Densité d’une loi

Définition 5 : Densité d’une loiDensité d’une loi

Soit(Ω, F, P) un espace de probabilités et soit Y une variable aléatoire

définie sur(Ω, F, P), à valeurs dans Rn, de loi loiVAY sur(Rp,Bp). On dit

que la loi PY admet une densité par rapport à la mesure de Lebesgue

dy sur(Rp,Bp) s’il existe une fonction pYdeRn dansR+, intégrable sur

Rn, telle que :

∀B ∈ Bn, P_Y(B) =



y∈B

p_Y(y)dy (B.7a)

Dans la définition précédente, la relation (B.7a) peut être remplacée par la sui-vante :

P_Y(dy) = pY(y)dy (B.7b)

La densité pYest encore appelée « densité de la probabilité de Y ». Cette densité

vérifie :  y∈Rn p_Y(y)dy = 1 (B.8) o. Biblio. Article : M odèle Article Comm. Fiabilité F ormu-lation Chapitr e

8

Bibliographie

[1] Nicos Makris. A half-century of rocking isolation. EARTHQUAKES AND

STRUCTURES, 7(6) :1187–1221, 2014.

[2] Matthew D Purvance, Abdolrasool Anooshehpoor, and James N Brune. Frees-tanding block overturning fragilities : Numerical simulation and experimen-tal validation. Earthquake Engineering & Structural Dynamics, 37(5) :791–808, 2008.

[3] A Dar, D Konstantinidis, and WW El-Dakhakhni. Seismic qualification of ro-cking objects in canadian nuclear power plants. In Proceedings of Tenth U.S. National Conference on Earthquake Engineering, 2014.

[4] G Housner. The behavior of inverted pendulum structures during

earth-quakes. Bulletin of the Seismological Society of America, 53(2) :403–417, 1963. [5] Peter Russell Lipscombe. Dynamics of rigid block structures. PhD thesis,

Uni-versity of Cambridge, 1990.

[6] F Peña, F Prieto, PB Lourenço, A Campos Costa, and JV Lemos. On the dy-namics of rocking motion of single rigid-block structures. Earthquake Engi-neering & Structural Dynamics, 36(15) :2383–2399, 2007.

[7] Tibor Winkler, Kimiro Meguro, and Fumio Yamazaki. Response of rigid body

assemblies to dynamic excitation. Earthquake engineering & structural dynamics, 24(10) :1389–1408, 1995.

[8] C Feau. Étude des phénomènes de glissement et

bascule-ment de blocs d’acier soumis au séisme. Technical Report

DEN/DANS/DM2S/SEMT/EMSI/RT/07-012/A, Commissariat à l’énergie atomique et aux énergies alternatives, 2007.

[9] JT Oden and JAC Martins. Models and computational methods for dynamic friction phenomena. Computer methods in applied mechanics and engineering, 52(1) :527–634, 1985.

[10] Desmond F Moore. Principles and applications of tribology, volume 1. Pergamon Press Oxford, 1975.

[11] Ernest Rabinowicz. Friction and wear of materials, volume 2. Wiley New York, 1965.

[12] Dan Stoianovici and Yildirim Hurmuzlu. A critical study of the applicability of rigid-body collision theory. Journal of engineering mechanics, 63(2) :307–316, 1996.

[13] U Andreaus and P Casini. On the rocking-uplifting motion of a rigid block in free and forced motion : influence of sliding and bouncing. Acta mechanica, 138(3-4) :219–241, 1999.

[14] M Aslam, WG Godden, and DT Scalise. Rocking and overturning response of rigid bodies to earthquake motions : A report of an analytical and experimen-tal study on the rocking and overturning response of rigid blocks to simulta-neous horizontal and vertical accelerations. Technical report, University of California, Berkeley, 11 1978.

[15] PR Lipscombe and S Pellegrino. Free rocking of prismatic blocks. Journal of engineering mechanics, 119(7) :1387–1410, 1993.

[16] JP Brossard. Mécanique générale - dynamique : théorie classique du choc. Techniques de l’ingénieur, TIB110DUO.(a1668), 1997.

[17] V Acary and B Brogliato. Coefficients de restitution et efforts aux impacts. Technical report, INRIA, 2004.

[18] KH Hunt and FRE Crossley. Coefficient of restitution interpreted as damping in vibroimpact. Journal of Applied Mechanics, 42 :440–445, 1975.

[19] Ioannis N Psycharis and Paul C Jennings. Rocking of slender rigid bodies allowed to uplift. Earthquake Engineering & Structural Dynamics, 11(1) :57–76, 1983.

[20] W Goldsmith. Impact. Edward Arnold, London, 1960.

[21] MN Chatzis and AW Smyth. Modeling of the 3d rocking problem. Interna-tional Journal of Non-Linear Mechanics, 47(4) :85–98, 2012.

