5.3 Strength verifications in Detail 3D

The different verifications required by ACI 318-19 are assessed based on the direct results provided by the model. Verifications are carried out for concrete strength, reinforcement strength, and anchorage (bond shear stresses).

Strength - Concrete

The concrete strength in compression is evaluated as the ratio between the maximum Equivalent principal stress f_c,eq (also σ_c,eq in previous text) obtained from FE analysis and the limit value f'_c,lim.

Equivalent Principal Stress expresses the equivalent uni-axial stress for a general tri-axial stress state.

$$f_{c,eq} = \sigma_{c3} - \sigma_{c1}$$

The f_c,eq value can, therefore, be directly compared with uniaxial strength limits. This expression is derived from the implementation of the Mohr-Coulomb plasticity theory, conservatively assuming the angle of internal friction φ = 0°.

Strength - Reinforcement

The strength of the reinforcement is evaluated in both tension and compression as the ratio between the stress in the reinforcement at the cracks f_s and the specified limit value f_y,lim.

$$f_{y,lim} = \phi_{s} \cdot f_{y}$$

Strength - Anchors

Anchors are checked for normal stresses in a similar way to reinforcement, where the limit value f_y,lim is determined. 

To make it easier to navigate the following text, we will first divide anchoring into three groups in terms of code-checking according to ACI or AISC.

Group 1

• Anchorage Types

  • Cast-in plate
  • Base plate - Stand-off = direct 
  • Base plate - Stand-off = Mortar joint - thickness of mortar less than 0.5 times the anchor diameter
  • Single anchor with projected length less than 0.5 times anchor diameter
• Anchor code-checks (ACI / AISC)

  • Tension/compression

    • All anchor types in tension – ACI 318-19 chap. 17.6.1.2  
    • All anchor types in compression – AISC 360-16 chap. E
  • Shear without lever arm

    • Bolt material – ACI 318-19 chap. 17.7.1.2 (b)
    • Headed studs – ACI 318-19 chap. 17.7.1.2 (a)
    • Reinforcement – ACI 318-19 chap. 17.7.1.2 (b)
  • Interaction of tension and shear - ACI 318-19 chap. 17.8

Group 2

• Anchorage Types

  • Base plate - Stand-off = Mortar joint - thickness of mortar more than 0.5 times the anchor diameter
• Anchor code-checks (ACI / AISC)

  • Tension/compression

    • All anchor types in tension – ACI 318-19 chap. 17.6.1.2  
    • All anchor types in compression – AISC 360-16 chap. E
  • Shear with lever arm

    • Bolt material – ACI 318-19 chap. 17.7.1.2 (b) + chap. 17.7.1.2.1.
    • Headed studs – ACI 318-19 chap. 17.7.1.2 (a) + chap. 17.7.1.2.1.
    • Reinforcement – ACI 318-19 chap. 17.7.1.2 (b) + chap. 17.7.1.2.1.
  • Interaction of tension and shear - ACI 318-19 chap. 17.8

Group 3

• Anchorage types

  • Base plate - Stand-off = gap
  • Single anchor with projected length more than 0.5 times the anchor diameter
• Anchor code-checks (ACI / AISC)

  • Tension/compression (with buckling)

    • All anchor types in tension – ACI 318-19 chap. 17.6.1.2
    • All anchor types in compression – AISC 360-16 chap. E3
  • Bending

    • For all anchor types – AISC 360-16 chap. F11
  • Shear

    • For all anchor types – AISC 360-16 chap. G
  • Interaction of axial force and bending

    • $\dfrac{N}{P_n}+\dfrac{M}{M_n}\le 1$ 

Tensile resistance of anchor according to ACI 318-19 chap. 17.6.1.2

$$\phi N_{sa}=\phi_{a,t}\,A_{se,N}\,f_{uta}$$

where:

• ϕ_a,t  – strength reduction factor for anchors in tension according to ACI 318-19 chap. 17.5.3 (a)
• A_se,N – tensile stress area (reduced by thread)
• f_uta – specified tensile strength of anchor steel and shall not be greater than 1.9 f_ya and 860 MPa

Shear resistance of anchor according to ACI 318-19 chap. 17.7.1.2 (a)

The steel strength in shear for headed studs is determined as:

$$\phi V_{sa}=\phi_{a,V}\,A_{se,V}\,f_{uta}$$

where:
ϕ_a,v – strength reduction factor for anchors in tension according to ACI 318-19 chap. 17.5.3 (a)
A_se,V – tensile stress area (reduced by thread)
f_uta – specified tensile strength of anchor steel and shall not be greater than 1.9 f_ya and 860 MPa

Shear resistance of anchor according to ACI 318-19 chap. 17.7.1.2 (b)

The steel strength in shear for anchors from bolt material and reinforcement is determined as:

$$\phi V_{sa}=\phi_{a,V}\,0.6\,A_{se,V}\,f_{uta}$$

where:

• ϕ_a,v  – strength reduction factor for anchors in tension according to ACI 318-19 chap. 17.5.3 (a)
• A_se,V – tensile stress area (reduced by thread)
• f_uta – specified tensile strength of anchor steel and shall not be greater than 1.9 f_ya and 860 MPa

Shear resistance of anchor connected to a base with mortar - ACI 318-19 chap. 17.7.1.2.1

If anchors are used with built-up grout pads (Group 2), the design strength calculated in accordance with 17.7.1.2 shall be multiplied by 0.80.

