Ewa Magnucka-Blandzi

Variational design of open cross-section thin-walled beam under stability constraints

Abstract A thin-walled beam is in pure bending subjected to couples M 0 . The open cross-section profile has two ribs, with cross-section A 0 and it is shaped symmetrically towards the plane perpendicular to the bending plane. The ribs are located at the profile ends. The shape of the profile line is searched for. Criterion is the minimal value of the cross-section area A 1 of the beam. The problem is described by means of variational calculus. Within the numerical calculations a Runge–Kutta method is used. The optimal shapes of beam profiles are shown graphically.

Approximate solutions of equilibrium equations of sandwich circular plate

The paper is devoted to a sandwich circular plate consisting of two facings and a core with variable mechanical properties. The simply supported plate is compressed in the middle plane. The mathematical and physical model of the plate is formulated, and also the field of displacements. The system of equilibrium equations is derived basing on the principle of the total potential energy. Then the system is approximately solved. Three formulas for an unknown function of displacements are assumed. The solutions (critical loads) are compared and presented in Figures and Tables.

Mathematical Modelling of Shear Effect of Sandwich Beam

This paper is devoted to simply supported sandwich beams with a metal foam core. The mechanical properties of the isotropic core of the beam are varied in a normal direction in relation to the middle plane of symmetry. Mathematical modelling of shear effect is presented. The fields of displacement for the flat cross section of the beam are defined. Basing on the principle of the stationary of the total potential energy the system of four partial differential equations is obtained. One way for solving the system of equilibrium equations is proposed.

Vibrations and Dynamic Stability of Cellular Plate

Subject of the paper is a circular plate under radial compression. The plate is made of the metal foam. Properties of the plate vary across its thickness. Middle plane of the plate is its symmetry plane. The field of displacement of any cross section of the plate, the 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 homogeneous circular plate and shown in Figures.

Mathematical Modeling of a Rectangular Sandwich Plate with a Non‐Homogeneous Core

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 core. Middle plane of the plate is its symmetry plane. The core is made of metal foam with properties vary across its thickness. The porous‐cellular metal as a core of 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 are used and a system of differential equations are formulated. This system is approximately solved. The forms of unknown functions are assumed, which satisfy boundary conditions for supports of...

Bending of five-layer beams with crosswise corrugated main core

The subject of the study is one orthotropic thin-walled sandwich beam with trapezoidal core and two-layer facings. The outer layers of facings are flat, but inner layers are trapezoidal corrugated. The main core of the beam is also trapezoidal corrugated – in perpendicular direction to the corrugation of inner layers of facings. The beam is with lengthwise corrugated layers and crosswise corrugated main core. The mathematical and physical model of this beam is formulated, and also the field of displacements. The system of equilibrium equations is analytically derived using the energy method. The obtained solutions will be verified numerically (FEM).

Mathematical modelling of the beam under axial compression force applied at any point – the buckling problem

The study is devoted to stability of simply supported beam under axial compression. The beam is subjected to an axial load located at any point along the axis of the beam. The buckling problem has been desribed and solved mathematically. Critical loads have been calculated. In the particular case, the Euler’s buckling load is obtained. Explicit solutions are given. The values of critical loads are collected in tables and shown in figure. The relation between the point of the load application and the critical load is presented.

Some approximation theorems for a general class of discrete type operators in spaces with a polynomial weight

The paper is devoted approximation of real-valued functions defined on an unbounded interval by linear and non-linear operators. The publication pays special attention to defining class of operators and examining their certain approximation properties. The form of the class of operators given in the present paper makes the achieved results more helpful from the computational point of view.

Three-point bending of seven layers beams - theoretical and experimental studies

Abstract The subject of the analytical and experimental studies therein is of two metal seven-layer beam - plate bands. The first beam - plate band is composed of a lengthwise trapezoidally corrugated main core and two crosswise trapezoidally corrugated cores of faces. The second beam - plate band is composed of a crosswise trapezoidally corrugated main core and two lengthwise trapezoidally corrugated cores of faces. The hypotheses of deformation of a normal to the middle surface of the beams after bending are formulated. Equations of equilibrium are derived based on the theorem of minimum total potential energy. Three-point bending of the simply supported beams is theoretically and experimentally studied. The deflections of the two beams are determined with two methods, compared and presented.

Mathematical Modeling of Three-layer Circular Plate under Uniformly Distributed Load (Pressure)

The paper is devoted to a three-layer circular plate consisting of two facings and a core with variable mechanical properties. The simply supported plate is subjected to pressure. The mathematical model of the plate is formulated, in particular the field of displacements. The nonlinear hypothesis is proposed. Basing on the principle of the total potential energy the system of equilibrium equations is derived. Then the system is solved using the method of Bubnov-Galerkin. The maximal deflections, normal stresses and shear stresses are calculated.