A A Novel Multilayer 3D Hexahedral Finite Element for Static and Free Vibration Analysis of Isotropic and Functionally Graded Structures

A Strain-Based Approach for Accurate 3D Structural Modeling

Abstract

This work presents, for the first time, a high-order three-dimensional quadrature element formulation for the hybrid membrane and bending analysis of plate structures under static and free vibration conditions. The proposed element represents a novel contribution to the field of computational mechanics, combining high-order polynomial interpolation with advanced quadrature numerical integration to accurately capture the full 3D stress and displacement fields within beam and plate elements. Unlike traditional 2D formulations, this new approach accounts for through-thickness deformation and stress variation, offering enhanced modeling capabilities for both thin and thick beams and plates. Separate yet consistent formulations are developed for in-plane (membrane) and out-of-plane (bending) behaviors, and their coupling is naturally integrated into the 3D framework. The use of extensive numerical examples demonstrates the accuracy, efficiency, and robustness of the proposed element, especially in representing complex boundary conditions, and higher vibration modes for isotropic and functionally graded (FG) material case. This pioneering work opens new perspectives for high-fidelity structural modeling using three-dimensional high-order quadrature finite elements.