# Lateral torsional buckling of members with plastic hinges (6.3.5)  Lateral-torsional buckling in bending: in steel design a beam subjected to bending is susceptible to lateral-torsional buckling

In steel design we distinguish between yield strength $f_y$ and ultimate strength $f_u$. We use also ultimate strength of a member which means that a member is plasticized at some part.

If a member is designed with plastic analysis and lateral-torsional buckling might be involved, then such buckling has to be prevented by restraints located at plastic hinges. Member has to be of class 1.

## Effective restraints  Class 1 member might be exposed to lateral-torsional buckling in bending. In this case (for example a cantilever beam) the upper flange is exposed to tension due to bending and the bottom flange is in compression due to bending. The loss of stability (lateral-torsional buckling) occurs in the compression zone.

If it is the bottom flange which is being under compression then the restraint is needed for the bottom flange (and vice versa).

To prevent local buckling in bending the constraint has to prevent compression flange from moving to the sides.  Restraint preventing lateral displacement of compression zone (that means of the bottom-compression flange)  If it is the upper flange which is threaten by a loss of stability then the slab might be use as a restraint for the compression flange against displacement; often the beam is attached to the slab to make sure that system is stable

The diagonal brace preventing lateral displacement is designed to resist 2.5 %of $N_{f,Ed}$. $N_{f,Ed}$ is axial force in compressed flange of the stabilized member at the plastic hinge location. That means that 2.5 % of the axial force is able to prevent local buckling to the side.

### Verification of stable length of a segment

Length between segments has to be not greater than the stable length. For I, H sections with $h/t_f\leq 40 \epsilon,\ \epsilon = \sqrt{235/f_y}$:

\begin{align} L_{stable} &= 35 \epsilon i_z \hskip2em &0.625 &\leq \psi \leq1 ,& \\ L_{stable} &= (60-40\psi) \epsilon i_z \hskip2em &-1 &\leq \psi \leq 0.625,& \\ \psi &= \frac{M_{Ed,\min}}{M_{pl,Rd}}, \end{align}

and $M_{Ed,\min}$ is minimum moment within the member, $M_{pl,Rd}$ is ultimate (plastic) resistance.  To explain how the stable length is being computed

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