# Punching Shear Design According to Eurocode 2 in RFEM

### Technical Article

For the structural components consisting of slabs, it is necessary to perform shear design on the locations with concentrated load introduction, applying the punching shear design rules according to Chap. 6.4 of EN 1992-1-1 [1]. The concentrated load introduction is present on the individual locations, for example by columns, concentrated load, or nodal supports. In addition, the end of linear load introduction on slabs is also regarded as the concentrated load introduction. This may include wall ends, wall corners, and ends or corners of line loads and line supports. You can perform the punching shear design for floor slabs or foundations, considering the available existing plate topology about the designed node of punching shear. The punching shear design according to EN 1992-1-1 checks that the acting shear force vEd does not exceed the resistance vRd.

#### Modeling Structure

RFEM 5 allows you to perform the punching shear design for both 2D slab and 3D structures. The RF-PUNCH Pro add-on module detects the locations of punching shear automatically and recommends them for the design. An integrated filter for finding the nodes of punching shear can be set individually. In this way, you can easily sort the designs, for example, by planes.

RF-PUNCH Pro automatically recognizes the type of punching node (single columns, wall ends or corners) as well as the punching node area (inner, edge or corner columns) from the RFEM model.

#### Basic Control Perimeter

The punching shear is performed in the so-called basic control perimeter. According to Chap. 6.4.2 of EC 2 [1], the control perimeter for slabs is located at a distance of 2 d (d = effective depth of the slab) from the load surface. Determination of the control perimeter geometry requires the consideration of column dimensions as well as slab openings up to a distance of 6 d from the load surface. RF-PUNCH Pro automatically recognizes the openings modeled during the FEM analysis. In addition, you can also define smaller openings in the module, which are negligible in the structural FEM calculation, for example. These can be considered when determining the basic control perimeter. The control perimeter geometry is displayed in the input windows of the add-on module even before you start the calculation.

Figure 01 - Basic Control Perimeter Around Column Considering Two Openings

In the case of floor slabs or foundations, the control perimeter is usually at a distance of 2 d from the column edge. According to 6.4.4 (2) [1], the determination of the control perimeter requires an iterative calculation. The German National Annex [2], NCI to 6.4.4 (2), allows a simplified calculation for floor slabs and slender foundations with λ = a_{λ} / d > 2 (where a_{λ} = foundation cantilever). In this case, the control perimeter can be set at a distance of 1 d. Generally, RF-PUNCH Pro performs an iterative calculation to find the control perimeter of foundations or floor slabs.

#### Related Shear Force v_{Ed}

The design shear force related to the control perimeter is calculated according to Eq. 6.38, EC 2 [1]:

$${\mathrm v}_\mathrm{Ed}\;=\;\mathrm\beta\;\cdot\;\frac{{\mathrm V}_\mathrm{Ed}}{{\mathrm u}_1\;\cdot\;\mathrm d}$$

where

u_{1} = length of the control perimeter

d = mean effective depth of the slab

β = load increment factor for considering asymmetrical distribution of shear force in the control perimeter

V_{Ed} = design value of the punching load

In order to consider the uneven rotational symmetric load, the punching load V_{Ed} is increased by the load increment factor β. For rigidly fixed structures where the adjacent spans do not differ in length by more than 25%, you may use the following values β in accordance with EN 1992-1-1, Figure 6.21N [1]:

β = 1.15 for internal columns

β = 1.4 for edge columns

β = 1.5 for corner columns

The German Annex [2], Figure 6.21N, complements the β factors with β = 1.35 for wall corners and β = 1.2 for wall ends, and specifies the recommended value for internal columns as β = 1.10.

A general method for determining the load increment factor β is described in Eurocode 2 [1], Chap. 6.4.3 (3). Here, the factor β is determined assuming the fully plastic shear stress distribution in the control perimeter. In accordance with EN 1992-1-1 [1], Eq. (6.39), you obtain:

$${\mathrm v}_\mathrm{Ed}\;=\;\mathrm\beta\;=\;1\;+\;\mathrm k\;\cdot\;\frac{{\mathrm M}_\mathrm{Ed}}{{\mathrm V}_\mathrm{Ed}}\;\cdot\;\frac{{\mathrm u}_1}{{\mathrm W}_1}$$

where

k = a coefficient dependent on the column dimensions, see Table 6.1 [1]

