D5.C.5.4.2 Detailed stress check

This method is used when the TORSION parameter is specified as two (2).

This method performs a detailed torsional analysis of a member depending on the torsion loading conditions and the support conditions at the member ends. This method is based on the SCI publication P057 and includes any warping stresses (direct warping stresses and warping shear stresses) depending on the end conditions of the member. This implementation considers seven different cases of loading and end conditions as given in publication P057 – Section 6. The loading/end conditions for a member are specified by the use of the CMT design parameter (Refer to D5.C.6 Design Parameters for parameter values and descriptions).

All the equations used to evaluate the torsional moments and associated stresses are as given in Appendix B of P057. The resultant stresses are evaluated at various sections along the length of the member and the following checks will be performed:

Clause 6.2.7(1) – Torsional resistance of the section.

In general, the torsion at any section T_Ed is resolved into two components, viz.

The pure torsional (St. Venant’s) moment (T_t,Ed) and

The warping torsional moment(T_w,Ed)

Therefore,

T_Ed = T_t,Ed + T_w,Ed = GJφ’ = EHφ’’’

[Ref SCI pub. P057]

where

φ’ and φ’’’

the first and third derivatives of twist (φ), respectively, and depend on the end conditions and loading. These are evaluated from the equations in Annex B of P057 and are based the specified CMT parameter.

Note: Although the equation given the NCCI document SN007b-EN-EU can be used to evaluate T_wrd, the NCCI does not give the eqn. to evaluate φ’’’. Therefore, Annex B of P057 is used.

The torsional resistance of the section is also considered as the sum of the pure torsion resistance and the warping torsion resistance. The pure torsion resistance (T_t,Rd) and the warping torsional resistance (T_w,Rd) are evaluated as:

For closed sections:

T_t,Rd = 2 · A_c · t · τ_max

where

A_c	=	the area enclosed by the mean perimeter
t	=	the max thickness
τ_max	=	the max. allowable shear stress = (fy/√3)/ Γm0

For open sections (I & channel):

T_t,Rd = τ_max · J / t

where

J	=	the torsion const
t	=	the max thickness

T_w,Rd = (f_y/ Γ_m0)· t · b² / 6

where

b	=	the width of the section
t	=	the thickness of the flange for I- sections; minimum of flange or web thickness channel sections

The check according to Cl 6.2.7(1) will then be performed to ensure that the following conditions are satisfied:

T_t,Ed / T_t,Rd ≤ 1

T_w,Ed / T_w,Rd ≤ 1

T_Ed / T_Rd ≤ 1

Clause 6.2.7(9) – Plastic shear resistance due to torsion

STAAD.Pro checks for shear resistance of a section based on Cl. 6.2.6 for EC3 and the plastic shear resistance (in the absence of torsion) is evaluated as:

V_{p l, R d} = \frac{A_{v} (f_{y} / \sqrt{3})}{γ_{M 0}}

where

A_v

as pre Cl.6.2.6 (3) for the various sections

When torsion is present, along with the shear force, the design shear resistance will be reduced to V_pl,T,Rd, where V_pl,T,Rd is evaluated as follows:

For I or H Sections:

$V_{p l, T, R d} = \sqrt{1 - \frac{τ_{t, E d}}{1.25 (f_{y} / \sqrt{3}) / γ_{M 0}}} V_{p l, R d}$
For Channel Sections:

$V_{p l, T, R d} = [\sqrt{1 - \frac{τ_{t, E d}}{1.25 (f_{y} / \sqrt{3}) / γ_{M 0}}} - \frac{τ_{w, E d}}{(f_{y} / \sqrt{3}) / γ_{M 0}}] V_{p l, R d}$
For Structural Hollow Sections:

$V_{p l, T, R d} = [1 - \frac{τ_{t, E d}}{(f_{y} / \sqrt{3}) / γ_{M 0}}] V_{p l, R d}$

where

τ_t,Ed	=	the shear stress due to direct (St. Venant’s) torsion
τ_w,Ed	=	the shear stress due to warping torsion

The various shear stresses due to torsion τ_t,Ed and τ_w,Ed are evaluated as follows:

For Closed sections:

The shear stresses due to warping can be ignored as they will be insignificant and hence:

τ_t,Ed = T_Ed/(2·A_c·t)

[Ref NCCI Sn007b-EN-EU]

where

T_Ed	=	the applied torsion
A_c	=	the area delimited by the mean perimeter
t	=	the thickness of the cross section
τ_w,Ed	=	0, since warping is ignored

For Open sections [I, H, Channel] sections:

For I and H sections, the web will not be subject to warping stresses and therefore warping shear can be ignored (τ_w,Ed=0).

