Electronic
Journal of Structural Engineering, Vol. 1, No. 2 (2001) 98110 
A. Parvin ^{1 }and Z. Ma ^{2}^{ }
^{1} A/Professor, Department of Civil Engineering, University of Toledo, OH 436063390 USAEmail:
aparvin@eng.utoledo.edu
Received 17 May 2001; revised 22 July 2001; accepted 24 July 2001
ABSTRACT
In this study a combination of helical springs
and fluid dampers are proposed as isolation and energy dissipation devices for
bridges subjected to earthquake loads. Vertical helical springs are placed
between the superstructure and substructure as bearings and isolation devices to
support the bridge and to eliminate or minimize the damage due to earthquake
loads. Additionally, horizontal helical springs are placed between the abutments
and bridge deck to save the structure from damage. Since helical springs provide
stiffness in any direction, a multidirectional seismic isolation system is
achieved which includes isolation in the vertical direction. To reduce the
response of displacement, nonlinear fluid dampers are introduced as energy
dissipation devices. Time history analysis studies conducted show that the
proposed bridge system is sufficiently flexible to reduce the response of
acceleration. The response of displacement due to provided flexibility is
effectively controlled by the addition of energy dissipation devices.
KEYWORDS
Seismic isolated structures; dynamic analysis;
vertical motion; helical spring.
Seismic
isolation reduces the response of a structure during an earthquake by
introducing flexibility and energy dissipation capabilities. Generally,
horizontal inertia forces cause the most damage to a structure during an
earthquake. Since the magnitude of the vertical ground acceleration component is
usually less than the horizontal ground acceleration component, vertical seismic
loads are not considered in the design of most structures. The vertical
acceleration is typically taken as two thirds of the horizontal acceleration
component for the same response spectral curve.
However,
recent observation and analysis of earthquake ground motion have shown that the
vertical motion in bridges should not be completely ignored. Researchers
compiled records and photographs of damage and failures of buildings and bridges
due to high vertical motion. Ratios
of peak verticaltohorizontal acceleration have been recorded as high as 1.6,
while the conventional design assumption is 0.67 [1]. Damage
from Kalamata, Greece (1986), Northridge, CA (1994), and Kobe, Japan (1995) due
to purely vertical effects is reported, along with the high vertical
tohorizontal acceleration ratios.
Not many
researchers address vertical motion in their studies. Among the few who have,
Button et al. [2] investigated the effect of vertical ground
acceleration on six bridge types and they recommended criteria for inclusion or
exclusion of vertical ground motion in the design and analysis of bridges. Their
study was limited to bridges with no base isolation and energy dissipation
devices. Waisman and Grigoriu [3] studied the influence of
the vertical seismic component on a frictionpendulum type baseisolated bridge.
Their model was limited to a single span, and single degreeoffreedom
system. Saadeghvaziri and Foutch [4] investigated the
behavior of reinforced concrete highway bridges that were not isolated and were
subject to combined vertical and horizontal earthquake motions. They concluded
that it is important to include the vertical component of ground acceleration
motion in the design of highway bridges.
Most
research studies in bridge isolation, where the vertical ground motion is
neglected, include theoretical and experimental analysis of various active and
passive isolators (for horizontal plane motions) that are not multidirectional
and are complex in some cases. Among
such recent studies, Xue et al. [5] proposed a new system
termed the Intelligent Passive Vibration Control (IPVC), which contains both
passive (isolation) and intelligent, or active (damping) elements. During small
earthquakes, only the passive system is utilized. During large earthquakes, the
active system is triggered by displacement limitations. Further experimental and
analytical results on passive/active control of a bridge, which employs sliding
bearings with recentering springs for isolation, and servohydraulic actuators
activated by absolute acceleration records, are reported by Nagarajaiah et al. [6].
This system allows for a sliding system with higher friction to be
implemented without fear of high acceleration response. Yang et al. [7]
presented analytical models for rubber and sliding bearings coupled with
actuators for bridges.
The
vertical motion in the bridge is crucial. The
uplift from the vertical motion may cause loss of contact followed by impact,
which is likely to lead to higher mode response and large axial forces in the
piers. Existing bridge bearings
including elastomeric bearings and lead rubber bearings among others are
designed to only provide isolation in the horizontal plane. For instance, in
some cases, using only horizontal isolation may provide sufficient protection
against an earthquake. However, in certain other cases, where vertical ground
acceleration is significant, a multidirectional isolation system, which
possibly employs helical springs may be required.
This
study involves novel bridge bearings consisting of helical springs and viscous
dampers to achieve a multidirectional seismic isolation system, which also
provides controlled flexibility in the vertical direction.
In the proposed configuration of the isolated bridge (Fig.1),
the deck and girders can be considered to be floating on helical spring
bearings. Helical springs, which have both vertical and shear
stiffness, are designed to support vertical loads, including the selfweight of
the bridge, providing the mechanism to accommodate movement in all directions.
To protect the bridge deck and abutment from damage by an earthquake in
the longitudinal direction, helical springs are also installed between the deck
and abutments. Additionally, fluid
dampers are added vertically at the locations of the interface between the
superstructure and its supporting pier and abutment to control the response of
displacement during an earthquake. The
combination of helical springs and fluid dampers is expected to provide an
efficient flexible seismic isolation and energy dissipation device that reduces
the response of the system.
The
following sections discuss the mathematical models for the damper and helical
spring of the bridge model. A numerical analysis study for the vertical response
of the bridge with the proposed isolation system, is then presented followed by
the conclusions.
Two fundamental isolation and energy dissipation elements presented in this section include the helical spring and the fluid damper, respectively. The helical spring possesses stiffness in all directions. Its stiffness can be customized according to design requirements. Compared to a nonisolated bridge, a springsupported bridge is relatively flexible in the vertical direction, allowing vibration in the vertical direction with no damage to the structure. The helical spring can greatly reduce the relative response of acceleration. Other advantages of using helical springs include high load carrying capacity, linear load versus deflection curve, nearly unlimited lifetime service (if provided with suitable corrosion protection), and constant properties with time [8]. These combined properties make the helical spring a very suitable elastic element with a restoring force. However, the helical spring has little damping effect [9]. If additional damping is required for practical purposes, supplemental damping devices can be combined with the springs or used separately.
The helical spring follows a linear relationship where elastic force is proportional to relative deformation. The relationship between static force and relative deformation of a helical spring is:

