seismic vibration control of bridges with nonlinear tuned
TRANSCRIPT
Seismic vibration control of bridges with
nonlinear tuned mass dampers.
Final Report
December 2020
Rajesh Rupakhety
Said Elias
Seismic vibration control of bridges with
nonlinear tuned mass dampers.
Final report
Rajesh Rupakhety
Said Elias
Report No. 2020-002
Selfoss 2020
Rajesh Rupakhety, Said Elias, Seismic vibration control of bridges with
nonlinear tuned mass dampers. Earthquake Engineering Research
Centre, University of Iceland, Report No. 2020-002 Selfoss, 2020
© Earthquake Engineering Research Centre, University of Iceland, , and
authors
All rights reserved. No part of this publication may be reproduced,
transmitted, or distributed in any form or by any means, electronic or
mechanical, including photocopying, without permission from the
Earthquake Engineering Research Centre or the authors.
Abstract
This study presents the design methodology and effectiveness of a
hysteretic tuned mass dampers in controlling seismic response of
structures. The hysteretic dampers (n-TVAs) consist of a mass attached
by a hysteretic spring to the host structure. The stiffness and strength of
the spring is tuned in such a way that the hysteretic energy dissipated by
the damper is equal to the viscous energy dissipated by a linear optimal
tuned mass damper. Due to the non-linear behaviour of the device,
optimal design is excitation dependent. A general procedure to design the
optimal parameters of the device is proposed in this study. The procedure
relies on an iterative solution of the equation of motion on a nonlinear
system. Analysis using some typical ground motions show that the
iterative procedure converges rapidly. The resulting n-TVAs are found
to be effective in controlling seismic response of a reinforced concrete
bridge structure, used here as a case study example. The proposed device
is at least as effective as existing TMD solutions. Investigation of the
proposed device using recorded and simulated ground motions
corresponding to a typical near-fault scenario in South Iceland shows that
the effectiveness of the device is highly dependent on the ground motion,
and for those ground motions which exert excessive demands on the
uncontrolled structure, the proposed devices can reduce the base shear
and mid-span displacement demands of the case study bridge by as much
as 25%.
Contents
1. Introduction ......................................................................................................................7
2. Design and evaluation methodology ................................................................................9
3. Numerical study ..............................................................................................................13 3.1. Force deformation curves ............................................................................................................. 14 3.2. System transfer function ................................................................................................................ 14 3.3. Energy assessment ........................................................................................................................... 16 3.4. Evaluation of the effectiveness of TVAs ................................................................................... 17
4. Evaluation of effectives is South Iceland ......................................................................22 4.1. Design of the n-TVAs ....................................................................................................................... 23 4.2. Robustness of the n-TVAs .............................................................................................................. 24
5. Conclusions ......................................................................................................................29
ACKNOWLEDGEMENTS ...............................................................................................31
DISCLAIMER ....................................................................................................................31
BIBLIOGRAPHY ...............................................................................................................32
List of figures
FIGURE 1. (A) MODEL OF CONTINUOUS BRIDGE; (B) REPRESENTATION OF THE TVA IN X
AND Y DIRECTIONS; (C) FORCE DEFORMATION CURVE OF THE N-TVA. 10 FIGURE 2. FLOW CHART FOR DESIGN OF THE NON-LINEAR SPRING OF THE N-TVAS. 14 FIGURE 3. FORCE DEFORMATION CURVES OF THE NON-LINEAR SPRING OF THE N-
TVAS SUBJECTED TO DIFFERENT GROUND MOTIONS. 15 FIGURE 4. NORMALIZED TRANSFER FUNCTIONS OF THE BRIDGE MID-SPAN
ACCELERATION TO WHITE NOISE GROUND ACCELERATION IN LONGITUDINAL
AND TRANSVERSE DIRECTIONS OF THE BRIDGE. 16 FIGURE 5. INPUT, DAMPING AND STRAIN ENERGY FOR DIFFERENT TVA SCHEMES
SUBJECTED TO THREE DIFFERENT GROUND MOTIONS. 17 FIGURE 6. TIME VARIATION OF THE NORMALIZED PIER BASE SHEAR UNDER THE 1940
IMPERIAL VALLEY, EARTHQUAKE GROUND MOTION FOR NC, C-TVAS, O-TVAS,
AND N-TVAS. 18 FIGURE 7. SAME AS IN FIGURE 6, BUT FOR THE 1989 LOMA PRIETA EARTHQUAKE GROUND
MOTION. 18 FIGURE 8. SAME AS IN FIGURE 6, BUT FOR THE 1995 KOBE EARTHQUAKE GROUND MOTION.
19 FIGURE 9. TIME VARIATION OF THE MID-SPAN DISPLACEMENT UNDER THE 1940 IMPERIAL
VALLEY EARTHQUAKE GROUND MOTION. 19 FIGURE 10. SAME AS IN FIGURE 9 BUT FOR THE 1989 LOMA PRIETA EARTHQUAKE GROUND
MOTION. 20 FIGURE 11. SAME AS IN FIGURE 9 BUT FOR THE 1995 KOBE EARTHQUAKE GROUND MOTION.
20 FIGURE 12. COMPARISON OF THE PERFORMANCE OF N-TVA WITH/WITHOUT DASHPOT TO
THAT OF O-TVA. THE TOP PANEL SHOWS MID-SPAN DISPLACEMENT IN THE LONGITUDINAL AND TRANSVERSE DIRECTIONS, AND THE BOTTOM PANEL SHOWS CORRESPONDING STROKE OF THE DEVICES. 22
FIGURE 13. ELASTIC RESPONSE SPECTRA (5% DAMPED) OF THE RECORDED AND SIMULATED GROUND MOTIONS USED IN THIS STUDY. THE RIGHT AND LEFT COLUMNS REPRESENT SEISMIC ACTION IN THE TRANSVERSE AND THE LONGITUDINAL DIRECTIONS OF THE BRIDGE, RESPECTIVELY. 23
FIGURE 14. BASE SHEAR DEMAND ON THE STRUCTURE (TOP: LONGITUDINAL AND BOTTOM: TRANSVERSE) OF THE UNCONTROLLED AND CONTROLLED STRUCTURE SUBJECTED TO SIMULATE GROUND MOTION NUMBER 2. 25
FIGURE 15. SAME AS IN FIGURE 13, BUT FOR GROUND MOTION RECORDED AT HELLA STATION. 26
FIGURE 16. SAME AS IN FIGURE 14 BUT FOR KALDARHOLT STATION. 26 FIGURE 17. SAME AS IN FIGURE 14 BUT FOR SELSUND STATION. 27 FIGURE 18. SAME AS IN FIGURE 14 BUT FOR THORSARBRU STATION. 27 FIGURE 19. DISPLACEMENT RESPONSE OF THE CONTROLLED AND UNCONTROLLED
STRUCTURE WHEN SUBJECTED TO GROUND MOTION RECORDED AT THE FLAGBJARNARHOLT STATION; THE TOP AND BOTTOM ROWS CORRESPOND TO LONGITUDINAL AND TRANSVERSE DISPLACEMENTS, RESPECTIVELY, AT THE MID-SPAN OF THE BRIDGE. 28
FIGURE 20. SAME AS IN FIGURE 19 BUT FOR HELLA STATION. 28 FIGURE 21. SAME AS IN FIGURE 19 BUT FOR KALDARHOLT STATION. 29
List of tables
TABLE 1. PEAK RESPONSES FOR DIFFERENT CONTROL SCHEMES. ..................................................... 21 TABLE 2. RESPONSE REDUCTION AND STOKE OF THE N-TVAS CORRESPONDING TO THE 8
DESIGN GROUND MOTIONS. ......................................................................................................................... 24 TABLE 3. BASE-SHEAR REDUCTION DUE TO N-TVAS COMPARED TO C-TVAS AND O-
TVAS ....................................................................................................................................................................... 24
1. Introduction
Lifeline structures like bridges need to be especially robust against natural hazards such as
earthquakes and strong winds. One approach to performance enhancement of bridges
against dynamic loads is through vibration control using passive, active, semi-active and
hybrid systems. Passive devices have the advantages of being simple to install and cost-
effective as compared to active, semi-active and hybrid systems. Generally, these controlled
structures assumed to remain in elastic range. An alternate, but time consuming, and
perhaps, not as insightful theoretically, is the Monte Carlo approach of response simulation.
