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1
Content available Bending of a five-layered composite beam with consideration of two analytical models
Magnucki Krzysztof
Archive of Mechanical Engineering
|
2024
|
LXXI, nr 1
27--46
EN
The subject of the work is a five-layered composite beam with clamped ends subjected to a uniformly distributed load along its length. Two analytical models of this beam are developed with consideration of the shear effect. The first model is formulated on the basis of the classical zig-zag theory. Whereas, the second model is developed using an individual nonlinear shear deformation theory with consideration of the classical shear stress formula (called Zhuravsky shear stress). The system of two differential equations of equilibrium for each beam model is obtained based on the principle of stationary total potential energy. These systems of equations are exactly analytically solved. The shear effect function and the maximum deflection are determined for each of these two beam models. Detailed calculations are carried out for exemplary beams of selected dimensionless sizes and material constants. The main goal of the research is to develop two analytical models of this beam, determine the shear effect function, the shear coefficient, and the maximum deflection in the elastic range for each model, as well as to perform a comparative analysis.
2
Content available Dynamic stability of a three-layer beam – generalisation of the sandwich structure theory
Magnucki Krzysztof, Magnucka-Blandzi Ewa
Acta Mechanica et Automatica
|
2024
|
Vol. 18, no. 1
1--7
EN
The work focuses on the dynamic stability problem of a simply supported three-layer beam subjected to a pulsating axial force. Two analytical models of this beam are developed: one model takes into account the non-linear hypothesis of cross-section deformation, and the other takes into account the standard "broken line" hypothesis. Displacements, strains and stresses for each model are formulated in detail. Based on the Hamilton principle, equations of motion are determined for each of these models. These systems of two differential equations for each model are approximately solved with the consideration of the axial pulsating force, and the fundamental natural frequencies, critical forces and the Mathieu equation are determined. Detailed studies are performed for an exemplary family of beams. The stable and unstable regions are calculated for the three pulsating load cases. The values of fundamental natural frequencies and critical forces of exemplary beams calculated from two models are compared.
3
Content available Free flexural vibrations of an expanded-tapered sandwich beam
Magnucki Krzysztof, Kustosz Joanna, Goliwąs Damian
Vibrations in Physical Systems
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2023
|
Vol. 34, nr 1
art. no. 2023114
EN
The paper is devoted to the analytical modelling of a simply supported expanded-tapered sandwich beam. The simplified analytical model of this beam with omitting the shear effect is elaborated. Based on Hamilton’s principle, the differential equation of motion of this beam is obtained. This equation is analytically solved with consideration of the deflection line of this beam subjected to its own weight. The fundamental natural frequencies for exemplary beams are derived. Moreover, the FEM model of the beam in the ABAQUS is developed. The calculation results of the fundamental natural frequency of exemplary beams of these two methods are presented in tables and figures.
4
Content available Strength of a bent sandwich beam with clamped ends
Kustosz Joanna, Magnucki Krzysztof, Goliwąs Damian
Pojazdy Szynowe
|
2023
|
Nr 1-2
3--7
EN
The subject of the work is a symmetrical sandwich beam with clamped ends under uniformly distributed load. The system of two equilibrium equations, formulated taking into account the literature, was solved analytically. The function of the shear effect and the maximum deflection of the beam were determined. The stress state at the clamped end of the beam is described in detail. The significant influence of the shear effect on the normal stresses in the outer layers of the beam near the clamped end was indicated. Exemplary calculations were made for the adopted family of beams. Moreover, the numerical FEM model of the beam was developed and calculations were made for this adopted family of beams. A comparative analysis of the obtained results was carried out.
5
Content available Three-point bending of a beam with non-homogeneous structure in the depth direction
Magnucki Krzysztof, Magnucka-Blandzi Ewa, Witkowski Dawid, Wittenbeck Leszek
Engineering Transactions
|
2023
|
Vol. 71, iss. 4
645--665
EN
This paper is devoted to the behavior of a non-homogeneous simply supported beam under three-point bending. The individual shear deformation function of a planar cross-section is adopted, and longitudinal displacements, strains, and stresses for two parts of the beam are explained. By applying the principle of stationary potential energy, a system of two differential equations of equilibrium is derived and solved analytically. The positions of the neutral axis, shear coefficients, and deflections are then calculated for three different beam families. Additionally, the bending problem of these beams is studied numerically using the finite element method (FEM). The results of both analytical and numerical calculations are presented in tables and figures. The main contribution of this paper lies in the development of an analytical model incorporating the individual shear deformation function and a numerical FEM model for this beam.
