#### This article explores the structural analysis and design of Corbels based on the guidance and recommendation of Eurocode 2 (part 1)

Corbels have long been an integral part of architectural design, dating back to ancient civilizations where they served both functional and decorative purposes. These structural elements, often made of stone, wood, or metal, project from a wall to support heavy loads such as beams, or even statues. Today, corbels are designed as structural elements and used to support heavy loads, often found in buildings, bridges, and other infrastructure.

Within the context of structural engineering, a corbel can be idealized as cantilever, albeit a short cantilever. However, there is stark difference between a corbel and a traditional cantilever. While a cantilever is resisting bending moment and shear force along its length, a corbel is usually too short, hence the transfer of forces is through a combination of axial forces.

Eurocode 2 (EN 1992-1) Design of Concrete Structures provides a comprehensive guideline for the design of concrete structures, including corbels. This article provides a comprehensive analysis of the key principles and design considerations involved in designing corbels to the above-referenced code.

### Designing a Corbel

In reinforced concrete design, concrete can withstand compressive stresses, while steel reinforcement can withstand tensile stresses. This concept is typically represented by concrete struts and steel ties, forming the core principle of the strut-and-tie design method. This method is applicable in designing various structures such as pile caps, deep beams, corbels, half-joints, and more structures. In specific scenarios, Eurocode 2 allows the utilization of the strut and tie method to analyze and design corbels. As stated in clause 6.5.1 of EN 1992-1-1:2004, this method is applicable in situations where there is a nonlinear stress distribution, such as near supports, concentrated loads, or areas experiencing plain stress.

This article will not delve into the core principle underlying the strut and tie design as a whole but would touch on it as it relates to the design of a corbel.

A corbel, as shown in *Figure 1* is considered to be a short cantilever when a_{c }< z where z is the lever arm and a_{c} are the distance from the face of the column to the bearing of the vertical force, F_{Ed}.

When the vertical load has a stiff bearing a_{c} may be measured to the edge of the bearing but where a flexible bearing is used a_{c} is measured to the vertical force.

### Strut & Tie Analogy for Corbels

The analysis of a corbel using the strut and tie method is illustrated in figure 1. In the figure the vertical load F_{Ed} at point B is resisted by the force F_{cd} in the inclined concrete strut CB and the force F_{td} in the horizontal steel tie AB.

Also, Figure 1 illustrates the corbel featuring an inclined strut BC at an angle θ to the horizontal tie AB. The strut carries force F_{Ed}, while F_{td} represents the force in the horizontal tie. Point B, situated at a distance a’ = (ae + 0.2aH) from the column face due to the horizontal force, H_{Ed} (= 0.2F_{Ed}).

From the triangle ABC’s geometry, the lever-arm depth is

z = (ae + 0.2aH) tan θ

**a) Concrete strut force, F**_{cd}

_{cd}

The design stress for the concrete strut, fed, is calculated as 0.34f_{ck}(1- f_{ck}/250). The vertical measurement of the concrete strut’s width is 2(d – z) from figure 1, with the width measured perpendicular to its axis given by Ws_{trut} = 2(d – z) cos θ. Thus, the force F_{Ed} in the concrete strut is:

F_{Ed} = f_{cd} × W_{strut} × b_w

= f_{cd} × 2(d - z) × b_w cos θ

Where: b_{w} represents the width of the corbel.

**(b) Inclination angle, θ, of the concrete strut**

From figure 1, resolving vertically at point B, we have:

F_{Ed} = f_{cd} sin θ \\= f_{cd}× 2(d - z) × b_w × cos θ × sin θ

=f_{cd} \times d \times b_w (1-[\frac{a'}{a}]tan \theta)sin2\theta

If we rearrange the above equation, we have:

\frac{F_{Ed}}{f_{cd}db_w} =(1-[\frac{a'}{d}]tan \theta)sin2 \theta

The above equation cannot be directly solved for θ, but Figure 2 (overleaf), derived from the above equation can be utilized for design purposes.

#### (c) Main tension steel, A_{s,main}

Resolving horizontally at B, the force F_{td }in the steel tie is:

F_{td} = F_{Ed} cos θ = F_{Ed} \frac{cos θ}{sin θ}

=F_{Ed}cot\theta

Therefore, the total force F’_{td} in the steel tie, accounting for the horizontal force of 0.2F_{Ed}, is

F'_{td}=F_{Ed}cot\theta+0.2F_{Ed}\\=F_{Ed}(cot\theta+0.2)

Thus, the area of main tension steel, A_{s,main}, is calculated as:

A_{s,main}=\frac{F'_{td}}{0.87f_{yk}}

### Design & Detailing

In addition to analysis of a corbel presented above, there are certain design and detailing requirements that need to be met. These requirements are outlined in six steps below.

