Stateczność dynamiczna płyty kołowej wykonanej z piany metalowej

The study is devoted to a radial compressed metal foam circular plate. Properties of the plate vary across its thickness. The middle plane of the plate is its symmetry plane. First of all, a displacement field of any cross-section of the plate was defined. Afterwards, the components of strain and stress states were found. The Hamilton principle allowed one to formulate a system of differential equations of dynamic stability of the plate. This basic system of equations was approximately solved. The forms of unknown functions were assumed and the system of equations was reduced to a single ordinary differential equation of motion. The equation was then numerically processed that allowed one to determine critical loads for a family of metal foam plates. The results of studies are shown in figures. They show the effect of porosity of the plate on the critical loads. The results obtained for porous plates were compared to homogeneous circular plates.Przedmiotem badań jest osiowo ściskana porowata płyta kołowa.Własności płyty są zmienne na jej grubości. Środkowa płaszczyzna płyty jest jej płaszczyzną symetrii. Zdefiniowano pole przemieszczeń dla dowolnego przekroju płyty. Ponadto wyznaczone są składowe stanów odkształceń i naprężeń. Układ równań stateczności dynamicznej płyty wyznaczono z zasady Hamiltona. Następnie równania te rozwiązano metodą Galerkina i otrzymano zwyczajne równanie różniczkowe typu Mathieu. Równanie to rozwiązano metodą Rungego-Kutty i wyznaczono obciążenia krytyczne dla rodziny płyt kołowych. Wyniki przedstawiono na wykresach. Wskazano wpływ porowatości materiału na obciążenia krytyczne płyty. Otrzymane wyniki porównano z odpowiednimi wielkościami dla płyty jednorodnej

Matematyczne modelowanie prostokątnej płyty trójwarstwowej z rdzeniem z pianki metalowej

The subject of the paper is a simply supported rectangular sandwich plate. The plate is compressed in plane. It is assumed that the plate under consideration is symmetrical in build and consists of two isotropic facings and a core. The middle plane of the plate is its symmetry plane. The core is made of a metal foam with properties varying across its thickness. The porouscellular metal as a core of the three layered plate is of continuous structure, while its mechanical properties are isotropic. Dimensionless coefficients are introduced to compensate for this. The field of displacements and geometric relationships are assumed. This non-linear hypothesis is generalization of the classical hypotheses, in particular, the broken-line hypothesis. The principle of stationarity of the total potential energy of the compressed sandwich plate is used and a system of differential equations is formulated. This system is approximately solved. The forms of unknown functions are assumed, which satisfy boundary conditions for supports of the plate. Critical loads for a family of sandwich plates are numerically determined. Results of the calculation are shown in figures.Przedmiotem pracy jest prostokątna płyta trójwarstwowa podparta przegubowo na czterech brzegach i ściskana w płaszczyźnie środkowej. Okładziny płyty są izotropowe i o takich samych właściwościach mechanicznych. Rdzeń wykonany z pianki metalowej jest również izotropowy, jego właściwości mechaniczne są zmienne na grubości. Płaszczyzna środkowa płyty jest jej płaszczyzną symetrii. Zdefiniowano pole przemieszczeń dla dowolnego punktu rdzenia oraz okładzin płyty. Sformułowano energię odkształcenia sprężystego płyty i pracę obciążenia. Następnie z zasady stacjonarności całkowitej energii potencjalnej otrzymano układ równań równowagi, który rozwiązano analitycznie w sposób przybliżony i wyznaczono obciążenie krytyczne płyty

Analiza nieliniowa stateczności dynamicznej płyty kołowej wykonanej z piany metalowej

