Modal rigidity center: it's use for assessing elastic torsion in asymmetric buildings

  • Georgoussis, George K. (Department of Civil and Construction Engineering, School of Pedagogical and Technological Education (ASPETE))
  • Received : 2009.12.11
  • Accepted : 2010.03.24
  • Published : 2010.06.25


The vertical axis through the modal center of rigidity (m-CR) is used for interpreting the code torsional provisions in the design of eccentric multi-story building structures. The concept of m-CR has been demonstrated by the author in an earlier paper and the particular feature of this point is that when the vertical line of the centers of mass at the floor levels is passing through m-CR, minimum base torsion is developed. For this reason the aforesaid axis is used as reference axis for implementing the code provisions required by the equivalent static analysis. The study examines uniform mixed-bent-type multistory buildings with simple eccentricity, ranging from torsionally stiff to torsionally flexible systems. Using the results of a dynamic response spectrum analysis as a basis for comparisons, it is shown that the results of the code static design are on the safe side in torsionally stiff buildings, but unable to predict the required strength of bents on the stiff side of systems with a predominantly torsional response. Suggestions are made for improving the code provisions in such cases.


  1. Anastassiadis, K., Athanatopoulou, A. and Makarios, T. (1998), "Equivalent static eccentricities in the simplified methods of seismic analysis of buildings", Earthq. Spectra, 14(1), 1-34.
  2. Cheung, V.W.T. and Tso, W.K. (1986), "Eccentricity in irregular multistory buildings", Can. J. Civil. Eng., 13, 46-52.
  3. Coull, A. and Puri, R.D. (1968), "Analysis of pierced shear walls", J. Struct. Eng. - ASCE, 94(1),71-82.
  4. Dempsey, K.M. and Tso, W.K. (1982), "An alternative path to seismic torsional provisions", Soil Dyn. Earthq. Eng., 1, 3-10.
  5. Georgoussis, G.K. (2006), "A simple model for assessing periods of vibration and modal response quantities in symmetrical buildings", Struct. Des. Tall Spec., 15, 139-151.
  6. Georgoussis, G.K. (2008), "Optimum design of multi-story uniform structures with simple eccentricity", Struct. Des. Tall Spec., 17(3), 719-738.
  7. Georgoussis, G.K. (2009), "An alternative approach for assessing eccentricities in asymmetric multistory buildings. 1 elastic systems", Struct. Des. Tall Spec., 18(2), 181-202.
  8. Goel, R.K. and Chopra, A.K. (1993), "Seismic code analysis without locating centers of rigitidy", J. Struct. Eng. - ASCE, 119(10), 3039-3055.
  9. Harasimowicz, A.P. and Goel, R.K. (1998), "Seismic code analysis of multi-storey asymmetric buildings", Earthq. Eng. Struct. D., 27, 173-185.<173::AID-EQE724>3.0.CO;2-W
  10. Heidebrecht, A.C. (1975), "Dynamic analysis of asymmetric wall- frame buildings", ASCE, National Structural Engineering Convention, New Orleans, LA.
  11. Heidebrecht, A.C. and Stafford Smith, B. (1973), "Approximate analysis of tall wall-frame structures", J. Struct. Div. - ASCE, 2, 169-183.
  12. Humar, J.L. (1984), "Design for seismic torsional forces", Can. J. Civil Eng., 12, 150-163.
  13. Jacobsen, L.S. and Ayre, R.S. (1958), Engineering vibrations, McGraw-Hill Book Company.
  14. Makarios, T. (2005), "Optimum torsion axis to multistory buildings by using the continuous model of the structure", Struct. Des. Tall Spec., 14(1), 69-90.
  15. Makarios, T. (2008), "Practical calculation of the torsional stiffness radius of multistorey tall buildings", Struct. Des. Tall Spec., 17(1), 39-65.
  16. Makarios, T. and Anastassiadis, K. (1998a), "Real and fictitious elastic axis of multi-storey buildings: theory", Struct. Des. Tall Build., 7(1), 33-45.<33::AID-TAL95>3.0.CO;2-D
  17. Makarios, T. and Anastassiadis, K. (1998b), "Real and fictitious elastic axis of multi-storey buildings: applications", Struct. Des. Tall Build., 7(1), 57-71.<57::AID-TAL96>3.0.CO;2-0
  18. Makarios, T., Athanatopoulou, A. and Xenidis, H. (2006), "Numerical verification of properties of the fictitious elastic axis in asymmetric multistorey buildings", Struct. Des. Tall Spec., 15(3), 249-276.
  19. Marino, E.M. and Rossi, P.P. (2004), "Exact evaluation of the location of the optimum torsion axis", Struct. Des. Tall Spec., 13, 277-290
  20. Newmark, N.M. and Rosenblueth, E. (1972), Fundamentals of earthquake engineering, Prentice-Hall, Inc.
  21. Pauley, T. and Priestley, M.J.N. (1992), Seismic design of reinforced and masonry buildings, Wiley Interscience.
  22. Pool, R.A. (1977), "Analysis for torsion employing provisions of NZRS 4203:1974", Bull. N. Zealand Soc. Earthq. Eng., 10, 219-225.
  23. Riddel, R. and Vasquez, J. (1984), "Existence of centers of resistance and torsional uncoupling of earthquake response of buildings", Proc. 8th World Conf. on Earthquake Engineering, Vol 4, 187-194.
  24. Smith, B.S. and Vezina, S. (1985), "Evaluation of centers of resistance in multistorey building structures", Proc. Instn. Civ. Engrs. Part 2, 79(4), 623-635.
  25. Tso, W.K. and Cheung, V.W.T. (1986), "Decoupling of equations of equilibrium in lateral load analysis of multistory buildings", Comput. Struct., 23(5), 679-684.
  26. Tena-Colunga, A. and Perez-Osornio, MA. (2005), "Assessment of shear deformations on the seismic response of asymmetric shear wall buildings", J. Struct. Eng. - ASCE, 131(11), 1774-1779.
  27. Tso, W.K. (1990), "Static eccentricity concept for torsional moment estimations", J. Struct. Eng. - ASCE, 116(5), 1199-1212.
  28. Tso, W.K. and Dempsey, K.M. (1980), "Seismic torsional provisions for dynamic eccentricity", Earthq. Eng. Struct. D., 8, 275-289.
  29. Zhu, T.J. and Tso, W.K. (1992), "Design of torsionally unbalanced structural systems based on code provisions II: strength distribution", Earthq. Eng. Struct. D., 21, 629-644.

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