D5.D.13.3 Clause 6.3.2.2 –Elastic critical moment and imperfection factors for LTB checks

The SW-NA recommends the use of Table 6.3 and 6.4 of DIN EN 1993-1-1:2005 to calculate the imperfection factors for Lateral Torsional Buckling (LTB) checks.

The calculation of the LTB reduction factor χ_LT, requires the calculation of the Elastic Critical Buckling Moment, M_cr. The SW-NA does not specify a particular method to calculate Mcr. Hence the calculation of Mcr has been based on the following NCCI documents:

Doubly symmetric sections

SN003a-EN-EU NCCI: Elastic critical moment for lateral torsional buckling provides equation used to calculate M_cr specifically for doubly symmetric sections:

M_{c r} = C_{1} \frac{π^{2} E I}{{(k L)}^{2}} {\sqrt{{(\frac{k}{k_{w}})}^{2} \frac{I_{w}}{I_{S}} + \frac{{(k L)}^{2} G I_{t}}{π^{2} E I_{S}} + {(C_{2} Z_{g})}^{2}} - C_{2} Z_{g}}

C₁ and C₂ are factors that depend on the end conditions and the loading conditions of the member. The NCCI provides values for C₁ and C₂ for the different cases as given in Table 3.1 and Table 3.2.

The NCCI considers three separate loading conditions:

Members with end moments
Members with transverse loading
Members with end moments and transverse loading.

STAAD.Pro accounts for the loading condition and the bending moment diagram through the CMM parameter. The values of C₁ and C₂ may also be directly specified using the C1 and C2 parameters, respectively (required for CMM = 7 or CMM = 8).

Mono-symmetric sections with symmetry about their weak axis

Annex D of SW-NA also provides a method to evaluate the elastic critical moment, M_cr, for uniform mono-symmetric sections that are symmetric about the weak axis. STAAD.Pro uses this method for the evaluating the elastic critical moment for Tee sections.

The equation to evaluate M_cr for mono symmetric sections is given as:

M_{c r} = C_{1} \frac{π^{2} E I_{z}}{{(k_{x} L)}^{2}} {\sqrt{{(\frac{k_{x}}{k_{w}})}^{2} \frac{I_{w}}{I} + \frac{{(k_{x} L)}^{2} G I_{T}}{π^{2} E I_{z}} + {(C_{2} z_{g} - C_{3} z_{1})}^{2}} - C_{2} z_{g} - C_{3} z_{1})}

The factors C₁, C₂, and C₃ are dependent on the end conditions and loading criteria. The program considers C₁, C₂, and C₃ as given in the tables 4.1 and 4.2 of the NCCI, based on the CMM parameter.

The default value of CMM = 0, which considers the member as a pin ended member with uniformly distributed load (UDL) along its span. This NCCI does not however consider the end moments and transverse loading condition. You use the C1, C2 and C3 parameters to input the required values for C₁, C₂, and C₃, respectively, to be used in calculating M_cr.

Note: If "MU" as well as C1, C2 and C3 have been specified, the program will ignore MU and use the user input values of C1, C2 and C3. STAAD.Pro obtains these values from Annex F of DD ENV version of 1993-1-1:1992.

Note: When CMM = 7 or CMM = 8, the values for C1, C2 and C3 parameters must be manually specified.

Both the NCCI documents mentioned above assume that the member under consideration is free to rotate on plan and that there are no warping restraints for the member ( k = kw = 1.0). STAAD.Pro takes into account of the end conditions using the CMN parameter for EC3. A value of K = kw =1 is indicated by a value of CMN = 1.0 in the design input. Hence the above methods will be used only for members which are free to rotate on plan and which have no warping restraints (i.e., CMN = 1.0). For members with partial or end fixities (i.e., CMN = 0.5 or CMN = 0.7), this implementation will fall back on to the method and coefficients in DD ENV 1993-1-1:1992 – Annex F.

For all cases that are not dealt with by the National Annex (or the NCCI documents) this implementation will use the method as per the DD ENV 1993-1-1:1992 code.

The term z_g in the equation to calculate M_cr refers to the distance between the point of application of load on the cross section in relation to the shear center of the cross section. The value of z_g is considered positive, if the load acts towards the shear center and is negative if it acts away from the shear center. By default, the program will assume that the load acts towards the shear center at a distance equal to (Depth of section/2) from the shear center. The use will be allowed to modify this value by using the ZG parameter. Specifying a value of ZG = 0 in the design input would indicate that the load acts exactly at the shear center of the section so that the term z_g in the equation will have a value of zero.

Note: The program does not consider the case of cantilevers.