One-dimensional bar or shear-type beam with additional energy dissipation devices is a distributed-parameter system with non-classical damping. For its dynamic analysis，the conventional way is to construct an equation of motion for each segment and obtain the dynamic response by using the real mode superposition method and the continuous condition. In essence，this method is a component mode synthesis based on undamped modes of substructure. Even though approximated dynamic responses can be estimated，it cannot consider the effect of damping on the dynamic behavior. To consider the discontinuity of damping and stiffness resulting from the additional damper，utilizing the generalized function theory，one non-dimensional equation of motion for the whole system is constructed in this paper. Then，using the Laplace integral transformation，the eigen function（complex mode）and eigenvalue equation are derived. Finally，the complex mode superposition method for distributed-parameter systems with non-classical damping is developed based on the derived orthogonality condition of eigen functions. In addition，the eigenvalue equation is a very complex transcendental equation，in order to get several natural frequencies，an equivalent polynomial method based on the Cauchy integral theorem is proposed，in which the eigenvalue equation is transformed into a set of linear equations such that their solutions can be obtained more easily. In the last section of this paper，the application of the proposed method is illustrated in a base-isolated shear-type beam and some useful information for the design of base-isolated structures is provided. To summarize，the complex mode superposition method is an extension of the conventional real mode superposition method for classically damped continuous and distributed-parameter systems，which is meaningful and valuable in the theory and application.