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.
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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Abbreviations
- 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
- A n :
-
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
- δσ x σ y :
-
direct stresses
- τ xy :
-
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
References
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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).
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With 1 Figure
On leave from M. A. College of Technology, Bhopal, 462007, India
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Sundara Raja Iyengar, K.T., Raman, P.V. Free vibration of rectangular beams of arbitrary depth. Acta Mechanica 32, 249–259 (1979). https://doi.org/10.1007/BF01379010
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DOI: https://doi.org/10.1007/BF01379010