In last post, we have learned how to check if the connection is a real hinge. Today we will treat our joint between steel members as semi-rigid and we will calculate connection rigidity. This is a very useful tool. It can be used to make an even more accurate design if necessary.

If you ever tried to calculate connection rigidity according to EN 1993-1-8 you already know how much work it is (if it is at all possible in your case). Here I will discuss a simpler possibility based on finite element method.

Introduction

This is a second post covering the topic of connection rigidity that was presented at first workshop for WrUT students I had a pleasure to organized with Ola Kociołek from Dlubal Poland. Today, we will discuss a FEA method of estimating connection rigidity.

Main issues with connection rigidity:

• How can I be sure that my connection is a hinge?
• If my connection is semi-rigid how to calculate it’s rigidity?
• How to take connection rigidity into account in static design?
• How to know if connection rigidity is important in my design?

What types of connections there are when it comes to rigidity

Before we get into the FEM calculations I think it is important to discuss what sort of connections rigidity there are. Understanding this will make our job later on so much easier. If we evaluate our connection properly, setting up an analysis will be easier.

• Rigid connection – let us take this out of the way. If our connection is rigid, there is nothing to calculate.
• Hinge – the same story here – if we are in the “free rotation” zone of our connection all the time, there is no point in doing anything more.
• Linear Semi-Rigid – this is the simplest of possibilities. You have to define a stiffness (or “spring constant” as it is sometimes called) in kNm/rad, the software will take that stiffness into account in the design. Most software solutions nowadays have this as a standard. Usually, connection stiffness is shown on a chart with between rotation (on the horizontal axis) and bending moment (on the vertical axis). Charts for all stiffness types are given below.
• Linear Semi-Rigid with slippage – second possibility, quite often seen in shear connections that can carry bending moment. Before the clearances are closed, a certain free rotation is possible. The closing of the clearances is often called slippage.
• Linear Semi-Rigid with rigid start and slippage – this is often met possibility in pre-loaded connections, even through after the “rigid start” ends (and usually slippage appears) codes treat such connection as “destroyed”. Pre-loaded connections carry moment due to friction between connected plates. At first, there is no movement at all (“rigid start”), and then when friction capacity is exceeded slippage appears (as the bolts in the connection are closing clearances). After that not only friction carry the force, the shear action of the bolts do it as well (theoretically increasing capacity, even through kinetic friction is lower than static friction in most cases). Since codes do not allow the post-slippage state of pre-loaded connections this is rarely used, but still, there is such a thing. Depending on material static and kinetic friction coefficient might have different values – in the chart below, it is assumed that the value is the same (so there is no drop in capacity when slippage appears). The second chart show more advanced model that takes into account difference between static and kinetic friction coefficient – it can be seen as a drop in capacity after friction capacity is reached.
• Nonlinear Semi-RigidIn the above examples you could notice that at the end of each chart (when shear action in the screws start) linear relation between rotation and bending moment is assumed. Obviously, in this case (and much more) this is not the case, even through this is a very popular simplification (usually you simply extend the first linear part of the chart, knowing you won’t reach the “softened part” – then linear simplification is just fine). In many situations stability path of the connection looks just like the one given below. This is one of the examples of nonlinearity (due to yielding) – just bear in mind that when you want to use the part of the curve after it “softens”, you have to very closely examine the screws and plates capacity first! There are many possibilities for great FEM analysis in nonlinear semi-rigid connections (some really complex)! Usually, they aren’t really needed (and simplifications given here are sufficient), but in rare cases, those come in handy – we will get there eventually 🙂

Of course, the above possibilities are just a tip of an iceberg. Since we are discussing connection rigidity, rather than connection nonlinearity (which is a much broader topic) things concerning different capacity in positive and negative rotational direction are omitted. Also, we won’t discuss here ineffectiveness of connection in any given direction, as well as limit loads etc.

How to calculate connection rigidity

In this post, I will use the same shear connection I used previously. I already know that this connection allows for certain free rotation, after which it will act in a way I will simplify as linear. With this in mind I know I’m aiming for “semi-rigid with slippage” curve as defined above.

I could, of course, create a nice model with contacts to get a full curve from one analysis run… but that is time-consuming, and I literally never do it this way for “semi-rigid with slippage” connections. It is far easier to estimate the free rotation as we did previously, and then separately calculate the rigidity in a much simpler model.

All we have to do is the implement all the plates in the connection, with screws being “rigid” elements. I usually fill the “holes” with infinitely rigid surfaces, representing screw diameter. Then the model is relatively simple both to create and calculate. It does “lie” however, since this rigid surface transfer compression and tension, while screw only transfers compression (by contact stresses)… this means that capacity obtained from such model shouldn’t be trusted, but for “fast” approximation of rigidity I’m fine with that. Of course, you could play with contact etc. to make it far more accurate… but the deeper you go in, the more it takes to finish. Usually, even such simplified method of rigidity verification is treated as “luxury” due to time restraints.

So let’s start with the geometry – please note that I have changed some parts of the cantilever beam section into plates (with thicknesses corresponding to web and flanges of the “removed” beam). Since connection requires changes in the cross section (top flange is removed along with part of the bottom flange) I can just as well take that into account – this obviously requires shell model. The “main” beam is made only with beam elements – but where the stiffener is a shape from rigid elements is made to “mimic” the real cross section  Also the connection region is being modeled that way.

Note that you need to have rigid elements on the edges of surfaces connected to the beam element (cantilever). You can, of course, connect a beam to the surfaces without them, but then all the forces and moments will be transferred through the middle of the web (instead of the whole cross-section) and this is of course greatly influence the outcomes as can be seen below:

The top example above shows connection without rigid elements, the central one with rigid elements only on the web, and the bottom line with rigid elements on entire cross-section outline. Of course, the same load was applied. Note the color scale (constant in all 3 examples) – stresses in the model with no rigid elements are at least 1500x higher than in the model with rigid elements.

