Predicting snap-through and post-buckling behaviour in thin shell structures remains challenging due to strong geometric nonlinearity, imperfection sensitivity, and the inability of standard Newton–Raphson load-control methods to traverse limit points. To address these challenges, this work presents the development and implementation of a general arc-length continuation solver within the MOOSE finite-element framework for robust tracking of nonlinear equilibrium paths. MOOSE (Multiphysics Object-Oriented Simulation Environment) is an open-source, LGPL-licensed, C++ finite-element framework developed at Idaho National Laboratory and widely used for large-scale multiphysics simulations, including nuclear engineering, solid mechanics, geophysics, and phase-field modeling. Its modular architecture and automatic differentiation capabilities make it well suited for extensible nonlinear solver development.The proposed solver employs a coupled constraint equation linking the load parameter and global displacement increment, enabling stable traversal of snap-through, snap-back, and multi-branch responses. The formulation is fully consistent with classical spherical arc-length continuation and is implemented in a modular manner compatible with existing MOOSE nonlinear solvers and executioners. The capabilities of the solver are demonstrated using MITC4 shell elements applied to thermome-chanically loaded structures exhibiting severe post-buckling behaviour, including temperature-dependent membrane–bending coupling. Benchmark examples illustrate the improved robustness and path-following capability of the arc-length approach compared to standard load-control procedures. This contribution establishes the first arc-length continuation capability for nonlinear structural mechan- ics in MOOSE, providing a foundation for future developments in shell instability analysis, imperfection sensitivity studies, and advanced multiphysics applications involving critical structural transitions.