dynamic analysis of stacked rigid blocks

Dynamic analysis of stacked rigid blocks

Pol D. Spanos*, Panayiotis C. Roussis, Nikolaos P.A. Politis

Department of Civil Engineering, George R. Brown School of Engineering, Rice University, PO Box 1892, Mail Stop 318, 6100 Main Street, Houston,

TX 77005-1892, USA

Accepted 11 May 2001

Abstract

The dynamic behavior of a structural model of two stacked rigid blocks subjected to ground excitation is examined. Assuming no sliding,

the rocking response of the system standing free on a rigid foundation is investigated. The derivation of the equations of motion accounts for

the consecutive transition from one pattern of motion to another, each being governed by a set of highly nonlinear differential equations. The

system behavior is described in terms of four possible patterns of response and impact between either the two blocks or the base block and the

ground. The equations governing the rocking response of the system to horizontal and vertical ground accelerations are derived for each

pattern, and an impact model is developed by conservation of angular momentum considerations. Numerical results are obtained by

developing an ad hoc computational scheme that is capable of determining the response of the system under an arbitrary base excitation.

This feature is demonstrated by using accelerograms from the Northridge, CA, 1994, earthquake. It is hoped that the two-blocks model used

herein can facilitate the development of more sophisticated multi-block structural models. q 2001 Published by Elsevier Science Ltd.

Keywords: Dynamics; Stacked rigid blocks; Nonlinearity; Impact problem; Patterns of motion; Seismic response

1. Introduction

It is a common practice in structural engineering when

dealing with structures under dynamic excitation, to assume

deformable continuum behavior. For a number of structural

systems, however, rigid body motion may well be a credible

dynamic behavior model. In fact, the seismic behavior of

block-like structures standing free on their foundation, such

as ancient monuments, petroleum storage tanks, water

towers, nuclear reactors, concrete radiation shields, compu-

ter-type equipment, and various artifacts, has been the object

of numerous studies for over a century. The approaches on

the subject involve both deterministic and stochastic studies.

Housner's landmark study [1] has provided the basic

understanding on the rocking response of a rigid block and

sparked modern scienti®c interest. The response of a rigid

block subjected to horizontal and vertical ground accelera-

tions, with the option of elastic tie-down rods and the assump-

tion of no sliding has been examined by Aslam et al. [2].

Various patterns, free-¯ight pattern included, have been

analyzed by Ishiyama [3]. A study on the dynamic behavior

of a rocking rigid block supported by a ¯exible foundation

which permits up-lift has been performed by Psycharis and

Jennings [4]. Spanos and Koh [5] have investigated the rock-

ing response of a rigid block subject to harmonic ground

motion, assuming no sliding. The linear and nonlinear equa-

tions of motion have been solved numerically assuming zero

initial conditions to identify likely steady-state patterns of

response. Allen et al. [6] have studied the dynamic behavior

of an assembly of two-dimensional rigid prisms. Furthermore,

the rocking response has been investigated both analytically

and experimentally by Tso and Wong [7]. Hogan [8], adopt-

ing the model, the analysis, and the response classi®cation of

Spanos and Koh [5], has performed a complete investigation

of the existence and stability of single-impact subharmonic

responses (1, n) (with n $ 1), as a function of the restitution

coef®cient b . Psycharis [9] has also presented an analysis of

the dynamic behavior of systems consisting of two blocks.

Moreover, the impact problem has been approached by Sino-

poli [10] by adopting a unilateral constraint, `kinematic

approach'. The in¯uence of nonlinearities associated with

impact on the behavior of free-standing rigid objects

subjected to horizontal base excitations has been studied by

Yim and Lin [11]. Furthermore, a general, two-dimensional

formulation for the response of free-standing rigid bodies to

base excitation has been presented by Shenton and Jones

[12,13]. Also, Augusti and Sinopoli [14] have presented a

formulation of dynamics and impact problem for a single

rigid body freely supported on rigid foundation; a review of

Soil Dynamics and Earthquake Engineering 21 (2001) 559±578

0267-7261/01/$ - see front matter q 2001 Published by Elsevier Science Ltd.

PII: S0267-7261(01)00038-0

www.elsevier.com/locate/soildyn

* Corresponding author. Tel.: 11-713-527-4909; fax: 11-713-285-5191.

E-mail address: [email protected] (P.D. Spanos).

the numerous studies performed on this subject has also been

presented there. Allen and Duan [15] have examined the

reliability of linearizing the equations of motion of rocking

blocks. The rocking and the overturning of precariously

balanced rocks by earthquake have been examined by Shi et

al. [16]. The criteria for initiation of slide, rock, and slide-rock

rigid-body modes have been presented by Shenton [17].

Scalia and Sumbatyan [18] have examined the slide rotation

of rigid bodies subjected to a horizontal ground motion.

Moreover, Pombei et al. [19] have studied the dynamics of

a rigid block subjected to horizontal ground motion, towards

formulating criteria that separate the various patterns of the

motion. The rocking response of free-standing blocks under

cycloidal pulses has been examined by Zhang and Makris

[20]. Furthermore, Makris and Zhang [21] have studied the

rocking response and the overturning of anchored blocks

under pulse type motions. Kim et al. [22] have investigated

experimentally the vibration properties of a rigid body placed

on sand ground surface.

A probabilistic approach to the problem of rocking of

rigid blocks has been pursued by Yim et al. [23]; the rocking

behavior of rigid blocks subjected to horizontal and vertical

accelerations, assuming no sliding has been studied there.

Koh et al. [24] have studied the behavior of a rigid block

rocking on a ¯exible foundation. Modulated white noise has

been used as a model of horizontal acceleration of the foun-

dation. The statistics of the rocking response have been

found by an analytical procedure which involves a combina-

tion of static condensation and stochastic linearization. Koh

and Spanos [25] have also presented an analysis of block

random rocking. Furthermore, Giannini and Masiani [26]

have tackled the problem of the dynamic response of a

rigid block oscillator to a Gaussian white noise excitation

process. An investigation by Dimentberg et al. [27] has

focused on the toppling failure of a free standing and an

anchored rigid block due to horizontal and vertical base

excitations. Expressions for the statistical properties and

probability distribution of the random toppling time have

been obtained; the excitation has been idealized as a white

noise. Moreover, Cai et al. [28] have examined the toppling

of a rigid body under random excitation by modeling the

base acceleration as an evolutionary process with a broad-

band spectrum. More recently, Lin and Yim [29] have exam-

ined the rocking behavior of slender rigid objects subjected

to periodic excitations with and without noise disturbance for

a better understanding of their response and stability. In a

companion paper [30], Lin and Yim have also examined, in a

probabilistic context, the responses of fully nonlinear rocking

systems subjected to combined deterministic and stochastic

excitations. Furthermore, Giannini and Masiani [31] have

used a random vibration approach to study the response of

a slender rigid block to seismic excitation.

