# RF-CONCRETE Members – Online Manual Version 5

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## RF-CONCRETE Members – Online Manual Version 5

# 2.1.2 Shear Force

### Shear Force

The check of shear force resistance is to be performed only in the ultimate limit state (ULS). The actions and resistances are considered with their design values. The general design requirement according to EN 1992-1-1, clause 6.2.1 is the following:

- V
_{Ed}≤ V_{Rd}

where

- V
_{Ed}: design value of applied shear force - V
_{Rd}: design value of shear force resistance

Depending on the failure mechanism, the design value of the shear force resistance is determined by one of the following three values.

- V
_{Rd,c}: design shear resistance of a structural component without shear reinforcement - V
_{ Rd,s}: design shear resistance of a structural component with shear reinforcement, limited by the yield point of shear reinforcement (failure of tie) - V
_{Rd, max}: design shear resistance limited by strength of concrete compression strut

If the acting shear force V_{Ed} remains below the value of V_{Rd,c}, no calculated shear reinforcement is necessary and the check is verified.

If the applied shear force V_{Ed} is higher than the value of V_{Rd,c}, a shear reinforcement must be designed.
The shear reinforcement must resist the entire shear force.
In addition, the bearing capacity of the concrete compression strut must be analyzed.

- V
_{Ed}≤ V_{Rd,s}and V_{Ed}≤ V_{Rd,max}

The various types of shear force resistance are determined according to EN 1992-1-1 as follows.

The design value for the design shear resistance V_{Rd,c} may be determined with:

${V}_{Rd,c}=\left[{C}_{Rd,c}\xb7k{\left(100{\sigma}_{l}\xb7{f}_{ck}\right)}^{\frac{1}{3}}-{k}_{1}\xb7{\sigma}_{cp}\right]{b}_{w}\xb7d$

Equation 2.1 EN 1992-1-1, Eq. (6.2a)

where

C | recommended value: 0.18 |

scaling factor for considering cross-section depth | |

ratio of longitudinal reinforcement | |

f | characteristic value of concrete compressive strength in [N/mm |

k | recommended value: 0.15 |

b | minimum cross-section width within tension zone in [mm] |

d | static effective depth of bending reinforcement in [mm] |

design value of concrete longitudinal stress in [N/mm |

It is allowed, however, to apply a minimum value of the shear force resistance V_{Rd,c,min}.

${V}_{Rd,c,\text{min}}=\left[{v}_{\text{min}}+{k}_{1}\xb7{\sigma}_{cp}\right]\xb7{b}_{w}\xb7d$

Equation 2.2 EN 1992-1-1, Eq. (6.2b)

where

The following applies for structural components with shear reinforcement running perpendicular to the component's axis (α = 90°):

${V}_{Rd,s}=\frac{{A}_{sw}}{s}\xb7z\xb7{f}_{ywd}\xb7\text{cot}\theta $

Equation 2.3 EN 1992-1-1, Eq. (6.8)

where

A | cross-sectional area of shear reinforcement |

s | spacing of links |

z | lever arm of the internal forces assumed with 0.9 d |

f | design yield strength of shear reinforcement |

θ | inclination of concrete compression strut |

The inclination of the concrete compression strut θ may be selected within certain limits depending on the loading. This way, the equation can take into account the fact that a part of the shear force is resisted by crack friction and the virtual truss is thus less stressed. The following limits are recommended in equation (6.7) of EN 1992-1-1:

- 1 ≤ cot θ ≤ 2.5

Thus, the compression strut inclination θ can vary between the following values:

| Minimum inclination | Maximum inclination |
---|---|---|

θ | 21.8° | 45.0° |

cot θ | 2.5 | 1.0 |

The following applies for structural components with shear reinforcement running perpendicular to the component's axis (α = 90°):

${V}_{Rd,\text{max}}=\frac{{\alpha}_{cw}\xb7{b}_{w}\xb7z\xb7{\nu}_{1}\xb7{f}_{cd}}{\text{cot}\theta +\text{tan}\theta}$

Equation 2.4 EN 1992-1-1, Eq. (6.9)

where

α | coefficient for considering stress state in compression flange |

b | cross-section width |

z | lever arm of the internal forces (precisely calculated in bending design) |

ν | reduction factor for concrete strength in case of shear cracks |

f | design value of concrete strength |

θ | inclination of concrete compression strut |