Random growth of crack with R-curve: Markov process model

A probabilistic model for the randomness of the progressive crack growth in a quasi-brittle material such as concrete is presented. The model consists of a Markov chain adapted to R-curve behavior. It yields the crack propagation probability in any loading step as well as the probability of failure at any stage of the fracture process. The R-curve is obtained from the given test data on the effect of structure size on the maximum loads. The standard deviation of the peak load is the minimum statistical information required. According to the available test results, this standard deviation is approximately a linear function of the crack propagation distance. The parameter estimation method is formulated and some applications are illustrated.     

Reliability Prediction of Composite Tubular Structure Under Mechanical Loading by Finite Element Method

The diversified use of filamentary composites in harsh marine environments, recorded in recent years, has prompted researchers to focus their work on the reliability prediction. Through failure criteria, Tsai–Wu and the maximum stress, the reliability of multilayer tubular structures under mechanical loading is the subject of this paper, where Monte Carlo method estimated the failure probability. A sensitivity analysis was performed in order to identify the influence of the different parameters, such as materials’ properties, geometry, manufacturing and loading, on the reliability of the composite cylindrical structure studied. To achieve a high accuracy of the results, we have carried out 105 simulations. The results showed great influence on pressure loading, ply thickness and finally winding angle of filament composite.     

A novel predictive model for mechanical behavior of single-lap GFRP composite bolted joint under static and dynamic loading

Mechanical connection of composite is critical due to its complicated meso-structure and failure mode, which has become a bottleneck on reliability of composite material and structure. Although many researches on composite bolted joints have been carried out, the theory and experiment on mechanical behavior of such a joint structure under dynamic loading were rarely reported. Here, we propose a novel predictive model for quasi-static and dynamic stiffness of composite bolted joint by introducing the strain rate dependent elastic modulus into the mass spring model. Combined with the composite laminate theory and Tsai-Hill theory, the present model was capable of predicting the strain rate dependent stiffness and strength of the composite bolted joint. Quasi-static and impact loading experiments were carried out by Zwick universal hydraulic testing machine and split Hopkinson tension bar, respectively. The stiffness and strength predicted by our model showed good accordance with the experiment data with errors below 12% under quasi-static loading and below 30% under impact loading. The results indicated that under impact loading, stiffness and strength of the composite bolted joint were significantly higher than their quasi-static counterparts, while the failure mode of the joint structure trended towards localization which was mainly bearing failure. Among various lay-up ratios studied, the optimal lay-up ratio for quasi-static and dynamic stiffness was 0:±45:90 = 3:1:1.     

Fatigue reliability evaluation using probability density evolution method

Probability density evolution method is extended to analysis of fatigue reliability. The joint probability density evolution equation of random parameters and fatigue damage is derived based on the principle of preservation of probability, and a finite difference algorithm in terms of TVD theory is presented. For a given damage threshold, cycle-dependent fatigue reliability can be calculated by the proposed method without assuming the probabilistic distribution of fatigue damage in advance. Two validation examples indicate that the proposed method is able to give reasonable results for constant-amplitude loading and variable-amplitude loading. The predicted fatigue reliability under constant-amplitude loading shows a considerable accuracy. In addition, reliability isolines of fatigue damage can be used to predict the fatigue life with a specified failure probability.     

Mechanical property variability in FRP laminates and its effect on failure prediction

Results are presented from an experimental study for the modeling of stochastic behavior of a unidirectional Glass/Polyester composite. An analytical approach is developed for the prediction of failure under general in-plane loading including the variability of strength, stiffness and the thermal expansion coefficients. Monte Carlo simulation and the first-order reliability method are used for comparison and the new method is proved to be in good agreement. Following international design codes, a direct comparison is also presented for failure locii at a specific reliability level as derived by the various probabilistic approaches. Results reveal that a serious overestimation of the reliability of the composite structure is being made when the stochastic nature of the material elastic properties is not taken into account.     

