J. Cent. South Univ. (2019) 26: 665-672

DOI: https://doi.org/10.1007/s11771-019-4037-3

Reliability analysis of a composite laminate using estimation theory

Esmail SADEGHIAN, Sina TOOSI

Department of Marine Engineering, Amirkabir University of Technology, Tehran, Iran

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: Composite laminates are made up of composite single-plies sequence. The plies generally have the same fiber and resin and their difference in fiber orientation results in a difference in various laminates' strength. Tsai-Hill failure criterion as a limiting state function to analyze structural reliability of a composite laminate and estimation theory in order to estimate statistical parameters of effective stress were utilized to construct probability box. Finally, we used the Monte Carlo simulation and FERUM software to calculate the upper and lower bounds of probability of failure.

Key words: composite laminate; structural reliability; probability box; estimation theory; stress analysis

Cite this article as: Esmail SADEGHIAN, Sina TOOSI. Reliability analysis of a composite laminate using estimation theory [J]. Journal of Central South University, 2019, 26(3): 665–672. DOI: https://doi.org/10.1007/s11771-019-4037-3.

1 Introduction

Composite laminate is widely used in structural elements in construction of the vessel and other mechanical systems. Composite vessels are usually exposed to various loadings in different environment. Therefore, in order to prevent from unknown failures, it is necessary to investigate and analyze the reliability of structures before their implementation.

In recent decades, many studies have been conducted on investigating the probable failure methods and its reliability in composite structures. First, YANG et al [1, 2] used this method to estimate the strength of composite laminate. They utilized Tsai-Hann failure criterion as well as random variables, including loading and strength of composite, to study ply level. Due to the complexity of failure concept, they used step by step estimating procedure of uniaxial stress reliability. Shortly after that, CEDERBAUM et al [3] employed B-method developed by HASOFER et al [4] to laminate plates. They investigated Hasin ply level failure criterion considering loading as random variable. By development of studies on composites, WETHERHOLD et al [5] comprised different methods of reliability calculation for a specific case using Tsai-Han and Tsai-Wu failure criteria while considering loading and strength as random variables. SOARES [6] established a general framework for calculating reliability of laminate surface. He provided an overall viewpoint about methods used for laminate using failure criteria of Tsai-Han and Tsai-Wu while taking into account loading and strength as random variables, as well as applying reliability equations system in limit state. The validity of Soares’s study results were also confirmed by DI SCIUVA et al [7]. In their study, a laminate was evaluated under distributed pressure in the middle section. The failure criterion of the study was buckling failure and loading; strength, rigidity and geometry were used as random variables. Their results had an acceptable level of accuracy.

The reliability based optimal design of geometrically non-linear composite structures using hierarchical genetic algorithms proposed by ANTONIO [8]. NGAH et al [9] provided an application of different methods of reliability estimation in a composite panel exposed to random loading and finally extracted covariance and probability density function. CARBILLET et al [10] chose second-moment-FORM (first-order reliability method) method to determine safety factors in the design of composite laminate with strongly non-linear behavior. NOH [11] utilized stochastic finite element method to analyze the laminated composite plate. He assumed five parameters of material that are random and used means of correlation functions to model the correlations these random parameters. WANG et al [12] carried out a series of compression and shear tests on a laminated composite and then investigated the reliability by Monte Carlo and improved first-order second-moment methods. LOPEZ et al [13] employed a full characterization method to assess the reliability of laminated composite plates and showed that this method has some advantages in comparison with the First-order reliability method (FORM), which has been widely use. PATEL et al [14] analyzed a composite plate under low-velocity impact and Gaussian response surface method used as a probability of failure function. Uncertainty parameters are properties of composite material, loading condition, and assessment of critical stress. Using sensitivity analysis, they concluded that shear strength and elastic modulus exert the most influence on the composite plate reliability. HAERI et al [15] applied an advanced Kriging model to approximate the mechanical model of the structure. They employed Tsai-Wu criterion as a limit state function in reliability analysis.

Several methods to calculate reliability in composite structures have been provided according to previous conducted studies; however, nonexistence of certain framework for utilization of these methods, their failure criteria, statistical analysis of mechanical parameters, and even their results is obvious. Hence, reliability analysis on composite requires further investigations due to the inherent variability in the behavior of materials.

In the present study, we analyzed first ply failure probability of composite laminate by utilizing the estimation theory to describe imprecise probability function of effective stress. Moreover, Monte Carlo simulation and FERUM software were applied to calculate the reliability index.

2 Reliability of composite structures

A need for applying uncertainty-based approaches in engineering topics has been known for a long time. Applying random parameters is very important in the structural design issues, especially in composite structures. In formulation of the structural reliability approach of composite laminates, strength parameters are modeled corresponding to each of the plies and at lower levels of mechanical properties of fibers, resins, orientation of fibers, and geometry of each ply as random variables with a degree of dependence.

