Design and Vibration Control by Friction Dampers in Truss
Bridges
L S MINSILI, ZHONG TIEYI and XIA HE
Northern Jiaotong University, School of Civil Engineering and Architecture
Beijing 100044, China
/H]LQ#FHQWHUQMWXHGXFQ
D E MANGUELLE
‘SADEG-Ingenieurs Conseils’
B.P. 7217 Yaounde Cameroon
VDGHJ#LFFQHWFPKWWSZZZLFFQHWFPVDGHJ
Abstract
Various types of control methods have been proposed from passive to active systems and a number of
hybrid methods combining both of them have found a major interest. It is likely that many devices can be
used for energy dissipation, however, with an appropriate installation, friction devices introduce a simple
and inexpensive solution and have been massively applied to buildings and related structures while there is
no particular application in bridge superstructures. Due to negative effects attributed to the rise of
acceleration in some structural elements, their massive application has been rejected by many contractors.
The objective of this work is to give a simple analytical method of controlling bridges vibration by means of
slotted bolted friction connections based on experimental studies by M. Constantinou et al (1990) and other
researchers. A first stage analysis is done on a through truss bridge with a sinusoidal excitation and later, as
an application, simulations are done with a running train and earthquake excitations. The obtained
analytical results show the effectiveness of friction dampers installed in the superstructure of a bridge and
exhibit in the same time the needs to modulate the rate of energy dissipation by adjustment of the friction
characteristics to meet the expected requirements.
Keywords: Bolted connections, friction dampers, energy dissipation, truss structure, vibration control.
INTRODUCTION
Friction process is one of the most important phenomena that we encounter in daily life in any of our
motions. If there is any movement between two surfaces, there should also be to a certain extent a friction
force to be overcome before the movement. From the car industry to the building design, vibration
mitigation systems have received particular interest in the recent past and are dealing with the concept of
mounting a system or some of its elements on surfaces, which permit sliding. Though the possibility of
using such devices and their applications is common in building industry, very few, if any, analytical studies
to date have been carried out to assess the merit of this idea for actual structural element of a truss system
applied to bridge engineering. Mostaghel (1983) studied the response of sliding structures to harmonic
support motion and established the effectiveness of sliding supports to control the acceleration level. Others
studies were done by R. Plunkett (1980), Jii-Song Tsai (1985), Ian D. Aiken (1993) and Pall A. et al
(1993). Fitz Gerald (1989) proved that slotted bolted connections can provide elastic and inelastic energy
dissipation, eliminating the need for inelastic member buckling, while the present author [Minsili (2000)]
studied effects of friction dampers with velocity dependant characteristics subjected to harmonic and
seismic loading.
Experimental and analytical studies have been carried out on existing bridges lately and good conclusions
have been made with a wide variation of dynamic loading, material properties and control methods. For
example research done on the design and implementation aspects of active control by S. J. Shelley et all
(1995) relates the testing and implementation of active-vibration-control system installed on the Big Darby
Creek Bridge, in Columbus Ohio, prior decommissioning in August 1992 due to concerns regarding its
serviceability. Knowing that many existing bridges in developing countries were build before 1970 with
traditional standards and that they are still in service with the growing of their loading rate, it is important to
take appropriate protection measures to avoid any unexpected eventual failure and any related social and
economic losses. Protection efforts on bridges are usually oriented to the protection of piers since damage
usually occurs in this part during strong earthquakes. Some solutions are directed towards responses’
minimization at the top of the pier by using structural devices such as bearings and sliding bearings while
others are resumed to the reinforcement of the overall structural properties of the bridge and thus to the
modification of its structural response [Zhang Hongjie (1998)]. It is important to see that reducing created
or induced responses in the bridge superstructure before its partial dissipation or consumption in the
substructure is an efficient way that should be studied with great emphasis. This paper addresses the design
and implementation of slotted bolted friction connections as a passive vibration control device installed in
through-truss railway bridge system. The frictional resistance of the interface is assumed to be velocity
dependent even if it can be taken as a Coulomb friction process. Included in this scope are: description of
the friction connection, analysis of the effectiveness of the proposed device under harmonic, train and
seismic excitations.
