4.4 8ACP and 4SCP models
4.8 Conclusion
The aim of this article has been to assess the capacity of a numerical model to represent, over time, the dynamic behavior of an imperfect rigid slender block subjected to seismic excitations. To this end, a series of experiments was per- formed, using a shaking table at CEA/Saclay, on 3 solid steel rigid blocks subjected
o.Biblio. Article:Modèle Chapitre
4
Article Comm.Fiabilité Formu-lationConclusion
to 2 samples of 100 unidirectional seismic accelerations. First the blocks were sub- jected to 100 stationary zero-mean Gaussian signals, defined by low-level Kanaï- Tajimi power spectral density (standard deviation: σth. = 0,02 g) preceded by an initial pulse to initiate movement. Then the blocks were subjected to 100 times the same theoretical acceleration corresponding to one of the realizations of the first series. These tests were analyzed and compared with two numerical models, one with symmetrical geometry (model 4SCP) and the other with asymmetrical geometry (model 8ACP), with the aim of reproducing out-of-plane behavior iden- tified both during release tests and unidirectional seismic tests. To perform these analyses, various indicators were specifically developed and are presented in the article.
Despite experimental uncertainties related to the control of the shaking table and inevitable measurement errors, and to the initial conditions of the blocks, the experimental results highlight a good reproducibility of block movement initi- ation following the sinusoidal pulse to which they were subjected. Thus it was observed that movement initiation was characteristic of block defects, as was the out-of-plane movement exhibited during the release tests. Furthermore, under unidirectional acceleration, these tests showed that a block whose defect spatial distribution induces a generally limited out-of-plane movement (block 2 in our case, see Table4.1) dissipates less energy during its movement than a block whose imperfections give rise to a major planarity defect (the case of block 1).
The numerical simulations corroborated these observations. In addition, they enabled a better comprehension of the dynamic behavior of the blocks, by show- ing:
— that on receiving a pulse, a perfect block acquires less energy than a block with a defect. In the present case, this was explained numerically by the fact that a block with a non-null defectδdiag. behaves, when set into motion by an excitation at its base, like a flawless block of greater slenderness;
— that, counter-intuitively, model 8ACP enables the prediction of the move- ment of a real steel block for a greater length of time than an experiment performed on a supposedly identical block;
— that despite experimental uncertainties the statistical match between the 8ACP model’s results and the experimental results was relatively good, both in terms of mean and standard deviation and in terms of the distribution of the crvand energy over time;
— that a flawless symmetrical model with a realistic slenderness correction, deduced from release tests, can provide an upper bound for overturn prob- ability for 1D excitations in the block axis, but not for 2D excitations.
crv(rads−1) Bloc1exp.=Ωm. (θ)
−0.5 0
graph 2
crv(rads−1) Bloc2exp.=Ωm. (θ)
−0.5 0 0.5
graph 3
crv(rads−1) Meanandstandarddeviation
−0.2 0 0.2
Bloc 1 exp. Bloc 2 exp.
Bloc 1 8ACP Bloc 4SCP Exemples
37 31
37 42
37 89 89
25 31
36 47 63 73 61
74 75
82 87
88
graph 4
cumulative distribution(–) P τAr.ΔΩ,τ>τ
0 0.2 0.4 0.6 0.8
1 Chutes
graph 5
crv(rads−1) Bloc18ACP=Ωc. (θ)
−0.5 0 0.5
graph 6
time (s)
Ωc.(rads−1) Bloc4SCP=Ωc. (θ)
−2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
−0.5 0 0.5
Figure 4.14 –Comparison of rotational velocities in the main axis for 100 tests with theoretically different accelerations (100=):
4.14graph 1: Deciles of measured experimental rotational velocity for block 1.
4.14graph 2: Deciles of measured experimental rotational velocity for block 2.
4.14graph 3: Means and standard deviations of the four 100-samples presented.
4.14graph 4: Proportion of blocks in motion over time.
4.14graph 5: Deciles of numerical rotational velocity calculated using model 8ACP.
4.14graph 6: Deciles of numerical rotational velocity calculated using model 4SCP.
crv(rads−1) Bloc1exp.=Ωm. (φ)
−0.1 0 0.1
graph 2
crv(rads−1) Bloc2exp.=Ωm. (φ)
−0.1 0 0.1
graph 3
crv(rads−1) Mean
−0.1
−0.05 0 0.05 0.1
Bloc 1 exp. Bloc 2 exp.
Bloc 1 8ACP Exemples graph 4
crv(rads−1) Standarddeviation
0 0.02 0.04 0.06 0.08
graph 5
time (s)
crv(rads−1) Bloc18ACP=Ωc. (φ)
−2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
−0.1 0 0.1
Figure 4.15 –Comparison of rotational velocities in the transverse axis for 100 tests with theoretically different accelerations (100=):
4.15graph 1: Deciles of measured experimental rotational velocity for block 1.
4.15graph 2: Deciles of measured experimental rotational velocity for block 2.
4.15graph 3: Means of the three 100-samples presented.
4.15graph 4: Standard deviations of the three 100-samples presented.
4.15graph 5: Deciles of numerical rotational velocity calculated using model 8ACP.
Note: it is not relevant to study the movement about the transverse axis for model 4SCP; in this ideal case there is strictly no movement.
energy(J) Bloc1exp.
0 0.5 1 1.5
graph 2
energy(J) Bloc2exp.
