3.1 Solution Techniques for Structural Reliability
3.1.2 Historical Developments of Probabilistic Analysis
As shown in Section 1.2 (Figure 1.3), the probabilistic methods include the stochastic finite element method, the first- and second-order reliability method, sampling methods, the utilization of stochastic expansions based on the random process concept, etc. Each method requires different computational effort and provides different insight into the response variability and levels of accuracy. In this section, we briefly discuss historical developments of the most widely used methods in probabilistic analysis and discuss the advantages and disadvantages of each method. The subsequent sections, 3.2 and 3.3, describe the details of Monte Carlo simulation, Latin Hypercube sampling methods, and the stochastic finite element method. Chapters 4 and 5 provide details of the first- and second-order reliability methods and the state-of-the-art method of stochastic expansion, which incorporates several probabilistic approaches.
First- and Second-order Reliability Method
Due to the curse of dimensionality in the probability-of-failure calculation (Equation 3. 9), numerous methods are used to simplify the numerical treatment of the integration process. The Taylor series expansion is often used to linearize the limit-state g(X) = 0. In this approach, the first- or second-order Taylor series expansion is used to estimate reliability. These methods are referred to as the First Order Second Moment (FOSM) and Second Order Second Moment (SOSM) methods, respectively. FOSM is also referred to as the Mean Value First Order Second Moment method (MVFOSM), since it is a point expansion method at the mean point and the second moment is the highest-order statistical result used in this analysis. Although the implementation of FOSM is simple, it has been shown that the accuracy is not acceptable for low probability of failure (Pf < 10-5) or for highly nonlinear responses [1]. In SOSM, the addition of a second-order term increases computational effort significantly, yet the improvement in accuracy is often minimal.
The safety index approach to reliability analysis given in the previous section is actually a mathematical optimization problem for finding the point on the structural response surface (limit-state approximation) that has the shortest distance from the origin to the surface in the standard normal space. Hasofer and Lind [15]
provide a geographic interpretation of the safety index and improve the FOSM method by introducing the Hasofer and Lind (HL) transformation. In the transformation procedure, the design vector X is transformed into the vector of standardized, independent Gaussian variables, U. Because of rotational symmetry and the HL transformation, the design point in U-space represents the point of greatest probability density or maximum likelihood as shown in Figure 3.2.
Because it makes the most significant contribution to the nominal failure probability Pf )(E), this design point is called the Most Probable failure Point (MPP).
x1G
x2G g(X)=0
X-Space fX(x)G
x1G
x2G
g(X)=0
(0,0)
ȼ
u1G u2G
MPP g(U)=0
u1G
\ u2G
U-Space
fU(u)G
(0,0)
MPP
Maximum PDF g(U)=0
Figure 3.2. Transformation and MPP
Different approximate response surfaces g(U)=0 correspond to different methods for failure probability calculations. If the response surface is approached by a first-order approximation at the MPP, the method is called the first-order reliability method (FORM); if the response surface is approached by a second- order approximation at the MPP, the method is called the second-order reliability method (SORM). Furthermore, if the response surface is approached by a higher order approximation at the MPP, the method is called the higher-order reliability Method (HORM). Historically, the HL transformation method is often referred to as FORM. It is also referred to as the advanced FOSM or extended FOSM, but the acronym FORM is more prevalent for the HL method, since probability distributions are no longer approximated by their first and second moments [21].
In FORM, the limit-state is approximated by a tangent plane at the MPP. The approximate FORM results are used to specify a bound based on the probability of failure. If the approximations of the limit-state at the most probable failure point are accurate, then the bounds will produce satisfactory results; otherwise, this method may result in large error. FORM gives inaccurate results when the failure surface is highly nonlinear. Thus, FORM sometimes oscillates and converges on
unreasonable values for probability of failure. The details of FORM and other common reliability methods are discussed in Chapter 4.
Stochastic Expansions
Stochastic expansion is an efficient tool for reliability analysis because the direct use of stochastic expansion, which is based on the concept of a random process, provides analytically appealing convergence properties for the stochastic analysis [4]. The purpose of the stochastic expansion is to better represent uncertainties of systems by introducing a series of polynomials aimed at characterizing the stochastic system being investigated.
