ACI 318-14 CR094/LB13-3 2 May 2013
CR094 – COMMENTARY TO CHAPTER 9 1
SECTIONAL STRENGTH 2
3
Background: 4
5
Chapter 9 was balloted by 318 three times (LB10-04, LB11-01, and LB11-03) and was approved 6
during the 318 meeting in Cincinnati in October 2011. The Commentary to Chapter 9 is being 7
balloted for the first time on LB13-03. 8
9
In order to facilitate balloting, the proposed sections of Commentary are interspersed with the 10
approved Code sections. The Code provisions are not part of this ballot, and ballot comments on 11
the Code will be considered not relevant (unless an error is identified). 12
13
Boxes have been placed around all proposed Commentary sections to assist with identification. 14
15
Minor editorial changes were introduced in the referencing of equations in 9.5.9.3.1 and 16
9.5.11.6.2 during the development of the Commentary. These changes are shown in red. 17
18
References to Code sections have been updated to reflect the latest version of 318-14. Reference 19
numbers used in the Commentary to Chapter 9 refer to the references in 318-11. The references 20
will be updated to (Author, Date) format after the Commentary is approved. 21
22
Ballot History: 23
24
The Commentary to Chapter 9 was balloted by Sub E on LB13E-01. Negatives by Bonacci, 25
Klein, Kuchma, Novak, Patel, Wood and Yañez were discussed and resolved at the Sub E 26
meeting in Minneapolis in March 2013. 27
28
29
30
CHAPTER 9 — SECTIONAL STRENGTH 31
9.1 — Scope 32
9.1.1 — This chapter provides minimum strength requirements for sections of members as 33
required by other chapters of this Code. Sectional strength requirements of this chapter shall be 34
satisfied unless the member or region of the member is designed in accordance with Chapter 18. 35
<8.1.1> <8.1.2> <9.1.3> 36
R9.1 — This Chapter applies where methods for determining strength at critical sections are 37
appropriate. Chapter 18 provides methods for designing discontinuity regions where section 38
based methods do not apply. <~> 39
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9.1.2 — Design strength at a section shall be taken as the nominal strength multiplied by the 40
applicable strength reduction factor, . <8.1.1> <9.1.1> <9.3.1> 41
9.1.3 — Nominal strength at a section of a member shall be calculated in accordance with: <~> 42
(a) 9.3 for flexure 43
(b) 9.4 for combined flexure and axial force 44
(c) 9.5 for one-way shear 45
(d) 9.6 for two-way shear 46
(e) 9.7 for torsion 47
9.2 — Design assumptions for flexural and axial strength 48
9.2.1 — Equilibrium and strain compatibility 49
R9.2.1 — The flexural and axial strength of a member computed by the strength design method 50
of the Code requires that two basic conditions be satisfied: (1) static equilibrium, and (2) 51
compatibility of strains. Equilibrium between the compressive and tensile forces acting on the 52
cross section at nominal strength should be satisfied. 53
Compatibility between the stress and strain for the concrete and the reinforcement at nominal 54
strength conditions should also be established within the design assumptions allowed by 9.2. 55
<R10.2.1> 56
9.2.1.1 — Conditions of equilibrium shall be satisfied at each section. <10.2.1> <10.3.1> 57
<18.3.1> 58
9.2.1.2 — Strain in concrete and nonprestressed reinforcement shall be assumed proportional 59
to the distance from neutral axis. <10.2.2> 60
R9.2.1.2— Many tests have confirmed that the distribution of strain is essentially linear across a 61
reinforced concrete cross section (plane sections remain plane), even near nominal strength. 62
The strain in both nonprestressed reinforcement and in concrete is assumed to be directly 63
proportional to the distance from the neutral axis. This assumption is of primary importance in 64
design for determining the strain and corresponding stress in the reinforcement. <R10.2.2> 65
9.2.1.3 — Strain in prestressed concrete and in bonded and unbonded prestressed 66
reinforcement shall include the strain due to effective prestress. <~> 67
9.2.1.4 — Changes in strain for bonded prestressed reinforcement shall be assumed 68
proportional to the distance from neutral axis. <18.3.2.1> 69
R9.2.1.4 – Changes in strain are caused by factored load combinations as defined in Chapter 7. 70
<~> 71
9.2.2 — Design assumptions for concrete 72
9.2.2.1 — Maximum strain at the extreme concrete compression fiber shall be assumed equal 73
to 0.003. <10.2.3> 74
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ACI 318-14 CR094/LB13-3 2 May 2013
R9.2.2.1 — The maximum concrete compressive strain at crushing of the concrete has been 75
observed in tests of various kinds to vary from 0.003 to higher than 0.008 under special 76
conditions. However, the strain at which nominal moments are developed is usually about 0.003 77
to 0.004 for members of normal proportions and materials. <R10.2.3> 78
9.2.2.2 — Tensile strength of concrete shall be neglected in flexural and axial strength 79
calculations. <10.2.5> <18.3.2.2> 80
R9.2.2.2 — The tensile strength of concrete in flexure (modulus of rupture) is a more variable 81
property than the compressive strength and is about 10 to 15 percent of the compressive strength. 82
Tensile strength of concrete in flexure is neglected in strength design. Neglecting tensile strength 83
of concrete is conservative when predicting the nominal strength of a section. The strength of 84
concrete in tension, however, is important in cracking and deflection considerations at service 85
loads. <R10.2.5> 86
9.2.2.3 — The relationship between concrete compressive stress and strain shall be 87
represented by a rectangular, trapezoidal, parabolic, or other shape that results in prediction of 88
strength in substantial agreement with results of comprehensive tests. <10.2.6> 89
R9.2.2.3 — This assumption recognizes the inelastic stress distribution of concrete at high stress. 90
As maximum stress is approached, the stress-strain relationship for concrete is not a straight line 91
but some form of a curve (stress is not proportional to strain). As discussed under R9.2.2.1, the 92
Code sets the maximum usable strain at 0.003 for design. 93
The actual distribution of concrete compressive stress within a cross section is complex and 94
usually not known explicitly. Research has shown that the important properties of the concrete 95
stress distribution can be approximated closely using any one of several different assumptions 96
for the shape of the stress distribution. 97
The Code permits any stress distribution to be assumed in design if shown to result in predictions 98
of nominal strength in reasonable agreement with the results of comprehensive tests. Many stress 99
distributions have been proposed. The three most common are the parabola, trapezoid, and 100
rectangle. <R10.2.6> 101
9.2.2.4 — The equivalent rectangular concrete stress distribution defined in 9.2.2.4.1 through 102
9.2.2.4.3 satisfies 9.2.2.3. <10.2.7> 103
R9.2.2.4 — For design, the Code allows the use of an equivalent rectangular compressive stress 104
distribution (stress block) to replace the more detailed approximation of the concrete stress 105
distribution. <R10.2.7> 106
— Concrete stress of 0.85 cf shall be assumed uniformly distributed over an 9.2.2.4.1107
equivalent compression zone bounded by edges of the cross section and a line parallel to 108
the neutral axis located a distance a from the fiber of maximum compressive strain: 109
<10.2.7.1> 110
1a c (9.2.2.4.1) 111
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R9.2.2.4.1 — In the equivalent rectangular stress block, an average stress of 0.85 cf is used with 112
a rectangle of depth 1a c . The equivalent rectangular stress distribution does not represent the 113
actual stress distribution in the compression zone at nominal strength, but does provide 114
essentially the same results as those obtained in tests.10.3
<R10.2.7> 115
— Distance from the fiber of maximum compressive strain to the neutral axis, 9.2.2.4.2116
c, shall be measured perpendicular to the neutral axis. <10.2.7.2> 117
— Values of 1 shall be in accordance with Table 9.2.2.4.3 <10.2.7.3> 9.2.2.4.3118
R9.2.2.4.3 — The values for 1 were determined experimentally. The lower limit of 1 for 119
concrete strengths greater than 8000 psi is based on experimental data from beams constructed 120
using high-strength concrete.10.1, 10.2
121
Table 9.2.2.4.3 — Values of 1 for equivalent rectangular concrete stress distribution 122
cf , psi 1
2500 cf 4000 0.85 (a)
4000 cf 8000 0 05 4000
0 851000
cf..
