(a) Flexural strength

(b) Axial strength or combined flexural and axial strength

(c) One-way shear strength

(d) Two-way shear strength

(e) Torsional strength

(f) Bearing

(g) Shear friction

**ϕ**given in Chapter 21.

*Equilibrium and strain compatibility*

**0.85**shall be assumed uniformly distributed over an equivalent compression zone bounded by edges of the cross section and a line parallel to the neutral axis located a distance

*f*'_{c}**from the fiber of maximum compressive strain, as calculated by:**

*a*a = β_{1}c | (22.2.2.4.1) |

**, shall be measured perpendicular to the neutral axis.**

*c***β**shall be in accordance with Table 22.2.2.4.3.

_{1}**Table 22.2.2.4.3—Values of β _{1} for equivalent rectangular concrete stress distribution**

f', psi_{c} | β_{1} | |
---|---|---|

2500 ≤ f' ≤ 4000_{c} | 0.85 | (a) |

4000 < f' < 8000_{c} | (b) | |

f' ≥ 8000_{c} | 0.65 | (c) |

**, shall be calculated in accordance with 20.3.2.3.**

*f*_{ps}**shall be calculated in accordance with 20.3.2.4.**

*f*_{ps}**shall be calculated in accordance with the assumptions of 22.2.**

*M*_{n}**.**

*f*_{y}**for composite slabs and beams, use of the entire composite section shall be permitted.**

*M*_{n}**for composite slabs and beams, no distinction shall be made between shored and unshored members.**

*M*_{n}**for composite members where the specified concrete compressive strength of different elements varies, properties of the individual elements shall be used in design. Alternatively, it shall be permitted to use the value of**

*M*_{n}**for the element that results in the most critical value of**

*f*'_{c}**.**

*M*_{n}**shall not exceed**

*P*_{n}**in accordance with Table 22.4.2.1, where**

*P*_{n,max}**is calculated by Eq. (22.4.2.2) for nonprestressed members and composite steel and concrete members, and by Eq. (22.4.2.3) for prestressed members.**

*P*_{o}**Table 22.4.2.1—Maximum axial strength**

Member | Transverse reinforcement | P_{n,max} | |
---|---|---|---|

Nonprestressed | Ties conforming to 22.4.2.4 | 0.80P_{o} | (a) |

Spirals conforming to 22.4.2.5 | 0.85P_{o} | (b) | |

Prestressed | Ties | 0.80P_{o} | (c) |

Spirals | 0.85P_{o} | (d) | |

Composite steel and concrete columns in accordance with Chapter 10 | All | 0.85P_{o} | (e) |

**shall be calculated by:**

*P*_{o}P = 0.85_{o}f'(_{c}A— _{g}A) + _{st}f_{y}A_{st} | (22.4.2.2) |

where ** A_{st}** is the total area of nonprestressed longitudinal reinforcement.

**shall be calculated by:**

*P*_{o}P = 0.85_{o}f'(_{c}A— _{g}A— _{st}A) + _{pd}f— (_{y}A_{st}f— 0.003_{se}E)_{p}A_{pt} | (22.4.2.3) |

where ** A_{pt}** is the total area of prestressing reinforcement, and

**is the total area occupied by duct, sheathing, and prestressing reinforcement; the value of**

*A*_{pd}**shall be at least**

*f*_{se}**0.003**. For grouted, post-tensioned tendons, it shall be permitted to assume

*E*_{p}**equals**

*A*_{pd}**.**

*A*_{pt}**, shall not be taken greater than**

*P*_{nt}**, calculated by:**

*P*_{nt,max}P = _{nt,max}f + (_{y}A_{st}f + Δ_{se}f)_{p}A_{pt} | (22.4.3.1) |

where (** f_{se} + Δf_{p}**) shall not exceed

**, and**

*f*_{py}**is zero for nonprestressed members.**

*A*_{pt}**, shall be calculated by:**

*V*_{n}V = _{n}V+ _{c}V_{s} | (22.5.1.1) |

(22.5.1.2) |

*V*shall be calculated in accordance with 22.5.10.

