Finite Element Analysis of Static Linear Euler-Bernoulli Microbeams Based on the Modified Strain Gradient Theory

Abstract

Abstract This paper develops a finite element formulation (FEF) for static linear Euler-Bernoulli microbeams using the modified strain gradient theory (MSGT). Remarkably, classical models often fail to predict microscale behaviour, e.g. micro-electromechanical systems (MEMS), by neglecting material length scale parameters (MLSPs). Thus, MSGT integrates into the beam formulation to address this. Subsequently, a finite element framework derives governing equations extending classical theory via MSGT and incorporating MLSPs into the stiffness matrix. For validation, a cantilever microbeam, a simple microbeams and a fixed-fixed microbeam are analysed, comparing results with classical theory and published analytical results, which reveals pronounced size effects e.g. increased stiffness, as beam dimensions approach MLSPs. The findings demonstrate that classical models inaccurately predict deflections and stresses, whereas MSGT-based FEF consistently aligns with microscale phenomena. Overall, this work establishes the formulation as a robust tool for MEMS design, emphasizing the necessity of gradient-enhanced models for precision in micro-engineering applications.