Since the introduction of the Spectral Stochastic Finite Element Method (SSFEM) by Ghanem and Spanos [9], Polynomial Chaos Expansion (PCE) has been successfully used to represent uncertainty in a variety of applications, including structural response. PCE employs orthogonal polynomials of random variables. Most commonly, the random variables are standard-normal, and Hermite polynomials are used in SSFEM. PCE is convergent in the mean-square sense, and any order PCE consists of orthogonal polynomials. This property can simplify the calculation of moments in statistical procedures.
Tatang [30] introduced the probabilistic collocation method in which the responses of stochastic systems are projected onto the PCE. Delta functions at each collocation point serve as the test functions in a Galerkin method. Tatang obtained coefficients of the PCE by using the model outputs at selected collocation points (roots of the polynomials). Isukapalli [16] pointed out the limitation of the probabilistic collocation method for large-scale models and suggested a stochastic response surface method that uses the partial derivatives of model outputs with respect to model inputs. To obtain the partial derivatives of model outputs, ADIFOR, a FORTRAN programming library, was used in the stochastic response surface method. Recently, PCE was applied to the buckling eigenproblem by evaluating coefficients of the PCE through Monte Carlo Simulation [26]. Xiu et al.
[32], [33] extended PCE to represent different distribution functions by using the Askey scheme. The Askey scheme, discussed in Chapter 5, classifies the hypergeometric orthogonal polynomials and indicates the limit transition relations between them. For instance, the Laguerre polynomials can be obtained from the Jacobi polynomials and can also be used to generate the Hermite polynomials.
Each of these methods has some limitations. In the case of the probabilistic collocation method, especially for many PCE degrees of freedom, the collocation points increase exponentially. Therefore, many collocation points are not sampled.
Consequently, the collocation points selected to obtain unknown coefficients of the PCE do not guarantee a space filling design (Figure 3.3a), one that fills up the available design space with specified sampling points according to suitably-defined design criteria (such as maximize minimum distance between points [25]). If we are interested in the tail regions of a probability density function (Figure 3.4), then the data selection procedure should be reconsidered when applying the probabilistic collocation method, because the selected design points are concentrated in the high probability region.
x2
x1
(a) Collocation Points (b) Stratified Sampling
Figure 3.3. Comparison of Design Points of Probabilistic Collocation Method and Stratified Sampling
Figure 3.4. Regions of Interest in Probability Density Function
According to the preceding methods, we can classify the usage of stochastic expansions, including PCE and Karhunen-Loeve (KL) expansions, into two approaches: the non-intrusive and intrusive formulation procedures, as shown in Figure 3.5. An intrusive formulation is one in which the representation of uncertainty is expressed explicitly within the analysis of the system. Conversely, the results from structural analysis, in which uncertainty is not explicitly represented, are used in a non-intrusive formulation to characterize stochastic system behavior. In practice, this means that intrusive methods require access to modification of the analysis codes, whereas non-intrusive methods may treat the analysis code as a “black box.” In one type of non-intrusive formulation, PCE is used to create the response surface without interfering with the finite element analysis procedure. Thus, this type of non-intrusive analysis is sometimes called the stochastic response surface method. Another type of non-intrusive formulation is the probabilistic collocation method. By contrast, the intrusive formulation method uses PCE and KL expansions to directly modify the stiffness matrix of a
x1
x2
PDFG
xG
Highest Probability RegionG
Region of InterestG
Stochastic Expansion ( PCE / KL Expansion )
Intrusive Formulation Non-intrusive Formulation
Spectral Stochastic FEM
Stochastic Galerkin FEM Stochastic Response Surface Method / Probabilistic Collocation Method
finite element analysis procedure. SSFEM and the stochastic Galerkin FEM [2] are both intrusive formulations.
The KL expansion can be applied to represent the characteristics of an uncertain system when its covariance function is known. However, if we do not have the information to form the covariance function, in the case of structural responses, then PCE can be used to represent this kind of uncertainty instead of the KL expansion. Chapter 6 includes a novel procedure using the non-intrusive formulation with an ANOVA and the Latin Hypercube Sampling (LHS) method.
This procedure can guarantee that each of the input variables has all portions of its range represented (Figure 3.3b).
Figure 3.5. Intrusive and Non-Intrusive Formulation