7. Probabilistic Analysis Examples via Stochastic Expansion
7.1 Gaussian and Non-Gaussian Distributions
7.1.3 Non-Gaussian Distribution Examples
This section presents the reliability analysis for non-Gaussian distributions to demonstrate the accuracy and efficiency of two existing techniques (the generalized PCE algorithm and the transformation method), which were described in Section 6.1.3. Assessments of the statistical characteristics, including the confidence interval, are demonstrated. Also, reliability analysis is applied to the joined-wing aircraft to demonstrate the procedure’s effectiveness in real engineering problems.
Pin-connected Three-bar Truss Structure
An indeterminate, asymmetric system consisting of a three-pin-connected truss structure is illustrated in Figure 7.8. The unloaded length, Lm, and orientation, Dm, of each member are deterministic. Young’s modulus, Em, of each member is also assumed to be deterministic. The load has random magnitude (P) and direction (T).
The cross-sectional area A for all members is also random. The random quantities are initially considered normally distributed as a baseline for comparison to a non- Gaussian distribution:
A~N(1 in2, 0.1 in2) P~N(1000 lb, 250 lb)
T~N(45o, 7.5o)
where the symbol x ~ N(Px,Vx) denotes that the random variable x is treated as a normal distribution and has the mean of Px and standard deviation of Vx.
The principle of virtual work is used to calculate the displacement vector v T
u, ]
[ of the joint at which the load is applied, and it is found by solving the following system of equations:
m m m m
m m
m L
A v E
u
P ¦3
1
2 cos sin )
cos (
cosT D D D (7.10)
m m m m
m m
m L
A u E
v
P ¦3
1
2 cos sin )
sin (
sinT D D D
The horizontal deflection of the structure should be u < 0.001 in.
Figure 7.8. Pin-connected Three-bar Truss Structure
Table 7.4. Comparison of Methods for Reliability Analysis
Pf Difference (%) MCS 0.0045 -
PCE 0.0044 2.2 FORM 0.0051 13.3
The results of this example are shown in Table 7.4, along with MCS and FORM results. To obtain the probability of failure,Pf , one million simulations were conducted to reach a converged result in MCS, and 100 samples of LHS were used to find thePf for the third-order PCE model. In this case, the nonlinear terms of PCE have significant effects on the regression model. The FORM result converged toPf = 0.0051. From the table, it is obvious that PCE is more accurate than FORM in this example. Due to the nonlinearity of the response, it is expected
that the use of PCE along with LHS instead of FORM is more applicable to this case.
Figure 7.9. Gamma Density Function of Random Variable A
Many practical engineering situations require a wide variety of skewed distributions, which are not bell-shaped and symmetric like the normal distribution.
The gamma distribution is a useful tool for representing skewed cases. For the preceeding example, the random variable A of the pin-connnected three-bar structure can be treated as a gamma distribution that has scale parameters D 10 and E 0.1 (Figure 7.9), with the other variables remaining unchanged. For the description of the gamma distribution, the mean and variance of the random variable A are P DE 1.0 and V2 DE2 0.1. This analysis was carried out using LHS with 200 samples for two non-Gaussian techniques. In the generalized PCE method, Laguerre polynomials were used to represent the gamma distribution of the random variable A, and in the transformation technique, the transformation function shown in Table 6.2 was used. In order to estimate the quality of the non- Gaussian simulation techniques, MCS was also conducted using one million samples.
Figure 7.10 shows the results of this case for two techniques, the transformation technique (TRANS) and the generalized PCE algorithm (GPCE). By comparison, the probability of failure using MCS was estimated as Pf= 0.0655. This value is quite close to the results of higher order (e.g., fourth through sixth) models for the two other techniques: in the higher-order models of PCE, a maximum difference of 4.27% and a minimum difference of 1.83% were detected. To show the
A
convergence of the results, we illustrated the results of the second through sixth- order models in Figure 7.10. The effects of fifth- and sixth-order PCE were not significant on this model, according to the F-statistics (Į = 0.10). The Pf of the fifth-order model in Figure 7.10 is slightly higher (1.0% to 2.1%) than the fourth and sixth-order model results, since the coefficients of fourth- and sixth-order PCE exhibit negative values, while the coefficient of the fifth-order PCE has a positive value. The bounded result is not significant in this case because the F-statistics indicate small effects of the fifth and sixth-order terms, and the statistics obtained in each random set of the pseudo-random number generator can be bounded with small intervals. The result of MCS also has a 1.2% bound. Figure 7.10 shows that the lower-order models of the generalized PCE algorithm have a more accurate result than the transformation technique; however, we can see that the two techniques have almost identical results for higher orders (e.g., fourth through sixth), which have sufficient accuracy compared to MCS.
Figure 7.10. Probability of Failure
Joined-wing Example
In the joined-wing problem of the previous section, all random variables were assumed to be Gaussian distributions. Now, we consider the non-Gaussian distribution case for the same configuration of the joined-wing (Figure 7.5).
Young’s moduli of five locations are modeled as uncorrelated random variables using Gamma distributions, withD 100 andE 6.9u108 (P DE 6.9u1010 Pa,
9 2 6.9u10 DE
V , COV = 0.1). The first and second buckling eigenvalues are O1=1.345 and O2=1.807 at the mean design point. The eigenvalue of 1.345
0.06 0.065 0.07 0.075 0.08 0.085 0.09
͢G ͣG ͤG ͥG ͦG ͧG ͨG
ΣΕΖΣ͑ΠΗ͑ʹͶG
ΗG
MCS TRANS GPCE
indicates a 34.5% margin of safety. To demonstrate the reliability analysis of the joined-wing, the condition O1 < 1.34 is chosen to represent failure.
Table 7.5. Comparison of Methods for Joined-wing Reliability Analysis
Approach Pf Difference (%) MCS (10,000) 0.1327 - LHS+PCE (200) 0.1343 1.25
Table 7.5 shows the probability of failure comparison between MCS and PCE.
The MCS result was obtained by conducting 10,000 NASTRAN simulations. The non-intrusive analysis was carried out using LHS with 200 samples. In the non- intrusive formulation, fourth-order PCE and the transformation technique (Section 6.1.3) were used without considering the interaction effects of each random variable. After obtaining the undetermined coefficients of PCE, residual analysis and ANOVA analysis were conducted to determine the model adequacy. The probability of failure of the given structural system was calculated using 10,000 MCS simulations on the surrogate model of PCE, which has sufficient adequacy.
The difference between the results of direct MCS and the current method was about 1.25%. This result shows that the non-intrusive formulation method, combined with the transformation technique and LHS of 200 full simulations, gives sufficient accuracy as compared to the 10,000-simulation result of MCS. A 95% confidence interval was computed by PCE for the mean response at the mean point using Equation 6.19; the resulting interval is 1.341dPd1.349. Once the approximation is constructed, many statistical properties of the response, such as probability of failure, can be estimated without incurring significant computational costs. The 95% confidence interval for MCS (1 of 10,000) is 1.336dP d1.352. We can interpret the interval calculated at the 95% level as 95% certainty that the true value of P is in this interval.