Seismic response of slender rigid structures with foundation …ssi.civil.ntua.gr/downloads/journals/2007-SDDE_Seismic... · 2013. 1. 15. · structures, uplifting and overturning

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Soil Dynamics and Earthquake Engineering 27 (2007) 642–654

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Seismic response of slender rigid structures with foundation uplifting

Marios Apostolou�, George Gazetas, Evangelia Garini

School of Civil Engineering, National Technical University, Athens, Greece

Received 22 June 2006; accepted 4 December 2006

Abstract

The rocking of rigid structures uplifting from their support under strong earthquake shaking is investigated. The structure is resting on

the surface of either a rigid base or a linearly elastic continuum. A large-displacement approach is adopted to extract the governing

equations of motion allowing for a rigorous calculation of the nonlinear response even under near-overturning conditions. Directivity-

affected near-fault ground motions, idealized as Ricker wavelets or trigonometric pulses, are used as excitation. The conditions under

which uplifting leads to large angles of rotation and eventually to overturning are investigated. A profoundly nonlinear rocking behavior

is revealed for both rigid and elastic soil conditions. This geometrically nonlinear response is further amplified by unfavorable sequences

of long-duration pulses in the excitation. Moreover, through the overturning response of a toppled tombstone, it is concluded that the

practice of estimating ground accelerations from overturning observations is rather misleading and meaningless.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Rocking; Foundation uplift; Overturning; Soil–structure interaction; Nonlinear dynamics; Near-fault motions

1. Introduction

While slender structural systems with a shallow founda-tion are generally considered as bonded to the ground,during strong seismic shaking uplifting from the support-ing soil is often practically unavoidable. Examples ofstructures that experienced uplifting from the supportingsoil have been reported in numerous earthquakes, includ-ing that of Chile 1960, Alaska 1964, San Fernando 1971,Kocaeli 1999, and Athens 1999. It is well documented inthe literature that uplifting changes the rocking behavior ina profoundly nonlinear sense and modifies the structuralresponse in most cases favorably. Apart from civilstructures, uplifting and overturning are some of the mostfamiliar phenomena for free-standing bodies (such asappended equipment, furniture, etc.) during strong earth-quakes.

Since the pioneering work of Milne and Perry in 1881[1,2] the uplifting and overturning response of rigid bodieshas attracted the interest of many earthquake engineersand seismologists for over a century [3]. Early analyti-

e front matter r 2007 Elsevier Ltd. All rights reserved.

ildyn.2006.12.002

ing author. Tel.: +302109515366.

ess: [email protected] (M. Apostolou).

cal and experimental studies conducted mostly in Japanhad been motivated by tombstones overturnings afterlarge earthquakes (Sagisaka, Inouye, Kimoura, Ikegamiamong others [3]). Housner [4] investigated in detail therocking behavior of rigid blocks subjected to baseexcitation. Using an energy approach he uncovered therole of the excitation frequency and of the block size onthe overturning potential. Makris and his co-workers[5,6] focused on the transient response of rigid blocksunder near-source ground shaking idealized as trigono-metric pulses, and derived the acceleration amplitudeneeded for overturning. Ishiyama [7] studied the slide-rocking motion of a rigid body on rigid floor andestablished criteria for overturning. Psycharis [8] intro-duced the compliance of the supporting soil with a visco-elastic Winkler foundation, extracted the linearized equa-tions of rocking motion, and addressed the structuralresponse of an uplifting system. Koh et al. [9] extendedPsycharis’ work on the linearized rocking response onflexible foundation by introducing the flexibility of thesuperstructure. Huckelbridge and Clough [10] carried out13�scale shaking table tests with 9-story steel moment framesand confirmed the beneficial role of transient uplift onstructural response.

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ARTICLE IN PRESSM. Apostolou et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 642–654 643

In the present study, two different systems of struc-tures undergoing rocking motion with uplift are examined(Fig. 1):

�

a

Fig

fou

a rigid block supported on undeformable ground, whichwill be referred to herein as ‘‘rigid foundation’’;

�
a rigid block founded on an elastically deformable
continuum in the form of a homogeneous halfspace or astratum over rigid bedrock.

The conditions under which uplifting of these simplesystems leads to large angles of rotation and eventually tooverturning are investigated, and minimum accelerationlevels for overturning are derived. Ground motion ismainly represented with two drastically different records(despite their nearly identical PGAs and not very differentstrong motion duration) obtained in the Athens andKocaeli earthquakes of 1999, as well as with idealizedRicker-wavelets and one-cycle sinusoidal pulses.

2. Uplifting and overturning on a rigid base

We consider first a rigid rectangular block with aspectratio b/h (half width over half height ratio) simplysupported on a rigid base, which is oscillating horizontally.The coefficient of friction is adequately large so that slidingis prevented. As long as the overturning moment (magh)about the base edge (where ag ¼ the base acceleration) doesnot exceed the restoring moment (mgb), the block remainsattached to the base and undergoes only horizontaloscillation. As soon as the restoring moment is exceededuplifting occurs setting the block on rocking motion. Thesystem configuration is illustrated in Fig. 1a. Under staticconditions, once uplifting is initiated about the cornerpoint, the body overturns. Thus, the critical uplifting

Rocking of a rigid block

on rigid foundation

Rocking of a 1-d

oscillator on rigid f

R

2 h

2 b 2 b

Rθ

θc

θ

b

. 1. The rocking systems considered in this paper. (a) Rocking of a rigid b

ndation (c) Rocking of a rigid block on elastic foundation.

acceleration of the base is identical with the minimumrequired to statically overturn the block in units of g(acceleration of gravity)

aover;stat ¼ ac ¼b

h. (1)

However, under dynamic base excitation the inertia force‘‘quickly’’ changes direction as the acceleration changessign, and overturning is avoided. Rocking oscillation takesplace with the two corner points, O and O0, beingalternately the pivot points. Between two successiveimpacts the governing equation of rocking motion can beexpressed in the compact form

€yðtÞ ¼ �p2ðsin½ycsgnyðtÞ � yðtÞ� þ ag cos½ycsgnyðtÞ � yðtÞ�Þ,(2)

where yðtÞo0ðor40Þ denotes the angle of rotation about O(or, respectively, about O0); yc ¼ arctanðb=hÞ is the angleshown in Fig. 1a; and p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimgR=Io

pis a characteristic

frequency parameter of the block; R is half the diagonal ofthe block. For a solid rectangular block the moment ofinertia about its pivot point is Io ¼ ð4=3ÞmR2 , andtherefore p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3g=4R

p.