Clearly, the dynamic behavior of multi-block structures

has not, to date, been exhaustively studied. Even for the

simplest case of multi-block structures involving two-

block assemblies, the rocking problem becomes very

complex. Such a con®guration, in which one block is placed

over the other, can be thought of as the model of ancient

Greek and Roman type structures composed of large heavy

members of a piece of machinery or statue placed on top of a

block-like base.

This paper focuses on the dynamic behavior of structures

consisting of two rigid blocks; one serving as a base and

another one on top of the base. Assuming rigid foundation,

large friction to prevent sliding, and point contact during a

perfectly plastic impact, the only possible response mechan-

ism under base excitation is rocking about the corners of the

blocks. The analytical formulation of this nonlinear problem

is rather challenging. Its complexity is associated with tran-

sitions from one pattern of motion to another, each one

being governed by a set of highly nonlinear equations.

The paper presents a derivation of the exact (nonlinear)

equations of motion for the system considered undergoing

base excitation and a treatment of the impact problem by

deriving expressions for the post-impact angular velocities.

Furthermore, it contains numerical results from the devel-

opment and use of ad hoc computer program for determin-

ing free vibration and seismic response of the system.

2. Formulation

2.1. Possible patterns of rocking response

From the outset, it is noted that the term `pattern' is

chosen to describe the various con®gurations of the relative

positions to each other of the two blocks; the term `mode' is

discarded to prevent confusion stemming from its use in

standard structural dynamics of MDOF systems.

Here, the dynamic behavior of systems consists of two

symmetric rigid blocks as shown in Fig. 1. Assuming no

sliding, the rocking response of the system standing free

on a rigid foundation is investigated. The top block rests

symmetrically on the base block, and the latter rests on a

rigid horizontal surface. The blocks have masses mi, and

centroid moments of inertia IGi, where i is the block index

with i� 1 for the base block, and i� 2 for the top block.

The two blocks have bases Bi� 2bi and heights Hi� 2hi.

The bottom right points of the block are denoted by Oi and

the left are denoted by O 0i. The bottom right and left points

of the top block are related to the bottom right point of the

base block by the distances l and l 0, and the angles between

the right vertical of the base block and the segments O1O2

and O1O 02;b and b 0, respectively. Finally, the center of

mass for each block is denoted by Gi, and its location is

determined by the distance from the bottom right point ri,

and the angle between the right vertical of the block and the

distance radius uci.

The system possesses two degrees of freedom, namely, u 1

and u 2, denoting the angles of rotation of the two blocks

with respect to the vertical. When subjected to a base

excitation, the system may exhibit four possible patterns

P.D. Spanos et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 559±578560

of rocking motion. Fig. 2 illustrates the classi®cation of the

four possible patterns with respect to the angles of rotation

u 1 and u 2. Patterns 1 and 2 involve a 2-DOF system

response, and re¯ect rotations of the two blocks in the

same or opposite direction. Patterns 3 and 4 re¯ect a

SDOF system response; in particular, pattern 3 describes

the motion of the system rocking as one rigid structure,

and pattern 4 concerns the case where only the top block

experiences rotation. Furthermore, each of the afore-

mentioned patterns is subdivided into two subcases that

account for opposite angle signs.

2.2. Initiation of motion

Before dealing with the equations of motion, it is impor-

tant to derive appropriate criteria for the initiation of motion

of the system when subjected to a base excitation with hori-

zontal and vertical components �xg and �yg, respectively.

Speci®cally, as can be seen in Fig. 3, the system may be

set into rocking either in pattern 3 or in pattern 4 when the

overturning moment of the horizontal inertia force about

one edge exceeds the restoring moment due to the weight(s)

of the block(s) and the vertical inertia force.

The criteria for motion initiation follows.

Transition from rest to pattern 3a requires that

2h �xg 2 b1 �yg . b1g; �1�where

h � m1h1 1 m2�2h1 1 h2�m1 1 m2

�2�

is the distance of the center of mass of the system from the

base of the base block.

Transition from rest to pattern 3b requires that

h �xg 2 b1 �yg . b1g: �3�Transition from rest to pattern 4a requires that

2h2 �xg 2 b2 �yg . b2g: �4�Transition from rest to pattern 4b requires that

h2 �xg 2 b2 �yg . b2g: �5�

P.D. Spanos et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 559±578 561

Fig. 2. Classi®cation of rocking patterns for a system of two stacked rigid blocks with respect to the angles of rotation.

Fig. 1. Geometric model of two stacked rigid blocks.

2.3. Equations of motion

The equations of motion for the possible patterns and

their subcases are derived by means of Lagrange's method.

The kinetic energy of the system is

T � T1 1 T2 � 1

2

X2

i�1

�miv2Gi

1 IGi_u 2

i �; �6�

while its potential energy is given by

V � V1 1 V2 �X2

i�1

mihGig: �7�

In the above equations, vGidenotes the velocity of the i-th

center of mass and hGidenotes the distance of the i-th center

of mass from the base of the base block.

The equations of motion for all possible patterns and

subcases are presented below.

Pattern 1:

�IO11 m2l2� �u 1 1 m2lr2cos�g1� �u 2 1 m2lr2sin�g1� _u 2

2

2 mxlgr1sin�u1 2 Su1uc1�2 m2glsin�u1 2 Su1

b�� 2�m1r1cos�u1 2 Su1

uc1�1 m2lcos�u1 2 Su1

uc1�� xg

1 �m1r1sin�u1 2 Su1uc1�1 m2lsin�u1 2 Su1

uc1�� yg; �8�

and

m2lr2cos�g1�1 IO2�u 2 2 m2lr2sin�g1� _u 2

1 2 m2gr2sin�u2 2 Su1uc2�

� m2r2� �ygsin�u2 2 Su1uc2�2 �xgcos�u2 2 Su1

uc2��;

�9�where IOi

are the moments of inertia with respect to the

points Oi, and

g1 � u1 2 u2 1 Su1�uc2

2 b�: �10�

The signum function Su1accounts for the two subcases of

the pattern under consideration; this will also be the case for

patterns 2 and 3, while the signum function Su2will account

for the two subcases of pattern 4. It is de®ned as

Sui�

1 if ui . 0

0 if ui � 0

21 if ui , 0

8>><>>: : �11�

Pattern 2:

�IO11 m2l 02� �u 1 1 m2l 0r2cos�g2� �u 2 1 m2l 0r2sin�g2� _u 2

2

2 m1gr1sin�u1 2 Su1uc1�2 m2gl 0sin�u1 2 Su1

b 0�� 2�m1r1cos�u1 2 Su1

uc1�1 m2l 0cos�u1 2 Su1

b 0�� xg

1 �m1r1sin�u1 2 Su1uc1�1 m2l 0sin�u1 2 Su1

b 0�� yg;

�12�


Fig. 3. Initiation of motion.

and

m2l 0r2cos�g2� �u 1 1 IO2�u 2 2 m2l 0r2sin�g2� _u 2

1

2m2gr2sin�u2 1 Su1uc2�

� m2r2� �ygsin�u2 1 Su1uc2�1 �xgcos�u2 2 Su1

uc2��;

�13�

where

g2 � u1 2 u2 2 Su1�uc2

1 b 0�: �14�Pattern 3:

IO�u 1 2 MgRsin�u1 2 Su1

uc� � 2MR� �xgcos�u1 2 Su1uc�

2 �ygsin�u1 2 Su1uc��;

�15�and

u1 � u2; �16�where M is the total mass of the system (i.e. M �m1 1 m2�;R; uc; and IO are the distance of the center of

gravity of the system from any base corner of the base

block, the angle between the right vertical of the base

block and the distance radius of the center of gravity of

the system, and the mass moment of inertia with respect

to any base corner of the base block, respectively.

Pattern 4:

u1 � 0 �17�and

IO2�u 2 2 m2gr2sin�u2 2 Su2

uc2� � m2r2� �xgcos�u2 2 Su2

uc2�

2 �ygsin�u2 2 Su2uc2��:�18�

Note that the above equations of motion are valid only in the

absence of impact. The impact problem is addressed later.

Furthermore, note that for small angles u 1 and u 2 the preced-

ing equations can be simpli®ed as follows.

Pattern 1:

�IO11 m2l2� �u 1 1 m2lr2cos�g1L� �u 2 1 Su1

m1gr1sin�uc1�

1 Su1m2glsin�b�

� 2�m1r1cos�uc1�1 m2lcos�b�� xg 2 Su1

�m1r1sin�uc1�

1 m2lsin�b�� yg;

�19�and

m2lr2cos�g1L� �u 1 1 IO2�u 2 1 Su2

m2gr2sin�uc2�

� 2m2r2� �ygSu2sin�uc2

�1 �xgcos�uc2��; �20�

where

g1L � Su1�uc2

2 b�: �21�

Pattern 2:

�IO11 m2l 02� �u 1 1 m2l 0r2cos�g1L� �u 2 1 Su1

m1gr1sin�uc1�

1 Su1m2gl 0sin�b 0�

� 2�m1r1cos�uc1�1 m2l 0cos�b 0�� xg 2 Su1

�m1r1sin�uc1�

1 m2l 0sin�b 0�� yg;

�22�and

m2l 0r2cos�g2L� �u 1 1 IO2�u 2 1 Su1

m2gr2sin�uc2�

� 2m2r2� �ygSu1sin�uc2

�2 �xgcos�uc2��; �23�

where

g2L � Su1�uc2

1 b 0�: �24�Pattern 3:

IO�u 1 1 Su1

MgRsin�uc� � 2MR� �xgcos�uc�1 �ygSu1sin�uc��;

�25�and

u1 � u2: �26�Pattern 4:

u1 � 0; �27�and

IO2�u 2 1 Su2

m2gr2sin�uc2� � 2m2r2� �xgcos�uc2

�1 �ygSu2sin�uc2

��:�28�

2.4. Transition between patterns

After the system is set to motion, it switches from one

pattern to another due to either an impact or a sudden change

in ground excitation. Transition criteria in the absence of

impact are presented below; the impact problem is studied

in Sections 3 and 4. Once motion is initiated, transition can

occur from either pattern 3 or pattern 4; the conditions for

transition from one pattern to another are derived by consid-

ering the overturning and restoring moments. Transition

from pattern 3a to pattern 1a is illustrated in Fig. 4, and

requires that

2cos�uc22 u1� �xg 2 sin�uc2

2 u1� �yg

. gsin�uc22 u1�1 dcos�uc2

1 v� _u 21

1 dsin�uc21 v�1

IG2

m2r2

� ��u 1; �29�

where

d ��2h1 1 h2�2 1 b2

1

q; �30�


and

v � atan2h1 1 h2

b1

� �: �31�

Transition from pattern 3a to pattern 2a is illustrated in

Fig. 5, and requires that

cos�uc22 u1� �xg 2 sin�uc2

2 u1� �yg

. gsin�uc21 u1�1 dcos�2u1 1 uc2

1 v� _u 21

1 dsin�2u1 1 uc21 v�1

IG2

m2r2

� ��u 1: �32�

Transition from pattern 3b to pattern 1b is illustrated in


cos�uc21 u1� �xg 2 sin�uc2

1 u1� �yg

. gsin�uc21 u1�1 dsin�v 2 uc2

� _u 21

2 dcos�v 2 uc2�2

IG2

m2r2

� ��u 1: �33�

Transition from pattern 3b to pattern 2b is illustrated in


2cos�uc22 u1� �xg 2 sin�uc2

2 u1� �yg

. gsin�uc22 u1�2 dsin�v 1 uc2

� _u 21

1 dcos�v 1 uc2�2

IG2

m2r2

� ��u 1: �34�

Transition from pattern 4a to pattern 1a is illustrated in


2�m1 1 2m2�h1 �xg 2 �m1b1 1 m2j� �yg

. m2r2�jsin�uc22 u2�1 2h1cos�uc2

2 u2�� u 2

1 m2r2�2h1sin�uc22 u2�2 jcos�uc2

2 u2�� _u 22 1 m1gb1;

�35�where

j � b1 2 b2: �36�Transition from pattern 4b to pattern 1b is illustrated in


�m1 1 2m2�h1 �xg 2 �m1b1 1 m2j� �yg

. 2m2r2�jsin�uc21 u2�1 2h1cos�uc2

1 u2�� u 2

1 m2r2�2h1sin�uc21 u2�2 jcos�uc2

1 u2�� _u 22 1 m1gb1:

�37�


Fig. 5. Transition from pattern 3a to pattern 2a in the absence of impact.


Transition from pattern 4b to pattern 2a is illustrated in


2�m1 1 2m2�h1 �xg 2 �m1b1 1 m2j0� �yg

. 2m2r2�j 0sin�uc21 u2�2 2h1cos�uc2

1 u2�� u 2

2 m2r2�2h1sin�uc21 u2�1 j 0cos�uc2

1 u2�� _u 22

1 m1gb1; �38�

where

j 0 � 2b1 2 j: �39�

Finally, transition from pattern 4a to pattern 2b is illustrated

in Fig. 11, and requires that

�m1 1 2m2�h1 �xg 2 �m1b1 1 m2j0� �yg

. m2r2�j 0sin�uc22 u2�2 2h1cos�uc2

2 u2�� u 2

2 m2r2�2h1sin�uc22 u2�1 j 0cos�uc2

2 u2�� _u 22

1 m1gb1: �40�

3. Accounting for impactÐpatterns 1 and 2

3.1. Preliminary remarks

In this section the impact problem is studied. During


Fig. 6. Transition from pattern 3b to pattern 1b in the absence of impact.



vibration the system switches from one pattern to another,

and impact between the two blocks or between the base

block and the ground can occur. Under the assumption of

inelastic impact (no bouncing), the only possible response

mechanism is pure rocking about the corners of the two

blocks (point-impact). Furthermore, the duration of impact

is assumed to be in®nitesimal (impulsive motion), implying

negligible changes in position and orientation, and instanta-

neous changes in angular velocities. In this regard, the

impact is analyzed using the principle of impulse and

momentum.

3.2. Impact in pattern 1

3.2.1. Base block impact with the ground

First, assume that the system is rocking in pattern 1a.

When the angle of rotation of the base block becomes

zero, an impact occurs yielding a change of the rotation

pole from O1 to O 01, that is, a switch to pattern 2b, see

Fig. 12.

This induces an instantaneous change in the angular

velocities of the blocks, whose post-impact values, _u11

and _u12 , need to be determined. During impact, assum-

ing point-impact it can be argued that the only forces

that exert a moment about O 01 are the weights of the

blocks (the impulse forces pass through the point O 01).Furthermore, the angular impulse due to these ®nite

forces about O 01 is quite small since the time of impact

is negligible. Thus, the angular momentum of the

system about O 01 is conserved. The angular momentum

of the system about O 01 immediately before impact is

simply the sum of angular momenta of each block about


Fig. 10. Transition from pattern 4b to pattern 2a in the absence of impact.

Fig. 11. Transition from pattern 4a to pattern 2b in the absence of impact.


the point under consideration. That is,

�h2O 0

1�system � �h2

O 01�base block 1 �h2

O 01�top block: �41�

The angular momentum of a rigid body about some

point A can be expressed in vector form as

hA � hG 1 rG=A £ mvG; �42�where vG is the velocity of the mass center G, rG/A is

the position vector of G relative to A, and hG is the

angular momentum of the body about its mass center.

The angular momentum is given by

hG � IGv; �43�with v being the angular velocity of the body, and IG

being the central mass moment of inertia of the body

about an axis perpendicular to the plane of motion. For

plane rigid body motion, the angular momentum vector

is always normal to the plane of motion so that Eq.(42)

can be rewritten as

hA � IGv 1 �mvG�d; �44�where d is the distance of the linear momentum vector

mvG from the point A. Eq. (44) is used to compute the

angular momentum of each block. Speci®cally, the

angular momentum of the base block is given by

�h2O 0

1�base block � IG1

_u21 1 �m1v2

G1�d; �45�

where IG1is the central mass movement of inertia of the

base block, _u21 and v2

G1are the angular velocity and

velocity of the mass center of the base block immedi-

ately before impact, respectively, and d is the distance

of the linear momentum vector m1v2G1

from the point

O 01.The angular momentum of each block, immediately

before and after impact, can be similarly determined.

Finally, the principle of conservation of angular momentum

for the system is expressed in the form

�h2O 0


O 01�system; �46�

where

�h2O 0

1�system � {IO1

2 2m1b21 1 m2l�r2cos�u2 2 uc2 1 b�

1 2h1cos�b�2 �2b1 2 j�sin�b��} _u 21 1 {IO2

1 m2r2�2h1cos�u2 2 uc2�1 �2b1 2 j�sin�u2

2 uc2��} _u22 ;

�47�and

�h1O 0

1�system � {IO1

1 m2l 0�r2cos�u2 2 uc2 2 b 0�1 2h1cos�b 0�

1 �2b1 2 j�sin�b 0��} _u 11 1 {IO2

1 m2r2�2h1cos�u2 2 uc2�1 �2b1 2 j�sin�u2

2 uc2��} _u12 :

�48�Eq. (46) involves the two unknown angular velocities, _u1

1

and _u12 , immediately after impact. A second equation may

be derived by considering the conservation of angular

momentum about O 02 for the top block alone. This is justi-

®ed by the fact that, since the top block point-contacts the

base block at O2, all impact forces are transmitted to the top

block through that point, so that the angular impulse of the

top block about O2 is zero (the angular impulse of a ®nite

force like the weight of the block can be neglected). Further-

more, since the motion is assumed impulsive, angular

impulse equals change in angular momentum, even about

the `moving' point O2. Therefore,

�h2O2�top block � �h1

O2�top block; �49�

where

�h2O2�top block � m2r2lcos�u2 2 uc2 1 b� _u2

1 1 IO2_u2

2 �50�and

�h1O2�top block � m2r2l 0cos�u2 2 uc2 2 b 0� _u1

1 1 IO2_u1

2 : �51�The system of Eqs. (46) and (49) is solved for the angular


Fig. 12. Transition from pattern 1a to pattern 2b through impact.

velocities immediately after impact to yield

_u11 � A _u2

1 ; �52�and

_u12 � B _u2

1 1 _u22 ; �53�

where

A � IO2C1 2 C2C3

C4IO22 C2C5

; �54�

and

B � C5C1 2 C4C3

C2C5 2 C4IO2

; �55�

with

C1 � IO12 2m1b2

1 1 m2�r2cos�u2 2 uc2 1 b�1 2h1cos�b�2 �2b1 2 j�sin�b��;

�56�

C2 � IO21 m2r2�2h1cos�u2 2 uc2�1 �2b1 2 j�sin�u2

2 uc2��; �57�

C3 � m2r2lcos�u2 2 uc2 1 b�; �58�

C4 � IO11 m2l 0�r2cos�u2 2 uc2 2 b 0�1 2h1cos�b 0�1 �2b1

2 j�sin�b 0��; (59)

and

C5 � m2r2l 0cos�u2 2 uc2 2 b 0�: �60�Note that, by symmetry, the same analysis also holds for the

case of transition from pattern 1b to pattern 2a.

3.2.2. Impact between the two blocks

Next, consider the case where the system is rocking in

pattern 1a. When the angle of rotation of the top block

becomes equal to that of the base block, an impact between

the blocks occurs and the top block may either start rocking

about O 02 (pattern 2a), or remain in contact with the base block,

implying that the system rocks as one rigid-body about O1

(pattern 3a); the two possible transitions are illustrated in

Fig. 13.

Assume that the system switches from pattern 1a to

pattern 2a. The forces generated during impact are applied at

the point O1, so that the angular impulse of the system about

that point is zero. Consequently, the angular momentum of the

system about O1 is conserved during the time of impact. That

is,

�h2O1�system � �h1

O1�system; �61�

with

�h2O1�system � {IO1

1 m2l��b2

1 1 �2h1 1 h2�2q

cos�z 2 b�} _u21

1 {IG21 m2r2

��b2

1 1 �2h1 1 h2�2q

cos�u2 2 u1

2 uc2 1 z�} _u22 ;

�62�and


1 m2l 0��b2

1 1 �2h1 1 h2�2q

cos�z 2 b 0�} _u21

1 {IG21 m2r2

��b2

1 1 �2h1 1 h2�2q

cos�u2 2 u1

1 uc2 1 z�} _u12 : (63)

A second equation may be derived by considering the conser-

vation of angular momentum about O 02, for the top block

alone. Since impact takes place at O 02, and neglecting the

angular impulse of the weight of the block, the angular impulse

of the top block about that point is zero. Furthermore, since the

motion is assumed is impulsive, the principle of conservation

of angular momentum is still valid about the `non-stationary'

point O 02. Therefore,

�h2O 0

2�top block � �h1

O 02�top block; �64�

with

�h2O 0

2�top block � m2r2lcos�u2 2 u1 1 uc2 1 b� _u2

1 1 IG2

1 m2r22cos�2uc2� _u2

2 ; �65�


Fig. 13. Transition from pattern 1a to either pattern 2a or pattern 3a through impact.

and

�h1O 0

2�top block � m2r2l 0cos�u2 2 u1 1 uc2 1 b 0� _u2

1 1 IO2_u2

2 : �66�Solving the system of Eqs. (61) and (64) for the angular

velocities immediately after impact yields

_u11 � A _u2

1 1 B _u22 ; �67�

and

_u12 � C _u2

1 1 D _u22 ; �68�

where

A � IO2E1 2 E2E3

E4IO22 E2E5

; �69�

B � IO2E6 2 E2E7

E4IO22 E2E5

; �70�

C � E5E1 2 E4E3

E2E5 2 E4IO2

; �71�

and

D � E5E6 2 E4E7

E2E5 2 E4IO2

; �72�

with

E1 � IO11 m2l

��b2

1 1 �2h1 1 h2�2q

cos�z 2 b�; �73�

E2 � IG21 m2r2

��b2

1 1 �2h1 1 h2�2q

cos�u2 2 u1 1 uc2 1 z�; �74�

E3 � m2r2lcos�u2 2 u1 1 uc2 1 b�; �75�

E4 � IO11 m2l 0

��b2

1 1 �2h1 1 h2�2q

cos�z 2 b 0�; �76�

E5 � m2r2l 0cos�u2 2 u1 1 uc2 1 b 0�; �77�

E6 � IG21 m2r2

��b2

1 1 �2h1 1 h2�2q

cos�u2 2 u1 2 uc2 1 z�;�78�

and

E7 � IG21 m2r2

2cos�2uc2�: �79�The preceding analysis was based on the assumption that, after

impact, the base and top blocks rock separately about O1 and

O 02, respectively (pattern 2a). If, however, _u12 . _u1

1 Ðwhich

is physically impossibleÐthen, after impact, the top block

remains in contact with the base block and the system rocks

as one rigid body about O1 (pattern 3a). In this case, _u12 � _u1

1 ,

and the proper value is derived by considering the conserva-

tion of angular momentum about O1 for the whole system.

That is,



with


1 m2l��b2

1 1 �2h1 1 h2�2q

cos�z 2 b�} _u21

1 {IG21 m2r2

��b2

1 1 �2h1 1 h2�2q

cos�u2 2 u1

2 uc2 1 z�} _u22 ;

�81�and

�h1O1�system � IO

_u11 ; �82�

where

IO � IO11 IG2

1 m2�b21 1 �2h1 1 h2�2� �83�

is the mass moment of inertia of the system about O1.

The ®nal relation is

_u11 � _u1

2 � B1

IO

_u21 1

B2

IO

_u22 ; �84�

where

B1 � IO11 m2l

��b2

1 1 �2h1 1 h2�2q

cos�z 2 b�; �85�

and

B2 � IG21 m2r2

��b2

1 1 �2h1 1 h2�2q

cos�u2 2 u1 2 uc2 1 z�:�86�

The same analysis also holds for the case of transition from

pattern 1b to pattern 2b or 3b.


3.3.1. Base block impact with the ground

First, note that if the system is rocking in pattern 2a when

the base block impacts the foundation, transition to pattern

1b occurs, see Fig. 14. In order to ®nd the angular velocities

after impact, conservation of angular momentum about O 01,for the system, and about O 02, for the top block, is ensured.

The equation concerning the conservation of angular

momentum about O 01 for the system is

�h2O 0


O 01�system; �87�

where

�h2O 0

1�system � {IO1

2 2m1b21 1 m2l 0�r2cos�u2 1 uc2 1 b 0�

1 2h1cos�b 0�2 jsin�b 0��} _u 21 1 {IO2

1 m2r2�2h1cos�u2 1 uc2�1 jsin�u2

1 uc2��} _u22 ;

�88�


and

�h1O 0

1�system � {IO1

1 m2l�r2cos�u2 1 uc2 2 b�1 2h1cos�b�

1 jsin�b��} _u 11 1 {IO2

1 m2r2�2h1cos�u2

1 uc2�1 jsin�u2 1 uc2��} _u12 :

�89�Conservation of angular momentum about O 02 for the top

block requires that

�h2O 0



where

�h2O 0

2�top block � m2r2l 0cos�u2 1 uc2 1 b 0� _u2

1 1 IO2_u2

2 ; �91�and

�h1O 0

2�top block � m2r2lcos�u2 1 uc2 2 b� _u2

1 1 IO2_u2

2 : �92�Solving the system of Eqs. (87) and (90) for the angular


_u11 � A _u2

1 1 B _u22 ; �93�

and

_u12 � C _u2

1 1 D _u22 ; �94�

where

A � IO2E1 2 E2E3

E4IO22 E2E5

; �95�

B � IO2E6 2 E2IO2

E4IO22 E2E5

; �96�

C � E5E1 2 E4E3

E2E5 2 E4IO2

; �97�

and

D � E5E6 2 E4IO2

E2E5 2 E4IO2

; �98�

with

E1 � IO12 2m2b2

1 1 m2l 0�r2cos�u2 1 uc2 1 b 0�1 2h1cos�b 0�

2jsin�b 0��;�99�

E2 � IO21 m2r2�2h1cos�u2 1 uc2�1 jsin�u2 1 uc2��; �100�

E3 � m2r2l 0cos�u2 1 uc2 1 b 0�; �101�

E4 � IO11 m2l�r2cos�u2 1 uc2 2 b�1 2h1cos�b�1 jsin�b��; �102�

E5 � m2r2lcos�u2 1 uc2 2 b�; �103�

and

E6 � IO21 m2r2�2h1cos�u2 1 uc2�1 j�sin�u2 1 uc2�: �104�

The preceding analysis is also valid for transition from

pattern 2b to 1a.

3.3.2. Impact between the blocks

Next, assume that the system is rocking in pattern 2a. If

the angle of rotation of the top block becomes equal to that

of the base block, an impact between the blocks occurs and

the top block tends to start rocking about O2 (pattern 1a). It

is also possible, see Fig. 15, that after impact the two blocks

remain in full contact so that the system rocks as one rigid

body about O1 (pattern 3a).

Consider the case where the system switches from

pattern 2a to pattern 1a. To ®nd the angular velocities

after impact, conservation of angular momentum about O1

for the whole system, and about O2 for the top block is

invoked. Speci®cally,




Fig. 14. Transition from pattern 2a to pattern 1b through impact.

where


1 m2l 0��b2

1 1 �2h1 1 h2�2q

sin�u1 1 z

2 b 0�} _u21 1 {IG2

1 m2r2

��b2

1 1 �2h1 1 h2�2q

sin�u2 1 uc2

1 z�} _u22 ;

�106�

and


1 m2l��b2

1 1 �2h1 1 h2�2q

sin�u1 1 z

2 b�} _u21 1 IO2

_u12 : �107�

Similarly,

�h2O2�top block � �h1

O2�top block; �108�

where

�h2O2�top block � m2r2l 0cos�u1 2 u2 1 uc2 2 b 0� _u2

1 1 �IG2

1 m2r22cos�u1 1 u2 2 uc2 2 b 0�� _u2

2 ;

�109�

and

�h1O2�top block � m2r2lcos�u1 2 u2 1 uc2 2 b� _u2

1 1 IO2_u2

2 :

�110�

Solving the system of Eqs. (105) and (108) for the angular

velocities immediately after impact gives

_u11 � A _u2

1 1 B _u22 ; �111�

and

_u12 � C _u2

1 1 D _u22 ; �112�

where

A � IO2E1 2 IO2

E2

E3IO22 IO2

E4

; �113�

B � IO2E5 2 IO2

E6

E3IO22 IO2

E4

; �114�

C � E4E1 2 E3E2

IO2E4 2 E3IO2

; �115�

and

D � E4E5 2 E3E6

IO2E4 2 E3IO2

; �116�

with

E1 � IO11 m2l 0

��b2

1 1 �2h1 1 h2�2q

sin�u1 1 z 2 b 0�; �117�

E2 � m2r2l 0cos�u1 2 u2 1 uc2 2 b 0�; �118�

E3 � IO11 m2l

��b2

1 1 �2h1 1 h2�2q

sin�u1 1 z 2 b�; �119�

E4 � m2r2lcos�u1 2 u2 1 uc2 2 b�; �120�

E5 � IG21 m2r2

��b2

1 1 �2h1 1 h2�2q

sin�u2 1 uc2 1 z� �121�and

E6 � IG21 m2r2

2cos�u1 1 u2 2 uc2 2 b 0�: �122�In the preceding analysis, it was assumed that, after impact,

the base and top blocks rock separately about O1 and O2,

respectively (pattern 1a). If this analysis leads to the physi-

cally impossible case where _u12 , _u1

1 , then, after impact,

the top block remains attached to the base block, and the

system rocks as one rigid body about O1 (pattern 3a). In this

case, _u12 � _u1

1 , and the associated value is derived by

ensuring the conservation of the angular momentum about

O1 for the system. Speci®cally,




Fig. 15. Transition from pattern 2a to either pattern 1a or pattern 3a through impact.

with


1 m2l 0��b2

1 1 �2h1 1 h2�2q

sin�u1 1 z

2 b 0�} _u21 1 {IG2

1 m2r2

��b2

1 1 �2h1 1 h2�2q

sin�u2 1 uc2

1 z�} _u22 ;

�124�and

�h1O1�system � IO

_u11 ; �125�

with IO given by Eq. (83). The relation for the angular velo-

cities is the same as in Eq. (84), but here B1� E1 and

B2� E5, with E1� E5 given as above. This same analysis

also holds for the case of transition from pattern 2b to

pattern 1b or 3b.

4. Accounting for impactÐpatterns 3 and 4


Consider the case where the two blocks are rocking

together as one body about O1 (pattern 3a). When impact

with the foundation occurs, the pole of rotation switches to

O 01, and the top block may either start rocking indepen-

dently about O 02 (pattern 1b) or remain in contact with the

base block so that the system rocks as one rigid body

about O 01 (pattern 3b). The two possibilities are illustrated

in Fig. 16.

Consider the case where the system switches from pattern

3a to pattern 1b. The conservation of angular momentum for

the system is expressed as

�h2O 0


O 01�system; �126�

where

�h2O 0

1�system � �IO 2 2Mb2

1� _u21 ; �127�

and

�h1O 0

1�system � {IO1

1 m2l��2h1 1 h2�cos�b�1 bsin�b��} _u11

1 {IG11 m2r2�bsin�uc2�1 �2h1

1 h2�cos�uc2��} _u12 :

�128�Similarly,

�h2O 0



where

�h2O 0

2�top block � {IO1

1 m2l 0�h2cos�b�2 b2sin�b 0��} _u21 ;

�130�and

�h1O 0

2�top block � m2l�h2cos�b�1 b2sin�b��} _u1

1 1 IO2_u1

2 :

�131�Solving the system of Eqs. (126) and (129) for the angular


_u11 � A _u2

1 ; �132�and

_u12 � B _u2

1 ; �133�where

A � IO2C1 2 C2C3

C4IO22 C2C5

; �134�

and

B � C5C1 2 C4C3

C2C5 2 C4IO2

; �135�

with

C1 � IO 2 2Mb21; �136�

C2 � IG11 m2r2�b1sin�uc2�1 �2h1 1 h2�cos�uc2��; �137�

C3 � IO11 m2l 0�h2cos�b 0�2 b2sin�b 0��; �138�

C4 � IO11 m2l��2h1 1 h2�cos�b�1 b1sin�b��; �139�


Fig. 16. Transition from pattern 3a to either pattern 1b or pattern 3b through impact.

and

C5 � m2l�h2cos�b�1 b2sin�b��: �140�In the preceding analysis, it was assumed that after impact

the base and top blocks rock separately about O 01 and O 02,respectively (pattern 1b). If _u1

2 . _u11 ; after impact the top

block remains attached to the base block, and the system

rocks as one rigid body about O 01 (pattern 3b). In this case,_u1

2 � _u11 ; and the associated value is derived by ensuring

conservation of the angular momentum about O 01 for the

system. Speci®cally,

�h2O 0


O 01�system; �141�

with

�h2O 0

1�system � �IO 2 2Mb2

1� _u21 ; �142�

and

�h1O 0

1�system � IO

_u11 : �143�

Eq. (141) is solved for the common post-impact angular

velocity yielding

_u11 � _u1

2 � IO 2 2Mb21

IO

_u21 : �144�

The above relations also hold for transition from pattern 3b

to pattern 1a or 3a.

4.2. Impact on pattern 4

Assume that the system is rocking in pattern 4a. If the

angle of rotation of the top block becomes zero, an impact

with the base block occurs, and the pole of rotation of the

top block switches to O 02. Furthermore, the base block may

either start rocking about O 01 in pattern 1b, or remain still as

in pattern 4b, see Fig. 17.

