rocking response of rigid blocks subjected to ground ...

AZORES 1998 - INTERNATIONAL SEMINAR ON SEISMIC RISK AND REHABILITATION OF STONE MASONRY HOUSING

ROCKING RESPONSE OF RIGID BLOCKS SUBJECTED TO GROUND MOTION C. Casapulla & A. Maione University of Naples Federico II, Department of “Costruzioni e Metodi matematici in Architettura”, Naples, Italy

SUMMARY This work deals with the rocking response of a rigid free-standing block under the action of ground motion, as basic reference for the seismic analysis of masonry structures. A new computational strategy for representing the seismic input is suggested in order to define such an amplitude resonance as representing the most disadvantageous conditions for the block. The rocking response to artificial accelerograms is then analysed by means of a simplified equation of motion between two impacts. The coefficient of restitution is assumed as a variable of the problem to account also for other damping effects (e.g. local plastic deformations). A stabilised phase of the motion is identified for which an upper-bound of the maximum rotation angle of the block can be defined in closed form. The results are plotted in a kind of resonance spectra which point out the influence of such meaningful parameters as the coefficient of restitution and the size and slenderness of the block.

1. INTRODUCTION Historically, the rocking response of rigid blocks subjected to earthquake ground motion has been a field of interest to researchers for over a century. Nevertheless, it was Housner (1963) who first gave the problem a modern treatment. He showed that despite the apparent simplicity of a single rocking block dynamics, a nontrivial behaviour was present and a number of unexpected results emerged. Basically, the stability of a block subjected to a particular ground motion does not necessarily increase monotonically with the increasing size or decreasing slenderness ratio. Nor does overturning of a block by a ground motion with particular intensity imply that the block will necessarily overturn under the action of more intense ground motion. Thus, in order to simplify the analysis Housner (1963) described the base acceleration as a rectangular or a halfsine pulse and expressions were derived for the minimum acceleration required to overturn the block, as function of the duration of the pulse. Experimental and numerical analysis were later developed by Yim et al. (1980), showing that, in contrast with the response to a single pulse, the response to more irregular but simplified accelerograms is very sensitive to the geometrical parameters of the block, as well as to the details of ground motions and to the coefficient of restitution. Therefore, they used a probabilistic approach to identify certain statistically recurrent properties of the response. Aslam et al. (1980) also analysed, both numerically and experimentally, dynamic behaviour under harmonic excitations and simulated accelerograms, confirming the difficulty in providing prediction criteria for the response. The works in this field also highlight the importance of the coefficient of restitution by means of the measurement of the energy loss due to impact, first theoretically provided by Housner (1963). Housner’s coefficient, depending on the slenderness ratio of the block, was then adopted by Giannini (1984), who also introduced the dependence on the application point of the resulting impulsive forces due to impact. Comparisons between numerical analysis based on the Giannini’s formulation and experimental tests have been carried out, among other authors, by Liberatore et al. (2001a, b). Prieto et al. (2004) introduced the damping effects no longer by means of a coefficient of restitution but understood as the presence of impulsive forces (Dirac-delta ones). And Sinopoli (1989) showed the importance of the possibility of sliding, together with rocking, during the impact, depending on the slenderness ratio of the block. However, where the study of the block is referred to the masonry block, further difficulties due to uncertainties about the structural behaviour of masonry should be taken into account. This implies high levels of uncertainty for the seismic analysis of masonry structures, for which the single block is the basic reference. As a 1


