A Timoshenko beam element based on the modified couple stress theory

• We developed a new modified couple stress Timoshenko beam element.
• The stiffness and mass matrices are derived using Hamilton’s principle.
• New element is comprehensive that recovers the formulations of other beam elements.
• New element is used to study static deflection and pull-in of microcantilevers.
• New element results are in excellent agreement with the experimental data unlike classical FEM.

Since the classical continuum theory is neither able to evaluate the accurate stiffness nor able to justify the size-dependency of micro-scale structures, the non-classical continuum theories such as the modified couple stress theory have been developed. In this paper, a new comprehensive Timoshenko beam element has been developed on the basis of the modified couple stress theory. The shape functions of the new element are derived by solving the governing equations of modified couple stress Timoshenko beams. Subsequently, the mass and stiffness matrices are developed using energy approach and Hamilton’s principle. The formulations of the modified couple stress Euler–Bernoulli beam element and also classical Timoshenko and Euler–Bernoulli beam elements can be recovered from the original formulations of the new Timoshenko beam element. By two examples, it is indicated that how the new beam element can be applied to deal with the real-case problems. The static deflection of a short microbeam and pull-in voltage of an electrostatically actuated microcantilever made of silicon are evaluated by employing the new beam element and the results are compared to the experimental data as well as the classical FEM results. It is observed that the results of the new beam element are in good agreement with the experimental findings while the gap between the classical FEM and experimental results is notable.