Combination rules and critical seismic response of steel buildings modeled as complex MDOF systems

  • Reyes-Salazar, Alfredo (Facultad de Ingenieria, Universidad Autonoma de Sinaloa, Ciudad Universitaria) ;
  • Valenzuela-Beltran, Federico (Facultad de Ingenieria, Universidad Autonoma de Sinaloa, Ciudad Universitaria) ;
  • de Leon-Escobedo, David (Facultad de Ingenieria, Universidad Autonoma del Estado de Mexico, Ciudad Universitaria) ;
  • Bojorquez-Mora, Eden (Facultad de Ingenieria, Universidad Autonoma de Sinaloa, Ciudad Universitaria) ;
  • Barraza, Arturo Lopez (Facultad de Ingenieria, Universidad Autonoma de Sinaloa, Ciudad Universitaria)
  • Received : 2013.12.27
  • Accepted : 2015.10.26
  • Published : 2016.01.25


The Maximum seismic responses of steel buildings with perimeter moment resisting frames (MRF), modeled as complex MDOF systems, are estimated for several incidence angles of the horizontal components and the critical one is identified. The accuracy of the existing rules to combine the effects of the individual components is also studied. Two and three components are considered. The critical response does not occur for principal components and the corresponding incidence angle varies from one earthquake to another. The critical response can be estimated as 1.40 and 1.10 times that of the principal components, for axial load and interstory shears, respectively. The rules underestimate the axial load but reasonably overestimate the shears. The rules are not always inaccurate in the estimation of the combined response for correlated components. On the other hand, totally uncorrelated (principal) components are not always related to an accurate estimation. The correlation of the individual effects (${\rho}$) may be significant, even for principal components. The rules are not always associated to an inaccurate estimation for large values of ${\rho}$, and small values of ${\rho}$ are not always related to an accurate estimation. Only for perfectly uncorrelated harmonic excitations and elastic analysis of SDOF systems, the individual effects of the components are uncorrelated and the rules accurately estimate the combined response. The degree of correlation of the components, the type of structural system, the response parameter under consideration, the location of the structural member and the level of structural deformation must be considered while estimating the level of underestimation or overestimation.


critical response;steel buildings;seismic design codes;combination rules;effect of individual components;correlation of effects;MDOF and SDOF systems