6
Content available Elastic buckling of a generalized cylindrical sandwich panel under axial compression
Magnucki Krzysztof, Magnucka Ewa, Wittenbeck Leszek
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2023
|
Vol. 71, nr 2
art. no. e144623
EN
The paper is devoted to buckling problem of an axially compressed generalized cylindrical sandwich panel and rectangular sandwich plate. The continuous variation of mechanical properties in thickness direction of the structures is assumed. The generalized theory of deformation of the straight line normal to the neutral surface is applied. The analytical model of this sandwich panel is elaborated. Three differential equations of equilibrium of this panel based on the principle of stationary potential energy are obtained. This system of equations is analytically solved and the critical load is derived. Moreover, the limit transformation of the sandwich panel to a sandwich rectangular plate is presented. The critical loads of the example cylindrical panels and rectangular plates are derived.
7
Content available Effective shaping of a stepped sandwich beam with clamped ends
Magnucki Krzysztof, Kustosz Joanna, Goliwąs Damian
Acta Mechanica et Automatica
|
2023
|
Vol. 17, no. 2
200--204
EN
The aim of this work is to propose a sandwich beam with stepped layer thickness in three parts along its length. The total depth, width of the cross-section and its mass are constant. The beam is under a uniformly distributed load. The system of two equilibrium equa-tions was formulated for each part based on the literature. This system was analytically solved for the successive parts of the beam and the functions of the shear effect and deflection were determined in them. The effective stepped layer thicknesses was determined on the basis of the adopted criterion for minimizing the maximum deflection of the beam. The example calculations were made for two elected beams. The effective shapes of these beams are shown in the figures. Moreover, FEM numerical calculations of the deflections of these beams are performed.
8
Content available Three models of a sandwich beam: bending, buckling, and free vibrations
Magnucki Krzysztof, Magnucka-Blandzi Ewa, Wittenbeck Leszek
Engineering Transactions
|
2022
|
Vol. 70, iss. 2
97--122
EN
This paper is devoted to the analytical modelling of a sandwich beam. Three models of this beam are elaborated. Two nonlinear individual shear theories of deformation of a plane cross-sections are proposed. Based on Hamilton’s principle, two differential equations of motion for each model are obtained. The bending, buckling and free flexural vibration problems of the simply-supported sandwich beam considering these three models are studied. The results of these analytical investigations are presented in tables.
9
Content available Bending of a stepped sandwich beam: the shear effect
Kustosz Joanna, Magnucki Krzysztof, Goliwąs Damian
Engineering Transactions
|
2022
|
Vol. 70, iss. 4
373--390
EN
This paper is devoted to the stepped sandwich beam with clamped ends subjected to a uniformly distributed load. The bending problem of the beam is formulated and solved with consideration of the classical sandwich beam of constant face thickness. Two differential equations of equilibrium based on the principle of the stationary potential energy of the classical beam are obtained and analytically solved. Moreover, numerical-FEM models of the beams are developed. Deflections for an exemplary beam family with the use of two methods are calculated. The results of the study are presented in figures and tables.
10
Content available Analytical investigation of a beam on elastic foundation with nonsymmetrical properties
Wstawska Iwona, Magnucki Krzysztof, Kędzia Piotr
Engineering Transactions
|
2022
|
Vol. 70, iss. 1
77--93
EN
The subject of presented analytical and numerical investigation is the stability of an axially compressed beam on an elastic foundation. The shape function of the foundation was assumed. The formula was supplemented with the offset parameter. The critical values of loads were calculated and presented as a function of geometric and mechanical properties of the beam and nonsymmetrical properties of the elastic foundation. The highest values of critical loads can be obtained for the highest values of shape parameter and the lowest values of amplitudes of shape function. The values of critical loads increase with the increase of the value of the offset parameter.
