# Torsion and Related Concepts: Some Steps Beyond Saint-Venant’s Principle

• Patrick Muller
Conference paper
Part of the Lecture Notes in Engineering book series (LNENG, volume 28)

## Abstract

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.

The torsional moment appears then to be the sum of three terms:
• a Saint-Venant’s contribution,

• a contribution from the shearing net force,

• a contribution Q* due to an eventual restriction to warping.

Attention is called on the plurality of centers which may be associated to a given cross-section (centroïd,principal center,center of torsion,center of flexure,center of shear) and the conditions of their coïncidence are discussed.

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.

## Keywords

Internal Force Torsional Moment Warping Function Internal Constraint Sinusoidal Mode
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.

## References

1. [1]
VLASOV B. Z.,1961, Thin-Walled Elastic Beams, 2nd ed., Israel Program for Scientific Translations, Jerusalem.Google Scholar
2. [2]
GERMAIN P., 1973, The method of vixXuat power in continuum mechan-Lc., S.I.A.M. J. Appl. Mech., 25, 556–575.
3. [3]
TREFFTZ E., 1935, Uebet den Sehubmittefpunht in einem dutch e-Lne Einzeffazt gebogenen Bafhen, Z.A.M.M, 15, 220–225.
4. [4]
STEPHEN N.G. and MALTBAEK J. C., 1979, The netation4h-Lp between the cen-ten-g 6texute and Zw2,st, Int.J.Mech.Sci., 21, 373–377.
5. [5]
MULLER P., 1982, Sun to tocati,sation dei, centnea de 4Pex-Lon et de -ton-eon, Mech. Res. Comm., 367–372.Google Scholar
6. [6]
TIMOSHENKO S. P., 1946, Strength of Materials, Part II, Advanced Theory and Problems, 2nd ed., D. Van Nostrand Co, New-York.Google Scholar
7. [7]
ARGYRIS J. H., 1954, The open tube, Aircraft Engineering, April 1954, 102–112.Google Scholar