Continuum Mechanics and Thermodynamics

, Volume 30, Issue 5, pp 977–993 | Cite as

Flexural torsional buckling of uniformly compressed beam-like structures

  • M. FerrettiEmail author
Original Article


A Timoshenko beam model embedded in a 3D space is introduced for buckling analysis of multi-store buildings, made by rigid floors connected by elastic columns. The beam model is developed via a direct approach, and the constitutive law, accounting for prestress forces, is deduced via a suitable homogenization procedure. The bifurcation analysis for the case of uniformly compressed buildings is then addressed, and numerical results concerning the Timoshenko model are compared with 3D finite element analyses. Finally, some conclusions and perspectives are drawn.


Beam-like structures Equivalent beam model Timoshenko beam Homogenization procedure Buckling analysis 


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The author is grateful to Prof. Angelo Luongo for his many insightful suggestions and for his kind support throughout the progress of the work.


  1. 1.
    Friedman, Z., Kosmatka, J.B.: An improved two-node timoshenko beam finite element. Comput. Struct. 47(3), 473–481 (1993)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane-curved beams. Math. Mech. Solids 21(5), 562–577 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bui, Q.-B., Hans, S., Boutin, C.: Dynamic behaviour of an asymmetric building: experimental and numerical studies. Case Stud. Nondestruct. Test. Eval. 2, 38–48 (2014)CrossRefGoogle Scholar
  4. 4.
    Kim, H.-S., Lee, D.-G., Kim, C.K.: Efficient three-dimensional seismic analysis of a high-rise building structure with shear walls. Eng. Struct. 27(6), 963–976 (2005)CrossRefGoogle Scholar
  5. 5.
    Liu, H., Yang, Z., Gaulke, M.S.: Structural identification and finite element modeling of a 14-story office building using recorded data. Eng. Struct. 27(3), 463–473 (2005)CrossRefGoogle Scholar
  6. 6.
    Chajes, M.J., Finch, W.W., Kirby, J.T.: Dynamic analysis of a ten-story reinforced concrete building using a continuum model. Comput. Struct. 58(3), 487–498 (1996)CrossRefGoogle Scholar
  7. 7.
    Chajes, M.J., Zhang, L., Kirby, J.T.: Dynamic analysis of tall building using reduced-order continuum model. J. Struct. Eng. 122(11), 1284–1291 (1996)CrossRefGoogle Scholar
  8. 8.
    Miranda, E.: Approximate seismic lateral deformation demands in multistory buildings. J. Struct. Eng. 125(4), 417–425 (1999)CrossRefGoogle Scholar
  9. 9.
    Miranda, E., Taghavi, S.: Approximate floor acceleration demands in multistory buildings. I: formulation. J. Struct. Eng. 131(2), 203–211 (2005)CrossRefGoogle Scholar
  10. 10.
    Taghavi, S., Miranda, E.: Approximate floor acceleration demands in multistory buildings. II: applications. J. Struct. Eng. 131(2), 212–220 (2005)CrossRefGoogle Scholar
  11. 11.
    Boutin, C., Hans, S.: Homogenisation of periodic discrete medium: application to dynamics of framed structures. Comput. Geotech. 30(4), 303–320 (2003)CrossRefGoogle Scholar
  12. 12.
    Hans, S., Boutin, C.: Dynamics of discrete framed structures: a unified homogenized description. J. Mech. Mater. Struct. 3(9), 1709–1739 (2008)CrossRefGoogle Scholar
  13. 13.
    Chesnais, C., Boutin, C. l., Hans, S.: Structural dynamics and generalized continua. Mech. Gen. Contin., pp. 57–76 (2011)Google Scholar
  14. 14.
    Cluni, F., Gioffrè, M., Gusella, V.: Dynamic response of tall buildings to wind loads by reduced order equivalent shear-beam models. J. Wind Eng. Ind. Aerodyn. 123, 339–348 (2013)CrossRefGoogle Scholar
  15. 15.
    Zalka, K.A.: A simplified method for calculation of the natural frequencies of wall-frame buildings. Eng. Struct. 23(12), 1544–1555 (2001)CrossRefGoogle Scholar
  16. 16.
    Zalka, K.A.: A simple method for the deflection analysis of tall wall-frame building structures under horizontal load. Struct. Des. Tall Spec. Build. 18(3), 291–311 (2009)CrossRefGoogle Scholar
  17. 17.
    Zalka, K.A.: Maximum deflection of asymmetric wall-frame buildings under horizontal load. Period. Polytech. Civ. Eng. 58(4), 387 (2014)CrossRefGoogle Scholar
  18. 18.
    Zalka, K.A.: Buckling analysis of buildings braced by frameworks, shear walls and cores. Struct. Des. Tall Spec. Build. 11(3), 197–219 (2002)CrossRefGoogle Scholar
  19. 19.