A well-controlled structure is not expected to be stressed too much beyond its elastic
capacity, and the general trend in analysis of vibration control methods has been to assume
structures responding elastically. Base isolation, a passive device, is one of the most widely
used technology to reduce seismic response of bridges (Nagarajaiah et al., 1991; Jangid,
2004; Sahasrabudhe and Nagarajaiah, 2005; Agrawal et al, 2009; Nagarajaiah et al., 2009;
Madhekar and Jangid, 2010; Attary et al., 2015; Elias and Matsagar, 2017, 2019). The other
simple passive vibration control devices are the so-called tuned vibration absorbers (TVAs).
TVAs are widely used in tall buildings and bridges around the world. For broadband
efficiency in seismic circumstances, one of the practical options is to use the multiple TVAs
(MTVAs) with distributed natural frequencies, which have been studied by Li (2000, 2002),
Li and Liu (2003). Studies have shown that both MTVAs and tuned tandem mass dampers
(TTMDI) are robust devices for vibration mitigation of structures (Li and Cao, 2019; Cao
and Li, 2019; Chang and Li, 2019). Robustness of the vibration control scheme is as off-
tuning can have adverse effects on the structure. Performance enhancement of TVAs is
gaining widespread research interest. Lin et al. (2009) reported that the TVAs are more
effective in vibration control of structures under impulse‐like ground motion with more
forward‐and‐backward cycles. Aldemir et al. (2012) compared the performance of
passive and active TVAs in seismic vibration control of structures. They found that
conventional optimally designed TVAs could reduce the displacement response of the
structure but were not effective in controlling acceleration response. Active TVAs were
more found to be more effective to suppress both displacement and acceleration responses.
Generally, TVAs are useful for i) wind response mitigation of long span bridges, ii)
mitigation of vertical vibration of bridges caused by vehicles, iii) vibration control of
pedestrian bridges, and iv) seismic response mitigation of bridges.
Effectiveness of TVAs in reducing wind-induced vibrations of long-span bridges has been
studied by Lin et al. (2000a and 2000b), Gu et al. (2001), Pourzeynali and Datta (2002),
Chen and Kareem (2003), and Kwon and Park (2004). These studies showed that TVAs are
effective in reducing buffeting response of such bridges. Chen and Wu (2008) studied the
efficiency of multiple TVAs (MTVAs) in controlling wind-induced vibration of bridges.
Casciati and Giuliano (2009) demonstrated the usefulness of MTVAs in reducing gust
response of towers in suspension bridges. Casalotti et al. (2014) estimated the parameters
of hysteretic TVAs for multi-mode flutter mitigation in long-span suspension bridges.
Bortoluzzi et al. (2015) reported significant reduction in mid-span vibration of TVA-
8
controlled bridges subjected to wind forces. Ubertini (2010) and Ubertini et al. (2015)
present a probabilistic approach for arranging TVAs for flutter suppression in long-span
bridges. Verstraelen et al. (2016) found that the effectiveness of the absorbers obtained from
wind tunnel tests was higher than mathematically predicted.
Similarly, many researchers (Yang et al., 1997; Moreno and Dos-Santos, 1997; Kwon et al.,
1998; Wang et al., 2003; Chen and Cai, 2004; Chen and Chen, 2004; Chen and Huang,
2004; Yau and Yang, 2004a; Yau and Yang, 2004b; Li et al., 2005; Lin et al., 2005; Wu and
Cai, 2007; Hijmissen and van Horssen, 2007; and Hijmissen et al., 2009) have demonstrated
the usefulness of TVAs in reducing vertical displacements, absolute accelerations, end
rotations, and train accelerations during resonant speeds in bridges. Improved performance
of the MTVAs as compared to the TVAs was also reported in these studies. Liu et al. (2012)
showed the efficiency of MTVAs in mitigating vibrations of bridges caused by high speed
trains. TVAs have also been found to be useful in reducing vibrations of footbridges
(Dallard et al., 2001a, 2001b; Poovarodom et al. 2001, 2002, 2003; Yang et al., 2007; and
Hoang et al., 2008). Li et al. (2010) conclude that MTVAs designed by random optimization
were more efficient than conventionally designed ones in minimizing dynamic response
during crowd-footbridge resonance, and that appropriate frequency spacing improvement
efficiently diminishes the off-tuning effect of the MTVAs. Daniel et al. (2012) studied the
use of MTVAs for multi-modal control of pedestrian bridges. They concluded that MTVAs
are more robust than TVAs. Andersson et al. (2014) studied the use of passive and adaptive
damping systems to mitigate vibrations in a railway bridge during resonance. It was reported
that passive TVA and pendulum dampers showed a significant improvement under the same
detuned conditions. Recently, van Nimmen et al. (2016) concluded that a simplified design
procedure based solely on the contribution of the resonant mode is not appropriate for
evaluating the dynamic performance of footbridges installed with TVAs. Lievens et al.