The stress due to pure torsion is evaluated as:

τ_t,Ed = G·t·φ’

[Ref SCI pub. P057]

where

G	=	the shear modulus
φ’	=	a function depending on the end condition and loading(T). This will be taken from section 6 and Annex B of P057.

Note: Although the maximum stress is at the thickest section of the profile, the program uses the web thickness for this clause (since the shear capacity is based on the web area) unless the load is parallel to the flanges, in which case the flange thickness is used.

For channel sections that are free to warp at the supports and, thus, are not subject to warping stresses:

The warping shear stress is evaluated as:

τ_w,Ed = E·S_w·φ’’’ / t

[Ref SCI pub. P057]

where

E	=	the elastic modulus
S_w	=	the warping statistical moment
φ’	=	a function depending on the end condition and loading(T). This will be taken from section 6 and Annex B of P057.

Clause 6.2.7(5) – Check for elastic verification of yield

Eurocode 3 gives yield criterion as per eqn. 6.1 and STAAD.Pro uses the yield criterion given in EC-3. When a member is subject to combined bending and torsion, some degree of interaction occurs between the two effects. The angle of twist caused by torsion is amplified by the bending moments and will induce additional warping moments and torsional shears. Account must also be taken of the additional minor axis moments produced by the major axis moments acting through the torsional deformations, including the amplifications mentioned earlier.

For members subject to bending and torsion, the stresses are evaluated as follows:

Direct bending stress (major axis): σ_bz = M_z / Z_z
Direct bending stress (minor axis): σ_by = M_y / Z_y
Direct stress due to warping: σw = E·W_ns· φ’’
Direct stress due to twist (min. axis): σ_byt = M_yt / Z_y
Direct stress due to axial load (if any): σ_c = P/ A

where

M_z	=	the major axis moment & My is the minor axis moment
φ’’	=	the differential function based on twist (ref P057 Annex B. & Table 6)
W_ns	=	the normalized warping function
M_yt	=	φ·M_z (see Appendix B of P057 to evaluate φ)

Shear stresses due to torsion and/or warping is evaluated as described above for Clause 6.2.7(9).

Check for yield (capacity checks) is then done according to Eqn 6.1 of EN 1993-1-1:2005, as described for the Basic Stress Check (TORSION = 1):

{(\frac{σ_{x, E d}}{f_{y} / γ_{M 0}})}^{2} + {(\frac{σ_{z, E d}}{f_{y} / γ_{M 0}})}^{2} - (\frac{σ_{x, E d}}{f_{y} / γ_{M 0}}) (\frac{σ_{z, E d}}{f_{y} / γ_{M 0}}) + 3 {(\frac{τ_{E d}}{f_{y} / γ_{M 0}})}^{2} \leq 1

Clause EC-3:6 App A – Check for combined Torsion and Lateral Torsional buckling

The interaction check due to the combined effects of bending (including lateral torsional buckling) and torsion will be checked using Annex A of EN 1993-6: 2007. Note that this interaction equation does not include the effects of any axial load.

CAUTION: At present, SCI advises that no significant work has been published for this case and work is still ongoing. So at present is advisable not to allow for torsion in a member with large axial load.

Members subject to combined bending and torsion will be checked to satisfy:

\frac{M_{y, E D}}{χ_{L T} M_{y, R K} / γ_{M 1}} + \frac{{C_{M Z} M}_{z, E d}}{M_{z, R K} / γ_{M 1}} + \frac{k_{w} k_{z w} {k_{α} T}_{w, E d}}{T_{w, R k} / γ_{M 1}} \leq 1

where

C_mz	=	the equivalent uniform moment factor for bending about the z-z axis, according to EN 1993-1-1 Table B.3.
k_w	=	$0.7 - \frac{0.2 T_{w, E d}}{T_{w, R k} / γ_{M 1}}$
k_zw	=	$1 - \frac{M_{z, E d}}{M_{z, R k} / γ_{M 1}}$
k_a	=	$\frac{1}{1 - M_{y, E d} / M_{y, c r}}$
M_y,Ed and M_z,Ed	=	the design values of the maximum moment about the y-y and z-z axis, respectively
M_y,Rk and M_z,Rk	=	are the characteristic values of the resistance moment of the cross-section about it y-y and z-z axis, respectively, from EN 1993-1-1, Table 6.7
M_y,cr	=	the elastic critical lateral-torsional buckling moment about the y-y axis
T_w,Ed	=	the design value of the warping torsional moment
T_w,Rk	=	the characteristic value of the warping torsional resistance moment
χ_LT	=	the reduction factor for lateral torsional buckling according to 6.3.2 of EN 1993-1-1

Note: For all of the above checks the effective length of the member to be used for torsion can be set by using the EFT design parameter.