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EN
The paper is concerned with damage detection in plates while using the Discrete Wavelet Transform (DWT). Rectangular plate structures resting on a Winkler or elastic half-space type foundation, with free boundaries are examined. Plate bending is described and solved by the Boundary Element Method in a direct approach. Defects are introduced by additional edges forming slots in relation to the basic plate domain. Numerical investigation is conducted basing on signal analysis of the structural static response and by taking advantage of multi-resolution analysis (MRA) of the signal function which can be represented in a multi-scale manner. The obtained signal is decomposed with the use of Daubechies or Coiflet wavelet families. The white noise generator is used to model measurement inaccuracy which is an inevitable element of a real experiment. The efficiency of DWT of the contaminated signal in damage detection is studied.
EN
In the present work, the defect detection while using Discrete Wavelet Transform in rectangular plate structures is investigated. The plate bending is described by using the Boundary Element Method with boundary integral equations formulated in a modified simplified approach. The boundary elements of a constant type in a non-singular approach are implemented. Defects are introduced by additional edges forming slots or holes in relation to the basic plate domain. Estimation of the defect position is performed while using wavelet coefficients of curvature and deformation signals as well as a newly proposed moving variance estimator.
EN
The stability of the element of a steel welded girder subjected to bending and shear forces is considered. The considered element is a rectangular plate supported on boundary. The type of a plate boundary conditions depend on the types (thickness) of the stiffeners. Considered plate is loaded by in-plane forces causing bending and shear effects. The Finite Element Method was applied to carry out the analysis. Additionally the Boundary Element Method in terms of boundary-domain integral equation was applied to evaluate the critical shear loading.
EN
A static and dynamic analysis of Kirchhoff plates is presented in this paper. The proposed approach avoids Kirchhoff forces at the plate corners and equivalent shear forces at a plate boundary. Two unknown variables are considered at the boundary element node. The governing integral equations are derived using Betti’s theorem. The rectilinear and curved boundary element of the constant type are used. The non-singular formulation of the boundary (static analysis) and boundary-domain (free vibration analysis) integral equations with one and two collocation points associated with a single constant boundary element located at a plate edge are presented. Additionally, the classic three-node isoparametric curved boundary elements are introduced in static analysis according to the non-singular approach. Static fundamental solution and B`ezine technique are applied to the free vibration analysis. To establish the plate inertial forces, a plate domain is divided into triangular or annular sub-domains associated with one suitable collocation point.
5
Content available remote Defect detection in plate structures using wavelet transformation
EN
This paper is concerned with defect detection in plate structures while considering the influence of external loads. The examined structures are based on Kirchhoff plate structures. Rectangular plate structures are considered. Plate bending is described using the boundary element method. The boundary and boundary-domain integral equations are formulated in a modified, simplified approach without the need of using a value known from the classical theory of Kirchhoff plate bending. Constant-type boundary elements in a non-singular approach are introduced. The plates are loaded with a single static concentrated force or dynamic moving force. External loading is applied at selected points along the direction parallel to one dimension of the plate. Defects are introduced by additional edges forming slots or holes in relation to the basic plate domain. Deflections and curvatures are taken into account as structural responses. Analysis of structural responses is conducted using the signal processing tool of wavelet transformation in its discrete form.
EN
The structures examined in this paper are bridge-type trusses that were previously used as railway viaduct support structures. The considered trusses are modelled as 2D and 3D structures. The lower chord bar of the considered structure can be loaded by external forces located outside the rigid nodes (the points where truss bars are connected). Hence, in the numerical experiment in terms of 2D approach, the truss structure consists of the set of two-node beam finite elements with three degrees of freedom per node and exact shape functions. According to 3D approach, the truss is described as the set of two-node beam elements with six degrees of freedom per node. Axial and twisting displacements of the element are described by linear shape functions and the bending is described by polynomials of the third order corresponding to Euler-Bernoulli beam fields of deformation. The defect (damage) in truss structure is modelled as the local stiffness reduction of one or two lower chord bars. The analysis of a structural response is carried out using the discrete wavelet transformation (DWT). The aim if this work is to detect the localization of damage provided that it exists in the considered structure and to examine whether the DWT will prove to be the effective tool to defect detection. It is expected that the disturbance of the response signal will appear in the vicinity of the point where the defect exists. The family of Daubechies 4 wavelet is implemented. Numerical investigation is executed based on signal analysis of structural static response. Some numerical examples are presented.
EN
The paper is concerned with free vibrations of geometrically non-linear elastic Timoshenko beams with immovable supports. The equations of motion are derived by applying the Hamilton principle. The approximate solutions are based on the negligence of longitudinal inertia forces but inclusion of longitudinal deformations. The Ritz method is used to determine non-linear modes and the associated non-linear natural frequencies depending on the vibration amplitude. The beam is discretized into linear elements with independent displacement fields. Consideration of the beams divided into the regular mesh enables one to express the equilibrium conditions for an arbitrary large number of elements in form of one difference equation. Owing to this, it is possible to obtain an analytical solution of the dynamic problem although it has been formulated by the finite element method. Some numerical results are given to show the effects of vibration amplitude, shear deformation, thickness ratio, rotary inertia, mass distribution and boundary conditions on the non-linear natural frequencies of discrete Timoshenko beams.
