Reliability of reinforced concrete beams in limit state of cracking— failure rate analysis approach

The reliability of a reinforced concrete beam with respect to the limit state of crack width under monotonically increasing short-time loading conditions is estimated using the Monte Carlo technique and failure rate analysis. Using both methods the reliabilities of 24 beams whose test results are available in the literature and three beams tested in the laboratory have been determined for a limiting crack width of 0.2 mm. A comparison of the reliabilities obtained using failure rate analysis with those obtained directly from Monte Carlo simulation shows good agreement. From failure rate analysis it is found that a two-parameter Weibull distribution can be used to describe the failure life of a reinforced concrete beam with respect to the limit state of crack width. The mean moment to failure of each beam determined from failure rate analysis is compared with the experimental moment to failure.     

Fatigue reliability analysis of kiln welded shell

This paper presents a probabilistic reliability method for the welded shell during crack growth. The crack growth model incorporated with a failure assessment diagram, which provides a better estimation of the critical crack depth, is developed to describe fatigue failure. All variables used of particular welded joints are studied. The stress variables are based on resultant stress calculated by using the finite element (FE) code ANSYS. Fatigue reliability analysis of the welded shell is performed by using the Monte Carlo simulation method. It is demonstrated that fatigue reliability analysis is significantly useful for the repair schedule of shell cracks.     

On crack-crack interaction and coalescence in fatigue

The effect of crack-crack interaction and coalescence in fatigue was studied for a single pair of cracks analytically and for multi-cracks based on real microstructure variations by means of Monte Carlo simulations. Crack-crack interaction and coalescence lower the mean fatigue lifetime. The magnitude of lowering increases with increasing crack density and becomes particularly important in the low failure probability regime. Results of this investigation provide insight for laying the foundation of quantitative predictive technology on crack interaction and coalescence aspects of fatigue.     

陶瓷基复合材料基体随机开裂的损伤模拟

首先介绍了描述复合材料出现损伤时细观应力场的剪滞模型;给出了模拟陶瓷基复合材料基体随机开裂的有关公式;然后采用Monte Carlo方法模拟了基体随机开裂的过程;最后通过计算机模拟了陶瓷基复合材料基体裂纹的随机演化过程,分析了有关模型参数对基体裂纹演化的影响,并与文献模拟结果进行了对比。研究表明：最终裂纹间距主要分布在 1～2倍滑移长度之间;初始开裂应力越大,最终名义裂纹间距越小;Weibull 模量越大,最终名义裂纹间距越小;热残余应力越大,最终名义裂纹间距越大;模拟长度对裂纹演化影响很小;MonteCarlo方法可以有效地模拟陶瓷基复合材料的随机开裂过程。     

Modeling of fatigue crack propagation using dual boundary element method and Gaussian Monte Carlo method

This paper studies the modeling of fatigue crack propagation on a multiple crack site of a finite plate using deterministic and probabilistic methods. Stress intensity factor has been calculated by the combined deterministic approach of the dual boundary element method (DBEM) and the probabilistic approach of the Gaussian Monte Carlo method. The Gaussian Monte Carlo method has been incorporated to simulate the random process of the fatigue crack propagation. A finite plate of aluminum alloy 2024-T3 with a thickness of 1.6 mm and 14 holes is analyzed and the fatigue life of the plate is predicted by following a linear elastic law of fracture mechanics. The results of fatigue life predicted by DBEM-Monte Carlo method are in good agreement with experimental ones. The same approach is also applied to two other engineering applications of a gear tooth and a bracket.     