The subject of the paper is a circular plate under radial compression. The plate is made of metal foam. Properties of the plate vary across its thickness. The middle plane of the plate is its symmetry plane. The field of displacement of any cross section of the plate, nonlinear components of the strain field and the stress field are defined. Basing on the Hamilton principle, a system of differential equations of dynamic stability of the plate is formulated. This basic system of equations is approximately solved. The results of the studies are compared to the homogeneous circular plate and shown in figures.Przedmiotem pracy jest płyta kołowa obciążona promieniowo. Płyta wykonana jest z piany metalowej. Właściwości mechaniczne płyty są zmienne na jej grubości. Płaszczyzna środkowa płyty jest jej płaszczyzną symetrii. Zdefiniowano pole przemieszczeń dla dowolnego przekroju poprzecznego płyty, nieliniowe odkształcenia oraz naprężenia. Układ równań różniczkowych stateczności dynamicznej płyty zdefiniowano na podstawie zasady Hamiltona. Układ ten rozwiązano w sposób przybliżony. Wyniki badań numerycznych tej płyty porównano z odpowiednimi wielkościami dla płyty jednorodnej i przedstawiono na rysunkach

Transverse shear modulus of elasticity for thin-walled corrugated cores of sandwich beams. Theoretical study

The subject of the paper are four corrugated cores in form of circular arcs, a sin wave, trapezoids and an odd function. Transverse shear modules of these corrugated cores are analytically determined. A comparative analysis of these transverse shear modules is pre- sented. Areas of cross sections of the corrugated cores are constant. The theoretical study shows considerable sensitivity of the shear modulus to shape of the corrugation

Buckling and post-buckling behaviour of shallow – nearly flat cylindrical panels under axial compression

The paper is devoted to buckling problem of axially compressed shallow cylindrical panels. Governing differential equations of the nonlinear theory of shallow cylindrical shells are analytically solved. Critical stresses and equilibrium paths of the panels with small curvatures are analytically studied. The formula of the critical stresses for almost flat, cylindrical panels is derived. The “shallowness” of the panel is given by the parameter α and formulae are derived for a range of this parameter. The range of values of sectorial angle for these panels is also defined

Transverse shear modulus of elasticity for thin-walled corrugated cores of sandwich beams. Theoretical study

The subject of the paper are four corrugated cores in form of circular arcs, a sin wave, trapezoids and an odd function. Transverse shear modules of these corrugated cores are analytically determined. A comparative analysis of these transverse shear modules is pre- sented. Areas of cross sections of the corrugated cores are constant. The theoretical study shows considerable sensitivity of the shear modulus to shape of the corrugation

Strength of the Three Layer Beam with Two Binding Layers

The paper is devoted to the strength analysis of a simply supported three layer beam. The sandwich beam consists of: two metal facings, the metal foam core and two binding layers between the faces and the core. In consequence, the beam is a five layer beam. The main goal of the study is to elaborate a mathematical model of this beam, analytical description and a solution of the three-point bending problem. The beam is subjected to a transverse load. The nonlinear hypothesis of the deformation of the cross section of the beam is formulated. Based on the principle of the stationary potential energy the system of four equations of equilibrium is derived. Then deflections and stresses are determined. The influence of the binding layers is considered. The results of the solutions of the bending problem analysis are shown in the tables and figures. The analytical model is verified numerically using the finite element analysis, as well as experimentally

Strength of a metal seven-layer rectangular plate with trapezoidal corrugated cores

The subject of analytical and numerical investigations in this paper is a metal seven-layer rectangular plate with a trapezoidal corrugated main core and two trapezoidal corrugated cores of faces. The hypothesis of deformation of the normal to the middle surface of the plate after bending and field of displacements is formulated. The plate is simply supported on all its edges and subjected to a uniform pressure. Equations of equilibrium are derived based on the theorem of minimum total potential energy and are solved with the use of the Galerkin method. The influence of the trapezoidal corrugation pitch of the cores on the deflection and the equivalent stress is analysed

Bending, buckling and free vibration of a beam with unsymmetrically varying mechanical properties

The subject of the paper is a beam with unsymmetrically varying mechanical properties in the depth direction. The nonlinear hypothesis of plane cross section deformation is assumed. Based on Hamilton’s principle, two differential equations of motion are obtained. The system of equations is analytically solved with a view to analyse the bending, buckling and free vibration problems of the beam. Moreover, the FEM model of the beam is developed and deflections, critical axial forces and natural frequencies of the beam are calculated. The results of these two methods are compared