Structural elements like bars, trusses, beams, frames, plates, and shells have long been used in structures and machines because of their large stiffness-to-weight ratios. The Euler–Bernoulli theory for beam elements is currently used in a wide range of engineering fields. Frames may essentially be considered to be a type of general beam with axial loads. In the analysis of a right-angle frame, the stiffness of a corner has been assumed to be infinite, which is allowable only when the frame is sufficiently slender. However, a comparison of the results of a finite element analysis showed that the assumption of rigid corner stiffness is unacceptable for most cases because of the considerable errors that result. To resolve this problem, we assumed that the stiffness of a corner in a right-angle frame was finite, which is mostly the case, and solved the problem of a right-angle frame with round corners under internal pressure. Using the derived formula based on the assumption of finite corner stiffness and the formula for the round corner stiffness, we analyzed the entire right-angle frame structure and compared the results to finite element analysis results. As a final attempt, the quasi-optimal dimension of the corner was found to exhibit the lowest von Mises equivalent stress. This proposed approach could be applied to many problems involving frames with various boundary conditions to improve the accuracy.