岸桥起重机随机风振疲劳可靠性分析

基于时域方法研究岸桥起重机的风振疲劳可靠性问题。采用谐波叠加法给出了符合Davenport风速谱的多维脉动风速时间历程,基于Bernoulli方程得到相应的风压时间历程,并将相应的风压荷载作用于有限元模型,采用雨流计数法处理结构关键点的应力响应。基于疲劳失效的Basquin方程、Miner线性累积损伤准则和Goodman平均应力修正方程导出疲劳累积损伤的概率模型。考虑平均风速的概率分布,提出了基于概率累积损伤机制的风振疲劳可靠度和可靠性寿命计算方法,为岸桥起重机的风振疲劳可靠性分析作了一些有益的探索和研究。     

A comparative study of failure criteria in probabilistic fields and stochastic failure envelopes of composite materials

One of the major objectives of this paper is to offer a practical tool for materials design of unidirectional composite laminates under in-plane multiaxial load. Design-oriented failure criteria of composite materials are applied to construct the evaluation model of probabilistic safety based on the extended structural reliability theory. Typical failure criteria such as maximum stress, maximum strain and quadratic polynomial failure criteria are compared from the viewpoint of reliability-oriented materials design of composite materials. The new design diagram which shows the feasible region on in-plane strain space and corresponds to safety index or failure probability is also proposed. These stochastic failure envelope diagrams which are drawn in in-plane strain space enable one to evaluate the stochastic behavior of a composite laminate with any lamination angle under multi-axial stress or strain condition. Numerical analysis for a graphite/epoxy laminate of T300/5208 is shown for the comparative verification of failure criteria under the various combinations of multi-axial load conditions and lamination angles. The stochastic failure envelopes of T300/5208 were also described in in-plane strain space.     

A reliability-based framework for fatigue damage prognosis of composite aircraft structures

The extensive use of lightweight composite materials in composite aircraft structures drastically increases the sensitivity to both fatigue- and impact-induced damage of their critical structural components during their service life. Within this scenario, an integrated hardware–software system that is capable of monitoring the composite airframe, assessing its structural integrity, identifying a condition-based maintenance, and predicting the remaining service life of its critical components is therefore needed. As a contribution to this goal, this paper presents the theoretical basis of a novel and comprehensive probabilistic methodology for predicting the remaining service life of adhesively bonded joints within the structural components of composite aircraft, with emphasis on a composite wing structure. Non-destructive evaluation techniques and recursive Bayesian inference are used to (i) assess the current state of damage of the system and (ii) update the joint probability distribution function (PDF) of the damage extents at various locations. A probabilistic model for future aerodynamic loads and a damage evolution model for the adhesive are then used to stochastically propagate damage through the joints and predict the joint PDF of the damage extents at future times. This information is subsequently used to probabilistically assess the reduced (due to damage) global aeroelastic performance of the wing by computing the PDFs of its flutter velocity and the velocities associated with the limit cycle oscillations of interest. Combined local and global failure criteria are finally used to compute lower and upper bounds for the reliability index of the composite wing structure at future times.     

Probabilistic model for fatigue crack growth and fracture of welded joints in civil engineering structures

This paper presents a probabilistic assessment model for linear elastic fracture mechanics (LEFM). The model allows the determination of the failure probability of a structure subjected to fatigue loading. The distributions of the random variables for civil engineering structures are provided, and the relative importance of these random variables is determined. An example of a bridge detail is provided in order to show the application of the model. Partial factors are derived for the case of fatigue of welded joints in civil engineering structures. The failure probability appears to be relatively insensitive to the failure criterion (attainment of a through-thickness crack or fracture) when considering the total fatigue life.     

The interval estimation of reliability for probabilistic and non-probabilistic hybrid structural system

The aim of this paper is to improve evaluation of the reliability of probabilistic and non-probabilistic hybrid structural system. Based on the probabilistic reliability model and interval arithmetic, a new model of interval estimation for reliability of the hybrid structural system was proposed. Adequately considering all uncertainties affecting the hybrid structural system, the lower and upper bounds of reliability for the hybrid structural system were obtained through the probabilistic and non-probabilistic analysis. In the process of non-probabilistic analysis, the interval truncation method was used. In addition, a recognition method of the main failure modes in the hybrid structural system was presented. A five-bar statically indeterminate truss structure and an intermediate complexity wing structure were used to demonstrate the new model is more suitable for analysis and design of these structural systems in comparison with the probabilistic model. The results also show that the method of recognition of main failure modes is effective. In addition, range obtained through interval estimation is shown to be more credible than certain results of other reliability models.     

