Boundary characteristic orthogonal polynomials are used as shape functions in the Rayleigh–Ritz method to investigate vibration and buckling of nanobeams embedded in an elastic medium. The present formulation is based on the nonlocal Euler–Bernoulli beam theory. The eigen value equation is developed for the buckling and vibration analyses. The orthogonal property of these polynomials makes the computation easier with less computational effort. It is observed that the frequency and critical buckling load parameters are dependent on the temperature, elastic medium, small scale coefficient,and length-to-diameter ratio. These observations are useful in the mechanical design of devices that use carbon nanotubes. The present formulation is based on the nonlocal Euler-Bernoulli beam theory. The eigen value equation is developed for the Buckling and vibration analyzes. The orthogonal property of these polynomials makes the computation easier with less computational effort. It is observed that the frequency and critical buckling load parameters are dependent on the temperature, elastic medium, small scale coefficient, and length-to-diameter ratio. These observations are useful in the mechanical design of devices that use carbon nanotubes.