges were in the range of 50 ft, average measuresd natural

 

29 See footnote on page 26.

72
frequency from 3.2 to 6.9 Hz, and average damping coefﬁcient ranged from 0.8 to 4.5%

of the the critical vaue. The dynamic load factors obtained from strain readings were in
the range 0.03 to 0.25, and those calculated from deﬂection readings were in the range

0.02 to 0.42.

3.3.12 Billing (1984) reports the results of the dynamic ﬁeld tests carried out on 27
highway bridges in Ontario, Canada. The tests were carried out to support and conﬁrm
the provisions for dynamic loading and dynamic response of bridges, designed using the
Ontario Highway Bridge Design Code (OHBDC) - based on the frequency of the bridge.
The bridges were of various conﬁgurations and consisted of 14 steel bridges with spans
from 22 to 122 m, 3 timber bridges each of about 5 m span, and 10 concrete bridges of
spans from 16 to 41 m. All the tests were performed according to a standardized
procedure. Data was collected for 100 or more vehicle passes over each bridge in order
that statistical parameters of dynamic response could be established to determine the load
factors of a limit states code.

The results of the study revealed that:

a. The timber bridges apparently did not exhibit vibration.

b. The bridges were generally considerably stiffer under the service loading of a

single truck than they were assumed to be in design.
c. The data generally conﬁrmed that the form and the values selected in the ﬁrst

edition of OHBDC were adequate, except requirement of a few minor changes.

73

3.3.13. To investigate the main parameters inﬂuencing the dynamic response of highway
bridges due to trafﬁc loads, a comprehensive ﬁeld test study was carried out in Ontario,
Canada [Wright and Green 1964]. A total of 52 bridges, consisting mostly of simple span
bridges, were selected to observe vertical motion at or near the mid-span, when subjected
to moving vehicles. The spans of the 47 bridges varied ﬁom 50 to 320 feet, and the
widths from 16 to 100 feet with the number of lanes varying from 1 to 8. The
fundamental natural frequencies of the unloaded bridges varied between 1.96 and 10.6
Hz, and those for loaded bridges, in the range of 2.1 to 10.6 Hz., the bridges with extreme
values being different in both cases. The study was not speciﬁcally undertaken for
determination of dynamic ampliﬁcation due to moving loads, nevertheless results of the
tests reﬂected the absolute maximum values of ‘dynamic increments’ for the deﬂection to

be between 0.23 and 1.16.

3.3.14 Shepherd and Aves (1973) have reported the results of ﬁeld tests carried out on 22
spans of 10 different concrete bridges along with comparison of the test results with that
of a proposed simple mathematical model. The bridges consisted of 7 simply supported
spans of lengths from 31 to 80 feet, and 3 continuous spans of lengths from 22.5 to 48
feet. The experimental and theoretical frequencies ranged from 5.72 to 12.5 Hz and 6 to
10.2 Hz, respectively for the simple spans, and from 9.5 to 11.4 Hz and 9 to 12 Hz
respectively for the continuous spans. The report compared the DLFs speciﬁed by the
bridge design codes (of that time) of AASHO, Australia, New Zealand, India, Hungary,

Germany and the United Kingdom.

74

On the basis of results of the ﬁeld and model studies, it was concluded that :

a. The code speciﬁed dynamic load allowances were frequently exceeded in case
of relatively short span simply supported bridges when the single vehicle load

represented a large proportion of the total design live load and the frequency
ratio of the vehicle and the bridge was relatively high.

b. The load variation was due mainly to the roughness of the approach, and the
roughness of the bridge deck had a smaller contribution. On the basis of the
limited experiment results, and a simple theoretical analysis, a formula had
been proposed for the determination of DLFs against those speciﬁed in the
bridge codes. It was, however, admitted that more theoretical research and ﬁeld

tests are needed to adopt the proposed formula for all type of bridges.

3.3.15 The report of Reto Cantieni (1984) summarizes the dynamic test results carried
out by Swiss federal Laboratories for Material Testing and Research (EMPA) from 1958
to 1981 on 226 beam and slab type highway bridges. The bridges tested consisted of
prestressed concrete (205), reinforced concrete (5), composite steel and concrete (14), and
prestressed lightweight concrete structures. The number of spans of these bridges varied
between 1 and 42, with an average of 4. Of these 226 bridges, 72% were multi-span
continuous beam type, and 12% were single-span simply supported beam type structures.
The minimum total length was 13 meters (43 ft) with maximum span equal to 11 meters

(36 ft). The maximum total length was 3147.5 meters (10,300 ft) with maximum span

75
equal to 118.8 meters (390 ft). A total of 109 bridges were straight, 97 skewed or curved,

and 20 both skewed and curved.

It was concluded that a highway bridge would exhibit a pronounced dynamic
response, provided that its frequency range is the same as that for one of the two vehicle
modes concerned, and provided also that the vehicle speed and the surface roughness
combine in such a way that corresponding mode of vibration of the vehicle would be

excited.

3.3.16 In England, a ﬁeld test study was performed by Leonard30 et al. (1974), under
Transportation and Road Research Laboratory, Crowthome. The study was carried out on
30 highway bridges using eight commercial vehicles with gross weights more than 32
tons. The dynamic load factor values from 0.09 to 0.75 were recorded. Further details not

available due to, non availability of the original report.

3.3.17 The report of Green31 (1977), regarding the study of the dynamic response of
bridge superstructure in Ontario, Canada, indicated dynamic load factors of 0.10 to 0.87
in the case of highway bridges. Further details not available due to, non availability of the

original report.

3.3.18 O’connor and Pritchard (1985) have reported the results of two dynamic test

studies performed on Six Mile Creek Bridge; a short span , steel and concrete highway

 

3° The original report was not available. The information presented was taken from Huang et. al. (1993).
3' See the footnote No.16.

76

bridge in Australia. Field strains were used to calculate the maximum mid-span bending
moments for 170 trucks in normal trafﬁc. For vthe gross vehicle weights of 27 to 44 ton,

the dynamic load factors ranged from -0.08 to 1.32.

3.3.19 Chan and O’connor (1990) have mentioned the results of some of the dynamic
test studies performed in various countries, indicating that the DLF values higher than
those given by certain bridge design codes [AASHTO 1977], have been obtained in
several of studies. A vehicle model has been proposed for the highway bridge dynamic
load investigations. Among others, it has been concluded that the natural frequencies of a
range of simply supported bridges, with spans from 10 to 32 m, satisfy a relationship of
the type Lf = constant, where L is the span and f is the fundamental frequency of the

bridge, with reasonable accuracy.

3.3.20 To study the effect of truck suspension type on the dynamic response of short-
span bridges was the main concern of the ﬁeld investigation carried out by R. J. Heywood
(1995). The study was carried out on three bridges consisting of:

- an R.C.C bridge having 3-spans of length 8.2 m each,

- a prestressed concrete bridge with 4-spans of length 9.14 m each, and

— a 3-span timber girder bridge with span lengths 7.62, 9.14, and 7.62 m

repectively.
Vehicle vibrations may generally be expressed in terms of body bounce and axle

hop frequencies. The body bounce frequencies for the vehicles were from 1.5 to 2 Hz for

77

air-suspended, and from 2.5 to 4 Hz for steel-suspension vehicles. The axle hOp

frequencies generally range between 8 and 15 Hz. These might couple dynamically with

with bridges having similar frequencies or the span lengths from 8 to 15 m.

3.4

The results of the study lead to the following conclusion:

a. Dynamic increments (the percent increase of dynamic deﬂection over the static
deﬂection) were small for speeds less than 40 km/hr.

b. Dynamic increments (DI’s) greater than 50% were relatively common. DI’s of
100% or more were observed. The largest DI recorded was 137% for an air
suspended vehicle excited by pavement repair at the prestressed concrete bridge.

c. DI for air-suspended vehicles was generally less than that for steel suspended
vehicles.

d. DI due to steel suspended vehicles tended to increase with the speed of the

vehicle.

Ellml'll '13

From the literature review, it follows that the dynamic response of highway

bridges to moving loads is a complex phenomenon, depending upon a number of

variables pertaining to both the load and the bridge. The main factors affecting the

dynamic response and their effects on the bridge behavior are given below:

78

3.4.1 11mm

The dynamic response decreases, in general, with increase in span length of the
bridge. Comparison with analytical results shows that the AASHTO - 1992 formula:

_ 50
L+125

 

--------- (24)

underestimates the dynamic effect of vehicle load for short span bridges whereas for other
spans [Fleming and Romualdi 1961; Huang et al. 1993], it is fairly accurate when the
vehicle is modeled as an unsprung load [Fleming and Romualdi 1961]. The above

formula overestimates the bridge response in the case of spans more than 80 feet [Huang

et a1. 1993].

3.4.2 Wick.

Dynamic response decreases with an increase in weight of the vehicle. For shorter
spans, the dynamic effect will increase more rapidly with the reduction in weight of the
vehicle [Huang et al. 1993]. An increase in mass of the load will result in an increase in
the velocity at which the maximum response will take place [Fleming and Romualdi
1961]. In the case of 12-30 m span simply supported steel girder bridges subjected to a
ﬁve-axle tractor trailer, the dynamic effect represented by the ratio of the dynamic to the
static deﬂection decreases with increase in truck weight. However the decrease is due to
the fact that the static deﬂection increases more than dynamic deﬂection, thereby

decreasing the ratio of the two [Hwang and Nowak 1991].

79

3-4-3 Sneednﬂhelehisls

The dynamic effect of the load does change with the velocity of the vehicle. The
response however, does not follow the same trend for vehicles with different gross

vehicle weights [Hwang and Nowak 1991].

3.4.4 NumhethoadedLanes
An increase in number of loading lanes will generally result in an increase in the
dynamic load factor in the case of short span bridges, but has little effect on the

maximum value of dynamic load factors [Huang et al. 1993].

3.4.5 WWW

In the case of multi-girder bridges and especially for short spans, the lateral static
and dynamic distributions are signiﬁcantly different. The dynamic load factor in this case

is inversely proportional to the static lateral distribution factor [Huang et al. 1993].

 

Uneven approaches cause the initial displacement and velocity in the vehicle
entering the bridge span. These two conditions can signiﬁcantly increase the dynamic
response of the bridge [I-Iumar and Kashif 1993]. According to the investigations carried
out at Massachusetts Institute of Technology (M.I.T) in 1953 (see section 3.2.8), the
vertical oscillation of the vehicle itself, caused by the bridge surface roughness, had the

main inﬂuence on the dynamic response of the bridge [Wright and green 1959]. The

80

presence of a settlement of area in the approach may lead to extremely large dynamic
effects in the bridges due to initial vibration of the load or vehicle. The value of the
dynamic load factor may be much higher than that caused by several other load and
structure parameters generally considered in the dynamic analysis of the bridges due to
moving loads [Fleming and Romualdi 1961]. The existence of initial conditions, that is,
initial vertical displacement and velocity of the vehicle before entering the bridge span
generally results in higher amplitudes of vibration when compared with zero initial
conditions. The effect is more signiﬁcant and pronounced in the case the load is acting at
or close to mid-span [Sundara Raja Iyengar and Jagadish 1970]. Initial displacement
signiﬁcantly increases the dynamic ampliﬁcation in the case of moving loads over that
without initial displacement [Shepherd and Aves 1973; Humar and Kashif 1993] .
Presence of a small irregularity in the shape of a small sized bump or depression in the
approach road introduces initial conditions and this may drastically increase the
amplitude of dynamic response. The difference between the values of the dynamic
response with and without initial conditions decreases, as the value of the speed
parameter increases [Humar and Kashif 1993]. It is believed [Tung et al. 1956] that the
initial vibration of the bridge would further increase the dynamic effect of the moving

load.

3.4.7 Numhemflehisleunjhsﬂridge

Generally the dynamic ampliﬁcation factors are lower for multiple presence of

vehicles on the bridge, than for a single vehicle. Dynamic ampliﬁcation due to two

81

parallel abreast (in phase) vehicles is signiﬁcantly greater than when one of them is
lagging behind the other (out of phase). For isotropic plate model, the ampliﬁcation
factors for the moment are generally less than those for the deﬂection [Humar and Kashif
1995]. The ratio of the dynamic to the static deﬂection for two side by side trucks is less

than that for a single truck [Hwang and Nowak 1991].

3.4.8 namn'urg

Field tests have shown that the vehicle damping ratio is fairly low e.g., for a 10-
ton dump truck [Biggs et al. 1959], the average value of damping is 2.2% of the critical
value. It has been noticed that although damping of vehicle does reduce the dynamic
response, but not by a large amount. The damping effect reduces the dynamic response
more signiﬁcantly in situations where the vehicle enters the bridge with initial conditions
particularly when coupled with small vehicle weights and low values of the speed

parameter [I-Iumar and Kashif 1993].

3-4-9 Suﬁnginuﬂhexehisle

There is not much difference in the dynamic response of the bridge for the sprung
and the unsprung moving loads, in absence of the initial conditions (i.e., no initial
displacement and velocity). However, when the initial conditions that result from the
irregularities in the approach road, are taken into account the dynamic effects are
considerably larger in the case of a sprung load, for the ranges of speed parameter and

the mass ratios considered [Tung et al. 1956]. Dynamic effects are higher for a single

I/‘NL

82

unsprung load than for a single sprung load regardless of consideration or neglecting the
effects of initial conditions [Fleming and Romualdi 1961]. According to the studies
carried out at the University of Illinois[1951-55] in the case of a sprung vehicle with zero
vertical displacement and velocity, the dynamic effect was found to be the same as for an
equivalent unsprung load [Wright and Green 1959]. From the nature of the dynamic
response for the vehicles of various gross weights as a function of the speed of the
vehicle, it follows that the characteristics of the suspension system depend upon the
weight and the speed of the vehicle [Hwang and Nowak 1991]. It has been reported
[Heywood 1995] that while investigating the effect of vehicle suspension, the dynamic
load factor values in excess of 50 percent were commonly observed. Values recorded
were more than 100 percent in some cases. The maximum recorded value was 137
percent for an air-suspension system vehicle that was excited by a pavement repair

[Heywodd 1995] .

\ J,

3.4.10 prenfﬁnan

It has been reported [Fleming and Romualdi 1961] that the change in the
maximum dynamic response due to change in mass might be signiﬁcantly larger for

continuous bridges than for single span bridges.

3.4.11 AxleSming

Axle spacing is an important factor in computing the static as well as the dynamic

response of a bridge subjected to moving loads. The dynamic effect represented by the

83

ratio of the dynamic to the static deﬂection behaves differently from span to span with the
axle spacing. For longer spans (24 and 30 meters in this case), it increases with the
increase in axle distance. Both the maximum mid-span static and the dynamic deﬂections
increase with increase in axle spacing. In the case of smallest span considered (that is, 12
m), static deﬂections are constant as the tractor dominates the response, and the dynamic
deﬂections are also nearly constant. [Hwang and Nowak 1991]. The dynamic deﬂection
response is signiﬁcantly affected by the axle spacing, particularly for higher values of the
speed parameter or (= V/2Lf ). Generally the response produced by a single-axle vehicle
is signiﬁcantly larger than a two-axle vehicle, when both have the same masses [Humar

and Kashif 1993].

