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.