Exact solution of Eringen's nonlocal integral model for vibration and buckling of Euler–Bernoulli beam

Although, differential form of Eringen nonlocal model has been broadly used during investigation of nanoscale structures, the discrepancies in the results of integral and differential form of nonlocal equation are demonstrated for static bending analysis, recently. With this motivation, the present study aims to examine the buckling and vibration characteristics of nonlocal Euler–Bernoulli beams by solving the original integral constitutive equation. To this end, Fredholm type integral governing equation is split into three parts that include two Volterra integral equations of the second kind, and the first-order governing differential equation system is solved by utilizing the Laplace transform method. The results are compared with those from the literature, and it is concluded that, the nonlocal parameter induced softening phenomena can be captured accurately by using the original integral form of nonlocal constitutive equation.