Statistical study of stability under seismic excitation

time step, taking into account some variations on the model’s parameters.

For each reference parameter given in table3.1, a variation of±0,5of the last significant digit (s.d.) is randomly considered. In Figure3.6, two results of calculations are presented, each one corresponding to a chromosome with 11 modified parameters (e.g.:K^(N)=6×10⁷N m⁻¹→K^(N)=6±0,5×10⁷N m⁻¹) ;

— the results obtained with the optimal values of the model’s parameters (see table3.1), taking into account different release angles which correspond to a retention wire pulled 1 mm more or less. In fact, these values, which cor- respond to an angle variation of±0,08°, seem realistic considering the ex- perimental conditions (see Figure3.7).

For this kind of problem, the well-known sensitivity of the response to the time step does not allow us to obtain convergent analytical results for the whole duration of interest, with a reasonable computational cost. Therefore, the time- step can be viewed as a source of uncertainty. As expected, this empirical analysis shows that reducing the time step does not alter the early stages of the motion (Figures3.5bto3.5d), but induces significant changes only after several main impacts. The resulting variability seems to be within the range of experimental variability.

Regarding the sensibility of the numerical model to the parameters’ values, the induced variability of the analytical results remains within the range of the experimental variability (Figure 3.3). This observation justifies the number of significant digits retained in the parameters’ values given in table3.1.

Finally, the variability on the initial release angle induces only a slight shift in time of the global response, similar to that observed during the experimental campaign (Figure3.3c).

dernièremiseàjour:10janvier2017

±γ^gr.(eq.3.3a)

acceleration(g) groundmotion

−0.2 0

0.2 axis N–S axis E–W

X-axis

−20 0 20

convectedrotationalvelocities(deg/s) Y-axis

−40

−20 0 20

40 8ACP 4SCP

time (s)

Z-axis

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

0 50

Figure 3.8 – Sample of realizations of the bidirectional excitation signals and analytical results corresponding to the asymmetrical and symmetrical models

H=2b

0.1 0.12 0.14

class A

class B

class C

8ACP 4SCP

γgr.(eq.(3.3a))

H=2bsin(α)

PGA (g)

maximumverticaldisplacementH(m)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.005 0.01 0.015

(a) Scatter plots of the stability index for the two analytical models according to the PGA. 350 points are displayed.

H=2b

0.1 0.12 0.14

8ACP 4SCP

class A class B class C

H=2bsin(α)

probability density

maximumverticaldisplacementH(m)

0 50 100 150 200 250 300

0 0.005 0.01 0.015

(b) Probability density function of the stability index for the two analytical models. The solid line refers to the 8ACP model and the dotted line to the 4SCP model.

Figure 3.9 – Evolution of the maximum exterior contact point vertical displacement as function of the PGA

a bidirectional excitation, the notions of main and transversal axes do not make sense, that is why they are noted X and Y in the figure.

To obtain a statistical estimate of how these defects influence the response of a rigid block under seismic excitations, we considered as a relevant index of rocking amplitude and even overturning, the maximum of the vertical displacement of the external contact pointsH:

H= max

τ∈[0,T]

imax∈1···4(Hiext.(τ))

(3.10) whereHiext.(τ)is the vertical displacement of the exterior pointiat timeτ.

This index can be compared to the value2b sin(α), which corresponds to the vertical displacement of one of the two contact points of Housner’s model, when

|θ(τ)|=α(figure3.1). This value is a necessary threshold for toppling.

The results of the empirical statistical study are shown in Figure3.9. The left part3.9apresents the scatter plot of the maximum vertical displacement versus the peak ground acceleration (PGA) of the two-directional excitation, defined as :

PGA= max

τ∈[0,T]

γ^N–S(τ)2

+

γ^E–W(τ)2

(3.11)

For the sake of better readability, only 350 randomly chosen points are dis- played. Moreover, the intermediate portion of the vertical axis is not shown be- cause of the considerable range of possible values of the maximum vertical dis- placement. In fact, when a block has overturned, the vertical displacement of at least one contact point has necessarily exceeded the threshold2b.

Figure3.9bshows the probability density functions (PDF) for three classes of PGA (A : 0,170 g à 0,194 g, B : 0,223 g à 0,251 g, C : 0,283 g à 0,317 g). Each one contains 350 pairs of accelerograms. These classes are shown by the three strips of color in Figure3.9a. The same colors are used for the plot of the PDFs in Fi- gure3.9b. The solid and dotted lines correspond to the 8ACP and 4SCP models respectively. It may be observed that the responses of the two models are different, especially for the first class. This class corresponds to weak excitation levels, yet, higher than the theoretical PGA which induces rocking motion (≈0,144 g). Higher contact point vertical displacement values,H, are obtained and even overturning is possible in the case of the asymmetrical model, contrary to the case of the sym- metrical model. One possible explanation is that the effective base width of the 8ACP model is smaller than that of the 4ACP and thus implies a higher effective slenderness. A similar result was observed in an example treated in [17] where a four corner modal results in a more stable response than a distributed Winkler- like compliance contact model. In fact, the 8ACP model could be thought as a very particular case of the Winkler model, where contact compliance is distri- buted only over the feet area which may be not flat. Even more, this distributed flexibility is discretized with only two points per foot. Both, the 8ACP model pre- sented here and the Winkler model in [17], increase the apparent slenderness and are more prone to rocking and toppling.

Table3.2compares the predictions according to Ishiyama’s criteria to the ana- lytical results of this study for both models. Green and orange cells correspond to the 4SCP and 8ACP models respectively. The first column on the left presents the number of signals for which Ishiyama’s criteria predict that the block does not move, could rock or could overturn. The three remaining columns present the behavior given by the analytical results. For instance, the orange cell at the bot- tom of the second column shows that amongst the 1717 signals which according to Ishiyama could have lead to overturning, 235 did not move the block at all. Ishiya- ma’s criteria are established for one-directional excitations and plane motions. To apply them in the case of a bi-directional excitation,γ(τ)andv_sol(τ)will be taken as the square root of the sum of the square (SRSS) of the ground acceleration and

o.Biblio. Chapitre

3

8ACP 4SCP Results of 3D numerical simulations :

stay still rock overturn

Ishiyama’s equations prediction for a 2D perfect block :

stay still 343 (16,2 %)

343 (16,2 %) 0 (0 %) 0 (0 %) 343 (16,2 %) 0 (0 %) 0 (0 %) rock

eq. (3.3a) 57 (2,7 %)

53 (2,5 %) 4 (0,2 %) 0 (0 %) 57 (2,7 %) 0 (0 %) 0 (0 %) overturn

eq. (3.3a) (3.3b) 1717 (81,2 %)

235 (11,1 %) 446 (21,1 %) 1036 (48,9 %) 410 (19,4 %) 351 (16,6 %) 956 (45,2 %) Table 3.2 – Comparison between Ishiyama’s criteria and the results of the nume- rical models.

velocity respectively :

γ(τ) =

γ^N–S(τ)²+γ^E–W(τ)² (3.12) The results in Table3.2show that, in general, the 8ACP model is more prone to rocking and overturning than the symmetrical flawless model. It is also noticed that Ishiyama’s criteria are conservative. Actually, as already mentioned, these cri- teria, which are not theoretically proven, are considered to be necessary but not sufficient conditions. These results are in agreement with the authors’ experience from past analytical studies on overturning of rigid blocks. In fact, they never met a case of overturning which could not have been predicted by Ishiyama’s overtur- ning criterion.