According to EN 1992-1-1:2004/AC:2010 - SECTIONS 6.4 and 9.4.3

Online application for punching shear proof - Shear load capacity for reinforced concrete slabs verification according to Eurocode (EN 1992-1-1:2004)

Available languages: English, Italian, Croatian

Available norms: EN 1991-1-1:2004, UNI EN 1992-1-1:2004 - Italian NA

Available languages: English, Italian, Croatian

Available norms: EN 1991-1-1:2004, UNI EN 1992-1-1:2004 - Italian NA

When a flat slab is exposed to a concentrated load larger than the capacity, the effect on the slab is referred to as punching shear. In these slabs, the shear force per unit length can become high close to the area of loading. If the capacity for shear punching in the slab is exceeded, a punching shear failure may occur within the discontinuity regions of the flat slab. This type of failure is a brittle failure mechanism, and may cause a global failure of the structure. Punching shear failure is a typical failure for slab-column connections.

Figure 1 - Basic control perimeter used for punching shear control in Eurocode 2

Punching shear failure is a local failure mechanism, where diagonal tensile cracks form a failure surface around the loaded area of the slab. The failure occurs along a truncated cone shape in the structure.

A control perimeter at some distance from the loaded area defines the section for punching shear calculations. This control perimeter varies in the different methods for calculating punching shear.

Normal design practice is to always control for punching shear in cases where the structure functions as a flat slab.

"Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings" (from now on referred to as EC2) defines rules for design for shear punching. The method in EC2 is based on experiments, and most of the formulas are therefore **empirical**. The next sections will briefly present the calculation method used in EC2. The formulas given in this section are gotten from Sections 6.4 and 9.4.3 in EC2, and are mainly given for cases with uniformly distributed loading. This section will only cover the effect of punching shear on slabs.

Figure 2 - Critical control section for punching shear design

Figure 3 - Control area for punching shear design

A calculation model for control of the punching shear capacity in the ultimate limit state is shown in Figures 2 and 3, where **A _{cont}** is the basic control area,

The critical control perimeter **u _{1}** can normally be evaluated at the distance

Figure 4 - Typical critical control perimeters around loaded areas

The effective thickness of the slab, **d _{eff}**, is assumed to be constant and is normally given by the expression in equation:

d_{eff} = d_{y} + d_{z} / 2

In equation above, **d _{y}** and

If the loaded area is close to an opening in the slab, and the distance between the edge of the loaded area and the edge of the opening does not exceed 6d, the critical control perimeter must be reduced. This reduction is done by assuming the part of the control perimeter contained between two tangents drawn from the centre of the loaded area to the outline of the opening to be ineffective, as shown in Figure 5.

Figure 5 - Critical control perimeter around loaded area close to slab opening

For columns situated near a corner or an edge, this must be taken into account when defining the critical control perimeter. Figure 6 shows the critical control sections for different situations where a column is situated near a corner or an edge. The section including continuation to the edge of the slab must be smaller than the one defined by Figure 6.

Figure 6 - Critical control perimeters around loaded areas close to edges or corners

Figure 7 shows the location of the control perimeter for a column with an enlarged circular column head. For **l _{H} < 2 ⋅ h_{H}**, as shown in Figure 7, the punching shear capacity must only be controlled at a critical control section outside the column head. The distance from the center of the cross-section to the critical section, rcont, can be defined as given in equation for a circular column:

r_{cont} = 2 ⋅ d + l_{H} + 0,5 ⋅ c

In equation, **l _{H}** is the distance from the column edge to the edge of the column head and

Figure 7 - Slab supported by column with enlarged column head, for l_{H} < 2h_{H}

For a rectangular column with a rectangular column head, **l _{H} < 2h_{H}** and edges with lengths

r_{cont} = 2 ⋅ d + 0,56 ⋅ (l_{1} ⋅ l_{2})^{0,5}

r_{cont} = 2 ⋅ d + 0,69 ⋅ l_{1}

Where:

l_{1} = c_{1} + 2 ⋅ l_{H1}

l_{2} = c_{2} + 2 ⋅ l_{H2}

and l_{1} ≤ l_{2}

For **l _{H} > 2h_{H}**, as shown in Figure 8, the punching shear capacity must be controlled at a critical section outside the column head as well as inside the column head.

Figure 8 - Slab supported by column with enlarged column head where l_{H} > 2 ⋅ (d + h_{H})

For circular columns the distance from the center of the cross-section to the control section within the column head can be assumed as given in equation:

r_{cont,int} = 2 ⋅ (d + h_{H}) + 0,5 ⋅ c

The distance from the center of the cross section to the control section outside the column head can be assumed as in equation:

r_{cont,ext} = l_{H} + 2 ⋅ d + 0,5 ⋅ c

The controls given in equations below must be performed for a slab with concentrated loading.

