The small modular floating lead-cooled fast reactor represents a typical fourth-generation reactor, featuring advantages such as compact structure, flexible layout, and inherent safety, making it suitable for providing power to oceanic islands and remote areas, and thus, is of significant importance for global energy security. This paper focuses on a 50MWth fully natural circulation small modular floating lead-cooled fast reactor, and conducts research on the multi-physics coupling of the reactor. Firstly, utilizing the OpenMC program, simplified model of the small lead-cooled fast reactor is established for simulation and calculation of the reactor's cross-section parameters, which will serve as input parameters for the GeN-Foam program. Subsequently, a full-core three-dimensional multi-physics field coupling model encompassing neutron physics, thermal-hydraulic, and structural mechanics is established using the GeN-Foam program. The model is fed with the cross-sectional parameters calculated by the OpenMC program and other relevant core parameters, and steady-state operational numerical simulation is conducted, yielding positive findings for parameters like core outlet temperature and maximum fuel temperature. Finally, simulations of transient emergency reactor shutdown scenarios are performed, obtaining the changing patterns of operational parameters such as coolant temperature distribution, core power density distribution, and fuel rod deformation with respect to control rod position and power level. This paper plays a significant role in the design and operation of small lead-cooled fast reactors.