on the values of the optimal deterministic design. Using the ARS, probabilities of failure
are calculated by Monte Carlo simulations
A quadratic polynomial of 17 variables has 171 coefficients (=18xl9/2). The
number of sampling points generated by LHS was selected to be twice the number of
coefficients. Table 8-17 shows that the quadratic response surfaces constructed from LHS
with 342 points offer good accuracy. Quadratic ARS over as +5% design perturbation is
employed to perform reliability analysis and design optimization.
Table 8-17. Quadratic analysis response surface approximation to the worst margins
using Latin Hypercube sampling of 342 points
Critical margins of load Critical margins of load
case 1 case 2
Rsquare adj. 0.9686 0.9772
RMSE predictor 0.01615 0.01080
Mean of response 0.4186 0. 1726
The deterministic design is then evaluated under material and manufacturing
uncertainties. The probability of failure of the deterministic optimum under uncertainties
is shown in Table 8-18. The dominant failure mode is in-plane shear and transverse
tensile strength failure for load case 1) of internal proof pressure of 35 psi. The high
failure probability can be reduced by using increased safety factor or reliability-based
design optimization.
Table 8-18. Probabilities of failure calculated by Monte Carlo simulation of 106 samples
(material and manufacturing uncertainties)
Probability of Probability of System Probabilistic
System probability
failure of load case failure of load case SfiinyFco
of failure SfiinyFco
1 (strength) 2 (buckling) for 10-4 prObability
89.78x10-4 0 89.78x10-4 0.837178
As shown by the aluminum panel design example, RBDO using a design surface
approximation around deterministic optimization yields a reliability-based design near