11
Content available Free axisymmetric flexural vibrations of circular plate with symmetrically varying mechanical properties supported on elastic foundation
Magnucki Krzysztof
Vibrations in Physical Systems
|
2020
|
Vol. 31, nr 2
art. no. 2020217
EN
The subject of the paper is a circular plate with clamped edge supported on elastic foundation. Mechanical properties of the plate symmetrically vary in its thickness direction. Free axisymmetric flexural vibration problem of the plate with consideration of the shear effect is analytically studied. Two partial differential equations of motion based on the Hamilton principle are obtained. The system of equations is analytically solved and the fundamental natural frequency of axisymmetric vibration for example plates is derived.
12
Content available Analytical, FEM-numerical and experimental studies of bending of a sandwich plate-strip with metal foam core
Magnucki Krzysztof, Alsdorf Dennis, Lewiński Jerzy, Kowalski Michał, Richter Alexander, Zbierski Andrzej
Pojazdy Szynowe
|
2020
|
Nr 3
1--10
EN
The subject of the study is a sandwich plate-strip subjected to a four-point load. An analytical model of the strip was developed, taking into account the classical zig-zag theory, namely the broken line hypothesis. Three parts of the plate-strip are distinguished: two of them are the edge parts, where bending and the shear effect is considered, the third one is the middle part subjected to pure bending. The total maximum deflection of the plate-strip and the maximum deflection of the selected middle part of the plate-strip are calculated. The FEM-numerical study is carried out similarly to the analytical approach. The experimental study was carried out on the test stand in the Institute of Rail Vehicles TABOR. The analytical, numerical and experimental results are compared each with other. The sandwich panels can be used as parts of the floor or rail vehicle paneling.
13
Content available Analityczne, numeryczne (MES) i doświadczalne badanie zginania trójwarstwowego pasma płytowego z rdzeniem z pianki metalowej
Magnucki Krzysztof, Alsdorf Dennis, Lewiński Jerzy, Kowalski Michał, Richter Alexander, Zbierski Andrzej
Pojazdy Szynowe
|
2020
|
Nr 3
11--19
PL
Przedmiotem badań jest trójwarstwowe pasmo płytowe poddane czteropunktowemu zginaniu. Opracowano analityczny model tego pasma, korzystając z klasycznej teorii linii łamanej nazywanej teorią Zig-Zag. W paśmie tym wyróżniono trzy przedziały: dwa brzegowe, w których występuje zginanie i ścinanie oraz jeden środkowy, w którym występuje czyste zginanie. Wyznaczono całkowite ugięcie maksymalne pasma płytowego oraz maksymalne ugięcie odcinka środkowego. Przeprowadzono obliczenia numeryczne metodą elementów skończonych (MES) dla takiego samego modelu pasma, jak wyżej wspomniany model analityczny. Próbę doświadczalną przeprowadzono na stanowisku badawczym w Instytucie Pojazdów Szynowych. Porównano wyniki badań analitycznych, numerycznych i doświadczalnych. Analizowane płyty warstwowe mogą być stosowane, m. in. jako części podłogi lub poszycia pojazdu szynowego.
14
Content available Three-point bending of an expanded-tapered beam with consideration of the shear effect
Magnucki Krzysztof, Lewiński Jerzy, Magnucka-Blandzi Ewa, Stawecka Hanna
Journal of Theoretical and Applied Mechanics
|
2020
|
Vol. 58 nr 3
661--672
EN
The paper is devoted to an expanded-tapered beam of rectangular cross section subjected to three-point bending. The analytical model of the beam is formulated with consideration of a non-linear hypothesis of the cross section deformation. The problem of shear stress distribution in the beam is analysed based on the above mentioned hypothesis. Moreover, a numerical FEM model (SolidWorks) is developed. Examplary computations have been carried out based on the analytical and numerical models.
15
Content available remote Bending of beams with consideration of a seventh-order shear deformation theory
Magnucki Krzysztof, Stawecki Włodzimierz, Magnucka-Blandzi Ewa
Engineering Transactions
|
2020
|
Vol. 68, iss. 2
119--135
EN
The subject of the paper is a simply-supported prismatic beam with bisymmetrical crosssections under non-uniformly distributed load. The shapes of the cross-sections and the nonuniformly distributed load are described analytically. The individual seventh-order shear deformation theory-hypothesis of the planar beam cross-sections is assumed. Based on the principle of stationary potential energy two differential equations of equilibrium are obtained. The system of the equations is analytically solved, and the shear and deflection coefficients of the beam are derived. Moreover, the shear stress patterns for selected cross-sections are determined and compared with stresses determined by Zhuravsky’s formula. The results of example calculations are presented in tables and figures.
