Received: 26 May 2019 Revised: 15 July 2019 Accepted: 22 July 2019 DOI: 10.1002eqe.3217
RESEARCH ARTICLE
Displacementbased analysis and design of rocking structures
Natalia Reggiani ManzoMichalis F. Vassiliou
DBAUG, ETH Zurich, Zurich, Switzerland
Correspondence
Michalis F. Vassiliou, Chair of Seismic Design and Analysis, DBAUG, ETH Zurich, Zurich, Switzerland.
Email: vassiliouibk.baug.ethz.ch
Funding information
ETH Zurich, GrantAward Number: ETH 10 181
Summary
The response of a rigid rocking block is traditionally described by its tilt angle. This is a correct description, but this paper suggests that describing rocking via displacements is more meaningful, because it uncovers that two geometrically similar blocks of different size will experience the same top displacement, pro vided that they are not close to overturn. The above is illustrated for both ana lytical pulse excitations and for recorded ground motions. Thus, the displacement demand of a ground motion on a rocking block is only a function of its slenderness, not of its size. This reduces the dimensionality of the prob lem and allows for the construction of sizeindependent rocking demand spectra.
KEYWORDS
dimensional analysis, dimensionality reduction, displacementbased design, rocking, uplifting structures
1INTRODUCTION
The systematic study of the rocking oscillator started with Housners seminal paper in 1963.1 Motivated by the surpris ing stability that tall slender golfballonatee structures presented in the 1960 Chilean earthquake, he showed that a out of two geometrically similar planar rigid objects, the larger one is harder to overturn dynamically, and b the overturning potential of a ground motion increases with its dominant period.
The interest on the rocking oscillator26 sources from its ability to describe systems that cannot be described ade quately by the classical elastic oscillator.7 Indeed, the rocking oscillator can be used to understand the behavior of masonry structures,813 the seismic behavior of unanchored equipment,1422 and to explain the stability of ancient GrecoRoman and Chinese temples that have been standing for more than 2500 years in earthquake prone regions.2226 Rocking motion has also inspired researchers to use inerters as seismic protection devices.27,28 What is not widely known in the western world is that rocking has been used since more than 40 years as a seismic isolation method in the USSR and now in former USSR countries.29 The Soviet system comprises an intentionally designed soft rocking story. The uplift of the rocking columns works as a mechanical fuse and limits the forces transmitted to the superstruc ture. Rocking has also been suggested as a seismic design method for bridges, either without3038 or with3941 a restraining system. In New Zealand, a 60mtall bridge designed to rock has already been built across the Rangitikei River in 1981,42,43 and very recently, the WigramMagdala restrained rocking bridge has been constructed.44 Moreover, a 33mtall chimney at the Christchurch airport has been designed to uplift,45 and three 30 to 38mtall chimneys in Piraeus, Greece, have been retrofitted by allowing them to uplift in case of an earthquake.
Makris and Vassiliou46 and Vassiliou and Makris47 have suggested that as the size of the rocking system increases, the restraining system can become obsolete and merely increases the design forces of both the superstructure and the
Earthquake Engng Struct Dyn. 2019;117. wileyonlinelibrary.comjournaleqe2019 John WileySons, Ltd. 1
2 REGGIANI MANZO AND VASSILIOU
foundation. In buildings, rocking walls have been suggested as a resilient design approach Makris and Aghagholizadeh48 and Aghagholizadeh and Makris49 and references therein.
One of the main challenges for the wider adoption of rocking systems stems from their response being absolutely uncorrelated to any elastic system. Therefore, the elasticbased research results are not applicable: eg, intensity mea sures, response spectra, motiontomotion variability, and design ground motions need to be redetermined. To this end, the rocking oscillator should be described with the minimum parameters needed.
This paper suggests that the current state of the art of using the tilt angleas the DOF of a rocking system is, of course, correct, but it is not the optimal. Using the top displacement of the oscillator, u, reduces the dimensionality of the problem. Then, the displacement demand on a rocking block becomes only slightly dependent on its size and is a function only of its slenderness.
2ROTATIONBASED DIMENSIONAL ANALYSIS OF THE ROCKING OSCILLATOR
The equation of inplane motion for a rigid rectangular rocking column Figure 1 with slendernessand a semi diagonal of length R Figure 1 is
where
u
14p2 sin ggcos ; 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p 14 3g4R 2
is the frequency parameter of the rocking column. The upper sign in front ofcorresponds to a positive, and the lower to a negative rocking anglewith respect to the defined coordinate system Figure 1.