MATHEMATICAL FORMULATION AND MODEL
The main objective in structural modeling and analysis is to provide the simplest mathematical formulation
of the true bridge behavior that satisfies a given assessment for the response determination. Thus the
analytical model as it follows, describes the geometric domain, the structural properties and the loading as
closely as possible to facilitate the engineering interpretation of numerical response quantities.
Formulation
Surface friction is an energy dissipative process which takes place with relative tangential displacement of
contacting solids in zones of real contact between them formed by the action of an external load. Depending
on the types of displacement, it is possible to distinguish three types of friction, i.e. sliding friction, rotating
friction and rolling friction. The friction force T is the resultant of the tangential resistance forces arising at
real contact points when one body slides along the surface of another, and is a non-potential force. In
passing from rest to sliding, there is a preliminary displacement region (fig 1 section OA). The tangential
resistance under preliminary displacement conditions is called the incomplete friction force, or better the
adhesion force since it is partially potential in character.
Figure 1: The friction process
Figure 2: a) Illustration of the friction device
b) Free-body diagram
The full stationary friction force corresponds to a transition from preliminary displacement to sliding (point
A), and is conditionally called static friction. After the preliminary displacement starts stable sliding,
characterized by a sliding friction force (line AB). Depending on the problem posed, friction can be
evaluated by a force or a moment. In engineering practice, friction is usually evaluated by means of a
dimensionless quantity f the coefficient of sliding friction, which is the ratio of the friction force to the
normal load
(1)
f =F N
In metal-forming process the ratio of the tangential resistance, in the contact zone between two bodies, to
the yield point of the weaker material is called the friction coefficient. In this case, in accordance with
plasticity theory, the instantaneous coefficient of friction cannot exceed 0.5. Since this coefficient is a
function of input factors (nature of sliding bodies, loads, lubricants, speed,…) and internal factors (changes
in roughness, mechanical properties, heat transfer,…), it should be considered as variable. Kragelsky
(1982) showed that the pressure has a significant effect on the magnitude of the friction coefficient and
suggested that the change in the magnitude of the friction coefficient as a function of the normal pressure is
determined by the ratio between these forces. The increase in friction force F as a function of the contact
time t is given by the equation
(2)
F = F α − (α − 1)e − βt
0
[
]
where F0 is the friction force at a constant time equal to zero, α = F∞ / F0 , F∞ being the friction force for an
infinite contact time. The value of D is independent of the pressure and is constant for the given friction
pair, E being a constant.
Yamada (1999) considered the damping coefficient of a variable damping device in its non-linearity relation
expressed by a power series:
η ( t ) = η0 + ∑i = 1
n
λi
u ( t )i
i!
(3)
with the condition that η ( t ) implicitly depends on the velocity and displacement u(t) of the structural
dynamics motion equation, λi are series coefficients and constitute parameters that adjust the non-linearity of
the considered damping.
Constantinou M. et al (1990) suggested to take the sliding coefficient of friction as a function of the sliding
velocity U for Teflon-steel interface in the following approximated equation
f s = f max − Df exp(− a U )
(4)
in which f max is the coefficient of sliding friction at large velocity, Df is the difference between f max and the
sliding value at very low velocity and a is a constant for a given normal pressure and condition of interface.
f max , Df and a was tested experimentally and equation (4) reproduced results with a good accuracy.
Considering the breakaway process that occurs before sliding from rest, and the fact that the same process
may happen whenever the response of the structure passes through the zero velocity, the present author
[Minsili (2000)] gave also the expression of the friction coefficient for steel-steel interface as
(5)
f s ( t ) = f sl + Df b.exp( − a U ( t ) ) , with U ( t ) ≠ 0.
where Df = f 0 − f SL , f 0 =0.4 is the coefficient of static friction (zero velocity) and f SL =0.15 the coefficient
of sliding friction under consideration at ‘very large velocity’. Since the most important is the theoretical
transition from static to sliding considering energy dissipated through the breakaway process and the related
coefficients of friction, coefficients a and b can be extrapolated from the work done by Kragelsky (1982)
and Constantinou (1990) and taken here as 5 and 0.2 respectively.
Let us now consider an axial element that carries primarily only axial loads with a mounted friction device
as illustrated in figure 2. Since during sliding additional forces due to friction occurs along the member, the
forces on each end may not be absolutely equal, the same thing with the related end displacements and
velocities. The friction force acting along the sliding surfaces is governed by
F ≤ f s (t ) N
(6)
where N is the applied normal force to the interface and is taken as constant here for bolted connections.