基于最终层失效的复合材料舱体稳健性协调优化

传统可靠性设计难以符合现代设计要求,对舱体进行稳健性优化设计,可提高其综合可靠性。基于基体破坏和纤维断裂两种失效模式,采用验算点法求解复合材料单层可靠度。基于最终层失效假设,提出把结构看作由串联子系统组成的并联系统的思想,结合材料刚度比率退化准则和单层可靠度理论,采用概率逐步失效分析方法,计算出主要失效链,从而得出结构的失效概率。复合材料舱体设计变量复杂,提出二级优化方法思想：一级为系统级布局优化,对加强筋截面形状、位置确定等参数进行优化;二级为子系统级尺寸优化,对筋截面尺寸、复合材料各铺层厚度等参数进行优化。采用自适应随机搜索遗传算法,以复合材料舱体质量最小为目标函数,以可靠度要求为约束条件,采用稳健性协调优化方法,对存在初始缺陷的复合材料舱体进行稳健性优化,为复合材料结构优化设计提供参考。     

Probabilistic analysis of multi-step failure process of a laminated composite in bending

This paper presents results of a theoretical and experimental analysis of the random bending strength of laminated composites. Multi-layered composites with a periodic structure of sublaminates are considered in a combined loading (bending action and plane stress) state. The random strength response of the composites is analyzed on the basis of a multi-step failure probabilistic model. Here, the multi-step failure is considered as a process in which the failure of some single structural elements (sublaminates, laminae) leads to load redistribution onto other intact elements. A numerical algorithm based on the Monte Carlo technique and respective computer code are offered. The effect of scatter of structure parameters and number of sublaminates on the bending strength distribution is analyzed in detail. Experimental confirmation is considered on an example of uniaxial bending of laminated carbon-fiber-reinforced plastics. A method for evaluation of reliability in a random combined loading is also suggested.     

Fatigue reliability study on T‐welded component considering load shedding

In this paper, a coupled reliability method for structural fatigue evaluation considering load shedding is first proposed based on probabilistic fracture mechanics in which the uncertainties of the structural parameters are taken into account. Then, the method is applied to predict the fatigue reliability of the T‐welded structure to the case of considering load shedding or not. The compared results show that by considering the load shedding, the structural fatigue reliability might be improved with less conservativeness. The influence rules of the load‐shedding coefficient on the fatigue failure probability of the T‐welded component are investigated, and some interesting results are obtained. That is, the influences of load‐shedding coefficient on the fatigue failure probability can be divided into three regions, namely the high, medium and low fatigue failure areas. The last area is the most intriguing when we try to design a T‐welded structure. The thickness of T‐welded structure along the crack propagation direction is found to be one of the important design variables for the design of fatigue reliability, in which the low‐fatigue failure zone is used as one of the reliability constraints. The basic design frame of T‐welded structure is established to constrain the fatigue failure probability within the low‐fatigue failure area.     

Theoretical framework for estimating a product's reliability using a variable‐amplitude loading spectrum and a stress‐based approach

An innovative approach for predicting the reliability of a structure that is subject to a variable‐amplitude dynamic load is presented. In this approach, a Gassner durability curve with its scatter is modelled using a 2‐parametric Weibull's probability density function (PDF). The trend of the Gassner durability curve is modelled with a general hyperbola equation in a log‐log scale. The hyperbola equation is applied to represent the durability curve for the 63.2% probability of fatigue failure that describes the dependency of the Weibull's scale parameter on the loading spectrum's maximum stress. Equations are derived to link the parameters of the hyperbola curve to the material's S‐N curve and the loading spectrum. The Weibull's shape parameter is estimated from the scatter of the material's S‐N curve. The proposed Gassner‐curve model is applied to calculate the fatigue reliability from the PDF of the loading spectrum's maximum stress and the PDF of the durability‐curve's amplitude stress for the selected number of loading‐cycles‐to‐failure.     

Non-binary decomposition trees—a method of reliability computation for systems with known minimal paths/cuts

A coherent system with independent components and known minimal paths (cuts) is considered. In order to compute its reliability, a tree structure T is constructed whose nodes contain the modified minimal paths (cuts) and numerical values. The value of a non-leaf node is a function of its child nodes' values. The values of leaf nodes are calculated from a simple formula. The value of the root node is the system's failure probability (reliability). Subsequently, an algorithm computing the system's failure probability (reliability) is constructed. The algorithm scans all nodes of T using a stack structure for this purpose. The nodes of T are alternately put on and removed from the stack, their data being modified in the process. Once the algorithm has terminated, the stack contains only the final modification of the root node of T, and its value is equal to the system's failure probability (reliability).     

