GravityEffect-on-EQResponse

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GRAVITY EFFECTS ON EARTHQUAKE

RESPONSE OF A FLEXURE BUILDING: A SHEAR

BUILDING COMPARISON

ERIC AUGUSTUS TINGATINGA

Institute of Civil Engineering, University of the Philippines

Diliman, Quezon City 1101, Philippines

[email protected]

HIDEJI KAWAKAMI

Geosphere Research Institute, Saitama University

255 Shimo-Ohkubo, Sakura-ku, Saitama, 338-8570, Japan

[email protected]

HIDENORI MOGI

Department of Civil and Environmental Engineering

Saitama University, 255 Shimo-OhkuboSakura-ku, Saitama, 338-8570, Japan

[email protected]

Received 14 March 2008Accepted 18 May 2009

An analytical building model including the nonlinear e®ects caused by gravity is presented inthis paper. Governing equations are derived for both single-degree-of-freedom (SDOF) andmulti-degree-of-freedom (MDOF) models with large displacements taken into account, and

solutions are obtained by direct integration and modal analysis. The response of typicalstructures subjected to harmonic ground excitation was expressed in exact and approximateforms, compared with the response of an equivalent shear building. Numerical examples showthat while gravity generally decreases the natural frequency of elastic SDOF systems with smalldisplacement approximations, actual natural frequency increases with ground motion. The

di®erence in the natural frequency and response of MDOF systems to the equivalent shearbuilding is not only due to gravity, but also caused by the geometry of the structure. Exactsolution shows that the frequency varies with ground motion amplitude.

Keywords : Flexure building; gravity e®ect; nonlinearity; analytical building; building model.

1. Introduction

It is an established fact in earthquake engineering that gravity is important when

structures undergo large displacements due to strong ground motions. When yielding

occurs, gravity becomes the dominant force in causing the structure to collapse.1À9

International Journal of Structural Stability and DynamicsVol. 10, No. 2 (2010) 187À203#.c World Scienti¯c Publishing CompanyDOI: 10.1142/S0219455410003488

187

http://dx.doi.org/10.1142/S0219455410003488

http://dx.doi.org/10.1142/S0219455410003488



In this paper, rotations and displacements will be used interchangeably when

comparing motions with shear buildings.

3. SDOF Systems

The SDOF °exure building model similar to the analytical model reported by

Jennings and Husid1 is shown in Fig. 1(a). This idealized lumped mass model can be

used to study the response of buildings, towers, and other similar types of structures

which support a heavy weight at the top.

3.1. Motion equation and vibration properties

The absolute position of the mass with respect to the given coordinate system shownin Fig. 1(a) can be expressed as

x ¼ xg þ h sin ; y ¼ h cos ; ð1Þand the components of acceleration of the mass can be written as

€x ¼ €x g þ hðcos :: À sin

:2Þ; ð2Þ€y ¼ Àhðsin

:: þ cos :2Þ: ð3Þ

The dynamic equilibrium of the mass suggests that summation of forces along the

x - and y -axes gives

S ¼ m€x g þmhðcos :: À sin

:2Þ À P ; ð4ÞT ¼ Àmhðsin

:: þ cos :2Þ þmg ; ð5Þ

(a) (b)

Fig. 1. Flexure building subjected to lateral forces P (t ) and earthquake-induced ground motion xg ðtÞ: (a)SDOF system (b) MDOF system.

Gravity E®ects on Earthquake Response of a Flexure Building 189



and that summation of moments about the mass center gives

S ðh cos Þ À T ðh sin Þ þ ~c: þ ~k ¼ 0: ð6Þ

Substitutions and algebraic manipulations yield the governing equation for themotion of a SDOF °exure building subjected to earthquake-induced ground motion

