Quadrilateral finite elements with five displacements per corner node built up from four triangular elements are developed by a discretization of the Mindlin plate bending theory and based on a modified form of the mixed variational principle for stresses and displacements. The modified principle separates the contribution of shear deformation by taking the transverse shear strains as free functions. It also gives discrete element governing equations in a canonical form. Similar elements are developed based on the potential energy formulation. The elements can be readily specialized to the Kirchhoff plate theory. The discrete shear strains are transformed back to normal-fiber rotations for enforcing general boundary conditions. The discretization procedure uses one-dimensional polynomials instead of two-dimensional shape functions. The discretization satisfies $C^1$ continuity for w, and $C^0$ continuity for the transverse shear strains. The Mindlin elements are tested with soft and hard boundary conditions. They are naturally free of shear-locking. They are also capable of reproducing the boundary layer that occurs with some soft boundary conditions. Mindlin elements results tend to those of the Kirchhoff theory as the thickness decreases. Extensive numerical results are obtained for square plates some of which are compared with exact results or results obtained by other methods.

中文翻译：

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基于混合变分原理的 Mindlin 和 Kirchhoff 板的有限元

由四个三角形单元构成的每个角节点有五个位移的四边形有限元是通过 Mindlin 板弯曲理论的离散化并基于应力和位移的混合变分原理的改进形式而开发的。修改后的原理通过将横向剪切应变作为自由函数来分离剪切变形的贡献。它还给出了规范形式的离散元素控制方程。基于势能公式开发了类似的元素。这些元素可以很容易地专门用于基尔霍夫板理论。离散剪切应变被转换回正常纤维旋转以强制执行一般边界条件。离散化过程使用一维多项式代替二维形状函数。$C^1$w 的连续性，和$C^0$横向剪切应变的连续性。Mindlin 单元使用软边界条件和硬边界条件进行测试。它们自然没有剪切锁定。它们还能够再现在某些软边界条件下出现的边界层。随着厚度的减小，Mindlin 单元的结果趋于基尔霍夫理论的结果。获得了方形板的广泛数值结果，其中一些与精确结果或通过其他方法获得的结果进行了比较。