Archive of Applied Mechanics

, Volume 88, Issue 1–2, pp 253–269 | Cite as

Hysteretic beam element with degrading smooth models



This work presents the development of a hysteretic beam element in the context of the finite element method, that is suitable for the inelastic dynamic analysis of framed structures. The formulation proposed is able to capture the main characteristics of hysteresis in structural systems and mainly accounts for stiffness degradation, strength deterioration and pinching phenomena, as well as for non-symmetrical yielding that often characterizes their behavior. The proposed formulation is based on the decoupling of deformations into elastic and hysteretic parts by considering additional hysteretic degrees of freedom, i.e., the hysteretic curvatures and hysteretic axial deformations. The direct stiffness method is employed to establish global matrices and determine the mass and viscous damping, as well as the elastic stiffness and the hysteretic matrix of the structure that corresponds to the newly added hysteretic degrees of freedom. All the governing equations of the structure, namely the linear global equations of motion and the nonlinear evolution equations at elemental level that account for degradations and pinching, are solved simultaneously. This is accomplished by converting the system of equations into state space form and implementing a variable-order solver based on numerical differentiation formulas (NDFs) to determine the solution. Furthermore, hysteretic loops and degradation phenomena are easily controlled by modifying the model parameters at the element level enabling simulations of a more realistic response. Numerical results are presented and compared against experimental results and other finite element codes to validate the proposed formulation and verify its ability to simulate complex hysteretic behavior exhibiting cyclic degradations.


Bouc–Wen Hysteresis Stiffness degradation Strength deterioration Pinching 



The authors would like to acknowledge the support from the “RESEARCH PROJECTS FOR EXCELLENCE IKY/SIEMENS”.


