Static and dynamic analysis of beams is widely used in engineering problems. In this paper, morphing of plane element to beam element will be used for static, buckling, and free vibration analysis. The formulation of the proposed plane element begins with the writing of Taylor's expansion of the strain field. Selecting a complete first-order polynomial for strain field will eliminate some common errors such as parasitic shear error and sensitivity to distortion and rotation of the coordinate axes. Furthermore, establishing equilibrium equations of the plane problem reveals dependencies between the strain states. These dependencies, in addition to reducing the number of unknowns, will improve the efficiency of proposed element. By calculating the conversion matrix, the proposed beam element is created based on the suggested plane element. This matrix helps to calculate the stiffness, mass, and geometric stiffness matrices of the proposed beam element. Several benchmark tests will use to demonstrate the efficiency of the new element. For comparison, results of other researchers’ good plane and beam elements will be available. In these numerical tests, the good accuracy of the proposed element in static, buckling, and free vibration analysis of beams with different boundary conditions and different aspect ratios will be proved.

中文翻译：

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用于静态、屈曲和自由振动分析的平面单元到梁单元的变形

梁的静动力分析广泛应用于工程问题中。在本文中，平面单元到梁单元的变形将用于静态、屈曲和自由振动分析。所提出的平面单元的公式化始于应变场的泰勒展开式的书写。为应变场选择完整的一阶多项式将消除一些常见误差，例如寄生剪切误差以及对坐标轴扭曲和旋转的敏感性。此外，建立平面问题的平衡方程揭示了应变状态之间的依赖性。这些依赖关系除了减少未知数之外，还将提高提议元素的效率。通过计算转换矩阵，基于建议的平面元素创建建议的梁元素。该矩阵有助于计算建议梁单元的刚度、质量和几何刚度矩阵。将使用几个基准测试来证明新元素的效率。为了进行比较，将提供其他研究人员良好的平面和梁单元的结果。在这些数值试验中，将证明所提出的单元在具有不同边界条件和不同纵横比的梁的静力、屈曲和自由振动分析中的良好精度。