Complex bending and initial destruction of hybrid timber beams

A mathematical model of the deformation of hybrid timber beams has been developed. By hybrid we mean bars, formed by rigid connection (gluing) on certain contact surfaces of a set of layers of different forms of cross-sections and different types of timber. In general, the bars are in conditions of complex bending with stretching-compression. The physical non-linearity of timber, as well as the different tensile and compression resistance, is taken into account. In the general case, the problem reduces either to solving a system of three nonlinear algebraic equations of the third degree with respect to generalized deformations of the cross section or to a system of three nonlinear ordinary differential equations with respect to the components of the displacement vector of the points of the axis of the rod. To solve the obtained algebraic equations the Newton method is used, the solution of the differential equations is performed using the Galerkin type method. An analytical approximation of the experimental tension-compression diagrams of timber along the fibers in the form of polynomials of the second and third degree is proposed. The coefficients of the approximating functions are determined in two ways: using the least squares method with the experimental deformation diagrams; by imposing certain requirements on the diagrams, using the basic mechanical characteristics of the timber (maximum stresses and deformations, moduli of elasticity). Numerical values of the approximation coefficients for 15 different types of timber are given. The above examples of calculations of hybrid timber beams have shown the possibility of the emergence of hidden mechanisms of destruction, as well as the strong influence of the rearrangement of layer materials on the stress-strain state of the structure. The method developed in the article for the calculation of hybrid rod-shaped timber structures offers great opportunities for solving optimization problems in the design, and allows rational use of various types of timber

Моделирование деформирования и разрушения железобетонных балок при четырехточечном изгибе

A new mathematical model for the four-point bending of reinforced concrete beams is developed and investigated. The model takes into account multi-modulus concrete behavior, nonlinear stress-strain relationships, and damage evolution. An algorithm for a numerical implementation of the model is proposed. The corresponding boundary value problem is solved by the hp-version of the least-squares collocation method in combination with an acceleration of an iterative process based on Krylov subspaces and parallelizing. Special attention is given to the influence of mathematical model parameters on the results of numerical simulation. The results are compared with experimental data and three-dimensional simulation. A satisfactory agreement is shownРазработана и исследована новая математическая модель четырехточечного изгиба железобетона с учетом разносопротивляемости бетона растяжению-сжатию, физической нелинейности и разрушения. Предложен алгоритм численной реализации модели. Соответствующая краевая задача решалась hp-вариантом метода коллокации и наименьших квадратов в комбинации с ускорением итерационного процесса, основанным на подпространствах Крылова, и распараллеливанием. Исследовано влияние параметров математической модели на результаты численного моделирования. Проведено сравнение результатов расчетов с экспериментальными данными и трехмерным моделированием и показано удовлетворительное согласие с ним