3.4.12 Sneedﬁarameter

Speed parameter is deﬁned as the ratio of the vehicle speed to the half period of
ﬁrst mode of vibration of the bridge. The speed parameter generally ranges up to 0.30
[Humar and Kashif 1993 and others]. In the case of the points on the line of loading, the
moment ampliﬁcation is generally smaller with large values of speed parameter, as
compared to the deﬂection ampliﬁcation. For larger values of speed parameter, maximum
deﬂection ampliﬁcation for the deﬂection takes place after the load moves away from the
mid-span. The maximum deﬂection ampliﬁcation factors are generally greater than the
maximum moment ampliﬁcation factors [Sundara Raja Iyengar and Jagadish 1970]. The
dynamic response is generally higher for higher values of the speed parameter as well as

the mass ratio (i.e., the ratio of the vehicle mass to that of the bridge), although the

84

maximum values are at different values of the two variables The dynamic response in
general increases with increase in the value of the speed parameter [Humar and Kashif
1993]. The dynamic effect for the maximum moment is small in the case of an unsprung
load. Considerable variation has been observed for ampliﬁcation factors, as a function of
the speed parameter [Tung et al. 1956]. The magnitude of the maximum dynamic effects

were proportional to the speed of the vehicle [Wen and Veletsos 1962].

3.4.13 EzequensLRatic

Assuming frequency of the vehicle to be 2.75 Hz, the general range of frequency
ratio (that is, ratio of the vehicle frequency to that of the bridge) is S 1.4 whereas if the
vehicle frequency is taken as 5 Hz, the frequency ratio reaches to about _<_ 2.5. The
maximum value of dynamic ampliﬁcation does not occur at a frequency ratio of 1.0, as
generally believed, but takes place for frequency ratios from 0.4 to above 1.0 for speed
parameter values between 0.10 and 0.30. Contrary to the case of the speed parameter, the
variation of the dynamic response with the frequency ratio does not follow a general trend
[Humar and Kashif 1993]. In case of slab or beam-and-slab bridges modeled as
orthotropic plates, large dynamic effects were observed for frequency ratios between 0.6

and 1.0 [Sundara Raja Iyengar and Jagadish 1970].

3.4.14 MassBatin
In the case of a single unsprung load, there is a small increase in the dynamic

effect with variation of mass ratio from zero to 1.0 [Fleming and Romualdi 1961]. In case

85

of slab or beam-and-slab bridges modeled as orthotropic plates, the dynamic
ampliﬁcation is generally directly proportional to the mass ratio (i.e., the ratio of the
vehicle mass to that of the bridge), although the effect is of secondary nature and less
pronounced [Sundara Raja Iyengar and Jagadish 1970]. For the commonly used heavy

vehicles, the mass ratio generally ranges to about 3 1.4 [Humar and Kashif 1993].

*************************

WW

4J an [IE f ”If

The differential equation of the elastic curve of a prismatic beam with x-axis
coincident with the longitudinal axis, loaded in the plane of a principal axis through the
shear center, the x-y plane, thus resulting in bending about the other, or perpendicular,
principal axis, that is the z-axis, is given by the expression

2
131ng = — M(x) --------- (a)

 

where E1 is the stiffness of the beam,
y is the deﬂection of the beam, and
M(x) is the bending moment, as a function of the distance, x, along
the length of the beam, due to the load acting on the beam

Differentiating Eq(a) successively with respect to x, the distance along the beam, one gets

 

 

d’y dM

EI =-——=—V ......... b
dx, 1 (x) ( )
d‘y dv

E1 = .. __ = .........
dx. dx 900 (c)

where V(x) is the shear force, and

q(x) is the load intensity per unit length of beam

86

87

In case of a beam oscillating in free vibration, the only force acting on the beam is the

inertia force, given by — m a = - m

 

where a is the acceleration
m is the mass per unit length of the beam = pA = yA/g.
p is the density per unit length
7 is the speciﬁc weight
g is the acceleration due to gravity = 32.2 ft. per second2

Substituting this force into Eq(c), one obtains

 

 

+m =0 --------- (1)

This partial differential equation is the general beam equation for an elastic
prismatic undamped beam in free vibration. The solution of general beam equation (Eq 1)
can be obtained by applying the method of separation of variables. Consequently, the

solution is of the form

VOL 0 = (P(X)Y(t) --------- (2)

Substituting Eq(2) into Eq(l), one obtains

Eltpr+mtpY=0 --------- (3)

or,

(p'VY+-gl—I(pY=0 --------- (4)

88

 

 

4
where (()'v is the 4th derivative of <p(x) with respect to ‘x’ = dxq:
oo 2
and Y is the second derivative of Y(t) with respect to ‘t’ = 2
t
Dividing both sides of Eq(4) by Eq(2), one gets
1V "
1L. + 32.1 = o ......... (5)
(p EI Y
1V
__ = ..EX = a4 ......... (6)
(p El Y
Thus, since each term is independent,
«p‘v(x)=a‘q>(x) --------- (7)
and
31 Y(t) + a’ Y(t) = 0 --------- (8)
E I
Restating Eq(8),
Y(t) + (1) 2Y(t) = 0 --------- (9)
where
(t)2=a4E :> a"=(r)2—t1 --------- (10)
m E I

The general solution of Eq(7) is
(p(x)=Al cos ax+A2 sin ax+A3 cosh ax+A4 sinh ax
Determination of the constants of integration A, depends upon boundary conditions (be)

but, in all cases, results in a series of roots 1.“, n = 1, 2, ..., co such that Eq(7) may be

restated as

89
9..“’(X)= K. 9.. (x) --------- (11)

where it = a4 --------- (d)

n 11

For example, in case of a simply supported single span beam, the solution is in a
sinusoidal form with an = 2LT:

where L = the span length,

4
and (pn (x) = Dn sin( 23) . In this case, a4 =(E) = con‘ 31.
L L El
4 2
Thus 01 2 =P—I(n—7£) 2 0) =(11—TC) -E—I --------- (e)
" m L " L m

Also, the frequency for the nth mode of vibration is given by the expression

00.. n21: ’EI
f = :: — .........
" 2n 2L2 m (t)

From the frequency equation, it can be concluded that in the case of beams having

 

 

similar geometric properties, the frequency of transverse vibration decreases

proportionally with increase in length squared, 117. Or frequency is inversely

proportional to length squared i.e., f 0: LIT.

4.2 Wins

In order to establish orthogonality conditions, Eq(l 1) is restated for modes ‘i’ and ‘j’,
(Piw = in (P. --------- (12)

(lev =7HCP, """"" (l3)

90
Multiplying Eq(12) by (p). and Eq(13) by (p, and integrating over their length yields

L L
[9."’9,dx = i». [9.9,dx --------- (14)
0 0
L L
I‘PJ‘W‘Pi dxzkj'l‘pj‘pi dx """"" (15)
0 0

At this point, it is necessary to integrate Eqs(14) and (15) by parts.
L
For J‘tpiWQDj dx, let u=(pj & dv= q)!" dx
0

and du=tp§ dx & v=(p.'"

Thus,

 

L

"I L I" '
[9.”9, dx=9,—9 ., -[9. 9, dx --------- (16)
0

L
Further, for Itpﬁ'kp} dx, let u=(p; and
0

du=rp;'dx & v=tp:'

So ﬁnally,
L L | L
[09%, dx = <1,- mz'1, — 939:1, + [cpz'cpg'dx
0 0
L
=7». [9.9,- dx --------- (17)
0
Similarly, it follows from Eq(l 5) that,

L L L L
93"9. I0 -<p;'<1>: I0 + [91'93'dx = k,- [9. 9, dx --------- (18)
0 0

El E121" 1,.

These are constraints or compatibility conditions imposed by the supports or end

conditions at x=i; a usually is equal to 0 or L ( ends of span). Thus at supports, such

as for the simply supported condition, displacement y at x = 0 and x = L is zero at all

times t leading to the space variable (p at x = 0 and x = L becoming equal to zero.

E121" ll .

1. Simply supported Displacement and

2. Fixed - end

3 . F rec-end

4. Guided-end

the moment at

support are zero.
Displacement and
the slope at support
are zero.

Moment and the
shear at support

are zero.

Slope and the

shear at support

are zero.

um)
y(x = §, t) = 0
yﬂx=io=0
y(x = x, t) = 0
y'(x = R, t) = 0
y"(x = x, t) = 0
y”'(x = i, t) = 0
y'(x = i, t) = 0
y"'(x = 32, t) = 0

and
(p (32) = 0
9NQ=0
9(b=0
'00?) = o
9"(8) = 0
(pm(;) = 0
91;) = 0
<0"'(§) = 0

Review of the boundary conditions and comparison with the ﬁrst two terms in

Eqs(17) & (18) shows that these are always zero. i.e.,

L_O . I ML_O
lo— ’ (pinho-

92

9,9’."I:=0 ; cpgtpg'|:=0
L L
93"9r IO =0 ; (pg'cp; |0 =0

therefore, from Eqs(17) & (18)
L L L
[0%, dx = [<91'93’dx = 7». [9. 9,- dx
0 0 0
and
L L
[vi'wj'dx = 1,- Icp. 9,- dx
0 0
Subtracting Eq(19) from Eq(20)
L
(2.]. —}ti)J‘(pi (pj dx =0
0

It follows that if A, at A]. ,

L
[e,(x)nj(x)dx=0, iatj
0

L

[cpz'(x)<p;'<x)dx=o, is: j

0

and

L
Itp'vr(x)(pj(x)dx= 0. iej
0

--------- (19)

--------- (20)

--------- (21)

--------- (22)

......... (23)

--------- (24)

Equations (22), (23), & (24) are the orthogonality conditions for elastic beams.

When i = j, Eqs(21) leads to

E“) 3(x) dx = 0ti

--------- (25)

93
where ori is a constant. In addition Eq(l7), after substituting i = j in Eq(19) and

comparing with Eq(25), transforms to

L L
[9.”9. dx=j(9.")’ dx=71roni --------- (26)
0 0

These are the normalization functions.

Now, Eq(l) may be rewritten as

m'y+Ely'V=0 --------- (27)

and, to account for all modes of vibration, Eq(2) may be rewritten as
y(x,t) = Emma) --------- (28)
Consequently, substitution of Eq(28) into Eq(27), results in
i(mcpI Yi+ElcpiIv Yi)=0 ......... (29)
1:1

Whereas the multiplication of Eq(29) by (pl. and subsequent integration over the entire

span, yields

an L L
2(m Y,[¢,rpjdx+1~:1v,]rpjrpgv dx)=0 _________ (30)
0 0

i=1
With the application of the orthogonality conditions i.e., Eqs(22) & (24), and the

normality conditions, i.e., Eqs(25), and (26), Eq(30) transforms to
.. L L
vi (mjtpizdx) + vi (131eri tpg" dx) = 0 ---------(31)
0 0

Now, deﬁning the ‘principal mass’, as

94

L

M“ = mjtpnzdx --------- (32)

0

and using Eq(26), the ‘principal rigidity’, as
L
R. =Elj9.9LVdX=EU~.a. --------- (33)
0

By the substitution of Eqs(32) and (33) to Eq(3l), it becomes
M" if, (t) + R, Yn(t) = 0 --------- (34)
However, since 3." = a: ,

0)an

 

Rn =EI}.n an = EI a4 an = El( E1 )orn =wn2 metn --------- (35)
L
So, Rn =00n2mjtp: dx=01n2Mn --------- (36)
0
Thus, MnYn(t)+(02nMnYn(t)=0
that is, Y, (t)+rn3,i(n (t) = 0 --------- (37)

and the equation is uncoupled into the form of a single degree of freedom equation by
modes. This is the equation of a simple harmonic motion whose solution is of the form

Yn(t)=An c0300n t + Bn sinh)n t --------- (37a)

and y=y(x.t)=:<p. Y. --------- (37b)

11 - I
Given the initial conditions(ic), i.e.,

yo(x,t = 0) = f1(X) and 90(XJ = 0) = f2(X) --------- (38)

yo=Z<p.Yo.--f.<x) and yo=Z¢.Yo.=f.(x) --------- <39)

95

multiplying both sides of Eqs(39) stated in mode ‘i’ by (p). and integrating over the span

L, yields

and

Q L L
ZYo.[9.<p,-dx= [9,f.(x)dx
i=1 0 0

an L L
ZY01I¢i¢jdx = I‘P,‘ f2(x)dx
I=| 0 o

......... (40)

--------- (41)

When i¢j, because of orthogonality as given by Eq(22), both the Eqs(40) and (41) equal

zero; however, when i= j= n, Eqs(32) & (36) yield

and

or,

and

Thus,

M L
YOn 4 = J¢n f|(x)dx
m 0

 

Y0 n

L
M" = [4.90m
m 0

L

[‘1’ n f1 (X) dx

"0

m

YOnzhd

 

L

If? 11 f2 0‘) dx

“0

m

Y:
0n M

“° Y
y(x,t) = ZIP“ (x)(Y0n cos a)“ t+$ sin (0" t)
(o

......... (42)

-------- (43)

-------- (44)

......... (45)

--------- (46)

96
4.3 W

In this case, Eq(l) becomes
0 ‘y(x,t) 6‘y(x,t) _
EI—(3x4 + m———atz — p(x,t)

or

E1 yIV + my = p(x,t) -------- (47)
Using

y(x.0=2<p.(x>Y.(t) -------- (48)

i=1
Eq(47) leads to
ZEItp”.(x)Y,(t) + Zrnrp,(x)i"{i (t) = p(x,t) ' --------- (49)

i-l i-I
Multiplying each term by (p J.(x) and integrating over the span,

L

so an L L
m2 Molt». «p,- dx+EIZ we [9.” 0,- dx = [9, p(x.t>dx --------- (50)

121 i=1
From the orthogonality conditions, Eq(22) & (24), all terms are equal to zero; therefore

wheni=j=n,

1?. (m [(6.0012 dx + Y. (BI [61”(x)¢.(x)dx= [9.00 p(x.t>dx -------- (51)

L
Letting m J'((i>,,(x))2 dx = Mn , and
0

El [01.“(x)<p.(x)dx = m, [(9.00)‘ p(x,t)dx

97

=m w: [(9.(X))‘dx

=00‘M

1'1 n

L
and deﬁning [ (p , (x) p(x, t) dx = P, ---------(52)
0

Eq(51) transforms to the typical equation of motion for the forced vibration case in

uncoupled coordinates

MnYn(t)+(02nM,Y,(t)=P, --------- (53)

where Mn is the principal mass, lb-secz/ft

4.4

1)

2)

Generatllmedune
In general, the procedure for ﬁnding the response of an elastic beam is as follows:
Apply boundary conditions (be) to the solution of Eq(7), i.e.,

(p(x)=A,sinax+A,cosax+A3sinh ax+A4coshax --------- (54)

where

2
m . . .
a: = 7. n = 1:21 , are the system ergenvalues, and the correspondrng functrons

 

(p, (x) , r1 = l, 2, ..., 00, are the corresponding system modes or mode shapes.
Note that the simply supported beam is an exception as in most of the cases, such
as for a cantilever beam, the frequency equation is transcendental and the shape
function, (p(x), fairly complicated.

