The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads. Reissner was elected a fellow by the american academy ofarts andsciences in 1950. In fact, the eulerbernoulli and timoshenko beam theories both have its counterpart in plate theory. If the basis functionsn a nare chosen as piecewise. However, it did not follow exactly the original derivation from reissner. From the generalized virtual work principle, where the unity constraint of the rotational quaternion is properly considered and the consistent energy complements of the rotational quaternions are employed, we derive the weak kinematic equations in the quaternionbased description. Aerospace engineering structural analysis wikipedia. Hans reissner followed a call from his alma mater, the technische hochschule berlin. Modified bolle reissner theory of plates including. On the differentiation of the rodrigues formula and its. On a variational theorem in elasticity reissner 1950. However, for a genuine nitestrain formulation, we should also adopt a consistent constitutive model and it is not immediately obvious how this could be implemented. Bernoulli beam theory, which exploits the slender shape of a beam.
Kirchhoff theory for plates eulerbernoulli theory for beams. Leung and mao 27 compared the solution between movable and immovable edges of simply supported rectangular plates using galerkins method. A fournode plate bending element based on mindlinreissner. A verification and validation of the geometrically exact beam.
The boundary layer for the reissnermindlin plate model. Large deformation analyses of spaceframe structures. Eightnode reissnermindlin plate element based on boundary. The key idea is to make use of the equilibrium equations set, in terms of stress resultants, in order to deduce on the energy conjugate strain measure through application of the virtual work principle. A higherorder bending theory for laminated composite and. A simple finite element method for the reissner mindlin plate model in the primitive variables is presented and analyzed.
A study of the edgezone equation of mindlinreissner. The structure of the solution of the reissner mindlin plate equations is investigated, emphasizing its dependence on the plate thickness. On the generalization of reissner plate theory to laminated. Mindlin, 1951 is applied for thick plates, where the shear deformation and rotary inertia effects are included. Professor reissner s work contributed to improved understanding of the mechanics of plates, shells, and beams, of the dynamics of structures, and of the theory of elasticity, as well as advancing knowledge of turbulence, aerodynamics, and wing theory. Page 72 f cirak the extension of timoshenko beam theory to plates is the reissner mindlin plate theory in reissner mindlin plate theory the outofplane shear deformations are nonzero in contrast to kirchhoff plate theory almost all commercial codes abaqus, lsdyna, ansys, use reissner mindlin type plate finite elements assumed displacements during loading. Pdf geometrically exact finite element formulations for curved. It will be shown that the geometrically exact kirchhofflove beam elements. A uniformly accurate finite element method for the. Reissner s work represents the foundation of the largedisplacement finitestrain theory of sheardeformable beams 9. Objectivity of strain measures in the geometrically exact three.
We apply the scheme for hyperbolic equations developed in 7, 10, 11. Hans reissner was professor of applied mechanics and the founder of the aerodynamics institute at the aachen technische hochschule. The derivation of this new plate theory was inspired from reissner s ideas. The extension of timoshenko beam theory to plates is the reissner mindlin plate theory in reissner mindlin plate theory the outofplane shear deformations are nonzero in contrast to kirchhoff plate theory almost all commercial codes abaqus, lsdyna, ansys, use reissner mindlin type plate finite elements.
Geometrically exact finite element formulations for curved slender beams. Mathematical and computer modelling of dynamical systems. Reissner, least work solutions of shear lag problems, journal of the. Vibration of a reissnermindlintimoshenko platebeam system. A plate theory takes advantage of this disparity in length scale to reduce the full threedimensional solid mechanics problem to a twodimensional problem. Reissnermindlin plate theory an overview sciencedirect. The stress resultants m,, m, and t, in figure 2 are defined with respect to coordinate directions n and s, normal to and along the boundary curve. This article deals with the nonlinear modelling of beams that are bent, sheared and. 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. This plate theory gives very good results with laminated plates 4 and was also extended to periodic plates 5, 6. Both mindlins and reissner s theories are sixth order. Modeling stenttype structures using geometrically exact. Reissner mindlin plate bending finite elements bo haggblad asea brown boveri corporate research, s721 78 vasterhs, sweden klausjurgen bathe massachusetts institute of technology, cambridge, ma 029, u. Simo reissner theory, archives of computational methods in engineering, 181, 2017.
The reissner s theory 11, which can be used either uate internal force concentrations due to loads distributed for thin or thick plates, takes into account the shear deforma over small regions that very often appear in practical prob tion effect, defining six boundary values. The beam theory is used in the design and analysis of a wide range of structures, from buildings to bridges to the loadbearing bones of the human body. The reissner variational principle is used in the updated lagrangian corotational reference frame, to derive an explicit expression for the. Beam theory continued jn reddy axial displacement is uncoupled from transverse displacement. In the article we introduce an analytical solution for reissners large deflection finitestrain planar beam subject to an end force and a bending. A refined zigzag beam theory for composite and sandwich beams. The geometrically exact beam theory in skew coordinates is derived in section 3.
