Principaux résultats et explication succincte de l’article
3.4 Numerical model
— The crv around the transverse axis (figure3.3c) has a quasi-constant, slightly convex envelope, and varies linearly between positive and negative values (figure3.3i).
— The crv around the longitudinal axis (figure3.3d), after a first peak quasi- concomitant with the main impact, exhibits a highly concave envelope which is almost equal to zero at the beginning and at the end of the interval. The crv varies from positive to negative values, following a curve composed of two segments (figure3.3j).
— The crv measured around the main axis is not smooth. A slight “sawtooth”
pattern is superposed to the main curve (figure3.3b) which disappears in some runs between the second and the third mis (figure3.3e).
Figures 3.3l,3.3mand3.3nreveal a correlation between the three compo- nents of the angular velocity. Actually, the peaks of the three components are concomitant. Moreover, the aforementioned sawtooth pattern of the crv around the main axis disappears only when the out-of-plane motion stops (figures3.3e, 3.3fand3.3g). This observation demonstrates the inter-dependence between the out-of-plane and in plane motions. Therefore, as already mentioned, the ques- tion of the influence of this inter-dependence on the earthquake response of rigid blocks arises and must be investigated.
Requirements for the numerical model
An accurate numerical model should be able to reproduce the observed ex- perimental response. As already mentioned, this response can be summarized in three main phases : (i) a 3D repeatable pattern (i.e.the same for each run) which appears to be quasi-periodic during the first 6 second à 7 second mis, (ii) a 3D part that seems to be more or less erratic and, (iii) eventually a quasi-2D behavior.
Given that all runs show a quasi-identical 3D motion until the second mi, the numerical model should predict, with high accuracy, the crvs during this time interval. Moreover, the above repeatable response suggests that the initiation of the out-of-plane motion has a deterministic cause, which is the same for all runs.
Since the block was not placed exactly on the same spot before each test, the au- thors assume that the 3D behavior is induced by an asymmetry of the block itself and not by local defects of its support plate.
present in Newmark’s explicit algorithms. Finally, the natural outputs of Simo’s algorithm are convected rotational velocities, which are also the quantities mea- sured experimentally.
Contact non-linearities
The four feet of the block are not modeled explicitly but are represented as nodes. Impact and friction are modeled by the penalty method [9, 40]. Never- theless, in this case, penalty parameters are not purely numerical artifices used to impose contact conditions. Instead, they reflect physical, local flexibility and dissipation of the feet that are not taken into account by the rigid body model of the block. For each node, the contact forceFis computed from a penetration lengthξ, its time derivative ˙ξand the sliding velocityvglis.(relative tangential velo- city between the support and the node). In the following, superscripts•(N)and•(T) denote respectively the normal and the tangential components with respect to the support plate.
In the present case, the problem of contact is quite complex since contact points can both impact the plate support instantaneously with a moderate velocity (bet- ween 0,01 and 10 m s−1) and remain in contact for some time. Actually, in the lite- rature, different laws exist to describe these two phenomena [18].
In this work, the Kelvin-Voigt law is used for the normal contact force com- ponent, with a small modification to avoid tensile force (whenξ≈0and ˙ξ <0). It uses two parameters, a normal stiffnessK(N)and a normal viscous dampingC(N):
ifξ≥0 F(N)=maxK(N)ξ+C(N)ξ,˙ 0 (3.5a)
ifξ <0 F(N)=0 (3.5b)
Regarding friction, this model uses Coulomb’s law with a static friction coeffi- cientμ0and a dynamic friction coefficientμ.
if vglis.=0 F(T)> μ0F(N) (3.6a) if vglis.>0 F(T)= −μF(N) vglis.
vglis. (3.6b)
where•denotes the usual Euclidean norm onR3.
The discontinuity of Coulomb’s law is usually regularized with an artificial stiff- ness K(T) and damping C(T). In this study, both stiffness and damping parame- ters corresponds to physical properties of the feet. Hence, their values will be set up based on an optimization procedure to fit the experimental results instead of choosing artificial very high values. In the case of a stiffness penalty regularization the algorithm is based, classically, on the elastoplastic analogy and a radial return mapping as in [40]. This algorithm is, in essence, the same as in [17]. In the case of both stiffness and damping penalties, a similar radial return mapping is used.
