PSI - Issue 59

Svyatoslav Gomon et al. / Procedia Structural Integrity 59 (2024) 559–565 Svyatoslav Gomon et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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2. Results and discussion To propose a mathematical apparatus for the calculation of wooden bending elements and structures using the deformation method, the following hypotheses and prerequisites are necessary: a) the “calculated cross - section” concept for the wooden element is introduced. This term involves the equilibrium conditions of the cross-section of the element, assumptions about the distribution of deformations and stresses in this cross-section and the criterion for the exhaustion of the load-bearing capacity; b) the calculated cross-section is normal to the longitudinal axis of the element, and a fold is formed in the compressed zone; c) for small deformations along the height of the calculated cross-section, the hypothesis of linear deformation distribution is valid; d) the relationship between stresses and strains in tensile wood is assumed to be a linear dependence: ; (3) e) The relationship between stresses and strains in compressed wood is assumed to be described by a transformed diagram, sufficiently represented by a second-degree polynomial: . (4) The coefficients of the function (4) can be determined using the expressions: (6) f) wooden elements, where force factors must be applied in a way that does not induce torsion, are considered; g) characteristic values of wood resistance in the wooden element are considered as design values. The adopted “calculated cross - section” allows assumptions about the stress distribution and its maximum ordinate in the accepted cross-section with a fold in the compressed zone of the wooden element. It is essential to note that an elastoplastic material with limited deformability is considered. The decisive criterion that needs to be considered in terms of material resistance is the fold because it is the weakest area. In the pre-destruction state, it is often visually challenging to detect a cross-section with a fold in the wooden element due to the instantaneous destruction of the structure. For the calculated cross-section with a fold in the compressed zone of the element, known laws of deformation distribution along the height can be applied both in the compressed and in the tension zones. This is possible because the apparatus of infinitesimally small quantities can be used, assuming the continuity of the function in the vicinity of the considered cross-section point. The use of the continuity apparatus in the mechanics of a solid body is justified by the assumption of the material’s continuity and homogeneity. These conditions are practically fulfilled in the cross-section with a fold in the compressed zone of the wooden element. Therefore, the calculated cross section represents a model that reflects the regularities of deformation and destruction of the wooden element. In the analysis of wooden elements using the full physical state diagram, several characteristic stages of the stress-strain state can be identified (Fig. 3): - stage I (Fig. 3a): the element operates without folds in the compressed zone within the limits of proportionality under stresses ; ; (5) ;