physical and numerical modeling of seismic soil-structure interaction in layered soils

Vol.11, No.3 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION September, 2012

Earthq Eng & Eng Vib (2012) 11: 343-358 DOI: 10.1007/s11803-012-0126-0

Earthquake induced pounding between adjacent buildings considering soil-structure interaction

Sadegh Naserkhaki1, Farah N.A. Abdul Aziz1 and Hassan Pourmohammad2

1. Department of Civil Engineering, Faculty of Engineering, Universiti Putra Malaysia, Serdang, Malaysia

2. Department of Civil Engineering, Faculty of Engineering, Islamic Azad University, Karaj Branch, Karaj, Iran

Abstract: Many closely located adjacent buildings have suffered from pounding during past earthquakes because they vibrated out of phase. Furthermore, buildings are usually constructed on soil; hence, there are interactions between the buildings and the underlying soil that should also be considered. This paper examines both the interaction between adjacent buildings due to pounding and the interaction between the buildings through the soil as they affect the buildings seismic responses. The developed model consists of adjacent shear buildings resting on a discrete soil model and a linear visco-elastic contact force model that connects the buildings during pounding. The seismic responses of adjacent buildings due to ground accelerations are obtained for two conditions: fi xed-based (FB) and structure-soil-structure interaction (SSSI). The results indicate that pounding worsens the buildings condition because their seismic responses are amplifi ed after pounding. Moreover, the underlying soil negatively impacts the buildings seismic responses during pounding because the ratio of their seismic response under SSSI conditions with pounding to those without pounding is greater than that of the FB condition.

Keywords: adjacent buildings; underlying soil; pounding; seismic response; fi xed-based (FB); structure-soil-structure interaction (SSSI)

Correspondence to: Sadegh Naserkhaki, Department of Civil Engineering, Faculty of Engineering, Universiti Putra Malaysia, Serdang, MalaysiaTel: +98-26-34425876; Fax: +98-26-34425876 E-mail: [email protected]

Research Assistant; Senior Lecturer; Assistant ProfessorReceived November 22, 2011; Accepted May 30, 2012

1 Introduction

Pounding is the impact of the adjacent buildings on each other when they vibrate out of phase and the separation gap between them is less than the minimum distance required for them to vibrate freely due to earthquake excitation. This phenomenon has caused building damage during most destructive earthquakes. For instance, pounding-incurred building damage happened during the 1985 Mexico City and 1989 Loma Prieta earthquakes, as reported by Rosenblueth and Meli (1986) and Kasai and Maison (1997), respectively. Even for recent earthquakes, there are several reports of building damage due to pounding despite great improvements in building codes (Wang, 2008; GRM, 2008, 2009).

Building codes in earthquake-prone areas typically assign preventive provisions to avoid pounding between the adjacent buildings (TBC, 1997; INBC, 2005; IBC, 2009). Despite these building code provisions, the risk of building pounding is still high because:

Building codes do not consider the out of

phase responses of the adjacent buildings (Kasai et al., 1996; Hao and Shen, 2001), and changes of the phase difference of seismic responses due to the underlying soil (Jeng and Kasai, 1996).

Building displacements can be larger than the displacements considered by building codes due to the underlying soil (Savin, 2003).

Many researchers have attempted to elucidate the effects of pounding on the seismic responses of the adjacent buildings, but many aspects of the subject are yet to be determined. The primary notable contributions in the study of pounding of the adjacent buildings are the studies conducted by Anagnostopoulos (1988), Maison and Kasai (1990), Anagnostopoulos and Spiliopoulos (1992) and Maison and Kasai (1992). Numerical models of the pounding of the adjacent buildings have been developed, and the effects of different parameters have been investigated. Pounding signifi cantly amplifi es the seismic responses of the adjacent buildings, particularly by increasing the story shear, which could lead to building damage. The most important conclusion from these studies is that neglecting pounding effects could result in inappropriate building design where the pounding potential is high.

More recent studies have attempted to account for different factors involved in the pounding of the adjacent buildings. Studies on the effects of the mass distribution on pounding structures (Cole et al., 2011), pounding of seismically isolated buildings (Ye et al., 2009;

344 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.11

Polycarpou and Komodromos, 2010), eccentric building pounding (Wang et al., 2009), heavier adjacent building pounding (Jankowski, 2008), mid-column building pounding, (Karayannis and Favvata, 2005) and corner building pounding (Papadrakakis et al., 1996) are some examples. Although these studies signifi cantly contribute to the fi eld, they did not account for the infl uence of the underlying soil on building pounding.

It is apparent that the underlying soil affects the seismic responses of buildings, whereas there are a limited number of studies on seismic pounding that consider the soil effects. The underlying soil affects the pounding of the adjacent buildings in two ways: spatially varying earthquakes and the soil-structure interaction (SSI). Hao et al. (2000) and Hao and Gong (2005) investigated the seismic responses of the adjacent buildings subjected to pounding due to spatially varying earthquakes. The attenuation of waves propagating through the soil and the associated time lag cause the buildings to experience different seismic responses. However, the infl uence of the spatial variation of earthquake ground motions is of secondary importance compared to the SSI because the adjacent buildings are close to each other.