(1a) 
where is the vertical stiffness of spring, and is the relative deflection in vertical direction. The shear stiffness is taken as 40% ~ 50% of the vertical stiffness [8,10]. In the shear direction, a similar relationship is taken as:

(1b) 
where is the shear stiffness of helical spring, is the relative deflection in shear direction, and is the ratio coefficient and is equal to . The spring stiffness is linear for static or dynamic analysis, which is a significant simplifying factor for the numerical analysis.
Fluid dampers have been used or proposed for structures as energy dissipation devices during the past three decades. Taylor and Constantinou [11] reported multiple episodes of high capacity fluid damping devices being used in buildings, bridges and related structures, which were originally invented and developed to attenuate the shock and blast effects in military equipment. Constantinou et al. [12] studied the effect of various passive energy dissipation systems used in buildings.
The fluid damping level can be up to 20%~50% of critical, thus greatly decreasing the response of displacement [11,13]. The output force of the fluid damper is insensitive to temperature. This property allows greater versatility in the application of these devices. In addition, there are also noteworthy advantages in installation, operation and maintenance of the fluid dampers and they have been proven to be reliable and cost effective.
The output force of the fluid damper at any time is typically represented as:

(2) 
where is the velocity of the piston rod, is the damping coefficient, and is the exponent coefficient, ranging from 0.1 to 1.8 as manufactured [14]. The piston rod stroke and the damping output force are mechanical characteristics of the fluid damper. The piston rod movement is limited to its stroke. Therefore, the displacement of the damped system should not be greater than the maximum stroke of the fluid damper. The fluid damper can provide damping in its axial direction only. To eliminate the damage caused by nonaxial forces to the damper, a roller is placed at the end of the piston rod.
Damping ratio is used as a measure to evaluate the damping level of a multimode damped system, and can be obtained for any mode as:

