Compliant mechanisms (CMs) are used to transfer motion, force, and energy, taking advantages of the elastic deformation of the involved compliant members. A branch of special type of elastic phenomenon called (post) buckling has been widely considered in CMs: avoiding buckling for better payload-bearing capacity and utilizing post-buckling to produce multi-stable states. This paper digs into the essence of beam’s buckling and post-buckling behaviors where we start from the famous Euler–Bernoulli beam theory and then extend the mentioned linear theory into geometrically nonlinear one to handle multi-mode buckling problems via introducing the concept of bifurcation theory. Five representative beam buckling cases are studied in this paper, followed by detailed theoretical investigations of their post-buckling behaviors where the multi-state property has been proved. We finally propose a novel type of bi-stable mechanisms termed as pre-buckled bi-stable mechanisms (PBMs) that integrate the features of both rigid and compliant mechanisms. The theoretical insights of PBMs are presented in detail. To the best of our knowledge, this paper is the first study on the theoretical derivation of the kinematic models of PBMs, which could be an important contribution to this field.