Derivation of element stiffness matrices by assumed strain distributions

SUMMARYThe degenerated approach for shell elements of Ahmad and co-workers is revisited in this paper. To avoid transverse shear locking e ects in four-node bilinear elements, an alternative formulation based on the enhanced assumed strain (EAS) method of Simo and Rifai is proposed directed towards the transverse shear terms of the strain ÿeld. In the ÿrst part of the work the analysis of the null transverse shear strain subspace for the degenerated element and also for the selective reduced integration (SRI) and assumed natural strain (ANS) formulations is carried out. Locking e ects are then justiÿed by the inability of the null transverse shear strain subspace, implicitly deÿned by a given ÿnite element, to properly reproduce the required displacement patterns. Illustrating the proposed approach, a remarkably simple single-element test is described where ANS formulation fails to converge to the correct results, being characterized by the same performance as the degenerated shell element. The adequate enhancement of the null transverse shear strain subspace is provided by the EAS method, enforcing Kirchho hypothesis for low thickness values and leading to a framework for the development of shear-lockingfree shell elements. Numerical linear elastic tests show improved results obtained with the proposed formulation.

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“…in replacement of (11). The next step refers to the extension of this equality over all the sample points used in the numerical integration.…”

Section: Analysis Of the Degenerated Elementmentioning

confidence: 99%

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

Sá

Jorge

Valente

et al. 2001

Numerical Meth Engineering

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“…As the shell thickness decreases, the convergence of the finite element solution rapidly deteriorates [8]. Researchers developed methods to overcome locking phenomena in flat triangular shell elements, such as the Mixed Interpolation of Tensorial Components method used in the MITC3 element [8], the line integration method used in TRIA3 element [42], the mixed or hybrid formulation [43][44][45], incompatible displacement methods [46][47], stabilization methods [29,[48][49][50][51][52][53][54], assumed strain methods [19,28,55], etc.…”

Section: Introductionmentioning

confidence: 99%

A 3-Node Co-Rotational Triangular Elasto-Plastic Shell Element Using Vectorial Rotational Variables

Li¹,

Izzuddin²,

Vu‐Quoc³

et al. 2017

Volume 13 Number 3

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A 3-node co-rotational triangular elasto-plastic shell element is developed. The local coordinate system of the element employs a zero-'macro spin' framework at the macro element level, thus reducing the material spin over the element domain, and resulting in an invariance of the element tangent stiffness matrix to the order of the node numbering. The two smallest components of each nodal orientation vector are defined as rotational variables, achieving the desired additive property for all nodal variables in a nonlinear incremental solution procedure. Different from other existing co-rotational finite-element formulations, both element tangent stiffness matrices in the local and global coordinate systems are symmetric owing to the commutativity of the nodal variables in calculating the second derivatives of strain energy with respect to the local nodal variables and, through chain differentiation with respect to the global nodal variables. For elasto-plastic analysis, the Maxwell-Huber-Hencky-von Mises yield criterion is employed together with the backward-Euler return-mapping method for the evaluation of the elasto-plastic stress state, where a consistent tangent modulus matrix is used. Assumed membrane strains and assumed shear strains---calculated respectively from the edge-member membrane strains and the edge-member transverse shear strains---are employed to overcome locking problems, and the residual bending flexibility is added to the transverse shear flexibility to improve further the accuracy of the element. The reliability and convergence of the proposed 3-node triangular shell element formulation are verified through two elastic plate patch tests as well as three elastic and three elasto-plastic plate/shell problems undergoing large displacements and large rotations.

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“…The main idea of such approach can be traced back to the ANS method [37][38][39] developed by many scientists for the isoparametric displacement-based and hybrid-mixed finite element formulations [22,23,28,40,41]. In contrast with above formulations, we treat the term "ANS" in a broader sense.…”

Section: Remarkmentioning

confidence: 99%

Exact geometry solid-shell element based on a sampling surfaces technique for 3D stress analysis of doubly-curved composite shells

Куликов

Mamontov

Плотникова

et al. 2015

Curved and Layered Structures

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A hybrid-mixed ANS four-node shell element by using the sampling surfaces (SaS) technique is developed. The SaS formulation is based on choosing inside the nth layer In not equally spaced SaS parallel to the middle surface of the shell in order to introduce the displacements of these surfaces as basic shell variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree In − 1 in the thickness direction for each layer permits the presentation of the layered shell formulation in a very compact form. The SaS are located inside each layer at Chebyshev polynomial nodes that allows one to minimize uniformly the error due to the Lagrange interpolation. To implement the efficient analytical integration throughout the element, the enhanced ANS method is employed. The proposed hybrid-mixed four-node shell element is based on the Hu-Washizu variational equation and exhibits a superior performance in the case of coarse meshes. It could be useful for the 3D stress analysis of thick and thin doubly-curved shells since the SaS formulation gives the possibility to obtain numerical solutions with a prescribed accuracy, which asymptotically approach the exact solutions of elasticity as the number of SaS tends to infinity.

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Derivation of element stiffness matrices by assumed strain distributions

Cited by 273 publications

References 9 publications

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

Development of shear locking‐free shell elements using an enhanced assumed strain formulation

A 3-Node Co-Rotational Triangular Elasto-Plastic Shell Element Using Vectorial Rotational Variables

Exact geometry solid-shell element based on a sampling surfaces technique for 3D stress analysis of doubly-curved composite shells

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