A new three-node element for bending, free vibration and buckling analysis of composite laminated beams based on FSDT theory

Abstract

In this paper, a new three-node element with three degrees of freedom per node is proposed for bending, free vibration and buckling analysis of laminated beams. The element's formulation is based on the first-order shear deformation theory (FSDT). For this aim, transverse displacement and rotation field of the element are selected from fifth and fourth order, respectively. Moreover, the shear strain is varied as quadratic function throughout the element. It is worth noting that the quadratic function can be used for axial displacement field. By employing equilibrium equation, curvature and shear strain relations of laminated beam with FSDT theory, the exact and explicit shape functions of the displacement fields are obtained. By utilizing the obtained shape functions, the explicit form of the stiffness matrix is calculated for the element. On the other hand, by using the governing equation of the free vibration and buckling of the beam, the explicit form of the translation and rotary mass matrices, and geometric stiffness matrix of the element are obtained. It should be mentioned, the proposed element is free of shear locking. Finally, several numerical tests fulfill to assess the robustness of the developed element. For this purpose, bending, free vibration and buckling analysis of laminated beams with different boundary conditions and aspect ratios are performed. The results of the numerical tests demonstrate high accuracy and efficiency of the proposed element for free vibration and buckling analysis of laminated beams.

Static and Free Vibration Analyses of Functionally Graded Plane Structures

“…Figure 5 represents the onedimensional quadratic element on its own reference in natural coordinates. Considering that a generic degree of freedom is approximated as a linear combination of the corresponding nodal degrees of freedom and the shape functions, the previous element stiffness 𝑲 𝒆 and mass 𝑴 𝒆 matrices can be represented in a compact form as follows [34,36]:…”

“…where 𝑩 𝒎𝒃 and 𝑩 𝒔 -the matrices that pair the deformations with the displacementsand 𝑳-the matrix of the Lagrange quadratic shape functions 𝑁 that are associated with the displacement field description-are, for the i-th node, constituted as follows: Considering that a generic degree of freedom is approximated as a linear combination of the corresponding nodal degrees of freedom and the shape functions, the previous element stiffness K e and mass M e matrices can be represented in a compact form as follows [34,36]:…”