(b)
cf 8000 0.65 (c)
9.2.3 — Design assumptions for nonprestressed reinforcement 123
9.2.3.1 — Deformed reinforcement used to resist tensile or compressive forces shall conform 124
to 6.2.1. <~> 125
9.2.3.2 — Stress-strain relationship and modulus of elasticity for deformed reinforcement 126
shall be idealized in accordance with 6.2.2.1 and 6.2.2.2. <10.2.4> 127
9.2.4 — Design assumptions for prestressing reinforcement 128
9.2.4.1 — For members with bonded prestressing reinforcement conforming to 6.3.1, stress at 129
nominal flexural strength, psf , shall be calculated in accordance with 6.3.2.3. <18.7.2> 130
9.2.4.2 — For members with unbonded prestressing reinforcement conforming to 6.3.1, psf 131
shall be calculated in accordance with 6.3.2.4. <18.7.2> 132
9.2.4.3 — If the embedded length of the prestressing strand is less than d , the design strand 133
stress shall not exceed the value defined in 21.4.8.3, as modified by 21.4.8.1(b). <12.9.3> 134
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9.3 —Flexural strength 135
9.3.1 — General 136
9.3.1.1 — Nominal flexural strength nM shall be calculated in accordance with the 137
assumptions of 9.2. <~> 138
9.3.2 — Strength reduction factor 139
9.3.2.1 — Strength reduction factor for flexural strength, , in nonprestressed members and 140
at sections in pretensioned members where strand embedment equals or exceeds the 141
development length shall be calculated in accordance with 9.4.2. <9.3.1> <9.3.2.1> <10.3.2> 142
<10.3.3> <10.3.4> <18.8.1> 143
9.3.2.2 — For sections in pretensioned members where strand where strand is not fully 144
developed, shall be calculated at each section based on the distance from the end of the 145
member in accordance with Table 9.3.2.2, where db is the longest debonded length at the 146
end of the member and tr is defined in Eq. (9.3.2.2), . sef is the effective stress in the 147
prestressed reinforcement after allowance of all losses, and d is defined in 21.4.8.1. 148
<9.3.2.7> <CE093> 149
3000
setr b
fd (9.3.2.2) 150
Table 9.3.2.2 — Strength reduction factor for sections near the end of pretensioned 151
members 152
Condition near
end of member
Stress in
concrete
under
service
load*
Distance from end of
member to section under
consideration
All strands
bonded
Not
applicable
tr 0.75 (a)
tr to d Linear interpolation
from 0.75 to 0.9†
(b)
One or more
strands
debonded
No tension
db tr 0.75 (c)
db tr to db d Linear interpolation
from 0.75 to 0.9†
(d)
Tension
calculated
db tr 0.75 (e)
db tr to 2db d Linear interpolation
from 0.75 to 0.9†
(f)
* Calculated stress in extreme concrete fiber of precompressed tensile zone under service loads, at any section
along the length of the beam, after allowance for all prestress losses using gross cross-sectional properties.
† For sections within the regions defined in rows (b), (d), and (f), it shall be permitted to use a strength
reduction factor of 0.75. <9.3.2.7> 153
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R9.3.2.2 — If a critical section occurs in a region where strand is not fully developed, failure 154
may be by bond slip. Such a failure resembles a brittle shear failure; hence, values are reduced 155
with respect to a section where strands are fully developed. For sections between the end of the 156
transfer length and the end of the development length, the value of may be determined by 157
linear interpolation, as shown in Fig. R9.3.2.2(a) and (b). 158
Where bonding of one or more strands does not extend to the end of the member, instead of a 159
more rigorous analysis, may be conservatively taken as 0.75 from the end of the member to 160
the end of the transfer length of the strand with the longest debonded length. Beyond this point, 161
may be varied linearly to 0.9 at the location where all strands are developed, as shown in Fig. 162
R9.3.2.2(b). Alternatively, the contribution of the debonded strands may be ignored until they are 163
fully developed. Embedment of debonded strand is considered to begin at the termination of the 164
debonding sleeves. Beyond this point, the provisions of 21.4.8 are applicable. <R9.3.2.7> 165
166
Fig. R9.3.2.2(a)—Variation of with distance from the free end of strand in pretensioned 167
members with fully bonded strands. 168
169
Fig. R9.3.2.2(b)—Variation of with distance from the free end of strand in pretensioned 170
members with debonded strands where 21.4.8.1(b) applies. 171
Use tr rather than 3000
seb
fd
.
Use tr rather than 23000
seb
fd
.
(This is an intentional change from
318-11 and was approved by 318 in
Toronto.)
Refer to Eq. (21.4.8.1)
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ACI 318-14 CR094/LB13-3 2 May 2013
9.3.3 — Prestressed concrete members 172
9.3.3.1 — Deformed reinforcement conforming to 6.2.1 used with prestressed reinforcement 173
shall be permitted to be considered to contribute to the tensile force and be included in 174
flexural strength calculations at a stress equal to yf . <18.7.3> 175
9.3.3.2 — Other nonprestressed reinforcement shall be permitted to be included in flexural 176
strength calculations if a strain compatibility analysis is performed to determine stresses in 177
such reinforcement. <18.7.3> 178
9.3.4 — Composite concrete members 179
9.3.4.1 — Provisions of 9.3.4 apply to concrete elements constructed in separate placements 180
but connected so that all elements resist loads as a unit. <17.1.1> 181
R9.3.4.1 — Composite structural steel-concrete beams are not covered in this chapter. Design 182
provisions for these types of composite members are covered in Reference 17.1. <R17.1.1> 183
9.3.4.2 — For calculation of nM for composite concrete slabs and beams, use of the entire 184
composite section shall be permitted. <17.2.1> 185
9.3.4.3 — For calculation of nM for composite concrete slabs and beams, no distinction shall 186
be made between shored and unshored members. <17.2.4> 187
R9.3.4.3 — Tests have indicated that the strength of a composite concrete member is the same 188
whether or not the first element cast is shored during casting and curing of the second element. 189
<R17.2.4> 190
9.3.4.4 — For calculation of nM for composite concrete members where the specified 191
concrete compressive strength of different elements vary, properties of the individual 192
elements shall be used in design. Alternatively, it shall be permitted to use the value of cf 193
for the element that results in the most critical value of nM .<17.2.3> 194
9.4 — Combined flexural and axial strength 195
9.4.1 — General 196
9.4.1.1 — Nominal combined flexural and axial strength shall be calculated in accordance 197
with the assumptions of 9.2. <18.11.1> 198
9.4.2 — Strength reduction factor 199
R9.4.2 — The nominal flexural strength of a member is reached when the strain in the extreme 200
compression fiber reaches the assumed strain limit 0.003. The net tensile strain t is the tensile 201
strain in the extreme tension steel at nominal strength, exclusive of strains due to prestress, creep, 202
shrinkage, and temperature. The net tensile strain in the extreme tension steel is determined from 203
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ACI 318-14 CR094/LB13-3 2 May 2013
a linear strain distribution at nominal strength, shown in Fig. R9.4.2, using similar triangles. 204
<R10.3.3> 205
206
Fig. R9.4.2—Strain distribution and net tensile strain in a nonprestressed member. 207
When the net tensile strain in the extreme tension steel is sufficiently large (equal to or greater 208
than 0.005), the section is defined as tension-controlled where ample warning of failure with 209
excessive deflection and cracking may be expected. When the net tensile strain in the extreme 210
tension steel is small (less than or equal to the compression-controlled strain limit), a brittle 211
failure condition may be expected, with little warning of impending failure. Flexural members 212
are usually tension-controlled, whereas compression members are usually compression-213
controlled. Some sections, such as those with small axial load and large bending moment, will 214
have net tensile strain in the extreme tension steel between the above limits. These sections are in 215
a transition region between compression- and tension-controlled sections. Section 9.4.2 specifies 216
the appropriate strength reduction factors for tension-controlled and compression-controlled 217
sections, and for intermediate cases in the transition region. 218