(3) 
where is the damping ratio of the ith mode, is the total energy dissipated by the fluid damper per cycle, and is the total elastic energy of the system per cycle for the ith mode. For a structure subjected to dynamic loading, the equivalent damping ratio throughout the complete duration of the loading is:

(4) 
where P_{j} is the damping force, F_{j} is the elastic force, and D_{j} is the response of displacement at any jth time step.
Equation (4) will be used in this study as the basic formula to evaluate the damping ratio, which will be an approximation if the forces and displacements are solved numerically. Harmonic motion is a special case of Equation (4). It is noted that for a linear fluid damper, the damping ratio is independent of amplitude of motion. For a nonlinear case, the damping ratio generally reduces with increasing amplitude of motion.
Structural system damping is another factor that affects the dynamic performance of the structure. This damping is defined as the resistance to motion provided by the internal friction of the materials. The friction develops as the molecules forming the materials are forced across one another when the structure moves relatively. However, evaluation of system damping cannot be easily performed in practice. Usually, some percentage of critical damping is taken instead [15].
For dynamic analysis, the isolated bridge (Fig.1)
can be modeled as a continuous beam for simplicity. Since this model is flexible
in the vertical direction, it cannot be considered as a rigid block in that
direction. The entire vertical load is carried by helical spring bearings.
Fluid dampers are placed at the location of the bearings and do not
support the vertical load. Their functions are to dissipate seismic energy, suppress
possible resonance, and limit displacements.
If a group of springs and dampers is employed at one location of the
bridge, the resultant stiffness as well as the damping of the springs and
dampers need to be calculated for a particular direction.
Fluid dampers provide damping in only one direction, while helical
springs have stiffness in all directions.
Figure 1. TwoSpan Box Girder Bridge Prototype 
The displacement method is used to construct the relationship between the force and deformation of a deformable body. The derivation follows the typical procedure of matrix structural analysis for the bridge model [16,17]. In this model, the total stiffness matrix K_{c} is found by adding the supporting spring stiffness to the diagonal element of the global stiffness matrix at the corresponding degreesoffreedom.
A consistent approach for mass accounts for translational, as well as rotational degreesoffreedom, while the lumped mass approach only considers the translational degreesoffreedom. Since the rotational component of earthquake ground motion is not considered in most cases, the motion in rotational degreesoffreedom would not be excited. Additionally, in the lumped mass bridge model, the amount of rotations compared to translations are insignificant. Hence, the rotational degreesoffreedom are excluded from the stiffness matrix.
The static stiffness equation, which is in matrix form, is partitioned as:

(5) 
where and represent translation and rotation, respectively.
If in Equation (5), then . Substituting into the first submatrix in Equation (5) yields:
or 
(6) 
where K= is the translation stiffness matrix. Only those degreesoffreedom related to translation are retained. Therefore, the condensed matrix becomes compatible for use with the diagonal lumped mass matrix M.
The system damping matrix is expressed as:

(7) 
where is the modal shape matrix and c_{m} is the generalized modal damping matrix.
The diagonal damping matrix when fluid dampers are placed at the bearings in the vertical direction is represented by C_{d} . The damping forces of fluid dampers are determined by the damping coefficient, the damping exponent, and the velocity of the piston.
The dynamic equation of the baseisolated bridge model in the vertical direction has the following nonlinear form:

(8) 
where is the acceleration of the earthquake ground motion, are the vector of vertical displacement, velocity and acceleration, respectively.
Among numerous direct integration methods to solve for the nonlinear response in Equation (8), the Newmark integration method appears to be the most effective with the smallest numerical errors. In the Newmark method, the acceleration is assumed to be linear for the time to . For the time interval following relations are assumed:
and 
(9a) 