€x g and a lateral force P as

mh2:: þ ~c

: þ ~k Àmgh sin ¼ Àmh cos €x g þ h cos P : ð7ÞWhen rotations are relatively small, i.e. sin ffi , this equation can be simpli¯ed as

mh2:: þ ~c

: þ ð~k À mghÞ ¼ Àmh€x g þ hP : ð8ÞThe natural frequency of the system can be obtained from the equation for the

corresponding free-undamped system as

!n ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

~k Àmgh

mh2

s : ð9Þ

Equations (7)À(9) are similar to the equations reported by Jennings and Husid.1 For

a stable physical structure, ~k is always greater than mgh because equivalence of these

two terms suggests an unstable structure that will fail by elastic buckling. If we

de¯ne the gravity e®ect parameter as

¼ ~k À mgh~k

; 0 < 1; ð10Þ

then we can write the natural frequency as

!n ¼ ffiffiffi

p Á !0; ð11Þwhere !0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~k=ðmh2Þ

q is the frequency when gravity is ignored. Sun et al.2 used a

dimensionless parameter similar to to describe the increase in elastic natural period

due to gravity.

3.2. Earthquake response of SDOF systems

The governing equation for the motion of SDOF system subjected to earthquake-

induced ground motion is given by Eqs. (7) and (8) with P ¼ 0. When subjected to

harmonic ground motion

xg ¼ xg 0 sinð!tÞ: ð12ÞEquations (7) and (8) become

mh2::

þ~c

:

þ~k

Àmgh sin

¼mh cos !2xg 0 sin

ð!t

Þ;

ð13

Þ:: þ 2!n

: þ ! 2n ¼ 1

h!2xg 0 sinð!tÞ; ð14Þ

respectively; where ¼ ~c=ð2mh2!nÞ is the damping ratio.

190 E. A. Tingatinga, H. Kawakami & H. Mogi



Before we proceed with solution of Eqs. (13) and (14), we ¯rst take note of a few

important concepts. First, it should be realized that Eq. (13) is nonlinear because of

the gravity term Àmgh sin . If this expression is linearized to include a third-order

term, then this equation reduces to the Du±ng11

equation. However, since Eq. (13) isexpressed in terms of angle, then we can say that the solution presented by Du±ng is

good only for relatively small values of rotation angles, say, less than 15 . Thus, a

more accurate elastic response of a °exure building undergoing large displacements

can only be obtained by integrating Eq. (13).

The solution of the linear equation in Eq. (14) can be obtained analytically. The

steady-state response of the model undergoing small rotations can be expressed as

¼ 0 sinð!tÀ rÞ; ð15Þwhere

0 ¼ xg 0

hÁ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 À 2Þ2 þ ð2 Þ2

q ; ð16Þ

r ¼ tanÀ1 2

1 À 2

: ð17Þ

Here, ¼ !=!n is the tuning ratio. The displacement of the mass relative to the

ground is u ¼ h and the absolute displacement of the mass isx ¼ xg þ u ¼ xg 0 d sinð!t À aÞ; ð18Þ

where d is the transmissibility12 for displacement and a is the phase angle

de¯ned by

d ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ ð2 Þ2

ð1 À 2Þ2 þ ð2 Þ2

s ; ð19Þ

a ¼ tanÀ1

2

ð2

Þð1 À 2Þ þ ð2 Þ2

; ð20Þ

respectively.

We note here that when the structure is subjected to large amplitude ground

motions, the magnitude of steady-state response angle 0 may exceed =2 but the

maximum absolute displacement is limited to xg 0 þ h.

3.3. E®ects of gravity on SDOF systems

To investigate the e®ects of gravity on the response of elastic structures, we comparethe responses of this simple model and its equivalent shear building whose motion is

governed by the equation

m€u þ cu: þ ku ¼ Àm€x g ; ð21Þ




where k and c are estimated from the response of the two systems undergoing small

displacements. The equivalent damping for the same damping ratio is c ¼ ~c=h2 and

the equivalent sti®ness for the same restoring moment is equal to

k ¼~k

h2À mg

h; ð22Þ

if gravity is taken into account. The value of k so obtained provides a ¯rst com-

parison of the shear building with the °exure building, because their natural

frequencies and the governing equations are the same, namely, Eq. (21) will be the

same as Eq. (8). If gravity is ignored, however, the elastic sti®ness will be

k ¼ ~k=h2; ð23Þ

and its natural frequency is equal to !0.