The SSI and its infl uence on the pounding of the adjacent buildings have been studied by Rahman et al. (2001), Chouw (2002) and Shakya and Wijeyewickrema (2009). Rahman et al. (2001) studied the pounding behavior of two 6-story and 12-story moment-resistant, reinforced concrete frame structures while considering the soil effects. In their studies, the adjacent buildings and underlying soil were modeled using the FEM-BEM method with the RUAUMOKO software package; the seismic responses to the El-Centro earthquake were obtained via an inelastic dynamic analysis, and the Hertz contact force model was used to represent the pounding between the adjacent buildings. Rahman et al. (2001) found that the shift of period due to the underlying soil altered the time at which the fi rst impact occurred, which had consequences on the subsequent poundings. However, they conservatively did not draw a unique conclusion; they instead recommended each case be evaluated individually for its particular confi guration, site condition and expected seismic hazard.

Chouw (2002) went further by stating that poundings could amplify the induced fl oor vibrations, while the SSI suppressed the induced vibrations. Amplifying the fl oor vibrations referred to higher mode vibrations, which signifi cantly infl uenced the secondary structures. Unlike previous studies, this research did not determine how the pounding affects the buildings structural members. Chouw (2002) also claimed that both the soil and long-period pulses in the ground excitation could increase the pounding potential of buildings.

Shakya and Wijeyewickrema (2009) analyzed non-equal story height buildings considering the underlying soil effects to study the mid-column pounding of the adjacent buildings. They used the SAP 2000 software

to model the adjacent buildings and the underlying soil. The buildings were connected by a combination of the Gap element and the Kelvin-Voigt model. The authors asserted that pounding forces, interstory displacements and normalized story shears were generally decreased when the underlying soil was considered.

The seismic responses of the adjacent buildings are generally subjected to several uncertainties in addition to the unknown characteristics of the earthquake. With one confi guration of the adjacent buildings, three fundamental periods are found: two for either building and one for the pounded buildings. Thus, more studies are necessary to create a more reliable conclusion. This research fi rst elucidates the effect of the underlying soil on the fundamental period of individual and pounded adjacent buildings. Then it develops an analytical model to analyze the seismic responses of the adjacent buildings resting on the soil subjected to earthquake induced pounding. Sinusoidal ground accelerations with a wide range of periods as well as real earthquake accelerations are applied to the model, and the resulting responses are analyzed and discussed. A parametric study is performed, and the effects of the underlying soil on the seismic responses of the adjacent buildings subjected to earthquake induced pounding are investigated.

2 Development of the analytical model

The analytical model comprises two sub-models: (1) the adjacent buildings resting on the soil, vibrating individually and freely, and (2) pounding forces, which are combined to create the analytical model for the pounding of the adjacent buildings resting on the soil.

2.1 Analytical model of the adjacent buildings resting on the soil

The adjacent buildings and underlying soil are modeled as shear buildings and discrete soil, respectively, with a concentrated mass, a viscous damper and a linear spring, as shown in Fig. 1(a). The analytical model of the adjacent buildings resting on the soil is shown in Fig. 1(b). The building and the underlying soil are connected through interaction forces with equal magnitudes but opposite directions. These interaction forces come from the inertial forces that correspond to the masses of the building and the underlying soil, called the inertial interaction (Clough and Penzien, 2003; Naserkhaki and Pourmohammad, 2011). Moreover, the adjacent buildings are coupled through the underlying soil, and the response of each building affects the other because they are located in near proximity, termed the structure-soil-structure interaction" or "SSSI" effect (Padron et al., 2009; Naserkhaki and Pourmohammad, 2011). The equation of motion for two adjacent buildings with the SSSI effect consideration due to earthquake acceleration of ug (t) is proposed by Naserkhaki and Pourmohammad (2011) as:

No.3 Sadegh Naserkhaki et al.: Earthquake induced pounding between adjacent buildings considering soil-structure interaction 345

M U C U K U M v vbsb bsb bsb bsb bsb bsb bsb + + = +( )bsb fbsb ug (t)(1)

where Mbsb, Cbsb and Kbsb are the mass, damping and stiffness matrices, respectively. U Ubsb bsb, , Ubsb, vbsb and vfbsb are the acceleration, velocity, displacement and the infl uence vectors of the buildings and underlying soil, respectively. This equation consists of two sets of equations corresponding to the two buildings, while these two sets of equations are coupled by the off-diagonal SSSI components of stiffness and damping matrices. The fi rst set includes n+2 coupled equations; n for the NDOF for the left building and 2 for the 2DOF for the underlying soil. Similarly, the second set includes m+2 coupled equations; m for the MDOF for the right building and 2 for the 2DOF for the underlying soil (n>m).

A more tractable version of Eq. (1) is created by its expansion, Eq. (2). The defi nitions of the variables in Eq. (1) are still valid for Eq. (2), while the subscripts l and r stand for the left and right buildings, respectively; b and s denote the building and underlying soil, respectively; and, bs (sb is the transpose of bs) and bsb indicate the SSI and the SSSI, respectively. The seismic responses of the adjacent buildings resting on the soil subjected to earthquake acceleration are obtained by

Eq. (2); however, the pounding between the adjacent buildings is not yet involved in this equation.

m m 0 0m m 0 0

0 0 m m0 0 m m

uu

ls lbs

lsb lb

rs rbs

rsb rb

ls

lb

uu

rs

rb

(2)

+

c 0 c 00 c 0 0

c 0 c 00 0 0 c

uuuu

ls bsb

lb

bsb rs

rb

ls

lb

rs

r

bb

+

k 0 k 00 k 0 0

k 0 k 00 0 0 k

uuuu

ls bsb

lb

bsb rs

rb

ls

lb

rs

rb

=

m m 0 0m m 0 0

0 0 m m0 0 m m

0v0

v

ls lbs

lsb lb

rs rbs

rsb rb

lb

rb

+

v0v0

ls

rs

u tg ( )

