Carbon nanotubes (CNTs) with their extraordinary properties have been recognized as one of the most promising nanomaterials ever discovered. This paper presents a density functional theory (DFT) study on the structural and elastic properties of CNTs with different chiralities containing Fe atoms (Fe-CNT). In the first step, the effects of Fe encapsulation on the radius and energy of CNTs are investigated. It is observed that, unlike armchair CNTs, the equilibrium radius of zigzag CNTs increases after Fe encapsulating. Different positions for the Fe atom inside CNTs are considered. The results reveal that depending on the size and chirality of CNTs, several possible stable sites for the Fe atom inside CNTs exist. In the next step, for the most stable Fe-CNT structure, Young’s modulus is computed and it is seen that the encapsulation of Fe atoms reduces the stiffness of CNTs.