It is assumed that energy is only dissipated during impact. Housner1 assumed that a the impact is instantaneous and b that the impact forces are concentrated on the impacting corner. Under these assumptions, the ratio of postim pact to preimpact rotational velocities is
FIGURE 1 Geometric characteristics of the rigid rocking block
REGGIANI MANZO AND VASSILIOU 3
r14after 1413sin2: 3before 2
Researchers including the senior author of this paper have critically evaluated the Housner modelespecially its damping assumptions.5054 Indeed, while assuming the impact to be instantaneous seems a reasonable assumption, there is no evident reason to assume that the impact forces act on the impacting corner. Given the large sensitivity of the time history response of the rocking oscillator to all the parameters that define it, Housners model might seem sim plistic. However, experimental testing shows that even though it cannot predict the response to an individual ground motion, it can predict the statistics of the response to a set of ground motions.55 Therefore, we consider it adequate within the scope of earthquake engineering.
By inspecting Equations 1 and 2, one can conclude that the rotational response of a rocking block to a ground motion is a function of
14f R;;g;u t: 4 max1 g
As the gravity acceleration, g, is constant, the rotational response to a given ground motion is a function of two param etersand R, similarly to the elastic oscillator, in which the response is a function of the eigenperiod, T, and damping ratio . Therefore, by keeping one parameter constant R or , one can construct rotational spectra for rocking struc tures. However, unlike the elastic oscillator, where, for usual structures, one parameter T is more influential than the other , in the case of rocking structures, both R andstrongly influence the rotational response.
Since ground motions containing distinguishable acceleration andor velocity pulses are particularly destructive Vassiliou and Makris56 and references therein, Zhang and Makris57 have studied the response of a planar rocking block to acceleration pulses given by analytical expressions. A pulse of a given waveform can be described by two parameters. Zhang and Makris57 chose the acceleration amplitude ap and the dominant cyclic frequency p. Then, the response will be a function of
max 14 f2 R;;g;ap;p : 5
Equation 5 involves six quantities with two reference dimensions Time and Length. Therefore, according to the VaschyBuckingham Theorem of Dimensional Analysis,58,59 the number of dimensionless parameters describing the problem is 624. There is not a unique solution for choosing these four parameters. Zhang and Makris57 sug gested describing the problem as
a
max141;p;p : 6
p is often called sizefrequency parameter and depends on the frequency of the excitation and on the size of the block. p
ap is usually called nondimensional acceleration, but it can also be perceived as a nondimensional strength param g tan
eter, since mgRsin is the moment that withstands uplift strength and mapRcos is the overturning moment. Therefore, dimensional analysis reduces the dimensionality of the problem from six to four. Hence, by keeping the slenderness parameterconstant, one can produce contour plots of the maximum tilt angleas a function of pp and apgtan, the socalled rocking spectra. It is worth mentioning that Dimitrakopoulos and DeJong60 have shown that for small values ofone can drop it as an independent parameter from Equation 6 as long as the coefficient of restitu
tion, r, is treated as an extra independent parameterhowever, in this section, r is not treated independently.
Figure 2 shows the rocking spectra of symmetric and antisymmetric Ricker wavelets. Ricker wavelets are defined as
the second and third derivative of the Gaussian:
p gtan
4
REGGIANI MANZO AND VASSILIOU
FIGURE 2
Nondimensional rocking spectra based on rotations. 0.1 Colour figure can be viewed at wileyonlinelibrary.com
!
1 22t2
ug14ap1T2 e2T2p; 7
where
22t2 p
!
ug14 23pffiffie p; 8 r 3Tp 3Tp
22 142t2 ap 4 t 2t 2 3T2
Tp 14 2 p
9
and r1.3801 to enforce that the function maximum is equal to ap.
The spectra confirm the remarkable observation that larger structures are harder to overturn dynamically and that
higher frequency pulses have a lower overturning potential. Interestingly, they show a heavy dependence of the response on both pp and apg tan.
3DISPLACEMENTBASED DIMENSIONAL ANALYSIS OF A ROCKING OSCILLATOR EXCITED BY ANALYTICAL PULSES
3.1Analysis based on the frequency parameter of the block p
The dimensional analysis of the previous section is one of the many correct solutions to describe the problem. It is based on rotations. This section, however, suggests that there is another, displacementbased basis of describing the problem, which is also mathematically correct and more convenient. The convenience does not lie only on the fact that earth quake engineers are more used to displacements than rotations: A displacementbased analysis further reduces the dimensionality of the problem allowing the construction of 2D rocking spectra.
Indeed, the rotationbased analysis of the problem is based on the recipe for similarity analysis described in Chap ter 5 of the wellknown Dimensional Analysis textbook of Barenblatt61: If the problem has an explicit mathematical formulation, the independent variables in the problem and the constant parameters that appear in the equations, boundary conditions and initial conditions, etc., are adopted as the governing parameters. As this section shows, choos ing the parameters that appear in the analytical equation might not be the most convenient way of describing this par ticular problem.
REGGIANI MANZO AND VASSILIOU 5 The top displacement of the rocking block can be obtained by a onetoone mapping on the rotations:
u 14 2R sin2R sin: 10
The upper sign in front ofcorresponds to a positive and the lower sign to a negative tilt anglewith respect to the defined coordinate system. If we use the top displacement as the single DOF of the problem, then the maximum response can be described as
umax 14 f3 R;;g;ap;p : 11
To numerically compute the response of the block, we will resort to Equation 1, which is given in terms of rotation . Then, using Equation 10, we compute the displacement response.
Applying Buckinghams theorem on Equation 11, one possible nondimensionalization is
umaxp2
u2a
maxp142;p;p : 12
ap p gtan p ap
as a function of p and g tan for a given 0.1. The remarkable obser linear on the nondimensional strength parameter ap but only loosely on the sizefrequency parameter p. When the
Figure 3 shows the contour plots of a
vation is that within the nonoverturning region, the nondimensional displacement depends heavily and strongly non
p
gtan p block is not close to overturning, the influence of p is practically negligible.
p
as a function of g tan for different values of p and a constant slenderness 0.1. For rea
umaxp2 ap
p
umax p 2 umax p 2
Figure 4 plots a
sons of figure clarity, only nonoverturning values of a are plot, ie, not plotting a means that the block has
p
pp umaxp2 p ap
as a function of p for different values of g tan and 0.1. Again, it is observed umaxp2 ap p
overturned. Figure 5 plots a
that, as long as the system is away from overturning, the dominant factor that influences a is g tan, not p . In
p
p
fact for small values of nondimensional acceleration ap , the response for all values of sizefrequency parameter p is
gtan p practically the same. The response starts to deviate only when the system is close to overturningor has overturned.
FIGURE 3 Nondimensional rocking spectra based on displacements Colour figure can be viewed at wileyonlinelibrary.com
6
REGGIANI MANZO AND VASSILIOU
FIGURE 4
umaxp2 ap p
a vs g tan plots for constant p . 0.1 Colour figure can be viewed at wileyonlinelibrary.com
p
umaxp2 p ap
a vs p plots for constant g tan and 0.1 Colour figure can be viewed at wileyonlinelibrary.com
p
FIGURE 5
In other words, a small and a large block, geometrically similar to each other and excited by analytical pulses, will have roughly equal top displacement, provided that the displacement is not enough to bring them close to overturn. A given pulse will induce the same displacement demand. The larger block is more stable simply because its displace ment capacity ie, the displacement needed to cause overturn, ie, its width is larger.
Therefore, using a displacement basis to describe the problem further decreases the number of parameters needed to define it. Practically, the displacement demand on a rocking oscillator excited by a pulse is only a function of its non
dimensional strength parameter ap , not of its size. g tan
umax p 2
The strongly nonlinear nature of rocking motion is also evident in Figures 4 and 5. a
tion of the rocking displacement to the ground motion displacement, does not depend monotonically on the strength
, which expresses the rela
p
REGGIANI MANZO AND VASSILIOU 7
parameter ap . In fact, the discontinuities of the p 14 2 line of Figure 4 convey that a block can survive a stronger gtan p
pulse and overturn in a weaker one.
Going back to dimensional quantities, Figure 6 plots the displacement response to a symmetric Ricker pulse with
ap1 g and Tp0.5 second and to an antisymmetric Ricker pulse with ap1 g and Tp1 second. The plots confirm that the displacement demand only loosely depends on the size, if the block is not close to overturning. The dominant factor is the slenderness. Therefore, we can define the displacement demand rocking spectrum of a ground motion as a unary function
udemand 14 fif udemand 2b; 13
which is computed via Equations 1 and 10 for a large enough block size. To check the stability of a block, one has to compute the maximum displacement demand via Equation 13 and compare it with the displacement capacity ie, the block width.
