D5.D.6.3 Clause 6.3.2.2 –Elastic critical moment and imperfection factors for LTB checks

The French NA recommends the use of Table 6.3 and 6.4 of NF 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 French NA gives a method to evaluate M_cr in its Annex MCR. This implementation will make use of this method to evaluate Mcr. Annex MCR however deals with the calculation of Mcr for doubly symmetric sections. Hence this implementation will use this method only for doubly symmetric sections. For mono symmetric sections that are symmetric about the minor axis (i.e Tee sections) this implementation will use the method from the NCCI document SN030a-EN-EU as given in the section below. For any other type of section that is not dealt with by the Annex, this implementation will use the method and tables given in Annex F of DD ENV 1993-1-1:1992.

Annex MCR

This document provides a method to calculate M_cr specifically for doubly symmetric sections only. Hence only doubly symmetric sections will be considered for this method in this implementation.

The equation to evaluate M_cr is given as:

M_{c r} = C_{1} \frac{π^{2} E I_{s}}{{(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_{s})}^{2}} - C_{2} z_{s}]

C₁ and C₂ are factors that depend on the end conditions and the loading conditions. The NCCI provides values for C₁ and C₂ for the different cases as given in Table1 and Table 2 of the Annex. Table 1 deals with the condition of a simply supported member with end moments and the value of C₁ is determined by the end moment ratio (Refer to the NA for details). Clause 3.2 of the National Annex however gives a formula to evaluate C₁ as:

C_{1} = \frac{1}{\sqrt{0.325 + 0.423 ψ + 0.252 ψ^{2}}}

This formula however does not match the values given in Table 1 of the NA. Hence this implementation will use the values of C1 from Table 1 if the end moment ration (ψ) is exactly equal to the values of ψ in the table. For all other cases this implementation will calculate the value of C1 from equation (6) in the Annex.

The value of C2 will be determined from Table 2 of the Annex based on the loading and end conditions (i.e the CMM parameter in STAAD.Pro).

You also have the option to specify specific values for C₁ and C₂ using the C1 and C2 parameters in the design input mode. Refer to D5.C.6 Design Parameters

The French NA considers three separate loading conditions:

Members with end moments
Members with transverse loading
Members with end moments and transverse loading: use CMM 7 or CMM 8 for this condition.

The load to moment ratio (μ) will then be used in the calculations will then be used to calculate C1 and C2 as given in section 3.5 of Annex MCR (See Annex MCR in the NA for details).

The parameter MU may be specified when using CMM = 7 or 8 to specify the load to moment ratio (μ) to be used in the calculations. For the French National Annex if CMM = 7 or 8 is been specified, you must also either specify a value for MU or values for the C1 and C2 parameters directly.

SN030a-EN-EU – Mono-symmetrical uniform members under bending and axial compression:

This document provides a method to evaluate the elastic critical moment (Mcr) for uniform mono symmetric sections that are symmetric about the weak axis. Hence for this implementation the elastic critical moment for "Tee-Sections" will be worked out using the method in this NCCI.

Note: Though this method could also be applicable to mono-symmetric built-up sections, STAAD.Pro currently does not have a means to specify/identify a mono-symmetric built-up section. Hence this implementation will use this method only 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_{s}}{{(k_{x} L)}^{2}} [\sqrt{{(\frac{k_{x}}{k_{w}})}^{2} \frac{I_{w}}{I_{s}} + \frac{{(k_{x} L)}^{2} G I_{T}}{π^{2} E I_{x}} + {({C_{2} z_{e} - C}_{3} z_{1})}^{2}} - ({C_{2} z_{e} - C}_{3} z_{1})]

The factors C₁, C₂, and C₃are dependent on the end conditions and loading criteria. This implementation will consider C₁, C₂, and C₃as given in the tables below:

Table 1. Values of C₁ and C₂ for end moment loading (for k_c = 1)
Ψ	C₁	C₂
+1.00	1.00	1.00
+0.75	1.14	0.99
+0.50	1.31	0.99
+0.25	1.52	0.98
0.00	1.77	0.94
-0.25	2.05	0.85
-0.50	2.33	0.68
-0.75	2.57	0.37
-1.00	2.55	0.00

Table 2. Values of factors C₁, C₂, and C₃ for cases with transverse loading (for k_c = 1)
C₁	C₂	C₃
1.13	0.45	0.52
2.57	1.55	0.75
1.35	0.63	1.73
1.68	1.64	2.64

The CMM parameter specified during design input will determine the values of C1, C2 and C3. The default value of CMM is 0, which considers the member as a pin ended member with UDL along its span. This NCCI does not however consider the "end moments and transverse loading" condition. The user however can use the new "C1", ‘C2’ and ‘C3’ parameters to input the required values for C1, C2 and C3 to be used in calculating Mcr.

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. The current implementation of EC3 in STAAD.Pro obtains these values from Annex F of DD ENV version of 1993-1-1:1992.

Also, the NCCI document and Annex MCR of the FR-NA 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 .i.e., CMN parameter =1.0). Hence the above methods will be used only for members which are free to rotate on plan and which have no warping restraints. For members with partial or end fixities (ie, 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.

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 "zg" in the equation to calculate Mcr 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 ‘zg’ 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 "zg" in the equation will have a value of zero.

Note: There is a separate method specified in the NCCI document "SN006a-EN-EU" to calculate Mcr for cantilever beams. Again this document does not give any specific formulae to evaluate the coefficients. Hence, this has not been implemented in STAAD.Pro.