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Shear Lag in Steel Structures

September 11, 2020 Posted by Parishith Jayan , , 5 comments
If we consider ourselves attending a design class in the undergraduate degree program, once the design philosophies are over, the first design lecture would be on "Design of Tension Members".

Have you ever wondered, why we are learning this thing first?

You can check any standard textbooks, once the author finished talking about introductions, prerequisites, design philosophies, the first actual design would be "Design of Tension Members". In fact, if you check the codebooks, they too follow the same order. (Attached the content page of IS-800:2007 to support my argument)

IS-800:2007, parishith jayan

To answer the question, WHY?

It is the simplest design involving not so complicated loading conditions and very few limit-states that are easy to understand.
To elaborate, it is easy to calculate the stress component in the case of tensile loading (Stress=Load/Cross-sectional Area), which is not so with transverse loading, since we need to calculate bending stresses.

To gain some insight regarding stresses and forces and why we calculate stress, read this article - Why do we calculate STRESS, when we have FORCES?

Regarding the limit states for tension members, the tension member can fail due to
  1. Gross section yielding
  2. Rupture of Net section
Kindly note, I have not listed the limit states for the failure of connections.

See, it is simple. Very few concepts and that is why it is placed first in the order.

Before moving to the topic, would like to ask one more question. What are the most widely used tension member sections?

As a designer, say, I am about to design a tension bracing member, the first section that comes to my mind would-be "Angle Sections" (Hoping, you too think of the same). 

When considering angle members, there is one important behavior or phenomenon which should be addressed or taken into account while determining its tensile capacity. It is nothing but "Shear Lag", which is the main concentration for today's article.

Contents

  • What is meant by "Shear Lag effect"?
  • Exercise to feel shear lag
  • How do we incorporate "Shear Lag" in the calculation of tension capacity?
    • As per AISC-360-10
    • As per IS-800:2007
  • Inference/Conclusion

What is meant by "Shear Lag effect"?

The attached image shows the gusset plate connection of typical bracing members. 

The most important thing to notice is the way in which they are connected. As we can clearly see, they are connected using bolts to the gusset plate, which in turn is welded to the main member. 

What is so special about this?
"Only one of the leg is connected". So what?

You can say, that if the bolts are properly designed to take care of the design force, then the connection should work fine.

The problem is not with the connection, it is with the behavior of the member. Since only one leg is connected, the entire tensile force will get transferred to that leg, to flow through the connection. Which totally cancels out our limit state checks for the tension member.

We will be calculating yielding strength based on the entire area of the member (including both the legs) and rupture strength based on the net area (including both legs neglecting the bolt hole). Since at the location of connection, all the forces are getting transferred to the connected leg, the area of concentration of force reduces, so both of these limit states are not valid.

Usually, the force which is acting in one of the legs gets transferred to the entire cross-section in the form of shear. At the location of the connection, the stress distribution is non-uniform due to the concentration of force on the connected leg. This "lag" in stress distribution in the tension member is termed as "SHEAR LAG".

Exercise to feel shear lag

Just try to make a replica of an angle section using paper which is around 50 to 75mm long. Let the sides of the angle be 10mm x 10mm. 

Ask your friend to hold one end of the angle by grabbing both the legs of the paper angle, which you made. And on the other side, you too do the same and slightly apply a pull force. You can feel the stress in the entire cross-section of the paper (i.e. you can feel, both of the legs are experiencing stress due to the pull). Now, you just hold only one leg of the angle and apply a little pull. As you can evidently see, the outstanding leg of the angle at your end just casually drops off. You can touch and verify, that there is no stress on the outstanding leg at the location of the connection. But, as we move away from the connection, the angle still would experience stress on both the legs. This lagging in stress distribution is what we call, "Shear Lag".

Hope I made it clear.

How do we incorporate "Shear Lag" in the calculation of tension capacity?

Due to the behavior of shear lag, the effective area of the tension member is getting reduces. So, this reduction in the area is applied in the form of "shear lag factor, 'U'" in AISC-360-10/16.

Herewith attached the snap from AISC-360-10 to determine the effective area and the table to determine the shear lag factor for different cross-sections.
effective net area, parishith jayan, shear lag

shear lag factor, parishith jayan

The shear lag factor can be determined based on case 2 of the table. As we closely look, the factor depends on two things, firstly the centroidal distance from the plane of connection, secondly the length of the connection.

So, if we are about to increase the effective area (which would ultimately increase the tensile capacity of the section), we should try to reduce the value of 'x' (centroidal distance from connection plane) and increase the length of the connection.

Keeping the concepts that we learned so far in front, let me ask you a question.

If you are using an unequal angle section (one leg will be longer than the other), which leg will u connect in order to get higher tensile capacity?

Obviously, the longer leg. Why do we do so? Because it will considerably reduce the value of "x" considering the short leg being connected. 

In IS-800:2007, we do follow the following steps to determine the tension capacity of the section.
shear lag, parishith jayan
What is happening in the above formula, the capacity of the connected leg and outstanding leg are separately calculated. The assumption here is that the outstanding leg will reach yielding stress when the connected leg probably would have attained the ultimate stress.

The net area is calculated for the connected leg to determine its capacity. The area of the outstanding leg is reduced with the reduction factor "β", which is our shear lag factor. It depends on the factors like the length (L) of the connection and shear lag width (bs).

Alternatively, a simpler approach is also provided in IS-800:2007 as shown in the snap below. 
shear lag, parishith jayan

Here, the net sectional area is multiplied with the reduction factor "α" to account for the shear lag based on the number of bolts used in the connection (Length of the connection).

Inference/Conclusion

As we have seen the concept of shear lag and the factors affecting it, the following are my inference
  1. The thickness of the member does not play a role in the determination of shear lag factor or it does not affect the tensile capacity of the section.
  2. If the distance between the plane of connection and the centroidal axis of the outstanding leg increases, it reduces the tension capacity of the section.
  3. From the experimental studies, it is proved that if the length of the connection is increased up to 4 bolts, the tensile capacity increases. And if we further increase the length it does not have a significant effect. (Kulak and Wu, 1997).

Lateral Torsional Buckling

September 02, 2020 Posted by Parishith Jayan No comments

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. 
Lateral Torsional Buckling

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.