(2016) studied robust TVAs in vibration mitigation of footbridges. Tubino et al. (2016)
studied MTVAs for vibration mitigation of footbridges subjected to several loading
conditions. They concluded that the MTVAs reduced the vibration amplitudes considerably
under almost all loading conditions considered in their study. Earthquake-induced vibration
is also a major concern in design, operation and maintenance of bridges. As reported by Lee
et al. (2013), many bridges in the United States of America (USA) were damaged or totally
collapsed by earthquakes during 1980-2012. The study showed that concrete bridges are
more vulnerable to earthquakes than steel bridges. Vestroni et al. (2014), and Carpineto et
al. (2014) reported experimental tests on a simply-supported beam with hysteretic TMDs
under harmonic and random base excitations. The hysteretic TMDs placed at the beam mid-
span was found to diminish vibrations in the first flexural mode by up to 98% for harmonic
excitations and 87% for random excitations. Pisal and Jangid (2016) conclude that, for
seismic excitation, optimized multiple tuned vibration friction absorbers (MTVFAs)
positioned at mid-span are more efficient in response reduction than optimal single TVFA
(STVFA) and MTVFAs distributed along the length of the bridge. Miguel et al. (2016a and
2016b) showed the effectiveness of robust TVAs in controlling the vibration of bridges
subjected to ground motion. They showed that classical analytical techniques could be
effectively used in robust optimal design of the STVA. Matin et al. (2014, 2017, and 2018)
9
presented multi-mode control of continuous concrete bridges using MTVAs. They reported
that MTVAs can reduce seismic response considerably, and that their performance is
dependent on their spatial placement on the bridge. In a recent study, Lu et al. (2018)
reported that the TVA damping coefficient was significantly increased by the introduction
of eddy‐current damping mechanism. A detail review of nonlinear dissipative devices is
available in Lu et al. (2018).
In most studies, TVAs installed on bridges are assumed to be elastic. This assumption
implies unlimited displacement capacity of such devices. In practice, such devices are
expected to yield under certain conditions. It is therefore realistic to model their inelastic
behavior. In fact, non-linear yielding behavior can be utilized to reduce displacement
demand on the structure. The main objective of the present study is to investigate design of
non-linear TVAs (n-TVAs) and their efficiency in reducing seismic response of bridges. In
this study, the springs of the TVAs are modelled with hysteretic behavior. Design of TVAs
with non-linear springs is formulated and the performance of the device is compared to their
linear counterparts, such as conventional linear TVSs (c-TVAs) and optimum linear TVAs
(o-TVAs).
2. Design and evaluation methodology
A reinforced concrete (RC) continuous span bridge, as shown in Figure 1, is considered in
this study. The piers and rigid abutments support the conventional bridge bearings as
illustrated in Figure 1(a). In Figure 1(b), kx and cx are the stiffness and damping coefficients
of the TVA in x(longitudinal) direction. Similarly, ky and cy are the stiffness and damping
coefficients in y (transverse) direction. The mass m is attached to both springs to control the
bidirectional responses of the bridge. Figure 1 shows the details of TVA for only one of the
spans of the bridge. Similar TVAs are provided in all the spans of the bridge. The springs
are inelastic with hysteresis curves like the one shown in Figure 1(c). The equations of
motion of the bridge equipped with linear TVAs is
s s s tM Q C Q K Q F+ + = (1)
where ][ sM , ][ sC , and ][ sK are the mass, damping, and stiffness matrices of the controlled
bridge, respectively of order )22()22( nNnN ++ ; N is the number of degrees of freedom
of the bridge model and n is the number of TVAs, respectively;
T
1 2 N 1 n 1 2 N 1 n, ,{ } { , , , , , }Q U U U u u V V V v v= , }{Q , and }{Q are the displacement, velocity, and
acceleration vectors, respectively. In the equation, {Ft} is the forcing function given by
s x gx s y gyM r a M r a− − , where xr and yr are the influence coefficient vectors in the
longitudinal and transverse directions, and gxa and gya are the corresponding ground
10
accelerations. Moreover, Ui and Vi are the displacements of the ith degree of freedom of the
bridge in the longitudinal and transverse directions, respectively, and iu , iv are those of the
ith TVA. When the TVAs have nonlinear springs, the equation of motion needs to be solved
numerically using iterative methods.
Figure 1. (a) Model of continuous bridge; (b) Representation of the TVA in X and Y
directions; (c) Force deformation curve of the n-TVA.
The design parameters of linear conventional TVAs (c-TVAs) are given by
2
x l xk m= (2-a)
2
y l yk m= (2-b)
where, kx and ky are the stiffness of the c-TVAs in longitudinal and transverse directions
respectively; ml is the mass of c-TVAs placed on each span; and x and
y are the
corresponding vibration frequencies, which are assumed to be the same as the fundamental
frequency of the bridge in the X and Y directions. For a given damping ratio d , the
damping coefficients (xc and
yc ) of the c-TVAs are calculated as:
2x d l xc m = (3-a)
2y d l yc m = (3-b)
In case of the optimum linear TVAs (o-TVAs), the frequency of the device in each direction
is a fixed proportion ( optf ) of the corresponding fundamental frequency of the bridge (1 1,x y
).
1 1, ,x y optx x opty yf f = (4-a)
The fixed proportion is known as the optimum tuning frequency ratio. This ratio, for lightly
damped structure, like the bridge being considered here, is given by Sadek et al. (1997)
11
1 1
x
optx optx
x x
f
= − + +
(4-b)
11
11
1 1
y
opty opty
y y
f
= − + +
(4-c)
where, the optimum damping ratios are given by
1 1
x
doptx x
= + + +
(5-a)
1 1
y
dopty y
= + + +
(5-b)
In Equations (4) and (5), is the ratio of the mass of the TVA to the total mass of the
bridge; x and y are the mode shape amplitudes of the bridge at the location of TVA in
longitudinal (X) and transverse (Y) directions, respectively. Similarly, x and y are the
damping ratios of the bridge in X and Y directions, respectively. Having obtained the
optimized frequencies from equations 4 and 5, the stiffness and damping coefficients of the
o-TVA can be computed from equations 2 and 3, respectively.
In case of non-linear TVAs (n-TVAs), the hysteretic behavior of the springs is assumed to
follow the Wen (1976) model. According to this model, the restoring force of the n-TVAs
in x and y directions are given as,
(1 )nx nx xx xF ak u a F Z= + − (6-a)
(1 )ny ny yy yF ak v a F Z= + − (6-b)
where, xxF , and
yyF are the yield strengths of the spring in x and y directions, respectively;
a is the ratio of post-yielding stiffness to initial stiffness, represented by nxk and
nyk ; u and
v are the displacements of the springs. xZ , and
yZ are the hysteretic displacement
components satisfying the following 1st order differential equations.
1p p
yield x x xu Z Au u Z u Z −
= − − (7-a)
1p p
yield y y yv Z Av v Z v Z −
= − − (7-b)
where, yieldu and yieldv are the yield displacement and , , p , and A are dimensionless
parameters of the model. Here, p controls the curve smoothness during transition from
elastic to plastic state, and its value is fixed at 10 to simulate bilinear behavior. The
parameters and control the shape of hysteresis loop; and A is the restoring force
amplitude (Wen 1976). In this study, and A are taken as 0.5, and 1, respectively. The n-
TVAs are designed in such a way that their effective stiffness (secant stiffness at maximum
displacement) is equal to the stiffness of the o-TVAs. Let us consider peak strokes of xD
12
and yD in X and Y directions. Then the ratio of yield displacement to peak stroke is defined
as /x yield xR u D= and /y yield yR v D= . This ratio controls the extent of plastic deformation in the
device and is reciprocal of ductility demand. A value of this ratio less than 1 implies inelastic
behavior. This can be treated as a design variable. Based on the assumption that the energy
dissipated by a o-TVA is the same as that dissipated by an elastic-plastic spring of the n-
TVA, an approximate value of R can be computed from the damping ratio of the o-TVA
by using the following equation.
