Torsion and Related Concepts: Some Steps Beyond Saint-Venant’s Principle
In many dynamical problems, one needs a refined theory of torsion for elastic beams where warping of the cross-section is taken into account.
A convenient choice of the warping function is the principal warping function of Saint-Venant. This function is associated with the principal center of the cross-section, the definition of which is purely geometrical.
a Saint-Venant’s contribution,
a contribution from the shearing net force,
a contribution Q* due to an eventual restriction to warping.
In the case of non-uniform torsion at equilibrium, one may neglect the effect of the warping moment Q*: this internal constraint implies that only Saint-Venant’s contribution -which appears as an “effective” torsional moment- may be observed; the warping moment is then determined from equilibrium conditions and may be taken ,under usual assumptions ,to be proportional to the third derivative of the torsional rotation.
The domain of interest of a more refined theory (to take on not to take into account the effect of the wanping moment Q*?) is then discussed by means of two adimensional parameters which characterize the geometry of the cross-section. In statics, it is shown that the effect of Q* may be important in the case of solid beams, but that its impact is concentrated in the small “extra-Saint-Venant’s zones” the internal constraint neglecting the effect of Q* may then be assumed. In dynamics , significant modifications may be brought to the values of natural frequencies of beams submitted to torsional vibrations when this effect is taken into account.
KeywordsInternal Force Torsional Moment Warping Function Internal Constraint Sinusoidal Mode
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