Real time hybrid simulation of steel structures with supplemental elastomeric dampers
Akbar Mahvashmohammadi, Richard Sause, James Ricles, Thomas Marullo
ATLSS Center, Department of Civil and Environmental Engineering, Lehigh University, Bethlehem, PA.
R. Michael, S. Sweeney
School of Engineering, Penn State Erie, The Behrend College, Erie, PA, USA
Ernest Ferro
Corry Rubber, Corry, PA, USA
Abstract
Passive damping systems can improve the seismic performance of buildings by reducing drift and
inelastic deformation demands on the primary lateral load resisting system, in addition to reducing the
velocity and acceleration demands on non-structural components. Recent research by the authors shows
that adding passive damper systems to steel moment resisting frames (MRFs) enables significant
reductions in the steel weight of the MRFs, while enhancing the seismic performance of the structure.
Large-scale elastomeric dampers, have been tested and models for nonlinear analysis developed by the
authors. A simplified design procedure (SDP) has been proposed by Lee et al [1-2] for elevating the
performance of moment resistant frames (MRFs) by incorporating the supplemental dampers into the
design process. SDP was used to design a 0.6 scaled steel structure using elastomeric dampers. A set of
20 ground motion were chosen that their median spectral acceleration matches the design spectrum in the
range of 0.2-2 (Sec). Nonlinear dynamic analysis was performed using OpenSees [3] and 3 records that
created results corresponding to median response were selected for experiments.
Real time hybrid simulation is an accurate and economical way to test and verify performance of new
earthquake resistant design instruments and design methodologies. This method divides the structure into
analytical and experimental sub structures which avoids costly experimental studies on members that are
well studied and allows modelling these parts analytically. CR integration developed by Chen et al [4]
was used as the integrator for hybrid testing. Effect of integration parameters on accuracy of simulations
was studied.
1. Introduction
Conventional seismic design of steel frames results in damage and residual drift under the design earthquake. Passive damping systems can improve the seismic performance of buildings by reducing drift and
inelastic deformation demands on the members of the primary lateral load resisting system, in addition to
reducing the velocity and acceleration demands on non-structural components.
Lee et al. [1-2] developed and tested elastomeric dampers made of ultra-high damping rubber. Lee et al.
found the behavior of these dampers to be less sensitive to frequency and ambient temperature compared
to viscoelastic dampers. Lee et al. proposed a simplified design procedure (SDP) for multi degree of
freedom frame systems with viscoelastic or high damping elastomeric dampers. Seismic performance
anticipated by the SDP was verified using nonlinear dynamic time history analyses (NDTHA), which
showed good agreement with the design demand predictions used in the SDP [2].
Kontopanos [5] investigated the cyclic behavior of an elastomeric damper made from an ultra-high
damped elastomer. The elastomeric material was pre-compressed into structural tubes which could
provide viscous like damping under small strains and friction like damping under large strains.
Karavalisis et al. [6] presented a rate dependent hysteretic model for compressed elastomer damper
consisting of a modified Bouc-Wen model in parallel with a nonlinear dashpot. He designed several
structural frames with elastomeric dampers using the SDP and verified their seismic performance using
NDTHA analyses. The NDTHA results showed that the use of elastomeric dampers can significantly
improve seismic performance. He came to the conclusion that when dampers are used, MRFs can be
designed for less than design base shear specified by design specifications without experiencing
noticeable performance change.
Sause et al. [7] tested a 2nd generation elastomeric damper. The damper was constructed in 0.6 scale using
50 duro compressed butyl blend elastomer. The damper had layers of elastomer with two different
thicknesses to provide both, a slip and non-slip layer, energy dissipation and stiffness while preventing
the dampers from being damaged under the design earthquake. They came to the conclusion that precompression force on slip layer was not enough and slipping layer did not contribute enough to damper
response.
Real-time hybrid testing combines experimental testing and numerical simulation, and provides a viable
alternative for the dynamic testing of structural systems. An integration algorithm is used in real-time
hybrid testing to compute the structural response based on feedback restoring forces from experimental
and analytical substructures. Explicit integration algorithms are usually preferred over implicit algorithms
as they do not require iteration and are therefore computationally efficient [8]. Chen et al [4] developed an
unconditionally stable explicit integrator called CR integrator. Chen et al [8] used CR integration for
hybrid testing of a 1st generation elastomeric damper.
This paper introduces a third generation elastomeric damper which is designed to overcome shortcomings
of the second generation elastomeric damper. Damper was characterized and a rigorous model was
developed to be used in nonlinear dynamic time history analysis (NDTHA). A 3 story structure was
designed using SDP. A set of 20 ground motions were chosen that cause a median response spectrum that
matches design spectrum at period range of 0.2-2.0(sec). 3 of those ground motions were chosen for real
time hybrid simulations. Sensitivity of CR integration to integration parameters was studied as well.
2. Damper Description
Sause et al designed, manufactured and characterized a second generation elastormic dampers [7]. Figure
1 shows components of an elastomeric damper consisting of an inner steel tube, outer steel tube,
elastomeric material and thin steel plates. The thin and thick layers of the elastomeric material are
chemically bounded during the curing process to the inner steel tube, and the thicker layer to thin steel
plates to form the inner assembly. The inner assembly is then pre-compressed into the outer steel tube.
The thin steel plates are bolted to the outer tube to prevent slip in the thicker layers of elastomer. The
thinner layer develops slip when the shear force in the layer exceeds the static frictional force resistance
between this layer and the inside surface of the outer steel tube. One damper consists of two or more outer
tubes welded together side by side. Test results revealed that target pre-compression force on thin layer
was not achieved and lead to negligible contribution from thin layer in damper behavior.
To overcome this problem a third generation elastomeric damper was developed by authors. A thicker
outer tube was used to increase pre compression force on elastomeric material. Also a duro 60 elastomer
material was used instead of 50 duro to increase stiffness and damping.
The combination of a slip and non-slip elastomer layer can provide ideal characteristics of the damper.
The slip in the thinner elastomeric layer causes energy dissipation to occur by friction while also limiting
the maximum force developed in the damper
3. Characterization Tests
Characterization tests of the damper were conducted at the Network of Earthquake Engineering
Simulation (NEES) Real Time Multi Directional (RTMD) facility located at Lehigh University [9]. The
test setup is shown in Figures 2 and 3. An actuator with 2300 kN load capacity and 840 mm/sec
maximum velocity is used to apply a predefined displacement history to the damper. The damper is
connected to a stiff damper support beam which is bolted to the laboratory strong floor. The damper
support beam has shear keys to prevent movement between the beam and the strong floor.
Previous experience has shown that elastomeric damper properties depend on applied displacement
amplitude, excitation frequency, and ambient temperature [2]. To capture the effects of these
dependencies, different tests should be conducted with different displacement amplitudes, frequencies and
temperatures. Harmonic displacement histories with amplitudes of 2.54, 6.35, 19.05, 25.4, 38.1, 50.8
(mm) were selected and applied to a damper at different frequencies of 0.1, 0.5,1,2 and 3 (Hz). To
investigate the effect of temperature the same displacement amplitudes were applied to a damper at 1 (Hz)
frequency and temperatures of 10, 20 and 30 C. Each time history displacement includes a total of 12
cycles, where the first-two cycles and last three cycles are used to ramp up and down, respectively, to the
targeted displacement amplitude. Figure 4 shows an example of an applied displacement history
associated with 25.4 mm of amplitude at 1 Hz and 20 C.
Figure 1. Damper Components
Figure 2. Schematic of elastomeric damper in test setup
Figure 3. Photograph of elastomeric damper in test setup
Displacement (mm)
30
20
10
0
-10
-20
-30
0
2
4
6
8
10
12
Time (s)
Figure 4. Applied displacement time history
4. Damper Behavior
Figure 5 shows full cycle hysteresis loops for displacement amplitudes between 2.54 and 50.8 (mm),
excitation frequency of 1(Hz) and ambient temperature of 20(C). Damper shows noticeable degradation at
displacement amplitudes of 2.54 and 6.35 (mm). Damper shows softening behavior at displacement
amplitude range of 12.7-38.1 (mm). At displacement amplitude of 50.8 (mm) damper begins to show
slight hardening behavior with more degradation.
200
150
100
Force (kN)
50
0
-50
-100
-150
-200
-50
-40
-30
-20
-10
0
10
Displacement (mm)
20
30
40
50
Figure 5. Experimental full cycle hysteresis loops, frequency=1(Hz), T=20(C)
5. Damper Modeling
5.1. Simple model for design purposes.
To design structures with elastomeric dampers using SDP, it is required to have a simple linear spring
dashpot model for damper, at different displacement amplitudes, excitation frequencies and ambient
temperatures.
