Skip to main content

Advertisement

SpringerLink
Log in

On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

The problem of free vibrations of the Timoshenko beam model is here addressed. A careful analysis of the governing equations allows identifying that the vibration spectrum consists of two parts, separated by a transition frequency, which, depending on the applied boundary conditions, might be itself part of the spectrum. For both parts of the spectrum, the values of natural frequencies are computed and the expressions of eigenmodes are provided. This allows to acknowledge that the nature of vibration modes changes when moving across the transition frequency. Among all possible combination of end constraints which can be applied to single-span beams, the case of a simply supported beam is considered. These theoretical results can be used as benchmarks for assessing the correctness of the numerical values provided by several numerical techniques, e.g. traditional Lagrangian-based finite element models or the newly developed isogeometric approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

\({\mathbf{A}}\) :

Coefficient matrix for the homogenous system

\({\mathbf{X}}\) :

Unknown column matrix for the homogenous system

\({\mathbf{0}}\) :

Right-hand side column matrix for homogeneous system

\({A}\) :

Cross-sectional area

\({A_{1}}\), \({A_{2}}\), \({A_{3}}\), \({A_{4}}\) :

Integration constants for \({V}\), first part of the spectrum

\({B}\) :

Cross-sectional depth (and width)

\({B_{1}}\), \({B_{2}}\), \({B_{3}}\), \({B_{4}}\) :

Integration constants for \({\varPhi}\), first part of the spectrum

\({C_{1}}\), \({C_{2}}\), \({C_{3}}\), \({C_{4}}\) :

Integration constants for \({V}\), transition frequency

\({D}\) :

Constant factor [see Eq. (B.3)]

\({D^{\star}}\) :

Differential operator d/dx

\({D_{1}}\), \({D_{2}}\), \({D_{3}}\), \({D_{4}}\) :

Integration constants for \({\varPhi}\), transition frequency

\({E}\) :

Young’s modulus

\({E_{1}}\), \({E_{2}}\), \({E_{3}}\), \({E_{4}}\) :

Integration constants for \({V}\), second part of the spectrum

\({E_{1n}}\), \({E_{2n}}\), \({E_{3n}}\), \({E_{4n}}\) :

Integration constants for the \({n}\)th eigenmode

\({F_{1}}\), \({F_{2}}\), \({F_{3}}\), \({F_{4}}\) :

Integration constants for \({\varPhi}\), second part of the spectrum

\({G}\) :

Shear modulus

\({H}\), \({K}\) :

Amplitude of eigenmodes for double eigenvalue

\({I}\) :

Cross-sectional mass moment of inertia

\({L}\) :

Beam length

\({\tilde{L}}\) :

Special value of beam length

\({M}\) :

Bending moment

\({T}\) :

Shear force

\({T^{\star}}\) :

Differential operator d/dt

\({V}\) :

Vibration mode for transversal displacement

\({a}\) :

Shear stiffness

\({b}\) :

Transversal inertia

\({\hat{b}}\), \({\hat{c}}\) :

Coefficients of biquadratic wave-numbers equation

\({b^{\star}}\), \({c^{\star}}\) :

Coefficients of biquadratic frequency equation

\({c}\) :

Bending stiffness

\({d}\) :

Rotary inertia

\({f_{\lambda}}\) :

Space frequency associated with wave-number \({\lambda}\)

\({f_{\lambda_{n}}}\) :

Space frequency associated with the \({n}\)th vibration mode

\({K}\), \({k_{1}}\), \({k_{2}}\) :

Integer values corresponding to wave-numbers of vibration modes

\({t}\) :

Time variable

\({v}\) :

Transversal displacement

\({x}\) :

Space variable (beam abscissa)

\({\hat{\varDelta}}\) :

Discriminant of wave-number equation

\({\varDelta^{\star}}\) :

Discriminant of frequency equation

\({\varPhi}\) :

Vibration mode for section rotation

\({\hat{\alpha}_{1}}\) :

Coefficient of eigenmode for generalized wave-number

\({\alpha_{1}}\), \({\alpha_{2}}\) :

Eigenmode coefficients for first/second wave-number

\({\tilde{\alpha}_{2}}\) :

