  Contents

# Chapter 22 Sectional Strength

### 22.1.1

This chapter shall apply to calculating nominal strength at sections of members, including (a) through (g):
(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

### 22.1.2

Sectional strength requirements of this chapter shall be satisfied unless the member or region of the member is designed in accordance with Chapter 23.

### 22.1.3

Design strength at a section shall be taken as the nominal strength multiplied by the applicable strength reduction factor ϕ given in Chapter 21.

### 22.2.1

Equilibrium and strain compatibility

### 22.2.1.1

Equilibrium shall be satisfied at each section.

### 22.2.1.2

Strain in concrete and nonprestressed reinforcement shall be assumed proportional to the distance from neutral axis.

### 22.2.1.3

Strain in prestressed concrete and in bonded and unbonded prestressed reinforcement shall include the strain due to effective prestress.

### 22.2.1.4

Changes in strain for bonded prestressed reinforcement shall be assumed proportional to the distance from neutral axis.

### 22.2.2.1

Maximum strain at the extreme concrete compression fiber shall be assumed equal to 0.003.

### 22.2.2.2

Tensile strength of concrete shall be neglected in flexural and axial strength calculations.

### 22.2.2.3

The relationship between concrete compressive stress and strain shall be represented by a rectangular, trapezoidal, parabolic, or other shape that results in prediction of strength in substantial agreement with results of comprehensive tests.

### 22.2.2.4

The equivalent rectangular concrete stress distribution in accordance with 22.2.2.4.1 through 22.2.2.4.3 satisfies 22.2.2.3.

### 22.2.2.4.1

Concrete stress of 0.85fc' 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 a from the fiber of maximum compressive strain, as calculated by:
 a = β1c (22.2.2.4.1)

### 22.2.2.4.2

Distance from the fiber of maximum compressive strain to the neutral axis, c, shall be measured perpendicular to the neutral axis.

### 22.2.2.4.3

Values of β1 shall be in accordance with Table 22.2.2.4.3.
Table 22.2.2.4.3—Values of β1 for equivalent rectangular concrete stress distribution
fc', psi β1
2500 ≤ fc' ≤ 4000 0.85 (a)
4000 < fc' < 8000 (b)
fc' ≥ 8000 0.65 (c)

### 22.2.3.1

Deformed reinforcement used to resist tensile or compressive forces shall conform to 20.2.1.

### 22.2.3.2

Stress-strain relationship and modulus of elasticity for deformed reinforcement shall be idealized in accordance with 20.2.2.1 and 20.2.2.2.

### 22.2.4.1

For members with bonded prestressing reinforcement conforming to 20.3.1, stress at nominal flexural strength, fps, shall be calculated in accordance with 20.3.2.3.

### 22.2.4.2

For members with unbonded prestressing reinforcement conforming to 20.3.1, fps shall be calculated in accordance with 20.3.2.4.

### 22.2.4.3

If the embedded length of the prestressing strand is less than d, the design strand stress shall not exceed the value given in 25.4.8.3, as modified by 25.4.8.1(b).

### 22.3.1.1

Nominal flexural strength Mn shall be calculated in accordance with the assumptions of 22.2.

### 22.3.2.1

Deformed reinforcement conforming to 20.2.1, provided in conjunction with prestressed reinforcement, shall be permitted to be considered to contribute to the tensile force and be included in flexural strength calculations at a stress equal to fy.

### 22.3.2.2

Other nonprestressed reinforcement shall be permitted to be considered to contribute to the flexural strength if a strain compatibility analysis is performed to calculate stresses in such reinforcement.

### 22.3.3.1

Provisions of 22.3.3 apply to members constructed in separate placements but connected so that all elements resist loads as a unit.

### 22.3.3.2

For calculation of Mn for composite slabs and beams, use of the entire composite section shall be permitted.

### 22.3.3.3

For calculation of Mn for composite slabs and beams, no distinction shall be made between shored and unshored members.

### 22.3.3.4

For calculation of Mn 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 fc' for the element that results in the most critical value of Mn.

### 22.4.1.1

Nominal flexural and axial strength shall be calculated in accordance with the assumptions of 22.2.

