An efficient three-node triangular Mindlin–Reissner flat shell element

Abstract

Shell elements are extensively used by engineers for modeling the behavior of shell structures. Among common shell elements, triangular shell elements are not influenced by element warping. This paper proposes a new three-node triangular flat shell element with six degrees of freedom per each node, named TMRFS. The element is formed by assemblage of new bending and membrane elements. The bending element is formulated based on the hybrid displacement function element method and Mindlin–Reissner plate theory. In this element, an assumed displacement function is employed as the trial function. The membrane component is an unsymmetric triangular membrane element with drilling vertex rotations. The membrane element employs two different types of displacement fields as the test and trial functions. The test function is a displacement field which is the same as one used in well-known Allman triangular element. Meanwhile, instead of displacement field, the analytical stress field is considered as the trial function. Numerical tests show that the accuracy of the proposed flat shell element is reasonable in comparison with some popular triangular elements and its performance is insensitive to geometry, load and boundary conditions. Moreover, the proposed element preserves the advantages of its formulation including free of membrane locking, shear locking and stiffness matrix singularity problems.

Appendix

Matrix ${\bar{\mathbf{N}}}_{\text{p}}$ can be defined through the Timoshenko’s beam functions and a linear function. Timoshenko’s beam functions are employed to determine the deflection $(w )$ and tangential rotation $(\theta_{s} )$ and the linear function is employed to determine normal rotation $(\theta_{n} )$, as follows

$$\begin{aligned} & w = \,(L_{1} + \upsilon L_{1} L_{2} (L_{1} - L_{2} ))w_{i} + \frac{{l_{ij} }}{2}(L_{1} L_{2} + \upsilon L_{1} L_{2} (L_{1} - L_{2} ))\theta_{si} \\ & \quad + (L_{2} + \upsilon L_{1} L_{2} (L_{2} - L_{1} ))w_{j} + \frac{{l_{ij} }}{2}( - L_{1} L_{2} + \upsilon L_{1} L_{2} (L_{1} - L_{2} ))\theta_{sj} \\ & \theta_{s} = \, - (6L_{1} L_{2} /l_{ij} )\mu w_{i} + L_{1} (1 - 3\mu L_{2} )\theta_{si} + (6L_{1} L_{2} /l_{ij} )\mu w_{j} \\ & \quad + L_{2} (1 - 3\mu L_{1} )\theta_{sj} \\ & \theta_{n} = (1 - s)\theta_{ni} + s\theta_{nj} \\ \end{aligned}$$

(57)

with

$$\mu { = }\frac{{5(1 - \upsilon )l_{ij}^{2} }}{{5(1 - \upsilon )l_{ij}^{2} + 12h^{2} }},$$

(58)

in which $l_{ij}$ is the length of edge ij, s is the coordinate along the edge ij, $L_{1} = 1 - s/l_{ij} \,{\text{and}}\,L_{2} = s/l_{ij} .$ It should be noted, $\theta_{n}$ and $\theta_{s}$ are the rotations in the local coordinates that should be transformed to global coordinates. The relationship between $(\theta_{n} ,\theta_{s} )$ and $(\theta_{x} ,\theta_{y} )$ can be defined as follows

$$\left\{ {\begin{array}{*{20}c} {\theta_{n} } \\ {\theta_{s} } \\ \end{array} } \right\} = \frac{1}{{l_{ij} }}\left[ {\begin{array}{*{20}c} {y_{j} - y_{i} } & {x_{i} - x_{j} } \\ {x_{j} - x_{i} } & {y_{j} - y_{i} } \\ \end{array} } \right]_{{}} \left\{ {\begin{array}{*{20}c} {\theta_{x} } \\ {\theta_{y} } \\ \end{array} } \right\}$$

(59)

where $\left( {x_{i} ,y_{i} } \right)$ and $(x_{j} ,y_{j} )$ are, respectively, the Cartesian coordinates of nodes i and j located on the edge ij. As was mentioned, matrix ${\bar{\mathbf{N}}}_{\text{p}}$ is an interpolation function that must be defined along each element edge, as follows

$$\begin{aligned} \bullet \,\,\,{\text{Along}}\,{\text{edge}}\,12:\,{\bar{\mathbf{N}}}_{\text{p}} = [{\mathbf{N}}^{\text{k}} {\mathbf{N}}^{\text{f}} {\mathbf{N}}^{\text{h}} ] \hfill \\ \bullet \,\,\,{\text{Along}}\,{\text{edge}}\,23:\,{\bar{\mathbf{N}}}_{\text{p}} = [{\mathbf{N}}^{\text{h}} {\mathbf{N}}^{\text{k}} {\mathbf{N}}^{\text{f}} ] \hfill \\ \bullet \,\,\,{\text{Along}}\,{\text{edge}}\,31:\,{\bar{\mathbf{N}}}_{\text{p}} = [{\mathbf{N}}^{\text{h}} {\mathbf{N}}^{\text{k}} {\mathbf{N}}^{\text{f}} ] \hfill \\ \end{aligned}$$

