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Hedi Boukraa, B.Sc.

2023-05-24

VE 1023 | Design of Flat Slab According to DIN EN 1992-1-1 with NA: Punching Shear Design

Description

The model is based on the example 4 of [1]: Point-supported slab. The internal forces and the required longitudinal reinforcement can be found the in verification example 1022. In this example, punching is examined in the axis B/2.

The required punching shear reinforcement is determined with manual calculation. Then, the results are compared with the RFEM results.

Verification Example 1022 | Slab Designed According to DIN EN 1992-1-1 with NA

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Verification Example 1023 | Flat Slab Designed According to DIN EN 1992-1-1 with NA: Punching Shear Design

Analytical Solution

The middle effective depth d_eff results from the concrete cover and the diameter of the longitudinal reinforcement:

d_{eff} = (d_{y} + d_{z}) / 2 = (18.0 + 20.0) / 2 = 19.0 cm

Effective Depth of Slab According to DIN EN 1992-1-1, 6.4.2 (1)

The basic control perimeter u₁ is then calculated:

u_{1} = 4 \cdot a + 2 \cdot 2.0 d \cdot π = 4 \cdot 45 + 2 \cdot 2.0 \cdot 19 \cdot π = 418.76 cm

Basic Control Perimeter in Case of Rectangluar Column According to DIN EN 1992-1-1, 6.4.2, Figure 6.13

With the coefficient ß=1.10, a shear force to be supported in the control perimeter is calculated:

v_{Ed} = \frac{β \cdot V_{Ed}}{u_{1} \cdot d} = \frac{1.10 \cdot 0.7873 MN}{4.19 m \cdot 0.19 m} = 1.088 MN / m

Maximum Shear Stress According to DIN EN 1992-1-1, 6.4.3 (3)

The punching shear resistance of the slab without shear reinforcement:

V_{Rd, c} = C_{Rd, c} \cdot k \cdot {(100 \cdot ρ_{l} \cdot f_{ck})}^{1 / 3} + k_{1} \cdot σ_{cp} \geq (v_{\min} + k_{1} \cdot σ_{cp}) = 0.12 \cdot 2.0 \cdot {(100 \cdot 0.0166 \cdot 35)}^{1 / 3} + 0.10 \cdot 0 = 0.929 MN / m^{2} \geq 0.586 MN / m^{2}

where

v_{\min} = (0.0525 / γ_{c}) \cdot k^{3 / 2} \cdot {f_{ck}}^{1 / 2} = (0.0525 / 1.5) \cdot 2 . 0^{3 / 2} \cdot 35^{1 / 2} = 0.586 MN / m^{2}

ρ_{l} = \sqrt{ρ_{ly} \cdot ρ_{lz}} = \sqrt{\frac{31.42 {cm}^{2} / m \cdot 10^{- 4}}{0.18 m} \cdot \frac{31.42 {cm}^{2} / m \cdot 10^{- 4}}{0.20 m}} = 0.0166 \leq 0.02

C_{Rd, c} = 0.18 / γ_{c} = 0.18 / 1.5 = 0.12

k_{1} = 1 + \sqrt{\frac{200 mm}{190 mm}} = 2.02 \leq 2.0

Punching Shear Resistance of Slab Without Shear Reinforcement According to DIN EN 1992-1-1, 6.4.4 (1)

Punching shear reinforcement is therefore necessary.

Maximum value of punching shear resistance:

v_{Rd, \max} = 1.4 \cdot v_{Rd, c} = 1.4 \cdot 0.929 = 1.300 MN / m^{2} \geq v_{Ed} = 1.088 MN / m^{2}

Maximum Value of Punching Shear Rresistance According to DIN EN 1992-1-1/NA, 6.4.5 (3)

Determination of the punching shear reinforced area:

V_{Rd, c, out} = C_{Rd, c} \cdot k \cdot {(100 \cdot ρ_{l} \cdot f_{ck})}^{1 / 3} + k_{1} \cdot σ_{cp} \geq (v_{\min} + k_{1} \cdot σ_{cp}) V_{Rd, c} = 0.10 \cdot 2.0 \cdot {(100 \cdot 0.0166 \cdot 35)}^{1 / 3} + 0.10 \cdot 0 = 0.774 MN / m^{2} \geq 0.586 MN / m^{2}

where

C_{Rd, c} = 0.15 / γ_{c} = 0.15 / 1.5 = 0.10

Design Value of Shear Resistance According to DIN EN 1992-1-1, 6.2.2 (1)

u_{out} = \frac{β \cdot V_{Ed}}{v_{Rd, c, out} \cdot d} = \frac{1.10 \cdot 0.787}{0.774 \cdot 0.19} = 5.889 m

