The kirchhofflove theory also known as the classical plate theory cpt is based on three basic assumptions. Kirchhoff love plate theory, thus extending the scope of application to thick plates. The theory assumes that a midsurface plane can be used to represent a three. 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. It is a refinement from kirchhoff love plate shell theory with. The kirchhofflove theory of plates is a twodimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to. The kirchhoff love theory also known as the classical plate theory cpt is based on three basic assumptions. The simplest approach to piezoelectric beams and plates incorporate effectives forces and moments induced. On the range of applicability of the reissnermindlin and kirchhoff.
This demo program solves the outofplane kirchhoff love equations on the unit square with uniform transverse loading and fully clamped boundary conditions. The theory assumes that a midsurface plane can be used to represent a threedimensional plate. For thick plates this theory is too restrictive, since also transverse shear deformations have to. 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. To take these facts into consideration, we have had to make many changes and additions. 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. 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. Nonlinear boundary conditions in kirchhofflove plate theory 65 2. Nonlinear boundary conditions in kirchhofflove plate theory. An accurate modelling of piezoelectric multilayer plates. Remark other contributions for the uncoupled cases follow.
In the late 1900s, the theory of finite elements was evolved which is the basis for all the. For thin plates, love and kirchhoff s plate bending theory can be used. Mindlinreissner theory is a plate shell theory for structures having one dimension much smaller than the two other dimensions. Kirchhoff plate theorybased electromechanicallycoupled. The twodimensional plate theories can be classified into two types. 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. Bending of rectangular plates are treated in chapter 3. International journal of civil, mechanical and energy. They considered shear deformation effects on the plate flexural behaviour by assuming linear variations for all. Classical thin plate theory is based upon the assumption which were initiated for beam by bernoulli.
What is the difference between the mindlinreissner theory. The proposed displacement field departs with respect to classical kirchhoff love plate theory 7 in the following. Introduction to the theory of plates stanford university. On the regularity properties of solutions of elliptic. Kirchhofflove theory an overview sciencedirect topics. Pdf extended finite element methods for thin cracked. Optimal design of a kirchhofflove plate of variable.
Kirchhoff love theory for kirchhoff plates 202014 aircraft structures kirchhofflove plates 19 ga 2 applied pressure 3. Love theory, especially for moderately thin plates, this assertion is not yet fully substantiated. 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. Lyly, rolf stenberg, theory manual of the stabilized mitc plate and flat. For thick plates,mindlinreissner plate bending theory can be used. Although the kirchhoff love plate theory and its extension mindlinreissner. It is the purpose of this note to show that in the asymptotic regime usually assumed in. The principal additions are 1 an article on deflection of plates due to transverse shear, 2 an article on stress. Plate theory kcpt thick plate theory reissner mindlin. 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.
On this kind of plate, the shear effect can generally be neglected and consequently the kirchhoff love model is mechanically appropriate. 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. Implementation of a highorder kirchhofflove shell lehrstuhl fur. It is assumed that a midsurface plane can be used to represent the threedimensional plate in twodimensional form. A posteriori estimates for conforming kirchhoff plate elements. First, the plate is assumed to be inextensible in the transverse direction.
We will consider the classical thin plate theory or kirchhoff plate theory. The thickness of the plate hx,y and transverse load mg pzx,y vary gradually we are away from the regions of stress concentration. The approach is a generalization of the onedimensional euler bernoulli beam theory, which exploits the slender shape of a. Although no doubt very effective for analysis of prob. 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. The thin shells undergoing kirchhoff love kl constraints are described using three. Merdaci slimane study and comparison of different plate theory. 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.
The mindlinreissner theory of plates is an extension of kirchhofflove plate theory that takes into account shear deformations throughthethickness of a plate. The more common model is based on the elementary beam or plate theories using the kinematic assumption of the love kirchhoff s theory. Kirchhofflove plate theory, the solution space of the problem requires c1 continuity while the. The classical laminated plate theory clpt of love kirchhoff it is called a thin plate when the arrow generated by the shear deformations remains negligible in front of the arrow generated by the. Later, the theory of elasticity was extended as there were many researchers working on the plate and the extended plate theory was formulated.
Kirchhoff s thin plate theory has been described in chapter 2. In this chapter, we derive the basic equations which describe the behavior of plates taking advantage of the plate s thin, planar character. Furthermore, for through the thickness cracks, the limit of the energy release rate of the threedimensional model can be expressed with the kirchhoff love. Inplane terms of the strain tensors from the general expression, eq. An accurate model helps the control engineer to understand. Existence of an optimal shape of the thin rigid inclusions in the kirchhoff love plate, journal of applied. The starting point in the development of both these elements is, in essence, a plate shell theory which includes shear deformations. The authors propose an original method for formulating problems of optimal design for plate structures of variable thickness. How twodimensional bending can extraordinarily stiffen. 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. The kirchhoff love theory is an extension of eulerbernoulli beam theory to thin plates. On the range of applicability of the reissnermindlin and. Similarily, displacement vector can be divided into two components. Kirchoff 5 is considered as the one who formulated the extended plate theory.
But w 0 at various points the deformation will be different right. This approach can directly be employed to establish simple shell elements and our dkt discrete kirchhoff trian gular element is one of that kind57. On the other hand, plate theory is concerned mainly with lateral loading. These three assumptions are the basis of the classical plate theory or the kirchhoff plate theory. Pdf nonlinear boundary conditions in kirchhofflove. This theory is an extension of eulerbernoulli beam theory and was developed in 1888 by love using assumptions proposed by kirchhoff. First, we notice, when the thickness vanishes, that the order of the friction force must be lower than that of the contact pressure. The theory was developed in 1888 by love using assumptions proposed by kirchhoff. An approximate thickness optimization of a rectangular kirchhoff love plate with variable stiffness under uniform load is performed in this paper. In this chapter we combine the threedimensional theory for large deformations, described in. On the regularity properties of solutions of elliptic differential equations. The term plate is usually reserved for flat structures, while the term shell refers to a curved plate. Theory is closely related to saintvenants compatibility condition in the elasticity theory 46. Institute of structural engineering page 1 method of finite elements i chapter 6 2d elements slidesarepreparedincollaborationwithdr.
The kirchhoff classical plate theory cpt for the pure bending case is based on the displacement field. Chapter 12 plate bending elements learning objectives to introduce basic concepts of plate bending. 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. Smart materials and intelligent system design prof. Mindlins first order shear deformation plate theory has been developed by mindlin 6, mindlin and deresiewicz 7 and mindlin et al 8.
1393 493 913 1325 181 659 443 374 1464 475 1297 1283 1044 296 219 264 1620 771 984 460 326 981 555 29 358 1399 1446 432 834 407