Heads up: There are no amended sections in this chapter.
This chapter shall apply to the design of nonprestressed and prestressed beams, including:

(a) Composite beams of concrete elements constructed in separate placements but connected so that all elements resist loads as a unit

(b) One-way joist systems in accordance with 9.8

(c) Deep beams in accordance with 9.9

Design properties for concrete shall be selected to be in accordance with Chapter 19.
Design properties for steel reinforcement shall be selected to be in accordance with Chapter 20.
Materials, design, and detailing requirements for embedments in concrete shall be in accordance with 20.7.
For cast-in-place construction, beam-column and slab-column joints shall satisfy Chapter 15.
For precast construction, connections shall satisfy the force transfer requirements of 16.2.
If a beam is not continuously laterally braced, (a) and (b) shall be satisfied:

(a) Spacing of lateral bracing shall not exceed 50 times the least width of compression flange or face.

(b) Spacing of lateral bracing shall take into account effects of eccentric loads.

In prestressed beams, buckling of thin webs and flanges shall be considered. If there is intermittent contact between prestressed reinforcement and an oversize duct, member buckling between contact points shall be considered.
In T-beam construction, flange and web concrete shall be placed monolithically or made composite in accordance with 16.4.
Effective flange width shall be in accordance with 6.3.2.
For T-beam flanges where the primary flexural slab reinforcement is parallel to the longitudinal axis of the beam, reinforcement in the flange perpendicular to the longitudinal axis of the beam shall be in accordance with 7.5.2.3.
For torsional design according to 22.7, the overhanging flange width used to calculate Acp, Ag, and pcp shall be in accordance with (a) and (b):

(a) The overhanging flange width shall include that portion of slab on each side of the beam extending a distance equal to the projection of the beam above or below the slab, whichever is greater, but not greater than four times the slab thickness.

(b) The overhanging flanges shall be neglected in cases where the parameter Acp2/pcp for solid sections or Ag2/pcp for hollow sections calculated for a beam with flanges is less than that calculated for the same beam ignoring the flanges.

For nonprestressed beams not supporting or attached to partitions or other construction likely to be damaged by large deflections, overall beam depth h shall satisfy the limits in Table 9.3.1.1, unless the calculated deflection limits of 9.3.2 are satisfied.

Table 9.3.1.1—Minimum depth of nonprestressed beams

Support condition Minimum h[1]
Simply supported /16
One end continuous /18.5
Both ends continuous /21
Cantilever /8

[1]Expressions applicable for normalweight concrete and fy = 60,000 psi. For other cases, minimum h shall be modified in accordance with 9.3.1.1.1 through 9.3.1.1.3, as appropriate.

For fy other than 60,000 psi, the expressions in Table 9.3.1.1 shall be multiplied by (0.4 + fy/100,000).
For nonprestressed beams made of lightweight concrete having wc in the range of 90 to 115 lb/ft3, the expressions in Table 9.3.1.1 shall be multiplied by the greater of (a) and (b):

(a) 1.65 — 0.005wc

(b) 1.09

For nonprestressed composite beams made of a combination of lightweight and normalweight concrete, shored during construction, and where the lightweight concrete is in compression, the modifier of 9.3.1.1.2 shall apply.
The thickness of a concrete floor finish shall be permitted to be included in h if it is placed monolithically with the beam or if the floor finish is designed to be composite with the beam in accordance with 16.4.
For nonprestressed beams not satisfying 9.3.1 and for prestressed beams, immediate and time-dependent deflections shall be calculated in accordance with 24.2 and shall not exceed the limits in 24.2.2.
For nonprestressed composite concrete beams satisfying 9.3.1, deflections occurring after the member becomes composite need not be calculated. Deflections occurring before the member becomes composite shall be investigated unless the precomposite depth also satisfies 9.3.1.
For nonprestressed beams with Pu < 0.10f'cAg, εt shall be at least 0.004.
Prestressed beams shall be classified as Class U, T, or C in accordance with 24.5.2.
Stresses in prestressed beams immediately after transfer and at service loads shall not exceed permissible stresses in 24.5.3 and 24.5.4.
Required strength shall be calculated in accordance with the factored load combinations in Chapter 5.
Required strength shall be calculated in accordance with the analysis procedures in Chapter 6.
For prestressed beams, effects of reactions induced by prestressing shall be considered in accordance with 5.3.11.
For beams built integrally with supports, Mu at the support shall be permitted to be calculated at the face of support.
For beams built integrally with supports, Vu at the support shall be permitted to be calculated at the face of support.
Sections between the face of support and a critical section located d from the face of support for nonprestressed beams and h/2 from the face of support for prestressed beams shall be permitted to be designed for Vu at that critical section if (a) through (c) are satisfied:

