Due to nonlocal and strain gradient effects with rigid and deformable boundary conditions, the thermal vibration behavior of perforated nanobeams resting on a Winkler elastic foundation (WEF) is examined in this paper. The Stokes transformation and Fourier series have been used to achieve this goal and to determine the thermal vibration behavior under various boundary conditions, including deformable and non-deformable ones. The perforated nanobeams’ boundary conditions are considered deformable, and the nonlocal strain gradient theory accounts for the size dependency. The problem is modeled as an eigenvalue problem. The effect of parameters such as the number of holes, elastic foundation, nonlocal and strain gradient, deformable boundaries and size on the solution is considered. The effects of various parameters, such as the length of the perforated beam, number of holes, filling ratio, thermal effect parameter, small-scale parameters and foundation parameter, on the thermal vibration behavior of the perforated nanobeam, are then illustrated using a set of numerical examples. As a result of the analysis, it was determined that the vibration frequency of the nanobeam was most affected by the changes in the dimensionless WEF parameter in the first mode and the changes in the internal length parameter when all modes were considered.

This study presents an analytical investigation into the free vibration behavior of perforated nanobeams resting on a Kerr-type elastic foundation within the framework of Eringen’s nonlocal elasticity theory. Specifically, Eringen’s nonlocal elasticity theory is employed to inherently capture small-scale effects, while the three-parameter Kerr model is utilized to provide a mathematically consistent representation of shear continuity and realistic surface interactions. In this context, the governing equations of motion for a perforated Euler–Bernoulli nanobeam are derived using Hamilton’s principle, incorporating both the nonlocal parameter and perforation geometric factors, namely, the filling ratio and the number of holes. The resulting equations are solved analytically via the Navier method for simply supported boundary conditions. The results indicate that the Kerr foundation model exhibits an intermediate behavior between the Winkler and Pasternak models, owing to the stiffness-reducing effect of its upper spring layer connected in series. A key finding is the “masking effect,” where high foundation stiffness significantly suppresses the frequency reduction caused by nonlocal small-scale effects. Furthermore, it is observed that in the absence of foundation support, the vibration behavior is governed by the competition between mass reduction and stiffness loss depending on the number of holes; however, foundation dominance stabilizes the system regardless of perforation geometry.

In this study, the buckling behavior of carbon nanotube (CNT) reinforced nanobeams on an elastic foundation is investigated using the nonlocal elasticity theory and nonlocal finite element method within the framework of Euler-Bernoulli beam theory. The effects of CNT volume fraction, nonlocal parameter, elastic foundation parameter and length-to-thickness ratio on the critical buckling load are analyzed. CNT reinforced nanobeam is modeled considering short and long CNT reinforcements, and mechanical properties are determined by the rule of mixtures. To determine the critical buckling loads of CNT reinforced nanobeams, stiffness matrices and force vectors, including nonlocal effects, are derived based on the Euler-Bernoulli beam theory and analyses are carried out accordingly. The results show that increasing the CNT volume fraction significantly increases the critical buckling load, suggesting that CNTs play an essential role in strengthening the nanobeams. On the other hand, the nonlocal parameter negatively affects the critical buckling load and decreases the buckling strength of the nanobeam. However, the effect of the nonlocal parameter on the buckling strength is negligible. The elastic foundation parameter positively affects the critical buckling load and increases the buckling resistance. This finding indicates that the elastic foundation plays an important role in improving the structural stability of the nanobeams. The length-to-thickness ratio is another important parameter, indicating that long and thin nanobeams are more prone to buckling and the critical buckling load decreases with increasing this ratio. Since the effects of the above-mentioned four parameters on the buckling strength were determined using the finite element method, the results obtained will guide the subsequent numerical modeling studies of nano-sized beams.

In this study, equal strength cantilever and simply supported beams made of functionally graded material (FGM) whose material properties vary though the height direction were investigated. These equal strength cantilever FGM beams were loaded with uniformly distributed load and a point load at the tip and simply supported FGM beams were loaded with uniformly distributed loads. They have all variable cross-section and straight axis. For calculating equivalent material properties of FGMs, power law distribution and Mori-Tanaka model were used. A computer program was developed for the analysis of the problem. The dimensionless deflection values for cantilever beams and simply supported beams were obtained for different materials by the help of developed computer program. Obtained results are presented in the form of tables and graphs which may be useful for the researchers.