A New Locking-Free Thick/Thin Shell Element
with Incompatible Approximation in a General
Orthogonal Curvilinear Coordinate System
Jingxu Chen (B) , Yongchang Cai, and Pengfei Yan
State Key Laboratory of Disaster Reduction in Civil Engineering, College of Civil Engineering,
Tongji University, Shanghai 200092, People’s Republic of China
jingxuchen@foxmail.com
Abstract. A new locking-free thick/thin shell element with incompatible approximation (SEIA) in a general orthogonal curvilinear coordinate system is proposed
for the analysis of the thick/thin shell. The key points for the development of
the present element include the definition of the incompatible polynomial displacement approximation in each independent element and the utilization of the
fictitious thin layers to ensure the displacement conformity between adjacent elements. Superior to most available shell elements, the SEIA avoids the shear and
membrane locking naturally without the adoption of numerical expediencies, has
concise theoretical derivation and easy numerical implementation, is insensitive to
element distortions and thus provides reliable solutions for the thick/thin shell. The
present work also initiates a methodology to develop locking-free thick/thin shell
elements. Numerical investigations and comparisons demonstrate the convergence
and robustness of the present element.
Keywords: Thick/thin shell · Locking-free · Incompatible approximation ·
Curvilinear coordinate
1 Introduction
Given its high load bearing capacity compared with the plate and small volume compared
with the solid, the curved thin-walled shell structure is widely used in the automotive,
mechanical, aerospace, biomedical and civil engineering industries [1–4]. The middle
surface of the shell is a curved surface where the in-plane displacement and the transverse
displacement usually occur simultaneously [5]. Due to the membrane-bending coupling
in the shell, the mechanical analysis of the shell is much more complicated than the
analysis of the plate and thus has received widespread attention recently [6]. At present,
the commonly used shell elements include the thin shell element based on the Kirchhoff/Love assumption and the thick/thin shell element based on the Reissner/Mindlin
assumption [7].
The Kirchhoff/Love assumption neglects the shear effect and has the C 1 continuity
requirement [8]. In the finite element method, the satisfaction of C 1 continuity is quite
difficult, and special skills are needed to construct the thin shell element [9]. Therefore,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. N. Atluri and I. Vušanovi´ c (Eds.): ICCES 2020, MMS 97, pp. 95–116, 2021.
https://doi.org/10.1007/978-3-030-64690-5_10
with Incompatible Approximation in a General
Orthogonal Curvilinear Coordinate System
Jingxu Chen (B) , Yongchang Cai, and Pengfei Yan
State Key Laboratory of Disaster Reduction in Civil Engineering, College of Civil Engineering,
Tongji University, Shanghai 200092, People’s Republic of China
jingxuchen@foxmail.com
Abstract. A new locking-free thick/thin shell element with incompatible approximation (SEIA) in a general orthogonal curvilinear coordinate system is proposed
for the analysis of the thick/thin shell. The key points for the development of
the present element include the definition of the incompatible polynomial displacement approximation in each independent element and the utilization of the
fictitious thin layers to ensure the displacement conformity between adjacent elements. Superior to most available shell elements, the SEIA avoids the shear and
membrane locking naturally without the adoption of numerical expediencies, has
concise theoretical derivation and easy numerical implementation, is insensitive to
element distortions and thus provides reliable solutions for the thick/thin shell. The
present work also initiates a methodology to develop locking-free thick/thin shell
elements. Numerical investigations and comparisons demonstrate the convergence
and robustness of the present element.
Keywords: Thick/thin shell · Locking-free · Incompatible approximation ·
Curvilinear coordinate
1 Introduction
Given its high load bearing capacity compared with the plate and small volume compared
with the solid, the curved thin-walled shell structure is widely used in the automotive,
mechanical, aerospace, biomedical and civil engineering industries [1–4]. The middle
surface of the shell is a curved surface where the in-plane displacement and the transverse
displacement usually occur simultaneously [5]. Due to the membrane-bending coupling
in the shell, the mechanical analysis of the shell is much more complicated than the
analysis of the plate and thus has received widespread attention recently [6]. At present,
the commonly used shell elements include the thin shell element based on the Kirchhoff/Love assumption and the thick/thin shell element based on the Reissner/Mindlin
assumption [7].
The Kirchhoff/Love assumption neglects the shear effect and has the C 1 continuity
requirement [8]. In the finite element method, the satisfaction of C 1 continuity is quite
difficult, and special skills are needed to construct the thin shell element [9]. Therefore,
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
S. N. Atluri and I. Vušanovi´ c (Eds.): ICCES 2020, MMS 97, pp. 95–116, 2021.
https://doi.org/10.1007/978-3-030-64690-5_10