Stochastic analysis of lattice, nonlocal continuous beams in vibration

In this paper, we study the stochastic behavior of some lattice beams, called Hencky bar-chain model and their non-local continuous beam approximations. Hencky bar-chain model is a beam lattice composed of rigid segments, connected by some homogeneous rotational elastic links. In the present stochastic analysis, the stiffness of these elastic links is treated as a continuous random variable, with given probability density function. The fundamental eigenfrequency of the linear difference eigenvalue problem is also a random variable in this context. The reliability is defined as the probability that this fundamental frequency is less than an excitation frequency. This reliability function is exactly calculated for the lattice beam in conjunction with various boundary conditions. An exponential distribution is considered for the random stiffness of the elastic links. The stochastic lattice model is then compared to a stochastic nonlocal beam model, based on the continualization of the difference equation of the lattice model. The efficiency of the nonlocal beam model to approximate the lattice beam model is shown in presence of rotational elastic link randomness. We also compare such stochastic function with the one of a continuous local Euler-Bernoulli beam, with a special emphasis on scale effect in presence of randomness. Scale effect is captured both in deterministic and non-deterministic frameworks.

Bending, Buckling and Free Vibration Analysis of Size-Dependent Nanoscale FG Beams Using Refined Models and Eringen’s Nonlocal Theory

In this study, a theoretical unification of twenty-one nonlocal beam theories are presented by using a unified nonlocal beam theory. The small-scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The present unified theory satisfies traction free boundary conditions at the top and bottom surface of the nanobeam and hence avoids the need of shearing correction factor. Hamilton’s principle is employed to derive the equations of motion. The present unified nonlocal formulation is applied for the bending, buckling and free vibration analysis of functionally graded (FG) nanobeams. The elastic properties of FG material vary continuously by gradually changing the volume fraction of the constituent materials in the thickness direction. Closed-form analytical solutions are obtained by using Navier’s solution technique. Non-dimensional displacements, stresses, natural frequencies and critical buckling loads for FG nanobeams are presented. The numerical results presented in this study can be served as a benchmark for future research.

Asymptotic derivation of nonlocal beam models from two-dimensional nonlocal elasticity

The paper is focused on the possible justification of nonlocal beam models (at the macroscopic scale) from an asymptotic derivation based on nonlocal two-dimensional elasticity (at the material scale). The governing partial differential equations are expanded in Taylor series, through the dimensionless depth ratio of the beam. It is shown that nonlocal Bernoulli–Euler beam models can be asymptotically obtained from nonlocal two-dimensional elasticity, with a nonlocal length scale at the beam scale (macroscopic length scale) that may differ from the nonlocal length scale at the material scale. Only when the nonlocality is restricted to the axial direction are the two length scales coincident. In this specific nonlocal case, the nonlocal Bernoulli–Euler model emerged at the zeroth order of the asymptotic expansion, and the nonlocal truncated Bresse–Timoshenko model at the second order. However, in the general case, some new asymptotically-based nonlocal beam models are built which may differ from existing references nonlocal structural models. The natural frequencies for simply supported nonlocal beams are determined for each nonlocal model. The comparison shows that the models provide close results for low orders of frequencies and the difference increases with the order.