Strain gradient beam element

The classical continuum theory is neither able to accurately model the mechanical behavior of micro/nano-scale structures nor capable of justifying the size-dependent behavior observed in these structures; so the non-classical continuum theories such as the strain gradient theory have been emerged and developed. In order to enable the finite element method (FEM) to more accurately deal with the problems in micro/nano-scale structures, a size-dependent Euler–Bernoulli beam element is developed based on the strain gradient theory. Compared to the classical Euler–Bernoulli beam element, the nodal displacement vector of the new Euler–Bernoulli beam element has an additional component, i.e. the nodal curvature, associated with the additional kinematic parameter existing at the boundaries of strain gradient beams. The mass and stiffness matrices of the new non-classical beam element are derived based on the Galerkin's method. In some examples, it is shown that how the new element can be employed to solve a real-case problem and the results are compared to the analytical and available experimental data as well as the results obtained by employing the classical beam elements. It is observed that there is a good agreement between the experimental and the strain gradient based FEM results while the difference between the experimental and the classical FEM results is significant. In addition, it is indicated that the new beam element can successfully capture the size-dependency and the structures modeled by this element show stiffer behavior than those modeled by the classical beam element. Moreover, by setting some material length scale parameters to zero the new beam element is able to recover the results of the classical theory and the modified couple stress theory (another non-classical continuum theory).

► We developed an Euler–Bernoulli beam element based on the strain gradient theory.
► We obtained nodal values, shape functions and stiffness and mass matrices.
► The results of a problem are compared to the experimental and classical results.
► A good agreement is observed between the experimental and the present results.
► The gap between the experimental and the classical FEM results is significant.