, ,1
2x y doptx doptyR
= − (8)
For the structure being studied here, the value of R is approximately 0.75 and is taken to
be equal in the X and Y directions. For a bilinear model, the initial stiffness of the n-TVA
is related to its effective stiffness by the following equations.
( )1
x
nx
x x
kk
R a R=
− + (9-a)
( )1
y
ny
y y
kk
R a R=
− +
(9-b)
Because the peak stroke of the device is not known a priori but depends on the excitation,
an iterative procedure needs to be followed to estimate the parameters of the n-TVAs. In
the proposed method, an initial value of the peak stoke is assumed. This value can be taken
as the peak stroke of the o-TVA for a given ground motion. With this peak stroke and given
R , an initial value of yield displacement is computed, which is then used to estimate the
yield force by multiplying with the initial stiffness. With these parameters established, non-
linear time history analysis is carried out with different ground motions and the actual peak
stroke corresponding to each ground motion is computed. In the next iteration the computed
peak stroke is used, and the process is repeated until the assumed peak stroke is equal to the
computed peak stroke within some reasonable tolerance. The design and iteration is carried
out independently in X and Y directions. The results, in each direction and for each ground
motion, an estimate of the yield displacement and yield force. For each direction, the final
yield force is taken as the maximum of the yield forces induced by the different ground
motions (see Figure 2). This implies that the design is optimal for the strongest ground
motion used in the analysis but might not dissipate as much hysteretic energy when
subjected to less demanding ground motions. In this study, three different ground motions
were used, as will be described subsequently, and it was found that the iterations converged
very fast. In practical scenarios, design ground motion specified by relevant seismic
provision can be used in the same manner as is described here. Unlike the o-TVAs, viscous
dashpots are not used with n-TVAs, where damping is provided by hysteretic energy
dissipation.
13
3. Numerical study
The responses of both controlled and uncontrolled bridges are computed for bidirectional
ground shaking. Seismic response of long-span structures is affected by wave propagation
and incoherency of ground motion (see, for example, Zerva 2016). In the example
considered here, the span length is only 30m and the fundamental frequency of the bridge
is ~2 Hz. In such cases, the lagged coherency is almost equal to 1 (see, for example,
Rupakhety and Sigbjörnsson, 2012 and AfifChaouch et al., 2016). In this study, for sake of
simplicity, wave passage and incoherency effects are not considered, and it is assumed that
the same ground motion is experienced by all the supports of the bridge. Further, standard
codes such as Eurocode 8 (2005); American Society of Civil Engineers (ASCE) standards
7-10 (2010); and Federal Emergency Management Agency-356 (FEMA-356, 2000) require
at least three earthquake ground motions for time history analysis of structures for seismic
design and evaluation. The ground motions used in this study are the horizontal components
of the 1940 Imperial Valley Earthquake (El Centro station); 1989 Loma Prieta Earthquake
(Los Gatos Presentation Centre); and the 1995 Kobe Earthquake (Japan Meteorological
Agency - JMA station). The peak ground acceleration (PGA) of the Imperial Valley, Loma
Prieta, and Kobe earthquake ground motions applied on the bridge are: 0.35g, 0.57g, and
0.86g in the longitudinal direction and 0.21g, 0.61g, and 0.82g in the transverse direction,
respectively, where g denotes the gravitational acceleration.
A three-span continuous reinforced concrete (RC) bridge is considered in this study. The
details of the RC bridge sections are given in Elias and Matsagar (2017). Cross-sectional
area of deck and piers are respectively, 3.57 m2 and 1.767 m2. The moment of inertia of the
deck in both principal directions of the cross-section are equal and considered to be 2.08
m4. Similarly, the moment of inertia of the piers is 0.902 m4 in both principal directions of
its cross-section. Young’s modulus of elasticity and mass per unit volume respectively are
3.6 × 107 kN/m2 and 23.536 kN/m3 for the RC material used in this bridge. Each span is 30
m long and pier height is 10 m. The fundamental frequency of the RC bridge is 1.86 Hz in
both longitudinal and transverse directions. Damping is assumed to be 5% of critical.
The c-TVAs, o-TVAs and n-TVAs are located at the center of each span. A mass ratio of
0.01 is assumed for all the types of TVAs. The total mass ratio considering TVAs in each
of the three spans is therefore 0.03. The stiffness, damping coefficient, and tuning frequency
ratio of the c-TVAs are 1.250 107 N/m, 2.133 105 N-sec/m, and 1, respectively. These
properties of the o-TVAs are 1.214 107 N/m), 3.132 105 N-sec/m. The properties of the n-
TVAs were designed in an iterative manner described above. The initial stiffness in the
longitudinal and transverse directions is 1.592 107 N/m and the corresponding yield
displacement are 18.4 cm and 9.6 cm. A flowchart of the design process of the n-TVA is
presented in Figure 2.
14
Figure 2. Flow chart for design of the non-linear spring of the n-TVAs.
3.1. Force deformation curves
The force deformation curves of the n-TVA springs in longitudinal and transverse directions
are shown in Figure 3 for three different ground motions. The forces are normalized by the
weight of the n-TVAs. The design of the n-TVAs was controlled by the Kobe Earthquake
ground motion in the transverse direction and Loma Prieta Earthquake ground motion in the
longitudinal direction. The hysteresis curves shown in Figure 3 show largest inelasticity for
these ground motions (see the middle panel in the top row and the right panel in the bottom
row). When subjected to the Imperial Valley Earthquake ground motion, the n-TVAs remain
elastic.
3.2. System transfer function
The differences in dynamic properties of the different control schemes can be studied by
comparing the transfer functions of the controlled structure to that of the uncontrolled
structure. This section presents the transfer function, relating ground acceleration to the
acceleration response of the bridge mid-span. If ( )AxS and ( )AyS denote the auto-power
spectral density of the acceleration response in longitudinal and transverse directions
respectively, and ( )FxS and ( )FyS denote auto-power spectral density of corresponding
ground acceleration, the transfer functions are defined as
( )( )
( )
2 Ax
x
Fx
SH
S
=
(10-a)
( )( )
( )
2 Ay
y
Fy
SH
S
=
(10-b)
Figure 6 shows transfer functions normalized by the maximum ordinate of the transfer
function of the uncontrolled structure. The excitation is taken as white noise (see, for
example, Elias et al., 2016; 2018). The effect of the TVAs is to reduce the amplitude of the
15
resonant peaks, making them wider, which is a consequence of additional damping provided
by the devices. The reduction in resonant peak is more prominent when using o-TVAs as
compared to the c-TVAs. The area under the transfer function, for a white noise excitation,
provides the ratio between the root mean square response and noise intensity. It is therefore
an indicator of the effectiveness of vibration transmission through a structure; lower area
implying reduction in vibration. These areas for different TVA schemes are indicated in
Figure 4, which shows that the n-TVAs are the most effective, although not very different
from o-TVAs. It should be noted that these conclusions apply only to white noise excitation
and reduction in root mean square response. The effectiveness of the devices in reducing
peak response is investigated in the following sections.
