The proposed displacement field departs with respect to classical kirchhoff love plate theory 7 in the following. We will consider the classical thin plate theory or kirchhoff plate theory. Similarily, displacement vector can be divided into two components. The long, slender wings of an aircraft can be analyzed, to a first approximation, as beams, but a more refined analysis will treat the upper and lower skins of the wing as thin plates. The kirchhofflove plate theory klpt 9 is used for the analysis of thin plates, and which is based on assumptions that are closely related to the eulerbernoulli. On the regularity properties of solutions of elliptic. How twodimensional bending can extraordinarily stiffen. Although the kirchhoff love plate theory and its extension mindlinreissner. The most widely used plate theory is the kirchhoff love plate theory, also called the classical thin plate theory 3. Pdf extended finite element methods for thin cracked. In this chapter we combine the threedimensional theory for large deformations, described in. Smart materials and intelligent system design prof. Nonlinear boundary conditions in kirchhofflove plate theory 65 2. An approximate thickness optimization of a rectangular kirchhoff love plate with variable stiffness under uniform load is performed in this paper.
Classical thin plate theory is based upon the assumption which were initiated for beam by bernoulli. Pdf nonlinear boundary conditions in kirchhofflove. Mindlins first order shear deformation plate theory has been developed by mindlin 6, mindlin and deresiewicz 7 and mindlin et al 8. The thickness of the plate hx,y and transverse load mg pzx,y vary gradually we are away from the regions of stress concentration. A posteriori estimates for conforming kirchhoff plate elements. Furthermore, for through the thickness cracks, the limit of the energy release rate of the threedimensional model can be expressed with the kirchhoff love. The thin shells undergoing kirchhoff love kl constraints are described using three. The kirchhoff classical plate theory cpt for the pure bending case is based on the displacement field. The theory was developed in 1888 by love using assumptions proposed by kirchhoff.
Kirchhoff love s method lendutan pelat segiempat rectangular slabs deflection x y z x y z m x m x m y m y persamaan umum pelat klasik. The midplane of the plate lies along the two long dimensions of the plate, whereas the normal to the plate extends along the shorter dimension. For thin plates, love and kirchhoff s plate bending theory can be used. In this chapter, we derive the basic equations which describe the behavior of plates taking advantage of the plate s thin, planar character. For thick plates,mindlinreissner plate bending theory can be used. Lyly, rolf stenberg, theory manual of the stabilized mitc plate and flat. But w 0 at various points the deformation will be different right. Kirchhoff s thin plate theory has been described in chapter 2. A new approach to mixed methods for kirchhofflove plates and. Constitutive equations based on kirchhoff plate theory since the piezoelectric patches are typically manufactured as a thin plate, peh skin can be modeled as a twodimensional kirchhoff plate analogous to the onedimensional euler bernoulli beam. Love plate problem and ability to derive the basic formulations related to the problem.
Kirchoff 5 is considered as the one who formulated the extended plate theory. Plate theory kcpt thick plate theory reissner mindlin. Love theory, especially for moderately thin plates, this assertion is not yet fully substantiated. Institute of structural engineering page 1 method of finite elements i chapter 6 2d elements slidesarepreparedincollaborationwithdr. The kirchhofflove theory also known as the classical plate theory cpt is based on three basic assumptions. On the other hand, plate theory is concerned mainly with lateral loading. Kirchhoff love plate theory, thus extending the scope of application to thick plates.
In the framework of the kirchhoff love asymptotic theory of elastic thin plates we consider the unilateral contact problem with friction for a plate on a rigid foundation signorini problem with friction. On this kind of plate, the shear effect can generally be neglected and consequently the kirchhoff love model is mechanically appropriate. Theory is closely related to saintvenants compatibility condition in the elasticity theory 46. On the range of applicability of the reissnermindlin and. Kirchhofflove theory an overview sciencedirect topics. Although no doubt very effective for analysis of prob. The theory assumes that a midsurface plane can be used to represent a threedimensional plate. First, the plate is assumed to be inextensible in the transverse direction.
