Vol 20, No 3 (2024)

The methodology and results of calculating the stress-strain state of metal frame structures when they are strengthened by attaching additional elements to the original ones. With such strengthening, additional assembly stresses emerge in the structure. This paper presents a mathematical model and a variational method for determining assembly displacements and stresses, where the equations for displacement of the system due to unit concentrated forces are not used in solving the problem. The proposed mathematical model and method can be used with equal success for solving linear and nonlinear problems. The mathematical model and the calculation method for analyzing the stress-strain state of a frame structure strengthened during operation allow to successively determine displacements and stresses in the structure from the effects of initial, assembly and additional operational loads. The basic hypotheses of the bar theory, taking shearing into account, and the Lagrange variational principle are applied. A distinguished feature of the calculation method is that in the process of solving the problem, constraints are imposed on the displacements of the original and strengthening structural elements and, taking into account these constraints, the assembly displacements and stresses due to the initial loads are calculated. This feature significantly simplifies the solution of the problem and allows to expand the range of questions under study, since it removes the limitations associated with the determination of assembly forces. The test problems have been solved. Comparison of the values of assembly displacements and stresses obtained in the test problems and determined by other methods demonstrate reliability and high accuracy of the calculations.

The effect of service life of a reinforced concrete building frame on its robustness parameters in the case of sudden failure of the outermost column has been investigated. The reinforced concrete frame of a philharmonic hall was chosen as the study subject. In order to evaluate its robustness, a relative robustness index, which is related to the parameters of the failure load for a system with and without initial local failure, has been utilized. Quasi-static modeling using the finite element method taking into account physical and geometric nonlinearity was performed as a part of the study. The physical nonlinearity of concrete, considering long-term operation of the structure, was accounted for by modified bilinear constitutive models of the material. Such models differed for elements with different stress-strain states in long-term operation. The parameters of the constitutive models were obtained using the integral deformation modulus proposed by Bondarenko. This approach has been employed to analyze the deformations and forces in the elements of the load-bearing system in the scenario of the outermost column failure. The curves for the percentage of destroyed elements of the load-bearing structure versus the parameters of the failure load have been plotted for the models with and without initial local failure of the outermost column, as well as for short-term and long-term operation. It is shown that the values of the failure load parameter and the relative robustness index decrease when the service life of the structure is accounted for.

At present, the maximum response spectrum is a basic concept in seismic engineering, providing a convenient means for representing the impact of earthquakes on structures. It also provides a practical approach to the application of structural dynamics principles to the design of structures and the development of requirements in building codes and regulations. Unfortunately, this concept is practically unknown to Russian designers and is not used in seismic resistance calculations. In Russian standards, the concept of dynamic coefficients is used in seismic resistance calculations, which has no physical meaning for earthquakes. The absence of the concept of response spectra in Russian regulatory documents on seismic resistance calculations of structures, in our opinion, is a serious mistake. The presentation of response spectra of maximum displacements, velocities and accelerations in logarithmic coordinates on a single graph made it possible to identify patterns in almost any seismic impacts, which has found wide application in regulatory documents of many countries.

Differential equilibrium equations of the momentless shell theory are very easily integrated in cases of cylindrical and right circular conical shells. Shells of zero Gaussian curvature defined in arbitrary curvilinear coordinates are more difficult to analyze, which was reaffirmed by the case of elliptical conical shells. For the first time, analytical expressions of normal and tangential internal forces in a momentless right elliptical conical shell defined in non-orthogonal conjugate system of curvilinear coordinates are obtained. The derived results can be used for approximation of the stress state of thin conical shells with elliptical base and also for the investigation of stability of these shells. Four internal tangential forces obtained by integration of the system of four equilibrium equations of a shell element contain two unknown integration functions, which are determined by satisfying given boundary conditions. The application of obtained analytical equations is demonstrated by an example of analysis of a truncated elliptical conical shell with free upper edge. A uniformly distributed surface load in the direction of the vertical axis of the shell was assumed as external load. The presented formulae are easily adapted for the analysis of a right circular conical shell.

A dynamic problem with negative flow of time is formulated. The conventional equations of motion with the addition of initial conditions are sufficient not only for the analysis of motion of a deformable system under the regular, forward flow of time, but they allow to restore the state of the system for past moments of time. The practical use of solving problems with negative time can be found primarily in testing numerical methods for integrating the equations of motion, since forward and backward algorithms are not identical. The proposed technique of testing numerical methods for solving dynamic problems can be applied virtually to any computational scheme of integrating the equations of motion. Two examples of numerical solution based on explicit computational scheme with Adams extrapolation are presented. The addressed problems deal with the plane deformation state of plates under large displacements. Plate regions are partitioned into triangular finite elements with uniform spacing for spatial meshing. The obtained curvilinear boundaries in this case are stepped. The results of the presented test cases demonstrated good accuracy of the tested method. Problems requiring a large number of integration steps (up to 1 million) were considered, and the system returned to the initial state with high accuracy. The second of the given numerical solutions had a computational scheme of 160 000 finite elements, and the dynamic solution of the problem has a pronounced wave-like behavior. In the examples, data on the recovery of elastic displacement, velocity and stress values are given. The main conclusion of the study is that the proposed technique of control of numerical methods can be effectively applied, especially for problems with wave-like solution properties.