Figure 3. Force deformation curves of the non-linear spring of the n-TVAs subjected to
different ground motions.
-3
-2
-1
0
1
2
3
-0.2 -0.1 0.0 0.1 0.2
Norm
aliz
ed F
orc
e
Longitudinal
Stroke (m)
Imperial Valley, 1940
PGA = 0.35g PGA = 0.57g PGA = 0.86g
PGA = 0.82gPGA = 0.61gPGA = 0.21g
-3
-2
-1
0
1
2
3
-0.2 -0.1 0.0 0.1 0.2
Stroke (m)
Loma Prieta, 1989-3
-2
-1
0
1
2
3
-0.2 -0.1 0.0 0.1 0.2
Stroke (m)
Kobe, 1995
-2
-1
0
1
2
-0.1 0.0 0.1
Norm
aliz
ed F
orc
e
Transverse
Stroke (m)
-2
-1
0
1
2
-0.1 0.0 0.1
Stroke (m)
-2
-1
0
1
2
-0.1 0.0 0.1
Stroke (m)
16
Figure 4. Normalized transfer functions of the bridge mid-span acceleration to white noise
ground acceleration in longitudinal and transverse directions of the bridge.
3.3. Energy assessment
Ground shaking imparts energy into a structure, part of which is dissipated through damping
and hysteresis in the structure and the remaining energy accounts for strain and kinetic
energy of the structure. The input energy at the ith time step normalized by the total mass of
the structure (Ei) is given by
( )
T T
i
1 1t
1 i i
i s x gx i s y gyE Q M r a Q M r aM
= − +
(13)
where iQ is the incremental displacement vector between two consecutive time steps. It
was reported by Elias et al. (2016) and Elias (2018) that the structures installed with the
TVAs were able to dissipate more energy than those without TVAs. The damping energy at
the ith time step per unit mass of the structure is
( )
T
1t
1 i
di i s i
i
E Q C QM =
= −
(14)
The strain energy per unit mass at the ith time step is
( )
T
t
1 1
2si i s iE Q K Q
M
=
(15)
In these calculations, the energy dissipated by the TVAs are not considered, i.e., the matrices
and vectors in equations 13-15 contain only the bridge degrees of freedom. Figure 6 shows
the accumulation of different types of energies defined above. The top panel shows damping
energy dissipated by the structure along with the input energy (dashed curve). The bottom
panel shows strain energy. The energy corresponding to different TVAs and ground motions
are marked as indicated in the figure. The results show that the damping energy
corresponding to the structural degrees of freedom is considerably reduced when TVAs are
used, which is a direct consequence of reduced vibration amplitudes. The reduction in
damping energy is up to about 50% for one of the ground motions considered here. The
1.6 1.8 2.0 2.20.0
0.2
0.4
0.6
0.8
1.0A
NC= 9.40
Longitudinal
No
rmal
ized
Tra
nsf
er F
un
ctio
n H
()
2 f
or
Acc
eler
atio
n a
t M
id-S
pan
(M
agn
itu
de)
Frequency (Hz)
NC
c-TVAs
o-TVAs
n-TVAs
Ac-TVAs
= 8.61
An-TVAs
= 7.95
Ao-TVAs
= 8.11
1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.30.0
0.2
0.4
0.6
0.8
1.0
Transverse
Frequency (Hz)
ANC
= 9.30
Ac-TVAs
= 8.44
An-TVAs
= 7.32
Ao-TVAs
= 7.77
17
performance of n-TVAs is much better than c-TVAs and comparable to that of o-TVAs.
Similarly, the strain energy of the bridge is considerably reduced by the TVAs.
Figure 5. Input, damping and strain energy for different TVA schemes subjected to three
different ground motions.
3.4. Evaluation of the effectiveness of TVAs
This section presents different response parameters of the uncontrolled structure and
different control schemes. Figure 6, Figure 7, and Figure 8 present base shear of the bridge
normalized by its weight for different ground motions and control schemes.
Displacement response at the mid-span is presented in Figure 9 through Figure 11. The results
show considerable decrease in base shear due to the TVAs. The peak base shears and
percentage in peak reductions (in bracket) are listed in Table 1 for all control schemes and
ground motions. The n-TVA induces up to 33.5% and 34% reduction in base shear in the
longitudinal and transverse directions, respectively. In some cases, the c-TVAs seem to
amplify base shear. Although the performance of o-TVAs and n-TVAs are, in general
comparable, n-TVAs are found to be more effective than o-TVAs for some ground motions.
In Figure 9 to Figure 111, it is observed that the displacement at mid-span of the RC bridge
is significantly reduced by o-TVAs and n-TVAs. It is evident from the results that, for the
ground motions considered here, a maximum response reduction of 22% is achieved by c-
0.00
0.09
0.18
0.27
0.36
0 10 20 30 40
50%
Ener
gy (
m2/s
ec2)
Damping Energy
Time (sec)
Imperial Valley, 1940
PGA = 0.35g PGA = 0.57g PGA = 0.86g
PGA = 0.82gPGA = 0.61gPGA = 0.21g
0.0
0.3
0.6
0.9
1.2
1.5
1.8
0 10 20 30 40 50
45%
Time (sec)
Loma Prieta, 19890.00
0.45
0.90
1.35
0 10 20 30 40
Time (sec)
Kobe, 1995
45%
0.00
0.03
0.06
0.09
0.12
0 5 10 15 20 25
Ener
gy (
m2/s
ec2)
Strain Energy
Time (sec)
0.00
0.09
0.18
0.27
0.36
10 15 20 25
NC
c-TVAs
o-TVAs
n-TVAs
Time (sec)
0.00
0.09
0.18
0.27
0.36
10 15 20 25
Time (sec)
18
TVAs. At the same time, the response is amplified by 6% for one of the ground motions.
Optimization of the TVAs based on the formulations proposed by Sadek et al. (1997)
improved the performance.
Figure 6. Time variation of the normalized pier base shear under the 1940 Imperial Valley,
Earthquake ground motion for NC, c-TVAs, o-TVAs, and n-TVAs.
Figure 7. Same as in Figure 6, but for the 1989 Loma Prieta Earthquake ground motion.
