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N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11
129
Focussed on Recent advances in “Experimental Mechanics of Materials” in Greece
Earthquake design for controlled structures
Nikos G. Pnevmatikos
Technological Educational Institution of Athens, Greece
pnevma@teiath.gr
George A. Papagiannopoulos
University of Patras, Greece
gpapagia@upatras.gr
George D. Hatzigeorgiou
Hellenic Open University, Greece
hatzigeorgiou@eap.gr
ABSTRACT. An alternative design philosophy, for structures equipped with control
devices, capable to resist an expected earthquake while remaining in the elastic
range, is described. The idea is that a portion of the earthquake loading is under-
taken by the control system and the remaining by the structure which is designed
to resist elastically. The earthquake forces assuming elastic behavior (elastic forces)
and elastoplastic behavior (design forces) are first calculated according to the
codes. The required control forces are calculated as the difference from elastic to
design forces. The maximum value of capacity of control devices is then compared
to the required control force. If the capacity of the control devices is larger than
the required control force then the control devices are accepted and installed in
the structure and the structure is designed according to the design forces. If the
capacity is smaller than the required control force then a scale factor, α, reducing
the elastic forces to new design forces is calculated. The structure is redesigned and
devices are installed. The proposed procedure ensures that the structure behaves
elastically (without damage) for the expected earthquake at no additional cost,
excluding that of buying and installing the control devices.
KEYWORDS. Response spectrum analysis; Structural control; Earthquake
engineering.
Citation: Pnevmatikos Ν., Papagiannopoulos
G., Hatzigeorgiou G., Earthquake design for
controlled structures, Frattura ed Integrità
Strutturale, 40 (2017) 129-136.
Received: 05.12.2016
Accepted: 13.03.2017
Published: 01.04.2017
Copyright: © 2017 This is an open access
article under the terms of the CC-BY 4.0,
which permits unrestricted use, distribution,
and reproduction in any medium, provided
the original author and source are credited.
INTRODUCTION
ver the past few decades various control algorithms and control devices have been developed, modified and
investigated by various groups of researchers. The works of Yao, 1975, Housner et al., 1994, Kobori et al., 1998,
and Soong 1998 are representative [1-4]. There have been some attempts to connect the control forces with the O
N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11
130
design codes. Yang et al. 2003 [5] suggested the maximum control force to be a percentage of the building weight, while
Cai et al., 1997, [6] give this force as a portion of the seismic force. Lee et al., 2004, [7] determined the upper limit of
control force based on the response spectrum of the external earthquake.
During the past years the design philosophy of new structure was to design stiff structures with high strength to resist the
earthquake in the elastic range. After that the design philosophy moves one step further. Using the ductility of the
material, structures were designed to resist lower level of earthquake forces within the elastic range but to have adequate
ductility in order to face the attack of stronger earthquakes and prevent them from collapse. This drives to lighter
structures compared to the previous structures and more economical. However, the capacity design and the reinforcement
details increased the cost. Taking in account the cost of repair of the retrofitting of structure, after a strong earthquake,
the design of ductile structures should be under consideration.
The design philosophy proposed here is to use control devices installed in the structure and provide a reservoir of
strength, stiffness or damping, necessary for preventing the structure from damage when an expected earthquake will
occur. Thus, the control system will drive the structure to behave in the elastic range when it is attacked by the expected
earthquake and no damages will occur. As far as the cost is concerned, it is possible to achieve substantial savings by
avoiding retrofit of structure during the lifetime of structure and utilize these savings for installing a control system. A
systematic procedure to achieve the above objective is proposed in this work.
DESIGN PROCEDURE FOR STRUCTURES EQUIPPED WITH A CONTROL SYSTEM
he evolution of the design philosophy of structures passes through different stages. Fist the engineers design stiff
and massive structures in order to behave elastically during the expected earthquake. As years passed and damages
were observed after earthquakes the design philosophy was moved from the resistance of structure to energy
dissipation capacity of the structural elements and design of structures with an overall ductile behavior. This drives
engineers to perform capacity design for structures. This philosophy is nowadays included in all current design regulations.
