# Size-Dependent Mechanical Behaviors of Defective FGM Nanobeam Subjected to Random Loading
###### Abstract  
Considering the uncertainties of the materials and loads, the nanobeam made of functionally graded materials were investigated based on the non-local elastic theory. The spline finite point method (SFPM) was established to analyze the bending behavior of the nanobeam-based Timoshenko theory. In comparison with finite element method (FEM), SFPM has higher accuracy. Further, the multi-source uncertainties are considered, material properties are quantified as interval parameters, and loads are taken as random parameters. To deal with the problems with two types of uncertainties coexisting, a hybrid uncertain analysis model was established, and the method of polynomial chaos expansion and dimensional wise (PCE-DW) analysis was proposed to predict the response of nanobeam in the hybrid uncertain system. Numerical examples ultimately illustrate the effectiveness of the model and solution techniques, compared with MCS. The results furtherly verify the efficiency and accuracy. hybrid uncertainty propagation; polynomial chaos; dimension wise; nanobeam; size-dependent effect +
Footnote †: journal: Applied  
the modified coupled stress theory. [PERSON] and [PERSON] [16] examined the size effect on dynamic stability of FGM Timoshenko nanobeam. [PERSON] and [PERSON] [10] analyzed the bending and buckling behaviors of the FGM Euler-Bernoulli and Timoshenko beams, taking the small-scale effect into account. [PERSON] et al. [17] derived the governing equation of the free vibration for the FGM nanobeam and computed the structural response utilizing the finite element method (FEM). The works mentioned above were based on the deterministic model. It can be said that the deterministic model, with deterministic properties, has been well-applied to analyze mechanical behaviors of the microstructures. However, the properties of the model are always uncertain [18; 19; 20], such as the material properties (Young's modulus, Poisson's ratio), geometric dimensions (width and height of section), external load (value and position of load), and boundary conditions (rigid constraint, elastic constraint), etc. For instance, a Stone-Thrower-Wales defect [21] and pentagon-octagon-pentagon defect [22] will lead to uncertainties, regarding the material properties. The uncertainties existing in reality are ignored in the deterministic model, which is simple, but unreasonable. To handle these uncertain problems, the probabilistic method is the most popular technique--by assuming the uncertain parameters, as random variables, obey a certain distribution form. These random problems can be solved by Monte Carlo simulation (MCS) method, which is a simple, but powerful, technique [23]. However, a large number of samples need to generate and calculate the response of the samples, which is very time consuming. The stochastic perturbation method [24] can be alternatively utilized to well-investigate the propagation of the randomness in the uncertain system, which reduces the computing burden. However, it is just as efficient for the problems with small uncertainty, due to the fact that it is a local approximate method based on first-order Taylor series expansion. The spectral stochastic method [25] is also based on the series expansion, but the response of uncertain system is expanded in the random space, not in a single point. The polynomial chaos expansion (PCE) is an one of the outstanding methods, thanks to its profound mathematical basis [26]. Unfortunately, the distribution information can be accurately known [27] in some cases. The lower and upper bounds of the uncertain parameters is easily obtained in practice, which can be well-quantified by the interval mathematics [28]. Therefore, the convex model, based on interval mathematics, can serve as a powerful supplement to this part [29]. In order to predict the bounds of the response, some methods have been developed, such as the perturbation method [30], [PERSON] expansion method and gradient method [31], and response surface method [32].