consequence, it is evident that the current seismic rules on masonry structures are lacking in scientific robustness and, therefore, many indications can be accepted only as prudential criteria, based on experience and on rough evaluations. For these reasons, it is necessary to tackle the problem again, devoting greater attention to the parameters that influence the block response. Particularly, it is necessary to tackle the difficulties in defining a reliable spectral response for the rigid block such as that for elastic systems. These difficulties are due to: 1) the lack of the natural frequency, with the consequence that an harmonic excitation cannot have the same important role as for elastic systems; 2) the lack of explicit damping factor, at least for rigid blocks, with the consequence that the response is always affected by the natural motion; 3) the load-displacement characteristic that is completely different from the more common structural systems. In other words the seismic response of rigid blocks is strictly influenced by the particular features of each earthquake ground motion. Thus the validated theories on the seismic behaviour of general structural systems cannot be applied directly to the safety of rigid systems subjected to overturning. With the goal of overcoming these difficulties, the present paper focuses on the construction of artificial accelerograms aimed at defining such an amplitude resonance as representing the most disadvantageous conditions for the block. These accelerograms only respect the self-balanced effect resulting from the null integrals of the displacements, velocities and accelerations of the ground over the time of their duration. This is mostly typical of the far-fault motions without fracture. Such accelerograms are represented as combinations of elementary schemes, consisting of three acceleration pulses and obeying this property of ground motion. The interval durations between two pulses are calibrated such as to define the most disadvantageous conditions for the block. The dynamic response of a rigid free-standing block is then analysed by means of a simplified equation of motion between two impacts. The dissipation of the motion due to impact of the block on the ground is represented by the coefficient of restitution which is assumed as a variable of the problem to account also for other damping effects (e.g. local plastic deformations). The rocking response to such artificial accelerograms becomes a natural motion characterised only by a given initial velocity due to each pulse. This kind of analysis provides an upper-bound of the maximum rotation angles of the block. A kind of resonance spectra shows the influence of such meaningful parameters as the size and slenderness of the block, the coefficient of restitution and the features of the acceleration pulses representing earthquakes.

2. PULSE SEQUENCES REPRESENTING EARTHQUAKE GROUND MOTION The need of treating new computational strategies for representing the seismic input rather than trying to generalize and parameterize the structural response of the block has already been highlighted in (Casapulla et al., 2007). Basically, some methodological criteria were identified for the construction of artificial accelerograms as combinations of elementary schemes of pulses. These are: a) regularity of the schemes so as to provide simple and clear solutions; b) property of representing the most disadvantageous conditions for the block in terms of amplitude resonance. On these criteria, a particular aspect of the far-fault earthquake ground motions without fracture was considered in the mentioned previous work as a reference point: the self-balanced effect of the earthquake resulting from the null integral of the displacements, velocities and accelerations of the ground over the time of its duration. Thus, such new artificial accelerograms previously proposed are herein used to define and discuss likely upper bound solutions for the most disadvantageous conditions for the block, which can represent a reliable threshold for the assessment of its seismic safety risk. More in detail, these accelerograms are represented as combinations of elementary schemes of pulses (Diracdelta’s type), consisting of three acceleration pulses and obeying such a property of ground motion mentioned above (Figure 1). The single pulse is I = &y&∆t and T is equal to the duration of the first half-cycle of the block, i.e. from t = 0 to the time of the first impact. The sequence of such self-equilibrated pulses showed in Figure 2a) has been identified so as to define the most disadvantageous conditions for the block. In fact, the pulse sequence resulting from this (Figure 2b) is characterised by the interval durations between two pulses Ti calibrated as the duration of the subsequent halfcycles of the block ti (the progressive time between two subsequent impacts). This increases the angular velocities at each half-cycle and the motion of the block between two impacts becomes of natural type, characterised only by a given initial velocity (the previous one damped by the coefficient of restitution C during impact plus the new one due to the active pulse). Also, the most disadvantageous condition is given by the 2


application of each pulse just after impact, because it is not damped by the coefficient of restitution, as already introduced in (Casapulla et al., 2006). Within the analysis developed in this paper the coefficient of restitution is assumed as a variable of the problem to account also for other damping effects (e.g. local plastic deformations). The so defined pulse sequence is still characterised by zero final velocity and displacement, as resulting by the sum of the effects. Therefore, this kind of induced amplitude resonance is considered as representing the most disadvantageous effect of an earthquake to the block under study as showed in the following sections.