3.4. Numerical comparison with shear building

Consider a building whose mass m is 104 kg and height h is 5 m subjected to a ground

motion xg ðtÞ ¼ xg 0 sin(!t), where xg 0 ¼ 0.1, 0.5, 1.0, and 2.0 m. The value of ~k will be

calculated from typical values of , e.g. 0.4 and 0.9 ( ¼ 0:4 is only used here for

comparison). A 5% damping ratio will be assumed.

The exact equation of motion of the system given in Eq. ( 13) will be solved

numerically and its steady-state response will be estimated after several cycles. Theresponse in terms of rotation angle, relative and absolute displacements and their

corresponding phase lags will be compared with the solution of Eq. (14) for small

displacements, i.e. Eqs. (16)À(20). Finally, we compare the response to the steady-

state response of the shear building with sti®ness given by Eq. (23). The frequency

response curves are plotted in Fig. 2.

Figure 2 shows that the natural frequency of the SDOF °exure building implied

by Eq. (14) decreases with , indicating that Eq. (11) is more reasonable compared

with the natural frequency implied by the equivalent shear building in Eq. (21). The

actual natural frequency of the system, whose equation is de¯ned by Eq. (13) andshown by the marked lines in Fig. 2, however, increases with the ground motion

amplitude.

If we rewrite Eq. (13) such that the restoring force is ~k Àmgh sin , then the

e®ective sti®ness as a function of can be approximated as

K ðÞ ffi~k À mgh sin

: ð24Þ

The resulting natural frequency as a function of the steady-state amplitude 0 can be

approximated by

!nð0Þ ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiK ð0Þmh2

r : ð25Þ




Exact

1.02.0

0.50.1

Flexure BuildingSmall displacements

Shear Building xg0, m

=0.4

0 1 2 3

m n

m n

nm

n

m

(rad/s)

m

n

R o t a t i o n a n g l e , 0

( d e g )

P h a s e ,

r

R e l a t i v e d i s p l a

c e m e n t ( m )

A b s o l u t e d i s p l a c e m e n t

( m )

P h a s e ,

a

φ

φ

θ

90

60

30

0

0

/2

0

6

8

2

4

0

6

8

2

4

0

/2

90

60

30

0

0

/2

0ππ

π

ω ω

η η

ππ

π

π π

6

8

2

4

0

6

8

2

4

0

/2

0 1 2 3

=0.9

(rad/s)

Fig. 2. Frequency response of SDOF °exure building with m

¼104 kg, h

¼5 m, and gravity-e®ect para-

meter equal to (a) 0.4 and (b) 0.9. The building is subjected to a harmonic ground motion xg ¼ xg 0 sin(!t )with xg 0 equal to 0.1, 0.5, 1.0, and 2.0 m, and the response is compared with the response of an equivalent

shear building. Damping ratio is assumed to be 5% for all models.




The amplitude-dependent natural frequency or the e®ective sti®ness increases when

undergoing large rotations, which is shown as dark-solid lines in Fig. 3(a). In this

¯gure, the actual natural frequencies estimated using the solution of Eq. ( 13) (shown

as peaks in the plots of Fig. 2) are plotted as discrete points for di®erent values of

ground displacements.

Finally, if yielding were taken into account and an elastoplastic restoring

moment-rotation relation P ð~k; Þ is assumed, then the e®ective sti®ness is estima-

ted as

K pðÞ ffi P ð~k; Þ Àmgh sin

: ð26Þ

The natural frequency predicted by Eq. (25) will decrease with the increase in

ground motion amplitude depending on the value of yield angle y, as shown by the

dark lines in Fig. 3(b). The actual natural frequencies are plotted as points for

di®erent values of ground motion amplitude xg 0.

We also observe from Fig. 2 that the sudden drop of approximately 2xg 0 in the

plots of absolute displacement is due to the superposition of ground motion. Andhence for the same rotation angle, the absolute displacement of the mass is larger

when the motion is in phase with the ground displacement than when it is out of

phase with the ground motion. The e®ect of the superposition of ground displace-

ment is shown in Fig. 4 for points m and n in Fig. 2(a).

