Key Engineering Materials Vol. 574 (2014) pp 43-51
© (2014) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/KEM.574.43
Design on Dynamic Performance of Highway Bridges to Moving
Vehicular Loads
Qingfei Gao1,a, Zonglin Wang1,b, Binqiang Guo2,c, Haoran Bu3,d, Wei Xiong4,e
1
School of Transportation Science and Engineering, Harbin Institute of Technology, Harbin, China,
150090
2
Zhejiang Provincial Institute of Communications Planning, Design & Research, Hangzhou, China,
310000
3
Tianjin Municipal Engineering Design & Research Institute, Tianjin, China, 300000
4
Shenzhen China Overseas Construction Limited, Shenzhen, China, 518001
a
gaoqingfei_1986@163.com, bwangzonglin@vip.163.com, cguobinqianglove@163.com
d
buhaoran123@126.com, ehitxw@foxmail.com
Keywords: Highway bridges; dynamic performance; design; moving vehicular loads
Abstract. Based on the survey of existing highway bridges, there are a large number of flaws induced
by moving vehicles. The most important cause of this phenomenon is the lack of design codes on
dynamic performance of highway bridges to moving vehicular loads. The existing theory of
vehicle-bridge interaction is reviewed. Then the home-code program VBCVA combined with finite
element program ANSYS is introduced to analyze the problem of vehicle-bridge interaction. Also,
the existing design indexes of dynamic performance are discussed, such as dynamic impact factor,
deflection limit, and acceleration. On the basis of above theory and program, the framework of design
on dynamic performance of highway bridges to moving vehicular loads is proposed.
1 Introduction
The dynamic response of a bridge to a moving vehicle is a complex problem affected by the
dynamic characteristics of both the bridge and the vehicle and by the road surface profile. Many of
these parameters interact with one another, further complicating the issue, and consequently, many
research studies have reported seemingly conflicting conclusions. As a result, there is considerable
variation in the treatment of dynamic load effects by bridge design and evaluation codes in different
countries. In another word, the dynamic performance of highway bridges to moving vehicular loads
cannot be rationally designed and evaluated.
In most current design codes, dynamic effects are taken into account by increasing the static design
stresses by an impact factor. This factor can be quite large – up to 70% for the case of the Eurocode.
There is increasing evidence to suggest that this factor is excessively high – some authors have
suggested that 10% is enough. This means that materials and money are being wasted in the
construction of new bridges and older existing bridges are being replaced when they are still perfectly
safe. In addition, for the control of vibrations, the OHBDC recognizes that limiting human perception
to the vertical vibration of bridges is a primary objective. Historically, quasi-empirical deflection
limits have been used to control vibrations. However, the 1979 OHBDC provisions are based on
limiting acceleration, which is a key parameter for human perception of motion. And the acceleration
limits for pedestrian bridges are simplified to a single linear relationship with the square root of
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Some Research Results on Bridge Health Monitoring, Maintenance and
Safety III
fundamental frequency in 1983 OHBDC. This relationship remains unchanged to now. Although
many provisions about dynamic effects of bridges under moving vehicles are included in most design
codes, lots of bridges are still damaged by moving vehicular loads. According to a survey of U.S. and
Canadian transportation agencies, approximately 75 percent of the agencies responding saying that
they had experienced possible problems in existing bridges, attributed to dynamic load effects from
vehicles. Common observed problems are expansion joint failures and fatigue cracking in the girders,
connections, bearings, and concrete decks of steel bridges [1].
In a word, it is extremely significant to study the design on dynamic performance of highway
bridges to moving vehicular loads. Additionally, it is better that the end result would be
recommendations for the design codes of bridges.
2 Theory of Vehicle-Bridge Interaction
Vehicles driving over a bridge create a complex dynamic phenomenon known as vehicle-bridge
interaction. Vehicle-bridge interaction refers to the dynamic coupling that occurs between a vehicle
and a bridge. When crossing a bridge, the vehicle vertically vibrates and pitches, leading to the
introduction of dynamic loading on the bridge. In turn, the vibrations of the bridge influence the
dynamics of the vehicle [2]. The interaction between a bridge and the vehicles moving over the bridge
is a coupled, nonlinear dynamic problem.
Table 1 Symbol conventions
Symbol
x
y
z
t
m
L
EI
g
Meaning
location
vertical displacement of bridge
vertical displacement of vehicle body
time
mass of per unit length
span length
stiffness of beam
acceleration of gravity
Symbol
P
M
M1
M2
v
k1
c1
c
Meaning
moving force
mass of vehicle
mass of tires
mass of vehicle body
velocity of moving force or vehicle
stiffness of suspension system
damping of suspension system
damping of bridge
The moving force model (Fig. 1) is the simplest model that can be conceived, which has been
frequently adopted by researchers in studying the vehicle-induced bridge vibrations. With this model,
the essential dynamic characteristics of the bridge caused by the moving action of the vehicle can be
captured with a sufficient degree of accuracy. However, the effect of interaction between the bridge
and the moving vehicle was just ignored. For this reason, the moving force model is good only for the
case where the mass of the vehicle is small relative to that of the bridge, and only when the vehicle
response is not of interest.
