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1

Rola homogenizacji w predykcji nośności opakowań z tektury falistej – przegląd metod i zastosowań

Garbowski Tomasz

Przegląd Papierniczy

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2024

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R. 80, nr 5

271--278

PL

Artykuł omawia kluczową rolę homogenizacji w predykcji nośności opakowań z tektury falistej, podkreślając metody i techniki stosowane w tym procesie. Homogenizacja pozwala na uproszczenie złożonej struktury tektury falistej, co prowadzi do zwiększenia efektywności obliczeń i precyzyjniejszej predykcji wytrzymałości. Przegląd przedstawia różnorodne metody homogenizacji, takie jak: metoda Voigta-Reussa-Hill, Mori-Tanaki, metoda samouzgodniona oraz ich zastosowanie w modelowaniu właściwości mechanicznych. W pracy omówiono także zastosowanie metod numerycznych, w tym metody elemen tów skończonych (MES), które porównano z metodami eksperymentalnymi. Praktyczne zastosowania homogenizacji zilustrowano za pomocą studiów przypadków, ukazując korzyści w projektowaniu i optymalizacji opakowań. Przegląd wskazuje również na obecne wyzwania i przyszłe kierunki badań, podkreślając znaczenie wyników dla przemysłu opakowaniowego, zwłasz cza w kontekście poprawy wytrzymałości, efektywności materiałowej oraz optymalizacji kosztów produkcji.

EN

This article discusses the crucial role of homogenization in predicting the load-bearing capacity of corrugated board packaging, emphasizing the methods and techniques used in this process. Homogenization simplifies the complex structure of corrugated board, leading to increased com putational efficiency and more precise strength predictions. The review presents various homogenization methods, such as the Voigt-Reuss-Hill, Mori-Tanaka, and self-consistent methods, and their application in modeling mechanical properties. The article also discusses the use of numerical methods, including the finite element method (FEM), and compares them with experimental methods. Practical applications of homogenization are illustrated through case studies, highlighting the benefits in the design and optimization of packaging. The review also identifies current challenges and future research directions, emphasizing the importance of the results for the packaging industry, particularly in terms of improving strength, material efficiency, and production cost optimization.

2

Problem modelowania przestrzennych konstrukcji drewnianych z masywnego drewna z wykorzystaniem modeli prętowych

Janas Agnieszka, Brol Janusz

Przegląd Budowlany

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2024

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R. 95, nr 3

34--38

PL

W artykule przedstawiono problem właściwego odzwierciedlenia w modelowaniu komputerowym z wykorzystaniem elementów prętowych, drewnianych schodów spiralnych jako elementów z belek drewnianych ułożonych wzajemnie na sobie. Podano sposób rozwiązania tego zagadnienia przez autorów artykułu na przykładzie dyplomowej pracy inżynierskiej pt.: „Projekt wieży widokowej o konstrukcji drewnianej” [1] wykonanej przez autorkę artykułu.

EN

The article presents the problem of proper representation, in computer modelling using bar elements, of wooden spiral stairs as elements made of wooden beams stacked on top of each other. The method of solving this issue by the authors of the article is presented on the example of the engineering diploma thesis titled: „Design of an observation tower with a wooden structure” [1] prepared by the author of the article.

3

Modelowanie numeryczne MES kompozytowej belki stalowo-OSB

Derlatka Anna, Rajczyk Marlena

Przegląd Budowlany

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2024

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R. 95, nr 4

29--33

PL

W pracy przedstawiono model numeryczny belki zespolonej stalowo-OSB. Belka została zbudowana z cienkościennych blach stalowych usztywnionych za pomocą płyt OSB. Oceniono wrażliwość modelu numerycznego na rodzaj elementów skończonych oraz ich rozmiar. Stwierdzono, że najbardziej korzystnym rozwiązaniem jest zastosowanie modelu zbudowanego z 27-węzłowych elementów skończonych o średnim rozmiarze oczka siatki wynoszącym 10 mm. Po walidacji modelu z wynikami badań eksperymentalnych oceniono, że opracowany model numeryczny belki zespolonej może być wykorzystany do gromadzenia danych służących do predykcji nośności belki za pomocą technik uczenia maszynowego.