The damper force-displacement hysteresis loop with the last full cycle of displacement is used to
determine the damper mechanical properties. In the SDP the damper mechanical properties of interest
include the equivalent stiffness Keq and loss factor ɳ. Keq is illustrated in Figure 6, and is associated with
maximum magnitudes of damper force fDmax and maximum damper displacement dmax. The loss factor η is
associated with the energy dissipation of the damper, and can be related to the equivalent viscous
damping in the SDP. The loss factor is defined as [10]:

1 ED
*
2 ES
(1)
In Eq. (1) ED is the dissipated energy and ES the strain energy associated with one cycle of displacement.
Formulas used to determine the mechanical properties for the damper are given below. Eq. (3) is used to
determine η in lieu of Eq. (1) since Keq, dmax, and dmin are conveniently obtained from the hysteresis loop
(see Figure 6).
K eq 
f D max  f D min
d max  d min
sin(  ) 
ED
d  d min 2
 .K eq .( max
)
2
  tan(  )
(2)
(3)
(4)
Keq and η are plotted as a function of displacement amplitude of the characterization tests, and shown in
Figure 7. The results shown in these figures are from characterization tests with a 20 C ambient
temperature, and include results for the various excitation frequencies. Effect of ambient temperature on
damper mechanical properties is shown in Figure 8.
Figure 6. Equivalent stiffness
(a)
(b)
Figure 7. Effect of amplitude and frequency on (a) equivalent stiffness and (b) loss factor; at T=20C
(a)
(b)
Figure 8. Temperature effect on (a) equivalent stiffness and (b) loss factor for tests with frequency=1 Hz
5.2. Rigorous model for nonlinear dynamic time history analysis (NDTHA)
Karavalisis et al. [6] presented a rate dependent hysteretic model for compressed elastomer damper with
softening behavior consisting of a modified Bouc-Wen model in parallel with a nonlinear dashpot.
𝑓(𝑡) = 𝛼𝑘𝑢(𝑡) + (1 − 𝛼)𝑘𝑢𝑦 𝑧(𝑡)
(5)
1
[𝐴𝑢̇ − 𝜐(𝛾|𝑢̇ ||𝑧|𝑛 𝑠𝑔𝑛(𝑧) − 𝛽𝑢̇ |𝑧|𝑛 )]
(6)
𝑧̇ =
𝑢𝑦
He used below equation to simulate degradation of tangent stiffness of the 1 st generation elastomeric
dampers:
𝑘 = 𝑘𝑎 𝑒
𝑢
− 𝑚𝑎𝑥
𝑢𝑟𝑒𝑓
+ 𝑘𝑏
(7)
where 𝑘1 , 𝑘2 𝑎𝑛𝑑 𝑢𝑟𝑒𝑓 are constants, and 𝑢𝑚𝑎𝑥 is the average of the maximum absolute deformation
amplitudes in compression (𝑢𝑚𝑎𝑥,𝑐 ) and tension (𝑢𝑚𝑎𝑥,𝑡 ), as follows:
𝑢𝑚𝑎𝑥 =
|𝑢𝑚𝑎𝑥,𝑐 | + 𝑢𝑚𝑎𝑥,𝑡
2
(8)
Bouc-wen model can only simulate either softening or hardening behavior. This model will lead to
softening behaviour if β + γ > 0 and hardening behaviour if 𝛽 + 𝛾 > 0 𝑎𝑛𝑑 β − γ < 0. However the
third generation damper shows both softening and hardening behaviour at different displacement
amplitudes. To overcome this problem a model was developed with elements described below:
1- A Bouc-Wen element with softening behavior was used to model thicker layer behaviour at
smaller amplitudes.
2- A Bouc-Wen element with zero post yielding stiffness and no stiffness degradation was used to
model behaviour of thin layer
3- A Bouc Wen element with hardening behaviour was used to model behaviour of thick layer at
higher displacements.
4- A nonlinear dashpot was used to model damper velocity sensitive response.
Figure 9 shows damper model components.
Bouc-Wen 1
Bouc-Wen 2
Bouc-Wen 3
𝑢𝑑 ,𝑓𝑑
Nonlinear dashpot
Figure 9. Damper model
It was assumed that:
𝐴1 = 𝜐1 = 𝛽1 + 𝛾1 = 𝐴2 = 𝜐2 = 𝛽2 + 𝛾2 = 𝐴3 = 𝛽3 − 𝛾3 = 1.0 , 𝑘𝑎2 = 𝛼2 = 0
(9)
So there is 20 independent variables to be found using particle swarm optimization method used by Ye et.
al (2007) . Parameters to be found are listed below.
𝑘𝑎1 , 𝑘𝑏1 , 𝛼1 , 𝑢𝑦1 , 𝑢𝑟𝑒𝑓1 , 𝛽1 , 𝑛1 , 𝑘𝑏2 , 𝛼2 , 𝑢𝑦2 , 𝛽2 , 𝑛2 , 𝑘𝑎3 , 𝛼3 , 𝑢𝑦3 , 𝑢𝑟𝑒𝑓3 , 𝛽3 , 𝑛3 , 𝑐𝑁𝐷 , 𝑎𝑁𝐷
Table 1. Bouc-Wen 1 parameters
𝑘𝑎1 (kN/mm)
𝑘𝑏1 (kN/mm)
7.13
4.19
𝛼1
0.1186
𝑢𝑦1 (mm)
20.43
𝑢𝑟𝑒𝑓1 (mm)
2.30
γ1
0.8725
n1
1.0
Table 2. Bouc-Wen 2 parameters
𝑘𝑏1 (kN/mm)
𝑢𝑦1 (mm)
4.79
1.22
Table 3. Bouc-Wen 3 parameters
𝑘𝑎3 (kN/mm)
𝑘𝑏3 (kN/mm)
5.48
0.03
γ1
0.55
n1
1.63
𝛼3
𝑢𝑦3 (mm)
𝑢𝑟𝑒𝑓3 (mm)
γ3
n3
0.097
6.15
14.86
-0.59
1.06
Table 4. Nonlinear dashpot parameters
𝑐𝑁𝐷 (𝑘𝑁/(𝑚𝑚/𝑠𝑒𝑐)𝑎 )
𝑎𝑁𝐷
3.188
0.4
Initial stiffness of the damper is summation of initial stiffness of all layers and is called 𝑘𝐵𝑊 .
𝑘𝑁
𝑘𝐵𝑊 = ∑3𝑖=1(𝑘𝑎𝑖 + 𝑘𝑏𝑖 ) = 21.62 (𝑚𝑚)
(10)
To perform numerical simulations using CR integration nonlinear dashpot should be linearized in the
range of small velocities. This linear damping coefficient is called 𝑐𝑡𝑟 . The threshold velocity, 𝑣𝑡𝑟 was
chosen to be 0.01 (m/s). 𝑘𝐵𝑊 and 𝑐𝑡𝑟 values will be used and referenced frequently at this paper.
𝑘𝑁
𝑐𝑡𝑟 = 𝑐𝑁𝐷 . 𝑣𝑡𝑟 𝛼𝑁𝐷 −1 = 802( 𝑚 )
(11)
𝑓𝑁𝐷
−𝑣𝑡𝑟
𝑐𝑡𝑟
𝑣𝑡𝑟
𝑣
Figure 10. Nonlinear dashpot model
Figure 11 shows agreement between experimental and analytical full cycle hysteresis loops. Figure 12
shows agreement between experimental and analytical hysteresis loops for displacement amplitudes of
50.8, 101.6 and 152.4(mm).
6. Design of a Steel MRF with Elastomeric Dampers
Figure 13 shows plan and elevation views of a prototype office located in Southern California. This study
focuses on 2D analysis of one quarter of this structure. The results are valid for 2D analysis of the whole
structure due symmetry. Structure is located on stiff soil. A smooth design response spectrum with
parameters 𝑆𝐷𝑆 = 1.0, 𝑆𝐷1 = 0.6, 𝑇0 = 0.12 sec and 𝑇𝑠 = 0.6 sec represents the DBE. MRFs were
designed by Dong [11] based on strength requirements of IBC2006 (2006) only without satisfying drift
requirements. A “lean-on” column is included in the model to include p-Δ effects. The focus on this paper
is on a 0.6 scaled version of this structure, because a 0.6 scaled of this structure is fabricated in Lehigh
University and will be used for net phases of hybrid simulations.
