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1
Content available Free axisymmetric flexural vibrations of circular plate with symmetrically varying mechanical properties supported on elastic foundation
100%
Magnucki K.
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2020
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tom 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.
2
Content available remote On stability problems of pressure vessel ellipsoidal heads.
63%
Magnucki K. , Szyc W.
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1999
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tom Vol. XLVI, nr 1
43-55
EN
The subject of the analysis are ellipsoidal heads of cylindrical tanks. The stability of these heads under external or internal pressure is investigated. Moreover, in both loadcases, linear and non-linear stability analysis of heads is realized, The calculations have been executed by means of finite element method, using the COSMOS/M system. The results have been compared with existing analytical solutions of ellipsoidal shells specified in literature. New formulae determining values of critical pressure are proposed.
PL
W pracy rozwiązano numerycznie problem stateczności den elipsoidalnych cienkościennych zbiorników ciśnieniowych obciążonych ciśnieniem zewnętrznym i wewnętrznym. Teoretyczny opis stateczności powłok walcowych i elipsoidalnych znany jest z literatury. Brak jest natomiast opisów konstrukcji złożonych np. zbiornika, który składa się z powłoki walcowej i dwóch powłok elipsoidalnych. Rozwiązanie stateczności konstrukcji złożonej zrealizowano za pomocą metody elementów skończonych w opisie liniowym oraz nieliniowym z uwzględnieniem skończonych przemieszczeń i liniowego wzmocnienia materiału. Wyniki obliczeń porównano ze znanymi z literatury wzorami. Zaproponowano pewne poprawki w tych wzorach, dzięki ktorym otrzymano nowe wyrażenia dla ciśnień krytycznych. W przypadku ciśnienia zewnętrznego, gdy utrata stateczności dna występuje w zakresie odkształceń spężystych, rozwiązania analityczne Mushtariego i Galimova uogólniono dla potrzeb praktycznych, poprzez małą zmianę wykładnika potęgi. W przypadku obciążenia ciśnieniem wewnętrznym dla den o małej wyniosłości toroidalno-kulistych lub elipsoidalnych utrata skuteczności występuje w zakresie odkształceń plastycznych. Z uwagi na brak rozwiązań teoretycznych do analizy przyjęto wzór Galletly'ego, który jest sformułowany na podstawie wyników badań doświadczalnych dla den toroidalno-kulistych. Przeprowadzono analizę parametrów geometrycznych znormalizowanych den toroidalno-kulistych i elipsoidalnych. Następnie zaproponowano dla potrzeb praktycznych wyrażenie określające ciśnienie krytyczne dla den elipsoidalnych obciążonych ciśnieniem wewnętrznym. Wyniki nieliniowej analizy stateczności den elipsoidalnych potwierdzają słuszność zaproponowanych wyrażeń do ciśnień krytycznych.
3
Content available remote Bending of beams with symmetrically varying mechanical properties under generalized load – shear effect
63%
Magnucki K. , Lewinski J.
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tom 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.
4
Content available Dynamic stability of a three-layer beam – generalisation of the sandwich structure theory
63%
Magnucki K. , Magnucka-Blandzi E.
Acta Mechanica et Automatica
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2024
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tom 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.
5
Content available Bending of a stepped sandwich beam: the shear effect
51%
Kustosz J. , Magnucki K. , Goliwąs D.
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tom 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.
6
Content available Free flexural vibrations of an expanded-tapered sandwich beam
51%
Magnucki K. , Kustosz J. , Goliwąs D.
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2023
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tom 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.
7
Content available remote A shear deformation theory of beams with bisymmetrical Cross-sections based on the Zhuravsky shear stress formula
51%
Magnucki K. , Lewinski J. , Magnucka-Blandzi E.
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tom 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.
8
Content available remote Approximate estimation of stability of homogeneous beam on elastic foundation
51%
Wstawska I. , Kędzia P. , Magnucki K.
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tom 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.
9
Content available remote Analytical and numerical studies of an unsymmetrical sandwich beam – bending, buckling and free vibration
51%
Magnucki K. , Magnucka-Blandzi E. , Lewiński J. , Milecki S.
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tom 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.
10
Content available Elastic buckling of a generalized cylindrical sandwich panel under axial compression
51%
Magnucki K. , Magnucka E. , Wittenbeck L.
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tom 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.
11
Content available Strength of a bent sandwich beam with clamped ends
51%
Kustosz J. , Magnucki K. , Goliwąs D.
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tom 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.
12
Content available Analytical investigation of a beam on elastic foundation with nonsymmetrical properties
51%
Wstawska I. , Magnucki K. , Kędzia P.
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2022
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tom 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.
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