Free vibration analysis of simply supported beams with solid and thin-walled cross-sections using higher-order theories based on displacement variables

•Refined 1D models for the free vibrations of thin-walled structures are developed.•The Carrera unified formulation is used to implement variable kinematics 1D models.•Lagrange polynomials are used on the beam cross-section.•A Navier-type solution is used to obtain closed-form solutions.•The efficiency of the proposed models is clear from the analyses of various problems.

Solutions for undamped free vibration of beams with solid and thin-walled cross-sections are provided by using refined theories based on displacement variables. In essence, higher-order displacement fields are developed by using the Carrera unified formulation (CUF), and by discretizing the cross-section kinematics with bilinear, cubic and fourth-order Lagrange polynomials. Subsequently, the differential equations of motion and the natural boundary conditions are formulated in terms of fundamental nuclei by using CUF and the strong form of the principle of virtual displacements. The second-order system of ordinary differential equations is then reduced into a classical eigenvalue problem by assuming simply supported boundary conditions. The proposed methodology is extensively assessed for different solid and thin-walled metallic beam structures and the results are compared with those appeared in published literature and also checked by finite element solutions. The research demonstrates that: (i) the innovative 1D closed form CUF represents a reliable and compact method to develop refined beam models with solely displacement variables; (ii) 3D-like numerically exact solutions of complex structures can be obtained with ease; and (iii) the numerical efficiency of the present method is uniquely robust when compared to other methods that provide similar accuracies.