This paper studies a variable length elastica with a fixed point constraint by an assembly method that regards the whole elastica as an assembly of two components, i.e., pinned-clamped elasticas. The pinned-clamped elastica is obtained based on the post-buckled deformed shape with one internal inflection point. Thus, multiple coexisting solutions can be located accurately, which reveals three distinct equilibrium paths for the complete load–displacement curves. Under displacement control, two critical points on two equilibrium paths are found at saddle-node bifurcations. Interestingly, a new critical point is located at the boundary point of one equilibrium path, where the shapes of two pinned-clamped elasticas are two different post-buckled deformed shapes. Changing the location of the fixed point constraint allows the position of the boundary point to be easily manipulated, and the associated snap-through phenomenon can occur on different equilibrium paths. This flexible generation of the snap-through phenomenon is useful for designing engineering systems that require controllable snap-through.