EN

The paper presents a numerical model of a steel-OSB composite beam. The beam was made of thin-walled steel sheets stiffened with OSB. FEM numerical calculations were performed using the ADINA System. The sensitivity of the numerical model to the type of finite elements and their size was assessed. It was found that the most advantageous solution is the use of the model built of 27-node finite elements with an average mesh size of 10 mm. After validation of the model with the results of experimental research, it was assessed that the developed numerical model of the composite beam can be used to collect data for predicting the load-bearing capacity of the beam using machine learning techniques.

4

Zastosowanie MES do oceny degradacji rur stalowych

Duszyńska Iwona, Krykowski Tomasz, Gremza Grzegorz, Stefanek Paweł, Bzówka Joanna

Materiały Budowlane

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2024

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nr 4

82--85

5

Modelowanie i walidacja wytrzymałościowa przegrody budowlanej w zabudowie szkieletowej

Gołębiowski Łukasz, Teodorczyk Michał, Fedorczyk Zbigniew, Piekarczuk Artur

Inżynieria i Budownictwo

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2024

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R. 80, nr 4

239--242

PL

W artykule zaprezentowano badania doświadczalne i analizy numeryczne przegród w zabudowie szkieletowej. Przedmiotem analizy były elementy badawcze reprezentujące fragment przegrody składającej się z profili stalowych zimnogiętych oraz okładzin gipsowo-kartonowych. Przeprowadzono test wytrzymałości w układzie czteropunktowego zginania, w którym rejestrowano ciągły zapis obciążenia od przemieszczenia elementu. Badania prowadzono do momentu zniszczenia. Następnie opracowano model obliczeniowy bazujący na metodzie elementów skończonych (MES), odwzorowujący doświadczenie. Wyniki badań uzyskane w doświadczalnym teście wytrzymałościowym porównano z wynikami uzyskanymi w analizie numerycznej. Przeprowadzona walidacja modelu numerycznego wykazała dużą zgodność w zakresie charakterystyki obciążenia od przemieszczenia. Zastosowanie analiz numerycznych stanowi alternatywę dla czasochłonnych badań doświadczalnych przy szacowaniu przemieszczeń przegród w zabudowie szkieletowej.

EN

The article presents experimental research and numerical analysis of a building partition in frame construction. The subject of the analysis were sections of the building partition consisting of cold-bent steel profiles and plasterboard cladding. The experiment was carried out using a four-point bending test and recording a continuous load-displacement curve until failure. Next, a computational model based on the finite element method was developed to simulate the experiment. The results from the experimental tests were then compared with that obtained from the numerical analysis. The validation of the numerical model demonstrated a high level of agreement in terms of load-displacement characteristics. The use of numerical analyses provides an alternative to time-consuming experimental tests for estimating displacements of building partitions in frame construction.

6

Performance evaluation of nonlinear viscoelastic materials using finite element method

Sabri Laith Abed, Al-Tamimi Adnan Naji Jameel, Alshamma Fathi A., Mohammed M. N., Salloomi Kareem N., Abdullah Oday I.

International Journal of Applied Mechanics and Engineering

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2024

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Vol. 29, no. 1

142--158

EN

This research paper applies the finite element method as a methodology to evaluate the structural performance of nonlinear viscoelastic solids. A finite element algorithm was built and developed to simulate the mathematical nonlinear viscoelastic material behavior based on incremental constitutive equations. The derived Equation of the incremental constitutive included the complete strain and stress histories. The Schapery’s nonlinear viscoelastic material model was integrated within the displacement-based finite element environment to perform the analysis. A modified Newton-Raphson technique was used to solve the nonlinear part in the resultant equations. In this work, the deviatoric and volumetric strain–stress relations were decoupled, and the hereditary strains were updated at the end of each time increment. It is worth mentioning that the developed algorithm can be effectively employed for all the permissible values of Poisson’s ratio by using a selective integration procedure. The algorithm was tested for a number of applications, and the results were compared with some previously published experimental results. A small percentage error of (1%) was observed comparing the published experimental results. The developed algorithm can be considered a promising numerical tool that overcomes convergence issues, enhancing equilibrium with high-accuracy results.