Calculate ‘the principal mass’, M, , from Eq(32).

98
3) Calculate ‘the generalized load’, P, , from Eq(52). This provides values for the

uncoupled equation of motion, Eq(53).
4) Solve Eq(53) using the Method of Undetermined Coefficients, to obtain the
particular solution.
5) Apply initial conditions (ic) to solve for the constants of integration. This yields
the solution for Y, (t).

6) Substitute the functions (p, (x) , found in step-l, and Y, (t), found in step-5, into

Eq(28) to obtain the complete solution.

4.5 Minimizing
For including the effect of damping, it may be assumed to be in modes so that the

expression
D, Y, (t) = 2 g, 00, M, Y, (t) --------- (55)

may be added to Eq(53) to yield

M, Y, (t)+D, Y, (0+6): M, Y, (t) = P, (t) ---------(56)
or, M, Y, (t)+2 g, 0), M, Y, (n+6): M, Y,(t)= P, (t)
or, Y. (0+2 12.0). Y. (0+9: mm? --------- (57)

This is the typical under-critically damped uncoupled equation of motion with

solution

Y, (t)=e'§"“""(A,sin c0,Dt+B, cos (amt) --------- (58)

99

where 00 ,D = 60 , ,/1—E, i , and E," is the fraction of critical damping in mode ‘n’.

In order to compute the bending moment, M, and the shear force, V, in an elastic

beam, following relationships may be used:

 

2
Moment: M(x,t) = —EIa—¥£:Ltl
a x
3
and Shear: V(x, t) = M = —El 6 y(x,t)
a x 6 x
B . [I | I' l E |
The following is a brief review of integration by parts:
Integration by parts is of the form
[udv = uv — Ivdu
Example:
Ixexdx
Let u = x and dv = e‘dx, then
du = dx and v = e" , substituting into Eq.(A),
[xe‘dx = xe" — Ic"dx =xe" —e" + C
4.6 MW

 

For a simply-supported (SS) beam, (p, (x) = sin DEX

and, with zero damping

--------- (59)

--------- (60)

--------- (A)

--------- (Ela)

100

y(x,t)=Z<p,(x)(A, cos (0, t+B, sin 00, t)

or, y(x, t) = 2 sin n E X

 

(A, cosw, t+B, sine), t) --------- (Elb)

To meet the orthogonality condition of Eq(32), i.e.,

 

 

 

L
M. = m [(6.60) de --------- (Bic)
0
L nrtx
that is M, = mjsin2 dx
0 L
L L L L
Integrating, [sin‘ 31’: dx= 35— sin2 npx =—
L 2 4np L 2
0 0
Therefore, M , = m_L ; 3L: 3 -------- (Eld)
2 M, L
With initial conditions Y0, and Y0, ,
°° . nrtx Yo, .
y(x, t) = Zsm (Y0, cos 0), t+— sm (0, t) -------- (Ele)
n -I 0) n

where the values for Y0, and Y0, are obtained from Eqs(44) & (45).
For a situation with an initial velocity, v, over the span except at the supports and
zero displacement; from Eq(3 8)
y0 =fl (x)=0, and y, =f2 (x)= v.
Therefore from Eq(44), 11,, =0 , and from Eq(45)

L

m 1¢.(x)f2(x)dx

Y =
0n Mao

 

101

2vL nrrx

 

Y0, = — sin dx
L 0
4v
= _, n= 1,3,5, ...,(odd only) -------- (E11)
r111

Substituting into Eq(46)

w

y(x,t)=i‘—I Z sinnnx(0+ 1 sinc),t)
n=l(oddonly) n0)"

 

 

4v °° l

 

. IITCX
S111

 

sin 00, t
n = |(oddon|y) n (on

4v 1 . RX. 1 . nx.
= — —-— srn—srnwlt +— srn—srnw3t + -------- (Elg)
1t 0), L (03
4.7 .2111‘ 1'1' 11'.“ 12-..!‘1 611% :111211 121

For a simply supported beam with ic=0, ﬁnd the response of a beam subjected to a

suddenly applied mid-span load p(t).

 

p(t) = 0 t < 0
= po t2 0
L
From Eq(52), that is, Itp, (x) p(x,t)dx = P, --------- (E2a)
0
L
P, = [sin r11: x p(x,t) dx=po sing; --------- (E2b)

0

Accordingly, from Eq(53),

Y, + 03,2Y, = P0 --------- (E2c)

where P, = _2_p_0_ sin 11t— --------- (E2d)
m L 2
Now, Y, (t) = Y,H (t) + Y,C (t) --------- (E2e)

where Y,H (t) is the homogeneous solution ( free vibration case ), and

Y,C (t) is the complimentary solution (forced vibration case ).

with Y,H (t) = A, cos 0), t + B, sin c), t --------- (E2f)
and Y,C (t)=C, + C2 t --------- (E2g)
such that

Y, (t) = A, cos 0), t + B, sin 0), t +Y,C (t)

Thus, substituting the complimentary or particular equation,
Y,C (t)=C, + C2 t
Ync (t) = C2

into the differential equation , E2c,

one obtains (02.. (C, + C2 t) = P0 --------- (E2b)

Setting constants equal, Eq(E2h) yields 0) 2 C, =

 

which in turn yields C, =

and setting coefﬁcients of ‘t’ equal yields C2 = 0

I)0
(.02

Thus, Y,C (t) =C, =

 

103
. P,
Therefore, Y, (t) = A, cos (0, t + B, srn (0, t +
00

 

, --------- (E2i)

Y,(t)=B,00,cosw, t-A,(0,sin 00, t

Applying the initial conditions i.e., Y, (t = 0)=0 , and Y, (t = 0)= 0 , we have

A,(1)+B,(0)+P°,=o :, A __ P0

 

0" 2
00, 0)

 

and, B,0),(l)—A,00,(0)=0 => B,=0

Substituting values of the coefﬁcients A, and B, , in Eq(E2i), one gets

 

 

P P
Y"(t)=—0)02 cosc),t + 0

11

2
n

P
= (00‘ (1— cos 0),t)

1'1

 

 

2p . nrt
Y t = 0 l—cosco t sm—

Finally. y(x,t)= i9. (X)Yn (0

no

2
= ——pl-—,-(1-cosm,Using-118mn 11: x
,_,mL00, 2 L

 

(D

2p0 1 . 111: . nnx
x,t = srn— l—cosw t srn
y< ) mg, .2 2( .1

 

 

 

01',

104

 

 

 

 

 

 

2
= p, [ lzsin—E(l—coso,t) sin—H
mL 0), 2 L
. . 21tx
+ 2srnrt (l—cosmzt) sm
2
. 3n, . 31tx
+ 2srn—i—(—cosod3t) srn

3

1 41tX ]
+...
0),, L

+

 

 

2 sin21r(l— cos 0)4 t) sin

713x

 

 

 

 

2 (1— cos 013t) sin3

2P6 1 . nx 1
x,t = l—cosm t sm——
Y( ) mL [w,( r) L

1 £03

 

 

 

l .51tx l .71tx
+m,(1-cosw5t)sm L — 2(l-cosc07t) sm L —...]

 

 

5 (07
2p0 °° l . krt . krrx
y(x,t) = [—srn—(l— cos 0) t) srn :I --------- (61)
mL k-l(§only) (I): 2 k L

This is response of an elastic beam, subjected to a quickly applied constant

. . . . 1
stationary force; a rapidly convergrng serles because of the term —2 .

4.8

 

Find the response of a simply supported beam subjected to the load p(x, t) = p0 sin (2 t at

x=awithic=0.

 

L
P, = (sing?p(x,t)dx = p0 sin Qt sin Mt a --------- (E3a)
0
Therefore, Y, + (0,2, Y, = P, sin Qt --------- (E3b)
_ 2
Where, P, = p0 sin 111: a --------- (E3c)

 

 

mL

105
Now, Yn(t)= YnH (t) + Yncm --------- (E3d)

with, Y,H (t) = A, sin 0) ,t + B, cos 00 nt --------- (E3c)

and for complimentary solution, assume that

Y,C(t)=C sin Qt --------- (E3f)
with Y,C(t)=CQ cos or --------- (E3g)
and Y,C(t)= - (:02 sin Qt --------- (E3h)

Substituting into the differential equation,
— CQ2 sin Qt + 03,2, C sin Qt=P, sith
Coefﬁcients of ‘ sin I ’ yield (00,2, - Q2 )C = P0

P1

 

 

Thus, C=C, =0), .92 """"" (B31)
and therefore,

Y,(t) =A, sin m,t + B, cos (0, t + C, sin Qt
and Y,(t)=A,00,cosm,t-B,0),sinco,t+C,Qcoth
Using zero initial conditions i.e.,

Y,(t=0)=0 :> B,(1)=0 B, =0
and Y,(t=0)=0 :5 A,co, +C,Q=O A, =—C"Q
0)"
F . .

so, Y, (t)=—2—'—(sm Qt - gsm 0), t) --------- (E3k)

(0 —Q2 00

Substituting (0, (x) and Y, (t) into Eq(28) yields

106

y(x,t) = in. (x) Y. (t)

n-l

 

 

 

= 22p, (sinnn a) {—2—1—7) (sin Qt — 2sin 00, t)sinnn x
L L 01,—Q a) L

 

 

 

2p0 ‘° 1 . Q . . nrra. nrrx
srn Qt——srn0) t srn srn
legi (mi-02X 0), " ) L L l

--------- (62)

Concentrating on Eq(62), it follows that if the frequency, f,,=:,:l , of the
it

 

. . . . 0)
dlsturblng force 1S equal to or even close to any of the natural frequencres, f, = 2 " , the
1:

corresponding term gets very large, leading to the phenomenon of resonance. Therefore,
to avoid resonance, the frequency of lowest mode of vibration i.e., when n = 1, must be
several times larger than the frequency of the force acting on the structure.

In case of a simply supported beam, a design engineer is generally interested in

the mid-span deﬂection. To check the effect on deﬂection, consider the case of a

2
disturbing force acting at the center of the span i.e., when a = L / 2. Letting 97: 1112 in

Eq(62),one obtains

107

 

y(x,t) = 2:0 [(1-1 2J—l—i-sin%(sith — w, sine), t)

 

 

 

W1 (”1
-( 1 2) Lzsin 31: x(sith — 013 sinm3t) ........ (63)
+(_1_2_) izsin 5n x(sith - 1115 since5 t)—...]
1““15 015

When the values of 111 are small i.e., when Q is very small as compared to 0) ,
only the ﬁrst term of the above equation approximates the steady state response with

good accuracy, leading to the conclusion that the ratio of the dynamic deﬂection, y ,y, ,

to the static deﬂection, y, , is

 

1 2 . For example, if the period of the disturbing force
— W

is half of that of the natural period of the beam i.e., Q: 0.50) , then dynamic deﬂection is

about 33 percent greater than the static deﬂection.

4.9 WM
Find the response of a simply supported beam subjected to load P(x, t) = P0 that moves
across the beam at a constant velocity v such that its position is given at any time t as x =

a=vt,0<t<L/v,ic=0
L

From Eq.(52), that is P, = [o,(x) p(x,t)dx --------- (E4a)
0

nrrx
L

 

For simply supported beam, cp, (x) = sin

108

 

 

 

L
P,=[sinnn xp(x,t) dx=p0 sinnn vt --------- (E4b)
0
leads to, Y, + 0) ,ZY, = 3&— sin n n v t -------- (E4c)
mL
Now, Y, (t) = Y,, (t) + Y,c (t) --------- (E4d)

where Y,, (t) is the homogeneous solution (free vibration case), and
Y,c (t) is the complimentary solution (forced vibration case ).
' with Y,c (t) = A, cos 0), t + B, sin 0), t --------- (E4c)

and for particular solution assume

 

 

 

 

 

. nrr v
Y,p (t) = C srn t --------- (E41)
therefore,
. 111: v 111: v
Y,p (t) = Ccos t --------- (E4g)
2 2 2
.. n 1: v , n1t v
and an (I) = - T- C srn t --------- (E4h)
substitute into the differential equation (E4c)
nzpzv2 ,npv 2 .npv 2p .npv
-——T— C ln—t+w, Csrn—t= —° srn—t
L L L mL
Coefﬁcients of sin 111: vt :
C ( 0) 2 __rﬁ_1t:_v_2_)= m

n L2

109

2po/mL L2

Therefore C = = 2 p0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

--------- E4'
" 2 112 p2 v2 mL wi-n‘p2 v2 ( I)
_ L2
. . nit v .
so that Y, (t)=A,cosw,t+B, srnw,t+C, srn t --------- (E4j)
and Y,(t)=—A,0),sinco,t+B,0),cosw,t+C,(n:v)cosn:vt
--------- (E4k)
so that at timet=0,
Y,(t=0)=A,.1+B,.0+C,.0=0 :> A,=0
and Y,(t=0)=—A,w,.0+B,w,.l+C,(n:v).l=0
C
or B, =——"[““ V) --------- (E41)
0), L
Y, (t)= C, [sinmt vt — _1_(n1r v) sine), t] --------- (E4m)
L (0, L
Substituting (E4i) and (E4m) in Eq.(28)
y(x,t) = 29.. (x)Y. (0
n=1
2poL°° l ( nnv [nnv], ],nrtx
x, t = s1n t — srnrt),t s1n
y‘ ) m Elimiﬁ -n2n2 v2) L M
--------- (E4n)
. .. VT,
Usrng definltlon or, = 2L , --------- (E4p)

where v is the velocity of the moving load,

110

and T, = 21 is the period of the beam in the mode n.
60

. T‘V . 2 2
Accordingly, a,=m ylelds n2v2=a, 0), L2, 1rv=or,co,L

l‘l

 

 

 

 

 

 

 

 

 

 

 

 

 

2 L 2 L
Thus, C, = Po/m = Po/m = 22130 2 2
2 nzrtzv2 2 nzorf1 (0,2,L2 mLc),(l-—n a,)
wn‘—2—_ “’n‘ 2
L L
--------- (E4q)
and B =_C_,_(n1rv) =_C,(na,m,L)
" 60, L 0), L
2
=—na,C,-— 2’0““; , --------- (E4r)
mL (0,(1— n (1,)
Therefore,
Y, (t)= C,(sinnﬂ vt ) +B,sin0),t
. V .
= C, (srn t] -n0t,C,51n0),t
= 2p, (sinna (0 t—na sinm t) --------- (E45)
me:(l—n2a,2, " " " "
and
2p0 ”(sinna,00,t —n0r,5inc0 t) . nnx
x,t = —— " 51 ——
y‘ ) mL§ o§(l—n2o§) L
maybewrittenas:
y(x t)- 2p° :( 1 )5inP—H(sinna (0 t-nor sinco t)
’ mL ,,,00§(1—n20r,‘,) L " " " "

lll

 

 

 

 

 

--------- (E4t)

2p0 . nx . .
x,t= srn—srnacot— 0t srnwt
Y( ) leme-af) L( 11 1 1)
l . 21tx
+ 2 2 srn
(020—402)
nv it v
where a,= , az— ,.
01,L (02L

At mid-span, i.e., x = {/2 ,

 

 

(sin2 azwzt —2 a2 sin mzt) +...:l

Po L3
- = --------- E4u
yam 48El ( )
Therefore, the dynamic load factor (DLF) is
DLF(t)=-ydi
Si
96 °° . . .
=——P% Z ( 2 12 2 )srnﬂ(srnna,w,t—n0r,srn00,t)
mL ,_,(,,,,d, 0),(1-n a,) 2
--------- (E4v)
nn 2 El
Also, using 00, =(—) —— --------- (E4w)
L m
or _ 1: v _ vL _________ (E4x)

 

 

"— 0),, L_n2 1t,/EI/m

Substituting the expression for or, and rewriting , Eq(E4v) modiﬁes to

112

 

 

 

 

mL ,8, 00:»(nrtv/L)2 L ,

y(x,t)=ﬂ[i ( l )(sinnn Vt- guisinmﬂ) sinntx] (64)

Concentrating on the time t, required for the force po to cross the beam span, one
may get a number of solutions for the beam deﬂection at any point along the beam,
depending upon the velocity v of the moving force. An important factor in this
connection is the ratio of this time t, to the periods ‘T, ‘ of modes of free vibration. For
example, if the time required by the force p0 to cross the entire length of the beam is

equal to half the period T, of the ﬁrst mode of free vibration, that is,

T

1!

