Exact and Approximate Stiffness Matrix and Nodal Load Vector for a Beam Finite Element with Linearly Varying Stiffness along Its Length

Introduction. Modern trends in construction, related to the optimization of weight and materials, require accurate methods for calculating the stress-strain state, particularly of beams with variable stiffness. Analytical calculation of the stressstrain state for such beams is fraught with considerable difficulties, limiting its practical application. Numerical methods, specifically the Finite Element Method (FEM), are widely used to solve these problems, where the law of stiffness change is typically approximated by a piecewise (discrete) function. This study is aimed at the development of an approach based on piecewise-linear approximation of stiffness. Linear stiffness approximation suggests an optimal balance of accuracy and computational resources. This approach provides significantly higher accuracy compared to the traditional discrete approximation with similar computational complexity, allowing for adequate modeling of both smooth stiffness gradients and its violent changes.

Materials and Methods. A first-approximation stiffness matrix for a one-dimensional beam finite element with linearly varying flexural stiffness was derived on the basis of a variational formulation of the problem. An exact stiffness matrix was obtained by direct integration of the differential equation for beam bending. In the calculation examples, an exact solution was obtained using the Maple software package. The numerical solution using FEM was implemented in the author's program written in Python.

Results. During the study, approximate and exact stiffness matrices of the beam finite element were obtained, as well as the vector of nodal reactions (loads) from distributed loads. The efficiency of the proposed approach was demonstrated by numerical examples. The results obtained by the FEM were verified using analytical calculations. Based on the performed calculations, recommendations and criteria for using the exact or approximate stiffness matrix were developed.

Discussion. Finite elements that account for linear change of stiffness along the length make it possible to increase the accuracy of the results and reduce the degree of discretization of the computational scheme by more than two times. The approximate matrix shows good convergence with a smooth change in stiffness along the length. In such cases, discrete approximation is also acceptable. The exact matrix allows for calculating cases where the stiffness within the beam changes by orders of magnitude with low error. The classical discrete approximation in this case does not ensure high accuracy of the calculation results.

Conclusion. The paper presents stiffness matrices for finite elements that account for linear change of stiffness along the length. Their derivation is performed by two methods: on the basis of a variational formulation of the problem, and by direct integration of the differential equation of bending. The resulting matrices enable more accurate stress-strain analysis of beams with variable stiffness. They have an analytical format that simplifies their integration into existing software systems. Further research will be directed towards applying the obtained matrices to the calculation of reinforced concrete beams, considering physical nonlinearity, as well as to solving problems of stability and dynamics of beams with variable stiffness.

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