This paper is devoted to the analysis of the efficiency of the reduced model constructed using the POD or the Karhunen-Loève, KL, method in nonlinear dynamics for continuous systems. We present a theory for continuous systems and we develop a numerical analysis based on the use of the finite element method applied to the weak formulation of different continuous problems. A nonlinear basis is constructed by the POD or the KL method. We prove the fundamental properties required to use such a basis of the admissible displacement field space in order to construct a reduced model. It is explained that if the nonlinear basis constructed with the POD or the KL method can be viewed as an optimal basis for representing the response of the system, it is not a priori an optimal nonlinear basis for reducing the model. A numerical solver of the generalized eigenvalue problem related to the POD or the KL method is proposed for large systems. Another solver is also proposed in introducing a reduced generalized eigenvalue problem based on a modification of the snapshot method.