**Part 2. Calculations of overlap for punching, assessment of the need to strengthen the overlap in the zone of punching**

During the operation of the building, cracks may appear on the section of the slab located directly around the column due to the punching of the floor slab. Punching occurs due to the shift of the plate relative to the loaded area located immediately around the column. The punching force F is assumed to be equal to the load transferred from the ceiling to the column (Fig. 1a). Figure 1b shows the layout of the transverse reinforcement in the area of the junction of the column with a flat floor slab. The force F pushing through the floor slab is perceived by the concrete of the slab F _{b} _{,} _{ult} and the transverse reinforcement F _{sw} _{,} _{ult} installed in the joint area. If the condition of the strength of the slab in terms of non-punching is not met, then cracks appear on the upper surface of the slab in the zone of the junction with the column along the perimeter of the punching (Fig. 1, view A).

Fig.1.

To determine the length of additional reinforcing bars (RS) in the above-column zone of the slab, it is taken into account that the rods must be inserted beyond the perimeter of the above-column zone for installing additional reinforcement to the length of the docking reinforcement bypass *l _{l}* . The zone for installing additional reinforcement can be assessed on the stress mosaic (Fig. 1). The dimensions of the above-column zone are determined by computer calculation of the slab and are 2800×2800 mm. The length of the reinforcement bypass is determined by the following formula:

**Calculations of a flat monolithic floor slab for punching in the zone of junction of the floor slab with columns**

In accordance with the design requirements for the location of transverse reinforcement in the punching zone, the step of transverse reinforcement (S _{w} ) must be equal to:

in the direction perpendicular to the design contour (pos. 2 in Fig. 2 and Fig. 3), – no more than S _{w} u003d h _{0} /3 and not closer than 300 mm (h _{0} u003d 166 mm, S _{w} u003d h _{0} /3 = 166/3 = 55.3 mm, S _{w} = 50 mm). It is allowed to increase the step up to S _{w} = h _{0} /2, but in this case, the most unfavorable location of the punching pyramid should be considered and when performing the calculation, only reinforcing bars crossing the punching pyramid should be taken into account;

the rods closest to the contour of the load application area (item 3 in Fig. 2 and Fig. 3) are located no closer than h _{0} /3 (h _{0} /3 = 166/3 = 55.3 mm) and no further than h _{0} / 2 (h _{0} /2 = 166/2 = 83 mm);

in the direction parallel to the sides of the design contour – no more than 0.25´a _{1} and no more than 0.25´b _{1} (a _{1} , b _{1} – the length of the side of the contour of the load application area). When the dimensions of the cross section of the column are 400´400 mm, the step of the transverse rods is taken as S _{w} = 0.25´400 = 100 mm.

The concentrated force F, pushing through the floor slab, is assumed to be equal to the load transferred from the floor to the column. (F = 18.5´6.4´6.4 = 757.76 kN). Figure 2 shows a diagram showing the calculated cross section (1), the contour of the calculated cross section (2) and the contour of the load application area (3).

*The calculation of elements without transverse reinforcement for punching under the action of a concentrated force F* is carried out from the condition: , where:

F – concentrated force from external load (F = 757.76 kN);

We carry out the calculation for concrete of class B25.

F _{b} _{,} _{ult} – ultimate force perceived by concrete (F _{b} _{,} _{ult} = g _{b} _{1} ´R _{bt} ´u´h _{0} , R _{bt} = 1050 kN/m ^{2} ;^{ }R _{bt} u003d 0.9´1050 kN / m ^{2} – design tensile strength of concrete, taking into account the coefficient of concrete working conditions g _{b} _{1} u003d 0.9;

u = 566´4 = 2.264 m – perimeter F _{b} _{,} _{ult} = (see Fig. 2);

h _{0} u003d 0.166 m – reduced working height of the slab section (h _{0} u003d h _{01} + h _{02} ).

F _{b} _{,} _{ult} = 0.9´1050´2.264´0.166 = 355.15 kN)

Condition (757.76 kN > 355.15 kN) is not performed.

Requires installation of transverse reinforcement.

*The calculation of elements with transverse reinforcement for punching under the action of a concentrated force F* is carried out from the condition: , where:

F – concentrated force from external load (F = 757.76 kN);

F _{b} _{,} _{ult} – ultimate force perceived by concrete (F _{b} _{,} _{ult} = 355.15 kN);

F _{sw} _{,} _{ult} – limiting force perceived by the reinforcement during punching.

When performing the calculation for punching, the area of the transverse reinforcement A _{sw} is taken into account, located within a distance of 0.5´h _{0} on both sides of the contour of the calculated cross-section, that is, in total within h _{0} (see Fig. 3).

Fig.2. Scheme for the calculation for punching with indication of the design cross section (1), the contour of the design cross section (2) and the contour of the area of application of the load (3) |

Figure 3 shows the arrangement of evenly distributed transverse reinforcement in the punching zone of the floor slab. The transverse reinforcement bars are marked, the area of which (A _{sw} ) is taken into account when calculating the floor slab for punching, and the pitch of the transverse reinforcement bars S _{w} is also shown.