Partial discretization, which is described in this paper, reduces the number. The kirchhofflove theory of plates is a twodimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to. Then, it is assumed that the transverse shear deformations are negligible. Introduction to the theory of plates stanford university. To take these facts into consideration, we have had to make many changes and additions. Remark other contributions for the uncoupled cases follow. An accurate modelling of piezoelectric multilayer plates. Existence of an optimal shape of the thin rigid inclusions in the kirchhoff love plate, journal of applied.
For thick plates this theory is too restrictive, since also transverse shear deformations have to. Kirchhoff love theory for kirchhoff plates 202014 aircraft structures kirchhofflove plates 19 ga 2 applied pressure 3. Chapter 12 plate bending elements learning objectives to introduce basic concepts of plate bending. The kirchhoff love theory also known as the classical plate theory cpt is based on three basic assumptions. On the range of applicability of the reissnermindlin and kirchhoff. Inplane terms of the strain tensors from the general expression, eq.
The theory assumes that a midsurface plane can be used to represent a three. The mindlinreissner theory of plates is an extension of kirchhofflove plate theory that takes into account shear deformations throughthethickness of a plate. The term plate is usually reserved for flat structures, while the term shell refers to a curved plate. Bending of rectangular plates are treated in chapter 3. It is a refinement from kirchhoff love plate shell theory with. A deep collocation method for the bending analysis of. The more common model is based on the elementary beam or plate theories using the kinematic assumption of the love kirchhoff s theory. Galerkinvlasov method for the flexural analysis of. The authors propose an original method for formulating problems of optimal design for plate structures of variable thickness. This theory is an extension of eulerbernoulli beam theory and was developed in 1888 by love using assumptions proposed by kirchhoff.
The kirchhoff love plate theory is based on the kirchhoff love hypothesis which makes assumptions similar to those in the bernoullinavier hypothesis used in the theory. What is the difference between the mindlinreissner theory. The extension of the eulerbernoulli beam theory to plates is the kirchhoff plate theory suitable only for thin plates the extension of timoshenko beam theory to plates is the reissnermindlin plate theory suitable for thick and thin plates as discussed for beams the related finite elements have problems if applied to thin problems in very thin. It is the purpose of this note to show that in the asymptotic regime usually assumed in. This demo program solves the outofplane kirchhoff love equations on the unit square with uniform transverse loading and fully clamped boundary conditions. So, if this fellow is subjected to some kind of a bending, if you imagine that there is some bending that is taking place in the system then in a very kind of a you know. Kirchhofflove plate theory, the solution space of the problem requires c1 continuity while the. This approach can directly be employed to establish simple shell elements and our dkt discrete kirchhoff trian gular element is one of that kind57. The thickness of the plate h x,y is an order of magni tude smaller than the dimensions in the x or y direc tion. An accurate model helps the control engineer to understand.
Merdaci slimane study and comparison of different plate theory. They considered shear deformation effects on the plate flexural behaviour by assuming linear variations for all. Implementation of a highorder kirchhofflove shell lehrstuhl fur. The present chapter focuses on another type of structural component, plates, which are defined as structures possessing one dimension far smaller than the other two. The twodimensional plate theories can be classified into two types. The kirchhoff love theory is an extension of eulerbernoulli beam theory to thin plates. Optimal design of a kirchhofflove plate of variable. The approach is a generalization of the onedimensional euler bernoulli beam theory, which exploits the slender shape of a. The principal additions are 1 an article on deflection of plates due to transverse shear, 2 an article on stress. Mindlinreissner theory is a plate shell theory for structures having one dimension much smaller than the two other dimensions. It is assumed that a midsurface plane can be used to represent the threedimensional plate in twodimensional form. International journal of civil, mechanical and energy.
Kirchhoff plate theorybased electromechanicallycoupled. In the late 1900s, the theory of finite elements was evolved which is the basis for all the. The kirchhoff love theory of plates is a twodimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. First, we notice, when the thickness vanishes, that the order of the friction force must be lower than that of the contact pressure. Later, the theory of elasticity was extended as there were many researchers working on the plate and the extended plate theory was formulated. The simplest approach to piezoelectric beams and plates incorporate effectives forces and moments induced.
128 410 427 343 932 186 1453 1042 937 1155 878 1072 1047 1327 413 1523 830 919 434 21 1303 818 216 1326 535 1086 1200 465