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Concrete Corbel Designer to EC2's banner
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Concrete Corbel Designer to EC2

Verified by the CalcTree engineering team on August 12, 2024

This calculator designs a reinforced concrete corbel using strut and tie method. It provides the required main tension steel and horizontal (or vertical) links in the corbel. It uses the strut-and-tie method for corbels provided in Reinforced Concrete Design to Eurocode 2 by J. H. Bungey, R. Hulse, and William Henry Mosley.
All calculations are performed in accordance with BS EN 1992-1-1: 2004 Eurocode 2: Design of concrete structures - Part 1-1 : General rules and rules for buildings. This code is typically referred to as "EC2".

Calculation

Assumptions

  1. Strut-and-tie system is used in the analysis.
  2. Flexible bearing plate is considered, hence
    
    is measured from face of column to the vertical force,
    
    .
  3. The horizontal (and/or vertical links) are provided as per UK NA and PD6687 publication, which is more onerous then that provided in Section J.3 of EC2.
  1. At the compression-tension node (under the bearing plate), the bearing stress is checked against the safe strut stress, however the concrete strut stress is not checked.
  2. The edge depth under the bearing plate is not explicitly checked, but should be
    
    .
  3. The compression strut is not checked for crushing.

Corbel symbols used in this calculator



Inputs

Load



FEd
:400 kN​



HEd
:{"mathjs":"Unit","value":80,"unit":"kN","fixPrefix":false}​



ac
:{"mathjs":"Unit","value":200,"unit":"mm","fixPrefix":false}​



Corbel Properties



h
:600 mm​



b
:{"mathjs":"Unit","value":350,"unit":"mm","fixPrefix":false}​



c
:50 mm​



Reinforcement

Main tension bars provided:


db
:20mm​



n
:3​

Preferred link size:


dl
:16mm​



Bearing Plate



t
:{"mathjs":"Unit","value":10,"unit":"mm","fixPrefix":false}​



b1
:{"mathjs":"Unit","value":350,"unit":"mm","fixPrefix":false}​



l
:{"mathjs":"Unit","value":120,"unit":"mm","fixPrefix":false}​



Material Properties

Concrete:


fck
:{"mathjs":"Unit","value":30,"unit":"MPa","fixPrefix":false}​



αcc
:0.85​



γc
:1.5​

Reinforcement:


fyk
:{"mathjs":"Unit","value":500,"unit":"MPa","fixPrefix":false}​



γs
:1.15​


Outputs

Dimension requirement check






d
:540​



Remark (i)
:OK!​



Bearing stress check

  1. 
    
    


σEd
:9.523809523809524​



σRd,max
:12.716​



Remark (ii)
:OK!​



Concrete strut check

  1. Strut angle,
    
    


θ
:61.194769848646466​



Remark (iii)
:OK! θ = 61 is used.​

  1. 
    
    


z
:389.1803992791089​



Remark (iv)
:OK!​



Main tension steel

  1. 
    
    


F'td
:299.9494120427436​



As,main prov
:942.4777960769375​



As,main req
:690 mm2 or 3-H20​



Remark (v)
:OK!​



Horizontal / vertical links

  1. If
    
    provide horiz links


ac/h
:0.3333333333333333​



Remark (vi)
:Provide horizontal links.​



As, per link req
:471.23889803846896​



As,link req
:4-H16 @ 75 mm​


Explanation

What is a corbel?

A corbel is a structural element provided whenever a horizontal member, such as a beam, is not directly supported by a column.
Concrete column with two corbels supporting beams

EC2 provides a couple different methods for the design of corbels:
  1. Shear approach - EC2 Section 6.2.2(6) or Section 6.2.3(8) - where the applied vertical shear force on the corbel within a certain distance from the column face can be reduced by
    
    for checking against the corbel's concrete shear resistance. See the referenced code clauses for more details.
  2. Strut and tie approach - EC2 Section 5.6.4, Section 6.5 and additional information in Annex J.3 - uses a truss analogy to simplify stress patterns in a concrete member into a triangulated model.
This calculator uses the strut and tie approach.

What is a strut and tie model?