At each time-step, first, a trial friction force corresponding to a sticking condition is considered. Then, if this trial force exceeds Coulomb’s sliding threshold, both the friction force amplitude and the contact point position are updated. The in- terested reader can find more details on the specific integration algorithm of the impact-friction law, in the case of both stiffness and damping penalties in [41].
Defects
As already mentioned, the observed out-of-plane motion can be explained only by introducing some defects in the model or eccentricities of the initial condi- tions. At the early stages of this study, several possible explanations were conside- red for a rigid body resting on four contact points only. These comprise out of plane initial angle of the pulling wire, mass eccentricity, unequal stiffness and/or
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Article:Modèle ArticleComm.Fiabilité Formu-lationConclusiondamping values of the four feet, different friction coefficients for each foot, dif- ferent contact normal directions at each foot (i.e. it has been assumed that though the four contact points lie on the same plane the base plate is not perfectly flat).
Combinations of these possible scenarios have also been investigated. However, these models failed to adequately represent, qualitatively and quantatively, the complex behavior of the experimental block. In particular, they were not able to represent the transition between the 3D and the quasi-2D motions observed at the end of the records and described in section 3.3. That is why, the numerical model was enriched by introducing four supplemental contact points (one sup- plemental contact point per foot, Figure 3.2b). This eight asymmetric contact points (8ACP) model is an effort to account, in a simplified fashion, for inevitable defects at the individual foot level, due to its finite size.
As shown in Figure3.2b, this model considers four external contact points at the four corners of the block and four internal contact points located on the two bisectors of the lower face of the block.
The four internal points are assumed to belong to the same plane. To reduce the number of parameters to be identified, the horizontal position of these points along the two bisectors is defined by a single parameterδdiag.. The vertical distance of each external point from the four internal points is defined by the parame- ter δlong.i , ibeing the number of the corner. It turns out that the identified values of these parameters are such that, during the motion, the exterior points are in contact with the base plate when the block tilt angle is big, whereas, the interior points are in contact with the support plate when the block is slightly tilted.
Parameter identification
The above geometrical defects, though large enough to cause a non-negligible out-of-plane motion, are too small to be measured by usual instruments. Moreo- ver, the values of the penalty parameters (normal and tangential stiffness and dam- ping for each contact point) are not knowna priori. Therefore the values of these parameters have been determined through an optimization procedure by means of an evolutionary algorithm [42].
This algorithm minimizes an error or cost function which is a measure of the divergence between the results given by a numerical model, itself defined by a set of parameters called “chromosome”, and a reference. In the present case, the nu- merical model is the above model of the block with defects and the reference is the experimentaly recorded response of a release test (Run 01). At each iteration, cal- led “generation”, the best chromosomes are used to create the new chromosomes that have to be tested. In [43], this technique was applied to find the optimal para- meters of a contact model in a similar case (large number of parameters and highly non-linear cost function). A lot of development in the field of numerical optimi- zation is still going on (e.g.[44]). For the problem in hand, the definition of the cost function is not an easy task and should be carried out empirically guided by the specificity of the physical processes and engineering judgment. In the present case, the evolutionary algorithm focuses on the “quasi-periodically” repeated out- of-plane motion (first two bullets in the description of the experimental results in section 3.3). Therefore, the main effort is concentrated to reproduce accurately the response of the block until the second mi. Amongst others, the cost function accounts for :
— the duration between the first two main impacts,
— the envelopes of the crv signals, measured around the transverse and lon- gitudinal axes, clipped between the two first main impacts ,
— the Fourier spectra of the same crvs signals, which give relevant informa- tions about both the frequency content and the pattern of the out-of-plane motion (e.g.linear or bilinear oscillations in Figures3.3iand3.3j. The resulting error function is not presented here because it cannot be expres- sed in a simple closed form. Furthermore, it should be acknowledged that there
is not a unique possible choice of the cost function. Several other cost functions could have been envisaged. The important point is not the precise form of the error norm but whether the so determined parameters’ values give satisfactory results. Actually, as it will be shown in the sequel, the chosen cost function leads to a good agreement between the analytical and experimental results.
