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# Why do Probabilistic Finite Element Analysis?

** The intention of this book is to provide an introduction to performing probabilistic finite element analysis. As a short guideline, the objective is to inform the reader of the use, benefits and issues associated with performing probabilistic finite element analysis without excessive theory or mathematical detail. **

This book is written for an experienced engineer or project manager who is familiar with finite element analysis but unfamiliar with probabilistic finite element analysis. In very broad terms, probabilistic finite element analysis can be viewed as the combination of probability theory and finite element methods. The basic concepts underlying probability theory are presented such that the reader will understand and appreciate the challenges and benefits from undertaking a probabilistic finite element analysis.

Numerous examples from various discipline areas are provided to demonstrate** how probabilistic finite element analysis is performed**, **what types of inputs are required, what types of results are produced, and how the results can be interpreted and used in practical applications. **The intention of providing a number of examples is to aid the reader in understanding the basic concepts, and hopefully inspire them to apply these concepts to their own problems.

## Who Has Written This Book?

This book represents the accumulated experience of the probabilistics mechanics and reliability group at Southwest Research Institute (SwRI), which has been involved with the development and application of probabilistic finite element analysis for over 20 years. The group has developed several advanced and efficient probabilistic analysis methods, computer software (NESSUS, DARWIN, etc.), and solved a wide variety of problems for government and industry clients. The author has personally worked with this group for over 18 years.

The information, ideas and opinions in the book come from a variety of materials (leading books, reports, and papers) that have been written on the subject as well as the experience that has been gained from applying probabilistic finite element analysis to practical problems.

## Introduction

Numerical simulation is now routinely used to predict the behaviour and response of complex systems. Simulation is also being increasingly relied upon as performance requirements for engineered systems increase and as a means of augmenting or reducing the need for full system testing.

Finite element analysis, arguably the most well known type of numerical simulation, has become a popular tool for simulating the behaviour and response of complex structures and mechanical systems. Fundamentally, Finite Element Analysis provides the capability of computing the response of a structure to applied loadings. At one end of the scale, reasonably simple finite element models are used to predict response and behaviour when analytical solutions are not possible, for example, when the geometry is nontrivial or nonlinear materials are involved. At the other end of the scale, finite element analysis of complete processes and systems are now being performed. These analyses include multiple interacting physics, such as solid mechanics, dynamics, hydrodynamics, heat conduction, fluid flow, transport, chemistry, and acoustics.

To simplify the discussion herein, this book focuses on solid mechanics, and even more specifically, on structures and mechanical systems. However, it should be kept in mind that the governing physics that can be treated in a probabilistic finite element analysis can be applied regardless of the underlying physics being simulated by the model.

Finite element analysis (FEA) is used to perform numerical investigations for a variety of reasons. In safety-critical situations where testing is either cost-prohibitive or impossible, FEA is used to increase confidence in the expected behaviour of the structure. Examples include the design of aircraft engines, spacecraft and orthopaedic implants. In manufacturing situations, FEA is used to help minimize costs. Examples include automobiles, electronic equipment and electric power.

The motivating need for FEA is either to increase safety, reduce cost, or both. The current state-of-practice is to use factors of safety to provide an adequate margin of safety against failure. For example, material properties that are typically used in a FEA are not average values, but actually an upper or lower percentile based on statistical analysis of property data. In an FEA, loadings are typically assumed to be a “maximum expected” value. In applications that allow it, the design incorporates a large amount of redundancy or ductility so that the structure can survive if a local failure or extreme loading event occurs.

In many high-performance applications, overly conservative assumptions cannot be tolerated due to the excessive cost or weight such assumptions produce. Thus, it can be seen that the safety factor approach leads to overconservative and uneconomical designs. Furthermore, the maximum potential of the design is never realized and the reliability is not quantified.

Quantifying the effect of uncertainties provides the analyst with an estimate of the true margin of safety for a particular design and allows alternative designs to be assessed on the basis of quantified reliability. Knowledge of the effect of uncertainties can also lead the analyst to drastically different conclusions regarding which input parameters are most important. It is for these reasons that probabilistic Finite Element Analysis is rapidly gaining widespread acceptance in design.

## Contents

**1 Overview**

1.1... Who Should Read This Booklet?

1.2... Who Has Written This Booklet?

1.3... Organization of this Booklet

**2 Introduction**

2.1... Uncertainties

2.2... Effect of Uncertainties on Structural Response

2.3... Benefits from a Probabilistic Finite Element Analysis

**3 Probabilistic Finite Element Analysis**

3.1... Basic Probability and Statistics

3.2... Probabilistic Analysis Methods

3.3... Selection of Random Variables

3.4... Multiple Failure Modes

3.5... Practical Issues

**4 Case Studies**

4.1... Crankshaft Fatigue Reliability

4.2... Aircraft Lever Fatigue Life

4.3... Probabilistic Analysis of an Aircraft Composite Wing

4.4... Automotive Crashworthiness

4.5... Cervical Spine Impact Injury Assessment

4.6... Dynamic Experiment Containment Vessel

4.7... Probabilistic Tunnel Vulnerability Assessment

4.8... Gas Turbine Engine Rotor Risk Assessment

4.9... Fracture Reliability of Space Shuttle Main Engine Flowliner

4.10. Probabilistic Space Shuttle Debris Impact and Damage

**5 Summary**

**6 Acknowledgements**

**7 References**

# About

B.H. Thacker

Published June 2008

Softback, 80 Pages