4.4 8ACP and 4SCP models
4.6 Representativity of models subjected to random excitations
a significant modification of the rest of the movement after the pulse.
Let us note that the experimental movement is always regular at its initiation. In this phase of the movement, the instability of the numerical behavior is thus one of the limits of the 8ACP model, caused by the geometry defined by discrete points and amplified by measurement noise.
Summary In conclusion we can say that a symmetrical model is unsatisfactory for this type of excitation, since it greatly underestimates the energy accumulated by the block during the initial pulse, which leads to an underestimation of the amplitude of the rest of the movement. The 8ACP model can provide a correct estimation of the beginning of the movement, but is very sensitive to measure- ment noise when rocking is initiated.
time (s)
cumulative distribution(–) P τ Div.Δy,Δτ,τ>τ
0.08 2 4 6 8
0.2 0.4 0.6 0.8 1
(a)
time (s)
cumulative distribution(–) P τ Div.Δy,Δτ,τ>τ
0.08 2 4 6 8
0.2 0.4 0.6 0.8 1
(b)
Key to Figures4.9aet4.9b:
Comparison between the ex- perimental main axis move- ment of block 1 and its simu- lation using the 8ACP model.
Same, but the signal lasts only for the period represented by the red zone .
Comparison between the ex- perimental main axis move- ment of block 1 and its simu- lation using the 4SCP model.
Envelope of the three curves comparing the experimental main axis movements of blocks 1 and 2, 2 and 3, then 1 and 3.
Parameters:
ΔΩ=0,03 rad s−1 Δτ=0,04 s τ=5δT=5 ms
Figure 4.9 –Instants of divergence, trajectory by trajectory:
4.9a: Distribution of the instant of divergenceτDiv.ΔΩ,Δτ,τ for series100 ≈.
4.9b: Distribution of the instant of divergenceτDiv.ΔΩ,Δτ,τ for series100 =.
time (s)
crv(rads−1) ≈100Ωm. (θ)et ≈100Ωc.(θ)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.5 0
0.5 Bloc 1 exp. Bloc 2 exp.
Bloc 1 8ACP Bloc 4SCP
Figure 4.10 –Comparison of experimental and numerical resultswhen the cal- culation initiates atτ=τ0.
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effect of the random signals on the estimation ofP(τDiv.> τ). To this end, the block 1 experimental results and the numerical simulation results corresponding to the post-pulse free movement of the 8ACP model are compared using indicatorτDiv.. Figure4.9presents this set of results. In general, it appears that the variation in P(τDiv. > τ)is more regular for the 100-sample100 =than for the 100-sample 100 ≈. This illustrates the expected dependency of this indicator with respect to the signal. This will be covered in detail in the following paragraphs.
Disparities in experimental results between the different blocks The gray area in Figure4.9represents the envelope of three curves corresponding to the estima- tions ofτDiv.with the experimental results (i) for block 1 and block 2; (ii) for block 2 and block 3; (ii) for block and block 3. This results provides a reference value:
the time necessary for experimental trajectories of two experimental blocks, sub- jected to the same real accelerationrAr.(τ), to diverge significantly. Thus it appears that beyond7 s, all the experimental trajectories have diverged.
Disparities in experimental results and predictions of different models The es- timations ofP(τDiv. > τ), evaluated from the numerical results of models 8ACP or 4SCP on the one hand and the experimental results for the block on the other hand, correspond respectively to the orange and green solid lines in Figure 4.9.
These estimations show that divergence between the experimental results and those of the 4SCP symmetrical model occurs very rapidly, despite the initial con- ditions imposed at instantτ0(equation4.5).
Figure4.10, which presents the crvin the main axis of experimental blocks 1 and 2, as well as those of models 8ACP and 4SCP, shows that the previous result is due to the fact that model 4SCP does not predict the “micro-impacts” which oc- cur between 0,1 and 0,22 s, unlike model 8ACP. Thus it appears that, as expected, model 4SCP is relevant in representing the behavior of a block having less trans- verse movement, like block 2.