Eigenmode coefficient for second wave-number at transition frequency

\({\kappa}\) :

Shear correction factor

\({\hat{\lambda}_{1}}\) :

Generalized wave-number (first part of the spectrum)

\({\lambda_{1}}\) :

First wave-number (second part of the spectrum)

\({\lambda_{2}}\) :

Second wave-number (first and second part of the spectrum)

\({\tilde{\lambda}_{2}}\) :

Second wave-number at transition frequency

\({\lambda_{1}^{\star2}}\) :

First root (squared) of wave-numbers equations

\({\lambda_{2}^{\star2}}\) :

Second root (squared) of wave-numbers equations

\({\nu}\) :

Poisson’s ratio

\({\xi}\) :

Dimensionless space variable (dimensionless beam abscissa)

\({\rho}\) :

Beam density (mass per unit volume)

\({\phi}\) :

Section rotation

\({\omega}\) :

Angular frequency

\({\tilde{\omega}}\) :

Angular frequency at the transition value (cutoff frequency)

\({\omega^{\star}}\) :

Limiting value (upper/lower bound) for angular frequency

\({\omega_{n}}\) :

Angular frequency (theoretical value) for \({n}\)th vibration mode

References

  1. Timoshenko S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. (Ser. 5) 41, 744–746 (1921)

    Article  Google Scholar 

  2. Timoshenko S.P.: On the transverse vibrations of bars of uniform cross-section. Philos. Mag. (Ser. 5) 43, 125–131 (1922)

    Article  Google Scholar 

  3. Stefan J.: Über die Transversalschwingungen eines elastischen Stabes. K.K. Hof- und Staatsdruckerei, Wien (1858)

    Google Scholar 

  4. Rayleigh J.W.S.B.: The Theory of Sound, vol. 1. Macmillan and Co., London (1877)

    Google Scholar 

  5. Han S.M., Benaroya H., Wei T.: Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib. 225, 935–988 (1999)

    Article  MATH  Google Scholar 

  6. Labuschagne A., van Rensburg N.F.J., van der Merwe A.J.: Comparison of linear beam theories. Math. Comput. Model. 49, 20–30 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Karnopp B.H.: Duality relations in the analysis of beam oscillations. Z. Angew. Math. Phys. (ZAMP) 18, 575–580 (1967)

    Article  MATH  Google Scholar 

  8. Tabarrok B., Karnopp B.H.: Analysis of the oscillations of the Timoshenko beam. Z. Angew. Math. Phys. (ZAMP) 18, 580–587 (1967)

    Article  MATH  Google Scholar 

  9. Eisley J.G.: Nonlinear vibration of beams and rectangular plates. Z. Angew. Math. Phys. (ZAMP) 15, 167–175 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  10. Traill-Nash R.W., Collar A.R.: The effects of shear flexibility and rotatory inertia on the bending vibrations of beams. Q. J. Mech. Appl. Math. 6, 186–222 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  11. Downs B.: Transverse vibration of a uniform, simply supported Timoshenko beam without transverse deflection. J. Appl. Mech. 43, 671–674 (1976)

    Article  MATH  Google Scholar 

  12. Abbas B.A.H., Thomas J.: The second frequency spectrum of Timoshenko beams. J. Sound Vib. 51, 123–137 (1977)

    Article  Google Scholar 

  13. Bhashyam G.R., Prathap G.: The second frequency spectrum of Timoshenko beams. J. Sound Vib. 76, 407–420 (1981)

    Article  Google Scholar 

  14. Levinson M., Cooke D.W.: On the two frequency spectra of Timoshenko beams. J. Sound Vib. 84, 319–326 (1982)

    Article  MATH  Google Scholar 

  15. Stephen N.G.: The second frequency spectrum of Timoshenko beams. J. Sound Vib. 80, 578–582 (1982)

    Article  Google Scholar 

  16. Prathap G.: The two frequency spectra of timoshenko beams—a re-assessment. J. Sound Vib. 90, 443–445 (1983)

    Article  Google Scholar 

  17. Levinson M.: Author’s reply. J. Sound Vib. 90, 445–446 (1983)

    Article  MATH  Google Scholar 

  18. Nesterenko V.V.: A theory for transverse vibrations of the Timoshenko beam. J. Appl. Math. Mech. 57, 669–677 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nesterenko V.V., Chervyakov A.M.: Parabolic approximation to the theory of transverse vibrations of rods and beams. J. Appl. Mech. Tech. Phys. 35, 306–309 (1994)