Pipe failure probability—the Thomas paper revisited

Almost twenty years ago, in Volume 2 of Reliability Engineering (the predecessor of Reliability Engineering and System Safety), a paper by H. M. Thomas of Rolls Royce & Associates Ltd. presented a generalized approach to the estimation of piping and vessel failure probability. The ‘Thomas-approach’ used insights from actual failure statistics to calculate the probability of leakage and conditional probability of rupture given leakage. It was intended for practitioners without access to data on the service experience with piping and piping system components.This article revisits the Thomas paper by drawing on insights from development of a new database on piping failures in commercial nuclear power plants worldwide (SKI-PIPE). Partially sponsored by the Swedish Nuclear Power Inspectorate (SKI), the R&D leading up to this note was performed during 1994–1999. Motivated by data requirements of reliability analysis and probabilistic safety assessment (PSA), the new database supports statistical analysis of piping failure data. Against the background of this database development program, the article reviews the applicability of the ‘Thomas approach’ in applied risk and reliability analysis. It addresses the question whether a new and expanded database on the service experience with piping systems would alter the original piping reliability correlation as suggested by H. M. Thomas.     

A probabilistic multiparameter framework for the modeling of fatigue crack growth in concrete

This paper presents a probabilistic multiparameter framework for the modeling of fatigue crack growth in three grades of concrete. The framework relies on the use of ranked fatigue crack growth rate data (with specified occurrence probability levels) in the formulation of multiparameter fatigue crack growth expressions. These relate ranked fatigue crack growth rates to crack driving force parameters such as the stress intensity factor range, maximum stress intensity factor, stress ratio and occurrence probability level. A probabilistic framework is then presented for the estimation of material reliability or failure probability due to fatigue crack growth. The probabilistic model is then validated for the available data.     

Fracture statistics of a unidirectional composite

We propose a model dealing with the prediction of the failure stress of a unidirectional composite 0°; it is based on a probabilistic micro-macro approach. Experimental tests have been carried out on specimens (unidirectional composite 0° T300/914) with different gauge lengths in order to estimate the scale effect in the failure probability distribution.The distribution of defects along the fibres was estimated through the multifragmentation and the single fibre test. The image analysis technique was used to estimate the local volume fraction of the fibres in the bulk of the material. The above physical information is introduced in the model based on a finite element analysis. The scale effect and the influence of the involved parameters on the failure of the material were studied at two different scales and a good agreement was found between the numerical predictions and the experimental results.     

Geotechnical system reliability of slopes

In slope stability analysis it is customary to search for the critical slip surface considering the conventional factor of safety as an index of stability. With the development of reliability analysis approaches within a probabilistic framework, alternative definitions of the critical slip surface can be adopted. Thus one may define a critical slip surface as one with the lowest reliability index or one with the highest probability of failure. However, it is important to consider the slope stability problem in terms of a system of many potential slip surfaces. For such a system, the calculation of system reliability is appropriate and desirable. In this paper, system reliability bounds are calculated within a probabilistic framework. The ‘system reliability’ or the ‘system probability of failure’ must be estimated for comparison with the corresponding reliability or probability of failure with respect to a ‘critical’ slip surface. The general slope stability problem involving non-zero internal friction angle involves a non-linear performance function. Moreover, the expression for factor of safety is usually inexplicit except for the ordinary method of slices which is not accurate except when ‘ø = 0’. This paper addresses the system reliability for inexplicit and non-linear performance functions as well as for linear and explicit ones. Any version of the method of slices may be used although the proposed approach is presented on the basis of the Bishop simplified method. It is shown that the upper bound system failure probability is higher than the failure probability associated with a critical slip surface. The difference increases as the coefficient of variation of the shear strength parameters increases.     

Reliability Analysis of a Composite Wind Turbine Blade Section Using the Model Correction Factor Method: Numerical Study and Validation

Reliability analysis of fiber-reinforced composite structures is a relatively unexplored field, and it is therefore expected that engineers and researchers trying to apply such an approach will meet certain challenges until more knowledge is accumulated. While doing the analyses included in the present paper, the authors have experienced some of the possible pitfalls on the way to complete a precise and robust reliability analysis for layered composites. Results showed that in order to obtain accurate reliability estimates it is necessary to account for the various failure modes described by the composite failure criteria. Each failure mode has been considered in a separate component reliability analysis, followed by a system analysis which gives the total probability of failure of the structure. The Model Correction Factor method used in connection with FORM (First-Order Reliability Method) proved to be a fast and efficient way to calculate the reliability index of a complex composite structure.