Figure 1: Assembly unit of pulses chosen to represent earthquake ground motion

Figure 2: a) Combination of elementary schemes. b) Artificial seismic input as resulting from the combination of elementary schemes

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3. THE UPPER-BOUND OF THE MAXIMUM ROTATION ANGLE FOR A RIGID FREE-STANDING BLOCK UNDER A SEISMIC ROCKING MOTION 3.1. The response of the block to the single pulse and to the pulse sequence In this work the analysis of the model introduced in (Casapulla et al., 2007) is further developed. Hence the main features of that model are herein summarised before discussing the results of the analysis. A rigid free-standing block (Figure 3) with a generic coefficient of restitution C is subject to such an artificial seismic input represented by a sequence of pulses acting after the impact of the block on the ground, as previously explained. This representation together with the hypothesis of the absence of sliding between the block and the supporting base allows to describe the motion of the block between two impacts as natural type. Therefore, the response to the single pulse is firstly analysed and then the case of the pulse sequence is simply derived from this.

Figure 3: Rocking rigid block system

3.1.1. The response to the single pulse The initial pulse I of the artificial accelerogram in Figure 2b) instantaneously causes the velocity α& 1 which is defined from the law of conservation of the angular momentum as:

α& 1 =

&& g I M I a λα = IO g

(1)

where M is the mass, IO = 4 MR2/3 is the corresponding moment of inertia (with respect to O), g is the && g is given by: acceleration of gravity, λ = a/b is the slenderness and α && g = α

3 b 3 1 g= g 2 2 4 a +b 4b λ2 + 1

(2)

For very small rotations of the block, which we are most interested in, a commonly used approximation of the equation governing the natural motion after the pulse is: && = 0 Mg b + I O α

(3)

Eq. (3) represents a uniform accelerated motion with the following initial conditions:

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α(0 ) = 0;

&& g α& (0 ) = α& 1 = λ α

I g

(4)

Hence, the maximum rotation angle is reached when α& = 0 and it is:

α1 max = 0,5

2 α& 12 I && g   = 0,5 λ2 α && g α g

(5)

while the time at which the block impacts the ground is:

t1 = 2

α& 1 I = 2λ && g α g

(6)

It derives from Eqs. (4), (5) and (6) that, keeping slenderness λ and the initial pulse I constant, the initial angular velocity and the maximum rotation angle decrease for increasing values of the block’s size for the first halfcycle, while its duration is the same. Nevertheless, such three parameters increase with the slenderness when the block’s size is kept constant. This can easily be extended to the case of the pulse sequence, as described below.

3.1.2. The response to the pulse sequence Immediately after the first impact the subsequent pulse of acceleration 2I (Figure 2b) acts on the block, increasing the initial velocity of the natural motion for the new half-cycle. As previously introduced, the dissipation of energy due to impact on the ground is represented by the coefficient of restitution C, which is assumed as a variable of the problem to account also for other damping effects (e.g. local plastic deformations). Thus, the angular velocity at the beginning of the half-cycle i (i = 2, 3,.....n) is:

   α& i = α& i −1C + 2α& 1 = α& 1 2 1 +    

i −1

∑ j =1

   C j  − C i −1     

(7)

Iterating Eq. (6), the time ti at the end of half-cycle i is:

α& λI ti = ti −1 + 2 i = 2 && g α g

i

∑ j =1

   2 1 +    

j −1

∑ k =1

   C k  − C j −1     

(8)

and the duration of half-cycle i is:

∆ti = ti − ti −1

  α& λI   = 2 i = 2 2 1+ && g α g    

i −1

∑ j =1

   C j  − C i −1     

(9)

Eq. (9) shows that the duration of the generic half-cycle i only depends on the slenderness of the block and on the number of the cycles occurred, keeping C fixed. Actually, the initial angular velocity at each half-cycle increases with the increasing of the number of the cycles already occurred because of the subsequent pulse 2I applied. This implies the increase of the rotation angle, without influencing the duration of such an half-cycle. As a consequence, this duration is independent of the size of the block. Moreover, in the mentioned previous work it has been showed that, iterating Eq. (7) for i →∞, the angular velocity does not increase indefinitely but converges to the limit value α& * given by:

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α& * = lim i →∞ α& i = 2α& 1

1 1− C

(10)

This case implies that the duration of the pulse sequence representing the seismic input is relatively long. From Eqs. (9) and (10) it cames that, excluding the case of overturning of the block, the duration of the halfcycles tends to stabilise to the value:

α& 1 1 λI = 4 && g 1 − C α g (1 − C )