    Article  MATH  Google Scholar 

  20. Olsson P., Kristensson G.: Wave splitting of the Timoshenko beam equation in the time domain. Z. Angew. Math. Phys. (ZAMP) 45, 866–881 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ekwaro-Osire S., Maithripala D.H.S., Berg J.M.: A series expansion approach to interpreting the spectra of the Timoshenko beam. J. Sound Vib. 240, 667–678 (2001)

    Article  MATH  Google Scholar 

  22. Stephen N.G.: The second spectrum of Timoshenko beam theory—further assessment. J. Sound Vib. 292, 372–389 (2006)

    Article  MATH  Google Scholar 

  23. Stephen N.G., Puchegger S.: On the valid frequency range of Timoshenko beam theory. J. Sound Vib. 297, 1082–1087 (2006)

    Article  Google Scholar 

  24. Bhaskar A.: Elastic waves in Timoshenko beams: the ‘lost and found’ of an eigenmode. Proc. R. Soc. A Math. Phys. Eng. Sci. 465, 239–255 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Senjanović I., Vladimir N.: Physical insight into Timoshenko beam theory and its modification with extension. Struct. Eng. Mech. 48(4), 519–545 (2013)

    Article  Google Scholar 

  26. van Rensburg N.F.J., van der Merwe A.J.: Natural frequencies and modes of a Timoshenko beam. Wave Motion 44, 58–69 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pilkey W.D.: Formulas for Stress, Strain, and Structural Matrices, 2nd edn. Wiley, Hoboken (2005)

    MATH  Google Scholar 

  28. Reddy J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edn. CRC Press, Boca Raton (2003)

    Google Scholar 

  29. Karnowsky I.A., Lebed O.I.: Formulas for Structural Dynamics—Tables, Graphs and Solutions. McGraw-Hill, New York (2001)

    Google Scholar 

  30. Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. J. Appl. Math. Mech./Z. Angew. Math. Mech. (ZAMM), pp. 1–25 (2016, accepted)

  31. Graff K.F.: Wave Motion in Elastic Solids. Oxford University Press, London (1975)

    MATH  Google Scholar 

  32. Cazzani, A., Stochino, F., Turco, E.: On the whole spectrum of Timoshenko beams. Part II: further applications. Z. Angew. Math. Phys. (ZAMP), pp. 1–21 (2016). doi:10.1007/s00033-015-0596-9

  33. Goens E.: Über die Bestimmung des Elastizitätsmoduls von Stäben mit Hilfe von Biegungsschwingungen. Ann. Phys. (Ser. 5) 403, 649–678 (1931)

    Article  MATH  Google Scholar 

  34. Föppl A.: Vorlesungen über technische Mechanik—Dritter Band: Festigkeitslehre. B.G. Teubner, Leipzig (1897)

    MATH  Google Scholar 

  35. Kaneko T.: On Timoshenko’s correction for shear in vibrating beams. J. Phys. D Appl. Phys. 8, 1927–1936 (1975)

    Article  Google Scholar 

  36. Jensen J.J.: On the shear coefficient in Timoshenko’s beam theory. J. Sound Vib. 87, 621–635 (1983)

    Article  MATH  Google Scholar 

  37. Renton J.D.: A note on the form of the shear coefficient. Int. J. Solids Struct. 34, 1681–1685 (1997)

    Article  MATH  Google Scholar 

  38. Méndez-Sánchez R.A., Morales A., Flores J.: Experimental check on the accuracy of Timoshenko’s beam theory. J. Sound Vib. 279, 508–512 (2005)

    Article  Google Scholar 

  39. Dong S.B., Alpdogan C., Taciroglu E.: Much ado about shear correction factors in Timoshenko beam theory. Int. J. Solids Struct. 47, 1651–1665 (2010)