M_{Ed} = moment about the main axis of the control perimeter

W_{1} = basic control perimeter modulus

Figure 02 - Fully Plastic Shear Stress Distribution

While Expression (6.39) of EN 1992-1-1 [1] specifies the calculation of β for uniaxial load eccentricity only, the German Annex [2] provides the expanded Expression (NA.6.39.1) for considering biaxial load eccentricity:

$$\mathrm\beta\;=\;1\;+\;\sqrt{\left({\mathrm k}_\mathrm x\;\cdot\;\frac{{\mathrm M}_{\mathrm{Ed},\mathrm x}}{{\mathrm V}_\mathrm{Ed}}\;\cdot\;\frac{{\mathrm u}_1}{{\mathrm W}_{1,\mathrm x}}\right)^2\;+\;\left({\mathrm k}_\mathrm y\;\cdot\;\frac{{\mathrm M}_{\mathrm{Ed},\mathrm y}}{{\mathrm V}_\mathrm{Ed}}\;\cdot\;\frac{{\mathrm u}_1}{{\mathrm W}_{1,\mathrm y}}\right)^2}$$

RF-PUNCH Pro includes both options for calculating β mentioned above. The standard method is to select the model considering the fully plastic shear stress distribution.

RF-PUNCH Pro takes the design value of shear force V_{Ed} directly from the FEM calculation to perform the punching shear design. In the case of the punching shear design for columns, nodal supports, and concentrated loads, you can determine the shear force on the basis of the column axial force, support force, or the load value of the acting concentrated force.

In addition, RF-PUNCH Pro allows you to create the control perimeter on a FEM model and determine the acting shear force V_{Ed} there. For this, there are the following two options:

- The existing shear forces in the control perimeter are integrated or smoothed by the same control perimeter. The resulting design shear force V
_{Ed}must then be multiplied by the load increment factor β (see Eq. 6.38 [1]). If the factor β is determined by using the fully plastic shear distribution model, both bending moments M_{Ed,x}and M_{Ed,y}are also determined by integrating the slab internal forces in the control perimeter set on the slab. - The maximum value of the existing shear force in the control perimeter is used for the punching shear design. This method takes into account the effect of the uneven rotational symmetric load by using the maximum value. Therefore, an additional increment of the shear force by the factor β can be omitted.

Although the utilization of the maximum shear force value in the control perimeter is the most accurate method for determining the design value of the punching load, it is also the method most susceptible to the singularity effects. In particular, you should pay attention to the sufficient refinement of the FE mesh in the punching shear areas when taking the shear forces directly from the control perimeter in the FEM calculation. It is recommended to arrange at least two or three elements between the punching nodes and the control perimeter by using the FE mesh refinement.

Figure 03 - Distribution of Shear Stress in Section

In the case of foundations and floor slabs, you can reduce V_{Ed} by the soil pressure within the iteratively determined control perimeter, see 6.4.2 (2) [1]. If you create the basic control perimeter by 1 d for slender foundations in accordance with the simplified method of German Annex [2], you can apply only 50% of the soil pressure. Both design forms can be selected in RF-PUNCH Pro.

#### Design Form

Before the punching shear design, the program checks whether the design can be performed without punching reinforcement.

#### Punching Shear Resistance Without Punching Shear Reinforcement

The punching shear resistance without shear reinforcement v_{Rd,c} will be determined according to 6.4.4 (1), EN 1992-1-1 [1] as follows:

v_{RD,c} = C_{RD,c} ∙ k ∙ (100 ∙ ρ_{l} ∙ f_{ck})^{1/3} + k_{1} ∙ σ_{cp} ≥ (v_{min} + k_{1} ∙ σ_{cp})

where

C_{Rd,c} = 0.18 / γ_{c} in the case of flat slabs

C_{Rd,c} = 0.15 / γ_{c} in the case of floor slabs or foundations

k = 1 + √(200 / d)

ρ_{l,x/y} = A_{sl,x/y} / (b_{w} · d_{x/y})

ρ_{l} = √(ρ_{l,x} ∙ ρ_{l,y}) ≤ 0,02

A_{sl} = area of tension reinforcement

k_{1} = 0.1

σ_{cp} = normal stress in the control perimeter

v_{min} = 0,035 · k^{3/2} · f_{ck}^{1/2}

In the German Annex [2] the parameters mentioned above are modified as follows:

C_{Rd, c} = 0.18 / γ_{c} in the case of flat slabs

C_{Rd,c} = 0.18 / γ_{c} ∙ (0.1 ∙ u_{0} / d + 0.6) in the case of internal columns of flat slabs with u_{0} / d < 4

C_{Rd,c} = 0.15 / γ_{c} in the case of floor slabs or foundations

ρ_{l} = √(ρ_{l,x} ∙ ρ_{l,y}) ≤ min [0,02;0,5f_{cd}/f_{yd}]

v_{min} = (0,00525 / γ_{c}) ∙ k^{3/2} ∙ f_{ck}^{1/2} for d ≤ 600 mm

v_{min} = (0,00375 / γ_{c}) · k^{3/2} · f_{ck}^{1/2} for d > 800 mm

The punching shear is fulfilled without additional punching reinforcement if there is v_{Ed} ≤ v_{Rd,c}. Due to the structurally difficult design of shear reinforcement, you can usually try to avoid using the punching reinforcement and to instead apply the maximum allowable longitudinal reinforcement ratio ρ_{l} for this purpose. RF-PUNCH Pro determines the required longitudinal reinforcement ratio in order to avoid the punching shear reinforcement. However, it is also possible to define the existing longitudinal reinforcement for the calculation of V_{Rd,c} manually.