4. MDOF Systems

4.1. Motion equations

The fundamental equations of an N -story °exure building, shown in Fig. 1(b), will

be derived when it is subjected to lateral forces P ¼ fP ig and earthquake-induced

ground motion xg . Linear rotational dampers, not shown in Fig. 1(b), are also

installed in each story.

xg0=0.01

0.0250.050.1

0.5

n

1

2

3ω

η η

φ

φ

φ

π

θ

η

4

5

0 /2

=0.9

=0.4

0, rad θ 0, rad0 0.5

y=0.1

y=0.03

y=0.01

elastic=0.9

Fig. 3. Variation in frequency with steady-state response amplitude for (a) elastic ( ¼ 0:4 and ¼ 0:9),and (b) inelastic SDOF systems (with

¼0:9).




The absolute positions of the masses can be expressed as

xn ¼ xg þXni¼1

hi sin i; yn ¼Xni¼1

hi cos i: ð27Þ

By considering the dynamic equilibrium of the °oors, one can apply the D'Alembert's

principle to obtain the following equations:

S n À S nþ1 þ P n Àmn €x n ¼ 0; ð28ÞT n À T nþ1 À mng À mn €y n ¼ 0; ð29Þ

~knðn À nÀ1Þ À

~knþ1ðnþ1 À nÞ þ ~cnð

:

n À

:

nÀ1Þ À ~cnþ1ð

:

nþ1 À

:

nÞþ S nhn cos n À T nhn sin n ¼ 0; ð30Þ

for n ¼ 1; . . . ;N ; such that ~kN þ1 ¼ ~cN þ1 ¼ 0 and S N þ1 ¼ T N þ1 ¼ 0. The components

of the acceleration of the center of each mass can be obtained by di®erentiating

Eq. (27) twice, which can then be substituted into Eqs. (28) and (29) to give

S n ¼ €x g XN i¼n

mi ÀXN i¼n

P i þXN i¼n

mi

Xi j¼1

h jð:: j cos j À

: 2

j sin jÞ; ð31Þ

T n ¼ g XN

i¼nmi À

XN

i¼nmi

Xi

j¼1

h jð:: j sin j þ :2 j cos jÞ: ð32Þ

Substituting these expressions for S n and T n into Eq. (30), we have

À~knnÀ1 þ ð~kn þ ~knþ1Þn À ~knþ1 nþ1 À ghn sin nXN i¼n

mi À ~cn:nÀ1

þ ð~cn þ ~cnþ1Þ:n À ~cnþ1

:nþ1

þ hnXN i¼n

miXi j¼1

h j½cosðn À jÞ:: j À sinðn À jÞ

: 2

j

¼ Àhn cos n €x g XN i¼n

mi þ hn cos nXN i¼n

P i: ð33Þ

1

2

53

4

6

7

8

+1-1

(a)

1

2

3

45

6

7

8

+1-1

(b)

Fig. 4. Steady-state response of SDOF °exure building with m ¼ 104 kg, h ¼ 5 m, and ¼ 0:4 to groundmotion xg ¼ 1:0sin(!t) at forcing frequencies equal to (a) 1.175 rad/s and (b) 1.32 rad/s. Numbers indicatesequence of response to one cycle of ground motion.




If we let the total mass supported by the i th column, M i as

M i

¼ XN

k¼imk;

ð34

Þthen we can write Eq. (33) for n ¼ 1; . . . ;N such that the governing equation of the

N -story °exure building can be written in matrix form as

¤µ µ:: þ "

µ;µ: þ rµ ¼ À®µ €x g þ N µP ; ð35Þ

where

µ ¼ fig; ð36Þ

¤µ ¼ ½ij; ij ¼ hih j cosði À jÞ ÁM i j

i

M j j > i(

; ð37Þ

"µ;µ

: ¼ f"ig; "i ¼ ~cið:i À

:iÀ1Þ À ~ciþ1ð

:iþ1 À

:iÞ À hi

XN k¼i

mk

Xkr¼1

hr sinði À rÞ: 2

r ;

ð38Þ

rµ ¼ ~Kµ À g ¿ µ; ¿ µ ¼ f ig; i ¼ hiM i sin i; ð39Þ®

µ ¼ fig

; i ¼

hiM

icos

i;

ð40

ÞN µ ¼ ½ ij; ij ¼

0 j < i

hi cos i j ! i

(; ð41Þ

for i ¼ 1; . . . ;N and j ¼ 1; . . . ;N . Here,

~K

¼

~k1 þ ~k2 À~k2 Á Á Á 0

À~k2~k2 þ ~k3 Á Á Á 0

... ... . .. ...