Consider the case where the system switches from pattern

4a to pattern 1b. Conservation of angular momentum for the

system is expressed as

�h2O 0


O 01�system; �145�

where

�h2O 0

1�system � {IG2

1 m2r2��2h1 1 h2�cos�uc2�2 b1sin�uc2��} _u22 ;

�146�and

�h1O 0

1�system � {IO1

1 m2l��2h1 1 h2�cos�b�1 b1sin�b��} _u11

1 {IG21 m2r2��2h1 1 h2�cos�uc2�

1 b1sin�uc2��} _u12 :

�147�Similarly,

�h2O 0



where

�h2O 0

2�top block � �IO2

2 2m2b22� _u2

2 : �149�

and

�h1O 0

2�top block � m2l�h2cos�b�1 b2sin�b��u1

1 1 IO2_u1

2 : �150�

Solving the system of Eqs. (145) and (148) for the angular

velocities immediately after impact gives

_u11 � A _u2

2 ; �151�

and

_u12 � B _u2

2 ; �152�

where

A � IO2C1 2 C2C3

C4IO22 C2C5

; �153�

and

B � C5C1 2 C4C3

C2C5 2 C4IO2

; �154�


Fig. 17. Transition from pattern 3a to either pattern 1b or pattern 4b through impact.

with

C1 � IG21 mx2r2��2h1 1 h2�cos�uc2�2 b1�; �155�

C2 � IG21 m2r2��2h1 1 h2�cos�uc2�1 b1sin�u2��; �156�

C3 � IO22 2m2b2

2; �157�

C4 � IO11 m2l��2h1 1 h2�cos�b�1 b1sin�b��; �158�

and

C5 � m2l�h2cos�b�1 b2sin�b��: �159�In the preceding analysis, it was assumed that after impact

the base and top blocks rock separately about O 01 and O 02,respectively (pattern 1b). If _u1

1 . 0, after impact the base

block remains still and only the top block continues rocking

(pattern 4b). In this case, _u12 is derived from

�h2O 0



with

�h2O 0

2�top block � �IO2

2 2m2b22� _u2

2 ; �161�and

�h1O 0

2�rtop block � IO2

_u12 : �162�

Eq. (160) is solved for the post impact angular velocity

yielding

_u12 �

IO22 2m2b2

2

IO2

_u22 : �163�

This relation holds for transition from pattern 4b to

pattern 1a or 4a.

5. Numerical applications

Due to the nonlinearity of the preceding equations of

motion, their numerical integration has been pursued by

using a four-order Runge±Kutta algorithm. The dynamic

response of the system for the case of free rocking and

ground excitation has been computed for various system

parameters.

First free rocking of the system is considered. It is

induced by rotating the system through an initial angular

displacement, say u10� u20

; releasing it, and letting it

rock back and forth about alternative corners until the

motion decays to rest.

In the following example, a system of two blocks having

slenderness ratios H1/B1� 1 and H2/B2� 2.5 with base

width B1� 1.25 m and B2� 1.0 m were used. The blocks

were assumed homogeneous of equal material densities

r 1� r 2� 2500 kg/m3. The time step for the Runge±Katta

method was 0.001 s.

Fig. 18 shows the results for u10� u20

� 0.15 rad, and

zero initial angular velocities. It can be seen that the angles

of rotation of the two blocks are decaying in an oscillatory

manner, with the one of the base block decreasing at a faster

rate.

The system starts rocking about point O1 (pattern 3a) and

remains in this pattern until an impact with the foundation

occurs. When such an impact takes place, the pole of

rotation switches to O 01, and the top block starts rocking


Fig. 18. Free vibration response of a two-block system.

independently about O 02 (pattern 1b). From this point on, the

system experiences a number of impacts before coming to

rest. Note that in free rocking impact is the only mechanism

for transition between patterns, with every impact being

accompanied by a pattern transition.

In this example, the base block comes temporarily to rest

very quickly, since the amplitude of the second cycle is

already quite small. This block experiences a few more

short periods of rocking, generally as a result of an impact

with the top block. For the cases where the base block can be

thought of as being still, the system may be assumed to rock

in pattern 4. However, the top block undergoes a number of

cycles before coming to rest, with decreasing amplitude in

each cycle, except for the second half cycle. At t� 1 s, the

angle of rotation reaches a value of 0.183 rad, exceeding the

initial value u20� 0.15 rad. This is not unreasonable, since

the angular velocity of the top block increases during the

®rst impact. Note also that, the period of vibration changes

signi®cantly in each cycle for both blocks.

The dynamics of the preceding system was also analyzed

by employing the linearized equations for small angles of

rotation. The results for both the linear and nonlinear formu-

lation are illustrated in Figs. 19 and 20 for the base and top

block, respectively. Examining these ®gures, it can be seen

that the linear formulation based results deviate signi®-

cantly, during non-negligible time segments from the results

obtained by integrating the nonlinear equations of motion.

As a second case, the response of the system to a hori-

zontal earthquake excitation was computed. Results are

presented here for an accelerogram recorded at Simi Valley

during the 1994 Northridge earthquake in California. This

accelerogram, shown in Fig. 21, exhibits a peak amplitude

of 0.71 g. The structural model described in the previous

example is considered again here. The system response is

shown in Fig. 22 for both the base and top blocks. It can be

seen that the response of the base block is, in general, much

less than that of the top block. In particular, the base block is

set into motion whenever a large peak on the excitation

occurs, or when it is hit by the top block, in which case

the system rocks as one rigid block (pattern 3a or 3b). Atten-

tion is also drawn to the fact that the maximum response of

the two blocks, which occur at different times, does not

follow immediately the peak amplitude of the ground accel-

eration. Furthermore, note that the rocking of the system

ceases before the end of the earthquake record.

6. Concluding remarks

The dynamic behavior of structures of two stacked rigid

blocks subjected to ground excitation has been examined.

Assuming no sliding, the rocking response of the system

standing free on a rigid foundation has been investigated.

The analytical formulation of this nonlinear problem has

proved challenging. Its complexity is associated with the

transition from one pattern of motion to another, each one

being governed by a set of highly nonlinear equations.

Furthermore, depending on the pattern, this transition can

happen either through impact, occurring between the

blocks or between the base block and the ground, or, in

the absence of impact, by an instantaneous separation of

the two blocks.


Fig. 19. Rotation angle for the lower block; linear versus nonlinear formulation.

The exact (nonlinear) equations governing the rocking

response of the system to horizontal and vertical ground

acceleration have been derived for each pattern. The

nonlinear equations of motion have been integrated for

each pattern by developing an ad hoc computational method

involving a fourth-order Runge±Kutta scheme, and

accounting for the various patterns. Numerical results

for both free vibration and seismic response have been

derived.