### 22.4.2.1

Nominal axial compressive strength Pn shall not exceed Pn,max in accordance with Table 22.4.2.1, where Po 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.
Table 22.4.2.1—Maximum axial strength
Member Transverse reinforcement Pn,max
Nonprestressed Ties conforming to 22.4.2.4 0.80Po (a)
Spirals conforming to 22.4.2.5 0.85Po (b)
Prestressed Ties 0.80Po (c)
Spirals 0.85Po (d)
Composite steel and concrete columns in accordance with Chapter 10 All 0.85Po (e)

### 22.4.2.2

For nonprestressed members and composite steel and concrete members, Po shall be calculated by:
 Po = 0.85fc'(Ag— Ast) + fyAst (22.4.2.2)
where Ast is the total area of nonprestressed longitudinal reinforcement.

### 22.4.2.3

For prestressed members, Po shall be calculated by:
 Po = 0.85fc'(Ag— Ast— Apd) + fyAst— (fse— 0.003Ep)Apt (22.4.2.3)
where Apt is the total area of prestressing reinforcement, and Apd is the total area occupied by duct, sheathing, and prestressing reinforcement; the value of fse shall be at least 0.003Ep. For grouted, post-tensioned tendons, it shall be permitted to assume Apd equals Apt.

### 22.4.2.4

Tie reinforcement for lateral support of longitudinal reinforcement in compression members shall satisfy 10.7.6.2 and 25.7.2.

### 22.4.2.5

Spiral reinforcement for lateral support of longitudinal reinforcement in compression members shall satisfy 10.7.6.3 and 25.7.3.

### 22.4.3.1

Nominal axial tensile strength of a nonprestressed, composite, or prestressed member, Pnt, shall not be taken greater than Pnt,max, calculated by:
 Pnt,max = fyAst + (fse + Δfp)Apt (22.4.3.1)
where (fse + Δfp) shall not exceed fpy, and Apt is zero for nonprestressed members.

### 22.5.1.1

Nominal one-way shear strength at a section, Vn, shall be calculated by:
 Vn = Vc+ Vs (22.5.1.1)

### 22.5.1.2

Cross-sectional dimensions shall be selected to satisfy Eq. (22.5.1.2). (22.5.1.2)

### 22.5.1.3

For nonprestressed members, Vc shall be calculated in accordance with 22.5.5, 22.5.6, or 22.5.7.

### 22.5.1.4

For prestressed members, Vc, Vci, and Vcw shall be calculated in accordance with 22.5.8 or 22.5.9.

### 22.5.1.5

For calculation of Vc, Vci, and Vcw, λ shall be in accordance with 19.2.4.

### 22.5.1.6

Vs shall be calculated in accordance with 22.5.10.

### 22.5.1.7

Effect of any openings in members shall be considered in calculating Vn.

### 22.5.1.8

Effect of axial tension due to creep and shrinkage in restrained members shall be considered in calculating Vc.

### 22.5.1.9

Effect of inclined flexural compression in variable depth members shall be permitted to be considered in calculating Vc.

### 22.5.2.1

For calculation of Vc and Vs in prestressed members, d 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 0.8h.

### 22.5.2.2

For calculation of Vc and Vs in solid, circular sections, d shall be permitted to be taken as 0.8 times the diameter, and bw shall be permitted to be taken as the diameter.

### 22.5.3.1

The value of used to calculate Vc, Vci, and Vcw for one-way shear shall not exceed 100 psi, unless allowed in 22.5.3.2.

### 22.5.3.2

Values of greater than 100 psi shall be permitted in calculating Vc, Vci, and Vcw 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.

### 22.5.3.3

The values of fy and fyt used to calculate Vs shall not exceed the limits in 20.2.2.4.

### 22.5.4.1

This section shall apply to members constructed in separate placements but connected so that all elements resist loads as a unit.

### 22.5.4.2

For calculation of Vn for composite members, no distinction shall be made between shored and unshored members.

### 22.5.4.3

For calculation of Vn 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 Vn.

### 22.5.4.4

If an entire composite member is assumed to resist vertical shear, it shall be permitted to calculate Vc assuming a monolithically cast member of the same crosssectional shape.

### 22.5.4.5

If an entire composite member is assumed to resist vertical shear, it shall be permitted to calculate Vs 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.