  1. Beyer, K. and Bommer J.J. (2007), "Selection and scaling of real accelerograms for bi-directional loading: a review of current practice and code provisions", J. Earthq. Eng., 11(1), 13-45.
  2. Bisadi, V. and Head, M. (2010), "Orthogonal effects in nonlinear analysis of bridges subjected to multicomponent earthquake excitation", Proceedings of the ASCE 2010 Structures Congress.
  3. Bisadi, V. and Head, M. (2011), "Evaluation of Combination Rules for Orthogonal Seismic Demands in Nonlinear Time History Analysis of Bridges", J. Bridge Eng., 16(6), 711-717.
  4. Bojorquez, E., Reyes-Salazar A., Teran-Gilmore A. and Ruiz, S.E. (2010), "Energy-based damage index for steel structures", Steel Compos. Struct., 10(4), 331-348.
  5. Building Officials & Code Administration International (12003), 2th Edition, National Building Code.
  6. Clough, R.W. and Penzien, J. (1993), Dynamics of Structures, Second Edition, McGraHill.
  7. Gao, L. and Haldar, A. (1995), "Nonlinear seismic response of space structures with PR connections", Int. J. Microcomput. Civ. Eng., 10, 27-37.
  8. Grant, D. (2011), "Response spectral matching of two horizontal ground-motion components", J. Struct. Eng., 137(3), 289-297.
  9. Der Kiureghian, A. (1981), "A response spectrum method for random vibration analysis of MDOF systems", Earthq. Eng. Struct. Dyn., 9(5), 419-435.
  10. Federal Emergency Management Agency (2000), State of the Art Report on Systems Performance of Steel Moment Frames Subjected to Earthquake Ground Shaking, SAC Steel Project, Report FEMA 355C.
  11. Hernandez, J.J. and Lopez, O.A. (2003), "Evaluation of combination rules for peak response calculation in three-component seismic analysis", Earthq. Eng. Struct. Dyn., 32(10), 1585-1602.
  12. International Code Council (2003), International Building Code (IBC), Falls Church, VA.
  13. Lopez, O.A., Chopra, A.K. and Hernandez, J.J. (2000) "Critical response of structures to multi-component earthquake excitation", Earthq. Eng. Struct. Dyn., 29(12), 1759-1778.<1759::AID-EQE984>3.0.CO;2-K
  14. Lopez, O.A. and Torres, R. (1995), "A clarification of orthogonal effects in a three-dimensional seismic analysis", Earthq. Spectra, 11(4), 659-666.
  15. Lopez, O.A., Chopra, A.K and Hernandez, J.J. (2004), "Adapting the CQC3 rule for three seismic components with different spectra", J. Struct. Eng., ASCE, 130(3), 403-410.
  16. Lopez, O.A., Chopra, A.K and Hernandez, J.J. (2001), "Evaluation of combination rules for maximum response calculation in multi-component seismic analysis", Earthq. Eng. Struct. Dyn., 30(9), 1379-1398.
  17. Lopez, O.A, Hernandez J.J., Bonilla, R. and Fernandez A. (2006), "Response spectra for multi-component structural analysis", Earthq. Spectra, 22(1), 85-113.
  18. Mackie, K.R. and Cronin K.J. (2011), "Response sensitivity of highways bridges to randomly oriented multi-component earthquake excitation", J. Earthq. Eng., 15(6), 850-876.
  19. Menun, C. and Der Kiureghian, A. (2000), "Envelopes for seismic response vectors. I: Theory", J. Struct. Eng., ASCE, 126(4), 467-473.
  20. Menun, C. and Der Kiureghian, A. (1998), "A replacement for the 30%, 40% and SRSS rules for multicomponent seismic analysis", Earthq. Spectra, 14(1), 153-156.
  21. Newmark, N.M. (1975) "Seismic design criteria for structures and facilities, Trans-Alaska pipeline system", Proceedings of the U.S. National Conference on Earthquake Engineering.
  22. Newmark, N.M. and Hall, W.J. (1982), Earthquake Spectra and Design, Monograph Series, Earthquake Engineering Institute, Berkeley CA, USA.
  23. Penzien, J. and Watabe, M. (1975), "Characteristics of 3-Dimensional earthquake ground motions", Earthq. Eng. Struct. Dyn., 3(4), 365-373.
  24. Reglamento de construcciones del Distrito Federal (2004), Normas Tecnicas Complementarias de Diseno por Sismo, Gaceta Oficial del Distrito Federal, Mexico City.
  25. Reyes-Salazar, A., Haldar, A. and Romero-Lopez, M.R. (2000), "Force reduction factor for SDOF and MDOF", Joint Specialty Conference on Probabilistic Mechanics and Structural, ASCE, Paper 063.
  26. Reyes-Salazar, A. (1997), "Inelastic seismic response and ductility evaluation of steel frames with fully, partially restrained and composite connections", Ph.D. thesis, Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, Arizona.
  27. Reyes-Salazar, A. and Haldar, A. (1999), "Nonlinear seismic response of steel structures with semi-rigid and composite connections", J. Constr. Steel Res., 51(1), 37-59.
  28. Reyes-Salazar, A. and Haldar, A. (2000), "Dissipation of energy in steel frames with PR connections", Struct. Eng. Mech., 9(3), 241-256.
  29. Reyes-Salazar, A. and Haldar, A. (2001a), "Energy dissipation at PR frames under seismic loading", J. Struct. Eng., ASCE, 127(5), 588-593.
  30. Reyes-Salazar, A. and Haldar, A. (2001b), "Seismic response and energy dissipation in partially restrained and fully restrained steel frames: An analytical study", Steel Compos. Struct., 1(4), 459-480.
  31. Reyes-Salazar, A., Juarez-Duarte, J.A., Lopez-Barraza, A. and Velazquez-Dimas, J.I. (2004), "Combined effect of the horizontal components of earthquakes for moment resisting steel frames", Steel Compos. Struct., 4(3), 89-209.
  32. Reyes-Salazar, A., Lopez-Barraza, A., Lopez-Lopez, L.A. and Haldar, A. (2008), "Multiple-components seismic response analysis-A critical review", J. Earthq. Eng., 12(5), 779-799.
  33. Rigato, A.B. and Medina, R.A. (2007), "Influence of angle of incidence on seismic demands for inelastic single-storey structures subjected to bi-directional ground motions", Eng. Struct., 29(10), 2593-2601.
  34. Rosenblueth, E. (1980), Design of Earthquake Resistance Structures, Pentech Press Ltd.
  35. Rosenblueth, E. and Contreras, H. (1977), "Approximate design for multi-component Earthquakes", J. Eng. Mech. Div., ASCE, 103(5), 895-911.
  36. Salmon, C.G., Johnson, J.E. and Malhas, F.A. (2009), Steel Structures Design and Behavior, Fifth Edition, Pearson, Prentice Hall, New Jersey.
  37. Smeby, W. and DerKiureghian, A. (1985), "Modal combination rules for multi-component earthquake excitation", Earthq. Eng. Struct. Dyn., 13(1), 1-12.
  38. Shi, G. and Atluri, S.N. (1988), "Elasto-plastic large deformation analysis of spaces-frames", Int. J. Numer. Meth. Eng., 26(3), 589-615.
  39. Tsourekas, A. and Athanatopoulou, A. (2013), "Evaluation of existing combination rules for the effects caused by three spatial components of an Earthquakes", KSCE J. Civ. Eng., 17(7),1728-1739.
  40. Uniform Building Code (1994), Structural Engineering Design Provisions Vol. 2, International Conference of Building Officials.
  41. Wilson, E.L., Suharwardy, I. and Habibullah, A. (1995), "A Clarification of the orthogonal effects in a threedimensional seismic analysis", Earthq. Spectra, 11(4), 659-666.
  42. Wilson, E.L., Der Kiureghian, A. and Bayo, E.P. (1981), "A replacement for the SRSS method in seismic analysis", Earthq. Eng. Struct. Dyn., 9(2), 187-194.