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1

Analytical solution for one-dimensional consolidation of unsaturated soils under dynamic load

Liu Zhi-Yi, Song Yu, Zhou Feng-Xi, Wang Li-Ye

Journal of Theoretical and Applied Mechanics

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2023

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Vol. 61 nr 1

11--22

EN

Firstly, in this paper, based on the theory of the porous elastic medium and combined with the effective stress principle of unsaturated soil, a set of governing equations is established to describe consolidation of the unsaturated soil. Secondly, an analytical expression under any dynamic loads is obtained with the help of Laplace integral transformation. Finally, analysis of numerical examples under specific boundary conditions is made to discuss one-dimensional consolidation characteristics under harmonic loads and the influence of factors on the consolidation characteristics of unsaturated soil, such as excitation frequency and initial saturation.

2

Influence of Nonlinear Shear Modulus Change of Elastomeric Shell of a Composite Tractive Element with a Damaged Structure on its Stress State

Belmas Ivan, Kolosov Dmytro, Onyshchenko Serhii, Bilous Olena, Tantsura Hanna

Inżynieria Mineralna

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2023

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no. 1

147--154

EN

Purpose. Determination of a dependency of a stress state for composite elastomer-cable tractive element with a broken structure on a nonlinear dependency of shear modulus on deformations in the elastomeric shell. Methods. Analytical solution of a model of a composite tractive element with disturbed structure and a deformation-dependent shear modulus of an elastomeric shell. Findings. Algorithm for determining a stress state of a composite tractive element with broken structure and a deformation-dependent shear modulus. Scientific novelty. Character of dependency for a stress state of a composite tractive element on a nonlinear dependency of shear modulus on deformations. Practical significance. A possibility to determine the dependency of a stress state of a composite elastomer-cable tractive element on a nonlinear shear modulus. Scientific novelty. Character of dependency for a stress state of a composite tractive element on a nonlinear dependency of shear modulus on deformations. Practical significance. A possibility to determine the dependency of a stress state of a composite elastomer-cable tractive element on a nonlinear shear modulus allows considering the effect of this phenomenon on the tractive element strength and ensures an increase of its operational safety.

PL

Artykuł przedstawia studium i analizę funkcjonalną wymagań światowego przemysłu metalurgicznego co do jakości rud żelaza w podziemnych kopalniach Ukrainy. Stwierdzono zależności wpływu kształtu i parametrów przestrzeni kompensacyjnych na ich stateczność i wskaźniki jakości rudy. Udowodniono, że komora wyrównawcza w kształcie trapezu pionowego charakteryzuje się największą stabilnością i jest stabilna w zakresie wszystkich rozważanych głębokości, nawet w rudach o twardości 3–5 punktów. Mniejszą stateczność wykazuje komora kompensacji pionowej o kształcie sklepionym z niewielkimi spadkami w przyczółku sklepienia komory w rudach o twardości 3–5 punktów na głębokości 2000 m. Komora z opadami o rożnym natężeniu występuje w dolnej części nachylonych odsłonięć namiotu w rudach o twardości 3–5 punktów na głębokości 1750 m lub większej. Pomieszczenie kompensacji poziomej ma najmniejszą stateczność; spadki występują w rudach o twardości 3–5 punktów na głębokości 1400 m, a na głębokościach 1750–2000 m pozostają stabilne tylko w rudach twardszych. Stwierdzono, że zastosowanie komór kompensacyjnych o dużej stabilności umożliwia osiągnięcie ich maksymalnej objętości, zwiększenie ilości wydobywanej czystej rudy, zmniejszenie jej rozrzedzenia, poprawę jakości wydobywanej masy rudy, a co za tym idzie, wzrost jej ceny i konkurencyjności rynkowej.

3

Free vibration analysis of composite face sandwich plate strengthens by Al2O3 and SiO2 nanoparticles materials

Al-Shablle Marwan, Njim Emad Kadum, Jweeg Mushin J., Al-Waily Muhannad

Diagnostyka

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2023

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Vol. 24, No. 2

art. no. 2023205

EN

The main novelty of the paper is that analytical, experimental, and numerical analyses are used to investigate the free vibration problem of a sandwich structure in which Nanocomposites skins (SiO2/epoxy and Al2O3/epoxy) at different densities are used as the face sheet. The volume fraction's of nanoparticle addition varies (0% to 2.5%). The present free vibration was derived based on Kirchhoff's theory and aspiration to obtain the natural frequency. The results show that in structures with SiO2 nanoparticles with a density of 1180 kg⁄m3, the optimum increase (VF = 2.5%) is 50% in Young's modulus and 22% in natural frequency, while at a density of 1210 kg⁄m3 is 56 % in Young's modulus and 24.5% in natural frequency. Furthermore, the same structures reinforced with Al2O3 Nano-particles show that at the density of 1180 kg⁄m3 , the optimum (VF=2.5%) parentage increase in Young's modulus is 41% and 19% in natural frequency, while at the density of 1210 kg⁄m3 is 46% in Young's modulus and 21% in natural frequency. A numerical investigation was used to validate the obtained results of the analytical solution. The findings also show an acceptable convergence between analytical and numerical techniques with a maximum discrepancy not exceeding 3%.

4

Refined energy method for the elastic flexural-torsional buckling of steel H-section beam-columns. Part I: Formulation and solution

Giżejowski Marian, Barszcz Anna, Wiedro Paweł

Archives of Civil Engineering

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2023

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Vol. 69, nr 1

513--537

EN

Closed form solutions for the flexural-torsional buckling of elastic beam-columns may only be obtained for simple end boundary conditions, and the case of uniform bending and compression. Moment gradient cases need approximate analytical or numerical methods to be used. Investigations presented in this paper deal with the analytical energy method applied for any asymmetric transverse loading case that produces a moment gradient. Part I of this paper is devoted entirely to the theoretical investigations into the energy based out-of-plane stability formulation and its general solution. For the convenience of calculations, the load and the resulting moment diagram are presented as a superposition of two components, namely the symmetric and antisymmetric ones. The basic form of a non-classical energy equation is developed. It appears to be a function dependent upon the products of the prebuckling displacements (know from the prebuckling analysis) and the postbuckling deformation state components (unknowns enabling the formulation of the stability eigenproblem according to the linear buckling analysis). Firstly, the buckling state solution is sought by presenting the basic form of the non-classical energy equation in several variants being dependent upon the approximation of the major axis stress resultant M𝑦 and the buckling minor axis stress resultant Mz. The following are considered: the classical energy equation leading to the linear eigenproblem analysis (LEA), its variant leading to the quadratic eigenproblem analysis (QEA) and the other non-classical energy equation forms leading to nonlinear eigenproblem analyses (NEA). The novel forms are those for which the stability equation becomes dependent only upon the twist rotation and its derivatives. Such a refinement is allowed for by using the second order out-of-plane bending differential equation through which the minor axis curvature shape is directly related to the twist rotation shape. Secondly, the effect of coupling of the in-plane and out-of-plane buckling forms is taken into consideration by introducing approximate second order bending relationships. The accuracy of the classical energy method of solving FTB problems is expected to be improved for both H- and I-section beam-columns. The outcomes of research presented in this part are utilized in Part II.

PL

Rozwiązania w postaci zamkniętej dla wyboczenia giętno-skrętnego (FTB) sprężystych belek-słupów można uzyskać tylko dla prostych warunków brzegowych oraz przypadku równomiernego zginania i ściskania. Przypadki zmiennego momentu zginającego wymagają zastosowania przybliżonych metod analitycznych lub numerycznych. Badania przedstawione w niniejszym artykule dotyczą analitycznej metody energetycznej, stosowanej dla dowolnych przypadków asymetrycznych obciążeń poprzecznych, wywołujących nierównomierny moment zginający. Cześć I prezentowanego artykułu jest w całości poświęcona badaniom teoretycznym nad energetyczną formułą utraty stateczności z płaszczyzny zginania i jej ogólnemu rozwiązaniu. Dla wygody obliczeń obciążenie i wykres momentów zginających przedstawiono jako superpozycje dwóch składowych: symetrycznej i antysymetrycznej. Opracowano podstawową postać nieklasycznego (udoskonalonego) równania energetycznego. Jest ono funkcjonałem zależnym od iloczynów odkształceń stanu przedwyboczeniowego, przemieszczeń osi pręta i ich pochodnych, odpowiednio - 𝑢0 i 𝑤0, oraz składowych stanu odkształcenia pokrytycznego, przemieszczenia z płaszczyzny zginania przedkrytycznego i kąta skręcenia, odpowiednio - 𝑣0 i 𝜙𝑥. Przemieszczenia przedwyboczeniowe 𝑢0 osi pręta i 𝑤0 w płaszczyźnie zginania są znane i mogą być powiązane z siłą osiową 𝑁 i momentem zginającym względem osi głównej 𝑀𝑦 otrzymanymi z analizy pierwszego rzędu (LA). Składowe stanu deformacji 𝑣0 i 𝜙𝑥 z płaszczyzny płaskiego stanu zginania oraz ich pochodne są niewiadomymi umożliwiającymi sformułowanie problemu stateczności jako problemu wartości własnych (LBA). W artykule, po pierwsze, poszukiwane jest rozwiązanie stanu wyboczenia poprzez przedstawienie podstawowej postaci nieklasycznego równania energetycznego w kilku wariantach, zależnych od aproksymacji momentu 𝑀𝑧 , a mianowicie klasycznego, prowadzącego do analizy liniowego problemu własnego (LEA) i kwadratowego problemu własnego (QEA) oraz innych form prowadzących do nieliniowych analiz problemów własnych (NEA). Nowe formy to te, dla których równanie stateczności zależy tylko od kąta skręcenia i jego pochodnych. Takie udoskonalenie jest możliwe, gdy do zginania z płaszczyzny zastosowane zostanie równanie różniczkowe drugiego rzędu, za pomocą którego krzywizna osi słabszej jest bezpośrednio powiązana z kątem skręcenia. Po drugie, uwzględniono efekt sprzężenia form wyboczenia w płaszczyźnie i z płaszczyzny zginania przedwyboczeniowego przez wprowadzenie przybliżonych zależności zginania drugiego rzędu. Dzięki uwzględnieniu tych efektów znacznie poprawiono dokładność klasycznej metody energetycznej rozwiazywania problemów FTB elementów ściskanych i zginanych w płaszczyźnie większej bezwładności przekroju, zarówno w wypadku przekroju dwuteowego H, jak i I. Wyniki tej części są wykorzystywane w Części II, dotyczącej porównania i weryfikacji rozwiązań uzyskanych w formie zamkniętej w Części I artykułu.

5

A new and simple model for predicting soil erosion based on hole erosion tests

Cai Weiling, Bora Manash Jyoti, Pekkat Sreeja, Bordoloi Sanandam, Garg Ankit, Sekharan Sreedeep

Acta Geophysica

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2023

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Vol. 71, no. 2

823--836

EN

Determination of erosion characteristics is of great significance to assess the erodibility of geomaterials that are subjected to seepage force. The erosion characteristics indicate soil particle removal in term of internal erosion that might occur in earthen structures. Hole erosion test (HET) is a simple and effective approach to determine erosion characteristics. It is noted that there are not many studies that focus on the development of a theoretical model describing the erosion characteristics and the associated process of soil particle detachment in HETs. The aim of this study is to propose a simple model based on Bernoulli’s principle to interpret erosion characteristics of geomaterials in HETs. An analytical equation was deduced from a physically based model incorporating Bernoulli’s principle and erosion constitutive law for internal erosion within a soil pipe driven by pressure gradient. The analytical equation could be applied to determine soil particle removal, radial erosion propagation, erosion coefficient, and critical shear stress. A series of HETs were performed under different flow rate to verify the proposed model. The obtained results demonstrated that the proposed model allowed for reasonably predicting the amount of soil particle removal and understanding erosion characteristics of soils through the HET.

6

Analysis, evaluation, and optimization of bio-medical thermo-resistive micro-calorimetric flow sensor using an analytical approach

Babaelahi Mojtaba, Sadri Somayyeh

Metrology and Measurement Systems

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2022

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Vol. 29, nr 1

109--125

EN

Sensitive MEMS-based thermal flow sensors are the best choice for monitoring the patient’s respiration prompt diagnosis of breath disturbances. In this paper, open space micro-calorimetric flow sensors are investigated as precise monitoring tools. The differential energy balance equation, including convection and conduction terms, is derived for thermal analysis of the considered sensor. The temperature-dependent thermal conductivity of the thin silicon-oxide membrane layer is considered in the energy balance equation. The derived thermal non-linear differential equation is solved using a well-known analytical method, and a finite-element numerical solution is used for the confirmation. Results show that the presented analytical model offers a precise tool for evaluating these sensors. The effects of flow and thin membrane film parameters on thermo-resistive micro-calorimetric flow sensors’ performance and sensitivity are evaluated. The optimization has been performed at different flow velocities using a genetic algorithm method to determine the optimum configuration of the considered flow sensor. The geometrical parameters are selected as a decision variable in the optimization procedure. In the final step, using optimization results and curve-fitting, the expressions for the optimum decision variables have been derived. The sensor’s optimum configuration is achieved analytically based on flow velocity with the analytical terms for optimum decision variables.

7

Analytical evaluation of the influence of adding rubber layers on free vibration of sandwich structure with presence of nano-reinforced composite skins

Al-Shablle M., Al-Waily M., Njim E.K.

Archives of Materials Science and Engineering

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2022

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Vol. 116, nr 2

57--70

EN

Purpose: Developing structural designs that offer superior vibration properties is still a major challenge, but they stay solid and lightweight simultaneously. Composite faces are frequently used in insulating constructions as an alternative to sheet metal roofs. Rubber overlays have been added to reduce waves' natural frequency and fade time. Design/methodology/approach: The mechanical properties and the natural frequency calculation of the materials that make up the composite structural panels designed for structural applications with the addition of rubber layers were studied in this study. Findings: The results showed the addition of rubber layers with SiO2 nanoparticles with a density of 1180 kg m3, and the optimal decrease (VF = 2.5%) is 38.5% in the natural frequency while at a density of 1210 kg/m3, it is 40.2% in the natural frequency. While the addition of rubber layers with Al2O3 nanoparticles shows a density of 1180 kg/m3, the optimum reduction (VF = 2.5%) is 41% in HF while at a density of 1210 kg/m3 36.8% in an NF 41% during a density of 1210 kg/m3 38.4%. Research limitations/implications: Certain hypotheses were used to apply Kirchhoff's theory to solve the mathematical model of the structure. Practical implications: The work was carried out on the faces of nanocomposites made of SiO2/epoxy and Al2O3/epoxy with different densities and polylactic acid core. The inclusion of nanoparticles as a percentage of the fraction size ranges from 0% to 2.50%. Originality/value: This study's results shed light on the fundamental behaviour of the components that make up the sandwich in the presence of rubber layers.

8

Analytical solution of the electro-mechanical flexural coupling between piezoelectric actuators and flexible-spring boundary structure in smart composite plates

Gohari Soheil, Mozafari F., Moslemi N., Mouloodi Saeed, Sharifi S., Rahmanpanah Hadi, Burvill Colin

Archives of Civil and Mechanical Engineering

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2021

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Vol. 21, no. 1

546--570

EN

An analytical solution has been developed developed in this research for electro-mechanical flexural response of smart laminated piezoelectric composite rectangular plates encompassing flexible-spring boundary conditions at two opposite edges. Flexible-spring boundary structure is introduced to the system by inclusion of rotational springs of adjustable stiffness which can vary depending on changes in the rotational fixity factor of the springs. To add to the case study complexity, the two other edges are kept free. Three advantages of employing the proposed analytical method include: (1) the electro-mechanical flexural coupling between the piezoelectric actuators and the plate’s rotational springs of adjustable stiffness is addressed; (2) there is no need for trial deformation and characteristic function—therefore, it has higher accuracy than conventional semi-inverse methods; (3) there is no restriction imposed to the position, type, and number of applied loads. The Linear Theory of Piezoelectricity and Classical Plate Theory are adopted to derive the exact elasticity equation. The higher-order Fourier integral and higher-order unit step function differential equations are combined to derive the analytical equations. The analytical results are validated against those obtained from Abaqus Finite Element (FE) package. The results comparison showed good agreement. The proposed smart plates can potentially be applied to real-life structural systems such as smart floors and bridges and the proposed analytical solution can be used to analyze the flexural deformation response.

9

Analysis of the transient flow of non Newtonian power law fluids in homogeneous reservoirs with the elastic outer boundary

Zhao Chaochao, Min Chao

Acta Geophysica

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2021

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Vol. 69, no. 5

1865--1875

EN

Since some oil fields have entered the late stage of high water-cut development, the use of polymer flooding and other oil recovery technologies makes the fluid be of non-Newtonian characteristics, but the existing elastic outer boundary percolation model ignores this point. Firstly, this paper establishes a non-Newtonian power-law fluid percolation model with the elastic outer boundary condition. Secondly, the analytical solution of the percolation model is obtained in the Laplace space by the structural method. Thirdly, the double logarithmic characteristic curves are drawn. Finally, the sensitivity analysis of the parameters is carried out. The results show that the wellbore storage coefficient and skin factor play important roles in the early stage of the double logarithmic curves. For the radial flow stage and the later flow stage, the main influence factors are the power-law index, the elastic coefficient and outer boundary radius, respectively. It is concluded that the percolation model established in this paper provides a theoretical basis for reasonably solving the problem that the previous data deviation could not explain the reservoir parameters well. Meanwhile, the application of the structural method can provide a scientific calculation method for the well test analysis of non-Newtonian power-law fluids.

10

The fourth-order ordinary differential equation with the fractional initial/boundary conditions

Siedlecki Jarosław

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

79--87

EN

The initial/boundary value problem for the fourth-order homogeneous ordinary differential equation with constant coefficients is considered. In this paper, the particular solutions an ordinary differential equation with respect to the set of boundary conditions are studied. At least one of the boundary conditions is described by a fractional derivative. Finally, a few illustrative examples of particular solutions to the considered problem are shown.

11

A new mathematical model to calculate the equilibrium scour depth around a pier

Gazi Ainal Hoque, Afzal Mohammad Saud

Acta Geophysica

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2020

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Vol. 68, no. 1

181--187

EN

This paper sheds light on the formulation of a new equilibrium local scour depth equation around a pier. The total bed materials removed from the scour hole due to the force exerted by the fowing fuid after colliding with the pier in the fow feld are estimated. At the equilibrium condition, the shape of the scour hole around the pier may take any form, viz. linear, circular, parabolic, triangular, or combination of diferent shapes. To consider that, two functions are assumed at the stoss and the lee sides of the pier. The total volume of bed materials removed from the scour hole of an arbitrary shape at the stoss and the lee sides of the pier is obtained by integrating the two functions. The equilibrium scour depth is formed by applying the energy balance theorem. An example problem is illustrated and the results are compared with the equations presented by Melville and Coleman (Bridge scour. Water Resources Publication, Colorado, 2000) and HEC-18 (Richardson and Davis in Evaluating scour at bridges, HEC-18. Technical report no. FHWA NHI, 2001).

12

Differential equation of y = f(y') and interpretation of the solution in Mathematica program

Czajkowski Andrzej Antoni, Oleszak Wojciech Kazimierz, Frączak Piotr Stanisław

Problemy Nauk Stosowanych

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2019

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T. 10

25--34

EN

Introduction and aims: The paper presents a method of solving y=f(y') equations. The main aim of the work is to show how to solve this type of differential equations. In addition, the purpose of the discussion is to present the appropriate algorithms in Mathematica program, which are used to present the geometric interpretation of the obtained solutions. Material and methods: The sources contain material on the subject of differential equations. The method of mathematical analysis has been used. Results: In the analysis of selected examples, the method of substitution of new variable t has been used and the solution of the studied differential equation has been obtained in the form of the system of equations x=x(t) and y=y(t). Conclusion The solution of the differential equation of the type y=f(y') in the form of a system of equations x=x(t) and y=y(t) can be interpreted graphically using an appropriately used algorithm in Mathematica numerical program.

PL

Wstęp i cele: W pracy przedstawiono metodę rozwiązywania równań typu y=f(y'). Głównym celem pracy jest pokazanie sposobu rozwiązywania tego typu równań różniczkowych. Ponadto celem rozważań jest przestawienie odpowiednich algorytmów w programie Mathematica, które służą do przedstawienia interpretacji geometrycznej otrzymanych rozwiązań. Materiały i metody: Źródła zawierają materiał dotyczący tematyki równań różniczkowych. Zastosowano metodę analizy matematycznej. Wyniki: W analizie wybranych przykładów zastosowano metodę podstawienia nowej zmiennej t i otrzymano rozwiązanie badanego równania różniczkowego w postaci układu równań x=x(t) i y=y(t). Wniosek: Rozwiązanie równania różniczkowego typu y=f(y') w postaci układu równań x=x(t) i y=y(t) można zinterpretować graficznie stosując odpowiednio zastosowany algorytm w programie numerycznym Mathematica.

13

Differential equation of x = f(y') and interpretation of the solution in Mathematica program

Czajkowski Andrzej Antoni, Oleszak Wojciech Kazimierz, Frączak Piotr Stanisław

Problemy Nauk Stosowanych

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2019

|

T. 10

15--24

EN

Introduction and aims: The paper presents a method of solving x=f(y') equations. The main aim of the work is to show how to solve this type of differential equations. In addition, the purpose of the discussion is to present the appropriate algorithms in Mathematica program, which are used to present the geometric interpretation of the obtained solutions. Material and methods: The sources contain material on the subject of differential equations. The method of mathematical analysis has been used. Results: In the analysis of selected examples, the method of substitution of new variable t has been used and the solution of the studied differential equation has been obtained in the form of the system of equations x=x(t) and y=y(t). Conclusion: The solution of the differential equation of the type x=f(y') in the form of a system of equations x=x(t) and y=y(t) can be interpreted graphically using an appropriately used algorithm in Mathematica numerical program.

PL

Wstęp i cele: W pracy przedstawiono metodę rozwiązywania równań typu y=f(y'). Głównym celem pracy jest pokazanie sposobu rozwiązywania tego typu równań różniczkowych. Ponadto celem rozważań jest przestawienie odpowiednich algorytmów w programie Mathematica, które służą do przedstawienia interpretacji geometrycznej otrzymanych rozwiązań. Materiały i metody: Źródła zawierają materiał dotyczący tematyki równań różniczkowych. Zastosowano metodę analizy matematycznej. Wyniki: W analizie wybranych przykładów zastosowano metodę podstawienia nowej zmiennej t i otrzymano rozwiązanie badanego równania różniczkowego w postaci układu równań x=x(t) i y=y(t). Wniosek: Rozwiązanie równania różniczkowego typu y=f(y') w postaci układu równań x=x(t) i y=y(t) można zinterpretować graficznie stosując odpowiednio zastosowany algorytm w programie numerycznym Mathematica.

14

Analytical design of selected geotechnical solutions which protect civil structures from the effects of underground mining

Misa Rafał, Sroka Anton, Tajduś Krzysztof, Dudek Mateusz

Journal of Sustainable Mining

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2019

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Vol. 18, iss. 1

1--7

EN

This paper presents the authors' computational methods based on Knothe's theory. The methods enable the estimation of the reduction coefficient for effects which originate from mining operations performed via the application of a longitudinal structure which is sunk in to the ground. It could be, for example, a partition, which as a structural gap fulfils the function of an expansion grout, or via breaking the subsoil continuity (e.g. because of creating a peat-filled ditch or using a natural gap). Demonstrative calculations have been carried out in a few cases, i.a. to protect a structure situated in the vicinity of a planned tunnel. Additionally, some examples of the discontinuity zone which impact the obtained deformation values have been presented. The calculation method has been tested in case studies. The results of the calculations clearly show the positive influence of the applied geotechnical solutions on the minimisation of mining damage.

15

Screw dislocations in piezoelectric laminates with four or more phases

Wang X., Chen L., Schiavone P.

Archives of Mechanics

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2019

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Vol. 71, nr 3

239--262

EN

We present an analytical solution to the problem of a screw dislocation in a four-phase piezoelectric laminate composed of two piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media. A new version of the complex variable formulation is proposed such that the 2 × 2 real symmetric matrix appearing in the formulation becomes dimensionless. Using analytic continuation, the original boundary value problem is reduced to the identification of a single 2D analytic vector function which is completely determined following rigorous solution of the resulting linear recurrence relations in matrix form. An explicit expression for the image force acting on the piezoelectric screw dislocation is obtained once the single 2 × 2 real matrix function is identified. We also discuss the solution for a screw dislocation in an N-phase piezoelectric laminate composed of N − 2 piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media.

16

Analytical and numerical solving of linear non-homogeneous differential equations of the first-order with changeable coefficients by using constant variation method and application of Mathematica program

Czajkowski A. A., Oleszak W. K., Dyrdał J.

Problemy Nauk Stosowanych

|

2018

|

T. 8

5--20

EN

Introduction and aim: The paper presents the analytical and numerical algorithm of solving linear nonhomogeneous equations of the first order with changeable coefficients. The aim of the work is to show the algorithms for solving equations both analytically and numerically. The additional aim is to show numerical algorithms and graphical interpretation of solutions. Material and methods: Some selected equations have been chosen from the subject literature. In the solutions the constant variation method has been presented. Results: The paper presents the selected linear non-homogeneous equations of the first order with changeable coefficients containing exponential, logarithmic, trigonometric and cyclometric functions. Conclusion: Taking into account the constant variation method it is possible to solve the first order linear nonhomogeneous differential equations with changeable coefficients. Using the Mathematica program it is possible quickly get a solution and create its graphical interpretation.

PL

Wstęp i cel: W pracy pokazano algorytmy analityczny i numeryczny rozwiązywania równań różniczkowych liniowych niejednorodnych pierwszego rzędu o zmiennych współczynnikach. Celem pracy jest pokazanie algorytmu rozwiązywania równań zarówno sposobem analitycznym jak i numerycznym. Ponadto również dodatkowym celem jest pokazanie algorytmów numerycznych oraz interpretacji graficznej rozwiązań. Materiał i metody: Wybrane równania zaczerpnięto z literatury przedmiotu. W rozwiązaniach równań zastosowano metodę wariacji stałej. Wyniki: W pracy opracowano wybrane równania różniczkowe liniowe niejednorodne pierwszego rzędu o zmiennych współczynnikach zawierających funkcje wykładnicze, logarytmiczne, trygonometryczne i arcus. Wniosek: Stosując metodę uzmienniania stałej jest możliwe rozwiązywanie równań różniczkowych liniowych niejednorodnych pierwszego rzędu o zmiennych współczynnikach. Wykorzystując program Mathematica można szybko uzyskać rozwiązanie oraz sporządzić jego interpretację graficzną.

17

Analytical and numerical solving of linear non-homogeneous differential equations of the second-order with changeable coefficients by using constant variation method and application of Mathematica program

Czajkowski A. A., Oleszak W. K., Skorny G. P., Udała R.

Problemy Nauk Stosowanych

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2018

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T. 8

21--38

EN

Introduction and aim: The paper presents the analytical and numerical algorithm of solving linear nonhomogeneous equations of the second order with changeable coefficients. The aim of the work is to show the algorithms for solving equations both analytically and numerically. The additional aim is to make some graphical interpretation of solutions. Material and methods: Some selected equations have been chosen from the subject literature. In the solutions the constant variation method has been presented. Results: The paper presents the selected linear non-homogeneous equations of the second order with constant coefficients containing linear, homographic, logarithmic and trigonometric functions. Conclusion: Taking into account the constant variation method it is possible to solve the second order linear non-homogeneous differential equations with changeable coefficients. Using the Mathematica program it is possible quickly get a solution and create its graphical interpretation.

PL

Wstęp i cel: W pracy pokazano algorytm analityczny i numeryczny rozwiązywania równań różniczkowych liniowych niejednorodnych drugiego rzędu o zmiennych współczynnikach. Celem pracy jest pokazanie algorytmu rozwiązywania równań zarówno sposobem analitycznym jak i numerycznym. Ponadto dodatkowym celem jest interpretacji graficznej rozwiązań. Materiał i metody: Wybrane równania zaczerpnięto z literatury przedmiotu. W rozwiażanich równań zastosowano metodę wariacji stałej. Wyniki: W pracy opracowano wybrane równania różniczkowe liniowe niejednorodne drugiego rzędu o zmiennych współczynnikach zawierających funkcje liniowe, homograficzne, logarytmiczne i trygonometryczne. Wniosek: Stosując metodę uzmienniania stałej jest możliwe rozwiązywanie równań różniczkowych liniowych niejednorodnych drugiego rzędu o zmiennych współczynnikach. Wykorzystując program Mathematica można szybko uzyskać rozwiązanie oraz sporządzić jego interpretację graficzną.

18

Analysis of delamination in two-dimensional functionally graded multilayered beam with non-linear behaviour of material

Rizov V. I.

Engineering Transactions

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2018

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Vol. 66, iss. 1

61--78

EN

An analytical study of delamination fracture in a two-dimensional functionally graded multilayered beam exhibiting material non-linearity is carried-out. The beam is made of adhesively bonded horizontal layers. The material is two-dimensional functionally graded in the cross-section of each layer. A delamination crack is located symmetrically with respect to the beam mid-span. The delamination is studied in terms of the strain energy release rate. The solution derived is compared with the J-integral for verification. The effects of material gradients, the crack location along the beam height and the material non-linearity on the delamination fracture are investigated. The distribution of the J-integral value along the crack front is analysed too.

19

Układ dwóch nieskończenie długich belek z wypełnieniem sprężystym pod obciążeniem ruchomym

Ataman M., Szcześniak W.

Autobusy : technika, eksploatacja, systemy transportowe

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2018

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R. 19, nr 12

285--288

PL

W artykule omówiono analityczne rozwiązanie zadania z dynamiki układu dwóch nieskończenie długich belek z wypełnieniem sprężystym. Układ spoczywa na podłożu Winklera i jest obciążony ruchomą siłą skupioną. Ponieważ zagadnienie jest stacjonarne dla obserwatora poruszającego się z obciążeniem, cząstkowe równania różniczkowe opisujące drgania układu transformowano na równania różniczkowe zwyczajne w układzie współrzędnych związanym z ruchomą siłą. Układ równań doprowadzono do jednego równania różniczkowego, zawyżonego rzędu, na ugięcie dolnej belki. Rozwiązanie zadania doprowadzono do prostej całki nieskończonej Fouriera. W pracy przedstawiono obszerny wykaz publikacji z literatury przedmiotu [1-45].

EN

The paper discussed the analytical solution of a dynamic problem of a system of two infinite beams separated by an elastic core. The beams’ system rests on the Winkler foundation and is loaded with a moving concentrated force. Because the problem is stationary for an observer moving with the load, partial differential equations, describing the vibrations of the system, were transformed into ordinary differential equations in the coordinate system related to the moving force. The system of equations was transformed to one differential equation of an eighth order. The equation defines deflection of the lower beam. The solution of the problem was resulted to the simple infinite Fourier integral. An extensive list of publications on the related literature is presented in the paper [1-45].

20

Free in-plane and out-of-plane vibrations of rotating thin ring based on the toroidal shell theory

Senjanović I., Ćatipović I., Alujević N., Čakmak D., Vladimir N.

Archives of Mechanics

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2018

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Vol. 70, nr 5

429--455

EN

In this paper rigorous formulae for natural frequencies of in-plane and out-of-plane free vibrations of a rotating ring are derived. An in-plane vibration mode of the ring is characterised by coupled flexural and extensional deformations, whereas an out-of-plane mode is distinguished by coupled flexural and torsional deformations. The expressions for natural frequencies are derived from a generalised toroidal shell theory. For the in-plane vibrations, the ring is considered to be a short top segment of a toroidal shell. For the out-of-plane vibrations, the ring is considered to be a side segment of the shell. Natural vibrations are analysed by the energy approach. The expressions for the ring strain and kinetic energies are deduced from the corresponding expressions for the torus. It is shown that the ring rotation causes bifurcation of natural frequencies of the in-plane vibrations only. Bifurcation of natural frequencies of the out-of-plane vibrations does not occur. Otherwise, for non-rotating rings, the derived formulae for the natural frequencies of the in-plane and the out-of-plane flexural vibrations are very similar. The derived analytical results are validated by a comparison with FEM and FSM (Finite Strip Method) results, as well as with experimental results available in the literature.