_{s}**.**

*V*_{n}**.**

*V*_{c}**.**

*V*_{c}**and**

*V*_{c}**in prestressed members,**

*V*_{s}**shall be taken as the distance from the extreme compression fiber to the centroid of prestressed and any nonprestressed longitudinal reinforcement but need not be taken less than**

*d***0.8**.

*h***and**

*V*_{c}**in solid, circular sections,**

*V*_{s}**shall be permitted to be taken as 0.8 times the diameter, and**

*d***shall be permitted to be taken as the diameter.**

*b*_{w}**,**

*V*_{c}**, and**

*V*_{ci}**for one-way shear shall not exceed 100 psi, unless allowed in 22.5.3.2.**

*V*_{cw}**,**

*V*_{c}**, and**

*V*_{ci}**for reinforced or prestressed concrete beams and concrete joist construction having minimum web reinforcement in accordance with 9.6.3.3 or 9.6.4.2.**

*V*_{cw}**for composite members, no distinction shall be made between shored and unshored members.**

*V*_{n}**for composite members where the specified concrete compressive strength, unit weight, or other properties of different elements vary, properties of the individual elements shall be used in design. Alternatively, it shall be permitted to use the properties of the element that results in the most critical value of**

*V*_{n}**.**

*V*_{n}**assuming a monolithically cast member of the same crosssectional shape.**

*V*_{c}**assuming a monolithically cast member of the same crosssectional shape if shear reinforcement is fully anchored into the interconnected elements in accordance with 25.7.**

*V*_{s}**shall be calculated by:**

*V*_{c}(22.5.5.1) |

unless a more detailed calculation is made in accordance with Table 22.5.5.1.

**Table 22.5.5.1—Detailed method for calculating V_{c}**

V_{c} | ||
---|---|---|

Least of (a), (b), and (c): | ^{[1]} | (a) |

(b) | ||

(c) |

^{[1]}*M _{u}* occurs simultaneously with

*V*at the section considered.

_{u}**shall be calculated by:**

*V*_{c}(22.5.6.1) |

unless a more detailed calculation is made in accordance with Table 22.5.6.1, where ** N_{u}** is positive for compression.

**Table 22.5.6.1—Detailed method for calculating V_{c} for nonprestressed members with axial compression**

V_{c} | ||
---|---|---|

Lesser of (a) and (b): | ^{[1]}Equation not applicable if | (a) |

(b) |

^{[1]}*M _{u}* occurs simultaneously with

*V*at the section considered.

_{u}**shall be calculated by:**

*V*_{c}(22.5.7.1) |

where ** N_{u}** is negative for tension, and

**shall not be less than zero.**

*V*_{c}**for post-tensioned and pretensioned members in regions where the effective force in the prestressed reinforcement is fully transferred to the concrete. For regions of pretensioned members where the effective force in the prestressed reinforcement is not fully transferred to the concrete, 22.5.9 shall govern the calculation of**

*V*_{c}**.**

*V*_{c}**,**

*A*≥ 0.4(_{ps}f_{se}*A*+_{ps}f_{pu}*A*)_{s}f_{y}**shall be calculated in accordance with Table 22.5.8.2, but need not be less than the value calculated by Eq. (22.5.5.1). Alternatively, it shall be permitted to calculate**

*V*_{c}**in accordance with 22.5.8.3.**

*V*_{c}**Table 22.5.8.2—Approximate method for calculating V_{c}**

V_{c} |
||
---|---|---|

Least of (a), (b), and (c): | ^{[1]} |
(a) |

(b) | ||

(c) |

^{[1]}*M _{u}* occurs simultaneously with

*V*at the section considered.

