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The need to assess the behaviour of structures whenthey rock arises in many applications in earthquakeengineering such as the overturning potential ofelectrical cabinets, the stability of stacked storage vesselsand the behaviour of freestanding and backfilled walls. Insome cases bridge piers are fitted with rocker bearings tode-tune the structure from resonant effects that wouldoccur in an earthquake if the pier was fixed at the base.

Contact, separation and sliding at the base of a rockingstructure can be simulated in dynamic finite elementmodels by applying a contact interface between the blockbase and the floor support. The contact interface in arocking model however does not experience large plasticdeformation as in a crash test and neither is it slowenough to neglect inertial effects and apply a non-linearstatic analysis.

When an earthquake time history is applied in a rockingmodel, the amplitude of rocking can build up progressivelyand damping in the model can have a significant effect onpeak displacements and assessment of overturningpotential. Damping in the system is affected by theflexibility of the rocking body and the flexibility of the basesupport and for this reason a purely kinematic rigid bodyapproach may not always be appropriate.

Analytical SolutionsIf the co-efficient of friction of the materials at the base ofthe structure is high, sliding action can be neglected andequations in [1] can be applied to estimate the rockingperiod. Much research has been carried out to establishrelationships between earthquake characteristics androcking behaviour [1] [2] [3] [4].

Unlike a spring system in which the period of vibration isconstant with increasing amplitude, in a rocking systemthe period of rocking increases as the rocking amplitudeincreases. Figure 2 shows this effect by applying therocking period equation in [1] over a range of peakrocking angles for a block with the dimensions in theexample Finite Element model described below.

The width and height dimensions of a block affect thetheoretical period of rocking, however density of a blockfor a given dimension has no influence. The period ofrocking is also reduced by rocking impacts reducing thesway amplitude over each half cycle.

Earthquake REarthquake RockingockingBehaviour in FBehaviour in FiniteiniteElement ModelsElement ModelsDr Rory Lennon, of Halcrow Special Structures, discusses the analysis of tall blocks, such as bridge piers, underearthquake conditions.

Figure 1: 3D rocking wall model

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Under horizontal seismic acceleration, rocking of a blockwill not begin until the acceleration is large enough tocause uplift. This occurs when the overturning moment isgreater than the self-weight restoring moment, and thisoccurs when the ground acceleration is greater than theblock horizontal to vertical aspect ratio x g.

Time Step ModelIn the equations of motion of a rocking block, inertialrotation about either corner at the base of the block (Io) isequated with the self-weight restoring moment and themoment applied by acceleration of the ground (ug) in anearthquake:

Io d2/dt2 = -mg R Sin( - ) - mug R Cos( - )

The implementation of the equations in a numericalroutine is complicated by the self weight restoring momentchanging sign each time the block rocks at the base fromone corner to the other. A non-linear rocking model witha single degree of freedom the angle of rotation canbe implemented by applying the equations of motion in atime-stepping routine with a conditional statement tochange the sign of the restoring moment and applyimpact damping at the point of impact in each half cycle.This type of model can be generated in a spreadsheet andcan respond to horizontal and vertical acceleration timehistories, giving a 2D rocking effect without sliding.

Figure 4 shows the rotation of a rocking block in a singledegree of freedom system, tilted initially to an angle equalto half the overturning angle. In this case no groundacceleration is applied, the block is tilted and released torock in free vibration. The pattern of displacement inFigure 4 appears at first glance to be damped harmonicmotion, however the rotation follows a hyperbolic path.

Angular velocity in Figure 5 corresponds to the rotations inFigure 5 and shows that angular velocity decreasesinstantaneously at each rocking impact. The time step ina dynamic finite element analysis has to be small enoughto model this sudden change.

In Figure 6 the sign of angular acceleration, alsocorresponding to the rotations in Figure 4, is reversed ateach impact as the block rocks from one corner toanother, changing the sign of the self-weight restoringmoment.

The magnitude of acceleration decreases betweenimpacts as the horizontal distance between the centre ofmass in the block and the supporting corner gets smaller,reducing the lever arm of the self-weight restoringmoment between impacts. If this lever arm distance goesto zero between impacts, the block is at the point ofoverturning.

Figure 2: Theoretical rocking period in the example model Figure 4: S.D.O.F. and Finite Element rocking model rotations

Figure 3: Rocking block analysis parameters

Figure 5: Angular velocity in the example rocking model

Finite Element ModelingCreating a dynamic finite element rocking model withcontact interfaces allows additional features of thesimulation to be included:

Sliding at the base can be modelled in addition torocking.

Soft/plastic soil support, concrete crushing at thecontact interface and other non-linear material effectscan be considered.

Geometric details in the rocking structure affectingcentre of mass and contact positions can be included.

The analysis can be carried out in 3D to include plandisplacements and rotations.