(60)

In which ${\mathbf{N}}^{\text{h}}$ is a $5 \times 3$ matrix with zero components and the components of ${\mathbf{N}}^{\text{k}}$ and ${\mathbf{N}}^{\text{f}}$ are as follows

$$\begin{aligned} & N_{11}^{k} = \, - \,6{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \mu {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij} }}{{l_{ij}^{3} }},\,N_{21}^{k} = 6{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij} }}{{l_{ij}^{3} }},\,N_{31}^{k} = 6{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:\left( {x_{ij}^{2} - y_{ij}^{2} } \right)}}{{l_{ij}^{3} }} \\ & N_{41}^{k} = - y_{ij} {\kern 1pt} \:\left( {\frac{s}{{l_{ij} }} + L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:\left( {2{\kern 1pt} \:\frac{s}{{l_{ij} }} - 1} \right) - 1} \right)l_{ij}^{ - 1} ,\,N_{51}^{k} = x_{ij} {\kern 1pt} \:\left( {\frac{s}{{l_{ij} }} + L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:\left( {2{\kern 1pt} \:\frac{s}{{l_{ij} }} - 1} \right) - 1} \right)l_{ij}^{ - 1} \\ & N_{12}^{k} = y_{ij}^{3} \left( {\frac{s}{{l_{ij} }} - 1} \right)l_{ij}^{ - 3} + x_{ij}^{2} y_{ij} {\kern 1pt} \:\left( {\frac{s}{{l_{ij} }} + 3{\kern 1pt} \:L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu - 1} \right)l_{ij}^{ - 3} ,\,N_{22}^{k} = - 3{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{{2{\kern 1pt} }} \:\mu {\kern 1pt} \:x_{ij}^{2} y_{ij} }}{{l_{ij}^{3} }} \\ & N_{32}^{k} = - x_{ij} {\kern 1pt} \:\left( {\frac{{x_{ij}^{2} }}{{l_{ij}^{2} }} - \frac{{y_{ij}^{2} }}{{l_{ij}^{2} }}} \right)\left( {\frac{s}{{l_{ij} }} + 3{\kern 1pt} \:L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu - 1} \right)l_{ij}^{ - 1} - 2{\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij}^{2} \left( {\frac{s}{{l_{ij} }} - 1} \right)l_{ij}^{ - 3} \\ & N_{42}^{k} = - {\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij} {\kern 1pt} \:\left( {\mu {\kern 1pt} \:l_{ij} - 2{\kern 1pt} \:\mu {\kern 1pt} \:s + l_{ij} } \right)}}{{2l_{ij}^{2} }},\,N_{52}^{k} = {\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:x_{ij}^{2} \left( {\mu {\kern 1pt} \:l_{ij} - 2{\kern 1pt} \:\mu {\kern 1pt} \:s + l_{ij} } \right)}}{{2l_{ij}^{2} }} \\ & N_{13}^{k} = 3{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij}^{2} }}{{l_{ij}^{3} }},\,N_{23}^{k} = - x_{ij}^{3} \left( {\frac{s}{{l_{ij} }} - 1} \right)l_{ij}^{ - 3} - x_{ij} {\kern 1pt} \:y_{ij}^{2} \left( {\frac{s}{{l_{ij} }} + 3{\kern 1pt} \:L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu - 1} \right)l_{ij}^{ - 3} \\ & N_{33}^{k} = 2{\kern 1pt} \:x_{ij}^{2} y_{ij} {\kern 1pt} \:\left( {\frac{s}{{l_{ij} }} - 1} \right)l_{ij}^{ - 3} - y_{ij} {\kern 1pt} \:\left( {\frac{{x_{ij}^{2} }}{{l_{ij}^{2} }} - \frac{{y_{ij}^{2} }}{{l_{ij}^{2} }}} \right)\left( {\frac{s}{{l_{ij} }} + 3{\kern 1pt} \:L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu - 1} \right)l_{ij}^{ - 1} \\ & N_{43}^{k} = - {\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:y_{ij}^{2} \left( {\mu {\kern 1pt} \:lij - 2{\kern 1pt} \:\mu {\kern 1pt} \:s + l_{ij} } \right)}}{{2l_{ij}^{2} }},\,N_{53}^{k} = \frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij} {\kern 1pt} \:\left( {\mu {\kern 1pt} \:l_{ij} - 2{\kern 1pt} \:\mu {\kern 1pt} \:s + l_{ij} } \right)}}{{2l_{ij}^{2} }} \\ & N_{11}^{f} = 6{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij} }}{{l_{ij}^{3} }},\,N_{21}^{f} = - 6{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij} }}{{l_{ij}^{3} }},\,N_{31}^{f} = - 6{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:\left( {x_{ij}^{2} - y_{ij}^{2} } \right)}}{{l_{ij}^{3} }} \\ & N_{41}^{f} = y_{ij} {\kern 1pt} \:\left( {\frac{s}{{l_{ij} }} + L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:\left( {2{\kern 1pt} \:\frac{s}{{l_{ij} }} - 1} \right)} \right)l_{ij}^{ - 1} ,\,N_{51}^{f} = - x_{ij} {\kern 1pt} \:\left( {\frac{s}{{l_{ij} }} + L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:\left( {2{\kern 1pt} \:\frac{s}{{l_{ij} }} - 1} \right)} \right)l_{ij}^{ - 1} \\ & N_{12}^{f} = \frac{{ - y_{ij} {\kern 1pt} \:\left( { - 3{\kern 1pt} \:L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:l_{ij} {\kern 1pt} \:x_{ij}^{2} + sx_{ij}^{2} + sy_{ij}^{2} } \right)}}{{l_{ij}^{4} }},\,N_{22}^{f} = - 3{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:x_{ij}^{2} y_{ij} }}{{l_{ij}^{3} }} \\ & N_{32}^{f} = x_{ij} {\kern 1pt} \:\left( {\frac{s}{{l_{ij} }} - 3{\kern 1pt} \:L_{{1{\kern 1pt} }} \:L_{2} {\kern 1pt} \:\mu } \right)\left( {\frac{{x_{ij}^{2} }}{{l_{ij}^{2} }} - \frac{{y_{ij}^{2} }}{{l_{ij}^{2} }}} \right)l_{ij}^{ - 1} + 2{\kern 1pt} \:\frac{{sx_{ij} {\kern 1pt} \:y_{ij}^{2} }}{{l_{ij}^{4} }} \\ & N_{42}^{f} = {\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij} {\kern 1pt} \:\left( { - \mu {\kern 1pt} \:l_{ij} + 2{\kern 1pt} \:\mu {\kern 1pt} \:s + l_{ij} } \right)}}{{2l_{ij}^{2} }},\,N_{52}^{f} = {\kern 1pt} \:\frac{{ - L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:x_{ij}^{2} \left( { - \mu {\kern 1pt} \:l_{ij} + 2{\kern 1pt} \:\mu {\kern 1pt} \:s + l_{ij} } \right)}}{{2l_{ij}^{2} }} \\ & N_{13}^{f} = 3{\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij}^{2} }}{{l_{ij}^{3} }},\,N_{23}^{f} = \frac{{x_{ij} {\kern 1pt} \:\left( { - 3{\kern 1pt} \:L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu {\kern 1pt} \:l_{ij} {\kern 1pt} \:y_{ij}^{2} + sx_{ij}^{2} + sy_{ij}^{2} } \right)}}{{l_{ij}^{4} }} \\ & N_{33}^{f} = y_{ij} {\kern 1pt} \:\left( {\frac{s}{{l_{ij} }} - 3{\kern 1pt} \:L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:\mu } \right)\left( {\frac{{x_{ij}^{2} }}{{l_{ij}^{2} }} - \frac{{y_{ij}^{2} }}{{l_{ij}^{2} }}} \right)l_{ij}^{ - 1} - 2{\kern 1pt} \:\frac{{sx_{ij}^{2} y_{ij} }}{{l_{ij}^{4} }} \\ & N_{43}^{f} = {\kern 1pt} \:\frac{{L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:y_{ij}^{2} \left( { - \mu {\kern 1pt} \:l_{ij} + 2{\kern 1pt} \:\mu {\kern 1pt} \:s + l_{ij} } \right)}}{{2l_{ij}^{2} }},\,N_{53}^{f} = \frac{{ - L_{1} {\kern 1pt} \:L_{2} {\kern 1pt} \:x_{ij} {\kern 1pt} \:y_{ij} {\kern 1pt} \:\left( { - \mu {\kern 1pt} \:l_{ij} + 2{\kern 1pt} \:\mu {\kern 1pt} \:s + l_{ij} } \right)}}{{2l_{ij}^{2} }} \\ \end{aligned}$$

(61)