E-1Elastic critical moment

E-1.1 General

The elastic critical moment is affected by:

• a) Moment gradient in the unsupported length,
• b) Boundary conditions at the lateral support points,
• c) Non-symmetric and non-prismatic nature of the member, and
• d) Location of transverse load with respect to shear centre.

The boundary conditions at the lateral supports have two components:

• a) Torsional restraint — Where the cross-section is prevented from rotation about the shear centre, and
• b) Warping restraint — Where the flanges are prevented from rotating in their own plane about an axis perpendicular to the flange.

The elastic critical moment corresponding to lateral torsional buckling of a doubly symmetric prismatic beam subjected to uniform moment in the unsupported length and torsionally restraining lateral supports is given by:

$M_{\mathrm{cr}}=\frac{\pi^{2} E I_{y}}{\left(L_{\mathrm{LT}}\right)^{2}}\left[\frac{I_{\mathrm{w}}}{I_{y}}+\frac{G I_{i}\left(L_{\mathrm{LT}}\right)^{2}}{\pi^{2} E I_{y}}\right]^{0.5}$

where

• $I_{y}, I_{w}, I_{t}=$ moment of inertia about the minor axis, warping constant and St.Venants torsion constant of the cross-section, respectively;
• $G=$ modulus of rigidity; and
• $\begin{aligned} L_{\mathrm{LT}} & =\text { effective length against lateral torsional } \\ & \text { buckling (see } 8.3)\end{aligned}$.

This equation in simplified form for I-section has been presented in 8.2.2.1.

While the simplified equation is generally on the safe side, there are many situations where this may be very conservative. More accurate calculation of the elastic critical moment for general case of unsymmetrical sections, loading away from shear centre and beams with moment gradient can be obtained from specialist literature, by using an appropriate computer programme or equations given below.

E-1.2 Elastic Critical Moment of a Section Symmetrical About Minor Axis

In case of a beam which is symmetrical only about the minor axis, and bending about major axis, the elastic critical moment for lateral torsional buckling is given by the general equation:

$M_{\mathrm{cr}}=c_{1} \frac{\pi^{2} E I_{y}}{\left(L_{\mathrm{LT}}\right)^{2}}\left[\left(\frac{K}{K_{\mathrm{w}}}\right)^{2} \frac{I_{\mathrm{w}}}{I_{y}}+\frac{G I_{i}\left(L_{\mathrm{LT}}\right)^{2}}{\pi^{2} E I_{y}}+\left(c_{2} y_{s}-c_{3} y_{j}\right)^{2}\right]$

$-\left(c_{2} y_{\mathrm{g}}-c_{3} y_{\mathrm{j}}\right)\}$

where

• $c_{1}, c_{2}, c_{3}=$ factors depending upon the loading and end restraint conditions (see Table 42).
• $K=$ effective length factors of the unsupported length accounting for boundary conditions at the end lateral supports. The effective length factor $K$ varies from 0.5 for complete restraint against rotation about weak axis to 1.0 for free rotate about weak axis, with 0.7 for the case of one end fixed and other end free. It is analogous to the effective length factors for compression members with end rotational restraint.
• $K_{\mathrm{w}}=$ warping restraint factor. Unless special provisions to restrain warping of the section at the end lateral supports are made, $K_{\mathrm{w}}$ should be taken as 1.0 .
• $\begin{aligned} y_{\mathrm{g}} & =y \text { distance between the point of } \\ & \text { application of the load and the shear } \\ & \text { centre of the cross-section and is positive } \\ & \text { when the load is acting towards the shear } \\ & \text { centre from the point of application. }\end{aligned}$
• $y_{\mathrm{j}}=y_{\mathrm{s}}-0.5 \int_{A}\left(z^{2}-y^{2}\right) y d A / I_{z}$

$y_{s}$

• = co-ordinate of the shear centre with respect to centroid, positive when the shear centre is on the compression side of the centroid.
• $y, z=$ co-ordinates of the elemental area with respect to centroid of the section.
• $y_{j}=$ can be calculated by using the following approximation: