**Suggested Citation:**"9 Uncertainty and Stochastic Processes in Mechanics." National Research Council. 1991.

*Research Directions in Computational Mechanics*. Washington, DC: The National Academies Press. doi: 10.17226/1909.

**9**

**UNCERTAINTY AND STOCHASTIC PROCESSES IN MECHANICS**

Probabilistic computational mechanics is the methodology that forms the basis of the structure reliability and risk analysis of mechanical components and systems. Reliability and risk analyses are of critical importance both to public safety and the competitiveness of products manufactured in the United States.

Reliability analysis applications to enhance public safety include the performance of structures when subjected to seismic loads, determination of inspection cycles for aging aircraft, and evaluation of existing infrastructure such as bridges and lifelines. In the design of mechanical components and systems where safety is not a crucial issue, reliability engineering is also important because it can provide cost-effective methods for enhanced fabrication and inspection.

The problem common in reliability engineering is that certain features of the problem are uncertain or stochastic in character. Two of the most important sources of uncertainty in reliability engineering are unavoidable defects in the structures such as cracks and the environment, which includes factors such as load and temperature.

**PROBABILISTIC FRACTURE MECHANICS**

Cracks, whose behavior is described by the field of fracture mechanics, are one of the most pervasive causes of failure and therefore play a critical role in reliability engineering. Many of the problems of aging structures and aircraft, component life, and behavior under extreme loads are due to the growth of minor defects into major cracks. The growth of cracks is, however, an inherently stochastic process. Both the sizes and locations of the initial defects that lead to major cracks are random, and the growth of a crack under cyclic loading is stochastic in character. Generally, the growth of a crack under cyclic loading is modeled by the Paris law, in which the length of the crack *a* is governed by where *n* is the number of load cycles, and Δ*K* is the range of the stress intensity factor in the load cycle; *D* and *m* are constants that are

**Suggested Citation:**"9 Uncertainty and Stochastic Processes in Mechanics." National Research Council. 1991.

*Research Directions in Computational Mechanics*. Washington, DC: The National Academies Press. doi: 10.17226/1909.

fit to experimental data and exhibit significant scatter, or randomness.

In current engineering practice the reliability of a structure against excessive crack growth is usually ascertained by performing linear stress analysis and then using *S*-*n* charts, which provide the engineer with the probability of failure due to fatigue fracture of a component subjected to *n* cycles to a maximum stress *S*. These *S*-*n* charts are usually based on a simple rod specimen subjected to a uniaxial state of stress, which may be quite different from the complex stress pattern encountered in an actual component. Furthermore, the assumption of a perfect cyclic character with an amplitude that does not vary with time is usually quite unrealistic.

Computational mechanics is now reaching the stage where the actual growth of cracks in structures can be modeled along with the uncertainties in the crack growth law, initial flaw size, and randomness in loading. These methods can be based on Monte Carlo procedures; however, they are often expensive in terms of computer cost. Alternatively, the approximations of first-and second-order moment methods may be used. As described later, the latter may not be of sufficient accuracy in cases where the underlying problem is strongly nonlinear. To make these advances useful to engineers, better methods and an improved understanding of the limitations of available methods for these problems is needed.

The stochasticity in parameters *D* and *m* in the Paris law is probably due to randomness in the strength or toughness of the material and the randomness of the microstructure of the material. These ideas have been examined only very cursorily. A better understanding and methodologies for treating these problems are urgently needed for the following reasons: The development of the Paris law data involves many tests, which are often not feasible when advanced, high-cost materials are considered, and the Paris law is directly applicable only to mode I crack growth (crack growth under tension) and is not applicable to cracks that do not remain rectilinear, as in the presence of shear or in three-dimensional crack models.

By computational studies of the stochastic character of materials and their failure in conjunction with experiments, it may be possible to develop more generally applicable crack growth laws.

The implications of such improved computational mechanics methodologies are quite startling. It would be

**Suggested Citation:**"9 Uncertainty and Stochastic Processes in Mechanics." National Research Council. 1991.

*Research Directions in Computational Mechanics*. Washington, DC: The National Academies Press. doi: 10.17226/1909.

possible to relate lifetimes of components to the size and distribution of defects that are introduced in the fabrication process and, thus, design fabrication processes and optimal cost effectiveness. Inspection cycle and nondestructive evaluation techniques for structures such as bridges, pipelines, and aircraft could be optimized for reliability and cost.

**UNCERTAINTY AND RANDOMNESS IN LOADS**

Loads are the second major source of uncertainty in reliability analysis. Loads, man-made or natural, acting on mechanical and structural systems are often difficult to predict in terms of their time of occurrence, duration, and intensity. The temporal and spatial load characteristics needed for detailed analysis are also subject to considerable uncertainty. Nowhere in the engineering field does this fact manifest itself more strongly than in earthquake engineering. In view of this, the uncertainty issues associated with earthquake engineering, particularly with earthquake ground accelerations as loads to mechanical and structural systems, are used as an example to demonstrate the complexity of the problem associated with the uncertainty in loading conditions.

There are many ways in which strong-motion earthquake phenomena can be modeled from the engineering point of view. Each model consists of a number of component models that address themselves to particular phenomena of seismic events. For example, a succession of earthquake arrival times at a site may be modeled as a stationary or nonstationary Poisson process, and the duration of significant ground motion in each earthquake may be modeled as a random variable with its distribution function specified. Also, temporal and spatial ground-motion characteristics may be idealized as a trivariate and three-dimensional nonstationary and nonhomogeneous stochastic wave with appropriate definitions of intensity. Although further study is definitely needed, the progress made in this area has been rather remarkable. Some of the current models are able to reflect the randomness in the seismic source mechanism, propagation path, and surface layer soil amplification.

The difference in ground motion and resulting structural response estimates arising from the use of various models represents modeling as well as parametric uncertainties, since each component model contains a certain number of parameters to which appropriate values must be assigned for numerical analysis. Hence, the total uncertainty consists of modeling uncertainty and parametric uncertainty.

**Suggested Citation:**"9 Uncertainty and Stochastic Processes in Mechanics." National Research Council. 1991.

*Research Directions in Computational Mechanics*. Washington, DC: The National Academies Press. doi: 10.17226/1909.

In fact, a number of methods are available and have been used to identify the extent of uncertainty of parametric origin. The process for modeling uncertainty appears to be limited by the extent of the plausible models that can be constructed and the ability to examine the variability of the results from these different models. The degree of variability expressed in terms of range or any other meaningful quantity may be seen as representative of modeling error when, for example, ''best estimates'' are used for parameters within each model.

**SYSTEM STOCHASTICITY**

The last several years have seen a resurgence of research interest in the area of system stochasticity. The problem of system stochasticity arises when, among other things, the stochastic variability of the system parameters must be taken into consideration for evaluation of the system reliability under specified loading conditions.

Indeed, the parameters that control the constitutive behavior crack growth and strength of the material tend to be intrinsically random and/or uncertain due to a lack of knowledge. The stochastic variability of these parameters is idealized in terms of stochastic fields, multivariate and multidimensional as appropriate, for continuous systems and by means of a multivariate random variable for discretized systems.

The resurgence of interest appears to have arrived at a time when the finite element method has finally reached its maturity, so that the finite element solution to the problem of system stochasticity can augment existing software packages and thus provide added value. The recent effort in this direction has led to the establishment of a *genre nouveau*, "stochastic finite elements." However, many important issues remain to be addressed. In fact, it was only recently that the basic accuracy and convergence issue arising from the various methods of approximation was addressed in the context of Neuman expansion or Born approximation, primarily when dealing with static problems. Not only that, but the issue of stochastic shape functions has never really been resolved.

From the purely technical point of view, the subsequent comments seem in order with respect to stochastic finite element methods. Exact analytic solutions are available only for simple structures subjected to static loads. Mean-centered perturbation methods are the most widely used, accurate only for small values of variability of the stochastic properties of

**Suggested Citation:**"9 Uncertainty and Stochastic Processes in Mechanics." National Research Council. 1991.

*Research Directions in Computational Mechanics*. Washington, DC: The National Academies Press. doi: 10.17226/1909.

the system and inadequate to deal with nonlinear and/or dynamic problems. Solutions based on the variability response function are accurate only for small values of variability of the stochastic properties of the system and inadequate to deal with nonlinear and/or dynamic problems. In the analysis of response variability arising from system stochasticity, however, introduction of the variability response function has provided conceptual and practical novelty. The same analytical procedure can be used as in random vibration analysis where the response variance is obtained from the integral over the frequency of the frequency response function squared multiplied by the spectral density function of the stationary random input function. Monte Carlo simulation techniques are accurate for any variability value of the system's stochastic properties, applicable to nonlinear and dynamic problems, less time consuming, and more efficient in static problems if than Neuman expansion methods are used, and are applicable to non-Gaussian fields.

**BOUNDING TECHNIQUES**

The primary difficulty associated with probabilistic models dealing with intrinsic randomness and other sources of uncertainty often lies in the fact that a number, for that matter usually a large number, of assumptions must be made in relation to the random variables and/or stochastic processes that analytically idealize the behavior of mechanical and structural systems. In this regard the following statement is in order: When uncertainty problems cloud the process of estimating the structural response, the use of bounding techniques permits estimation of the maximum response, which depends on only one or two key parameters of design and analysis. The maximum response thus estimated provides a good idea as to the range of the structural response variability. Although the applicability of bounding techniques is, at this time, limited to less complicated load and structural models, a strong case can be made for the use and further development of this technique.

The bounding techniques indicated here for system stochasticity are of great engineering significance because these bounds can be estimated without knowledge of the spatial autocorrelation function of the stochastic field, which is difficult, if not impossible, to establish experimentally.