-0.6
-0.3
0.0
0.3
0.6
0 5 10 15 20 25
PGA = 0.35g
Bas
eshea
r/W
eight
of
th B
ridge
(W)
Longitudinal
Time (sec)
-0.6
-0.3
0.0
0.3
0.6
0 5 10 15 20 25
Time (sec)
Imperial Valley, 1940
-0.6
-0.3
0.0
0.3
0.6
0 5 10 15 20 25
Time (sec)
-0.4
-0.2
0.0
0.2
0.4
0 5 10 15 20 25
PGA = 0.21g
Bas
eshea
r/W
eight
of
th B
ridge
(W)
Transverse
Time (sec)
-0.4
-0.2
0.0
0.2
0.4
0 5 10 15 20 25
Time (sec)
c-TVAs
o-TVAs
NC
c-TVAs
o-TVAs
n-TVAs-0.4
-0.2
0.0
0.2
0.4
0 5 10 15 20 25
Time (sec)
-1.6
-0.8
0.0
0.8
1.6
0 5 10 15 20 25
PGA = 0.57g
Bas
eshea
r/W
eight
of
th B
ridge
(W)
Longitudinal
Time (sec)
-1.6
-0.8
0.0
0.8
1.6
0 5 10 15 20 25
Time (sec)
Loma Prieta, 1989-1.6
-0.8
0.0
0.8
1.6
0 5 10 15 20 25
Time (sec)
-0.6
-0.3
0.0
0.3
0.6
0 5 10 15 20 25
PGA = 0.61g
Bas
eshea
r/W
eight
of
th B
ridge
(W)
Transverse
Time (sec)
-0.6
-0.3
0.0
0.3
0.6
0 5 10 15 20 25
Time (sec)
c-TVAs
o-TVAs
NC
c-TVAs
o-TVAs
n-TVAs
-0.6
-0.3
0.0
0.3
0.6
0 5 10 15 20 25
Time (sec)
19
Figure 8. Same as in Figure 6, but for the 1995 Kobe Earthquake ground motion.
Figure 9. Time variation of the mid-span displacement under the 1940 Imperial Valley
Earthquake ground motion.
-1.6
-0.8
0.0
0.8
1.6
0 5 10 15 20 25
PGA = 0.86g
Bas
eshea
r/W
eight
of
th B
ridge
( W)
Longitudinal
Time (sec)
-1.6
-0.8
0.0
0.8
1.6
0 5 10 15 20 25
Time (sec)
Kobe, 1995-1.6
-0.8
0.0
0.8
1.6
0 5 10 15 20 25
Time (sec)
-0.8
-0.4
0.0
0.4
0.8
0 5 10 15 20 25
PGA = 0.82g
Bas
eshea
r/W
eight
of
th B
ridge
(W)
Transverse
Time (sec)
-0.8
-0.4
0.0
0.4
0.8
0 5 10 15 20 25
Time (sec)
c-TVAs
o-TVAs
NC
c-TVAs
o-TVAs
n-TVAs-0.8
-0.4
0.0
0.4
0.8
0 5 10 15 20 25
Time (sec)
-0.06
-0.03
0.00
0.03
0.06
0 5 10 15 20 25
PGA = 0.35g
Dis
pla
cem
ent
at M
id-S
pan
(m
)
Longitudinal
Time (sec)
-0.06
-0.03
0.00
0.03
0.06
0 5 10 15 20 25
Time (sec)
Imperial Valley, 1940
-0.06
-0.03
0.00
0.03
0.06
0 5 10 15 20 25
Time (sec)
-0.04
-0.02
0.00
0.02
0.04
0 5 10 15 20 25
c-TVAs
o-TVAs
PGA = 0.21g
Dis
pla
cem
ent
at M
id-S
pan
(m
)
Transverse
Time (sec)
-0.04
-0.02
0.00
0.02
0.04
0 5 10 15 20 25
NC
c-TVAs
Time (sec)
-0.04
-0.02
0.00
0.02
0.04
0 5 10 15 20 25
o-TVAs
n-TVAs
Time (sec)
20
Figure 10. Same as in Figure 9 but for the 1989 Loma Prieta Earthquake ground motion.
Figure 11. Same as in Figure 9 but for the 1995 Kobe Earthquake ground motion.
The maximum displacement response reduction due to o-TVAs was in order of 18 % for
longitudinal direction and 33% for transverse direction. The maximum response reduction
by n-TVAs is 33% along the longitudinal and transverse directions. The n-TVAs, in general,
are at least as effective as the o-TVAs, with an exception of transverse response caused by
the Imperial Valley Earthquake ground motion. It is interesting to note that, when subjected
to the Kobe Earthquake ground motion, which produces the largest response of the three
-0.12
-0.06
0.00
0.06
0.12
0 5 10 15 20 25
PGA = 0.57g
Dis
pla
cem
ent
at M
id-S
pan
(m
)
Longitudinal
Time (sec)
-0.12
-0.06
0.00
0.06
0.12
0 5 10 15 20 25
Time (sec)
Loma Prieta, 1989-0.12
-0.06
0.00
0.06
0.12
0 5 10 15 20 25
Time (sec)
-0.06
-0.03
0.00
0.03
0.06
0 5 10 15 20 25
c-TVAs
o-TVAs
PGA = 0.61g
Dis
pla
cem
ent
at M
id-S
pan
(m
)
Transverse
Time (sec)
-0.06
-0.03
0.00
0.03
0.06
0 5 10 15 20 25
NC
c-TVAs
Time (sec)
-0.06
-0.03
0.00
0.03
0.06
0 5 10 15 20 25
o-TVAs
n-TVAs
Time (sec)
-0.12
-0.06
0.00
0.06
0.12
0 5 10 15 20 25
PGA = 0.86 g
Dis
pla
cem
ent
at M
id-S
pan
(m
)
Longitudinal
Time (sec)
-0.12
-0.06
0.00
0.06
0.12
0 5 10 15 20 25
Time (sec)
Kobe, 1995-0.12
-0.06
0.00
0.06
0.12
0 5 10 15 20 25
Time (sec)
-0.08
-0.04
0.00
0.04
0.08
0 5 10 15 20 25
c-TVAs
o-TVAs
PGA = 0.82g
Dis
pla
cem
ent
at M
id-S
pan
(m
)
Transverse
Time (sec)
-0.08
-0.04
0.00
0.04
0.08
0 5 10 15 20 25
NC
c-TVAs
Time (sec)
-0.08
-0.04
0.00
0.04
0.08
0 5 10 15 20 25
o-TVAs
n-TVAs
Time (sec)
21
ground motions, the n-TVAs are considerably more efficient than the o-TVAs. It is also
noteworthy that the n-TVAs are more efficient than the o-TVAs for the ground motions for
which they were designed.
Table 1. Peak Responses for different control schemes.
