This article originally appeared in the June 1991 issue of *Technology
Today®,* published by Southwest Research Institute®.

*Software for probabilistic analysis provides a means for
improving safety and reliability of complex systems *

On July 19, 1989, United Airline flight 232 crashed while attempting an emergency landing
in Sioux City, Iowa, killing 111 people of the 298 on board. The reason: A critical
component failed in the primary engine, rendering the aircraft uncontrollable. An equally
tragic example of mechanical disaster was the failure of the seals in the space shuttle
Challenger's solid rocket booster that also resulted in loss of life. These and numerous
other failures have prompted the attention of the technical community to develop better
ways to engineer more reliable and safe designs.

NESSUS |

Quantifying the reliability or, conversely, the probability of failure of a design is a difficult task. In the process of designing a complex system, many possible designs are proposed and, after careful analysis, a final "best" design is selected. Criteria for selecting or discarding designs include cost, weight, feasibility, and reliability. With regard to reliability, the procedure usually entails comparing analysis results to those of a design guide or, for more conventional structures, a building code. The problem with this approach is that it does not take into account in a quantifiable manner the fact that there is inevitably some element of uncertainty in the basic design parameters, such as material properties, tolerances, and loadings.

To illustrate the concept of uncertainty, take, for example, the calculation of strength for a given material. A piece of the actual material, known as a coupon, is subjected to a known load in a test fixture and the displacement recorded. Using this information, the strength can be calculated. Now repeat the test several more times. By virtue of the fact that the strength will differ from test to test, we conclude that it is impossible to predict with complete certainty what it will be before each test is actually run. In fact, after many experiments have been run we will see that the strength is not a single value, but rather a scattering of values. Thus, we refer to it as being a random variable.

Coauthor Harry Millwater (seated) discusses NESSUS analysis with Manager Dr. Justin Wu, whose work in the development of advanced methods for probabilistic analysis has played a key role in the evaluation of the program. |

For analytical purposes, random variables can be characterized using statistical models. One such model is called the probability density function, or PDF. The PDF provides the ability to predict values of a random variable, such as strength, with some associated probability. Another closely related model for a random variable is the cumulative distribution function, or CDF. Mathematically, the CDF is obtained by integrating the PDF. Thus, the CDF value corresponding to a particular value of strength represents the probability that the strength will be less than or equal to this value.

One traditional design approach uses safety factors to simulate worst-case conditions in an attempt to account for uncertainties caused by these random variables. In this deterministic approach, variables that affect the loads acting on structure, for example, would be increased relative to their nominal values, by some factor of safety, and variables that affect the strength of the structure would be decreased by a factor of safety. The net effect is a design that is probably safe. But how safe is it? Probabilistic structural analysis attempts to answer this question.

The idea behind probabilistic structural analysis is to use the information about the probability of random variables along with the structural behavior in order to quantify the scatter in the structural response. The result from the analyses is a distribution, or a PDF and CDF, of the structural responses, not just a single number. Thus, the analysis gives a more complete picture of the actual situation. With this method, the engineer can compute the probability of achieving a certain level of structural response and can efficiently design structures according to these probabilities. This provides a safer, more cost-effective design.

Prior to the development of new probabilistic computer programs at SwRI, there existed only limited tools that could be used to answer this question in a general sense. The traditional method of quantifying the reliability of a structure, Monte Carlo simulation, requires thousands or hundreds of thousands of simulations of the structural model subjected to the random variables. This method, while highly accurate in the limit, is clearly impractical for large and complex structures where even a single computer simulation may require several hours of computer time. Thus, there is a strong need for new, innovative computer tools to assess the variability of complex structures in a practical amount of computer time with acceptable accuracy.

Broad application of probabilistic techniques is envisioned for prediction and prevention of failures, such as the cracking which led to rupture of this blade from a gas turbine engine. |

As a result, through the sponsorship of the National
Aeronautics and Space Administration (NASA), SwRI developed a sophisticated computer
program called NESSUS^{TM} (numerical evaluation of stochastic structures under
stress).[1] This work was done as part of
the probabilistic structural analysis method (PSAM) project, a 10-year effort now in its
seventh year. NESSUS, designed specifically for predicting structural response caused by
uncertain basic variables such as loads, material properties, part geometry, and boundary
conditions, is used by NASA to assist in the evaluation of the existing critical space
shuttle components, as well as the next generation space shuttle main engine. Therefore,
the primary thrust of the PSAM program is to develop a probabilistic capability for the
analysis of large, complex structural systems, usually requiring some type of
computationally expensive numerical analysis.