ν_{Ed} ≤ ν_{Rd,max}

ν_{Ed} ≤ ν_{Rd,c}

In equations above, **ν _{Ed}** is the maximum shear stress from the concentrated loading,

For an eccentric support reaction force, **V _{Ed}**, the value of

ν_{Ed} = β ⋅ V_{Ed} / (u_{i} ⋅ d)

Where, **u _{i}** is the length of the considered control section, and

β = 1 + k ⋅ (M_{Ed} / V_{Ed}) ⋅ (u_{1} / W_{1})

In equation above, **u _{1}** is the length of the critical control perimeter,

W_{1} is given by equation:

where **dl** is a length increment of the perimeter and **e** is the distance from **dl** to the axis where the moment M_{Ed} acts.

Figure 9 - Shear distribution from an unbalanced moment at connection between slab and inner column

Values of **k** for rectangular loaded areas:

c_{1} / c_{2} |
≤0.5 | 1.0 | 2.0 | ≥3.0 |

k |
0.45 | 0.60 | 0.70 | 0.80 |

The expression for **W _{1}**, and therefore

Where the adjacent spans do not differ in length by more than 25%, and the lateral stability does not depend on the frame action between the slab and the column, simplified expressions for **β**, as presented in Figure 10, may be used.

Figure 10 - Recommended simplified values of β for internal column, edge column and corner column

The punching shear capacity in slabs without punching shear reinforcement, **ν _{Ed}**, is to be controlled at the critical section, as described in Section 2.4.2, and can be calculated by equation:

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

The value given for **C _{Rd,c}** varies for different countries. It is set to be as in equation:

C_{Rd,c} = 0,18 / γ_{c}

The values for k, l, σ_{cp} and ν_{min} are given in following equations:

k_{1} = 1 + (200 / d)^{0,5} ≤ 2 ⋅ d (d is in mm)

ρ_{l} = (ρ_{lx} ⋅ ρ_{ly})^{0,5} ≤ 0,02

σ_{cp} = (σ_{cx} ⋅ σ_{cy}) / 2 = (N_{Ed,x} / A_{cx} + N_{Ed,y} / A_{cy}) / 2

ν_{min} = 0,035 ⋅ k^{3/2} ⋅ f_{ck}^{1/2}

**ρ _{lx}** and

In cases where punching shear reinforcement is shown to be necessary, the design value of the punching shear stress resistance for a slab with punching shear reinforcement, **ν _{Rd,cs}**, can be calculated by the expression given in the equation:

ν_{Rd,cs} = 0,75 ⋅ ν_{Rd,c} + 1,5 ⋅ (d / s_{sr}) ⋅ A_{sw} ⋅ f_{ywd,ef} ⋅ 1 / (u_{1} ⋅ d) ⋅ sinα

In the equation above, **A _{sw}** is the area of one perimeter of shear reinforcement around the column,

f_{ywd,ef} = 250 + 0,25 ⋅ d ≤ f_{ywd}

At the edge of a column, the punching shear capacity must be smaller than **ν _{Rd,max}**, as given in the equation:

ν_{Ed} = β ⋅ V_{Ed} / (u_{0} ⋅ d) ≤ ν_{Rd,max}

In the equation above, **β** is as given in Section 2.4.2 and **u _{0}** is the control perimeter at the edge of the column. The value of

ν_{Rd,max} = min{ 0,4 ⋅ ν ⋅ f_{cd} ; 1,6 ⋅ ν_{Rd,c} ⋅ u_{1} / (β ⋅ u_{0}) }

The control section where shear reinforcement is not necessary is given by the equation:

u_{out,ef} = β ⋅ V_{Ed} / (ν_{Rd,c} ⋅ d) ≤ 1,6 ⋅ ν_{Rd,c} ⋅ u_{1} / (β ⋅ u_{0})

If punching shear reinforcement is shown to be necessary, the punching shear reinforcement should be placed between the loaded area and the length **k ⋅ d** within the perimeter where shear reinforcement is not necessary, **u _{out,ef}**.

The value of

Link legs should be provided in at least two perimeters, and the spacing of the link leg perimeters should not exceed **s _{r,max} = 0,75 ⋅ d**. The spacing of link legs around a perimeter,

Figure 11 - Spacing of link legs

Where punching shear reinforcement is provided, the area of a link leg is given by the equation:

A_{sw,min} ⋅ (1,5 ⋅ sinα + cosα) / (s_{r} ⋅ s_{t}) ≥ 0,008 ⋅ f_{ck}^{0,5} / f_{yk}

Here, **α** is the angle between the shear reinforcement and the main reinforcement, **s _{r}** is the spacing of shear links in radial direction and