    Article  MATH  Google Scholar 

  40. Cowper G.R.: The shear coefficient in Timoshenko’s beam theory. J. Appl. Mech. 33, 335–340 (1966)

    Article  MATH  Google Scholar 

  41. Higuchi S., Saito H., Hashimoto C.: A study of the approximate theory of an elastic thick beam. Can. J. Phys. 35(6), 757–765 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  42. Hutchinson J.R.: Shear coefficients for Timoshenko beam theory. J. Appl. Mech. 68, 87–92 (2001)

    Article  MATH  Google Scholar 

  43. Stephen N.G.: On the variation of Timoshenko’s shear coefficient with frequency. J. Appl. Mech. 45, 695–697 (1978)

    Article  Google Scholar 

  44. Stephen N.G.: On “A check on the accuracy of Timoshenko’s beam theory”. J. Sound Vib. 257, 809–812 (2002)

    Article  Google Scholar 

  45. Chan K.T., Lai K.F., Stephen N.G., Young K.: A new method to determine the shear coefficient of Timoshenko beam theory. J. Sound Vib. 330, 3488–3497 (2011)

    Article  Google Scholar 

  46. Rosinger H.E., Ritchie I.G.: On Timoshenko’s correction for shear in vibrating isotropic beams. J. Phys. D Appl. Phys. 10(08), 1461–1466 (1977)

    Article  Google Scholar 

  47. Doyle P.F., Pavlovic M.N.: Vibration of beams on partial elastic foundations. Earthq. Eng. Struct. Dyn. 10, 663–674 (1982)

    Article  Google Scholar 

  48. Eisenberger M., Yankelevsky D.Z., Adin M.A.: Vibrations of beams fully or partially supported on elastic foundations. Earthq. Eng. Struct. Dyn. 13, 651–660 (1985)

    Article  Google Scholar 

  49. Cazzani A.: On the dynamics of a beam partially supported by an elastic foundation: An exact solution-set. Int. J. Struct. Stab. Dyn. 13:1350045 (2013). doi:10.1142/S0219455413500454

  50. Levinson M.: Free vibrations of a simply supported, rectangular plate: an exact elasticity solution. J. Sound Vib. 98, 289–298 (1985)

    Article  MATH  Google Scholar 

  51. Stephen N.G.: Mindlin plate theory: best shear coefficient and higher spectra validity. J. Sound Vib. 202, 539–553 (1997)

    Article  Google Scholar 

  52. Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane-curved beams. Math. Mech. Solids (2014). doi:10.1177/1081286514531265

  53. Cazzani, A., Malagù, M., Turco E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Contin. Mech. Thermodyn. 28(1–2), 139–156 (2016)

  54. Cazzani, A., Malagù, M., Turco, E., Stochino, F.: Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math. Mech. Solids 21(2), 182–209 (2016)

  55. Chiozzi, A., Malagù, M., Tralli, A., Cazzani A.: ArchNURBS: NURBS-based tool for the structural safety assessment of masonry arches in MATLAB. J. Comput. Civ. Eng. (2015). doi:10.1061/(ASCE)CP.1943-5487.0000481

  56. Cuomo M., Contrafatto L., Greco L.: A variational model based on isogeometric interpolation for the analysis of cracked bodies. Int. J. Eng. Sci. 80, 173–188 (2014)

    Article  MathSciNet  Google Scholar 

  57. Greco L., Cuomo M.: An implicit G 1 multi patch B-spline interpolation for Kirchhoff-Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Greco L., Cuomo M.: B-spline interpolation of Kirchhoff–Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)

    Article  MathSciNet  Google Scholar 

  59. Cazzani A., Garusi E., Tralli A., Atluri S.N.: A four-node hybrid assumed-strain finite element for laminated composite plates. Comput. Mater. Contin. 2, 23–38 (2005)

    MATH  Google Scholar 

  60. Bilotta A., Formica G., Turco E.: Performance of a high-continuity finite element in three-dimensional elasticity. Int. J. Numer. Methods Biomed. Eng. (Commun. Numer. Methods Eng.) 26, 1155–1175 (2010)

    Article  MATH  Google Scholar 

  61. Turco E., Caracciolo P.: Elasto-plastic analysis of Kirchhoff plates by high simplicity finite elements. Comput. Methods Appl. Mech. Eng. 190, 691–706 (2000)

    Article  MATH  Google Scholar 

  62. Rizzi N., Varano V., Gabriele S.: Initial postbuckling behavior of thin-walled frames under mode interaction. Thin Walled Struct. 68, 124–134 (2013)

    Article  Google Scholar 

  63. Gabriele, S., Rizzi, N., Varano, V.: A 1D higher gradient model derived from Koiter’s shell theory. Math. Mech. Solids (2014). doi:10.1177/1081286514536721

  64. Rizzi N., Varano V.: The effects of warping on the postbuckling behaviour of thin-walled structures. Thin Walled Struct. 49(9), 1091–1097 (2011)

    Article  Google Scholar 

  65. Zulli D., Luongo A.: Bifurcation and stability of a two-tower system under wind-induced parametric, external and self-excitation. J. Sound Vib. 331(2), 365–383 (2012)

    Article  Google Scholar 

  66. Pignataro M., Rizzi N., Ruta G., Varano V.: The effects of warping constraints on the buckling of thin-walled structures. J. Mech. Mater. Struct. 4(10), 1711–1727 (2009)

    Article  Google Scholar 

  67. Ruta G.C., Varano V., Pignataro M., Rizzi N.L.: A beam model for the flexural–torsional buckling of thin-walled members with some applications. Thin Walled Struct. 46(7–9), 816–822 (2008)

    Article  Google Scholar 

  68. Presta F., Hendy C.R., Turco E.: Numerical validation of simplified theories for design rules of transversely stiffened plate girders. Struct. Eng. 86(21), 37–46 (2008)

    Google Scholar 

  69. Piccardo G., Ranzi G., Luongo A.: A complete dynamic approach to the generalized beam theory cross-section analysis including extension and shear modes. Math. Mech. Solids 19(8), 900–924 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  70. 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. Nonlinear Mech. (2015). doi:10.1016/j.ijnonlinmec.2015.07.013

  71. Del Vescovo D., Giorgio I.: Dynamic problems for metamaterials: review of existing models and ideas for further research. Int. J. Eng. Sci. 80, 153–172 (2014)

    Article  MathSciNet  Google Scholar 

  72. dell’Isola F., Madeo A., Placidi L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. J. Appl. Math. Mech./Z. Angew. Math. Mech. (ZAMM) 92(1), 52–71 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  73. Berezovski, A., Giorgio, I., Della Corte, A.: Interfaces in micromorphic materials: wave transmission and reflection with numerical simulations. Math. Mech. Solids (2015). doi:10.1177/1081286515572244

  74. Madeo A., Neff P., Ghiba I.-D., Placidi L., Rosi G.: Band gaps in the relaxed linear micromorphic continuum. J. Appl. Math. Mech./Z. Angew. Math. Mech. (ZAMM) 95(9), 880–887 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  75. Madeo A., Neff P., Ghiba I.D., Placidi L., Rosi G.: Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Contin. Mech. Thermodyn. 27(4–5), 551–570 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  76. Placidi, L., Rosi, G., Giorgio, I., Madeo, A.: Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Math. Mech. Solids (2013). doi:10.1177/1081286512474016

  77. dell’Isola, F., Lekszycki, T., Pawlikowski, M., Grygoruk, R., Greco, L.: Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z. Angew. Math. Phys. (ZAMP) (2015). doi:10.1007/s00033-015-0556-4

  78. Andreaus U., Giorgio I., Madeo A.: Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids. Z. Angew. Math. Phys. (ZAMP) 66(1), 209–237 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  79. Grillo A., Federico S., Wittum G.: Growth, mass transfer, and remodeling in fiber-reinforced, multi-constituent materials. Int. J. Nonlinear Mech. 47(2), 388–401 (2012)

    Article  Google Scholar 

  80. Federico S., Grillo A., Imatani S., Giaquinta G., Herzog W.: An energetic approach to the analysis of anisotropic hyperelastic materials. Int. J. Eng. Sci. 46, 164–181 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  81. Alibert J.-J., Della Corte A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Z. Angew. Math. Phys. (ZAMP) 66, 2855–2870 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  82. 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)