7

Advanced Method for Determining Land Plot’s Physical Surface

Malashevskyi Mykola, Malashevska Olena

Geomatics and Environmental Engineering

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2024

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Vol. 18, no. 4

73--96

EN

This study examines the issue of the accuracy of physical surface calculations for land plots with complex configurations and reliefs. The goal of the study was to develop a methodology for determining the surface area of land plots with complex configurations and reliefs. The presented model was based on the finite element method. The developed method allows one to evaluate a relief’s complexity by using a dimensionless mean physical surface complexity factor; a Fortran program was developed for the methodology. Experiments that proved the effectiveness of the methodology and a comparative analysis of those areas that were calculated by the presented method and TIN model were carried out. The research findings proved the practicability of the methodology for calculating the physical surfaces of land plots with complex configurations. The presented methodology can be used for flood modeling, landscape and vertical planning, etc.

8

Advancing computational approaches for geometry optimization of steel structures

Lax Artur, Milewski Sławomir

Engineering Transactions

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2024

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Vol. 72, iss. 2

159--202

EN

The primary objective of this study is to develop and assess computational methods for optimizing the geometry of specific building structures modeled through parametric description. The focus is on steel bar structures, including trusses and beams, subjected to varying load conditions with fixed and uncertain parameters. The decision variables in the single- or multicriteria non-linear optimization problem correspond to selected geometric features of these structures. The proposed methodology revolves around dividing the entire construction into distinct structural patterns. This allows for addressing separate local optimization problems with a reduced number of decision variables, followed by a global optimization considering the interactions between these patterns. This approach is versatile, serving both the design of objects meeting required architectural and structural conditions and constraints, and the optimization of all or specific parameters, incorporating diverse economic (e.g., material usage) and engineering criteria (e.g., limit states).

9

Strength analysis of welded joints in support structures of heavy machinery subjected to changeable loading conditions

Smolnicki Tadeusz, Sokolski Piotr

Eksploatacja i Niezawodność

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2024

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Vol. 26, no. 1

art. no. 175889

EN

The durability of bearing units of large machines depends mainly on the condition of their welded joints. With this in mind, we developed numerical models of the analyzed bearing units, for which we performed FEM simulations of the stresses in welded joints inseveral basic load cases. In each of the respective variants the technical condition of the bearing nodes was different and it corresponded to the severity of the degradation processes. Different positions of the superstructure in relation to the undercarriage were also taken into account. The simulations used the hot-spot method dedicated to FEM analyses of complex welded structures. We discovered that the loads have a significant influence on the values and distribution of von Mises principal stresses and their axial components. Based on the carried out analyses, we identified the most unfavorable load cases that generate the highest stresses in the welded joints of the assessed nodes. We also demonstrated that the applied method effectively assesses the stresses of welded joints subjected to variable working loads.

10

Numerical model of interaction of a mobile lift chain with a main gas pipeline pipe in the process of repair work

Dutkiewicz Maciej, Andrusyak Andriy, Kychma Andrii, Vytvytskyi Vasyl, Velychkovych Andrii

Eksploatacja i Niezawodność

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2024

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Vol. 26, no. 3

art. no. 187159

EN

The paper proposes an original design of a mobile lifting device for diagnosing and repairing support nodes of above-ground segments of gas pipelines without stopping gas transportation. When using such a device, the rollers of the lifting chain interact with the surface of the pipe, which can possibly cause additional stresses in the gas pipeline. At the first stage of studying this problem, a model of a typical gas pipeline overhead crossing for the Carpathian region was built and the maximum operating force that should be developed by the mobile lift during repair operations was determined. At the second stage, a finite-element model of the interaction of the mobile lift chain with the main gas pipeline pipe was built. The additional equivalent stresses arising from the contact of the rollers of the lifting chain with the pipe surface were estimated. Recommendations are given to ensure the safe operation of the gas pipeline when using a mobile lift for local lifting of the pipe above the support of above-ground segments of gas pipelines.