### 22.5.5.1

For nonprestressed members without axial force, Vc shall be calculated by: (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 Vc
Vc
Least of (a), (b), and (c): (a) (b) (c)
Mu occurs simultaneously with Vu at the section considered.

### 22.5.6.1

For nonprestressed members with axial compression, Vc shall be calculated by: (22.5.6.1)
unless a more detailed calculation is made in accordance with Table 22.5.6.1, where Nu is positive for compression.
Table 22.5.6.1—Detailed method for calculating Vc for nonprestressed members with axial compression
Vc
Lesser of (a) and (b): Equation not applicable if (a) (b)
Mu occurs simultaneously with Vu at the section considered.

### 22.5.7.1

For nonprestressed members with significant axial tension, Vc shall be calculated by: (22.5.7.1)
where Nu is negative for tension, and Vc shall not be less than zero.

### 22.5.8.1

This section shall apply to the calculation of Vc 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 Vc.

### 22.5.8.2

For prestressed flexural members with Apsfse ≥ 0.4(Apsfpu + Asfy), Vc 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 Vc in accordance with 22.5.8.3.
Table 22.5.8.2—Approximate method for calculating Vc
Vc
Least of (a), (b), and (c): (a) (b) (c)
Mu occurs simultaneously with Vu at the section considered.

### 22.5.8.3

For prestressed members, Vc shall be permitted to be the lesser of Vci calculated in accordance with 22.5.8.3.1 and Vcw calculated in accordance with 22.5.8.3.2 or 22.5.8.3.3.

### 22.5.8.3.1

The flexure-shear strength Vci shall be the greater of (a) and (b):
 (a) (22.5.8.3.1a) (b) (22.5.8.3.1b)
where dp need not be taken less than 0.80h, the values of Mmax and Vi shall be calculated from the load combinations causing maximum factored moment to occur at section considered, and Mcre shall be calculated by: (22.5.8.3.1c)

### 22.5.8.3.2

The web-shear strength Vcw shall be calculated by: (22.5.8.3.2)
where dp need not be taken less than 0.80h, and Vp is the vertical component of the effective prestress.

### 22.5.8.3.3

As an alternative to 22.5.8.3.2, it shall be permitted to calculate Vcw as the shear force corresponding to dead load plus live load that results in a principal tensile stress of at location (a) or (b):
(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.

### 22.5.8.3.4

In composite members, the principal tensile stress in 22.5.8.3.3 shall be calculated using the cross section that resists live load.

### 22.5.9.1

When calculating Vc, the transfer length of prestressed reinforcement, tr, shall be assumed to be 50db for strand and 100db for wire.

### 22.5.9.2

If bonding of strands extends to the end of the member, the effective prestress force shall be assumed to vary linearly from zero at the end of the prestressed reinforcement to a maximum at a distance tr from the end of the prestressed reinforcement.

### 22.5.9.3

At locations corresponding to a reduced effective prestress force in 22.5.9.2, Vc shall be calculated in accordance with (a) through (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 Vcw in 22.5.8.3.
(c) The value of Vc calculated using 22.5.8.2 shall not exceed the value of Vcw calculated using the reduced effective prestress force.

### 22.5.9.4

If bonding of strands does not extend to the end of the member, the effective prestress force shall be assumed to vary linearly from zero at the point where bonding commences to a maximum at a distance tr from that point.

### 22.5.9.5

At locations corresponding to a reduced effective prestress force according to 22.5.9.4, Vc shall be calculated in accordance with (a) through (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 Vc in accordance with 22.5.8.3.
(c) The value of Vc calculated using 22.5.8.2 shall not exceed the value of Vcw calculated using the reduced effective prestress force.

### 22.5.10.1

At each section where Vu > ϕVc, transverse reinforcement shall be provided such that Eq. (22.5.10.1) is satisfied. (22.5.10.1)

### 22.5.10.2

For one-way members reinforced with transverse reinforcement, Vs shall be calculated in accordance with 22.5.10.5.

### 22.5.10.3

For one-way members reinforced with bent-up longitudinal bars, Vs shall be calculated in accordance with 22.5.10.6.

### 22.5.10.4

If more than one type of shear reinforcement is provided to reinforce the same portion of a member, Vs shall be the sum of the Vs values for the various types of shear reinforcement.