Therefore, the reduction of the dimension of the problem follows two steps: a applying Buckinghams theorem and b observing that the displacement demand is roughly independent of the size. The first step is exact and follows from dimensional analysis. The second step is approximate and in this section is illustrated for analytical pulses. Blochlinger62 gave a first indication that the approximation also works for recorded ground motions. Further evidence supporting this approximation and highlighting its limitations are given in a next section of this paper.
3.2Analysis based on the base width of the block b
The previous section chooses the frequency parameter p and the slenderness of the block, , as the two parameters to define it. However, p has a physical meaning that is totally unrelated to rocking. It is the natural frequency that the block would have had if it was hanging from its corner.63 But this is merely a coincidence, rocking blocks have no nat ural frequency,1 and the use of p often creates misunderstandings. In this section, we propose to describe the block with two physical parameters that have a clear physical meaning, directly related to the rocking problem. The slendernessis retained, as it controls the uplift of the structure and could be parallelized with the strength of a system, but p is replaced by b, which is the halfwidth of the base and exactly equal to one half of its displacement capacity. Then the displacement response will be
Using Buckinghamstheorem, we get
umax 14 f4 b;;g;ap;p : 14
FIGURE 6 Maximum block displacement as a function of block size for symmetric and antisymmetric Ricker excitation Colour figure can be viewed at wileyonlinelibrary.com
8
REGGIANI MANZO AND VASSILIOU
umaxp2ap bp2
ap bp2
The term g tan would be the reciprocal of the nondimensional strength, a
p
143 gtan; a ; : pp
15
would be the nondimensional displace
ment capacity, andtaken as an independent parameter controls damping, because it controls the coefficient of restitution.
Figure 7 plots displacement spectra according to the suggested nondimensionalization. One can observe that for both pulses, a base ie, a displacement capacity2b of roughly nine times the length scale of the pulse Le 14 ap is enough to
p2
keep the block stable, no matter what the nondimensional strength parameter is.
4DISPLACEMENTBASED ANALYSIS OF A ROCKING OSCILLATOR
EXCITED BY RECORDED GROUND MOTIONS
Analytical pulses can be used to qualitatively study the rocking oscillator. However, as the rocking problem is very sen sitive to all of its parameters, pulses would not suffice to prove that the displacement demand on a rocking structure depends only on its slenderness and not on its size. Therefore, this section examines the displacement response of a rocking block excited by recorded ground motions.
FIGURE 7 Displacementbased nondimensional rocking spectra using the width of the block to characterize its size Colour figure can be viewed at wileyonlinelibrary.com
FIGURE 8 Rocking oscillators of equal height
REGGIANI MANZO AND VASSILIOU 9
4.1FEMA P695 ground motions
There is no consensus in the engineering community on what ground motions should be used in time history analysis. Several approaches exist including using recorded scaled or unscaled, artificial, or synthetic ground motions. In this paper, we choose to use the three sets of ground motions proposed by FEMA P69564 farfield, nearfield pulselike, and nearfield nonpulselike only as a means to illustrate our rockingrelated argument, without taking stance on the debate around ground motions. It is evident that any ground motion selection method based on the response of an elastic system is in principle not applicable in the case of the rocking oscillator, as the elastic and rocking oscillator are uncorrelated. More information on the FEMA P695 ground motions can be found in FEMA.64
FIGURE 9 Displacement of a rocking oscillator as function of its slendernessColour figure can be viewed at wileyonlinelibrary.com
10 REGGIANI MANZO AND VASSILIOU
4.2Equal displacement rule for rocking structures and displacement demand spectra
Vassiliou et al65 have proven that rigid rocking oscillators of equal height attached to massless foundations of the same size behave identically, no matter what their actual column width is Figure 8. Therefore, the design question of a rocking structure would be: Find the size, 2B, of the foundation for a given oscillator height 2H. Hence, it is more meaningful to use H as a size parameter instead of R, even if the former does not explicitly appear in the equation of motion.