_{u}**shall be permitted to be the lesser of**

*V*_{c}**calculated in accordance with 22.5.8.3.1 and**

*V*_{ci}**calculated in accordance with 22.5.8.3.2 or 22.5.8.3.3.**

*V*_{cw}**shall be the greater of (a) and (b):**

*V*_{ci}(a) | (22.5.8.3.1a) |

(b) | (22.5.8.3.1b) |

where ** d_{p}** need not be taken less than

**0.80**, the values of

*h***and**

*M*_{max}**shall be calculated from the load combinations causing maximum factored moment to occur at section considered, and**

*V*_{i}**shall be calculated by:**

*M*_{cre}(22.5.8.3.1c) |

**shall be calculated by:**

*V*_{cw}(22.5.8.3.2) |

where ** d_{p}** need not be taken less than

**0.80**, and

*h***is the vertical component of the effective prestress.**

*V*_{p}**as the shear force corresponding to dead load plus live load that results in a principal tensile stress of at location (a) or (b):**

*V*_{cw}(a) Where the centroidal axis of the prestressed cross section is in the web, the principal tensile stress shall be calculated at the centroidal axis.

(b) Where the centroidal axis of the prestressed cross section is in the flange, the principal tensile stress shall be calculated at the intersection of the flange and the web.

**, the transfer length of prestressed reinforcement,**

*V*_{c}**, shall be assumed to be**

*ℓ*_{tr}**50**for strand and

*d*_{b}**100**for wire.

*d*_{b}**from the end of the prestressed reinforcement.**

*ℓ*_{tr}**shall be calculated in accordance with (a) through (c):**

*V*_{c}(a) The reduced effective prestress force shall be used to determine the applicability of 22.5.8.2.

(b) The reduced effective prestress force shall be used to calculate ** V_{cw}** in 22.5.8.3.

(c) The value of ** V_{c}** calculated using 22.5.8.2 shall not exceed the value of

**calculated using the reduced effective prestress force.**

*V*_{cw}**from that point.**

*ℓ*_{tr}**shall be calculated in accordance with (a) through (c):**

*V*_{c}(a) The reduced effective prestress force shall be used to determine the applicability of 22.5.8.2.

(b) The reduced effective prestress force shall be used to calculate ** V_{c}** in accordance with 22.5.8.3.

(c) The value of ** V_{c}** calculated using 22.5.8.2 shall not exceed the value of

**calculated using the reduced effective prestress force.**

*V*_{cw}**, transverse reinforcement shall be provided such that Eq. (22.5.10.1) is satisfied.**

*V*> ϕ_{u}*V*_{c}(22.5.10.1) |

**shall be calculated in accordance with 22.5.10.5.**

*V*_{s}**shall be calculated in accordance with 22.5.10.6.**

*V*_{s}**shall be the sum of the**

*V*_{s}**values for the various types of shear reinforcement.**

*V*_{s}(a) Stirrups, ties, or hoops perpendicular to longitudinal axis of member

(b) Welded wire reinforcement with wires located perpendicular to longitudinal axis of member

*V*for shear reinforcement in 22.5.10.5.1 shall be calculated by:

_{s}(22.5.10.5.3) |

where ** s** is the spiral pitch or the longitudinal spacing of the shear reinforcement, and

**is given in 22.5.10.5.5 or 22.5.10.5.6.**

*A*_{v}*V*for shear reinforcement in 22.5.10.5.2 shall be calculated by:

_{s}(22.5.10.5.4) |

where **α** is the angle between the inclined stirrups and the longitudinal axis of the member, ** s** is measured parallel to the longitudinal reinforcement, and

**is given in 22.5.10.5.5.**

*A*_{v}**shall be two times the area of the bar or wire within spacing**

*A*_{v}**.**

*s***, all bent the same distance from the support,**

*A*_{v}**shall be the lesser of (a) and (b):**

*V*_{s}(a) V = _{s}Asinα_{v}f_{y} | (22.5.10.6.2a) |

(b) | (22.5.10.6.2b) |

where **α** is the angle between bent-up reinforcement and longitudinal axis of the member.