0 0 Á Á Á ~kN

266666664

377777775

:

ð42

Þ

4.2. Exact solution

The exact solution to the motion of an N -story °exure building subjected to earth-

quake-induced ground motion and lateral forces can be obtained by rewriting Eq. (35)

in terms of the angular velocity - to form 2N ordinary di®erential equations as

µ: ¼ -; ð43Þ-: ¼ À¤

À1µ ð"µ;- þ rµ þ ®µ €x g À N µP Þ: ð44Þ




These equations can be solved using any practical numerical method for solving the

initial value problems. In this paper, a fourth-order RungeÀKutta method with an

integration step width equal to 10À3 is used.

4.3. Small displacement approximations

When masses undergo small displacements, i.e. sin ffi and cos ffi 1, the governing

equation in Eq. (35) becomes

¤µ:: þ E µ

: þ V µ ¼ À® €x g þ NP ; ð45Þwhere

¤ ¼

h21M 1 h1h2M 2

Á Á Áh1hN M N

h22M 2 Á Á Á h2hN M N

. .. ..

.

sym Á Á Á h2N M N

266666664

377777775; ð46Þ

E ¼

~c1 þ ~c2 À~c2 Á Á Á 0

À~c2 ~c2 þ ~c3 Á Á Á 0

... ... . .. ...

0 0 Á Á Á ~cN

2

6666664

3

7777775; ð47Þ

V ¼

~k1 þ ~k2 ÀM 1gh1 À~k2 Á Á Á 0

À~k2~k2 þ ~k3 À M 2gh2 Á Á Á 0

..

. ... . .

. ...

0 0 Á Á Á ~kN ÀM N ghN

266666664

377777775

; ð48Þ

® ¼

M 1h1

M 2h2

..

.

M N hN

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

; ð49Þ

N ¼ ½ ij; ij ¼0 j < i

hi j ! i

(: ð50Þ




The natural vibration frequencies !n and modes Ãn of the system can be obtained

from Eq. (45) with the right-hand side set equal to zero and E ¼ 0. The dynamic

response of the system can be obtained using the numerical method outlined in

Eqs. (43) and (44). Equation (45) can also be solved by the well-known modalanalysis such that the dynamic response of the system can be expressed in terms of

the modal coordinates as

µ ¼XN n¼1

Ãnq n; ð51Þ

where

q ::n þ 2 n!nq

:n þ ! 2

nq n ¼ ÀÀn €x g : ð52Þ

Here, Àn is the modal participation factor.

4.4. E®ects of gravity on MDOF systems

It is not evident in Eqs. (35) and (45) how the gravity or ground motion amplitude

a®ect the natural frequency and the response of the °exure building. To resolve this

issue, we shall investigate a N -story °exure building with equal story heights h .

4.4.1. Natural frequencies and modes

The frequency and modes shapes of the N -story °exure building can be obtained usingEq. (45). In terms of the inter-story rotation angle ' (from Fig. 1(b), µ ¼ G' where

G is a lower triangular transformation matrix whose nonzero elements are all 1),

Eq. (45) becomes

¤̂ ':: þ V̂ ' ¼ À® €x g ; ð53Þ

when E ¼ 0 and P ¼ 0, where

¤̂ ¼ ½̂ij; ̂ij ¼ h2 ÁM i

ði

Àj

Þ þXN

k¼iM k j < i;

XN k¼ j

M k j ! i;

8>>>>><>>>>>:

ð54Þ

V̂ ¼ ½v̂ij; v̂ij ¼

ÀghM i j < i;

~ki À ghM i j ¼ i;

À~k j j ¼ iþ 1;

0 otherwise;

8>>>>><>>>>>:

ð55Þ

® ¼ fig; i ¼ hM i: ð56Þ




On the other hand, the governing equation of the shear building in terms of the

displacement of the i th °oor with respect to the ground ui can be expressed in terms of

the inter-story drifts j,

ui ¼Xi j¼1

j or u ¼ G±; ð57Þ

and can be written as

m1 0 Á Á Á 0

m2 m2 Á Á Á 0

..