Figure 9 offers the displacement of a rocking oscillator as function of its slenderness , and for 2H2, 4, 10, 20, 80, and 1000 m, for a selection of the FEMA P695 ground motions. The 2H1000 m is offered only for reasons of math ematical completeness, to study the limit case of H. For reasons of plot clarity, each line is plotted only for crit, where crit is the minimum slenderness angle for which the block overturns. We observe that all blocks of the same slenderness angle present roughly the same displacement, as long as they are not close to overturning. The same obser vation holds for all the ground motions tested.
As analysis and design of a rocking structure would not involve a single ground motion, but a set of design motions, it makes sense to study the problem by applying sets of multiple excitations and comparing the statistics of the results eg, the median displacement among all the ground motions of the excitation set. Figure 10 plots displacement spectra
FIGURE 10 Median displacement spectra for nearfield pulselike record set Colour figure can be viewed at wileyonlinelibrary.com
REGGIANI MANZO AND VASSILIOU 11
of the median of the displacement for seven variations of the nearfield pulselike FEMA P695 set: a unscaled ground motions, b scaled so that their PGA is equal to 0:5PGA, or PGA, or 2PGA, c scaled so that their PGV is equal to 0:5PGV, or PGV, or 2PGV, where PGA and PGV are defined as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PGA 14 median PGAixPGAiy ; 16 i141…N
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PGV 14 median PGVixPGViy ; 17 i141…N
where N is the number of the ground motions and x and y are the two components of each ground motion. Note that each horizontal component of each ground motion is treated as an independent motion. Figures 11 and 12 plot the same spectra for the farfield and nearfield nonpulselike ground motions.
The following observations can be made:
FIGURE 11 Median displacement spectra for nearfield nonpulselike record set Colour figure can be viewed at wileyonlinelibrary.com
12 REGGIANI MANZO AND VASSILIOU
FIGURE 12 Median displacement spectra for farfield record set Colour figure can be viewed at wileyonlinelibrary.com
a. Themedianspectraaresmoother,likewisedesignelasticspectrathatwerederivedbystatisticalprocessingofelastic spectra of single ground motions are smoother than single ground motion spectra
b. Aslongasthesystemisnotclosetooverturning,thedisplacementdoesnotdependonthesizeoftheblock.Forthis part of the spectrum, instead of computing a different spectrum for each block size, one can compute the design spectrum for 2H 2H1000 m seems an adequate value and use it to calculate the displacement demand on any rocking structure ie, umaxf. We name the above finding equal displacement rule for rocking structures.
c. Asthesystemgetsclosertooverturning,theequaldisplacementruledoesnotapply:Smallersystemspresentlarger displacements than larger ones. Moreover, as the system approaches overturning, the slope of the spectrum increases dramatically, ie, a small decrease in tan leads to very large increase in displacement. This trend dictates that a rational design of a rocking structure would require that this steep part of the spectrum be avoided, because an earthquake slightly stronger than the design one would cause a tremendous increase in displacement. Therefore, the equal displacement rule applies to the rational design region.
d. The form of the spectrum for all three sets of ground motions presents some repetitive pattern: 3
i. Astends to zero, umax tends to a finite value. For spectra of individual ground motions, this value is 2PGD.
An explanation for this is offered in the next section.
REGGIANI MANZO AND VASSILIOU 13
ii. Asincreases from zero, the displacement demand amplifies 22.5 times and reaches a plateau. iii. Further increase ofleads to a monotonic decrease of the displacement demand.
iv. Naturally, when tan reaches PGAg, the displacement demand becomes zero, as there is no uplift.
4.3Preliminary design based on the equal displacement rule
If not for a final design, the equal displacement rule can be used for preliminary calculations. Indeed it is not an exact method, but a preliminary design method that does not aim at being exact, but at providing a tool for initial calcula tions, that for certain cases and required degree of accuracy can be enough. The same holds for yielding structures, where the equal displacement rule is used for many structural systems, while for more complicated systems, it is used only for preliminary design and then more refined methods are applied. It could be stated that the findings of this paper constitute the generalization of equal displacement rule from yielding to rocking systems. This section proposes a meth odology to design a rocking structure based on the equal displacement rule:
a. On the umaxtan curve, we plot the capacity line uC2Htan.
b. We determine the intersection of the capacity line and the 2H line. We define the abscissa of this point as
tank.
c. We use a multiplier of 2.5 to determine the design slenderness: tanD2.5tank. The multiplier serves as a safety
factor to move the design point away from the steep part of the spectrum.