However, observing the damages to the structures that were designed with the latter philosophy and making calculations,
the repair cost of the capacity design emerged and came into consideration. The answer to the previous consideration is
the new and proposed design philosophy where the structure is oriented to capacity design equipped with control devices
that will absorb a portion of seismic energy induced to the structure and as a result to keep the structure in the elastic
range. The three design philosophies are depicted in Fig. 1. The proposed design procedure for the spectrum is calculated
in such a way that one portion of earthquake forces is taken by the structure and the remaining ones are taken by the control
devices.
Figure 1: The three design philosophies of design of structures.
T
N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11
131
Finalized design
accepted
Control system
installation
Calculation of Elastic and Design forces
No
Yes
Redesign of structure
Estimate α>(1/q)
Reduced elastic spectrum by α
Elastic spectrum
New design spectrum
a
Design initially, accepted
Time history control analysis
of structure, with saturation
and time delay
Response
satisfies the
elastic
criteria?
No
Yes
Design of structure, according to Eurocodes
Elastic spectrum
Design spectrum
1/q
Is the difference of
elastic to design forces
lower than the device
capacity?
Redesign of structure
with higher α
Initially, the controlled structure is designed based on a design spectrum provided by the pertinent code (Eurocodes) with
a specific level of ductility. The required control forces that will take a portion of earthquake forces are calculated as the
difference from forces obtained from the elastic spectrum to those obtained from the design spectrum. The maximum
value of capacity of the control devices is compared with the required control force. If the capacity of the control devices
is larger than the required control force then the control devices are accepted and installed into the structure. If the
capacity is smaller than the required control force then a control device with larger capacity should be chosen or more
devices per floor should be installed
In case the maximum available control device capacity is smaller than the required control force, or, there is a limitation to
the number of control devices, then using an iterative procedure, a scale factor, α, higher than the value 1/q that reduces
the elastic response spectrum is calculated. The structure is redesigned based on the new reduced spectrum by scale factor, a,
and then the devices are installed into the structure. The flow chart of the procedure is shown in Fig. 2 with a solid line.
Figure 2: The flow chart of the proposed design procedure.
Estimation of scale factor, a.
From the elastic seismic forces and the maximum capacity of the control device a scale factor α is obtained and applied on
the elastic spectrum. Knowing the mass and initial stiffness of the structure the eigenmodes Φi, eigenperiods Ti or
eigenfrequencies fi and the corresponding damping ratios ξi of the uncontrolled system are obtained.
The participation factor ψi, and elastic seismic forces Fq,el,i for the ith eigenmode are given as:
T
i
i T
i i
ψ = , i=1,...,n
Φ ME
Φ MΦ
(1)
, , , ( , ), 1,...,q el i i i e i i iS T i n F MΦ (2)
where E is the direction matrix for the earthquake and Se,i(Ti, ξi) is the elastic spectral acceleration. The maximum elastic
seismic forces Fq,e for each degree of freedom are obtained combining the Square Root of Sum Squares method (SRSS)
the elastic seismic forces from each eigenmode, thus:
N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11
132
n
2
q,e q,e,i
i
= F F (3)
If Fd,max is the maximum control device capacity (maximum possible control force), then the maximum control force that
can be applied on the system is:
d,max f d,maxFF E (4)
where Ef is the location matrix for the control devices on the structure. Assuming that one part, up to Fd,max, of elastic
seismic forces are carried by control devices, the remaining seismic forces which go directly and force the structural
elements are:
q,e q d,max q d,max
q, new
q d,max
sign( ) if
=
0 if
F F F F F
F
F F
(5)
Fq new is a vector with n forces, where n is the degree of freedom of the system. These forces correspond to a reduced
spectral acceleration. From Eq. (2) this new spectral acceleration Sd,i,new(Ti, ξi), corresponding to new seismic forces, can be
obtained:
-1 Ti q,new
d,i,new i i
i
S ( T ,ξ ) , i=1,...,n
ψ
MΦ F
(6)
The reduction factor α can be obtained by dividing the new spectral acceleration Sd,i,new(Ti, ξi) by the corresponding initial
one:
d,i,new i i
i
e,i i i
S ( T ,ξ )
α , i=1,...,n
S ( T ,ξ )
(7)
The elastic spectrum is scaled using the maximum value of αi and the structure is redesigned based on the reduced
spectrum. The value of a is:
iα max(α ) (8)
In order to ensure a linear behavior of the structure, dynamic control analysis is performed for a range of earthquakes
(high and low frequency characteristics), with saturation control and time delay. If the response satisfies the elastic criteria,
then the value of α is accepted, otherwise it is slightly increased and the above procedure is repeated. The flow chart of
this procedure is shown in Fig. 2 with a dashed line.