Interaction in tension and shear according to ACI 318-19 chap. 17.8

It shall be permitted to neglect the interaction between tension and shear if (a) or (b) is satisfied.
(a) N_ua/(ϕN_n) ≤ 0.2
(b) V_ua/(ϕV_n) ≤ 0.2 

If N_ua/(ϕN_n) > 0.2 for the governing strength in tension and V_ua/(ϕV_n) > 0.2 for the governing strength in shear, then Eq. (17.8.3) shall be satisfied.

$$\frac{N_{ua}}{\phi N_n}+\frac{V_{ua}}{\phi V_n}\le 1.2$$

Compressive resistance of anchor according to AISC 360-16 chap. E3

$$P_n =\phi_{a,c}\, F_{cr}\, A_{g}$$

where:

• ϕ_a,t  – the strength reduction factor for anchors in compression according to AISC 360-16 chap. E1
  
• (a) When: $\dfrac{L_c}{r} \le 4.71\sqrt{\dfrac{E}{F_y}}\quad$  or     $\dfrac{F_y}{F_e}\le 2.25$

  • $F_{cr}=\left(0.658^{\,F_y/F_e}\right)F_y$
    
• (b) When: $\dfrac{L_c}{r} > 4.71\sqrt{\dfrac{E}{F_y}}\quad$  or     $\dfrac{F_y}{F_e}> 2.25$

  • $F_{cr}=0.877F_e$
    
• A_g​ – gross cross-sectional area of member
• E – modulus of elasticity of steel
• $F_e=\dfrac{\pi^2 E}{\left(\dfrac{L_c}{r}\right)^2}$ - elastic buckling stress
• F_y – specified minimum yield stress of the type of steel being used
• $r=\sqrt{\dfrac{I}{A_s}}$ – radius of gyration
• $I=\dfrac{\pi d_s^4}{64}$ – moment of inertia of the bolt 

Bending resistance of anchor according to AISC 360-16 chap. F11

$$M_n=\phi_{a,b}\, Z\, F_y\, \le 1.6\,\phi_{a,b}\, S_x\, F_y$$

where:

• $Z=\dfrac{d_s^{3}}{6}$ – plastic section modulus of the bolt
• $S_x=\dfrac{2I}{d_s}$ – elastic section modulus of the bolt

Shear resistance of anchor according to AISC 360-16 chap. G

$$V_n=\phi_{a,v}\,0.6\,A_v\,F_y$$

where:

• A_V = 0.844As – the shear area
• A_s – the bolt area reduced by threads

Concrete crushing at the anchor–concrete interface

The anchor shear resistance is also limited from the point of view of concrete crushing at the anchor–concrete interface. The limit values and the method for determining them are described in detail in the article - Shear behaviour of anchors in reinforced concrete. Once the contact force reaches this limit, the stop criterion is triggered, and the analysis is terminated before the resistance is exceeded.​ 

Pull-out check for headed anchors (Washer plates and Headed studs)

For headed anchors, an additional stop criterion is implemented to check the concrete bearing (crushing) above the anchor head - pull-out. During the analysis, the compressive force transferred through the head-to-concrete contact is monitored and compared with the limit value given by ACI 318-19, Clause 17.6.3.2.2a (pull-out failure of headed fastenings).

$$N_{pn} = \Phi \cdot \Psi_{c,p} \cdot 8 \cdot A_{brg} \cdot f'_c$$
​

where:

• $\Phi$ is the strength reduction factor - Table 17.5.3(c)
• A_brg net bearing area of the head of stud, anchor bolt, or headed deformed bar (without the shank area). 
• f'_c is the specified compressive strength of concrete
• $\Psi_{c,p}$ is the pullout cracking factor according to 17.6.3.3, and is always taken as 1.0, i.e. the value for cracked concrete. This is consistent with the CSFM approach used in Detail, where the tensile strength of concrete is neglected and the concrete is assumed to be cracked in tension.

Once the contact force reaches this code-based limit, the stop criterion is triggered, and the analysis is terminated before the pull-out resistance is exceeded.​ 

Anchorage -  Bond stress

The bond shear stress is evaluated independently as the ratio between the bond stress τ_b calculated by FE analysis and the bond strength f_bu.

