A probabilistic method has been developed for use in designing a composite-material structure to achieve a balance between maximum reliability and minimum cost. This method accounts for all naturally occurring uncertainties in properties of constituent materials, fabrication variables, geometry, and loading conditions. Heretofore, it has been common practice to use safety factors (also called "knockdown factors") to reduce design loads on composite structures in the face of uncertainties. Safety factors often dictate designs of structures substantially heavier than they would otherwise be, but provide no quantifiable measures of reliability. The present method involves a quantitative approach to reliability; the equations of the method are formulated to yield a design that is optimum in the sense that it minimizes a reliability-based cost.
The derivation of the equations includes the definition of a probabilistic sensitivity that quantifies the change in reliability relative to a change in each random variable (design parameter). The probability of failure for a given performance is given by
Pf = Φ(–β), (1)
where β is a reliability index and Φ is the cumulative distribution function of a normally distributed random variable. The probabilistic sensitivity factor for the ith random variable Xi is defined by
SFi = ∂β / ∂Xi = ui*/ β (2)
where ui* is the most probable failure point of a limit-state function in a unit normal probability space. The sensitivity of the reliability index to the mean mi of the normally distributed random variable Xi with standard deviation σi is given by
∂β / ∂mi = – SFi / σi (3)
Similarly, the sensitivity of the reliability parameter to the standard deviation is given by
∂β / ∂σi = – SFiui* / σi = – (ui*)2 / βσi (4)
The reliability-based total cost function, CT, is the criterion that enables one to achieve the balance between reliability and cost. This function is given by
CT = CI + Pf CF , (5)
where CI is the cost of manufacture and CF is the cost incurred in event of failure of the structure. The cost of manufacture can be expressed as
where pj is a distribution parameter (which can be either mj or σj), Cj(pj) is the manufacturing cost associated with the jth distribution parameter, and C0 is a constant cost. The total cost can be minimized when
∂CT / ∂pj = 0 (7)
for all j from 1 to N.
Then after substitution of terms from equations 1, 5, and 6 and use of the chain rule for derivatives, equation 7 becomes
for all j from 1 to N.
For a normally distributed random variable, ∂β/∂pj can be calculated by equations 3 and 4. Equation 8 represents a system of N nonlinear equations that, if solved, yield a design with an optimum tradeoff between reliability and cost.
This method can be considered a special case of method for comprehensive probabilistic assessment of composite structures. The comprehensive method is implemented in the Integrated Probabilistic Assessment of Composite Structures (IPACS) computer code. [The comprehensive method was described from a slightly different perspective, with emphasis on computation of structural responses and fatigue lives, in "Probabilistic Analysis of Composite-Material Structures" (LEW-16092), NASA Tech Briefs,Vol. 21, No. 2 (February 1997), page 58.]
The method was demonstrated in test case in which the objective was to minimize the reliability-based cost of a lower side panel of a composite (graphite-fiber/epoxy-matrix) fuselage structure, using, as a design parameter, the coefficient of variation (COV) of the modulus of longitudinal elasticity of the graphite fibers. For the case studied, the minimum normalized total cost for a normalized failure cost of $15,000/lb ($33,000/kg) was found to occur at COV = 0.05. The optimum COV as a function of the normalized failure cost was also computed (see figure).
This work was done by Christos C. Chamis of Lewis Research Centerand Michael C. Shiao and Surendra N. Singhal of NYMA, Inc. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com under the Materials category.
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Refer to LEW-16580.