Fig. 1 Analytical model of adjacent buildings resting on the soil

Ilncln kln

Ilm mlm

clm klm

Ili mli

cli kli

Il1 ml1

cl1 kl1

Irm mrm

crm krm

Iri mri

cri kri

Irl mrl

crl krl

ml, mlf kbsb mr, mrfkl, klf kbsbf kr, krfcl, clf cbsb cr, crf cbsbf

ulf+Hnul +uln

ulf+Hmul +ulm

ulf+Hiul +uli

ulf+H1ul +ul1

urf+Hmur +urm

urf+Hiur +uri

urf+H1ur +ur1

Hn

Hm

Hi

H1

u tg ( )(a) Adjacent buildings resting on the soil (b) Discrete model


The matrices and vectors of Eq. (2) corresponding to the left building are introduced in the following (those corresponding to the right building are the same as those for the left building, except that the subscripts l and n are replaced by r and m, respectively).

mm

mlb =

l

ln

1 0 00

SYM. (3)

klb =

+

+

+

+

k k kk k k

k k

k n

l1 l2 l2

l2 l3 l3

l3 l4

l( -1)SYM.

0 0 00 00 0

kk kk

n n

n

l l

l

(4)

mli and kli are mass and stiffness of ith fl oor of the left building. clb is Rayleigh damping matrix of the left building proportional to mass and stiffness matrices.

ulbT = { }u u nl1 l (5) ulbT = { }u u nl1 l (6)ulb

T= { }u u nl1 l (7)

vlbT

= { }1 1 (8)u il , u il and uli are acceleration, velocity and displacement of ith fl oor of the left building.

mls =+ +

+

= =

= =

m H m I H m

H m m m

ii

nii

n

ii

nf ii

n

l2

li li1 i l1

li1 l l1

(9)

m mlbs lsbT

= =

H m H mm m

n n

n

1 l1 l

l1 l

(10)

cls =

cc f

l

l

00

(11)

kls =

kk f

l

l

00

(12)

Hi is the height of ith fl oor from the center of gravity of the underlying soil and Ili is mass moment of inertia of the ith fl oor. Mass components of the discrete soil model (ml and mlf) are virtual masses of underlying soil plus mass of the rigid foundation (subscripts and f are the notation for rocking and horizontal components of soil deformation, respectively). Damping (cl and clf) and stiffness (kl and klf) coeffi cients of the discrete soil model are frequency dependent parameters that can be

described in the time domain by the basic constants of soil including its shear modulus (G), shear wave velocity (Vs) and Poissons ratio () and the width of the foundation (a).

ulsT = { }u u fl l (13) ulsT = { }u u fl l (14)uls

T= { }u u fl l (15)

vlsT

= { }0 lm f (16)where ul , ul and ul are acceleration, velocity and displacement corresponding to rocking deformation of the underlying soil, respectively, and u fl , u fl and ulf are acceleration, velocity and displacement corresponding to the horizontal deformation of the underlying soil, respectively

cbsb =

cc f

bsb

bsb

00

(17)

kbsb =

kk f

bsb

bsb

00

(18)

Values of SSSI damping (cbsb and cbsbf) and stiffness (kbsb and kbsbf) coeffi cients can be obtained from basic constants of soil as proposed by Mulliken and Karabalis (1998).

2.2 Analytical model of the pounding

Pounding of the adjacent buildings can be simplifi ed as pounding of different masses corresponding to each building at the same level. The stereomechanical and contact force models are two available pounding models in structural analysis. The momentum conservation principle is used in the former method by considering the coeffi cient of restitution to model the pounding. One of the signifi cant infl uence factors is the duration of pounding, which is ignored in this method, thereby preventing force from being derived by this method.

However, the contact force model provides the advantages of considering the pounding duration and pounding force that has been used widely in the numerical analysis of pounding of the adjacent buildings. In this force-based model, a spring and a viscous damper are introduced to model the pounding force. The characteristics of the associated spring and viscous damper could be either linear or nonlinear; depending on the linearity and damping, four models have been proposed: (i) linear elastic; (ii) linear visco-elastic; (iii) nonlinear elastic; and iv. nonlinear elastic with nonlinear damping. Additionally, eliminating damping from the linear visco-elastic and nonlinear elastic with nonlinear damping models reduces them to the linear elastic and nonlinear elastic models, respectively. Studies by


Jankowski (2005) and Muthukumar and DesRoches (2006) revealed that the differences between different contact force models were not signifi cant; however, the nonlinear elastic with nonlinear damping, and after that, the linear visco-elastic models, gave more accurate results. Note that there is a general agreement among researchers that the contact force model does not signifi cantly affect the seismic responses of a building.

The advantages of the linear visco-elastic contact force model are as follows: it is effi cient and practical and provides the pounding force while considering the energy dissipation during pounding, thus, it is used in this research. The relationship between the pounding force and displacement is shown in Fig. 2. Its only

defi ciency is that it provides tension forces at the end of pounding with no physical meaning, which is ignorable.

