Junction rule at any node junction in an electrical circuit, the sum of. Thin plate formulation this is similar to the beam formula, but since the plate is very wide we have a situation similar to plain strain. It takes into account the circuit topology seriesparallel, multiple sources, sources of different types, and components of different types. Following the assumptions of kirchhoff plate theory 25, the coordinate system is selected in a way that the axes x and y are in the middle plane of the laminated plates, whereas the axis z is. Usually, three assumptions are used to reduce the equations of three dimensional theory of elasticity to two dimensions. Two novel kirchhoff plate finite elements for the modified. Kirchhoff bending rectangular plate finite elements for msgt are developed as novel aspects.
Usually, three assumptions are used to reduce the equations of three dimensional theory of. In fact, the eulerbernoulli and timoshenko beam theories both have its counterpart in plate theory. Introduction the classical germainkirchhoff theory of thin elastic plates is known to be the leading term of an interior or outer asymptotic expunsion in. 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.
Jun 22, 2014 2012 10 10 mostafa abdalla lecture 1, about plate theory. The kirchhofflove theory is an extension of eulerbernoulli beam theory to thin plates. The kirchhofflove 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. The kirchhoff elastic rod, the nonlinear schrodinger equation. All this process describes how to derive the elastic equations for. Kirchhoff s law is a method of finding the voltage across every device and the current through every device. Kirchhoff plate theorybased electromechanicallycoupled. Although no doubt very effective for analysis of prob.
Modified strain gradient theory msgt is considered to be one of the most accurate methods for this, considering both theoretical compatibility and numerical accuracy. These equations are achieved via a transformation of the reference system from rectangular to polar. The transverse strain in kirchhoffs theory are assumed to be zero, while strain displacement relations implies that lateral deflection is. Static analysis of nanoplates based on the nonlocal kirchhoff.
Kirchhoff plate theory an overview sciencedirect topics. Introduction to the theory of plates stanford university. On the missing modes when using the exact frequency. Overview of plate theories in analogy to beams there are several different plate theories 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. This theory is an extension of eulerbernoulli beam theory and was developed in 1888 by love using assumptions proposed by kirchhoff. Pdf new look at kirchoffs theory of plates researchgate. Based on kirchhoff assumptions, at any point p, due to a small rotation displacement in the x direction u. Kantorovich variational method for the flexural analysis of. The second assumption can be relaxed to develop a more exact theory see 6. We will consider the classical thinplate theory or kirchhoff plate theory. The kirchhoff elastic rod, the nonlinear schrodinger. These three assumptions are the basis of the classical plate theory or the kirchhoff. Study on free vibration analysis of rectangular plate. This relationship is used to explicitly express the displacements of a plate particle according to the firstorder shear deformation theory in terms of the deflection of a homogeneous kirchhoff plate.
The thickness h is much smaller than the typical plate dimension, h. Batra 2 r department of modem mechanics university of science and technology of china hefei, anhui 230026, p. The kirchhofflove theory of plates is a twodimensional mathematical model that is used to. In other words, since rigid pavement thickness is very lesser than other two dimensions, transverse shear deformation is insignificant and can be neglected. His family were lutherans in the evangelical church of prussia. It is assumed that a midsurface plane can be used to represent the threedimensional plate in twodimensional form. This is the second part of a twopart paper on plate bending elements with shear effects included. Kirchhoffs law is a method of finding the voltage across every device and the current through every device. The starting point in the development of both these elements is, in essence, a plate shell theory which includes shear deformations.