He received the eminent scientist award from the wessex institute of technology in 1985. Geometrically exact finite element formulations for. The reissner variational principle 1953 see also atluri and reissner 1989 is used. Structural analysis of arches in plane with a family of. Haim baruh for a thin plate or beam, if the deformation is on the order of the thickness and. As presented there, the variational indicator of a mindlin reissner plate is, in linear elastic static analysis, where. Although the theoretical basis of shearfree kirchhofflove beam formulations has a much longer tradition than the simo reissner theory of. On finite element implementation of geometrically nonlinear. Pdf a bem formulation based on reissners theory to. On the analysis ofthe timoshenko beam theory with and without internal damping. On a consistent theory, and variational formulation of. Both theories include inplane shear strains and both are extensions of kirchhofflove plate theory incorporating firstorder shear effects. Objectivity of strain measures in the geometrically exact threedimensional beam theory and its niteelement implementation. Reissner type beam element formulation by simo and vuquoc.
Plate theory kcpt thick plate theory reissner mindlin. Geometrically exact finite element formulations for slender beams. The reissner theory is slightly different and is a static counterpart of the uflyandmindlin theory. This is a wikipedia book, a collection of wikipedia articles that can be easily saved, imported by an external electronic rendering service, and ordered as a printed book. Cosserat continua based on a geometrically nonlinear beam theory. The theory is often referred to as the geometrically exact finitestrain beam theory. Pdf introduction to plate bending theory nirajan paudel. The plate element obtained from our general 4node shell element is based on the mindlinreissner plate theory and represents an extension of the formulation given in reference 2, pp. Pdf a bem formulation based on reissners theory to perform. The three bending equations in this theory are often reformulated in terms of a potential function and the trans.
We study the behavior of solutions of ve di erent boundary value problems for the reissner mindlin plate model emphasizing the structure of the dependence of the solutions on the plate thickness. Mindlins theory is displacement based, whereas reissner s is stress based. Specifications of boundary conditions for reissnermindlin. Analytical solutions are obtained for the interior and edgezone equations of mindlin reissner plate theory in bending of composite circular sector plates laminated of transversely. The reissner mindlin theory does not require the crosssection to be perpendicular to the axial axes after deformation, as shown in figure 2. Finite element methods for timoshenko beam, circular arch. Apr 01, 1995 threedimensional reissner s beam theory the threedimensional beam theory discussed here has first been presented by reissner 11. Sep 16, 2008 in the present paper, we present a continuum mechanics based derivation of reissner s equations for largedisplacements and finitestrains of beams, where we restrict ourselves to the case of plane deformations of originally straight bernoullieuler beams. Unified approach for beam to beam contact, computer methods in applied mechanics and engineering, 315.
In section 4, we apply a spatial discretization based on. Relevant formulas valid in this natural system can be found in the appendix. Geometrically exact finite element formulations for highly. Oplinger mechanics and structures branch june 1991j 4uj2 71991 approved for public release. Reissner defined the onedimensional 1d strains in terms of virtual displacement and virtual rotation. A continuummechanics interpretation of reissners nonlinear. The theory was proposed in 1951 by raymond mindlin.
Thick plate theory the mindlin reissner theory of plates is an extension of kirchhofflove plate theory that considers shear deformations throughthethickness of a plate. In fact, timoshenko beam theory 14 is an extension of the eulerbernoulli beam theory to allow for the effect of transverse shear deformation by relaxing the normality assumption. Whitney, analysis of interlaminar mode ii bending specimens using a higher order beam theory, journal of reinforced plastics and composites, 10. Several alternative plate theories have appeared in the literature, among which those of mindlin 1951 and reissner 1945 are the most well known. While at the university of california at san diego, reissner researched principals of elasticity and problems concerning the behavior of elastic beam plates and shells. Summary plate bending finite elements based on the reissner mindlin theory offer improved possibilities to pursue. In all these papers, classical plate and beam theories are used.
However, simo reissner beam theory is unable to account for large strain until. Martinez modified bolle reissner theory of plates including transverse shear deformations latin american journal of solids and structures 12 2015 295316 a survey of various shear deformation theories on plates can be found in the works of reddy and. Professor reissner was born in aachen, germany, in 19, where his father dr. In this paper, the basic finite element formulation of a newly developed family of variable thickness, curved, c 0 continuity mindlin reissner model curved beam elements which include shear deformation and rotatory inertia effects is presented. Modeling stenttype structures using geometrically exact beam. Finite element implementation of the threedimensional finitestrain beam theory of reissner is considered in this work. The beam element with only linear shape functions appears not to be ideal for very thin beams the problem is caused by nonmatching u 3 and. The accuracy, convergence and efficiency of these newly developed curved beam elements are explored through a series of analyses of arch structures. Numerical results for shearlock free finite elements based. So for the reissner mindlin plate problem, it seems meaningful only to use lower order elements. Comparisons with exact solutions show that only the bendinggradient theory captures second order e. Lets try with using only one integration point for integrating the element shear stiffness matrix. The present work focuses on geometrically exact finite elements for highly slender beams. The new finite elements based on the timoshenko beam theory, reissner mindlin plate theory developed by me using special shape functions and standard finite element procedure are applied to the analysis of beams and plates.
