Fibers embedded into solid continua can act as reinforcements with applications such as composite mate- rials in aerospace engineering, steel-reinforced concrete structures in civil engineering or collagen fibers in biological tissue. For such systems, mathematical modeling and numerical simulation are key to fully assess and design their properties, especially when physical experimentation is infeasible. Various finite element method (FEM)-based techniques exist for modeling reinforced materials, each with its own limitations, highlighting the need for more efficient and accurate approaches. Previously, we have developed a mixed-dimensional 1D/3D approach for beam-solid interaction employing mortar methods [1,2] along with a scalable approximate block factorization preconditioner [3]. In the existing work, the coupling constraints are imposed through a penalty approach. For constraint imposition using Lagrange multipliers, questions around the stability of the discretized problem remain largely unexplored so far. It is well known though, that the choice of Lagrange multiplier shape functions is crucial for the mathematical properties of the discretized system, as the discrete Lagrange multiplier bases must satisfy an inf-sup condition with the discretization of the displacement field [4]. In this talk, we outline the formulation and discretization of beam-solid coupling using Lagrange multi- pliers for constraint enforcement. We discuss the spatial convergence behavior and examine the influence of shape functions, modeling assumptions and discretization on the stability of the discrete Lagrange multiplier formulation and supplement our findings with suitable numerical examples. REFERENCES: [1] I. Steinbrecher, M. Mayr, M. J. Grill, J. Kremheller, C. Meier, and A. Popp., A mortar-type finite element approach for embedding 1D beams into 3D solid volumes., Comput. Mech., 66(6):1377– 1398, 2020. [2] I. Steinbrecher, A. Popp, and C. Meier. Consistent coupling of positions and rotations for embed- ding 1D Cosserat beams into 3D solid volumes. Comput. Mech., 69:701–732, 2022. [3] M. Firmbach, I. Steinbrecher, A. Popp, and M. Mayr. An approximate block factorization pre- conditioner for mixed-dimensional beam-solid interaction. Comput. Methods Appl. Mech. Engrg., 431:117256, 2024. [4] Boffi D, Brezzi F, Fortin M. Mixed Finite Element Methods and Applications, 1st edn. Springer, Berlin, 2013.