Unless unusual amounts of ductility are required, the 0.005 limit will provide ductile behavior 219
for most designs. One condition where greater ductile behavior is required is in design for 220
redistribution of moments in continuous members and frames. Section 8.6.5 permits 221
redistribution of moments. Because moment redistribution is dependent on adequate ductility in 222
hinge regions, moment redistribution is limited to sections that have a net tensile strain of at least 223
0.0075. 224
For beams with compression reinforcement, or T-beams, the effects of compression 225
reinforcement and flanges are automatically accounted for in the computation of net tensile strain 226
t . <R10.3.4> 227
9.4.2.1 — Strength reduction factor for combined flexural and axial strength, , shall be in 228
accordance with Table 9.4.2.1. <9.3.1> <10.3.2> <10.3.3> <10.3.4> <18.8.1> 229
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Table 9.4.2.1 — Strength reduction factor for flexural and axial strength 230
Net tensile stain,
t
Classification
Transverse reinforcement
Spirals conforming to 9.4.3.5 Other
t ty Compression
controlled 0.75 (a) 0.65 (b)
0.005 ty t Transition
0.75 0.15
0.005
t ty
ty
(c)
*
0.65 0.250.005
t ty
ty
(d)
*
t 0.005 Tension
controlled 0.90 (e) 0.90 (f)
* For sections classified as transition, it shall be permitted to use the strength reduction factor corresponding to
compression-controlled conditions. <9.3.2.2>
231
R9.4.2.1 —A lower -factor is used for compression-controlled sections than is used for 232
tension-controlled sections because compression-controlled sections have less ductility, are more 233
sensitive to variations in concrete strength, and generally occur in members that support larger 234
loaded areas than members with tension-controlled sections. Members with spiral reinforcement 235
are assigned a higher than tied columns because they have greater ductility or toughness. 236
<R9.3.2.2> 237
For sections subjected to axial load with flexure, design strengths are determined by multiplying 238
both nP and nM by the appropriate single value of . Compression-controlled and tension-239
controlled sections are defined in 9.4.2.1 as those that have net tensile strain in the extreme 240
tension steel at nominal strength less than or equal to the compression-controlled strain limit, and 241
equal to or greater than 0.005, respectively. For sections with net tensile strain t in the extreme 242
tension steel at nominal strength between the above limits, the value of may be determined by 243
linear interpolation, as shown in Fig. R9.4.2.1. The concept of net tensile strain t is discussed 244
in R9.4.2. <R9.3.2.2> 245
Prior to ACI 318-14, the compression controlled strain limit, was explicitly defined for Grade 60 246
reinforcement and all prestressed reinforcement as 0.002, not for other reinforcement. In ACI 247
318-14, the compression controlled strain limit, ty , was defined as in 9.4.2.2. <~> 248
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ACI 318-14 CR094/LB13-3 2 May 2013
249
Fig. R9.4.2.1—Variation of with net tensile strain in extreme tension steel, t , and tc d for 250
Grade 60 reinforcement and for prestressing steel. 251
Because the compressive strain in the concrete at nominal strength is assumed in 9.2.2.1 to be 252
0.003, the net tensile strain limits for compression-controlled members may also be stated in 253
terms of the ratio tc d , where c is the depth of the neutral axis at nominal strength, and td is 254
the distance from the extreme compression fiber to the extreme tension steel. The tc d limits for 255
compression-controlled and tension-controlled sections are 0.6 and 0.375, respectively. The 0.6 256
limit applies to sections reinforced with Grade 60 steel and to prestressed sections. Figure 257
R9.4.2.1 also gives equations for as a function of tc d . <R.9.3.2.2> 258
9.4.2.2 — For deformed reinforcement, ty shall be taken as y sf E . For Grade 60 259
deformed reinforcement, it shall be permitted to take ty equal to 0.002. <10.3.3> 260
9.4.2.3 — For all prestressed reinforcement, ty shall be taken as 0.002. <10.3.3> 261
9.4.3 — Maximum axial strength 262
9.4.3.1 — Nominal axial strength of compression member, nP , shall not be taken greater than 263
,maxnP , as defined in Table 9.4.3.1, where oP is defined in 9.4.3.2 for nonprestressed and 264
composite members and in 9.4.3.3 for prestressed members. <10.3.6> <10.3.6.1> <10.3.6.2> 265
<10.3.6.3> 266
The equations with the term 1 tc d
are confusing. The term 1 tc d or
td c should be used.
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Table 9.4.3.1— Maximum axial strength 267
Member Transverse
Reinforcement ,maxnP
Nonprestressed Ties conforming to 9.4.3.4 0 80 oP. (a)
Spirals conforming to 9.4.3.5 0 85 oP. (b)
Prestressed Ties 0 80 oP. (c)
Spirals 0 85 oP. (d)
Composite conforming to Chapter 14 All 0 85 oP. (e)
268
R9.4.3.1 —To account for accidental eccentricity, the design axial strength of a section in pure 269
compression is limited to 85 or 80 percent of the nominal strength. These percentage values 270
approximate the axial strengths at eccentricity-to-depth ratios of 0.05 and 0.10, for the spirally 271
reinforced and tied members, respectively. The same axial load limitation applies to both cast-in-272
place and precast compression members. 273
For prestressed members, the design axial strength in pure compression is computed by the 274
strength design methods of Chapter 9, including the effect of the prestressing force. 275
Compression member end moments should be considered in the design of adjacent flexural 276
members. In nonsway frames, the effects of magnifying the end moments need not be considered 277
in the design of the adjacent beams. In sway frames, the magnified end moments should be 278
considered in designing the flexural members, as required in 8.6.4.1. <R10.3.6 and R10.3.7> 279
9.4.3.2 — For nonprestressed and composite members, oP shall be calculated as: 280
0.85o c g st y stP f A A f A (9.4.3.2) 281
where stA is the total area of nonprestressed longitudinal reinforcement. 282
9.4.3.3 — For prestressed members, oP shall be calculated as: 283
0.85 0.003o c g st pd y st se p ptP f A A A f A f E A (9.4.3.3) 284
where ptA is the total area of prestressing reinforcement, pdA is the total area occupied by 285
duct, sheathing, and prestressing reinforcement, and the value of sef shall not be taken less 286
than 0 003 p. E . For grouted, post-tensioned tendons, it shall be permitted to assume 287
pd ptA A . <~> <<Balloted as CE091>> 288
9.4.3.4 — Tie reinforcement for compression members shall satisfy provisions for lateral 289
support of longitudinal reinforcement in 14.7.6.2 and detailing provisions in 21.8.2. <7.10.5> 290
Reference to 14.7.6.2 per LB12-10. 291
9.4.3.5 — Spiral reinforcement for compression members shall satisfy provisions for lateral 292
support of longitudinal reinforcement in 14.7.6.3 and detailing provisions in 21.8.3. <7.10.4> 293
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Reference to 14.7.6.3 per LB12-10. 294
9.5 — One-way shear strength 295
9.5.1 — General 296
9.5.1.1 — Nominal one-way shear strength at a section, nV , shall be calculated as:<11.1.1> 297
n c sV V V (9.5.1.1) 298
R9.5.1.1—In a member without shear reinforcement, shear is assumed to be carried by the 299
concrete web. In a member with shear reinforcement, a portion of the shear strength is assumed 300
to be provided by the concrete and the remainder by the shear reinforcement through the 301
mechanism of the truss analogy. 302
The shear strength provided by concrete cV is assumed to be the same for beams with and 303
without shear reinforcement and is taken as the shear causing significant inclined cracking. 304
These assumptions are discussed in References 11.1, 11.2, and 11.3. 305
Chapter 18 allows the use of strut-and-tie models in the shear design of any structural concrete 306
member, or discontinuity region in a member. Traditional shear design procedures are 307
acceptable in B-regions. <R11.1> 308
9.5.1.2 — Cross-sectional dimensions shall satisfy Eq. (9.5.1.2). <11.4.7.9> 309
8u c c wV V f b d (9.5.1.2) 310
R9.5.1.2 – Eq. (9.5.1.2) ensures that the member cross section is sufficiently large to prevent 311
diagonal concrete compression failure. <~> 312
9.5.1.3 — For nonprestressed members, cV shall be calculated in accordance with 9.5.6, 313
9.5.7, or 9.5.8. <~> 314
9.5.1.4 — For prestressed members, cV , ciV , and cwV shall be calculated in accordance with 315