The equation of motion of a controlled structural system with n degrees of freedom subjected to an earthquake excitation
ag in the state space approach is:
g g f
a X AX B B F (9)
The matrixes X, A, Bg, Bf are given by
new
g f1 1 1
f2 1 2 12 2 2 1
, , ,
nx nxnx n nx
0 I 0U 0
X A B B
U EM K M C M E
(10)
N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11
133
where M and C denote the mass and damping matrices of the structure, respectively, Knew is the new stiffness matrix of
the redesigned structure, and F is the control force matrix. The control force F is determined by linear state feedback as
follows:
1 2 1 2
U
F G U G U G G GX
U
(11)
G is the gain matrix, which will be calculated by pole assignment method and according to the desired poles of the
controlled system.
If the response obtained for the controlled system satisfies the design criteria, then the reduction by q or by a scale factor,
α, is accepted. In this work a representative design criterion was used, that the story drift does not exceed h/300 (where h
is the story height). This value does not cause member yielding. In a similar way, additional design criteria concerning the
rotation and strength of structural members can be used. The above procedure was tested for a number of numerical
simulations, and some representative examples are presented next.
RESULTS AND DISCUSSION
he proposed approach is demonstrated by means of numerical example where an eight-story building, described in
the work of Yang et al, 1995 [7], is analyzed. Initially the elastic and design spectra are calculated based on
Eurocode 8 (EC8) seismic code.
Based on those spectra and on dynamic characteristics of building the seismic forces Fq,i for each eigenmode and their
combination are calculated for both elastic and design spectrum. The seismic forces which are obtained from elastic and
design spectrum and their differences are shown in Fig. 3(a). Assuming that the control devices are installed on each floor
and the maximum capacity is 1000kN, following the proposed procedure the scale factor α is calculated to be equal to
0.49 or the equivalent reduction from the elastic spectrum 1-α which is equal to 51%. The elastic and design spectra and
the reduced spectrum by 51% from the elastic spectrum, for which the structure will be redesigned, are illustrated in Fig.
3(b).
In order to ensure that the structure remains in the elastic range after redesigning, dynamic time control analysis history,
with saturation control and time delay, for a wide range of earthquakes should be performed. The numerical simulations
were performed in Simulink toolbox of Matlab software. The numerical simulation of the control scheme is described in
Fig. 3(c).
The response (displacement and acceleration) of the system subjected to Athens earthquake 1999 were calculated. From
the numerical results it was seen that full compensation of the displacements was achieved. According to the work of
Yang et al. (2003) when one control force corresponds for each degree of freedom then complete compensation of the
response can be achieved and the response state vector can be reduced to zero. Another reason that the relative
displacements are near to zero is that the elastic response spectrum of the Athens earthquake are lower than the elastic
spectrum that was used initially for the design procedure. The acceleration is equal to the external signal and the building
behaves like executing a rigid body motion. The control forces are identical, with maximum value at 917 kN and rms
value at 134 kN, because the mass of each story is the same. The storey drift between the floors was not exceeded the
limit value h/300=10 mm. Time history of displacement and the acceleration from 8th floor for the controlled and
uncontrolled structure is shown in Fig. 4.
