Journal ArticleDOI

# A simple procedure to approximate slip displacement of freestanding rigid body subjected to earthquake motions

10 Apr 2007--Vol. 36, Iss: 4, pp 481-501
TL;DR: In this paper, a simple calculation procedure for estimating absolute maximum slip displacement of a freestanding rigid body placed on the ground or floor of a linear/nonlinear multi-storey building during an earthquake is developed.
Abstract: A simple calculation procedure for estimating absolute maximum slip displacement of a freestanding rigid body placed on the ground or floor of linear/nonlinear multi-storey building during an earthquake is developed. The proposed procedure uses the displacement induced by the horizontal sinusoidal acceleration to approximate the absolute maximum slip displacement, i.e. the basic slip displacement. The amplitude of this horizontal sinusoidal acceleration is identical to either the peak horizontal ground acceleration or peak horizontal floor response acceleration. Its period meets the predominant period of the horizontal acceleration employed. The effects of vertical acceleration are considered to reduce the friction force monotonously. The root mean square value of the vertical acceleration at the peak horizontal acceleration is used. A mathematical solution of the basic slip displacement is presented. Employing over one hundred accelerograms, the absolute maximum slip displacements are computed and compared with the corresponding basic slip displacements. Their discrepancies are modelled by the logarithmic normal distribution regardless of the analytical conditions. The modification factor to the basic slip displacement is quantified based on the probability of the non-exceedence of a certain threshold. Therefore, the product of the modification factor and the basic slip displacement gives the design slip displacement of the body as the maximum expected value. Since the place of the body and linear/nonlinear state of building make the modification factor slightly vary, ensuring it to suit the problem is essential to secure prediction accuracy. Copyright © 2006 John Wiley & Sons, Ltd.

### INTRODUCTION

• In recent highly developed and complicated social system the damage of structures no longer represents total effects of earthquakes.
• The former investigators clearly pointed out a need for investigation into seismic behavior of nonstructural components in order to assess their vulnerability [1].
• The senior author pointed out that the period of horizontal base excitation makes an important contribution to elongation of the slip displacement of the body in addition to the friction coefficient and peak horizontal acceleration [7].
• The prediction accuracy of the slip displacement of the body on the building floor may deteriorate because filtering effects of structure enhance the advent of a certain wave component in the floor response.
• The use of the root mean square value of the vertical acceleration at the peak horizontal acceleration [11, 12] is proposed as the uniform downward acceleration.

### DESCRIPTION OF THE PROBLEM

• An equation of motion that governs the slip behavior of the body subjected to simultaneous horizontal and vertical acceleration is given as follows.
• Figure 1(a) and Figure 1(b) shows a mechanical model of the body placed on the ground and set on the building floor respectively.
• If the authors replace the shaking motion according to the problem, the subjects are essentially the same.
• Hz&& vz&& g , μ and ν are gravitational acceleration, static and kinetic friction coefficients, respectively.

### Simplification of the problem

• A simple treatment of the vertical acceleration is necessary to mathematically obtain the slip displacement of the body induced by the horizontal sinusoidal acceleration, i.e., the basic slip displacement of the body.
• Therefore, this study introduces a monotonous reduction in friction force while the body slips [13].
• The probability density of this ratio was modeled by the normal distribution and the product of σ and PVGA gave the root mean square value of the vertical ground acceleration.
• In case of the body set on the ground, this paper uses PVGA of each vertical accelerogram.
• In contrast, η is the magnification factor for the vertical floor response given by the quotient of the peak vertical floor response acceleration (VFRA) by PVGA.

### Horizontal sinusoidal acceleration

• In case of the body set on the ground, this paper uses PPHA determined by the period which maximizes a pseudo-acceleration spectrum of each horizontal accelerogram.
• In actual case scenario, PPHA is determined by the natural period of soil, since it will be predominant over other T wave period in the earthquake.
• Ref. [14] gives the calculation of the natural period of soil.
• In case of the body set on building floor, this paper uses PPHA determined by FFT analysis of each horizontal floor acceleration.
• For the convenience of actual case scenario, the calculation of the predominant period of the horizontal floor response acceleration is proposed latter.

### Mathematical solution

• Solve Eqs. (5) to (7) mathematically to obtain the basic slip displacement of the body.
• Figures 1(d) and 1(e) depict the typical time history of slip motion of the body subjected to the horizontal sinusoidal acceleration with =7m/sgAgx 2, T =6.28s and μ =0.4.
• Since Eq. (12) is, however, the transcendental function in terms of the cosine function, this study tries to find its approximate solution employing the Taylor’s series.
• The modification factor for the basic slip displacement to calculate the design slip displacement is statistically examined.

### Earthquake records

• This study uses 104 accelerograms observed around Japan [15].
• Maintaining the relation between PHGA and PVGA of each pair of accelerograms, the time history of vertical acceleration is also normalized and scaled in a same manner.
• The absolute maximum slip displacement of the body induced by the earthquake wave, Max Eq vhx , , is numerically computed and compared with the corresponding basic slip displacement, , induced by the horizontal sinusoidal acceleration with the nominal friction coefficient, μ Sin vhx , ′ .
• Figure 3(e) shows the probability density of the slip ratio, vh,β vh,β vh,β vh,β vh,β , of all results.
• The product of the modification factor, ob vhβ Pr , ob vhβ Pr , , and the basic slip displacement, , gives the design slip displacement, , as the maximum expected value.

### Methodologies

• The following section tries to apply the proposed procedure to estimate the slip displacement of the body set on the building floor.
• The modification factor for the basic slip displacement is statistically examined.
• Figures 6(a) to 6(c) show the predominant period of HFRA observed at the specified floor of linear building while Figures 6(d) to 6(f) show that of nonlinear building.
• The abscissa of these figures shows the predominant period of earthquake input to the building model determined by a pseudo-acceleration spectrum of each earthquake.
• Figure 7(g) is the probability density of the slip ratio, vh,β vh,β vh,β , of all results.

### CONCLUSION

• This paper proposes the use of the horizontal sinusoidal acceleration to approximate the slip displacement of the body wherever it is set on during an earthquake.
• A mathematical solution of the slip displacement of the body induced by the horizontal sinusoidal acceleration, i.e., the basic slip displacement, is presented in a practical form.
• The modification factor is quantified as an estimation error between the basic slip displacement and the absolute maximum slip displacement based on the probability of the nonexceedence of a certain threshold.
• It forms a certain probability density despite the acceleration intensity and the friction coefficient.
• The modification factors are slightly different according to the place of the body and linear/nonlinear state of building.

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A SIMPLE PROCEDURE TO APPROXIMATE SLIP DISPLACEMENT OF FREESTANDING RIGID
BODY SUBJECTED TO EARTHQUAKE MOTIONS
Tomoyo TANIGUCHI
1
and Takuya MIWA
2
1
Associate Professor, Department of Civil Engineering, Tottori University. 4-101 Koyama-Minami Tottori, 680-
8552, Japan. Tel +81-857-31-5287, Fax +81-857-28-7899, Email: t_tomoyo@cv.tottori-u.ac.jp
2
Engineer, Bridge Design Department, TTK Corporation. 1020 Ooaza-Simotakai, Toride, 302-0038, Japan. Tel
+81-297-78-1119, Fax +81-0297-78-5344, Email: Takuya_Miwa@ttk-corp.co.jp
(Former graduate student, Department of Civil Engineering., Tottori University)
SUMMARY
A simple calculation procedure for estimating absolute maximum slip displacement of a freestanding rigid body
placed on the ground or floor of linear/nonlinear multi-story building during an earthquake is developed. The
proposed procedure uses the displacement induced by the horizontal sinusoidal acceleration to approximate the
absolute maximum slip displacement, i.e., the basic slip displacement. The amplitude of this horizontal
sinusoidal acceleration is identical to either the peak horizontal ground acceleration or peak horizontal floor
response acceleration. Its period meets the predominant period of the horizontal acceleration employed. The
effects of vertical acceleration are considered to reduce the friction force monotonously. The root mean square
value of the vertical acceleration at the peak horizontal acceleration is used. A mathematical solution of the basic
slip displacement is presented. Employing over one hundred accelerograms, the absolute maximum slip
displacements are computed and compared with the corresponding basic slip displacements. Their discrepancies
are modeled by the logarithmic normal distribution regardless of the analytical conditions. The modification
factor to the basic slip displacement is quantified based on the probability of the nonexceedence of a certain
threshold. Therefore, the product of the modification factor and the basic slip displacement gives the design slip
displacement of the body as the maximum expected value. Since the place of the body and linear/nonlinear state
of building make the modification factor slightly vary, ensuring it to suit the problem is essential to secure
prediction accuracy.
Keywords; horizontal sinusoidal acceleration, maximum expected slip displacement, coincident vertical
acceleration, predominant period, peak horizontal ground acceleration, peak horizontal floor response
acceleration.
INTRODUCTION
In recent highly developed and complicated social system the damage of structures no longer represents total
effects of earthquakes. The functional loss of the social system as a result of the damage of nonstructural
component should be taken into account. An appropriate design for nonstructural components to maintain
minimum function in the event of earthquakes prevents from such systematic disorder. The former investigators
clearly pointed out a need for investigation into seismic behavior of nonstructural components in order to assess

their vulnerability [1]. An important class of nonstructural components, such as mechanical/electrical equipment,
which is essentially modeled as a rigid body, is of interest to this study.
The effects of base excitation on a freestanding rigid body have been investigated by many researchers. They
clarified five modes of the body (rest, slide, rock, slid-rock and jump), equations of motion and relation of wave
properties of base excitation to these modes [2-6]. The senior author pointed out that the period of horizontal
base excitation makes an important contribution to elongation of the slip displacement of the body in addition to
the friction coefficient and peak horizontal acceleration [7].
Shao and Tung [8] prepared a chart which enabled to determine the mean-plus-standard deviation of the
maximum sliding distance of an unanchored body. However, the study does not consider vertical excitation and
its applicability to the body placed on the building floor. Including the vertical excitation, Lopez Garcia and
Soong [1] presented the fragility information for sliding related failure modes. They classified the fragility
curves according to the friction coefficient and peak horizontal ground acceleration. Although it gives the
appropriate slip displacement if the body is on the ground, its prediction accuracy deteriorates when the body is
on the building floor. Historically, Newmark [9] presented a simple formula to determine the sliding distance of
a freestanding body subjected to a single rectangular acceleration pulse at the base concerning earthquake
response of embankments. Choi and Tung [10] concluded that Newmark’s formula could be used if an
adjustment factor consisting of the friction coefficient and peak horizontal base acceleration was applied.
However, the study is limited to the action of horizontal base excitation. In addition to that, prediction accuracy
deteriorates if the body is on the building floor with a long natural period.
As mentioned above, the previous procedures for estimating the slip displacement of the body do not consider
the period of horizontal excitation. Consequently, the prediction accuracy of the slip displacement of the body on
the building floor may deteriorate because filtering effects of structure enhance the advent of a certain wave
component in the floor response. This suggests the necessity of considering the period of horizontal excitation to
approximate the slip displacement of the body set on the building floor.
An objective of this study is to develop a simple procedure that can approximate the absolute maximum slip
displacement of the body during the earthquake wherever it is set on, i.e., the design slip displacement. In order
to improve the deficiencies in the previous research, this study introduces the use of the single horizontal
sinusoidal acceleration. Its amplitude is identical to either the peak horizontal ground acceleration or the peak
horizontal floor response acceleration. Its period meets the predominant period of the horizontal acceleration
employed. Generally, the peak horizontal ground acceleration is not always induced by the wave component
with the predominant period of earthquake. On the other hand, due to filtering effects of structure, the peak
horizontal floor response acceleration tends to be induced by the wave component with the predominant period
of floor response. This is the advantage of the use of the horizontal sinusoidal acceleration.
The slip displacement of the body induced by the horizontal sinusoidal acceleration, i.e., the basic slip
displacement, is used as the first approximation to the absolute maximum slip displacement of the body, since it

gives an upper bound of the slip displacement under the single action of horizontal base acceleration. The
discrepancy between the basic slip displacement and absolute maximum slip displacement are compiled to find
the modification factor for the basic slip displacement. Since it is statistically examined based on the probability
of nonexceedence of a certain threshold, the design slip displacement is given as the maximum expected value.
In contrast, Shao and Tung [8] suggested that the effects of the vertical ground motion on the mean-plus-
standard deviation of the maximum sliding distance was small but it was indispensable to compute the slip
displacement of the body accurately. In view of developing a simple calculation procedure and having a safety
margin in slip displacement, this study introduces a monotonous reduction in friction force as if the uniform
downward acceleration lasts while the body slips under action of the horizontal sinusoidal acceleration. The use
of the root mean square value of the vertical acceleration at the peak horizontal acceleration [11, 12] is proposed
as the uniform downward acceleration.
The fist part of the paper briefly describes the problem and the second part defines the horizontal sinusoidal
acceleration and derives a mathematical solution of the basic slip displacement. The third and forth part
examines the modification factor for the basic slip displacement of the body set on the ground and the building
floor respectively. Three buildings with different stories and spans and liner/nonlinear state are considered.
DESCRIPTION OF THE PROBLEM
An equation of motion that governs the slip behavior of the body subjected to simultaneous horizontal and
vertical acceleration is given as follows. Figure 1(a) and Figure 1(b) shows a mechanical model of the body
placed on the ground and set on the building floor respectively. If we replace the shaking motion according to
the problem, the subjects are essentially the same.
i) while slip
(
)
xsignzgzx
vh
&
&&&&
&&
)(
+
=
ν
(1)
ii) while stationary
0
=
x
&&
(2)
iii) slip commencement condition
)(
vh
zgz
&&&&
+>
μ
(3)
iv) slip termination condition
0
=
x
&
(4)
where and are a pair of horizontal and vertical acceleration of either the ground or building floor,
respectively.
h
z
&&
v
z
&&
g
,
μ
and
ν
are gravitational acceleration, static and kinetic friction coefficients, respectively. In
order to simplify the problem, the static and kinetic friction coefficients are assumed to be the same and denoted
as
μ
thereafter. The function gives the sign of variable.
()
xsign
&
BASIC SLIP DISPLACEMENT OF THE BODY
Simplification of the problem
A simple treatment of the vertical acceleration is necessary to mathematically obtain the slip displacement of the
body induced by the horizontal sinusoidal acceleration, i.e., the basic slip displacement of the body. Employing
Eqs. (1) to (4) and all accelerograms listed on Table 1, the absolute maximum slip displacement of the body
with/without the vertical ground acceleration is numerically computed. Each pair of horizontal and vertical
accelerograms is scaled to 9 m/s
2
in the peak horizontal ground acceleration (PHGA). The friction coefficient is

assumed to be 0.3 to 0.5. The abscissa of Figure 1(c) is the ratio of the slip displacement induced by the
simultaneous horizontal and vertical ground acceleration to the slip displacement induced by horizontal ground
acceleration. The ordinate is the probability density of the frequency. Although the vertical ground acceleration
does not always increase the slip displacement, it is better to ensure an adequate safety margin against the slip
displacement. Since the slip motion of the body discontinuously occurs during the earthquake and lasts a short
time, the effects of varying vertical acceleration on the slip motion of the body have few contributions.
Therefore, this study introduces a monotonous reduction in friction force while the body slips [13]. To calculate
the basic slip displacement of the body, Eqs. (1) and (3) are rewritten as follows.
(
)
xsignPVGAgzx
h
&
&&
&&
)(
η
σ
=
(5)
)PVGAg(z
h
ησμ
>
&&
(6)
where is the horizontal sinusoidal acceleration defined by Eq. (7).
h
z
&&
σ
is the standard deviation of the ratio of
the vertical ground acceleration to the peak vertical ground acceleration (PVGA) at the instant of PHGA. The
senior author(s) compiled this ratio with 144 accelerograms. The probability density of this ratio was modeled by
the normal distribution and the product of
σ
and PVGA gave the root mean square value of the vertical ground
acceleration. A pair of this vertical ground acceleration and PHGA was used for verification of the onset of slip
of the unanchored flat-bottom cylindrical shell tank [11]. In addition, the probability density of this ratio
computed with the ground acceleration was almost the same as that computed with the building floor response
acceleration [12]. Therefore, this paper uses 0.46 as the value of
σ
irrespective of the location of the body
placed on. In case of the body set on the ground, this paper uses PVGA of each vertical accelerogram. In actual
case scenario, although the value of PVGA is usually not provided by design codes, it can be determined by the
ratio of PVGA to PHGA [14] or the well-known empirical rule PVGA = 0.5 or 0.66 PHGA. In contrast,
η
is the
magnification factor for the vertical floor response given by the quotient of the peak vertical floor response
acceleration (VFRA) by PVGA. This paper uses the peak VFRA of each result of time history analysis as the
value of
η
PVGA
. In actual case scenario,
η
is determined by the response amplitude. Therefore,
η
is 1.0 if the
body is set on the ground.
Horizontal sinusoidal acceleration
The amplitude of the horizontal sinusoidal acceleration, , is identical to PHGA if the body is placed on the
ground. In contrast, it should be identical to the peak horizontal floor response acceleration (HFRA) if the body
is set on the building floor.
gA
gx
= t
T
gAz
gxh
π
2
sin
&&
(7)
where is the predominant period of the horizontal acceleration (PPHA) of either the horizontal ground
acceleration or floor response acceleration. In case of the body set on the ground, this paper uses PPHA
determined by the period which maximizes a pseudo-acceleration spectrum of each horizontal accelerogram. In
actual case scenario, PPHA is determined by the natural period of soil, since it will be predominant over other
T

