In this study, stability loss research has been made for composite material containing a locally curved one double-walled carbon nanotube and one triple-walled carbon nanotube separately. The research has been made within the scope of piecewise-homogenous body model by the use of three-dimensional linearized theory of stability (TDLTS). The carbon nanotubes are resistant to tension, but as they contain large gaps in their structure, their resistance to compression is very low. For this reason, searching the behaviors of nanotubes under compression is very important. The van der Waals forces between the walls of the carbon nanotube have been considered. It has been thought that the referred composite material is under the effect of uniformly distributed normal forces in the direction of carbon nanotube at infinity. In the model, ideal contact conditions have been used between the nanotube and the matrix. In addition, the effect of increasing the number of layers of the nanotube on the critical load has been investigated. The results obtained from this study will be able to guide in the implementations relevant to modeling of mechanical behaviors of the addressed composite material. Thus, the limits required to be considered in the production of composite material containing double-walled and triple-walled nanotubes with local curvature have been obtained.