9.5.9 or 9.5.10. <~> 316
9.5.1.5 — For calculation of cV , ciV , and cwV , shall be determined in accordance with 317
5.2.4. <~> 318
9.5.1.6 — sV shall be calculated in accordance with 9.5.11. <~> 319
9.5.1.7 — Effect of any openings in members shall be considered in calculating nV . 320
<11.1.1.1> 321
R9.5.1.7 — Openings in the web of a member can reduce its shear strength. The effects of 322
openings are discussed in Section 4.7 of Reference 11.1 and in References 11.4 and 11.5. Strut-323
and-tie models can be used to design members with openings, see Chapter 18. <R11.1.1.1> 324
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9.5.1.8 — Effects of axial tension due to creep and shrinkage in restrained members shall be 325
considered in calculating cV . <11.1.1.2> 326
9.5.1.9 — Effect of inclined flexural compression in variable depth members shall be 327
permitted to be considered in calculating cV . <11.1.1.2> 328
R9.5.1.9 — In a member of variable depth, the internal shear at any section is increased or 329
decreased by the vertical component of the inclined flexural stresses. <R11.1.1.2> 330
9.5.2 — Strength reduction factor 331
9.5.2.1 — Strength reduction factor for one-way shear, , shall be 0.75. <9.3.2.3> 332
9.5.3 — Geometric assumptions 333
9.5.3.1 — For calculation of cV and sV in prestressed members, d shall be taken as the 334
distance from extreme compression fiber to centroid of prestressed and any nonprestressed 335
longitudinal reinforcement but need not be taken less than 0.8h. <11.3.1> <11.4.3> 336
R9.5.3.1 — Although the value of d may vary along the span of a prestressed beam, studies11.2
337
have shown that, for prestressed concrete members, d need not be taken less than 0 80. h . The 338
beams considered had some straight tendons or reinforcing bars at the bottom of the section and 339
had stirrups that enclosed the steel. <R11.4.3> 340
9.5.3.2 — For calculation of cV and sV in solid, circular sections, d shall be permitted to be 341
taken as 0.8 times the diameter and wb shall be permitted to be taken as the diameter. 342
<11.2.3> <11.4.7.3> 343
R9.5.3.2 — Shear tests of members with circular sections indicate that the effective area can be 344
taken as the gross area of the section or as an equivalent rectangular area.11.1, 11.14, 11.15
<R11.2.3> 345
Although the transverse reinforcement in a circular section may not consist of straight legs, tests 346
indicate that Eq. (9.5.11.5.3) is conservative if d is taken as defined in 9.5.3.2.11.14, 11.15
347
<R11.4.7.3> 348
9.5.4 — Limiting material strengths 349
9.5.4.1 — The value of used to calculate cV , ciV , and cwV for one-way shear shall not 350
exceed 100 psi, except as allowed in 9.5.4.1. <11.1.2> 351
R9.5.4.1 — Because of a lack of test data and practical experience with concretes having 352
compressive strengths greater than 10,000 psi, the Code imposes a maximum value of 100 psi on 353
cf for use in the calculation of shear strength of concrete beams, joists, and slabs. Exceptions 354
to this limit are permitted in beams and joists if the transverse reinforcement satisfies an 355
increased value for the minimum amount of web reinforcement. There are limited test data on the 356
two-way shear strength of high-strength concrete slabs. Until more experience is obtained for 357
cf
'
Page 8727
ACI 318-14 CR094/LB13-3 2 May 2013
two-way slabs built with concretes that have strengths greater than 10,000 psi, it is prudent to 358
limit cf to 100 psi for the calculation of shear strength. <R11.1.2> 359
9.5.4.2 — Values of greater than 100 psi shall be permitted in calculating cV , ciV , and 360
cwV for reinforced or prestressed concrete beams and concrete joist construction having 361
minimum web reinforcement in accordance with 13.6.3.3 or 13.6.4.2. <11.1.2.1> 362
R9.5.4.2 — Based on the test results in References 11.7, 11.8, 11.9, 11.10, and 11.11, an 363
increase in the minimum amount of transverse reinforcement is required for high-strength 364
concrete. These tests indicated a reduction in the reserve shear strength as cf increased in beams 365
reinforced with the specified minimum amount of transverse reinforcement, which is equivalent 366
to an effective shear stress of 50 psi. <R11.1.2.1> 367
9.5.4.3 — The values of yf and ytf used to calculate sV shall not exceed the limits in 368
6.2.2.4. <11.4.2> 369
9.5.5 — Composite concrete members 370
9.5.5.1 — Provisions of 9.5.5 apply to concrete elements constructed in separate placements 371
but connected so that all elements resist loads as a unit. <17.1.1> 372
R9.5.5.1 — The scope of Chapter 9 is intended to include all types of composite concrete 373
flexural members. In some cases with fully cast-in-place concrete, it may be necessary to design 374
the interface of consecutive placements of concrete as required for composite members. 375
Composite structural steel-concrete beams are not covered in this chapter. Design provisions for 376
such composite members are covered in Reference 17.1. <R17.1.1> 377
9.5.5.2 — For calculation of nV for composite concrete members, no distinction shall be 378
made between shored and unshored members. <17.2.4> 379
R9.5.5.2 — Tests have indicated that the strength of a composite member is the same whether or 380
not the first element cast is shored during casting and curing of the second element. <R17.2.4> 381
9.5.5.3 — For calculation of nV for composite concrete members where the specified 382
concrete compressive strength, unit weight, or other properties of different elements vary, 383
properties of the individual elements shall be used in design. Alternatively, it shall be 384
permitted to use the properties for the element that results in the most critical value of nV . 385
<17.2.3> 386
9.5.5.4 — If an entire composite concrete member is assumed to resist vertical shear, cV shall 387
be permitted to be calculated assuming a monolithically cast member of the same cross-388
sectional shape. <17.2.1> <17.4.1> 389
9.5.5.5 — If an entire composite concrete member is assumed to resist vertical shear, sV shall 390
be permitted to be calculated assuming a monolithically cast member of the same cross-391
cf
'
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ACI 318-14 CR094/LB13-3 2 May 2013
sectional shape provided that shear reinforcement is fully anchored into the interconnected 392
elements in accordance with 21.8. <17.4.1> <17.4.2> 393
9.5.6 — cV for nonprestressed members without axial force 394
9.5.6.1 — For nonprestressed members without axial force, cV shall be calculated in 395
accordance with Eq. (9.5.6.1), unless a more detailed calculation is made in accordance with 396
Table 9.5.6.1. <11.2.1> <11.2.1.1> <11.2.2.1> 397
2 c c wV f b d (9.5.6.1) 398
Table 9.5.6.1 — Detailed method for calculating cV 399
cV
Least of (a), (b),
and (c):
1 9 2500 uc w w
u
V d. f b d
M
(a)
*
1 9 2500c w w. f b d (b)
3 5 c w. f b d (c)
* uM occurs simultaneously with uV at the section considered
400
R9.5.6.1 — The expression in row (a) in Table 9.5.6.1 contains three variables, cf (as a 401
measure of concrete tensile strength), w , and u uV d M , which are known to affect shear 402
strength11.3
, although some research data11.1, 11.2
indicate that this expression overestimates the 403
influence of cf and underestimates the influence of w and u uV d M . Further information 11.3
404
has indicated that shear strength decreases as the overall depth of the member increases. 405
The expression in row (b) in Table 9.5.6.1 limits cV near points of inflection. For most designs, 406
it is convenient to assume that the second term in the expressions in rows (a) and (b) of Table 407
9.5.6.1 equals 0 1. cf and use cV equal to 2 c wf b d , as permitted in Eq. (9.5.6.1). 408
<R11.2.2.1> 409
The shear strength is based on an average shear stress on the full effective cross section wb d . 410
9.5.7 — cV for nonprestressed members with axial compression 411
9.5.7.1 — For nonprestressed members with axial compression, cV shall be calculated in 412
accordance with Eq. 9.5.7.1, unless a more detailed calculation is made in accordance with 413
Table 9.5.7.1, where uN is positive for compression. <11.2.1> <11.2.1.2> <11.2.2.2> 414
2 12000
uc c w
g
NV f b d
A (9.5.7.1) 415
Page 8729
ACI 318-14 CR094/LB13-3 2 May 2013
Table 9.5.7.1 — Detailed method for calculating cV 416
cV
Lesser of
(a) and (b):
1 9 2500
4
8
uc w w
u u
V d. f b d
h dM N
(a)*,†
3 5 1500
uc w
g
N. f b d
A (b)
* uM occurs simultaneously with uV at the section considered.
