09 Fujii Paper

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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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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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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

+

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