11

Modeling the thermo-mechanical response and phase changes in metallic additive manufacturing (MAM) processes using a dissipative phase-field model

Darabi Roya, Azinpour Erfan, Reis Ana, Sa de Jose Cesar

Computer Methods in Materials Science

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2024

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Vol. 24, No. 2

5--25

EN

Additive manufacturing (AM) has emerged as a highly promising manufacturing technique, offering unprecedented possibilities for creating complex geometries and functional structures. However, harnessing the full potential of AM requires the development of a robust computational framework capable of capturing the intricate multi-scale and multi-physics nature of the process. The constitutive and structural responses encountered in AM are particularly challenging to reproduce due to the complex behavior of the material involved. This research aims to address these challenges by presenting a comprehensive computational approach that incorporates a material model capable of accurately representing the behavior of different phases occurring during AM. To achieve this, the finite element method, using the Lagrangian framework in the implicit time scheme, is employed through the widely adopted ABAQUS software. Computational implementation is facilitated using the FORTRAN programming language. By employing weakly coupled thermal and mechanical constitutive equations, the framework enables the analysis of thermal stresses, strains, and displacements during realistic solidification processes, which inherently involve highly nonlinear constitutive relations. Through a series of numerical examples, the capabilities of the proposed model are demonstrated across various computational scales, particularly during the rapid melting and solidification phases. These simulations reveal the formation of residual stresses, which can lead to part distortion and have detrimental effects on the mechanical properties of the manufactured components. This research contributes to the advancement of additive manufacturing by providing a reliable computational tool that integrates the complex interplay between thermal and mechanical phenomena. The developed framework enhances our understanding of the AM process, offering valuable insights into the factors influencing the structural integrity and performance of additively manufactured parts.

12

Functionally graded porous material and its application in sandwich beams for bending and vibration behaviors

Ton That Hoang Lan

Computer Methods in Materials Science

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2024

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Vol. 24, No. 1

15--24

EN

The article describes a functionally graded porous material in an application for sandwich beams. The bending and vibration behaviors of this structure are studied using the finite element method based on a simple beam model. The influences of some parameters, e.g., the porosity factor or the exponent graded, are also studied in this article. Finally, the numerical results are presented with some discussion.

13

Varying coefficients in parallel shared-memory variational splitting solvers for non-stationary Maxwell equations

Łoś Marcin, Woźniak Maciej, Paszyński Maciej

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2024

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Vol. 72, nr 3

art. no. e149179

EN

Direction-splitting implicit solvers employ the regular structure of the computational domain augmented with the splitting of the partial differential operator to deliver linear computational cost solvers for time-dependent simulations. The finite difference community originally employed this method to deliver fast solvers for PDE-based formulations. Later, this method was generalized into so-called variational splitting. The tensor product structure of basis functions over regular computational meshes allows us to employ the Kronecker product structure of the matrix and obtain linear computational cost factorization for finite element method simulations. These solvers are traditionally used for fast simulations over the structures preserving the tensor product regularity. Their applications are limited to regular problems and regular model parameters. This paper presents a generalization of the method to deal with non-regular material data in the variational splitting method. Namely, we can vary the material data with test functions to obtain a linear computational cost solver over a tensor product grid with non-regular material data. Furthermore, as described by the Maxwell equations, we show how to incorporate this method into finite element method simulations of non-stationary electromagnetic wave propagation over the human head with material data based on the three-dimensional MRI scan.

14

Finite element analysis of femur bone fracture modelling and analysis

Sarmah Pranjal, Kumar Ravi, Thakur Amrit, Sharma Mohit, Yadav Surendra Kumar, Kumar Virendra

Archives of Thermodynamics

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2024

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Vol. 45, no 2

261--268

EN

Femoral fractures are frequent in adolescents and children, and most fractures occur within the centre of the bone, typically referred to as the femur shaft. Plate and screws are ideal fixation methods for femoral fractures close to the articular surfaces. When using plates and screws, estimating the load on the plates and screws before starting treatment is important. The primary focus of this paper is the examination of fixation plates utilized in the treatment of femur bone fractures. The study employs the finite element method to conduct this analysis. Initial modelling of the femur bone is executed through the utilization of CATIA V5 software. Subsequently, the investigation transitions to the ANSYS R14.5 environment, where more in-depth analysis is carried out. The modelling of the fracture fixation plates is done on commercially available CAD software CATIA V5. The stress distribution of different biomaterials in the bone plate system is calculated when the system is subjected to compressive loads with varying healing times. Here we have used stainless steel (SS316-L), titanium alloy (Ti6Al4V) and magnesium alloy (AZ31). More focus was given to the magnesium alloy. Here a fracture gap of 1mm gap was taken for analysis. A comprehensive compressive force amounting to 750 N was applied to the bone-plate assembly during the simulation. This force magnitude corresponds to the approximate weight of an average human body.

15

Nonlinear free vibration of functionally graded graphene nanoplatelet-reinforced composite rectangular plates using the full layerwise finite element method

Tayebi M. S., Jedari Salami S., Tavakolian M.

Archives of Mechanics

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2024

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Vol. 76, nr 3

225--252

EN

This investigation is the first effort to study the nonlinear free vibration response of functionally graded (FG) graphene nanoplatelet (GPL)-reinforced composite rectangular plates using the full layerwise (LW) finite element method. The innovation of this article, which has not been investigated thus far, is a nonlinear vibration analysis with equivalent precision to three-dimensional (3D) elasticity while benefiting from decreased computational cost, ease of mesh adjustment, and faster achievement of the element stiffness matrix due to preserving the 2D structure. The modified Halpin–Tsai model and the rule of mixtures are employed to specify the effective material properties for composite plates with three different arrangements of GPLs. After confirming the results and formulation, an exhaustive parametric study is executed to examine how various characteristics of GPLs and the plate affect the nonlinear-to-linear frequency (NTL) ratio of the FG GPL-reinforced composite plate. The noteworthy finding is that inserting a small percentage of GPLs in pure epoxy changes its NTL ratio significantly. The effect of the GPL dimension on the NTL ratio is insignificant when the thickness-to-length ratio of GPLs is smaller than 10-3.

16

Characteristics of polymer ring springs

Wróbel G., Kaczmarczyk J.

Archives of Materials Science and Engineering

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2024

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Vol. 126, nr 1

22--33

EN

Purpose: The article presents a procedure for modelling and analysing a model system of ring and cone springs, in their particular material solution, using selected polymeric materials. Design/methodology/approach: Subsequently, physical and mathematical models were presented, allowing the formulation of general design assumptions for the target products. The material models made were subjected to experimental tests to evaluate the correctness of the analytical models. Commercially available types of pure polyethylene and polypropylene were used as materials for machined parts based on catalogue properties. The material models were tested under variable static and periodically varying dynamic loads. The constructed finite element model was subjected to verification of the compatibility of the results of the numerical analysis with the results of simple experiments in order to assess the correctness of the model. Practical implications: The correctness and adequacy of the computational model, confirmed in terms of simple load cases, will allow extending the scope of numerical simulation studies to systems that differ in material and geometric design features. Originality/value: The performed studies have proved the advisability of using polymeric materials in the area of the design of ring springs. It allows for the significant expansion of the area of static and dynamic characteristics, which opens new fields of application for similar solutions. Due to the developed and verified numerical model, it becomes possible to analyse similar structural elements in terms of materials and geometry. In particular, interesting results can be expected if the research area is extended to composite materials. Also, the specific properties of plastics make it possible to expand the rationale field for similar systems. The prices of the tested materials and, above all, the cost of processing in the presented structural solutions are competitive in cases where it is possible to obtain similar technical characteristics as steel structures. In addition, a wide range of design requirements can be met exclusively by polymer or composite springs. The tools presented here open up new possibilities for computer-aided design processes.

17

Influence of the radiofrequency applicators arrangement on the sizes of ablative zones inside hepatic tumor

Gas Piotr, Miaskowski Arkadiusz

Archives of Electrical Engineering

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2024

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Vol. 73, nr 3

557--571

EN

Radiofrequency (RF) ablation is a popular therapeutic technique for heating solid tumors that are medically unsuitable for resection or other treatments. Thermal ablation applicators create high-frequency electromagnetic fields (EMFs) within the tumor site, which causes heating, coagulation, and ultimately death of the cancer cells. The aim of this study is the numerical analysis of the temperature distributions, ablation zones, and specific absorption rates (SAR) during RF ablation in relation to an ellipsoidal shaped tumor placed in the model of liver tissue. The source of heat is a three-element system of RF needle applicators operating at a frequency 100 kHz, with a given electrode potential, inserted into the tumor. In order to obtain an appropriate temperature distribution in the target area, the Laplace equation coupled with the Pennes equation were solved using the finite element method (FEM). The arrangement effect of three needle-type applicators on the resultant thermal profiles and the volumes of ablation zones were analyzed and compared. In addition, the ablation zones for various angles of the RF applicator placed in the center of the tumor were analyzed. The paper shows that in order to control temperature distribution and ablation zones the proposed system of RF applicators and the arrangement of electrodes can be successfully applied in hepatocellular carcinoma treatment.

18

Homogenization of foil windings with globally supported polynomial shape functions

Bundschuh Jonas, Späck-Leigsnering Yvonne, De Gersem Herbert

Archives of Electrical Engineering

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2024

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Vol. 73, nr 1

77--85

EN

In conventional finite element simulations, foil windings with thin foils and with a large number of turns require many mesh elements. This renders models quickly computationally infeasible. This paper uses a homogenized foil winding model and approximates the voltage distribution in the foil winding domain by globally supported polynomials. This way, the small-scale structure in the foil winding domain does not have to be resolved by the finite element mesh. The method is validated successfully for a stand-alone foil winding example and for a pot inductor example. Moreover, a transformer equipped with a foil winding at its primary side is simulated using a field-circuit coupled model.

19

Design of a six-phase surface mounted permanent magnet synchronous motor with application to electric trucks

Vuong Dang Quoc, Cong Trinh Truong

Archives of Electrical Engineering

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2024

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Vol. 73, nr 3

721--741

EN

As we have known, permanent magnet synchronous motors (PMSMs) have garnered widespread interest across various industrial applications thanks to their advantages such as high efficiency, reliable performance, simple structure, and adaptability to various shapes and sizes. Due to characteristics of the high torque and low speed, the PMSMs make particularly well-suited for traction applications such as trucks, ship propulsion, mining, and more. In this context, a combination of the analytical method and finite element method (FEM) is proposed for designing and simulating a six-phase surface-mounted PMSM. Firstly, a model of the six-phase PMSM is analytically design to make required/main dimensions. The FEM is then applied to analyse and verify electromagnetic parameters such as of the current waveform, back electromagnetic force (EMF), magnetic flux density in the air gap, flux linkage, torque, cogging torque, torque ripple and harmonic components. Via the obtained results, the research will give a contribution of valuable insights for optimizing the design, performance and reliability for this motor in diverse industrial applications.

