On the Theory of Bending of Elastic Plates

Reissner, E.

doi:10.1002/sapm1944231184

Cited by 642 publications

(265 citation statements)

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“…For example, in the Hellinger-Reissner (HR) mixed-variational principle, the strain energy is expressed in complementary form in terms of in-plane and transverse stresses, and Cauchy's 3D equilibrium equations are introduced as constraints via Lagrange multipliers. Reissner [26,27] was the first to use the HR principle to derive a new firstorder theory for isotropic plates based on linear throughthickness assumptions of the in-plane displacement and stress fields, and inherently equilibrated quadratic transverse shear stresses.…”

Section: Mixed-variational Theoriesmentioning

confidence: 99%

“…The formulation of this theory is an extension of Reissner's original first-order approach for isotropic plates [26,27] to anisotropic composite laminates. The approach by Cosentino and Weaver [30] is less general than the one proposed by Batra and Vidoli [28] as the in-plane and transverse stress assumptions are based on the same set of functional unknowns in order to minimise the computational cost.…”

Section: Mixed-variational Theoriesmentioning

confidence: 99%

“…26) When the displacements are transformed from (û x ,û y ,û z ) to (û n ,û s ,û z ) and tractions transfomed from (t x , t y , t z ) = (σ nx , σ ny , σ nz ) to (t n , t s , t z ) = (σ nn , σ ns , σ nz ) using Eq. (A.25), the first variation of the work done on the boundary surface Eq.…”

Section: Appendix a Derivation Of Hr Governing Equationsmentioning

confidence: 99%

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A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

Groh

2016

Composite Structures

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Current consensus suggests a trade-off for predicting accurate 3D stress fields in multilayered plates. Relatively expensive layerwise models facilitate accurate stress predictions from underlying models assumptions, whereas more efficient equivalent single-layer theories often rely on variationally inconsistent transverse stresses from recovery steps. In contrast, we show that variationally consistent 3D stress fields are predicted from an equivalent single-layer model using a contracted form of the Hellinger-Reissner functional. Notably, this procedure facilitates computationally efficient analysis of 3D heterogeneous plates, i.e. laminates comprising layers with material properties that differ by multiple orders of magnitude and that can also vary continuously in-plane. The model includes effects of higher-order transverse shearing and zig-zag deformations by expressing the in-plane stress field as a Taylor series expansion of global and local higher-order stress resultants. Equilibrated transverse stress assumptions are derived from Cauchy's 3D equilibrium equations, which identically satisfy the interfacial and surface traction equilibrium conditions when applied within a variational statement that enforces the 3D equilibrium equations as constraints, hence the Hellinger-Reissner mixed-variational statement. By using inherently equilibrated stress fields as model assumptions, it is shown that only the classical membrane and bending equilibrium equations have to be enforced explicitly. As a result, a contracted Hellinger-Reissner functional emerges that requires fewer displacement Lagrange multipliers than when independent stress fields are assumed. The governing equations are derived in a generalised framework such that the order of the model, and hence its accuracy, increases automatically when implemented in a computer code. By refining the order of the stress assumptions, the model is adaptable to plate-like structures ranging from thin engineering laminates to highly heterogeneous, thick laminates comprising straight-fibre or tow-steered variable-stiffness laminates, foam, honeycomb or other compliant layers.

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Section: Mixed-variational Theoriesmentioning

confidence: 99%

Section: Mixed-variational Theoriesmentioning

confidence: 99%

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A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

Groh

2016

Composite Structures

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“…This formulation of the displacement functions was first given by Reisner [12,13] and Mindlin [14] in their plate theories. Some examples for theories of even higher orderincluding not only the shear deformation effects but also, e. g., warping of crossections -can be found in [15].…”

Section: Generalized Plate Equation For Thick-walled Laminatesmentioning

confidence: 99%

“…The investigated parameter combinations are shown in Table 1, the material parameter are chosen according to (12). With this choice of M x in the case of classical laminate theory the strain fields on the laminate top and bottom respectively are independent of the thicknesses and therefore should be comparable.…”

Section: Influence Of the Laminate Thicknessmentioning

confidence: 99%

Stress concentration analysis of thick-walled laminate composites with a loaded circular cut-out by using a first-order shear deformation theory

Hufenbach

Gottwald

Grüber

et al. 2008

Composites Science and Technology

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Plates and Shells: Asymptotic Expansions and Hierarchic Models

Dauge

Faou

Yosibash

2004

Encyclopedia of Computational Mechanics

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Concerning thin structures such as plates and shells, the idea of reducing the equations of elasticity to twodimensional models defined on the mid-surface seems relevant. Such a reduction was first performed thanks to kinematical hypotheses about the transformation of normal lines to the mid-surface. As nowadays, the asymptotic expansion of the displacement solution of the three-dimensional linear model is fully known at least for plates and clamped elliptic shells, we start from a description of these expansions in order to introduce the twodimensional models known as hierarchical models: These models extend the classical models, and pre-suppose the displacement to be polynomial in the thickness variable, transverse to the mid-surface. Because of the singularly perturbed character of the elasticity problem as the thickness approaches zero, boundary-or internal layers may appear in the displacements and stresses, and so may numerical locking effects. The use of hierarchical models, discretized by higher degree polynomials (p-version of finite elements) may help to overcome these severe difficulties.

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On the Theory of Bending of Elastic Plates

Cited by 642 publications

References 0 publications

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

Stress concentration analysis of thick-walled laminate composites with a loaded circular cut-out by using a first-order shear deformation theory

Plates and Shells: Asymptotic Expansions and Hierarchic Models

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