In this research, the vibration of a beam treated with a viscoelastic constrained-layer-damping has been studied and the effects of thermal variations and the attached lumped mass on the variation of the optimal design of the constrained layer have been investigated. For modeling the core, the second and third order polynomials were used respectively for out-of-plane and in-plane displacements, and for outer layers, the Euler-Bernoulli beam theory was used. With this modeling, the effect of the through-the-thickness normal strain in the mid-layer (core) can be included in the analyses, and the model will be applicable for studying the cases with moderately thick cores. The finite element method with 3-node elements has also been used for the solution purpose. Moreover, the viscoelastic material is assumed to be isotropic and its constitutive behavior is described by a complex shear modulus dependent on temperature and frequency. This dependence on frequency and temperature has been obtained by using the graphs of the experimental results presented in the relevant references. Numerical studies have been carried out to investigate the variation of the damping and harmonic response amplitude with the thickness of the core and the constraining layer at different temperatures. The results showed that the thermal variation could considerably change the region associated with the optimal design and the maximum damping. This implies that the range of thermal variations in the operating environment of the structure should be considered in designing a viscoelastic-damping layer. In the numerical studies, the effect of added rigid masses on changing the optimal design was investigated. The results show the necessity to consider all the added masses before designing the constrained layer damping.