Support thickness – Bending moment reduction and bending critical design section

September , 5th 2023 | Author: Prontubeam
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When
doing a **hand-made calculation of a structure** or when **modelling it in
a finite element program**, it is needed to do some **modelling assumptions**.
For example, it is quite common to **model** the structure at the **neutral
axis**. This type of modelling will end up with bigger spans than the real
ones and loose of stiffness for some cases. However, the modelled span and the
design bending moment can be considered differently, different from the neutral
axis, in function of the support conditions. In this article it will be
summarised the **EC-2 requirements/recommendations for span modelling** and **bending
moment calculation**. It will also be **compared against hand-made calculations
and non-linear ones**, using IDEA StatiCa (thanks to Construsoft for their
support).

**Figure ****1****. Span and bending moment
calculation (left) – Non linear results (right)**

The
**first point of this article** is to present and **summarise the EC-2
requirements for the effective span definition** provided in **section
§5.3.2.2** and presented in the following image:

**Figure ****2****. Section §5.3.2.2 (1) of
EC-2 – Effective span definition**

The following figure defines the different parameters required to calculate the total span, in function of the different support conditions:

**Figure ****3****. Figure 5.4 of section
§5.3.2.2 (1) of EC-2 – Effective span definition**

In general, except for the case where a bearing support is provided, the span has to be considered as the distance between the inner faces plus a value which is calculated as the minimum between the thickness of the support and the thickness of the connected beam/slab.

Additionally,
there is an important point in the same section §5.3.2.2, where it is specified
that when the **connection** between the slab and the supports it is **monolithic**,
**the critical moment has to be calculated at the face of the support**:

**Figure ****4****. Section §5.3.2.2 (3) of
EC-2 – Critical design moment section**

This is only applicable when the connection between the supports and the slab is monolithic.

**Figure ****5****. Section §5.3.2.2 (3) of
EC-2 – Effective span and critical design moment section**

In any case, the following rule could be applied:

**Figure ****6****. Section §5.3.2.2 (4) of
EC-2 – Bending moment reduction where no rotation restraint produced by the
support**

This means that if the modelled span is from axis to axis and the supporting support does not provide any rotation restraint, design bending moment could be reduced following the above formula. Sometimes it is difficult to prove that the support does not provide any restraint to the rotation as it depends on the support stiffness. The EC-2 provides as an example a wall but depends, for example, on the height and thickness of this wall.

**Calculations performed: spans, model and loads
definitions**

**a) ****Possible spans**

As explained above, different spans can be defined in function of the support conditions:

· Axis span: distance between the load and the support axis or the distance between the two support axis. This is the most common one, used generally for finite element models

· EC-2 span: defined following §5.3.2.2. It is similar for the previous one when the thickness of the two connected elements (H and B) are equal but differs when these are different

· Internal face span: distance between the load and the support internal face or the distance between the two supports internal faces. This is important when there is a monolithic connection between the slab/beam and the support

These spans are reflected in the following figure:

**Figure ****7****. Different span definition
used in this article**

In this article we will calculate the different moments, for a cantilever beam, produced by the 3 possible spans presented above and compare them with the results produced by the non-linear calculation.

**b) ****Load cases**

Two different loads will be used: a punctual one and a uniform distributed one:

**Figure ****8****. Load cases definition**

**c) ****Model geometry, reinforcement and materials**

The calculated beam is a 0.5x0.4 beam reinforced with 4C20 on the top and bottom face, closed stirrups C12@200 for the shear, supported by a variable width support (0.3m, 0.5m, 1m and 1.5m):

**Figure
****9****. Geometry and reinforcement
definition**

A full non-linear for the concrete (C40, fcd=fck/1.5=26.6MPa) and steel (fyd=fyk/1.15=434MPa and k=1.08) materials, with yielding cracking and crushing properties. This will be performed with the software IDEA StatiCa Detail. The following figure presents the material laws, for further details refer to IDEA StatiCa material definition:

**Figure ****10****. Non-linear material laws**

** **

**Results**

**a) ****Results: Interaction diagram**

The
first point is to calculate the maximum resisted bending moment of the beam considering
the geometry and reinforcement provided above and the EC-2 diagram
simplifications (equivalent rectangular stress distribution (C40, fcd=fck/1.5=26.6MPa)
and reinforcement considered yielded (fyd=fyk/1.15=434MPa, k=0). As presented
in the following figure, calculated with the *Interaction
diagram online tool of Prontubeam*, the maximum
resisted bending moment, without any tension/compression on the section, is
228kN.m:

**Figure ****11****. Interaction diagram for the
studied beam – 228kN.m Maximum resisted moment with N=0 **

**b) ****Results: handmade calculations**

The following table presents the hand-made calculated moments for each of the 3 spans presented above for each of the support thickness ad load configurations. These are compared against the resisted moment 228kN.m, to calculate the ratio between the acting moment and the resisted one:

**Figure ****12****. Hand-made calculated
moments for each load type, support thickness and span definition **

(*)
Nota: EC-2 moment refers to the moment calculated following the *EC-2
§5.3.2.2 figure 5.4* span definition

These results will be, later, compared against the non-linear model calculation to extract some conclusions.

**c) ****Results: Non-linear model**

The following picture presents the results obtained for the different cases calculated with the non-linear model. In particular, presents the ratio between the stress on the reinforcement for the given load and the maximum resisted stress (fyd=1.08*500/1.15 – Refer to IDEA StatiCa material law, bilinear with inclined upper branch):

**Figure ****13****. Non-linear results –
Punctual load – Ratio stress/limit stress on the top reinforcement **

**Figure ****14****. Non-linear results –
Distributed load – Ratio stress/limit stress on the top reinforcement **

One common point appears in all of them: the maximum is produced at the internal face of the support, following §5.3.2.2 (3) of the EC-2. These results are summarised in the following table, compared with the hand-made calculated moments, for each of the considered spans:

**Figure ****15****. Comparison - Non-linear
results – Hand-made calculated results **

As it can be observed again, the results confirm that the design bending moment has to be considered at the face of the support. The differences between the hand-made results and the IDEA StatiCa ones comes from the fact that the maximum resisted moment has been calculated equal to 228kN.m considering a bilinear steel law with horizontal plasticised branch while IDEA StatiCa permits to reach fyd=1.08*500/1.15 with plasticised strains.

**Maximum bending moment for the 1m thick support case –
Theoretical vs. Non linear**

The maximum resisted bending moment and its applied load is calculated using IDEA StatiCa and compared to the maximum one calculated with the interaction diagram.

Two values are calculated, one considering the concrete and steel with non-linear material laws presented below and a second one considering the classical bilinear material laws.

**Figure ****16****. Non-linear results –
maximum admissible distributed load considering non-linear material laws**

The following results are found:

**Figure ****17****. Non-linear results –
maximum admissible distributed load for both material possibilities**

The following table presents the calculated bending moments for each of the spans when applying both loads obtained with IDEA StatiCa:

**Figure ****18****. Hand-made and non-linear
results – Comparison for the maximum limit load**

As it can be seen, when the load of 33.6 kN.m is applied, which corresponds to the case where bilinear material load is considered, the bending moment at the face of the support is 236kN.m which is quite similar to the maximum resisted one calculated with the interaction diagram (228kN.m). It is proved that if the bending moment is calculated at the axis of the support or considering the theoretical span given by the EC-2, this gives an over-dimensioned bending moment which will not exist in reality if the connection between support and slab is monolithic. This confirms the EC-2, section §5.3.2.2 (3).

** **

**Conclusions**

The following conclusions are extracted from this article:

· In general, except for the case where a bearing support is provided, the effective span has to be modelled as the distance between the inner faces plus a value which is calculated as the minimum between the thickness of the support and the thickness of the connected beam/slab.

· In section §5.3.2.2 it is specified that when the connection between the slab and the supports it is monolithic, the critical moment has to be calculated at the face of the support. This is only applicable when the connection between the supports and the slab is monolithic

· In this article it has been proved that, as proposed by the EC, when the connection between the support and the slab/beam is monolithic, the bending moment has to be calculated at the internal face of the support

· If the modelled span is from axis to axis and the supporting support does not provide any rotation restraint, design bending moment could be reduced following the §5.3.2.2 (4) formula. Sometimes it is difficult to prove that the support does not provide any restraint to the rotation as it depends on the support stiffness. The EC provides as an example a wall but depends on the height and thickness of this wall

It is noted that all the information here presented is to be considered for information only. It is the engineer in charge of the calculation who needs to verify the calculation and ensure that is complient with the project standards.

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