Finite Element Modeling for Strain Gradient Beams and Planar Frames

Abstract

Size-dependent mechanical behavior becomes pronounced when structural dimensions approach intrinsic material length scales, rendering classical elasticity inadequate. Strain gradient elasticity (SGE) theories address this limitation by incorporating higher-order strain measures and associated nonclassical stress resultants, enabling a consistent representation of size effects. Despite their potential, applications of SGE to realistic structural systems—particularly planar frames—remain scarce due to analytical and numerical complexity. This study presents a comprehensive finite element formulation for strain gradient beams and planar frame structures within the SGE framework. A gradient Euler–Bernoulli beam theory is first established, including governing equations, stress resultants, and both classical and nonclassical boundary conditions. A new gradient beam element (GBE), characterized by a single material length-scale parameter and exact shape functions, is then developed. By combining this element with an existing gradient truss element (GTE), a gradient plane frame element (GPFE) is constructed, along with consistent equivalent nodal stress resultants for distributed loads. A simplified yet accurate approximation suitable for practical engineering use is also proposed. Numerical and experimental investigations—including analyses of determinate and indeterminate beams and scaled reinforced concrete frames—demonstrate the influence of size effects on stiffness and deformation and enable identification of the characteristic material length. The results highlight the capability of strain gradient formulations to bridge advanced continuum theories and structural-scale analysis of frame systems.