. ... . .

. ...

mN mN Á Á Á mN

2

66666664

3

77777775

::

1

::

2

..

.

::N

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

þ

k1 Àk2 Á Á Á 0

0 k2 Á Á Á 0

..

. ... . .

. ...

0 0 Á Á Á kN

2

66666664

3

77777775

1

2

..

.

N

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

¼ À

m1

m2

..

.

mN

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

€x g :

ð58Þ

And when the shear forces exerted by the °oor above is expressed in terms of the

inertia forces of the upper °oors,

S n ¼ kn n ¼ ÀXN i¼n

mið€x g þ €u iÞ ¼ À€x g XN i¼n

mi ÀXN i¼n

mi

Xi j¼1

:: j; ð59Þ

then Eq. (58) can be written as

M 1 M 2 Á Á Á M N

M 2 Á Á Á M N

. .. ...

sym M N

266666664

377777775

::

1

::

2

...

::N

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

þ

k1 0 Á Á Á 0

0 k2 Á Á Á 0

... ... . .. ...

0 0 Á Á Á kN

266666664

377777775

1

2

...

N

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>; ¼ À

M 1

M 2

...

M N

8>>>>>>><>>>>>>>:

9>>>>>>>=>>>>>>>;

€x g :

ð60Þ

Or, simply, we can pre-multiply both sides of Eq. (58) by GT to obtain Eq. (60).

4.4.2. Numerical example

We consider a two-story building with mn ¼ m, kn ¼ k, and~kn ¼

~k. If we are tocompare the frequency of the °exure building with that of an equivalent shear

building, then we write the equations of a °exure building with g ¼ 0. The governing

equations of the two-story °exure building for g ¼ 0 may be written using Eqs. (45)




and (53) as

mh22 1

1 1" #

::1

::2

8<: 9=;þ~k

2 À1

À1 1" # 1

2( ) ¼ À

mh2

1( )€x

g ; ð61

Þ

mh23 1

2 1

" #’::

1

’::

2

( )þ ~k

1 À1

0 1

" #’1

’2

¼ Àmh

2

1

( )€x g ; ð62Þ

with natural frequencies equal to 0:414 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

~k=ðmh2Þq

and 2:414 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

~k=ðmh2Þq

.

The governing equations of a two-story shear building in Eqs. (60) and (58)

reduces to

m2 1

1 1

" # ::

1

::

2

8<:

9=;þ k

1 0

0 1

" # 1

2

( )¼ Àm

2

1

( )€x g ; ð63Þ

m1 0

1 1

" # ::

1

::

2

8<:

9=;þ k

1 À1

0 1

" # 1

2

( )¼ Àm

1

1

( )€x g ; ð64Þ

and have natural frequencies equal to 0:618

ffiffiffiffiffiffiffiffiffiffik=m

p and 1:618

ffiffiffiffiffiffiffiffiffiffik=m

p . Comparison of

Eqs. (61) and (63) or Eqs. (62) a n d (64) with ~k

¼h2k account for the di®erence in the

natural frequencies of the systems even when gravity is ignored. That is, for the samerestoring force, i.e. values of sti®ness matrix in Eqs. (62) and (64), the motions di®er

for both models (i.e. values of mass matrix). Conversely, when masses undergo similar

accelerations, the restoring forces are di®erent, e.g. by comparing Eqs. (61) and (63).

Finally, if we let m ¼ 104 kg, h ¼ 5m, and ~k ¼ 4:905 Â 106 Nm/rad, then the

di®erence in the natural frequencies due to the combined e®ect of gravity and geo-

metry of the two building models are shown in Fig. 5(a), even though each layer

model (i.e. one-story) is equivalent.

Figure 5(b) shows the di®erence in the mode shapes of the two models, which

illustrates that the response at the top of the °exure building (both when g ¼ 9:81 m/s2

and g ¼ 0) according to the fundamental frequency is greater than that of its equiv-

alent shear building.