∆t * = 4

(11)

that clearly depends on the slenderness of the block and on the coefficient of restitution only, while the number of cycles is now irrelevant. This is because the angular velocity is now constant at each half-cycle as for Eq. (10). * This means that the motion of the block becomes of periodic type, with period T* = 2∆t and, also, the maximum rotation angle is limited to the value:

α* = 2

α& 12 && g (1 − C )2 α

= 2

&& g  I  2 λ2 α   (1 − C )2  g 

(12)

As for the size of the block, Eqs. (4), (5), (10) and (12) clearly show that this strictly influences the angular velocity and displacement, but not the duration of the half-cycles, as already highlighted and evident from Eqs. (6), (9) and (11). Finally, matching Eqs. (12) and (5), the response of the block to the pulse sequence with respect to the single pulse in terms of maximum rotation angle can be expressed by the ratio:

ξ=

α* α1 max

=

4

(1 − C )2

(13)

which clearly depends on the coefficient of restitution only. Particularly, it sharply increases with the increasing of C.

3.2. Discussion of the results In this section the results of the analysis developed above are discussed with particular reference to the stabilised phase of the response, when the duration of the cycles of the motion becomes constant and the block exhibits a periodic motion, with period T*=2∆t*. This phase excludes the case of overturning of the block and implies the assumption of unlimited duration of the pulse sequence representing earthquake ground motion. The purpose of this choice is to represent a kind of resonance spectra for the rigid block in terms of the ratio α*/αlim between the maximum rotation angle for the stabilised phase and the limit one given by αlim = arctan(1/λ), corresponding to overturning. This kind of spectra is characterised by the fact that the period T*of the stabilised cycles only depends on the slenderness of the block and on the coefficient of restitution C, as already observed within Eq. (11). Figure 4 shows the relation between the slenderness λ and the stabilised period T*, for different values of C. Obviously, as this relation is representative of the stabilised period, it is meaningful only if the overturning does not occur and this dipends on the size of the block for each given C, as it will be shown later within the resonance spectra. Moreover, it is evident from Figure 4 that, for a fixed C, T* increases for increasing values of λ, while T* correspondent to a particular value of λ increases as C increases. An increasing value of C implies that the damping effect on the velocity due to the impact on the ground decreases and therefore the initial velocity of the cycles within the stabilised phase and their duration increase, as evident from Eqs. (10) and (11). It means that stabilised period increases as damping effects decrease and this is the exact opposite of what happens for elastic systems if T* is assumed as the fundamental period. 6


Figure 4: Relation between the slenderness λ and the stabilised period T*, for different values of C Thus, in order to represent the results with reference to the same range of λ (3 ≤ λ ≤12), the spectra showed in the following Figures 5, 6 and 7 for different values of C (C = 0.6, 0.7, 0.8) are related to different intervals of periods. These figures show the results in terms of α*/αlim for a fixed C and for different sizes of the base, to take into account the influence of scaling effect. All the results refer to a sequence of pulses with intensity 2 I = &y&∆t = 0.025 g , being ∆t = 0.1 s and &y& = 0.25 g . The curves in Figure 5 firstly show that for C = 0.6 all the considered blocks reach the stabilised phase without overturning. However, the ratio α*/αlim, which describes the stabilised response, is strictly influenced by the size of the base and the slenderness of the block. Specifically, this ratio increases both for decreasing size of the base and proportionally with the slenderness of the block and this occurs for any value of C, as showed in all the plotted results (Figures 5, 6 and 7).

Figure 5: Resonance spectra in terms of the ratio α*/αlim for C = 0.6. Influence of the size of the base On the other hand, Figures 6 and 7 show that when C > 0.6 the scaling effect not only implies a higher ratio α*/αlim for a given T*, but also has a determining role on the possibility of attainment of the stabilised phase. In fact, for C = 0.7 the blocks with base 2b = 0.4 m reach the stabilised phase only if the period T* is less than 2.5 s (λ < 7.5), while overturning in the other cases. For C = 0.8 the scaling effect on the stability is accentuated: in fact only the blocks with 2b = 1.2 m are safe from overturning for any value of the slenderness/period 7


considered. The blocks with 2b < 1.2 m reach the stabilised phase only for values of slenderness decreasing with the decreasing of the base. For 2b = 0.4 m, for example, the stability is possible only if T* < 1.75 s (λ < 3.5).