200
150
Model
Experiment
100
Force (kN)
50
0
-50
-100
-150
-200
-50
-40
-30
-20
-10
0
10
Displacement (mm)
20
30
40
50
Figure 11. Experimental and analytical full cycle hysteresis loops.
200
150
100
100
50
50
Force (kN)
Force (kN)
200
Model
Experiment
150
0
-50
0
-50
-100
-100
-150
-150
-200
-30
-20
-10
0
Displacement (mm)
10
20
-200
-30
30
(a) Ramp up and full cycles, amplitude=25.4(mm)
-20
-10
0
Displacement (mm)
10
20
30
(b) full cycles and Ramp down, amplitude=25.4(mm)
250
250
Model
Experiment
200
Model
Experiment
200
150
150
100
100
50
50
Force (kN)
Force (kN)
Model
Experiment
0
-50
0
-50
-100
-100
-150
-150
-200
-200
-250
-40
-30
-20
-10
0
10
Displacement (mm)
20
30
40
-250
-40
(c) Ramp up and full cycles, amplitude=38.1(mm)
-30
-20
-10
0
10
Displacement (mm)
20
30
40
(d) full cycles and Ramp down, amplitude=38.1(mm)
250
Model
Experiment
200
200
Model
Experiment
150
100
100
50
50
Force (kN)
Force (kN)
150
0
-50
0
-50
-100
-100
-150
-150
-200
-200
-250
-50
-40
-30
-20
-10
0
10
Displacement (mm)
20
30
40
50
(e) Ramp up and full cycles, amplitude=50.8(mm)
-50
-40
-30
-20
-10
0
10
Displacement (mm)
20
30
40
50
(f) full cycles and Ramp down, amplitude=50.8(mm)
Figure 12. Experimental and analytical hysteresis loops, frequency=1.0(Hz), temperature=20C
SDP procedure developed by Lee et al (2003) was used to design damped braced frame (DBF), find
required number of dampers in each story and predict drift ratio under median DBE earthquake. It was
assumed that the inherent damping ratio of frame with no damper is 2%. To satisfy target drift of 1.5% it
is required to put 4, 3 and 2 dampers in the 1st, 2nd and 3rd stories. Table 5 compares structure properties
with and without dampers. Damper can effectively increase damping ratio of the structure and decrease
drift ratios.
𝐷𝑒𝑠𝑖𝑔𝑛 𝐶𝑎𝑠𝑒
D1
D2
𝐹𝑟𝑎𝑚𝑒𝑠
MRF
MRF + DBF
Table 5. Summary of designed cases
𝑉𝑏 (Kips)
𝑇 (𝑠𝑒𝑐)
𝜁 (%)
1.074
2.0
272.19
0.67
10.6
293.75
𝜃1 (%)
2.495
1.04
𝜃2 (%)
3.331
1.37
𝜃3 (%)
3.67
1.28
Figure 13. Plan and elevation view of the prototype building (Figures by Baiping Dong)
7. Finite element models
A finite element model of the designed structure with dampers was built in Hybrid FEM [12] program.
Hybrid FEM is a finite element program developed in Lehigh University for Hybrid simulations. The
model includes 588 DOFs. Displacement based nonlinear beam column element was used to model beam
and columns. More elements were used in parts of the structure that yielding is expected (for example
RBS sections and 1st story column base) to capture moment gradient correctly. A nonlinear panel zone
element was used to model shear and symmetric column bending deformations [13]. P-Δ effect was
modelled using a lean on column to model 2nd order displacements. It should be noted that Hybrid FEM
[12] program does not include shear flexibility of beam column members.
It was decided to make models with 2 different sets of damper arrangement. 1 set of damper arrangement
is what was shown in chapter 6 with 4, 3 and 2 dampers in the 1st, 2nd and 3rd stories respectively. This is
the ideal damper arrangement for design that satisfies target drift. Regarding that there was only 1 damper
and 1 actuator available at the time of testing, only the 3rd story damper forces were found experimentally
in hybrid experiments of these tests and damper forces of the 1st and 2nd stories were found analytically
using the damper model described in section 5.2. It was assumed that damper forces of the 2 dampers in
the 3rd story are equal. Ground motions for this damper arrangement were scaled according to section 8.
The 2nd arrangement did not include any analytical damper and there was only was experimental damper
in the 3rd story. This damper arrangement is not interesting for design purposes, but its advantage is that it
doesn’t include any analytical damper and possible shortcomings of damper model won’t violate
experiment results. For this case ground motions were scaled to cause about 1.5% maximum story drift.
In addition to hybrid simulation using Hybrid FEM (HSHF), pure numerical simulations using Hybrid
FEM (NSHF) were also run in this study. The only difference between HSHF and NSHF is that in NSHF
there is no experimental damper and all dampers are analytical dampers that were derived in section 5.2.
Comparing NSHF and HSHF results show effect and extent of accuracy of damper model in accuracy of
numerical predictions.
The same structural models were also built in OpenSees software. OpenSees model uses implicit CR
integration scheme. Comparison between OpenSees and NSHF results can verify accuracy of CR
integration and Hybrid FEM software. Dampers were modelled using rigorous damper model of section
5.2. Shear flexibilities of beams and columns were not modelled. Figure 15 shows structural models used
for hybrid simulations.
𝑀3
Rigid floor diaphragm
RBS
Panel zone
element
Experimental
dampers
𝑀2
Analytical
dampers
𝑀1
RBS
Rigid floor diaphragm
𝑀3
Experimental
dampers
𝑀2
Panel zone
element
𝑀1
Analytical
dampers
MRF
DB
DBF
(a)
(b)
F
Figure 14. Hybrid FEM model for hybrid simulations (a) with 4 and 3 analytical dampers in the 1st and 2nd
stories and 2 experimental dampers in the 3rd story, (b) with one experimental damper in the 3rd story
MRF
8. Ground motion selection and scaling
A set of 20 ground motions was selected from PeerNGA [14] data base which creates a median spectral
acceleration that matches design spectrum in the range of 0.2-2 (sec). The ground motion set only
included ground motion on sites with type D soil. Average scaling method [15] was used to scale ground
motions. Figure 15 shows response spectrum of selected ground motions, median response spectrum and
design spectrum. Table 6 shows the list of selected ground motions.
2.5
20 ground motions
Median spectrum
Design spectrum
Spectral Acceleration (g)
2
1.5
1
0.5
0
0.2
0.4
0.6
0.8
1
1.2
Period (s)
1.4
1.6
1.8
2
Figure 15. Response spectrum of selected ground motions
Table 6. Summary of selected ground motion
Record
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Earthquake
Northridge 1994
Northridge 1994
Northridge 1994
Northridge 1994
Northridge 1994
Kocaeli 1999
Kocaeli 1999
CHI-CHI 1999
CHI-CHI 1999
CHI-CHI 1999
CHI-CHI 1999
CHI-CHI 1999
CHI-CHI 1999
CHI-CHI 1999
Imperial Valley
1979
Imperial Valley
1979
West Moreland
1981
Superstition Hills
1987
Northridge 1994
Northridge 1994
Station
Component
Newhall
Saticoy
Saticoy
Roscoe
Roscoe
Duzce
Yarimca
CHY015
CHY036
CHY047
HWA019
HWA037
ILA013
TCU042
Chihuahua
NWH360
STC090
STC180
ROS3000
ROS3090
DZC180
YPT060
CHY015-W
CHY036-E
CHY047-W
HWA019-E
HWA037-E
HWA013-N
TCU042-E
H_CHI012
Magnitude
(Richter)
6.69
6.69
6.69
6.69
6.69
7.51
7.51
7.62
7.62
7.62
7.62
7.62
7.62
7.62
6.53
Distance
(km)
26.78
17.83
17.83
21.42
21.42
99.52
25.07
69.91
44.74
55.51
81.49
72.86
134.93
78.78
21.35
Scale
factor
0.562
1.381
0.741
1.761
1.217
1.346
1.361
2.104
1.244
2.673
2.525
2.748
2.133
1.985
1.93
EC
Meloland
Parachute
Facility
West
Moreland
Camarillo
Canyon
Country
EMO000
6.53
21.84
1.231
PTS315
5.9
20.6
2.110
WSM090
6.54
21.49
2.504
CMR180
LOS000
6.69
6.69
51.4
31.75
2.848
1.162
Scaled ground motions were applied to D2 design case of Table 5. Figure 16 compares story drifts
predicted by SDP and median of maximum story drifts found by NDTHA. NDTHA of DBE level
earthquakes showed that ground motions 6,7 and 14 cause responses close to median DBE response.
These 3 records were chosen for hybrid experiments.