In view of the aforementioned problems, wang and his coauthors 59 initiated studies to relate the solutions of kirchhoff or thin plate theory and mindlin or thick plate theory. Mindlin element based on mindlinreissner plate theory and assumed shear strain fieldspart ii. Kirchhoffs laws govern the conservation of charge and energy in electrical circuits. 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. This chapter addresses a thin plate theory, often called the classical plate theory cpt, or the kirchhoff plate theory, as well as the first order shear deformation theory for thick plates known as the reissnermindlin plate theory reissner, 1945. Again, this is similar to an assumption of the beam theory. In other words, since rigid pavement thickness is very lesser than other two dimensions, transverse shear deformation is insignificant and. For a unit width beam, flexural rigidity dei et 312. Kirchhoff plate theory and fe rigid pavement can be idealized using kirchhoff theory, which is applicable to thin plates cook et al, 1989. The term plate is usually reserved for flat structures, while the term shell refers to a curved plate. Thin plate b smallest side thick plate t 20 b 2000 1 10 1. The kirchhofflove theory also known as the classical plate theory cpt is based on three basic assumptions. Page 54 f cirak overview of plate theories in analogy to beams there are several different plate theories 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. Circuit theorykirchhoffs law wikibooks, open books for an.
The term plate is usually reserved for flat structures, while the term shell refers to. Circuit theorykirchhoffs law wikibooks, open books for. Sample plate geometry showing the midplane, or middle surface, and typical cartesian coordinate axes. Ac cording to the rules of topology, such a linking number defi. Kirchhofflove theory an overview sciencedirect topics. 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.
This law is also called kirchhoffs point rule, kirchhoffs junction rule or nodal rule, and kirchhoffs first rule. Shearing deformations xe shearing deformations classical thin plate theory is based on all three approximations and leads to the development of a fourth order partial differential equation in terms of the normal displacement of the plate. 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. Straindisplacement relations for nonlinear plate theory the chief characteristic of a thin flat plate is it flexibility. For the kirchhoff theory, the unknown quantities are the displacement components u0, v0, and w0. Example kirchhoffs voltage law kvl consider a simple one loop circuit voltages are numbered by the element name eg. Classical thin plate theory is based upon the assumptions initiated for the beams by bernoulli but it is first applied to plates and shells by love and kirchhoff. The displacement and rotation field is expanded by introducing some shear constraints. Based on the kirchhoff plate theory, the transverse deflection w0 of the levy solution may be written as. Deflection relationships between the homogeneous kirchhoff plate theory and different functionally graded plate theories zq. Note that pure plate elements are usually not available in most commercial. Solution from kirchhoffs theory in a simple text book problem of bending of simply supported square plate under doubly sinusoidal vertical load is reexamined concerning exact solution of the.
This theory is an extension of eulerbernoulli beam theory and was developed in 1888 by love 1 using assumptions proposed by kirchhoff. The theory assumes that a midsurface plane can be used to represent a three. The second assumption can be relaxed to develop a more exact. 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. This thin plate theory is also called the kirchhoff.
Kirchhoff plate theory kirchhoff plate theory 6 makes it easy to drive the basic equations for thin plates. These three assumptions are the basis of the classical plate theory or the kirchhoff plate theory. Kirchhoff theory for plates eulerbernoulli theory for beams. Thus, the kirchhoff plate theory can be considered a special case of the mindlin plate theory. Static analysis of nanoplates based on the nonlocal. The starting point in the development of both these elements is, in essence, a plateshell theory which includes shear deformations.
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 supported by ribs and longerons or stiffeners. 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. Linear and nonlinear plate theory references brush and almroth, buckling of bars, plates and shells, chp. These points, 1 6, are the foundation for plate bending theory that is usually referred to as the classical or kirchhoffs plate theory. First, the plate is assumed to be inextensible in the transverse direction. Gustav kirchhoff, german physicist who, with the chemist robert bunsen, firmly established the theory of spectrum analysis a technique for chemical analysis by analyzing the light emitted by a heated material, which kirchhoff applied to determine the composition of the sun. Multiscale homogenization in kirchhoffs nonlinear plate theory 3 and we seek to identify the e. Whereas the assumptions of beam theory deal with the kinematics of the crosssection of the beam, the assumptions of kirchhoff plate. All this process describes how to derive the elastic equations for circular thin plates. Development of the plate bending element basic behavior of geometry and deformation consider the thin plate in the xy plane of thickness t measured in the z direction shown in the figure below.
1309 1406 478 986 349 442 1198 1055 376 1011 325 389 1582 1386 1341 315 447 1014 1316 981 1119 683 422 702 299 104 1181 729