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EN
The paper deals with the plane problem of an elastic layer resting on a Winkler foundation with an emptiness. The stresses in the layer are caused by a given normal loading on its upper boundary plane. The mathematical formulation of the problem leads to a mixed boundary value problem and it is solved using Fourier transform methods and Fredholm integral equation of the second kind. The detailed analysis is derived analytically and numerically for an elliptic distribution of boundary loadings. The results for the normal displacement and the stresses on the lower boundary of the layer are presented in figures.
PL
Przeanalizowano wpływ przyjętego modelu podłoża na wyniki obliczeń ugięć i momentów zginających w lawach i płytach fundamentowych. Analizowano podłoża: winklerowskie, półprzestrzeni sprężystej i uwarstwione. Obliczenia przeprowadzono, wykorzystując program komputerowy. Wskazano na konieczność krytycznego oceniania otrzymywanych wyników.
EN
Different models used for the description of foundation elastic supports are analyzed and their influence on the displacement and bending moment predictions in strip and slab foundations are presented. Winkler, elastic half-space and multi-layer models are dealt with. Calculations are made with use of commercial computer software. A necessity for the critical evaluation of computer results is pointed out.
3
Friction Between a Body with an Elastic Layer and Vibrating Surface
EN
This paper presents the behavior of an object that contacts a vibrating plane through an elastic layer. The Coulomb model of friction is used. The aim of this paper is to show how the vibration and elasticity of the contact zone change the magnitude of the coefficient of friction. Consequently, an equivalent friction coefficient is defined. During the analysis and simulation for different parameters of vibration, the presence of stick-slip phenomenon for a specific range of the pushing force is noted. For some ranges of vibration of the base, dry friction can be presented as an equivalent viscous damping.
EN
Dynamic behaviour of an elastic layer with periodically undulating boundaries and resting on a rigid base is investigated. To this end a certain simplified averaged 2D-model of the layer is proposed as a tool of the analysis. The obtained model is applied to study some wave propagation problems. It is shown that the periodic shape of the boundaries leads to the dispersion of waves propagating along the unbounded layer.
PL
Praca dotyczy analizy dynamicznej warstwy sprężystej z periodycznie pofałdowanymi brzegami. W tym celu zaproponowano jako narzędzie analizy nowy uśredniony 2D-model warstwy. Wykazano, że periodyczne pofałdowanie brzegów powoduje efekty dyspersyjne dla fal propogujących się wzdłuż warstwy.
EN
Simplified models of a thermoelastic layer resting on a rigid substratum are obtained under tie assumption that the displacement distribution across the layer thickness is approximated using the concept of a decay function, [1], In the present contribution the above approach is applied to a heat conduction problem. The main aim of the approach is to investigate the proper choice of a decay function. The conditions under which the proposed simplified models can be applied are shown.
PL
Uproszczone modele warstwy sprężystej na sztywnym podłożu otrzymuje się przy założeniu, że rozkład przemieszczeń na grubości warstwy jest aproksymowany za pomocą funkcji zanikania, W pracy zastosowano to podejście w zagadnieniu rozkładu temperatury. Głównym celem tego opracowania jest wskazanie odpowiedniego sposobu doboru funkcji zanikania. Podano warunki, przy których zaproponowane modele mogą być stosowane.
EN
The aim of this paper is the discussion on the applicability of some rectangular elements to plane strain boundary value problems. Four different elements were considered: 4-node, 5-node, Serendipity 8-node and Lagrangian 9-node. Two cases: the material layer loaded by a concentrated vertical force and the same layer loaded by a symmetrical rigid punch were discussed. An elastic material was used to avoid the influence of the constitutive model on solutions. To model interface behaviour on the contact surface a Coulomb friction condition was applied. The use of the 4- and 5-node elements resulted in the prediction of the 'island' pattern of stress and strain tensors distributions and their non-applicability was proved independently from the boundary condition. The 8-node element predicted erroneous distributions of nodal forces and should be avoided in the case of contact problems. Among the discussed group of elements only the 9-node element turned out to be applicable for boundary value problems under plane strain condition.
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