Residual error estimation for anisotropic kirchhoff plates. Solution of clamped rectangular plate problems robert l. What links here related changes upload file special pages permanent link page. Balch division of mechanics and computation department of mecanical engineering stanford university stretching and bending of plates fundamentals introduction a plate is a structural element which is thin and. The construction of a linear theory for the extensional and. Love shell theory is recasted in the frame of the tdc including all relevant mechanical aspects. Although no doubt very effective for analysis of prob. Summary the equations that govern kirchhoff love plate theory are solved using quadratic powellsabin bsplines. Galerkinvlasov method for the flexural analysis of. To take these facts into consideration, we have had to make many changes and additions. Formulation of problems in the general kirchhofflove theory of. We have presented kirchhoff love shell formulations for compressible and incompressible nonlinear hyperelastic materials. Singular shearforce states in elementary plate theory. Kirchhofflove plate theory theoretical physics physics.
It is wellknown from kirchhoff that, on the edge boundary of the plate, the specific shearforces and the curvegradient of the specific twistingmoments, measured per unit length, are. Kirchhoff theory for plates eulerbernoulli theory for beams. Kirchhofflove shell formulation based on triangular isogeometric. The shell kinematics are completely described by the midsurface metric and curvature variables while the thickness stretch is statically condensed using the plane stress condition. The theory for plates can be regarded as an extension of beam theory, in the sense that a beam is a 1d specialization of 2d plates.
In fact, the eulerbernoulli and timoshenko beam theories both have its counterpart in plate theory. The thickness is usually constant but may be variable and is measured normal to the middle surface of the plate, fig. Statics of plate o classical linear theory of plate o plane. The equation of motion of the plate is established as. Love shell is given and taken as the starting point to derive the weak form. The proposed model is capable of handling plates with complex geometries and boundary. The theory was developed in 1888 by love using assumptions proposed by kirchhoff. Classical plate theory is the thin plate theory based on love kirchhoff s hypothesis which makes assumptions similar to those made by the bernoullinavier hypothesis used in the theory of. Mindlins first order shear deformation plate theory has been developed by mindlin 6, mindlin and deresiewicz 7 and mindlin et al 8. We show that the most classical kirchhoff love theory of thin plates is compatible with the occurrence of a specific singular shearforce state in the interior of the body.
Nonlinear boundary conditions in kirchhofflove plate theory. Geometrically exact finite element formulations for highly. The weak formulation features functions from the sobolev space h 2. Plates a plate is a flat solid body whose thickness is small compared to the other dimensions and is subjected to bending loads. Pdf analysis of simply supported rectangular kirchhoff. The kirchhoff love plate theory, is based on three assumptions kirchhoff s hypothesis which reduce the equations of the three dimensional theory. Kirchoff 5 is considered as the one who formulated the extended plate theory. On the range of applicability of the reissnermindlin and. 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. Pdf kirchhoff love and reissner mindlin plates models.
Pdf extended finite element methods for thin cracked. 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 theory assumes that a midsurface plane can be used to represent a three. A rigorous derivation of the equations for the clamped. The kirchhoff love theor y of plates is a twodimensional mathematical model that is used to determine the stress es and deformation s in thin plate s subjected to force s and moment s.
It is the purpose of this note to show that in the asymptotic regime usually assumed in. This theory is an extension of eulerbernoulli beam theory and was developed in 1888 b y love 1 using assumptions proposed by kirchhoff. Mindlin plate theory the deformations and strains are therefore given by. Among these theories, kirchhoff love is an appealing theory for developing efficient and reliable finite shell elements formulation. This theory is an extension of eulerbernoulli beam theory and was developed in 1888 by love using assumptions proposed by kirchhoff. Mindlinreissner plate theory mindlin 1951 is an extension of kirchhoff love plate theory love 1888 which takes into account shear deformations through the thickness of the plate.
In this paper a new kirchhoff plate model is developed for the static analysis of isotropic micro plates with arbitrary shape based on a modified couple stress theory containing only one material length scale parameter which can capture the size effect. Love theory, especially for moderately thin plates, this assertion is not yet fully substantiated. Consider a square plate of length 1 m, width 1 m which is made of an isotropic material with youngs modulus 200 gpa and poissons ratio 0. This demo program solves the outofplane kirchhoff love equations on the unit square with uniform transverse loading and fully clamped boundary conditions. Linear and nonlinear plate theory references brush and almroth, buckling of bars, plates and shells, chp. It is assumed that a midsurface plane can be used to represent the threedimensional plate. Kirchhoff love hypothesis material property change. The proposed formulations are the first of this category that account for curved 3d initial. Bending analysis of simply supported and clamped circular plate p.