In the free rocking regime the frequency of vibrationdepends strongly on the amplitude of rotation. Hence, theabove frequency parameter p is not the eigenfrequency ofthe system, but merely a measure of the dynamiccharacteristics of the block. Table 1 summarizes the mostimportant parameters of the problem and explains theirsymbols.When a rigid body is rocking back and forth about its

pivot points, it impacts on the ground and loses a part of itskinetic energy, even in a purely elastic impact. Its angularvelocity right after the impact (at time tþo ) is a fraction ofthat just prior to impact (at time t�o )

of rigid

oundation

Rocking of a rigid block on

elastic foundation

h2 h

R

2 bE

θc

θc

θ

c

lock on rigid foundation (b) Rocking of a 1-dof rigid oscillator on rigid

ARTICLE IN PRESS

Table 1

Geometric characteristics and dynamic parameters considered (Nomen-

clature)

Parameter Symbol

Angle of rotation yCritical angle of rotation ycAspect ratio b/h ¼ tan ycSize parameter R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ h2

pFrequency parameter p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimgR=I0

pPeriod parameter Tp ¼ 2p/pGround acceleration A ¼ agUplifting acceleration Auplift ¼ auplift g ¼ (b/h)gOverturning acceleration Aover ¼ aover gExcitation dominant period (frequency) TE(fE)

Coefficient of restitution r

Soil modulus of elasticity E

Effective unit weight of the block weff

M. Apostolou et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 642–654644

_y2ðtþo Þ ¼ r_y

2ðt�o Þ, (3)

where r is known in the literature as the coefficient ofrestitution (often with the symbol e2). Applying theprinciple of momentum preservation and neglecting energyloss during impact, we obtain for the coefficient ofrestitution the well-known expression [3]

r ¼ 1� 32sin2yc

� �2 ¼ e2. (4)The value of the coefficient of restitution calculated by

Eq. (4) is the maximum possible for a block with criticalangle yc set on rocking motion, under elastic impactconditions. In reality, some additional energy is lost,depending on the nature of the materials at the impactsurface. For a block with angle yc ¼ 0.4 rad (e.g., b ¼ 5mand h ¼ 12m), the maximum value of r with elastic impactis 0.60. While blocks with small aspect ratio tend toconserve most of their angular velocity, less slender blocksexhibit more geometrically ‘‘plastic’’ behavior duringimpact. Eventually for a value yc ¼ 0.95 rad (E541) thecoefficient of restitution vanishes even under purely elasticimpact conditions and no rocking can be maintained afterthe first impact!

The governing equation of motion (Eq. (2)) along withthe impact condition (Eq. (3)) can prescribe the rockingmotion of a rectangular block (or any rigid structure) on arigid base. A special case of practical interest is the rockingoscillator of Fig. 1b where the mass of the system isconcentrated at point C. For a negligible rotational inertiaof the mass, the moment of inertia about the rotation pointO yields Io ¼ mR2. Hence, the frequency parameter is now

p ¼ffiffiffiffiffiffiffiffiffig=R

p. (5)

Apparently, the system of Fig. 1b, a 1-dof rockingoscillator corresponds to the rigid rectangular block, ofFig. 1a if

R1dof ¼ 43Rblock. (6)

For all the analyses of a rigid rocking structure presented

next, the equation of motion was integrated numericallyusing the explicit algorithm and a time increment no lessthan 10�4 s.

2.1. Minimum acceleration levels for overturning under

dynamic conditions

Under dynamic base excitation, exceeding the ‘‘critical’’acceleration will simply initiate rocking. Whether the blockwill eventually overturn or not depends on its size andslenderness, as well as the kinematic characteristics andintensity of ground shaking. The major outcome of thenonlinear nature of rocking motion is that the requiredpeak ground acceleration for overturning is a sensitivefunction of both the block size and the excitationfrequency. This has been recognized by many researchersin the last 50 years [1–7]. Within the assumptions of smallrotations and slender structures, Kirkpatrick quantified theeffects of the two afore-mentioned parameters on theoverturning acceleration under sinusoidal excitation,through the following simplified formula [11]:

aover ¼b

h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ o

2E

p2

s. (7)

Recent studies by Makris et al. [5,6] unveiled thedetrimental role of long-period pulses inherent in near-fault ground motions. According to these studies a rockingblock subjected to one-cycle trigonometric pulse mayoverturn either with one impact (mode 1) or withoutimpact at all (mode 2), as explained in Fig. 2. A firstobservation is that as the frequency of excitation fEincreases, higher levels of acceleration are required toproduce overturning after one impact. Much largeraccelerations will lead to overturning without a singleimpact. The complicated nonlinear nature of the problem isalso revealed from this figure: When the excitationfrequency fE is larger than a critical value fc, the blockmay topple only without impact whereas for fEofc any oneof the two overturning modes could occur. In the lattercase, overturning with one impact is the critical one as it isensued always by lower levels of base acceleration.Counter-intuitively, the nonlinear nature of the problemreveals a ‘‘safe region’’ between the two modes, meaningthat while the block overturns for a certain level of shaking,it surprisingly remains standing when the amplitudeincreases; for even higher levels of shaking, overturningoccurs again without impact.While cycloidal pulses are reasonable idealizations of

near-fault ground motions, they cannot fully capture theeffect of a slight asymmetry inherent to near-fault pulses.Ricker wavelets (time-histories and response spectra arepresented in Fig. 3) have a distinct advantage in thisrespect. Thus, such wavelets are employed here to excitethe rectangular block of Fig. 2, and to bring it to rockingoscillations under elastic impact conditions (r ¼ 0:89). Asseen in the overturning spectra plotted in Fig. 4 more

ARTICLE IN PRESS

(g)

on

,αle

rati

ern

ing

Acc

ert

u

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2

Excitation Frequency, fE (hz)

Ov

over

Elastic impact ( r = 0.89 )

Inelasticimpact ( r = 0.8 )

static conditions:aover = 0.2 g

.5 0 0.5 1 1.5 2 2


Elastic impact (r = 0.89)

Inelastic impact ( r = 0.8 )

static conditions: a over = 0.2 g

.5

a b

static conditions: a over = 0.2 g static conditions: aover = 0.2 g

Fig. 2. Overturning spectra of a rectangular block with 2b ¼ 1m, and 2h ¼ 5m subjected to (a) a one-cycle-sinus and (b) one-cycle-cosinus excitation. Thecoefficient of restitution is 0.89 (elastic impact) and 0.8 (slightly inelastic impact). The line delineating the ‘‘overturn without impact’’ region from the safe

region does not depend on the coefficient of restitution and thus has no circle or triangle indication.

0

0.2

0.4

0.6

0.8

1

0

Sa (

g)

0

20

40

60

80

100

Sv (

cm

/sec)

-0.4

-0.2

0

0.2

0.4

a (

g)

-40

-20

0

20

40

v (

cm

/sec)

-10

-5

0

5

10

0 0.5 1 1.5 2.5 3

Time, t (sec)

d (

cm

)

0

4

8

12

16

0 0.4 0.8 1.2 1.6 2

Period, T (sec)

Sd (

cm

)

2

Fig. 3. Time-histories and response spectra of a Ricker-wavelet having PGA ¼ 0.3 g and predominant frequency f E ¼ 1:3Hz.

M. Apostolou et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 642–654 645

failure loops ‘‘appear’’ in this case. Also, there is nodistinction between overturning with one or withoutimpact as derived from the time-histories of Fig. 5. Forlong-duration motions (fEo0.6Hz) the overturning accel-

eration levels between the Ricker and the cosine pulse arealmost identical.An important question following the foregoing discus-

sion is whether high-rise buildings and tall bridge piers may

ARTICLE IN PRESS(g

)O

ve

rtu

rnin

g A

cce

lera

tio

n,

αov

er

0

0.5

1

1.5

22

0 0.5 1 1.5 2 2.5


Ricker wavelet ( r = 0.89 )


0 0.5 1 1.5 2 2.5


One-cycle cosinus ( r = 0.89 )


Fig. 4. Overturning spectra of a rectangular block with 2b ¼ 1m and 2h ¼ 5m subjected to a Ricker-wavelet excitation. The coefficient of restitution is0.89 (elastic impact). Note some similarity in the shape of the various curves with the one-cycle cosinus curves of Fig. 2.

0.27 g

loop 1:

overuturn with one

impact (to the left)

0.269 g

-0.4

-0.2

0

0.2

0.4

no overuturning

0.365 g

no overuturning

0.369 g

loop 2:

overuturn without

impact (to the right)

0.832 g

0 8 12 16

t (sec)

loop 3:

overuturn with two

impacts (to the

left)

0.801 g

no overuturning

0.8 g

-0.4

-0.2

0

0.2

0.4

0 8 12 16

t (sec)

θ (r

ad

)θ

(ra

d)

loop 2:

overuturn with an

impact (to the right)

0.862 g

0 8 12 16

t (sec)

no overuturning

4 0 8 12 16

t (sec)

4 4 4

Fig. 5. Time-histories of the rocking response for a rectangular block with 2b ¼ 1m and 2h ¼ 5m subjected to a Ricker-wavelet excitation off E ¼ 0:53Hz. The coefficient of restitution is 0.89 (elastic impact).


safely uplift from their foundation under very strongshaking. The beneficial effect of the block size to over-turning response has long been known [3,4]. This effect canbe illustrated with the overturning spectrum of a rigidrocking block under a specific ground motion in terms ofthe frequency parameter p. As portrayed in Fig. 6 for one-cycle sinusoidal pulses with periods of 0.40 and 0.80 s, thesize of the structure strongly affects the minimum accel-eration levels required for toppling. Thus, small structurestopple more easily than larger ones of equal slenderness.Moreover, for values of the frequency parameter p lowerthan unity the likelihood of even a very slender block(h=b ¼ 5) to overturn is negligible even under extremelystrong and long-period motions!

The interplay between slenderness and size of thestructure on the overturning potential is further clarified

by computing the response of a rectangular block with aconstant half-width b. In the plots of Fig. 7, the height ofthe block is gradually increasing so that both its slender-ness (h/b) and its frequency parameter (p) keep rising.Initially, a block of b ¼ 0:5m and h ¼ 1:0m is set onrocking under a long-duration one-cycle sinus pulse ofTE ¼ 0:8 s; to overturn, a peak ground acceleration of 0.7 gis needed. We next increase only its half-height h by 1mand the overturning acceleration drops rapidly down to0.35 g—an example of detrimental influence of slenderness.However, as the half-height of the structure is furtherincreased, the decrease of the critical acceleration di-minishes and the beneficial effect of the size parametergradually takes over. Eventually the overturning accelera-tion reaches a minimum (about 0.18 g) and thereafter tendsto slightly increase at higher values of h. The size effect has

ARTICLE IN PRESSM. Apostolou et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 642–654 647

now overshadowed the influence of the slenderness andbecome the prevailing parameter on the overturningresponse. Hence for a sufficiently large height, the moreslender a block the less vulnerable to overturning! Thus, wecan explain why large slender structures survive topplingeven under severe seismic shaking. In the experimentalwork of Huckelbridge and Clough [10] it was made clearthat for a practical building, transient uplifting responsewould in no way imply imminent toppling.