The power of NESSUS and its ability to balance the efficiency and accuracy required in large, complex structural applications lies in the synergistic coupling of general purpose structural analysis methods and approximate probabilistic algorithms. Probabilistic algorithms have been under development for the last 20 years and have now reached the stage where their use, in conjunction with complex structural models, is now a practical option. The principal advantage of using these methods is that estimates of the structural reliability and the identification of the important random variables can be obtained with relatively few structural analyses. Although accuracy cannot be guaranteed in general, experience has shown the results for nearly all problems to be excellent.

The individual components of NESSUS and their relationships to each other is shown in an accompanying figure. The finite element module, NESSUS/FEM, is used to compute the structural behavior of interest and the sensitivities of the structure with respect to each random variable. The fast probability integration (FPI) module uses the structural sensitivity data and the random variables statistics to quickly compute the probabilistic response, i.e., the PDF and CDF, and the probabilistic sensitivity factors. The probabilistic sensitivity factors define the contribution of each random variable to the structural reliability. This information provides a means to modify designs most efficiently to improve reliability.

The NESSUS code is designed to be very general in both the definitions of the random variables inputs and the computed outputs. Virtually any quantity of the structure can be random. Some examples include: geometric quantities, such as plate thicknesses, cross-sectional dimensions, beam lengths, and hole radii; material properties such as stiffness, thermal expansion coefficient, density, and damping; and loading, such as forces, pressures, temperatures, and accelerations. Some examples of output quantities include deflection, stress and strain at any location, natural, frequencies, buckling loads, and fatigue life.

NESSUS is based on the concept of a limit state that reflects the designer's definition of failure. Some examples of failures include (1) a stress or displacement exceeding some limiting value, (2) the natural vibrating frequency of a structure being too close to the operating frequency of a mechanical component attached to the structure, and (3) the number of cycles for a crack to grow to a critical length being less than that expected while in service. In other words, given the definition of a limit state, NESSUS computes the probability of failure and the statistical models of all input random variables.

A key output from NESSUS is a CDF of the performance measure, such as a critical stress or buckling load, associated with a particular mode of failure. In a design scenario, the CDF can be used in conjunction with a target reliability to obtain the required level of performance. Alternatively, in an analysis scenario, the reliability corresponding to a particular level of performance can be computed. A major advantage over the deterministic approach to design is now apparent: The additional information obtained using a probabilistic analysis gives the designer a prediction of the risk of failure, as opposed to a simple "go-no go" evaluation.

Use of the NESSUS probabilistic computer software is illustrated by a structural analysis performed on a turbine blade of the space shuttle main engine. A finite element model of a blade is pictured at left. The turbine blade, particularly vulnerable to high-cycle fatigue cracking because of its severe thermal shock exposure during rapid start and short-duration runs of the engine, was analyzed to determine stresses at various critical locations. Random variables considered in the analysis, indicated in the illustration, included loadings, material properties, and geometry. A typical result, shown in graph form, is the cumulative distribution function (CDF) of the effective stress at one critical location. This result shows the probability of reaching different levels of stress for this particular design configuration and set of input parameters. In contrast to the simple "go-no go" evaluation produced by a traditional analytical approach, probabilistic analysis provides a means of assessing the probability that stress might exceed a certain critical value, and, therefore, a means of predicting the risk of failure. The probabilistic sensitivity factors, shown with the graph, reveal the variables for which tighter control would result in a more reliable blade design. (In this case, hot gas seal leakage clearly dominates other considerations.) Conversely, the sensitivity factors also expose which variables are relatively unimportant in determining blade reliability, information important in establishing design and manufacturing controls for maximum cost effectiveness as well as structural reliability. |

Another key output from NESSUS is the relative importance of the input random variables to the performance measure under consideration. The probabilistic sensitivity factors (PSF) identify to the designer which features of a design lead to risk and give some indication as to how to minimize this risk. Thus, using the PSFs, the designer can effectively allocate resources to optimize the structure to provide maximum reliability at minimum cost.