Figure 16. Comparison of maximum story drifts found by NDTHA and SDP
9. Introduction to real time testing using CR integration:
Real-time hybrid testing, also known as real-time substructure testing; is a viable and economic technique
for investigating the dynamic response of structural systems. It divides a structural system into
experimental substructure(s) and analytical substructure(s), and enables the complete structural system to
be considered. A number of seismic hazard mitigation devices have been developed to enhance the
seismic performance of structural systems during earthquakes. Many of these devices are load-rate
dependent. The development of performance-based design procedures for structures with these devices
requires that the device’s behavior be well understood, accurate analytical models must exist, the
effectiveness of the devices, and the performance of the structural system with the devices be evaluated,
and the design procedure be verified. These requirements can be economically met by performing realtime hybrid tests of structural systems with these devices to acquire reliable experiment data [8]
An integration algorithm is used in real-time hybrid testing to compute the structural response based on
feedback restoring forces from experimental and analytical substructures. Explicit integration algorithms
are usually preferred over implicit algorithms as they do not require iteration and are therefore
computationally efficient. [8]. Chen et al [4] developed an unconditionally stable explicit integration
method called CR integration. Chen et al [4] showed that for a linear elastic system CR integration has the
same accuracy (in terms of period elongation and numerical damping) as implicit constant acceleration
Newmark method. Displacement and velocities of each step could be found using equations 12 and 13,
where 𝛼1 and 𝛼2 are integration parameters and can be calculated by equation 14. This formulation is
unconditionally stable for all softening systems.
𝑋̇𝑖+1 = 𝑋̇𝑖 + 𝛼1 . 𝛥𝑡. 𝑋̈𝑖
(12)
𝑋𝑖+1 = 𝑋𝑖 + 𝛥𝑡. 𝑋̇𝑖 + 𝛼2 . 𝛥𝑡 2 . 𝑋̈𝑖
(13)
2
−1
𝛼1 = 𝛼2 = 4. (4. 𝑀 + 2. 𝛥𝑡. 𝐶 + 𝛥𝑡 . 𝐾) . 𝑀
(14)
Ramp generators are used to smoothly impose the displacement command to the experimental
Substructure [8]. A linear ramp generator is used for the implementation of the CR integration algorithm
for real-time testing in this paper. The integration time step Δt is divided into n sub steps, i.e. 𝛥𝑡 = 𝑛. 𝛿𝑡.
The command displacement to be sent to the servo-controller for the (i+1)th time step is interpolated by
the linear ramp generator [8]. Chen et al [4] showed that CR integration method is unconditionally stable
for linear and softening systems. Chen et al [8] used CR integration for hybrid testing of a 1st generation
elastomeric damper.
The nonlinear finite element program HybridFEM has been developed and implemented into the real-time
integrated control system at the NEES RTMD Facility. A digital servo controller with a 1024Hz clock
speed controls the motion of the servo-hydraulic actuators and is integrated with several workstations that
are part of the real-time integrated control system for conducting real-time hybrid simulations using a
shared common RAM network (SCRAMNet). These workstations are shown in Figure 17. The
SCRAMNet has a communication rate of about 180ns which enables the transfer of data among the
integrated workstations in real-time with minimal communication delay. HybridFEM has been developed
in a manner that enables the integration algorithm, analytical substructure modeling, servo-hydraulic
actuator control law, and actuator delay compensation method to be integrated into a single Simulink
model on the Simulation Workstation and then downloaded onto the Real-time Target Workstation using
Mathworks xPC Target Software [16]. [17].
10. Effect of integration parameters on accuracy of results from CR integration
In hybrid testing it is desired to not update damping and stiffness matrices. Elastomeric dampers can show
highly nonlinear behavior. The elastic stiffness and damping coefficients used in assembling stiffness and
damping matrix of the structure for calculation of integration parameters are called 𝑘0 and 𝑐0
respectively. This section investigates effect of using different values of 𝑘0 and 𝑐0 on stability and
accuracy of CR integration method.
As mentioned in section 7 there were 2 finite element models built. One model in OpenSees program and
one model in a software developed in Lehigh University called HybridFEM which is used for hybrid
testing. The 2 models were identical. The model in OpenSees uses implicit constant acceleration
Newmark beta method and the model in HFEM uses CR integration. It was desired to verify the HFEM
model and accuracy of CR integration by comparing its results with OpenSees model that uses implicit
constant acceleration Newmark beta method. To achieve this goal some numerical simulations were run.
This section only uses the structure model with 1 damper in the 3rd story and no dampers in the 1st and 2nd
stories. There is no experimental damper in numerical simulations of this section. section 10.1 uses
simple damper model of section 5.1 and section 10.2 uses rigorous damper model of section 5.2. Only
ground motion 6 was used in this section and it was scaled to create maximum story drift about 1.5% in
the structure.
Figure 17. RTMD integrated control system architecture [8]
10.1. Simple Linear spring dashpot model
A simple linear spring dashpot damper model described in section 5.1 was used to calculate natural
periods of the 2 models. Damper stiffness and damping coefficient are called 𝑘 ′ and c respectively. Table
6 reports natural periods of the 2 models. Periods of the 2 models are the same with 4 significant digits.
Table 6. Comparison between natural periods of OpenSees and HFEM models
Mode number
Period (sec)
OpenSees Hybrid FEM
1
0.8264
0.8264
2
0.2502
0.2502
3
0.1186
0.1186
It was found that by using 𝑘0 and 𝑐0 values less than 𝑘 ′ and c, the numerical simulation using HFEM
software (CR integration) will go unstable. Numerical simulations were run using 𝑘0 = 𝑘 ′ and 𝑐0 = 𝑐.
The ground motion used was record number 6. The ground motion was scaled to create maximum story
drift about 1.5%. The structure was kept elastic in one case and was allowed to become inelastic in
another case to investigate effect of yielding in accuracy of CR integration as well. 2% Rayleigh damping
was assigned to the 1st and 2nd modes. Initial tangent stiffness (after applying P-Δ loads) was used to
construct damping matrix. Table 7 shows the RMS error between damper displacements of the 2 models.
It can be seen that CR integration creates higher error when structure is nonlinear. The reason is that when
structure becomes nonlinear, tangent stiffness of the structure changes and it won’t be same as the initial
stiffness that was used to calculate integration parameters. However this additional RMS error seems to
be very small. Figure 18 shows comparison between damper displacements of OpenSees and HFEM
model. Results seem to agree very well for both models.
Table 7. RMS error between damper displacements of the 2 models using simple damper model
Analysis Case
Damper Model
Structure Model Damper displacement RMS error
1
Simple linear spring dashpot
Elastic
0.8%
2
Simple linear spring dashpot
Inelastic
1.91%
40
40
NSHF
NSHF
OpenSees
30
30
20
20
Damper displacement (mm)
Damper displacement (mm)
OpenSees
10
0
-10
10
0
-10
-20
-20
-30
0
5
10
15
Time (s)
20
25
-30
0
5
10
15
20
25
Time (s)
(a)
(b)
Figure 18. Comparison between damper displacement of OpenSees and numerical simulation suing
HyrbidFEM (NSHF), using simple linear spring dashpot damper model and (a) linear structure, (b)
nonlinear structure.
10.2. Rigorous damper model
The rigorous damper model introduced earlier in this paper was used in this section. These maximum
tangent stiffness and damping coefficients of the damper model are called 𝑘𝐵𝑊 and 𝑐𝑡𝑟 respectively, as
shown previously. 2 analysis cases were run again. In one case structure was kept elastic and in the other
case structure was allowed to become nonlinear. 𝑘0 and 𝑐0 values used in assembling stiffness and
damping matrices for calculation of integration parameters were equal to 𝑘𝑏𝑤 and 𝑐𝑡𝑟 respectively. Table
8 shows RMS error between damper displacements of HFEM and OpenSees models.
Comparison between analysis case 1 of tables 7 and 8 (with simple and rigorous damper model) shows
that there is more RMS error when rigorous damper model is used. The reason is that rigorous damper
model is nonlinear and tangent stiffness at some displacement range could be very different with 𝑘0 that
was used for calculating integration parameters. Figure 19 shows damper displacements using rigorous
damper model. The agreement between damper displacements is very good. It should be noted that the
agreement between global response parameters are always better than damper displacements as will be
shown later.
It was noticed that when 𝑘0 and 𝑐0 values are less than 𝑘𝑏𝑤 and 𝑐𝑡ℎ , the damper displacement will be
noisy and frequency of the noise will be Nyquist frequency but simulation will be stable. Figure 3 shows
damper displacement noise when 𝑘0 = 0.25. 𝑘𝐵𝑊 and 𝑐0 = 𝑐𝑡𝑟 . Noise is more obvious at the first 6
seconds of the response and only this time interval is shown in Figure 20.