[22] JC Simo and N Tarnow. The discrete energy-momentum method. conser-ving algorithms for nonlinear elastodynamics. Zeitschrift für angewandte Ma-thematik und Physik ZAMP, 43(5) :757–792, 1992.

[23] Ioannis Politopoulos. Sur l’application de quelques

al-gorithmes en dynamique non-linéaire. Technical Report

DEN/DANS/DM2S/SEMT/EMSI/RT/05-037/A, Commissariat à l’énergie atomique et aux énergies alternatives, 2005.

[24] Chik-Sing Yim, Anil K Chopra, and Joseph Penzien. Rocking response of rigid blocks to earthquakes. Earthquake Engineering & Structural Dynamics, 8(6) :565– 587, 1980.

[25] ManYong Jeong, Kohei Suzuki, and Solomon Yim. Chaotic rocking behavior of freestanding objects with sliding motion. Journal of sound and vibration, 262(5) :1091–1112, 2003.

[26] Harry W Shenton III and Nicholas P Jones. Base excitation of rigid bodies. ii : Periodic slide-rock response. Journal of engineering mechanics, 117(10) :2307– 2328, 1991.

[27] Dimitrios Konstantinidis and Nicos Makris. Seismic response analysis of multidrum classical columns. Earthquake engineering & structural dynamics, 34(10) :1243–1270, 2005.

[28] Hongjian Zhang, Bernard Brogliato, and Caishan Liu. Study of the planar rocking-block dynamics with coulomb friction : critical kinetic angles. Journal of Computational and Nonlinear Dynamics, 8(2) :021002, 2013.

[29] Pol D Spanos and Aik-Siong Koh. Analysis of block random rocking. Soil Dynamics and Earthquake Engineering, 5(3) :178–183, 1986.

[30] Baoping Shi, Abdolrasool Anooshehpoor, Yuehua Zeng, and James N Brune. Rocking and overturning of precariously balanced rocks by earthquakes. Bul-letin of the Seismological Society of America, 86(5) :1364–1371, 1996.

[31] Dimitrios Konstantinidis and Nicos Makris. The dynamics of a rocking block in three dimensions. In Proceedings of the 8th HSTAM International Congress on Mechanics, pages 12–14, 2007. o. Biblio. Article : M odèle Article Comm. Fiabilité F ormu-lation Chapitr e

8

[32] Daniele Zulli, Alessandro Contento, and Angelo Di Egidio. 3d model of rigid block with a rectangular base subject to pulse-type excitation. International Journal of Non-Linear Mechanics, 47(6) :679–687, 2012.

[33] Nicos Makris and Yiannis Roussos. Rocking response and overturning of equipment under horizontal pulse-type motions. Technical report, Pacific Earthquake Engineering Research Center, 1998.

[34] AN Kounadis. Parametric study in rocking instability of a rigid block under harmonic ground pulse : A unified approach. Soil Dynamics and Earthquake Engineering, 45 :125–143, 2013.

[35] Y Ishiyama. Motions of rigid bodies and criteria for overturning by earth-quake excitations. Earthearth-quake Engineering & Structural Dynamics, 10(5) :635– 650, 1982.

[36] MJN Priestley, BJ Davidson, GD Honey, DC Hopkins, RJ Martin, G Ramsey, JV Vessey, and JH Wood. Seismic design of storage tanks. Recommendations of a study group of the New Zealand National Society for Earthquake Engineering, 17, 1986.

[37] WT Fielder, LN Virgin, and RH Plaut. Experiments and simulation of over-turning of an asymmetric rocking block on an oscillating foundation. Euro-pean journal of mechanics. A. Solids, 16(5) :905–923, 1997.

[38] Raphael JY Greenbaum, Andrew W Smyth, and Manolis N Chatzis. Monocu-lar computer vision method for the experimental study of three-dimensional rocking motion. Journal of Engineering Mechanics, page 04015062, 2015. [39] Juan Carlos Simo and Kachung Kevin Wong. Unconditionally stable

algo-rithms for rigid body dynamics that exactly preserve energy and momen-tum. International Journal for Numerical Methods in Engineering, 31(1) :19–52, 1991.

[40] Djordje Perić and DRJ Owen. Computational model for 3-d contact problems with friction based on the penalty method. International journal for numerical methods in engineering, 35(6) :1289–1309, 1992.

[41] Charlie Mathey. Étude du comportement sous séisme de blocs rigides (may be modi-fied). PhD thesis, Université de Clermont-Ferrand, 2016.

[42] David B Fogel. An introduction to simulated evolutionary optimization. Neu-ral Networks, IEEE Transactions on, 5(1) :3–14, 1994.

[43] Søren Emil Sørensen, Michael Rygaard Hansen, Morten Kjeld Ebbesen, and Ole Ø Mouritsen. Implicit identification of contact parameters in a continuous chain model. Modeling, Identification and Control (Online Edition), 32(1) :1–15, 2011.

