Document Type : Article

**Authors**

School of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.

10.24200/sci.2019.21429

**Abstract**

A new higher-order triangular plane element with drilling degrees of freedom is proposed by assumption of second-order strain field. In addition to inclusion of drilling degrees of freedom and utilization of higher-order assumes strains, satisfaction of equilibrium equations improves performance of the suggested element in comparison with many of the other available elements. After proposition of the new element, a series of benchmark problems are solved to evaluate performance of the suggested element. Accuracy and efficiency of the suggested element is compared with other strain-based plane elements. Detailed discussions are offered after each benchmark problem. Finally, based on the attained results, a final conclusion about characteristics of robust membrane elements is made.

**Keywords**

Refrences:

1.Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method for Solid and Structural Mechanics, Elsevier (2005).

2. Hughes, T.J.R., Taylor, R.L., and Kanoknukulchai, W. A simple and e_cient _nite element for plate bending", Int. J. Numer. Meth. Eng., 11(10), pp. 1529-1543 (1977).

3. Reddy, J.N., An Introduction to the Finite Element Method, New York, USA: McGraw-Hill (1993). 4. Rezaiee-Pajand, M. and Gharaei-Moghaddam, N. Analysis of 3D Timoshenko frames having geometrical and material nonlinearities", Int. J. of Mech. Sci., 94, pp. 140-155 (2015). 5. Rezaiee-Pajand, M. and Gharaei-Moghaddam, N. Frame nonlinear analysis by force method", Int. J. Ste. Str., 17(2), pp. 609-629 (2017). 6. Rezaiee-Pajand, M. and Gharaei-Moghaddam, N. Using co-rotational method for cracked frame analysis", Meccanica, 53(8), pp. 2121-2143 (2018). 7. Rezaiee-Pajand, M. and Gharaei-Moghaddam, N. Force-based curved beam elements with open radial edge cracks", Mech. Adv. Mater. Struc., pp. 1-13 (2018). DOI: 10.1080/15376494.2018.1472326 8. Rezaiee-Pajand, M. and Gharaei-Moghaddam, N. Vibration and static analysis of cracked and non-cracked non-prismatic frames by force formulation", Eng. Str., 185, pp. 106-121 (2019). 9. Sabir, A.B. A rectangular and triangular plane elasticity element with drilling degrees of freedom", 2nd Int. Conf. on Var. Meth. in Engrg., Southampton, UK, pp. 17-25 (1985). 10. Sabir, A.B. and Sfendji, A. Triangular and rectangular plane elasticity _nite elements", Thin. Wall. Struct., 21(3), pp. 225-232 (1995). 11. Tayeh, S.M., New Strain-Based Triangular and Rectangular Finite Elements for Plane Elasticity Problems, The Islamic University of Gaza (2003). 12. Belarbi, M.T. and Bourezane, M. On improved Sabir triangular element with drilling rotation", Rev. Europ. G_en. Civ., 9(9-10), pp. 1151-1175 (2005). 13. Belarbi, M.T. and Bourezane, M. An assumed strain based on triangular element with drilling rotation", Cour. Sav., 6, pp. 117-123 (2005). 14. Belarbi, M.T. and Maalem, T. On improved rectangular _nite element for plane linear elasticity analysis", Rev. Europ. El_em., 14(8), pp. 985-997 (2005). 15. Rezaiee-Pajand, M. and Yaghoobi, M. Formulating an e_ective generalized four-sided element", Eur. J. Mech. A-Solid, 36, pp. 141-155 (2012). 16. Rezaiee Pajand, M. and Yaghoobi, M. A free of parasitic shear strain formulation for plane element", Res. Civ. Env. Eng., 1, pp. 1-24 (2013). 17. Rebiai, C. and Belounar, L. A new strain based rectangular _nite element with drilling rotation for linear and nonlinear analysis", Arch. Civ. Mech. Eng., 13(1), pp. 72-81 (2013). 18. Rezaiee-Pajand, M. and Yaghoobi, M. A robust triangular membrane element", Lat. Amer. J. Sol. Struc., 11(14), pp. 2648-2671 (2014). 2274 M. Rezaiee-Pajand et al./Scientia Iranica, Transactions A: Civil Engineering 26 (2019) 2258{2275 19. Rezaiee-Pajand, M. and Yaghoobi M. An e_cient formulation for linear and geometric non-linear membrane elements", Lat. Amer. J. Sol. and Struc., 11(6), pp. 1012-1035 (2014). 20. Rebiai, C. and Belounar, L. An e_ective quadrilateral membrane _nite element based on the strain approach", Measurement, 50, pp. 263-269 (2014). 21. Rebiai, C., Saidani, N., and Bahloul, E. A new _nite element based on the strain approach for linear and dynamic analysis", Res. J. Appl. Sci. Eng. Tech., 11(6), pp. 639-644 (2015). 22. Rezaiee-Pajand, M. and Yaghoobi, M. Two new quadrilateral elements based on strain states", Civ. Eng. Inf. J., 48(1), pp. 133-156 (2015). 