Ground
Motion
Control
scheme
Base
shear/Weight of
the Bridge
Mid-Span
Displacement
(m)
Stroke (m)
X Y X Y X Y
Imper
ial
Val
ley,
1940 NC 0.848 0.386 0.068 0.046 - -
c-TVAs 0.753 0.318 0.060 0.036 0.133 0.079
o-TVAs 0.705 0.297 0.056 0.031 0.107 0.065
n-TVAs 0.641 0.298 0.051 0.032 0.118 0.076
Lom
a
Pri
eta,
1989 NC 1.722 0.741 0.136 0.074 - -
c-TVAs 1.493 0.585 0.118 0.062 0.289 0.133
o-TVAs 1.372 0.525 0.108 0.056 0.277 0.109
n-TVAs 1.156 0.490 0.091 0.049 0.245 0.117
Kobe,
1995 NC 1.534 0.900 0.122 0.083 - -
c-TVAs 1.336 0.908 0.107 0.088 0.265 0.159
o-TVAs 1.252 0.826 0.100 0.079 0.209 0.135
n-TVAs 1.019 0.788 0.081 0.077 0.238 0.128
From practical considerations, it might be desirable to control the stroke of the device. The
results indicate that the n-TVAs result in similar stroke as o-TVAs. The stroke in n-TVAs
might be large than that in o-TVAs if the excitation is not strong enough to cause inelastic
deformation because the n-TVAs are not equipped with viscous dashpots. A solution to
control the stroke of the n-TVAs is to reduce the value of R . It is to be noted that the its
value used in this study assumes that the energy dissipated by o-TVA is the same as that
done by n-TVA. If more energy dissipation is desired, a lower R value can be used. A few
values of R in the range of 0.25 to 0.75 were investigated and it was found that designing
with lower R values is very efficient in controlling the stroke. However, the efficiency of
the device in controlling structural response reduced with decreased value of R , in some
cases, its performance being worse than that of o-TVA. We then investigated other solutions
to control stroke, and the most promising one seems to be an addition of viscous dashpot to
the n-TVAs. Dashpots like the ones used with o-TVAs were added to the n-TVAs. The
results obtained with o-TVA, n-TVA, and nd-TVA denoting n-TVA with added dashpot are
shown in Figure 12
The results show that that the dashpots are effective in controlling the stroke. However, with
the inclusion of the dashpots, efficiency of the device in controlling structural response
decreases slightly, but its performance is better than that of o-TVAs.
22
Figure 12. Comparison of the performance of n-TVA with/without dashpot to that of o-
TVA. The top panel shows mid-span displacement in the longitudinal and transverse
directions, and the bottom panel shows corresponding stroke of the devices.
4. Evaluation of effectives is South Iceland
As a preliminary evaluation of the effectiveness of the proposed devices in hazard scenarios
corresponding to the South Iceland lowland. For this purpose, 5 accelerograms recorded
during the 17 June 2000 earthquakes are considered. These accelerograms were recorded at
Flagbjarnaholt, Hella, Kaldarholt, Selsund, and Thorsarbru. In addition, 9 artificial
accelerograms whose 5% damped response spectra are compatible to the average response
spectra of the recorded accelerograms are simulated.
23
Figure 13. Elastic response spectra (5% damped) of the recorded and simulated ground motions used
in this study. The right and left columns represent seismic action in the transverse and the
longitudinal directions of the bridge, respectively.
4.1. Design of the n-TVAs
To design the nTVS, 8 out of these 14 ground motions are considered, and the rest of the
ground motions are used to check the robustness of the design methodology. In iterative
methodology is used for each ground motion until the stroke of the TMD device converges
as explained in previous sections. The resulting strokes and response reduction factors
corresponding to the 8 ground motion used in the design process are presented in Table 2.
It can be observed that the stroke of the device caused by these ground motions are very
variable. This is due to the variability of the ground motions used here. The fundamental
period of vibration of the bridge being studied is around 0.6s, at which the spectral
displacement (see Figure 13) varies between about 2 o 10 cm. Therefore, although the
ground motion has similar spectral content on the average, the record to record variability
is quite high, as is expected close to earthquake sources. It is also apparent from Table 2
that the response reduction is higher when the stroke of the device is higher. This
corresponds to larger energy dissipation and therefore more effective vibration control.
Small values of peak stroke might indicate lack of yielding in the n-TVAs and therefore
lesser energy dissipation. In general, when the demand on the structure is high (peak stroke
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0S
pec
tral
Acc
elra
tion (
g)
Period (sec)
Ground Motion
Average
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Spec
tral
Acc
elra
tion (
g)
Period (sec)
Ground Motion
Average
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
Spec
tral
Dis
pla
cem
ent
(m)
Period (sec)
Ground Motion
Average
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
Sp
ectr
al D
isp
lace
men
t (m
)
Period (sec)
Ground Motion
Average
24
is higher), the reduction in base shear and mid-span displacement is up to 25%. However,
the reduction is not significant when the demand on the structure is low.
Table 2. Response reduction and stoke of the n-TVAs corresponding to the 8 design ground motions.
Ground
Motion
Control
scheme
Base shear
Reduction (%)
Mid-Span
Displacement (%) Stroke (m)
X Y X Y X Y
Sim
ula
ted
1
n-TVAs
25.13 12.75 24.28 23.28 0.215 0.218
2 18.31 8.78 18.41 8.66 0.184 0.199
3 10.02 12.17 10.45 12.79 0.200 0.244
South Iceland
2000 (station
Flagbjarnarholt)
6.81 15.67 5.00 16.47 0.042 0.179
4 20.57 5.48 19.23 4.66 0.160 0.118
5 10.17 15.89 2.14 9.32 0.210 0.104
6 12.55 15.45 11.84 19.09 0.210 0.102
7 20.30 19.79 20.23 10.66 0.189 0.142
4.2. Robustness of the n-TVAs
The n-TVAs and their effectiveness depend on the excitation. For practical purposes, a
suitable design needs to be selected, which needs to be robust against variability of ground
motion from the use used in designing the device. To test the robustness of vibration control
devices, the properties of the n-TVAs are taken as the average of the design properties
corresponding to the 8 design ground motions listed in Table 2. The effectiveness of this
design is then compared to other TMD schemes for the 14 ground motions. Reduction in
base shear by different control schemes when subjected to these 14 ground motions are
shown in Table 3. The results show that the n-TVAs are marginally better than the other
schemes. There is a large variability in the effectiveness against different ground motions
in all schemes. In general, the performance of the n-TVAs is better when the ground motion
is very demanding on the uncontrolled structure.