### 22.5.10.5.1

In nonprestressed and prestressed members, shear reinforcement satisfying (a), (b), or (c) shall be permitted:
(a) Stirrups, ties, or hoops perpendicular to longitudinal axis of member
(b) Welded wire reinforcement with wires located perpendicular to longitudinal axis of member

### 22.5.10.5.2

Inclined stirrups making an angle of at least 45 degrees with the longitudinal axis of the member and crossing the plane of the potential shear crack shall be permitted to be used as shear reinforcement in nonprestressed members.

### 22.5.10.5.3

Vs for shear reinforcement in 22.5.10.5.1 shall be calculated by: (22.5.10.5.3)
where s is the spiral pitch or the longitudinal spacing of the shear reinforcement, and Av is given in 22.5.10.5.5 or 22.5.10.5.6.

### 22.5.10.5.4

Vs for shear reinforcement in 22.5.10.5.2 shall be calculated by: (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 Av is given in 22.5.10.5.5.

### 22.5.10.5.5

For each rectangular tie, stirrup, hoop, or crosstie, Av shall be the effective area of all bar legs or wires within spacing s.

### 22.5.10.5.6

For each circular tie or spiral, Av shall be two times the area of the bar or wire within spacing s.

### 22.5.10.6.1

The center three-fourths of the inclined portion of bent-up longitudinal bars shall be permitted to be used as shear reinforcement in nonprestressed members if the angle α between the bent-up bars and the longitudinal axis of the member is at least 30 degrees.

### 22.5.10.6.2

If shear reinforcement consists of a single bar or a single group of parallel bars having an area Av, all bent the same distance from the support, Vs shall be the lesser of (a) and (b):
 (a) Vs = Avfysinα (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.

### 22.5.10.6.3

If shear reinforcement consists of a series of parallel bent-up bars or groups of parallel bent-up bars at different distances from the support, Vs shall be calculated by Eq. (22.5.10.5.4).

### 22.6.1.1

Provisions 22.6.1 through 22.6.8 apply to the nominal shear strength of two-way members with and without shear reinforcement. Where structural steel I- or channel-shaped sections are used as shearheads, two-way members shall be designed for shear in accordance with 22.6.9.

### 22.6.1.2

Nominal shear strength for two-way members without shear reinforcement shall be calculated by
 vn = vc (22.6.1.2)

### 22.6.1.3

Nominal shear strength for two-way members with shear reinforcement other than shearheads shall be calculated by
 vn = vc+ vs (22.6.1.3)

### 22.6.1.4

Two-way shear shall be resisted by a section with a depth d and an assumed critical perimeter bo as defined in 22.6.4.

### 22.6.1.5

vc for two-way shear shall be calculated in accordance with 22.6.5. For two-way members with shear reinforcement, vc shall not exceed the limits in 22.6.6.1.

### 22.6.1.6

For calculation of vc, λ shall be in accordance with 19.2.4.

### 22.6.1.7

For two-way members reinforced with single- or multiple-leg stirrups, vs shall be calculated in accordance with 22.6.7.

### 22.6.1.8

For two-way members reinforced with headed shear stud reinforcement, vs shall be calculated in accordance with 22.6.8.

### 22.6.2.1

For calculation of vc and vs for two-way shear, d shall be the average of the effective depths in the two orthogonal directions.

### 22.6.2.2

For prestressed, two-way members, d need not be taken less than 0.8h.

### 22.6.3.1

The value of used to calculate vc for two-way shear shall not exceed 100 psi.

### 22.6.3.2

The value of fyt used to calculate vs shall not exceed the limits in 20.2.2.4.

### 22.6.4.1

For two-way shear, critical sections shall be located so that the perimeter bo is a minimum but need not be closer than d/2 to (a) and (b):
(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

### 22.6.4.1.1

For square or rectangular columns, concentrated loads, or reaction areas, critical sections for two-way shear in accordance with 22.6.4.1(a) and (b) shall be permitted to be defined assuming straight sides.

### 22.6.4.1.2

For a circular or regular polygon-shaped column, critical sections for two-way shear in accordance with 22.6.4.1(a) and (b) shall be permitted to be defined assuming a square column of equivalent area.

### 22.6.4.2

For two-way members reinforced with headed shear reinforcement or single- or multi-leg stirrups, a critical section with perimeter bo located d/2 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 bo.