† (a) is not applicable if 4
08
u u
h dM N .
417
R9.5.7.1— The expressions in rows (a) and (b) of Table 9.5.7.1, for members subject to axial 418
compression in addition to shear and flexure, are derived in the Joint ACI-ASCE Committee 326 419
report11.3
. Values of cV for members subject to shear and axial load are illustrated in Fig. 420
R9.5.7.1. The background for these equations is discussed and comparisons are made with test 421
data in Reference 11.2. <R11.2.2.2> 422
423
Fig. R9.5.7.1—Comparison of shear strength equations for members subject to axial load. 424
9.5.8 — cV for nonprestressed members with significant axial tension 425
9.5.8.1 — For nonprestressed members with significant axial tension, cV shall be taken as 426
zero unless a more detailed calculation is made in accordance with Eq. 9.5.8.1, where uN is 427
negative for tension, and cV shall not be taken less than zero. <11.2.1> <11.2.2.2> 428
Figure must be updated.
Page 8730
ACI 318-14 CR094/LB13-3 2 May 2013
2 1500
uc c w
g
NV f b d
A (9.5.8.1) 429
R9.5.8.1—Equation (9.5.8.1) may be used to compute cV for members subject to significant 430
axial tension. Shear reinforcement may then be designed for n cV V . The term “significant” is 431
used to recognize judgment is required in deciding whether axial tension needs to be considered. 432
Low levels of axial tension often occur due to volume changes, but are not important in 433
structures with adequate expansion joints and minimum reinforcement. It may be desirable to 434
design shear reinforcement to carry total shear if there is uncertainty about the magnitude of 435
axial tension. <R11.2.2.3> 436
9.5.9 — cV for prestressed members 437
9.5.9.1 — The provisions of 9.5.9 shall govern the calculation of cV for post-tensioned 438
members and for pretensioned members in regions where the effective force in the prestressed 439
reinforcement is fully transferred to the concrete. For regions of pretensioned members where 440
the effective force in the prestressed reinforcement is not fully transferred to the concrete, the 441
provisions of 9.5.10 shall govern the calculation of cV . <~> 442
9.5.9.2 — For prestressed flexural members with effective prestress force that is at least 40 443
percent of the tensile strength of the flexural reinforcement, cV shall be calculated in 444
accordance with Table 9.5.9.2, but need not be taken less than the value obtained from Eq. 445
(9.5.6.1). Alternatively, it shall be permitted to determine cV in accordance with 9.5.9.3. 446
<11.3.2> 447
Table 9.5.9.2 — Approximate method for calculating cV 448
cV
Least of (a), (b),
and (c):
0 6 700u p
c wu
V df b d
M.
(a)
*
0 6 700c wf b d. (b)
5 c wf b d (c)
* uM occurs simultaneously with uV at the section considered
449
R9.5.9.2 — Section 9.5.9.2 offers a simple means of computing cV for prestressed concrete 450
beams.11.2
It may be applied to beams having prestressed reinforcement only, or to members 451
reinforced with a combination of prestressed reinforcement and nonprestressed deformed bars. 452
The expression in row (a) of Table 9.5.9.2 is most applicable to members subject to uniform 453
loading and may give conservative results when applied to composite girders for bridges. 454
In applying the expression to simply supported members subject to uniform loads, u p uV d M455
can be expressed as 456
Page 8731
ACI 318-14 CR094/LB13-3 2 May 2013
xx
xd
M
dV p
u
pu
2
457
where is the span length and x is the distance from the section being investigated to the 458
support. For concrete with cf equal to 5000 psi, cV from 9.5.9.2 varies as shown in Fig. 459
R9.5.9.2. Design aids based on this equation are given in Reference 11.6. <R11.3.2> 460
461
Fig. R9.5.9.2—Application of expression in row (a) of Table 9.5.9.2 to uniformly loaded 462
prestressed members. 463
9.5.9.3 — For prestressed members, cV shall be permitted to be taken as the lesser of ciV 464
calculated in accordance with 9.5.9.3.1 and cwV calculated in accordance with 9.5.9.3.2 or 465
9.5.9.3.3. <11.3.3> 466
R9.5.9.3 — Two types of inclined cracking occur in concrete beams: web-shear cracking and 467
flexure-shear cracking. These two types of inclined cracking are illustrated in Fig. R9.5.9.3. 468
Web-shear cracking begins from an interior point in a member when the principal tensile stresses 469
exceed the tensile strength of the concrete. Flexure-shear cracking is initiated by flexural 470
cracking. When flexural cracking occurs, the shear stresses in the concrete above the crack are 471
increased. The flexure-shear crack develops when the combined shear and flexural-tensile stress 472
exceeds the tensile strength of the concrete. 473
Equations (9.5.9.3.1(a)) and (9.5.9.3.1(b)), and (9.5.9.3.2) may be used to determine the shear 474
forces causing flexure-shear and web-shear cracking, respectively. The nominal shear strength 475
provided by the concrete cV is assumed equal to the lesser of ciV and cwV . The derivations of 476
Eq. (9.5.9.3.1(a)) and (9.5.9.3.2) are summarized in Reference 11.17. <R11.3.3> 477
Page 8732
ACI 318-14 CR094/LB13-3 2 May 2013
478
Fig. R9.5.9.3—Types of cracking in concrete beams. 479
— The flexure-shear strength, ciV , shall be taken as the greater of (a) and (b): 9.5.9.3.1480
<11.3.3.1> 481
(a) 0 6max
. i crec w p dci
V Mf b d V
MV (9.5.9.3.1a) 482
(b) 1 7. wi cc b dV f (9.5.9.3.1b) 483
where pd need not be taken less than 0 80h. , the values of Mmax and iV shall be 484
calculated from the load combinations causing maximum factored moment to occur at 485
section considered, and creM shall be calculated as: 486
6cre c pe dt
IM f f f
y
(9.5.9.3.1c) 487
488
R9.5.9.3.1 — In deriving Eq. (9.5.9.3.1(a)) it was assumed that ciV is the sum of the shear 489
required to cause a flexural crack at the point in question given by 490
max
crei
M
MVV
491
plus an additional increment of shear required to change the flexural crack to a flexure-shear 492
crack. The externally applied factored loads, from which iV and maxM are determined, include 493
superimposed dead load, earth pressure, and live load. In computing creM for substitution into 494
Eq. (9.5.9.3.1(a)), I and t are the properties of the section resisting the externally applied loads. 495
For a composite member, where part of the dead load is resisted by only a part of the section, 496
appropriate section properties should be used to compute df . The shear due to dead loads, dV , 497
and that due to other loads, iV , are separated in this case. dV is then the total shear force due to 498
unfactored dead load acting on that part of the section carrying the dead loads acting prior to 499
Page 8733
ACI 318-14 CR094/LB13-3 2 May 2013
composite action plus the unfactored superimposed dead load acting on the composite member. 500
The terms iV and maxM may be taken as 501
dui VVV 502
dumax MMM 503
where uV and uM are the factored shear and moment due to the total factored loads, and dM is 504
the moment due to unfactored dead load (the moment corresponding to df ). 505
For noncomposite, uniformly loaded beams, the total cross section resists all the shear and the 506
live and dead load shear force diagrams are similar. In this case, Eq. (9.5.9.3.1) reduces to 507
'0.6 u ctci c w
u
V MV f b d
M
508
where 509
'/ 6ct t c peM I y f f 510
The symbol ctM in the two preceding equations represents the total moment, including dead 511
load, required to cause cracking at the extreme fiber in tension. This is not the same as creM in 512
Eq. (9.5.9.3.1(a)) where the cracking moment is that due to all loads except the dead load. In Eq. 513
