System reliability analysis of a circular rock tunnel using a first order approximation

2014 ◽  
pp. 787-791
Author(s):  
P Zeng ◽  
R Jiménez
Keyword(s):  
Reliability Analysis ◽  
System Reliability ◽  
Order Approximation ◽  
First Order ◽  
System Reliability Analysis ◽  
First Order Approximation ◽  
Rock Tunnel

A Weighted Saddlepoint Approximation Method for System Reliability Analysis

2016 ◽  
Vol 23 (04) ◽  
pp. 1650014
Author(s):  
Huahan Liu ◽  
Wei Jiang
Keyword(s):  
Reliability Analysis ◽  
Linear Correlation ◽  
System Reliability ◽  
Saddlepoint Approximation ◽  
Weight Coefficient ◽  
Analysis Method ◽  
First Order ◽  
System Reliability Analysis ◽  
The Relationship ◽  
Weight Coefficients

The relationship between the reliability probabilities of the component and the system is hard to get. If this relationship can be obtained easily, the reliability of the system can be calculated by using the reliability structures of the components. The common method to express this relationship is using the linear correlation index, which only shows the linear correlation between the components failure rather than the relationship between high nonlinear functions. In order to describe this relationship accurately and calculate the system reliability using the component reliability structures, a Uniform Design (UD)-Saddlepoint Approximation (SA)-based system reliability analysis method is proposed. The system reliability analysis method is decomposed to three simple steps: (1) calculating the weight coefficient which represents the contribution rate of each component to system reliability, (2) approximating Cumulant Generation Function (CGF) of each component, (3) calculating CGF of the system and approximating the system reliability with SA method. The weight coefficient of each component is derived from UD method, and a variable interval selection method is developed to decrease the required number of samples and increase the accuracy of the weight coefficients. First-Order Saddlepoint Approximation (FOSA) method or Mean-Value First-Order Saddlepoint Approximation (MVFOSA) method is used to analyze the CGF of a component performance function. Then the CGF of the system can be obtained by the weighted addition law by combining the CGFs of components performance functions with the weight coefficients. Finally, the system reliability can be approximated by SA method. Four examples are employed to demonstrate that the new method outperforms other methods for system reliability analysis in terms of efficiency and accuracy.

Improved first-order approximation of eigenvalues and eigenvectors

AIAA Journal ◽  
1998 ◽  
Vol 36 ◽  
pp. 1721-1727
Author(s):  
Prasanth B. Nair ◽  
Andrew J. Keane ◽  
Robin S. Langley
Keyword(s):  
Order Approximation ◽  
Eigenvalues And Eigenvectors ◽  
First Order ◽  
First Order Approximation

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

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