Experimental campaign

Test equipment

The tests carried out in this study were performed using three solid steel blocks, identical within manufacturing tolerances. The dimensions of these blocks are given in figure4.1a. The blocks were rectangular cuboids of slenderness 6,9, each with four cuboid feet machined from the solid metal. Note that for reasons of space, only the results concerning the first two blocks will be detailed here.

4m m

5m m

690

mm 100 mm

100 mm 5 mm

Bottom of the block

corner 1 corner 2 corner 3

corner 4

(a)

δ^long._i ≈50μm δ^diag. ≈2 mm

8ACP 4SCP

(b)

Figure 4.1 –Geometry of the numerical and experimental blocks:

4.1a: Theoretical block dimensions and angle numbering.

4.1b: Contact point positions for the numerical models and defect order of mag- nitude.

The blocks were instrumented with convective rotational velocity sensors (mea- suring the velocity around the axes of the block’s mobile reference frame, called crv and notated Ω). These sensors have a measurement range of ±200°s⁻¹ and an accuracy of±0,01°s⁻¹. The movement of the blocks was observed around the three axes of the chosen convective reference frame. The acquisition of accel- eration and rotational velocity measurements was performed at 1000 Hz,with an anti-aliasing filter set to 500 Hz.

Release tests

Before performing the dynamic tests on the shaking table, release tests were carried out for each block. These tests consisted of positioning and maintaining the blocks immobile, balanced on two feet (in an unstable state), then releasing them to allow a free rocking movement. The tests enabled us to determine the characteristics of the numerical models used in the study, applying the method- ology proposed by Mathey et al.[51]. For space reasons, the results of these tests are not presented in this article; nevertheless, section 4.4will present the values obtained for the numerical model parameters.

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Shaking table tests

To assess the representativity of the numerical models, from both the deter- ministic point of view and that of statistical analysis under dynamic excitation, two series of 100 tests were performed simultaneously on the three blocks placed on the Vésuve unidirectional shaking table at CEA Saclay.

In, the first series of tests, the blocks were subjected to 100 different accelera- tion signals, each one generated by the random process defined in the following paragraph. In the rest of this paper, this series of tests will be referred to by the symbol=.

In the second test series, the blocks were subjected 100 times to the same signal, chosen from among those used previously. These repeatability tests are more dis- criminating for the numerical models, since they minimize excitation variability.

They therefore enable a specific evaluation of the “intrinsic quality” of a model as regards its aptitude to represent a given real behavior. In this study, we will refer to this series using the symbol≈.

Note that before each test the blocks were repositioned manually along a line marked on the shaking table in order to guarantee the closest possible angular conditions and initial position from one test to another (see Figure4.2).

Dynamic excitation characteristics In order to study the behavior of a block over time, we chose to use unidirectional stationary seismic excitation signals which were perpendicular to two lateral faces of the block (see Figure 4.2). Each signal is a realization, truncated over a time interval of 30 s, of a zero mean sta- tionary Gaussian process defined by Kanaï-Tajimi power spectral density with a characteristic frequency of 2,95 Hz and a damping coefficient of 55 %.

In general two types of behavior can be envisaged, depending on the value of the standard deviation σ_th.of the process. For a theoretical standard deviation, evaluated empirically asσ_th.≤0,05 g, most realizations do not achieve the acceler- ation limit which would enable rocking to be initiated. Indeed, the acceleration level required to enable a perfect block of slenderness 6,9 to initiate a rocking movement isγ^gr.= 1/6,9 g≈0,145 g([4] and Figure4.7a). Furthermore, it has been observed that if the standard deviation of the realizations is σ_th. ≥ 0,03 g, all the blocks that have initiated a rocking movement overturn rapidly (after a few “main impacts” (mi)), which does not enable their behavior to be studied over time. Note that a main impactoccurs in the case of unidirectional excitation when the block angle about the main axis becomes null (see figure 4.2), which is identified as a local extremum of the convective rotational velocity about this same axis.