LIFE PREDICTION BY SIMULATION OF CRACK GROWTH IN NOTCHED COMPONENTS WITH DIFFERENT MICROSTRUCTURES AND UNDER MULTIAXIAL FATIGUE

A modelling procedure was developed which is applicable to crack growth in notched components subjected to multiaxial fatigue for materials with different microstructures. An algorithm for crack growth, in a microstructure that was modelled as hexagons, was established as a competition between growth by crack linkages during the crack initiation and propagation stages and the propagation of a dominant crack as a single crack. Analytical results simulated by using the developed model were compared with experimental results from fatigue tests which had been conducted using notched specimens of pure copper, carbon steel and two kinds of titanium alloy. Cracking morphology, which was experimentally observed to depend on the microstructure and the loading mode, was well simulated using the present model. The fatigue failure life of a notched specimen was statistically estimated by a Monte Carlo procedure based on the model. The simulated life with a statistical scatter-band almost coincided with the experimental data.     

Life prediction based on analysis of crack coalescence in low cycle fatigue

The crack coalescence in some materials is known to make influence on the failure life significantly in low cycle fatigue as well as corrosion fatigue. A model was proposed to simulate crack linking during fatigue process. The crack distribution observed experimentally was idealized by a two-dimensional model with straight cracks. Some assumptions were made for the coalescence criterion in the model. A procedure for estimating residual life was developed based on the proposed model. The procedure was applied to fatigue process of a copper, in which the growth due to crack linkings was observed to be dominant through fatigue life. A good agreement was shown between the experimental and the predicted results. The statistical properties of life distribution were also discussed based on a Monte Carlo simulation.     

Life prediction by ferrite–pearlite microstructural simulation of short fatigue cracks at high temperature

This paper dealt with fatigue behavior simulation based on ferrite–pearlite microstructure modeled by correctional Voronoi-polygons. The model took grain size, grain orientation and the percentage of pearlite and ferrite into consideration. The basal energy was proposed to represent the inherent energy for slip-band and grain boundary to cracking. The driving force for crack initiation and propagation caused by load condition was considered as the energy increment of slip-band and grain boundary. The fatigue behavior including crack initiation, propagation, coalescence and interference were simulated based on Monte Carlo method. The simulation results show a satisfying agreement with the experimental ones.     

A Bayesian evaluation of simulation models of multiple-site fatigue cracks

The rivet holes along the longitudinal top row of the outer skin of the fuselage over a two-bay length are considered as the independent structural unit for the simulated multiple-site fatigue cracks. Models of multiple-site fatigue cracks are proposed. The models are divided into several phases with some uncertain parameters. These phases are incorporated sequentially into a computer code with the Monte Carlo simulation method. The Bayesian estimation of uncertain parameters in the model can be identified on visual inspections by the Bayesian procedure from in-service inspection data measuring crack lengths of each rivet hole. In summary, this study evaluates effects of differences in the simulation models for the crack coalescence and failure phase for the distribution of inspection data measuring crack lengths with the Bayesian estimation of uncertain parameters from simulated in-service inspection data.     

A probabilistic simulation of fluid leakage in multiple cracks situation

ABSTRACT In pressure vessels and piping the leak‐before‐break (LBB) assessment method is employed to avoid any catastrophic failure prior to a detectable leakage. One of the most important parameter, the leak rate, is investigated in the present paper by means of the Monte Carlo method. A brief review is carried out with emphasis on aspects such as crack growth, crack size with detectable leakage, crack opening area and leak rate. Issues concerning the property and behaviour of multiple cracks are also covered along with a review of the characteristics of leak rate through distributed multiple cracks using a statistical simulation method. The simulation results show that the effect of multiple cracks is quite significant to the LBB concept. Both the relationship between leak rate and crack length (or time) and the statistical characteristics of the leak rate are considerably different for different initial crack conditions.     

Sigmoidal crack growth rate curve: statistical modelling and applications

The present paper proposes a statistical model for describing sigmoidal crack growth rate curves. Major novelties are: a) exploitation of the maximum likelihood principle for obtaining material estimates by pooling together experimental data belonging to the different crack propagation regions; b) a general formulation which allows to adopt different sigmoidal models and any kind of statistical distribution for the model variables; c) fatigue life predictions through numerical integration of analytical functions with no need of Monte Carlo simulations. Experimental data taken from NASGRO database are used to check the validity of the statistical model in estimating material parameters included in the crack growth NASGRO algorithm. Illustrative plots of number of cycles to failure and crack length after a given number of cycles are presented, showing good agreement between the proposed statistical model and NASGRO results.     