### 22.6.4.3

If an opening is located within a column strip or closer than 10h from a concentrated load or reaction area, a portion of bo 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.

### 22.6.5.1

For nonprestressed two-way members, vc shall be calculated in accordance with 22.6.5.2. For prestressed two-way members, vc shall be calculated in accordance with (a) or (b):
(b) 22.6.5.5, if the conditions of 22.6.5.4 are satisfied

### 22.6.5.2

vc shall be calculated in accordance with Table 22.6.5.2.
Table 22.6.5.2—Calculation of vc for two-way shear
vc
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 αs is given in 22.6.5.3.

### 22.6.5.3

The value of αs is 40 for interior columns, 30 for edge columns, and 20 for corner columns.

### 22.6.5.4

For prestressed, two-way members, it shall be permitted to calculate vc using 22.6.5.5, provided that (a) through (c) are satisfied:
(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 fpc in each direction is not less than 125 psi

### 22.6.5.5

For prestressed, two-way members conforming to 22.6.5.4, vc shall be permitted to be the lesser of (a) and (b):
 (a) (22.6.5.5a) (b) (22.6.5.5b)
where αs is given in 22.6.5.3; the value of fpc is the average of fpc in the two directions and shall not exceed 500 psi; Vp is the vertical component of all effective prestress forces crossing the critical section; and the value of shall not exceed 70 psi.

### 22.6.6.1

For two-way members with shear reinforcement, the value of vc calculated at critical sections shall not exceed the limits in Table 22.6.6.1.
Table 22.6.6.1—Maximum vc for two-way members with shear reinforcement
Type of shear reinforcement Maximum vc at critical sections defined in 22.6.4.1 Maximum vc at critical section defined in 22.6.4.2
Stirrups (a) (b)
Headed shear stud reinforcement (c) (d)

### 22.6.6.2

For two-way members with shear reinforcement, effective depth shall be selected such that vu calculated at critical sections does not exceed the values in Table 22.6.6.2
Table 22.6.6.2—Maximum vu for two-way members with shear reinforcement
Type of shear reinforcement Maximum vu at critical sections defined in 22.6.4.1
Stirrups (a)
Headed shear stud reinforcement (b)

### 22.6.7.1

Single- or multiple-leg stirrups fabricated from bars or wires shall be permitted to be used as shear reinforcement in slabs and footings satisfying (a) and (b):
(a) d is at least 6 in.
(b) d is at least 16db, where db is the diameter of the stirrups

### 22.6.7.2

For two-way members with stirrups, vs shall be calculated by: (22.6.7.2)
where Av 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.

### 22.6.8.1

Headed shear stud reinforcement shall be permitted to be used as shear reinforcement in slabs and footings if the placement and geometry of the headed shear stud reinforcement satisfies 8.7.7.

### 22.6.8.2

For two-way members with headed shear stud reinforcement, vs shall be calculated by: (22.6.8.2)
where Av 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 s is the spacing of the peripheral lines of headed shear stud reinforcement in the direction perpendicular to the column face.

### 22.6.8.3

If headed shear stud reinforcement is provided, Av/s shall satisfy: (22.6.8.3)

### 22.6.9.1

Each shearhead shall consist of steel shapes fabricated with a full penetration weld into identical arms at right angles. Shearhead arms shall not be interrupted within the column section.

### 22.6.9.2

A shearhead shall not be deeper than 70 times the web thickness of the steel shape.

### 22.6.9.3

The ends of each shearhead arm shall be permitted to be cut at angles of at least 30 degrees with the horizontal if the plastic flexural strength Mp of the remaining tapered section is adequate to resist the shear force attributed to that arm of the shearhead.

### 22.6.9.4

Compression flanges of steel shapes shall be within 0.3d of the compression surface of the slab.

### 22.6.9.5

The ratio αv between the flexural stiffness of each shearhead arm and that of the surrounding composite cracked slab section of width (c2 + d) shall be at least 0.15.

### 22.6.9.6

For each arm of the shearhead, Mp shall satisfy: (22.6.9.6)
where ϕ corresponds to tension-controlled members, n is the number of shearhead arms, and v is the minimum length of each shearhead arm required to satisfy 22.6.9.8 and 22.6.9.10.