Vertical stacks of more than one block can be modelled(Figure 1)

MeshingIn an explicit dynamic analysis acceptable element sizescan be determined from the usual wave speed rules. If animplicit dynamic analysis is carried out, element sizesshould be small enough to capture modes of vibration ofthe rocking block and floor support that arise from impactat the base.

DampingIt is not always necessary to model the floor beneath theblock, grounded contact elements could be applied at thebase of the block instead. However, with Raleigh stiffnessdamping applied in a Finite Element model energy isdissipated in the regions that vibrate on impact in the floorsupport and the rocking block and modelling the floor withbeams, shells or solids allows greater control overdamping in the model.

The stiffness (beta) component of Raleigh damping canbe applied to control the level of amplitude decay in arocking block. The mass (alpha) component of Raleighdamping is associated with damping inertial movementsand unless the structure is affected by fluid drag, onlystiffness damping is needed.

Stiffness damping in the supporting floor can be tuned toa prescribed level consistent with structural dampingbased on the floor natural frequency. If it is possible toapply different levels of Raleigh stiffness damping betweenthe floor support and the rocking block, stiffness dampingin the rocking part of the model can be adjusted to giveoverall damping consistent with experimental test data orother targets.

Rocking amplitude decay can be used to establish theoverall system damping with the following formula [2]:

= 1/n [ln(o/n)]Where:

is the proportion of critical dampingn is the number of impactso is the starting tilt anglen is the peak rocking angle after impact n

Contact interface propertiesThe normal contact stiffness applied at the block to floorinterface influences the overall behaviour of the rockingsystem. A low normal contact stiffness gives greatercontact penetration, a longer impact time and a higherbounce effect at the base and this may not be realistic formodels with stiff materials.

A higher contact stiffness requires a smaller time step sizeand can also affect system damping. To refine theseparameters a sensitivity study checking the effect of time-step size and contact stiffness on amplitude decay may beneeded. A low coefficient of friction in a Mohr-Coulombfriction model at the contact interface generates slidingand at the point of impact in a rocking cycle. This canhave the effect of reducing rocking amplitudes as shownin the example model.

Example Finite Element ModelThe example FiniteElement model in Figure 7is meshed from plainstrain quad elements togive a concrete blocksection 6m high by 1.2mwide and a floor support1m high by 4m wide, fullyfixed at the ends. 2Dnode to node contactelements are applied atthe five nodal positionscoincident in the base ofthe block and the floor.The normal contactstiffness is 108N/m. Thetransient analysis uses aconstant time step of1/1000 of a second over 5seconds.

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Figure 6: Angular acceleration in the example rocking model

Figure 7: 2D example rocking model

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Comparison of S.D.O.F and Finite Element ModelsFigure 4 shows a close agreement between rotations inthe single degree of freedom spreadsheet model and theFinite Element model with small differences in thedistribution of amplitude decay near the start and end ofthe tests.

Effect of Base Friction on Rocking BehaviourThe rocking response of two similar models are comparedin Figure 8. The no sliding model has a coefficient offriction of 1.0 at the contact interface, effectivelypreventing horizontal sliding at the base. The slidingmodel has a coefficient of friction of 0.1 giving lessresistance to sliding. The effect of sliding at the base onreducing rocking amplitudes is clear in the rotation timehistories in Figure 8.

SummaryRocking behaviour in tall blocks under earthquake loadingcan be analysed to various levels of detail. Rockingperiod, peak displacements and overturning potential canbe investigated with established equations andprocedures. However the rotation of the block in responseto ground movement changes at different angles ofrotation in the rocking cycle and to investigate rockingresponse for a specific earthquake condition requires atime-history analysis.

If base sliding can be neglected in the model a singledegree of freedom time-step method can be used tosimulate rotation under horizontal and verticalacceleration time history loadings.

Creating a dynamic Finite Element model allowsadditional details to be included in the analysis such as 3Ddisplacement, greater damping control, sliding and softbase support conditions. Conditioning of the contactinterface in a rocking Finite Element model can have asignificant effect on the rocking behaviour.

References[1] Housner, G.W, The behaviour of Inverted PendulumStructures During Earthquakes, Bulletin of the SeismologicalSociety of America, Vol.53, No.2, February 1963.

[2] Priestly, J.N., Evison, R.J., Carr, A.J., Seismic response ofstructures free to rock on their foundations, Bulletin of the NewZealand National Society for Earthquake Engineering,September 1978

[3] C. Sim, A. K. Chopra and J. Penzien, Rocking response ofrigid blocks to earthquakes, Earthquake Engineering andStructural Dynamics, Vol. 8, 565-587, 1980

[4] Ishiyama, Y, Motions of rigid bodies and criteria foroverturning by earthquake excitations, Earthquake Engineeringand Structural Dynamics, Vol. 10, 635-650, 1982

[5] Advanced Contact Analysis in ANSYS, CAD-FEM GmbH /IDAC Ireland course notes 2003

ContactRory LennonHalcrow Special StructuresE lennonr@halcrow.com

Figure 8: Effect of friction at the base of the block