While the overturning hazard may not be the key issue inthe seismic response of slender structures (at least if stiffsoils support them), it is usually addressed in engineeringpractice for two different reasons: (a) toppling of non-structural elements are in many cases of special interest inseismic design procedures (for example appended equip-

TE=0.4sec

0

1

2

3

4

0 4 8 12 16 20

Half-height, h (m)

Overt

urn

ing A

ccele

ration,

� ove

r (g)

b = 1m

b = 0.5m

Fig. 7. Overturning spectra with respect to the half-height h for blocks with hal

and 0.8 s (right).

0

0.5

1

1.5

2

0 2 6

Frequency Parameter, p (rad/s)

Over

turn

ing A

ccele

ration, � o

ver (

g)

h / b = 3.33

h / b = 5

TE = 0.4 sec

TE = 0.8 sec

84

Fig. 6. Overturning spectra with respect to the frequency parameter p for

blocks with aspect ratio b/h 0.2 and 0.3, subjected to an one-cycle sinus-

type excitation.

ment, electrical transformers and so on [6]) and (b) fornearly a century the engineering community analyzedoverturning failures observed after an earthquake to obtainrough estimates of the true intensity of (unrecorded)seismic shaking. To demonstrate how difficult it is toobtain reliably such estimates, we study the toppling ofcemetery tombstones in the Athens earthquake of 1999(Fig. 8). We had hoped that back analysis of theoverturning would reveal the intensity of the unknownground motion at this location, 2 km away from thecausative fault [3,12,13].Two different earthquake records are used as the basis of

our analyses:

�

f-wi

The accelerogram of Sepolia station, recorded in theAthens 1999 earthquake, as a typical stiff-soil record ofa moderate (Ms 5.9) magnitude event, at a distance ofabout 9 km from the ruptured normal fault zone. Therecord has a PGA ¼ 0.36 g and dominant periods in therange of 0.15–0.25 s

�
The accelerogram of Düzce in the Kocaeli 1999 earth-
quake, which is typical of a large (Ms 7.4) magnitudeevent whose strike-slip rupture is directed towards therecording soil site, and stops a few kilometers before it.The strong forward-directivity effect has given theDüzce record a characteristic long duration accelerationpulse. Its PGA ¼ 0.37 g is similar to the one of the SPLBrecord, but its significant periods range from about 0.40to at least 1.50 s.

Minimum acceleration levels required to topple the tombare computed by numerical integration of Eq. (2) afterscaling up or down each record. Throughout the analysiselastic impact conditions are considered leading to acoefficient r ¼ 0:928. In the case of the Sepolia-typeexcitation the block can sustain rocking motion withoutoverturning until the accelerogram is increased so that it

TE=0.8sec

0

0.2

0.4

0.6

0.8

1

0 4 8 12 16 20

Half-height, h (m)

b = 1 m

b = 0.5 m

dth 0.5 and 1m subjected to a one-cycle sinus pulse of period 0.4 s (left)

ARTICLE IN PRESS

2h= 1.27m, 2b= 0.20m, R = 0.66m (R = (H2+B2)1/2)

rupture zone

10 km

ATHENS

Ano Liosia Adames

Fig. 8. A typical free-standing tombstone that toppled after the Athens earthquake.

Sepolia- 0.85 g

-1

0

1

0 2 4 6 8 1

-0.4

0

0.4

0 2 4 6 8 1

0.85 g

0.84 g

-0.4

0

0.4

0 2 4 6 8

0.27 g

0.26 g

Duzce- 0.27 g

-1

0

1

0 4 6 8 10

� (g

)�

(rad )

10 0

0

t (sec) t (sec)

2

Fig. 9. The down-scaled and up-scaled accelerograms of Düzce/Kocaeli and Sepolia/Athens with the corresponding time histories of induced rotation just

leading to overturning of the tombstone of Fig. 8.


acquires a PGA of 0.85 g (about 2.5 times the recordedvalue). By contrast, the Düzce excitation must be scaleddown to a PGA of 0.27 g for overturning to occur (about0.73 times the recorded value). Ground acceleration androtation time histories for marginal overturning for the tworecords are plotted in Fig. 9. Evidently, the long-durationpulse in the Duzce record tends to reduce the overturningacceleration towards its static value (0.20/1.27 gffi0.16 g).

The rocking response of the tomb under the Sepolia-typemotion is revisited next. Now the time increment of thisaccelerogram is artificially increased by 10% and by 20%.This leads to an increase of the predominant period ofmotion from 0.26 to 0.29 s and to 0.31 s, respectively. Theslight modification of the excitation period has a dramaticeffect on its rocking response: the overturning accelerationis reduced from 0.85 g down to 0.61 g and to 0.58 g for thetwo modified records! A 2-s detail of each modified time-history along with the original time-history (each onescaled to the critical acceleration) is plotted in Fig. 10.

The two distinct modes of overturning for trigonometricpulses as discussed by Makris et al. [6] are now extractedfor the tombstone and plotted in the overturning spectrumof Fig. 11. For relatively low values of the excitation periodTE, a rocking block such as the tomb of Fig. 8 will notoverturn even for peak ground acceleration 4 or 5 times the

pseudo-static critical acceleration (0.16 g). For values of TEexceeding about 0.3 s the minimum PGA to overturn theblock is rapidly decreasing. Eventually for sufficiently largeperiods (TE40.70 s) the minimum overturning accelerationapproaches the pseudo-static value. As seen in Fig. 11 thereal records and the sinusoidal pulses give fairly similarresults for the overturning response.Concluding, the peak ground acceleration that toppled

the cemetery block could vary from about 0.20 to 0.80 gwithin a period range 0.25–0.5 s. The former period iscloser to the records of the Athens 1999 earthquake, whichhowever were far-field. It is evident that the practice ofestimating ground acceleration from observations oftoppled and untoppled slender blocks, which has for acentury been utilized to assign levels of design accelerationin many parts of the world, is meaningless in view of thestrong frequency- and detail-dependence and the trulychaotic nature of rocking behavior.

2.2. Estimation of uplift

Base uplifting may have a strong influence on theperformance of a structural system. Thus, the likelihood ofbase uplift and the estimation of the maximum rotationbecome of great interest. An interesting way of portraying

ARTICLE IN PRESS

Sepolia

-1

0

1

4 4.5 5 5.5 6�

(g)

� (g

)�

(g)

-0.4

0

0.4

0 2 4 6 8 10

0 2 4 6 8 10

� (r

ad

)�

(ra

d)

� (r

ad

)

Sepolia modified

(Δt* = Δt x 1.1)

Sepolia modified

(Δt* = Δt x 1.2)

-1

0

1

4.4 4.9 5.4 5.9 6.4

-1

0

1

4.8 5.3 5.8 6.3 6.8

t (sec)

0.85 g

0.84 g

-0.4

0

0.4

0.61 g

0.60 g

-0.4

0

0.4

0 2 4 6 8 10

t (sec)

0.58 g

0.57 g

0.58 g

0.61 g

0.85 g

Fig. 10. Three scaled accelerograms based on the Sepolia (Athens) original record with the corresponding time histories of induced rotation just leading to

overturning of the tombstone of Fig. 8. The time scale of the second and third motion has been slightly increased from that of the original record, to see the

sensitivity of the toppling to the frequency content of the motion.


the response of a rigid body under rocking vibration is inthe form of the Rotation Response Spectrum or simply‘‘Rocking Spectrum’’, as introduced by Makris andKonstantinidis [14]. In this, the amplitude of rotation isplotted as a function of the period parameter Tp ¼ 2p=p fora certain value of the slenderness ratio h/b or conversely ofthe critical angle yc. For a rectangular block the periodparameter is Tp ¼ 4p

ffiffiffiffiffiffiffiffiffiffiffiR=3g

pffi 2:3

ffiffiffiffiRp

whereas for a rigid1-dof oscillator is Tp ¼ 2p

ffiffiffiffiffiffiffiffiffiR=g

pffi 2

ffiffiffiffiRp

. Note that thelatter period is equal to the eigenperiod of a linearizedpendulum with length R. Extending the concept, one canevaluate the rocking spectrum where the response axis isnow the dynamic ‘‘eccentricity’’ of the vertical load on thefoundation due to uplift, normalized to the half width b, ofthe base

e

b¼ 1� sinðyc � yÞ

sin yc(8a)

which, for slender blocks simplifies asymptotically to

e

bffi y

yc. (8b)

The rocking spectra for a value of critical angleyc ¼ 0.2 rad (h=bffi 5) and for elastic impact (coefficientof restitution r ¼ 0:89) are computed with the records ofSepolia and Düzce (Fig. 12). The far greater destructive-ness of the Düzce record is evident, attributed to both itshigher dominant periods and its long-duration acceleration

pulse, which is associated with large incremental velocity(in excess of 100 cm/s)—the end result of forward rupturedirectivity. Despite its equally large PGA, the Sepoliarecord is much easier for a slender block to safely undergo.The Düzce record can topple all rectangular blocks withRo1:2m while under the Sepolia excitation even smallerblocks (with R up to 0.2m) would exhibit rocking motionwithout toppling. Moreover, the amplitude of rotation forall non-toppled blocks is larger under a Düzce-type groundmotion.The detrimental influence of a long-duration excitation

pulse and the beneficial effect of the block size can also beunveiled with the use of a Ricker wavelet as excitation [15].A comprehensive parameter study is performed to this end.The response is measured in terms of the angle of rotationyðpycÞ; the only problem parameters on which it dependsare: the critical angle yc ¼ arctanðb=hÞ, or equivalently theuplifting acceleration Ac ¼ acg with ac ¼ b=h; the char-acteristic period Tp; and the period and peak accelerationof the Ricker excitation, TE and A ¼ ag, respectively. Aunique relationship has been found between the dimension-less parameter

Py ¼yyc

� �Tp

TE

� �2¼ YO2 (9)

and the ratio ac=a ¼ A�1 which could be interpreted as theinstantaneous factor of safety against uplifting. Y ¼ y=yc

ARTICLE IN PRESS

r = 0.89

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6

�(ra

d)

Sepoliarecord

Duzce record

Ricker wavelet (TE= 1.3 sec, PGA =0.28g)

pseudostatic overturning

Period Parameter, TP(sec) R∝

θc= 0.2 rad,

Fig. 12. Rocking spectra for blocks with h=b ¼ 5 (critical angleyc ¼ 0.2 rad) subjected to the unscaled records of: (a) Düzce, in theKocaeli Earthquake and (b) Sepolia, in the Athens Earthquake.

Earthquake Records

Sepolia (Athens,1999)

Kede-x (Athens,1999)

Kede-y (Athens,1999)

Shinkobe (Kobe, 1995)

Fukiai (Kobe,1995)

Duzce-ew (Kocaeli,1999)

Duzce-ns (Kocaeli,1999)

Sylmar (Northridge,1994)

Pacoima (S. Fernando, 1971)

Monastiraki (Athens,1999)

Adm. Bldg (Kalamata, 1986)

Pyrgos (Pyrgos, 1993)

Temblor (Parkfield, 1966)

Lefkada (Lefkada,2003)

0 0.4 0.8 1.2 1.6

Excitation Dominant Period, TE(sec)

0

0.2

0.4

0.6

0.8

1

Ove

rtu

rnin

g A

cce

lera

tio

n,

a(g

)

one-cycle sinus pulse:overturn with impact

one-cycle sinus pulse:overturn without impact

pseudo-static overturning

Fig. 11. Overturning spectra of the cemetery tomb (h=b ¼ 6:35) for one-cycle sinus-type and for numerous ground motions used as earthquake excitation.

