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EQUIVALENT SDOF MODELFOR ASYMMETRIC BUILDINGS
CONSIDERING BI-DIRECTIONAL EXCITATION
Kenji Fujii
Chiba Institute of Technology, Department of Architecture and Civil Engineering,
Narashino-Shi, Chiba, Japan, [email protected]
ABSTRACT
The estimation of nonlinear response of buildings subjected to a strong groundmotion is a key issue for the rational seismic design of new buildings and the seis-
mic evaluation of existing buildings. For this purpose, the nonlinear time-history
analysis of Multi-Degree-Of-Freedom (MDOF) models might be a solution, but it
is often too complicated whereas the results are not necessarily more reliable due to
uncertainties involved in input data. To overcome such shortcomings, several re-
searchers have developed nonlinear static procedures (NSP). This approach is a
combination of a nonlinear static analysis of MDOF model and a nonlinear dy-
namic analysis of the equivalent Single-Degree-Of-Freedom (SDOF) model. How-
ever, only a few investigations concerning the extension of the NSP for asymmetric
buildings under bi-directional excitation have been made.
This paper focuses on the theoretical background of equivalent SDOF model for
asymmetric buildings considering bi-directional excitation. In this paper, the prin-
cipal direction of the modal response is defined based on the mode vectors; two
independent equivalent SDOF models are formulated assuming that the asymmet-
ric buildings oscillate predominantly in the first mode in one direction, and in the
second mode in other orthogonal direction. The numerical examples are shown by
using single-storey asymmetric building models.
KEYWORDS
Nonlinear static procedure (NSP), equivalent SDOF model, bi-directional excita-
tion, single-storey asymmetric building.
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102 K. Fujii
1 INTRODUCTIONThe estimation of nonlinear response of buildings subjected to a strong ground
motion is a key issue for the rational seismic design of new buildings and the seis-
mic evaluation of existing buildings [1]. For this purpose, the nonlinear time-
history analysis of Multi-Degree-Of-Freedom (MDOF) model might be one solu-
tion, but it is often too complicated whereas the results are not necessarily morereliable due to uncertainties involved in input data. To overcome such shortcom-
ings, several researchers have developed simplified nonlinear analysis procedures
[2, 3, 4]. This approach is often referred to as Nonlinear Static Procedure (NSP)
[1]. It consists of a nonlinear static analysis of MDOF model and a nonlinear dy-
namic analysis of the equivalent Single-Degree-Of-Freedom (SDOF) model, and it
would be a promising candidate as long as buildings oscillate predominantly in the
fundamental mode. These procedures have been more often applied to planar frame
analyses; investigations concerning the extension of them to asymmetric buildings
under bi-directional excitation have been made only by Fajfar et al. [5] and the
author [6].
The concept of the NSP proposed by the author in [6] is shown in Figure 1. In this
procedure, two simplified models, equivalent single-storey model and equivalentSDOF model, are used to estimate the response of multi-storey asymmetric frame
building model. In the first step, the degree of freedoms is reduced from 3N to 3 by
converting multi-storey frame building model (Fig. 1a) to equivalent single-storey
model (Fig. 1b). In the next step, the degree of freedoms is reduced from 3 to 1 by
converting to two independent equivalent SDOF models (Fig. 1c), assuming that it
oscillates predominantly in the single mode in each orthogonal direction. The con-
version process from multi-storey frame building model (Fig. 1a) to the equivalent
single-storey model (Fig. 1b) is similar to those from planer frame to the equivalent
SDOF model; however the conversion process from the equivalent single-storey
model (Fig. 1b) to the equivalent SDOF model (Fig. 1c) would be the difficult part,
because of the bi-directional excitation.
Fig. 1: Concept of the proposed NSP [6]
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Equivalent SDOF model for asymmetric buildings 103
This paper focuses on the theoretical background of equivalent SDOF model for
asymmetric buildings considering bi-directional excitation used in the NSP. The
principal direction of the modal response is defined based on the mode vectors.
Based on this discussion, two independent equivalent SDOF models are formulated
assuming that the asymmetric buildings oscillate predominantly in the first mode in
one direction, and in the second mode in other orthogonal direction. The numerical
examples are shown by using single-storey asymmetric building models. The ap-
plication example for multi-storey asymmetric frame building models can be found
in reference [6].