EN
An initial stability of Kirchhoff plates by the Boundary Element Method (BEM) is presented in the paper. A plate is subjected by external in-plane normal and tangential conservative loadings acting in two perpendicular directions. The Betti’s theorem is used to derive the boundary-domain integral equations. The direct version of the Boundary Element Method is presented with combination to simplified boundary conditions. The singular and non-singular approach of the boundary integrals derivation is used.
EN
An initial stability of Kirchhoff plates supported on boundary and resting on internal supports is analysed in this paper. The internal supports are understood to be part of a plate surface or a line belonging to the plate. The proposed approach avoids Kirchhoff forces at the plate corner and equivalent shear forces at the plate boundary. Two unknown and independent variables are always considered at a boundary element node depending on the type of a plate edge such as the shear force and bending moment for a clamped edge, and the shear force and angle of rotation in normal direction for a simply-supported edge. For a free edge, the deflection and angle of rotation in normal direction are considered as two independent variables with additional angle of rotation in tangent direction which depends on boundary deflections. The two governing integral equations are derived using Betti’s theorem. These equations have the form of boundary-domain integral equations. The constant type of boundary element is used. The singular and non-singular formulations of the boundary-domain integral equations with one and two collocation points associated with a single boundary element located slightly outside of a plate edge are presented. To establish a plate curvature by double differentiation of the basic boundary-domain integral equation, the plate domain is divided into rectangular subdomains associated with suitable collocation points. According to the alternative approach, a plate curvature is also established by considering three collocation points located in close proximity to each other along a line parallel to one of the two axes of global coordinate system and establishment of appropriate difference operators.
10
Content available remote Historia Średzkiej Kolei Powiatowej
PL
W pracy przedstawiono zarys historii Średzkiej Kolei Powiatowej będącej przykładem linii kolejowej znaczenia lokalnego. Budowa tego typu połączeń kolejowych przypadła na koniec XIX i początek XX wieku i cechowała się największą intensywnością na ziemiach polskich ówczesnego zaboru pruskiego. Koleje lokalne stanowiły podstawowy środek transportu wszelkich towarów, a dla okolicznych mieszkańców były jedynym połączeniem ze stolicą powiatu (Środa Wlkp), a często regionu (Poznań). Koleje te przyczyniły się do intensywnego rozwoju rolnictwa i przemysłu związanego z produkcją rolną, a w szczególności przemysłu cukrowniczego. Średzka Kolej Powiatowa stanowiąca alternatywne połączenie Poznania ze Środą Wielkopolską stanowi interesujący przykład linii kolejowej lokalnego znaczenia, która do czasu przejęcia przez PKP posiadała dwie szerokości toru i realizowała przewozy osób i towarów również na terenie miasta Poznania.
EN
The paper presents a brief history of the Środa District Railway being an example of the importance of the local railway line. The construction of this type of rail fell at the end of the nineteenth and early twentieth century was characterized by the highest intensity on the Polish territories then annexed by Prussia. These railways had a basic meaning of transport of goods, and for the locals were the only connection to the capital of the district (Środa), and often the region (Poznań). Railways contributed to the intensive development of agriculture and related industries with agricultural production, particularly the sugar industry. Środa District Railway was an alternative connection of Środa Wielkopolska with Poznań too. It is an interesting example of the local railway line, which until it was acquired by the PKP, had two gauges and realized the transport of passengers and goods on the premises of the city of Poznan.
EN
A static analysis of circular and elliptic Kirchhoff plates resting on internal elastic supports by the Boundary Element Method is presented in the paper. Elastic support has the character of Winkler-type elastic foundations. Bilateral and unilateral internal constraints are taken into consideration. The Betti’s theorem is used to derive the boundary domain integral equation. The direct version of the boundary element method is presented and simplified boundary conditions, including curvilinear boundary elements, are introduced. The collocation version of boundary element method with non-singular approach is presented.
EN
An initial stability of Kirchhoff plates is analysed in the paper. Proposed approach avoids Kirchhoff forces at the plate corner and equivalent shear forces at a plate boundary. Two unknown variables are considered at the boundary element node. The governing integral equations are derived using Betti theorem. The integral equations have the form of boundary and domain integral equations. The constant type of boundary element are used. The singular and non-singular formulation of the boundary-domain integral equations with one and two collocation points associated with a single boundary element located at a plate edge are presented. To establish a plate curvature by double differentiation of basic boundary-domain integral equation, a plate domain is divided into rectangular sub-domains associated with suitable collocation points. A plate curvature can also be establish by considering three collocation points located in close proximity to each other along line pararel to one of the two axes of global coordinate system and establishment of appropriate differential operators.