(9.5.9.3.1(a)), the dead load shear is added as a separate term. 514
uM is the factored moment on the beam at the section under consideration, and Vu is the 515
factored shear force occurring simultaneously with uM . Because the same section properties 516
apply to both dead and live load stresses, there is no need to compute dead load stresses and 517
shears separately. The cracking moment ctM reflects the total stress change from effective 518
prestress to a tension of 6 cf , assumed to cause flexural cracking. <R11.3.3> 519
— The web-shear strength, cwV , shall be calculated as: <11.3.3.2> 9.5.9.3.2520
3 5 0 3cw c pc w p pV f f b d V. . (9.5.9.3.2) 521
where pd need not be taken less than 0 80h. and pV is the vertical component of the 522
effective prestress. 523
R9.5.9.3.2 — Equation (9.5.9.3.2) is based on the assumption that web-shear cracking occurs 524
due to the shear causing a principal tensile stress of approximately 4 cf at the centroidal axis 525
of the cross section. pV is calculated from the effective prestress force without load factors. 526
<R11.3.3.2> 527
— As an alternative to 9.5.9.3.2, cwV shall be permitted to be calculated as the 9.5.9.3.3528
shear force corresponding to dead load plus live load that results in a principal tensile 529
stress of 4 cf at the location indicated in (a) or (b): <11.3.3.2> 530
Page 8734
ACI 318-14 CR094/LB13-3 2 May 2013
(a) If the centroidal axis of the prestressed cross section is in the web, the principal 531
tensile stress shall be calculated at the centroidal axis. 532
(b) If the centroidal axis of the prestressed cross section is in the flange, the principal 533
tensile stress shall be calculated at the intersection of the flange and the web. 534
— In composite members, the principal tensile stress defined in 9.5.9.3.3 shall 9.5.9.3.4535
be calculated using the cross section that resists live load. <11.3.3.2> 536
9.5.10 — cV for pretensioned members in regions of reduced prestress force 537
9.5.10.1 — When calculating cV , the transfer length of prestressed reinforcement, tr , shall 538
be assumed to be 50 bd for strand and 100 bd for wire. <11.3.4> <11.3.5> 539
R9.5.10.1 — The effect of the reduced prestress near the ends of pretensioned beams on the 540
shear strength should be taken into account. Section 9.5.10.1 relates to the shear strength at 541
sections within the transfer length of prestressing steel when bonding of prestressing steel 542
extends to the end of the member. <R11.3.4 and R11.3.5> 543
9.5.10.2 — If bonding of strands extends to the end of the member, the effective prestress 544
force shall be assumed to vary linearly from zero at the end of the prestressed reinforcement 545
to a maximum at a distance tr from the end of the prestressed reinforcement. <11.3.4> 546
9.5.10.3 — At locations corresponding to a reduced effective prestress force according to 547
9.5.10.2, the value of cV shall be calculated in accordance with (a), (b), and (c): <11.3.4> 548
(a) The reduced effective prestress force shall be used to determine the applicability of 549
9.5.9.2. 550
(b) The reduced effective prestress force shall be used to calculate cwV in 9.5.9.3. 551
(c) The value of cV calculated using 9.5.9.2 shall not exceed the value of cwV calculated 552
using the reduced effective prestress force. 553
9.5.10.4 — If bonding of strands does not extend to the end of the member, the effective 554
prestress force shall be assumed to vary linearly from zero at the point where bonding 555
commences to a maximum at a distance tr from that point. <11.3.5> 556
R9.5.10.4 — This section relates to the shear strength at sections within the length over which 557
some of the prestressing steel is not bonded to the concrete, or within the transfer length of the 558
prestressing steel for which bonding does not extend to the end of the beam. <R11.3.4 and 559
R11.3.5> 560
9.5.10.5 — At locations corresponding to a reduced effective prestress force according to 561
9.5.10.4, the value of cV shall be calculated in accordance with (a), (b), and (c): <11.3.5> 562
(a) The reduced effective prestress force shall be used to determine the applicability of 563
9.5.9.2. 564
Page 8735
ACI 318-14 CR094/LB13-3 2 May 2013
(b) The reduced effective prestress force shall be used to calculate cV in accordance with 565
9.5.9.3. 566
(c) The value of cV calculated using 9.5.9.2 shall not exceed the value of cwV calculated 567
using the reduced effective prestress force. 568
9.5.11 — One-way shear reinforcement 569
9.5.11.1 — At each section where u cV V , transverse reinforcement shall be provided such 570
that Eq. (9.5.11.1) is satisfied. <11.4.7.1> 571
us c
VV V
(9.5.11.1) 572
9.5.11.2 — For one-way members reinforced with stirrups, ties, hoops, crossties, or spirals, 573
sV shall be calculated in accordance with 9.5.11.5. <~> 574
9.5.11.3 — For one-way members reinforced with bent-up longitudinal bars, sV shall be 575
calculated in accordance with 9.5.11.6. <~> 576
9.5.11.4 — If more than one type of shear reinforcement is provided to reinforce the same 577
portion of a member, sV shall be calculated as the sum of the sV values calculated for the 578
various types of shear reinforcement. <11.4.7.8> 579
9.5.11.5 — One-way shear strength provided by stirrups, ties, hoops, crossties, and 580
spirals 581
R9.5.11.5 — Design of shear reinforcement is based on a modified truss analogy. The truss 582
analogy assumes that the total shear is carried by shear reinforcement. However, considerable 583
research on both nonprestressed and prestressed members has indicated that shear reinforcement 584
needs to be designed to carry only the shear exceeding that which causes inclined cracking, 585
provided the diagonal members in the truss are assumed to be inclined at 45 degrees. 586
Equations (9.5.11.5.3), (9.5.11.5.4), and (9.5.11.6.2(a)) are presented in terms of nominal shear 587
strength provided by shear reinforcement sV . When shear reinforcement perpendicular to axis of 588
member is used, the required area of shear reinforcement vA and its spacing s are computed by 589
u cv
yt
V VA
s f d
590
Research 9.xx,9.xx
has shown that shear behavior of wide beams with substantial flexural 591
reinforcement is improved if the transverse spacing of stirrup legs across the section is reduced. 592
<R11.4.7> 593
Page 8736
ACI 318-14 CR094/LB13-3 2 May 2013
— Shear reinforcement satisfying (a), (b), or (c) shall be permitted in 9.5.11.5.1594
nonprestressed and prestressed members: <11.4.1.1> 595
(a) Stirrups, ties, or hoops perpendicular to longitudinal axis of member 596
(b) Welded wire reinforcement with wires located perpendicular to longitudinal axis 597
of member 598
(c) Spiral reinforcement 599
— Inclined stirrups making an angle of at least 45 degrees with the 9.5.11.5.2600
longitudinal axis of the member and crossing the plane of the potential shear crack shall 601
be permitted to be used as shear reinforcement in nonprestressed members. <11.4.1.2> 602
— sV for shear reinforcement complying with 9.5.11.5.1 shall be calculated as: 9.5.11.5.3603
<11.4.7.2> 604
v yts
A f dV
s (9.5.11.5.3) 605
where s is the spiral pitch or the longitudinal spacing of the shear reinforcement and vA 606
is defined in 9.5.11.5.5 or 9.5.11.5.6. 607
— sV for shear reinforcement complying with 9.5.11.5.2 shall be calculated 9.5.11.5.4608
as: <11.4.7.4> 609
v yts
A f dV
s
sin cos (9.5.11.5.4) 610
where is the angle between the inclined stirrups and the longitudinal axis of the 611
member, s is measured parallel to the longitudinal reinforcement, and vA is defined in 612
9.5.11.5.5. 613