(9b) 
where and are parameters used to achieve the integration accuracy and stability. In the case of and , the constantaverageacceleration method will yield unconditional stability in the iteration procedure.
From Equations (9a) and (9b), and can be solved in terms of as follows:
and 
(10a) 

(10b) 
To obtain the solution for displacement, velocity and acceleration at time , the equilibrium Equation (8) is rewritten as:

(11) 
Since is a nonlinear term, substituting Equations [10a] and [10b] into Equation [11] will not yield linear simultaneous equations with respect to . Hence cannot be solved directly. To avoid using the iteration technique to solve the displacement vector at each time step, the nonlinear term is expanded at time by a Taylor series as shown in the following equation:

(12) 
where it is assumed that the high order terms can be neglected without loss of acceptable accuracy and is an operator to diagonalize a vector to a matrix.
By substituting Equations (10a), (10b), and (12) in Equation (11), a linear equation with respect to at each time step is obtained as:

(13) 






and 



After
is solved from Equation (13),
and
can be obtained from Equations
(10a) and (10b), respectively. Since the velocity and acceleration at time
have been expressed in terms of
their previous values at time
after the displacement at time
is known, the iteration procedure
can be performed stepbystep with any given initial conditions.
In the
above analysis, it can be shown that the nonlinear problem has been simplified
to be an approximately linear problem by employing a Taylor series expansion to
the nonlinear term of the damping force at each time step. Next the
displacements at each time step are solved directly, and therefore the
stepbystep direct integration method can be implemented.
A
flexible system will be less susceptible to damage when subjected to an
earthquake excitation. However, there is concern over the issue of isolation for
an ideally flexible system. In practice, the bridges need to be designed with
sufficient amount of strength and stiffness to resist the service load. The
basic factors including the spring stiffness involved in engineering design are
taken into consideration for the twospan bridge model in this study. The span
length is based on the continuous beam model. Once the span length is decided,
the size of the cross section can be calculated by applying traffic load as a
live load plus the dead load of the bridge model, assuming the material is
concrete. Note that the deflection under normal bridge loading must be
controlled and can be the determinant for the bridge stiffness. From the point
of seismic isolation, the bearings are expected to be as flexible as possible.
Since the large displacement due to flexibility can be effectively reduced by
fluid dampers, the spring stiffness will be mainly determined by operational
loading. The stiffness of the springs can be calculated based on the reaction at
the bearing and the static settlement limit (fluid dampers are not accounted for
carrying the load). Also the
kinetic deflection change between traffic load on and off the bridge should be
considered as a factor to determine the spring stiffness.
Based
on the above analysis, a twospan continuous concrete slab and box girder bridge
illustrated in Fig.1 is employed as a model to demonstrate
the isolation effects of helical springs and fluid damper systems. This bridge
is supported and isolated by helical springs positioned in the vertical
direction as bearings. Helical springs are also installed longitudinally at both
ends between the abutment and the deck to protect the bridge from impact load.
Fluid dampers are located between the superstructure and substructure in the
vertical direction where necessary. The
length of the bridge model is 58.5 m (192 ft) for each span.
AASHTO HS2044 truck loading is used for the live load.
The dead weight of the bridge superstructure is estimated to be 2.138x10^{4}
kg/m (14.357 kip/ft). The typical crosssection of the bridge box girder has a
width of 12.95m (42.5 ft) and a height of 2.36 m (7.75 ft). The moment of
inertia in the vertical direction is 66.4 m^{4} (1.594x10^{8}
in^{4}).
Under the presumed total load, the maximum deflection of the bridge is
approximately 100 mm (4 in), and the deflection change between traffic on and
off the bridge is limited to less than 6 mm (0.24 in). The stiffness of the
helical springs is defined by the following, where the shear stiffness is
assumed to be 40% of the vertical stiffness:
Vertical stiffness of the spring placed at end bearing (32. 905 ).
Vertical stiffness of spring placed at the middle bearing is (98.716 ).
Vertical stiffness of the spring placed at the abutment is (0. 672 ).
Fig.2 illustrates this twospan bridge, which is simplified as a fivelumpedmass model when analyzed numerically. The appropriate damping force relation, in accordance with the manufacturer provided data for the damper, is selected as (Fig.3)

