Abstract The studies on the higher-order beam theory and its application to functionally graded (FG), laminated composite and sandwich beams have received much research attention over recent years. While significant progress has been made, there is an issue with the selection of the appropriate distribution function for describing section warping due to transverse loading. In the present study, a rational approach is proposed for the determination of correct warping functions for symmetric cross sections. Two new conditions on warping function are suggested: first, the warping function should vanish on the neutral plane; and second, the first derivative of warping function should take unity value on the neutral plane. A rigorous procedure is then developed by using the equilibrium condition and the three conditions. An exact higher-order theory is then developed for creating accurate analysis models to consider both material and geometry variations over the beam cross section. A finite element analysis model is presented and a two-node beam element with bubble displacement modes is implemented. Results for sandwich and FG beam problems are shown and numerical results are compared with those of plane stress continuum models to demonstrate the correctness and application of the new theory and finite element model.

中文翻译：

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高阶剪切变形梁理论的合理方法

摘要 近年来，高阶梁理论及其在功能梯度（FG）、层合复合材料和夹层梁中的应用研究受到了广泛的关注。虽然已经取得了重大进展，但在选择适当的分布函数来描述由于横向载荷引起的截面翘曲时存在问题。在本研究中，提出了一种合理的方法来确定对称横截面的正确翘曲函数。提出了两个关于翘曲函数的新条件：第一，翘曲函数应该在中性平面上消失；其次，翘曲函数的一阶导数在中性平面上取统一值。然后通过使用平衡条件和三个条件制定严格的程序。然后开发精确的高阶理论来创建精确的分析模型，以考虑梁横截面上的材料和几何变化。提出了有限元分析模型，并实现了具有气泡位移模式的二节点梁单元。显示了夹层梁和 FG 梁问题的结果，并将数值结果与平面应力连续模型的结果进行了比较，以证明新理论和有限元模型的正确性和应用。