0 0.5 1 1.5 2
graph 3
energy(J) Mean
0 0.5
1 Bloc 1 exp. Bloc 2 exp.
Bloc 1 8ACP Bloc 4SCP Exemples
graph 4
energy(J) Standarddeviation
0 0.2 0.4 0.6
graph 5
energy(J) Bloc18ACP
0 0.5 1 1.5 2
graph 6
time (s)
energy(J) Bloc4SCP
−2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
0 0.5 1 1.5 2
Figure 4.16 –Energy comparison for the 100 tests with theoretically different accelerations (100=):
4.16graph 1: Deciles of experimental energy for block 1.
4.16graph 2: Deciles of experimental energy for block 2.
4.16graph 3: Means of the four 100-samples presented.
4.16graph 4: Standard deviations of the four 100-samples presented.
4.16graph 5: Deciles of energy calculated using model 8ACP.
4.16graph 6: Deciles of energy calculated using model 4SCP.
crv(rads−1) Bloc1exp.≈Ωm. (θ)
−0.5 0
graph 2
crv(rads−1) Bloc2exp.≈Ωm. (θ)
−0.5 0 0.5
graph 3
acc.(g) ≈rAm.
−0.05 0 0.05
graph 4
crv(rads−1) Mean
−0.4
−0.2 0 0.2 0.4
Bloc 1 exp. Bloc 2 exp.
Bloc 1 8ACP Bloc 4SCP graph 5
crv(rads−1) Standarddeviation
0 0.1 0.2 0.3
graph 6
crv(rads−1) Bloc18ACP≈Ωc. (θ)
−0.5 0 0.5
graph 7
time (s)
crv(rads−1) Bloc4SCP≈Ωc. (θ)
−2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
−0.5 0 0.5
Figure 4.17 –Comparison of rotational velocities in the main axis for the 100 tests with theoretically identical accelerations (100≈):
4.17graph 1: Deciles of experimental rotational velocity for block 1.
4.17graph 2: Deciles of experimental rotational velocity for block 2.
4.17graph 3: Mean of the 100 measured accelerations.
4.17graph 4: Means of the four 100-samples presented.
4.17graph 5: Standard deviations of the four 100-samples presented.
4.17graph 6: Deciles of rotational velocity calculated using model 8ACP.
4.17graph 7: Deciles of rotational velocity calculated using model 4SCP.
crv(rads−1) Bloc1exp.≈Ωm. (φ)
−0.1 0 0.1
graph 2
crv(rads−1) Bloc2exp.≈Ωm. (φ)
−0.1 0 0.1
graph 3
acc.(g) ≈rAm.
−0.05 0 0.05
graph 4
crv(rads−1) Mean
−0.05 0 0.05
Bloc 1 exp. Bloc 2 exp.
Bloc 1 8ACP Bloc 4SCP graph 5
crv(rads−1) Standarddeviation
0 0.02 0.04 0.06 0.08
graph 6
cumulative distribution(–) P τAr.ΔΩ,τ>τ
0 0.2 0.4 0.6 0.8
1 ∅Chutes
graph 7
time (s)
crv(rads−1) Bloc18ACP≈Ωc. (φ)
−0.1 0 0.1
Figure 4.18 –Comparison of rotational velocities in the transverse axis for the 100 tests with theoretically identical accelerations (100≈):
4.18graph 1: Deciles of experimental rotational velocity for block 1.
4.18graph 2: Deciles of experimental rotational velocity for block 2.
4.18graph 3: Mean of the 100 measured accelerations.
4.18graph 4: Means of the four 100-samples presented.
4.18graph 5: Standard deviations of the four 100-samples presented.
4.18graph 6: Proportion of blocks in motion over time.
4.18graph 7: Deciles of rotational velocity calculated using model 8ACP.
energy(J) Bloc1exp.
0 0.5 1 1.5
graph 2
energy(J) Bloc2exp.
0 0.5 1 1.5 2
graph 3
acc.(g) ≈Am.
−0.05 0 0.05
graph 4
energy(J) Mean
0 0.5 1
Bloc 1 exp. Bloc 2 exp.
Bloc 1 8ACP Bloc 4SCP graph 5
energy(J) Standarddeviation
0 0.2 0.4 0.6
graph 6
energy(J) Bloc18ACP
0 0.5 1 1.5 2
graph 7
time (s)
energy(J) Bloc4SCP
−2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
0 0.5 1 1.5 2
Figure 4.19 – Energy comparison for the 100 tests with theoretically identical accelerations (100≈):
4.19graph 1: Deciles of experimental energy for block 1.
4.19graph 2: Deciles of experimental energy for block 2.
4.19graph 3: Mean of the 100 measured accelerations.
4.19graph 4: Means of the four 100-samples presented.
4.19graph 5: Standard deviations of the four 100-samples presented.
4.19graph 6: Proportion of blocks in motion over time.
4.19graph 7: Deciles of energy calculated using model 8ACP.
Chapitr e
5 Discussion
Les résultats décrits dans les deux articles précédents méritent encore d’être discutés. Leur format obligeant à la concision, nous proposons dans ce chapitre deux commentaires supplémentaires.
Tout d’abord, il sera question de la propagation des erreurs expérimentales de la campagne de 2014. Nous tâcherons de référencer les différentes sources d’erreur ou d’approximation, et la façon dont elles se propagent.
Ensuite, nous reviendrons sur la reproductibilité des essais de basculement de blocs rigide et élancés sous un signal sismique déterminé.