Figure 13 outlines the design procedure applied for a rocking bridge with columns of 6.7 m height 2H6.7 m. Based on Makris and Vassiliou,31 the response of the frame is equal to the response of a solitary block of 2H10 m. For this bridge, 21 design scenarios are explored, corresponding to the 21 spectra of Figures 10 to 12. Tables 1, 2, and 3 and Figure 14 summarize the findings for the 21 design scenarios and compare the displacement
FIGURE 13 Design procedure Colour figure can be viewed at wileyonlinelibrary.com
TABLE 1 Nearfield pulselike FS2.5
tan D
2H1000 m 2H10 m
Unscaled 0:5PGA PGA 2PGA 0.2839 0.1378 0.2671 0.4618
0.05 0.11 0.08 0.12
0.06 0.15 0.07 0.13
0:5PGV PGV 2PGV 0.0596 0.1094 0.1909 0.10 0.07 0.08 0.10 0.10 0.08
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REGGIANI MANZO AND VASSILIOU
TABLE 2
Nearfield nonpulselike FS2.5
Unscaled 0:5PGA PGA 2PGA
tan D 2H1000m 2H10 m
0.1524 0.0708 0.1356 0.2446 0.34 0.22 0.44 0.44 0.41 0.26 0.41 0.69
0:5PGV PGV 2PGV 0.0916 0.1892 0.3405
0.19 0.27 0.36 0.21 0.30 0.38
0:5PGV PGV 2PGV 0.0675 0.1320 0.2249 0.17 0.28 0.52 0.19 0.29 0.49
TABLE 3
Farfield FS2.5
tan D 2H1000m 2H10 m
Unscaled 0:5PGA PGA 2PGA 0.1228 0.0620 0.1215 0.2411 0.30 0.21 0.38 0.52 0.33 0.20 0.37 0.50
FIGURE 14
Comparison of the displacement response at the design point. Predictions based on the 2H1000 m and on the 2H10 m spectra Colour figure can be viewed at wileyonlinelibrary.com
predicted by the demand spectrum 2H1000 m to the displacement predicted by the 2H10 m spectrum. We observe that in all but two cases nearfault pulselike scaled to 0:5PGA and nearfault nonpulselike scaled to 2PGA, the error in predicting the median displacement is less than 20. In all cases, the error is smaller than 40, and no system overturned.
5INTERPRETATION OF THE EQUAL DISPLACEMENT RULE BASED ON THE EQUATION OF MOTION
The equal displacement rule can be interpreted by properly manipulating the equation of motion. Assuming small rota tion angles sinxx and cosx1, Equation 1 gives
For small angles, u2H. Then
3g u
144Hgg : 16
3g uu
u141g : 17
2 2bg
When u2b is small ie, the block is not close to overturning, the other terms dominate the response and u becomes a
REGGIANI MANZO AND VASSILIOU 15
function only of . Furthermore, when ug1, then u 14 3ug. Therefore, as 0 which can only happen for blocks g 2
with H, umax32 PGD. 6CONCLUSIONS
The widely used description of the rocking block via its rotation is correct, but not optimal. It reveals that larger blocks are more stable and that higher frequency pulses present less overturning potential. However, it does not reveal the equal displacement rule of rocking structures, namely, that a large and a small block of the same aspect ratio will present the same top displacement, if they both are not close to overturning. Not being close to overturning is a design necessity anyway; therefore, for the scope of design, we can claim that the displacement demand is the same, and it only depends on the slenderness, not on the size of the block. The above is illustrated for both analytical pulse excitations and for sets of recorded ground motions. Based on the above, a design method that uses a sizeindependent rocking spectrum is suggested. This should be taken into account when intensity measures for rocking structures6669 designed not to get close to overturning are explored.
ACKNOWLEDGEMENTS
This work was supported by the ETH Zurich under grant ETH10 181. The methods, results, opinions, findings, and conclusions presented in this report are those of the authors and do not necessarily reflect the views of the funding agency.
ORCID
Michalis F. Vassiliou https:orcid.org0000000245902126 REFERENCES
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How to cite this article: Reggiani Manzo N, Vassiliou MF. Displacementbased analysis and design of rocking structures. Earthquake Engng Struct Dyn. 2019;117. https:doi.org10.1002eqe.3217
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