Concepts and applications of finite element analysis, 1996. Later, the theory of elasticity was extended as there were many researchers working on the plate and the extended plate theory was formulated. They are based on an improved version of the first. In this section, predictions from ansystm are compared with exact solutions for a pressure load of 100 kpa. 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. Love plate problem and ability to derive the basic formulations related to the problem. Institute of structural engineering page 1 method of finite elements i chapter 6 2d elements slidesarepreparedincollaborationwithdr. Powellsabin bsplines and unstructured standard tsplines. Request pdf kirchhoff love plate theory in this chapter we will discuss the kirchhoff love plate model, where the current director vector a3 is forced to be of unit length and orthogonal. For the kirchhoff theory, the unknown quantities are the displacement components u0, v0, and w0. Kirchhoff plate method of fundamental solutions meshless methods abstract in this paper a new kirchhoff plate model is developed for the static analysis of isotropic micro plates with arbitrary shape based on a modi. The theory assumes that a midsurface plane can be used.
Kirchhoff love plate theory, thus extending the scope of application to thick plates. They considered shear deformation effects on the plate flexural behaviour by assuming linear variations for all. In this paper we study the procedure of reducing the threedimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate. Eigenvalue analysis of thin plate with complicated shapes. Kirchhoff and mindlin plates university of british columbia. Isogeometric kirchhofflove shell formulations for general. Bending analysis of simply supported and clamped circular. Floors, ceilings, windows, disc brakes, ship decks, truck beds the learning objectives understand the theory of thin plate bending, its limitations, and its applications in design and analysis. Kirchhofflove plate theory free download as pdf file. Straindisplacement relations for nonlinear plate theory the chief characteristic of a thin flat plate is it flexibility. Plate theory kcpt thick plate theory reissner mindlin.
Following a short period of controversy, the complete theory, nowadays known as the kirchhoff love equations for bending of. Then, boundary terms for the relevant boundary conditions of kirchho. The kirchhofflove theory of plates is a twodimensional mathematical model that is used to. Taylory,and sanjay govindjeez structural engineering, mechanics, and materials department of civil and environmental engineering university of california, berkeley berkeley, ca, usa summary in this brief note we present an e cient scheme for determining very accurate solutions to the clamped. We will consider the classical thin plate theory or kirchhoff plate theory.
A new kirchhoff plate model based on a modified couple. Similarily, displacement vector can be divided into two components. Therefore, a conforming finite element fe ansatz has to incorporate ansatz functions from h 2 as well. The kirchhoff love theory is an extension of eulerbernoulli beam theory to thin plates. On the shell theory on the nanoscale with surface stresses hal.
When the thickness of the plate is 125 m, kirchhoff love theory for thin plates is applicable. 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 starting point in the development of both these elements is, in essence, a plate shell theory which includes shear deformations. The kirchhoff love plate theory applies to the bending of thin plates and results in a partial differential equation of fourth order. Introduction to the theory of plates stanford university. An accurate model helps the control engineer to understand the fundamental states of the system, such as locations of. It is assumed that a midsurface plane can be used to represent the threedimensional plate in twodimensional form. The thin shells undergoing kirchhoff love kl constraints are described using three. Radiation heat transfer radiation differs from conduction and convection heat t transfer mechanisms, in the sense that it does not require the presence of a material medium to occur. Chapter 12 plate bending elements learning objectives to introduce basic concepts of plate bending. In the late 1900s, the theory of finite elements was evolved which is the basis for all the. Read nonlinear boundary conditions in kirchhofflove plate theory, journal of elasticity on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. In the kirchhoff love plate model, there are two fracture modes.
1268 1073 214 625 851 585 1417 1631 481 1063 939 940 286 1300 679 64 107 963 378 178 573 1072 1315 3 651 1626 1002 1108 647 652 590 852 1458 1284 1149 116 815 1020 133 461 1025 1478 835 1374