wave period in the earthquake. Ref. [14] gives the calculation of the natural period of soil. In case of the body set
on building floor, this paper uses PPHA determined by FFT analysis of each horizontal floor acceleration. For
the convenience of actual case scenario, the calculation of the predominant period of the horizontal floor
response acceleration is proposed latter.
Mathematical solution
Solve Eqs. (5) to (7) mathematically to obtain the basic slip displacement of the body. Firstly, assume an absence
of the vertical acceleration to simplify the problem. Figures 1(d) and 1(e) depict the typical time history of slip
motion of the body subjected to the horizontal sinusoidal acceleration with =7m/s
gA
gx
2
,
T
=6.28s and
μ
=0.4.
When the body undergoes the horizontal base acceleration, the body begins to slip when the horizontal inertia
force overcomes the friction force. From Eqs. (6) and (7), the time, , for the onset of slip is calculated as:
0
t
gx
A
Sin
T
t
μ
π
1
0
2
=
(8)
Despite the time history of horizontal base acceleration the slip acceleration of the body is uniform. The velocity
of the body at arbitrary time is calculated as:
1
Cgtx
+
=
&
(9)
Similarly, the velocity of the base is calculated as:
t
T
T
gAz
gxh
π
π
2
cos
2
=
&
(10)
where the initial velocity of the base is assumed to be
π
2gTA
gx
to derive Eq. (10). In contrast, in Eq. (9) is
the constant of integration and should be determined as that the body has the same velocity of the base at the
onset of slip.
1
C
+=
gx
gx
gx
A
SingA
A
gSin
T
C
μμ
μ
π
11
1
cos
2
(11)
From Fig. 1(e), the slip terminates when the velocity of the body and base become the same. The time is
calculated by Eqs. (9), (10) and (11) as follows.
t
T
T
gA
A
SingA
A
Sing
T
gt
gx
gx
gx
gx
π
π
μμ
μ
π
μ
2
cos
2
cos
2
11
=
+
(12)
Solve Eq. (12) for
t
to find the slip termination time, . Since Eq. (12) is, however, the transcendental function
in terms of the cosine function, this study tries to find its approximate solution employing the Taylor’s series.
From Figs. 1(d) and 1(e), since the slip terminates around
t
=
1
t
2T
, the Taylor’s series of the cosine function
around
t
=
2T
is used.
L
2
2
2
2
2
1
2
cos
+
T
t
T
t
T
ππ
(13)
Substitution of Eq. (13) into Eq. (12) yields the slip termination time, , as follows.
1
t

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• ...Historically, Newmark [9] presented a simple formula to determine the sliding distance of a freestanding body subjected to a single rectangular acceleration pulse at the base concerning earthquake response of embankments....

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Journal ArticleDOI
TL;DR: In this paper, the British Geotechnical Society for the opportunity of visiting London' again and for the honour of appearing before you in the home of the Institution of Civil Engineers, of which I am so proud to be a member.
Abstract: I wish to thank the British Geotechnical Society for the opportunity of visiting London’ again and for the honour of appearing before you in the home of the Institution of Civil Engineers, of which I am so proud to be a member. Several years ago I transmitted some preliminary notes on the topic of earthquake effects on dams to the late Karl Terzaghi, whose invaluable advice and suggestions regarding those notes were freely used in the preparation of this Paper. I wish also to acknowledge the comments and suggestions I have had from time to time concerning the subject from my colleague at the University of Illinois, Dr Ralph B. Peck ; from my associate in several consulting assignments, Dr Laurits Bjerrum ; and from my colleague for several months, while he was visiting the University of Illinois, Dr Tu’. N. Ambraseys. Finally, I should like to acknowledge the assistance on some of the calculations for this lecture that were made by two of my associates at the University of Illinois, Dr John W. Melin, and Mr Mohammad Amin.