= ( � / � c) ( T P / T E)2

p < 1.2 rad / sec (R > 5.1 m)

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8

Factor of Safety, �c / �

No

rma

lize

d

An

gle

of

Ro

tatio

n,

�

�

Fig. 13. Normalized angle of rotation for rigid blocks subjected to Ricker

pulse-type excitation with respect to the factor of safety for uplift: ac/a.Note that yc ¼ arctan ac. (The coefficient of restitution for all analyses isr ¼ 0:80.)


and O ¼ ðTp=TEÞ ¼ ðoE=pÞ are, respectively, the dimen-sionless uplifting rotation and frequency. Fig. 13 plots thisunique relationship Py ¼ Pyðac=aÞ.

The rocking amplitude developed under a Ricker-typeexcitation is compared next with two typical near-fault

earthquake accelerograms: (a) the aforementioned Düzcerecord and (b) the Pacoima dam record from the SanFernando earthquake (1971). The rocking spectra areplotted in Fig. 14 for four values of the frequencyparameter p. Evidently the use of low-frequency Ricker

ARTICLE IN PRESS

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1p = 1 rad/sec

Ricker wavelet (TE = 1.2 sec)

Duzce (1999)

p = 0.67 rad /sec

p= 3 rad/sec

Ricker wavelet (TE = 0.45sec)

Pacoima Dam (1971)

0

0.04

0.08

0.12

0.16

0.2

p = 1 rad/sec

�/� c

�/� c

�c /� �c /�

Fig. 14. Comparison of rocking spectra for blocks subjected to Ricker wavelets and two near-fault earthquake records. The coefficient of restitution is

r ¼ 0:80.


wavelets in representing near-fault ground motions isjustified.

For design purposes the peak rotational angle of astructure subjected to near-fault ground motion can beestimated conservatively with the following expressions:forac=a40:3

Py ffi 20 1�ac

a

� �(10a)

from which

yycffi 20 1� ac

a

� � TETp

� �2(10b)

for 0:15oac=ao0:3Py ffi 14 (10c)from which

yycffi 14 TE

TP

� �2. (10d)

(Note that these two equations apply only to sufficientlylarge structures: po1:2 rad=s). It is obvious that for a givenslenderness ratio, the response ratio y/yc increases with thesquare of the dominant excitation period but decreases inproportion to the size R of the block.

3. Uplifting and overturning on elastic soil

Consider now the system of Fig. 1(c): a rigid blocksupported on an elastic homogeneous half space ofYoung’s modulus E, Poisson’s ratio n, and damping ratiox, subjected to horizontal base excitation AðtÞ ¼ aðtÞg. Due

to soil compliance, the block can now undergo rotationalmotion without uplift (so long as rotational amplitudesremain below the critical angle). For large amplitudes, therocking response alternates between the modes of full-contact and uplift. The critical angle for uplift is given bythe following expression:

yuplift ¼Muplift

KR, (11)

where the uplifting moment Muplift is a fraction of themoment capacity Mult ( ¼ Nb) of the foundation–soilsystem

Muplift ¼ ZMult. (12)

Note that for a rigid beam on Winkler foundation theuplifting moment is Muplift ¼ Nb=3. Therefore, it is Z ¼ 13from Eq. (12). However, with an elastic continuum forrepresenting the soil and plane strain conditions of loadingthis coefficient increases up to 0.5 [16]. In this case andrecalling [17,18] the expression for rocking stiffness of astrip foundation on a homogeneous half-space ðKR ¼pGb2=2ð1� nÞÞ, the rotation at incipient uplifting becomes

yuplift ¼Nb

2KR¼ 4weffhð1� nÞ

pG¼ 8weffhð1� n

2ÞpE

. (13)

Due to soil flexibility the following parameters should betaken into account:

�

the soil properties E, n, r,

�
the effective unit weight of the block: weff ¼ N=4bh,
where N is the block’s weight (per unit length),

�

the presence of bedrock at a shallow depth.

ARTICLE IN PRESSM. Apostolou et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 642–654652

In the present study the dynamic analysis of the rockingresponse is implemented with a finite element discretization

using Abaqus [19]. The structure and the underlying soil arerepresented with plane-strain elements. An advancedcontact algorithm has been adopted to incorporatepotential slipping or uplifting of the foundation, consider-ing purely elastic impact. Wherever the supporting soil istreated as a homogeneous halfspace, two-dimensionalinfinite elements are applied to model the boundaryconditions.

The compliance of the supporting soil introducesadditional modes of deformation. The structure can nowrotate without necessarily uplifting (the linear componentof the motion). In addition, uplifting (the nonlinearcomponent of rocking) may also take place. What is more,in soft soil the impact during rocking is more ‘‘absorbing’’as radiation and hysteretic damping are generated in thesoil. Thus attenuation of the motion is faster. Fig. 15illustrates the two components of rocking for a slenderh=b ¼ 5 block supported on three different elastic soils,having E ¼ 100MPa (very stiff), 20MPa (moderately stiff),and 5MPa (very soft). Two values of the ac=a ratio, 0.20and 0.50 are considered; since ac ¼ 0:20 g the impliedRicker peak accelerations a are, respectively, 1.0 and 0.4 g.The gray curves stand for the time history of total rotation,defined as Dtotal/2b, while the black curves are for therotation component due to uplifting, defined as Duplift/2b.Dtotal ¼ Duplift+Delastic is the vertical distance between the

total rotation

Es = 100 MPa,αc / α = 0.2

Es = 20 MPa,αc / α = 0.2

Es = 5 MPa,αc / α = 0.2

-0.06

0.01

0.08

0 2 4 6 8 10

(rad)

-0.06

0.01

0.08

0 2 4 6 8 10

-0.06

0.01

0.08

0 2 4 6 8 10

t (sec)

θ(r

ad)

θ(r

ad)

