# Size-Dependent Mechanical Behaviors of Defective FGM Nanobeam Subjected to Random Loading
###### Abstract
Using the vertex method (VM), the exact solution will be obtained if the monotonicity of the function is guaranteed [33]. The basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations. However, all the possible combination of each uncertain parameter vertex will be calculated, which leads to the computational costs increasing exponentially, due to the increase in the number of uncertain parameters. Alternatively, based on the interval mathematic and perturbation theories, [PERSON] et al. [34] developed a matrix perturbation method that is widely used to determine the range of the structural response because it is simple, yet efficient. Unfortunately, the accuracy of this approach cannot be guaranteed by augmenting the uncertain parameter number [31]. In order to avoid the problem, [PERSON] et al. [35] presented a parameter perturbation method by using the partial derivatives information. Although it is successful to use the first order interval approach to analyze the response bounds, it will not work well when the uncertain range is not small enough. The uncertain problems with random and unknown, but bounded, parameters can be well-described by the probabilistic and interval method, respectively. However, the random parameters may always coexist with the unknown, but bounded, ones. Thus, the behavior of defective nanobeam subjected to random loads can be not described via a single type of uncertain parameters. It is necessary to construct a hybrid uncertain model, with both probabilistic and interval parameters, simultaneously, which is one of motivation of this paper. In the aspect of a solution, the numerical model is utilized to calculate the deterministic analysis, whose accuracy directly affects predicting the statistical properties or bounds of the nanostructure's response. More precise responses can be obtained by utilizing more accurate numerical model. Moreover, the efficiency of numerical model is another important factor in the prediction of the response. With high accuracy, meshless methods (MMs) were developed to avoid some of the difficulties in FEM, which is an alternative to method to the finite element method (FEM), due to its unique features. Further, [PERSON] [36] provided a brief review on the advances and trends in the MMs formulations. However, the huge amount of computation is still one of the urgent problems for MMS. In order to deal with the problem of computational consumption and accuracy, the spline-based numerical methods, such as the spline strip [37; 38; 39; 40; 41] and B-spline finite point (B-SFPM) [42] methods, have been proposed to structural analysis. Moreover, B-SFPM would be more promising to utilize, with high accuracy and efficiency, because the basic function is cubic B-spline function, which has compact support and explicit form. Therefore, the spline finite point method is introduced to conduct the deterministic analysis of nanobeam, considering the size-dependent effect. To sum up, this is a summary of the work performed in this paper. First, the mechanical behavior of the FGM nanobeam is investigated, considering the size-dependent effect, based on the nonlocal elastic theory. The spline finite point method (SFPM) is proposed to construct the deterministic framework analyzing the bending behavior. In order to be more objective, the material defect and load randomness are taken into account, which are quantified as interval and random parameters, respectively. Furtherly, a hybrid model, including random and interval, is presented to predict the response of the defective FGM nanobeam subjected to random load. To solve this hybrid model, the polynomial chaos expansion and dimension wise methods are combined. At last, three cases of the simple supported nanobeam numerical example are examined to verify the proposed approach, followed by some conclusions.