Elastic Critical Moment

Within each unbraced segment the points of maximum moment and shear and the segment end points are all checked against the segment capacity.

When calculating the member capacity for an unbraced segment the lateral torsional buckling must be considered. In these cases it is necessary to calculate the elastic critical moment (Mcr).

While BS EN1993-1-1:2005 does not provide exact information on how to compute M_cr, the program calculates the elastic critical moment using procedures, described in these articles:

SN003a-EN-EU NCCI: Elastic critical moment for lateral torsional buckling
SN030a-EN-EN NCCI: Mono-symmetrical uniform members under bending and axial compression
M.A Serna, A.Lopez, I. Puente, D.J. Yong "Equivalent uniform moment factors for lateral-torsional buckling of steel members" (2006) Journal of Constructional Steel Research, 62.
Rubrique du Praticien, "Abaques De Deversement Pour Profiles Lamines", Construction Metallique no 1 - Mars 1981.

The program assumes that the factors k and kw are equal to 1.0. The following procedures are implemented to calculate the C factors, depending on the beam major axis loading:

Members with end moments only—according to the French Annex:

C 1 = 1 / \sqrt{0.325 + 0.423 \cdot ψ + 0.252 \cdot ψ^{2}}

C2 = 0.0

C3 = 0.5(1.0 + ψ)·C1

SN030a-EN-EU, Table 4.2 shows values for factors C1, C2, and C3 for theoretical pinned and fixed end beams with uniform and concentrated midspan loads. Checks for symmetry, shape, and relative magnitudes of the end and midspan moments of the moment diagram are made to see if these cases are applicable. Beams having end moments within about 1 kN-m of zero are assumed to be pinned. End moments within 5% of the expected theoretical end value based on midspan moment are assumed to be fixed. For example, consider a symmetric, linear moment diagram with a +100 kN-m midspan moment. End moments of -95 kN-m are within tolerance for a fixed end case with a concentrated load midspan and Table 4.2 Case D will be assumed.

Table 1. Values of factors C1, C2, and C3 for cases with transverse loading (for k_z = 1)
Case	C1	C2	C3
A	1.13	0.45	0.52
B	2.57	1.55	0.75
C	1.35	0.63	1.73
D	1.68	1.64	2.64

Note: Negative loads inducing a moment diagram of the opposite sign to those indicated in the preceding table assume the same respective values for C1, C2, and C3. For example, Case A with a upward uniform load uses the same values of C1 as for the download load diagramed.

If end moments are not zero and shape of moment is symmetric and there is no straight line between center and ends of segment use French Annex formulas based on the procedure detailed in the French Journal, Rubrique du Praticien, "Abaques De Deversement Pour Profiles Lamines", Construction Metallique no 1 - Mars 1981:

C1 = C₁⁰|Mmax/M|

C2 = 4/π²·|μ|·C₁⁰

C3 = 0.525 if ends pinned, 0.753 if ends fixed, 0.64 for one end pinned, other fixed. where

C₁⁰	=	$\sqrt{1 + γ + (\frac{1}{3} + \frac{1}{2 π^{2}}) (γ^{2} - 8 μ) - 2 γ μ (1 - \frac{3}{π^{2}}) + 8 μ^{2} (\frac{2}{5} - \frac{2}{π^{2}} + \frac{3}{π^{4}})}$
γ	=	4μ + β - 1
μ	=	$\frac{f L^{2}}{8 M}$
f	=	Uniform load
M	=	Maximum end moment
L	=	Member Length

For all other cases use formula (13) from M.A. Serna article "Equivalent uniform moment factors for lateral - torsional buckling of steel members":

\frac{\sqrt{\sqrt{k} A_{1} + {[\frac{1 - \sqrt{k}}{2} A_{2}]}^{2} + \frac{1 - \sqrt{k}}{2} A_{2}}}{A_{1}}

C2 = 0.0

C3 = 0.0

where

A₁	=	$\frac{M_{\max}^{2} + 9 k M_{2}^{2} + 16 M_{3}^{2} + 9 k M_{4}^{2}}{(1 + 9 k + 16 + 9 k) M_{\max}^{2}}$
A₂	=	$\| \frac{M_{\max} + 4 M_{1} + 8 M_{2} + 12 M_{3} + 8 M_{4} + 4 M_{5}}{37 M_{\max}} \|$

Coefficient k is related to the lateral bending and warping prevention at end supports. It is equal to 1 if lateral bending and warping are free and equal to 0.5 if lateral bending and warping are prevented. Moments M₁ and M₅ are begin and end moments respectively, moment M₃ is the moment at the middle of the span, moment M₂ is moment on L/4 position and moment M₄ is the moment on 3L/4 position. Moments M₁ through M₅ must have its corresponding signs.

Note: Values of C for cantilevers are calculated as for all other members as the program assumes the ends of a cantilever to be laterally braced.

The output shows the values of: class, the design shear and moments at the indicated beam location, the segment unbraced length (Lb), the type of moment that controls the member capacity (Mb = buckling, Mc = plastic capacity, Mv = is shear reduced capacity) and the member capacity. Note that the controlling condition may not correspond to any of the maximum or minimum moment conditions. This indicates that the controlling condition occurs in a segment with a lesser moment but greater unbraced length.