### 22.6.9.7

Nominal flexural strength contributed to each slab column strip by a shearhead, Mv, shall satisfy: (22.6.9.7)
where ϕ corresponds to tension-controlled members. However, Mv shall not exceed the least of (a) through (c):
(a) 30 percent of Mu in each slab column strip
(b) Change in Mu in column strip over the length v
(c) Mp as given in 22.6.9.6

### 22.6.9.8

The critical section for shear shall be perpendicular to the plane of the slab and shall cross each shearhead arm at a distance (3/4)[v — (c1/2)] from the column face. This critical section shall be located so that bo is a minimum, but need not be closer than d/2 to the edges of the supporting column.

### 22.6.9.9

If an opening is located within a column strip or closer than 10h from a column in slabs with shearheads, the ineffective portion of bo shall be one-half of that given in 22.6.4.3.

### 22.6.9.10

Factored shear stress due to vertical loads shall not be greater than on the critical section given in 22.6.9.8 and shall not be greater than on the critical section closest to the column given in 22.6.4.1(a).

### 22.6.9.11

Where transfer of moment is considered, the shearhead shall have adequate anchorage to transmit Mp to the column.

### 22.6.9.12

Where transfer of moment is considered, the sum of factored shear stresses due to vertical load acting on the critical section given in 22.6.9.8 and the shear stresses resulting from factored moment transferred by eccentricity of shear about the centroid of the critical section closest to the column given in 22.6.4.1(a) shall not exceed .

### 22.7.1.1

This section shall apply to members if Tu ≥ ϕTth, where ϕ is given in Chapter 21 and threshold torsion Tth is given in 22.7.4. If Tu < ϕTth, it shall be permitted to neglect torsional effects.

### 22.7.1.2

Nominal torsional strength shall be calculated in accordance with 22.7.6.

### 22.7.1.3

For calculation of Tth and Tcr, λ shall be in accordance with 19.2.4.

### 22.7.2.1

The value of used to calculate Tth and Tcr shall not exceed 100 psi.

### 22.7.2.2

The values of fy and fyt for longitudinal and transverse torsional reinforcement shall not exceed the limits in 20.2.2.4.

### 22.7.3.1

If Tu ≥ ϕTcr and Tu is required to maintain equilibrium, the member shall be designed to resist Tu.

### 22.7.3.2

In a statically indeterminate structure where Tu ≥ ϕTcr and a reduction of Tu can occur due to redistribution of internal forces after torsional cracking, it shall be permitted to reduce Tu to ϕTcr, where the cracking torsion Tcr is calculated in accordance with 22.7.5.

### 22.7.3.3

If Tu 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.

### 22.7.4.1

Threshold torsion Tth 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 Nu is positive for compression and negative for tension.
Table 22.7.4.1(a)—Threshold torsion for solid cross sections
Type of member Tth
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 Tth
Nonprestressed member (a)
Prestressed member (b)
Nonprestressed member subjected to axial force (c)

### 22.7.5.1

Cracking torsion Tcr shall be calculated in accordance with Table 22.7.5.1 for solid and hollow cross sections, where Nu is positive for compression and negative for tension.
Table 22.7.5.1—Cracking torsion
Type of member Tcr
Nonprestressed member (a)
Prestressed member (b)
Nonprestressed member subjected to axial force (c)

### 22.7.6.1

For nonprestressed and prestressed members, Tn shall be the lesser of (a) and (b):
 (a) (22.7.6.1a) (b) (22.7.6.1b)
where Ao shall be determined by analysis, θ shall not be taken less than 30 degrees nor greater than 60 degrees; At is the area of one leg of a closed stirrup resisting torsion; A is the area of longitudinal torsional reinforcement; and ph is the perimeter of the centerline of the outermost closed stirrup.

### 22.7.6.1.1

In Eq. (22.7.6.1a) and (22.7.6.1b), it shall be permitted to take Ao equal to 0.85Aoh.