The linear visco-elastic contact force model consists of a linear spring representing the stiffness of the contact and a viscous damper representing the energy dissipation during pounding. Figure 3(a) demonstrates the analytical model of the adjacent buildings connected by the linear visco-elastic contact force model. The contact force model is inactive when the buildings vibrate individually and freely; however, it is activated when the separation gap is closed, causing the adjacent buildings to pound together. Pounding forces develop immediately after pounding by the relationship:

F k u c ui i i i ip sg sg= + (19)where ui and ui are the relative displacement and relative velocity at the ith fl oor, respectively, and ksgi and csgi are the contact stiffness and damping of the ith fl oor, respectively. The relative displacement is given by

u u H u u u H u u ui f i i f i i i= + +( ) + + +( )l l l r r r sg(20)

Fig. 2 Pounding force-displacement relationship (Naserkhaki, 2011)

Linear elasticLinear visco-elasticSeparation gap

Poun

ding

forc

e

Relative displacement

Fig. 3 Analytical model of pounding

(a) Linear visco-elastic pounding model (b) Free body diagram of pounding forces

where usgi is the separation gap at the ith fl oor. In a similar way, relative velocity is calculated from:

u u H u u u H u ui f i i f i i= + +( ) + +( )l l l r r r (21)Contact stiffness is a term without a special calculation

procedure. It has been proposed to be proportional to the axial stiffness of the pounded diaphragm by some authors (Maison and Kasai, 1990; Maison and Kasai, 1992; Zhu et al., 2002; Ruangrassamee and Kawashima, 2003; Muthukumar and DesRoches, 2006). Although other researchers believe that this parameter is better described by the lateral stiffness of the pounded fl oor, e.g., values equal to 50-100 times and 20 times the lateral stiffness of the pounded fl oor were proposed by Anagnostopoulos (1988) and Naserkhaki (2011), respectively.

Luckily, a rational method exists to acquire the contact damping value. It can be determined from the mass and stiffness of the building in the pounding state, given by Eq. (22) (Anagnostopoulos, 1988):

c k m mm mi i

i i

i isg sg

l r

l r

2=+

(22)

Separation gap

ksgmusgm

usgi

usg1

csgm

ksgi

csgi

ksg1

csg1

ksgm(ulf+Hmul+ulm(urf+Hmur+urm+usgm))

csgm(u

lf+Hmu l+

ulm(

urf+Hm

u r+u

rm))

ksgi(ulf+Hiul+uli(urf+Hiur+uri+usgi))

csgi(u

lf+Hiu l+

uli(

urf+Hi

u r+u

ri))

ksg1(ulf+Hlul+ull(urf+H1ur+ur1+usgi))

csg1(u

lf+Hlu l+

uli(

urf+H1

u r+u

r1))


where:

=+

- ln ln 2 2e

e (23)

e is the coeffi cient of restitution, the ratio of the contact and separation velocities (i.e., the start and end velocity) of the pounding:

eu uu u

=

( )( )

l r start

l r end

(24)

The coeffi cient of restitution (e) ranges between 1 (pure elastic) and 0 (pure plastic poundings). Typical values in various applications for metals are between 0.6 and 1.0 (Nguyen et al., 1986). Rajalingham and Rakheja (2000) found the coeffi cients of restitution less than 1.0 and greater than 0.49 are acceptable in decreasing pounding force while values less than 0.3 are undesirable in constructive engineering applications. For typical building material, an interval of 0.5


3 Numerical study

Obtaining the seismic responses of the adjacent buildings during an earthquake requires solving a second order linear ordinary differential equation (Eq. (31)). However, this equation is conceptually nonlinear because its characteristics are changed periodically from the no-pounding to the pounding states and vice versa during the analysis. The step by step Newmark (1959) method, which makes use of integration from the initial to the fi nal condition for each time step with a linear acceleration history, is employed for the solution. The equations and solutions were implemented by a written computer program code to assist with the analyses. Different time steps were taken during the analyses to ensure the computational effi ciency of the computer program code; for the no-pounding state, time steps were 0.02 s, 0.01 s and 0.005 s, depending on the time step for which the earthquake acceleration was recorded; and for the pounding state, the time step was equal to 0.001 s, which guarantees the accuracy of the results.

The underlying soil primarily affects the dynamic properties of the buildings and thus alters the seismic responses of the buildings. The dynamic properties of the buildings and the seismic responses of the buildings are studied for four cases:

(1) N-FB, adjacent buildings with FB condition do not pound together.

(2) P-FB, adjacent buildings with FB condition pound together.

(3) N-SSSI, adjacent buildings with SSSI condition do not pound together.

(4) P-SSSI, adjacent buildings with SSSI condition pound together.

The results are presented and discussed in the following subsections.

3.1 Dynamic properties of the adjacent buildings

All dynamic characteristic factors of the studied buildings including height, material, stiffness and mass are combined into the most signifi cant dynamic property of the building called the building period, which helps to predict the buildings seismic behavior during the earthquake. The underlying soil essentially causes an elongation of the building period. Finding the building period requires solving the matrix eigenvalue problem:

K M = (33)where M and K are the mass and stiffness matrices, respectively. Moreover, and are eigenvalues and eigenvectors, respectively. The eigenvalues i are the roots of the characteristic equation:

f ( ) = [ ] =det 0K M (34)where f() is a polynomial of order equal to the number

of DOFs of the building/buildings. The solution method for the eigenvalue problem must be iterative in nature because it requires fi nding the roots of the polynomial f(). The inverse vector iteration method is used in this study to obtain the fundamental period of the building. Higher modal periods, which contribute less in building vibration than the fundamental period, are not presented in this study. The underlying soil increases both the fundamental and 2nd modal periods, while the rate of change for the fundamental period is greater than that of the 2nd modal period (Naserkhaki and Pourmohammad, 2011).

