Arabian Journal for Science and Engineering

, Volume 44, Issue 5, pp 4703–4724 | Cite as

Development of Eight-Node Curved-Side Quadrilateral Membrane Element Using Chain Direct Integration Scheme (SCDI) in Area Coordinates (MHCQ8-DI)

  • Guoxiang Zhang
  • Min WangEmail author
Research Article - Civil Engineering


This paper proposes an integration scheme using successively direct integration and conventional numerical integration to solve the element stiffness matrix. First, construct the integral formula for the element stiffness matrix by conducting direct integration for global coordinate. Second, solve the element stiffness matrix by converting the integration formula of element stiffness matrix for global coordinate into one for local coordinate. In view of the sequence and dependencies of the mixed use of the two integration schemes, the scheme is named as Chained Direct Integration Scheme (SCDI for short). This scheme can be applied when the global coordinate cannot be directly integral because integral interval of global coordinate cannot be determined. To better illustrate the specific operation of the scheme, an eight-node curved-side quadrilateral membrane element (MHCQ8-DI for short) with general interpolation points is constructed. The new element has the advantages of anti-distortion, good convergence, and the excellent performance to fit curved boundaries. The solution process of stiffness matrix of MHCQ8-DI presents that the scheme can reduce the integration dimension and computational workload. The numerical results show that the simulation accuracy for the new scheme was not lower than one for merely using conventional numerical integration. Besides, the theoretical derivation is given to exclude zero energy mode.


Chain direct integration scheme High accuracy Curved-side element Second-order completeness General interpolation points Zero energy mode 


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We would like to acknowledge the helpful comments on this paper received from our reviewers.


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Department of Civil Architectural Engineering SchoolCentral South UniversityChangshaChina

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