Structural analysis using a dipolar elastic Timoshenko beam

In recent years, several non-local beam theories have emerged in trying to model stiffness and strength phenomena observed in micro-beams used in micro-mechanical systems (MEMS) and sensors. Of particular importance are the theories that treat in a unified way both long and short beams, through Timoshenko beam kinematics. At the same time and in combination with experiments, beam models have been used to estimate material lengths that are necessary for the assessment of non-local elasticity theories. Aside from the particular details of the non-local assumptions used, boundary conditions play a considerable role in such problems, especially the non-classical boundary conditions that are required by the non-local theories. It is astonishing how few and simple beam configurations have been presented so far. They fall in the category of statically determinate beams and their key results are often given in the form of Fourier series, making it hard for the designers of MEMS to use these results and expand them into more complex beam systems like frames, grids, etc. It is the scope of this work to present closed-form solutions to the Timoshenko beam problem, based on dipolar strain gradient elasticity and show how to treat statically indeterminate problems in a consistent manner.