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# PERMAS-RA Reliability Analysis with PERMAS

In the classical approach to structural analysis a deterministic model is used to predict the behavior of the design under various loading conditions. The results of such calculations are compared to typical limiting constraints such as a maximal stress or deflection under the consideration of safety margins. This is called deterministic approach to the problem of structural safety and the Finite Element Method has become a widespread tool in such procedures.

In contrast to this method, the stochastic analysis of a design assumes some properties of a structure or the loads to be uncertain knowing only the characteristics of their probability distributions. The limiting constraints on the design will usually be of the same kind as in the deterministic approach. However, the results from the probabilistic analysis will yield the probability of failure with respect to these constraints and the sensitivity of this probability with respect to the uncertain properties of the model.

This module combines the Finite Element Analysis with the well known COMREL software developed by RCP GmbH, Munich. So, the expereience comprised in both software systems could be merged in a single application simplifying the approach to the Stochastic Finite Element Method.

The procedure in reliability analysis comprises the following three steps:

• Definition of uncertain quantities in structural analysis (like geometrical or load parameters) by basic variables with an assigned distribution function.
• Definition of limit state functions (or failure functions) related to result quantities of a structural analysis.
• Calculation of the probability of failure for each limit state function.

The following quantities can be used as basic variables:

• Design parameters (like geometrical data or coordinates)
• Material parameters
• Parameters of the limit state functions
• Parameters of other basic variables

More than 20 different types of distribution functions are available to describe the basic uncertain variables.

The stochastic analysis performs an assessment of the failure parameters for the following analysis types:

• Linear static analysis
• Contact analysis
• Dynamic eigenvalue analysis

For this purpose, a number of methods are available:

• Efficient sensitivity based methods as First/Second Order Reliability Methods (FORM/SORM)
• Response surface methods
• Monte Carlo simulation using adaptive sampling
• Crude Monte Carlo simulation

The reliability analysis allows taking into account several loading cases as well as different boundary conditions using different failure functions.

• The definition of Failure functions is made using
• General functions
• Dependent on
- results (displacements, stresses, etc.)
- basic variables
- constant values
• The primary Results of such an analysis are
• Probability of failure for each limit state function,
• Parameter sensitivites of the limit state functions,
• Result sensitivities for basic variables (elasticities),
• Selected data of each iteration for Monte Carlo simulations.