(14) 
where is the velocity of the piston rod [18]. Fig.4 gives the diagram of the damping forcedisplacement relationship for the case of harmonic motion with period and . The enclosed loop areas are the total energies that can be dissipated by the damper in one cycle of harmonic motion. For different periods, the damper has a different area. When the system is subjected to the earthquake motion, the loops will not be symmetrical shapes. However, the energy dissipation capability indicated by the diagram could be useful in selecting the parameters for the fluid damper.
Figure 2. Lumped Mass Bridge Model 
Figure 3. Damping ForceVelocity Relationship of Selected Fluid Damper 
Figure 4. Harmonic Damping ForceDisplacement Relationship 
Among the software tools to perform the required numerical analysis for structures with protective devices against earthquakes, the 3D BASIS by Nagarajaiah et al. [19] is noteworthy. This tool has the capability to analyze various hybrid isolation configurations for threedimensional seismic motion. Although this numerical analysis software covers modeling of various combinations of isolators and energy dissipation devices, it assumes the isolation devices are rigid in the vertical direction. This model of isolation devices is more applicable for approximation of elastomeric bearing behavior, which is almost incompressible, and has large vertical stiffness, while the shear stiffness is not more than one percent of the vertical stiffness.
In our study, the MATLAB^{TM} software package is employed to obtain the responses of the bridge model illustrated in Fig.1. The bridge is subjected to the vertical components of the Northridge and El Centro earthquakes. Initially, isolated systems with various damping ratios are considered to observe the effect of damping level on the system. During the next step, vertical response of the isolated and undamped, isolated and damped, and nonisolated bridge models are compared to observe the isolation effect of coil springs in the bridge model with or without dampers.
In the bridge model, the natural frequency of the first mode in the vertical direction is 1.56 . The Northridge vertical ground motion, with peak acceleration of 0.799 , and El Centro vertical ground motion with peak acceleration of 0.210 are used as input excitations (Fig. 5). The system damping ratio is assumed to be two percent of the critical.
(a) El Centro 
(b) Northridge 
Figure 5. Vertical Ground Accelerations 
Initially, the effect of fluid dampers placed at the bridge bearing on the vertical response of the isolated bridge system was studied. Maximum responses and damping level of all the undamped and da
mped cases for Northridge and El Centro earthquakes are compared in Table 1. For the Northridge earthquake, the undamped responses at midspan of the bridge model are illustrated in Fig.6. The maximum response of acceleration at midspan is 0.796 and the dynamic deflection is over 78 (3 ). To reduce the displacement response, fluid dampers are used for the bridge model. For the damped case with one fluid damper placed at each bearing, responses of the bridge model are shown in Fig.7. The maximum response of midspan displacement in the damped case reduces 40.5% from 78. 6 (3 ) to 46. (1.84 ) of the undamped case. The maximum response of midspan velocity and acceleration also decreases 23.7% and 34.3%, respectively, when the fluid damping level is 9.16%. Fluid dampers exhibit excellent damping effects for the bridge model. The damping force versus displacement loop of the damper at the middle bearing is illustrated in Fig.8. Unlike the curves in Fig.3, the shape of the loop is not symmetric. The acceleration versus displacement hysteresis of the lumped mass at midspan of damped case bridge model is shown in Fig. 9. The maximum response of displacement is reduced to 46.8 (1.84 ) and the maximum response of acceleration is also as small as 0.523g .
Table 1 Maximum Vertical Responses of the Bridge Model
Earthquake 
Location 
Damping 
Displacement
(mm) 
Velocity (mm/s) 
Acceleration
(g) 
Fluid
Damping (%) 
Northridge 

