Effect of an opening in the punching calculation

This is not the first article I write about punching shear calculation. In the article “Contribution of longitudinal reinforcement in punching resistance” I discuss of the importance of the longitudinal reinforcement when calculating the resistance of a slab against the punching failure. In this article, I would like to discuss the effect of an opening in the slab on the shear distribution and its punching resistance. It is also discussed the EC-2 recommendation of reducing the resistant control perimeter due to the presence of this opening and the effect of a moment in the punching calculation (so called beta value). The recommendations of the EC-2 are checked against a FEM model (Ansys model).

Model presentation

For this article, we will use a simplified ANSYS FEM model composed of one 0.5mx0.5m column in the middle of a simply supported 0.5m thick slab with a size-varying opening at 2m from the centre of the column. The model will be loaded only with its self-weight. The following picture shows an example of the FEM element model under the self-weight of the slab:

Figure 1. Finite Element Model (FEM) – Model presentation

To avoid as much as possible the influence of the perimeter supports, they have been modelled at a distance of 10m from the column.

The following figure shows a top and bottom view of the FEM model mesh, with the distances and elements. The column is modelled using Ansys BEAM44 elements and the slab SHELL43 elements. The size of the slab elements is 0.2mx0.2m but this mesh does not influence the results if we get far enough from the column node (central node).

Figure 2. Finite Element Model (FEM) - Mesh 

Studied cases

To study the influence of the opening in the shear distribution, we will increase the dimension of the opening (always square opening). The following figure shows the nine cases studied in this article (the size of the opening is given in bolt black letter on the top of each studied case):

(*) The column is placed in the middle, below the slab

Figure 3. Nine studied cases – Variable opening size 

Punching shear – Control perimeter

As we know, the punching shear is produced in a theoretical failure perimeter, called "reference control perimeter - u1”. The EC-2, in the section §6.4.2, provides the definition of this reference control perimeter. This control perimeter is placed at a distance 2d from the support footprint, except if there are punctual loads of stiff areas close to the support. The following picture provides, for our particular case, this control perimeter u1. In this article, as a simplification, d is considered as 0.9*h = 0.9*500mm = 450mm (2*d=0.9m).

Figure 4. Reference control perimeter shape

The point (3) of the section §6.4.2 of the EC-2 explains that this control perimeter is reduced in the presence of an opening.

(*) NOTA: It is noted that in this EC-2 particular section (§6.4.2 (3)), the EC-2 refers to the “control perimeter” and not the “reference control perimeter u1” while the rest of the section §6.4.2 always refers to the “reference control perimeter u1”.

The following picture, extracted from the EC-2, explains how to reduce the control perimeter in the presence of an opening:

Figure 5. U1 control perimeter reduction - EC-2 Section §6.4.3

As we are studying the impact of the opening on the perimeter, we have to consider this reduction in our hand-made calculations, to check it against the FEM results. The following picture shows an example of how the control perimeter U1 is reduced (U1*) to consider the influence of the opening at a distance less than 6d. It is reminded that d in this calculated is considered as 0.9*h = 0.9*500mm = 450mm.

Figure 6. Control perimeter reduction based on EC-2 (§6.4.3) 

Punching shear – Stresses and parameters calculations

According to the EC-2, section §6.4.3 (3), the calculation of the shear stress produced by the punching force is calculated as follows:

Figure 7. Punching shear stress - EC-2 Section §6.4.3 (3)

Where the VEd is the punching shear force, Ui is the control perimeter and the d is the effective depth, calculated as 0.9*h (where h is the thickness of the slab, 500mm). The beta value is calculated according to the following formula, extracted from §6.4.3 of the EC-2:

Where the W₁ value is calculated as follows:

Reading the EC-2 It is not totally clear if the u1 perimeter and the W1 value has to be calculated using the reduced control perimeter u1* (ui in the formula above) or the whole control perimeter (without reduction). In this article, we have calculated the beta value considering the reduction of the perimeter in both, u1 and W₁ values (conservative approach). The beta value also depends on the ratio M/Fz of our calculation. The bigger the opening is the more unequal is the shear force arriving to the column, producing a bigger bending moment in the column and, therefore, a bigger beta value.

FEM Model calculation – Column results and parameter calculations

The following table summarises, for the different nine studied cases, the compression force and bending moment resisted by the column extracted from the FEM model, the reduced control perimeter and the W₁ and beta values calculated as explained above:

Table 1. FEM results – beta value and reduced control perimeter

(*)Note that, the bigger the opening is, the smaller is the total load applied on the model as the quantity of mass (accelerated to produce the self-weight load) is reduced. However, a bending moment will appear due to the unequal load due to this opening.

Observing the previous table, we can raise the first conclusion: The impact of the opening on the beta value is negligible. However, the reduction of the control perimeter cannot be neglected.