**Step 1**

The bearing stress of the load on the corbel directly under the load should not exceed:

0.48(1 - f_{ck}/250)_{fck}

**Step 2**

A horizontal force *H _{Ed} *must be resisted. This force acts at the level of the top of the bearing, a distance

*a*above the horizontal tie.

_{8}H_{Ed} = 0.2F_{Ed}

**Step 3**

The main tension steel, A_{s,main }must be fully anchored into the column and the other end of these bars must be welded to an anchorage device or loops of reinforcing bars.

**Step 4**

The angle of inclination, *θ *of the compression struts must be within the following limits.

68° \ge \theta \ge 45 \quad or \quad 2.5 \ge \tan \theta \ge 1.0

**Step 5**

The following equation must be satisfied.

f_{cd} \le 0.34f_{ck}(l - f_{ck}/250

Where: f_{cd }is the design stress of the concrete strut

**Step 6**

Closed horizontal links each of area *A*_{s,link} >= *0.5 A*_{s,main} should be provided to confine the concrete in the compression strut and I: *A*_{s,link} >= *A*_{s,main} when *a _{c}* <

*0.5h*If

_{c}.*a*>

_{c}*0.5h*then closed vertical links will alternatively be required.

_{c}### Worked Example

Determine the quantity of reinforcement required for the depicted corbel in Figure 2. This corbel possesses a width of b = 450mm and sustains an ultimate load of F_{Ed} = 550kN positioned at a distance ac = 200mm from the column face. The bearing is adaptable and situated a distance aH = 75mm above the tension tie, measuring 450mm by 200mm. The characteristic material strengths are: concrete compressive strength fck = 30N/mm² and yield strength of reinforcement steel fyk = 460N/mm².

#### Verify Bearing Stress

We must first determine that the actual bearing stress does not exceed the safe bearing stress of the corbel geometry.

**Safe Bearing Stress **

0.48(1 - f_{ck}/250)_{fck} \\= 0.48(1-25/250)25=10.8Mpa

**Actual Bearing Stress**

=\frac{550\times10^3}{450\times200} =6.1Mpa

Since actual bearing stress (6.1Mpa) is less than the safe bearing stress, we can adopt the corbel geometry and proceed to determine the concrete struts.

#### Determine Concrete Struts

The effective depth of the corbel is taken as d =550mm. And the distance a’ from figure 1 will be given as:

a'=(200+0.2\times75)=215mm

Therefore a’/d:

\frac{a'}{d}=\frac{215}{550} =0.39

f_{cd}=0.34f_{ck}(1-f_{ck}/250)\\=0.34\times25(1-25/250)\\=7.65Mpa

\frac{F_{Ed}}{f_{cd}db_w}=\frac{550\times10^3}{7.65\times550\times450} =0.29

Hence, from Figure 2 we can determine the value of θ = 59.5^{o}

z=a'tan\theta=215tan59.5\\=365mm (>a_c=200mm)

#### Main Tension Steel

F'_{td} =F_{Ed}(cot\theta+0.2)\\=550(cot59.5+0.2)\\=434kN

A_{s,min}=\frac{F'_{td}}{0.87f_{yk}}

=\frac{434\times10^3}{0.87\times460} =1084.5mm^2

*Provide 3T25mm Bars (A _{s_prov} =1473mm^{2})*

#### Horizontal Links

Because *a _{c}/h_{c} *= 200/600 = 0.33 (< 0.5) closed horizontal links will be provided. The total area of both legs of a link,

*A*

_{s,link,}is given by:

A_{s,link} > 0.5A_{s,main} = 0.5\times1084.5\\=542.3 mm^2

Provide four 3H16 links, each with area A_{s,link} = 603mm^{2}.

#### Design Resume

Figure 4 shows the detailing of the reinforcement in the corbel.

See: **Design for Shear using Bent-up Bars **

**Sources & Citations **

*Mosley W. H., Hulse R. and Bungay J.H. (2012) Reinforced Concrete Design to Eurocode 2 (7th ed.) Basingstoke, UK: Palgrave MacMillan*