Undamped 
76.3 
700.4 
0.749 
0 

End bearing 
1 damper 
44.8 
521.6 
0.503 
9.16 


2 dampers 
32.3 
432.4 
0.463 
19.35 


Undamped 
78.6 
727.9 
0.796 
0 

Midspan 
1 damper 
46.8 
555.3 
0.523 
9.16 


2 dampers 
36.1 
504.7 
0.473 
19.35 


Undamped 
75.9 
696.6 
0.748 
0 

Center bearing 
1 damper 
45.2 
546.0 
0.541 
9.16 


2 dampers 
36.5 
515.0 
0.603 
19.35 
El
Centro 

Undamped 
23.0 
233.1 
0.225 
0 

End bearing 
1 damper 
10.7 
109.0 
0.128 
13.30 


2 dampers 
7.6 
75.5 
0.115 
28.36 


Undamped 
23.7 
239.5 
0.240 
0 

Midspan 
1 damper 
11.9 
118.0 
0.139 
13.30 


2 dampers 
9.0 
101.5 
0. 153 
28.36 


Undamped 
22.8 
233.1 
0.223 
0 

Center bearing 
1 damper 
11.9 
120.3 
0.147 
13.30 


2 dampers 
9.3 
96.0 
0.163 
28.36 
Figure 8 Damping ForceDisplacement Loop at Middle Bearing (Northridge) 
Figure 9 Damping ForceDisplacement Loop at Middle Bearing (Northridge) 
The damping ratio is increased by placing two fluid dampers at each bearing. The reduction in response is minimal when the number of dampers was doubled. The fluid damping ratio for this case is 19.35%. Therefore, to reduce the response by adding more fluid dampers to the system is almost of no use when damping ratio has reached a certain value. The reduction of responses cannot be solely achieved by dampers.
The analysis for the case of the El Centro vertical component leads to similar phenomena. The only difference is that the maximum responses are less significant due to the smaller peak ground acceleration. The maximum undamped response of displacement at midspan is 23.7 (0.93 ) and the acceleration is 0.240 . The response of displacement, velocity and acceleration for the case with one damper placed at each bearing decreases by 49.8%, 50.7%, and 42.1%, respectively, where the fluid damping level is 13.3%. When two dampers were used at each bearing, the response of displacement and velocity at midbearing had minimal reduction while the acceleration at midbearing slightly increased. This result is expected, since damping is effective in reducing the displacement. As a tradeoff, the response of the acceleration would increase with a decrease in displacement.
As mentioned previously, damping is more efficient in reducing the displacement of an undamped system. For the damped system, using more than one damper decreases the response of displacement very little and even increases the response of acceleration. Therefore, to achieve an ideal effect one can use a more flexible system such as helical springs in combination with dampers to reduce the response.
Of particular interest is the response at the midspan of the bridge subjected to vertical earthquake ground motion. Conventional bearings have too much stiffness in the vertical direction and cannot resist vertical motion. The vertical response of acceleration at midspan would be significant, especially in the event of the Northridge earthquake with its large vertical ground acceleration. The spring bearings show the promise of providing the flexibility needed in the vertical direction.
Next, to show the isolation effects of the spring bearing, the isolated and nonisolated responses at the center bearing and at the midspan of bridge model subjected to Northridge and El Centro vertical ground acceleration are compared in Table 2. In the nonisolated case, the bridge girder becomes flexible relative to the rigid supports. There is a significant difference between the response of acceleration at the bearing and at the midspan. For the nonisolated case, the response of the acceleration can be over 2g for the bridge when subjected to a strong ground motion such as the Northridge earthquake. In addition, the maximum acceleration at midspan is also larger than the undamped isolated acceleration response. Even in the event of an earthquake with small peak vertical ground acceleration, such as El Centro, the acceleration at the bearing is as low as 0.2 while the acceleration at the midspan is over 0.5 .
Table 2. Maximum Vertical Responses of Isolated and NonIsolated Bridge Model
Equake 
Location 
Damping 
Displacement
(mm) 
Velocity (mm/s) 
Acceleration
(g) 
Fluid
Damping (%) 
Northridge 