23. Hamadi, D., Ayoub, A., and Maalem, T. A new strain-based _nite element for plane elasticity problems", Eng. Comp., 33(2), pp. 562-579 (2016). 24. Rezaiee-Pajand, M. and Yaghoobi, M. Geometrical nonlinear analysis by plane quadrilateral element", Sci. Ira., 25(5), pp. 2488-2500 (2018). 25. Rebiai, C. Finite element analysis of 2-D structures by new strain based triangular element", J. Mech., 35(3) pp. 1-9 (2018). 26. Rezaiee Pajand, M., Gharaei Moghaddam, N., and Ramezani, M.R. Two triangular membrane elements based on strain", Int. J. Appl. Mech., 11(1), p. 1950010 (2019). 27. Belounar, L. and Guenfoud, M. A new rectangular _nite element based on the strain approach for plate bending", Thin Wall. Struct., 43(1), pp. 47-63 (2005). 28. Hamadi, D., Abderrahmani, S., Maalem, T., and Temami, O. E_ciency of the strain based approach formulation for plate bending analysis", Int. J. Mech. Aero. Ind. Mechatr. Manufac. Eng., 8(8), pp. 1408- 1412 (2014). 29. Abderrahmani, S., Maalem, T., and Hamadi, D. On improved thin plate bending rectangular _nite element based on the strain approach", Int. J. Eng. Res. Afr., 27, pp. 76-86 (2016). 30. Abderrahmani, S., Maalem, T., Zatar, A., and Hamadi, D. A new strain based sector _nite element for plate bending problems", Int. J. Eng. Res. Afr., 31, pp. 1-13 (2017). 31. Belarbi, M.T. and Charif, A. Novel sector element based on strain with in-plane rotations", Rev. Europ. El_em., 7(4), pp. 439-458 (1998) (In French). 32. Belounar, A., Benmebarek, S., and Belounar, L. Strain based triangular _nite element for plate bending analysis", Mech. Adv. Mater. Struc., pp. 1-13 (2018). DOI: 10.1080/15376494.2018.1488310 33. Ashwell, D.G. and Sabir, A.B. A new cylindrical shell _nite element based on simple independent strain functions", Int. J. Mech. Sci., 14(3), pp. 171-183 (1972). 34. Djoudi, M.S. and Bahai, H. Strain based _nite element for the vibration of cylindrical panels with openings", Thin wall. Struct., 42(4), pp. 575-588 (1972). 35. Hamadi, D., Temami, O., Zatar, A., and Abderrahmani, S. A comparative study between displacement and strain based formulated _nite elements applied to the analysis of thin shell structures", Int. J. Civ. Env. Struc. Const. Arch. Eng., 8(8), pp. 875-880 (2014). 36. Mousa, A. and Djoudi, M. New strain based triangular _nite element for the vibration of circular cylindrical shell with oblique ends", Int. J. Civ. Env. Eng., 15(5), pp. 6-11 (2015). 37. Rezaiee-Pajand, M. and Yaghoobi, M. An e_cient at shell element", Meccanica, 53(4-5), pp. 1015-1035 (2018). 38. To, C.W.S. and Liu, M.L. Hybrid strain based threenode at triangular shell elements", Finite Elem. Anal. Des., 17(3), pp. 169-203 (1994). 39. Rezaiee-Pajand, M. and Yaghoobi, M. A hybrid stress plane element with strain _eld", Civ. Eng. Inf. J., 50(2), pp. 255-275 (2017). 40. Belounar, L. and Guerraiche, K. A new strain based brick element for plate bending", Alex. Eng. J., 53(1), pp. 95-105 (2014). 41. Guerraiche, K., Belounar, L., and Bouzidi, L. A new eight nodes brick _nite element based on the strain approach", J. Solid Mech., 10(1), pp. 186-199 (2018). 42. Messai, A., Belounar, L., and Merzouki, T. Static and free vibration of plates with a strain based brick element", Eur. J. Comp. Mech., pp. 1-21 (2018). DOI: 10.1080/17797179.2018.1560845 43. Alvin, K., Horacio, M., Haugen, B., and Felippa, C.A. Membrane triangles with corner drilling freedoms-I. The EFF element", Finite Elem. Anal. Des., 12(3-4), pp. 163-187 (1992). 44. Allman, D.J. A quadrilateral _nite element including vertex rotations for plane elasticity analysis", Int. J. Numer. Meth. Eng., 26(3), pp. 717-730 (1988). 45. Cook, R.D. A plane hybrid element with rotational DOF and adjustable sti_ness", Int. J. Numer. Meth. Eng., 24(8), pp. 1499-1508 (1987). 46. MacNeal, R.H. and Harder, R.L. A re_ned fournoded membrane element with rotational degrees of freedom", Comput. Struct., 28(1), pp. 75-84 (1988). 47. Cook, R.D. Some options for plane triangular elements with rotational degrees of freedom", Finite Eleme. Anal. Des., 6(3), pp. 245-249 (1990). 48. Cook, R.D. Modi_ed formulations for nine-dof plane triangles that include vertex rotations", Int. J. Numer. Meth. Eng., 31(5), pp. 825-835 (1991). 49. Felippa, C.A. A study of optimal membrane triangles with drilling freedoms", Comput. Meth. Appl. M., 192(16-18), pp. 2125-2168 (2003). 50. Timoshenko, S.P. and Goodier, J.N., Theory of Elasticity, 3rd Edn., McGraw-Hill: New York, U.S. (1934).

Transactions on Civil Engineering (A)

July and August 2019Pages 2258-2275