To overcome this difficulty we considered, for each excitation signal, the con- catenation of a deterministic trigonometric pulse and a truncated realization of the stationary process defined above. The pulse provides the blocks with an un- balanced initial condition (non-null angle and rotational velocity) before they are solicited by the random excitation. This pulse is modeled by a truncated sinu- soidal signal, over one period, of amplitude 0,2 g and a duration of 0,4 s. This amplitude is sufficient to initiate rocking in the blocks. The duration is however sufficiently short for the blocks not to overturn. The standard deviation of the process was chosen to be σ_th. = 0,02 g so that the blocks remain in motion for a relatively long period without overturning. Note, however, that the amplitude of the 100 realizations generated for the tests is so low that the maxima, in absolute values (peak ground acceleration (PGA)), never exceeded 0,10 g. As a consequence, a block which becomes totally immobile during a test cannot in theory resume its rocking movement.

Theoretical, real, measured and calculated signals In practice, there are two challenges to performing and analyzing this type of experiment. The first comes from the servo-control of the shaking tables, which does not allow a target ac- celeration signal to be reproduced with accuracy. The second comes from the

positioning line

z

y

x

z

y

θ: main axis

z

x

φ: transverse axis

y

x

ψ: longitudinal axis Figure 4.2 – Angle notation conventions

accelerometers, which inevitably produce noisy signals. Thus, for clarity’s sake, we distinguish:

— the target accelerationsA_t.which correspond to the signals generated by the procedure described in the previous paragraph;

— the accelerations really generated by the tableAr.. Hypothetically, these are the accelerations experienced by the blocks;

— the accelerations measured by the sensorsAm..

First of all, let us stipulate that instant 0 of these signals was chosen to be eas- ily identifiable from the raw measurements. Since the table is displacement con- trolled, instantτ0 = 0 s is defined as the instant at which the displacement of the table with respect to its initial position attains 10 mm, during the first pulse.

To improve the accuracy of the acceleration measurements, 4 sensors were placed on the shaking table. For each realization r of an excitation signal, with r∈1,100, the 4 measurements_rAⁱ_m., i∈1,4are averaged, and the resulting sig- nal is filtered by a windowed “Low Pass Filter Impulse Response” numerical filter, with as its parameters Fs = 1000 Hz, Fc = 55 Hz, order = 200and a “Flat-top” type window. These averaged signals, filtered and denoted_rA_m.will be used as input data for the numerical models.

Figure 4.3a presents in graphs 1 and 2 five realizations of these signals, and in graphs 3 et 4 the means and standard deviations of two series of realizations (series 100 ≈and series 100 =). In this figure, we can observe that the standard deviation atτ <0is non-null, which highlights residual measurement noise, per- sisting despite appropriate numerical processing to reduce it. We can also observe high-amplitude spikes in the acceleration (for example atτ=0,55 sin zoom A, fig- ure4.3b). These are due to shock waves produced by the impacts of the blocks on the table during their movement. This phenomenon is also visible in zoom B (fig- ure4.3c). However, the curve of series100 = presents higher-frequency content

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than that of series100 ≈, which is more regular. This difference can be explained by the fact that in series100 ≈, from one test to another, the block movements are almost identical during the first seconds, since the accelerations experienced by the blocks are also nearly identical. The impacts of the blocks on their base occur at approximately the same moment for all the tests. This is evinced by the wide peaks on the curve representing the standard deviation of the measured acceler- ation signals. Conversely, since the signals of series 100 =are constructed differ- ently, the block movements are also different from one test to another, even in the first few seconds. The impacts of the blocks thus occur at different moments for each test, giving rise to measurement perturbations which are more evenly spread over time for series 100 =, to such an extent that the standard deviation of the measured signals largely exceeds the target standard deviation σ_th.=0,02 g throughout the duration of the test.

Still in zoom B, it can be observed that the standard deviation of the measured signal in series 100 ≈is higher than the standard deviation of the residual mea- surement noise even away from the above-mentioned spikes. However, the target accelerations of this part are identical, since^≈_rAt.=⁼₁At., r∈1,100. This highlights the experimental difficulty in exactly reproducing a given target signal.

To conclude, let us note that a notation similar to that used for accelerations was used to distinguish real rotational velocities_rΩr., their measured values_rΩm., and the rotational velocities calculated by the various numerical models_rΩc.. Fur- thermore, to distinguish the crv of the different axes, these are notated Ω^(θ) for the main axis, Ω^(φ) for the transverse axis andΩ^(ψ) for the longitudinal axis (see Figure4.2).