isolated+undamped 
76.3 
700.4 
0.749 
0.00 

Bearing 
isolated+damped 
45.2 
546.0 
0.541 
9.16 


nonisolated 
0.0 
0.0 
0.792 
0.00 


isolated+undamped 
78.6 
727.9 
0.796 
0.00 

Midspan 
isolated+damped 
46.8 
555.3 
0.523 
9.16 


nonisolated 
7.8 
451.0 
2.300 
0.00 
El Centro 

isolated+undamped 
23.0 
233.1 
0.225 
0.00 

Bearing 
isolated+damped 
11.9 
120.3 
0.147 
13.30 


nonisolated 
0.0 
0.0 
0.208 
0.00 


isolated+undamped 
23.7 
239.5 
0.240 
0.00 

Midspan 
isolated+damped 
11.9 
118.0 
0.139 
13.30 


nonisolated 
1.9 
127.9 
0.560 
0.00 
Therefore, if the bridge is not flexible enough in the vertical plane, the response of acceleration can be greater than the ground acceleration, even though the response of displacement at midspan is negligible. For the isolated undamped case, the acceleration responses are reduced over 50% at midspan of the bridge compared to the nonisolated case for both earthquakes, resulting in reduced inertia force on the bridge structure. The isolated damped case is of more practical significance as the response of displacement and velocity due to the increase in the flexibility from the isolation device can be reduced over 50%. This is done by the addition of over 10% damping ratio without causing overdamping as may be the case in a linear damped system.
For the isolated cases, the maximum response of acceleration throughout the bridge span is limited to 0.54 for Northridge and 0.2 for El Centro. Hence, the bridge is protected from vertical ground motion by introducing spring bearings which result in a more flexible bridge system. The large response of displacement due to spring flexibility can effectively be reduced by damping.
Fig.10 illustrates three cases of maximum vertical dynamic deflection shape throughout the bridge model occurring at the same time. It can be observed that the vertical deflection throughout the span length of isolated bridge model has less variation than the nonisolated one, and that the relative deflection is reduced which is likely to result in reduction of stresses.
Figure 10.
Vertical Deflection of Bridge Model Throughout Span 
In order to achieve better protection for the bridge subjected to strong vertical ground motion, helical springs are used as bearings with fluid dampers as energy dissipators. The bridge model is supported on spring bearings in lieu of conventional rigid bearings. The springsupported bridge is modeled to be much more flexible in the vertical direction. It is concluded that the response of acceleration in an isolated damped bridge model, particularly at the midspan, has been greatly reduced up to 75% compared to the nonisolated case. Therefore, the inertia forces induced by acceleration response in the bridge structure are also reduced which in turn benefits the structural design. In addition, the damage from the deflection gap between the inspan and bearing is alleviated because of the flexibility due to the spring bearings.
The larger response of displacement of the bridge throughout the span, as a tradeoff of the flexible spring bearings, can be effectively reduced by adding fluid dampers to the isolation system. In general, bridge structures have damping levels less than 10% critical. The damping level of a structural system isolated by fluid dampers could be over 20% with more energy absorbed, offering a dramatic reduction in deflection at no cost of increase in base shear.
It is noted that extra damping becomes less efficient at higher damping levels. Reduction of response will hardly be achieved by only adding more dampers, especially in a less flexible bridge system. The stiff system is more sensitive to the ground acceleration excitation and the response of acceleration is greater than the flexible system. Use of helical springs as bearings will certainly provide additional flexibility for the bridge structural system in which the displacements are controlled by dampers. The proposed bridge system is effective in greatly reducing the structural responses and relative deflection.
Financial support of this project has been provided by the National Science Foundation, Grant No. CMS9502772. All earthquake ground motion records were made available from the University of Southern California via World Wide Web, and from the University of California at Berkeley via Gopher.
10.
Waller, R.A. 1969. Building on
Springs. Pergamon Press, UK.
15.
Clough, R.W., and Penzien, J. 1993.
Dynamics of Structures. 2nd
Edition, McGrawHill, New York.
16.
Azar, J.J. 1972. Matrix
Structural Analysis. Pergamon Press, UK.
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Journal of Structural Engineering, Vol. 1, No.2 (2001) 98110 