Punching Shear Check Example 2

In the example calculation, a punching calculation will be made for a column whose dimensions are 100x50. Other parameters are listed as follows.

• Slab thickness : 27 cm

• Slab cover: 3 cm

• Material : C35/S420

• f_ctd = 1380.42 kN/m² (Concrete design tensile strength)

The slab useful height, d , is calculated;

d = ( Slab thickness - Slab cover ) = 27 - 3 = 24 cm

Column dimensions

c₁ = 100 cm c₂ = 50 cm

The punching circumference (u_p) is calculated at a distance d/2 from the column surface and is shown in the picture below. In this case, the following operations are performed to find the edges (b₁, b₂) of the punching.

In this case, the punching circumference (u _p ), and the punching area (A _z ) obtained by multiplying the punch perimeter by the slab useful height, d, is calculated as shown below.

u_p = 2*(b₁ + b₂) = 2*(124+74) = 396 cm = 3.96 m

A _z = u _p * d = 3.96 * 0.24 = 0.95040 m ²

The shear stresses plotted below are the punch stresses perpendicular to the floor plane.

J values ​​are the sum of the polar inertia and second moments of inertia of the surfaces forming the punching area (A _z ). Also TBDY Equation 7.28 according γ _f is calculated according to the loading direction coefficient taken into account. Here, separate calculations are made in the X and Y directions of the Column for both values.

Strong Axis (Major Aspects) to J and γ _f Coefficient Account

The following steps are followed for the calculation of the coefficient _{j f (maj)} and γ _{v (maj)} .

To find the value of J _(maj) , the following way is followed.

c _(maj) is the center of gravity distance perpendicular to the moment vector, considered when _{finding the} J value (J _(maj) ) on the strong axis of the section . Since the punching circumference is rectangular;

c_(maj) = b₁ / 2 = 124/2 = 62 cm = 0.62 m c'_(maj) = b₁ - c_(maj) = 124 - 62 = 62 cm = 0.62 m

In this case, the polar inertia and the two moments of inertia are respectively as follows.

In this case, the sum of the polar inertia and the second moments of inertia J _(maj) with respect to the major axis of the section is found as follows.

J _(maj) = J _{1 (maj)} + J _{2 (maj)} = 0.079122 + 0.1365388 = 0.21566080 m ⁴ = 21566080 cm ⁴

Weak Axis (Minor Direction) to J and γ _f Coefficient Account

For the calculation of γ _{f (min)} and γ _{v (min)} coefficient, the following steps are followed.

The following method is used to find the J _(min) value.

c _(min) is the center of gravity distance perpendicular to the moment vector, which is considered when finding the J value (J _(min) ) on the strong axis of the section . Since the punching circumference is rectangular;

c_(min) = b₁ / 2 = 74/2 = 37 cm = 0.37 m c'_(min) = b₁ - c_(min) = 74 - 37 = 37 cm = 0.37 m

In this case, the polar inertia and the two moments of inertia are respectively as follows.

In this case, the sum of the polar inertia and the second moments of inertia J _(min) with respect to the minor axis of the section is found as follows.

J _(min) = J _{1 (min)} + J _{2 (min)} = 0.017914 + 0.081483 = 0.0993968 m ⁴ = 9939680 cm ⁴

Finding Punching Stresses

The forces to be considered for punching stresses and the values ​​obtained from hand geometry are given below.

V _d = 679.31 kN DM _{d (maj)} = 394.624 kNm DM _{d (min)} = 65.3914 kNm

A _z = 0.95040 m ²

γ _{f (maj)} = 0.537 γ _{v (maj)} = 0.463 γ _{f (min)} = 0.660 γ _{v (min)} = 0.340

J _(maj) = = 0.21566080 m ⁴ , J _(min) = = 0.0993968 m ⁴ ,

c _(maj) = 0.62 m, c _(min) = 0.37 m, c ' _(maj) = 0.62 m, c _(min) = 0.37 m

If we substitute all the values ​​we found above for the stress formulas together with the internal forces based on the punching design;

Among the values ​​of τ _{pd, 1} , τ _{pd, 2, the} absolute value is τ _{pd, 1} = 1323.0 kN / m ² . If we compare this value with the concrete design tensile strength f _ctd value;

τ _{pd, 1} = 1323.0 kN / m ² <f _ctd = 1380.42 kN / m ²

We reach the result. From here, we can conclude that the punching strength of this column is sufficient. These values ​​can also be compared with the report results.