Stiffness matrices of beams embedded in an elastic medium and subjected to axial forces are considered. Both the bending and the axial deformations have been incorporated. Two approaches for deriving the element stiffness matrix analytically have been proposed. The first approach is based on the direct force–displacement relationship, whereas the second approach exploits shape functions within the finite element framework. The displacement function within the beam is obtained from the solution of the governing differential equation with suitable boundary conditions. Both approaches result in identical expressions when the exact transcendental displacement functions are used. Exact closed-form expressions of the elements of the stiffness matrix have been derived for the bending and axial deformation. Depending on the nature of the axial force and stiffness of the elastic medium, seven different cases are proposed for the bending stiffness matrix. A unified approach to the non-dimensional representation of the stiffness matrix elements and system parameters that are consistent across all the cases has been developed. Through Taylor-series expansions of the stiffness matrix coefficients, it is shown that the classical stiffness matrices appear as an approximation when only the first few terms of the series are retained. Numerical results shown in the paper explicitly quantify the error in using the classical stiffness compared to the exact stiffness matrix derived in the paper. The expressions derived here gives the most comprehensive and consistent description of the stiffness coefficients, which can be directly used in the context of finite element analysis over a wide range of parameter values.

考虑嵌入在弹性介质中并受到轴向力的梁的刚度矩阵。弯曲和轴向变形都被合并了。已经提出了两种用于分析地推导单元刚度矩阵的方法。第一种方法基于直接的力-位移关系，而第二种方法利用有限元框架内的形状函数。梁内的位移函数是从具有合适边界条件的控制微分方程的解中获得的。当使用精确的超越位移函数时，两种方法都会产生相同的表达式。对于弯曲和轴向变形，已经导出了刚度矩阵元素的精确封闭形式表达式。根据轴向力的性质和弹性介质的刚度，为弯曲刚度矩阵提出了七种不同的情况。已经开发了一种统一的方法来对所有情况下一致的刚度矩阵元素和系统参数进行无量纲表示。通过刚度矩阵系数的泰勒级数展开，表明当仅保留级数的前几项时，经典刚度矩阵表现为近似值。与论文中推导出的精确刚度矩阵相比，论文中显示的数值结果明确量化了使用经典刚度的误差。这里导出的表达式给出了刚度系数的最全面和一致的描述，