θ

Fig. 15. Uplifting behavior of a block with b ¼ 1m and h ¼ 5m induced byblack curves the uplifting part of the response. Plane-strain conditions prevail

two corners of the foundation (O and O0). It is seen that inthe very stiff, nearly undeformable soil (E ¼ 100MPa),rotation is due almost exclusively to uplifting: the twocurves almost coincide. On the other end of the spectrum,on soft soil (E ¼ 5MPa) uplifting contributes only part ofthe total rotation of the block, and essentially only duringthe strong excitation pulse. The ensuing free oscillationsare simply the rotational vibrations of the block on ahomogeneous elastic layer, at periods (ffi2 s) well above thecut-off period of the soil stratum: hence neither rotationnor hysteretic damping are present, and the free vibrationscontinue unattenuated [18]. Note also in Fig. 15 that forac=a ¼ 0:2 the peak uplifting angle (nonlinear component)is 0.08 rad in case of E ¼ 100MPa, while for E ¼ 20 and5MPa the uplifting response decreases to 0.06 and 0.04 rad,respectively. On the contrary, the linear component of themotion (due to soil compliance) becomes larger as soilcompliance increases, compensating to a larger extent forthe reduced uplifting.To investigate the effect of soil compliance on rocking

response, the peak angle of rotation is plotted in Fig. 16 fora range of E-values (5–1000MPa) and three different blocksizes (R ¼ 2.8, 3.5, and 5.1m). For very high values of themodulus of elasticity, the amplitudes of rotation convergeto the limiting case of the amplitude on rigid base(yrigidffi0.03 rad, ffi0.05 rad, and ffi0.08 rad for each ofthe aforementioned three R-values, respectively). Upondecreasing E, the effect of soil deformability leads under-

uplifting rotation

Es = 100 MPa,αc / α = 0.5

Es = 20 MPa,αc / α = 0.5

Es = 5 MPa,αc / α = 0.5

-0.06

0.01

0.08

0 2 4 6 8 10

-0.06

0.01

0.08

0 2 4 6 8 10

-0.06

0.01

0.08

0 2 4 6 8 10

t (sec)

a Ricker pulse of frequency 1Hz. Gray curves show the overall response;

.

ARTICLE IN PRESS

0

0.05

0.1

0.15

0.2

1 10 100 1000 10000

E (MPa)

� (r

ad)

5.1 Duzce

5.1 Ricker

3.5 Ricker

2.5 Ricker

R (m) Excitation

Fig. 16. Amplitudes of rotation for three rectangular block-type

structures with constant aspect ratio (h=b ¼ 5 or yc ¼ 0.2 rad). Theexcitation is a Ricker wavelet (PGA ¼ 0.3 g, TE ¼ 1.3 s) and the Duzcerecord (PGA ¼ 0.37 g).

E = 5 MPa, weff = �w

E = 5 MPa, weff = 0.25 �w

E = 20 MPa, weff = 0.25 �w

E = 20 MPa, weff = �w

0.05

0.04

0.03

0.02

0.01

0

Factor of Safety, �c /�

(ra

d)

0 0.4 0.8 11.2

Fig. 17. Influence of the soil stiffness and the unit block weight on the

rocking spectra for a block with b ¼ 1m and h ¼ 5m. The excitation is aRicker pulse of characteristic frequency 1Hz.


standingly to greater values of the maximum angle, whichcan go up more than 2 times the rigid base value. For evensmaller values of E, less than about 10–15MPa, theincreased softening of the soil is beneficial, leading tosmaller y-values! In all these cases (E45MPa), thestructure oscillates in rocking without overturning, despitethe pseudo-statically predicted toppling. However, for verysmall values of E, less than about 2–5MPa, the trendchanges again and y increases with increasing E. Failure isnow possible since the large deformability of the soil leadsto significant rotation that triggers deleterious P–d effects.A quite interesting rocking behavior is revealed whensmaller structures of equal slenderness are considered asalso shown in Fig. 16. In this way two smaller blocksare considered with base widths 1.4m and 1.0m, andheights 7.0m and 5.0m, respectively; therefore the criticalangle of rotation remains constant. Only the dimensionsof each block, described through the half–diagonal R ¼ðb2 þ h2Þ1=2, change (from 2.5 to 5.1m). The followingtrends are worthy of note in this figure: (a) the overall sizeof the block affects strongly its rotation; the smallest of thethree blocks undergoes the largest rotation for all values ofE and it in fact overturns for E ffi 15MPa; and (b) thevariation of ymax with respect to the soil modulus is notmonotonic; it exhibits a peak at Effi15–30MPa where therocking period TR ¼ 2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiJb=KR

pis tuned to the excitation

period (1.3 s), and tends again to become very large as Etends to zero. A secondary peak is also noticed atEffi150–200MPa. Nevertheless, the maximum rockingangle in case of soft soil would in most cases be not morethan 1.5–2 times the corresponding ‘‘rigid-base’’ value.

Finally, Fig. 17 shows the dependence of the peak angleof rotation, y on the ratio ac=a , for combinations of twovalues of soil Young’s modulus, E ¼ 20 and 5MPa, andtwo values of the effective unit weight of the block,weff ¼ gw and gw=4, where gw ¼ the unit weight of water.The latter value of weff is typical for the unit weight of abuilding, while the former value represents a heavier

structure, such as a water tank. The following trends areworthy of note:

�

The values of y do not vanish for ac=a ¼ 1 due to‘‘elastic’’ rotation of the foundation.

�
The softer soil leads to small overall rotation, since in
this particular case the natural rotational frequency ofthe block on elastic soil is much smaller than thedominant frequencies of the Ricker excitation. (With thestiffer soil, the corresponding frequencies are closer tothe excitation frequencies.)

�
Increasing the unit weight weff of the structure also
reduces the peak rotation, since the block’s naturalrotational frequency further decreases. In fact, in thiscase y is roughly proportional to E and inverselyproportional to weff .