Initially, there were 54 parameters processed by the optimization algorithm.
Actually, there are six physical properties parameters per contact point (the tan- gential and normal stiffness and damping constants and the static and sliding co- efficients of friction), five geometrical parameters (δlong.i for i = [1· · ·4] andδdiag.) and the initial tilt angleθ^0.
Reducing the number of parameters
Monitoring of the evolution of the best chromosomes for some generations reveals that certain parameters evolve in a similar fashion. For instance, this is the case for the stiffness of the four contact points. Using this observation, the number of parameters was reduced as follows in order to speed up the convergence of the evolutionary algorithm.
First, two sets of parameters of physical properties are considered, one for all internal contact points (subscript•int.) and another for all external contact points (subscript•ext.). The normalK(N)and tangential stiffnessK(T)are assumed to have the same value for internal and external contact points. Hence :
K(T)int. =K(T)ext.=K(T)
K(N)int. =K(N)ext.=K(N) (3.7) The viscous damping parameters,C(T)andC(N), are respectively defined by the critical damping ratiosβ(T)int.,β(T)ext.,β(N)int. andβ(N)ext.:
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
C(T)int.=2β(T)int.√ K(T)m C(T)ext.=2β(T)ext.
√K(T)m C(N)int.=2β(N)int.√
K(N)m C(N)ext.=2β(N)ext.√
K(N)m
(3.8)
where m is the mass of the block (54 kg). Moreover, the same critical damping ratio is used in the tangential and normal directions whereas different values are assigned to internal and external contact points.
β(T)int. =β(N)int.=βint.
β(T)ext.=β(N)ext.=βext. (3.9) Regarding the friction coefficients, there is no distinction between static and dynamic friction coefficients, but different values are considered for internal,μint.
and external,μext.contact points.
Finally, the 8ACP model requires six contact parameters, five geometrical de- fect parameters, and one initial condition. Hence, the chromosome is a vector with 12 components. The numerical analysis was carried out with a time stepδT=10−5s, and the chromosome shown in table3.1was retained after 250 generations of 40 chromosomes (around 10 000 simulations). This chromosome results in the lo- west value of the cost function after a large number of iterations. The employed method is heuristic and does not provide any proof that the given set of parame- ters corresponds to the absolute minimum of the cost function.
The number of significant digits in this table corresponds to the precision re- quired to have no significant changes in the response of the numerical model if a variation of the same order of magnitude is considered, separately for each para- meter, while the others remain unchanged.
1. The signals used in these representations correspond to two mediocre results (and not to a result and the actual reference) in order to better understand the process graphically.
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Article:Modèle ArticleComm.Fiabilité Formu-lationConclusionI:Constructiond’unmodèlemécaniquedéterministe
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Figure 3.4 –Comparison between numerical and experimental results of a release test.For all Figures, abscissae are the time (s) and ordinates are the convected rotational velocities (deg/s) around the axis written on the left. Housner’s model is only presented around the main axis.
3.4a: global results and contact indicis plot.
3.4bto3.4eand3.4hto3.4j: Zoom in on Figure3.4a. 3.4fand3.4g: Zoom in on Figures3.4cand3.4d.
3.4hto3.4j: the vertical red lines represent the main impacts.
45bis
Parameter Value K(N) 6×107N m−1 K(T) 5×107N m−1
βint. 0,15
βext. 0,18
μint. 0,30
μext. 0,25
Parameter Value
δlong.1 0,010 mm
δlong.2 0,040 mm
δlong.3 0,092 mm
δlong.4 0,058 mm
δdiag. 6,0 mm
^θ0 0,1170 rad
Table 3.1 – Optimal parameter values found by the evolutionary algorithm by comparison with the first experimental run, using a time-step for the numerical integration equal toδT=10−5s.