**shall be calculated by Eq. (22.5.10.5.4).**

*V*_{s}v = _{n}v_{c} | (22.6.1.2) |

v = _{n}v+ _{c}v_{s} | (22.6.1.3) |

**and an assumed critical perimeter**

*d***as defined in 22.6.4.**

*b*_{o}*v*for two-way shear shall be calculated in accordance with 22.6.5. For two-way members with shear reinforcement,

_{c}**shall not exceed the limits in 22.6.6.1.**

*v*_{c}**shall be calculated in accordance with 22.6.8.**

*v*_{s}**and**

*v*_{c}**for two-way shear,**

*v*_{s}**shall be the average of the effective depths in the two orthogonal directions.**

*d***need not be taken less than**

*d***0.8**.

*h***for two-way shear shall not exceed 100 psi.**

*v*_{c}**is a minimum but need not be closer than**

*b*_{o}**to (a) and (b):**

*d*/2(a) Edges or corners of columns, concentrated loads, or reaction areas

(b) Changes in slab or footing thickness, such as edges of capitals, drop panels, or shear caps

**located**

*b*_{o}**beyond the outermost peripheral line of shear reinforcement shall also be considered. The shape of this critical section shall be a polygon selected to minimize**

*d*/2**.**

*b*_{o}**10**from a concentrated load or reaction area, a portion of

*h***enclosed by straight lines projecting from the centroid of the column, concentrated load or reaction area and tangent to the boundaries of the opening shall be considered ineffective.**

*b*_{o}*v*shall be calculated in accordance with Table 22.6.5.2.

_{c}**Table 22.6.5.2—Calculation of v_{c} for two-way shear**

v_{c} | ||
---|---|---|

Least of (a), (b), and (c): | (a) | |

(b) | ||

(c) |

Note: β is the ratio of long side to short side of the column, concentrated load, or reaction area and α_{s} is given in 22.6.5.3.

**using 22.6.5.5, provided that (a) through (c) are satisfied:**

*v*_{c}(a) Bonded reinforcement is provided in accordance with 8.6.2.3 and 8.7.5.3

(b) No portion of the column cross section is closer to a discontinuous edge than four times the slab thickness *h*

(c) Effective prestress ** f_{pc}** in each direction is not less than 125 psi

**shall be permitted to be the lesser of (a) and (b):**

*v*_{c}(a) | (22.6.5.5a) |

(b) | (22.6.5.5b) |

where **α _{s}** is given in 22.6.5.3; the value of

**is the average of**

*f*_{pc}**in the two directions and shall not exceed 500 psi;**

*f*_{pc}**is the vertical component of all effective prestress forces crossing the critical section; and the value of shall not exceed 70 psi.**

*V*_{p}**calculated at critical sections shall not exceed the limits in Table 22.6.6.1.**

*v*_{c}**Table 22.6.6.1—Maximum v_{c} for two-way members with shear reinforcement**

Type of shear reinforcement | Maximum v at critical sections defined in 22.6.4.1_{c} | Maximum v at critical section defined in 22.6.4.2_{c} | ||
---|---|---|---|---|

Stirrups | (a) | (b) | ||

Headed shear stud reinforcement | (c) | (d) |

**calculated at critical sections does not exceed the values in Table 22.6.6.2**

*v*_{u}**Table 22.6.6.2—Maximum v_{u} for two-way members with shear reinforcement**

Type of shear reinforcement | Maximum v at critical sections defined in 22.6.4.1_{u} | |
---|---|---|

Stirrups | (a) | |

Headed shear stud reinforcement | (b) |

(a) ** d** is at least 6 in.

(b) ** d** is at least

**16**, where

*d*_{b}**is the diameter of the stirrups**

*d*_{b}**shall be calculated by:**

*v*_{s}(22.6.7.2) |

where ** A_{v}** is the sum of the area of all legs of reinforcement on one peripheral line that is geometrically similar to the perimeter of the column section, and

*s*is the spacing of the peripheral lines of shear reinforcement in the direction perpendicular to the column face.

