Textile-reinforced concrete is becoming more and more popular. The material enables the realization of very thin structures and shells, often with organic shapes. However, a problem with this reinforcement occurs when the structure is bent (contains a corner), and the flexural stiffness around this bent area is required. This article presents the design, solution, and load-bearing capacity of an L-shaped rigid frame made of textile-reinforced concrete. Basic material parameters of concrete matrix and carbon textile reinforcement were supplemented by a four-point bending test to calibrate fracture energy Gf, critical compressive displacement Wd, solver type, and other parameters of a numerical model created by Atena Engineering in specialized non-linear structural analysis software for reinforced concrete structures. The calibrated numerical model was used to evaluate different variants of carbon textile reinforcement of the L-shaped frame. The carbon textile reinforcement was homogenized using epoxy resin to ensure the interaction of all fibers, and its surface was modified with fine-grained silica sand to increase the cohesion with the concrete matrix. Specimens were produced based on the most effective variant of the L-shaped frame reinforcement to be experimentally tested. Thanks to the original shaping and anchoring of the reinforcement in the corner area, the frame with composite textile reinforcement is rigid and can transmit the bending stresses in both positive and negative directions. The results of the mechanical loading test on small experimental specimens correspond well to the results of numerical modeling using Atena Engineering software.