— For each rectangular tie, stirrup, hoop, or crosstie, vA shall be taken as the 9.5.11.5.5614
effective area of all bar legs or wires within spacing s. <~> 615
— For each circular tie or spiral, vA shall be taken as two times the area of the 9.5.11.5.6616
bar or wire within spacing s. <11.4.7.3> 617
R9.5.11.5.6 — Although the transverse reinforcement in a circular section may not consist of 618
straight legs, tests indicate that Eq. (9.5.11.5.3) is conservative if d is taken as defined in 619
9.5.3.2.11.14, 11.15
<R11.4.7.3> 620
9.5.11.6 — One-way shear strength provided by bent-up longitudinal bars 621
— The center three-fourths of the inclined portion of bent-up longitudinal bars 9.5.11.6.1622
shall be permitted to be used as shear reinforcement in nonprestressed members if the 623
angle between the bent-up bars and the longitudinal axis of the member is at least 30 624
degrees. <11.4.7.7> 625
— If shear reinforcement consists of a single bar or a single group of parallel 9.5.11.6.2626
bars having an area vA , all bent the same distance from the support, sV shall be taken as 627
the lesser of (a) and (b): <11.4.7.5> 628
Page 8737
ACI 318-14 CR094/LB13-3 2 May 2013
(a) sins v yA fV (9.5.11.6.2a) 629
(b) 3 cs wfV b d (9.5.11.6.2b) 630
where is the angle between bent-up reinforcement and longitudinal axis of the 631
member. 632
— If shear reinforcement consists of a series of parallel bent-up bars or groups 9.5.11.6.3633
of parallel bent-up bars at different distances from the support, sV shall be calculated in 634
accordance with Eq. (9.5.11.5.4). <11.4.7.6> 635
9.6 — Two-way shear strength 636
R9.6 – Factored shear stress in two-way members due to shear and moment transfer is calculated 637
in accordance with the requirements of 12.4.4. Section 9.6 provides requirements for 638
determining factored shear strength, either without shear reinforcement or with shear 639
reinforcement in the form of stirrups, headed shear studs or shearheads. Factored shear demand 640
and capacity are calculated in terms of stress, permitting superposition of effects from direct 641
shear and moment transfer. <~> 642
9.6.1 — General 643
9.6.1.1 — Provisions 9.6.1 through 9.6.9 define the nominal shear strength of two-way 644
members with and without shear reinforcement. Where structural steel I- or channel-shaped 645
sections are used as shearheads, two-way members shall be designed for shear in accordance 646
with 9.6.10. <~> <11.11.4> 647
9.6.1.2 — Nominal shear strength for two-way members without shear reinforcement shall be 648
calculated in accordance with Eq. 9.6.1.2. <~> <11.11.1> <11.11.7.2> 649
n cv v (9.6.1.2) 650
9.6.1.3 — Nominal shear strength for two-way members with shear reinforcement other than 651
shearheads shall be calculated in accordance with Eq. 9.6.1.3. <11.1.1> <11.11.7.2> 652
n c sv v v (9.6.1.3) 653
9.6.1.4 — Two-way shear shall be resisted by a section with a depth d and an assumed critical 654
perimeter ob that extends completely or partially around the column, concentrated load, or 655
reaction area. <~> 656
9.6.1.5 — cv for two-way shear shall be calculated in accordance with 9.6.6. For two-way 657
members with shear reinforcement, cv shall not exceed the limits in 9.6.7.1. <~> <11.11.3.1> 658
9.6.1.6 — For calculation of cv , shall be determined in accordance with 5.2.4. <~> 659
9.6.1.7 — For two-way members reinforced with single- or multi-leg stirrups, sv shall be 660
calculated in accordance with 9.6.8. <~> <11.11.3.1> 661
Page 8738
ACI 318-14 CR094/LB13-3 2 May 2013
9.6.1.8 — For two-way members reinforced with headed shear stud reinforcement, sv shall 662
be calculated in accordance with 9.6.9. <~> <11.11.5.1> 663
9.6.2 — Strength reduction factor 664
9.6.2.1 — Strength reduction factor for two-way shear, , shall be 0.75. <9.3.2.3> 665
9.6.3 — Effective depth 666
9.6.3.1 — For calculation of cv and sv for two-way shear, d shall be taken as the average of 667
the effective depths in the two orthogonal directions. <~> 668
9.6.3.2 — For prestressed, two-way members, d need not be taken less than 0.8h. <11.3.1> 669
<11.4.3> 670
9.6.4 — Limiting material strengths 671
9.6.4.1 — The value of used to calculate cv for two-way shear shall not exceed 100 psi. 672
<11.1.2> <11.1.2.1> 673
R9.6.4.1 — See R9.5.4.1 674
9.6.4.2 — The value of ytf used to calculate sv shall not exceed the limits in 6.2.2.4. 675
<11.4.2> 676
9.6.5 — Critical sections for two-way members 677
9.6.5.1 — For two-way shear, critical sections shall be located so that the perimeter ob is a 678
minimum but need not be closer than d/2 to: <11.11.1.2> <15.5.2> 679
(a) Edges or corners of columns, concentrated loads, or reaction areas 680
(b) Changes in slab or footing thickness, such as edges of capitals, drop panels, or shear 681
caps 682
R9.6.5.1 — The critical section for shear in slabs subjected to bending in two directions follows 683
the perimeter at the edge of the loaded area.11.3
The shear stress acting on this section at factored 684
loads is a function of cf and the ratio of the side dimension of the column to the effective slab 685
depth. A much simpler design equation results by assuming a pseudocritical section located at a 686
distance 2d from the periphery of the concentrated load. When this is done, the shear strength 687
is almost independent of the ratio of column size to slab depth. For rectangular columns, this 688
critical section was defined by straight lines drawn parallel to and at a distance 2d from the 689
edges of the loaded area. Section 9.6.9.1 allows the use of a rectangular critical section. 690
For slabs of uniform thickness, it is sufficient to check shear on one section. For slabs with 691
changes in thickness, such as the edge of drop panels or shear caps, it is necessary to check shear 692
at several sections. 693
cf
'
Page 8739
ACI 318-14 CR094/LB13-3 2 May 2013
For edge columns at points where the slab cantilevers beyond the column, the critical perimeter 694
will either be three-sided or four-sided. <R11.11.1.2> 695
9.6.5.2 — For two-way members reinforced with single- or multi-leg stirrups, a critical 696
section with perimeter ob located d/2 beyond the outermost line of stirrup legs that surround 697
the column shall also be considered. <11.11.7.2> 698
9.6.5.3 — For two-way members reinforced with headed shear stud reinforcement, a critical 699
section with perimeter ob located d/2 beyond the outermost peripheral line of shear 700
reinforcement shall also be considered. <11.11.5.4> 701
9.6.5.4 — For square or rectangular columns, concentrated loads, or reaction areas, critical 702
sections for two-way shear shall be permitted to be calculated assuming straight sides. 703
<11.11.1.3> 704
9.6.5.5 — For a circular or regular polygon-shaped column, critical sections for two-way 705
shear shall be permitted to be calculated assuming a square column of equivalent area. <15.3> 706
9.6.5.6 — If an opening is located within a column strip or closer than 10h from a 707
concentrated load or reaction area, a portion of ob enclosed by straight lines projecting from 708
the centroid of the column, concentrated load or reaction area and tangent to the boundaries of 709
the opening shall be considered ineffective. <11.11.6> 710
R9.6.5.6 — Provisions for design of openings in slabs (and footings) were developed in 711
Reference 11.3. The locations of the effective portions of the critical section near typical 712
openings and free edges are shown by the dashed lines in Fig. R9.6.5.6. Additional research11.61
713
has confirmed that these provisions are conservative. <R11.11.6> 714
715
