The manuscript is devoted to graph-based mathematical simulation of a widespread steelmaking process, in which cored wire is injected into treatment ladle with molten steel stirred by argon blowing. Despite the process has better effectiveness than other methods its accuracy can be increased further using mathematical modelling. For example there is melt’s depth, where an addition needs to be released after melting of cored wire. Due complexity of wire dynamics many models in literature are oversimplified and don’t take important features. A wire has bending strains, remained after its unpacking, which leads to slightly curved shape of the lightweight wire. As consequence when it reaches a weighty melt there is a risk of swirling on the melt’s surface instead of diving inside. Also ladle wall and hydrodynamic drag of molten steel can significantly influence wire motion. In this work cored wire is replaced by a mechanical system of rigid rods with initially equal length and rotational springs between them. One of the peculiarities is a variable count of rods in the system during injection. A vector-network graph is used to decompose multirod system to basic mechanical elements (rods, spherical joints, and springs) and their relationships with information about forces and torques. An incidence matrix has repetitive nature and it is created using the graph in a way that simplifies getting of cutset and circuit equations. Using sequential substitutions the equations of motion are obtained in accordance to works of other scientists, which do it for systems with kinematically linked two and three bodies. For a computer realisation the semi-implicit Euler’s method is used to numerically solve SLAE. There is a parameters table of conducted numerical experiments and a chart with depth of wire’s tip depending on time, which has good correspondence with a real process. Presented computer simulation helps to find an optimal injection speed (3–5 m/s) of considered cored wire in a ladle. Alternatively a wire with higher elasticity can be taken to immerse it deeper, because increasing of the speed is not always effective, especially when it is already high.