# Shear capacity of concrete

# Regions cracked in bending

In prestressed single span members without shear reinforcement, the shear resistance of the regions cracked in bending may be calculated using expression below (EN1992-1-1 eq. 6.2 a & b).

V_{Rd,c} = [C_{Rd,c} k (100 ρ_{1} f_{ck})^{1/3} + k_{1} σ_{cp}] b_{w} d

with a minimum of

V_{Rd,c} = [υ_{min}+ k_{1} σ_{cp}] b_{w} d

where

C_{Rd,c} = 0,18 / γ_{c}

k_{1} =0,15

f_{ck} is in MPa

k = 1 + (200 / d)0,5 ≤ 2,0, with d in mm

ρ_{1} = A_{sl} / (b_{w} d)

A_{sl} is the area of the tensile reinforcement, which extends ≥ (l_{bd} + d) beyond the section considered (see figure below),

b_{w} is the smallest width of the cross-section in the tensile area (mm).

σ_{cp} = N_{Ed} / A_{c} < 0,2 (MPa)

where

N_{Ed} is the axial force in the cross-section due to loading or pre stressing (in N) (N_{Ed} > 0 for compression). The influence of imposed deformations on N_{E} may be ignored.

A_{c} is the area of concrete cross-section (mm^{2}),

V_{Rd,c} is in (N)

υ_{min} = 0,035 k^{3/2} f_{ck}^{1/2}

# Regions uncracked in bending

In regions uncracked in bending (where the flexural tensile stress is smaller than f_{ctk,0,05}/gamma_{c}) the shear resistance should be limited by the flexural strength of the concrete. In these regions the shear resistance is given by EN1992-1-1 eq. 6.4:

V_{Rd,c} = I * b_{w} / S * ((f_{ctd})^{2}+α_{1}*σ_{cp}*f_{ctd})^{0.5}

Where

I is the second moment of intertia

b_{w} is the width of the cross-section at the centroidal axis

S is the first moment of the area above and about the centroidal axis

α_{1} is a factor taking transmission length into account, in PRE-Stress this value is included in σcp

σ_{cp} is the concrete compressive stress at the centroidal axis due to axial loading and/or prestressing.

The shear force V_{Ed} should always satisfy the condition

V_{Ed} ≤ 0,5 b_{w} d υ f_{cd}

where

0,6 (1 - f_{ck} / 250) (f_{ck} in MPa)

# Regions uncracked in bending (hollow core slab)

For hollow core slabs, alternative formulas for calculating shear resistance is given in EN1168 (Precast concrete products).

There are two alternatives to the equation 6.4 in EN1992-1-1, one simplified expression (4.3.3.2.2.3) and one more advanced (4.3.3.2.2.2).

So, for hollow core slabs there are three alternatives for calculating shear resistance in uncracked regions, choice of method is made in the calculation settings, 'Method for web shear failure'.

For the advanced method the following formula is used:

Where

The simplified expression is a variant of eq. 6.4 from EN1992-1-1 with two constants added:

# Shear tension capacity of a hollow core slab with a topping

Should the hollow core slab have a topping, EN1168 provides an annex F.2 in which a composite shear resistance can be calculated replacing any of the three versions above.

This formula is based on the simplified expression and takes the topping into account, this is a shear tension control rather than a shear force control.

Whether or not to use this composite shear resistance is chosen in the calculation settings, 'Composite web shear capacity according to EN1168 Annex F.2.2'.

Where

# Shear tension capacity of a hollow core slab with a number of filled cores

Should there be core fillings the calculation section in which there fillings are active an addition to the shear capacity is calculated according to EN1168 Annex F.3.

Where

# Punching shear capacity

Should there be point loads applied to a hollow core slab PRE-Stress allows for a punching shear capacity to calculated according to EN1168 4.3.3.2.4 in the ultimate limit state. This control can enabled in the calculation settings, 'Perform Punching check (EN1168)'.

Where

For point loads positioned close to a free edge of floor bay the capacity is reduced by a factor 2 during certain circumstances: the load is acting by more than 50% on the outermost web and placed reinforcement is insufficient. Sufficient reinforcement consist of there being at least one strand in the outermost web and transverse reinforcement is considered (at least 1.2 m fully anchored strips designed for a tensile force equal to the point load). The reduction is applied based on whether or not the reduction option is checked in the calculation settings.