Chapter 22 Sectional Strength

Equilibrium shall be satisfied at each section.

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

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

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

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

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

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.

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.

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

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.

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

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

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

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

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 f_y.

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.

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

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

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

For calculation of 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 f_c' for the element that results in the most critical value of M_n.

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

Nominal axial compressive strength P_n shall not exceed P_n,max in accordance with Table 22.4.2.1, where P_o 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.

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)

For nonprestressed members and composite steel and concrete members, P_o shall be calculated by:

Po = 0.85fc'(Ag— Ast) + fyAst

(22.4.2.2)

For prestressed members, P_o shall be calculated by:

Po = 0.85fc'(Ag— Ast— Apd) + fyAst— (fse— 0.003Ep)Apt

(22.4.2.3)

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

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

Nominal axial tensile strength of a nonprestressed, composite, or prestressed member, P_nt, shall not be taken greater than P_nt,max, calculated by:

Pnt,max = fyAst + (fse + Δfp)Apt

(22.4.3.1)

Nominal one-way shear strength at a section, V_n, shall be calculated by:

Vn = Vc+ Vs

(22.5.1.1)

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

(22.5.1.2)

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

For prestressed members, V_c, V_ci, and V_cw shall be calculated in accordance with 22.5.8 or 22.5.9.

For calculation of V_c, V_ci, and V_cw, λ shall be in accordance with 19.2.4.

V_s shall be calculated in accordance with 22.5.10.

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

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

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

For calculation of V_c and V_s 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.

For calculation of V_c and V_s in solid, circular sections, d shall be permitted to be taken as 0.8 times the diameter, and b_w shall be permitted to be taken as the diameter.

The value of used to calculate V_c, V_ci, and V_cw for one-way shear shall not exceed 100 psi, unless allowed in 22.5.3.2.

Values of greater than 100 psi shall be permitted in calculating V_c, V_ci, and V_cw 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.

The values of f_y and f_yt used to calculate V_s shall not exceed the limits in 20.2.2.4.

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

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

For calculation of 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.

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

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

For nonprestressed members without axial force, V_c shall be calculated by:

(22.5.5.1)

Vc
Least of (a), (b), and (c):	[1]	(a)
		(b)
		(c)

For nonprestressed members with axial compression, V_c shall be calculated by:

(22.5.6.1)

Vc
Lesser of (a) and (b):	[1] Equation not applicable if	(a)
		(b)

For nonprestressed members with significant axial tension, V_c shall be calculated by:

(22.5.7.1)

This section shall apply to the calculation of 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.

For prestressed flexural members with A_psf_se ≥ 0.4(A_psf_pu + A_sf_y), V_c 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.

Vc
Least of (a), (b), and (c):	[1]	(a)
		(b)
		(c)

For prestressed members, V_c shall be permitted to be the lesser of V_ci calculated in accordance with 22.5.8.3.1 and V_cw calculated in accordance with 22.5.8.3.2 or 22.5.8.3.3.

When calculating V_c, the transfer length of prestressed reinforcement, ℓ_tr, shall be assumed to be 50d_b for strand and 100d_b for wire.

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 ℓ_tr from the end of the prestressed reinforcement.

At locations corresponding to a reduced effective prestress force in 22.5.9.2, V_c shall be calculated in accordance with (a) through (c):

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 ℓ_tr from that point.

At locations corresponding to a reduced effective prestress force according to 22.5.9.4, V_c shall be calculated in accordance with (a) through (c):

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

(22.5.10.1)

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

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

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

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.

Nominal shear strength for two-way members without shear reinforcement shall be calculated by

vn = vc

(22.6.1.2)

Nominal shear strength for two-way members with shear reinforcement other than shearheads shall be calculated by

vn = vc+ vs

(22.6.1.3)

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

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

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

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

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

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

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

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

The value of f_yt used to calculate v_s shall not exceed the limits in 20.2.2.4.

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

For two-way members reinforced with headed shear reinforcement or single- or multi-leg stirrups, a critical section with perimeter b_o 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 b_o.

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

For nonprestressed two-way members, v_c shall be calculated in accordance with 22.6.5.2. For prestressed two-way members, v_c shall be calculated in accordance with (a) or (b):

v_c shall be calculated in accordance with Table 22.6.5.2.

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

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

For prestressed, two-way members, it shall be permitted to calculate v_c using 22.6.5.5, provided that (a) through (c) are satisfied:

For prestressed, two-way members conforming to 22.6.5.4, v_c shall be permitted to be the lesser of (a) and (b):

(a)	(22.6.5.5a)
(b)	(22.6.5.5b)

For two-way members with shear reinforcement, the value of v_c calculated at critical sections shall not exceed the limits in Table 22.6.6.1.

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)

For two-way members with shear reinforcement, effective depth shall be selected such that v_u calculated at critical sections does not exceed the values in Table 22.6.6.2

Type of shear reinforcement	Maximum vu at critical sections defined in 22.6.4.1
Stirrups		(a)
Headed shear stud reinforcement		(b)

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):

For two-way members with stirrups, v_s shall be calculated by:

(22.6.7.2)

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.

For two-way members with headed shear stud reinforcement, v_s shall be calculated by:

(22.6.8.2)

If headed shear stud reinforcement is provided, A_v/s shall satisfy:

(22.6.8.3)

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.

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

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

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

The ratio α_v 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.

For each arm of the shearhead, M_p shall satisfy:

(22.6.9.6)

Nominal flexural strength contributed to each slab column strip by a shearhead, M_v, shall satisfy:

(22.6.9.7)

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)[ℓ_v — (c₁/2)] from the column face. This critical section shall be located so that b_o is a minimum, but need not be closer than d/2 to the edges of the supporting column.

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

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).

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

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 .

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

Nominal torsional strength shall be calculated in accordance with 22.7.6.

For calculation of T_th and T_cr, λ shall be in accordance with 19.2.4.

The value of used to calculate T_th and T_cr shall not exceed 100 psi.

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

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

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

If T_u 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.

Threshold torsion T_th 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 N_u is positive for compression and negative for tension.

Type of member	Tth
Nonprestressed member		(a)
Prestressed member		(b)
Nonprestressed member subjected to axial force		(c)

Type of member	Tth
Nonprestressed member		(a)
Prestressed member		(b)
Nonprestressed member subjected to axial force		(c)

Cracking torsion T_cr shall be calculated in accordance with Table 22.7.5.1 for solid and hollow cross sections, where N_u is positive for compression and negative for tension.

Type of member	Tcr
Nonprestressed member		(a)
Prestressed member		(b)
Nonprestressed member subjected to axial force		(c)

For nonprestressed and prestressed members, T_n shall be the lesser of (a) and (b):

(a)		(22.7.6.1a)
(b)		(22.7.6.1b)

Cross-sectional dimensions shall be selected such that (a) or (b) is satisfied:

(22.7.7.1a)

(22.7.7.1b)

For hollow sections where the wall thickness is less than A_oh/_Ph, the term (T_uPh/1.7A_oh²) in Eq. (22.7.7.1b) shall be taken as (T_u/1.7A_oht), where t is the thickness of the wall of the hollow section at the location where the stresses are being checked.

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

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

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.

Design bearing strength shall satisfy:

ϕBn ≥ Bu

(22.8.3.1)

Nominal bearing strength B_n 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.

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)

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.

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

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

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

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.

Design shear strength across the assumed shear plane shall satisfy:

ϕVn ≥ Vu

(22.9.3.1)

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

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)

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)

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)

The value of V_n 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 f_c' shall be used in Table 22.9.4.4.

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)

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

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.

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