INVESTIGATION OF STRESS-STRAIN STATE OF TRANSVERSELY ISOTROPIC PLATES UNDER BENDING USING EQUATION OF STATICS {1,2} –APPROXIMATION

The study examined the construction of the fundamental solution for the equations of statics {1,2} – approximation for transversely isotropic plates under bending with the action of concentrated force. Equations {1,2} -approximation were obtained by the decomposition method in the thickness coordinate using the Legendre polynomials. These equations take into account all the components of the stress tensor, including the transverse shear and normal stresses. Since the classical theory of Kirchhoff-Love doesn’t take account of these stresses, the study on the basis of refined theories of stress-strain state of transversely isotropic plates under the action of concentrated force effects is an important scientific and technical problem. The fundamental solution of obtained equations results using a two-dimensional Fourier integral transform and inverse treatment techniques, built with the help of a special G-function. This method allows reducing the system of resolving differential equations for statics of flat plates and shells to a system of algebraic equations. After that, the inverse Fourier transform restores the fundamental solution. The work was carried out numerical studies that demonstrate patterns of behavior of components of the stress-strain state, depending on the elastic constants of transversely isotropic material. The results play a decisive role in the study of boundary value problems in the mechanics of thin-walled elements of constructions, including under the influence of concentrated and local diverse forces

Аналіз фундаментальних розв’язків рівнянь статики, побудованих для трансверсально-ізотропних пластин

In present research, we examined and analyzed the fundamental solutions of the equations of statics for the transversal-isotropic plates, which were built using generalized theory of the {m,n}-approximation. The methods for reducing the three-dimensional problems of the theory of elasticity to the two-dimensional ones are explored. In the study, we analyzed results, obtained on the basis of theory of the {m,n}-approximation, for the purpose of determining the refinement, which is introduced by the retention of a large number of terms in the expansion series of the desired functions. This theory is the most preferable for reducing the three-dimensional equations of the theory of elasticity to the two-dimensional ones since it is not based on any hypotheses, but employs the method of I. N. Vekua for the expansion of the desired functions into the Fourier series by the Legendre polynomials. This approach makes it possible to examine not only the thin plates, but also the plates of medium and large thickness, and allows us to consider transverse shearing and normal stresses. Since the classical theory of Kirchhoff–Love does not take these stresses into account, then examining on the basis of the refined theories of the stressed-strained state of transversal-isotropic plates under the action of concentrated force impacts is a relevant scientific and technical task. We carried out numerical studies that make it possible to determine the refinement, which is introduced by the retention of a large number of terms in the expansion series of the desired functions and to analyze the character of behavior of internal force factors of the zero spin stressed state and the state of bending, obtained with the use of generalized theory of the {1,0}- and {1,2}-approximation.The obtained results play a decisive role when exploring different boundary problems of the mechanics of thin-walled elements of structures, including those exposed to the concentrated and local diverse actions.Рассмотрены задачи статики трансверсально-изотропных пластин, которые находятся под действием сосредоточенной силы. Проведен анализ фундаментальных решений, полученных с использованием различных приближенных теорий. С помощью рассмотренных теорий трехмерная задача теории упругости сведена к двумерной. Рассмотрены безмоментное напряженное состояние и состояние изгиба для трансверсально-изотропных пластин. Проанализированы фундаментальные решения уравнений статики трансверсально-изотропных пластин, построенные на базе обобщённой теории {m,n}-аппроксимации для разных приближенийРозглянуто задачі статики трансверсально-ізотропних пластин, які знаходяться за дії зосередженої сили. Проведено аналіз фундаментальних розв’язків, отриманих із використанням різних наближених теорій. За допомогою розглянутих теорій тривимірна задача теорії пружності зведена до двовимірної. Розглянуто безмоментний пружний стан і стан вигину для трансверсально-ізотропних пластин. Проаналізовано фундаментальні розв’язки рівнянь статики трансверсально-ізотропних пластин, що побудовані на базі узагальненої теорії {т,п}-аппроксімаціі для різних наближен

Removal of heavy metals using food industry waste as a cheap adsorbent

Due to the high capacity of food factories, a lot of waste is generated. Due to the presence of nutrients in them can increase pollution in the sewerage network and cause environmental problems. To animal feed, grain waste can be used to produce fertilizer, compost, fuel, soil cover, etc. The natural pigment lycopene, thickeners, proteins, etc., can be produced from the seeds and skin of the tomato paste factory waste. In the present paper, soybean oil residue was used to remove cadmium, zinc, and lead ions from aqueous solutions. Discontinuous experiments to investigate the effect of initial concentration (100-300 ppm), solution pH (1-5), contact time (1-60 min), adsorbent amount (0.02-2 g) on the uptake of cadmium, zinc, and lead ions by soybean oil residues. The results showed that with increasing the pH, the value of the contact time, the amount of adsorbent, and the adsorption rate increases with increasing the initial concentration of metal ions, the adsorption rate decreases. According to the results, the best pH for adsorption of metal ions is about 3-5, and the equilibrium time for cadmium ions is 40 minutes, for zinc ions is 20 minutes, and for lead, the ion is 10 minutes