L
t,=——
V

(Di

. . . L
then in thls case, usrng — = —— ,
v 0),

00,2 =1: —— --------- (E31)

Consequently, the ﬁrst term of the series solution, Eq(64), becomes inﬁnity, and using a
special technique [Timoshenko et al. (1974)], can be represented by the relation

pot nrrx

Y(X,t) z—
m‘n V

 

cos—“EL sin --------- (65)

This expression will yield the maximum value for the deﬂection y at the instance

the time t = L / v, i.e., when the force will reach the far end of the beam, given by

 

 

--------- (66)

Ymax = 2

Now, using the relation

 

 

 

2 = = 1r — ------- E3m
(0‘ n m L’ L2 ( )
and substituting
E I
v2 = n2 --------- E3n
m L2 ( )
in Eq(66), one obtains
p0 L3 . n x
— srn — --------- 67
ym... 3 E, L ( )

From the above equation, the maximum deﬂection of the beam under the action of

moving force, p0 , is found to be about 1.5 times the static deﬂection, ys, ,

_ P0 L3

_ --------- 68
y.. 48131 ( )

caused by the same force while acting statically at mid-span of the beam. Considering the
practical aspect i.e., the bridge response to the moving vehicles, the actual time ‘t,’ taken

by a vehicle to cross the bridge span is generally large as compared to the period of the

2 2
ﬁrst natural mode of vibration, making the ratio a2 =

 

2 2 small. Now considering
0), L

only the forced vibration response , and supposing that in the most unlikely case the
forced and free vibration responses may come into phase, giving rise to the phenomenon

of resonance, the maximum deﬂection is given by the expression

114

 

 

 

 

 

2p0 pv 1
= 1+
y'm mL( 0.), L)(m,2—(pv/L)2)

2 L3 1

= p, +0r2 _________ (69)
p El l—or

_2p0 L3 l
p‘EI l—a)

This equation leads to the conclusion that the maximum deﬂection of a beam,

. . 1 . . .
acted upon by a force movrng at a slow speed, 15 about 1—— tlmes the maxrmum static
-0r

deﬂection produced by the same force. It may be noted here that or is:
a. directly proportional to velocity v of the load, and

b. inversely proportional to the span L as well as the ﬁrst fundamental frequency

0), of the beam. Thus the term I—L might be the origin of the DLF.
—or

4.10 W

It is an established fact that the load effect caused by a vehicle crossing a bridge
with certain speeds is larger than when it is stationary. This is due to the dynamic effect,
caused by virtue of vehicle motion. Consequently, the deformation of the bridge under
the action of a moving vehicle, generally termed dynamic response, is a complex
phenomenon because of a number of factors capable of affecting the vehicle-bridge
interaction and consequently, the dynamic response. The dynamic characteristics of a

highway bridge is a function of the mass, stiffness, damping, and load parameters of the

115

system, including both the vehicle and the bridge. In a practical sense assuming a
constant speed:

a. the stiffness is a function of the bridge only,

b. the mass and the damping are ﬁlnctions of both the vehicle and the bridge, and

c. the loading ﬁmction depends on the vehicle velocity and the pavement surface

condition of the approaches as well as the bridge.

The major causes responsible for the dynamic load effects, in the case of highway
bridges, are given in section 1.3.

In the simplest caseI of the analysis of the dynamic effect caused by the live load
effect of a moving load, a design engineer may be interested in following two important
situations:

I) When the mass of the moving load or vehicle is large compared to the mass of

the bridge structure itself.

11) When the mass of the moving load, compared to mass of the structure, is

small.

In the former situation, which is generally true in case of short-span bridges, the
mass of the structure, that is, the beam can be neglected in the analysis. Accordingly, the
deﬂection of the beam, caused by the moving load, is proportional to the pressure , P(t),
caused by the unsprung mass of the moving load rnu [Timoshenko 1965] on the beam

shown in Figure 4.1, shown below.

 

' The discussion in the following paragraphs has been taken mainly from Timoshenko (1965).

116

 

 

 

   

1’ x
F >lm"

FIG. 4.1: Load as a moving mass

 

The deﬂection in this case is given by the Eq(2) of the section 3.2.1. that is the

relation

2 2 2 2
v d y x (L—x)
= 1____ ......... 70
y m“gl gdle 3LEI ( )

 

It is an expression for the beam deﬂection due to a mass moving with constant
velocity v across the beam. For a load moving along center line of the beam, an

approximate solution of Eq(70) ( First established by R. Willis2 in 1849 as reported by

Mu gx2 (L—x)2
3LEI

 

[Timoshenko 1965]) can be obtained by replacing y with on the right

side of that equation. In this case, y will be maximum when the load is at the mid-span.

Whereas, the maximum pressure is given by the expression

M L
= M 1+v‘ _L_] ......... 71

In this case, the maximum mid-span deﬂection , y,“ , is given by a relation

similar to Eq(7l), which is

 

2 The original reference not available. The information has been taken mainly from Timoshenko (1965).

117

- ( 1+ 1: M“ L]
Ymax yst g 3E1

 

--------- (72)

From the comparison of this approximate solution with the results of an exact
solution of Eq(70), obtained by G. G. Stokes, it is concluded that the approximate
solution is sufficiently accurate for practical situations [Timoshenko 1965].

In the latter case, that is, when the mass of the load is small compared to the mass
of the bridge, which is generally true in the case of large span bridges, the moving load
can be represented by a moving force without loss of signiﬁcant precision, and the
results obtained in the previous discussion for the case of a moving force can be applied
in this case as well.

It is pointed out here that in the foregoing discussion, the results obtained are
those for a massless force or moving mass. In practical situations, the real moving loads
are actually rolling masses, causing change in the natural frequency of the bridge with
every change in position of the load moving over the bridge. This variation in the natural
frequency of the vehicle-bridge system is advantageous from structural point of view, as
in this situation, the oscillating moving load and the bridge will not remain in phase
during all of the time taken by the load in crossing over the bridge, thereby reducing the
phenomenon of resonance (it was shown in section 4.9 that resonance can occur in the
case of a non-oscillating load as well, as for a given beam, it is basically a function of
velocity of the load). Accordingly, both the dynamic load effect and the corresponding
magniﬁcation of static deﬂection of the bridge, will be less than that predicted by the

aforementioned theory.

118

In addition to above, certain other factors like unevenness caused by deterioration
of the riding surface, low and high spots on the approach surface, and imperfect wheel
balancing of the passing vehicles may also cause dynamic effects that may be quite high,
particularly in the case of short span bridges. It therefore can be concluded that these
additional dynamic effects are responsible for emphasis in the bridge design codes on the
use of comparatively high dynamic load factors for the design of short-span bridges.

Theoretical solutions of the various problems to obtain the response of beams,
representing highway bridges, under the different kinds of moving loads considered
above readily lead to the conclusion that beam or bridge response is generally dominated
by the ﬁrst ftmdamental mode of vibration, provided the fundamental frequencies are
well separated. Accordingly in such cases, the bridge response under the action of a
moving load can be reasonably approximated by modeling a bridge as a single-degree-of-
freedom (SDOF) system, and taking into account the ﬁrst mode of vibration only. This
will lead to the simpliﬁcation of the real problem, which then can be represented by an
ordinary differential equation along with simultaneous consideration of masses of the

bridge as well as the moving load.

*******************#*****

S l D. .
5.1 W

Since the recognition of the existence of dynamic effects of moving loads on the
railway bridges in the middle of 19th century, and the start of major research work on the
problem of dynamic effects of moving loads on highway bridges in early 19305, a
number of the analytical as well as experimental studies have been carried out to identify
the main factors or variables affecting the bridge response. Most of these studies were
concentrated on relating the bridge response directly to the basic properties of the vehicle
and the bridge, such as span and natural frequency of the bridge, and the speed and the
weight of the vehicle. Initially the dynamic response was generally expressed as a
function of the bridge span only [Waddell 1918; Final Report 1931]. Some of these
investigations addressed the problem by using a slightly different approach. Instead of
using the basic properties of the vehicle and the bridge, such as the natural frequencies
and masses of both the vehicle and the bridge and the speed of the vehicle separately, the
ratios of these factors were used in the investigations. The dimensionless ratios of the

corresponding vehicle and bridge properties consisted of :
a the 5 ed arameter = l
' pe p 2Lf

where V is the speed of vehicle,
L is the span length, and

f is the natural frequency of the bridge.

119

120
b. the frequency ratio i.e., the ratio of the vehicle frequency to that of the bridge

0. the mass or the weight ratio i.e., the ratio of the mass or weight of the vehicle to

that of the bridge.
Several studies have been carried out using these dimensions parameters [University of
Illinois 1951-1955; Wen and Veletsos 1962; Tan and Shore 1968a, b; Humar and Kashif

1993, 1995; Yang et al. 1995].

5. 2 W
In the dynamic response studies described in chapter 3, the moving load has been
essentially modeled in one or more of the following ways:
1. a single force moving at constant speed
2. a single unsprung mass moving at constant speed
3. a multi-axle unsprlmg load
4. a single sprung load
5. a combination of sprung and unsprung loads
6. a multi-axle sprung load
Whereas the bridge has been idealized as:
1. an elastic beam,
2. a plate (isotropic or orthotropic) structure, or

3. a combination of beam and plate structures.

121
5.3 W113

The ranges of the dynamic load factors observed during various studies are given

in the following table [Huang et al. 1993]:

Table 5.1: Comparison of Observed DLF Values

 

 

 

 

 

 

 

 

 

 

 

Agency/Country Study Reference Range of DLF Remarks
1. AASHTO, USA Fenves et al. 1962 0.03 - 0.25 From strain readings
(1958-60) 0.02 - 0.42 From deﬂections
2. TRRL, England Leonard et al. 1974 0.09 - 0.75 From 30 bridges
3. New Zealand Shepherd and Aves 0.10 - 0.70 Mostly for a stand-
1973; Wood and ard 2-axle truck, for
Shepherd 1979 14 bridges
4. Ontario, Canada Green 1977 0.10 - 0.87 Continuous bridges
0.07 - 0.75 Other bridges
5. EMPA, Cantieni 1984 Maximum: 0.70 Normal Conditions
Switzerland Maximum: 2.50 Over a 50 mm thick
Plank 300x5000mm
6.(a)Australia O’Connor and Prit- -.03 - 1.32 For Six Mile Creek
chard 1985; Chan bridge
and O’Connor 1989
(b) - do - Heywood 1995 -.10 - 1.37 Off the approach
pavement repair
7. Thailand Ibanathan and Wie- 0.26 - 0.36 Moving force case
land 1987 0.28 - 0.33 Moving mass case
8. Florida, USA Huang et al. 1993 -.01 - 0.41 Multiple Criteria
9. South Korea Chang and Lee 1994 0.00 - 0.66 Surface roughness

 

 

 

 

 

122
5.4 C ll. LE”! ,. ,. I: I S .ﬁ ,.

When speaking of the progress made in determining the behavior of bridges due
to static loads and the sad state of affairs in understanding dynamic effects, C. E. Inglis
(1934) stated:

“This stage of arrested development is largely due to the blighting inﬂuence of
impact factors, which prescribe that the dynamical effect of a moving load shall be taken
into account by multiplying it by a factor depending upon the span of the bridge, and
usually on nothing else. The simplicity of this process, which relieves bridge designers
from intellectual effort, has ensured its popularity in the past, but the imposition of these
purely empirical factors, which gain their simplicity only at the cost of ignoring all the
considerations of real importance, has most effectively discouraged the enterprise bridge
engineers might otherwise have shown in developing their science to deal intelligently
with dynamical stresses.”

Before discussing the correlation of various code speciﬁcations, given in chapter 2
and the appendix, with results of theoretical and test studies described in chapters 3 and 4,
it should be made clear that the purpose of design codes is not to predict the response of
a structure but to ensure the structural safety. This is generally done by specifying
guidelines to limit the response (deﬂections, moments and stresses) of a given structure
under the various loading conditions within prescribed acceptable limits e.g. by
specifying:

a. load factors to increase the actual loads by certain amount to account for

the possible overloading of the structure while put to actual use,

123

b. resistance factors to decrease the calculated strength or resistance of materials
of construction because of possible variation in properties of actual materials to
be used in construction, and
c. span-to-depth ratios for various structural components to limit their deﬂections.
The purpose of this discussion or correlation is to see with reference to
theoretical and test investigations discussed earlier in this study, the extent to which the
codes under consideration are ensuring structural safety with respect to the dynamic
effect of trafﬁc loading in highway bridges. The discussion is as under:
1. Although under normal conditions the dynamic effect represented by the DLF
should not exceed 25% [Schilling 1982], it is generally the unusual circumstances such as
the existence of an area of settlement in the approach road [Fleming and Romualdi 1961]
or the presence of a bump at a critical location that result in much higher dynamic effects
[Schilling 1982] which may be, in some cases as high as 1.37 [Heywood 1995].
Moreover, several investigations [Fleming and Romualdi 1961; Ibanathan and Wieland
1987; Galdos et al. 1993; Huang et al. 1993; Yang et al. 1995] have shown that the
AASHTO (1992) formula to account for the dynamic effect of trafﬁc loads
underestimates the dynamic effect in certain cases (e.g., in the case of short span bridges)

and overestimates it in others.

124

54.1 W
The formula used by the American Association of State Highway and Transport
Ofﬁcials (AASHTO) design speciﬁcations for highway bridges 1992, to account for the

dynamic effect of moving loads is:

_ 50
L+125

 

where

I = impact fraction (maximum 30 percent);

L = length in feet of the portion of the span that is loaded to produce the
maximum stress in the member.