Comparison between experimental and analytical results
Figure3.4shows the experimental and numerical crvs as well as the crv around the main axis of Housner’s 2D model . In addition to the crvs, the contact time- histories for each point of the 8ACP model are shown (a line is drawn as long as a point remains in contact).
Figure 3.4a also confirms that, as expected, Housner’s model gives a good approximation of the main (in plane) motion.
As already mentioned in section 3.3, the analytical model should be able to reproduce accurately the response between the first and the second main impacts which was quasi-identical for all runs. The proposed analytical model meets this requirement. In fact :
— The transverse motion is well reproduced (figure3.4c). The numerical and experimental motions follow the same sawtooth like pattern, share the same slightly convex envelope and have the same frequency and phase. One can notice that the experimental transverse motion which starts, “surprisingly”, 0,02 s before the mi (see at 0,51 s in Figure3.4f), is accurately represented by the model. The plot showing the time-histories of the contact indices gives an explanation to this phenomenon (contact plots of Figure 3.4a).
Actually, it can be seen that the motion around the transverse axis (Y) is initiated by the transition between adjacent, exterior and interior contact points (transition from1ext.and2ext.to 1int. and2int.). Then, a higher ampli- tude rocking motion starts when transition from contact point3ext.and4ext.
occurs. A similar interpretation can be made for the motion around the lon- gitudinal axis (Z) (figure3.4d). However, the fact that the first experimental peak, at 0,55 s, is not predicted by the model must be acknowledged. Mo- reover, the crv around this axis is slightly overestimated by the numerical model3.4g)
— The “sawtooth” pattern of the main motion (figure3.4b) is accurately re- produced. It has the same frequency as the motion around the axis Y and Z, and the amplitudes of the oscillations are the same for the numerical and experimental motions. Of course, Housner’s 2D model cannot predict this particularity which is due to the existence of a secondary, out-of-plane mo- tion.
After the second mi, the experimental and numerical time-histories may diverge more. Nevertheless, this divergence is within the range of variability of the results of the various runs of the experimental campaign (figure3.3a). In particular, the quasi-periodicity of the pattern described in the previous section is well reprodu- ced between the second and the sixth mi. Furthermore, the instant of transition between the 3D behavior and the quasi-2D behavior is also well reproduced.
A very good correlation between the numerical and experimental time-histories can also be noticed between the ninth and eleventh main impacts (figures 3.4h to3.4j). This holds for the three axis simultaneously, demonstrating that the un- derlying physics are well represented by the analytical model.
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Figure 3.5 –Sensitivity to the time-step.For all figures, abscissae are the time (s) and ordinates are the convected rotational velocities (deg/s) measured around the axis written on the left.
3.5bto3.5d: zoom in on Figure3.5a.
Empirical sensitivity analysis
As mentioned previously in the introduction, the rigid block rocking motion is well-known to be very sensitive to small variations of the experimental or nu- merical conditions. Although not statistically representative, because of the small number of runs studied, the experimental variability presented in Figure 3.3 gives a rough estimate of the expected level of the model’s variability. To gain insight into the sensitivity of the numerical model, an empirical sensitivity ana- lysis was carried out. The results of this analysis are shown in Figure3.7, 3.6 and3.5. They correspond to :
— the results obtained with the optimal values of the model’s parameters (see table3.1), taking into account three different values of the time-step used for the numerical integration (see Figure3.5) ;
— the results obtained with the initial angle given in table3.1and the original
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Article:Modèle ArticleComm.Fiabilité Formu-lationConclusionb
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Figure 3.6 –Sensitivity to the values of physical parameters.For all figures, abs- cissae are the time (s) and ordinates are the convected rotational velocities (deg/s) measured around the axis written on the left.
3.6bto3.6d: zoom in on Figure3.6a.
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^θ0+0,08° reference
^θ0−0,08°
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Figure 3.7 – Sensitivity to the initial tilt angle.For all figures, abscissae are the time (s) and ordinates are the convected rotational velocities (deg/s) measured around the axis written on the left.
3.7bto3.7d: zoom in on Figure3.7a.
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Article:Modèle ArticleComm.Fiabilité Formu-lationConclusiontime 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).