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Formulation and evaluation of a new four-node quadrilateral element for analysis of the shell structures

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Abstract

Shell structures are lightweight constructions which are extensively used by engineering. Due to this reason presenting an appropriate shell element for analysis of these structures has become an interesting issue in recent decades. This study presents a new rectangular flat shell element called ACM-SQ4 obtained by combining bending and membrane elements. The bending element is a well-known plate bending element called ACM which is based on the classical thin-plate theory and the membrane element is an unsymmetric quadrilateral element called US-Q4θ, the test function of this element is improved by the Allman-type drilling DOFs and a rational stress field is used as the element’s trial function. Finally, some numerical benchmark problems are used to evaluate the performance of the proposed flat shell element. The obtained results show that despite its simple formulation, the proposed element has reasonable accuracy and acceptable convergence in comparison with other shell elements.

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Correspondence to Hamed Ghohani Arab.

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Appendix A

Appendix A

Natural coordinates for the plate bending element are shown in Fig. 14.

Fig. 14
figure 14

Geometry of the plate bending element in Local coordinate system

The shape function of the ith degree of freedom in the natural coordinate system is

$$ \begin{aligned} N_{1} (\xi ,\eta ) = (1/8)(1 - \xi )(1 - \eta )(2 - \xi - \eta - \xi^{2} - \eta^{2} ) \hfill \\ N_{2} (\xi ,\eta ) = (b/8)(1 - \xi )(1 - \eta )( - (1 - \eta^{2} )) \hfill \\ N_{3} (\xi ,\eta ) = (a/8)(1 - \xi )(1 - \eta )(1 - \xi^{2} ) \hfill \\ N_{4} (\xi ,\eta ) = (1/8)(1 + \xi )(1 - \eta )(2 + \xi - \eta - \xi^{2} - \eta^{2} ) \hfill \\ N_{5} (\xi ,\eta ) = (b/8)(1 + \xi )(1 - \eta )( - (1 - \eta^{2} )) \hfill \\ N_{6} (\xi ,\eta ) = (a/8)(1 - \xi )(1 - \eta )( - (1 - \xi^{2} )) \hfill \\ N_{7} (\xi ,\eta ) = (1/8)(1 + \xi )(1 + \eta )(2 + \xi + \eta - \xi^{2} - \eta^{2} ) \hfill \\ N_{8} (\xi ,\eta ) = (b/8)(1 + \xi )(1 + \eta )(1 - \eta^{2} ) \hfill \\ N_{9} (\xi ,\eta ) = (a/8)(1 + \xi )(1 + \eta )( - (1 - \xi^{2} )) \hfill \\ N_{10} (\xi ,\eta ) = (1/8)(1 - \xi )(1 + \eta )(2 - \xi + \eta - \xi^{2} - \eta^{2} ) \hfill \\ N_{11} (\xi ,\eta ) = (b/8)(1 - \xi )(1 + \eta )(1 - \eta^{2} ) \hfill \\ N_{12} (\xi ,\eta ) = (a/8)(1 - \xi )(1 + \eta )(1 - \xi^{2} ). \hfill \\ \end{aligned} $$
(27)

The element stiffness matrix in the natural coordinate system is calculated by Eq. (28)

$$ {\mathbf{K}} = \int\limits_{V} {{\mathbf{B}}^{\text{T}} {\mathbf{DB}}{\text{d}}V} , $$
(28)

where \( {\text{d}}V \) is the differential volume and its value in the natural coordinates of the element is equal to \( abtd\xi d\eta \) and B is a \( 3 \times 12 \) matrix shown in Eq. (29).

$$ B = \left[ {\begin{array}{*{20}c} {B_{1,1} } & {B_{1,2} } & {B_{1,3} } & {B_{1,4} } & {B_{1,5} } & {B_{1,6} } & {B_{1,7} } & {B_{1,8} } & {B_{1,9} } & {B_{1,10} } & {B_{1,11} } & {B_{1,12} } \\ {B_{2,1} } & {B_{2,2} } & {B_{2,3} } & {B_{2,4} } & {B_{2,5} } & {B_{2,6} } & {B_{2,7} } & {B_{2,8} } & {B_{2,9} } & {B_{2,10} } & {B_{2,11} } & {B_{2,12} } \\ {B_{3,1} } & {B_{3,2} } & {B_{3,3} } & {B_{3,4} } & {B_{3,5} } & {B_{3,6} } & {B_{3,7} } & {B_{3,8} } & {B_{3,9} } & {B_{3,10} } & {B_{3,11} } & {B_{3,12} } \\ \end{array} } \right]. $$
(29)

