{"id":50195,"date":"2025-11-14T15:22:45","date_gmt":"2025-11-14T14:22:45","guid":{"rendered":"https:\/\/www.gt-engineering.it\/?post_type=pubblicazione&p=50195"},"modified":"2026-01-03T09:52:14","modified_gmt":"2026-01-03T08:52:14","slug":"securite-fonctionnelle-calcul-du-pfd-1-ere-partie","status":"publish","type":"pubblicazione","link":"https:\/\/www.gt-engineering.it\/fr\/publications\/approfondissements-articles-tuttomisure\/securite-fonctionnelle-calcul-du-pfd-1-ere-partie\/","title":{"rendered":"P4: S\u00e9curit\u00e9 Fonctionel – Calcul du PFD"},"content":{"rendered":"\n\n \n \n \n
\n\n \n
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\n \n \n

\n Introduction <\/h2>\n <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
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Une distribution de probabilit\u00e9 continue est indiqu\u00e9e par\u00a0<\/span>f(x)<\/span><\/em>\u00a0et est g\u00e9n\u00e9ralement appel\u00e9e\u00a0<\/span>fonction de densit\u00e9 de probabilit\u00e9 (PDF).<\/span><\/strong>\u00a0Elle est exprim\u00e9e par une \u00e9quation et peut \u00eatre repr\u00e9sent\u00e9e comme dans la figure 1.5 {1.4.2.1}. La courbe en cloche n’est qu’un exemple de PDF possible.<\/span><\/p>\n

La principale propri\u00e9t\u00e9 d’un PDF est que :<\/span><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
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La probabilit\u00e9 que\u00a0<\/span>x<\/span><\/em>\u00a0prenne des valeurs comprises entre a et b est \u00e9valu\u00e9e comme l’int\u00e9grale suivante de la\u00a0<\/span>fonction de densit\u00e9 de probabilit\u00e9 :<\/span><\/strong><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
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Cette probabilit\u00e9 est illustr\u00e9e dans la figure 1.6 {1.4.2.2}.<\/span><\/p>\n

La fonction de densit\u00e9 de probabilit\u00e9 est \u00e9galement appel\u00e9e\u00a0<\/span>densit\u00e9 de d\u00e9faillance<\/span><\/strong>\u00a0ou\u00a0<\/span>distribution de dur\u00e9e de vie.<\/span><\/strong><\/p>\n

La distribution d’une variable continue peut \u00e9galement \u00eatre d\u00e9crite par la\u00a0<\/span>fonction de distribution cumulative<\/span><\/strong>\u00a0. Celle-ci donne la probabilit\u00e9 que la variable al\u00e9atoire prenne une valeur inf\u00e9rieure ou \u00e9gale \u00e0 x. Son expression est\u00a0:<\/span><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

Pour \u2013 \u221e< x <+ \u221e.<\/span><\/p>\n

F(x) est une fonction non d\u00e9croissante : F(-\u221e)=0 et F (+\u221e)=1 , donc :<\/span><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
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La d\u00e9riv\u00e9e de la fonction de distribution cumulative est la\u00a0<\/span>fonction de densit\u00e9 de probabilit\u00e9<\/span><\/strong>\u00a0(ou densit\u00e9 de d\u00e9faillance) de la variable al\u00e9atoire\u00a0<\/span>X<\/span><\/em>\u00a0:<\/span><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
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La relation entre la\u00a0<\/span>fonction de distribution cumulative F(x)<\/span><\/strong>\u00a0et la\u00a0<\/span>fonction de densit\u00e9 de probabilit\u00e9 f(x)<\/span><\/strong>\u00a0est illustr\u00e9e dans la figure 1.8 {1.4.3.1}.<\/span><\/p>\n

Ces d\u00e9finitions pour F(x) permettent d’exprimer comme suit :<\/span><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
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Puisque\u00a0<\/span>nous raisonnons en termes de temps<\/span><\/strong>\u00a0et que le temps est une variable al\u00e9atoire positive, la\u00a0<\/span>fonction de distribution cumulative<\/span><\/strong>\u00a0peut s’\u00e9crire de la mani\u00e8re suivante :<\/span><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n
\n \n
\n \n \n

\n La fonction de fiabilit\u00e9 R(t) <\/h2>\n <\/div>\n <\/div>\n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
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R(t)<\/span><\/em>\u00a0est la probabilit\u00e9 qu’aucune d\u00e9faillance de l’\u00e9l\u00e9ment ne se produise dans l’intervalle (0 t].<\/span><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
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En d’autres termes, R(t) est la probabilit\u00e9 qu’un \u00e9l\u00e9ment fonctionne \u00ab sans d\u00e9faillance \u00bb dans l’intervalle de temps (0, t], tandis que la d\u00e9faillance se produira dans (t, . Connaissant la fonction de densit\u00e9 de probabilit\u00e9 f(x), nous avons :<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
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Si le syst\u00e8me ne peut \u00eatre trouv\u00e9 que dans deux \u00e9tats, soit un fonctionnement correct, soit une d\u00e9faillance, nous pouvons d\u00e9finir la fonction de non-fiabilit\u00e9 F(t) comme compl\u00e9mentaire de R(t), c’est-\u00e0-dire :<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
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La fonction de densit\u00e9 f(t) peut maintenant \u00eatre exprim\u00e9e comme :<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
\n