1,999 citations

Journal ArticleDOI
TL;DR: In this article, a computer program was developed to simulate the motions of rigid bodies subjected to horizontal and vertical ground motions, numerically solving the non-linear equations of motion, transitions of motion and motions after impact between the body and the floor.
Abstract: This investigation deals with motions of rigid bodies on a rigid floor subjected to earthquake excitations, and criteria for overturning of the bodies. In order to study any motions of a body in a plane, the motions are classified into six types, i.e. (1) rest, (2) slide, (3) rotation, (4) slide rotation, (5) translation jump and (6) rotation jump. Then, the following are studied: the equations of motion, transitions of motion, and motions after impact between the body and the floor. One of the features of this investigation is the introduction of the tangent restitution coefficient which enables us to estimate the magnitude of the tangent impulse at the instant of impact. A computer program was developed to simulate the motions of bodies subjected to horizontal and vertical ground motions, numerically solving the non-linear equations of motion. Several types of simulation were carried out and the following conclusions were found. The coefficient of friction must be greater than the breadth—height-ratio in order for the body to rock. The motions after impact from translation jump are greatly influenced by the normal and tangent restitution coefficients. As criteria for overturning of bodies, at least two factors must be taken into account: the horizontal acceleration and the velocity of the floor. Then it is possible to estimate the motions of the floor from the overturning of bodies in a more reliable manner than before.

307 citations

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the lower limits of the maximum horizontal acceleration and velocity of the input excitations, from the overturning of rigid bodies on a rigid floor subjected to sinusoidal and earthquake excitations and concluded that the horizontal velocity and acceleration must be taken into account as criteria for overturning.
Abstract: This investigation deals with motions of rigid bodies on a rigid floor subjected to sinusoidal and earthquake excitations, and overturning of the bodies. Experiments and simulations of frequency sweep tests were conducted, and it is concluded that the horizontal velocity as well as the acceleration must be taken into account as criteria for overturning. Simulations by earthquake excitations show that the criteria are also applicable to the earthquake excitations. Therefore it is possible to estimate the lower limits of the maximum horizontal acceleration and velocity of the input excitations, from the overturning of bodies.

274 citations

### "A simple procedure to approximate s..." refers background in this paper

• ...They clarified five modes of the body (rest, slide, rock, slid-rock and jump), equations of motion and relation of wave properties of base excitation to these modes [2-6]....

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Journal ArticleDOI
TL;DR: In this article, it was shown that a slide-rock mode can also be initiated from rest, and that the friction required to initiate a rock mode increases with ground acceleration, as would be expected.
Abstract: The motion of a plane rigid body that rests unrestrained on a rigid foundation that is accelerating is described by five modes of response: rest, slide, rock, slide-rock, and free flight. Determining the correct initial mode of response is essential to the analysis of the generalized behavior. In the past it has been generally assumed that only a slide or rock mode can be initiated from rest. It is demonstrated in the present paper that a slide-rock mode can also be initiated from rest. Criteria governing the initiation of the slide, rock, \Iand\N slide-rock modes are derived. Results show that it is incorrect to assume that a rock mode ensues if static friction is simply greater than the width-to-height ratio of the body; a slide-rock mode is initiated when friction is greater than the width-to-height ratio, but less than a critical value that is a function of the ground acceleration. The friction required to initiate a rock mode increases with ground acceleration, as would be expected. The results demonstrate a more natural transition of the governing mode of response, from sliding to slide-rock and finally to pure rocking, as friction is increased for a given foundation acceleration. The analysis is restricted to a rectangular block, assuming Coulomb friction and horizontal ground acceleration.

195 citations

### "A simple procedure to approximate s..." refers background in this paper

• ...They clarified five modes of the body (rest, slide, rock, slid-rock and jump), equations of motion and relation of wave properties of base excitation to these modes [2-6]....

[...]