Figure 6: Resonance spectra in terms of the ratio α*/αlim for C = 0.7. Influence of the size of the base

Figure 7: Resonance spectra in terms of the ratio α*/αlim for C = 0.8. Influence of the size of the base The comparison between the spectral responses of blocks related to different value of C is possible if we refer to the slenderness to identify the period of the blocks. As an example, for a block with 2b = 0.6 m and λ = 5, the periods T* are 1.25 s, 1.67 s, 2.25 s when C is 0.6, 0.7, 0.8, respectively. Comparing the response of this block represented through the spectra in Figures 5, 6 and 7 it results that the ratio α*/αlim is increasing for increasing values of C. It is, in fact, α*/αlim = 0.25 for C = 0.6, α*/αlim = 0.42 for C = 0.7 and α*/αlim = 0.95 for C = 0.8. The last one is almost four time the value related to C = 0.6. Lastly, Figure 8 shows the influence of C and λ on the time ts required for reaching the stabilised phase, according with Eq. (8). It is obvious that the less time is needed to reach the stabilised phase the more reliable the results are. Therefore, from Figure 7 emerges that the response of thicker blocks with lower coefficient of

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restitution are also more reliable besides more stable. In particular, for C = 0.8 only the blocks with λ = 3 reach the stabilised phase.

Figure 8: Influence of the coefficient of restitution on the relation between the time required to reach the stabilised phase and the slenderness.

In conclusion, such an induced amplitude resonance analysed in this work is considered as representing the most disadvantageous effect of an earthquake to the block and therefore would be compared to that classical results from the elastic linear single-degree-of-freedom oscillator. The final goal is to evaluate the difference in amplifications of the response of the two systems in relation to the characteristics of the seismic actions, e.g. impulsive forces or harmonic forces. On these bases further studies are required to point out new important guidelines for the construction of more reliable response spectra for masonry structures and for improving the current seismic codes in this field.

4. CONCLUSIONS Many difficulties arise in defining reliable response spectra for rigid free-standing blocks subject to earthquake ground motions and these are well known in the literature. The line suggested in this paper as an attempt in overcoming these difficulties is focused on the construction of artificial accelerograms aimed at defining such an amplitude resonance as representing the most disadvantageous conditions for the block. These accelerograms only respect the self-balanced effect resulting from the null integral of the displacements, velocities and accelerations of the ground over the time of its duration. Therefore, the general earthquake ground motion can be represented by a combination of elementary schemes characterised by zero final velocity and displacement and by interval durations between two pulses calibrated such as to define the most disadvantageous conditions for the block. Two assembly units, consisting of three acceleration pulses, obeying this property of ground motion are used for the dynamic analysis of a rigid free-standing block by means of a simplified equation of motion between two impacts. The dissipation of the motion due to impact of the block on the ground is represented by the coefficient of restitution which is assumed as a variable of the problem to account also for other damping effects (e.g. local plastic deformations). The rocking response to such artificial accelerograms becomes a natural motion characterised only by a given initial velocity due to each pulse. Assuming unlimited duration of the pulse sequence and excluding overturning, all the results of this analysis refer to the stabilised phase of motion, for which an upper-bound of the maximum rotation angle of the block has been defined in closed form. The meaningful parameters that influence the stabilised response are the size and slenderness of the block, the coefficient of restitution and the features of the acceleration pulses representing earthquakes. 9


The results are plotted in such a kind of diagrams as representing resonance spectra, characterised by the fact that the stabilised period only depends on the slenderness of the block and on the coefficient of restitution and is independent of the size of the block. It means that, for a given coefficient of restitution, the block in condition of resonance can be represented whether by the slenderness or by the stabilised period. The latter, therefore, can be considered as the fundamental period, on the analogy of elastic systems. This aspect will be further investigated in relation to the characteristics of the seismic actions, e.g. impulsive forces or harmonic forces. The numerical analysis herein developed for a given intensity of the pulses, highlights that the stability of the block always decreases for: 1) decreasing size of the base; 2) proportional increasing of the slenderness; 3) increasing of the coefficient of restitution. This result is of great importance when considering that generally the stability of the block is extremely sensitive to little changes in natural accelerograms and the structural response cannot easily be generalised and parameterised. Finally, the reliability of the results is evaluated in terms of time required to reach the stabilised phase. It emerges that the response of thicker blocks with lower coefficient of restitution are also more reliable besides more stable. On these bases, further studies are currently under development to point out new important guidelines for the construction of more reliable response spectra for masonry structures and for improving the current seismic codes in this field.