Table 8. RMS error between damper displacements of the 2 models using rigorous damper model
Analysis Case
Damper Model
Structure Model Damper displacement RMS error
1
Rigorous damper model
Elastic
4.70%
2
Rigorous damper model
Inelastic
5.33%
Regarding that the actual maximum tangent stiffness and damping coefficients of the damper could be
larger than maximum tangent stiffness and damping coefficient of the model, effect of using 𝑘0 and 𝑐0
values larger than 𝑘𝐵𝑊 and 𝑐𝑡𝑟 needs to be investigated as well. Table 9 summarizes RMS error between
damper displacement from numerical simulation using HybridFEM and damper displacement using
OpenSees when 5 different sets of integration parameters were used in CR integration. Increasing 𝑘0 and
𝑐0 values in calculation of integration parameters can increase RMS error. In other words increasing 𝑘0
and 𝑐0 values can decrease accuracy of CR integration.
40
40
NSHF
NSHF
OpenSees
OS
30
30
Damper displacement (mm)
Damper displacement (mm)
20
10
0
-10
20
10
0
-10
-20
-30
0
5
10
15
20
25
-20
0
5
10
15
20
25
Time (s)
Time (s)
(a)
(b)
Figure 19. Comparison between damper displacement of OpenSees and numerical simulation suing
HyrbidFEM (NSHF), using rigorous damper model and (a) linear structure (b) nonlinear structure.
1
x 10
0.8
-3
NSHF
OpenSees
Damper displacement (mm)
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
1
2
3
Time (s)
4
5
6
Figure 20. Noise in damper displacement when 𝑘0 = 0.25. 𝑘𝑏𝑤 and 𝑐0 = 𝑐𝑡ℎ
Analysis
Case
1
2
3
4
5
Table 9. Effect of 𝑘0 and 𝑐0 values in accuracy of CR integration
Damper Model
Structure
Damper displacement
𝑘0
𝑐0
Model
RMS error
Rigorous damper model
Inelastic
5.33%
𝑘𝐵𝑊
𝑐𝑡𝑟
Rigorous damper model
Inelastic
6.8𝑘𝐵𝑊
8.99%
𝑐𝑡𝑟
Rigorous damper model
Inelastic
1.75𝑐𝑡𝑟
8.77%
𝑘𝐵𝑊
Rigorous damper model
Inelastic
6.8𝑘𝐵𝑊 1.75𝑐𝑡𝑟
12.72%
Rigorous damper model
Inelastic
46.8𝑘𝐵𝑊 1.75𝑐𝑡𝑟
34.99%
11. Maximum damper stiffness and damping coefficient
It was mentioned that when 𝑘0 and 𝑐0 values are less than 𝑘𝐵𝑊 and 𝑐𝑡𝑟 , the damper displacement will be
noisy and frequency of the noise will be Nyquist frequency but simulation will be stable. That won’t be
the case in laboratory because noise in model command will be amplified by compensator and simulation
will go unstable. In laboratory testing 𝑘0 and 𝑐0 values should be larger than the maximum tangent
stiffness and damping coefficients of the damper to prevent having noise in model command. Also it
should be noted that the actual maximum tangent stiffness and damping coefficient of the damper, 𝑘𝑚𝑎𝑥
and 𝑐𝑚𝑎𝑥 are different with maximum tangent stiffness and damping coefficient of the model 𝑘𝐵𝑊 and
𝑐𝑡𝑟 , because model was not calibrated to capture damper behavior in very small displacements very well.
To find the maximum stiffness, a very slow test with displacement amplitude of 2.54(mm) and excitation
frequency of 0.01(Hz) was performed. Figure 21 shows damper force displacement hysteresis loops for
this test and the maximum tangent stiffness, 𝑘𝑚𝑎𝑥 . This results in
𝑘𝑁
𝑘𝑚𝑎𝑥 = 46.75(𝑚𝑚) = 2.16𝑘𝑏𝑤
(15)
An experiment with excitation frequency of 1.0(Hz) and displacement amplitude of 2.54(mm) was used to
find 𝑐𝑚𝑎𝑥 . To find maximum tangent damping coefficient we can write:
𝑑𝑓 = 𝑘𝑡𝑎𝑛 . 𝑑𝑢 + 𝑐𝑡𝑎𝑛 . 𝑑𝑣
(16)
Where 𝑑𝑓, 𝑑𝑢 and 𝑑𝑣 are incremental damper force, incremental damper displacement and incremental
damper velocity. 𝑘𝑡𝑎𝑛 and 𝑐𝑡𝑎𝑛 are tangent damper stiffness and damping coefficient. When 𝑑𝑢 = 0 we
have 𝑣 = 0 and we can write:
𝑑𝑓 = 𝑐𝑡𝑎𝑛 . 𝑑𝑣
(17)
Which means slope of damper fore velocity hysteresis loops at zero velocity is tangent damping
coefficient. If we assume that like a nonlinear dashpot, the maximum tangent damping coefficient
happens at zero velocity then 𝑐𝑚𝑎𝑥 can be found as shown in Figure 22. This results in
𝑘𝑁
𝑐𝑚𝑎𝑥 = 834 ( 𝑚𝑚
) = 1.04𝑐𝑡ℎ
𝑠𝑒𝑐
(18)
30
2.5
2
20
1
10
0.5
Force (kN)
Displacement (mm)
1.5
0
-0.5
0
-10
-1
-1.5
-20
-2
-2.5
0
20
40
60
80
100
Time (s)
120
140
160
180
-30
-2.5
200
-2
-1.5
-1
-0.5
0
0.5
Displacement (mm)
1
1.5
2
2.5
(a) Damper displacement time history
(b) Damper hysteresis loops
Figure 21. Experiment with displacement amplitude of 2.54(mm) and excitation frequency of 0.01 (Hz)
2.5
40
2
30
1.5
10
0.5
Force (kN)
Displacement (mm)
20
1
0
-0.5
0
-10
-1
-20
-1.5
-30
-2
-2.5
0
2
4
6
Time (s)
8
10
12
-40
-15
-10
-5
0
velocity (mm/s)
5
10
15
(a) Damper displacement time history
(b) Damper force velocity hysteresis loops
Figure 22. Experiment with displacement amplitude of 25.4(mm) and excitation frequency of 0.1 (Hz)
12. Real time hybrid simulation experiments
12.1. Preliminary experiments without compensator
A preliminary HFEM experiment was run. The model includes 1 experimental damper in the 3rd story.
There is no analytical damper in the structure. Structure model is the nonlinear model described
previously. Model includes 2% Rayleigh damping assigned to initial stiffness which creates 2% Rayleigh
10
1
damping on the 1st and 2nd modes. 𝛥𝑡, n and 𝛿𝑡 were 1024 , 10 and 1024. CR integration parameters were
found using 𝑘0 and 𝑐0 values larger than 𝑘𝑚𝑎𝑥 and 𝑐𝑚𝑎𝑥 found in previous chapter (𝑘0 = 4𝑘𝑏𝑤 and 𝑐0 =
1.5𝑐𝑡𝑟 ). There was no compensator used for this preliminary experiment.
Figure 23 shows achieved damper displacement of this experiment. Although used 𝑘0 and 𝑐0 values were
larger than maximum tangent stiffness and damping coefficient of the damper, noise is obvious in damper
displacement during the first 5 seconds. FFT transform of damper displacement showed that frequency of
this noise was about 18(Hz). It was mentioned before that when 𝑘0 and 𝑐0 values less than 𝑘𝐵𝑊 and 𝑐𝑡𝑟
are used in pure numerical simulation, damper displacement will have a noise at Nyquist frequency.
However this experiment shows a noise at a frequency different with Nyquist frequency. This indicates
that the origin of this noise in not small 𝑘0 and 𝑐0 values. A lot of investigations were performed to find
region of this noise and ways to overcome it. The noise is probably due to natural frequency of some
member in actuator. To achieve good experimental results, a compensator should be used and
compensators amplify high frequency noise and cause instability, so it is important to remove this noise.
For example the same test was performed by using ATS compensator [17] and experiment went unstable.
40
Displacement (mm)
30
20
10
0
-10
-20
0
5
10
15
20
25
Time (s)
Figure 23. Damper displacement for HFEM experiment using 𝑘0 = 4. 𝑘𝑏𝑤 and 𝑐0 = 1.5𝑐𝑡𝑟
Experiments showed that using very large values of 𝑘0 and 𝑐0 in calculation of integration parameters can
remove noise in damper displacement. However as it was shown in Table 9, increasing 𝑘0 and 𝑐0 values
can increase error between HFEM numerical simulation and OpnSees results. In other words increasing
𝑘0 and 𝑐0 values in calculation of CR integration parameters, decreases accuracy of CR integration. Also
the noise in damper displacement only exists at the beginning and to some extent at the end of each
ground motion, where damper displacement is smaller. To overcome instability problem without over
sacrificing accuracy, it was decided to use two sets of integration parameters for each experiment. One set
of integration parameters that are calculated based on very large 𝑘0 values to be used at the beginning and
at the end of each experiment to overcome the noise and prevent instability as a result of amplified noise
by compensator. And another set of integration parameters which are calculated based on more
reasonable values of 𝑘0 and 𝑐0 values to be used for main part of ground motion. So each experiment
starts using the first set of integration parameters, then after a switching time (𝑡1 ); 2nd set of integration
parameters will be used and again after the 2nd switching time (𝑡2 ); 1st set of integration parameters will
be used.
Table 10 shows that using high value of 𝑘0 and 𝑐0 , at the beginning and at the end of a pure numerical
simulation using HybridFEM (NSHF) doesn’t change RMS error noticeably. Analysis cases 1 and 2 of
this table use 1 set of integration parameter and analysis case 3 uses 2 sets of integration parameters. RMS
error of case 3 is very close to RMS error of case 1.
Table 10. Effect of 𝑘0 and 𝑐0 values in accuracy of CR integration
Analysis
Case
Damper
Model
Structure
Model
𝒕𝟏
(sec)
𝒕𝟐
(sec)
𝒌𝟎 and 𝒄𝟎
𝒖𝒅 RMS
error
t<𝒕𝟏 or t>𝒕𝟐
𝒕𝟏 <t<𝒕𝟐
OpenSees vs
NSHF
1
Rigorous
damper model
Inelastic
-
−
−
6.8𝑘𝑏𝑤 1.75𝑐𝑡ℎ
12.72%
2
Rigorous
damper model
Inelastic
-
-
−
46.8𝑘𝑏𝑤 1.75𝑐𝑡ℎ
34.99%
3
Rigorous
damper model
Inelastic
6.0
20.0
46.8𝑘𝑏𝑤 1.75𝑐𝑡ℎ
6.8𝑘𝑏𝑤 1.75𝑐𝑡ℎ
13.31%
12.2. Main experiments with compensator
12.2.1 Adaptive inverse transfer function compensator [17]
Accurate actuator control is a one of the critical issues to achieve a successful real-time hybrid
simulation since it affects the stability of the simulation. The dynamics of servo-hydraulic actuators
can cause a time delay and amplitude change in the response of the actuator to command
displacements. To obtain accurate experimental results in a real-time hybrid simulation, the time
delay and amplitude error need to be appropriately compensated whereby the target displacement 𝑥 𝑡
is achieved by the actuator. [17]
The procedure to minimize actuator delay is shown conceptually in Figure 24(a), where a
compensated displacement command 𝑢𝑐 is sent to the actuator to attempt to have measured
displacement 𝑥 𝑚 match target displacement 𝑥 𝑡 . [17]. In general, the combined servo-hydraulic
actuator system and experimental substructure can exhibit nonlinear behavior due to the complexity
of the servo-valve dynamics as well as any nonlinearity in the experimental substructure, resulting in
a variable time delay and amplitude error. [17]. To overcome this problem Chae et al [17] developed
an adaptive 2nd order time series (ATS) compensator. The concept for the ATS compensator is shown
schematically in Figure 24(b). Compensated signal at each time step is calculated by equation 15.
𝑢𝑘𝑐 = 𝑎0𝑘 𝑥𝑘𝑡 + 𝑎1𝑘 𝑥̇ 𝑘𝑡 + 𝑎2𝑘 𝑥̈ 𝑘𝑡
(15)
Where 𝑘 is the time index and velocity and acceleration of each step will be estimated by finite
difference method, using equation 16
𝑥 𝑡 −𝑥 𝑡
𝑥 𝑡 −2𝑥 𝑡
+𝑥 𝑡
𝑘−1
𝑘−2
𝑥̇ 𝑘𝑡 = 𝑘 𝛥𝑡𝑘−1, 𝑥̈ 𝑘𝑡 = 𝑘 𝛥𝑡
2
and 𝑎0 , 𝑎1 and 𝑎2 are compensator parameters found by equation 17
−1
𝐴 = (𝑋𝑚 𝑇 𝑋𝑚 ) 𝑋𝑚 𝑇 𝑈𝑐
𝑚
𝑚
𝑚
where 𝐴 = [𝑎0𝑘 𝑎1𝑘 𝑎2𝑘 ]𝑇 , 𝑋𝑚 = [𝑥 𝑚 𝑥̇ 𝑚 𝑥̈ 𝑚 ] , 𝑥 𝑚 = [𝑥𝑘−1
𝑥𝑘−2
… 𝑥𝑘−𝑞
] and 𝑈𝑐 =
𝑐
𝑐
𝑐
[𝑢𝑘−1 𝑢𝑘−2 … 𝑢𝑘−𝑞 ] .
(16)
(17)
The ATS compensator was used in this study and lead to very good agreement between desired and
achieved actuator displacement.
(a)
(b)
Figure 24. Schematic of actuator delay compensation: (a) without feedback; (b) with feedback (ATS
compensator). [17]
12.2.2 Experiments and results
3 sets of experiments were performed. In all experiments the 3 ground motions that are picked in ground
motion selection chapter were used. Modeling of different parts of the structure other than dampers is the
same in 3 models. In experiment set 1 there is 1 experimental damper in the 3rd story and there is no
analytical damper. Experiment set 2 includes 2 experimental dampers in the 3rd story, 3 and 4 analytical
dampers in the 2nd and 1st stories respectively. This damper arrangement, as shown before, is the desired
arrangement to satisfy target drift for a DBE level earthquake. There is still only one experimental damper
on floor in lab and it was assumed that both dampers in the 3rd story will have the same force time history.
Ground motions in this experiment set were scaled to DBE level. Experiment set 3 is same as experiment
set 2 but ground motions are scale to MCE level.
Table 10 compares agreement of damper displacement (𝑢𝑑 ) results between OpenSees, NSHF and HSHF.
Times of switching integration parameters (𝑡1 and 𝑡2 ) and 𝑘0 and 𝑐0 values at each time range are
reported. The last column reports RMS error between experimental damper force and damper force (𝑓𝑑 )
obtained by model when it is subjected to experimental damper displacement. There is good agreement
between damper displacement results of OpenSees and NSHF, and HFHS and NSHF, which means both
CR integration and damper model, are working properly. Table 11 reports 1st, 2nd and 3rd floor
displacements, 𝑢1 , 𝑢2 and 𝑢3 . Agreement between “OpenSees and NSHF” and “NSHF and HSHF” is
better in global response parameters. Figures 25, 26 and 27 shows results of experiments 1, 2 and 3.
Tables 12 and 13 report results of the 2nd set of experiments and Tables 14 and 15 report results of the 3rd
set of experiments. Figures 28, 29 and 30 show results of experiments 4, 5 and 6 and Figures 31, 32and
33 show results of experiments 7, 8 and 9.