(a) Support reaction, in direction of applied shear, introduces compression into the end region of the beam

(b) Loads are applied at or near the top surface of the beam

(c) No concentrated load occurs between the face of support and critical section

Unless determined by a more detailed analysis, it shall be permitted to take the torsional loading from a slab as uniformly distributed along the beam.
For beams built integrally with supports, Tu at the support shall be permitted to be calculated at the face of support.
Sections between the face of support and a critical section located d from the face of support for nonprestressed beams or h/2 from the face of support for prestressed beams shall be permitted to be designed for Tu at that critical section unless a concentrated torsional moment occurs within this distance. In that case, the critical section shall be taken at the face of the support.
It shall be permitted to reduce Tu in accordance with 22.7.3.
For each applicable factored load combination, design strength at all sections shall satisfy ϕSnU including (a) through (d). Interaction between load effects shall be considered.

(a) ϕMnMu

(b) ϕVnVu

(c) ϕTnTu

(d) ϕPnPu

ϕ shall be determined in accordance with 21.2.
If Pu < 0.10f'cAg, Mn shall be calculated in accordance with 22.3.
If Pu ≥ 0.10f'cAg, Mn shall be calculated in accordance with 22.4.
For prestressed beams, external tendons shall be considered as unbonded tendons in calculating flexural strength, unless the external tendons are effectively bonded to the concrete along the entire length.
Vn shall be calculated in accordance with 22.5.
For composite concrete beams, horizontal shear strength Vnh shall be calculated in accordance with 16.4.
If Tu < ϕTth, where Tth is given in 22.7, it shall be permitted to neglect torsional effects. The minimum reinforcement requirements of 9.6.4 and the detailing requirements of 9.7.5 and 9.7.6.3 need not be satisfied.
Tn shall be calculated in accordance with 22.7.
Longitudinal and transverse reinforcement required for torsion shall be added to that required for the Vu, Mu, and Pu that act in combination with the torsion.
For prestressed beams, the total area of longitudinal reinforcement, As and Aps, at each section shall be designed to resist Mu at that section, plus an additional concentric longitudinal tensile force equal to Afy, based on Tu at that section.
It shall be permitted to reduce the area of longitudinal torsional reinforcement in the flexural compression zone by an amount equal to Mu/(0.9dfy), where Mu occurs simultaneously with Tu at that section, except that the longitudinal reinforcement area shall not be less than the minimum required in 9.6.4.
For solid sections with an aspect ratio h/bt ≥ 3, it shall be permitted to use an alternative design procedure, provided the adequacy of the procedure has been shown by analysis and substantial agreement with results of comprehensive tests. The minimum reinforcement requirements of 9.6.4 need not be satisfied, but the detailing requirements of 9.7.5 and 9.7.6.3 apply.
For solid precast sections with an aspect ratio h/bt ≥ 4.5, it shall be permitted to use an alternative design procedure and open web reinforcement, provided the adequacy of the procedure and reinforcement have been shown by analysis and substantial agreement with results of comprehensive tests. The minimum reinforcement requirements of 9.6.4 and detailing requirements of 9.7.5 and 9.7.6.3 need not be satisfied.
A minimum area of flexural reinforcement, As,min, shall be provided at every section where tension reinforcement is required by analysis.
As,min shall be the greater of (a) and (b), except as provided in 9.6.1.3. For a statically determinate beam with a flange in tension, the value of bw shall be the lesser of bf and 2bw.

(a)

(b)

If As provided at every section is at least one-third greater than As required by analysis, 9.6.1.1 and 9.6.1.2 need not be satisfied.
For beams with bonded prestressed reinforcement, total quantity of As and Aps shall be adequate to develop a factored load at least 1.2 times the cracking load calculated on the basis of fr defined in 19.2.3.
For beams with both flexural and shear design strength at least twice the required strength, 9.6.2.1 need not be satisfied.
For beams with unbonded tendons, the minimum area of bonded deformed longitudinal reinforcement As,min shall be:
As,min = 0.004Act (9.6.2.3)

where Act is the area of that part of the cross section between the flexural tension face and the centroid of the gross section.

A minimum area of shear reinforcement, Av,min, shall be provided in all regions where Vu > 0.5ϕVc except for the cases in Table 9.6.3.1. For these cases, at least Av,min shall be provided where Vu > ϕVc.