The problem is described by the equation
EI
∂ 4 y ( x, t )
∂ 2 y ( x, t )
∂y ( x, t )
m
+
+c
= δ ( x − vt ) P( x, t )
4
2
∂x
∂t
∂t
(1)
Key Engineering Materials Vol. 574
45
P(x, t)
x
y(x, t)
m, EI
x=vt
x
y(x, t)
x=vt
M, g
m, EI
L
L
y
y
Fig. 1 Moving force model
Fig. 2 Moving mass model
For cases where the inertia of the vehicle cannot be regarded as small, a moving mass model (Fig.
2) should be adopted instead. One drawback with the moving mass model is that it excludes
consideration of the bouncing action of the moving mass relative to the bridge. Such an effect is
expected to be significant in the presence of pavement roughness, or for vehicle moving at rather high
speeds. Occasionally, it may be necessary to consider the separation and recontact of the moving
vehicle with the bridge for some very bad road conditions, in which the bouncing action of the
vehicles plays a decisive role in the separation-recontact process.
The problem is described by the equation
∂ 4 y ( x, t )
∂ 2 y ( x, t )
∂y ( x, t )
∂ 2 y ( x, t )
EI
)
+m
+c
= δ ( x − vt ) M ( g −
∂x 4
∂t 2
∂t
∂t 2
(2)
The vehicle model can still be enhanced through consideration of the elastic and damping effects of
the suspension systems. The simplest model in this case is a moving mass supported by a
spring-dashpot unit, the so-called sprung-mass model (Fig. 3).
The problem is described by the equations
M 2 ɺɺ
z (t ) + k1 ( z (t ) − y ( x, t ) x =vt ) + c1 ( zɺ(t ) −
EI
∂y ( x, t )
)=0
∂t x =vt
∂ 4 y ( x, t )
∂ 2 y ( x, t )
∂y ( x, t )
∂ 2 y ( x, t )
d 2 z (t )
+
+
=
−
+
−
−
m
c
δ
(
x
vt
)[(
M
M
)
g
M
M
]
1
2
1
2
∂x 4
∂t 2
∂t
∂t 2
dt 2
(3)
(4)
M2
z(t)
x=vt
k1
c1
M1
y(x, t)
x
m, EI
L
y
Fig. 3 Moving sprung-mass model
Because of the emergence of high-performance computers and the advance in computation
technologies, it becomes feasible to have a more realistic modeling of the dynamic properties of the
various components constituting a moving vehicle [3, 4].
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Some Research Results on Bridge Health Monitoring, Maintenance and
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3 Program of Vehicle-Bridge Interaction
The dynamic effect of the vehicle is an important problem in bridge design and evaluation. In
recent years, the dynamic analysis of vehicle-bridge interaction system has been carried out for lots of
cases to ensure the safety of bridge structure and running vehicles and the riding comfort of
passengers [5].
The numerical simulation of bridge/vehicle interaction is the basis of this research topic. Three
methods of numerical simulation have been studied. They are time-step iteration, inter-system
iteration, and modal synthesis method. After overall consideration, modal synthesis method has been
adopted in this research.
To analyze the dynamic interaction between bridge and vehicles accurately, the dynamic equations
of vehicle-bridge coupling system is deduced by the modal synthesis method. The sophisticated
preprocessing and post processing functions in ANSYS and a home-code program VBCVA (Vehicle
Bridge Coupling Vibration Analysis) are integrated into a general tool [6,7]. Also, it has been proved
that this program is valid and effective enough to analyze the vehicle-bridge interaction vibration
system [8].
3.1 Vehicle Model
From the dynamic analysis point of view, the vehicle is composed of the body, tires and suspension
system [9].
A vehicle body is represented by a distributed mass subjected to rigid-body motions. Vertical
displacements and pitching rotations are considered. Pitching moments of inertia of the body are
calculated from the weight distribution and dimensions of body. Tires are treated as linear elastic
spring components.
The virtual-work equation for any virtual displacement can be expressed in the matrix form as
[ M v ]{ɺɺ
zv } + [Cv ]{zɺv } + [ K v ]{zv } = {P}
(5)
where [Mv], [Cv], and [Kv] are the matrices of the sprung mass of the vehicle, damping constant in the
vehicle model ,and the stiffness of vehicle spring, respectively, and {zv} is the displacement vector.