2 FORMULATION OF EQUIVALENT SDOF MODEL CONSIDERINGBI-DIRECTIONAL EXCITATION
2.1 Principal axis of modal responseThe equation of motions of a single-storey asymmetric building model (1-mass
3-DOFs model) subjected to a bi-directionally horizontal ground motion can be
written as equation (1), considering a set of orthogonal U- and V- axes in X-Y
plane and an angle as shown in Fig. 2.( )gU gV a a+ + = +R U VMd Cd f M && & (1)
{ } { }, X Y Z x y R R M = =
Rd f (2)
0 0
0 0 ,
0 0
XX XY X
YX YY Y
X Y
m C C C
m C C C
I C C C
= =
M C (3)
r I m= (4)
{ } { }cos sin 0 , sin cos 0 = =
U V (5)
where M is the mass matrix and C is the damping matrix and assumed proportionalto the stiffness matrix, fR is the vector representing restoring force at the centre of
mass of floor diaphragm, m andIare the mass and moment of inertia, respectively,
ris the radius of gyration about the centre of mass, agUand agVare ground accelera-
tion in U- and V-direction, respectively.
The principal direction of modal response is defined as the direction of incidence
that produces the largest i-th modal mass ratio. The i-th modal mass in U- and V-
direction,MiU*,MiV
*are expressed by equations (6) and (7), respectively.
( )( )
( )
2
* 2
22 2
cos sinXi Yi
iU iU
Xi Yi i
M mr
= =
+ +T
i i M (6)
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104 K. Fujii
Fig. 2: Single-storey asymmetric building model
( )( )
( )
2
* 2
22 2
sin cosXi YiiV iV
Xi Yi i
M mr
+= =
+ +T
i i M (7)
,iU iV
= =T T
i U i V
T T
i i i i
M M
M M(8)
{ }Xi Yi i =
i (9)
where i is the i-th natural mode. From equations (6) and (7), the following inter-
esting relationship is obtained as equation (10).
( )
2 2* *
22 2.Xi YiiU iV
Xi Yi i
M M m const r
++ = =
+ +(10)
By differentiating MiU*, equation (6), with respect to and equating to zero, the
principal direction of the i-th modal response is obtained and its tangent is given by
equation (11).
tani Yi Xi
= (11)
Note that the principal direction of the i-th modal response obtained by equation
(11) is identical with the conclusion shown by Gonzlez [7]. Considering the firstmode and substituting 1 into equations (6) and (7), equation (12) is obtained.
( )
2 2* *1 1
1 122 2
1 1 1
, 0X YU VX Y
M m M r
+= =
+ +(12)
Equation (12) indicates that the ground motion in V-direction has no contributions
in the first modal response.
2.2 Equation of Motion for Equivalent SDOF ModelIn the equation of motion of single-storey asymmetric building model, equation (1),
it is assumed that the U-axis is taken as the principal axis of the first modal re-
sponse, and the difference between V-axis and the principal axis of the second
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Equivalent SDOF model for asymmetric buildings 105
modal response is negligibly small. From this assumption and equation (12), equa-
tion (13) is obtained.* * * *
1 1 2 20, 0V U U V M M M M (13)
It is assumed that the building oscillates predominantly in the first mode under U-
directional (unidirectional) excitation, and it oscillates predominantly in the second
mode under V-directional excitation. Under bi-directional excitation, it is assumed
that d and fR can be written in the form of equation (14), even if the building re-
sponses beyond the elastic range.
( )
* *
1 1 2 2
* *
1 1 2 2
U U V V
U U V V
D D
A A
= +
= +
1 2
R 1 2
d
f M (14)
* * * *
1 1 1 2 2 2,U U U V V V D M D M = = T T
1 2 Md Md (15)
* * * *
1 1 1 2 2 2,U U U V V V A M A M = = T T
1 R 2 R f f (16)
whereD1U*,D2V
*are the first and second modal equivalent displacement, and A1U
*,
A2V*
are the first and second modal equivalent acceleration, respectively. By substi-
tuting equation (14) into equation (1) and by multiplying 1U T1 from the left side,and considering the orthogonal condition, equation (17), and the relation expressed
in equation (18), equation (19) is obtained.
0, 0= T T1 2 1 2 M C (17)* *
1 1 1 1V U V U M M = (18)
* ** * *1 1
1 1 1* *
1 1
U VU U U gU gV
U U
C MD D A a a
M M
+ + = +
&& & (19)
( )* 21 1U UC = T
1 1 C (20)
where C1U* is the first modal damping. In equation (19), the first term in the right-hand side expresses the contribution ofagU, and the second term expresses the con-
tribution ofagV. Considering equation (13), the contribution ofagV can be assumed
zero and equation (19) can be rewritten as equation (21).*
* * *1
1 1 1*
1
U
U U U gU
U
CD D A a
M+ + = && & (21)
Equation (22) can be also obtained similarly by substituting equation (14) into
equation (1) and by multiplying 2VT
2 from the left side and considering equation
(13).*
* * *2
2 2 2*
2
V
V V V gV
V
C
D D A aM+ + = && &
(22)
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106 K. Fujii
( )* 22 2V VC = T
2 2 C (23)
where C2V*
is the second modal damping. Equations (21) and (22) are the equation
of motion of equivalent SDOF models.