20

Analysis of deformations of road embankments founded on displacement columns improving soft subsoil

Szajna Waldemar, Bondareva Liudmyla, Szatanik Bartosz

Archives of Civil Engineering

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2024

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Vol. 70, nr 1

19--36

EN

Drilled displacement columns, constructed in the form of unreinforced or reinforced concrete elements, are currently a very commonly used method of improving soft subsoil, creating an alternative to more expensive pile foundations. A frequently used solution for improving soft soils of road or railway embankments is to design a regular pattern of columns of relatively small diameter. Columns along the perimeter of the improved area are reinforced with rigid steel profiles, while the internal ones are made as concrete elements. Column heads are usually covered with a load transfer platform (layer of compacted granular fill) which is additionally reinforced with geosynthetics. The application of soil improvement with displacement columns is not always successful. It is due to the errors and shortcomings occurring at the design stage, including simplifications in modelling, to construction faults, which may include insufficient experience of contractors and/or improper supervision. Referring to the real object that failed, the article provides the results of numerical parametric analyses taking into account the influence of the key design parameters such as: the stiffness of the load transfer layers, the amount and stiffness of the geosynthetic reinforcement as well as the column spacing. The article presents comparisons of numerical results obtained with the finite element analyses for various approaches to geometry modelling (axisymmetric, 2D and 3D). The simulations indicate that the use of the axisymmetric model of a single column in routine design may lead to the deformations exceeding the serviceability limit states.

