A generalized, geometrically rigorous theory of the stability of compressed rods is presented, based on the exact Euler–Bernoulli beam model without linearization of curvature expressions and without a priori assumptions about the buckling shape. As the baseline configuration, the deformation of a simply supported beam into a circular arc under a constant bending moment is examined, with the identified geometric pattern extended to other buckling modes. A universal analytical expression for the critical force is derived, depending on the central angle and the corresponding critical eccentricity that emerges as a result of the geo-metric transformation of the deformed axis. It is shown that Euler’s classical formula is a special case of the proposed solution and overestimates the critical force by 23.37%. For the first time, an exact formula for the critical force of a cantilever beam is rig-porously obtained, and the validity of the effective length coefficient is confirmed. One of the key consequences of the new theory is the possibility of applying the principle of superposition to stability problems, allowing for the combined influence of multiple transverse loads with various application schemes. The proposed approach covers a wide range of boundary conditions and loading types, and completes the construction of a geometrically rigorous stability theory within the Euler–Bernoulli model.