07-05-2013, 12:37 PM
BEAMS SUBJECTED TO TORSION AND BENDING
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INTRODUCTION
When a beam is transversely loaded in such a manner that the resultant force passes through the longitudinal shear centre axis, the beam only bends and no torsion will occur. When the resultant acts away from the shear centre axis, then the beam will not only bend but also twist.
When a beam is subjected to a pure bending moment, originally plane transverse sections before the load was applied, remain plane after the member is loaded. Even in the presence of shear, the modification of stress distribution in most practical cases is very small so that the Engineer’s Theory of Bending is sufficiently accurate.
If a beam is subjected to a twisting moment, the assumption of planarity is simply incorrect except for solid circular sections and for hollow circular sections with constant thickness. Any other section will warp when twisted. Computation of stress distribution based on the assumption of planarity will give misleading results. Torsional stiffness is also seriously affected by this warping. If originally plane sections remained plane after twist, the torsional rigidity could be calculated simply as the product of the polar moment of inertia (Ip = Ixx + Iyy) multiplied by (G), the shear modulus, viz. G. (Ixx + Iyy). Here Ixx and Iyy are the moments of inertia about the principal axes. This result is accurate for the circular sections referred above. For all other cases, this is an overestimate; in many structural sections of quite normal proportions, the true value of torsional stiffness as determined by experiments is only 1% - 2% of the value calculated from polar moment of inertia.
UNIFORM AND NON-UNIFORM TORSION
Shear Centre and Warping
Shear Centre is defined as the point in the cross-section through which the lateral (or transverse) loads must pass to produce bending without twisting. It is also the centre of rotation, when only pure torque is applied. The shear centre and the centroid of the cross section will coincide, when section has two axes of symmetry. The shear centre will be on the axis of symmetry, when the cross section has one axis of symmetry.
Classification of Torsion as Uniform and Non-uniform
As explained above when torsion is applied to a structural member, its cross section may warp in addition to twisting. If the member is allowed to warp freely, then the applied torque is resisted entirely by torsional shear stresses (called St. Venant's torsional shear stress). If the member is not allowed to warp freely, the applied torque is resisted by St. Venant's torsional shear stress and warping torsion. This behaviour is called non-uniform torsion.
Uniform Torsion in Non-Circular Sections
When a torque is applied to a non-circular cross section (e.g. a rectangular cross section), the transverse sections which are plane prior to twisting, warp in the axial direction, as described previously, so that a plane cross section no longer remains plane after twisting. However, so long as the warping is allowed to take place freely, the applied load is still resisted by shearing stresses similar to those in the circular bar. The St.Venant’s torsion (Tsv) can be computed by an equation similar to equation (1) but by replacing Ip by J, the torsional constant.
CONCLUSIONS
Analysis of a beam subjected to torsional moment is considered in this chapter. Uniform torsion (also called St.Venant’s torsion) applied to the beam would cause a twist. Non-uniform torsion will cause both twisting and warping of the cross section. Simple methods of evaluating the torsional effects are outlined and discussed.