Finally, Figure 4.9 also shows that for test series 100 =, model 8ACP almost exactly predicts the experimental behavior of block 1 for at least 1 s, in half the cases for 2 s and sometimes up to 5,5 s. In the case of series 100 ≈, divergence between the trajectories of model 8ACP and block 1 occurs principally between 2 and 2,2 s.
Disparities between experimental results and post-pulse free movement Ad- ditionally, given the chosen nature of the acceleration signals, the influence of the energy supplied by the sinusoidal pulse on the estimation ofP(τDiv. > τ)was studied. To this end, 100 numerical calculations were performed using the 8ACP model, considering accelerationsrAsin, r∈1,100defined by:
rAsin(τ) =rAm.(τ) , τ∈[τ0,0,2] (interval in red in Figure4.9)
rAsin(τ) =0 , τ∈[0,2, T] (4.6)
It is to be noted that inτ0, the initial conditions applied for the numerical calcu- lations are those defined by equation4.5. The estimation ofP(τDiv. > τ)evaluated from the numerical and experimental results is traced by the dotted line in Fig- ure4.9. Thus, for the two test series100 =and100 ≈, it appears that the instant of divergence between the experimental results and the free movement is on the whole 1 to 2 seconds earlier than the instant of divergence corresponding to the random excitation results. It is thus established that the initial energy supplied by the sinusoidal pulse contributes only weakly to the prediction capacity of model 8ACP estimated in the previous paragraph. The chosen accelerations are thus rel- evant in studying the capacity of the model to predict the behavior of a block subjected to a 1D seismic excitation.
Summary The comparison of these three case studies enables us to conclude that under random excitation, despite acceleration measurement uncertainties,
the numerical 8ACP model provides an estimation of the movement of block 1 which is more accurate than that provided by an identical specimen (blocks 2 and 3, identical within manufacturing tolerances).
Statistical suitability
To test statistically the suitability of the models, we consider the 100-samples corresponding respectively to the experimental movements of block 1 and block 2, to the responses of model 8ACP with the block 1 parameters (defects: ses Table4.1;
initial conditions: see Eq. (4.5)), and to the responses of model 4SCP with the same initial conditions as those applied to model 8ACP. For each case, the 100-samples considered are100 =and100 ≈.
Since some excitation realizations led to the numerical or experimental blocks’
overturning, the first post-processing step was to identify these overturning events.
We recall that the comparison indicators are conditional on the fact that the tra- jectories did not cause the blocks to overturn. Thus, the overturning instants and test numbers are reported in Figures4.14and4.18. Note that there were no numer- ical or experimental overturns for series100 ≈. Shown in the same Figures are (i) the mean, (ii) the standard deviation and (iii) the deciles (from the procedure indi- cated in section4.3) corresponding to the following values: (a) rotational velocity about the main axis, (b) rotational velocity about the transverse axis and (c) energy (calculated according to the procedure explained in section4.3). In addition, the Figures show the variation over time in the proportion of blocks in movement (as presented in section4.3). Superimposed on the curves relating to sample100 =are the traces of realizations which correspond to the same testr=100. The results for the100 =samples are shown in Figures4.14,4.15and4.16, and those for the100 ≈ series are shown in Figures4.17,4.18and4.19. We recall that the axes are indicated in Figure 4.2. The principal interpretations of these results are presented in the following paragraphs.
Differences between the experimental results of block 1 and block 2 For these two series of excitations, two major differences in experimental behavior can be noted between block 1 and block 2: block 2 has less out-of-plane movement than block 1, and takes longer to become motionless. In the graph of experimental block 2 in Figure4.15, the first and last deciles appear during the first mi; the sec- ond and penultimate deciles appear during the second mi, and the other deciles barely appear. This indicates that block 2 exhibits out-of-plane behavior between the 1st and the 2nd impact in less than 20 % of the tests, and between the 2nd and the 3rd in less than 40 % of the tests. However, the 10 deciles appear from the first main impact(mi) for block 1, which means that the block had out-of-plane behav- ior from the first impact in at least 80 %of the tests. This observation holds for the standard deviation: the 3rdgraph in Figure4.15shows that the rotational velocity in the transverse axis of block 2 has a lower standard deviation than the rotational velocity of block 1 over the first 7 seconds. These considerations remain valid for the 100 tests of the same theoretical acceleration (see Figure4.18).
