Nonlinear primary resonance of nano beam with axial initial load by nonlocal continuum theory

•The nonlinear primary resonance of the nano beam is investigated.•Both the axial initial load and the damping effect are taken into account.•The initial tension load will result in the reduction of the hardening nonlinearity but the compression one can strengthen this characteristic.

Based on the nonlocal continuum theory, the nonlinear primary resonance of nano beam with the axial initial load is investigated. The amplitude–frequency response for the primary resonance is derived with the multiple scale method and the stability is analyzed. The nonlinear primary resonance of nano beam is discussed with the influences of small scale effect, axial initial load, mode number, Winkler foundation modulus and the ratio of the length to the diameter. From the results, the typical hardening nonlinearity can be observed. Moreover, some significant and interesting nonlinear phenomena can be found for the primary resonance of nano beam. This work is expected to be useful for the design and analysis for the nonlinear dynamic behaviors of structures at nano scales.