PL

Rozwój infrastruktury komunikacyjne, ze względu na wymaganą geometrię szlaków, powoduje konieczność wznoszenia obiektów na słabych gruntach. Stosowanie wzmocnień w postaci różnego rodzaju kolumn, które przenoszą obciążenia na głębsze, nośne warstwy gruntu jest w takich sytuacjach powszechne. Wśród tej grupy technologii szczególnie często stosowane są wiercone kolumny przemieszczeniowe, stanowiące wsparcie nasypów. Pomimo dostępności wielu wyników badan nasypów na kolumnach wzmacnianych geosyntetykami i dużych doświadczeń w ich realizacji, awarie nadal się zdarzaj ą. Motywacją do podjęcia tematu jest przypadek drogi na niskim nasypie, wykonanym na słabym podłożu organicznym wzmocnionym kolumnami (rys. 2) na której wystąpiły duże nierównomierne osiadania nawierzchni. Różnice osiadania stref nad kolumnami i pomiędzy kolumnami miały bardzo różniące się od siebie wartości nawet w sąsiednich polach i wahały się w granicach od 5,5 do 23 mm. Rozmieszczenie kolumn w planie, pokazano na rys. 3a. Rysunek pokazuje także obszar gruntu przypadaj ący na jedną kolumnę As, co pozwala na wyznaczenie tzw. współczynnika wzmocnienia ARR = AcAs, gdzie Ac jest polem przekroju poprzecznego kolumny. Na rys. 3b pokazano zastępcze rozmieszczenie kolumn, ułatwiające analizy 2D. Celem pracy jest zbadanie wrażliwości układu na zmiany wartości istotnych parametrów projektowych, takie jak: geometria i sztywność warstwy transmisyjnej, ilość i sztywność zbrojenia geosyntetycznego oraz rozstaw kolumn, a także na sposób przestrzennego modelowania geometrii konstrukcji. Zadanie wykonano w formie analiz parametrycznych wykorzystując MES. Przedstawiono porównanie wyników uzyskanych dla modeli pojedynczych kolumn w stanie osiowej symetrii (AX), modeli 2D i modeli 3D. Analizując wyniki skupiono się na przemieszczeniach. Zestawienie analizowanych wariantów modeli przedstawiono w tab. 1. Modele, których parametry są zgodne z parametrami rzeczywistego obiektu nazwano bazowymi i oznaczono dodatkowym symbolem ‘b’. Parametry gruntów wykorzystane w modelach numerycznych zamieszczono w tab. 2. Obliczenia wykonano programem Plaxis. Do symulacji warstwy transmisyjnej (ang. LTP) oraz platformy roboczej wykorzystano model Coulomba–Mohra. Do symulacji pozostałych warstw gruntu zastosowano model Hardening Soil. Obliczenia obejmowały konsolidację gruntu obciążonego w warunkach z drenażem. W analizach 2D i 3D kolumny modelowano z wykorzystaniem elementów ‘embedded beams’. Pomiędzy warstwy zbrojenia geosyntetycznego i otaczający je grunt wprowadzono warstwy elementów ‘interface’. Rysunek 4 przedstawia porównanie wyników osiadania uzyskanych przy zastosowaniu modelu osiowej symetrii. Wzrost rozstawu kolumn zwiększa osiadania nad głowicami oraz różnice osiadania. Interesujące jest porównanie krzywych odpowiadających modelom AX-b i AX-1, gdzie niewielki wzrost ARR spowodował bardzo dużą redukcję osiadania i różnicy osiadania. Zastanawiający jest przebieg krzywej uzyskany przy braku zbrojenia geosyntetycznego. Wyjaśnienia uzyskanych tutaj małych różnic osiadania wymaga jednak dodatkowych analiz. Rysunki 5 do 8 odnoszą się do rozwiązań uzyskanych w modelach 2D. Rozkład osiadania w modelu bazowym 2D-b przedstawiono na rys. 5. Kolejny rysunek przedstawia porównanie wpływu rozstawu kolumn na osiadania. W tym przypadku, podobnie jak w modelu AX, niewielki wzrost ARR (z 3,2% na 3,61%) spowodował znaczne ograniczenia osiadania i różnicy osiadania. Sugeruje to, że istnieje pewna graniczna wartość współczynnika wzmocnienia, powyżej której rozwiązanie konstrukcyjne nabiera pozytywnych cech z punktu widzenia użytkowego. Wpływ ilości zbrojenia geosynetycznego, jego rozmieszczenia oraz sztywności, pokazano na rys. 7. Z rys. 7a wynika, że nawet dwukrotny wzrost sztywności zbrojenia nie wpływa jakościowo na redukcję deformacji układu. Problemy nadmiernych deformacji nasypu, wynikające z dużego rozstawu kolumn, trudno jest także zrekompensować sztywnością warstwy transmisyjnej, co pokazano na rys. 8. Rysunek 9 przedstawia zasadnicze elementy modelu 3D oraz uzyskane osiadania nawierzchni. Na rys. 10 zamieszczono odpowiadające sobie rozwiązania bazowe uzyskane w modelach AX, 2D i 3D. Rysunek ten potwierdza wcześniejsze spostrzeżenia, że w rozwiązaniu osiowosymetrycznym uzyskuje si ę największe osiadania i największe różnice osiadania, jednakże model ten nie uwzględnia wpływu zginania na wytężenie kolumn. Próbę wyjaśnienia wpływu rozstawu kolumn na deformacje nasypu przedstawia rys. 11. Pokazano na nim trajektorie naprężeń głównych uzyskane w rozwiązaniach 2D-b (ARR = 3; 2%) oraz 2D-s4 (ARR = 4; 18%). W pierwszym przypadku, duży rozstaw kolumn, przy ograniczonej wysokości nasypu uniemożliwił pełne rozwinięcie się efektu przesklepienia. Na rys. 12 zestawiono wszystkie poprzednio omówione rozwiązania, wyliczając dla każdego z nich efektywność transferu obciążenia na kolumnę, zdefiniowaną formułą (4.1) w funkcji różnicy osiadania. Z przeprowadzonych symulacji wynika, że sposób modelowania geometrii (AX, 2D i 3D) jest istotnym czynnikiem wpływającym na uzyskane wyniki opisujące deformacje nasypu. W niskich nasypach bardzo istotnym parametrem projektowym jest rozstaw kolumn i związany z nim współczynnik wzmocnienia ARR. Zbyt mała wartość tego współczynnika uniemożliwia pełne rozwinięcie się efektu przesklepienia, redukując efektywność rozwiązania i prowadząc do dużych wartości osiadania i różnic osiadania. Błędy na tym etapie projektowania trudno jest zrekompensować przez ilość i sztywność zbrojenia geosyntetycznego, czy tez zagęszczenie (sztywność) warstwy transmisyjnej. W rozważanym przypadku krytyczna wartość parametru ARR, powyżej której deformacje układu ulegają znacznej redukcji wynosi około 3,6%. Przeprowadzone analizy dotyczą tylko jednego nasypu i należy zachować ostrożność przy próbie uogólniania przedstawionych wniosków.