We typically design structural elements based on Euler–Bernoulli beam theory which has the fundamental assumption that plane sections remain plane after bending. Strut and tie is an alternative approach to beam theory, used when non-linear strain distribution exists (e.g. deep beams, at supports). A common rule-of-thumb is to use strut and tie to analyse a concrete element when it's

.
The strut and tie model assumes the stress distribution in a structural element is resolved as a theoretical truss, consisting of a concrete strut in compression and two steel ties in tension formed by the longitudinal and shear reinforcement.
Strut and tie model of a corbel

As per EC2 clause 5.6.4, the forces in the elements of a strut-and-tie model should be determined by maintaining the equilibrium with the applied loads in the ultimate limit state.

How to design a corbel using strut and tie?

As per Annex J.3 of EC2, the corbel is considered as a short cantilever, and hence the strut and tie method is valid, when the distance

.
Strut-and-tie system diagram of a corbel

Considering the diagram above, the ultimate vertical load,

will be resolved into a strut and tie triangulated model which has components both resisted by the concrete and the main steel reinforcement.
The concrete resistance in compression is called a compression strut

and

is the angle of inclination of the strut. The tension resistance will be provided by the main steel reinforcement which acts as the horizontal tension tie

.
The equations for

and

are not explicitly provided in EC2. Design guides and textbooks often provide a detailed procedure of using strut and tie.
Explore the toggles below for derivation of the equations and the required design checks, based on textbook Reinforced Concrete Design to Eurocode 2.

Compression strut,



The compression strut,

is given by the following equation:

Fcd=νfcd×2(d−z)×b×cos⁡θF_{cd} = \nu f_{cd} \times 2 ( d - z ) \times b \times \cos θ
Where:
  1. 
    
    is the design stress of compression strut, where
    
    as per equation 6.6N of EC2.
  1. 
    
    is the effective depth of the corbel where
    
    is the cover to the main tension bar and
    
    is the size of the main tension bar
  1. 
    
    = breadth of the corbel
  1. 
    
    = angle of inclination of the strut
  1. 
    
    is the lever arm of internal forces, ; the corbel is assumed to consist of compressive and tensile zones separated by a lever arm
  1. 
    
    is the distance of the compression-tension node from the face of the column, because of the horizontal force
    
    
  2. 
    
    is the distance the horizontal force
    
    acts from the horizontal tie
  1. 
    
    = width of the concrete strut measured vertically
The above equation can be derived from the geometry of the below image.
derivation [adapted from Source]


Inclination angle of strut,



As per Annex J.3 of EC2,

is limited by

or

.
Angle

, can be solved by resolving vertically at point A in the image below.

FEd=Fcd×sin⁡θ=νfcd×2(d−z)×b×cos⁡θsin⁡θ=νfcd×(d−a′tan⁡θ)×b×sin⁡2θ=νfcd×d×b×(1−a′dtan⁡θ)sin⁡2θF_{Ed} = F_{cd} \times \sin θ= \nu f_{cd} \times 2 ( d - z ) \times b \times \cos θ \sin \theta \\ \hspace{2.72cm}= \nu f_{cd} \times (d - a' \tan \theta ) \times b \times \sin 2θ \\ \hspace{2.72cm}= \nu f_{cd} \times d \times b \times (1-\dfrac{a'}{d} \tan θ) \sin 2θ
Rearranging gives:

FEdfcd d b=[1−(a′d)tan⁡θ]sin⁡2θ \dfrac{F_{Ed}}{f_{cd} \ d \ b} =[ 1-(\dfrac{a'}{d}) \tanθ] \sin2θ
The equation above cannot be solved directly for

. This calculator determines

for different values of

and

. Then linear interpolation is used to find

for the actual value of

and

.
derivation [adapted from Source]


Tension tie,



The tension ties can be solved by resolving vertically at point A in the image below.
The initial force in the steel tie,

, considering the vertical load alone, and the resolved force

, considering both vertical and horizontal loads are given by:

