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Acta Mechanica

, Volume 32, Issue 4, pp 249–259 | Cite as

Free vibration of rectangular beams of arbitrary depth

  • K. T. Sundara Raja Iyengar
  • P. V. Raman
Contributed Papers

Summary

The state space approach is extended to the two dimensional elastodynamic problems. The formulation is in a form particularly amenable to consistent reduction to obtain approximate theories of any desired order. Free vibration of rectangular beams of arbitrary depth is investigated using this approach. The method does not involve the concept of the shear coefficientk. It takes into account the vertical normal stress and the transverse shear stress. The frequency values are calculated using the Timoshenko beam theory and the present analysis for different values of Poisson's ratio and they are in good agreement. Four cases of beams with different end conditions are considered.

Keywords

Free Vibration Transverse Shear Beam Theory Timoshenko Beam Space Approach 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notation

2h

depth of beam

k

Timoshenko shear constant

L

length of the beam

n

mode number

u, v

displacement inx, y directions

A

area of cross section

An

coefficient in series representation

E

modulus of elasticity

G

modulus of rigidity

I

moment of inertia aboutz-axis

ϱ

mass density

μ

Poisson's ratio

r

\(\sqrt {I/A} /L\)

θ

r×n

δσxσy

direct stresses

τxy

shear stress

η

eigenvalue of square matrix

ω

frequency of harmonic vibration

λ

eigenvalue=\(\sqrt {\frac{\varrho }{G}} \omega L\)

Ω

frequency parameter=\(\sqrt {\frac{{\varrho A}}{{EI}}} \frac{{\omega L^2 }}{{n^2 }}\)

Ω*

frequency parameter=Ω×θ

Freie Schwingungen rechteckiger Balken beliebiger Höhe

Zusammenfassung

Die Zustandsraum-Technik wird auf zweidimensionale elastodynamische Probleme ausgedehnt. Die Formulierung ist besonders geeignet für die Aufstellung von Näherungstheorien beliebigen Grades. Freie Schwingungen von Rechteckbalken beliebiger Höhe wurden mit Hilfe dieser Technik untersucht. Das Verfahren umgeht den Begriff des Schubbeiwertsk. Es berücksichtigt die senkrechte Normalbeanspruchung und die Querkraft. Die Frequenzwerte werden mit Hilfe der Balkentheorie von Timoshenko und der vorliegenden Analyse berechnet, und zwar für verschiedene Werte der Querdehnzahl. Die berechneten Werte befinden sich in guter Übereinstimmung. Vier Fälle von Balken mit verschiedenen Endbedingungen werden untersucht.

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References

  1. [1]
    Lord Rayleigh: Theory of Sound. Vols. I and II. Dover Pub. 1945.Google Scholar
  2. [2]
    Timoshenko, S.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philosophical Magazine41, 744–746 (1921).Google Scholar
  3. [3]
    Timoshenko, S.: On the transverse vibration of bars of uniform cross section. Philosophical Magazine43, 125–131 (1922).Google Scholar
  4. [4]
    Cowper, G. R.: The shear coefficient in Timoshenko's beam theory. J. App. Mech.33, 335–340 (1966).Google Scholar
  5. [5]
    Vlasov, V. Z., Leontev, U. N.: Beams, Plates and Shells on Elastic Foundations. Israel Prog. Scientific Translation. 1966.Google Scholar
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    Das, Y. C., Setlur, A. V.: Method of initial functions in two dimensional elastodynamic problems. J. App. Mech.37, 137–140 (1970).Google Scholar
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    Bahar, L. Y.: A state space approach to elasticity. J. Franklin Institute299, 33–41 (1975).Google Scholar
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    Sundara Raja Iyengar, K. T., Chandrasekhara, K., Sebastian, V. K.: Thick rectangular beams. J. Eng. Mech. Div. (ASCE)100, 1277–1282 (1974).Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • K. T. Sundara Raja Iyengar
    • 1
  • P. V. Raman
    • 1
  1. 1.Department of Civil EngineeringIndian Institute of ScienceBangaloreIndia

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