#### Maximum Punching Shear Resistance v_{Rd,max}

If the punching shear design is not possible without punching reinforcement, the maximum punching shear resistance v_{Rd,max} is calculated in the next step.

According to 6.4.5 (3) of EN 1992-1-1 [1], the maximum punching shear resistance is performed on a column periphery. The considered periphery length u_{0} is enclosing the basic control perimeter and can be determined directly on the load surface. The maximum punching shear resistance on the column periphery v_{Rd,max} is determined according to 6.4.5. (3), EN 1992-1-1 [1], as follows:

v_{Rd,max} = 0,4 · ν · f_{cd}

where ν = 0,6 · (1 - f_{ck} / 250) (f_{ck} in [N/mm²])

The acting design shear force on the column periphery results from:

v_{Ed,u0} = β · V_{Ed} / (u_{0} · d)

The design is fulfilled if there is v_{Ed,u0} ≤ v_{Rd,max}.

In the German National Annex [2], the design of the maximum punching shear resistance is not performed on the column periphery, but in the basic control perimeter u_{1} according to Expression NA6.53.1 as follows:

v_{Ed,u1} ≤ v_{Rd,max} = 1.4 · v_{Rd,c,u1}

#### Punching Shear Resistance with Punching Shear Reinforcement

If the design of v_{Rd,max} was performed successfully, the required punching shear reinforcement is determined in the next step. The required punching shear reinforcement is calculated according to the adjusted Expression 6.52 of EN 1992-1-1 [1]. The required reinforcement A_{sw} results from the following equation:

$${\mathrm A}_\mathrm{SW}\;=\;\frac{({\mathrm v}_\mathrm{Ed}\;-\;0.75\;\cdot\;{\mathrm v}_{\mathrm{Rd},\mathrm c})\;\cdot\;\mathrm d\;\cdot\;{\mathrm u}_1}{1.5\;\cdot\;{\displaystyle\frac{\mathrm d}{{\mathrm s}_\mathrm r}}\;\cdot\;{\mathrm f}_{\mathrm{ywd},\mathrm{ef}}\;\cdot\;\sin\;\mathrm\alpha}$$

where

v_{Rd,c} = design shear resistance without punching shear reinforcement

d = mean of the effective depths

s_{r} = radial spacing of perimeters of shear reinforcement

f_{ywd,ef} = 250 + 0.25 d ≤ f_{ywd}

α = angle between the shear reinforcement and the slab plane

Figure 04 - Punching Shear Reinforcement

According to DIN EN 1992-1-1 / NA [2], the amount of reinforcement in the first shear reinforcement perimeter must be increased by the factor κ_{sw,1} = 2.5, and by κ_{sw,2} = 1.4 in the second shear reinforcement perimeter.

The punching shear reinforcement should be placed up to a distance of 1.5 d from the outermost perimeter. The required length of the outmost perimeter is u_{out,ef}, which is defined according to Eq. 6.54 of EC 2 [1]:

$${\mathrm u}_{\mathrm{out},\mathrm{ef}}\;=\;\mathrm\beta\;\cdot\;\frac{{\mathrm V}_\mathrm{Ed}}{{\mathrm v}_{\mathrm{Rd},\mathrm c}\;\cdot\;\mathrm d}$$

#### Summary

The provisions for punching shear design according to Eurocode 2 cannot be applied efficiently without a software solution. An example is the calculation of the load increment factor β based on the model with fully plastic shear force distribution in the control perimeter, or the iterative calculation of the basic control perimeter area of foundations. Also, ground plans of buildings are designed in a more free and complex way, so it is not possible to follow the simplification rules and thus, they also cannot be applied. The RF-PUNCH Pro add-on module of the structural FEA software RFEM allows you to take all the data required for the geometrical determination of the basic control perimeter and design loads for the punching shear design directly from the FEM entries or FEM calculation. Thus, the punching shear design of columns, wall corners, and wall ends can be performed very easily and in an efficient way. For columns, there is an additional option to consider an enlarged column head. The results of the punching shear design are displayed in clearly arranged result tables, including all intermediate results relevant for the individual designs. The results as well as the required punching shear reinforcement, shear force distribution, and punching shear resistances can be displayed graphically in the RFEM graphic window.

Reference

[1] *Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings*; EN 1992-1-1: 2011-01

[2] *National Annex - Nationally determined parameters – Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings*; DIN EN 1992-1-1 / NA: 2013-04

[2] *Manual RFEM 5*. (2013). Tiefenbach: Dlubal Software. Download...

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