As mentioned previously holes for bolts were filled with rigid surfaces, and those were connected with rigid elements (representing screws). Using rigid elements as screws actually make sense – you could implement real cross-section of the screw easily, but since the middle surface of the web and stiffener are around 15mm apart, this would lead to a situation where the screw would be bent. In reality, this will not occur, and as such rigid elements are actually more accurate.

With our model ready we can perform a simple static test of the connection, and observe final deformations:

Now let’s analyze what influences the deformation we have obtained:

• Stiffness of the connection
• Deflection of the cantilever
• Rotational deformation of the perpendicular beam

We have to decide which elements from the above mentioned we wish to take into account. For one, cantilever deflection is never required, but also with such loads it will be minimal. If we want to be more accurate, we can remove this influence by replacing the beam of the cantilever with a rigid element as shown below:

Now we have to decide if the torsional stiffness of the beam should be considered.  This depends on whether we wish to calculate the rigidity of the connection between those beams or to calculate the rigidity of the support the cantilever gets in the beam with this connection. For connection rigidity you never want to include deformations of the connected elements into the rigidity of the connection itself: it makes no sense since if you implement a proper rigidity of the connection in the model, elements are still there and will deform again. This means that you would obtain too high deformation from it.

On another hand, you have to take into account the deformations of elements you will remove from the model. This means that if we would like to add a support for our cantilever instead of perpendicular beam (to simplify the model), then we should take into account the rotation of the perpendicular beam, simply because it won’t be in the model any longer, and it’s rotation is real and should be represented somehow in the design (here as a part of connection rigidity).

When calculating deformation in connection rigidity design, you should only take into account the influence of elements that will be later removed from the model for simplification. Other elements should be treated as rigid, since their deformations will be calculated in the new model again, and you do not want to duplicate the influence.

Rigidity of the support

Firstly, let’s assume that the goal here is to calculate the rigidity of the support of the cantilever. Then the rotational stiffness of the perpendicular beam is important (since the beam itself will be deleted from the final simplified model). Based on the calculations we have made it is easy to calculate the bending moment applied to the connection. For cantilever, this is a very simple static scheme, and hand calculations are sufficient. Applied force is 1kN, arm for this force is 1.175m, so the applied bending moment is 1.175kNm.

Angle of rotation is also known (simple trigonometry, length is 1.175m, one end moved down for 6.9mm so tan(6.9 / 1175) =  0.350 → angle is 0.336deg (or 0.00587 rad). This allows for calculate the connection rigidity as 1.175 / 0.336 = 3.50 kNm/deg (or 200.1 kNm/rad). With such rigidity we can substitute everything with one support as shown below:

Note that the deformation is 7.1mm, as in the model with a real cross section of the cantilever.

Rigidity of the connection

If we would like to implement a hinge with a proper rigidity to the model containing all beams from the previous example, then we would have to exclude torsional deformation of the perpendicular beam. Since in the final model this perpendicular beam also will be present, it will have a chance to deform, and as such taking that into account in connection rigidity would be wrong. We can either change the perpendicular beam for a rigid element or simplify the model by removing the entire beam and supporting the stiffener on its edges that will be welded to the beam. The second approach was chosen and the model is below:

You can clearly see that the torsional deformation of the beam has a really big impact on the total deformation. Without it, only a small fraction of deformation is achieved with the identical load. Also, note that a rigid element is used instead of a cantilever beam as previously.

With a value 0.4 (so basically anything between 0.35 to 0.45) accuracy of estimation would be small. It is best to increase the number of digits software shows as an output, and with that, the more accurate value can be achieved (0.3570mm in this case):

At this point, we can simply calculate the rigidity of the connection itself, as we have done previously. The outcome now is: 3867 kNm/rad. Setting this up allows for receiving correct deformations of the model, that consists of only 2 beams without any surfaces (but with the rigid member that allows for an eccentricity of the connection to be implemented):

Since the connection has a proper rigidity, we do not have to model it with plate elements, simply one “number” is sufficient for the outcome to be correct. This method is especially important if there are a lot of connections like that in the model – since it can greatly reduce computing time and increase the accuracy of the design.

At this stage, it is quite easy to forget about the slippage in the connection. We have the rigidity, but it happens only when screws actually are in contact with the edges of the holes in both elements (stiffener in perpendicular beam and web in cantilever). We have previously discussed how to calculate this value – in this example, the free rotation angle is 1.1721 deg (0.0205 rad). This means that the connection has a certain free rotation, before the rigidity we have just calculated will be present. This can be set in nonlinearity settings of the hinge connection, if done properly this will give such results in this case:

In such simple example, it is easy to verify the outcome by hand calculation. Maximal deflection now is 31.2mm, and previously it was 7.1mm. The difference is 24.1mm, as much as the end of the rigid 1.175m beam will move if rotated by 0.0205 rad.

Summary

Today I have shown you the method that I use when I have to calculate the rigidity of the connection. As I mentioned, this is only a simplified solution with certain drawbacks, but it is very time-effective. As can be seen in this example, if slippage is possible this cannot be omitted, as it will have a big impact in design. In the following posts, I will discuss how the rigidity of the connections impact static designs and when connection rigidity is important in design.

Of course, this simplified method won’t work everywhere, since in end-plate connections you need to model compressive contact somehow, prestress of screws should be addressed, etc. There are more advanced models one can use, and I will get there eventually (perhaps even create a course on how to deal with that).

Hey! I have a surprise for you : ) If you are interested you can subscribe below to get my free FEA essentials course.