Pdf optimal design of a kirchhofflove plate of variable. Deformation of a thin plate highlighting the displacement, the midsurface red and the normal to the midsurface blue. Boundary element bending analysis of moderately thick. This value is reported in table 3 for different adjacent boundary conditions. Application of phtsplines in bending and vibration. Kantorovich variational method for the flexural analysis of cscs. Navier 3 can be considered as the originator of the modern theory. Mod12 lec12 theory of flat plate collectors liquid based a duration. Again, this is similar to an assumption of the beam theory. In order to avoid locking, some special treatments are mandatory, for example selective reduced integration. Kirchhoff love plate theory, is applied for thin plate modelling. This theory is an extension of eulerbernoulli beam theory and was developed in 1888 by love using assumptions proposed by kirchhoff. Solution of clamped rectangular plate problems robert l.
Isogeometric collocation for kirchhofflove plates and. Singular shearforce states in elementary plate theory. The same dimension reduction procedure is eventually applied to processes which are sensitive to delamination modes, namely opening vs. By letting the thickness of the plates go to zero, a rateindependent delamination model for a laminated kirchhoff love plate is obtained as limit of these quasistatic processes. The firstorder shear deformation plate theory fsdpt, described in detail in chapter 4, is an extension of the wellknown timoshenko beam theory andor the mindlinreissner plate theory that is applied to laminated composite plates.
Theory of rectangular plates part 1 introduction video. Sophie germain, the first mathematician who succeeded in obtaining a theory of the flexure of a thin plane plate was poisson. We deal with the kirchhoff love model for a bending thin plate with physical boundary conditions. Justification of the nonlinear kirchhofflove theory of. The principal additions are 1 an article on deflection of plates due to transverse shear, 2 an article on stress. Concepts and applications of finite element analysis, 1996. 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. A new approach is introduced for deriving a mixed variational formulation for kirchhoff plate bending problems with mixed boundary conditions involving clamped, simply supported, and free boundary parts. Nonlinear theory of elastic plates 1st edition elsevier. Geometrically exact finite element formulations for. Kirchhoff s laws and circuit analysis ec 2 circuit analysis. Nonlinear boundary conditions in kirchhofflove plate theory 65 2.
We select only the first three flexural modes of the em thin plate which. Pdf analysis of simply supported rectangular kirchhoff. Electrical circuit theory and technology ches iitm. We propose here a new mixed formulation, based on a decomposition of the bending moment. The kirchhofflove theory of plates is a twodimensional mathematical model that is used to. Kirchhoff love and reissner mindlin plate s models. Fiber reinforcements and general theory of composites. Kirchhoff hypothesis an overview sciencedirect topics. Free edge vertical deflection and their derivatives.
The mindlinreissner theory of plates is an extension of kirchhofflove plate theory that takes. The small free vibrations and deformation of a thin. The theory assumes that a midsurface plane can be used to represent a threedimensional plate in twodimensional form. 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. Pdf kirchhoff love and reissner mindlin plates models. The extension of the eulerbernoulli beam theory to plates is the kirchhoff plate theory. Pdf quasistatic delamination of sandwichlike kirchhoff.
This theory is an extension of eulerbernoulli beam theory and was developed in 1888 b y love 1 using assumptions proposed by kirchhoff. A decomposition result for kirchhoff plate bending. Wikiproject civil engineering rated cclass this article is within the scope of wikiproject civil engineering, a collaborative effort to improve the coverage of civil engineering on wikipedia. Mindlin element based on mindlinreissner plate theory and assumed shear strain fieldspart ii. Electrical circuit theory and technology john bird. Then later on the theory of elasticity was formulated. This choice will allow us to improve the numerical solution by employing local adaptive refinement near the crack tip and crack faces, while preserving the geometry exactness. Stochastic analysis of moderately thick plates using the generalized polynomial chaos and element free galerkin method. Finitedifference timedomain analysis of structureborne sound using a plate model based on the kirchhoff love plate theory.
An approximate thickness optimization of a rectangular kirchhoff love plate with variable stiffness under uniform load is performed in this paper. Unlimited viewing of the articlechapter pdf and any associated supplements and figures. Shear deformation and higher order theories of plates in bending are generally based on plate element equilibrium equations derived either through variational principles or other methods. These three assumptions are the basis of the classical plate theory or the kirchhoff plate theory. The second assumption can be relaxed to develop a more exact theory. The kirchhoff love theory is an extension of eulerbernoulli beam theory to thin plates. To take these facts into consideration, we have had to make many changes and additions. 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. 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. Then it was the german physicist chladni 2 who discovered the various modes of free vibrations. In this paper we follow the approach of ciarlet and destuynder, as presented in the textbook 22 and rigorously develop equations for a poroelastic plate. Two different rotation interpolation schemes with strong. The present work focuses on geometrically exact finite elements for highly slender beams.
Bending of the plate will cause material above and below this midplane to. Nonlinear boundary conditions in kirchhofflove plate theory. Introduction to the theory of plates stanford university. Norton since 1960, there have been two major theoretical advances in the earth sciences. Indeed, it is demonstrated in, that at a corner of a kirchhoff love plate, when its surrounding is loaded by a pressure not a point force, the exact deflection has a critical regularity r u 3. Mindlin plate theory the deformations and strains are therefore given by. The benefits of such an approach are evident, since the time taken from design to analysis is greatly. In the present work, we will pair nurbs geometries with a phtsplines solution basis to model fracture mechanics problems under the kirchhoff love plate theory. Similarily, displacement vector can be divided into two components. In the framework of isotropic homogeneous nonlinear elasticity for a st.
The middle plane of the plate remains free of inplane stressstrain. It is assumed that a midsurface plane can be used to represent the threedimensional plate in twodimensional form. 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 statically interrelated. They involve coupling of flexure with torsion torsiontype problem and if applied vertical load is along one face of the plate, coupling even with extension problem. Venant kirchhoff material, we consider a threedimensional plate of thickness. 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. Deformation of a plate highlighting the displacement, the midsurface red and the normal to the midsurface blue. Flexural wave band gaps in a multiresonator elastic. In his memoir he obtains the differential equation for the deflection of the plate, which is generally admitted, and certain boundaryconditions. Thin plate theory kirchoffs classical plate theory kcpt thick plate theory reissner mindlin. The theory was developed in 1888 by love using assumptions proposed by kirchhoff. An accurate model helps the control engineer to understand. Our strategy of deriving the poroelastic plate model is similar to that for.
Bending moment mixed method for the kirchhofflove plate. 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. The theory assumes that a midsurface plane can be used to represent a three. The formulation is developed based on the modified couple stress theory in conjunction with kirchhoff love plate theory. A plate is a flat solid body whose thickness is small compared to the other dimensions and is subjected to bending loads. These compatibility conditions take the form of variational equations with divergence free tensor fields as testfunctions, by contrast with saintvenant compatibility conditions, which take the form of systems of. Using a new method that we call the singular inverse method, we prove the existence of a solution rescaled uniformly in. Pdf nonlinear boundary conditions in kirchhofflove. Geometrically nonlinear plate theory crucial such as the one introduced for buckling of plates physical. Isogeometric analysis iga is a fundamental step forward in computational mechanics that offers the possibility of integrating methods for analysis into computer aided design cad tools and vice versa. This suggests that this locking effect is due to the xfem enrichment. Hirschberg eindhoven university of technology 28 nov 2019 this is an extended and revised edition of iwde 9206. Boundary element bending analysis of moderately thick plates. This study deals with forced vibration analysis of a microplate subjected to a moving load.