Table 10. Damper results of the 1st set of experiments
Experiment Rec
#
#
𝒕𝟏
(sec)
𝒕𝟐
(sec)
𝒌𝟎 and 𝒄𝟎
𝒖𝒅 RMS error
𝒇𝒅 RMS
error
t<𝒕𝟏 or
t>𝒕𝟐
𝒕𝟏 <t<𝒕𝟐
OpenSees
vs NSHF
HSHF vs
NSHF
Model vs
Exp
1
6
6.0
20.0
36.8𝑘𝑏𝑤 1.75𝑐𝑡ℎ
6.8𝑘𝑏𝑤 1.75𝑐𝑡ℎ
13.31%
16.93%
22.46%
2
7
8.5
27.0
36.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
6.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
9.22%
18.35%
18.87%
3
14
25.0
56.0
36.8𝑘𝑏𝑤 1.75𝑐𝑡ℎ
6.8𝑘𝑏𝑤 1.75𝑐𝑡ℎ
16.92%
23.42%
21.89%
Table 11. Global response parameters for the 1st set of experiments
𝒖𝟑 RMS error
Experiment Rec
#
#
𝒖𝟐 RMS error
𝒖𝟏 RMS error
OpenSees
vs NSHF
HFHS vs
NSHF
OpenSees
vs NSHF
HFHS vs
NSHF
OpenSees
vs NSHF
HFHS vs
NSHF
1
6
9.56%
9.32%
9.70%
9.24%
11.21%
9.74%
2
7
5.51%
12.19%
5.56%
12.5%
7.06%
12.98%
3
14
7.80%
14.99%
7.60%
15.19%
11.75%
16.69%
40
40
NSHF
HSHF
30
Opensees
20
Displacement (mm)
Displacement (mm)
30
10
0
-10
NSHF
20
10
0
-10
-20
-20
0
5
10
15
20
25
0
5
10
Time (s)
(a)
20
25
(b)
40
200
Model
Model command
30
150
Actuator Displacement
Experiment
100
20
Force (kN)
Displacement (mm)
15
Time (s)
10
0
50
0
-50
-10
-100
-20
0
5
10
Time (s)
15
20
-150
-20
25
-10
0
10
20
30
(d)
100
100
NSHF
HSHF
OpenSees
NSHF
50
Displacement (mm)
50
Displacement (mm)
40
Displacement (mm)
(c)
0
-50
-100
0
-50
-100
0
5
10
15
20
25
0
5
10
15
Time (s)
Time (s)
(e)
(f)
20
Figure 25. Results of experiment 1. (a) comparison of damper displacement between OpenSees and
NSHF. (b) Comparison of damper displacement between HSHF and NSHF. (c) Comparison of damper
displacement model command and achieved actuator displacement. (d) Comparison between
experimental and analytical damper hysteresis loops. (e) Comparison between roof displacement of
OpenSees and NSHF. (f) Comparison between roof displacement of HSHF and NSHF.
25
30
30
NSHF
HSHF
20
Opensees
10
Displacement (mm)
Displacement (mm)
20
0
-10
-20
NSHF
10
0
-10
-20
-30
-30
0
5
10
15
20
25
30
0
5
10
Time (s)
15
20
25
30
10
20
30
Time (s)
(a)
(b)
30
200
Model
Model command
20
Experiment
Actuator Displacement
10
Force (kN)
Displacement (mm)
100
0
-10
0
-100
-20
-30
0
5
10
15
20
25
-200
-30
30
-20
-10
Time (s)
0
Displacement (mm)
(c)
(d)
100
100
NSHF
HSHF
OpenSees
NSHF
50
Displacement (mm)
Displacement (mm)
50
0
-50
-100
0
-50
-100
0
5
10
15
Time (s)
(e)
20
25
30
0
5
10
15
20
25
Time (s)
(f)
Figure 26. Results of experiment 2. (a) comparison of damper displacement between OpenSees and
NSHF. (b) Comparison of damper displacement between HSHF and NSHF. (c) Comparison of damper
displacement model command and achieved actuator displacement. (d) Comparison between
experimental and analytical damper hysteresis loops. (e) Comparison between roof displacement of
OpenSees and NSHF. (f) Comparison between roof displacement of HSHF and NSHF.
30
40
40
NSHF
30
HSHF
30
Opensees
NSHF
20
Displacement (mm)
Displacement (mm)
20
10
0
-10
-20
10
0
-10
-20
-30
-30
0
10
20
30
40
50
60
70
80
0
10
20
30
40
Time (s)
(a)
60
70
80
(b)
40
200
Model
Model command
30
Experiment
Actuator Displacement
100
20
Force (kN)
Displacement (mm)
50
Time (s)
10
0
-10
0
-100
-20
-30
0
10
20
30
40
50
60
70
-200
-30
80
-20
-10
0
Time (s)
10
20
30
(c)
(d)
100
100
NSHF
HSHF
OpenSees
NSHF
50
Displacement (mm)
50
Displacement (mm)
40
Displacement (mm)
0
-50
-100
0
-50
-100
0
10
20
30
40
50
60
70
80
0
10
20
30
40
Time (s)
Time (s)
(e)
(f)
50
60
70
80
Figure 27. Results of experiment 3. (a) comparison of damper displacement between OpenSees and
NSHF. (b) Comparison of damper displacement between HSHF and NSHF. (c) Comparison of damper
displacement model command and achieved actuator displacement. (d) Comparison between
experimental and analytical damper hysteresis loops. (e) Comparison between roof displacement of
OpenSees and NSHF. (f) Comparison between roof displacement of HSHF and NSHF.
Table 12. Damper results of the 2nd set of experiments
Experiment Rec
#
#
𝒕𝟏
(sec)
𝒕𝟐
(sec)
𝒌𝟎 and 𝒄𝟎
𝒖𝒅 RMS error
𝒇𝒅 RMS error
t<𝒕𝟏 or
t>𝒕𝟐
𝒕𝟏 <t<𝒕𝟐
OpenSees
vs NSHF
HSHF
vs NSHF
Model vs
Experiment
4
6
6.0
20.0
35.5𝑘𝑏𝑤 1.25𝑐𝑡ℎ
5.5𝑘𝑏𝑤 1.25𝑐𝑡ℎ
10.65%
27.91%
26.96%
5
7
8.5
27.0
35.5𝑘𝑏𝑤 1.25𝑐𝑡ℎ
5.5𝑘𝑏𝑤 1.25𝑐𝑡ℎ
9.46%
20.46%
18.74%
6
14
25.0
56.0
36.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
6.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
24.81%
28.7%
21.91%
Table 13. Global response parameters for the 2nd set of experiments
𝒖𝟑 RMS error
Experiment Rec
#
#
𝒖𝟐 RMS error
𝒖𝟏 RMS error
OpenSees
vs NSHF
HFHS vs
NSHF
OpenSees
vs NSHF
HFHS vs
NSHF
OpenSees
vs NSHF
HFHS vs
NSHF
4
6
8.24%
14.37%
8.750%
13.70%
9.87%
13.99%
5
7
6.22%
14.08%
6.44%
14.87%
7.64%
15.28%
6
14
10.48%
18.18%
10.39%
17.58%
16.24%
17.72%
30
30
NSHF
NSHF
20
Displacement (mm)
Displacement (mm)
HSHF
Opensees
20
10
0
-10
-20
10
0
-10
-20
0
5
10
15
20
25
0
5
10
Time (s)
20
25
(b)
30
200
Model
Model command
150
Actuator Displacement
20
Experiment
100
10
Force (kN)
Displacement (mm)
15
Time (s)
(a)
0
50
0
-50
-10
-100
-20
0
5
10
15
20
-150
-20
25
-15
-10
Time (s)
-5
0
5
10
15
20
30
(d)
(c)
80
80
NSHF
60
HSHF
60
OpenSees
NSHF
40
Displacement (mm)
Displacement (mm)
25
Displacement (mm)
20
0
-20
-40
40
20
0
-20
-40
-60
-60
0
5
10
15
Time (s)
(e)
20
25
0
5
10
15
20
25
Time (s)
(f)
Figure 28. Results of experiment 4. (a) comparison of damper displacement between OpenSees and
NSHF. (b) Comparison of damper displacement between HSHF and NSHF. (c) Comparison of damper
displacement model command and achieved actuator displacement. (d) Comparison between
experimental and analytical damper hysteresis loops. (e) Comparison between roof displacement of
OpenSees and NSHF. (f) Comparison between roof displacement of HSHF and NSHF.
30
30
HSHF
NSHF
20
Opensees
10
Displacement (mm)
Displacement (mm)
20
0
-10
NSHF
10
0
-10
-20
-20
-30
-30
0
5
10
15
20
0
30
25
5
10
15
20
25
30
10
20
30
Time (s)
Time (s)
(b)
(a)
200
30
Model
Model command
20
Experiment
Actuator Displacement
Force (kN)
Displacement (mm)
100
10
0
-10
0
-100
-20
-200
-30
-30
0
5
10
15
20
30
25
-20
-10
0
Displacement (mm)
Time (s)
(c)
(d)
100
100
HSHF
NSHF
NSHF
OpenSees
50
Displacement (mm)
Displacement (mm)
50
0
-50
0
-50
-100
-100
0
5
10
15
Time (s)
(e)
20
25
30
0
5
10
15
20
25
30
Time (s)
(f)
Figure 29. Results of experiment 5. (a) comparison of damper displacement between OpenSees and
NSHF. (b) Comparison of damper displacement between HSHF and NSHF. (c) Comparison of damper
displacement model command and achieved actuator displacement. (d) Comparison between
experimental and analytical damper hysteresis loops. (e) Comparison between roof displacement of
OpenSees and NSHF. (f) Comparison between roof displacement of HSHF and NSHF.
