The nonlinear propagation of the dust-acoustic (DA) waves in a strongly coupled dusty plasma containing strongly correlated arbitrarily (positively or negatively) charged dust and weakly correlated Boltzmann electrons and ions has been investigated by employing the generalized hydrodynamic model and the reductive perturbation method. It has been shown that the strong correlation among the charged dust is a source of dissipation and is responsible for the formation of the DA shock waves in such a strongly coupled dusty plasma. It has also been found that the DA shock waves with positive (negative) potential are formed for positively (negatively) charged dust. The basic features of such DA shock waves have been identified. It has been suggested that a laboratory experiment be performed to test the theory presented here.