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Three stress-based triangular elements

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Abstract

To obtain proper stresses , three new triangular elements are formulated in this study. First, a complementary energy functional is used within an element for the analysis of plane problems. In this energy manifestation, the Airy stress function will be applied as a functional variable. Then, some basic analytical solutions are assigned for the stress functions. These trial functions are matched with each element number of degrees of freedom. The result is a number of equations with anonymous constants. Subsequently, according to the principle of minimum complementary energy, the unknown constants can be expressed in terms of displacements. Finally, this system can be rewritten in terms of the nodal displacement. In this way, three new triangular elements are formulated. To validate the results, extensive numerical studies are performed. The findings clearly demonstrate accuracies of structural displacements as well as stresses.

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Abbreviations

AT6:

Accurate triangular element

t :

Thickness of the element

U :

Displacement vector along element boundaries

\( \mu \) :

Poisson’s ratio

\( {D_{ij}} \) :

Elastic modulus

\( l \) and m :

Direction cosines of the outer normal

\( {{\varPi }}_C^* \) :

Complementary energy within the element

\( \sigma \) :

The stress vector of the element

C :

Elastic flexibility matrix

\( \varphi \) :

Airy stress function

\( {C_{ij}} \) :

Elastic compliances

\( {u_i} \) :

Nodal displacements belong to x

\( V_C^* \) :

Complementary energy along the element boundaries

T :

Surface force vector on the element boundaries

\( E \) :

Young’s modulus

\( {q^e} \) :

Elemental nodal displacement vector

x′ and y′:

Axes of material symmetry

v i :

Nodal displacements belong to y

\( N_i^0\left( {{\xi_1},{\xi_2}} \right) \) :

Shape function

References

  1. Fu XR, Cen S, Li CF, Chen XM (2008) Analytical trial function method for development of new 8-node plane element based on the variational principle containing airy stress function. Eng Comput Int J Comput Aided Eng Softw 27(4):442–463

    MATH  Google Scholar 

  2. Stricklin JA, Ho WS, Richardson EQ, Haister WE (1977) ‘On isoparametric vs linear strain triangular elements’. Int J Numer Methods Eng 11:1041–1043

    Google Scholar 

  3. Lee NS, Bathe KJ (1993) Effects of element distortion on the performance of isoparametric elements. Int J Numer Methods Eng 36:3553–3576

    MATH  Google Scholar 

  4. Kikuchi F, Okabe M, Fujio H (1999) Modification of the 8-node serendipity element. Comput Methods Appl Mech Eng 179:91–109

    MATH  Google Scholar 

  5. Li LX, Kunimatsu S, Han XP, Xu SQ (2004) The analysis of interpolation precision of quadrilateral elements. Finite Elem Anal Design 41:91–108

    Google Scholar 

  6. Rajendran S, Liew KM (2003) ‘A novel unsymmetric 8-node plane element immune to mesh distortion under a quadratic displacement field’. Inte J Numer Methods Eng 58:1713–1748

    MATH  Google Scholar 

  7. Li CJ, Wang RH (2006) A new 8-node quadrilateral spline finite element. J Comput Appl Math 195:54–65

    MathSciNet  MATH  Google Scholar 

  8. Long YQ, Xu Y (1994) Generalized conforming triangular membrane element with vertex rigid rotational freedom. Finite Elem Anal Design 17:259–271

    MATH  Google Scholar 

  9. Cen S, Chen XM, Fu XR (2007) Quadrilateral membrane element family formulated by the quadrilateral area coordinate method’. Comput Methods Appl Mech Eng 196(41–44):4337–4353

    MATH  Google Scholar 

  10. Soh AK, Long YQ, Cen S (2000) Development of eight-node quadrilateral membrane elements using the area coordinates method. Comput Mech 25(4):376–384

    MATH  Google Scholar 

  11. Cen S, Fu XR, Zhou MJ (2011) 8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes. Comput Methods Appl Mech Eng 200:2321–2336

    MathSciNet  MATH  Google Scholar 

  12. Cen S, Zhou MJ, Fu XR (2011) A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom less sensitive to severe mesh distortions. Comput Struct 89(5–6):517–528

    Google Scholar 

  13. Sergei N, Viacheslav K, Antti B, Niemi H (2016) Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems. Comput Methods Appl Mech Eng 308:182–211

    MathSciNet  MATH  Google Scholar 

  14. Albocher U, Oberai AA, Barbone PE, Harari I (2009) Adjoint-weighted equation for inverse problems of incompressible plane-stress elasticity. Comput Methods Appl Mech Eng 198(30–32):2412–2420

    MATH  Google Scholar 

  15. Cen S, Fu XR, Zhou GH, Zhou MJ, Li CF (2011) Shape-free finite element method: the plane hybrid stress-function (HS-F) element method for anisotropic materials. Sci China Phys Mech Astron 54(4):653–665

    Google Scholar 

  16. Zhou P, Cen S (2015) A novel shape-free plane quadratic polygonal hybrid stress-function element. Math Probl Eng. https://doi.org/10.1155/2015/491325

    Article  MathSciNet  MATH  Google Scholar 

  17. Babaniyi OA, Oberai AA, Barbone PE (2017) Direct error in constitutive equation formulation for plane stress inverse elasticity problem. Comput Methods Appl Mech Eng 314:3–18

    MathSciNet  MATH  Google Scholar 

  18. Cervera M, Chiumenti M, Capua DD (2012) Benchmarking on bifurcation and localization in J2 plasticity for plane stress and plane strain conditions. Computer Methods Appl Mech Eng Vol 241–244:206–224

    MATH  Google Scholar 

  19. Artioli E, Miranda ED, Lovadina C, Patruno L (2017) A stress/displacement virtual element method for plane elasticity problems. Comput Methods Appl Mech Eng 325:155–174

    MathSciNet  MATH  Google Scholar 

  20. Rezaiee-Pajand M, Karkon M (2016) Geometrical nonlinear analysis of plane problems by corotational formulation. J Eng Mech 142(10):04016073

    Google Scholar 

  21. Madeo A, Zagari G, Casciaro R (2012) An isostatic quadrilateral membrane finite element with drilling rotations and no spurious modes. Finite Elem Anal Design 50:21–32

    Google Scholar 

  22. Madeo A, Casciaro R, Zagari G, Zinno R, Zucco G (2014) A mixed isostatic 16 dof quadrilateral membrane element with drilling rotations based on Airy stresses. Finite Elem Anal Design 89:52–66

    MathSciNet  Google Scholar 

  23. Fosdick R, Schuler K (2003) Generalized airy stress functions. Meccanica 38:571–578

    MathSciNet  MATH  Google Scholar 

  24. Zavelani Rossi A (1974) Finite element techniques in plane limit problems. Meccanica 9(4):312–324

    MATH  Google Scholar 

  25. Benedetti D, Maier G, Zavelani Rossi A (1972) A finite element approach to optimal design of plastic structures in plane stresses. Meccanica 7(1):37–38

    MATH  Google Scholar 

  26. Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method: its basis and fundamental, 6th edn. Heinemann, Butterworth

    Google Scholar 

  27. Lee NS, Bathe KJ (1993) Effects of element distortion on the performance of isoparametric elements. Int J Numer Methods Eng 36:3553–3576

    MATH  Google Scholar 

  28. Taylor RL, Beresford PJ, Wilson EL (1976) A non-conforming element for stress analysis. Int J Numer Methods Eng 10:1211–1219

    MATH  Google Scholar 

  29. Ibrahimbegovic A, Wilson EL (1991) A modified method of incompatible modes. Commun Appl Numer Methods 7:187–194

    MATH  Google Scholar 

  30. Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29:1595–1638

    MathSciNet  MATH  Google Scholar 

  31. Hughes TJR (1980) Generalization of selective integration procedures to anisotropic and nonlinear media. Int J Numer Methods Eng 16:1413–1418

    MathSciNet  MATH  Google Scholar 

  32. Chen J, Li CJ, Chen WJ (2010) A family of spline finite elements. Comput Struct 88:718–727

    Google Scholar 

  33. Ooi ET, Rajendran S, Yeo JH (2004) A 20-node hexahedron element with enhanced distortion tolerance. Int J Numer Methods Eng 2004(60):2501–2530

    MATH  Google Scholar 

  34. Long ZF, Li JX, Cen S (1999) Some basic formulae for area coordinates used in quadrilateral elements. Commun Numer Methods Eng 15:841–852

    MATH  Google Scholar 

  35. Chen XM, Cen S, Long YQ (2003) Membrane elements insensitive to distortion using the quadrilateral area coordinate method. Comput Struct 82:35–54

    Google Scholar 

  36. Chen XM, Cen S, Fu XR (2008) A new quadrilateral area coordinate method (QACM-II) for developing quadrilateral finite element models. Int J Numer Methods Eng 73:1911–1941

    MATH  Google Scholar 

  37. Zhang X, Liu XH, Song KZ (2001) Least-squares collocation meshless method. Int J Numer Methods Eng 51:1089–1100

    MathSciNet  MATH  Google Scholar 

  38. Zhang JM, Yao ZH, Li H (2002) A hybrid boundary node method. Int J Numer Methods Eng 53:751–763

    MathSciNet  Google Scholar 

  39. Taylor RL, Beresford PJ, Wilson EL (1976) A non-conforming element for stress analysis. Int J Numer Methods Eng 10:1211–1219

    MATH  Google Scholar 

  40. MacNeal RH, Harder RL (1985) A proposed standard set of problems to test finite element accuracy. Finite Elem Anal Des 1(1):3–20

    Google Scholar 

  41. Pian THH, Sumihara K (1984) Rational approach for assumed stress finite elements. Int J Numer Methods Eng 20:1685–1695

    MATH  Google Scholar 

  42. Allman DJ (1988) A quadrilateral finite element including vertex rotations for plane elasticity analysis. Int J Numer Methods Eng. 26:717–730

    MATH  Google Scholar 

  43. MacNeal RH, Harder RL (1988) A refined four-nodded membrane element with rotational degrees of freedom. Comput Struct 28:75–84

    MATH  Google Scholar 

  44. Choi N, Choo YS, Lee BC (2006) A hybrid Trefftz plane elasticity element with drilling degrees of freedom. Comput Methods Appl Mech Eng 195:4095–4105

    MATH  Google Scholar 

  45. Long YQ, Xu Y (1994) Generalized conforming quadrilateral membrane element with vertex rigid rotational freedom. Comput Struct 52(4):749–755

    MATH  Google Scholar 

  46. Pimpinelli G (2004) An assumed strain quadrilateral element with drilling degrees of freedom. Finite Elem Anal Des 41:267–283

    Google Scholar 

  47. Zhang H, Kuang JS (2008) Eight-node membrane element with drilling degrees of freedom for analysis of in-plane stiffness of thick floor plates. Int J Numer Methods Eng 76:2117–2136

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

    Mohammad Rezaiee-Pajand & Arash Karimipour

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  1. Mohammad Rezaiee-Pajand
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  2. Arash Karimipour
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Correspondence to Mohammad Rezaiee-Pajand.

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Rezaiee-Pajand, M., Karimipour, A. Three stress-based triangular elements. Engineering with Computers 36, 1325–1345 (2020). https://doi.org/10.1007/s00366-019-00765-6

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  • DOI: https://doi.org/10.1007/s00366-019-00765-6

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