# Shear Resistance Vc According to ACI 318-19

### Technical Article

With the most recent ACI 318-19 standard, the long-term relationship to determine the concrete shear resistance, V_{c}, is redefined. With the new method, the member height, the longitudinal reinforcement ratio, and the normal stress now influence the shear strength, Vc. This article describes the shear design updates, and the application is demonstrated with an example.

#### Introduction

In the previous standard ACI 318-14 [2], eight equations are specified for the calculation of the shear strength V_{c} - without considering the application limits. The user can choose between a simplified and an exact calculation method. One of the objectives of the new concept in ACI 318-19 was to reduce the design equations for V_{c}. Furthermore, the concept should consider the influence of the component height, the longitudinal reinforcement ratio, and the normal stress.

#### Shear Resistance V_{c} According to ACI 318-19

For non-prestressed reinforced concrete beams, the shear resistance V_{c} is calculated according to ACI 318-19 [1] with the Equations a) to c) from Table 22.5.5.1. With the new Equations b) and c), the member height, the longitudinal reinforcement ratio, and the normal stress now influence the shear strength, V_{c}. The Equation a) was basically taken from ACI 318-14 [2].

The determination of the shear resistance V_{c} according to Table 22.5.5.1 [1], depends on the inserted shear reinforcement A_{v}. If the minimum shear reinforcement A_{v,min} according to 9.6.3.4 is available or exceeded, the calculation of V_{c} may be performed either according to Equation a)

${\mathrm{V}}_{\mathrm{c},\mathrm{a}}=\left[2\xb7\mathrm{\lambda}\xb7\sqrt{{\mathrm{f}}_{\mathrm{c}}^{\text{'}}}+\frac{{\mathrm{N}}_{\mathrm{u}}}{6{\mathrm{A}}_{\mathrm{g}}}\right]\xb7{\mathrm{b}}_{\mathrm{w}}\xb7\mathrm{d}$

V_{c,a} |
Concrete shear resistance according to Equation a) from Table 22.5.5.1 |

λ |
Factor for standard or lightweight concrete |

f'_{c} |
Concrete compressive strength |

N_{u} |
Design axial force |

A_{g} |
Cross-sectional area |

b_{w} |
Width of cross-section |

d |
Effective depth |

or Equation b)

${\mathrm{V}}_{\mathrm{c},\mathrm{b}}=\left[8{\xb7\mathrm{\lambda}\mathrm{}\left({\mathrm{\rho}}_{\mathrm{w}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\xb7\sqrt{{\mathrm{f}}_{\mathrm{c}}^{\text{'}}}+\frac{{\mathrm{N}}_{\mathrm{u}}}{6{\mathrm{A}}_{\mathrm{g}}}\right]\xb7{\mathrm{b}}_{\mathrm{w}}\xb7\mathrm{d}$

V_{c, b} |
Shear resistance of the concrete according to equation b) from Table 22.5.5.1 |

λ |
Factor for standard or lightweight concrete |

ρ_{w} |
Longitudinal reinforcement ratio of the tension reinforcement |

f'_{c} |
Concrete compressive strength |

N_{u} |
Design axial force |

A_{g} |
Cross-sectional area |

b_{w} |
Width of cross-section |

d |
Effective depth |

from Table 22.5.5.1 [1].

If you compare the two equations shown above, you can see that in Equation b), the factor 2 λ has been replaced by the term 8 λ (ρ_{w} )^{1/3}. The longitudinal reinforcement ratio ρ_{w} influences the calculation of the shear resistance V_{c}. Figure 01 shows the distribution of 8 λ (ρ_{w})^{1/3} as a function of ρ_{w} (with λ = 1).

For λ = 1.0, 8 λ (ρ_{w})^{1/3} becomes equal to the value 2 λ for a longitudinal reinforcement ratio ρ_{w} = 1.56%. When calculating V_{c, }Equation a) for λ= 1 and a longitudinal reinforcement ratio ρ_{w} < 1.56% and Equation b) for λ= 1 and ρ_{w} > 1.56% results in the greater concrete shear resistance. The standard allows the application of both equations. Therefore, the maximum value from Equations a) and b) can be used for a cost-effective design.

For beams with shear reinforcement A_{v} < A_{v,min}, Equation c) of Table 22.5.5.1 [1] is to be used according to ACI 318-19 [1].

${\mathrm{V}}_{\mathrm{c},\mathrm{c}}=\left[8\xb7{\mathrm{\lambda}}_{\mathrm{s}}\xb7{\mathrm{\lambda}\mathrm{}\xb7\left({\mathrm{\rho}}_{\mathrm{w}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\xb7\sqrt{{\mathrm{f}}_{\mathrm{c}}^{\text{'}}}+\frac{{\mathrm{N}}_{\mathrm{u}}}{6{\mathrm{A}}_{\mathrm{g}}}\right]\xb7{\mathrm{b}}_{\mathrm{w}}\xb7\mathrm{d}$

V_{c, c} |
Shear resistance of the concrete according to equation c) from Table 22.5.5.1 |

λ_{s} |
Factor for considering the component height |

λ |
Factor for normal or lightweight concrete |

ρ_{w} |
Longitudinal reinforcement ratio of the tension reinforcement |

f '_{c} |
Concrete compressive strength |

N_{u} |
Design axial force |

A_{g} |
cross-sectional area |

b_{w} |
Width of the cross-section |

d |
Effective depth |

Except for variable λ_{s}, Equation c) is similar to Equation b) discussed above. For structural components with little or no shear reinforcement, the concrete shear resistance V_{c } decreases with increasing structural component height. By introducing the factor λ_{s}, the "Size Effect" is taken into account. The factor λ_{s} is determined according to Equation 22.5.5.1.3 [1] as follows.

${\mathrm{\lambda}}_{\mathrm{s}}=\sqrt{\frac{2}{1+\raisebox{1ex}{$\mathrm{d}$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}}\le 1$

λ_{s} |
Factor for considering the component height |

d |
Effective depth |

The reduction of the shear resistance V_{c, c} by the factor λ_{s} is only effective for structural heights d > 10in. Figure 02 shows the distribution of term 8 λ_{s} λ (ρ_{w})^{1/3} for the different effective depths d.

#### Example: Calculate the Required Shear Reinforcement According to ACI 318-19

The following section describes how to determine the required shear reinforcement according to the new concept of ACI 318-19 [1] for a reinforced concrete beam, which was designed in a previous Knowledge Base Article according to ACI 318-14 [2]. Figure 03 shows the structural model and the design load.

The rectangular cross-section has the dimensions 25in · 11in. The concrete has a compressive strength of f'_{c} = 5,000 psi. The yield strength of the reinforcing steel used is f_{y} = 60,000 psi. The effective depth of the tension reinforcement is applied with d = 22.5in. The design value of the acting shear force V_{u} at a distance d from the support is 61.10 kips.

The determination of the shear resistance V_{c} according to Table 22.5.5.1 [1] depends on the height of the inserted shear reinforcement A_{v}. The prerequisite for using Equations a) and b) is that the minimum shear reinforcement according to 9.6.3.4 [1] is applied. For this reason, a check is performed in the first step whether a minimum reinforcement has to be considered according to 9.6.3.1 [1].

${\mathrm{V}}_{\mathrm{u}}\mathrm{\lambda}\xb7\sqrt{{\mathrm{f}}_{\mathrm{c}}^{\text{'}}}\mathrm{}\xb7{\mathrm{b}}_{\mathrm{w}}\xb7\mathrm{d}$

V_{u} |
Design load of the shear force |

λ |
Factor for normal or lightweight concrete |

f '_{c} |
Concrete compressive strength |

b_{w} |
Width of the cross-section |

d |
Effective depth |

61.10 kips > 13.13 kips

This requires a minimum shear reinforcement. This is calculated according to 9.6.3.4 [1], as follows.

${\mathrm{a}}_{\mathrm{v},\mathrm{min}}=\frac{{\mathrm{A}}_{\mathrm{v}}}{\mathrm{s}}=\mathrm{max}\left[0,75\xb7\sqrt{{\mathrm{f}}_{\mathrm{c}}^{\text{'}}}\xb7\frac{{\mathrm{b}}_{\mathrm{w}}}{{\mathrm{f}}_{\mathrm{y}}};50\xb7\frac{{\mathrm{b}}_{\mathrm{w}}}{{\mathrm{f}}_{\mathrm{y}}}\right]$

a_{v,min} |
Minimum shear reinforcement |

A_{v} |
Cross-sectional area of the shear reinforcement |

s |
Distance of the stirrups |

f'_{c} |
Concrete compressive strength |

b_{w} |
Width of cross-section |

f_{y} |
Yield strength of reinforcing steel |

a_{v,min} = 0.12 in²/ft

When considering the minimum shear reinforcement, the concrete shear resistance V_{c }can now be determined with Equations a) or b) of Table 22.5.5.1 [1].

The shear resistance V_{c,a} according to Equation a) is calculated as V_{c,a} = 35.0 kips.

To apply Equation b), it is necessary to know the longitudinal reinforcement ratio ρ_{w}. To be able to compare the calculated shear reinforcement with the calculation result of RF-CONCRETE Members, ρ_{w }is determined with the required longitudinal reinforcement at the distance d from the support. A bending moment of M_{y,u} = 1533 kip-in results in a longitudinal reinforcement of A_{s,req} = 1.33 in², which is ρ_{w} = 0.536%. Figure 01 shows the influence of the longitudinal reinforcement ratio ρ_{w} on the calculation of V_{c,b}. Since ρ_{w} < 1.5% here, Equation b) will result in a lower shear resistance V_{c,b} than Equation a) and we can skip determining V_{c,b}. However, we calculate V_{c,b }to show it.

V_{c,b} = 24.52 kips

As expected, Equation b) provides a lower shear resistance than Equation a).

Additionally, the shear resistance V_{c} is limited to the maximum value V_{c,max} according to 22.5.5.1.1 [1].

${\mathrm{V}}_{\mathrm{c},\mathrm{max}}=5\xb7\mathrm{\lambda}\xb7\sqrt{{\mathrm{f}}_{\mathrm{c}}^{\text{'}}}\xb7{\mathrm{b}}_{\mathrm{w}}\xb7\mathrm{d}$

V_{c, max} |
Maximum value of the shear resistance of the concrete according to Equation 22.5.5.1.1 |

λ |
Factor for standard or lightweight concrete |

f'_{c} |
Concrete compressive strength |

b_{w} |
Width of cross-section |

d |
Effective depth |

V_{c,max} = 87.5 kips

Finally, the calculation of the required shear reinforcement results in the following applicable concrete shear force resistance V_{c}.

V_{c} = max [V_{c,a}; V_{c,b} ] ≤ V_{c,max}

V_{c} = [35.0 kips; 24.5 kips] ≤ 87.5 kips

V_{c} = 35.0 kips

The required shear reinforcement req a_{v} is calculated as follows:

${\mathrm{req}\mathrm{}\mathrm{a}}_{\mathrm{v}}=\frac{{\mathrm{V}}_{\mathrm{u}}-\mathrm{\Phi}\xb7{\mathrm{V}}_{\mathrm{c}}}{{\mathrm{\Phi}\xb7\mathrm{d}\xb7\mathrm{f}}_{\mathrm{y}}}{\ge \mathrm{a}}_{\mathrm{v},\mathrm{min},9.6.3.4}$

req a_{v} |
Required shear reinforcement |

V_{u} |
Design load of the shear force |

Φ |
Partial safety factor for shear force design according to Table 21.2.1 |

V_{c} |
Shear resistance of the concrete according to Table 22.5.5.1 |

d |
Effective depth |

f_{y} |
Yield strength of reinforcing steel |

a_{v,min,9.6.3.4} |
Minimum shear reinforcement according to 9.6.3.4 |

req a_{v} = 0.41 in²/ft ≥ 0.12 in²/ft

The reinforced concrete design according to ACI 318-19 [1] can be performed with RFEM. The RF-CONCRETE Members add-on module also calculates a required shear reinforcement of 0.41 in²/ft at the distance d from the support (see Figure 04).

Finally, the maximum load capacity of the concrete compression strut of the shear truss is verified according to Section 22.5.1.2.

${\mathrm{V}}_{\mathrm{u}}\le {\mathrm{V}}_{\mathrm{c}}+8\xb7\sqrt{{\mathrm{f}}_{\mathrm{c}}^{\text{'}}}\xb7{\mathrm{b}}_{\mathrm{w}}\xb7\mathrm{d}$

V_{u} |
Design load of the shear force |

V_{c} |
Shear resistance of the concrete according to Table 22.5.5.1 |

f'_{c} |
Concrete compressive strength |

b_{w} |
Width of cross-section |

d |
Effective depth |

61.10 kips ≤ 175.0 kips

The shear design according to ACI 318-19 is fulfilled.

#### Summary

ACI 318-19 [1] introduced a new concept to determine the shear resistance V_{c}. It was possible to reduce the number of potential design equations from the previous version to three equations while taking into account the influence of the normal stress, the component height, and the longitudinal reinforcement ratio. This simplifies the calculation of the shear resistance V_{c}.

#### Author

#### Dipl.-Ing. (FH) Alexander Meierhofer

Head of Product Engineering Concrete & Customer Support

#### Keywords

Reinforced concrete Shear design ACI 318-19 Shear force Shear resistance Vc

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