30
30
NSHF
HSHF
20
Opensees
10
Displacement (mm)
Displacement (mm)
20
0
-10
-20
NSHF
10
0
-10
-20
-30
-30
0
10
20
30
40
50
60
70
80
0
10
20
30
40
Time (s)
50
60
70
80
Time (s)
(b)
(a)
30
200
Model command
20
Model
Actuator Displacement
Experiment
10
Force (kN)
Displacement (mm)
100
0
-10
0
-100
-20
-30
0
10
20
30
40
50
60
70
-200
-30
80
-20
-10
0
Time (s)
Displacement (mm)
(c)
(d)
100
10
20
30
100
NSHF
HSHF
OpenSees
NSHF
50
Displacement (mm)
Displacement (mm)
50
0
-50
-100
0
-50
-100
0
10
20
30
40
50
60
70
80
0
10
20
30
Time (s)
40
50
60
70
80
Time (s)
(e)
(f)
Figure 30. Results of experiment 6. (a) comparison of damper displacement between OpenSees and
NSHF. (b) Comparison of damper displacement between HSHF and NSHF. (c) Comparison of damper
displacement model command and achieved actuator displacement. (d) Comparison between
experimental and analytical damper hysteresis loops. (e) Comparison between roof displacement of
OpenSees and NSHF. (f) Comparison between roof displacement of HSHF and NSHF.
Table 14. Damper results of the 2nd set of experiments
Experiment Rec
#
#
𝒕𝟏
𝒕𝟐
𝒌𝟎 and 𝒄𝟎
𝒖𝒅 RMS error
𝒇𝒅 RMS error
t<𝒕𝟏 or
t>𝒕𝟐
𝒕𝟏 <t<𝒕𝟐
OpenSees
vs NSHF
HSHF
vs
NSHF
Model vs
Experiment
7
6
6.0
20.0
36.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
6.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
14.0%
30.15%
30.72%
8
7
8.5
27.0
36.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
6.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
10.85%
21.11%
21.96%
9
14
25.0
56.0
36.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
6.8𝑘𝑏𝑤 1.5𝑐𝑡ℎ
31.65%
34.67%
32.76%
Table 15. Global response parameters for the 2nd set of experiments
𝒖𝟑 RMS error
Experiment Rec
#
#
𝒖𝟐 RMS error
𝒖𝟏 RMS error
OpenSees
vs NSHF
HFHS vs
NSHF
OpenSees
vs NSHF
HFHS vs
NSHF
OpenSees
vs NSHF
HFHS vs
NSHF
7
6
9.76%
16.83%
9.84%
14.5%
12.37%
13.63%
8
7
7.13%
10.88%
7.48%
10.83%
9.56%
12.02%
9
14
19.18
17.45
17.05
14.31
21.36
14.66
50
50
NSHF
40
HSHF
40
Opensees
NSHF
30
Displacement (mm)
Displacement (mm)
30
20
10
0
-10
10
0
-10
-20
-20
0
5
10
15
20
25
0
5
10
15
Time (s)
Time (s)
(a)
(b)
50
20
25
300
Model
Model command
40
Actuator Displacement
Experiment
200
30
Force (kN)
Displacement (mm)
20
20
10
100
0
0
-100
-10
-20
0
5
10
15
20
-200
-20
25
-10
0
Time (s)
10
20
30
40
(c)
(d)
150
150
NSHF
HSHF
OpenSees
NSHF
100
Displacement (mm)
100
Displacement (mm)
50
Displacement (mm)
50
0
-50
50
0
-50
0
5
10
15
20
25
0
5
10
15
Time (s)
Time (s)
(e)
(f)
20
Figure 31. Results of experiment 7. (a) comparison of damper displacement between OpenSees and
NSHF. (b) Comparison of damper displacement between HSHF and NSHF. (c) Comparison of damper
displacement model command and achieved actuator displacement. (d) Comparison between
experimental and analytical damper hysteresis loops. (e) Comparison between roof displacement of
OpenSees and NSHF. (f) Comparison between roof displacement of HSHF and NSHF.
25
40
40
NSHF
NSHF
20
Displacement (mm)
Displacement (mm)
HSHF
Opensees
20
0
-20
-40
-60
0
-20
-40
-60
0
5
10
15
20
25
30
0
5
10
15
Time (s)
25
200
Model
Model command
Experiment
Actuator Displacement
20
100
Force (kN)
0
-20
0
-100
-40
-60
0
5
10
15
20
25
-200
-50
30
-40
-30
-20
Time (s)
-10
0
10
20
30
Displacement (mm)
(c)
(d)
100
100
NSHF
HSHF
OpenSees
NSHF
50
Displacement (mm)
50
Displacement (mm)
30
(b)
40
Displacement (mm)
20
Time (s)
(a)
0
-50
-100
-150
0
-50
-100
-150
0
5
10
15
20
25
30
0
5
10
15
Time (s)
Time (s)
(e)
(f)
20
25
Figure 32. Results of experiment 8. (a) comparison of damper displacement between OpenSees and
NSHF. (b) Comparison of damper displacement between HSHF and NSHF. (c) Comparison of damper
displacement model command and achieved actuator displacement. (d) Comparison between
experimental and analytical damper hysteresis loops. (e) Comparison between roof displacement of
OpenSees and NSHF. (f) Comparison between roof displacement of HSHF and NSHF.
30
60
60
NSHF
NSHF
40
Displacement (mm)
40
Displacement (mm)
HSHF
Opensees
20
0
-20
-40
20
0
-20
-40
0
10
20
30
40
50
60
70
80
0
20
30
40
Time (s)
(a)
(b)
60
50
60
70
Model
Actuator Displacement
40
Experiment
200
Force (kN)
20
0
-20
100
0
-100
-40
0
10
20
30
40
50
60
70
-200
-40
80
-30
-20
-10
0
Time (s)
10
20
30
40
50
Displacement (mm)
(d)
(c)
150
150
NSHF
HSHF
OpenSees
NSHF
100
Displacement (mm)
100
Displacement (mm)
80
300
Model command
Displacement (mm)
10
Time (s)
50
0
-50
-100
50
0
-50
-100
0
10
20
30
40
50
60
70
80
0
10
20
30
40
Time (s)
Time (s)
(e)
(f)
50
60
70
80
Figure 33. Results of experiment 9. (a) comparison of damper displacement between OpenSees and
NSHF. (b) Comparison of damper displacement between HSHF and NSHF. (c) Comparison of damper
displacement model command and achieved actuator displacement. (d) Comparison between
experimental and analytical damper hysteresis loops. (e) Comparison between roof displacement of
OpenSees and NSHF. (f) Comparison between roof displacement of HSHF and NSHF.
13. Summary and Conclusions
This paper introduced a third generation elastomeric damper which was designed to overcome
shortcomings of the second generation elastomeric damper. Damper was characterized and simple and
rigorous damper models were developed to be used for design and nonlinear dynamic time history
analysis (NDTHA) of structures with elastomeric dampers. A 3 story structure was designed using SDP to
satisfy a target drift of 1.5%. A set of 20 ground motions were chosen that cause a median response
spectrum that matches design spectrum at period range of 0.2-2.0(sec). 3 of those ground motions that
correspond to median DBE earthquake were chosen for real time hybrid simulation experiments.
CR integration developed by Chen and Ricles [4] was used for hybrid simulation experiments. Numerical
studies were performed to investigate effect of integration parameters on accuracy of CR integrator. To
overcome instability while keeping accuracy of CR integration acceptable, it was decided to use 2 sets of
integration parameters. Recently developed ATS compensator [17] was used to control the actuator which
led to excellent actuator control. Experiments were performed and accuracy of damper model and CR
integration was investigated. Good agreement was noticed between predicted numerical simulations and
experimental results. Damper model captures damper behavior very well, however experimental results
could be used to update the damper model as well.
Acknowledgements
This paper is based upon research conducted at the NEES Real-Time Multi-Directional (RTMD) Earthquake Simulation Facility located at the ATLSS Center at Lehigh University. The research was supported by grants from the Pennsylvania Department of Community and Economic Development through
the Pennsylvania Infrastructure Technology Alliance, and by the National Science Foundation, Award
No. CMS-0936610, in the George E. Brown, Jr. Network for Earthquake Engineering Simulation
Research (NEESR) program, and Grant No. CMS-0402490 within the George E. Brown, Jr. Network for
Earthquake Engineering Simulation Consortium Operation. The compressed elastomer dampers were
manufactured and donated to the project by Corry Rubber. Any opinions, findings, and conclusions
expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.
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