PL
W pracy przedstawiono zastosowanie transformacji falkowej do wykrywania uszkodzeń w konstrukcjach inżynierskich na przykładzie belki spoczywającej na podłożu sprężystym. Sformułowano i rozwiązano zadania statyki belki Eulera-Bernoulliego z uszkodzeniami oraz zlokalizowano te uszkodzenia przy zastosowaniu dyskretnej transformacji falkowej. Uszkodzenia belek modelowano przez lokalną redukcję sztywności. W wybranym punkcie belki (punkcie kontrolnym) śledzono parametry geometryczne (przemieszczenie pionowe, kąt obrotu przekroju) w funkcji poruszającego się po belce obciążenia skupionego. Tak przygotowany sygnał odpowiedzi konstrukcji poddano dyskretnej transformacji falkowej.
EN
The method of localization of damage is proposed in the paper. The method is based on the analysis of the response signal of a structure which is induced by moved set of external loading. The basis for discussion is the assumption that any damage (e.g. as a fatigue crack, corrosion) produces some disturbance in the registered signal (e.g. deflection or acceletarion). The main advantage of presented approach of damage detection is exertion of the response signal of "damaged" structure and measuremet is realized in one selected point. In the future this procedure is planned to implement to the identification of bridge structures. The paper presents the application of wavelet transformation to detect defects in engineering structures on the example of a beam resting on elastic foundation. The aim of the study was the formulation and solution of the problem of static Euler-Bernoulli beam with damage and localization of its by using discrete wavelet transformation. The local damage is modeled as local stiffness reduction. A continuous beam is modelled by two-node finite elements with twho degree of freedom per node where the bending is described by polinominals of third order corresponding to Euler-Bernoulli beam fields of deformation. In a selected point of a beam (control point for example the node of a finite element) the geometric parameters (deflection, angle of rotation) were monitored as the functions of moving set of the concentrated loading. The results of calculations using discrete wavelet analysis are presented in graphical form.
EN
Free vibration analysis of Kirchhoff plate by the Boundary Element Method is presented in the paper. The boundary integral equation are derived according to the Bettie theorem. The collocation version of BEM with non-singular approach with one and double collocation points is used. The constant type of element is introduced. Boundary suport at selected point is modelled as support in vicinity of point along single boundary element.
EN
A static analysis of Kirchhoff and Reissner plates by the boundary element method has been presented in the paper. The Betti’s theorem has been used to derive the boundary integral equation. The direct version of the boundary element method has been presented.
EN
A static analysis of Kirchhoff plates using the boundary element method is presented in the paper. In this approach, physical boundary conditions are imposed. The Bettie theorem is used to derive the boundary integral equation. The collocation version of the boundary element method is presented. Typical and simplified, curved constant boundary elements are introduced.
17
Content available remote Application of simplified curved boundary elements to the plate analysis. Part 2
EN
Static analysis of Kirchhoff plate by the Boundary Element Method is presented in the paper. The Bettie theorem is used to derive the boundary integral equation. Simplified curved elements are introduced. Modified approach of boundary integral equation formulation is adopted in which there is no need to introduce the equivalent shear forces at the boundary and concentrated forces at the plate corners. Two unknown and independent variables are considered at the boundary element node. The collocation version of boundary element method with singular and non-singular approach is presented.
18
Content available remote Application of simplified curved boundary elements to the plate analysis. Part 1
EN
Static analysis of Kirchhoff plate by the Boundary Element Method is presented in the paper. The Bettie theorem is used to derive the boundary integral equation. Simplified curved elements are introduced. Modified approach of boundary integral equation formulation is adopted in which there is no need to introduce the equivalent shear forces at the boundary and concentrated forces at the plate corners. The collocation version of boundary element method with singular and non-singular approach is presented.
EN
An initial stability of Kirchhoff plates is presented in the paper. Using proposed approach, there is no need to introduce Kirchhoff forces at the plate corner and equivalent shear forces at a plate boundary. Two unknown and independent variables are considered at the boundary element node. The Bettie theorem is used to derive the boundary integral equation. The collocation version of boundary element method with "constant" type of elements is presented. The source points are located slightly outside a plate boundary, hence the quasi diagonal integrals of fundamental functions are non-singular. To describe a plate curvature, the set of internal collocation points is introduced.
EN
A static analysis of Kirchhoff plates rested on the elastic internal supports has been discussed in the paper. The Finite Strip Method and Boundary Element Method have been used as an engineering tool in the analysis. Suitable fundamental solutions are applied in these method. Using BEM modified approach, there is no need to introduce the Kirchhoff forces at the plate corner and equivalent shear forces at the plate boundary. Two unknown and independent variables are considered at the boundary element node. The collocation points are located slightly outside the plate boundary, hence the quasidiagonal integrals of fundamental functions are non-singular. The constant type of boundary element has been used. According to the finite strip method a continuous structure is divided into a set of identical elements simply supported on opposite edges. The unknowns are the deflections and the transverse slope amplitudes along the nodal lines. The difference equation formulation is applied to express the equilibrium conditions of the discrete system. This reduces the number of degrees of freedom to be analyzed. The solution of one equilibrium difference equation yields the fundamental function of the considered plate strip. The fundamental solution derived in this way, can be used to solve the static problem of finite plate in analogically as in the boundary element method for continuous systems.
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