The aim of this work is to develop a new generalized formula and a numerical computation program for evaluating the energy form coefficient of a complex and arbitrary cross section for full and thin-walled cross section with respect to any central axis, for the bending of beams of small lengths in comparison with the transverse dimension of the section. This coefficient plays a very important role in the calculation of the deformation energy of beams subjected to bending under the effect of a shearing force for short beams. It also enters in the formulation of FEM bending model, in order to calculate the stresses and the strains due to the external forces. The application is made for complex sections used in various fields of construction and in particular for airfoils designed for aerospace construction. A method is developed to calculate this coefficient as a function of the rotation of the central axes. The calculation of the area, the moments, and the product of inertias with respect to the central axes is necessary. The formula for calculating this coefficient is presented as a definite integral of a non-analytical function determined point by point along the direction of the application of the shear force. This function is based on the calculation of the partial static moments. The calculation of the latter is based on the development of a technique by subdividing the upper part of the section into adjacent common triangles at one point for the full solid section or by segments on the boundary for the thin-walled section. To speed up the process of numerically calculating this integral with high precision and reduced time, Gauss Legendre quadrature of order 40 is used. The calculation of the distribution of the tangential stress as well as its maximum value is determined. A shear shape coefficient is therefore determined. In the second part of this work, an application is made for the static calculation by the FEM of a hyper static beam with a view to determining the influence of this coefficient on all the parameters of resistance and bending stiffness as a correction of the classical model of bending by the FEM. A study of the error made by the classical bending model on our shear effect model is presented. A coefficient of efficiency of a section is presented.

这项工作的目的是开发一个新的广义公式和数值计算程序，用于评估复杂和任意横截面的能量形式系数，用于相对于任何中心轴的全壁和薄壁横截面，用于梁的弯曲与截面的横向尺寸相比，长度较小。该系数在计算短梁受剪力作用下弯曲变形能时起着非常重要的作用。它也进入了FEM的公式弯曲模型，以计算由于外力引起的应力和应变。该应用适用于各种建筑领域中使用的复杂部分，特别是为航空航天建筑设计的翼型。开发了一种方法来计算该系数作为中心轴旋转的函数。必须计算面积、力矩和相对于中心轴的惯性积。计算该系数的公式表示为沿剪切力施加方向逐点确定的非解析函数的定积分。该功能基于部分静力矩的计算。后者的计算基于一种技术的发展，即通过将截面的上部在一个点处细分为相邻的公共三角形（对于完整的实体截面）或通过在边界上的段（对于薄壁截面）。为了加快以高精度和缩短时间对该积分进行数值计算的过程，使用了 40 阶的高斯勒让德求积。确定切向应力分布的计算及其最大值。因此确定剪切形状系数。在这项工作的第二部分，应用程序进行静态计算 为了加快以高精度和缩短时间对该积分进行数值计算的过程，使用了 40 阶的高斯勒让德求积。确定切向应力分布的计算及其最大值。因此确定剪切形状系数。在这项工作的第二部分，应用程序进行静态计算 为了加快以高精度和缩短时间对该积分进行数值计算的过程，使用了 40 阶的高斯勒让德求积。确定切向应力分布的计算及其最大值。因此确定剪切形状系数。在这项工作的第二部分，应用程序进行静态计算超静力梁的FEM旨在确定该系数对阻力和弯曲刚度的所有参数的影响，作为对FEM经典弯曲模型的修正。研究了经典弯曲模型对我们的剪切效应模型造成的误差。给出了部分的效率系数。