Although the bond strength is not explicitly defined in ACI 318-19, the calculation of the development length can be found in Section 25.4.2. However, since the bond strength is the basic input for determining the development length, see R25.4.1.1 and ACI Committee 408 1966, the bond strength can be calculated as follows:

Let us assume that if we anchor the reinforcement bar into a concrete block to the development length l_d or greater, pulling out the reinforcement will lead to rupture of the reinforcement and not to pulling out of the concrete. This can be written with the following formula.

$$\pi\cdot d_{b} \cdot l_{d} \cdot f_{bu}=f_{y}\cdot A_{s}$$

where:

d_b is the diameter of the reinforcement bar, l_d is the development length, f_bu is the bond strength, f_y is the yield strength of the reinforcement, and A_s is the area of the reinforcement rebar.

From the preceding, the formula for calculating bond strength can be easily derived:

$$f_{bu}=\frac{f_{y}\cdot A_{s}}{\pi\cdot d_{b} \cdot l_{d} }$$

The development length l_d is then determined according to ACI 318-19 Table 25.4.2.3 as follows:

$$l_{d}=\left( \frac{f_{y}\cdot\psi_{t}\cdot\psi_{e}\cdot\psi_{g}}{C\cdot\lambda\sqrt{f'_{c}}} \right)\cdot d_{b}$$

where:

C = 25 (2.1 for metric) for no. 6 and smaller bars and deformed wires, C = 20 (1.7 for metric) for no. 7 and larger bars, λ = 1.0 for normal weight concrete, ψ_t, ψ_e, ψ_g are determined according to ACI 318-19 Table 25.4.2.3. 

Only uncoated or zinc-coated (galvanized) reinforcement is supported, so ψ_e = 1.0. ψ_g is automatically determined from the reinforcement grade, and ψ_t is automatically derived from the position of the reinforcement in the model and from the direction of concreting that can be set in the application for each project item as follows.

$$\textsf{\textit{\footnotesize{Fig. 5.3.1\qquad Direction of concreting}}}$$

These verifications are carried out with respect to the appropriate limit values for the respective parts of the structure (i.e., in spite of having a single grade both for concrete and reinforcement material, the final stress-strain diagrams will differ in each part of the structure due to tension stiffening and compression softening effects).

Anchorage -  Total force

Total force F_tot and limit force F_lim

The total force F_tot is a result of the finite element analysis and can be defined in two ways.

$$F_{tot}=A_{s} \cdot f_{s}$$

where A_s is the area of the reinforcement bar and f_s is the stress in the bar.

Or as a sum of the anchorage force F_a and the bond force F_bond.

$$F_{tot}=F_{a}+F_{bond}$$

where F_a is the actual force in the anchorage spring and F_bond is the bond force that can be obtained by integrating the bond stress τ_b along the length of reinforcement bar l.

$$F_{bond}=C_{s} \cdot \int_{0}^{l}\tau_{b}\left( x \right)dx$$

C_s is the circumference of the reinforcement bar.

The limit force F_lim is the maximum force in the element of the rebar considering the strength of the rebar and also anchoring conditions (bond between concrete and reinforcement and anchorage hooks, loops, etc.).

$$F_{lim}=min\left( F_{lim,bond}+F_{au},F_{u} \right)$$

$$F_{u}=f_{y,lim}\cdot A_{s}$$

$$F_{au}=\beta\cdot f_{y,lim}\cdot A_{s}$$

$$F_{lim,bond}=C_{s}\cdot l \cdot f_{bu}$$

where C_s is the circumference of the reinforcement bar, and l is the length from the beginning of the rebar to the point of interest.

$$\textsf{\textit{\footnotesize{Fig. 5.3.2\qquad Definition of the limit force Flim}}}$$

$$F_{lim,2}=F_{lim,1}+F_{lim,add}$$

where F_lim,add is the additional force calculated from the magnitude of the angle between neighboring elements. F_lim,2 must be always lower than F_u.

The available anchorage types in CSFM include a straight bar (i.e., no anchor end reduction), 90-degree hook, 180-degree hook, perfect bond, and continuous bar. All these types, along with the respective anchorage coefficients β, are shown in Fig. 5.3.3 for longitudinal reinforcement. The values of the adopted anchorage coefficients are derived from the comparison of the equation from section ACI 318-19 25.4.3.1 and equations taken from section ACI 318-19 25.4.2.3. It should be noted that, in spite of the different available options, CSFM distinguishes three types of anchorage ends: (i) no reduction in the anchorage length, (ii) a reduction of 30% of the anchorage length in the case of a normalized anchorage, and (iii) perfect bond.

$$\textsf{\textit{\footnotesize{Fig. 5.3.3\qquad  Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in CSFM:}}}$$

$$\textsf{\textit{\footnotesize{(a) straight bar; (b) 90-degree hook; (c) 180-degree hook; (d) perfect bond; (e) continuous bar}}}$$

The anchorage coefficient for stirrups is always - β = 1.0.

In order to comply with ACI, the anchorage spring should be used in the calculation, the anchorage spring is modified by the β coefficient so the user must use one of the available anchorage types when defining the reinforcement start and end conditions.