Matrix B elements in the natural coordinate system are defined as

$$ \begin{array}{*{20}c} \begin{aligned} B_{1,1} = - \frac{3}{{4a^{2} }}z\xi (1 - \eta ) \hfill \\ B_{1,2} = 0 \hfill \\ B_{1,3} = - \frac{1}{4a}z(\eta - 1)(1 - 3\xi ) \hfill \\ B_{1,4} = - \frac{3}{{4a^{2} }}z\xi (\eta - 1) \hfill \\ \end{aligned} & \begin{aligned} B_{2,1} = - \frac{3}{{4b^{2} }}z\eta (1 - \xi ) \hfill \\ B_{2,2} = \frac{1}{4b}z(\xi - 1)(1 - 3\eta ) \hfill \\ B_{2,3} = 0 \hfill \\ B_{2,4} = - \frac{3}{{4b^{2} }}z\eta (1 + \xi ) \hfill \\ \end{aligned} & \begin{aligned} B_{3,1} = \frac{2}{8ab}z(3\xi^{2} + 3\eta^{2} - 4) \hfill \\ B_{3,2} = \frac{2}{8a}z( - 3\eta^{2} + 2\eta + 1) \hfill \\ B_{3,3} = - \frac{2}{8b}z( - 3\xi^{2} + 2\xi + 1) \hfill \\ B_{3,4} = - \frac{2}{8ab}z(3\xi^{2} + 3\eta^{2} - 4) \hfill \\ \end{aligned} \\ \begin{aligned} B_{1,5} = 0 \hfill \\ B_{1,6} = - \frac{1}{4a}z(1 - \eta )(1 + 3\xi ) \hfill \\ B_{1,7} = \frac{3}{{4a^{2} }}z\xi (\eta + 1) \hfill \\ B_{1,8} = 0 \hfill \\ B_{1,9} = - \frac{1}{4a}z(1 + \eta )(1 + 3\xi ) \hfill \\ B_{1,10} = \frac{3}{{4a^{2} }}z\xi (\eta + 1) \hfill \\ B_{1,11} = 0 \hfill \\ B_{1,12} = \frac{1}{4a}z(1 + \eta )(1 - 3\xi ) \hfill \\ \end{aligned} & \begin{aligned} B_{2,5} = - \frac{1}{4b}z(\xi + 1)(1 - 3\eta ) \hfill \\ B_{2,6} = 0 \hfill \\ B_{2,7} = \frac{3}{{4b^{2} }}z\eta (1 + \xi ) \hfill \\ B_{2,8} = \frac{1}{4b}z(\xi + 1)(1 + 3\eta ) \hfill \\ B_{2,9} = 0 \hfill \\ B_{2,10} = \frac{3}{{4b^{2} }}z\eta (1 - \xi ) \hfill \\ B_{2,11} = - \frac{1}{4b}z(\xi - 1)(1 + 3\eta ) \hfill \\ B_{2,12} = 0 \hfill \\ \end{aligned} & \begin{aligned} B_{3,5} = - \frac{2}{8a}z( - 3\eta^{2} + 2\eta + 1) \hfill \\ B_{3,6} = \frac{2}{8b}z(3\xi^{2} + 2\xi - 1) \hfill \\ B_{3,7} = \frac{2}{8ab}z(3\xi^{2} + 3\eta^{2} - 4) \hfill \\ B_{3,8} = \frac{2}{8a}z(3\eta^{2} + 2\eta - 1) \hfill \\ B_{3,9} = - \frac{2}{8b}z(3\xi^{2} + 2\xi - 1) \hfill \\ B_{3,10} = - \frac{2}{8ab}z(3\xi^{2} + 3\eta^{2} - 4) \hfill \\ B_{3,11} = - \frac{2}{8a}z(3\eta^{2} + 2\eta - 1) \hfill \\ B_{3,12} = \frac{2}{8b}z( - 3\xi^{2} + 2\xi + 1). \hfill \\ \end{aligned} \\ {} & {} & {} \\ \end{array} $$
(30)

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Sangtarash, H., Arab, H.G., Sohrabi, M.R. et al. Formulation and evaluation of a new four-node quadrilateral element for analysis of the shell structures. Engineering with Computers 36, 1289–1303 (2020). https://doi.org/10.1007/s00366-019-00763-8

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