**shall be calculated by:**

*v*_{s}(22.6.8.2) |

where ** A_{v}** is the sum of the area of all shear studs on one peripheral line that is geometrically similar to the perimeter of the column section, and

**is the spacing of the peripheral lines of headed shear stud reinforcement in the direction perpendicular to the column face.**

*s***of the remaining tapered section is adequate to resist the shear force attributed to that arm of the shearhead.**

*M*_{p}**0.3**of the compression surface of the slab.

*d***α**between the flexural stiffness of each shearhead arm and that of the surrounding composite cracked slab section of width (

_{v}**) shall be at least 0.15.**

*c*_{2}+*d***, shall satisfy:**

*M*_{v}(22.6.9.7) |

where **ϕ** corresponds to tension-controlled members. However, ** M_{v}** shall not exceed the least of (a) through (c):

(a) 30 percent of ** M_{u}** in each slab column strip

(b) Change in ** M_{u}** in column strip over the length

*ℓ*_{v}(c) ** M_{p}** as given in 22.6.9.6

**(3/4)**[

**from the column face. This critical section shall be located so that**

*ℓ*— (_{v}*c*_{1}/2)]**is a minimum, but need not be closer than**

*b*_{o}**to the edges of the supporting column.**

*d*/2**, where**

*T*≥ ϕ_{u}*T*_{th}**ϕ**is given in Chapter 21 and threshold torsion

**is given in 22.7.4. If**

*T*_{th}**, it shall be permitted to neglect torsional effects.**

*T*< ϕ_{u}*T*_{th}**and**

*T*_{th}**shall not exceed 100 psi.**

*T*_{cr}**and**

*f*_{y}**for longitudinal and transverse torsional reinforcement shall not exceed the limits in 20.2.2.4.**

*f*_{yt}**and**

*T*≥ ϕ_{u}*T*_{cr}**is required to maintain equilibrium, the member shall be designed to resist**

*T*_{u}**.**

*T*_{u}**and a reduction of**

*T*≥ ϕ_{u}*T*_{cr}**can occur due to redistribution of internal forces after torsional cracking, it shall be permitted to reduce**

*T*_{u}**to**

*T*_{u}**ϕ**, where the cracking torsion

*T*_{cr}**is calculated in accordance with 22.7.5.**

*T*_{cr}**is redistributed in accordance with 22.7.3.2, the factored moments and shears used for design of the adjoining members shall be in equilibrium with the reduced torsion.**

*T*_{u}**shall be calculated in accordance with Table 22.7.4.1(a) for solid cross sections and Table 22.7.4.1(b) for hollow cross sections, where**

*T*_{th}**is positive for compression and negative for tension.**

*N*_{u}**Table 22.7.4.1(a)—Threshold torsion for solid cross sections**

Type of member | T_{th} | |
---|---|---|

Nonprestressed member | (a) | |

Prestressed member | (b) | |

Nonprestressed member subjected to axial force | (c) |

**Table 22.7.4.1(b)—Threshold torsion for hollow cross sections**

Type of member | T_{th} | |
---|---|---|

Nonprestressed member | (a) | |

Prestressed member | (b) | |

Nonprestressed member subjected to axial force | (c) |

**shall be calculated in accordance with Table 22.7.5.1 for solid and hollow cross sections, where**

*T*_{cr}**is positive for compression and negative for tension.**

*N*_{u}**Table 22.7.5.1—Cracking torsion**

Type of member | T_{cr} | |
---|---|---|

Nonprestressed member | (a) | |

Prestressed member | (b) | |

Nonprestressed member subjected to axial force | (c) |

**shall be the lesser of (a) and (b):**

*T*_{n}(a) | (22.7.6.1a) | |

(b) | (22.7.6.1b) |

where ** A_{o}** shall be determined by analysis,

**θ**shall not be taken less than 30 degrees nor greater than 60 degrees;

**is the area of one leg of a closed stirrup resisting torsion;**

*A*_{t}**is the area of longitudinal torsional reinforcement; and**

*A*_{ℓ}**is the perimeter of the centerline of the outermost closed stirrup.**

*p*_{h}(a) For solid sections

(22.7.7.1a) |

(b) For hollow sections

(22.7.7.1b) |

ϕB ≥ _{n}B_{u} | (22.8.3.1) |

for each applicable factored load combination.