FEM Model calculation – Shear stresses

To understand the shear stress distribution, we will focus on the surrounding column FEM model area, far from the perimeter supports. It is also extracted the shear forces in a simplified punching perimeter with a radius R=(half of the column + 2.d). The following figure shows the part of the model that will be studied and the shape and location of the simplified punching perimeter. The shear force is calculated as (Fx²+Fy²)^0.5. Note that the colour scale has been adapted to present in grey those elements around the column that do not need to be studied (mesh size dependent elements, too close to the column):

Figure 8. Studied FEM areas – Example

Before focusing on the area close to the column, where the punching failure will be produced, we are going to look to the global shear stress on the slab on each case and the effect of adding an opening:

Figure 9. FEM Results – Shear stresses

When the opening appears, there is a shear stress perturbation around it, reducing the shear force going to the column and increasing the shear force going to the right/top/bottom supports.

The following pictures show a zoom of the area close to the column and provide the control perimeter where the Ansys results will be extracted (black thin circumference around the column).

Figure 10. Detailed FEM Results – Shear stresses

The shear force is not uniformly distributed through the entire perimeter; therefore, it seems reasonable to consider a control perimeter reduction. Part of the effect of a non-uniform shear distribution should be reflected by the beta coefficient but, as presented in Table 1, the beta coefficient does not represent completely this effect.

We are going to check in detail the effect of the opening in the shear stress distribution around the punching control perimeter. We have obtained from the FEM model the shear forces in the simplified control perimeter and compared with the EC-2 reduction proposal.

The next figure shows, on the left, the reduction of the control perimeter for the 0.8mx0.8m opening and presents, on the right, the shear forces in the FEM simplified control perimeter with the opening projection lines to compare the shear stress diminution in the opening projection area:

Figure 11. FEM Results – 0.8mx0.8m opening

As in the previous figure, the next one presents on the left, the reduction of the control perimeter for the 3.2mx3.2m opening and presents, on the right, the shear forces in the FEM simplified control perimeter with the opening projection lines to compare the shear stress diminution in the opening projection area:

Figure 12. FEM Results – 3.2mx3.2m opening

In the same way, we have obtained the Ansys shear forces on each punching perimeter for all the studied cases. These punching perimeters, as well as the opening projection lines (presented in black), are provided in the next figure. The stress scale has been adapted to compare all the cases directly.

Figure 13. Punching perimeters – Results

It is observed that the opening produces a reduction of the shear force transmission. However, the EC-2 seems to be, in this case, too conservative as it considers that a part of the perimeter does not contribute at all to the shear force transmission, while, in reality, it does. In fact, it is verified that in the opening affected area the minimum shear force is, in most of the cases, more than 50% of the maximum shear force on the rest of the perimeter and it is really far from being zero. The following table summarises the results obtained using the EC-2 and the FEM results:

(*) The perimeter in Ansys where the results are obtained has been simplified to a circumference placed at a distance of (half of the column + 2.d)

Table 2. Results – Theoretical and FEM results

As the shear force depends on the finite element mesh of the FEM, the results in the control perimeter are not uniform. This is the reason why there are two FEM values (maximum and minimum value) for the case without opening (while, in reality, an unique constant value should appear). For the other cases, the minimum Ansys value corresponds to the part of the perimeter closer to the column and the maximum one to the rest of the perimeter, where the maximum is produced.

Figure 14. Theoretical and FEM Results – Graph of shear force (*)

(*) As explained above, it is noted that the maximum value obtained from Ansys, for the case where there is no opening, is slightly bigger than the case calculated manually following the EC-2. This is due to the control perimeter simplification. The control perimeter is not exactly a circumference as shown in Figure 4, therefore, for the case without opening, the applicable Ansys result is “Ansys minimum value”, 218kN/m, which almost matches with the EC-2 calculation, 215kN/m .

Conclusions

·         For the case without opening the results obtained with the FEM model matches almost exactly with the EC-2 value

·         The EC-2 beta value, whose purpose is to consider that due to the presence of a bending moment the shear force is not uniformly distributed in the control perimeter, is almost not impacted by the presence of an opening in the studied cases (value of 1.06, meaning a, increase of 6% of the shear force produced only by the axial force)

·         The EC-2 control perimeter reduction proposal as shown in Figure 5 seems quite real in the “shape” but not in the “value”. This means that there is a strong shear reduction in the area given by the EC-2 (“shape”) but considering that the perimeter does not contribute at all seems too conservative (“value”). It is verified that, for the studied cases, at least the minimum shear force is 45% of the maximum shear force, far from the zero value from the EC-2

·         In the studied cases, the bigger the opening is, the more conservative the EC-2 seems to be. However, this conclusion cannot be applied for all the cases as we have studied a very particular case with a simplified geometry. The engineer is in charge of studying each case in detail and to properly apply the EC-2 recommendations.

It is reminded that this is a just a simple case and that the engineer is the responsible of his calculations regardless of the information here above presented.