Block 2 also seems to be less stable. The average energy of the blocks in the100 = series (see Figure 4.16) shows that blocks 1 and 2 receive on the whole the same amount of energy from the sinusoidal pulse. However, in the seconds follow- ing the pulse, the energy of block 2 wanes less quickly. As a corollary, this phe- nomenon is also present in the proportion of motionless blocks; it can be observed in Figure4.14that block 2 generally ceases moving later than block 1. This obser- vation also applies to the deciles of the rotational velocity about the main axis (see Figure4.14): the various bands converge towards zero earlier for block 1 than for block 2. In addition, it can be noted that block 2 overturned 4 times, whereas block 1 overturned only once, although this is not statistically significant. This re- sult should be understood (from a statistical point of view) on average for various realizations of the process under consideration. It can indeed be noted that in the
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case of the acceleration chosen for the (100 ≈) repeatability tests, block 2 generally became motionless sooner (see Figure4.18).
Capacity of model 8ACP to represent the experimental movement of block 1 Variability in the results for series100 =is regular over time. The mean, standard deviation and maximum values of the rotational velocity in the main movement axis of block 1 are correctly found by model 8ACP. The proportion of blocks be- coming motionless over time is also quite well identified.
For the series100 ≈, the variability of movement in the main axis for block 1 can be broken down into three phases (see Figure4.17):
— between 0 and 2 seconds, very low variability,
— from 2 to 9 seconds, moderate variability,
— from 9 seconds to the end, strong variability of the same order as that for the results of series100 =.
Each transition is signaled by a sudden increase in standard deviation or a marked divergence of the deciles. The existence of these three phases, the transition in- stants between these phases and the values of the mean and standard deviations in each of the phases are well identified by model 8ACP. These aspects are also clearly visible in terms of energy (see Figure 4.19). However, the proportion of blocks becoming motionless with time is not perfectly retrieved by the model (see Figure 4.18). Furthermore, it can be noted that the first transition, at τ= 2 s, corresponds to a particular instant of divergence between the trajectories (see Fig- ure4.9a). In the third phase, the mean of the rotational velocities in the main axis is nil (see Figure 4.17), and the variability of these trajectories is quite similar to that of the trajectories of series=.
Concerning rotational velocity in the transverse axis, the model largely under- estimates variability in the first seconds. The authors attribute this mainly to the initial condition onψdescribed in equation4.5. This equation is not very realistic, since experimentally the blocks are placed manually on the shaking table. Thus, if experimentally the mean crv measured about the transverse axis is nil for the duration of the test, it requires 3 s for series 100 ≈and 1 s for100 = for the mean crvpredicted by the model to become nil. We can consider that at the end of this time the average behavior of the transverse movement is no longer influenced by the initial state.
Differences between the predictions of models 4SCP and 8ACP As regards the one hundred theoretically different signals (100 =), the movement of the symmet- rical block predicted by the model 4SCP is statistically different from that pre- dicted by model 8ACP. First of all, this model predicts many more overturns (12 overturns, see Figure 4.14). In Figure4.16, the energy of model 4SCP is overall greater than that of model 8ACP, throughout the duration of the test. Similarly, in Figure4.14, the 4SCP blocks remain in motion longer than the 8ACP blocks.
Of course, the symmetrical model predicts no movement in the transverse axis, since the seismic excitation applied to the model of the block is 1D and perfectly aligned with the block’s axis (see Figure4.2).
With respect to the one hundred theoretically identical signals (100 ≈), the re- sults of the two models are still statistically different. The symmetrical block con- tinues to acquire more energy and rotational velocity in the main axis than the 8ACP block under a sinusoidal pulse, but they dissipate more quickly than those of the 8ACP block. This can also be observed for rotational velocity, energy and the proportion of motionless blocks. In particular, it can be seen in Figure 4.19 that for the interval 4 s to 5 s, the experimental variability of the movements of block 1 is low, whereas the energy of the blocks is particularly high (this is less the case for block 2). This particularity is correctly retrieved by model 8ACP, but not at all by the symmetrical model, which predicts that the block can already come to rest (see Figure4.15).