[44] Jean-Laurent Duchaud, Sami Hlioui, François Louf, and Mohamed Gabsi. Electrical machine optimization using a kriging predictor. In International Conference on Electrical Machines and Systems, 2014.

[45] YK Lin and Yan Yong. Evolutionary kanai-tajimi earthquake models. Journal of engineering mechanics, 113(8) :1119–1137, 1987.

[46] PE Pinto and I. Vanzi. Base isolation : reliability for different design criteria. Tenth world conference on earthquake engineering, 1992.

[47] Serge M Prigarin. Spectral models of random fields in Monte Carlo methods. VSP, 2001.

[48] PG Mertens and A Preumont. Improved generation and application of ar-tificial time-histories and psd functions. In Proceedings of SMiRT, volume 14, 1997.

[49] CM Wong and WK Tso. Steady state rocking response of rigid blocks part 2 : Experiment. Earthquake engineering & structural dynamics, 18(1) :107–120, 1989.

[50] Mohamed A ElGawady, Quincy Ma, John W Butterworth, and Jason Ingham. Effects of interface material on the performance of free rocking blocks. Ear-thquake Engineering & Structural Dynamics, 40(4) :375–392, 2011.

[51] Charlie Mathey, Cyril Feau, Ioannis Politopoulos, David Clair, Laurent Baillet, and Michel Fogli. Behavior of rigid blocks with geometrical defects under seismic motion : an experimental and numerical study. Earthquake Enginee-ring & Structural Dynamics, 2016. eqe.2773.

[52] HP Mouzakis, IN Psycharis, DY Papastamatiou, PG Carydis, C Papantono-poulos, and C Zambas. Experimental investigation of the earthquake res-ponse of a model of a marble classical column. Earthquake engineering & struc-tural dynamics, 31(9) :1681–1698, 2002.

[53] Reuven Y Rubinstein and Dirk P Kroese. Simulation and the Monte Carlo method, volume 707. John Wiley & Sons, 2011.

[54] Christian Robert and George Casella. Monte Carlo statistical methods. Springer Science & Business Media, 2013.

[55] Murray Rosenblatt. Remarks on a multivariate transformation. The annals of mathematical statistics, pages 470–472, 1952.

[56] Ove Ditlevsen and Henrik O Madsen. Structural reliability methods, volume 178. Wiley New York, 1996.

[57] Siu-Kui Au and James L Beck. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 16(4) :263–277, 2001.

[58] Henri Procaccia and Patrick Morilhat. Fiabilité des structures des installations industrielles : théorie et applications de la mécanique probabiliste. Éd. Eyrolles, 1996. [59] C Allin Cornell. A probability-based structural code*. In ACI Journal

Procee-dings, volume 66. ACI, 1969.

[60] E Rosenblueth and L Esteva. Reliability basis for some mexican codes. ACI Special Publication, 31, 1972.

[61] AM Hasofer, NC (Niels Christian) Lind, and University of Waterloo. Solid Me-chanics Division. An exact and invariant first-order reliability format. University of Waterloo, Solid Mechanics Division,., 1973.

[62] R Rackwitz. Practical Probabalistic Approach to Design : By R. Rackwitz. Technical University of Munich, Institut fur Bauingenieurwesen III, 1976.

[63] Bernd Fiessler, Rudiger Rackwitz, and Hans-Joachim Neumann. Quadratic limit states in structural reliability. Journal of the Engineering Mechanics Divi-sion, 105(4) :661–676, 1979.

[64] Karl Breitung. Asymptotic approximations for multinormal integrals. Journal of Engineering Mechanics, 110(3) :357–366, 1984.

[65] M Hohenbichler, S Gollwitzer, W Kruse, and R Rackwitz. New light on first-and second-order reliability methods. Structural safety, 4(4) :267–284, 1987. [66] Prestandard FEMA. Commentary for the seismic rehabilitation of buildings.

FEMA-356, Federal Emergency Management Agency, Washington, DC, 2000. [67] Vitali˘ı Ogorodnikov. Numerical modelling of random processes and fields :

algo-rithms and applications.

[68] Ray W Clough and Joseph Penzien. Dynamics of structures. Technical report, 1975.

[69] Christian P Robert. Méthodes de Monte Carlo par chaînes de Markov. Economica, 1996. o. Biblio. Article : M odèle Article Comm. Fiabilité F ormu-lation Chapitr e

8

[70] George Fishman. Monte Carlo : concepts, algorithms, and applications. Springer Science & Business Media, 2013.

[71] J-M Hammersley, D-C Handscomb, and Françoise Rostand. methodes de monte-carlo.[les]. 1964.

[72] J.-M. Bourinet, C. Mattrand, and V. Dubourg. A review of recent features and improvements added to ferum software. In Proc. of the 10th International Conference on Structural Safety and Reliability (ICOSSAR’09), Osaka, Japan, 2009.