Verallgemeinerte Theorie der Plattenstatik

Kromm, Alexander

doi:10.1007/bf00538133

Cited by 62 publications

(26 citation statements)

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“…Theories based on this method were worked out by E. Reissner [9,10,11], Bollé [12,13], Hencky [14], Mindlin [15], Kromm [16], B. Vlasov [17], Ambarcumyan [18], Mushtari [19], Panc [20], Levinson [21] et al All these theories are connected with certain engineering assumptions which have a influence on the applicability. But in general, strong limits for the application cannot be defined.…”

Section: Introductionmentioning

confidence: 99%

On the determination of transverse shear stiffnesses of orthotropic plates

Altenbach

2000

Z. angew. Math. Phys.

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Since the classical Kirchhoff plate theory neglects the transverse shear strains, it does not include constitutive equations for the transverse shear force-transverse shear strain interactions. In this case, the transverse shear forces can be calculated from the equilibrium equations.For thick, laminated and sandwich plates, the transverse shear strains should be taken into account and the relevant constitutive equations have a significant influence on the correct estimation of the deflections or the stress resultants, and due to this fact we need refined theories. There are two different possibilities to formulate such refined theories which are able to describe both homogeneous and inhomogeneous in the thickness direction plates. The first is based on the assumption of an equivalent single layer, the second on a layerwise formulation. Equivalent single layer theories, which limit the analysis to the estimation of global characteristics, are mostly prefered. The simplest refinement of the classical plate theory are the so-called first order shear deformation theories. The quality of such theories depends on the correct determination of the effective stiffnesses. Many theories result in identical stiffnesses for bending, tension/compression, in-plane shear and torsion. Differences can be obtained for the transverse shear stiffnesses.In this contribution, different proposals for the transverse shear stiffnesses with values based on a deformable directed surface theory will be compared. In the case of homogeneous isotropic plates, the transverse shear stiffness estimated due to the deformable surface theory is in agreement with Mindlin's proposal. On the other hand, for isotropic symmetric sandwiches with thin face sheets and a very soft core Reissner's approximation can be obtained. (2000). 73K10, 73K20. Mathematics Subject Classification

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Section: Introductionmentioning

confidence: 99%

On the determination of transverse shear stiffnesses of orthotropic plates

Altenbach

2000

Z. angew. Math. Phys.

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“…Although it is well-known that the concentrated corner reactions of classical plate theory are due to the simplifying assumptions of that theory, it has been presumed that a thick plate theory should spread such vertical forces over portions of the edges near the comers, cf., e.g., [10]. Such results would seem to be due to the fact that even thick plate theories overly constrain the possible deformations of plate elements.…”

Section: Rm (34)mentioning

confidence: 93%

The simply supported rectangular plate: An exact, three dimensional, linear elasticity solution

Levinson

1985

J Elasticity

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An exact three dimensional solution for the problem of a transversely loaded, simply supported rectangular plate of arbitrary thickness is presented within the linear theory of elastostatics. The solution, obtained in a semi-inverse fashion, satisfies all the boundary conditions of the problem in a pointwise manner and is in the form of a double Fourier sine series. The classical Navier solution for the problem is shown to be the limit of the present solution as the plate thickness aspect ratio approaches zero. It is noted that the solution presented provides a benchmark against which approximate theories of transversely loaded plates may be measured.The new elasticity solution also provides a heuristic basis for a novel theory of thick plates of arbitrary planform and edge support recently given by the author.

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“…Numerical values of the critical time are presented there. Moreover, the paper [22] presents results obtained using approximate plate theories based on the Kirchhoff-Love and Timoshenko-type hypothesis [67,68]. The numerical values of the critical time obtained with the TDLTSDB are compared with those produced by the refined plate theory [67,68].…”

Section: Fibrous Compositesmentioning

confidence: 97%

“…The corresponding boundary-value problem for the Laplace transforms of the sought quantities is solved by employing the semi-analytical FEM. The boundary-value problems of these approximate theories were formulated by integrating the TDLTSDB equations over the thickness and using the Kirchhoff-Love hypothesis and the hypothesis from [67,68]. Specific numerical investigations were conducted for the case where the plate material consists of alternating layers of two isotropic, homogeneous materials, and these layers are located in planes parallel to the middle plane of the plate.…”

Section: Fibrous Compositesmentioning

confidence: 99%

Three-dimensional instability problems for viscoelastic composite materials and structural members

Akbarov

2007

Int Appl Mech

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Investigations of viscoelastic composite materials and structural members carried out using the TDLTSDB and initial-imperfection method are reviewed. The investigations address the internal and surface loss of stability of layered and fibrous composites, the loss of stability of plates, and the delamination (buckling) of plates with cracks. Each of these problems is reviewed separately. New areas of further research are proposed. The review focuses on the investigations carried out by the author and his students Keywords: buckling, crack, delamination, plate, instability, viscoelastic composite material 1. Introduction. As in [33,[41][42][43][44][45] and many other publications, by three-dimensional instability problems will be understood here instability problems investigated employing the three-dimensional linearized theory of stability of deformable bodies (TDLTSDB). The investigations regarding the construction of the TDLTSDB are detailed in the paper [46] and in the monographs [33,[41][42][43]45]. It was noted there that the tendencies to construct the theory, which allows us (i) to describe the buckling phenomenon for a well-defined three-dimensional stress-strain state and (ii) to obtain reliable data on the critical parameters for thin-walled and massive structural elements, contributed to the creation of the TDLTSDB. Historically, the TDLTSDB began to be developed in the first half of the 20th century, and the basic TDLTSDB relations were obtained in [30, 31-33, 35, 38, 45, 63, 71-73, 80, 84, 85] using either a strict and exact approach based on the linearization of the exact and strict relations of the nonlinear three-dimensional theory or an approximate approach based on a physical and, in part, linearization of certain relations. Development and detailed analyses of the results from these investigations were presented in the monographs [41,42,45].In the second half of the 20th century, the main attention of researchers was focused on the solution to separate classes of TDLTSDB problems. At the same time, general aspects of the TDLTSDB were also developed. It should be noted that the results of these studies were the subject of many monographs such as [41][42][43][44][45] and were reviewed in many papers such as [28,46,47,49,52,55,56,58].It follows from the search of the foregoing and other numerous inquiries that the TDLTSDB was mainly applied to analyze the stability of structural members made of time-independent materials. Moreover, these investigations used the bifurcation (Euler) approach. The Euler approach is known to be generally unsuitable for investigating instability problems for structural elements made of time-dependent materials and subjected to external static loading. This is why a dynamic investigation method was suggested in [40,45] to examine these problems. Note that this method represents the sought quantities with a multiplier exp(iWt) and determines the critical parameters of the problems from the requirement ImW = 0. However, if the dynamic investigation method is applied with...

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Verallgemeinerte Theorie der Plattenstatik

Cited by 62 publications

References 0 publications

On the determination of transverse shear stiffnesses of orthotropic plates

On the determination of transverse shear stiffnesses of orthotropic plates

The simply supported rectangular plate: An exact, three dimensional, linear elasticity solution

Three-dimensional instability problems for viscoelastic composite materials and structural members

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