Table 9.6.3.1—Cases where Av,min is not required if 0.5ϕVc < Vu ≤ ϕVc

Beam type Conditions
Shallow depth h ≤ 10 in.
Integral with slab h ≤ greater of 2.5tf or 0.5bw
and
h ≤ 24 in.
Constructed with steel fiber-reinforced normalweight concrete conforming to 26.4.1.5.1(a), 26.4.2.2(d), and 26.12.5.1(a) and with f'c ≤ 6000 psi h ≤ 24 in.
and
One-way joist system In accordance with 9.8
If shown by testing that the required Mn and Vn can be developed, 9.6.3.1 need not be satisfied. Such tests shall simulate effects of differential settlement, creep, shrinkage, and temperature change, based on a realistic assessment of these effects occurring in service.
If shear reinforcement is required and torsional effects can be neglected according to 9.5.4.1, Av,min shall be in accordance with Table 9.6.3.3.

Table 9.6.3.3—Required Av,min

Beam type Av,min/s
Nonprestressed and prestressed with Apsfse< 0.4(Apsfpu + Asfy) Greater of: (a)
(b)
Prestressed with Apsfse≥ 0.4(Apsfpu+ Asfy) Lesser of: Greater of: (c)
(d)
(e)
A minimum area of torsional reinforcement shall be provided in all regions where Tu ≥ ϕTth in accordance with 22.7.
If torsional reinforcement is required, minimum transverse reinforcement (Av + 2At)min/s shall be the greater of (a) and (b):

(a)

(b)

If torsional reinforcement is required, minimum area of longitudinal reinforcement Aℓ,min shall be the lesser of (a) and (b):

(a)

(b)

Concrete cover for reinforcement shall be in accordance with 20.6.1.
Development lengths of deformed and prestressed reinforcement shall be in accordance with 25.4.
Splices of deformed reinforcement shall be in accordance with 25.5.
Bundled bars shall be in accordance with 25.6.
Minimum spacing s shall be in accordance with 25.2.
For nonprestressed and Class C prestressed beams, spacing of bonded longitudinal reinforcement closest to the tension face shall not exceed s given in 24.3.
For nonprestressed and Class C prestressed beams with h exceeding 36 in., longitudinal skin reinforcement shall be uniformly distributed on both side faces of the beam for a distance h/2 from the tension face. Spacing of skin reinforcement shall not exceed s given in 24.3.2, where cc is the clear cover from the skin reinforcement to the side face. It shall be permitted to include skin reinforcement in strength calculations if a strain compatibility analysis is made.
Calculated tensile or compressive force in reinforcement at each section of the beam shall be developed on each side of that section.
Critical locations for development of reinforcement are points of maximum stress and points along the span where bent or terminated tension reinforcement is no longer required to resist flexure.
Reinforcement shall extend beyond the point at which it is no longer required to resist flexure for a distance equal to the greater of d and 12db, except at supports of simply-supported spans and at free ends of cantilevers.
Continuing flexural tension reinforcement shall have an embedment length at least d beyond the point where bent or terminated tension reinforcement is no longer required to resist flexure.
Flexural tension reinforcement shall not be terminated in a tension zone unless (a), (b), or (c) is satisfied:

(a) Vu ≤ (2/3)ϕVn at the cutoff point

(b) For No. 11 bars and smaller, continuing reinforcement provides double the area required for flexure at the cutoff point and Vu ≤ (3/4)ϕVn

(c) Stirrup or hoop area in excess of that required for shear and torsion is provided along each terminated bar or wire over a distance 3/4d from the termination point. Excess stirrup or hoop area shall be at least 60bws/fyt. Spacing s shall not exceed d/(8βb)

Adequate anchorage shall be provided for tension reinforcement where reinforcement stress is not directly proportional to moment, such as in sloped, stepped, or tapered beams, or where tension reinforcement is not parallel to the compression face.
Development of tension reinforcement by bending across the web to be anchored or made continuous with reinforcement on the opposite face of beam shall be permitted.
At simple supports, at least one-third of the maximum positive moment reinforcement shall extend along the beam bottom into the support at least 6 in., except for precast beams where such reinforcement shall extend at least to the center of the bearing length.
At other supports, at least one-fourth of the maximum positive moment reinforcement shall extend along the beam bottom into the support at least 6 in. and, if the beam is part of the primary lateral-load-resisting system, shall be anchored to develop fy at the face of the support.
At simple supports and points of inflection, db for positive moment tension reinforcement shall be limited such that d for that reinforcement satisfies (a) or (b). If reinforcement terminates beyond the centerline of supports by a standard hook or a mechanical anchorage at least equivalent to a standard hook, (a) or (b) need not be satisfied.

