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Corbels – Simplified design

January , 30th 2024 | Author: Prontubeam (@Prontubeam_en) Read: 949 times

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The terms Brackets or Corbels are often used interchangeably. Basically, corbels are a special type of cantilever beams, connected to columns. Logically, we will discuss this in more detail later in this article. They are commonly used in industrial buildings, for the installation of cranes and in precast reinforced concrete structures (see Figures 2 to 4).

Typical-Corbel-example

Figure 1. Typical Corbel example

Moment and shear transfer at a bearing corbel

Figure 2. Moment and shear transfer at a bearing corbel

 

Figure 3. Columns with corbel for beam support.

 

Figure 4. Columns with corbel for beam support.

In this article, we will give some comments and recommendations that we use in our office and in our professional practice regarding corbels, in a simplified way, presenting expressions to obtain the dimensions that they should have in our design, and the amount and tentative reinforcement arrangement that should be placed.

 

Brief Introduction

Let's start, for the moment, by commenting a little on what they are, their use in engineering and the way they should be analyzed and designed:

Let's assume a structure, made up of a reinforced concrete column and a rectangular beam, connected to it, and made of the same material:

Figure 5. Column with cantilever beam 

 

In this case, the cantilever measures 2.3 m (2,300 mm) from the edge of the beam to the edge of the column, and is 0.60 m (600 mm high). Let's call "h" the height of the beam, and "a" the length of the cantilever. The ratio between the two, for our case, would be:

In this case, given a point load at the far end of the element, we could obtain the value of the bending moment (at the column face), simply by using the equations of statics. Once this is calculated, all that remains is to perform the conventional design of the reinforced concrete beam for the determination of its reinforcement.

However, if the recently mentioned ratio "a/h" is less than or equal to 1, it is no longer possible to carry out the design in the same way as in the previous case. Statics will always be valid. But the problem lies in the non-compliance with one of the strongest hypotheses used for the flexural design of reinforced concrete beams: "Maintenance of plane sections after deformation (Bernoulli-Navier)".

For instance, for a case like the following, where we will have:

Figura 6. Column with corbel. 

 

In such cases, we find a particular crack pattern, as shown in the following image:

Figure 7. Cracks and stresses in cantilever [2]

In the example shown in Figure 8, a reinforced concrete column and corbel support a pretensioned concrete beam. The figure shows there are 10 different force vectors in the connection as follows (designated with letters, “A” through “K”):

A: diagonal compression strut in corbel

B: horizontal compone reaction to force at A

C: vertical component reaction to force at A

D: internal diagonal resultant to force B and C

E: diagonal compression strut in beam

F & G: horizontal component reaction to force E

H: tension field to force E & F

J: horizontal friction force caused by relative movement of beam and corbel

K: horizontal membrane reaction to beam rotation due to eccentric prestressing

Figure 8. Force vectors 

 

Analyzing this in more detail, we can comment on the characteristic failures of this type of structure. The most common are the following:

Figure 9. Cracks and Failures. Tensions and compressions in elements [2].

 

·         Type 1: direct shear failure at the interface between the corbel and the element on which it rests.

·         Type 2: yielding of the reinforcement tensioned by the moment and direct traction.

·         Type 3: Bearing of the compressed internal concrete strut.

·         Type 4: localized failure by bearing or shear under the loaded area.

It is always important to understand the types of cracks and failures of structures in order to be able to develop resistant mechanisms and expressions consistent with this at the time of the design of our structure.

From all this it follows that, for corbels, we require different formulations to those used in the conventional flexural design of reinforced concrete elements. The treatment that is usually performed, accepted worldwide, is a methodology of great simplicity and very good results. The famous "Strut - Tie" method. Which consists of the conformation of resistant mechanisms in the interior of our structure, using compressed (strut) and tractioned (ties or tensors), pinned elements, similar to what is analyzed with trusses. In this article it is not our intention to go into this subject, but we can close the idea, mentioning that it is the theory used for the famous "D" type zones (discontinuous), which differ from the B zones (Bernoulli's, where the hypothesis of flat sections are valid) by the methodology used for their design.

In the following, we will present some useful tips for estimating the dimensions, quantity, arrangement and diameter of the reinforcement in the corbels.