This formula was originally adopted in the “Speciﬁcations for Steel Highway
Bridges”, issued by the joint Conference Committee of American Association of State
Highway and Officials (AASHO) and American Railway Engineering Association
(AREA) in 1927. The formula was included in the ﬁrst edition of AASHO standard
speciﬁcations published in 1931 [ASCE 1958]. The formula remained an integral part of
AASHTO highway speciﬁcations up to and including the 15th edition published in 1992
[AASHTO 1992].

The formula is generally based on the results of dynamic load tests carried out by
A. H. Fuller during 1922-1925 at the Iowa Engineering Experiment Station on railroad
bridges. The general level and the upper limit of the impact value depicts the inﬂuence of
A. H. Fuller’s work [Fuller 1922, 1925; ASCE 1958].

A special committee of ASCE working on the dynamic effect in highway bridges,
proposed a similar dynamic load formula in its ﬁnal report published in ASCE
Transactions in 1931 [Final Report 1931]. The Committee, in the case of girders of
highway bridges recommended that:

a. for spans less than 40 ft, the impact increment i.e., the dynamic effect of the

stress be taken as 25% of the live load stress, and

125
b. for spans equal to or than 40 ft, the fraction of the dynamic effect of the

moving load be calculated from the formula:

_ 50
L+160

 

where
I = ‘impact’ fraction
L = span length in feet
Although under normal conditions the dynamic effect represented by the DLF,

should not exceed 25% [Schilling 1982], it is generally the unusual circumstance such as
the existence of an area of settlement in the approach road [Fleming and Romualdi 1961]
or the presence of a bump at a critical location that result in much higher dynamic effects
[Schilling 1982] which may be, in some cases as high as 1.37 [Heywood 1995] in terms
of the DLF. Moreover, several investigations [Fleming and Romualdi 1961; Ibanathan
and Wieland 1987; Galdos et a1. 1993; Huang et al. 1993; Yang et al. 1995] have shown
that the AASHTO (1992) formula to account for the dynamic effect of traffic loads
underestimates the dynamic effect in certain cases (e.g., in the case of short span bridges)
and overestimates it in others.

In the light of above discussion, it follows that the dynamic effect provisions of
the AASHTO (1992) bridge design code are inadequate, and unable to account for the
dynamic effects of real trafﬁc loads on highway bridges, thus not always adequately

ensuring structural safety properly.

126
5.4.2 W

The dynamic load allowance (DLA) provided in AASHTO LRF D code 1994 is
generally based on data and recommendations of Page'(l976). The code commentary
mentions following two sources of dynamic effects caused by the trafﬁc loads:

1. The dynamic response due to hammering effect of wheel loads, mainly caused

by surface discontinuities, crack potholes and delarninations

2. The response caused by the vehicle by virtue of its motion, caused by either

long undulations in the road surface like settlement of ﬁll, or by the resonance
effects due to similarity in the bridge and vehicle frequencies of vibration.

Although as indicated in the code commentary, under normal conditions the
dynamic effect represented by the DLF should not exceed 25% , it is generally the
unusual circumstance such as the existence of a bump at a critical location that result in
much higher dynamic effects [Schilling 1982]. Both of the above sources contribute to
the possibility of unusually higher dynamic response of a bridge to moving vehicles.
However, it is thought that the ﬁrst one is the major contributor and more responsible for
the provision of a dynamic load allowance of 75%, the likelihood of frequency match
between the vehicle and the bridge being low in the majority of practical cases.
Comparing the provision of a maximum value of 75% for the DLA by the code to the
results of analytical and ﬁeld investigations, it is observed that where several researchers
[Fenves et al. 1962; Shepherd and Aves 1973; Leonard et al. 1974; Cantieni 1984;

Ibanathan and Wieland 1987; Huang et al. 1993; Chang and Lee 1994] have recorded the

 

' The original reference is not available. The information presented has been taken mainly from the
Commentary of AASI-ITO LRFD (1994).

127
maximum values of DLFs less than or equal to 0.75, higher values up to 1.37 have also

been reported [Green 1977; O’Connor and Pritchard 1985; Chan and O’Connor 1989;

Heywood 1995]. The general value of 33% DLA speciﬁed by the code, like that provided
for unusual conditions (75%), also seems justiﬁed and reasonable in view of remarks
made in the code commentary regarding the use of a combination of design truck and
lane loading versus the design truck only [Schilling 1982]. Thus, there is a good

correlation between ﬁeld results and the provisions of AASHTO LRFD 1994 for the

dynamic effect of vehicle loads on highway bridges.

 

OHBDC 1991 speciﬁes the dynamic load allowance with respect to the number of
axles of the vehicle, the maximum value being 0.40 for a single axle load and the
minimum 0.25 for vehicles with three or more axles. For a 2-axle vehicle, the dynamic
load allowance speciﬁed is 0.30.

Keeping in view the remarks given in the case of AASHTO LRF D 1994 above,
and the fact that the dynamic effect of a two-axle vehicle is generally less than that of an
equivalent single axle vehicle [Humar and Kashif 1993], the dynamic load allowance
provisions seem reasonable for normal conditions [Schilling 1982]1(of loading and
smooth road surface). An unusual condition like the settlement of an area in the approach
road or the presence of a bump at a critical location may result in very high dynamic
effects [Schilling 1982], which may be, in some cases as high as 1.37 [Heywood 1995]

in terms of DLF. However, the code commentary has some discussion on that. OHBDC

128
1991 DLA provisions are based in part on test results [Billing 1984], and a comparative

study made of the average weight of the vehicles used in the above study (300-400 kN)
and the OHBD design truck (740 kN). It was found that for the OHBD truck, using “a
more-or-less inverse linear relationship” between the vehicle weight and DLF, the
maximum value of DLA for the whole truck was less than 0.2, no matter how close the
natural frequencies of the vehicle and the bridge are to the resonance sensitive range
[OHBDC 1991]. It has been clariﬁed in the Commentary [OHBDC 1991] that in
specifying the dynamic allowance values the approach road and the bridge deck have
been assumed to be paved to acceptable standards. Accordingly a conservative and
minimum value of 0.25 has been speciﬁed for vehicles with 3 or more axles. In the case:

a. the approach road has not been paved for a considerable period of time or

b. the expansion joints between the approach road and the bridge super structure

are not generally ﬂush with the riding surface,

it has been recommended that that the DLA be increased from 0.4 to 0.5 within three
meters or one-tenth of the span length, whichever is greater, from the location of the joint.
This distance has been stated to correspond to one half cycle of the axle spring for a
vehicle traveling at approximately 100 km/h (65 mph). Moreover, the design engineer has
been cautioned to be careful in using design vehicles lighter than the OHBD truck, as the
DLA values may not be conservative in the case of lighter vehicles.

In the case of normal load and smooth road surface, provisions of OHBDC 1991
seem to be quite reasonable for heavier design loads comparable to OHBD Truck (gross

weight 740 kN a 166.4 kips, the AASHTO HSZO-44 design truck weighs 72 kips) in the

129
case of normal loading and smooth riding surface conditions [Schilling 1982]. Although

in the case of unusual conditions, OHBDC provisions for the DLA apparently seem
inadequate on the basis of the DLF investigation results mentioned in sections 5.4.1 and
5.4.2 above, for design vehicles with gross weights comparable to that of OHBD truck
these may not be inappropriate. Comparing the corresponding weights of OHBD truck
and the AASHTO HS20-44 truck, and considering the fact that the DLF multiplied by the
static live load gives the load caused by the dynamic effect, it follows that in the case of a
3-axle vehicle such as HS20-44, the dynamic effect indicated by OHBDC 1991 is about
(166.4 / 72 * 0.25 =) 0.81 within 3 m of the expansion joint, and 0.58 for the remaining
span. Whereas in the case of 2-axle vehicle, the corresponding values are 0.92 and 0.69
respectively. Values for vehicles to be treated as single-axle vehicles for the design
purpose, the DLF values are still higher. Thus for the vehicles with gross weights close or
higher than that of OHBD truck, provisions of OHBDC 1991 seem to have good
correlation with ﬁeld investigations. However, for lighter vehicles their adequacy might

be questionable.

5.4.4 WWW

To account for the dynamic effects caused by moving loads, the formula used by
CPHB 1967 is of the same type as that used by AASHTO code 1992. The formula used
by CPHB is:

_15
L+20

 

130

where I is the impact fraction (maximum 30 percent), and

L is the span length in feet.

From the great resemblance in the general layout and the fact that CPHB 1967 was
prepared by an American consultant ﬁrm, it follows that it is essentially based on the
same principles as that of AASHTO code in use in 19605.

Although under normal conditions, the upper limit of 30% speciﬁed by the code
seems generally acceptable [Schilling 1982], it is of concern that as was shown in the
case of the AASHTO ‘impact formula’ used in 1992 and the earlier editions [Fleming and
Romualdi 1961; Ibanathan and Wieland 1987; Huang et al. 1993; Yang et al. 1995], the
CPHB formula would most probably underestimate the dynamic response in certain
normal cases and especially for the unusual conditions discussed in the case of AASHTO
1992 code. In the case of Pakistan, these unusual conditions are much more likely to exist
due to:

a. general deterioration of highways mostly as a result of general economic

conditions prevailing in the country,

b. modest truck load limitations, and

c. lack of proper and timely repair and maintenance of highways and bridges.

Thus, from above discussion, it appears that CPHB 1967 of Pakistan does not
have a desirable correlation with the results of ﬁeld investigations performed in various

parts of the world to determine the dynamic effects of trafﬁc loads on highway bridges.

131

5.5 W
55.1 W

The following is a comparison of the values of dynamic effect of trafﬁc loading
obtained using the highway bridge design codes referred to in this report as well as
theoretical values using a moving mass model for the vehicle load.

The dynamic effect of trafﬁc loading is generally represented in two different
ways. Theoretically, as in chapter 4, it is expressed as a dynamic load factor (DLF) given

as

Ydyn _ (1+ DI)YSI
ySt Yst

DLF =

 

=1+DI

where y ,y, is the maximum dynamic deﬂection, and
y,, is the maximum static deﬂection

DI is the dynamic load effect or the increase in the static deﬂection due to
motion of the load , expressed as a percentage of the static deﬂection.
The dynamic load effect of the moving load, in this case, is equal to the simple product of
the static live load and the DLF obtained using above relation. On the other hand, the
bridge design codes generally give values for dynamic effect, represented by ‘DI’ in
above expression, only. Thus, in the case of code-speciﬁed values of the dynamic effect,
one (1) is added to the value of the dynamic effect such as impact [AASHTO 1992;
CPHB 1967] or dynamic load allowance [OHBDC 1991; AASHTO LRFD 1994]
obtained from the code directly, before multiplying with the speciﬁed static live load to

determine the total effect of the moving load. In the following, only the dynamic effect

132

due to moving load has been calculated to compare the values obtained from various
sources. Thus one (1) has to be added to the indicated values before multiplying with the
static load of relevant design vehicle to obtain the total load effect due to moving vehicle.

The values have been calculated using the design2 data of a single span simply

supported composite reinforced concrete-deck-steel girder bridge with following design

data:

5.5.1.1 mmdﬂmiienies

o Span (center-to-center of supports) = 70 ft.

0 Width = 28 ft.

0 Live loading AASHTO HS20-44
. Wearing surface 30 lb./ft2.

0 Concrete strength, f,’ 3000 lb./in2.

0 A36 Steel beam (W36x150) weight = 150 1b./ﬁ.

0 Grade 40 steel reinforcement

. Modulus of Elasticity of Steel = 29x103 kip/inz.

o Modulus of Elasticity of Concrete

0 Modular ratio, 11

 

= 4.176x106 kip/ftZ.
= 3.122x106 kip/inz.
= 4.496x105 kip/ftz.

=9

2 The design data has been taken from Example 13.1 and the composite bridge design example, given in
section 19.7 of A. H. Nilson and G. Winter (1986). “Design of Concrete Structures” 10ed, McGraw-Hill

Book Company, New York.

133

 

 

R.C.C Dock Slab

 

 

W36X1 50

 

 

. s i

Bridge Span (center-to—center of supports) = 70 feet

 

FIG. 5.1: W

 

 

134

5.5.1.2 W

In calculating the properties of the composite section, the concrete deck slab area
has been transformed into equivalent steel area using the modular ratio. Accordingly, the
material properties used in the calculation are essentially those of the steel, rather than of
concrete. The properties of the composite section are given below:

0 Self weight of the girder:

 

Slab weight (78x6.5/l44x0.15) = 0.528 kip/ft.
Wearing surface (78/12x0.03) = 0.195 kip/ﬂ.
mm = 0.15 kip/ft.
Total weight = 0.873 kip/f1.
. Moment of Inertia (15016 in") = 0.72415 ft“.

0 Mass of the girder per unit length, m

[0.873 (kip/ft.) / 32.2 (ft./sec2.)] = 0.027112 kip-secz/ftz.

5.5.2 Wham

5.5.2.1 W
In AASHTO -1992 code, the basic load combination Group IA requires the
ultimate capacity of the structure component to be not less than
1.3 [D + 2.20 (L + 1)]

where D is the dead load,

135

L is the live load, and
I is the dynamic effect of live load.

The value of I is given by the expression

_ 50
L+125

 

Using the data of above design example, that is L = 70 ft, I is found to be 0.256.
Thus, the dynamic load effect for the current example is 0.256 times the load of the

AASHTO HS20—44 design truck weighing 72 kips.

5.5.2.2 W

The AASHTO LRFD - 1994 code, in the case of “Strength-I” load combination
speciﬁes the factored design load to be not less than

1.25 DC + 1.75 (LL + 1M) +...

where DC is the dead load of the component to be designed,

LL is the live load ( design truck, tandem or lane load), and

[M is the dynamic load allowance (not applicable to the lane load).

Accordingly, using speciﬁcations given in the Appendix for AASHTO LRFD
1994, the dynamic effect of live load is found to be 0.75 for the deck joints, and 0.33 for
other components in the case of Strength Limit State. The governing live load for the

present example, is again H820-44 design truck as before.

136

5.5.2.3 W

In the OHBDC - 1991 code, the design load in the case of Ultimate Limit State

Combination-1 requires the design load to be not less than
1.2 D + 1.4 L
where D is the dead load of the structure, and

L is the live load inclusive of the dynamic load allowance (DLA).

Accordingly, using speciﬁcations given in the Appendix for OHBDC 1991, the
dynamic effect of live load is represented by DLA . The design live load in this case is
the OHBD truck weighing 740 kN (about 166.4 kips). For comparison with AASHTO
codes, it is observed that the OHBD truck is about 2.31 times heavier than HS20-44
truck. Since the dynamic load effect is a percentage of the live load, it follows that total
dynamic effect speciﬁed in the OHBDC - 1991 is actually 2.31xDLA when compared
with AASHTO HS20-44 truck. In the case of vehicles required by the OHBDC to be
considered as single axle ones, the total dynamic effect is then found to be 2.31x0.5 =
1.155 for the portions up to 3 m or 10% of the span whichever is greater; from the
location of the expansion joint and 2.31x0.4 = 0.924 for the others. The corresponding
values are 0.924 and 0.693 for two-axle vehicles; and 0.809 and 0.578 for vehicles with

three or more axles.