Int\u00e9gration de l’\u00e9quation sup\u00e9rieure dans le temps :<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
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Les taux d\u2019\u00e9chec et leurs incertitudes peuvent \u00eatre estim\u00e9s \u00e0 partir des retours d\u2019exp\u00e9rience sur le terrain, en utilisant des statistiques conventionnelles.<\/span><\/p>\n

Le mod\u00e8le le plus r\u00e9pandu et le plus connu pour le taux de d\u00e9faillance est la courbe dite \u00ab en baignoire \u00bb. Dans la\u00a0<\/span>phase initiale<\/span><\/strong>\u00a0de la dur\u00e9e de vie du composant, \u03bb(t) diminue rapidement avec le temps\u00a0; ce ph\u00e9nom\u00e8ne r\u00e9sulte de l’existence d’une fraction \u00ab\u00a0faible\u00a0\u00bb de la population dont les d\u00e9fauts provoquent une d\u00e9faillance dans un court laps de temps \u00e0 compter de leur apparition.<\/span><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
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Durant la p\u00e9riode appel\u00e9e\u00a0<\/span>dur\u00e9e de vie utile<\/span><\/strong>\u00a0, \u03bb(t) est approximativement constant, par exemple dans le cas des composants \u00e9lectroniques. Pour les composants \u00e9lectrom\u00e9caniques, \u03bb(t) est une fonction du temps et, durant cet intervalle, il augmente constamment.<\/span><\/p>\n

La derni\u00e8re p\u00e9riode est caract\u00e9ris\u00e9e par une\u00a0<\/span>usure<\/span><\/strong>\u00a0, avec un taux de d\u00e9faillance \u03bb(t) en augmentation rapide caus\u00e9 par l’usure, le vieillissement et la fatigue.<\/span><\/p>\n

Pendant la dur\u00e9e de vie utile d’un composant \u00e0\u00a0<\/span>taux de d\u00e9faillance constant<\/span><\/strong>\u00a0, en consid\u00e9rant comme condition initiale que la fiabilit\u00e9 \u00e0 l’instant 0 est maximale et qu’elle est \u00e9gale \u00e0 1, nous avons :<\/span><\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n \n \n
<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n
\n \n
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La fonction de fiabilit\u00e9 R(t) est repr\u00e9sent\u00e9e dans la Figure 1.13a {1.5.2a} et les fonctions de densit\u00e9 de probabilit\u00e9 f(t) dans la Figure 1.13b {1.5.2b}, dans le cas \u03bb = constante.<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
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Le tableau 1.3 {1.5.1} pr\u00e9sente un r\u00e9sum\u00e9 des quatre fonctions d\u00e9crites jusqu\u2019\u00e0 pr\u00e9sent.<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n\n\n\n\n \n \n \n
\n\n \n \n \n \r\n\r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n \r\n <\/picture>\r\n\r\n \r\n\r\n \r\n \r\n <\/div>\n \n
\n \n
\n

La figure 1.14 {1.5.3} montre la relation entre F(t) et R(t)<\/p>\n<\/div> <\/div>\n \n \n

<\/div>\n \n\n <\/div>\n <\/section>\n\n \n","protected":false},"excerpt":{"rendered":"

A continuous probability distribution is indicated with f(x) and is usually called Probability Density Function (PDF). It is expressed by an equation and it can be represented as in the Figure 1.5 {1.4.2.1}. The bell curve is just an example of a possible PDF.<\/p>\n","protected":false},"author":1,"featured_media":41083,"parent":49611,"menu_order":3,"template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-50195","pubblicazione","type-pubblicazione","status-publish","has-post-thumbnail","hentry"],"acf":[],"yoast_head":"\nP4: S\u00e9curit\u00e9 Fonctionel - Calcul du PFD - Gt-Engineering<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.gt-engineering.it\/fr\/publications\/approfondissements-articles-tuttomisure\/securite-fonctionnelle-calcul-du-pfd-1-ere-partie\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"P4: S\u00e9curit\u00e9 Fonctionel - Calcul du PFD - Gt-Engineering\" \/>\n<meta property=\"og:description\" content=\"A continuous probability distribution is indicated with f(x) and is usually called Probability Density Function (PDF). It is expressed by an equation and it can be represented as in the Figure 1.5 {1.4.2.1}. 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