Table 3. Base-shear reduction due to n-TVAs compared to c-TVAs and o-TVAs
Ground Motion c-TVAs o-TVAs n-TVAs
X Y X Y X Y
1 26.67 11.47 24.45 10.17 25.19 12.68
2 15.50 9.28 12.50 8.28 18.19 8.89
3 11.82 12.99 10.50 11.49 10.34 12.68
South Iceland
2000 (station
Flagbjarnarholt) 2.08 11.26 1.79 9.31 5.04 10.97
4 4.60 14.84 5.47 14.29 13.02 15.01
5 1.13 16.71 1.35 15.88 1.92 16.83
6 10.97 13.53 10.89 13.19 12.66 18.87
25
7 20.09 21.34 19.50 16.54 20.39 21.34
8 18.11 32.09 16.75 27.94 19.84 32.08
9 16.63 -0.25 15.69 0.87 16.21 3.76
South Iceland
2000 (station
Hella)
-0.05 23.31 0.32 21.80 0.21 22.62
South Iceland
2000 (station
Kaldarholt)
-1.04 22.62 0.11 20.68 0.87 26.14
South Iceland
2000 (station
Selsund)
16.88 3.56 16.17 2.40 14.19 7.26
South Iceland
2000 (station
Thjorsarbru)
26.76 26.26 25.37 22.00 30.31 30.06
Some examples of time history response of the bridge with different control schemes are
presented in Error! Reference source not found..
Figure 14. Base shear demand on the structure (top: longitudinal and bottom: transverse) of the
uncontrolled and controlled structure subjected to simulate ground motion number 2.
26
Figure 15. Same as in Figure 13, but for ground motion recorded at Hella station.
Figure 16. Same as in Figure 14 but for Kaldarholt station.
27
Figure 17. Same as in Figure 14 but for Selsund station.
Figure 18. Same as in Figure 14 but for Thorsarbru station.
28
Figure 19. Displacement response of the controlled and uncontrolled structure when subjected to
ground motion recorded at the Flagbjarnarholt station; the top and bottom rows correspond to
longitudinal and transverse displacements, respectively, at the mid-span of the bridge.
Figure 20. Same as in Figure 19 but for Hella station.
29
Figure 21. Same as in Figure 19 but for Kaldarholt station.
5. Conclusions
The study presents the effectiveness of using inelastic tuned vibration absorbers (n-TVAs)
in seismic response mitigation of reinforced concrete (RC) bridges. We present a
methodology to design the parameters of nonlinear springs used in n-TVAs. The method
assumes that the n-TVAs without additional viscous damping dissipate the same energy as
the energy dissipated by viscous damping in o-TVAs. The design is an iterative process
starting with an initial guess of peak displacement, which is revised in consecutive steps
until convergence is achieved. In this study, the initial peak displacement was based on the
peak displacement of the o-TVAs. In practical applications, the initial peak displacement
can be estimated from elastic analysis using design ground motion.
Optimization of n-TVAs is not straightforward because the energy dissipated by hysteresis
of the device depends on excitation. This implies that the device cannot be optimized for all
potential ground motions. A device that is designed based on a given ground motion might
not yield when subjected to a weaker ground motion, and therefore hysteretic energy
dissipation is not achieved. In such situations, the performance of the n-TVAs might be
lower than that of o-TVAs, which dissipate energy due to viscous damping. This might
seem, at first look, as a limitation of n-TVAs. However, the efficiency of the device in
reducing structural response is more crucial at stronger ground motions which have the
potential to damage the structure or cause functional loss. The results of the case study
presented here show that the proposed n-TVAs are more efficient in controlling structural
displacements and base shear when subjected to design ground motion. In summary, the n-
30
TVAs are at least as effective as the o-TVAs, and are preferable because they do not require
expensive viscous dashpots like the o-TVAs.
In practical applications, unlimited stroke of TVAs cannot be accommodated by the host
structure. Therefore, reducing device stroke can become a design problem. It was observed,
from the results of simulations presented here, that n-TVAs might increase device stroke
slightly, especially when the excitation is not strong enough to cause hysteretic energy
dissipation in the device. Bagheri and Rahmani-Dabbagh (2018) present the performance
of inelastic TMD and conclude that they perform better than optimal TMD. They also point
out that the performance of inelastic TMDs may be further improved by introducing
additional viscous damping into the device. Our results indicate that addition of viscous
dashpot to the n-TVA can result in reduced performance in controlling structural response.
However, this additional damping was found to be effective in controlling device stroke. It
seems then that there is a trade-off between controlling structural response and device
stroke, which need to be simultaneously optimized in practical design applications based on
allowable limits of these parameters. It is also to be noted that the n-TVAs used here were
optimized assuming that there is no viscous damping in the device. Optimization of n-TVAs
with additional viscous damping is an important issue to be further investigated.
The foregoing arguments support the use of n-TVAs for loading scenarios controlling the
design of the structure, for example, life safety and ultimate limit states when it comes to
seismic design. The modern design philosophy however leans towards performance-based
design, where different performance levels corresponding to different levels of excitation
are defined. In such a philosophy, allowable limits on response parameters are defined for
different intensities of ground motion. It appears then that a n-TVA designed for load
corresponding to ultimate limit state might not be effective in controlling the response at
lower performance levels, such as serviceability and immediate occupancy. This is because
at weaker excitations, n-TVAs behave as o-TVAs but without viscous dashpots. To account
for these scenarios, and if there is a need to control structural response at lower performance
levels, it is advisable to provide additional dashpots with the n-TVAs. In this way, the
devices can be at least as effective as o-TVAs for low intensity ground shaking. This can
result in slight performance reduction during strong shaking, but also helps to control device
stroke during very strong ground shaking. Future studies need to focus on optimization of
n-TVA parameters with additional dashpot, and its robustness against de-tuning effects
caused by shifts in natural frequencies of the structure.
Examination of the effectiveness of the devices in a seismic hazard scenario corresponding
to the South Iceland lowland shows that the effectiveness of the control devices depends, to
a large degree, on the ground motion being used. This applies not only to the n-TVAs, which
are inherently dependent on excitation, but also to other control schemes, which although
are independent of excitation, their effectiveness depends on excitation. This variability
seems to be very large and is a direct consequence of the variability of ground motion. Such
variability, as has been observed in recent earthquakes, is very high close to the earthquake
faults. Nevertheless, n-TVAs designed in the average sense were found to be robust against
31
different ground motions, and although their performance was variable, they did not result
in unwanted effects such as response amplification. It is theoretically possible to increase
the effectiveness of the n-TVAs by allowing for more inelastic deformation, i.e., by
increasing the ductility capacity of the device. This would imply higher energy dissipation
even during relatively milder shaking. This approach might be problematic in a highly active
seismic zone because repeated yielding of the device during frequent small earthquakes
might warrant for frequent maintenance and/or replacement of the devices. Then there are
other considerations such as stroke of the device which can be practically accommodated
by the host structure. In this sense, future studies in vibration control of structures with n-
TVAs should focus on a performance-based approach where different design constraints
and hazard scenarios are pre-defined and practical constraints are properly addressed.
ACKNOWLEDGEMENTS
We acknowledge financial support from the Vegagerðin research grant which was used to
finance part of the involvement of the second author in this study
DISCLAIMER
The authors of the present report are responsible for its contents. The report and its findings
should not be regarded as to reflect the Icelandic Road Authority’s guidelines or policy, nor
that of the respective author’s institutions.
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