137

5.5.2.4 W
CPHB - 1967 speciﬁes in the case of simple span prestressed concrete bridge
structures of moderate lengths, the ultimate load capacity to be not less than
1.5 D+2.5 (L+I)
where D is the dead load,

L is the design live load, and

15

I is the dynamic effect of live load, given by I:—
L + 20

in which
= impact fraction (maximum 30 percent)

L = length of span in feet

The design truck-train load has a gross weight of 122 kips which is 1.694 times
the weight of HS20-44 truck. For the present example, I = 15/(70+20) = 0.167. Thus the
total dynamic load effect in this case turns out to be 1.694x0.167 = 0.283.

It is added that in case of the moving force model [Eq(E4v) of section 4.9], the
values of the dynamic load effect are the same for all codes because in this case the mass
of the vehicle is neglected and the calculation of the dynamic effect is dependent only on
the properties of the bridge which are constant. Moreover, the dynamic effect values
obtained from the ‘moving force’ and the ‘moving mass’ models cannot be compared

because both models have different background. The moving force model neglects the

138

effect of vehicle mass, and the mass model, ignores the effect of the bridge weight on the

dynamic response of the bridge.

A comparison of dynamic load effect obtained using the four highway bridge

design codes discussed in this report, and theoretical mass model are given in Table 5.2:

Table 5.2: Comparison of Dynamic Effect Using Design Example

 

 

 

 

 

 

Equivalent’ Code Moving Mass
Codes Speciﬁed Values Model4 Remarks

AASHTO LRFD 0.75 0.238 Deck joints
1994 (USA). 0.33 Other components
AASHTO 1992 0.256 0.238
(USA)
OHBDC 1991 1.155 (0.924)3 0.549 Single-axle vehicle
(Ontario, Canada) 0.924 (0.693) Double-axle vehicle

0.809 (0.578) Multi-axle vehicles
CPHB 1967 0.283 0.403
(Pakistan)

 

 

 

 

 

3 These values include the effect of the gross weight of the design vehicle load used by different codes as a

ratio of the gross weight of the AASHTO HS20-44 design vehicle (72 kips).

’ These values have been calculated from Eq(74) of section 4.10.
5 The ﬁgures in the parentheses give dynamic effect values for the bridge portion beyond 3 meters from the
expansion joint (between the approach road and the bridge); and those outside the parentheses, the values

up to 3 meters from the expansion joint.

 

139
In the second column of Table 5.2 marked ‘Equivalent Code Speciﬁed Values’,

the values of the dynamic load effect have been scaled on the basis of AASHTO HSZO-44
design truck. This simply means that the dynamic load effect obtained directly from a
code has been multiplied by the ratio of gross weight of the design truck of that particular
code to the gross weight of AASHTO HSZO-44 design truck (72 kips). In other words, if
a bridge is designed, using a vehicle similar in conﬁguration, gross weight, and axle load
distribution to the AASHTO HS20-44 design truck and calculate the dynamic load effect
speciﬁed by each of above mentioned design codes, then the actual dynamic effect
accounted for in the design would look like that given in the second column of Table 5.2.
Thus, in these terms, OHBDC - 1991 gives the highest dynamic load effects (Multi-axle
vehicles; 0.809 within 3 meters of the expansion joint, 0.578 for the remaining portion),
followed by AASHTO LRFD - 1994, CPHB-1967 and AASHTO-l992 in descending
order.

In the case of ‘Moving Mass’ model, the mass of the vehicle is accounted for
while calculating the dynamic load effect of moving vehicles. Accordingly, in this case,
the dynamic load effect is directly proportional to the gross vehicle weight. Thus, it is
seen that the dynamic load effect is the greatest in the case of OHBDC-1991, which has
the heaviest design vehicle among the above codes, and the smallest in the case of

AASHTO, having the lightest design vehicle of them.

140

5.5.3 WWII/1011211

From the Table 5.3, it is observed that :

1. In general, the contribution of third and higher modes of vibration to overall value
of DLF is very small. For example at t = 0.6 sec, DLF = 38.69 for the ﬁrst mode, 0.23 for
the third mode (only 0.59% of that for the ﬁrst mode), and 0.05 for the 5th mode (only
0.12% of that for the ﬁrst mode). Thus, it can be concluded that while calculating the
response of a bridge subjected to a moving load, using moving force model as above, the
effect of higher modes of vibration is negligible and the dynamic response can predicted
with fairly accurately using ﬁrst mode of vibration only.

2. A close look at Eq(E4v) (section 4.9), which is the basis of the Table 5.3, reveals
that in the case of ﬁrst mode of vibration the term in the denominator, does not become
zero for the given parameters of the bridge- vehicle system, and therefore the
phenomenon of resonance does not take place. A side calculation shows that at vehicle
speed of about 75.44 ft/sec (51.43 miles per hour), the denominator term of the Eq(E4v)
becomes zero for the system parameters considered in above calculations, thus seemingly
causing an unstable condition or resonance. However, a closer inspection would reveal
that this apparently resonance-type phenomenon could occur only if the system is
undamped and that high amplitudes approach inﬁnity only with inﬁnite time, obviously
not possible since the beam is of ﬁnite length deﬁning the duration of the exciting

phenomenon.

141
3. A further examination of Eq(E4v), Table 5.3, and the graph for the ﬁrst mode

only, indicates that the DLF values are increasing with each cycle, however this increase
is still ﬁnite, and of course neglecting the effect of damping. Otherwise, DLF values will
practically be less than indicated above, because of certain amount of damping and time
required to reach the amplitude shown which has to be present in the real structures.

Table 5.3 is on the following page.

 

142

1,, 53,, ., 3., 0' E I
LNilsnnandJMnteL-Jm

 

Ser
Tel
No.

Time
(899)

Dynamic Load Factor (DLF) Considering

 

Mode 1

Mode 3
only

Mode 5 only

Modos1 8:
3

Modes1,3
315

 

wwwNNNNNNNNNN-I-A-t—b-s-I—t-t—s-L
ggggg‘o’igggnaorocoxlmotawro-sococn‘rounAmmao‘om‘o’mbwb"

 

0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
1.95
2.00

only
0
0.048682
0.3837934
1 .26371 51
2.8929640
5.4004934
8.82391 33
13.1 00881
1 8.068437
23.470466
28.972885
34.185573
38.689576
42.067686
43.936250
43.975922
41 .959127
37.772171
31 .430299
23.084464
13.019098
1 .6408255
-10.541 35
-22.94330
-34.938650
45.895489
65.214333
-62. 364985
-66.91 9881
68.581692
67.203342
62.799076
65.545786
45.774435
-33.952099
-20.655632
6.538751 9
7.7055282
21 .384655
33.846001
44.513554

 

0
6.2897588
6.3880283
0.76993809
0.08072533
6.7344771
0.19004214
0.23708615
0.23074736
6.2708628
6.5750710
0.72887394
0.22840561
-0.6241 1 51
0.00896188
0.1 1708325
0.43360824
6.1866278
6.7243580
0.62010027
0.33674797

6.450725
6.1302589
6.0561050
0. 58458202
6.0528300
-0.81 39086
0.461 59146
0.38582549
6.2347953
6.2090817
6.2603936
0.66689132
0.1074821 1
6.8296888
0.27791068
0.364221 14
6.0025791
6.2179893
6.4688507
0.67329996

 

 

0
6.07343278
0.03120893
6.025471 32
-0.10490460
0.00665270
-0.02526652
-0. 1 045531 1
0.0141 8826
0.00684677
6.07447843
0.04345093
0.05285684
-0.03833978
0.0671 7071
0.08396744
6.02352261
0.061 79512
0.0821 8368
6.04051 391
0.02532023
0.05377137
6.07501657
-0.02109868
0.02445598
-0.09874059
-0.04897148
0.01997821
-0.09074483
6.04381830
0.04664906
605325778
-0.01 596383
0.08654823
-0.01 006314
0.00734560
0.1 109821 1
0.01 1 33512
0.0025643
0.10206793
0.00033948

 

0
6.2410722
6.0042349
2.03365326
2.97368937
4.66601630
9.01 395548
1 3. 3379677
1 8.2991 852
23.1996040
28.3978142
34.9144474
38.91 79824
41 .4435717
43.9452120
44.0930057
42. 3927360
37.5855432
30.7059412
23.7045642
13.3558463
1 .19010032
-10.6716‘18
~22.999407
64.354068
45.948319
66.028242
61 .903393
66.534056
68.816487
67.412424
63.059470
64.878895
45.666957
64.781788
-20.377721
6.1745308
7.70294906
21 .1666657
33.3771 51 1
45.1868545

 

0
6.3145050
0.02697398
2.0081 8194
2.86878477
4.67266899
8.98868896
1 3.2334146
1 8.31 33734
23.2064508
28.3233357
34.9578983
38.9708393
41 .4052320
44.0123827
44.1 769731
42.3692134
37.6473388
30.7881248
23.6640503
1 3.381 1665
1 .24387169
-10.746634
-23.020505
64.329612
46.047060
66.077213
61 .883415
66.624800
68. 860306
67.365775
63.1 12728
64.894858
45.580409
64.791851
-20.370376
6.0635486
7.71428418

21 . 169230
33.4792190
45.1871940

 

 

I43

 

 

~20

 

O

'—

0

FIG. 5.2: Time Va Dynamic Load Factor (DLF)

 

o ' . 7 s ‘
0 s
o o
O. ‘ o
O ‘ ’
\‘ ,
o
' a
I
0‘ ' v
Q 0
O
a \ '
’l o 0
O
' a '. o
. . . ‘ .
t:—i~»-- +-—~'¢——4 6 —¢ - #4 ~+i rF-~+—+—¢——+ o —4~ L -+——+———+—+—+—+—+—+—+——+——+—+—+—'—+—t—+———

8.833838.

0 O O O O O O

gs'géssssas’sa

v- v- v- T v- N

 

 

 

Tima (Sac)

 

 

 

 

 

 

 

FIG. 6.3: Time Va Dynamic Load Factor (DLF)

 

 

 

  

. .-O.“'Ove’““0.0-0.“...2p656 0-0.9.01

°. 8

   

docv-

  

 

 

Tina (Sac)

 

 

—O—Modo1ordy
---O--~Moda30niy
—oo—om5m
—x—m1a3
—*-Modaa1.335

 

 

 

 

 

W
6.1 ﬁndings
1. A review of analytical, model and ﬁeld test studies performed to determine

dynamic response of bridges subjected to trafﬁc loads has been made. The investigations
review goes back to 1849 when the ﬁrst study to determine the response of railway

bridges was performed.

2. Theoretical basis of derivation of dynamic deﬂection due to the action of simple
load models such as a constant moving force, a suddenly applied mid-span load, and a
time variant force have been derived in the form of examples to illustrate the theoretical

concepts to determine the dynamic effects of moving loads.

3. AASHTO 1992, and CPHB 1967 highway bridge design codes, in general, do not
have good agreement with the results of ﬁeld investigations conducted to determine the
dynamic response of highway bridges subjected to trafﬁc loads in the case of unusual
loading and approach road conditions. Thus, the corresponding provisions of both codes
may be insufficient in some circumstances. The maximum values for the dynamic load
factors speciﬁed by these two codes are correct only to the extent that loading conditions

are normal and the approach as well as the riding surface are fairly smooth.

144

145

4. The provisions of OHBDC 1991 seem to be reasonable. This code uses heavier
design truck loads (Their design truck has a gross weight of 740 kN a 166.4 kips as
compared to the AASHTO HSZO-44 design truck weighing 72 kips) and a unique design
philosophy. For lighter design vehicles and vehicles with 3- or more axles the indicated

DLF may be less conservative.

5. AASHTO LRFD 1994 is based on a probabilistic approach as far as load and
resistance factors are concerned. The dynamic load allowance provided in this code
seems generally justiﬁed in view of most of the results of ﬁeld test studies carried out to

date. (The data primarily belongs to the United States.)

6. It has been remarked [Humar and Kashif 1993] that two bridges with the same
spans or fundamental natural frequencies may have different values for the speed
parameter, the frequency ratio and the mass ratio. Their dynamic responses will
accordingly be different too. To provide for this situation, design curves have been
proposed [Humar and Kashif 1993] to compute a more rational dynamic “ lbad factor
(DLF). The procedure is systematic if not very simple. From a preliminary design of the
bridge, the weight and the frequency of the bridge can be determined.»Using the weight of
the design vehicle, mass ratio can be calculated. The DLF generally increases with the
speed parameter, so only maximum value of the latter needs to be considered. The

response however does not vary systematically with the frequency ratio. In this case, the

146

response can be calculated for several possible values of frequency ratios and the one

leading to the maximum DLF be used in the design [Humar and Kashif 1993].

7. Simple formulas have been proposed for calculating dynamic load factors, based
on analytical research, in terms of the speed parameter only [Yang et al. 1995]. These

formulas may be helpful in predicting the response of various type of bridges.

8. From the review of investigations made in chapter 3, and results of theoretical
derivations made in chapter 5, and test studies given in chapter 5, it follows that the
determination of dynamic response of highway bridges subjected to moving loads is a
very complex problem. Use of high strength materials (both conventional and newly
developed ones) , and the adoption of ultimate strength design criteria permitting bridge
designers to use more slender sections in the design of bridges may lead to the bridge
structure being more susceptible to vibrations. The situation becomes more aggravated
keeping in view the trend and possible use of heavier and faster vehicles on the highways.
This leads to the real need of development of a rational and more reliable procedure to
determine the dynamic response due to trafﬁc loads. Use of dynamic load factors can
certainly make the designing process simple but does not help the designer understand the
real effect of traffic loads, which is dynamic rather than static as it is generally treated in

design.

147

6.2 Recommendations

Keeping in view the knowledge gained through the techniques used by various

investigators in the study of the dynamic response of the highway bridges subjected to

moving loads and other engineering knowledge, the following options are suggested for

the design of a reasonable and reliable bridge:

a.

b.

Detailed dynamic analysis for vibration sensitive systems

A “quasi-static” approach more simpliﬁed than ‘a’ but rational based on
partial dynamic analysis results and recommendations of theoretical,
experimental, or ﬁeld investigations carried out by various researchers
[Humar and Kashif 1993; Yang et al. 1995] for most bridges that are
neither dynamically sensitive nor insensitive.

Use of the static method typiﬁed by AASHTO LRFD 1994 or OHBDC

1991 for bridges which may be dynamically insensitive.

Further work is required to deﬁne the parameters representing dynamic

sensitivity. The above would be enhanced or inﬂuenced by:

d.