Before determining the limit force F _{sw} _{,} _{ult} , perceived by the reinforcement during punching, it is necessary to set the diameter and class of the transverse reinforcement (Æ8А400), determine the cross-sectional area of the transverse reinforcement within the spacing of the bars S _{w} = 0.1 m (А _{sw} = 1.01´10 ^{-4} m ^{2} for two rods with a diameter of 8 mm), determine the design resistance of the transverse reinforcement to tension R _{sw} = 28´10 ^{4} kN / m ^{2} according to the assigned reinforcement class (A400), determine the distributed force in the transverse reinforcement.

The distributed force in the transverse reinforcement is determined by the formula:

The ultimate force F _{sw} _{,} _{ult} , perceived by the reinforcement during punching, taking into account the perimeter of the calculated cross-sectional contour u = 2.264 m, is determined by the formula:

.

757.76 kN<355.15 kN+512.2 kN. 757.76 kN<867.35 kN.

The strength of the floor slab for punching under the action of a concentrated force F and with a uniformly distributed transverse reinforcement in the punching zone is ensured.

But, in addition to checking the condition of punching strength of a slab with transverse reinforcement according to the formula , it is also necessary to evaluate the punching strength of the plate outside the reinforcement zone. In this case, the calculation is carried out without taking into account the transverse reinforcement, and at the same time, the contour of the calculated cross section is considered, indicated as position 1 in Figure 3. The perimeter of the contour is 6.1 m ( u = (0.083 + 0.2 + 0.08 + 0, 4+0.08+0.2+0.083)´4 = 6.1 m).

The ultimate force perceived by concrete F _{b} _{,} _{ult} , is 956.91 kN (F _{b} _{,} _{ult} = F _{b} _{,} _{ult} = g _{b} _{1} ´R _{bt} ´u´h _{0} = 0.9´1050´6.1´0.166 = 956.91 kN).

The condition of non-punching along the contour of the calculated cross section without taking into account the transverse reinforcement in the calculation: , 757.76<956.91 - in progress.

Fig.3. Arrangement of evenly distributed transverse reinforcement in the punching zone (1-contour of the design cross-section without taking into account the transverse reinforcement in the calculation, 2-contour of the design cross-section, 3-contour of the load application area) |

Another option for the location of the transverse reinforcement in the punching zone is the cruciform arrangement of the transverse reinforcement. Figure 4 shows a cruciform arrangement of transverse reinforcement, the pitch of which is taken similar to that shown in Figure 3.

With a cruciform arrangement of transverse reinforcement in the punching zone, the perimeter of the contour *u* for calculating the ultimate bearing capacity of transverse reinforcement is taken according to the actual length of the sections of the location of the transverse reinforcement *L _{swx}* and

*L*along the design punching contour (see Fig. 4).

_{swy} Contour perimeter u = 2´( *L _{swx}* +

*L*= 1.6 m. . Ultimate force perceived by concrete F

_{swy})_{b}

_{,}

_{ult}= 355.15 kN.

Strength condition .

757.76 kN>355.15 kN+361.98 kN. 757.76 kN>717.13 kN – the condition is not met.

To fulfill the strength condition, it is necessary to increase the class of concrete by two steps. For class B35 concrete R _{b} _{t} = 1300 kN/m ^{2} , F _{b} _{,} _{ult} = 0.9´1300´2.264´0.166 = 436.7 kN.

436.7 kN + 361.98 kN = 798.68 kN > 757.76 kN – the condition is met.

Rice. 4. Cross-shaped arrangement of transverse reinforcement in the punching zone (1-contour of the design cross-section without taking into account the transverse reinforcement in the calculation, 2-contour of the design cross-section, 3-contour of the load application area) |

When calculating elements with transverse reinforcement for punching under the action of a concentrated force F, it is also necessary to ensure that the following requirements are met:

the force that the concrete and reinforcement perceives when calculating for punching F _{b} _{,} _{ult} + F _{sw} _{,} _{ult} should be no more than 2´F _{b} _{,} _{ult} , that is, 798.68 kN < 2´436.7 = 873.4 kN - condition performed;

when calculating for punching, transverse reinforcement can be taken into account in the calculation if F _{sw} _{,} _{ult} ³0.25´ F _{b} _{,} _{ult} , 361.98 kN ³ 0.25´436.7 kN = 109.18 kN – the condition is met.

The assessment of the strength of the slab for punching outside the reinforcement zone is carried out taking into account the perimeter u = 4´(0.4+0.697) = 4.39 m of the calculated cross section of contour 1 (see Fig. 5).

Strength condition: . F _{b} _{,} _{ult} = 0.9´1300´4.39´0.166 = 852.6 kN >757 kN – the condition is met.

When calculating a floor slab for punching, in addition to the action of a concentrated force F, the bending moment M can be taken into account in the calculation.

Rice. Fig. 5. Layout of reinforcing cages K1 and K2, including transverse reinforcement in the punching zone (1 – contour of the calculated cross section without taking into account the transverse reinforcement in the calculation, 2 – contour of the calculated cross section, 3 – contour of the load application area) |

**Re-calculation for punching at F _{1} =1.3F**

Describe measures to strengthen the overlap in the punching zone.

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