Deterministic and probabilistic creep–fatigue–oxidation crack growth modeling

Fatigue, creep, oxidation or their combinations have long been recognized as the principal mechanisms in many high-temperature failures in power plant components, turbine engines, and exhaust systems in vehicles. Depending on the specific materials and loading conditions and temperature, the role of each damage mechanism may change significantly, ranging from independent development to competing and combined creep–fatigue, fatigue–oxidation, and creep–fatigue–oxidation. In this paper a new linear superposition theory is proposed to model the cycle-dependent and time-dependent creep–fatigue–oxidation crack growth phenomena. The model can be reduced to creep–fatigue and fatigue–oxidation crack growth models previously developed by the authors as well as, under some assumptions, the current widely used linear superposition theory. The limits of the current superposition theory and the advantages of the new theory are clearly demonstrated with several worked examples. A general probabilistic analysis procedure is also proposed by introducing the uncertainties of parameters in fatigue, creep, and oxidation crack growth laws with the help of the Monte Carlo simulation.     

Data-driven prediction of the probability of creep–fatigue crack initiation in 316H stainless steel

Stainless steel components in advanced gas-cooled reactors (AGRs) are susceptible to creep–fatigue cracking at high temperatures. Quantifying the probability of creep–fatigue crack initiation requires probabilistic numerical simulations; these are complex and computationally intensive. Here, we present a data-driven approach to develop fast probabilistic surrogate models of creep–fatigue crack initiation in 316H stainless steel. We perform a set of Monte Carlo simulations based on the R5V2/3 high temperature assessment procedure and determine the sensitivity of the probability of crack initiation to loads and operating conditions. The data are used to train different supervised machine learning models considering Bayesian hyperparameter optimization. We discuss the relative performance of such models and show that a gradient tree boosting algorithm results in surrogate models with the highest accuracy.     

Reliability analysis of large structural systems

Brute force Monte Carlo simulation methods can, in principle, be used to calculate accurately the reliability of complicated structural systems, but the computational burden may be prohibitive. A new Monte Carlo based method for estimating system reliability that aims at reducing the computational cost is therefore proposed. It exploits the regularity of tail probabilities to set up an approximation procedure for the prediction of the far tail failure probabilities based on the estimates of the failure probabilities obtained by Monte Carlo simulation at more moderate levels. In this paper, the usefulness and accuracy of the estimation method is illustrated by application to a particular example of a structure with several thousand potentially critical limit state functions. The effect of varying the correlation of the load components is also investigated.     

Curvilinear fatigue crack reliability analysis by stochastic boundary element method

In this paper, the stochastic boundary element method, which combines the mixed boundary integral equations method explored in Reference 1 with the first-order reliability method, is developed to study probabilistic fatigue crack growth. Due to the high degree of complexity and non-linearity of the response, direct differentiation coupied with the response-surface method is employed to determine the response gradient. Three random processes, the mode I and mode II. stress intensity factors and the crack direction angle, are included in the expression of the response gradient. The sensitivity of these random processes is determined using a first-order response model. An iteration scheme based on the HL-RF method² is applied to locate the most probable failure point on the limit-state surface. The accuracy and efficiency of the stochastic boundary element method are evaluated by comparing the cumulative distribution function of the fatigue life obtained with Monte Carlo simulation. The reliability index and the corresponding probability of failure are calculated for a fatigue crack growth problem with randomness in the crack geometry, defect geometry, fatigue parameters and external loads. The response sensitivity of each primary random variable at the design point is determined to show its role in the fatigue failure. The variation of each primary random variable at the design point with the change of probability of failure is also presented in numerical examples.     