Each building confi guration possesses three distinct fundamental periods: (1) the fundamental period of the left building, (2) the fundamental period of the right building and (3) the fundamental period of the pounded buildings. To obtain the fundamental period of the left and right buildings in the no-pounding state, M and K are replaced by Mbsb and Kbsb, respectively. In the no-pounding state, the adjacent buildings with the FB condition are vibrating individually and freely, and the fundamental period of each building is not affected by the other building. The adjacent buildings with the SSSI condition are coupled through the soil, so they have interaction through the underlying soil during the vibration and the fundamental period of each building is slightly affected by the other building. In the pounding state, regardless of the FB or SSSI condition, the adjacent buildings interact together through the pounded fl oors, so the vibration and the fundamental period of each building is substantially affected by the other building. When the adjacent buildings are pounded together, M is again replaced by Mbsb, while K is replaced by (Kbsb+Kp). To obtain the fundamental period of the pounded buildings, all components of Kp are effective, meaning that all adjacent fl oors collide together. The fundamental period of pounded adjacent buildings is almost equal for both cases of collision between all adjacent fl oors and the collision between only the top fl oor of the short building and the adjacent fl oor of the tall building (i.e., the difference is minor).

The buildings under study here are residential buildings with a mass of 100 tons per story and a lateral stiffness in compliance with INBC (2005). The fundamental period of the buildings can be found from their mass and stiffness. Additionally, a wide range of soil types are chosen, from soft to hard soils, with a shear wave velocity ranging from 140 to 750 m/s.

The variation of the fundamental periods of three confi gurations of the adjacent buildings (i.e., 10-story vs. 8-story, 10-story vs. 6-story, and 10-story vs. 4-story) are shown in Fig. 4. In the fi gure, there are three lines corresponding to each confi guration; the upper and the lower lines indicate the variation of the fundamental periods of the no-pounding state of individual buildings, and the middle line indicates the variation of the fundamental periods of pounding states in the pounded buildings.


Figure 4 indicates that longer fundamental periods are obtained for buildings resting on softer underlying soils with lower shear wave velocities, while the longest fundamental period is referred to the softest underlying soil. Furthermore, the rate of the fundamental period increment is higher for the taller buildings, with the maximum (25%) increment being that of the 10-story building.

Figure 4 also indicates that the fundamental period of pounded buildings in the pounding state falls between the fundamental periods of the individual buildings in the no-pounding state, regardless of the foundations condition. In the pounding state, pounded buildings are combined into a new structural system with a new fundamental period that is stiffer than an individual tall building but is more fl exible than an individual short building. This result shows that, after pounding, the tall

buildings fundamental period shifts toward the rigid zone, while the short buildings fundamental period shifts toward the fl exible zone. However, the effect of this period shift is smaller for the tall building than for the short building, particularly if the latter building is too short. The nearer the fundamental period of pounded buildings is to the fundamental period of the tall building, the more the tall building dominates the seismic responses of buildings during pounding.

3.2 Seismic responses of the adjacent buildings to sinusoidal excitations

This section discusses the effects of the underlying soil on the seismic responses of the confi guration of a 10-story building adjacent to a 5-story building due to ground accelerations without and with pounding. The buildings are residential buildings with a mass of 100 tons per story and a lateral stiffness of 171 MN/m. These properties are adopted from INBC (2005), which gives the fundamental periods as tabulated in Table 1. These building properties provide a relatively fl exible behavior for a 10-story building, whereas a 5-story building exhibits a relatively stiff behavior; thus, the 10-story building is called the fl exible building and the 5-story building is called the stiff building.

The contact stiffness is equal to 10,000 MN/m, and the coeffi cient of restitution is taken as 0.65. When the SSSI condition is considered, the underlying soil has a shear wave velocity equal to 140 m/s, a shear modulus equal to 32.34 MN/m2 and a Poissons ratio equal to 0.35. The ground excitations applied to these models are artifi cial sinusoidal ground accelerations with periods ranging from 0.1 s to 10 s. This range of excitation period is wide enough to capture the buildings responses due to any real earthquake (the seismic responses of the adjacent buildings due to real earthquakes will be discussed in the next section).

Figures 5 and 6 show the results spectrum of seismic induced maximum displacements and story shears of the top fl oors of the buildings, respectively, because the top fl oors experience the most critical condition. Additionally, the peak displacements and story shears of the top fl oors of the buildings are summarized in Tables 2 and 3, respectively, for comparison. The peak response (displacement/story shear) of each case is the largest response among all maximum responses of that case and occurs in the resonance period. The peak responses of the buildings occur within the ground acceleration periods ranging between 0.5 s and 1.5 s. Meanwhile, the local peak responses occur at smaller ground acceleration periods, corresponding to the higher modes of vibration but with minor, negligible effects for structural members.

