To study the complex interactions between underwater shocks and composite structures, a strongly coupled Eulerian-Lagrangian numerical solver is developed. The coupled numerical solver consists of an Eulerian fluid solver, a Lagrangian solid solver, a one-fluid cavitation model, and an interface capturing scheme. The interface capturing scheme features a fluid characteristics method and a modified ghost fluid method (MGFM). The MGFM is reformulated for fluid-solid coupling by treating simultaneously the fluid characteristics equation and the solid equation of motion to determine the interface variables, leading to a strongly coupled Eulerian-Lagrangian scheme. Various components of the numerical solver are first individually tested and validated. The strongly coupled solver is then applied to realistic shock-structure interaction problems involving composite structures. The accuracy of the coupled solver is demonstrated via comparison with numerical predictions and experimental observations available in literature. Finally, the validated coupled numerical solver is utilized to study the effectiveness of a proof-of-concept shock mitigation scheme.