The Finite Element Analysis of Buckling of Laminated Rectangular Reinforced Concrete Plates with Circular Hole

In this investigation an approach buclcling analysis for simply supported rectangular reinforced concrete plates under uniaxial compression are developed.The objective of this investigation is to compare buclcling load of the reinforced plate with circular hole and without hole. A laminated composite plate is considered.The finite element models for with central circular hole and without hole are designed. The composite materials and composite laminates materials approach are discussed. The solution of the problem by computer program coded in FORTRAN is explained and numerical example is given. The buckling behaviour of thin plates with a hole and made of composite materials are research topics of great practical importance. For example, composite plate-like subcomponents with holes are being considered for use in many types of structural analysis of building due to various reasons. The need for a hole in a subcomponent is typically required by practical concerns. For example, windows,air conditioned hole, etc. In some applications, these structural elements are required to resist buckling. Thus an understanding of their buckling behaviour is needed for their design. Plate-like subcomponents come in many forms such as a rectangular plate with a circular hole. The present study focuses on rectangular plates with a single hole. Developing a through understanding of the behaviour of this subcomponent is a fundamental step toward understanding the behaviour of complex structures with holes such as building structures. Knowledge of the basic response of the subcomponent provides information useful for the preliminary design of complex structures. In addition, this basic knowledge provides valuable insight into modelling complex structures with general purpose finite element codes. Furthermore, knowledge of the subcomponent response is very useful in identifying erroneous results that may occur because of improper finite element modelling. The present study is to describe the results of research that has been conducted on the buckling behaviour of rectangular composite plates with holes. It is necessary to establish some convenient parameters and terminology for descnlJing the plate geometry,loading and boundary conditions and material composition.

The buckling behaviour of thin plates with a hole and made of composite materials are research topics of great practical importance.For example, composite plate -like subcomponents with holes are being considered for use in many types of structural analysis of building due to various reasons.The need for a hole in a subcomponent is typically required by practical concerns.For example, windows,air conditioned hole, etc.In some applications, these structural elements are required to resist buckling.Thus an understanding of their buckling behaviour is needed for their design.Plate -like subcomponents come in many forms such as a rectangular plate with a circular hole.The present study focuses on rectangular plates with a single hole.Developing a through understanding of the behaviour of this subcomponent is a fundamental step toward understanding the behaviour of complex structures with holes such as building structures.Knowledge of the basic response of the subcomponent provides information useful for the preliminary design of complex structures.In addition, this basic knowledge provides valuable insight into modelling complex structures with general purpose finite element codes.Furthermore, knowledge of the subcomponent response is very useful in identifying erroneous results that may occur because of improper finite element modelling.The present study is to describe the results of research that has been conducted on the buckling behaviour of rectangular composite plates with holes.
It is necessary to establish some convenient parameters and terminology for descnlJing the plate geometry,loading and boundary conditions and material composition.The loading conditions discussed herein are uniaxial compression loads.For a plate with a hole, there are different ways of applying these loads that generally correspond to different deformation states.These deformation states are associated with the application of displacement or stress boundary conditions to introduce the loads.The compression loads are applied to a plate by either uniformly displacing or uniformly stressing two opposite exterior plate edge as illustrated in figure .3.
The compression displacement loadings are particularly important because they are representative of the load transmission that occurs between a plate -like subcomponent and an adjacent support structure that has a much higher relative in plane stiffuess.When a plate does not have a hole, the distinction between displacement and stress loadings is unnecessary.
Plates that are simply supported on all edges are referred to herein concisely as simply supported respectively.In all cases considered herein the hole boundary is a free edge.The classical buckling problem for a laminated composite plate requires satisfying the governing differiantial equations derived and the boundary conditions.The deflection and the bending moment along the boundary are zero.
where v is poisson's ratio, E. is modulus of elasticity for steel, Eoj is modulus of elasticity for concrete.
[Dl=lÑ The finite element method have proved to be extremely powerful and versatile tools for the analysis of a wide variety of plate problems.In figure 5,the model of a symmetrical plate with a central circular hole is shown.It is consists of 69 nodes and 13 elements.In figure 6, the model of a symmetrical plate without hole is composed of 81 nodes and 16 elements.In the models are used the nine -node isoparametric element.

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" " " .0 0 " 0 0 " " " " " 10•'0 The basic theory, incorporated into the computer program used for this investigation, can be found.Computer generated results are presented in tabular forms.The computer program used is coded in.FORTRAN.The program is developed using the formulation of the equation of orthotropic and isotropic plates.In this program, finite elements meshes are automatically found by subprogram MESH.Two dimensional finite element model is used.This program have nine -node plane isoparametric element.The stiffness matrix and the load vector are evaulated by subroutines presented here.There are five degrees of freedom per node in the element.The program containes only ninenode quadrilateral isoparametric elements.Two examples (one is with hole another is without hole) of application of the computer program are given below.
The plates are assumed to be under uniform axial loads the rectangular edges of the plates and the circular hole is free.Effects of hole size on critical buckling loads of the orthotropic rectangular simple plates under uniaxial compressions are shown in Tablel.Finite element meshes of the plate with and without hole are shown in figure 5 and 6.The rectangular reinforced plate simply supported at the edges, with dimensions 3m.x 3m.x O.2m.,was discussed.Number of laminates is 6.Buckling load o/the plate without hole is -144.3N/mm.

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figure Z. Be and E. are modulus of elasticity concrete and steel ' g are transformations of bending and geometric stiffuess matrices.
The presence of hole significantly affects the buckling behaviour and optimum design of laminated plates.Finite element results were obtained for simply supported stress loaded plates loaded by uniaxial compression.Results are presented that predict the effects of hole size.A basic characteristic of compression -loaded rectangular orthotropic plates with hole is that under certain circumtances they exhibit higher buckling loads than corresponding plates without hole.The effect deals with the plate bending stiffuess.Inherently associated with a centrally located hole is a loss in bending stif'fuess in the central region of a plate that grows in importance as the hole size increases.The increase in loss of bending stif'fuess due to increase in hole size yields a reduction in buckling resistance.l.LIN,C.C.,KUO,C.S.,1987.Buckling of Laminated with Holes.Journal of Composite Materials.Vol.23 -June 1989.2.AKBULUT,H.,KARAKUZU,R.,SAYMAN,O.,1995.Dikdortgen delikli kompozit plaklarda burkulma katsayJ1annm bulunmaSl.9.Ulusa1 Mekanik Kongresi.3. SAYMAN,O.,AKSOY,S.Kompozit Malzemeler.E.U.izMiR.4.TURVEY,G.J.,MARSHALL,I.H.,1995.Buckling and Postbuckling of Composite Plates.ChapmanHall.LONDON.5.SZlLARD,R.,1974.Theory and Analysis ofPlates.D.S.A.