Outer Perimeter According to DIN EN 1992-1-1, 6.4.5 (4)

distance from the outer perimeter to the edge of the load surface:

a_{out} = (u_{out} - u_{0}) / 2 π = (5.889 - 4 \cdot 0.45) / 2 π = 0.651 m

Distance from Outer Perimeter to Load Surface Edge

The required number of shear reinforcement rows:

n = \frac{a_{out} - 0.5 d - 1.5 d}{0.75 d} + 1 = \frac{0.651 - 0.5 \cdot 0.19 - 1.5 \cdot 0.19}{0.75 \cdot 0.19} + 1 = 2.90 \to 3 rows

Required Number of Shear Reinforcement Rows

With 3 rows of reinforcement, the radial spacing s_r is:

s_{r} = \frac{a_{out} - 0.5 d - 1.5 d}{n - 1} = \frac{0.651 - 0.5 \cdot 0.19 - 1.5 \cdot 0.19}{3 - 1} = 0.135 m

Radial Spacing of Perimeters of Shear Reinforcement

Reinforcement per row:

A_{sw} = \frac{v_{Ed} - 0.75 \cdot v_{Rd, c}}{1.5 \cdot (\frac{d}{s_{r}}) \cdot f_{ywd, ef} \cdot (\frac{1}{u_{1} \cdot d}) \cdot \sin (α)} = \frac{1.088 - 0.75 \cdot 0.929}{1.5 \cdot (\frac{0.190}{0.135}) \cdot 297.5 \cdot (\frac{1}{4.188 \cdot 0.190}) \cdot \sin (90^{\circ})} \cdot 10^{4} = 4.98 {cm}^{2}

Punching Shear Resistance with Shear Reinforcement According to DIN EN 1992-1-1, 6.4.5 (1)

This results in the following punching shear reinforcement in the individual reinforcement rows:

Reinforcement row	Distance from column edge	Factor k_sw	Punching shear reinforcement
-	-	-	A_sw*k_sw
1	0.5d=0.095 m	2.5	2.5*4.98cm²=12.45cm²
2	0.085+0.99=0.194 m	1.4	1.4*4.98cm²=6.97cm²
3	0.085+2*0.99=0.293 m	1.0	1.0*4.98cm²=4.98cm²

Results

In the following tables, the results of RFEM 6 are compared with the analytical solution.

Punching shear design
Designation	Variable	Unit	RFEM result	Analytical solution	Ratio
Design shear force	V_Ed	[kN]	787.3	787.3	1.0
Basic control perimeter	u₁	[m]	4.188	4.188	1.0
Maximum shear stress	v_Ed	[MN/m²]	1.088	1.088	1.0
Punching shear resistance without shear reinforcement	v_Rd,c	[MN/m²]	0.929	0.929	1.0
Maximum value of punching shear resistance	v_Rd,max	[MN/m²]	1.300	1.300	1.0
Outer perimeter	u_out	[m]	5.889	5.889	1.0
Distance from the outer perimeter to the edge of the load surface	a_out	[m]	0.651	0.651	1.0
Required number of shear reinforcement rows	n	[-]	3	3	1.0
Punching shear resistance with shear reinforcement	A_sw	[cm₂]	4.98	4.98	1.0
First reinforcement row	A_sw,1	[cm₂]	12.46	12.45	1.0
Distance from the outer perimeter to the edge of the load surface	l_sw,1	[m]	0.095	0.095	1.0
Second reinforcement row	A_sw,2	[cm₂]	6.98	6.97	1.0
Distance from the outer perimeter to the edge of the load surface	l_sw,2	[m]	0.194	0.194	1.0
Third reinforcement row	A_sw,3	[cm₂]	4.98	4.98	1.0
Distance from the outer perimeter to the edge of the load surface	l_sw,3	[m]	0.293	0.293	1.0

Verification Example 1023 | Design of Flat Slab According to DIN EN 1992-1-1 with NA: Punching Shear Reinforcement

437x

26x

Verification Example 1023 | Flat Slab Designed According to DIN EN 1992-1-1 with NA: Punching Shear Design

References

German Concrete and Structural Engineering Association E. V, examples for design according to Eurocode 2. Volume 1: Building construction, Berlin: Ernst & Sohn 2012, 1st prize corrected reprint of 1. Edition, 978-3-433-01877-4

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