Probabilistic Approach : A detailed analysis could possibly be carried out
to determine the probabilities of simultaneous occurrence of various
factors responsible for the ampliﬁcation of dynamic effects in highway
bridges caused by the trafﬁc loading. Based on these probabilities and the

maximum effects caused by each individual factor, a suitable value would

 

148
then be ﬁxed for the dynamic load factor for use in the design of highway

bridges.

In view of the complex nature of the dynamic response, and highly variable and
non-deterministic nature of trafﬁc loading to which highway bridges are subjected,
probabilistic approach, as now would be in the OHBDC 1991 or the AASHTO LRFD
1994 codes, is recommended to be more widely adopted in the years to come for the
reliable and cost-effective design of highway bridges. Although with modern high-speed
digital computers and quality software even a very detailed design might be reasonable,
nevertheless the design process should not be overly cumbersome, at least in the case of
dynamically insensitive structures. Accordingly, it is suggested that the code provisions
should be simple, and concise for the bridges of most common types of span lengths, for
example, simply supported short and medium span bridges. For other type of structures, a
comparatively detailed dynamic analysis procedure is recommended. Until, they are
improved upon, the AASHTO LRFD 1994 or OHBDC 1991 may be adopted for general
design. Use may also be made of the dynamic response curves developed [Humar and
Kashif 1993] or the formulas proposed [Yang et al. 1995] by researchers for comparison.

###############################

 

APPENDIX

ClS'ﬁl' D .11!"
A.1 HSHIQIBEDB'I D' S '[i I' 422'

A.1.1 American Association of State Highway and Transport Ofﬁcials (AASHTO)
LRFD Bridge Design Speciﬁcations - 1994 deﬁnes the dynamic load effect of the moving
loads as ‘Dynamic Load Allowance”, denoted by IM. The relevant code provision i.e., the
Article 3.6.2.1 states:

“Unless otherwise permitted in Articles 3.6.2.2 ( Buried Components ) and 3.6.2.3
(Wood Components), the static effects of the design truck or tandem shall be increased by
the percentage speciﬁed in Table l for dynamic load allowance.”

“The factor to be applied to the static load shall be taken as: (l + IM/lOO).”

“The dynamic load allowance shall not be applied to pedestrian loads or to the

design lane loa .”

 

 

 

Table 3.6.2.1-l - Dynamic Load Allowance, IM
Component IM
Deck joints - All Limit States 75%
All Other Components

 

 

* Fatigue and Fracture Limit State 15%

 

 

 

 

 

* All Other Limit States 33%

 

“Dynamic load allowance need not be applied to:
- retaining walls not subject to vertical reactions from the superstructure, and

- foundation components which are entirely below ground level.”

149

150

“The dynamic load allowance may be reduced for components, other than joints,
if justiﬁed by suﬂicient evidence, in accordance with the provisions of Article 4.7.2.1.”
A.1.2 According to the code Commentary, some of the above provisions were based on
the report of Page (1976). The corresponding Commentary article C3.6.2.1 states:

“The dynamic load allowance (IM) in table 1 is an increment to be applied to the
static wheel load to account for wheel load impact from moving vehicles.”

“Dynamic effects due to moving vehicles may be attributed to two sources:

- hammering effect is the dynamic response of the wheel assembly to riding
surface discontinuities, such as deck joints, cracks potholes and
delaminations, and

- dynamic response of the bridge as a whole to passing vehicles which may
be due to long undulations in the roadway pavement, such as those caused
by settlement of ﬁll, or to resonant excitation as a result of similar
frequencies of vibration between bridge and vehicle.”

“Field tests indicate that in the majority of highway bridges, the dynamic
component of the response does not exceed 25% of the static response to vehicles. This is
the basic for dynamic load allowance with the exception of deck joints. However, the
speciﬁed live load combination of the design truck and lane load, represents a group of
exclusion vehicles which are at least 4/3 of those caused by the design truck alone on
short-and medium-span bridges. The speciﬁed value of 33% in Table 1 is the product of

4/3 and the basic 25%.”

151

“This article recognizes the damping effect of soil when in contact with some
buried structural components, such as footings. To qualify for relief from impact, the
entire component must be buried. For the purpose of this article, a retaining type

component is considered to be buried to the top of the ﬁll”

A.2 HSHIQH'I 3.! D . 5 'ﬁ |° 4222
The provisions of the AASHTO Highway Bridge Design Speciﬁcations - 1992,

for impact (dynamic load) allowance are as follows:
“3.8 IMPACT
3.8.1 Application

Highway Live Loads shall be increased for those structural elements in group A,
below, to allow for dynamic, vibratory and impact effects. Impact allowances shall not be
applied to items in group B. It is intended that impact be included as part of the loads
transferred from the superstructure to substructure, but shall not be included in

loads transferred to footings nor to those parts of piles or columns that are below ground.”

“3.8.1.1 Group A - Impact shall be included.

(1) Superstructure, including legs of rigid frames.

(2) Piers, (with or without bearings regardless of type) excluding footings and those
portions below the groundline.

(3) The portions above the groundline of concrete or steel piles that support the
superstructure.”

152

“3.8.2 Impact Formula

3.8.2.1 The amount of the impact allowance or increment is expressed as a
fraction of the live load stress, and shall be determined by the formula:

_ SO
L+125

 

(3 - 1 )
in which

I = impact fraction (maximum 30 percent);

L = length in feet of the portion of the span that is loaded to produce the

maximum stress in the member.”

“3.8.2.2 For uniformity of application, in this formula, the loaded length, L, shall be as
follows:

(a) For roadway ﬂoors: the design span length.

(b) For transverse members, such as ﬂoor beams: the span length of member
center to center of supports.

(c) For computing truck load moments: the span length, or for cantilever arms the
length from the moment center to the furthermost axle.

(d) For shear due to truck loads: the length of the loaded portion of span from the
point under consideration to the far reaction; except, for cantilever arms, use a 30 percent
impact factor.

(e) For continuous spans: the length of span under consideration for positive

moment, and the average of two adjacent loaded spans for negative moment.”

“ 3.8.2.3 For culverts with cover
0’ to 1’ - 0” inc. 1= 30%

153

1’ - 1” to 2’ -0” inc.l=20%
2’ - 1” to 2 - 11” inc. I = 10%”

A.3 [ll .11.] 3.. D . C 1 [01111111211221

A.3.1 Ontario Highway Bridge Design Code-1991 (OHBDC 1991) allows for the

impact of moving vehicles in the following way:

“24.3.2 Dynamic Load Allowance

24.3.2.1 Highway live loads shall be increased by the dynamic load allowance
unless otherwise speciﬁed.

24.3.2.2 The values speciﬁed for the dynamic load allowance shall apply unless
alternative values based on tests or dynamic analysis are approved.

24.3.2.3 The dynamic load allowance applied to a Truck shall be as given in Table

24.3.2.3 for the number of axles considered in the design lane.”

 

Table 24.3.2.3 Dynamic load allowance

 

 

 

Number of axles DLA

1 0.40

2 0.30

3 or more 0.25
“24.3.2.4 The dynamic load allowance for the uniformly-distributed portion
of the Lane Load shall 0.10.
24.3.2.5 For soil-steel structures, the dynamic load allowance shall be 0.40,

reduced according to Clause 24.3.2.6

l54

24.3.2.6 When there is a depth of earth cover, DE , between the riding
surface and the top of the structure, the dynamic load allowance shall be multiplied by the

reduction factor:
( 1 — 05 Db. )
Aﬁer this reduction, the dynamic load allowance shall not be less than 0.10. For

arch and soil-steel structures, the ﬁll depth shall be measured at the crown.”

“24.3.2.7 For wood components, the dynamic load allowance resulting from Clauses
24.3.2.3, 24.3.2.4, and 24.3.2.6 shall be multiplied by 0.70.

24.3.2.8 The dynamic load allowance shall be included as part of loads transferred
from the superstructure to the substructure, but shall not be included in loads transferred
to footings that are surrounded with earth nor to those parts of piles that are below
ground.”

A.3.2 The relevant sections of the OHBDC code Commentary state:

“C24.3.2.1 The dynamic load allowance is an equivalent static load, expressed as a
function of the OHBD Load, which is considered for design purposes to be equivalent to
the dynamic and vibratory effects of the interaction of the moving vehicle and the bridge.
Dynamic load allowance is not required for centrifugal, braking, collision or pedestrian

loads. The maintenance vehicle load... includes an allowance for dynamic effects.”

“C24.3.2.2 The use of advanced dynamic analyses , or tests, either on structural

models or existing structures, is subject to approval.”

“€243.23 Dynamic load is caused by a combination of :

l55

- bumps in the riding surface or expansion joints which result in direct impact to
the bridge deck;

- dynamic variation in axle loads due to undulations and roughness in the riding
surface; and

- dynamic response of the main longitudinal bridge components to the moving
vehicle loads.

The effects of these factors are generally not independent, but the relative
contribution from each may vary signiﬁcantly depending upon the component loaded.
The dynamic load allowance for a truck, or part thereof, is speciﬁed according to the
number of axles involved in generating the load effect, as follows:

i) loading by a single axle load;

ii) loading by two axle loads;

iii) loading by three or more axles, including loading by the entire Truck.

a) Components governed by a single axle unit, or part thereof, may include slabs,
such as concrete deck slabs, pan ﬁll ﬂoors, steel orthotropic decks and short-span
supporting elements. Dynamic variation in axle load due to roughness in the riding
surface, and impact at bumps, is directly transferred to the component, which results in a
high dynamic load. The dynamic interaction of such components with the moving load is
generally very small, because vibration of the component has a short period as compared

to the duration of loading. the speciﬁed values of dynamic load allowance are based on

156

test results..., assuming that the approach riding surface and bridge deck have been paved
to acceptable standards. If the approach is unlikely to be paved for an extended period of
time, or if expansion joints between the superstructure and approach pavement are not
generally ﬂush with the roadway riding surface, vehicles will enter the bridge in a state of
excitation. To allow for this, it is recommended that the dynamic load allowance be
increased from 0.4 to 0.5 within 3 m or one-tenth of the span length, whichever is greater,
from the location of the joint. This distance corresponds approximately to one half cycle
of the axle spring for a vehicle travelling at approximately 100 km/h (62.15 mph).

b) For longitudinal components whose design is governed by two axle loads, the
axles are not generally in phase so that the impact effect is moderated. the dynamic
response of the component to moving loads results in some dynamic load ampliﬁcation,
but spans of such components are substantially less than 20 m and the frequencies are
usually suﬁiciently high that a frequency match between component and vehicle is
unlikely. a consideration of either effect however, would load to about the same value of
DLA.

c) For longer spans, for which the critical loading is due to three or more axles, a
frequency match with a vehicle suspension is possible and the largest dynamic load is
likely to involve dynamic interaction between the structure and the vehicle. The effect of
axle load impact is not likely to be critical. the duration of loading is longer, and the
bridge and vehicle may interact for more than one cycle of vibration if the bridge

frequency. There is increased dynamic response of bridge superstructure having natural

 

157

frequencies in the range 2 to 5 Hz, a range typical of the bounce frequencies of vehicles.
The increase is because of interaction between the vehicle and the bridge, and has many

of the characteristics of resonance in a simple oscillator.”

A.4 Code of Practice Highway Bridge: 1262
The existing Code of Practice Highway Bridges (CPHB) 1967 of Pakistan

provides the following speciﬁcations to account for the impact effect in the design of

highway bridges:

“2.9- IMPACT
Live Loads stresses produced by the standard Truck-train loading shall be increased for
items Group A, by allowance as stated therein, for dynamic, vibratory and impact effects.

Impact shall not be applied to items in group B.”
“(A) Group A
(l) Superstructure, including steel or concrete supporting columns, steel

towers, legs of rigid frames and generally those portions of the structure which extend

down to the main foundation.

(2) The portions above the groundline of concrete or steel piles which are

rigidly connected to the superstructure as in rigid frame or continuous designs.”
“(B) Group B

(1) Abutments, retaining walls, piers, piles except Group A(2)

158

(2) Foundation pressures and footings
(3) Timber structure
(4) Sidewalk loads

(5) Culverts and structures having cover of 3 feet or more”

“(C) Impact Formula
The amount of this allowance or increment is expressed as a fraction of the live

load stress, and shall be determined by the formula:

15

1:
L+20

 

in which
I = impact fraction (maximum 30 percent)

L = length of span in feet”

“For uniformity of application, the span length “L” shall be especially considered
as follows:

For roadway ﬂoors, use the design span length.

For transverse members, such as ﬂoor beams, use the span length of member
centre to centre of supports. .

For computing truck load moments use the span length, except for cantilever arms
use the length from the moment centre to the fartherrnost axle.

For continuous spans use the length of span under consideration for positive
moment, and use the average of two adjacent loaded spans for negative moment.

For bridges having cantilever arms with suspended spans, use the span (length)

159

of the cantilever arm plus half the length of the suspended span when designing the
cantilever and the span length of the suspended span (when) designing the latter.
For shear due to truck loads use the length of the loaded portion of span from the
point under consideration to the far reaction.”
“For culverts with cover 0’ to 1’ - 0” inc.I = 30%
1’ - 1” to 2’ - 0” inc. I =20%
2’ - 1” to 2 - 11” inc. I = 10%”

*t*********¢****#****

LISILQEREEERENCES

1. Akin', J. E., and Moﬁd, M. (1989). “Numerical Solution for Response of Beams
With Moving Mass.” Journal of Structural Division, ASCE, 115(1), 121-131.

2. American Association of State Highway and Transportation Officials
(AASHTO)(1931-1992)2. Standard Speciﬁcations for Highway Bridges, Washington, DC

3. American Association of State Highway and Transportation Officials, Load and
Resistance Factor Design (AASHTO LRFD)(1994). Standard Speciﬁcations for Highway
Bridges, Washington, DC

4. Biggs, J. M., Suer, H. S., and Louw, J. M. (1959). “Vibrations of Simple-Span
Highway Bridges.” Transactions. ASCE, 124, 291-318.

5. Biggs, J. M., and Suer, H. S. (1956) “Vibration Measurements on Five Simple
Span Bridges.” Highway Research Board Bulletin No. 124, Washington DC, 1-15.

6. Billing, J. R. (1984). “Dynamic loading and testing of bridges in Ontario.”
Canadian Journal of Civil Engineering, 1 1(4), 833-843.

7. Blejwas3, T. 13., Feng, C. C., and Ayre, R. S. (1979). “Dynamic interaction of
moving vehicles and structures.” Journal of Sound and Vibration, 67(4), 513-521.

8. Blumenschein", E. W. (1951). “Can reliance be placed on natural bond between
concrete and steel?” Civil Engineering, New York, p.404.

9. Boehning, R. H. (1953) “Single and Tandem Axle Dynamic Effects on a Highway
Bridge Model.” Highway Bridge Impact Investigations. Third Progress Report, Civil
Engineering Studies, Structural Research Series No. 57, University of Illinois, Urbana,
Ill.

10. Cantieni, R. (1984). “Dynamic Load Testing of Highway Bridges.”
Transportation Research Record 950, Vol. 2, Transportation Research Board,
Washington, DC

 

' Reported by Yang and Lin (1995).

2 Speciﬁcations for several years before 1992 were not available.
3 Same as footnote No. l.

‘ Reported by Wright and Green (1959).

160

161

11. Chan, T. H. T. and O’Connor, C.(1990); “Vehicle Model for Highway Bridge
Impact”; ASCE Journal of Structural Engineering, Vol. 116, No. 7, 1772-1793.

12. Chang, D. and Lee, H. (1994). “Impact Factors for Simple-span Highway
Bridges”; ASCE Journal of Structural Engineering, Vol. 120, No. 3, 704-715.

13. Cheung, Y. K., and Cheung, M. S. (1972). “Free Vibration of Curved and Straight
Beam-Slab or Box-Girder Bridges.” International Association of Bridge and Structural
Engineering (IABSE), 32(11), 41-52.

14. “Code of Practice Highway Bridges (CPHB) 1967.” Highway Department,
Government of (West) Pakistan.

15. Committee on Deformation Limitations of Bridges. (1958). “Deﬂection
Limitation of Bridges.” Journal of Structural Division, ASCE, 84(1), 1-20.

16. Edgerton, R. C., and Beecroft, G. W. (1956). “Dynamic Studies of Two
Continuous Plate Girder Bridges.” Highway Research Board Bulletin No. 124, 1956,
Washington DC, 3346.

17. Eichmann, E. S.(l954). “Inﬂuence of Vehicle Suspensions on Highway Bridge
Impact.” Highway Bridge Impact Investigations. Fourth Progress Report, Civil
Engineering Studies, Structural Research Series No. 70, University of Illinois, Urbana.

18. Fenves, S. J., Veletsos, A. S., and Seiss, C. F. (1962). “Dynamic Studies of the
AASHO Road Test Bridges.” Highway Research Board Special Report No. 73,
Washington DC, 83-96.

19. Final Report of the Special Committee (1931). “Impact in Highway Bridges”;
Transactions. American Society of Civil Engineers, 95, 1089-1117.

20. Fleming, J. F., and Romualdi, J. P. (1961). “Dynamic Response of Highway
Bridges.” Journal of Structural Division, ASCE, 87(1), 31-61.

21. Foster, G. M. and Oehler, L. T. (1956). “Vibration and Deﬂection of Rolled-
Beam and Plate-Girder Bridges.” Highway Research Board Bulletin No, 124, Washington
D. C., 79-110.

22. Fryba, L. (1972). Vibration of solids and structures under moving loads.
Noordhoff International Publishing, Groningen, the Netherlands.

23. Fuller, A. H. (1922); “Preliminary Impact Studies - Skunk River Bridge on the
Lincoln Highway Near Ames, Iowa.”; Engineering Experiment Station Bulletin 63, Vol.
XXI, No. 26, Ames, Iowa.

162

24. Fuller, A. H. (1925). “Experimental Impact Studies on Highway Bridges”;
Engineering Experiment Station Bulletin 75, Vol. XXIV, No. 30, Ames, Iowa.

25. Galdos, N. H., Schelling, D. R., and Sahin, M. A. (1993); “ Methodology for
Impact Factor of Horizontally Curved Box Bridges”, Journal of Structural Division,
ASCE, Vol. 119, No. 6, 1917-1934.

26. Greens, R. (1977). “Dynamic Response of Bridge Superstructures - Ontario
Observations.” Supplementary Report SR 275, Transportation and Road Research Lab.,
Crowthome, England, 40-48.

27. Hayes, J. M., and Sbarounis, J. A. (1956) “Vibration Study of Three Span
Continuous I-beam Bridge.” Highway Research Board Bulletin No. 124, Washington DC,
47-78.

28. Heywood, R. J. (1995) “Are Road-Friendly Suspensions Bridge-Friendly? OECD
DIVINE”; Proceedings Fourth International Bridge Engineering Conference, 1995, Vol.
1, 281-295; Transportation Research Board, National research Council, Washington, D.
C.

29. Highway Bridge Impact Investigations. F ifth6 Progress Report, Engineering
Experiment Station, University of Illinois, Urbana, September 1955.

30. Highway Bridge Impact Investigations. First7 Progress Report, Department of
Civil Engineering, University of Illinois, Urbana, May 1951.

31. Highway Bridge Impact Investigations. Second Progress Report, Civil
Engineering Studies, Structural Research Series No. 24, University of Illinois, Urbana,
April 1952.

32. Huang, D., Wang, T-L, and Shahawy, M. (1995);”Vibration of Thin-Walled Box-
Girder Bridges Excited by Vehicles”, Journal of Structural Division, ASCE, Vol. 121,
No. 9,1330-1337.

33. Huang, D., Wang, T-L, and Shahawy, M. (1993); “Impact Studies of Multigirder
Concrete Bridges”, Journal of Structural Division, ASCE, Vol. 119, No. 8, 2387-2402.

34. Huang, D., Wang, T-L, and Shahawy, M. (1992); “ Impact Analysis of
Continuous Multigirder Bridges Due to Moving Vehicles”, Journal of Structural
Division, ASCE, Vol. 118, No. 12, 3427-3443.

 

’ Reported by Huang et al. (1993).
6 Reported by Wright and Green (1959).
’ Same as footnote No. 6.

163

35. Humar, J. L., and Kashif, A. M. (1995).; “Dynamic Response Analysis of Slab-
Type Bridges”, Journal of Structural Division, ASCE, Vol. 121, No. 1, Jan. 1995, 48-62.

36. Humar, J. L., and Kashif, A. M. (1993). “Dynamic Response of Bridges Under
Traveling Loads.” Canadian Journal of Civil Engineering, Ottawa, Canada, 20(2), 287-
298.

37. Hwang, E-S and Nowak, A. S.(1991) “ Simulation of Dynamic Load for
Bridges”; ASCE Journal of Structural Engineering, Vol. 117, No. 5, 1413-1434.

38. Ibanathan and Wield (1987).”Bridge Vibrations Due to Vehicle Moving Over
Rough Surface.” Journal of Structural Division, ASCE, 113(9), 1994-2008.

39. Inglis, C. E. (1934). A Mathematical Treatise on Vibrations in Railway Bridges.
Cambridge University Press, London, Great Britain.

40. Joness, M. A. (1972). “Dynamic Analysis of Box Girder Bridges.” Ph.D.
Dissertation, Illinois Institute of Technology, Chicago, Ill.

41. Kashif, A. M. (1992). “Dynamic Response of Highway Bridges to Moving
Vehicles.” Ph.D. thesis, Carleton University, Ottawa, Ontario, Canada. (Abstract)

42 Leonardg, D. R., Grainger, J. W., Eyre, R. (1974). “Loads and Vibrations Caused
by Eight Commercial Vehicles with Gross Weights Exceeding 32 Tons.” Laboratory
Report LR 52, Transportation and Road Research Lab., Crowthome, England.

43. Looney, C. T. G. (1958). “High-Speed Computer Applied to Bridge Impact”
Journal of Structural Division, Proceedings of ASCE, (Paper No. 1759), ST5, Vol. 84, 1-
41.

44. Nieto-Ramirez, J. A., and Veletsos, A. S. (1966). “Response of Three-Span
Continuous Highway Bridges to Moving Vehicles.” Engineering Experiment Station
Bulletin No. 489, University of Illinois, Urbana, Illinois.

45. Nowak, A. S., and Hong, Y-K. (1991). “Bridge Live-Load Models”, Journal of
Structural Division, ASCE, Vol. 117, No. 9, 2757-2767.

46. O’Connor, C. and Pritchard, R. W. (1985). “Impact Studies on Small Composite
Girder Bridge.” Journal of Structural Division, ASCE, 111(3), 641-653.

 

' The original report not available. Reported by Huang et al. (1995).
9 Same as footnote No. 8.

 

164

47. “Ontario Highway Bridge Design Code (OHBDC).” (1983). Ministry of
Transportation and Communication, MTO, Highway Engineering Division, Ontario,
Canada.

48. “Ontario Highway Bridge Design Code (OHBDC).” (1991). Ministry of
Transportation and Communication, MTO, Highway Engineering Division, Ontario,
Canada.

49 Oran"), C., and Veletsos, A. S. (1961). “Analysis of Static and Dynamic Response
of Simple Span Multi-Girder Highway Bridges.” Civil Engineering Studies, Structural
Research Series No. 221, University of Illinois, Urbana, Ill.

50. Page, J. (1976). “Dynamic Wheel Load Measurements on Motorway Bridges.”
Transportation and Road Research Lab., Crowthome, England.

51. Rabizadeh, R., and Shore, S. (1975). “Dynamic Analysis of Curved Box Girder
Bridges.” Journal of Structural Division, ASCE, 101, 1898-1912.

52. Saadeghvaziri, M. A. (1993). “Finite Element Analysis of Highway Bridges
Subjected to Moving Loads” Computers & Structures, Vol. 49, No. 5, 837-842.

53. Scheffy, C. F. (1956) “Dynamic Load Analysis and Design of Highway Bridges.”
Highway Research Board Bulletin No. 124, Washington, DC, 16-32.

54. Schilling, C. G. (1982). “Impact Factors for Fatigue Design.” Journal of
Structural Division, ASCE, Vol. 108, ST9, 2034-2044.

55. Shepherd, R., and Aves, R. J. (1973). “Impact Factors of Simple Concrete
Bridges” Proceedings, Institution of Civil Engineers, Part 2, 55, 191-210. (Paper No.
7548).

56. Sridharan, N. and Mallik, A. K. (1979). “Numerical Analysis of Beams Subjected
to Moving Loads.” Journal of Sound and Vibration, 65(1), 147-150.

57. Stokes", G. G. (1869). “Discussion of a Differential Equation Relating to the
Breaking of a Railway Bridge.” Transactions. Philosophical Society, Cambridge,
England.

58. Sundara Raja Iyengar, K T., and Jagadish, K. S. (1968). “The Response of Beam
and Slab Bridges to Moving Forces.” International Association of Bridge and Structural
Engineering (IABSE), 28(11), 69-86.

 

'° The original report not available. Reported by Humar and Kashif ( 1993).
" Same as footnote No. 6.

165

59. Sundara Raja Iyengar, K T., and Jagadish, K. S. (1970). “Dynamic Response of
Highway Bridges to Moving Loads.” International Association of Bridge and Structural
Engineering (IABSE), 30(II), 57-76.

60. Sundara Raja Iyengar”, K. T., Jagadish, K. S. and Narayana Iyengar, R.. “Free
Vibration of Beam and Slab Bridges”. The Journal of the Institute of Engineers (India),
Vol. XLVIII, No. 1, Pt CI, September 1967 (Special).

61. Tan, C. P., and Shore, S. (1968a). “Dynamic Response of a Horizontally Curved
Bridge.” Journal of Structural Division, ASCE, 94(3), 761-781.

62. Tan, C. P., and Shore, S. (1968b). “Response of Horizontally Curved Bridge to
Moving Load.” Journal of Structural Division, ASCE, 94(9), 2135-2151.

63. Tiedeman, J. L., Albrecht P. and Lloyd R. Cayes, L. R. (1993) “Behavior of Two-
Span Continuous Bridge Under Truck Axle Loading”, Journal of Structural Division,
ASCE, Vol. 119, No. 4, 1243-1250.

64. Timoshenko, S. P., and Young, D. H. ((1965); “Theory of Structures”, Second
Edition, McGraw-Hill Book Co., New York; pp. 609-617 (Chapter 12)

65. Timoshenko, S., Young, D. H., and Weaver, Jr., W. (1974); “Vibration Problems
in Engineering”; Fourth Edition, John Wiley & Sons, NEW YORK; pp. 414453
(Chapter 5)

66. Tung, E. P., Goodman, L. E., Chen, T. Y., and , N. M. “Highway Bridge Impact
Problems.” Highway research Board Bulletin No. 124, 1956, Washington, DC, 111-134.

67. Veletsos, A. S., and Huang, T. (1970). “Analysis of Dynamic Response of
Highway Bridges.” Journal of Engineering Mechanics Division, ASCE, 96(5), 593-620.

68. Waddell, J. A. L. (1918). “New Impact Formulas Needed in Designing Bridges of
Various Types.” Engineering News-Record, Vol. 81, No. 21, 924-928.

69. Walker, W. H. (1968). “Model Studies of the Dynamic Response of a Multigirder
Highway Bridge.” Engineering Experiment Station Bulletin No. 495, University of
Illinois, Urbana, Illinois.

70. Walker, W. H., and Veletsos, A. S. (1966); “Response of Simple Span Highway
Bridges to Moving Vehicles.” Engineering Experiment Station Bulletin No. 486,
University of Illinois, Urbana, Illinois.

 

'2 The original report not available. Repoted by Sundara Raja Iyengar and Jagadish (1968).

 

 

166

71. Wang, T. L., and Huang, D. (1992). “Cable-Stayed Bridge Vibration Due to Road
Surface Roughness.” Journal of Structural Division, ASCE, 118(5), 1354-1374.

72. Warburton”, G. B. (1976). The dynamic behavior of structure, 2nd ed., Pergamon
Press, Oxford, England.

73. Wen, R. K. and Veletsos, A. S. (1962), “Dynamic Behavior of Simple-Span
Highway Bridges.” Highway Research Board Bulletin No. 315, National Academy of
Sciences, National Research Council, Washington, DC

74. Wood”, J. H., and Shepherd, R. (1979). “Vehicle Induced Vibrations.” Bulletin
No. 44, Road Res. Unit, Auckland, New Zealand.

75. Wright, D. T., and Green, R. (1959). “Highway Bridge Vibrations, Part I: Review
of Previous Studies.” Research Report No. 4, Ontario Joint Highway Research
Programme, Queen’s University, Kingston, Ontario, Canada.

76. Wright, D. T., and Green, R. (1964). “Highway Bridge Vibrations, Part 11: Ontario
Test Programme.” Research Report No. 5, Ontario Joint Highway Research Programme,
Queen’s University, Kingston, Ontario, Canada.

77. Yang, Y-B, and Lin, B-H. (1995); “Vehicle-Bridge Interaction Analysis by
Dynamic Condensation Method”, Journal of Structural Engineering, ASCE, Vol. 121,
No. 11,1636-1643.

78. Yarnada", Y. and Veletsos, A. s. (1958). “Free Vibration Of Simple Span I-Bearn
Bridges.” Eighth Progress Report, Par B, Highway Bridge Impact Investigations,
University of Illinois, Urbana, Ill.

79. Yang, Y-B, Liao, S-S, and Lin, B-H. (1995). “Impact Formulas for Vehicles
Moving over Simple and Continuous Beams”, Journal of Structural Engineering, ASCE,
Vol.121,No. 11, 1644-1650.

80. Yoshida'”, D. M., and Weaver, W. (1971). “Finite-Element Analysis of Beams

And Plates With Moving Loads.” International Association of Bridge and Structural
Engineering (IABSE), 31(1), 179-195.

******#********¥*****

 

‘3 Reported by Yang and Lin (1995).

u The original report not available. Reported by Huang et al. (1993).

'5 The original report not available. Reported by Humar and Kashif (1993).
'6 Reported by Humar and Kashif (1993).