5. ACKNOWLEDGMENTS The authors acknowledge the sponsorship of the Italian Civil Protection, through the three-year RELUIS Project - Line 1 (2005-2008). The first author is involved as a co-investigator.

6. REFERENCES Aslam, M., Godden, W.G. and Scalise, D.T. (1980) Earthquake rocking response of rigid bodies. Journal of Engineering Mechanics ASCE 106, 377-392. Casapulla, C. and Maione, A. (2005) Dynamic analysis of a sdof model of a masonry lintel and peers system. Proc. International Conference “250th Anniversary of the 1755 Lisbon Earthquake”, Lisbon, Portugal, 534-539. Casapulla, C., Jossa, P. and Maione, A. (2006) Issues about the dynamic behaviour of rigid free-standing blocks under earthquake ground motion. Proc. V Intern. Conference on “Structural Analysis of Historical Constructions”, New Delhi, India, vol. 2, 1151-1158. Casapulla, C., Jossa, P. and Maione, A. (2007) Un riferimento di base per la sicurezza delle strutture murarie sotto sisma. Proc. XII National Conference “L’Ingegneria Sismica in Italia”, Pisa, Italy. Giannini, R. (1984) Analisi dinamica di un sistema di blocchi sovrapposti. Proc. II National Conference “L’Ingegneria Sismica in Italia”, Rapallo, Italy, 647-662. Giannini, R. and Masiani, R. (1991) Risposta sismica del blocco rigido. Proc. V National Conference “L’Ingegneria Sismica in Italia”, Palermo, Italy, 491-500. Housner, G.W. (1963) The Behaviour of Inverted Pendulum Structures During Earthquakes. Bulletin of the Seismological Society of America 53 (2), 403-417. Ishiyama, Y. (1982) Motions of rigid bodies and criteria for overturning by earthquake excitation. Earthquake Engineering and Structural Dynamics 10(5), 635-650. Liberatore, D., Spera, G., D’Alessandro, G. and Nigro, D. (2001a) Risposta di blocchi snelli soggetti a un moto sismico alla base. Proc. X National Conference “L’Ingegneria Sismica in Italia”, Potenza/Matera, Italy. Liberatore, D. and Spera, G. (2001b) Oscillazioni di blocchi snelli: valutazione sperimentale della dissipazione di energia durante gli urti. Proc. X National Conference “L’Ingegneria Sismica in Italia”, Potenza/Matera, Italy. Prieto, F., Lourenço, P.B. and Oliveira, C.S. (2004) Impulsive Dirac-delta forces in the Rocking Motion. Earthquake Engineering and Structural Dynamics 33, 839–857. Spanos, P.D. and Koh, A.S. (1984) Rocking of rigid blocks due to harmonic shaking. Journal of Engineering Mechanics ASCE 110(11), 1627-1642. Sinopoli, A. (1989) Kinematic approach in the impact problem of rigid bodies. Applied Mechanic Review ASME 42(11), 233-244. Shenton H.W. III, (1996) Criteria for initiation of slide, rock, and slide-rock rigid-body modes. Journal of Engineering Mechanics ASCE 122(7), 690–693.

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Tso, W.K., Wong, C.M. (1989a) Steady state rocking response of rigid blocks. Part 1: Analysis. Earthquake Engineering and Structural Dynamics 18, 89-106. Wong, C.M., Tso, W.K. (1989b) Steady state rocking response of rigid blocks. Part 2: Experiment. Earthquake Engineering and Structural Dynamics 18, 107-120. Yim, C.K., Chopra, A.K. and Penzien, J. (1980) Rocking response of rigid blocks to earthquakes. Earthquake Engineering and Structural Dynamics 8, 565-87.

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