Static and free vibration analyses for Timoshenko micro-beam problems using finite element method

Abstract The objective of the work is to examine the static and dynamic behavior of micro beams using a beam finite element, with an emphasis on size-dependent effects. The model employs a Timoshenko beam formulation and is based on the Modified Couple Stress Theory (MCST). The main characteristic of the new element is that it incorporates a length scale parameter into the element matrices to account for size effects. The element is simplified to the typical Timoshenko beam element when the parameter is set to zero. Static and free vibration studies are conducted under different boundary conditions to assess the accuracy of the model. The comparison confirms the reliability of the proposed model by demonstrating that the newly created element beam in this study agrees well with the reference material. Additionally, the MATLAB algorithm programmed in this paper simplifies the ability to cyclically compute the matrices for four types of beam element: Classical Euler-Bernoulli, micro Euler-Bernoulli, classical Timoshenko, and micro Timoshenko beam, by incorporating theoretical conversion parameter. As a result, the relationships between these beam types are clarified and applied flexibly in beam analysis.