Using equation 4, the friction force can be written as
(7)
F < f s (t ) N
and
u1 − u2 = 0
F = f s ( t ) N sgn( u1 − u 2 )
(8)
in static and sliding mode respectively, where sgn denotes the signum function.
Analytical Model
The considered through-truss bridge consists of the superstructure and substructures. The latter will not be
considered in analysis and are supposed to be rigid so that their actions on friction analysis can be ignored.
The expected responses of non-friction members are in the elastic range and a quantifiable basis for ground
motion input variations along the truss length and movement joint characterization can be established.
However, the friction members’ responses as it was said are also in the elastic range in non-sliding mode
and behave plastically in sliding mode according to conditions established in equations (7) and (8). Since
the aim of this study is to assess the gain of the addition of friction elements in a bridge structure, we can
consider only one of both bridge trusses and its related plan loading. We then neglect all effects such as
transverse flexure and torsion even if they are important in bridge analysis but out of the scope of this work.
The truss system is isolated with the left support fixed while the right one is movable and can be represented
with or without a friction pendulum bearing.. We suppose that all truss elements will be subjected only to
tension or compression with the condition that the structure without friction elements has been statically
submitted to fatigue and stability checking according required standards and norms. As it is seen in figure 3,
the added friction element in the panel does not intercede with the primary diagonal element of the structure.
Each friction element is divided into two parts: one part connected (bolted or welded) to two cover plates.
Figure 3:
a.) Bridge model with incorporated friction devices.
b.) Panel representation with the friction connection
Passing over the primary diagonal, these plates are connected (slotted bolted) to the “movable” part that can
slide under specified conditions. The slot may be installed in the diagonal friction element flange or in the
cover plate. To allow the motion of the “movable” part of the friction element, a clear distance should be
provided depending on the expected dynamic camber (panel drift). For SDOF structure the computation of
response characteristics such as ∆u , ∆u needed for their assessment is easy in nature. For plan structures
individual elements can not only have axial elongation, but can also rotate during any time of the vibration
history. This is why it is important to escape from the traditional computation of these values and apply
more efficient ones. A form of co-rotational technique for truss elements is used to compute incremental
displacements and velocities between nodes of element at each step. The derivation uses a plan corotational truss element (spatial are applicable too) with the Kirchhoff theory and the key element in such
derivation is the introduction of the variation of the local transformation matrices, from which can be
derived the mid-point formula according to Bathe K.J. (1983):
∆u =
G
G
2
1 G
( S 21 + d 21 ) T d 21
ln + l0
2
(9)
G
G
G
G
where S 21 = S 2 − S1 , are vector difference between the two end-points positions in the general S and
G
varying local d coordinate systems.; l0 , ln are primary and deformed element. Using the initial basic form
for the element, the section is assumed constant and the axial element force is N = EA∆u / l 0 for non-friction
members. By discussing the dynamic behavior of the truss model according to the free-body diagram
represented in figure 2b, we can assume that the restoring force is proportional to the displacement. The
dissipation of energy through the theoretical viscous damping is done by the damping force and is
proportional to the velocity. In addition, the mass in the model is always considered to be unchanging with
time. As a consequence of these assumptions, the dynamic equilibrium in the system is established by
equating to zero the sum of the components in figure 2b, i.e. FI (t ) = mu(t ) , FS (t ) = − f S (t ) N sgn u (t ) , P(ti ) ,
Fk (t ) = ku (t ) and FD (t ) = cu (t ) are respectively the inertial, the friction (static or sliding), the excitation, the
spring and damping forces at instant t. Hence at time ti the equilibrium of these forces is expressed as
FI (ti ) + FD (ti ) + FK (ti ) = FS (ti ) + P (ti )
(10)
and at short time-step ∆t later this equation can be transformed by considering the corresponding
incremental displacements and forces in the incremental friction equation as
m∆ui + c∆ui + kui = ∆FSi + ∆Pi
(11)
where all coefficients dependent to variables such as velocity or friction phase corresponding to time ti are
assumed to remain constant during the incremental time step ∆t . Since in general the external force and
friction force do not remain constant for the time increment, this last equation is an approximate equation.