(a) d ≤ (1.3Mn/Vu + a) if end of reinforcement is confined by a compressive reaction

(b) d ≤ (Mn/Vu + a) if end of reinforcement is not confined by a compressive reaction

Mn is calculated assuming all reinforcement at the section is stressed to fy, and Vu is calculated at the section. At a support, a is the embedment length beyond the center of the support. At a point of inflection, a is the embedment length beyond the point of inflection limited to the greater of d and 12db.

At least one-third of the negative moment reinforcement at a support shall have an embedment length beyond the point of inflection at least the greatest of d, 12db, and n/16.
External tendons shall be attached to the member in a manner that maintains the specified eccentricity between the tendons and the concrete centroid through the full range of anticipated member deflections.
If nonprestressed reinforcement is required to satisfy flexural strength, the detailing requirements of 9.7.3 shall be satisfied.
Post-tensioned anchorage zones shall be designed and detailed in accordance with 25.9.
Post-tensioning anchorages and couplers shall be designed and detailed in accordance with 25.8.
Length of deformed reinforcement required by 9.6.2.3 shall be in accordance with (a) and (b):

(a) At least n/3 in positive moment areas and be centered in those areas

(b) At least n/6 on each side of the face of support in negative moment areas

If torsional reinforcement is required, longitudinal torsional reinforcement shall be distributed around the perimeter of closed stirrups that satisfy 25.7.1.6 or hoops with a spacing not greater than 12 in. The longitudinal reinforcement shall be inside the stirrup or hoop, and at least one longitudinal bar or tendon shall be placed in each corner.
Longitudinal torsional reinforcement shall have a diameter at least 0.042 times the transverse reinforcement spacing, but not less than 3/8 in.
Longitudinal torsional reinforcement shall extend for a distance of at least (bt + d) beyond the point required by analysis.
Longitudinal torsional reinforcement shall be developed at the face of the support at both ends of the beam.
Transverse reinforcement shall be in accordance with this section. The most restrictive requirements shall apply.
Details of transverse reinforcement shall be in accordance with 25.7.
If required, shear reinforcement shall be provided using stirrups, hoops, or longitudinal bent bars.
Maximum spacing of shear reinforcement shall be in accordance with Table 9.7.6.2.2.

Table 9.7.6.2.2—Maximum spacing of shear reinforcement

Vs Maximum s, in.
Nonprestressed beam Prestressed beam
Lesser of: d/2 3h/4
24
Lesser of: d/4 3h/8
12
Inclined stirrups and longitudinal bars bent to act as shear reinforcement shall be spaced so that every 45-degree line, extending d/2 toward the reaction from mid-depth of member to longitudinal tension reinforcement, shall be crossed by at least one line of shear reinforcement.
Longitudinal bars bent to act as shear reinforcement, if extended into a region of tension, shall be continuous with longitudinal reinforcement and, if extended into a region of compression, shall be anchored d/2 beyond mid-depth of member.
If required, transverse torsional reinforcement shall be closed stirrups satisfying 25.7.1.6 or hoops.
Transverse torsional reinforcement shall extend a distance of at least (bt + d) beyond the point required by analysis.
Spacing of transverse torsional reinforcement shall not exceed the lesser of ph/8 and 12 in.
For hollow sections, the distance from the centerline of the transverse torsional reinforcement to the inside face of the wall of the hollow section shall be at least 0.5Aoh/ph.
Transverse reinforcement shall be provided throughout the distance where longitudinal compression reinforcement is required. Lateral support of longitudinal compression reinforcement shall be provided by closed stirrups or hoops in accordance with 9.7.6.4.2 through 9.7.6.4.4.
Size of transverse reinforcement shall be at least (a) or (b). Deformed wire or welded wire reinforcement of equivalent area shall be permitted.

(a) No. 3 for longitudinal bars No. 10 and smaller

(b) No. 4 for longitudinal bars No. 11 and larger and for longitudinal bundled bars

Spacing of transverse reinforcement shall not exceed the least of (a) through (c):

(a) 16db of longitudinal reinforcement

(b) 48db of transverse reinforcement

(c) Least dimension of beam

Longitudinal compression reinforcement shall be arranged such that every corner and alternate compression bar shall be enclosed by the corner of the transverse reinforcement with an included angle of not more than 135 degrees, and no bar shall be farther than 6 in. clear on each side along the transverse reinforcement from such an enclosed bar.
For beams along the perimeter of the structure, structural integrity reinforcement shall be in accordance with (a) through (c):

(a) At least one-quarter of the maximum positive moment reinforcement, but not less than two bars or strands, shall be continuous

(b) At least one-sixth of the negative moment reinforcement at the support, but not less than two bars or strands, shall be continuous

(c) Longitudinal structural integrity reinforcement shall be enclosed by closed stirrups in accordance with 25.7.1.6 or hoops along the clear span of the beam

For other than perimeter beams, structural integrity reinforcement shall be in accordance with (a) or (b):

(a) At least one-quarter of the maximum positive moment reinforcement, but not less than two bars or strands, shall be continuous.