  1. 1.
    Andriotis, C., Gkimousis, I., Koumousis, V.K.: Modeling reinforced concrete structures using smooth plasticity and damage models. J. Struct. Eng. (2016). doi: 10.1061/(ASCE)ST.1943-541X.0001365 Google Scholar
  2. 2.
    Baber, T.T., Noori, M.N.: Random vibration of degrading pinching systems. J. Eng. Mech. 11(8), 1010–1026 (1985)CrossRefGoogle Scholar
  3. 3.
    Baber, T.T., Wen, Y.K.: Random vibration of hysteretic degrading systems. J. Eng. Mech. 107(6), 1069–1087 (1981)Google Scholar
  4. 4.
    Bathe, K.J.: Finite Element Procedures. Prentice Hall Engineering, New York (2007)MATHGoogle Scholar
  5. 5.
    Bouc, R.: Forced vibration of mechanical system with hysteresis. In: Proceedings of the 4th Conference on Nonlinear Oscillations, Prague (1967)Google Scholar
  6. 6.
    Charalampakis, A.E., Dimou, C.K.: Identification of Bouc–Wen hysteretic systems using particle swarm optimization. Comput. Struct. 88(21–22), 1197–1205 (2010)CrossRefGoogle Scholar
  7. 7.
    Charalampakis, A.E., Koumousis, V.K.: Identification of Bouc–Wen hysteretic systems by a hybrid evolutionary algorithm. J. Sound Vib. 314(3–5), 571–585 (2008)CrossRefMATHGoogle Scholar
  8. 8.
    Charalampakis, A.E., Koumousis, V.K.: A Bouc–Wen model compatible with plasticity postulates. J. Sound Vib. 322, 954–968 (2009)CrossRefGoogle Scholar
  9. 9.
    Chatzi, E.N., Smyth, A.W.: The unscented Kalman filter and particle filter methods for nonlinear structural system identification with non-collocated heterogeneous sensing. Struct. Control Health Monit. 16(1), 99–123 (2009)CrossRefGoogle Scholar
  10. 10.
    Christos, S.D., Vlasis, K.K.: Plane stress problems using hysteretic rigid body spring network models. Comput. Part. Mech. (2016). doi: 10.1007/s40571-016-0128-1 Google Scholar
  11. 11.
    Clough, R.W.: Effects of stiffness degradation on earthquake ductility requirements. Technical Report No. 66-16. University of California, Berkeley (1966)Google Scholar
  12. 12.
    Erlicher, S., Bursi, O.S.: Bouc–Wen-type models with stiffness degradation: thermodynamic analysis and applications. J. Eng. Mech. 134(10), 843–855 (2009)CrossRefGoogle Scholar
  13. 13.
    Erlicher, S., Point, N.: Thermodynamic admissibility of Bouc–Wen type hysteresis models. C. R. Mec. 332(1), 51–57 (2004)CrossRefMATHGoogle Scholar
  14. 14.
    FEMA: Effects of strength and stiffness degradation on seismic response. Report No. FEMA-P440A, Prepared for the Federal Emergency Management Agency by the Advanced Technology Council, Washington, DC (2009)Google Scholar
  15. 15.
    Foliente, G.C.: Hysteresis modelling of wood joints and structural systems. J. Struct. Eng. 121(6), 1013–1022 (1995)CrossRefGoogle Scholar
  16. 16.
    Gkimousis, I.A., Koumousis, V.K.: Inelastic mixed fiber beam element for steel cyclic behavior. Eng. Struct. 106, 399–409 (2016)CrossRefGoogle Scholar
  17. 17.
    Ikhouane, F., Hurtado, J.E., Rodellar, J.: Variation of the hysteresis loop with the Bouc–Wen model parameters. Nonlinear Dyn. 48(4), 361–380 (2007)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ikhouane, F., Rodellar, J.: On the hysteretic Bouc–Wen model. Part I: forced limit cycle characterization. Nonlinear Dyn. 42, 63–78 (2005)CrossRefMATHGoogle Scholar
  19. 19.
    Ikhouane, F., Rodellar, J.: On the hysteretic Bouc–Wen model. Part II: robust parametric identification. Nonlinear Dyn. 42, 79–95 (2005)CrossRefMATHGoogle Scholar
  20. 20.
    Ismail, M., Ikhouane, F., Rodellar, J.: The hysteresis Bouc–Wen model, a survey. J. Arch. Comput. Methods Eng. 16, 161 (2009). doi: 10.1007/s11831-009-9031-8 CrossRefMATHGoogle Scholar
  21. 21.
    Kottari, A.K., Charalampakis, A.E., Koumousis, V.K.: A consistent degrading Bouc–Wen model. Eng. Struct. 60, 235–240 (2014)CrossRefGoogle Scholar
  22. 22.
    Ma, F., Zhang, H., Bockstedte, A., Foliente, G.C., Paevere, P.: Parameter analysis of the differential model of hysteresis. J. Appl. Mech. 71, 342–349 (2004)CrossRefMATHGoogle Scholar
  23. 23.
    McKenna, F., Fenves, G.L., Scott, M.H.: Open System for Earthquake Engineering Simulation. University of California, Berkeley (2000)Google Scholar
  24. 24.
    Moysidis, A.N., Koumousis, V.K.: Hysteretic plate finite element. J. Eng. Mech. (2015). doi: 10.1061/(ASCE)EM.1943-7889.0000918 Google Scholar
  25. 25.
    Ninakawa, T., Sakino, K.: Inelastic behavior of concrete filled circular steel tubular columns subjected to uniform cyclic bending moment. In: Proceedings 11th World Conference Earthquake Engineering No. 1358, Acapulco (1996)Google Scholar
  26. 26.
    Park, Y.J., Reinhorn, A.M., Kunnath, S.K.: IDARC: inelastic damage analysis of reinforced concrete frame-shear wall structures. National Center for Earthquake Engineering Research, State University of New York, Buffalo, Technical Reports NCEER-87-0008 (1987)Google Scholar
  27. 27.
    Simeonov, V.K., Sivaselvan, M.V., Reinhorn, A.M.: Nonlinear analysis of structural frame system by the state-space approach. Comput. Aided Civ. Infrastruct. Eng. 15, 76–89 (2000)CrossRefGoogle Scholar
  28. 28.
    Sireteanu, T., Giuclea, M., Mitu, A.M.: Identification of an extended Bouc–Wen model with application to seismic protection through hysteretic devices. Comput. Mech. 45(5), 431–441 (2010)CrossRefMATHGoogle Scholar
  29. 29.
    Sivaselvan, M.V., Reinhorn, A.M.: Hysteretic models for deteriorating inelastic structures. J. Eng. Mech. 126, 633–640 (2000)CrossRefGoogle Scholar
  30. 30.
    Takeda, T., Sozen, M.A., Nielsen, N.N.: Reinforced concrete response to simulated earthquakes. ASCE J. Struct. Div. 96, 2557–2573 (1970)Google Scholar
  31. 31.
    Talatahari, S., Kaveh, A., Mohajer Rahbari, N.: Parameter identification of Bouc–Wen model for MR fluid dampers using adaptive charged system search optimization. J. Mech. Sci. Technol. 26, 2523–2534 (2012). doi: 10.1007/s12206-012-0625-y CrossRefGoogle Scholar
  32. 32.
    Triantafyllou, S., Koumousis, V.: Small and large displacement analysis of frame structures based on hysteretic beam elements. J. Eng. Mech. 138, 36–49 (2012)CrossRefGoogle Scholar
  33. 33.
    Triantafyllou, S.P., Koumousis, V.K.: An hysteretic quadrilateral plane stress element. Arch. Appl. Mech. 82, 1675 (2012). doi: 10.1007/s00419-012-0682-9 CrossRefMATHGoogle Scholar
  34. 34.
    Valanis, K.C.: A theory of viscoplasticity without a yield surface. Arch. Mech. Stosow. 23, 517–534 (1971)MathSciNetMATHGoogle Scholar
  35. 35.
    Visintin, A.: Differential Models of Hysteresis. Springer, Berlin (1994)CrossRefMATHGoogle Scholar
  36. 36.
    Wang, C., Foliente, G.C.: Hysteretic models for deteriorating inelastic structures. J. Eng. Mech. 127(11), 1200–1202 (2001)CrossRefGoogle Scholar
  37. 37.
    Wen, Y.K.: Method for random vibration of hysteretic systems. J. Eng. Mech. Div. 102, 249–263 (1976)Google Scholar
  38. 38.
    Wen, Y.K.: Equivalent linearization for hysteretic system under random excitation. J. Appl. Mech. 47, 150–154 (1980)CrossRefMATHGoogle Scholar
  39. 39.
    Xie, L.L., Lu, X.Z., Guan, H., Lu, X.: Experimental study and numerical model calibration for earthquake-induced collapse of RC frames with emphasis on key columns, joints, and the overall structure. J. Earthq. Eng. 19(8), 1320–1344 (2015). doi: 10.1080/13632469.2015.1040897 CrossRefGoogle Scholar
  40. 40.
    Yu, Q.S., Gilton, C., Uang, C.M.: Cyclic response of RBS moment connections: loading sequence and lateral bracing effects. Report No. SSRP-99/13. University of California, San Diego (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institute of Structural Analysis & Aseismic ResearchNational Technical University of AthensAthensGreece

Personalised recommendations