Note that the shift in the peak responses toward the fl exible zone due to the underlying soil effect is visible in both fi gures. In the no-pounding cases, the peak responses of the buildings with the SSSI condition

Fig. 4 Variation of fundamental periods of adjacent buildings with soil shear wave velocity


Fig. 5 Maximum displacement of the top fl oor of the buildings against ground acceleration periods under sinusoidal excitation

Fig. 6 Maximum story shear of the top fl oor of the buildings against ground acceleration periods under sinusoidal excitation

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Dis

plac

emen

t (m

)

0.160.140.120.100.080.060.040.02

0

Dis

plac

emen

t (m

)

N-FB

P-FB

N-SSSI

P-SSSI

N-FB

P-FB

N-SSSI

P-SSSI

0.1 1 10Period (s), logarithmic scale

(a) Top fl oor of 10-story building


(b) Top fl oor of 5-story building


(a) Top fl oor of 10-story building


(b) Top fl oor of 5-story building

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0

Stor

y sh

ear (

MN

)

3.0

2.5

2.0

1.5

1.0

0.5

0

Stor

y sh

ear (

MN

)

N-FB

P-FB

N-SSSI

P-SSSI

N-FB

P-FB

N-SSSI

P-SSSI

Table 1 Fundamental periods of the buildings with different conditions

Fundamental period (s)10-story individual building Pounded adjacent buildings 5-story individual building

FB 1.02 0.95 0.53SSSI 1.27 1.11 0.61


occur for periods 25% and 15% longer than the period in which the peak responses of the buildings with the FB condition occur for the individual fl exible (Figs. 5(a) and 6(a)) and stiff buildings (Figs. 5(b) and 6(b)), respectively. For the pounding cases, the peak response of both buildings occurs at periods 17% longer for the SSSI condition than for the FB condition (Figs. 5 and 6).

Furthermore, Figs. 5 and 6 show that pounding causes a shift in the peak response toward the rigid zone for the fl exible building, whereas it shifts toward the fl exible zone for the stiff building. The shift of the peak responses due to pounding is less remarkable for the fl exible building compared to the stiff building. The peak responses of the fl exible building for the pounding case occur in periods 7% (FB) and 13% (SSSI) shorter than the no-pounding case, while for the stiff building, the peak responses for the pounding case occur at periods 79% (FB) and 82% (SSSI) longer than the no-pounding case because the fl exible building shares larger mass during pounding than the stiff building.

In addition to the shift in the peak responses, pounding causes the buildings to experience different values of responses due to seismic excitations (Figs. 5 and 6; Tables 2 and 3). The peak displacements are reduced 0.41-fold (FB) and 0.56-fold (SSSI) in the fl exible building due to pounding (Fig. 5(a)). The comparison between the SSSI and FB conditions reveals that the underlying soil causes larger displacements in the building; the increment is up to 1.16-fold and 1.57-fold for the no-pounding and pounding cases, respectively. The larger displacements of the SSSI buildings are due to the additional displacements imposed on the fl exible building by the underlying soil, mostly due to the rocking component of the underlying

soil for upper stories. Unlike the fl exible building, the peak displacements of a stiff building are increased 1.17-fold (FB) and 1.55-fold (SSSI) due to pounding (Fig. 5(b)). The peak displacements of the stiff building are 1.10-fold and 1.46-fold larger for the no-pounding and pounding cases, respectively, when considering the underlying soil. While the underlying soil causes larger displacements in all cases, the rate of increment is greater for the pounding cases.

The effect of pounding on the story shears of both buildings is incremental where the buildings experience larger story shears due to pounding (Fig. 6). The peak story shear is increased 1.34-fold (FB) and 2.00-fold (SSSI) for the fl exible building and 2.14-fold (FB) and 2.89-fold (SSSI) for the stiff building due to pounding. It is notable that the underlying soil develops contradictory effects on building story shears for the no-pounding and pounding cases. The story shear is reduced 0.74-fold for the fl exible building and 0.77-fold for the stiff building for the no-pounding case, whereas for pounding case, it is increased 1.11-fold for the fl exible building and 1.03-fold for the stiff building. The reduction of the story shear for the top fl oors of the buildings in the no-pounding cases is due to a reduction of the story drift of the top fl oors because of the rocking component from the underlying soil. However, pounding suppresses the rocking effect and causes the buildings to experience larger story drifts and, thus, larger story shears.

The effects of pounding on the seismic responses of other fl oors of the buildings in terms of displacement ratio and story shear ratio are shown in Figs. 7 and 8, respectively (displacement/story shear ratio of each fl oor refers to the ratio of its peak displacement/story shear in the pounding case compared to its peak displacement/story shear in the no-pounding case). In general, the

Table 3 Peak story shears of the top fl oor of the buildings

Story shear of 10-story (MN) Story shear of 5-story (MN)

No-pounding PoundingRatio of

pounding to no-pounding

No-pounding PoundingRatio of

pounding to no-pounding

FB 2.165 2.893 1.336 1.227 2.628 2.142SSSI 1.608 3.216 1.999 0.941 2.716 2.887Ratio ofSSSI to FB 0.743 1.112 - 0.767 1.033 -

Table 2 Peak displacements of the top fl oor of the buildings

Displacement of 10-story (m) Displacement of 5-story (m)No-pounding Pounding Ratio of

pounding to No-pounding

No-pounding Pounding Ratio ofpounding to no-

poundingFB 0.564 0.233 0.413 0.089 0.104 1.168SSSI 0.653 0.365 0.559 0.097 0.151 1.554Ratio ofSSSI to FB 1.157 1.567 - 1.098 1.460 -


results show that the seismic responses of the buildings can be categorized into following three types, depending on the building fl oor whose response is of concern:

(1) Above-pounding fl oors for the fl exible building with the extreme condition at the top fl oor,

(2) Through-pounding fl oors for the fl exible building with the extreme condition at the mid fl oor,

(3) Through-pounding fl oors for the stiff building with the extreme condition at the top fl oor.

