PSI - Issue 47

Mohammed Lamine Moussaoui / Procedia Structural Integrity 47 (2023) 932–937 Mohammed Lamine Moussaoui / Structural Integrity Procedia 00 (2023) 000–000

935

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and � � � � � � The objective function becomes: � � � using the generalized variables ∗ � . � � � � � ∗ � � � � ∗ � � � � � � . � � � � . � � � � � � � 2 � � To get the optimal values, the gradient of the function in Eq. 9 must satisfy the condition: � � , ⋯ , � � � � ⌊ 0, ⋯ ,0 ⌋ � 2 � � 2 � � 0 ⟹ � � � Let be: ����� � � and: The probabilistic law parameters can then be efficiently computed with: �� � ����� � � One has to solve: �

� 9 �

� 10 � � 11 � � 12 � � 13 � � 14 �

�� �� � � � �

3. Numerical example It is useful for the engineer to compute in practice the reliability and to get the failure probability from the information of the structural failures recorded data as it is cited by Moussaoui � 2021 � . Therefore this allows plotting the � , � �� graph. The available data in Table 1 Bellaouar and Beleulmi � 2013 � give the corresponding values presented in Table 2 . Table 1 . Failure probability vs. time 1 2 3 4 5 � 10 20 30 50 70 � � � 0.005 0.030 0.160 0.400 0.800 Table 2 . Parametric data 1 2 3 4 5 � 2.303 2.996 3.401 3.912 4.248 � � 5.296 � 3.491 � 1.747 � 0.672 � 0.476 Then one gets the following linear system of equations to be solved:

� 16.5860 1569..186906 �� � � ��� � 10.730 � 29.204 