Deterministic and Reliability-Based Optimization of Composite Laminates for Cryogenic Environments

Abstract

Designs of composite laminates are investigated for hydrogen tanks in cryogenic environments. Large residual strains, which can develop due to thermal mismatch between matrix and bers, result in matrix cracking at cryogenic temperatures and increase hydrogen leakage through the tank wall. To reduce thermal mismatch, ply angles need to be close to each other, but this leads to a substantial weight increase under biaxial loading. First deterministic optimization is used to investigate possible weight reduction measures. Reducing axial loads on walls by auxiliary stiffening mechanisms led to signi cant weight reduction. Reliability-based optimizations were performed to identify the uncertainties in composite material properties with the largest in uences on the optimum design. Then measures for reducing uncertainty in important parameters are examined. The results indicate that the most effective measure for reducing thickness is quality control. Nomenclature E; ¾ , CV = mean value, standard deviation, and coef cient of variation (¾=E) of the random variables E 1 , E 2 , G 12 = elastic modulus along and transverse to ber direction and shear modulus of a composite ply h = total laminate thickness T zero = stress-free temperature t 1 , t 2 = thickness of plies with angles µ 1 and µ 2 , respectively ® 1 ; ® 2 = coef cient of thermal expansion along and transverse to ber direction " 1 ; " 2 ;°1 2 = strains along and transverse to ber direction and shear strain of a composite ply µ ; µ 1 ; µ 2 = ply orientation angles ¹ 12 = major Poisson ratio of a composite ply Subscripts l = lower limit u = upper limit

Using Cross Validation to Design Conservative Surrogates

“…The cryogenic operating temperatures are responsible for large residual strains due to the different coefficients of thermal expansion of the fiber and the matrix, which is challenging in design. The deterministic and probabilistic design optimizations of the barrel were performed by Qu et al [46], using response surface approximations for probability of failure calculations. In this section, the probabilistic optimization problem of [46] is considered for investigating conservative surrogates.…”

“…The deterministic and probabilistic design optimizations of the barrel were performed by Qu et al [46], using response surface approximations for probability of failure calculations. In this section, the probabilistic optimization problem of [46] is considered for investigating conservative surrogates. The geometry, material parameters, and loading conditions are taken from their paper.…”

“…E 2 , G 12 , 1 , and 2 are functions of the temperature. Since the design must be feasible for the entire range of temperature, strain constraints are applied at 21 different temperatures, which are uniformly distributed from 20 to 300 K. Details on the analysis and the temperature dependence of the properties are given in [46]. The solutions for the deterministic optimization problem are summarized in Table 4.…”

“…Here, the critical strain is known to be the transverse strain on the first ply (direction 2 in Fig. 12), with the effect of other strains on the probability of failure being negligible [46]. Hence, failure of the system occurs when the difference G between the critical strain and the failure strain is positive:…”

Error Estimation and Error Reduction in Separable Monte-Carlo Method

“…(2)]. References [14][15][16] discuss reliability-based optimization of composite laminates in detail. For complex structures, the stress is calculated through finite element analysis and it may be computationally expensive.…”