D5.D.10.2 Clause 6.3.2.2 –Elastic critical moment and imperfection factors for LTB checks

The NBN-NA recommends the use of Table 6.3 and 6.4 of 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 NBN-NA gives a method to calculate M_cr in Annex E, which is used by STAAD.Pro. Annex E, however, only deals with the calculation of M_cr for doubly symmetric sections and mono symmetric sections that are symmetric about the minor axis (i.e, Tee sections). For any other type of section that is not dealt with by Annex E, STAAD.Pro uses the method and tables given in Annex F of DD ENV 1993-1-1:1992:

Doubly symmetric sections

Annex D of NBN-NA 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} + \frac{{(k L)}^{2} G I_{t}}{π^{2} E I} + {(C_{2} Z_{g})}^{2}} - C_{2} Z_{g}}

C₁ & C₂ are factors that depend on the end conditions and the loading conditions. The Annex provides values for C₁ & 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 calculate C₁ as:

C₁ = 1.77 - 1.04ψ + 0.27ψ² ≤ 2.60

The value of C₂ is determined based on the Table 2 of the Annex, based on the loading and end conditions as specified using the CMM parameter.

This NBN-NA 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.

Mono-symmetric sections with symmetry about their weak axis

Annex D of NBN-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. Hence for this implementation the elastic critical moment for Tee-Sections is evaluated using the method in this Annex.

Note: Though this method could also be applicable to mono-symmetric built-up sections, STAAD.Pro 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_{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_{j})}^{2}} - C_{2} z_{g} - C_{3} z_{j})}

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:

End moments and support conditions

Table 1. Critical moment coefficients for singly symmetric sections with end moments
Bending moment diagram	k_z	Value of coefficients
		C₁	C₃
		C₁	ψ_f ≤ 0	ψ_f > 0
ψ = +1	1.0	1.00	1.000
ψ = +1	0.5	1.05	1.019
ψ = +3/4	1.0	1.14	1.000
ψ = +3/4	0.5	1.19	1.017
ψ = +1/2	1.0	1.31	1.000
ψ = +1/2	0.5	1.37	1.000
ψ = +1/4	1.0	1.52	1.000
ψ = +1/4	0.5	1.60	1.000
ψ = 0	1.0	1.77	1.000
ψ = 0	0.5	1.86	1.000
ψ = -1/4	1.0	2.06	1.000	0.850
ψ = -1/4	0.5	2.15	1.000	0.650
ψ = -1/2	1.0	2.35	1.000	1.3 - 1.2ψ_f
ψ = -1/2	0.5	2.42	0.950	0.77 - ψ_f
ψ = -3/4	1.0	2.60	1.000	0.55 - ψ_f
ψ = -3/4	0.5	2.45	0.850	0.35 - ψ_f
ψ = -1	1.0	2.60	-ψ_f	-ψ_f
ψ = -1	0.5	2.45	0.125 - 0.7ψ_f	-0.125 - 0.7ψ_f

Note: According to Section 3(1): C₂z_g = 0

Table 2. Value of coefficients
Load and support conditions	Bending moment diagram	k_z	Value of coefficients
Load and support conditions	Bending moment diagram	k_z	C₁	C₂	C₃
		1.0	1.12	0.45	0.525
		0.5	0.97	0.36	0.478
		1.0	1.35	0.59	0.411
		0.5	1.05	0.48	0.338
		1.0	1.04	0.42	0.562
		0.5	0.95	0.31	0.539

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 uniformly distributed load (UDL) along its span. This NCCI does not however consider the "end moments and transverse loading" condition. You can use the parameter MU to describe the moment distribution for cases where the end moments vary (i.e., CMM 7 or CMM 8). Alternatively, 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 = k_w = 1.0). STAAD.Pro takes into account of the end conditions using the CMN parameter for EC3. A value of K = k_w =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.

For the term z_j, please refer to Annex E of NBN NA 2018.

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.