联合分布函数蒙特卡罗模拟及结构可靠度分析

针对复杂极限状态方程可靠度计算问题,提出了基于理论联合分布函数以及2 种近似联合分布函数的结构失效概率蒙特卡罗模拟方法,并给出了计算流程图.采用2 个算例证明了所提方法的有效性.结果表明:所提的失效概率模拟方法的计算精度很高,尤其适用于复杂极限状态方程的可靠度计算问题.2 种联合分布函数近似构造方法得到的失效概率精度相当,近似方法与精确方法结果的差异随失效概率的减小而增大,而且随着变量间相关性的增加而增加.当失效概率小于10-3时,近似方法的失效概率误差较大.     

A probabilistic model for the onset of High Cycle Fatigue (HCF) crack propagation: Application to hydroelectric turbine runner

A fatigue reliability model for hydroelectric turbine runners is presented in this paper. In the proposed model, reliability is defined as the probability of not exceeding a threshold above which HCF contribute to crack propagation. In the context of combined LCF–HCF loading, the Kitagawa diagram is used as the limit state threshold. Two types of crack geometries are investigated: circular surface flaw and embedded flaw in a semi-infinite medium. The accuracy of FORM/SORM approximations was considered acceptable for engineering purpose in our application given the minimal numerical burden posed by such a method compared to Monte Carlo simulations. Our results show that the probability of an embedded flaw close to the surface has a major influence on reliability. Furthermore, we observe that the assumption that crack geometrical characteristics are independent leads to non-conservative results.     

A modification of the Monte Carlo method for steady heat conduction problems

The unique capability of calculating single points by Monte Carlo Methods is used in an attempt to develop an efficient Monte Carlo method for whole field calculations of heat conduction problems. Rather than calculate the whole field by Monte Carlo, only select points around the boundaries of inscribed figures of standard shapes are are so calculated. Interior points of the inscribed figures are then calculated by analytical (non-numerical) techniques. The procedure is described and shown to be more efficient than previously published Monte Carlo modifications but not as efficient as the Gauss–Seidel method. This new procedure will most likely be competitive for subregion calculations rather than for whole field calculations (for which standard numerical procedures are most efficient) or for single point calculations (for which other Monte Carlo methods are most efficient).     

Simulation of creep crack growth in ceramic composites

The elevated temperature response resulting from tensile creep of fiber reinforced ceramic composites was modeled using Monte Carlo simulation. The model consisted of a uniaxially loaded fiber tow aligned with the direction of applied load, and modeled the growth of matrix cracks resulting from creep failure of bridging fibers. A creep strain rate consisting of primary and steady state components was assumed, and each component was modeled by a power law relationship. Power law creep exponents in the range of 2.0–2.5 for a selected SiC/SiC system at stress levels ranging from 60 MPa to 200 MPa were evaluated. Fatigue-like behavior was predicted as a result of tensile creep, and a fatigue exponent of 3.03 ± 0.07 was predicted for nominal stress levels less than 200 GPa. The influence of initial crack length on failure lifetime was also studied, but was found to have little influence on the predicted lifetime. The predicted failure response suggested a stress dependent creep process could be used to model experimental data and evaluate the failure mechanism of reinforced composites.     

Stochastic critical stress intensity factor response of single edge notched laminated composite plate using displacement correlation method

Abstract

The second-order statistics of critical stress intensity factor (SIF) of single edge notched fiber reinforced composite plates with random system properties and subjected to uniaxial tensile loadings is investigated. This paper is an extension of reference (Lal and Kapania, 2013) by the present authors by considering more number of input random system parameters for higher accuracy. A C⁰ finite element method based on a higher-order shear deformation plate theory using displacement correlation method via isoparametric quarter point element is proposed for basic formulation. A stochastic finite element method using first-order perturbation technique and Monte Carlo simulation (MCS) is employed to examine the mean, coefficient of variance, and probability density faction of critical first mode SIF. The effect of different fiber orientations, crack length, plate thickness, a number of layers, and the lamination schemes with random system properties on the statistics of SIF of single edge crack laminated composite plate is evaluated. The tensile failure load is predicted using Hashin’s failure criteria. The present approach is validated with results available in literature and by employing independent MCS.