It is widely recognized that the contact geometry at organic molecule-metal interfaces has tremendous effects on electron transport through molecule-metal junctions. Understanding this fundamental issue directly correlates to the success of building molecular electronics devices. The current-voltage (I-V) characteristics for benzendithiolate (BDT) molecule sandwiched between two gold electrodes have been studied both experimentally and theoretically. However, a significant discrepancy in I-V values between theoretical predictions and experimental is still not well understood. Since in the quantum mechanical calculations for the electron transport, the geometry of the metal-molecule-metal contacts has to be assumed a priori, it is valuable to seek insights into the local bonding structures of BDTs on Au nanowires via classical molecular simulations to identify the detailed Au-BDT-Au contact molecular configurations. In this study, we combine grand canonical Monte Carlo (GCMC) with molecular dynamics (MD) simulations to investigate the dynamic elongations of gold nanowires in the presence of BDT molecules. We focus on the development of a simulation strategy to deal with the chemical adsorption and desorption of BDT molecules on Au nanowire by GCMC, followed by MD simulations for the elongations. Simulation results demonstrate that the existence of BDT molecules significantly affect the ductile elongations compared with those obtained in vacuum. The density of BDT and the number of nonbonded and bonded BDT molecules in the simulation box is monitored during the entire elongation process. In addition, the detailed contact geometry and elongation force variations are obtained. Our research result will help to resolve the original discrepancy between experimental and theoretical studies.