Modelling a lintel - ¿Beam or Shell?

August , 12th 2018 | Author:
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More than once we have
encountered the problem of modelling a lintel with finite elements (FEM) and **we
have asked ourselves which type of element would be more suitable to model it. Beam
elements or shell elements?** In this article we study in detail the
difference of modelling the lintels with beam elements or with shell elements.
We also study, in the case of using beam elements, different possible cases and
which one would represent better the reality.

Figure 1. Different studied cases – Results advancement

To answer this question we have modelled a concrete wall in several ways, first with shell elements (elements Shell43 of Ansys) and then with beam elements (Beam44 of Ansys). The beam elements have the geometry of the lintel, that is to say, they are elements with a section of 1m of high and 0.5 of width. The following image shows the geometry of the model:

Figure 2. Geometry of the model – Concrete wall

The difference between the different cases modelled with beam elements is the depth that we have introduced the beam elements inside the shell elements looking for the correct behaviour in the embedment (correct embedding moment). The following table shows the studied cases:

Figure 3. Studied cases - FEM models

All the cases have been loaded in the same way: A lateral load applied on the 5 upper nodes on the left side of the wall with a load of 1kN at each node. In total there are 5kN acting on one side of the wall.

Figure 4. Applied loads

**Lintel FEM model using beam elements without embedding
the elements inside the wall**

What this article aims to demonstrate is that it seems not to be correct to model the lintel with beam elements by joining it to the wall through one single node instead of introducing a certain number of elements at each end, in the connection between the lintel and the wall. The following image shows the bending moments in the lintel in this case, without introducing the beam elements into the lateral walls.

Figure 5. Results of the lintel modelled with beam element with no embedment at each end. Bending moments

There is one important thing to
be highlighted in this first studied case: **THERE ARE NO BENDING MOMENTS IN
THE LINTEL.** **We can imagine that this result does not seem correct... Let's
explain why**.

**Lintel FEM model using shell elements**

The first thing we do is to solve the shell element model and, through the Ansys FSUM command in different sections, we obtain the bending moments law on the lintel. We see that the lintel presents bending moments of 1025N.m at the left end and -1052N.m at the right one.

Figure 6. Results of the lintel modelled with shell element. Bending moments

We will compare this result with those obtained when we model the lintel with beam elements. The following image summarizes the results for each of the first 4 cases studied in which the lintel has been modelled with beam elements. As the scale is not totally readable, is has been provided the maximum moment in each case. For all the cases it occurs at the left side of the lintel:

Figura 7. Results using beam elements for the lintel modelling. Bending moments

It is verified that, as we embed
the beam elements inside the wall, the moment at the connection increases. The
question is: **which is the correct beam embedment length inside the shell
elements’ wall? **To answer this question we have made one last case, case 5,
introducing the beam elements beyond the middle of the wall, more specifically,
1.75m at each end:

Figure 8. Results in the lintel modelled with beam elements embedded 1.5m at each end. Bending moments

We see that the moment has barely increased with this new modelling (from 1033N.m to 1057N.m).

In view of the results it seems that when we embed the beam elements a distance of 1.25m (4 elements), the moments in the lintel modelled with shell elements and modelled with beam elements are remarkably close:

To determine the suitability of the results we have made an additional check. We have simplified the model by approaching it to a typical frame whose results are known under concentrated lateral load:

Figure 9. Frame model – Simplified model

Figure 10. Frame results under lateral load

The following table summarises the results of the main 3 cases:

Looking at the results we know that if we do not model embedded beam elements inside the shell elements we are not capturing the rotation in the upper part of the wall, in the junction between the wall and the beam, and it is this rotation which produces the bending moments. The following image shows the deformed structure where it is seen that the lintel does not accompany the rotation.

Figure 11. Model deformation under the studied load

We are going to study one last interesting case. If we assume that we accept Navier's theory that a plane section of a beam remains plane after its deformation (* we will see this point later) we need the nodes of the section to behave "rigidly", forcing the condition that the section remains plane. To do this, we are going to model infinitely rigid beam elements in the section where the real lintel would be connected to the rest of the structure.

Figure 12. Additional study – Rigid beam elements in the junction

We are going to study this configuration and we will compare it with the results obtained previously. The first thing is to check the bending moments on the lintel:

Figure 13. Lint beam model results modelled with rigid connection elements. Bending moments

Interesting result, we get 1190.8kN for the bending moment which is very close to the calculated simplified frame one and also close enough to the result with the beam elements embedded 1.25m inside the wall.

(*) First of all, a small clarification has to be done: We get back to the hypothesis that the section remains plane after the deformation. We have discovered that in this case this hypothesis is not fulfilled at all. It is possible that the span of the lintel (1.5m) compared to the height (1m) it is not enough to consider the linear elastic distribution in the section.

Figure 14. Lintel section deformation on the junction lintel-wall. Plane section remains plane?

To finish this study we will compare the stresses in the three cases that have turned out to be the most similar and the case of the not-embedded lintel:

· Lintel modelled with shell elements

· Lintel modelled with beam elements – Embedded 1.25m length (half of the wall)

· Lintel modelled with beam elements – Rigid elements representing the connection lintel-wall

· Lintel modelled with beam elements – Beam elements not embedded

__Stresses in X
direction__

Figure 15. Stresses in X direction (N.m)(*)

**(*) The 4 pictures have the
same stress scale**

__Stresses in Y
direction__

Figura 16. Stresses in Y direction (N.m)(*)

**(*) The 4 pictures have the
same stress scale**

It is generally the model with rigid elements the one that best represents the distribution of the efforts and is the model of the lintel without embedding the worst one.

**Vibration frequencies**

Something that usually disturbs us when we simplify a model is its behaviour against earthquakes, more particularly their vibration frequencies. Here below are presented, highlighted in red, the frequencies related to the modes of vibration in the direction of the load (horizontal, X direction) for the 3 most relevant studied cases.

Figure 17. Vibration frequencies - Comparison

** **

**CONCLUSIONS**

First of all, we want to emphasize that this is a study of a very simplified reality and the results are not applicable in all the cases. It is the engineer's responsibility to verify the applicability and consistency of the results in each specific case.

Below it is summarised results obtained:

• We have seen that the model in which the beam elements are not embedded inside the shell elements on both sides does not properly represent the reality

• The model with shell elements and the simplified frame calculation show reasonable similarity

• The model in which the beam elements are embedded within 4 shell elements (1.25m) is the model that most closely approximates the embedding moments of the model with shell elements.

• From a certain value (4 shell elements in this case), embedding more beam elements inside the shell ones do not represents better the reality. After a certain embedment distance the behaviour of the lintel does not change significantly and may start to be unreal

• The model that uses rigid elements in the connection seems to represent reasonably well the reality for both, the embedment moments and stress in the whole model. Even so, we recommend looking for more information about this type of modelling

• When we model the lintel with beam elements, the vibration frequencies move away from those obtained with the shell element model. As the Hz of the vibration frequencies increase, the difference is bigger. It is the FEM model with the rigid junction whose vibration frequencies are more similar to the ones of the shell element model

I hope the results shown in this article could guide the reader when modelling lintels but in no case should these results be taken as applicable to all models without further studies.

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