From strength perspective, the failure of a composite laminate is divided into two categories [16]. One is last ply failure resulted by crack in the resin field, and the other one is the first ply failure resulted by separation of the layers, crack in the resin and fiber failure.

In the present study, the first ply failure approach has been used to analyze structural reliability of a composite laminate.

3 Limit state function and random variables

The final strength of a composite laminate depends on the strength of each of the single layers and their arrangement. Over the last 30 years, several failure criteria have been introduced and developed to describe the failure mechanism of composite laminates. Tsai-Hill failure criterion for composite elements under stress in the main direction is as follows [17].

           (1)

In Eq. (1), each of variables, X₁, X₂ and Y is determined based on first ply failure stress tensor:

                    (2)

                    (3)

                    (4)

Therefore, the limit state function is formulated based on the introduced failure criterion.

 (5)

Finally, the failure occurrence condition is defined as regarding the concept of limit state function. The failure condition is unsafe regions for occurrence, which have been defined based on the general distinction between resistance and strength of structure. The geometric explanation of limit state function has been provided in Figure 1.

Figure 1 Separation of safe and unsafe (failure) regions by limit state function [18]

4 Stress analysis of a composite laminate

A simple example of composite laminate under axial tensile loading has been analyzed in order to investigate the structural reliability of composite elements. The strength statistical characteristics of failed ply failure have been determined based on selecting the first ply failure as the failure criterion to reliability analysis and Tsai-Hill limit state function using effective stress method. Before determining effective stress statistical characteristics, progressive failure analysis of composite laminate is needed in order to determine the first failed ply. Its procedure has been shown in the following flowchart.

Figure 2 Flowchart of stress analysis in composite laminate

A four-UD-ply with configuration of [0/90]_s under tensile loading is considered. The laminate failure has been analyzed with the assumption of plane stresses. By considering this assumption, the stiffness matrix alters from six-in-six form to orthotropic plane matrix which is three-in-three matrix.

Table 1 shows the geometric and mechanical properties of composite laminate under study.

Table 1 Properties of composite sheet and imposed loads

The composite plate with above properties was modeled in MARC finite element software (see Figure 3). Failure analysis results on the main direction are shown in Figure 4.

As it can be seen from above diagram and according to Tsai-Hill failure theory, the ply with 90-degree fiber orientation would fail sooner than the other plies.

5 Effective stress and modifying limit state function

After analyzing failure stress of considered laminate and determining failure stress tensor of 90° ply, it was observed that the component of shear stress is equal to zero, so the term of in the limit sate function can be eliminated. Another assumption of the present study was utilizing effective stress criterion in order to reduce the number of random variables. To determine the equivalent stress at the first failed ply, the potential function theory is used [19].

          (6)

where N_ij is the non-dimensional parameter used in the laminate; R_ij is the mean residual stress of material used in the sample resin. The micro-mechanical relations were used to calculate these parameters. The symbol σ_ij represents arrays of plane stress tensors that have been created due to the external loads in the sample. Therefore, according to this theory, the effective stress is defined as follows.

Figure 3 Finite element model of laminate under tensile load

Figure 4 Failure index in two composite layers with fiber orientation of 90°and 0°

                              (7)

σ₁ and σ₂ stresses in the Tsai-Hill limit state function are replaced by calculated effective stress. However, it should be noted that the definition of random behavior for the variables is needed to analyze the probability of failure in considered ply. This is because the component effective stress is uncertain due to the existence of various uncertainties inherent in the problem and, therefore, statistical parameters such as mean and standard deviation should be used to show it. For this purpose, a method named Stochastic FEM has been used in recent decades, which was ignored in the present studies due to its complexities. In this work, we used estimation theory to determine the bound of mean and standard deviation of effective stress. Adjusting proper probability distribution function for effective stress of 90° ply, the overall form of the limit state function was modified as follows:

     (8)

6 Estimation theory

Since stochastic finite element method (SFEM) was not conducted in the present study, the estimation theory was used to assign a proper probability distribution function to the effective stress random variable. The aim of estimation theory is to estimate the parameters of statistical population by having a limited number of samples. The estimation is carried out using components named estimator. Estimators are divided into two categories of point and distance. The point estimator has a large number of errors which depends on the volume of the sample. Generally, if sample size of random sample is less than 30 and the distribution of the population is normal and the population variance is considered equal to the sample variance, then confidence level of (1–α)100%, the population mean would be in below range:

              (9)

According to Eq.(9),if α=5%, then μ can be considered the population of effective stresses of 90° ply with confidence level of 95% at below range:

            (10)

In a same way, the estimation of σ² variance of a population with confidence level of (1–α)100% for σ² from a normal population, when a random sample would be selected with the size of n<30, is as follows.

                  (11)

where S² is the variance of selected sample; andare chi-squared distribution with v=n–1

degree of freedom. In this section, statistical population sample is required to estimate the confidence level of mean and standard deviation. For this purpose, the force-based diagram of coefficient changes of Tsai-Hill failure was used. In this way, the corresponding stress tensor of each force and consequently the considered effective stress tensor were calculated. Data gathering was continued to reach effective stress of the failure which has been estimated based on Tsai-Hill. In Table 2, the gathered samples based on diagram of Figure 4 have been represented.

Table 2 Statistical sample of effective stresses imposed to 90° ply based on Tsai-Hill criterion

According to the sample size (n=8) and provided definitions about the concept of distance estimation, mean and standard deviation of the statistical population, the effective stresses which lead to failure are defined at confidence level of 95% and based on Tsai -Hill criterion as follows:

9.32 MPa<σ<28.69 MPa

22.97 MPa<μ<49.31 MPa

Determined bound based on classic estimate theory shows variation of statistical parameters; therefore, imprecise probability distribution function for effective stress occurrence in this bound can be defined based on Figure 5 which is p-box.

Figure 5 Effective stress p-box

According to the occurred stress tensor in 90° ply and Eqs. (2), (3) and (4), the values of statistical parameters of each one of the random variables have been provided in Table 3.

Table 3 Values of statistical parameters and distribution type of them

So far, the range of changes in mean and standard deviation of failure effective stress of glass/epoxy composite plate with [0/90]_s layering under uniaxial loading, as well as the values of other random variables of Tsai-Hill limit sate function, has been calculated. Therefore, the effect of selecting mean and standard deviation of effective stress on failure probability in glass/epoxy composite laminate has been investigated using FERUM software code. Results were compared to that of Monte Carlo simulation.

7 Results and discussion

Since stochastic finite element method (SFEM) was not conducted in analyzing the effective stress of first failure ply and, consequently, random field of effective stress was not available to approximate its statistical parameters, it tried to investigate the differences of failure probability values and confidence coefficient through selecting estimated mean values and standard deviation in the distance with confidence level of 95%. The parameters P_f and β have been calculated using FERUM software code and its failure probability analysis methods such as subset simulation and FORM. Results are represented in Table 4.

Table 4 Failure probability and reliability index based on changes in mean and standard deviation of effective stress

The results can also be represented as Figure 6.

According to Figure 6 and as it was expected, the failure probability is increased with effective stress’s mean values increasing in constant standard deviation and as a result, the reliability index has a downward trend.

Figure 6 Changes in reliability index based on changes of mean effective stress

In order to investigate the accuracy of calculations, Monte Carlo simulation has been derived to calculate the failure probability of composite laminate under axial loading. For this purpose, first, values of stochastic variables based on assigned distribution type are generated; second, values of limit state function are calculated.Figure 7 represents the upper and lower bounds of limit state CDF.

Figure 7 Upper and lower bound of limit state CDF

As shown, lower bound of Tsai-Hill limit state function using one million iteration doesn't reach to failure condition . It means that failure probability is very low.For upper bound failure of Monte Carlo simulation, failure probability and reliability index are 7.14×10^–3 and 2.45, respectively, which are in good agreement with the results in Table 4. In order to demonstrate the generated values of state function for upper and lower bound according to upper and lower of effective stress occurrence, Figure 8 is presented.

Figure 8 shows that some of limit state function values for upper bound are less than zero. In this situation failure occurs according to Tsai-Hill failure criterion.

There is notable gap between limit state function values for lower bound and failure condition, meaning that failure probability is very low, shown in Figure 9.

8 Conclusions

Estimation theory is one of the useful and appropriate means in determining bounds of statistical parameters variation of a random variable. In this paper, inherent uncertainties in determining exact values of mean and standard deviation of effective stress are simulated using estimation theory and calculation of the minimum number of data as a sample. Simulation is carried out using Monte Carlo and FERUM software, resulting in determination of variation bound of failure probability and reliability index. One of the notable conclusions of this paper is achieving reasonable results with the minimum number of data from a random variable, requiring the least time and cost. It should be noted that in the case of existence of several numbers as random variables with imprecise probability distribution function, full interval analysis is required, an expensive and time-consuming activity concerning estimated widespread bound from estimation theory.

Figure 8 Variation of limit state function values in proportion to effective stress of upper bound

Figure 9 Variation of limit state function values in proportion to effective stress of lower bound

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(Edited by DENG Lü-xiang)

中文导读

基于估计理论的复合材料层合板可靠性分析

摘要：复合层压板由复合单层序列组成。同一层通常具有相同的纤维和树脂，其纤维取向上的差异将直接导致各种层合物强度的不同。本文利用Tsai-Hill失效准则作为分析复合材料层合结构可靠性的极限状态函数和估算理论来估计有效应力的统计参数，构建概率盒。最后，使用蒙特卡罗模拟和FERUM软件来计算失效概率的上限和下限。

关键词：复合层压板；结构可靠性；概率框；估算理论；压力分析

Received date: 2017-05-27; Accepted date: 2018-11-10

Corresponding author: Sina TOOSI; Tel: +98-9120175930; E-mail: Sinatoosi@aut.ac.ir; ORCID: 0000-0002-3544-1450