The linear isoparametric triangular element: Theory and application

Talaslidis, Demosthenes; Wempner, Gerald

doi:10.1016/0045-7825(93)90129-l

Cited by 9 publications

(9 citation statements)

References 17 publications

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“…An alternative to the averaging of the transverse strain‐displacement matrix has already been suggested by Talaslidis and Wempner, 18 where the energy of the transverse shear strains was averaged (their Equation (46a,b)) in order to achieve isotropic response. Therefore, the alternative formulation would calculate the contribution of the transverse shear strain to the stiffness matrix as

{[K^{s}]}_{E} = \frac{A}{3} \sum_{j = 1}^{3} {[B_{o_{j}}^{s}]}_{E}^{T} {[D^{s}]}_{E} {[B_{o_{j}}^{s}]}_{E} .

The matrix

{[B^{s}]}_{E}

aggregates the nodal strain‐displacement matrices

{[B_{o_{j}}^{s}]}_{E} = [\begin{array}{ccc} {([], ^{1} {boldB}_{o_{j}}^{s})}_{E}, {([], ^{2} {boldB}_{o_{j}}^{s})}_{E}, {([], ^{3} {boldB}_{o_{j}}^{s})}_{E} \end{array}] .

We can refer to the transverse shear stiffness matrix (26) as the averaged‐K formulation.…”

Section: Elastic Properties Of the Element In Its Framementioning

confidence: 99%

“…An alternative to the averaging of the transverse strain-displacement matrix has already been suggested by Talaslidis and Wempner, 18 where the energy of the transverse shear strains was averaged (their Equation (46a,b)) in order to achieve isotropic response. Therefore, the alternative formulation would calculate the contribution of the transverse shear strain to the stiffness matrix as…”

Section: Transverse Shear Actionmentioning

confidence: 99%

“…This stiffness matrix has a rank of 8: it has six rigid body modes, one higher order torsional mode, 18 and three zero energy modes due to the drilling rotations being associated with no stiffness. The torsional mode corresponds to the rotation of the bottom surface of the shell relative to the top surface about the normal to the element surface.…”

Section: Properties Of the Elementwise Stiffness Matrixmentioning

confidence: 99%

“…The overall stiffness matrix of the element in the local element coordinates is obviously

{[K]}_{E} = {[K^{m}]}_{E} + {[K^{b}]}_{E} + {[K^{s}]}_{E} .

This stiffness matrix has a rank of 8: it has six rigid body modes, one higher order torsional mode, 18 and three zero energy modes due to the drilling rotations being associated with no stiffness.…”

Section: Elastic Properties Of the Element In Its Framementioning

confidence: 99%

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Robust flat‐facet triangular shell finite element

Krysl

2022

Numerical Meth Engineering

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A flat facet finite element with three nodes and six degrees of freedom per node for linear static and dynamic analysis of thick and thin shells is presented. The membrane response is modeled with constant strain triangles. The bending response treats the transverse shear with an improved version of the discrete-shear-gap technique. The main novelty of the present approach is the robust treatment of the drilling rotations: the element internally responds to five degrees of freedom per node, but externally it naturally possesses six degrees of freedom per node. The drilling degree of freedom is decoupled from the bending and twisting, and the undesirable user-selectable arbitrary stabilization is avoided. The element is shown to be robust for extremely thin shells, and passes tests such as the Raasch hook problem. Performance is illustrated with static and dynamic examples including branched and folded shells.

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{[K^{s}]}_{E} = \frac{A}{3} \sum_{j = 1}^{3} {[B_{o_{j}}^{s}]}_{E}^{T} {[D^{s}]}_{E} {[B_{o_{j}}^{s}]}_{E} .

The matrix

{[B^{s}]}_{E}

aggregates the nodal strain‐displacement matrices

{[B_{o_{j}}^{s}]}_{E} = [\begin{array}{ccc} {([], ^{1} {boldB}_{o_{j}}^{s})}_{E}, {([], ^{2} {boldB}_{o_{j}}^{s})}_{E}, {([], ^{3} {boldB}_{o_{j}}^{s})}_{E} \end{array}] .

We can refer to the transverse shear stiffness matrix (26) as the averaged‐K formulation.…”

Section: Elastic Properties Of the Element In Its Framementioning

confidence: 99%

Section: Transverse Shear Actionmentioning

confidence: 99%

Section: Properties Of the Elementwise Stiffness Matrixmentioning

confidence: 99%

“…The overall stiffness matrix of the element in the local element coordinates is obviously

{[K]}_{E} = {[K^{m}]}_{E} + {[K^{b}]}_{E} + {[K^{s}]}_{E} .

Section: Elastic Properties Of the Element In Its Framementioning

confidence: 99%

See 2 more Smart Citations

Robust flat‐facet triangular shell finite element

Krysl

2022

Numerical Meth Engineering

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“…Furthermore, the discrete approximation is drawn in a consistent manner from the general theory of the continuum and the mechanical behavior of the finite element, without resorting to special manipulations or computational procedures. In addition, it has been shown [47,51,52] that essential prerequisites for the achievement of these goals are: the identification of constant and higher-order deformational modes which are contained in the displacement/rotation assumptions, the realization that the constant terms are necessary for convergence, and that higher-order terms reappear in different strain components. Therefore, our approximation does not need to retain the higher-order terms in two different strain components (they are needed only to inhibit a mode).…”

Section: Parametrization Of the Geometrymentioning

confidence: 99%

Static and free vibration analysis of functionally graded carbon nanotube reinforced skew plates

García‐Macías

Castro-Triguero

Flores

et al. 2016

Composite Structures

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Highlights • Invariant definition of the transversely isotropic constitutive tensor based on representation theorem of functionally graded materials (FGM) in oblique coordinates. • Study of influence of fiber direction and skew angle on natural frequencies of (FGM) nanocomposite skew plates. • Consideration of one asymmetric and three symmetric distributions of functionally graded nanocomposite skew plates. • The underlying shell theory is formulated in oblique coordinates, based on the Hu-Washizu principle and first-order shear deformation theory (FSDT). • Independent approximations of displacements, strains and stresses.

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A rational approach to mass matrix diagonalization in two‐dimensional elastodynamics

Paraskevopoulos

Talaslidis

2004

Numerical Meth Engineering

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SUMMARYA variationally consistent methodology is presented, which yields diagonal mass matrices in twodimensional elastodynamic problems. The proposed approach avoids ad hoc procedures and applies to arbitrary quadrilateral and triangular finite elements. As a starting point, a modified variational principle in elastodynamics is used. The time derivatives of displacements, the velocities, and the momentum type variables are assumed as independent variables and are approximated using piecewise linear or constant functions and combinations of piecewise constant polynomials and Dirac distributions. It is proved that the proposed methodology ensures consistency and stability.

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The linear isoparametric triangular element: Theory and application

Cited by 9 publications

References 17 publications

Robust flat‐facet triangular shell finite element

Robust flat‐facet triangular shell finite element

Static and free vibration analysis of functionally graded carbon nanotube reinforced skew plates

A rational approach to mass matrix diagonalization in two‐dimensional elastodynamics

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