SUMMARY AND CONCLUSIONS
procedure to design a structure equipped with control devices is described. The structure is designed based on a
reduced spectrum. A scale factor α which multiplies the elastic spectrum and produces a reduced spectrum is
proposed. The design philosophy is that one part of seismic forces are taken by control devices and to the rest of
earthquake forces taken up from the structure. The numerical results indicate that reduction of the spectrum can be
achieved using control devices. The cost of repairing the post-earthquake damages of an uncontrolled structure which was
design based on ductility demand can be considered as a motivation to install a control system which will keep the
structure in the elastic range. The control system is acceptable if the results obtained from the dynamic control analysis
T
A
N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11
134
0 500 1000 1500 2000 2500 3000 3500 4000
0
1
2
3
4
5
6
7
8
Earthquake Force (kN)
S
to
ri
e
s
500 (kN)
1000 (kN)
1500 (kN)
1850 (kN)
2000 (kN)
2200 (kN)
2700 (kN)
2800 (kN)
Earthquake force (kN)
St
o
ri
es
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
1
2
3
4
5
6
7
8
Period (sec)
A
c
c
e
le
ra
ti
o
n
(
m
/s
e
c
2
)
Period (sec)
A
cc
el
er
at
io
n
(m
/s
ec
2 )
keep the structure within the elastic limit. Design criteria such as inter-story drift which shouldn’t exceed a specific value
that causes yielding of the structural members could be used in order to ensure elastic behavior.
(a)
(b)
(c)
Figure 3: The difference between elastic and design forces for each story, (a), the elastic and design spectrum, (solid lines), and the
reduced elastic spectrum (dash line) for the structure with control devices, (b). Model and control scheme in Simuling toolbox (c).
Earthquake Model Response
G Saturation
Time delay
N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11
135
0 5 10 15 20 25 30 35 40
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Time (sec)
D
is
p
la
c
e
m
e
n
t
8
th
(
m
)
(a)
(b)
Figure 4: Displacement and the acceleration from 8th floor for the controlled and uncontrolled structure.
The proposed procedure was applied to 8-story building and the numerical results show the effectiveness of the
procedure. The control system helps the structure not only to reduce the maximum response (displacements and
accelerations) and keep it in the elastic range, but also to perform at much lower level than the maximum response values.
This is proved by comparison of the root mean square (rms) values with the maximum values of response. The proposed
design procedure seems to be an effective tool for designing controlled structures although further numerical research and
experimental verification are needed. Additionally, the methodology should be applied to a space and irregular structures
with significant participation of higher modes and with torsional effects.
0 5 10 15 20 25 30 35 40
-15
-10
-5
0
5
10
15
Time (sec)
A
c
c
e
le
ra
ti
o
n
8
th
(
m
/s
e
c
2
)
N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11
136
REFERENCES
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Mechanics, 123(9) (1997) 897–971.
[3] Kobori, T., Inoue, Y., Seto, K., Iemura, H., Nishitani, A., Procedings 2nd World Conf. on Structural Control, Kyoto,
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[4] Soong, T.T., Active structural control: Theory and practice, Longman Scientific &Technical/Wiley London/New
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[5] Yang, J.N., Wu, J.C., Agrawal A.K., Hsu S.Y., Sliding mode control of seismically excited linear structures, Journal of
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[6] Cai, G.P., Huang, J.Z., Sun, F., Wang, C., Modified sliding mode bang-bang control for seismically excited linear
structure, Earthquake Engineering and Structural Dynamic, 29 (1997) 1647-1657.
[7] Lee, S.H., Min, K.W., Lee, Y.C., Chung, L., Improved design of sliding mode control for civil structures with
saturation problem, Earthquake Engineering and Structural Dynamics, 33 (2004) 1147-1164.
[8] Yang, J.N., Wu, J.C., Agrawal, A.K., Hsu, S.Y., Sliding mode control for non linear and hysteretic structures, Journal
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/UKR <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>
/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice