 # D1.A.5.6 Design for Torsion

Stresses due to torsion in non-HSS sections are considered per AISC 360-10 Section H3.3. This section states that the available torsional strength for non-HSS members shall be the least value obtained according to the limit states of yielding under normal stress, shear yielding under shear stress, or buckling:

 ϕt = 0.9 (LRFD), Ωt = 1.67 (ASD)

For the limit state of yielding under normal stress (H3-7):

Fn = Fy

For the limit state of shear yielding under shear stress (H3-8):

Fv = 0.6Fy

The calculation of Fv and Fn is based on AISC Design Guide 9 Torsional Analysis of Structural Steel Members (DG-9). In general terms, in case of shear stress, Fv will comprise of components of shear stress due to shear about both axes, warping shear stress and shear stress due to pure torsion. In case of normal stress Fn , stress due to axial force and stress due to flexure about both axes is considered. For some sections, like Single Angles and Tees, the component due to warping is negligible with respect to stress for pure torsion (Ref. Section 4.2 and 4.3 of Design Guide 9).

## Notes

• STAAD.Pro will perform these torsion design checks when the TORSION parameter has been set to 1 (these are not checked by default).
• When torsion checks are performed, TRACK 3 output may be used to provide detailed torsion design output for Design Guide 9 checks.
• The torsion checks per Design Guide 9 require additional analysis to calculate Θ (rotation of the element due to applied torsion; refer to the following sections) at the 13 design segments along the member. Thus there is a performance cost for each torsion check. Therefore, it is recommended that torsion checks only be performed on the necessary members (rather than all members).

## Pure Torsional Shear Stress

These shear stresses are always present in the cross-section of a member subjected to torsional moment. They are in plane shear stresses which vary linearly along the thickness of an element.

 τt = GtΘ'

 where

where
 τt = the pure torsional shear stress at the element edge G = the shear modulus of elasticity of steel t = thickness of the element Θ' = first derivative of rotation expressed as a function of local x (distance from start end to point where rotation is calculated)

In the case of an element with a rectangular cross section:

τt = Tut/J

In the case of a hollow circular element or a pipe section with an inner radius of R:

τt = TuR/J

In the case of a tube section:

 τt = Tu/(2bht)

 where

where
 Tu = the total torsional moment action at any location along the beam

## Shear Stress Due to Warping

When warping in a member is restrained, in plane shear stresses are developed which are constant along thickness of the element but vary along the length of the element.

 τws = -ESwsΘ'''/t

 where

where
 τws = shear stress at a point,s, due to warping E = the modulus of elasticity of steel t = thickness of the element Sws = warping statical moment at a point, s Θ''' = third derivative of rotation expressed as a function of local x (distance from start end to point where rotation is calculated)
Note: The shear stress due to warping is neglected for angle, tee, tube, or pipe sections.

## Normal Stress Due to Warping

When warping in a member is restrained, direct stress acting perpendicular to the cross-section of the element is generated. These stresses are constant along the cross-section but vary along the length of the member.

σns = EWnsΘ''

where

where
 σns = normal stress at a point,s, due to warping E = the modulus of elasticity of steel Wns = normalized warping function at a point, s Θ'' = second derivative of rotation expressed as a function of local x (distance from start end to point where rotation is calculated)
Note: Point s refers to a point on the cross section area of a particular section as explained in Section 3.2.2 of Design Guide 9.

## Combined Stresses due to Axial, Bending, and Torsional Stresses

Under section 4.6 of Design Guide 9, the combined stress in a section due all the stresses as explained in sections 4.1, 4.2, 4.3, and 4.4 is

 fn = σa + σbz + σby + σs

 fv = τsz + τsy + τt + τws

These stresses are calculated at 13 sections along the beam length.

Only member loads of the following types are considered in these checks:

• concentrated torque (moment about local x axis)
• concentrated force eccentric from the member shear center
• uniformly distributed torque (full or partial)
• uniformly distributed force eccentric to the member shear center
• end torques (only considered when end supports are fixed)

Linearly varying torque is not considered in the torsion checks. Joint loads are also not considered in the torsion checks.

The boundary conditions for torsional analysis and the method to calculate rotation,Θ , and its derivatives are used as described in DG-9.

STAAD.Pro calculates the stresses due to flexure, pure torsion, and warping torsion at 13 different sections along the member length. The total stress is the vector summation at each location.