As shown in Fig. 7(a), the displacement ratios for the fl exible building are less than one, so it experiences smaller peak displacements after pounding. The displacement reduction due to pounding is justifi ed through all fl oors of the fl exible building and is approximately half of that in the no-pounding case. Although both the FB and SSSI conditions demonstrate a similar pattern, the SSSI condition has greater displacement ratios than the FB condition, meaning that the underlying soil causes larger displacements in the fl exible building in the pounding case, which does not favor the fl exible building during pounding.

Despite reducing the displacements, the relative displacements (i.e., story drifts) are not reduced through

all fl oors of the fl exible building. When the adjacent buildings pound together, through-pounding fl oors in the fl exible building are prevented from moving further by the adjacent building, while its above-pounding fl oors move freely, which causes a sudden jump between the displacements of the through-pounding and above-pounding fl oors. While the relative displacements of the through-pounding fl oors are reduced, this sudden jump causes a sharp increment of relative displacement in the above-pounding fl oors, which is like a whiplash behavior.

Because the story shears are produced due to relative displacements, the story shears of the fl exible building are decreased in the through-pounding fl oors, while they are dramatically increased in the above-pounding fl oors (Fig. 8(a)). The story shear ratios of the fl exible building are greater for the SSSI than for the FB condition, which means that if the fl exible building is pounded to the adjacent building, then it experiences larger story shears because of the underlying soil. Therefore, the underlying soil is again unfavorable for the fl exible building.

For the stiff building, both the displacement ratios and story shear ratios of all fl oors are greater than unit

Fig. 7 Displacement ratios of the buildings after pounding

Fig. 8 Story shear ratios of the buildings after pounding

-2.5 -1.5 -0.5 0.5 1.5 2.5Displacement ratio

(a) 10-Story

109876543210

109876543210

Floo

r lev

el

Floo

r lev

el FBSSSI

FBSSSI

-2.5 -1.5 -0.5 0.5 1.5 2.5Displacement ratio

(b) 5-Story

-2.5 -1.5 -0.5 0.5 1.5 2.5Story shear ratio

(a) 10-story

109876543210

109876543210

Floo

r lev

el

Floo

r lev

el FBSSSI

FBSSSI

-2.5 -1.5 -0.5 0.5 1.5 2.5Story shear ratio

(b) 5-story


(Figs. 7(b) and 8(b)), meaning that they are amplifi ed after pounding. The stiff building is pushed away by the fl exible building during pounding, which not only produces larger displacements but also creates larger story drifts and, consequently, larger story shears. Furthermore, the SSSI condition has greater displacement and story shear ratios than the FB condition. Therefore, the underlying soil is also detrimental to the stiff building because it causes it to experience larger displacement and story shears in the pounding case.

In summary, pounding causes smaller displacements but larger story shears for the fl exible building. Considering the underlying soil effect (SSSI condition), the displacements and story shears produced in the fl exible building due to pounding are both larger than those in the FB condition. For the stiff building, the displacements and story shears both are larger in the pounding case than in the no-pounding case. Both the displacements and story shears produced in the stiff building due to pounding are amplifi ed when the underlying soil is considered. Thus, pounding worsens the adjacent buildings conditions because their responses are amplifi ed due to the pounding and the rate of amplifi cation is higher by considering the underlying soil. The effects of pounding should be considered in the building design and the adjacent buildings should be modeled together with the underlying soil to maximize the safety of the building design.

3.3 Responses of the adjacent buildings to real earthquakes

The acceleration records of real earthquakes are applied to the building confi guration to investigate their seismic responses. The characteristics of the earthquake records are summarized in Table 4. The results are shown in terms of envelopes of maximum displacements and story shears of the buildings, the dominant factors in building seismic design. The structural elements are then designed to resist the story shear from earthquake excitation and their displacements are checked to remain in allowable limits.

The envelopes of maximum displacements and story shears of the buildings produced due to different earthquakes are shown in Figs. 9 and 10, respectively. Figures 9 (a), (c), (e) and (g) show that the maximum displacements of the fl exible building are reduced throughout all fl oors during pounding. The changes in the story shear are not the same as the changes in the

displacements due to earthquake induced pounding. While the story shears of the fl exible building are reduced in the through pounding fl oors, they are increased dramatically in the above pounding fl oors (Fig. 10 (a), (c), (e) and (g)) due to the whiplash effect. For the stiff building, the critical condition happens in the no-pounding side of the building where both the displacements (Fig. 9 (b), (d), (f) and (h)) and story shears (Fig. 10 (b), (d), (f) and (h)) are increased because the fl exible building pushes the stiff building away during pounding. The patterns of seismic responses are similar for both the FB and SSSI conditions; however, Figures 9 and 10 show that the SSSI condition provides larger values of responses, particularly after pounding.

Though the patterns of seismic responses for the buildings are similar, each building demonstrates a unique response to each earthquake with different values and ratios of responses. The top fl oor of the fl exible building on the pounding side and the top fl oor of the stiff building on the no-pounding side suffer the most from earthquake induced pounding, leading to a discussion of these fl oors responses in the following paragraphs.

The top fl oor of the fl exible building with the FB condition experiences the largest displacement of 0.12 m after pounding under the El Centro earthquake while it experiences the largest displacement of 0.18 m after pounding under the Kobe earthquake if its condition is changed to SSSI. The maximum displacement ratio of the top fl oor of the fl exible building with the FB condition is 0.73 under the Victoria earthquake, while for the SSSI condition this ratio is 0.74 under the El Centro earthquake. The largest story shear produced at the top fl oor of the fl exible building after pounding is 2.22 MN for the FB condition and 2.52 MN for the SSSI condition, both under the El Centro earthquake. The maximum story shear ratio of the top fl oor of the fl exible building is 3.36 for the FB condition and 4.88 for the SSSI condition under the Victoria earthquake and the El Centro earthquake, respectively.

