Coordinative optimization method of composite laminated structures based on system reliability

Abstract

This paper develops the coordinative optimization method based on system reliability for laminated structures. The proposed method improves the rough RBO based on first layer failure (FLF) criterion for composite laminates, and the coupling optimization method of thickness and sequence in traditional RBO strategy based on last layer failure criterion (LLF) is improved. In this paper, the finite element analysis is used to obtain the response for the failure based on two-dimension Hashin failure criterion (the limit function). Obviously, the stiffness of composite materials will decline due to destruction of elements. Therefore, stiffness degradation is considered to describe the process of damage evolution. Subsequently, combining with the branch-bound method (B&B), we can complete the search of main failure sequences and calculate the system reliability with the help of the second-order upper bound theory. In order to guarantee the efficiency and accuracy of optimization, the adaptive GA algorithm is introduced in the whole optimization procedure. After the proposed optimization policy is given in detail, two laminated structures are presented and the results are compared with the traditional optimal method based on safety factor, which demonstrates the validity and reasonability of the developed methodology.

Appendix

In order to validate the accuracy of the proposed method for system failure probability of laminates, the numerical approach for system failure probability based upon Monet Carlo simulation is given below.

• Step1 Initialize the numbers of generated samples and failed samples as N_generated = 1e6 ${N}_T^{fiber}$, ${N}_C^{fiber}$, ${N}_T^{matrix}$and ${N}_C^{matrix}$, which respectively represents FT failure, FC failure, MT failure and MC failure mode;

• Step2 Conduct the response analysis by the finite element method. Substitute the data σ₁, σ₂, τ₁₂ into the four state functions Eq. 1, record sensitivity information of failure modes and restore the failed samples as follow rule:

  If the FT mode happens, then ${N}_T^{fiber}={N}_T^{fiber}+1$, the FT failure probability is calculated as ${P}_{FT}=\frac{N_T^{fiber}}{N_{generated}}$; if the FC failure mode happens, then ${N}_C^{fiber}={N}_C^{fiber}+1$, the FC failure probability is calculated as ${P}_{FC}=\frac{N_C^{fiber}}{N_{generated}}$; If the MT mode happens, then ${N}_T^{matrix}={N}_T^{matrix}+1$, the MT failure probability is calculated as ${P}_{MT}=\frac{N_T^{matrix}}{N_{generated}}$; If the MC mode happens, then ${N}_C^{matrix}={N}_C^{matrix}+1$, the MC failure probability is calculated as ${P}_{MC}=\frac{N_C^{matrix}}{N_{generated}}$;

• Step3 Compare the size of four failure modes and update the stiffness matrix based on the materials degradation criteria, further complete the search of main failure sequences with help of B&B method:

  • (i) The max failure possibility is obtained P_max, therefore the max^th layer can be considered as destroyed firstly and the stiffness is deteriorated correspondingly, then go to step 1 and 2 for structural reanalysis;

  • (ii) P_i stands for each ply failure probability, if the relation P_i/P_max ≥ 0.3, the i − th layer is record as branching point, which represents the i − th layer can also be destroyed firstly, and then the stiffness is deteriorated correspondingly, then go back to step 1 and 2;

  • (iii) Above analysis procedure is end until total stiffness matrix becomes singular ∣ det K_T/ det K₀ ∣  ≤ λ(λ → 0), and the main failure sequences are found.

• Step4 According with sensitivity information obtained in step 2, calculate the correlation factor by Eq. 13, the failure probability of single main sequence ${P}_f^{sys}$ is computed, and the system failure probability P_f is acquired with help of the weakest sequence.