A new rectangular beam theory

Levinson, M.

doi:10.1016/0022-460x(81)90493-4

Cited by 465 publications

(171 citation statements)

References 6 publications

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“…These theories extend from first-order shear deformation theories by Mindlin [10] and Yang, Norris and Stavsky [11] to higher-order Levinson-Reddy-type shear deformation models that enforce vanishing shear strains at the top and bottom surfaces in the displacement field a priori [12,13], and further to generalised higher-order theories that do not make this initial assumption and may account for transverse normal deformation, i.e. thickness stretching [14,15].…”

Section: Displacement-based Theoriesmentioning

confidence: 99%

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: Displacement-based Theoriesmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

“…For example, Levinson (1981) suggested a third-order rectangular beam model by taking the in-plane warping of cross section into account. Based on the same idea, Murthy (1981) used another definition of rotation, leading to the shear correction factor 5/6 rather than 2/3 by Levinson (1981). Following the kinematics of Levinson (1981), Bickford (1982) proposed a variationally consistent beam theory and Reddy (1984aReddy ( , 1984b further developed a higher-order differential governing equation for plates.…”

Section: Introductionmentioning

confidence: 99%

“…Based on the same idea, Murthy (1981) used another definition of rotation, leading to the shear correction factor 5/6 rather than 2/3 by Levinson (1981). Following the kinematics of Levinson (1981), Bickford (1982) proposed a variationally consistent beam theory and Reddy (1984aReddy ( , 1984b further developed a higher-order differential governing equation for plates. The comprehensive numerical investigations on the accuracy of various HSD models were conducted by Rohwer (1992) which showed that the Murthy's (1981) and Reddy's (1984aReddy's ( , 1984b theories are the best choices.…”

Section: Introductionmentioning

confidence: 99%

“…Rohwer (1992) indicated that the kinematics and the rotation play important roles in the HSD model. For the kinematics, besides the third-order shear deformation model (Levinson, 1981;Murthy, 1981), the trigonometric model Civalek, 2014a, 2014b;Karama et al, 1998;Mantari et al, 2012;Touratier, 1991), the hyperbolic sine model (Akgöz and Civalek, 2015;Akavci and Tanrikulu, 2008;El Meiche et al, 2011;Soldatos, 1992) and the exponential model (Aydogdu, 2009;Karama et al, 2003) have been developed. For the rotation, two definitions are usually adopted in the existing work.…”

Section: Introductionmentioning

confidence: 99%

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The Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models

Duan

2016

Lat. Am. j. solids struct.

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The unified solution is studied for a beam of rectangular cross section. With the rotation defined in the average sense over the cross section, the kinematics with higher-order shear deformation models in axial displacement is first expressed in a unified form by using the fundamental higher-order term with some properties. The shear correction factor is then derived and discussed for the four commonly used higher-order shear deformation models including the third-order model, the sine model, the hyperbolic sine model and the exponential model. The unified solution is finally obtained for a beam subjected to an arbitrarily distributed load. The relation with that from the conventional beam theory is established, and therefore the difference is reasonably explained. A very good agreement with the elasticity theory validates the present solution.

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Unified Finite Elements Based on the Classical and Shear Deformation Theories of Beams and Axisymmetric Circular Plates

Reddy

Wang

Lam

1997

Commun. Numer. Meth. Engng.

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SUMMARYIn this paper a uni®ed ®nite element model that contains the Euler±Bernoulli, Timoshenko and simpli®ed Reddy third-order beam theories as special cases is presented. The element has only four degrees of freedom, namely de¯ection and rotation at each of its two nodes. Depending on the choice of the element type, the general stiness matrix can be specialized to any of the three theories by merely assigning proper values to parameters introduced in the development. The element does not experience shear locking, and gives exact generalized nodal displacements for Euler±Bernoulli and Timoshenko beam theories when the beam is homogeneous and has constant geometric properties. While the Timoshenko beam theory requires a shear correction factor, the third-order beam theory does not require speci®cation of a shear correction factor. An extension of the work to axisymmetric bending of circular plates is also presented. A stiness matrix based on the exact analytical form of the solution of the ®rst-order theory of circular plates is derived.

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A new rectangular beam theory

Cited by 465 publications

References 6 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

The Unified Solution for a Beam of Rectangular Cross-Section with Different Higher-Order Shear Deformation Models

Unified Finite Elements Based on the Classical and Shear Deformation Theories of Beams and Axisymmetric Circular Plates

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