This figure was only meant to be an example of theinterplay among the various soil and structure parameters.The results should not be unduly generalized.

4. Conclusions

The paper investigates the rocking and overturningresponse of slender rigid structures allowed to uplift. Thefollowing concluding remarks can be drawn:

(1)

Under static conditions only the slenderness of astructure is the decisive parameter for toppling. Indynamic terms however, the size and the slenderness ofthe structure, as well as the nature of base shakingaffect the overturning potential. For large structures,size effects are prevailing in such a way that a largeslender block can safely undergo a certain excitationwhile a less slender but smaller block overturns. Thisexplains why large structures survive toppling evenunder very strong seismic shaking, far greater than thepseudostatically required for overturning. Moreover, itis not only the dominant frequency but also the natureand especially the asymmetry of a base excitation thathave a strong effect on the overturning potential.

ARTICLE IN PRESSM. Apostolou et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 642–654654

Therefore, the practice of estimating ground shakinglevels by analyzing observations of toppled and un-toppled slender blocks after an earthquake is mean-ingless.

(2)
For relatively large structures on rigid foundation,seismic uplifting response can be reliably predicted. Insuch a case the rocking amplitude can be normalizedand estimated through suitable dimensionless charts.An example is given in this study for pulse-typemotions idealized as Ricker wavelets. The seismicresponse of smaller structures however, is rather‘‘chaotic’’ and can hardly be predicted with confidenceif the details of the base motion are not known withaccuracy.
(3)
For an elastic supporting soil, there is no definitiverelation between rocking response and the ac=a ratio.This is especially true with soft soils when the linearcomponent of the response (full-contact regime due tosoil compliance) becomes important. This is one of theconsequences of the strong geometric nonlinearity ofthe problem. The additional complication arising fromnonlinear material behavior of soils and the mobiliza-tion of bearing capacity mechanisms is beyond thescope of this paper, but it has been introduced in Refs.[20,21].
Acknowledgements

Most of the present work was part of the researchproject ‘‘Thales’’ funded by the National TechnicalUniversity of Athens (2003–2005).

References

[1] Milne J. Experiments in observational seismology. Trans Seismol Soc

Jpn 1881;3:12–64.

[2] Perry J. Note on the rocking of a column. Trans Seismol Soc Jpn

1881;3:103–6.

[3] Ishiyama Y. Review and discussion on overturning of bodies by

earthquake motions. Building Research Institute, Research paper 85,

Ministry of Construction, Japan, 1980.

[4] Housner GW. The behavior of inverted pendulum structures during

earthquakes. Bull Seismol Soc Am 1963;53(2):404–17.

[5] Makris N, Roussos Y. Rocking response of rigid blocks under near-

source ground motions. Géotechnique 2000;50(3):243–62.

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

cycloidal pulses. J Eng Mech ASCE 2001;127(5):473–83.

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

earthquake excitations. Earthquake Eng Struct Dyn 1982;10:635–50.

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

uplift. Earthquake Eng Struct Dyn 1983;11:57–76.

[9] Koh AS, Spanos P, Roesset JM. Harmonic rocking of rigid block on

flexible foundation. J Eng Mech ASCE 1986;112(11):1165–80.

[10] Huckelbridge AA, Clough RW. Seismic response of uplifting building

frame. J Struct Eng ASCE 1978;104(8):1211–29.

[11] Kirkpatrick P. Seismic measurements by the overthrow of columns.

Bull. Seismol. Soc. Am. 1927;17(2):95–109.

[12] Shi B, Anooshehpoor A, Zeng Y, Brune J. Rocking and overturning

of precariously balanced rocks by earthquake. Bull Seismol Soc Am

1996;86(5):1364–71.

[13] Apostolou M, Anastasopoulos J, Gazetas G. Back-analysis of sliding

and toppled structures for the estimation of ground motion during

the Athens Earthquake (1999). In: Proceedings of the 2nd Greek

conference on earthquake engineering and engineering seismology,

Technical Chamber of Greece, Thessaloniki, Greece, 2001 (in Greek).

[14] Makris N, Konstantinidis D. The rocking spectrum and the

limitation of design guidelines. In: Proceedings of the 15th ASCE

engineering mechanics conference, Columbia University, New York,

NY, 2002.

[15] Gerolymos N, Apostolou M, Gazetas G. Neural network analysis of

overturning response under near-fault type excitation. Earthquake

Eng Eng Vibration 2005;4(2):213–28.

[16] Cremer C, Pecker A, Davenne L. Modeling of nonlinear dynamic

behaviour of a shallow strip foundation with macro-element. J

Earthquake Eng 2002;6(2):175–211.

[17] Gazetas G. Simple physical methods for foundation impedances. In:

Benerjee PK, Butterfield R, editors. Dynamics of foundations and

buried structures. Elsevier Applied Science; 1987. p. 44–90.

[18] Gazetas G. Analysis of machine foundation vibrations: state of the

art. Soil Dyn Earthquake Eng 1983;2(1):2–42.

[19] Abaqus 6.1. Standard user’s manual. Rhode Island: Hibbit, Karlsson

and Sorensen Inc.; 2001.

[20] Gazetas G, Apostolou M. Nonlinear soil–structure interaction:

foundation uplift and soil yielding. In: Todorovska M, Celebi M,

editors, Proceedings of the 3rd US–Japan workshop on soil–structure

interaction, USGS, Menlo Park, CA, 2004.

[21] Gazetas G, Apostolou M, Anastasopoulos J. Seismic bearing

capacity and uplifting of foundations: Adapazari 1999. In: Prakash

S, editor, Proceedings of 5th international conference on case

histories in geotechnical engineering, 2004 (in cd-rom).

Seismic response of slender rigid structures with foundation upliftingIntroductionUplifting and overturning on a rigid baseMinimum acceleration levels for overturning under dynamic conditionsEstimation of uplift

Uplifting and overturning on elastic soilConclusionsAcknowledgementsReferences

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