(b) Longitudinal reinforcement shall be enclosed by closed stirrups in accordance with 25.7.1.6 or hoops along the clear span of the beam.

Longitudinal structural integrity reinforcement shall pass through the region bounded by the longitudinal reinforcement of the column.
Longitudinal structural integrity reinforcement at noncontinuous supports shall be anchored to develop fy at the face of the support.
If splices are necessary in continuous structural integrity reinforcement, the reinforcement shall be spliced in accordance with (a) and (b):

(a) Positive moment reinforcement shall be spliced at or near the support

(b) Negative moment reinforcement shall be spliced at or near midspan

Splices shall be full mechanical, full welded, or Class B tension lap splices.
Nonprestressed one-way joist construction consists of a monolithic combination of regularly spaced ribs and a top slab designed to span in one direction.
Width of ribs shall be at least 4 in. at any location along the depth.
Overall depth of ribs shall not exceed 3.5 times the minimum width.
Clear spacing between ribs shall not exceed 30 in.
Vc shall be permitted to be taken as 1.1 times the value calculated in 22.5.
For structural integrity, at least one bottom bar in each joist shall be continuous and shall be anchored to develop fy at the face of supports.
Reinforcement perpendicular to the ribs shall be provided in the slab as required for flexure, considering load concentrations, and shall be at least that required for shrinkage and temperature in accordance with 24.4.
One-way joist construction not satisfying the limitations of 9.8.1.1 through 9.8.1.4 shall be designed as slabs and beams.
If permanent burned clay or concrete tile fillers of material having a unit compressive strength at least equal to f'c in the joists are used, 9.8.2.1.1 and 9.8.2.1.2 shall apply.
Slab thickness over fillers shall be at least the greater of one-twelfth the clear distance between ribs and 1.5 in.
For calculation of shear and negative moment strength, it shall be permitted to include the vertical shells of fillers in contact with the ribs. Other portions of fillers shall not be included in strength calculations.
If fillers not complying with 9.8.2.1 or removable forms are used, slab thickness shall be at least the greater of one-twelfth the clear distance between ribs and 2 in.
Deep beams are members that are loaded on one face and supported on the opposite face such that strut-like compression elements can develop between the loads and supports and that satisfy (a) or (b):

(a) Clear span does not exceed four times the overall member depth h

(b) Concentrated loads exist within a distance 2h from the face of the support

Deep beams shall be designed taking into account nonlinear distribution of longitudinal strain over the depth of the beam.
Strut-and-tie models in accordance with Chapter 23 are deemed to satisfy 9.9.1.2.
Deep beam dimensions shall be selected such that:
(9.9.2.1)
Distributed reinforcement along the side faces of deep beams shall be at least that required in (a) and (b):

(a) The area of distributed reinforcement perpendicular to the longitudinal axis of the beam, Av, shall be at least 0.0025bws, where s is the spacing of the distributed transverse reinforcement.

(b) The area of distributed reinforcement parallel to the longitudinal axis of the beam, Avh, shall be at least 0.0025bws2, where s2 is the spacing of the distributed longitudinal reinforcement.

The minimum area of flexural tension reinforcement, As,min, shall be determined in accordance with 9.6.1.
Concrete cover shall be in accordance with 20.6.1.
Minimum spacing for longitudinal reinforcement shall be in accordance with 25.2.
Spacing of distributed reinforcement required in 9.9.3.1 shall not exceed the lesser of d/5 and 12 in.
Development of tension reinforcement shall account for distribution of stress in reinforcement that is not directly proportional to the bending moment.
At simple supports, positive moment tension reinforcement shall be anchored to develop fy at the face of the support. If a deep beam is designed using Chapter 23, the positive moment tension reinforcement shall be anchored in accordance with 23.8.2 and 23.8.3.
At interior supports, (a) and (b) shall be satisfied:

(a) Negative moment tension reinforcement shall be continuous with that of the adjacent spans.

(b) Positive moment tension reinforcement shall be continuous or spliced with that of the adjacent spans.

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