The probabilistic algorithms within the FPI code are primarily
the work of Dr. Justin Wu, manager of the Probabilistic Mechanics and Reliability Section
of the SwRI Mechanical and Materials Engineering Division. The development of advanced
probabilistic methods in FPI began as Dr. Wu's Ph.D. research and has progressed over the
last eight years to a state-of-the-art probabilistic analysis program. FPI is designed to
estimate, very quickly, the probability of failure and is often thousands of times more
efficient than Monte Carlo simulation. FPI has proven so successful that it is being
applied in a number of other areas besides structural analysis. Some of these areas
include rock mechanics, rotor dynamics, turbo-machinery, and nuclear waste management.[2]

Diagram shows the principal components of NESSUS and their relationships. |

Recognizing the value of this technology to industry, the Institute initiated an effort in late 1989 to ready NESSUS for commercial use. An objective of the NESSUS commercialization effort is to take NESSUS from a research to a design applications environment. The commercialization effort is intended to provide a reasonable level of software support required by external use and to provide other industries the opportunity to explore and utilize this technology.

In November 1989, SwRI conducted a two-day on-site workshop for the NASA Lewis Research Center, and in April 1990, sponsored a short course on probabilistic structural analysis methods to participants representing a broad range of industries. Feedback from direct NESSUS was then requested to help determine what major deficiencies of the code existed and to rank them in order of importance. Many shortcomings identified as a result of this evaluation have been resolved as part of the commercialization effort.

Key elements of the commercialization efforts include:

- an improved user interface,

- enhanced documentation,

- added graphics capabilities (A program known as P/NESSUS was
developed to translate geometrical data and results information between NESSUS and a
graphics program, PATRAN
^{TM}, selected because of its widespread use.[3]),

- a quality assurance procedure, and

- a configuration management system.

The development of technology within NESSUS is continuing. For example, research is currently under way to define methods for computing system reliability, modeling the probabilistic progressive fracture process, and computing more accurate confidence bounds. System reliability computations involve computing the reliability of a structure composed of multiple modes of failure. In probabilistic progressive fracture, the growth of a crack in the structure is progressively modeled through the structure according to a probability-based fracture path. The effects of statistical uncertainty, such as the case in which little or no data is available to characterize a given random variable, can be accounted for using confidence bounds on the computed CDF results. In short, the NESSUS code is being continually enhanced to maintain its position as the state of the art in probabilistic structural analysis.

The margin for variance in the performance of large, complex structural systems is necessarily becoming tighter as performance requirements for such systems increase. Uncertainties, or random variables, however, are present in all engineered designs and cannot be avoided or ignored. Therefore, an analysis that can systematically account for uncertainties must be performed. Using the new methods implemented in the NESSUS computer software, this capability is now possible, and provides the engineer with a tool that can be used to design-in or improve the safety and reliability of complex structural systems. Nevertheless, many questions and concerns remain. Foremost among them is the inertia among engineers to change from the tried and true traditional factor of safety design methods to the probabilistic way of thinking. While the transition will be slow, the overwhelming power of this new information will almost certainly change the way design, analysis, and the associated testing will be performed on future structures.

NESSUS has been used to analyze critical space shuttle engine components other than the turbine blade, including high-pressure piping system, an injector component, and a hot gas transfer tube liner. This illustration is a graphic depiction of computed stress magnitudes under conditions of near collapse of the hot gas tube liner, a thin-walled structure subjected to severe thermal and pressure loadings. |

Dr. Ben Thacker, a senior research engineer in the Probabilistic Mechanics and Reliability Section within the Mechanical and Materials Engineering Division, specializes in probabilistic finite element methods and software development. His current activities include serving as manager for the NESSUS commercialization effort and for a major program to apply probabilistic methods to predict the uncertain response of deep underground tunnels subjected to ground shock. |

Dr. Harry Millwater, a senior research engineer in the Probabilistic Mechanics and Reliability Section within the Mechanical and Materials Engineering Division, specializes in software development and applications in computational mechanics. Currently, he is involved in the development and enhancement of NESSUS in the areas of system reliability and fracture mechanics. |

[1] A registered trademark of Southwest Research Institute

[2] See *Technology Today*, Vol. X, No.
3, 1989

[3] A registered trademark of PDA Engineering

*Published in the June 1991 issue of Technology
Today®, published by Southwest Research Institute. For more information, contact
Joe Fohn.*