a b s t r a c tIn this paper, elastic beam model using nonlocal elasticity theory is developed for the bending analysis of microtubules (MTs) based on the Euler-Bernoulli beam theory. The size effect is taken into consideration using the Eringen's non-local elasticity theory. The derivation of governing equation of bending from shear and moment resultants of the beam and stress-strain relationship of the one-dimensional nonlocal elasticity model is presented. The model is then applied on the studies of static analysis of microtubules using the method of differential quadrature (DQ). After the developed DQ method is numerically validated, detailed numerical analyses about the effects of boundary conditions and load types are conducted and the influence of nonlocal parameter on the static response of MTs is discussed. It is hoped that the results in the manuscript may present a benchmark in the study of bending in microtubules.

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The longitudinal free vibration problem of a micro-scaled bar is formulated using the strain gradient elasticity theory. The equation of motion together with initial conditions, classical and non-classical corresponding boundary conditions for a micro-scaled elastic bar is derived via Hamilton’s principle. The resulting higher-order equation is solved for clamped-clamped and clamped-free boundary conditions. Effects of the additional length scale parameters on the frequencies are investigated. It is observed that size effect is more significant when the ratio of the microbar diameter to the additional length scale parameter is small. It is also observed that the difference between natural frequencies predicted by current and classical models becomes more prominent for both lower values of slenderness ratio of the microbar and for higher modes.

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Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson's ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.

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