In this paper, the three-dimensional (3D) stress analysis of plate-type structures in instability conditions is presented. The displacement-based and hybrid-mixed four-node quadrilateral elements are developed taking the advantages of the sampling surfaces (SaS) method. The SaS formulation is based on considering inside the plate N not equally spaced SaS parallel to the middle surface to specify the displacements of these surfaces as primary plate unknowns. The displacements, strains and stresses are assumed to be distributed through the thickness using Lagrange polynomials of degree N–1 that lead to a well-set higher-order plate theory. The locations of SaS are based on the use of Chebyshev polynomial nodes that allow us to minimize uniformly the error due to Lagrange interpolation. To circumvent shear locking and have no spurious zero energy modes, the assumed transverse shear strains are employed. The nonlinear equilibrium equations are solved by the Newton–Raphson iterative method combined with the Crisfield arc-length algorithm. The accuracy and efficiency of both elements in different conditions such as coarse and distorted meshes are investigated. The developed assumed-natural strain (ANS) elements can be useful for the 3D stress analysis of thin and thick plates in whole states of equilibrium path involving bifurcation, snap-through, and/or snap-back phenomena.

本文提出了在不稳定条件下板型结构的三维应力分析。利用采样表面（SaS）方法的优势，开发了基于位移的混合混合四节点四边形单元。SaS公式是基于考虑板N内部平行于中间表面的SaS的间距不相等而将这些表面的位移指定为主要板未知数。假设使用N级的拉格朗日多项式将位移，应变和应力分布在整个厚度上–1导致设置良好的高阶板理论。SaS的位置基于Chebyshev多项式节点的使用，这些节点使我们能够均匀地最小化由于Lagrange插值引起的误差。为了规避剪切锁定并且没有虚假的零能量模式，采用了假定的横向剪切应变。非线性平衡方程是通过Newton-Raphson迭代方法结合Crisfield弧长算法求解的。研究了两种元素在不同条件下（例如粗糙和扭曲的网格）的准确性和效率。所开发的假定自然应变（ANS）元素可用于在平衡路径的整个状态下，涉及分叉，穿通和/或回跳现象的薄板和厚板的3D应力分析。