**shall be calculated in accordance with Table 22.8.3.2, where**

*B*_{n}**is the loaded area, and**

*A*_{1}**is the area of the lower base of the largest frustum of a pyramid, cone, or tapered wedge contained wholly within the support and having its upper base equal to the loaded area. The sides of the pyramid, cone, or tapered wedge shall be sloped 1 vertical to 2 horizontal.**

*A*_{2}**Table 22.8.3.2—Nominal bearing strength**

Geometry of bearing area | B_{n} |
||
---|---|---|---|

Supporting surface is wider on all sides than the loaded area | Lesser of (a) and (b) | (a) | |

2(0.85f'_{c}A_{1}) |
(b) | ||

Other cases | 0.85f'_{c}A_{1} |
(c) |

**, shall be calculated in accordance with 22.9.4. Alternatively, it shall be permitted to use shear transfer design methods that result in prediction of strength in substantial agreement with results of comprehensive tests.**

*A*_{vf}**used to calculate**

*f*_{y}**for shear friction shall not exceed the limit in 20.2.2.4.**

*V*_{n}ϕV ≥ _{n}V_{u} | (22.9.3.1) |

for each applicable factored load combination.

V _{n}= µA _{vf}f_{y} | (22.9.4.2) |

where *A _{vf}* is the area of reinforcement crossing the assumed shear plane to resist shear, and µ is the coefficient of friction in accordance with Table 22.9.4.2.

**Table 22.9.4.2—Coefficients of friction**

Contact surface condition | Coefficient of friction µ^{[1]} | |
---|---|---|

Concrete placed monolithically | 1.4λ | (a) |

Concrete placed against hardened concrete that is clean, free of laitance, and intentionally roughened to a full amplitude of approximately 1/4 in. | 1.0λ | (b) |

Concrete placed against hardened concrete that is clean, free of laitance, and not intentionally roughened | 0.6λ | (c) |

Concrete placed against as-rolled structural steel that is clean, free of paint, and with shear transferred across the contact surface by headed studs or by welded deformed bars or wires. | 0.7λ | (d) |

^{[1]}λ = 1.0 for normalweight concrete; λ = 0.75 for all lightweight concrete. Otherwise, λ is calculated based on volumetric proportions of lightweight and normalweight aggregate as given in 19.2.4, but shall not exceed 0.85.

V _{n}= A(µsinα _{vf}f_{y}+ cosα) | (22.9.4.3) |

where **α** is the angle between shear-friction reinforcement and assumed shear plane, and µ is the coefficient of friction in accordance with Table 22.9.4.2.

**across the assumed shear plane shall not exceed the limits in Table 22.9.4.4. Where concretes of different strengths are cast against each other, the lesser value of**

*V*_{n}**shall be used in Table 22.9.4.4.**

*f*'_{c}**Table 22.9.4.4—Maximum V_{n} across the assumed shear plane**

Condition | Maximum V_{n} | ||
---|---|---|---|

Normalweight concrete placed monolithically or placed against hardened concrete intentionally roughened to a full amplitude of approximately 1/4 in. | Least of (a), (b), and (c) | 0.2f'_{c}A_{c} | (a) |

(480 + 0.08f')_{c}A_{c} | (b) | ||

1600A_{c} | (c) | ||

Other cases | Lesser of (d) and (e) | 0.2f'_{c}A_{c} | (d) |

800A_{c} | (e) |

**, the force in the shear-friction reinforcement, to calculate required**

*A*_{vf}f_{y}**.**

*A*_{vf}**on both sides of the shear plane.**

*f*_{y}