The Loma Prieta earthquake causes the largest displacement of 0.10 m for the FB condition and 0.13 m for the SSSI condition at the top fl oor of the stiff building after pounding. The maximum displacement ratio of the top fl oor of the stiff building is 1.17 for the FB condition under the El Centro earthquake and 1.39 for the SSSI condition under the Kobe earthquake. The largest story shear of the top fl oor of the stiff building is 1.63 MN for the FB condition after pounding, which is

Table 4 Characteristics of records of earthquakes

Earthquake Year Record/Component PGA (g) Site condition (USGS)Kobe 1995 KOBE/KAK090 0.345 DEl Centro 1940 IMPVALL/I-ELC180 0.313 CVictoria, Mexico 1980 VICT/CPE315 0.587 BLoma Prieta 1989 LOMAP/G01090 0.473 A


Fig. 9 Envelopes of maximum displacements of the buildings under different real earthquakes

-2.5 -1.5 -0.5 0.5 1.5 2.5Displacement (m)

(a) Kobe (10-story)

Floo

r lev

el

Floo

r lev

el N-FBP-FBN-SSSIP-SSSI

-2.5 -1.5 -0.5 0.5 1.5 2.5Displacement (m)(b) Kobe (5-story)

Floo

r lev

el

Floo

r lev

el-2.5 -1.5 -0.5 0.5 1.5 2.5

Displacement (m)(c) El Centro (10-story)

-2.5 -1.5 -0.5 0.5 1.5 2.5Displacement (m)

(d) El Centro (5-story)

-2.5 -1.5 -0.5 0.5 1.5 2.5Displacement (m)

(e) Victoria (10-story)

-2.5 -1.5 -0.5 0.5 1.5 2.5Displacement (m)

(f) Victoria (5-story)

Floo

r lev

el

Floo

r lev

el

Floo

r lev

el

Floo

r lev

el

-2.5 -1.5 -0.5 0.5 1.5 2.5Displacement (m)

(g) Loma Prieta (10-story)

-2.5 -1.5 -0.5 0.5 1.5 2.5Displacement (m)

(h) Loma Prieta (5-story)


Fig. 10 Envelopes of maximum story shears of the buildings under different real earthquakes

-6 -3 0 3 6Story shear (MN)

(a) Kobe (10-story)

Floo

r lev

el

Floo

r lev

el

N-FBP-FBN-SSSIP-SSSI

-6 -3 0 3 6Story shear (MN)(b) Kobe (5-story)

Floo

r lev

el

Floo

r lev

el


(c) El Centro (10-story)


(d) El Centro (5-story)


(e) Victoria (10-story)


(f) Victoria (5-story)

Floo

r lev

el

Floo

r lev

el

Floo

r lev

el

Floo

r lev

el


(g) Loma Prieta (10-story)


(h) Loma Prieta (5-story)


produced under either the El Centro earthquake or the Loma Prieta earthquake. The largest story shear of the top fl oor of the stiff building with the SSSI condition is 1.98 MN after pounding under the El Centro earthquake. The maximum story shear ratio of the top fl oor of the stiff building is 1.88 for the FB condition and 3.44 for the SSSI condition, both under the Kobe earthquake.

Not only do the values of the maximum displacements and story shears increase but the ratios also increase when the underlying soil is considered. These increments happen for all earthquakes, although the rate of increment is different for each earthquake. Although the pattern of the earthquake induced response of buildings is the same, the underlying soil has a unique effect for each building confi guration. Each case possesses three distinct fundamental periods (i.e., two for either individual building and one for pounded buildings); the underlying soil shifts each fundamental period to a new one. Thus, different responses are expected under the same earthquake because of different foundation conditions (i.e., FB and SSSI), which increases the uncertainty of the problem. Therefore, each case must be evaluated specifi cally considering the buildings together with the underlying soil since ignoring the underlying soil can underestimate the design and lead to detrimental consequences.

4 Conclusions

This study develops a numerical model of the adjacent buildings resting on the soil with the buildings connected by the visco-elastic contact force model during pounding. The contact force model is activated when the buildings pound together. The dynamic properties of the adjacent buildings as well as their seismic responses are presented and discussed in this paper.

Each building confi guration possesses three distinct fundamental periods; two for either building and one for the pounded buildings where the fundamental period of the pounded buildings falls between the individual fundamental periods of the buildings, closer to the tall and fl exible building. The underlying soil causes a lengthening of all three fundamental periods.

The seismic responses of the adjacent buildings subjected to sinusoidal ground accelerations as well as the accelerations of different earthquakes are calculated. The results show that pounding causes smaller displacements but larger story shears in the fl exible building, while the displacements and story shears are increased in the stiff building due to pounding. The underlying soil (SSSI) increases the displacements and story shears produced in both buildings due to pounding compared to those seen under the FB condition. In conclusion, pounding worsens the adjacent buildings conditions, which is amplifi ed by the underlying soil. Pounding effects should be considered in the building design and the buildings should be modeled together with the underlying soil to create the safest design

profi le; ignoring the effects of the underlying soil may result in unrealistic and unconservative designs with detrimental consequences.

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