    Article  MathSciNet  MATH  Google Scholar 

  83. Yang Y., Ching W., Misra A.: Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film. J. Nanomech. Micromech. 1(2), 60–71 (2011)

    Article  Google Scholar 

  84. dell’Isola, F., Della Corte, A., Giorgio, I., Scerrato, D.: Pantographic 2D sheets: discussion of some numerical investigations and potential applications. Int. J. Nonlinear Mech. (2015). doi:10.1016/j.ijnonlinmec.2015.10.010

  85. Rahali Y., Giorgio I., Ganghoffer J.F., dell’Isola F.: Homogenization à à la Piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)

    Article  MathSciNet  Google Scholar 

  86. Rahali Y., Goda I., Ganghoffer J.-F.: Numerical identification of classical and nonclassical moduli of 3D woven textiles and analysis of scale effects. Compos. Struct. 135, 122–139 (2016)

    Article  Google Scholar 

  87. Misra, A., Poorsolhjouy, P.: Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics. Math. Mech. Solids (2015). doi:10.1177/1081286515576821

  88. Carcaterra A., dell’Isola F., Esposito R., Pulvirenti M.: Macroscopic description of microscopically strongly inhomogenous systems: A mathematical basis for the synthesis of higher gradients metamaterials. Arch. Ration. Mech. Anal. 218(3), 1239–1262 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  89. Neff P., Ghiba I.-D., Madeo A., Placidi L., Rosi G.: A unifying perspective: the relaxed linear micromorphic continuum. Contin. Mech. Thermodyn. 26(5), 639–681 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  90. Eremeyev V.A., Pietraszkiewicz W.: Local symmetry group in the general theory of elastic shells. J. Elast. 85, 125–152 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  91. Eremeyev V.A., Pietraszkiewicz W.: Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct. 49, 1993–2005 (2012)

    Article  Google Scholar 

  92. Challamel N., Kocsis A., Wang C.M.: Discrete and non-local elastica. Int. J. Nonlinear Mech. 77, 128–140 (2015)

    Article  Google Scholar 

  93. Roveri N., Carcaterra A.: Damage detection in structures under travelling loads by the Hilbert–Huang transform. Mech. Syst. Signal Process. 28, 128–144 (2012)

    Article  Google Scholar 

  94. Bilotta A., Turco E.: A numerical study on the solution of the Cauchy problem in elasticity. Int. J. Solids Struct. 46, 4451–4477 (2009)

    Article  MATH  Google Scholar 

  95. Bilotta, A., Turco E.: Numerical sensitivity analysis of corrosion detection. Math. Mech. Solids (2014). doi:10.1177/1081286514560093

  96. Alessandrini G., Bilotta A., Morassi A., Turco E.: Computing volume bounds of inclusions by EIT measurements. J. Sci. Comput. 33(3), 293–312 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  97. Buffa, F., Cazzani, A., Causin, A., Poppi, S., Sanna, G.M., Solci, M., Stochino, F., Turco E.: The Sardinia radio telescope: a comparison between close range photogrammetry and FE models. Math. Mech. Solids (2015). doi:10.1177/1081286515616227

  98. Stochino, F., Cazzani, A., Poppi, S., Turco, E.: Sardinia radio Telescope finite element model updating by means of photogrammetric measurements. Math. Mech. Solids (2015). doi:10.1177/1081286515616046

Download references

Author information

Authors and Affiliations

  1. DICAAR—Department of Civil and Environmental Engineering and Architecture, University of Cagliari, 2, via Marengo, 09123, Cagliari, Italy

    Antonio Cazzani

  2. DADU—Department of Architecture, Design and Urban Planning, University of Sassari, Asilo Sella, 35, via Garibaldi, 07041, Alghero, SS, Italy

    Flavio Stochino & Emilio Turco

Authors
  1. Antonio Cazzani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Flavio Stochino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Emilio Turco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Cazzani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cazzani, A., Stochino, F. & Turco, E. On the whole spectrum of Timoshenko beams. Part I: a theoretical revisitation. Z. Angew. Math. Phys. 67, 24 (2016). https://doi.org/10.1007/s00033-015-0592-0

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-015-0592-0

Mathematics Subject Classification

Keywords

Search

Navigation