# A Probable Approach to Reliability Assessment of Reinforced Plates
## 1 Introduction  
Studying the stress state and failure of composite materials, in particular those reinforced with random rigid linear inclusions, is an urgent task. Probability assessment of the failure state under a given loading is an important stage in calculations of the strength and reliability of machine parts and structural elements. The strength and reliability of the material depend on structural defects and are always some random value. Therefore, when solving the problems of predicting the limited state of structural materials, the complex application of probabilistic statistical methods and deterministic solutions in fracture mechanics is relevant. These methods, applied to the study of the strength of materials with structural defects, in particular to composites, have been developed recently. The paper [1] presents a simulation of the damage growth in open-hole laminates under compression. It used the developed numerical scheme with discontinuous elements of the hard shell and enriched connected elements. The authors hypothesized that due to localized bending damage, compressive loading could lead to the formation of oblique cracks in the matrix. According to the modeling technology, an assumed vertical connected fracture was formulated with a coordinate system of an inclined fracture. Representation of damage in the form of discrete cracks makes it possible to more accurately predict models of brittle failure of composite materials. In article [2], an assessment of the fatigue reliability of welded joints in orthotropic steel bridge decks was carried out, taking into account the effects of loading sequences. The effect of initial damage and traffic growth on fatigue reliability was studied. It was concluded that while evaluating the fatigue life of orthotropic steel decks equipped with long-term deformation control systems, initial damage reduces the fatigue life of orthotropic steel decks significantly, and the importance of considering load sequence effects is also shown. The study [3] proposed a model of a symmetrical multilayer plate, which contains elliptical cutouts, under conditions of a flat tensile loading. An analysis of stress change and progressive failure was carried out. The failure of the first and lastlayers of the composite depends on various parameters. The influence of the parameters was analyzed too. Using the complex variable of [PERSON]'s formula enables one to write analytical representations of layer-by-layer stresses around elliptical cuts. Progressive fracture prediction is based on the Tsai-Hill and Tsai-Wu strength criteria. Accordingly, before the studies were carried out, the most significant influence on the tensile strength of the laminate under a given loading was exerted by the stacking sequence of the fibers and their orientation. Conclusions were drawn about the effectiveness of using the method of complex variables and conformal mapping to study the failure of a composite plate with an elliptical cut. The article [4] shows the importance of joint use of the deterministic approach and taking into account stochastic effects in the design of structures made of composite materials. All geometric and material parameters of the investigated composite layered plates were supposed to be stochastic. To study the stochastic effect of material properties, orientation, and layer thickness on the failure of multilayer composite plates under the action of static loadings, Monte Carlo simulation was used. A classical lamination theory was proposed to calculate strength factors using maximum stress, Tsai-Wu, and Tsai-Hill strength criteria for plates of three different materials in different stacking sequences. It was shown that an increase in the variation coefficient of the input variables leads to an increase in the variation in the distribution of failure layers. Conclusions were drawn about a significant degree of dependence between the distribution of failure layers and the probability of failure from a combination of laminate laying sequences.