16
Content available remote A shear deformation theory of beams with bisymmetrical Cross-sections based on the Zhuravsky shear stress formula
Magnucki Krzysztof, Lewinski Jerzy, Magnucka-Blandzi Ewa
Engineering Transactions
|
2020
|
Vol. 68, iss. 4
353--370
EN
This paper is devoted to simply supported beams with bisymmetrical cross-sections under a generalized load. Based on the Zhuravsky shear stress formula, the shear deformation theory of a planar beam cross-section is formulated. The deflections and the shear stresses of exemplary beams are determined. Moreover, the numerical-FEM computations of these beams are carried out. The results of the research are shown in figures and tables.
17
Content available Stress state in the tapered beam bending : analytical and numerical FEM studies
Magnucki Krzysztof, Lewiński Jerzy, Stawecka Hanna
Pojazdy Szynowe
|
2019
|
Nr 2
1--8
EN
The paper is devoted to comparative analysis of the stress state in bending of a tapered cantilever beam, calculated analytically and numerically (FEM). The analytical model is described based on bibliography, moreover, the numerical FEM model is developed with the use of the SolidWorks software. The results i.e. the stresses obtained by analytical and numerical calculation are compared and specified in Tables and Figures.
PL
Praca przestawia analizę porównawczą stanu naprężenia w zginanej belce wspornikowej o zmiennej wysokości. Przeprowadzono obliczenia analityczne i numeryczne metodąelementów skończonych. Model analityczny został opisany na podstawie literatury, na-tomiast model do obliczeń metodą elementów skończonych opracowano z zastosowaniem systemu SolidWorks. Wyniki, tzn. naprężenia wyznaczone analitycznie i numerycznie zostały porównane i zamieszczone w tablicach oraz zilustrowane na rysunkach.
18
Content available remote Bending of beams with symmetrically varying mechanical properties under generalized load – shear effect
Magnucki Krzysztof, Lewinski Jerzy
Engineering Transactions
|
2019
|
Vol. 67, iss. 3
441--457
EN
The paper is devoted to simply supported beams with symmetrically varying mechanical properties in the depth direction. Generalized load of the beams includes the load types from uniformly distributed to point load (three-point bending). This load is analytically described with the use of a certain function including a dimensionless parameter. The value of the parameter is decisive for the load type. The individual nonlinear “polynomial” hypothesis is applied to deformation of a planar cross section. Based on the definitions of the bending moment and the shear transverse force the differential equation of equilibrium is obtained. The equation is analytically solved and the deflections are calculated for an exemplary beam family. The results of the study are specified in tables.
19
Content available remote Approximate estimation of stability of homogeneous beam on elastic foundation
Wstawska Iwona, Kędzia Piotr, Magnucki Krzysztof
Engineering Transactions
|
2019
|
Vol. 67, iss. 3
429--440
EN
The paper deals with a proposition of obtaining an analytical solution for a beam on elastic foundation. The main objective of presented work was stability analysis of the axially compressed beam. The analytical model was proposed. Shape function for inhomogeneous properties of the foundation was assumed. The Galerkin method was used to calculate the values of critical forces. Main conditions have been defined. The critical loads as a function of geometric and mechanical properties of the beam as well as inhomogeneous properties of the elastic foundation have been calculated.
20
Content available remote Analytical and numerical studies of an unsymmetrical sandwich beam – bending, buckling and free vibration
Magnucki Krzysztof, Magnucka-Blandzi Ewa, Lewiński Jerzy, Milecki Szymon
Engineering Transactions
|
2019
|
Vol. 67, iss. 4
491--512
EN
The subject of the paper is an unsymmetrical sandwich beam. The thicknesses and mechanical properties of the beam faces are different. Mathematical model of the beam is formulated based on the classical broken-line hypothesis. The equations of motions of the beam is derived on the ground of the Hamilton’s principle. Bending, buckling and free-vibration are studied in detail for exemplary unsymmetrical structure of the beam. The values of deflection, critical force and natural frequency are determined for the selected beam cases. Moreover, the same examples are computed with the use of two FEM systems, i.e. SolidWorks and ABAQUS, in order to compare the analytical and numerical calculation. The results are presented in tables and figures.
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