Vol 22, No 3 (2026)

A mathematical model is proposed to describe nonlinear deformation of concrete under uniaxial and multiaxial loading. The material model takes into account the difference in the material's resistance to tension and compression. Accounting for orthotropy reflects the directional nature of concrete microcracking. The model accounts for the nonlinearity, multiaxiality, and anisotropy of concrete deformation using a locally orthotropic hyperelastic material model with orthotropic axes coinciding with the principal stress directions. The orthotropic material model takes into account all seven possible joint invariants of the strain tensor and anisotropy tensors. The case of a piecewise quadratic approximation of the elastic potential is studied in detail. Due to the existence of the potential, this model has improved convergence in the numerical solution of nonlinear boundary value problems. A comparison of the results of the proposed locally orthotropic hyperelastic deformation model with experimental data and calculation results using N.I. Karpenko’s orthotropic model demonstrated good prediction accuracy under uniaxial (difference from experiments less than 1%) and multiaxial (difference from experiments less than 15%) loading.

Coastal erosion in the Mekong Delta has intensified due to climate change and rising sea levels, necessitating robust protective structures. This study evaluates the stability of a coastal revetment in Duyen Hai, Vinh Long province, using both two-dimensional (2D) and three-dimensional (3D) numerical simulations. Vinh Long, following its 2025 administrative merger, faces significant pressure from the East Sea’s irregular tidal regime and monsoon-driven wave energy. The stability analysis was conducted using the Simplified Bishop Method within the Limit Equilibrium Method (LEM) framework via the GeoStudio software suite (SLOPE/W and SLOPE3D). Results indicate that the safety factors (FS) for all considered load combinations, basic and extreme, comply with the Vietnamese national standard TCVN 9901:2023. Notably, the 3D simulations yielded FS values approximately 11.2% higher than the 2D counterparts, with the 3D model providing a more realistic representation of soil-structure interaction by accounting for lateral resistance and interaction forces between cross-sections. Furthermore, under an extreme loading combination representing the rainy season, where upstream discharge artificially elevates the landside water level by 1.5 m, the safety factors decrease due to increased pore water pressure within the embankment. This research demonstrates that while 2D analysis remains a conservative and safe approach for design, 3D modeling offers critical insights for cost-effective design refinement and a more accurate understanding of complex failure mechanisms in soft-soil coastal foundations.

Reinforced timber structures, particularly continuous beams, offer a number of advantages that contribute to their widespread use. The economic benefits of rational reinforcement of the cross-section of multi-span reinforced timber beams in terms of material consumption, as well as the lack of similar known contemporary works, determine the relevance of the research topic. The object of the study is a three-span reinforced timber beam with equal spans, loaded with a uniformly distributed load. To achieve the goal of the study, the following tasks were solved: a review of existing methods for calculating flexural reinforced timber elements and the selection of a calculation methodology for the studied structure; an analysis of the distribution of bending moments along the beam length and the development of design solutions; and the selection of a rational option, with the relative consumption of reinforcement serving as the efficiency criterion. Design solutions with symmetrical and asymmetrical reinforcement with the placement of bars along the entire length of the beam and part of its length were considered. It was found that the difference in bending moments in the middle span and the design moments at the supports allows to have no reinforcement in a part of the middle span. Furthermore, it is possible to reduce the cross-section dimensions. The most efficient reinforcement option for a three-span beam is symmetrical reinforcement, with the reinforcement placed along part of the beam's length.

A numerical simulation of the three-dimensional “shell - soil” system was performed, comprising a T-connection of cylindrical shells and the surrounding soil body, using contact elements to describe the gap between the structure and the soil. The study was carried out for soil fragments of various widths in order to evaluate their influence on the magnitude and shape of the surface settlement trough. Eight models were considered, in which the distance L from the edges of the main and adjoining shells to the lateral faces of the soil body varied from 1D to 8D in steps of D , where D is the diameter of the main shell. The surrounding soil body was modeled with three-dimensional finite elements (SOLID186, SOLID187), the cylindrical shell - with shell elements (SHELL181); the soil behavior was described by the elastic-plastic Mohr-Coulomb model. Families of surface settlement trough curves were obtained in the transverse and longitudinal directions relative to the axis of the adjoining shell. It was established that for the soil body width from 1D to 5D both the settlement magnitudes and the shape of the settlement trough change noticeably, whereas with a further increase of the model domain ( L ≥ 5D ) the change in the shape and depth of the trough becomes insignificant. Based on the stress analysis of the system, it is recommended for engineering calculations to adopt a soil body width of L ≥ 5D , with L = 5D being preferable, providing sufficient accuracy of settlement prediction without an excessive increase in the number of finite elements. The obtained results help maintain a balance between computational speed and reliability of the mathematical modeling, which is especially important in multivariate analysis.

This study presents a comparative analysis of the stiffness characteristics of brick masonry based on experimental data and an orthotropic model. The main focus is on the influence of mortar joint strength on the anisotropy of the elastic properties of masonry. For three series of samples KRO-1, KRO-2, KRO-3 with different mortar strengths, experimental compression tests were conducted, along with numerical modeling in the Abaqus software, including micro-modeling and macro-modeling based on an orthotropic model. The results demonstrate that the ratio of the elastic moduli of brick and mortar  E_b/ E_m significantly affects the distribution of strains and stresses in the masonry. An asymmetry in the stiffness matrix D₁₂ ≠ D₂₁ was observed, indicating the need to account for micromechanical effects in the “brick - mortar” contact zones. The highest anisotropy was found in samples with low-strength mortar series KRO-3, where the anisotropy coefficient D₁₁/D₂₂ reached 1.253. The study confirms the validity of using an orthotropic model to describe the stiffness characteristics of masonry but highlights the necessity of its modification to account for structural heterogeneity and edge effects. The obtained results have practical significance for the design of masonry structures under complex stress conditions.