Response analysis of rigid structures rocking on viscoelastic foundation

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2008; 37:1039–1063Published online 17 March 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.800

Response analysis of rigid structures rockingon viscoelastic foundation

Alessandro Palmeri1,§ and Nicos Makris2,∗,†,‡

1School of Engineering, Design & Technology, University of Bradford, Bradford, U.K.2Department of Civil Engineering, University of Patras, Patras, Greece

SUMMARY

In this paper the rocking response of slender/rigid structures stepping on a viscoelastic foundation isrevisited. The study examines in depth the motion of the system with a non-linear analysis that complementsthe linear analysis presented in the past by other investigators. The non-linear formulation combines thefully non-linear equations of motion together with the impulse-momentum equations during impacts.The study shows that the response of the rocking block depends on the size, shape and slenderness ofthe block, the stiffness and damping of the foundation and the energy loss during impact. The effect of thestiffness and damping of the foundation system along with the influence of the coefficient of restitutionduring impact is presented in rocking spectra in which the peak values of the response are compared withthose of the rigid block rocking on a monolithic base. Various trends of the response are identified. Forinstance, less slender and smaller blocks have a tendency to separate easier, whereas the smaller the angleof slenderness, the less sensitive the response to the flexibility, damping and coefficient of restitution ofthe foundation. Copyright q 2008 John Wiley & Sons, Ltd.

Received 9 December 2005; Revised 9 January 2008; Accepted 9 January 2008

KEY WORDS: uplifting; rocking; overturning; viscoelastic foundation; seismic stability; stepping bridges

INTRODUCTION

Under sufficiently strong ground shaking, slender/rigid structures may enter into rocking motion.The possibility of tall structures to separate from their foundation limits the earthquake-generatedforces and moments and in most cases leads to more economical designs assuming that stability

∗Correspondence to: Nicos Makris, Department of Civil Engineering, University of Patras, Patras 26500, Greece.†E-mail: [email protected]‡Professor of Structures and Applied Mechanics.§Lecturer in Civil Engineering.

Contract/grant sponsor: European Social Fund (ESF)Contract/grant sponsor: Operational Program for Educational and Vocational Training II (EPEAEK II)Contract/grant sponsor: Program PYTHAGORAS II

Copyright q 2008 John Wiley & Sons, Ltd.

1040 A. PALMERI AND N. MAKRIS

prevails. For instance, the piers of the South Rangitikei Viaduct in New Zealand [1] (Figure 1,top left), the Rio Vista Bridge in Sacramento, California [2] (Figure 1, top right) and the NorthApproach Viaduct of the Lions Gate Bridge in Vancouver, British Columbia [3] (Figure 1, bottom)are allowed to rock on their foundation.

The analysis of the rocking response of a rigid block on a rigid foundation is well known in theliterature. Early studies on the rocking response of a rigid block supported on a base undergoinghorizontal motion were presented by Housner [4]. Following Housner’s seminal paper, a largenumber of studies have been presented to address the complex dynamics of the free-standing block([5–14], among others). During the same period, a handful of studies examined the rocking responseof structures stepping on a non-rigid foundation. Psycharis and Jennings [15] presented a linearanalysis of the rocking response of a rigid block stepping on a viscoelastic foundation; Yim andChopra [16] presented a linear analysis of a flexible structure supported on a base allowed to uplift,whereas Gazetas [17] has documented the permanent tilting of buildings due to foundation failure.More recently, Apostolou et al. [18] investigated with finite element (FE) analyses the conditions

Figure 1. South Rangitikei Viaduct in New Zealand (top left), Rio Vista Bridge in Sacramento, California(top right) and North Approach Viaduct of the Lions Gate Bridge in Vancouver, British Columbia (bottom).

Copyright q 2008 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2008; 37:1039–1063DOI: 10.1002/eqe

RIGID STRUCTURES ROCKING ON VISCOELASTIC FOUNDATION 1041

under which a rigid block founded on an elastically deformable continuum may experience largeangle of rotations. An and Kiyomiya [19] and Chen et al. [20], in two independent studies, exploredanalytically and experimentally the feasibility of utilizing a rocking mechanism for earthquakeprotection of slender viaduct piers, showing in both cases a significant positive effect in mitigatingthe seismic response. Finally, Cheng [21] reported the results of free vibration tests carried out onstepping precast bridge columns, in which energy is dissipated through impacts and slips on thefoundation plate.

In this paper we present a comprehensive study on the rocking response of a rigid blockwhich steps on viscoelastic supports. It is assumed that the foundation system cannot take tension;therefore, separation occurs at a height close to the static deflection, which is the position where thedamping force and the spring force sum up to zero. This paper examines the motion of the systemwith a fully non-linear analysis that complements the linear analysis presented by Psycharis andJennings [15]. The developed methodology is then applied to the response analysis of a bridge pierthat is allowed to step; and the study reaches a conclusion on the effect that non-rigid foundationshave on the rocking response of tall/slender structures.

STATEMENT OF THE PROBLEM AND MODEL PARAMETERS

We consider the rigid block shown in Figure 2 with size R0 and slenderness �, which can pivotabout points L and R when it is set to rocking. Once in contact with the left/right spring supports,points L and R are allowed to move only vertically. When the motion is weak, the body is incontinuous contact with both supports at pivot points L and R (stages B, C and D in Figure 2,bottom). When the vertical uplift of the heel of the body exceeds the static deflection of thesupporting springs, �s=Mg/(2K ), the separation occurs and the body is supported only at its toe.Accordingly, there are three regimes of motion: (i) full contact at both points L and R (stagesB, C and D) (ii) contact only at point L (stage A—tilting to the left); and (iii) contact only atpoint R (stage E—tilting to the right). There is a fourth regime of motion, when the rigid bodyseparates from both supports, which is not the subject of this study. When this happens our analysisterminates.

The element that distinguishes the response of rocking structures from other vibrating systemsis the energy loss at the impact points. In order for a slender block to sustain rocking motion ona rigid foundation (no jumps), a finite amount of energy needs to be lost at every impact. Theminimum amount of energy that needs to be lost is dictated from the conservation of angularmomentum, which yields that the angular velocity after the impact is a fraction of the angularvelocity before the impact:

�a=e �b, 0<e�emax<1 (1)

where e is a coefficient of restitution that expresses the energy loss during impact in terms ofthe angular velocity of the entire block. The value of max emax is a function of the physicalcharacteristics of the rocking block:

emax= I0+MR20 cos(2�)

I0+MR20

= 1+�cos(2�)

1+�(2)



A B C D E

Figure 2. Schematic of a rigid block rocking on viscoelastic foundation.

where �=MR20/I0 is the shape factor of the rigid block, M and I0 being the mass and the moment

of inertia with respect to the centre of mass, respectively. For rectangular blocks, Equation (2)reduces to emax=1−3sin2(�)/2 [4].

In most practical cases, where pure rocking happens (no jumps), the value of the coefficient ofrestitution, e, is close to emax; therefore, the rocking response of a rigid block on a rigid foundationis fully described by two parameters only—its slenderness angle, �, and the frequency parameter,p, which results from its size, R0, its shape, �, and the intensity of the surrounding gravitationalfield, g:

p=√

�

�+1

g

R0(3)

for rectangular blocks, �=3 and p=√3g/(4R0). Accordingly, for a rigid block rocking on a rigid

foundation the response � depends on the two aforementioned parameters and the excitation:

�= f (�, p,excitation) (4)



In the case of a rigid block stepping on viscoelastic supports with vertical stiffness K =�2vM/2

and frequency-independent damping, C=�vM�v, the response depends also on the ratio of thevelocity of the imminent pivot point before the impact of the support to the velocity of the pivot pointafter the impact at the support. This ratio defines an alternative coefficient of restitution, �= vaL/vbL=vaR/vbR, which has been introduced by Psycharis and Jennings [15]. Given the above-mentionedparameters, the response of a rigid block rocking on a two-support viscoelastic foundation isexpressed by

�= f (�,�, p,�,�v,�v,excitation) (5)

The coefficient of restitution � introduced by Psycharis and Jennings [15] can be related to thecoefficient of restitution e introduced by Housner [4] via the equation:

e= 1+��+(1−�)�cos(2�)

1+�(6)

The full derivation of Equation (6) is offered in the report by Palmeri and Makris [22].Figure 3 shows the relationship between e and � for different values of the slenderness � of

a rectangular block supported at the edges of the base (�=3). The selection of the value of �is arbitrary. The lower limit, �=0, corresponds to rocking on a rigid base, whereas the upperlimit, �=1, corresponds to purely elastic transition, where no energy is dissipated when the blocks

Figure 3. Relation of the coefficient of restitution e introduced by Housner [4] with the coefficient ofrestitution � introduced by Psycharis and Jennings [15].



alternate pivot points. Interestingly, the coefficient of restitution e in less slender blocks (larger �)is more sensitive to the value of the coefficient of restitution �.

Equation (5) indicates that there are six parameters of the rocking structure–foundation systemthat influence its response other than the characteristics of the ground excitation. Before proceedingwith a dimensionless analysis of this problem, it is of interest to examine the range of theseparameters covered in civil engineering applications. All the results presented in this study are forrectangular blocks with supports placed at the end corners of the block. In this case, �=3 and thesix parameters of the rocking structure in Equation (5) reduce to five.

For a bridge tower, the frequency parameter can be as low as p=0.5rad/s or even smaller,whereas for a household refrigerator p=2rad/s. The slenderness of a 10m tall classical columncan be as low as �=10◦ =0.17rad; whereas, the slenderness of a three-storey building that is8m wide is approximately �=36◦ =0.62rad. The vertical frequency, �v, can originate from thevertical stiffness of an elastomeric pad or the stiffness of a conventional foundation, dependingon the configuration of the rocking structure. Accordingly, practical values of interest for the ratio�=�v/p are in the range 10<�<100. Of course, there may be special cases where ratio � liesoutside this range. Finally, the values of the coefficient of restitution, �, and viscous damping ratio,�v, are in the range 0��1 and 0��v�1, respectively.

NON-LINEAR EQUATIONS OF MOTION

Equations of motion in the full-contact regime

With reference to Figure 4 (top), which shows the free-body diagram of the block in full contactwith the supporting springs, dynamic equilibrium gives:

Mu = −Mug(t)+H

M v = −M[g+ vg(t)]+NL+NR

I0� = −HR0 cos(�)cos(�)

+NLR0[sin(�)cos(�)+cos(�)sin(�)]−NRR0[sin(�)cos(�)−cos(�)sin(�)]

(7)

where M is the mass of the block, I0 is its moment of inertia with respect to the centre of mass O,� and � are the rotation and the angular acceleration of the block, respectively, u and v are thehorizontal and the vertical accelerations of the centre of mass, respectively, R0 is the distancebetween the centre of mass and the pivot points L and R; � is the slenderness angle, NL andNR are the vertical reaction forces at the left and right supports, respectively, H =HL+HR is theresultant of the horizontal components of the reaction forces at the supports, which can be thoughtto be applied at the centre of the base O′, ug(t) and vg(t) are the time histories of the horizontaland vertical components of the ground acceleration, respectively, whereas g is the acceleration ofgravity.

It is assumed that the supports can be described by the viscoelastic Kelvin–Voigt model, i.e. bya frequency-independent elastic spring K =M�2

v/2 in parallel with a viscous dashpot C=�vM�2v.

Accordingly, the vertical component of the reaction force at the I th support, with I =L,R, is

NI =K (�s−vI )−C vI (8)



Figure 4. Free-body diagram of rocking block during the full-contact regime resting on viscoelasticfoundation (top) and details of the contraction of the support (bottom).

where vI is the vertical displacement of the I th point with respect to the rest and �s=Mg/(2K )

is the static deflection of the supports under the weight of the block (see Figure 4, bottom).Notice that in the full-contact regime, the inequality NI�0 has to be satisfied, i.e. the supports

are assumed to not take any tension, and the separation at the I th support happens at the timeinstant in which NI =0 and NI<0.

Since the rigid block is not allowed to slide, the system has only two degrees of freedom.Among the several possible choices, we select the vertical displacement of the centre of mass, v,and the rotation of the block, �. Moreover, in the full-contact regime the displacement field canbe thought as the superposition of a vertical translation and a rotation around the centre of thebase, O′. According to the kinematics of the systems, then, one can derive the expressions of the



displacements of interest:

u = R0 cos(�)sin(�)

vO′ = v+R0 cos(�)[1−cos(�)]vL = vO′ +R0 sin(�)sin(�)

vR = vO′ −R0 sin(�)sin(�)

(9)

The substitution of Equations (9) into (8) and then into Equations (7) furnishes the set of thesecond-order non-linear differential equations governing the motion in the full-contact regime,which in terms of the dimensionless parameters �=MR2

0/I0, �=�v/p and �v=C/(M�v) takethe form:

v+2�v�pv+�2 p2v+ ��2g

1+�cos(�)

{1−cos(�)+ 2�v p

�sin(�)�

}+ vg(t)=0

[1+�cos2(�)cos2(�)]�−�cos2(�)cos(�)sin(�)�2

+��v�p[1−cos(2�)cos(2�)]�

+��2 p2[cos2(�)− 1+�

��2cos(�)−cos(2�)cos(�)

]sin(�)

+ (1+�)�2 p2

gcos(�)sin(�)

[v+ 2�v p

�v

]+ (1+�)p2

gcos(�)cos(�)ug(t)=0

(10)

Equations of motion when rigid block is tilting to one side

When the block is in contact only at the left support (Figure 5, left), dynamic equilibrium gives

Mu = −Mug(t)+HL

M v = −M[g+ vg(t)]+NL

I0� = −HLR0 cos(�+�)−NL[−R0 sin(�+�)](11)

whereas the kinematic relations between the displacements of interest are

u = uL+R0 sin(�+�)

vL = v−R0[cos(�+�)−cos(�)](12)

The substitution of Equations (12) into Equations (8) and then into Equations (11) furnishes theequations of motion when the rigid block is tilting to the right. Similar equations are derived whenthe block is in contact at the right support (Figure 5, right). Therefore, the equations of motionwhen the rigid block is tilting to one side, in terms of the dimensionless parameters �, � and �v,



Figure 5. Free-body diagrams of rocking block when contact with one support is lost.

can be arranged in the compact form:

v+�v�pv+ �2 p2

2v− g

2

{��2

1+�

[cos(�±�)−cos(�)∓ 2�v

�psin(�±�)�

]−1

}+ vg(t)=0

[1+�cos2(�±�)]�∓ �

2sin[2(�±�)]�2+��v�p sin

2(�±�)�

±��2 p2

2

[cos(�)− 1+�

��2−cos(�±�)

]sin(�±�)

± (1+�)�2 p2

2gsin(�±�)

[v+ 2�v

�pv

]+ (1+�)p2

gcos(�±�)ug(t)=0

(13)

where in the case of double signs, the upper one corresponds to the block tilting to the left andthe lower one to the right (more details can be found in the report by Palmeri and Makris [22]).

Equations governing the impact on the left support

Along with the differential equations governing the motion in the three regimes considered above,the equations governing the impacts when the full contact is re-established have to be derived. Inparticular, the instantaneous variation in the field of velocities when the block impacts on the leftis governed by (i) the kinematics of the system before the impact, when the block is tilting to theright:

ub= R0 cos(�−�)�b (14)



(ii) the kinematics after the impact, when the block is in full-contact regime:

ua= R0 cos(�)cos(�)�a (15)

(iii) the impulse-momentum equations, which relate the variations of the horizontal, vertical andangular velocities of the centre of mass to the time integrals of the horizontal and vertical compo-nents of the impulsive force associated with the impact at the left support:

M[ua− ub] =∫ a

bHLdt

M[va− vb] =∫ a

bNLdt

I0[�a− �b] = −R0 cos(�+�)

∫ a

bHL dt+R0 sin(�+�)

∫ a

bNL dt

(16)

where the limits of integration ‘b’ and ‘a’ denote time instants immediately before and after theimpact, respectively.

(iv) The additional equation that controls the reduction of the vertical velocity at the impactpoint:

vaL=�vbL ⇒ va+R0 sin(�+�)�a=�[vb+R0 sin(�+�)�b] (17)

where �, with 0��1, is the coefficient of restitution.The solution of Equations (14)–(17) furnishes the values of the vertical and angular velocities

va and �a after the impact as functions of the corresponding values before the impact:

va = J11(�)vb+ J12(�)�b

�a = J21(�)vb+ J22(�)�b(18)

where

J11(�) = �I0+[�A(�)+B(�)]MR20

I0+[A(�)+B(�)]MR20

J12(�) = (�−1)I0R0+[�A(�)+cos(�−�)cos(�+�)]MR30

I0+[A(�)+B(�)]MR20

sin(�+�)

J21(�) = (�−1)MR0

I0+[A(�)+B(�)]MR20

sin(�+�)

J22(�) = I0+[�B(�)+cos(�−�)cos(�+�)]MR20

I0+[A(�)+B(�)]MR20

(19)

A(�) and B(�) being the following functions of the rotation �:

A(�) = cos(�)cos(�)cos(�+�)

B(�) = sin2(�+�)(20)



At the limiting case of a rigid block rocking on a rigid foundation, the rotation during impactis zero (�=0), and the vertical velocity of the pivot point after the impact is also zero, i.e. �=0.Accordingly, the last two of Equations (19) reduce as

J21 = − MR0

I0+MR20

sin(�)=− �

1+�

sin(�)

R0

J22 = I0+MR20 cos

2(�)

I0+MR20

= 1+�cos2(�)

1+�

(21)

Furthermore, when the foundation is rigid, the rocking motion is described by just a single degreeof freedom; hence, the vertical velocity of the centre of mass can be expressed as a function ofthe angular velocity only. In particular, before the impact:

vb= R0 sin(�)�b (22)

Substitution of Equations (21) and (22) into the second of Equation (18) leads to

�a= 1+�cos(2�)

1+��b (23)

Since for a rigid foundation the ratio e is given by

e= �a

�b(24)

the substitution of Equation (23) into (24) gives

e= 1+�cos(2�)

1+�(25)

By taking into account that cos(2�)=1−2sin2(�), the latter coincides with the expression of emaxwhich has been derived in the literature by applying the conservation of momentum about theimpact point on a rigid foundation just before the impact and right after the impact [4]:

emax= 1+�[1−2sin2(�)]1+�

(26)

which for rectangular blocks (�=3) simplifies to

emax=1− 32 sin

2(�) (27)

Equations governing the impact on the right support

In an analogous way, one can obtain

va = J11(−�)vb− J12(−�)�b

�a = −J21(−�)vb+ J22(−�)�b(28)

in which the functions J are those given by Equations (19).



Equations (10), (13), (18) and (28) are considered in the companion paper [23], where a linearizedformulation is presented which is an improvement to the unconservative equations published earlierin the literature [15].

RESPONSE ANALYSIS TO TRIGONOMETRIC PULSES

The ability of distinct pulses to generate a structural response that resembles the earthquake-induced response has been examined in past studies ([5, 13, 18, 24–30], among others). Some ofthe proposed pulses are physically realizable motions with zero final ground velocity and finiteacceleration, whereas some other idealizations violate one or both of the above requirements.Physically realizable pulses can adequately describe the impulsive character of near-fault groundmotions, both qualitatively and quantitatively. The input parameters that describe the pulses havean unambiguous physical meaning. The minimum number of parameters is two, which are theacceleration amplitude ap (or velocity amplitude vp) and the duration Tp [26, 27]. The moresophisticated model by Mavroeidis and Papageorgiou [30] involves four parameters. Pulse-typeexcitations are of key importance in this study because they allow the introduction of dimensionlessparameters, which uncover the underlying physics of the response. Of particular interest are theforward motions that can be approximated with a one-sine acceleration pulse, and the forward-and-back motions that can be approximated with a one-cosine acceleration pulse [13, 27].

Non-linear time history analysis

Figure 6 plots the response to a one-sine pulse of a rigid block with �=0.25rad (∼=14.3◦), �=3(rectangular shape) and p=1rad/s (R0=7.36m) rocking on a rigid foundation (solid lines) anda two-support viscoelastic foundation (dashed lines), with values of �=�v/p=20,40,60 and 80.The coefficient of restitution is assumed to be �= va/vb=0 and the viscous damping ratio �v=0.1.The peak values of the response on the viscoelastic foundation are slightly less than the peakvalue of the response on a rigid base. This is because when the block is rocking on a viscoelasticfoundation, energy is lost not only at the instant when the pivot points alternate but also becauseof the vertical motion of the support points, which engages the dashpots. This trend is confirmedin the paper by Apostolou et al. [18], where a non-linear FE model is used.

In Figure 6, a shaded band is used to indicate the interval of the rotation [−�,�] in which rigidblock and two-support foundation are in full contact under a quasi-static rocking motion:

�=arcsin

[�s

R0 sin(�)

]=arcsin

[1+�

��2 sin(�)

](29)

It is worth noting that the value of � is independent of the frequency parameter p, that is, the sizeof the rigid block, R0, does not influence the value of the rotation for which the base separatesfrom one of the supports.

Figure 7 shows the effect of the parameter �. As the parameter � increases, the mechanicsof rocking becomes more linear, and the energy lost during impacts is less, leading to a slowerdecaying response. Complete separation is also easier.



Figure 6. Rotation and angular velocity time histories of a rigid block with parameters �=0.25rad andp=1rad/s rocking on a viscoelastic foundation with parameters �v=0.1, �=0 and various values of �

when subjected to a one-sine pulse.

Figure 8 shows the time histories, for different combinations of the parameters � and �v, ofthe total energy of the system, E , which is the sum of the potential energy, V , and of the kineticenergy, T , given by

V = Mgv+ 12 [kv2LH(−r)+kv2RH(r)]

T = 12 I0�

2+ 12M(u2+ v2)

(30)

where H(·) is the Heaviside step function:

H(x)={1, x�0

0, x<0(31)

and where the variable r denotes the regime of motion, that is, r =0 in the full-contact regime,r =−1 when the block is tilting to the left and r =+1 when it is tilting to the right. Asa consequence, in the full-contact regime both terms in the square brackets contribute to thepotential energy, with H(−r)=H(r)=1. On the contrary, when the rigid block is tilting tothe left (right) only the left (right) term contributes, with H(−r)=1 and H(r)=0 (H(−r)=0and H(r)=1).



Figure 7. Rotation and angular velocity time histories of a rigid block with parameters �=0.25rad andp=1rad/s rocking on a viscoelastic foundation with �v=0.1, and �=0 (solid line), �=0.4 (dashed line)

for various values of parameter �.

Figure 8 shows that for low values of � (nearly inelastic impact), the energy dissipated with theviscoelastic foundation is larger than the energy dissipated with a rigid foundation since additionalenergy is dissipated in the dashpots. Only in the case when �=0.4 (more elastic impacts) and�v=0.1, the energy dissipated with the viscoelastic foundation is less than the energy dissipatedwith a rigid foundation.

In Figures 6 and 7, it is observed that as the stiffness of the foundation increases (largervalues of �=�v/p) there is complete separation of the rocking structure from both its supports.This behaviour depends also on the value of the viscous damping ratio and the slenderness ofthe structure. The range of parameters that influence the complete separation of a rocking blockis examined herein for the case of free vibration due to an initial angular velocity, �(0). Theminimum angular velocity needed to bring a block rocking on a rigid foundation to the verge



Figure 8. Time history of the total energy of a rigid block on a rigid base (solid line) andviscoelastic base (dashed line).

of overturning is computed by equating the kinetic energy immediately after the initiation ofmotion:

T (0)= 1

2(I0+MR2

0)�(0)2= 1+�

2�MR2

0 �(0)2 (32)

with the potential energy at the verge of overturning, �(tov)=0, �(tov)=�:

V (tov)=MgR0[1−cos(�)]∼= �2

2MgR0 (33)

Equating the results from Equations (32) and (33), the minimum angular velocity needed to bringa rocking block to the verge of overturning is

�(0)=√√√√√ V (tov)

1+�

2�MR2

0

=�p (34)

Figure 9 shows the minimum amount of damping needed to avoid complete separation of rockingblocks with slenderness � for different values of the vertical stiffness ratio �=�v/p, when thefree vibrations start with the initial conditions �(0)=0 and �(0)=0.95�p. The plots show thatless slender blocks on a more stiff foundation tend to separate more. These conclusions are in



Figure 9. Minimum amount of viscous damping, �v, needed in the supports to avoid complete separation(fly off) of rocking blocks with slenderness � and various values of the vertical stiffness parameter,

�=�v/p, when the free vibrations start with the initial conditions �(0)=0 and �(0)=0.95�p.

agreement with the conclusions by Psycharis and Jennings [31], which are also confirmed in thispaper for recorded accelerograms.

Rocking spectra

The response of rigid blocks to one-sine and one-cosine acceleration pulses rocking on viscoelasticfoundation is summarized in the rocking spectra plotted in Figures 10 and 11. The smaller theangle of slenderness �, the less sensitive the response of the block to the flexibility, dampingand coefficient of restitution of the foundation. Furthermore, less slender and smaller blockshave a tendency to separate easier. In these graphs, a vertical line defines the minimum valueof the dimensionless quantity �p/p for which the rigid block does not experience completeseparation.



Figure 10. Comparison of rotation spectra of a rigid block rocking on a rigid foundation (solidlines) and on a viscoelastic foundation (other lines) for two levels of acceleration amplitude of

a one-sine pulse with duration Tp=2/�p.

RESPONSE ANALYSIS TO EARTHQUAKE LOADING

Following the response analysis to trigonometric pulses, our study proceeds with the responseanalysis to earthquake loading after selecting four strong ground motions chronologically listed in



Figure 11. Comparison of rotation spectra of a rigid block rocking on a rigid foundation (solidlines) and on a viscoelastic foundation (other lines) for two levels of acceleration amplitude of

a one-cosine pulse with duration Tp=2/�p.

Table I. All of them can be approximated with simple trigonometric pulses. A type-A cycloidalpulse approximates a forward motion; a type-B cycloidal pulse approximates a forward-and-backmotion, whereas a type-Cn pulse approximates a recorded motion that exhibits n main cycles inits displacement history. Table I offers the velocity amplitude and the duration of the main pulses



TableI.Inform

ationpertinentto

thestrong

recordsselected

inthis

studyandthekinematic

characteristicsof

theassociated

pulses.

Makrisand

Mavroeidisand

coworkers

[13,22–

25]

Papageorgiou

[30]

Earthqu

ake

Station

DPG

DPG

VPG

APu

lse

T pvp

a pT p

vp

(date)

(com

ponent

∗ )M

w(km)

(m)

(m/s)

(m/s2

)type

(s)

(m/s)

(m/s)2

(s)

(m/s)

SanFernando

,California

Pacoim

aDam

6.7

5.0

0.35

1.13

1.23

C1

1.30

0.85

4.11

1.47

1.15

(9Fe

bruary

1971

)(164

orFN

)

Erzikan,Turkey

Erzikan

6.7

2.0

0.28

0.84

0.52

C1

1.80

0.70

2.44

2.44

0.67

(13March

1992

)(N

S)

Northridg

e,California

Sylm

arconverterstation

6.7

7.1

0.30

1.22

0.73

C2

2.30

0.60

1.64

3.00

0.80

(17Janu

ary19

94)

(FN)

Northridg

e,California

Rinaldi

receivingstation

6.7

7.1

0.28

1.66

0.84

A0.80

1.75

6.87

1.25

1.42

(17Janu

ary19

94)

(228

orFN

)

∗ FN,fault-no

rmal;NS,

north-south.



Figure 12. Rocking spectra of rigid slender block with slenderness �=0.25rad when subjected to thePacoima Dam motion recorded during the 1971 San Fernando (California) earthquake and to the motion

recorded during the 1992 Erzikan (Turkey) earthquake.

of the recorded motions as identified by Makris and coworkers [13, 26–29] and by Mavroeidisand Papageorgiou [30].

The response to earthquake loading is summarized in rotation spectra depicted in Figures 12 and13, which plot peak rotation values of various blocks with different sizes and the same slenderness,�=0.25rad (=14.32◦), versus the ratio 2/p. For each recorded ground motion, the left plots showthe response when only the horizontal recorded acceleration is induced, whereas the right plotsshow the response when both horizontal and vertical recorded accelerations are induced. As 2/pincreases one moves to larger blocks. In some occasions, the peak rotation of the blocks rockingon viscoelastic foundation is larger than the peak rotation of a free-standing block rocking on arigid base, and the difference can be as high as 50% (see Figure 13). In other cases, the oppositeoccurs (see Figure 14, left). Interestingly, only when �v=0.4 (high viscous damping), �=0 (purelyinelastic impacts) and the vertical component of the ground acceleration is neglected (bottom rightgraphs), the complete separation is avoided for any value of the frequency parameter p.

APPLICATION TO A BRIDGE TOWER

The non-linear formulation developed in this paper is used to estimate the response of a bridgetower that resembles the geometry of the tower of the South Rangitikei Viaduct in New Zealand,



Figure 13. Rocking spectra of rigid slender block with slenderness �=0.25rad when subjected to theSylmar converter station motion recorded during the 1994 Northridge (California) earthquake and to the

Rinaldi receiving station motion recorded during the 1994 Northridge (California) earthquake.

Figure 14. Maximum rotation values of the bridge tower when subjected to the strong ground motionslisted in Table I. Histograms: viscoelastic foundation with different values of the frequency ratio

(�=40,60,100); dotted-dashed lines: rigid base.



as presented in the paper by Beck and Skinner [1]. The frequency parameter of the tower isp=0.38rad/s and its slenderness is �=0.15rad. Each leg of the frame steps on a concrete pilefoundation through a thin elastomeric pad that is connected via metallic plates. At each leg thereis a pair of hysteretic energy absorbers mounted between the leg and the supporting foundation.The schematic of the support system of the fame presented in the paper by Beck and Skinner [1]does not offer any dimensions. Hence, our analysis assumes a range of parameters.

The vertical stiffness of an elastomeric pad is given by

K = EcA

tr(35)

where Ec is the compression modulus of the pad, A is the loaded area and tr is the rubber thicknessthat is free to bulge. For a circular pad, the compression modulus is given by Ec=6GS2 [32],where S= A/(dtr) is the shape factor, with G and d being the shear modulus of the elastomericmaterial and the diameter of the bearing, respectively. Assuming a circular pad, Equation (35) gives

K =6GS2A

tr=6GS2dS=6dGS3 (36)

Using a typical value for G=1.0MPa and a value for d=2m [1], the vertical stiffness for S=5assumes the value of K =4710MN/m, where for S=10 it assumes the value of K =37700MN/m.Given that the corresponding mass of the system is approximately M=700000kg, the value ofthe vertical frequency is �v=√

2K/M=11.6rad/s when S=5, and in this case �=�v/p=305;whereas �v=328rad/s when S=10, and in this case �=863.

In a more recent study, Martinez-Rodrigo and Luco [33] proposed the isolation of the samebridge tower through a new stepping mechanism with stiffness K =158kN/m, which gives �v=21.2rad/s and �=55.8.

Figure 14 plots the peak normalized rotations of the tower of interest when subjected to thefour ground motions listed in Table I. They are ordered with increasing peak ground acceleration.The most interesting observation is that for the first two ground motions, a stiffer device produceslarger rotations, whereas in the last three ground motions the opposite happens. Nevertheless, whatis important to note is that a two order of magnitude increase in the stiffness of the stepping systemhas a small effect on the value of the peak rotation. The tower while very slender (�=0.15rad)experiences small rotations because it has such a large size (p=0.36rad/s). Thus, given therandomness of seismic loadings, one can conclude that, in the design of stepping bridges the use ofsoft isolation devices seems to be preferable, since this solution does not increase the peak rotationexcessively, whereas the probability to have a complete separation is considerably reduced.

CONCLUSIONS

In this paper, the rocking response of slender/rigid structures stepping on a viscoelastic foundationhas been examined in depth. First, the fully non-linear formulation of the problem has beendeveloped, which involves also the impulse-momentum equation that prevails during impact. Thenon-linear response analysis of this study complements the linear analysis presented in the pastby other investigators and shows how the response of the rocking block depends on its size, shapeand slenderness, together with the stiffness and damping of the foundation and the energy lossduring impacts.



The effect of the stiffness and damping of the foundation system along with the influence ofthe coefficient of restitution during impact are presented in terms of rocking spectra together withthe peak response of the rigid block rocking on a monolithic base. Various trends of the responseare identified:

(a) For low values of the parameter �= va/vb (nearly inelastic impacts), the energy dissipatedwith a viscoelastic foundation is larger than the energy dissipated with a rigid foundationsince in addition to the energy loss during impacts energy is dissipated in the dashpots;when the impacts become more elastic (larger values of �) the situation reverses.

(b) Less slender and smaller blocks on a stiffer foundation have a tendency to separate easier;the smaller the angle of slenderness, the less sensitive the response to the flexibility, dampingand coefficient of restitution of the foundation.

(c) The angle of separation, �=arcsin[(1+�)/(��2 sin(�))], is independent of the size of theblock.

NOTATION

The following symbols are used in this paper:

C viscous damping of each foundation dashpote coefficient of restitution in terms of the angular velocity of the entire blockg gravitational accelerationHL,HR horizontal component of the reaction force at the pivot points L and R, respectivelyI0 moment of inertia of the rigid block with respect to the centre of massK elastic stiffness of each foundation springM mass of the rigid blockNL,NR vertical component of the reaction force at the pivot points L and R, respectivelyp frequency parameter of the rigid blockR0 size of the rigid blocku horizontal displacement of the centre of massug(t) horizontal component of the ground motionv vertical displacement of the centre of massvL,vR vertical displacements of the pivot points L and R, respectivelyva, vb vertical velocity of the centre of mass after and before an impact, respectivelyvg(t) vertical component of the ground motion� slenderness angle of the rigid block� frequency ratio�s static deflection of the supporting springs� coefficient of restitution in terms of the velocity of the imminent impact point� rotation of the rigid block�a, �b angular velocity of the rigid block after and before an impact, respectively� shape factor of the rigid block�v viscous damping ratio of the vertical vibration of the rigid block (full contact

condition)�v circular frequency of the vertical vibration of the rigid block (full-contact condition)



ACKNOWLEDGEMENTS

We thank the European Social Fund (ESF), Operational Program for Educational and Vocational TrainingII (EPEAEK II) and, particularly, the Program PYTHAGORAS II for funding the above work.

REFERENCES

1. Beck JL, Skinner RI. The seismic response of a reinforced concrete bridge pier designed to step. EarthquakeEngineering and Structural Dynamics 1974; 2:343–358.

2. Yashinsky M, Karshenas MJ. Fundamental of seismic protection for bridges. EERI Monograph MNO-9, 2003.3. Dowdell DJ, Hamersley BA. Lions Gate Bridge North approach—seismic retrofit. In Behaviour of Steel Structures

in Seismic Areas—STESSA 2000. Proceedings of the Third International Conference, Montreal, Canada, MazzolaniFM, Tremblay R (eds). 2000; 319–326.

4. Housner GW. The behaviour of inverted pendulum structures during earthquakes. Bulletin of the SeismologicalSociety of America 1963; 53:404–417.

5. Yim CK, Chopra AK, Penzien J. Rocking response of rigid blocks to earthquakes. Earthquake Engineering andStructural Dynamics 1980; 8:565–587.

6. Ishiyama Y. Motion of rigid bodies and criteria for overturning by earthquake excitations. Earthquake Engineeringand Structural Dynamics 1982; 10:635–650.

7. Aslam M, Godden WG, Scalise DT. Earthquake rocking response of rigid bodies. Journal of the EngineeringMechanics Division (ASCE) 1980; 106:377–392.

8. Psycharis IN. Dynamic behavior of rocking structures allowed to uplift. Report No. EEERL 81-02, 1981.9. Spanos PD, Koh AS. Rocking of rigid blocks due to harmonic shaking. Journal of the Engineering Mechanics

Division (ASCE) 1984; 110:1627–1642.10. Hogan SJ. On the dynamics of rigid-block motion under harmonic forcing. Proceedings of the Royal Society of

London, Series A 1989; 425:441–476.11. Shenton III HW. Criteria for initiation of slide, rock and slide–rock rigid-body modes. Journal of Engineering

Mechanics (ASCE) 1996; 122:690–693.12. Shi B, Anooshehpoor A, Zeng Y, Brune JN. Rocking and overturning of precariously balanced rocks by

earthquakes. Bulletin of the Seismological Society of America 1996; 86:1364–1371.13. Makris N, Roussos Y. Rocking response of rigid blocks under near source ground motions. Geotechnique 2000;

50:243–262.14. Zhang J, Makris N. Rocking response and overturning of free-standing blocks under cycloidal pulses. Journal

of Engineering Mechanics (ASCE) 2001; 127:473–483.15. Psycharis IN, Jennings PC. Rocking of slender rigid bodies allowed to uplift. Earthquake Engineering and

Structural Dynamics 1983; 11:57–76.16. Yim CK, Chopra AK. Dynamics of structures on two-spring foundation allowed to uplift. Journal of Engineering

Mechanics (ASCE) 1984; 110:1124–1146.17. Gazetas G. Overturning and settlement in Adapazari during the 1999 Izmit (Turkey) Earthquake. Proceedings of

the 1st International Conference on the Kocaeli Earthquake of 17 August 1999, Instabul, Turkey, 1999.18. Apostolou M, Gazetas G, Garini E. Seismic response of slender rigid structures with foundation uplifting. Soil

Dynamics and Earthquake Engineering 2007; 27:642–654.19. An T, Kiyomiya O. Dynamic response analyses and model vibration tests on seismic isolating foundation of

bridge pier. Structural Engineering/Earthquake Engineering (JSCE) 2006; 23:195s–214s.20. Chen Y-H, Liao W-H, Lee C-L, Wang Y-P. Seismic isolation of viaduct piers by means of a rocking mechanism.

Earthquake Engineering and Structural Dynamics 2006; 35:713–736.21. Cheng C-T. Energy dissipation in rocking bridge piers under free vibration tests. Earthquake Engineering and

Structural Dynamics 2007; 36:503–518.22. Palmeri A, Makris N. Response analysis of rigid structures rocking on viscoelastic foundation. Report No. EEAM

2005-02, 2005.23. Palmeri A, Makris N. Linearization and first-order expansion of the rocking motion of rigid blocks stepping on

viscoelastic foundation. Earthquake Engineering and Structural Dynamics 2008; DOI: 10.1002/eqe.799.24. Velestos AS, Newmark NM, Chelepati CV. Deformation spectra for elastic and elastoplastic systems subjected

to ground shock and earthquake motions. Proceedings of the 3rd World Conference on Earthquake Engineering,vol. 2, Wellington, New Zealand, 1965; 663–682.



25. Hall JF, Heaton TH, Halling MW, Wald DJ. Near-source ground motion and its effects on flexible buildings.Earthquake Spectra 1995; 11:569–605.

26. Makris N, Chang SP. Effect of viscous, viscoplastic and friction damping on the response of seismic isolatedstructures. Earthquake Engineering and Structural Dynamics 2000; 29:85–107.

27. Makris N, Black CJ. Dimensional analysis of rigid-plastic and elastoplastic structures under pulse-type excitations.Journal of Engineering Mechanics (ASCE) 2004; 130:1006–1018.

28. Makris N, Black CJ. Dimensional analysis of bilinear oscillators under pulse-type excitations. Journal ofEngineering Mechanics (ASCE) 2004; 130:1019–1031.

29. Makris N, Black CJ. Evaluation of peak ground velocity as a ‘good’ intensity measure for near-source groundmotions. Journal of Engineering Mechanics (ASCE) 2004; 130:1032–1044.

30. Mavroeidis GP, Papageorgiou AS. A mathematical representation of near-fault ground motions. Bulletin of theSeismological Society of America 2003; 93:1099–1131.

31. Psycharis IN, Jennings PC. Upthrow of objects due to horizontal impulse excitation. Bulletin of the SeismologicalSociety of America 1985; 75:543–561.

32. Kelly JM. Earthquake-resistant Design with Rubber. Springer: London, U.K., 1997.33. Martinez-Rodrigo MD, Luco JE. Seismic base-isolation by use of a telescopic stepping mechanism. Earthquake

Engineering and Structural Dynamics 2005; 34:227–245.


Response analysis of rigid structures rocking on viscoelastic foundation

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