D5.D.8.3 Clause 6.3.2.2 –Elastic critical moment and imperfection factors for LTB checks

The Polish NA recommends the use of Table 6.3 and 6.4 of PN 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 Polish National Annex does not specify a particular method to calculate M_cr. Hence the calculation of M_cr has been based on the following NCCI documents:

SN003a-EN-EU – Elastic critical moment for Lateral torsional Buckling

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. The equation to evaluate M_cr is given in the NCCI 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 of the member. The NCCI provides values for C₁ and C₂ for the different cases as given in the tables below:

Table 1. Values of C₁ for end moment loading (for k=1)
ψ	C₁
+1,00	1,00
+0,75	1,14
+0,50	1,31
+0,25	1,52
0,00	1,77
-0,25	2,05
-0,50	2,33
-0,75	2,57

This 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.

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

This document provides a method to evaluate the elastic critical moment (M_cr) for uniform mono symmetric sections that are symmetric about the weak axis. Hence, 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 C₃ are dependent on the end conditions and loading criteria. This implementation will consider C₁, C₂, and C₃ as given in the tables below:

Table 2. 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	0.38
-1.00	2.55	0.00

Table 3. 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. You can use the C1, C2, and C3 parameters to input the required values for C₁, C₂, and C₃ 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.

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). Other values of CMN (i.e., CMN = 0.5 or CMN = 0.7) are not applicable to the Polish NA.

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.