 

Simplified design

 

Personally, I have encountered some situations in which I have to make a quick, simple and effective pre-design of the element.

At more than one moment, the professional may be asked to make an estimate of dimensions and assembly scheme for, for example, coordination with other disciplines, initial calculation to anticipate costs, etc. Other times, it is necessary to control the design made by someone else (either someone who is in our charge, or simply the supervision of a project made by a third party). In these situations, it is when the recommendations that we will give can be useful. This should be taken as an initial estimate, or pre-dimensioning. Remember that the Strut and Tie method must be used (with all that this implies).

It should be noted that, in this opportunity, we are talking exclusively about the corbel and we will not pay attention to the column (nor to its additional longitudinal reinforcement to be considered, as a result of the moment produced by the eccentric load on the corbel).

Without further ado, let's start with some dimensional recommendations:

 

Corbels geometry

Note that in order to consider that these structures behave as D (discontinuous) regions, the ratio between "a" and "h" must be:

Speaking with a little more propriety, we call "a" the distance between the point of application of the load, and the plumb (vertical face) of the column. The height "h" is the total height of the corbel.

Figure 10. Distance from load to column head. 

 

The minimum area (in plan) to ensure that the requesting stress is less than the allowable stress is:

Where:

fck: concrete characteristic resistance.

Ps: service vertical load.

Figure 11. Axonometric. Plan dimensions of corbel. 

 

Therefore, it must be satisfied that:

Dimensions in elevation (view):

As mentioned above, in order to consider these structures as corbels, we must be faced with the following condition:

In the case of a corbel with a sloped edge, the following ratio is recommended:

Figure 12. Elevation. Corbel Dimensions. 

 

Reinforcement arrangements

 

For the corbel reinforcement we can consider two types of rebars. These may have their variants in terms of shape, but in general they are: Vertical reinforcement, "As" and reinforcement in the form of horizontal stirrups (parallel to the column stirrups), "Ash":

Figure 13.  Elevation. Types of reinforcement in corbels. 

 

Figure 14.  Axonometric. Types of corbel reinforcement. 

 

Let's start with the approximate calculation of the "As" reinforcement:

Figure 15.  Elevation. Load and angle of application. 

We make a first simplification, in terms of the angle shown in Figure 15.

Θ: Angle between load and axis of column reinforcement

The value of this angle, in this instance, can be estimated at:

Therefore:

Then, the force that we will use for our calculation (considering a factor of 1.5) will be:

Pu: Estimated increased load

With this, we are now in a position to determine the required area of vertical reinforcement in the corbel, "As":

Being:

fyk: Steel yield stress

fyd: steel design stress

With this, we can distribute the steel area obtained in a certain amount of rebars, along the corbel, as shown in the following image:

Figure 16.  As reinforcement. 

Generally, the area obtained from the reinforcement "As" is uniformly distributed over the entire width "c". It must be anchored from the column axis, as shown in Figure 18.

Then, to estimate the reinforcement "Ash", we can use the following simplified expression:

Figure 17.  Ash reinforcement. 

 

This value is a minimum. From there, we can place higher amounts of stirrups in the corbel. For our calculation, we count two legs for each stirrup.

It is recommended to distribute the area of the "Ash" stirrups over a length that is at least 2/3 of h.

Figure 18.  Anchoring and reinforcement distribution. 

 

With this, we have fairly approximate and useful dimensions of the two main reinforcements of the corbel.

To this we must add the checks of the nodes when applying the Strut-Tie method. But that would be a matter of more exact (and always necessary) checks, which are not the purpose of this presentation.

We hope to have been of help with these tips.

Greetings.

 

References

[1] Números gordos en el proyecto de estructuras – Juan Carlos Arroyo Portero.

[2] Introducción al cálculo de Hormigón Armado -  Ing. Rodolfo Orler.

[3] Hormigón armado y pretensado -  Carlos Larsson.

[4] Método de bielas y tirantes – ACHE 2003.

[5] Eurocódigo 2.

[6] EHE 1998.

[7] Precast concrete structures – Kim Elliot.

 

 

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Joaquín Guraiib . Civil Engineer Structural Engineer & Project Manager Structural analysis | Architectural Design | BIM Co-founder & Manager at Teknik-id Linkedin profile
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