Ftd=Fcdcos⁡θ=FEdsin⁡θcos⁡θ=FEd×cot⁡θandFtd′=FEd×cot⁡θ+HEdF_{td} = F_{cd} \cos \theta \\ \hspace{0.9cm}= \dfrac{F_{Ed}}{\sin \theta} \cos \theta \\ \hspace{1.05cm}= F_{Ed} \times \cot θ\\ \text{and}\\ F_{td}' = F_{Ed} \times \cot θ + H_{Ed}
Where:
  1. 
    
    is the ULS design vertical force on the corbel
  2. 
    
    is the ULS design horizontal force on the corbel
derivation [adapted from Source]

This calculator considers the following design checks for a corbel:

1) Dimension Requirement Check

The distance from the vertical load to the face of the column,

should be less then the effective depth of the corbel,

given by:

acd<1.0\dfrac{a_c}{d}<1.0
Corbel dimension check


2) Bearing Stress Check

As per Section 6.5.4 of EC2, the compressive capacity of a compression-tension node with anchored ties is given by:

σRd,max=k2v′fcd\sigma_{\text{Rd,max}}=k_2 v' f_{cd}
Where:
  1. 
    
    is the recommended value in EC2
  2. 
    
    as per Section 6.5.2 of EC2
Therefore, bearing stress on the corbel from the vertical force

distributed by the bearing plate, must satisfy:

Bearing stress<0.85(1−fck/250)αccfckγc→bearing stress=FEdb1×l\text{Bearing stress}<0.85(1-f_{ck}/250)\dfrac{\alpha_{cc}f_{ck}}{\gamma_c}\\\rightarrow \text{bearing stress} = \dfrac{F_{Ed}}{b_1\times l}
Bearing check at compression-tension node of strut & tie model


3) Concrete Strut Check

The below two checks confirm that the concrete strut is OK.
As per Annex J.3 of EC2, corbels may be designed using strut-and-tie models. A concrete element is defined as a corbel when:

ac≤za_c \leq z
Also as per Annex J.3 of EC2, the inclination of the strut is limited by:

1.0≤tan⁡θ≤2.5or68o≥θ≥45o 1.0 ≤ \tanθ ≤ 2.5\\\text{or}\\68^o \geq \theta \geq 45^o
Corbel concrete strut checks


4) Main Tension Steel Design

Total force in the main tension steel,

is given by:

Ftd′=FEd×cot⁡θ+HEdF_{td}' = F_{Ed} \times \cot θ + H_{Ed}
As per IStructE Manual for EC2, if

the minimum horizontal force is given by:

HEd=0.2 FEdH_{Ed}=0.2 \space F_{Ed}
The main tension steel required,

is then given by:

As,main=Ftd′/fcdA_{\text{s,main}}=F'_{td}/f_{cd}

5) Horizontal or Vertical Links Design

As per Section J.3 of EC2, UK NA and PD6687 publication, in addition to the main tension reinforcement required to resist

, closed horizontal or vertical links should also be provided to confine the concrete in the compression strut, such that if:
  1. 
    
    then provide horizontal links with total area:

As, horizontal link≥0.5×As,main providedA_{\text{s, horizontal link}} \geq 0.5\times A_{\text{s,main provided}}
  1. 
    
    then provide vertical links with total area:

As, vertical link≥0.5×FEd/fydA_{\text{s, vertical link}}\geq 0.5\times F_{Ed}/f_{yd}

Also, refer to the toggle below for corbel detailing requirements.

Detailing Requirements

  1. Main tension steel to be fully anchored into column. Ends of bars to be welded to an anchorage device or loops of reinforcing bars.
  2. Horizontal links to be provided within
    
    of the corbel.
Detailing requirements of a corbel

  1. The edge depth under the bearing plate should be
    
    , as shown in the image below.
Dimension requirement of corbel


Acknowledgements

This calculation was built in collaboration with Eric Castillo. Learn more.

References

  1. Reinforced Concrete Design to Eurocode 2 by J. H. Bungey, R. Hulse, and William Henry Mosley
  2. Strut-and-tie Models by The Concrete Centre
  1. IStructE Manual to EC2

Related Resources

  1. Concrete Column Design Calculator to AS3600
  2. Concrete Beam Design Calculator to AS3600
  3. RC Rectangular Beam Calculator (IMP) ACI 318-19
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