Abstract Increasing applications of laminated composite structures necessitate the development of equivalent single layer (ESL) models that can achieve similar accuracy but are more computationally efficient than 3D or layer-wise models. Most ESL displacement-based models do not guarantee interfacial continuity of shear stresses within laminates. A possible remedy is the enforcement of interlaminar equilibrium in variational formulations, for example, in the framework of the Hellinger-Reissner variational principle, leading to a mixed force/displacement model. In this paper, the governing equations for bending and stretching of laminated beams, comprising only seven stress resultants and two displacement functionals, are obtained using global fifth-order and a local linear zigzag kinematics. As a strong-form solution technique, the differential quadrature method (DQM) is an efficient tool which can provide excellent convergence with relatively few number of grid points. However, in dealing with high-order differential equations, the conventional DQM can incur considerable errors due to the nature of numerical differentiation. Therefore, a mixed inverse differential quadrature method (iDQM) is proposed herein to solve the governing fourth-order differential equations for bending and stretching of laminated beams. This approach involves approximating the first derivatives of functional unknowns, thereby reducing the order of differentiation being performed. Using a non-uniform Chebychev-Gauss-Lobatto grid point profile, numerical results show that the accuracy of stress predictions is improved by using iDQM compared to DQM. In addition, the Cauchy’s equilibrium condition is satisfied more accurately by iDQM, especially in the vicinity of boundaries.

中文翻译：

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基于Hellinger-Reissner混合变分公式的恒变刚度层合梁静力分析的混合逆微分正交法

摘要 越来越多的层压复合结构的应用需要开发等效单层 (ESL) 模型，这些模型可以实现相似的精度，但比 3D 或分层模型在计算上更有效。大多数基于 ESL 位移的模型不能保证层压板内剪切应力的界面连续性。一种可能的补救措施是在变分公式中实施层间平衡，例如，在 Hellinger-Reissner 变分原理的框架内，导致混合力/位移模型。在本文中，使用全局五阶和局部线性曲折运动学获得了仅包含七个应力合力和两个位移函数的层合梁弯曲和拉伸的控制方程。作为一种强形式求解技术，差分正交法 (DQM) 是一种有效的工具，可以在相对较少的网格点数下提供出色的收敛性。然而，在处理高阶微分方程时，由于数值微分的性质，传统的 DQM 会产生相当大的误差。因此，本文提出了一种混合逆微分正交法 (iDQM) 来求解层压梁弯曲和拉伸的控制四阶微分方程。这种方法涉及逼近函数未知数的一阶导数，从而减少正在执行的微分顺序。使用非均匀 Chebychev-Gauss-Lobatto 网格点轮廓，数值结果表明，与 DQM 相比，使用 iDQM 提高了应力预测的准确性。此外，