# Structures Simplified

## Lateral Torsional Buckling

September 02, 2020 Posted by Parishith Jayan

## What happens to a member, when it is subjected to transverse loading?

A member subjected to transverse loading will undergo bending and start to deflect along the plane in which it is loaded. It seems so simple, right?

Now, what will happen if we keep on increasing the load?

It will further bend until it reaches its maximum moment capacity and fails. This is what we call a bending behavior of the laterally supported beam. If a beam needs to behave this way, there are certain conditions that the beam should satisfy. They are,
1. There should not be any local buckling in the elements.
2. The compression flange of the member should be restrained (i.e.) The beam should be restrained in the lateral direction so that it cannot move laterally.
What happens when the first condition fails? The beam would fail due to local buckling of the web or flange even before attaining its full moment capacity.

What if the second condition fails? The beam will fall under the laterally unsupported beam category. And when the load is increased, the beam will displace laterally (to be more precise, the compression flange moves laterally) and the failure would occur due to the combination of lateral displacement and bending. It considerably reduces the moment capacity of the section. This phenomenon is called "Lateral Torsional Buckling". For our ease, let's call it LTB.

## Whether all laterally unsupported beam exhibits LTB?

No. When the beam is too short, even though the beam is laterally unsupported, the beam won't buckle laterally. It will fail after the attainment of full moment capacity like a laterally supported beam.

## The behavior of Lateral Torsional Buckling

As we see, the short beam won't exhibit LTB, what about long beams?

Let's consider a simply supported (but laterally unsupported) beam with a concentrated load at the midspan. If we keep on increasing the load. Initially, the beam will undergo bending and deflects vertically. Because of this, compressive stress develops in the top flange and tensile stress develops in the bottom flange.

When the load keeps on increasing, the stress developed will the beam section also increases. In order to relieve the compressive stress generated, the compression flange needs to get elongated (The stress created can be relieved in the form of displacement). Since the beam is oriented in such a way that its major axis taking up the loading, it can't elongate in the axis of loading. So, now the compression flange makes use of the lateral direction, which is unrestrained. It tries to displace in that direction, causing the whole section to twist about the axis of loading.

The key point to note in this behavior is that the cross-sectional shape won't change. The beam will undergo twist alone. This twisting makes the compression flange move away from the actual line of the beam resembling the buckling behavior of the column sections.

## How does the position of the load affect LTB?

Based on the position of the load, the moment resistance of the beam is enhanced or decreased. Say for example, if the load is applied at the top flange of the section, the effect of LTB will be more and if the load is applied at the bottom flange of the section, then the effect of LTB will be less.

What is the myth behind this?

In the first case, the load is applied above the shear center of the beam, when the top flange starts to buckle laterally, the applied load will be at some eccentricity now, which would cause some additional twisting moment. This load is called the "Destabilizing Load"

Contrarily when the load is applied at the bottom flange, the moment generated tries to stabilize the beam, thereby reducing the torsional buckling. This load is called the "Stabilizing load".

## Whether LTB occurs when an I-beam is positioned in H-shape and the load is applied along the minor axis?

In this case, the beam would be positioned to look like "H" and when the load is applied. Initially, the beam will bend in its minor axis. Comparing the bending capacity of the minor and major axis, the minor axis holds very low bending capacity. So, in this case, the beam will reach its maximum bending capacity in the minor axis before, the section displaces laterally. LTB won't occur.

## Whether LTB can be neglected?

The lateral-torsional buckling can be neglected in certain scenarios, they are as follows.
• As we already discussed, when the beam is loaded along its minor axis. The beam will generate its full bending capacity even before LTB could initiate.
• In sections like a square hollow tube and circular hollow tube. They have the same moment of inertia value in both the axis.
• When the beam is too short (this limit will be provided in design codes), the member will fail by creating its full moment capacity rather than LTB.

## Conclusion

It is evident that the behavior of lateral-torsional buckling largely influences the bending capacity of the laterally unsupported beam. Neglecting this would result in over-estimating the capacity of the section.

In order to reduce the impact of LTB, the long beams should be braced laterally at appropriate locations, to reduce the effective length of the member in a lateral direction, which would considerably increase the capacity.