# D5.D.12.2 Clause 6.3.2.2 –Elastic critical moment and imperfection factors for LTB checks

The DE-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 DE-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:

C_{1} and C_{2} are factors that depend on the end conditions and the loading conditions of the member. The NCCI provides values for C_{1} and C_{2} 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 C_{1} and C_{2} 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 DE-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:

The factors C_{1}, C_{2}, and C_{3} are dependent on the end conditions and loading criteria. The program considers C_{1}, C_{2}, and C_{3} 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 C_{1}, C_{2}, and C_{3}, respectively, to be used in calculating M_{cr}.

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.