Abstract
In this paper, a geometrically nonlinear formulation for a six-node triangular shell element is proposed. Total Lagrangian formulation is utilized to consider large displacements and rotations in the shell analysis. To avoid shear and membrane locking, a proper interpolation function for the strain field is implemented. Both algorithm and flowchart of the nonlinear solution, which are utilized in the author’s computer program, are presented. To validate the suggested formulation, several popular benchmark problems are solved. Moreover, the obtained results are compared with those of the other well-known elements. Findings demonstrate the ability of the suggested shell element.
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Abbreviations
- \(a_{i}\) :
-
The thickness of node i
- \({B}_{{ij}}\) :
-
The linear strain–displacement matrix
- \(e_{ij}\) :
-
Linear part of strain tensor
- \({e}_{1},{e}_{2}, {e}_{3}\) :
-
The unit vector in global Cartesian system
- \(h_{i}(r, s, t)\) :
-
The two-dimensional interpolation functions
- \({N}_{ij}\) :
-
The nonlinear strain–displacement matrix
- \(Q^{i}\) :
-
Rotation matrix
- r, s, t :
-
The convected coordinates
- \(\vec {u}\) :
-
The nodal displacement vector in global Cartesian system
- \(\vec {U}\) :
-
Vector of incremental nodal displacements
- u, v, w :
-
The nodal displacements
- \(\vec {u}_{l}\) :
-
The linear part of nodal displacement
- \(\vec {u}_{q}\) :
-
The quadratic part of nodal displacement
- \(\vec {V}_1^{i}, \vec {V}_2^{i}\) :
-
The unit vectors orthogonal to \(V^{i}_{n}\) and to each other
- \(\vec {V}_n^{i}\) :
-
The director vector of node i
- \(\vec {x}_{i}\) :
-
The position vector of node i
- \(\alpha _{i}, \beta _{i}\) :
-
The rotation of \(V^{i}_{n}\) about \(V^{i}_{1}, V^{i}_{2}\)
- \(\varepsilon _{ij}\) :
-
Green–Lagrange strain tensor
- \(\eta _{ij}\) :
-
Nonlinear part of strain tensor
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Rezaiee-Pajand, M., Arabi, E. & Masoodi, A.R. A triangular shell element for geometrically nonlinear analysis. Acta Mech 229, 323–342 (2018). https://doi.org/10.1007/s00707-017-1971-8
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DOI: https://doi.org/10.1007/s00707-017-1971-8