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CivilComp Proceedings
ISSN 17593433 CCP: 75
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and Z. Bittnar
Paper 36
Triply Coupled Vibrations of ThinWalled Open CrossSection Beams including Shear Effect A. Arpaci, E. Bozdag and E. Sunbuloglu
Department of Mechanical Engineering, Istanbul Technical University, Turkey A. Arpaci, E. Bozdag, E. Sunbuloglu, "Triply Coupled Vibrations of ThinWalled Open CrossSection Beams including Shear Effect", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 36, 2002. doi:10.4203/ccp.75.36
Keywords: coupled vibration, thinwalled beam, shear effect.
Summary
Beams of thinwalled open crosssections are widely used in structural design. In
general, the centroid and the shear centre do not coincide; hence the flexural and the
torsional vibrations are coupled. The case that the flexural vibrations in one direction are
coupled with the torsional vibrations is extensively studied by many researchers. But, the
case that the flexural vibrations in two mutually perpendicular directions and the torsional
vibrations are all coupled is not sufficiently dealt with. Gere and Lin [1] have determined
the triply coupled free vibration characteristics of uniform, open crosssection channels
with simply supported ends by employing the Rayleigh  Ritz method. Yaman [2] has
formulated the mathematical model by using the wave propagation approach and has
presented the wave numbers for undamped and structurally damped channels. Arpaci et al. [3]
have studied effect of rotary inertia on triply coupled vibrations of thinwalled beams.
To the authors' knowledge, no work considering the shear effect on triply coupled
vibrations of such beams appears in literature. The objective of the present study is to
investigate the effect of shear on triply coupled vibrations of beams with thinwalled open
crosssections having no axis of symmetry. The effects of crosssectional warping are also
included.
Equations of Motion

where and are deflections of shear centre in and directions respectively, is angle of rotation of crosssection, and are principal centroidal moments of inertia, is warping constant, is torsion constant, and are Young's and shear moduli respectively, is polar moment of inertia about shear centre , and are coordinates of centroid , is cross sectional area, is mass density, and are shear correction factors, are bending slopes and is time.
Assuming , , and five coupled equations are reduced to five independent equations by a procedure analogous to the solution of simultaneous algebraic equations where is radian frequency. The result is an ordinary differential equation of 12 th order for each function. These latter equations are solved by taking a solution in the form .
Clamped end:  
Free end:  ; 
Hinged end: 
Numerical evaluation is performed for natural frequencies and the shear effect is discussed in detail. It is shown that the accuracy of the calculations of natural frequencies is considerably disturbed in higher modes when the shear effect is not taken account of.
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