A Hybrid Stress Plane Element with Strain Field

Document Type: Research Papers

Authors

Ferdowsi University of Mashhad

Abstract

In this paper, a plane quadrilateral element with rotational degrees of freedom is developed. Present formulation is based on a hybrid functional with independent boundary displacement and internal optimum strain field. All the optimality constraints, including being rotational invariant, omitting the parasitic shear error and satisfying Fliepa’s pure bending test, are considered. Moreover, the static equilibrium equations are satisfied in this scheme. Authors’ element has only four nodes and twelve degrees of freedom. For the boundary displacement field, Alman’s second-order displacement function is employed. The validities of the proposed element are demonstrated by eleven numerical examples: thick curved beam, thin cantilever beam, Cooke’s skew beam, thin curved beam, cantilever beam with distortion parameter, high-order patch test, cantilever beam with five and four irregular mesh, Mc Neal’s thin cantilever beam and cantilever shear wall with and without openings. When utilizing the coarse and irregular meshes, numerical tests show the high accuracy, rapid convergence and robustness of the suggested element. Less sensitivity to distortion is another property of the new element.

Keywords

Main Subjects


Canhui, Z. and Suong, V.H. (2014). "A systematic and quantitative method to determine the optimal assumed stress fields for hybrid stress finite elements", Finite Elements in Analysis and Design, 80, 41-62.

Cen, S., Chen, X.M. and Fu, X.R. (2007). "Quadrilateral membrane element family formulated by the quadrilateral area coordinate method", Computer Methods in Applied Mechanics and Engineering, 196(41-44), 4337-4353.

Cen, S., Chen, X.M., Li, C.F. and Fu, X.R. (2009). "Quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method (QACM-II)", International Journal for Numerical Methods in Engineering, 77, 1172-1200.

Cen, S., Fu, X.R. and Zhou, M.J. (2011a). "8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes", Computer Methods in Applied Mechanics and Engineering, 200, 2321-2336.

Cen, S., Zhou, M.J. and Fu, X.R. (2011b). "A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom less sensitive to severe mesh distortions", Computers and Structures, 89, 517-528.

Cena, S., Zhoua, M.J. and Fub, X.R. (2011). "A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom less sensitive to severe mesh distortions", Computers and Structures, 89(5-6), 517-528.

Choi, N., Choo, Y.S. and Lee, B.C. (2006). "A hybrid Trefftz plane elasticity element with drilling degrees of freedom", Computer Methods in Applied Mechanics and Engineering, 195, 4095-4105.

Choo, Y.S., Choi, N. and Lee, B.C. (2006). "Quadrilateral and triangular plane elements with rotational degrees of freedom based on the hybrid Trefftz method", Finite Elements in Analysis and Design, 42, 1002-1008.

Felippa, C.A. (2006). "Supernatural QUAD4: a template formulation", invited contribution to J.H. Argyris Memorial Issue, Computer Methods in Applied Mechanics and Engineering, 195, 5316-5342.

Fu, X.R., Cen, S., Li, C.F. and Chen, X.M. (2010). "Analytical trial function method for development of new 8-node plane element based on the variational principle containing Airy stress function", Engineering Computations, 27(4), 442-463.

Huang, M., Zhao, Z. and Shen, C. (2010). "An effective planar triangular element with drilling rotation", Finite Elements in Analysis and Design, 46, 1031-1036.

Long, Y.Q. and Xu, Y. (1994). "Generalized conforming quadrilateral membrane element with vertex rigid rotational freedom", Computers and Structures, 52(4), 749-755.

Long, Z.F., Cen, S., Wang, L., Fu, X.R. and Long, Y.Q. (2010). "The third form of the quadrilateral area coordinate method (QACM-III): theory, application, and scheme of composite coordinate interpolation", Finite Elements in Analysis and Design, 46(10), 805-818.

Madeo, A., Casciaro, R., Zagari, G., Zinno, R. and Zucco, G. (2014). "A mixed isostatic 16 DOF quadrilateral membrane element with drilling rotations, based on Airy stresses", Finite Elements in Analysis and Design, 89, 52-66.

Madeo, A., Zagari, G. and Casciaro, R. (2012). "An isostatic quadrilateral membrane finite element with drilling rotations and no spurious modes", Finite Elements in Analysis and Design, 50, 21-32.

McNeal, R.H. and Harder, R.L. (1985). "A proposed standard set of problems to test finite element accuracy", Finite Elements in Analysis and Design, 1(1), 3-20.

Prathap, G. and Senthilkumar, V. (2008). "Making sense of the quadrilateral area coordinate membrane elements", Computer Methods in Applied Mechanics and Engineering, 197(49-50), 4379-4382.

Rezaiee-Pajand, M. and Karkon, M. (2014). "Hybrid stress and analytical functions for analysis of thin plates bending", Latin American Journal of Solid and Structures, 11, 556-579.

Rezaiee-Pajand, M. and Yaghoobi, M. (2012). "Formulating an effective generalized four-sided element", European Journal of Mechanics A/Solids, 36, 141-155.

Rezaiee-Pajand, M. and Yaghoobi, M. (2013). "A free of parasitic shear strain formulation for plane element", Research in Civil and Environmental Engineering, 1, 1-27.

Rezaiee-Pajand, M. and Yaghoobi, M. (2014). "An efficient formulation for linear and geometric non-linear membrane elements", Latin American Journal of Solid and Structures, 11, 1012-1035.

Rezaiee-Pajand, M. and Yaghoobi, M. (2015). "Two new quadrilateral elements based on strain states", Civil Engineering Infrastructures Journal, 48(1), 133-156.

Rojasa, F., Andersonb, J.C. and Massonea, L.M. (2016). "A nonlinear quadrilateral layered membrane element with drilling degrees of freedom for the modeling of reinforced concrete walls", Engineering Structures, 124, 521-538.

Santos, H.A.F.A. and Moitinho de Almeida, J.P. (2014). "A family of Piola­Kirchhoff hybrid stress finite elements for two­dimensional linear elasticity", Finite Elements in Analysis and Design, 85, 33­49.

Wisniewski, K. and Turska, E. (2006). "Enhanced Allman quadrilateral for finite drilling Rotations", Computer Methods in Applied Mechanics and Engineering, 195, 6086-6109.

Wisniewski, K. and Turska, E. (2008). "Improved four-node Hellingere Reissner elements based on skew coordinates", International Journal for Numerical Methods in Engineering, 76, 798-836.

Wisniewski, K. and Turska, E. (2009). "Improved 4-node Hu-Washizu elements based on skew coordinates", Computers and Structures, 87, 407-424.

Xing, C. and Zhou, C. (2016). "A singular planar element with rotational degree of freedom for fracture analysis", Theoretical and Applied Fracture Mechanics, 86, 239-249.

Zouaria, W., Hammadib, F. and Ayadc, R. (2016). "Quadrilateral membrane finite elements with rotational DOFs for the analysis of geometrically linear and nonlinear plane problems", Computers and Structures, 173, 139-149.