4.4.3. Frequency response

To investigate the e®ect of ground motion amplitude on the natural frequency, we use

an example of the response of a typical two-story building to harmonic ground motion.

The frequency response of the same two-story °exure building with m

¼104 kg,

hn ¼ 5 m, and ~k ¼ 4:905 Â 106 Nm/rad subjected to ground motion amplitudes equalto 0.1 and 0.5 m will be investigated for a damping ratio equal to 5%.

Figure 6 shows the response of the 2DOF system in terms of the amplitudes of

steady-state rotation angle, displacement relative to the ground and its phase, and




absolute displacement and the corresponding phase angle of each °oor. The response

of the equivalent shear building model with kn ¼ ~kn=h2 is also computed and shown

with light lines.

Figure 6 clearly shows the di®erence in the natural frequencies and displacement

amplitude of the two models. The peak response of the shear building appears to bedi®erent when compared with the peak response of the °exure building. It is also worth

noting that similar to the SDOF system, the fundamental frequency estimated using

Eq. (35) with P ¼ 0 increases with the ground motion amplitude, when compared

Number of floors, N 1 2 3

6

3

12

9

ω

15

0

1st

mode

2nd

3

rd

mode

mode

(a) Markers § and ¤ denote the frequencies estimated from the solutionof Eq. (35) with P ¼ 0 for x g 0 ¼ 0:1 and 0.5 m, respectively.

N =2-1

N =3+1 -1 +1

Flexure Building (Small displacements, g=9.81 m/s2)

Flexure Building (Small displacements, g=0)

Shear Building

Marked lines

1st

2nd 1

st

2nd

3rd

(b)

Fig. 5. Comparison of (a) natural frequencies and (b) mode shapes of °exure building (with g ¼ 0 andg ¼ 9:81 m/s2) and the equivalent shear building.


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with the frequency of small displacement approximations of Eq. (45). The second

natural frequency decreases with the ground motion amplitude. The natural

frequencies estimated using the peaks of plots in Fig. 6 are plotted as discrete points in

Fig. 5 to show the e®ect of ground motion amplitude (xg 0 ¼ 0:1 and 0.5 m).

Exact

0.50.1

Flexure BuildingSmall displacements Shear Building

xg0, m

0 P h a s e ,

r

/2

(rad/s)

0 2 4 6 8 10 12 14

R o t a t i o n a n g l e

0

( d e g )

60

40

20

0

0 R e l a t i v e d i s p l a c e m e n t ( m )

3

5

1

2

4

0 P h a s e ,

a

/2

0 A b s o l u t e d i s p l a c e m

e n t ( m )

3

5

1

2

4

0

60

40

20

0

0

3

5

1

2

4

0

0

3

5

1

2

4

/2

/2

0 2 4 6 8 10 12 14

(rad/s)

ππ

ππ

ππ

ππ

ω ω

φ

φ

θ

Fig. 6. Frequency response of 2DOF °exure building with mn ¼ 104 kg, hn ¼ 5 m, ~kn ¼ 4:905Â 106 Nm/rad. The building is subjected to harmonic ground motion xg ¼ xg 0sin(!t ) with ground motion amplitudes

xg 0 ¼ 0:1, 0.5 m.




5. Concluding Remarks

The fundamental equations governing the motion of a °exure building subjected to

earthquake-induced ground motion and lateral loads are presented in this paper. To

understand how gravity a®ects the response of structures, the natural frequencies

and response to harmonic ground motions of typical structures modeled as °exure

building are compared with an equivalent shear building. Numerical examples show

that while gravity generally decreases the natural frequency of elastic SDOF systems

subject to small displacement approximations, actual natural frequency increases

with the ground motion. The di®erence in the natural frequency and response of

MDOF systems to the equivalent shear building is not only due to gravity, but also

caused by (using Eqs. (61)À(64) and Fig. 5) the geometry of the structure. Exact

solution shows that the frequency varies with the ground motion amplitude.

Acknowledgments

Mr. Tingatinga would like to acknowledge the support from Japanese Government

(Monkasho) during his three-year doctoral studies in Saitama University.

References

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