The second problem that occurs with that same equation is that the friction force term is velocity dependent
and is not known before we get the velocity ui at that time increment ∆t , therefore an appropriate method
should then be found.
Material property matrices
The problem of defining structural properties of any structure can be reduced basically to the evaluation of
its stiffness, damping and mass matrices. These properties are found by evaluating first the properties
associated with individual finite elements and the related degrees of freedom of their respective nodes. The
mass of any element is assumed to be concentrated at nodes and the distribution of the distributed
superstructure mass to these points is determinate by static according to the Chinese Bridge Codes. It is
evident that from the lumped mass theory the mass matrix has a diagonal form without zero diagonal
elements since only transnational degrees of freedom are defined.
m =0
if
i≠ j
M = mij with ij
(12)
mij ≠ 0
if
i= j
[ ]
Structural stiffness in structures with friction damping systems are affected whenever one or many of its
element conditions are changing from one friction mode to another. This effect is clear for single degree of
freedom systems and according to figures 1 and 2, individual stiffness of friction element tends to reach a
zero-value in the sliding mode. The stiffness of the overall structure is then between a minimum value K m
and a maximum value K M when all sliding elements are in sliding mode and in stick mode respectively.
Thus the stiffness matrix depends not only on the configuration of the structure but also on its loading
conditions. It is then better to partition the global stiffness matrix in friction-dependent stiffness submatrices K f , K fo , Kof and friction non-dependent stiffness K0 , that is
 K0 # K0 f 


K=" # "
K

 f 0 # Kf 
(13)
The non-linearity of systems with added friction damping does not allow us to apply any superposition
method using the uncoupled model responses. But for simplicity in this work the damping matrix will be
proportional only to mass matrix according to the generalized modal damping value with the viscous
damping ratio 0.02.
Integration of the friction equation and solution algorithm
Among the many methods available for the solution of the nonlinear equation of motion, one of the most
effective is the step-by-step integration method. In this method, the response is evaluated at successive
increments ∆t of time, usually taken of equal length of time for computational convenience. At the
beginning of each interval, the condition of dynamic equilibrium is established. Then, the response for a
time increment is evaluated approximately on the basis that variable stiffness if any remain constant during
the interval. The nonlinear characteristics of these variables are considered in the analysis by reevaluating
these coefficients at the beginning of each time increment. The response is then obtained using the
displacement and velocity calculated at the end of the time interval as the initial conditions for the next time
step. There are many procedures available for performing the step-by-step integration of equation 11, two
of the most popular methods are the constant acceleration method and the linear acceleration method. As the
names of these methods imply, in the first, the acceleration is assumed to remain constant during the time
interval. This second method has been found to yield excellent results with relatively little computational
effort. To tackle the second problem mentioned earlier for the velocity-dependence of the friction force
FS (ti +1 ) at the right hand of each time interval, additional numerical approximations are required to reach
the pseudo-friction force. Initializing data of (i+1)-th interval step by the i-th interval’s except for excitation
loads P(t) which is known a priori, and with the velocity-dependent expression of other variables, we define
new results of (i+1)-th interval through iterative procedures that is stopped when a tolerance error
( ε = uk −1 − uk ) between old and new increments of k procedures in one interval step [ti , ti +1 ]. For the first
interval i=1, the friction force is considered to be at the stick mode, that is at t0 =0, |u( t0 )|=0 according to
conditions of equation 7 and 8 in the static mode. An analogue iterative solution-algorithm is given in the
paper of the same author [Minsili (2000)].
NUMERICAL RESULTS
The primary factor to be considered in selecting a step-by-step procedure is efficiency, which concerns the
computational effort required to achieve the desired level of accuracy over the range of time for which the
response is needed. Accuracy alone cannot be a criterion for method selection because, in general, any
desired degree of accuracy can be obtained by any method if the time step is made short enough. As with
any numerical integration method the accuracy of the considered step-by-step method depends on the length
of the time increment ∆t . Three factors are considered in selection of this interval: (1) the rate of variation
of the applied loads (input excitation and the friction), (2) the complexity of the nonlinear stiffness
properties, and (3) the natural period T =0.298 sec. of vibration of the considered structure. In general the
material-property variation is not a critical factor; however, if a significant sudden change takes place as in
the present stick-sliding transition modes, a special treatment to tackle accurately this change is considered
in computations. As an effort to verify the adequacy of the proposed numerical procedure, the harmonic
response is first investigated. This harmonic excitation is expressed as P(t)=Gsinwt with G=0.4g being the
amplitude of ground acceleration and w the excitation frequency taken as 0.6 times the value of the first
natural frequency of the truss in non-resonance case. Assuming that the friction mechanism is a velocity
dependant type as described earlier, the middle span (node 8-Y) vertical displacement of the bottom chord is
depicted in figure 4 for resonance and non-resonance cases, while in figure 5 is given the response
comparison also on node 16-X.