The model used in this paper, besides demonstrating the

complexity of even a simple two-block system rocking on a

rigid foundation problem, can also be used to incorporate

some of the characteristics of a ¯exible foundation. For

example, by inserting an elastic layer between the base

block and the rigid ground, the complex impact problem

may be eliminated. Also, it is possible to undertake a

comprehensive numerical investigation of the system

response for the actual ground motions, utilizing the compu-

ter program developed in this work. In the same context,

using an ensemble of synthetic earthquake ground motions,

response statistics may be obtained for a range of system

parameters, from which possible systematic trends in the

system response may be identi®ed. Experimental investiga-

tion of the problem can also be expected in future work; this

attempt will be essential in validating the analytical results

for the nonlinear equations of motion. Finally, dynamic

analysis of multi-block structures, consisting of more than

two blocks, can be performed using a method analogous to

the one described in this work. This effort could perhaps

be expedited by incorporating in the analysis concepts of

the discrete elements technique as in the study of Azevedo

et al. [32].


Fig. 20. Rotation angle for the upper block; linear versus nonlinear formulation.

Fig. 21. Accelerogram from the Northridge, CA, 1994, earthquake.

Acknowledgements

The support of this work by a grant from the National

Science Foundation is gratefully acknowledged. Further, the

assistance of Dr M. Kokolaras with the creation of an initial

electronic template for the paper is appreciated.

References

[1] Housner G. The behavior of inverted pendulum structures during

earthquakes. Bulletin of the Seismological Society of America

1963;53(2):403±17.

[2] Aslam M, Godden W, Scalise D. Earthquake rocking response on

rigid bodies. Journal of the Structural Division of the ASCE

1980;106(ST2):377±92.

[3] Ishiyama Y. Motions of rigid bodies and criteria for overturning by

earthquake excitations. Earthquake Engineering and Structural

Dynamics 1982;10(5):635±50.

[4] Psycharis I, Jennings P. Rocking of slender rigid bodies allowed to

uplift. Earthquake Engineering and Structural Dynamics 1983;11:57±

76.

[5] Spanos PD, Koh AS. Rocking of rigid blocks due to harmonic shak-

ing. Journal of Engineering Mechanics 1984;110:1627±42.

[6] Allen R, Oppenheim I, Parker A, Bielak J. On the dynamic response

of rigid body assemblies. Earthquake Engineering and Structural

Dynamics 1986;14:861±76.

[7] Tso W, Wong C. Steady state rocking response of rigid blocks. Part 1:

analysis. Earthquake Engineering and Structural Dynamics

1989;18:89±106.

[8] Hogan S. The many steady state responses of a rigid block under

harmonic forcing. Earthquake Engineering and Structural Dynamics

1990;19:1057±71.

[9] Psycharis I. Dynamic behaviour of rocking two-block

assemblies. Earthquake Engineering and Structural Dynamics

1990;19:555±75.

[10] Sinopoli A. Dynamics and impact in a system with unilateral

constraints. The relevance of dry friction. Journal of the Italian Asso-

ciation of Theoretical and Applied Mechanics 1990;22:210±15.

[11] Yim CS, Lin H. Nonlinear impact and chaotic response of slender

rocking objects. Journal of Engineering Mechanics, ASCE

1991;117(9):2079±100.

[12] Shenton H, Jones N. Base excitation of rigid bodies. 1: Formulation.

Journal of Engineering Mechanics, ASCE 1991;117(EM10):2286±

306.

[13] Shenton H, Jones N. Base excitation of rigid bodies. 2: Periodic

slide-rock response. Journal of Engineering Mechanics, ASCE

1991;117(EM10):2307±28.

[14] Augusti G, Sinopoli A. Modeling the dynamics of large block struc-

tures. Mechanica 1992;27:195±211.

[15] Allen R, Duan X. Effects of linearizing on rocking block. Journal of

Structural Engineering 1995;12(7):1146±9.

[16] Shi B, Anooshehpoor A, Zeng Y, Brune JN. Rocking of rigid blocks

due to harmonic shaking. Journal of Engineering Mechanics

1984;110:1627±42.

[17] Shenton III HW. Criteria of initiation of slide, rock, and slide-rock

rigid-body modes. Journal of Engineering Mechanics, ASCE

1996;122:690±3.

[18] Scalia A, Sumbatyan MA. Slide rotation of rigid bodies subjected to a

horizontal ground motion. Earthquake Engineering and Structural

Dynamics 1996;25:1139±49.

[19] Pombei A, Scalia A, Sumbatyan MA. Dynamics of rigid block due to

horizontal ground motion. Journal of Engineering Mechanics, ASCE

1998;124(7):713±7.

[20] Zhang J, Makris N. Rocking response of free-standing blocks under

cycloidal pulses. Journal of Engineering Mechanics, ASCE

2001;127(5):473±83.


Fig. 22. Response to earthquake excitation.

[21] Makris N, Zhang J. Rocking response of anchored blocks under pulse-

type motions. Journal of Engineering Mechanics, ASCE

2001;127(5):484±93.

[22] Kim Y-S, Miura K, Miura S, Nishimura M. Vibration characteristics

of rigid body placed on sand ground. Soil Dynamics and Earthquake

Engineering 2001;21:19±37.

[23] Yim CS, Chopra A, Penzien J. Rocking response of rigid blocks to

earthquakes. Earthquake Engineering and Structural Dynamics

1980;8(6):567±87.

[24] Koh A-S, Spanos PD, Roesset J. Harmonic rocking of rigid body

block on ¯exible foundation. Journal of Engineering Mechanics,

ASCE 1986;112(11):1165±80.

[25] Koh A-S, Spanos PD. Analysis of block random rocking. Soil

Dynamics and Earthquake Engineering 1986;5:178±83.

[26] Giannini R, Masiani R. Random vibration of the rigid block. Compu-

tational Stochastic Mechancis 1991:741±52.

[27] Dimentberg M, Lin Y, Zhang R. Toppling of computer-type equip-

ment under base excitation. Journal of Engineering Mechanics, ASCE

1993;119(1):145±60.

[28] Cai G, Yu J, Lin Y. Toppling of rigid block under evolutionary

random base excitations. Journal of Engineering Mechanics

1995;121(8):924±9.

[29] Lin H, Yim CS. Nonlinear rocking motions. 1: Chaos under noisy

periodic excitations. Journal of Engineering Mechanics

1996;122(8):719±27.

[30] Lin H, Yim CS. Nonlinear rocking motions. 2: Overturning under

random excitations. Journal of Engineering Mechanics

1996;122(8):728±35.

[31] Giannini R, Masiani R. Non-Gaussian solution for random rocking of

slender rigid block. Probabilistic Engineering Mechanics 1996;11:87±96.

[32] Azevedo J, Sincraian G, Lemos JV. Seismic behavior of blocky

masonry structures. Earthquake Spectra 2000;16(2):337±65.


dynamic analysis of stacked rigid blocks

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