 Shear Resistance Vc According to ACI 318-19

Technical Article on the Topic Structural Analysis Using Dlubal Software

• Technical Article

With the most recent ACI 318-19 standard, the long-term relationship to determine the concrete shear resistance, Vc, 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 , eight equations are specified for the calculation of the shear strength Vc - 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 Vc. Furthermore, the concept should consider the influence of the component height, the longitudinal reinforcement ratio, and the normal stress.

Shear Resistance Vc According to ACI 318-19

For non-prestressed reinforced concrete beams, the shear resistance Vc is calculated according to ACI 318-19  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, Vc. The Equation a) was basically taken from ACI 318-14 .

The determination of the shear resistance Vc according to Table 22.5.5.1 , depends on the inserted shear reinforcement Av. If the minimum shear reinforcement Av,min according to 9.6.3.4 is available or exceeded, the calculation of Vc may be performed either according to Equation a)

Shear Resistance Vc According to ACI 318-19, Table 22.5.5.1, Equation a)

$$Vc,a = 2 · λ · fc' + Nu6 Ag · bw · d$$

 Vc,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 Nu Design axial force Ag Cross-sectional area bw Width of cross-section d Effective depth

or Equation b)

Shear resistance Vc according to ACI 318-19, Table 22.5.5.1, Equation b)

$$Vc,b = 8 · λ (ρw)13 · fc' + Nu6 Ag · bw · d$$

 Vc, b Shear resistance of the concrete according to equation b) from Table 22.5.5.1 λ Factor for standard or lightweight concrete ρw Longitudinal reinforcement ratio of the tension reinforcement f'c Concrete compressive strength Nu Design axial force Ag Cross-sectional area bw Width of cross-section d Effective depth

from Table 22.5.5.1 .

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 (ρw )1/3. The longitudinal reinforcement ratio ρw influences the calculation of the shear resistance Vc. Figure 01 shows the distribution of 8 λ (ρw)1/3 as a function of ρw (with λ = 1).

For λ = 1.0, 8 λ (ρw)1/3 becomes equal to the value 2 λ for a longitudinal reinforcement ratio ρw = 1.56%. When calculating Vc, Equation a) for λ= 1 and a longitudinal reinforcement ratio ρw < 1.56% and Equation b) for λ= 1 and ρw > 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 Av < Av,min, Equation c) of Table 22.5.5.1  is to be used according to ACI 318-19 .

Shear resistance Vc according to ACI 318-19, Table 22.5.5.1, Equation c)

$$Vc,c = 8 · λs · λ · (ρw)13 · fc' + Nu6 Ag · bw · d$$

 Vc, c Shear resistance of the concrete according to equation c) from Table 22.5.5.1 λs Factor for considering the component height λ Factor for normal or lightweight concrete ρw Longitudinal reinforcement ratio of the tension reinforcement f 'c Concrete compressive strength Nu Design axial force Ag cross-sectional area bw Width of the cross-section d Effective depth

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

Factor λs for considering the component height according to ACI 318-19, Equation 22.5.5.1.3

$$λs = 21 + d10 ≤ 1$$

 λs Factor for considering the component height d Effective depth

The reduction of the shear resistance Vc, c by the factor λs is only effective for structural heights d > 10in. Figure 02 shows the distribution of term 8 λs λ (ρw)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  for a reinforced concrete beam, which was designed in a previous Knowledge Base Article according to ACI 318-14 . 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 fy = 60,000 psi. The effective depth of the tension reinforcement is applied with d = 22.5in. The design value of the acting shear force Vu at a distance d from the support is 61.10 kips.

The determination of the shear resistance Vc according to Table 22.5.5.1  depends on the height of the inserted shear reinforcement Av. The prerequisite for using Equations a) and b) is that the minimum shear reinforcement according to 9.6.3.4  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 .

Requirement of minimum shear reinforcement according to ACI 318-19, 9.6.3.1

$$Vu > λ · fc' · bw · d$$

 Vu Design load of the shear force λ Factor for normal or lightweight concrete f 'c Concrete compressive strength bw 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 , as follows.

$$av,min = Avs = max 0,75 · fc' · bwfy ; 50 · bwfy$$

 av,min Minimum shear reinforcement Av Cross-sectional area of the shear reinforcement s Distance of the stirrups f'c Concrete compressive strength bw Width of cross-section fy Yield strength of reinforcing steel

av,min = 0.12 in²/ft

When considering the minimum shear reinforcement, the concrete shear resistance Vc can now be determined with Equations a) or b) of Table 22.5.5.1 .

The shear resistance Vc,a according to Equation a) is calculated as Vc,a = 35.0 kips.

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

Vc,b = 24.52 kips

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

Additionally, the shear resistance Vc is limited to the maximum value Vc,max according to 22.5.5.1.1 .

Maximum value of shear resistance of concrete Vc, max according to ACI 318-19, 22.5.5.1.1

$$Vc,max = 5 · λ · fc' · bw · d$$

 Vc, 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 bw Width of cross-section d Effective depth

Vc,max = 87.5 kips

Finally, the calculation of the required shear reinforcement results in the following applicable concrete shear force resistance Vc.

Vc = max [Vc,a; Vc,b ] ≤ Vc,max

Vc = [35.0 kips; 24.5 kips] ≤ 87.5 kips

Vc = 35.0 kips

The required shear reinforcement req av is calculated as follows:

Required Shear Reinforcement req av According to ACI 318-19

$$req av = Vu - Φ · VcΦ · d · fy≥ av,min,9.6.3.4$$

 req av Required shear reinforcement Vu Design load of the shear force Φ Partial safety factor for shear force design according to Table 21.2.1 Vc Shear resistance of the concrete according to Table 22.5.5.1 d Effective depth fy Yield strength of reinforcing steel av,min,9.6.3.4 Minimum shear reinforcement according to 9.6.3.4

req av = 0.41 in²/ft ≥ 0.12 in²/ft

The reinforced concrete design according to ACI 318-19  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.

Check of maximum resistance of the concrete compression strut of the shear truss according to ACI 318-19, 22.5.1.2

$$Vu ≤ Vc + 8 · fc' · bw · d$$

 Vu Design load of the shear force Vc Shear resistance of the concrete according to Table 22.5.5.1 f'c Concrete compressive strength bw 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  introduced a new concept to determine the shear resistance Vc. 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 Vc.

Author Dipl.-Ing. (FH) Alexander Meierhofer

Head of Product Engineering Concrete & Customer Support

Mr. Meierhofer is the development leader of the programs for concrete structures and is available for the customer support team in the case of the questions related to reinforced and prestressed concrete design.

Reference

  ACI 318-19, Building Code Requirements for Structural Concrete and Commentary  ACI 318-14, Building Code Requirements for Structural Concrete and Commentary 