Table 1: Maximal displacements and accelerations for harmonic excitations.
Node number
Displacements, (m)
Acceleration (g)
and direction
Without friction With friction
Without friction With friction
8-X
0.66
0.11
0.18
1.27
8-Y
2.37
0.92
0.6
1.94
16-X
1.22
0.29
0.3
1.76
0.15
0.02
0.10
0.05
U(t) , (m)
U(t) , (m)
0.01
0.00
-0.01
0.00
-0.05
-0.10
-0.02
0
2
4
6
8
10
-0.15
0
2
4
t , (s)
Non-resonance
t , (s)
Resonance
Figure 4: Middle span vertical displacement history
6
Without
With Friction
0.02
0.015
Without
With Friction
0.010
0.005
U(t) , (m)
U(t), (m)
0.01
0.00
-0.01
0.000
-0.005
-0.010
-0.02
0
5
t , (s)
10
-0.015
0
2
4
6
8
10
t , (s)
Middle span, node 8-Y
Left abutment, node 16-X
Figure 5: Sinusoidal response comparison diagrams for nodes 8-Y and 16-X for different cases
These results show that friction devices incorporated in a truss structures display important characteristics
that need to be considered in detail and verified experimental for their validation. When compared to the
dynamic behavior of the same structure without friction devices, we can see that the dynamic response of the
present structure in resonance and non-resonance is not very different if we consider only the rise of the
amplitude and its stabilization. But a very important fact is that since each friction force is in phase-lag with
the velocity-difference at the element nodes, effect of these added friction forces is displayed in the change
of the frequency component of the structure (figure 5). We can also see that insertion of friction devices in
the structure clearly reduces the displacement at nodal points for the whole structure sometime up to 70 %.
In the other side there is also a negative effect in the rise of the acceleration at almost any nodal coordinate
as seen in table 1 only for specific nodes. The simulation of a running train on the same structure is also
done with the help of a specific computer software developed in the department on the basis of the
experimental train Shao Shan-9 (designed speed 200 km/h) on the Qinshen Railway Line in China
(Qinhuangdao- Shenyang). The bridge load is taken as a vertical continuous load which is a random variable
both with regard to the coordinate x and to the time t and moves at a constant speed along the truss
deck[Friba (1977)], transverse and longitudinal effects are neglected. The train vertical load vector P(x,t) is
then formulated as
P( x, t ) = [p + εpˆ ( s)][1 + rˆ(t )]
(14)
where p=E[f(x,t)] is the constant mean value of the continuous load,
pˆ ( s )
rˆ(t )
s
is the centred random component of the load depending on the moving coordinate s
is the centred random component of the load depending on time t,
is the length coordinate with the origin moving at the speed c, ε << 1 .
Analysis of the structure responses under the running train for various speeds is done only for comparative
purpose to assess effect of the friction devices and obtained results are shown in figures 6.
Table 2: Maximal displacements and accelerations on node 8-Y under running train excitation.
Train speed
Displacements (cm)
Acceleration (g)
(km/h)
Without friction With friction
Without friction With friction
60
2.76
2.29
0.38
0.93
100
2.94
2.28
0.87
0.75
160
2.86
2.31
1.09
1.04
200
2.94
2.38
2.27
1.59
Figure 6: Responses’ comparison under train excitation at various speeds (start)
0.030
0.030
Without
With F1
0.025
0.025
Without
With F1
0.020
U(t), (m)
U(t), (m)
0.020
0.015
0.010
0.005
0.015
0.010
0.005
0.000
0.000
0
5
10
15
20
25
30
0
5
10
t , (s)
15
20
25
30
t , (s)
a) Train speed 60 km / h
b) Train speed 100 km / h
Figure 6: Displacement comparison under train excitation at various speeds (continued)
0.030
Without
With F1
0.025
0.025
0.020
0.020
U(t), (m)
U(t), (m)
0.030
0.015
0.010
0.005
Without
With F1
0.015
0.010
0.005
0.000
0.000
0
5
10
15
20
25
30
0
5
10
t , (s)
15
20
25
30
t , (s)
c) Train speed 160 km / h
d) Train speed 200 km / h
Figure 7: Vertical displacement history on node 8-Y under earthquake excitations.
0.02
0.02
0.01
U(t), (m)
0.01
U(t), (m)
Without
With F1
Without
With F1
0.00
0.00
-0.01
-0.01
-0.02
0
20
40
t , (s)
Borrego%4
60
0
10
20
30
40
t , (s)
El-Centro
The results obtained thus far indicate the proposal to use slotted bolt connections as energy absorbers during
the passage of a train at various speeds may not be very effective. This effect is mostly attributed to the nonpotential nature of the friction process itself. Since in the system involving friction, as long as there is
motion, the magnitude of the friction force is constant and its direction is opposite to that of the sliding
velocity. Reduction of the middle span displacement can be notified when the excitation load reaches its
maximum allowing also maximum energy dissipation. But as the maximum train excitation remains nearly
constant as it is seen in figure 6, there is no practical difference between the responses of primary truss
structure and those of the friction-damped truss. The reverse situation is also seen during the fall of the
maximum train excitation when the last wagon of the train crosses the right support, and the friction seems
to add energy to the structure even the considered amount is negligible. The general displacement
reduction, obtained mainly with the change in the excitation magnitude, for each simulated speed is around
20 % while the reduction in acceleration (Table 2) is not quite defined and still remains one of the most
important problem to be considered with great care.
The performances of friction connections in the truss under seismic loading was calculated using the dual
recorded earthquake motions of Borrego Mountain (1968) with magnification factor 4, and of El Centro
(1940) at the Imperial Valley station. The horizontal and the vertical components are jointly considered. An
examination of obtained results shows that, compared with the structure without friction device, inserted
friction connection reduced the displacement by 45 % under El Centro excitation and only by 32 % under
Borrego as it is seen in figure 7. This discrepancy is in particular associated to the frequency component of
each excitation and to the behavior of the truss structure under related excitations. Since the dissipated
energy is also related to the effect of the sudden developed drag force at element nodes by the excitation,
effect of the represented friction may be simply neglected if the input force is too big. In the other way, it’s
possible to find a suitable value of the represented friction force to counter effect of the excitation force.
One solution can be obtained by varying the magnitude of the force normal to the sliding interface.
CONCLUSION
Under a variable cyclic loading of an earthquake ground motion causing deformation into the sliding plate
region, there is a certain resistance force developed in the sliding interface at rest different to the original
position. It is quite easy for the return excitation response in the opposite direction (due to elastic
displacement accumulated by non-friction truss members) to push the structure back to a lower deflection
level, and so with the connected gusset plates of the friction device. In an ordinary braced frame, there
would simply be inelastic buckling deformation in the bracing elements. If there is no complete return to the
original ‘zero’ position, the accumulated energy in the connection can simply be released by unscrewing all
related bolts. From the time history analysis performed nodal displacements and nodal accelerations were
assessed for a through truss bridge structure without and with a friction isolation device inserted as an
additional cross diagonal slotted bolted element. The following observations can be made based on the
peak responses: 1) Introduction of friction device into a truss structure reduces peak displacement values of
the overall structure, this is a straight consequence of the internal panel drift reduction during the friction
force process. 2) The presence of a friction force limits the relative displacement to small values but the
acceleration response may be undesirably large as it is seen in tables 1 and 2. 3). As a direct result of the
first two observations, there is a need to find an optimal friction device system with appropriate variable
characteristics that will meet the exploitation expected performances of the structure.
Knowing that the present analysed control system will be subjected to changes with the variation of some of
its characteristics, how fare is it possible to control its responses under such conditions? The answer of this
question can find a straight answer if we try to find an optimal number of bolts for each element with which
the structure will have an optimal expected performances.
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