### 22.7.6.1.2

In Eq. (22.7.6.1a) and (22.7.6.1b), it shall be permitted to take θ equal to (a) or (b):
(a) 45 degrees for nonprestressed members or members with Apsfse < 0.4(Aps fpu + As fy)
(b) 37.5 degrees for prestressed members with Apsfse ≥ 0.4(Aps fpu + As fy)

### 22.7.7.1

Cross-sectional dimensions shall be selected such that (a) or (b) is satisfied:
(a) For solid sections (22.7.7.1a)
(b) For hollow sections (22.7.7.1b)

### 22.7.7.1.1

For prestressed members, the value of d used in 22.7.7.1 need not be taken less than 0.8h.

### 22.7.7.1.2

For hollow sections where the wall thickness varies around the perimeter, Eq. (22.7.7.1b) shall be evaluated at the location where the term is a maximum.

### 22.7.7.2

For hollow sections where the wall thickness is less than Aoh/Ph, the term (TuPh/1.7Aoh2) in Eq. (22.7.7.1b) shall be taken as (Tu/1.7Aoht), where t is the thickness of the wall of the hollow section at the location where the stresses are being checked.

### 22.8.1.1

Section 22.8 shall apply to the calculation of bearing strength of concrete members.

### 22.8.1.2

Bearing strength provisions in 22.8 shall not apply to post-tensioned anchorage zones or strut-and-tie models.

### 22.8.2.1

Factored compressive force transferred through bearing shall be calculated in accordance with the factored load combinations defined in Chapter 5 and analysis procedures defined in Chapter 6.

### 22.8.3.1

Design bearing strength shall satisfy:
 ϕBn ≥ Bu (22.8.3.1)
for each applicable factored load combination.

### 22.8.3.2

Nominal bearing strength Bn shall be calculated in accordance with Table 22.8.3.2, where A1 is the loaded area, and A2 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.
Table 22.8.3.2—Nominal bearing strength
Geometry of bearing area Bn
Supporting surface is wider on all sides than the loaded area Lesser of (a) and (b) (a)
2(0.85fc'A1) (b)
Other cases 0.85fc'A1 (c)

### 22.9.1.1

This section shall apply where it is appropriate to consider shear transfer across any given plane, such as an existing or potential crack, an interface between dissimilar materials, or an interface between two concretes cast at different times.

### 22.9.1.2

The required area of shear-friction reinforcement across the assumed shear plane, Avf, 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.

### 22.9.1.3

The value of fy used to calculate Vn for shear friction shall not exceed the limit in 20.2.2.4.

### 22.9.1.4

Surface preparation of the shear plane assumed for design shall be specified in the construction documents.

### 22.9.2.1

Factored forces across the assumed shear plane shall be calculated in accordance with the factored load combinations defined in Chapter 5 and analysis procedures defined in Chapter 6.

### 22.9.3.1

Design shear strength across the assumed shear plane shall satisfy:
 ϕVn ≥ Vu (22.9.3.1)
for each applicable factored load combination.

### 22.9.4.1

Value of Vn across the assumed shear plane shall be calculated in accordance with 22.9.4.2 or 22.9.4.3. Vn shall not exceed the value calculated in accordance with 22.9.4.4.

### 22.9.4.2

If shear-friction reinforcement is perpendicular to the shear plane, nominal shear strength across the assumed shear plane shall be calculated by:
 Vn = µAvf fy (22.9.4.2)
where Avf 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
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)
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.

### 22.9.4.3

If shear-friction reinforcement is inclined to the shear plane and the shear force induces tension in the shear-friction reinforcement, nominal shear strength across the assumed shear plane shall be calculated by:
 Vn = Avffy(µsinα + 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.

### 22.9.4.4

The value of Vn 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 fc' shall be used in Table 22.9.4.4.
Table 22.9.4.4—Maximum Vn across the assumed shear plane
Condition Maximum Vn
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) 0.2fc'Ac (a)
(480 + 0.08fc')Ac (b)
1600Ac (c)
Other cases Lesser of (d) and (e) 0.2fc'Ac (d)
800Ac (e)

### 22.9.4.5

Permanent net compression across the shear plane shall be permitted to be added to Avf fy, the force in the shear-friction reinforcement, to calculate required Avf.

### 22.9.4.6

Area of reinforcement required to resist a net factored tension across an assumed shear plane shall be added to the area of reinforcement required for shear friction crossing the assumed shear plane.

### 22.9.5.1

Reinforcement crossing the shear plane to satisfy 22.9.4 shall be anchored to develop fy on both sides of the shear plane.