    Zalka, K.A.: Mode coupling in the torsional-flexural buckling of regular multistorey buildings. Struct. Des. Tall Spec. Build. 3(4), 227–245 (1994)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zalka, K.A.: Global Structural Analysis of Buildings. CRC Press, Boca Raton (2002)Google Scholar
  21. 21.
    Potzta, G., Kollár, L.P.: Analysis of building structures by replacement sandwich beams. Int. J. Solids Struct. 40(3), 535–553 (2003)CrossRefzbMATHGoogle Scholar
  22. 22.
    Piccardo, G., Tubino, F., Luongo, A.: A shear-shear torsional beam model for nonlinear aeroelastic analysis of tower buildings. Z. Angew. Math. Phys. 66(4), 1895–1913 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Piccardo, G., Tubino, F., Luongo, A.: Equivalent nonlinear beam model for the 3-d analysis of shear-type buildings: Application to aeroelastic instability. Int. J. Non-Linear Mech. 80, 52–65 (2016)ADSCrossRefGoogle Scholar
  24. 24.
    Piccardo, G., Tubino, F., Luongo, A.: Equivalent timoshenko linear beam model for the static and dynamic analysis of tower buildings. Appl. Math. Model., Submitted (2017)Google Scholar
  25. 25.
    Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9(5), 241–257 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of gabrio piola. Math. Mech. Solids 20(8), 887–928 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Berrehili, Y., Marigo, J.-J.: The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers. Math. Mech. Complex Syst. 2(2), 181–207 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1d nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)CrossRefGoogle Scholar
  30. 30.
    Giorgio, I., Rizzi, N., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A 473, 20170636 (2017)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Alibert, J.-J., Della Corte, A., Giorgio, I., Battista, A.: Extensional elastica in large deformation as \(\varGamma \)-limit of a discrete 1d mechanical system. Z. Angew. Math. Phys. 68(2), 42 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Alibert, J.-J., Della Corte, A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Z. Angew. Math. Phys. 66(5), 2855–2870 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Chatzigeorgiou, G., Javili, A., Steinmann, P.: Unified magnetomechanical homogenization framework with application to magnetorheological elastomers. Math. Mech. Solids 19(2), 193–211 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Carinci, G., De Masi, A., Giardinà, C., Presutti, E.: Super-hydrodynamic limit in interacting particle systems. J. Stat. Phys. 155(5), 867–887 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Carinci, G., De Masi, A., Giardinà, C., Presutti, E.: Hydrodynamic limit in a particle system with topological interactions. Arab. J. Math. 3(4), 381–417 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Luongo, A., Zulli, D.: Mathematical Models of Beams and Cables. Wiley, Hoboken (2013)CrossRefGoogle Scholar
  37. 37.
    Antman, S.S.: The theory of rods. In: Linear Theories of Elasticity and Thermoelasticity, pp 641–703. Springer (1973)Google Scholar
  38. 38.
    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, Berlin (2005)zbMATHGoogle Scholar
  39. 39.
    Capriz, G.: A contribution to the theory of rods. Riv. Mat. Univ. Parma 7(4), 489–506 (1981)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Lacarbonara, W.: Nonlinear Structural Mechanics: Theory, Dynamical Phenomena and Modeling. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  41. 41.
    Altenbach, H., Bîrsan, M., Eremeyev V.A.: Cosserat-type rods. In: Generalized Continua from the Theory to Engineering Applications, pp. 179–248. Springer (2013)Google Scholar
  42. 42.
    Pignataro, M., Rizzi, N., Luongo, A.: Bifurcation, Stability and Postcritical Behaviour of Elastic Structures. Elsevier, Amsterdam (1990)Google Scholar
  43. 43.
    Bažant, Z.P., Cedolin, L.: Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories. World Scientific, Singapore (2010)zbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.International Research Center on Mathematics and Mechanics of Complex SystemsUniversity of L’AquilaL’AquilaItaly
  2. 2.Department of Civil, Construction-Architectural and Environmental EngineeringUniversity of L’AquilaL’AquilaItaly

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