Embedding 1D Euler Beam in 2D Classical Continua

In this contribution, the classical Cauchy first-gradient elastic theory is used to solve the equilibrium problem of a bidimensional (2D) reinforced elastic structure under small displacements and strains. Such a 2D first-gradient continuum is embedded with a reinforcement, which is modeled as a zero-thickness interface endowed with the elastic properties of an extensional Euler–Bernoulli 1D beam. Modeling the reinforcement as an interface eliminates the need for a full geometric representation of the reinforcing bar with finite thickness in the 2D model, and the associated mesh discretization for numerical analysis. Thus, the effects of the 1D beam-like reinforcements are described through proper and generalized boundary conditions prescribed to contiguous continuum regions, deduced from a standard variational approach. The novelty of this work lies in the formulation of an interface model coupling 1D and 2D continua, based on weak formulation and variational derivation, capable of accurately capturing stress distributions without requiring full geometric resolution of the reinforcement. The proposed framework is therefore illustrated by computing, with finite element simulations, the response of the reinforced structural element under uniform bending. Numerical results reveal the presence of jumps for some stress components in the vicinity of the reinforcement tips and demonstrate convergence under mesh refinement. Although the reinforcement beams possess only axial stiffness, they significantly influence the equilibrium configuration by causing a redistribution of stress and enhancing stress transfer throughout the structure. These findings offer a new perspective on the effective modeling of fiber-reinforced structures, which are of significant interest in engineering applications such as micropiles in foundations, fiber-reinforced concrete, and advanced composite materials. In these systems, stress localization and stability play a critical role.