Mindlin reissner theory is also referred to as the firstorder shear deformation plate theory fsdt. Thick plate theory reissner mindlin plate theory mpt 5 w. Analytical derivation of a general 2d nonprismatic beam. Its application to a square beam lattice reveals that classical reissner mindlin theory cannot properly model such microstructures. Pdf geometrically exact finite element formulations for slender. For the transverse displacement, rotation, and shear stress. A refined zigzag beam theory for composite and sandwich beams alexander tessler structural mechanics and concepts branch nasa langley research center mail stop 190, hampton, virginia, 23681 2199, u. Other mechanisms, for example twisting of the beam, are not allowed for in this theory. The accuracy, convergence and efficiency of these newly developed curved beam elements are explored. Timoshenko beam theory is not always more accurate than elementary beam theory nicholson, j.
Stress distribution in terms of displacement field. The numerical results show that accurate solution is obtained for less number of elements. The theory is often referred to as the geometrically exact nitestrain beam theory reissner 1972, 1973, 1981. Specifications of boundary conditions for plate bending 98 5 here p px, y is the applied pressure load. In a horizontal beam supported at the ends and loaded downwards in the middle, the material at the overside of the beam is compressed while the. University of california, san diego search for more papers by this author. A continuum mechanics based derivation of reissners large. In the quasistatic case, the amount of bending deflection and the stresses that develop are assumed not to change over time.
Journal of engineering mechanics june 2000 647 study of edgezone equation of mindlin reissner plate theory by asghar nosier,1 arash yavari,2 and shahram sarkani,3 member, asce abstract. The firstorder shear deformation plate theory fsdpt, described in detail in chapter 4, is an extension of the wellknown timoshenko beam theory andor the mindlin reissner plate theory that is applied to laminated composite plates. Geometrically exact threedimensional beam theory compmech. Journal of applied mathematics and physics zamp 23, 795804 1972. A flat plate, like a straight beam carries lateral load by bending. A supported beam loaded by a force and a distribution of pressure it is convenient to show a twodimensional crosssection of the threedimensional beam together with the beam cross section, as in fig. We will develop a twodimensional plate theory which employs the inplane coordinates x and y in see plate and associated x, y, z coordinate system as independent variables. Eulerbernoulli beam theory in the development of a reissner mindlin plate element. A similar, but not identical, theory had been proposed earlier by eric reissner in 1945. According to the timoshenko beam theory, the inplane bending of a clamped uniform beam.
A continuummechanics interpretation of reissners non. For help with downloading a single wikipedia page as a pdf, see help. Combining the reissnermindlin plate model and the timoshenko beam model can be seen as a first step towards a better model, while still. View the article pdf and any associated supplements and figures for a period of 48 hours. Marco di sciuva, marco gherlone1 department of aeronautics and space engineering politecnico di torino corso duca degli abruzzi 24. On the analysis of the timoshenko beam theory with and. D the equation isused to analyze an infinitebeam loaded with i a concentrated. A beam deforms and stresses develop inside it when a transverse load is applied on it. Geometrically exact finite element formulations for slender.
Modified bolle reissner theory of plates including transverse. Classical bernoullieuler beam theory is the earliest and simplest approximation used for analysis of. The method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise. The exact definitions for kinematic relationships have been derived reissner 1973, 1981, while the expressions for boundary conditions. Reissner formulated a onedimensional largestrain beam theory for plane. Objectivity of strain measures in the geometrically exact. Mar 21, 20 in the paper, we present the reissner simo beam theory in which the rotations are represented by quaternions. Request pdf on the differentiation of the rodrigues formula and its significance for the vector. A closed form solution for reissners planar finitestrain beam. Analytical derivation of a general 2d nonprismatic beam model based on the hellinger reissner principle angela beltempoa. The form of uflyandmindlin plate theory that is most commonly used is actually due to mindlin. A continuummechanics interpretation of reissner s nonlinear sheardeformable beam theory. A study of the edgezone equation of mindlinreissner plate.
Representative numerical examples are given in section 5. Large deformation analyses of spaceframe structures, using. Hetnarski the timoshenko beam equation in terms ofvariable w is derived where w is the deflection due to thebending ofa beam. Refined zigzag theory for homogeneous, laminated composite. Finite element methods for timoshenko beam, circular arch and. Simply supported for eg beam vertical deflection 0 and moment 0 wx, y0 simply supported. For the latter case of extensible elastica, we succeed in attaching a continuum mechanics meaning to the stress resultants and to all. Introduction to the theory of plates stanford university.
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