Fig. 3a and 3b show the first mode shape of the single-storey asymmetric building
model and corresponding equivalent SDOF model, respectively. In Fig. 3b, the
equivalent SDOF model consists of a concentrated mass M1U*
located at A, the
rigid body OA that is pin-connected to the base at O, and the nonlinear rotational
spring. Denoting that the distance from the centre of mass of single-storey asym-
metric building model G toA is e1, equation (24) is obtained from Fig. 3:
( )1 1 1 1 1U Ue + = (24)
( )
( )
2
1 1 1 1
1 1 22 2
1 1 1
cos sinX YU U
X Yr
=
+ +(25)
( )1 1 1 1 1 1cos sinX Y = (26)
where 1 is the distance from G to O. Since cos1 and sin1 can be obtained by
equation (27), equations (25) and (26) can be rewritten as equations (28), (29).
2 2 2 2
1 1 1 1 1 1 1 1cos ,sinX X Y Y X Y = + = + (27)
( )
2 2
1 11 1 22 2
1 1 1
X YU U
X Yr
+ =
+ +(28)
2 2
1 1 1 1X Y = + (29)
Substituting equation (28) into equation (24) and considering equation (29), equa-
tion (30) is obtained.
( )1 1e r r = (30)
Fig. 3: Equivalent SDOF model for the first mode
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108 K. Fujii
Iare 1524 t, 1.075 105
t m2, respectively. Fig. 5 shows the hysteresis model of the
frame element, and Table 1 shows the properties of each frame element (elastic
stiffness KE, yield strength Qy, secant stiffness ratio at yield point y and post-yielding stiffness degradation ratio 2), which is determined based on the planerpushover analysis of each frame in original building model. The envelopes are
assumed symmetric in both positive and negative loading directions. Torsional
stiffness of member is neglected. No second order effect (ex. P- effect) is consid-
ered. Muto hysteretic model [8] is employed for each frame with one modificationas shown in Fig. 5(b): the unloading stiffness after yielding stage is modified as it
decreases with proportional to -0.5 (: ductility ratio of frame element), to repre-sent the degradation of unloading stiffness after yielding of R/C members as the
model employed by Otani [9].
The damping matrix is assumed proportional to the instant stiffness matrix and 3%
of the critical damping for the first mode. Fig 6 shows the mode shape and natural
periods of (a) Model-1 and (b) Model-2. In this figure, the principal directions of
modal response obtained by equation (11) are also shown. As shown in this figure,
the difference (1 2) of Model-1 is 94.6 degrees, and that of Model-2 is 86.5degrees; therefore the principal axes of the first and the second modes are almost
mutually orthogonal.
Table 1 Properties of each frame
Model-1 Model-2
Frame KE
(MN/m)
Qy
(kN)y 2
KE
(MN/m)
Qy
(kN)y 2
Y1 946.9 4661 0.218 0.005 539.9 3494 0.181 0.006
Y2 569.6 3760 0.231 0.007 129.5 1591 0.292 0.022
Y3 129.5 1591 0.292 0.022 129.5 1591 0.292 0.022
Y4 114.1 1472 0.280 0.023 114.1 1472 0.280 0.023
X1 858.7 3524 0.305 0.004
X2-X5 80.5 1001 0.287 0.024
X6 70.7 942 0.261 0.024
Same as Model-1
Fig. 5: Hysteretic model for frame element
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Equivalent SDOF model for asymmetric buildings 109
Fig. 6: Mode shape of models in elastic range
Fig. 7: Elastic response spectra Fig. 8: Orbit of artificialground motion
3.2 Ground Motion
In this study, the earthquake excitation is considered bi-directional in X-Y plane,
and six sets of artificial ground motions are used. The first 60 seconds of two hori-
zontal components (major and minor horizontal components) of the following re-
cords are used to determine phase angles of the ground motion: El Centro 1940
(referred to as ELC), Taft 1952 (TAF), Hachinohe 1968 (HAC), Tohoku University
1978 (TOH), JMA Kobe 1995 (JKB) and Fukiai 1995 (FKI). Target elastic spec-
trum of major components with 5% of critical damping SA(T, 0.05) is determined
by the following equation
( )
( )
24.8 45 m/s 0.16s
,0.05 12.0 0.16s 0.576s
12.0 0.576 0.576s
A
T T
S T T
T T
+