{P} is the global nodal force vectors, due to interaction between the bridge and vehicle.
3.2 Bridge Model
A general-purpose finite-element program, ANSYS, designed specifically for advanced analysis
applications is used to develop the finite-element model. ANSYS allows for the comprehensive
coverage of both linear and nonlinear behavior for both concrete and steel. All material properties
used in the analytical modeling, including the properties of reinforcement steel and concrete, are
based on actual data. In this program, the bridge structure is modeled as an assembly of grillage
members, consisting of longitudinal beams and transverse elements. It is simple and widely
applicable for various types of bridges with satisfactory accuracy.
The dynamic equations of a bridge subjected to wheel loads can be easily established from the
global mass [M] and stiffness [K] matrices as well as damping [C] matrix, that is
[ M ]{ ɺɺ
y} + [C ]{ yɺ} + [ K ]{ y} = {P}
(6)
Once the nodal displacements are obtained, the displacements at any points on the bridge can be
derived from the beam-bending displacement functions. Moreover, the strains at any point on the
bridge can also be obtained from the derivatives of the beam displacement function [10].
Key Engineering Materials Vol. 574
47
3.3 Model of Bridge Surface
It is assumed that a bridge profile is a realization of a random process that can be described by a
power spectral density (PSD) function. Typical PSD function can be approximated by an exponential
function
S (γ ) = αγ − n , γ a < γ < γ b
(7)
where α = roughness coefficient; n = spectral shape index; and γ = spatial frequency, with γa and γb
being the lower and upper limits. The bridge profile is modeled as a stationary Gaussian random
process. Therefore, it can be generated by an inverse Fourier transform as follows:
I
X (t ) = ∑ 4 S (ωi )∆ω cos(ωi t − θi )
(8)
i =1
where S(ωi) = PSD function; ωi = circular frequency; and θi = random number uniformly distributed
from 0 to 2π. Bridge profiles are different depending on the random numbers θi used in Eq. 8. For
each case, 50 bridge profiles are generated. The mean dynamic load is calculated as an average of
these 50 realizations.
3.4 Numerical Methods
The equations of motion of the vehicle are nonlinear, while those of the bridge are considered
linear. Considering the different characteristics of the equations of motion, we apply Newmark β
method to solve the equations of motion of the vehicle, while the solutions of those of the bridge were
determined by the mode-superposition procedure based on the subspace-iteration method. The main
procedure for dynamic analysis of the bridges is shown in Fig. 4.
This program can be used for the cases of multi-lane, multi-vehicle, and unequal speed. It has been
proved that this program is valid and effective enough to analyze the vehicle-bridge interaction
vibration system.
Fig. 4 Flow diagram of modal synthesis method
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Some Research Results on Bridge Health Monitoring, Maintenance and
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4 Designs on Dynamic Performance of Highway Bridges to Moving Vehicular Loads
The dynamic response of bridges due to vehicular loading has been a subject of interest to
engineers for more than 100 years. Over the last 40 years, a significant amount of research has been
conducted in the area of bridge dynamics, and this research has been both analytical and experimental
in scope. The dynamic response of bridges is complicated by a number of independent but interacting
factors and, as a consequence, many of the studies have produced seemingly conflicting results and
conclusions. Correspondingly, there is considerable variation in the treatment of dynamic loads by
bridge design codes in different countries.
4.1 Dynamic Impact Factors
Loads associated with a vehicle crossing a bridge consist of the live loading resulting from the
weight of the vehicle and the dynamic forces due to oscillations of the vehicle on its suspension
system as well as those forces induced by the dynamic response of the bridge. For design, both the
vehicle weight and the dynamic effects must be treated in a consistent manner that accounts for the
variability and uncertainty in the forces and which includes an appropriate factor of safety against an
overload condition occurring. The live loading can be reasonably quantified based on the static
weights of actual and design vehicles. In comparison, determining the precise dynamic forces
induced by vehicles is complex, somewhat abstract, and difficult to quantify.
Most bridge design codes typically specify the dynamic loading from vehicles as a fraction of the
design live loading that is added to the static load. Traditionally, this dynamic loading fraction has
been referred to as the “impact factor”, although the term “dynamic load allowance” is considered
more descriptive and encompassing and thus is becoming more popular. Failure to properly account
for dynamic loading can lead to excessive bridge stress that may cause failure in parts or in all of the
bridge.
As we know, there are four types of formulas about dynamic impact factors.
i) Constant value, such as AASHTO2007, and OHBDC1991.
ii) Function of the span length L, such as JTJ021-89, AASHTO1996, and Japan.
iii) Function of the fundamental frequency f, such as JTG D60-2004, and OHBDC1979.
iv) Function of other parameters (the ratio of dead load to live load), such as France.
0.4
0.6
µ=
+
(9)
1 + 0.2 L 1 + 4 G
P
where µ is dynamic impact factor, G is dead load, and P is live load.
It has been shown that the dynamic response of bridges under vehicular loading is influenced by
many parameters. In particular, the dynamic characteristics of the vehicle, the dynamic characteristics
of the bridge, and variations in the surface conditions of the bridge and approach roadways have all
been shown to have a strong influence on the dynamic loading.
Therefore, it is extremely significant to establish the rational formula of dynamic impact factor.
4.2 Deflection Limitation
The original source of the present AASHTO deflection limits is traceable to the 1905 American
Railway Engineering Association (AREA) specification where limits to the span-to-depth ratio, L/D,
of railroad bridges were initially established. Actual limits on allowable live-load deflection appeared
in the early 1930’s when the Bureau of Public Roads conducted a study that attempted to link the
objectionable vibrations felt on a sample of bridge built in that era.
Key Engineering Materials Vol. 574
49
It has been shown that deflection limits should be derived by considering human reaction to
vibration rather than structural performance. The important parameters that effect human perception
to vibration are the acceleration, deflection and period (or frequency) of the response.
The AASHTO Standard Specification limits live-load deflections to L/800 for ordinary bridges and
L/1000 for bridges in urban areas that are subjected to pedestrian use.
Both the Canadian Standard and the Ontario Highway Bridge Code use a relationship between
natural frequency and maximum superstructure static deflection to evaluate the acceptability of a
bridge design for the anticipated degree of pedestrian use.
Most European Common Market countries base their design specifications upon the Eurocodes. In
general, the full live-loads are factored with a “vibration factor” to account for extra stress due to
vibrations in European bridge codes. No additional checks (frequency, displacements etc.) are then
required. So deflection limits are not normally applied in European bridge design.
In China (Code for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and
Culverts, 2004), the live-load deflection limit is L/600 for simplified span and continuous span, and
that is L/300 for cantilever span.
It has been shown that the existing deflection limit was initially instituted to control bridge
vibration, but deflection limits are not a good method for controlling bridge vibration. In another
word, to limit the live-load deflection is not effective enough for bridge design.
4.3 Acceleration
As we know, the acceleration induced by moving vehicles is closely related to the inertia force of
bridges. However, it has been ignored in most of bridge design codes.
Human reactions to vibrations are classified as either physiological or psychological.
Psychological discomfort results from unexpected motion, but physiological discomfort results from
a low frequency, high amplitude vibration such as seasickness. Vertical bridge acceleration is of
primary concern, since it is associated with human comfort.
A 1971 study conducted by the American Iron and Steel Institute (AISI) reviewed AASHTO
criteria and recommended relaxed design limits based on veritical acceleration to control bridge
vibrations [11]. The proposed criteria require that:
i) Dynamic component of acceleration, a (in/sec2)
a = DI δ s (2π fb ) 2
(10)
ii) The acceleration limit is that [a] = 100 in./sec2=0.26g.
In BS5400 (British Standard), the simplified formula of maximum vertical acceleration is that
a = 4π 2 f 0 2 ys Kψ
(11)
where f0 is fundamental natural frequency, ys is static live-load defection, K is shape parameter, and Ψ
is dynamic response parameter which is related to damping.
In OHBDC1995, to make sure the safety of bridges and the comfort of users, the acceleration limit
is determined as
[a ] = 0.5 f 0
(12)
There are some simplified formulas of acceleration and the limits of acceleration. However, they
are not rational enough to apply for bridges in China. In addition, they are not the same to each other.
Therefore, it is necessary to do more studies on this design index, acceleration.
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Some Research Results on Bridge Health Monitoring, Maintenance and
Safety III
4.4 Development Tendency
To supplement existing design codes, lots of design provisions on dynamic performance of bridges
have to been studied in future. Many key issues are listed as follows. The framework of design on
dynamic performance of highway bridges under moving vehicles can be seen in Fig. 5.
i) Loads, including load form and load class.
ii) Serviceability limits state.
iii) Design index, including the selection of index and the establishment of simplified formula.
iv) Limitation of index, mainly based on the security of bridges and the comfort of users.
Fig. 5 Framework of design on dynamic performance of highway bridges
5 Conclusions
The existing theory of vehicle-bridge interaction was reviewed, including moving force model,
moving mass model, and moving sprung-mass model. Then the home-code program VBCVA
combined with finite element program ANSYS was introduced to analyze the problem of
vehicle-bridge interaction. Also, the existing design indexes of dynamic performance were discussed,
such as dynamic impact factor, deflection limit, and acceleration. On the basis of above theory and
program, the framework of design on dynamic performance of highway bridges to moving vehicular
loads was proposed.
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