Fig. R9.6.5.6—Effect of openings and free edges (effective perimeter shown with dashed lines). 716
Page 8740
ACI 318-14 CR094/LB13-3 2 May 2013
9.6.6 —Two-way shear strength provided by concrete 717
9.6.6.1 — For nonprestressed, two-way members, cv shall be calculated in accordance with 718
Table 9.6.6.1, where is the ratio of long side to short side of the column, concentrated 719
load, or reaction area and s is defined in 9.6.6.2. <11.11.2.1> 720
Table 9.6.6.1 — Calculation of cv for two-way shear 721
cv
Least of (a), (b), and (c):
4 cf (a)
42 cf
(b)
2 sc
o
df
b
(c)
722
R9.6.6.1 — For square columns, the shear stress due to factored loads in slabs subjected to 723
bending in two directions is limited to '4 cf . However, tests
11.61 have indicated that the value 724
of '4 cf is unconservative when the ratio of the lengths of the long and short sides of a 725
rectangular column or loaded area is larger than 2.0. In such cases, the actual shear stress on the 726
critical section at punching shear failure varies from a maximum of about '4 cf around the 727
corners of the column or loaded area, down to '2 cf or less along the long sides between the 728
two end sections. Other tests11.62
indicate that cv decreases as the ratio d/bo increases. 729
Equations (9.6.6.1(b)) and (9.6.6.1(c)) were developed to account for these two effects. 730
For shapes other than rectangular, is taken to be the ratio of the longest overall dimension of 731
the effective loaded area to the largest overall perpendicular dimension of the effective loaded 732
area, as illustrated for an L-shaped reaction area in Fig. R9.6.6.1. The effective loaded area is 733
that area totally enclosing the actual loaded area, for which the perimeter is a minimum. 734
<R11.11.2.1> 735
Page 8741
ACI 318-14 CR094/LB13-3 2 May 2013
736
Fig. R9.6.6.1—Value of for a nonrectangular loaded area. 737
9.6.6.2 — The value of s is 40 for interior columns, 30 for edge columns, and 20 for corner 738
columns. <11.11.2.1> 739
R9.6.6.2 — The words “interior,” “edge,” and “corner columns” in 9.6.6.2 refer to critical 740
sections with four, three, and two sides, respectively. <R11.11.2.1> 741
9.6.6.3 — For prestressed, two-way members, it shall be permitted to calculate cv using 742
9.6.6.4 provided that (a), (b), and (c) are satisfied. <11.11.2.2> 743
(a) Bonded reinforcement is provided in accordance with 12.6.2.3 and 12.7.5.5 744
(b) No portion of the column cross section is closer to a discontinuous edge than four 745
times the slab thickness h 746
(c) Effective prestress, pcf , in each direction is not less than 125 psi 747
R9.6.6.3 — For prestressed slabs and footings, modified forms of the expressions in rows (b) 748
and (c) of Table 9.6.6.1 are specified for two-way shear strength. [Note, an error was introduced 749
in the commentary in 318-02. Reference to the expressions in rows (b) and (c) of Table 9.6.6.1 750
is consistent with 318-95 and earlier versions.] Research11.63, 11.64
indicates that the shear strength 751
of two-way prestressed slabs around interior columns is conservatively predicted by the 752
expressions in 9.6.6.4, where cv corresponds to a diagonal tension failure of the concrete 753
initiating at the critical section defined in 9.6.5.1. The mode of failure differs from a punching 754
shear failure around the perimeter of the loaded area predicted by the expression in row (b) of 755
Table 9.6.6.1. Consequently, the term is not included in the expressions in 9.6.6.4. Values for 756
c'f and pcf are restricted in design due to limited test data available for higher values. When 757
calculating pcf , loss of prestress due to restraint of the slab by shear walls and other structural 758
elements should be taken into account. <R11.11.2.2> 759
Page 8742
ACI 318-14 CR094/LB13-3 2 May 2013
9.6.6.4 — For prestressed, two-way members conforming to 9.6.6.2, cv shall be permitted to 760
be calculated as the lesser of (a) and (b): <11.11.2.2> 761
(a) 3 5 0 3p
c pco
V. f . f
b d 762
(b) 1 5 0 3ps
c pco o
Vd. f . f
b b d
763
where s is defined in 0, the value of pcf is the average of pcf in the two directions and shall 764
not be taken greater than 500 psi, pV is the vertical component of all effective prestress forces 765
crossing the critical section, and the value of cf shall not exceed 70 psi. 766
9.6.7 — Maximum shear for two-way members with shear reinforcement 767
R9.6.7 – Critical sections for two-way members with shear reinforcement are defined in 9.6.5.1 768
for the section immediately adjacent to the column, concentrated load or reaction area and 769
9.6.5.2 and 9.6.5.3 for the section located just beyond the outermost peripheral line of stirrup or 770
headed shear stud reinforcement. Values of maximum cv for these critical sections are given in 771
Table 9.6.7.1. Values of maximum uv for these critical sections are given in Table 9.6.7.2. 772
Note that maximum cv and maximum uv values at the innermost critical section (defined in 773
9.6.5.1) when headed shear stud reinforcement is provided are higher than when stirrups are 774
provided. Compared with a leg of a stirrup having bends at the ends, a stud head exhibits smaller 775
slip, and thus results in smaller shear crack widths. The improved performance results in larger 776
limits for shear strength and spacing between peripheral lines of headed shear stud 777
reinforcement. Maximum cv values at the critical section beyond the outermost peripheral line 778
of shear reinforcement are independent of the type of shear reinforcement provided. <~> 779
9.6.7.1 — For two-way members with shear reinforcement, value of cv calculated at the 780
critical sections defined in 9.6.5 shall not exceed the values in Table 9.6.7.1. <11.11.3.1> 781
<11.11.5.1> <11.11.5.4> 782
Table 9.6.7.1 — Maximum cv for two-way members with shear reinforcement 783
Type of shear
reinforcement
Maximum cv at
critical sections defined
in 9.6.5.1
Maximum cv at
critical section defined
in 9.6.5.2 and 9.6.5.3
Stirrups 2 cf (a) 2 cf
(b)
Headed shear stud
reinforcement 3 cf (c) 2 cf
(d)
9.6.7.2 — For two-way members with shear reinforcement, effective depth shall be selected 784
such that uv calculated at the critical sections defined in 9.6.5.1 does not exceed the values in 785
Table 9.6.7.2. <11.11.3.2> <11.11.5.1> <11.11.7.2> 786
Page 8743
ACI 318-14 CR094/LB13-3 2 May 2013
Table 9.6.7.2— Maximum uv for two-way members with shear reinforcement 787
Type of shear
reinforcement
Maximum uv at critical
sections defined in 9.6.5.1
Stirrups 6 cf (a)
Headed shear stud
reinforcement 8 cf (b)
9.6.8 — Two-way shear strength provided by single- or multiple-leg stirrups 788
9.6.8.1 — Single- or multiple-leg stirrups fabricated from bars or wires shall be permitted to 789
be used as shear reinforcement in slabs and footings conforming to (a) and (b): <11.11.3> 790
(a) d is at least 6 in. 791
(b) d is at least 16 bd , where bd is the diameter of the stirrups 792
9.6.8.2 — For two-way members with stirrups, sv shall be calculated as: <11.11.3.1> 793
<11.4.7.2> 794
v yts
o
A fv
b s (9.6.8.2) 795
where vA is the sum of the area of all legs of reinforcement on one peripheral line that is 796
geometrically similar to the perimeter of the column section, and s is the spacing of the 797
peripheral lines of shear reinforcement in the direction perpendicular to the column face. 798
9.6.9 — Two-way shear strength provided by headed shear stud reinforcement 799
R9.6.9 — Tests11.69
show that vertical studs mechanically anchored as close as possible to the 800
top and bottom of slabs are effective in resisting punching shear. The critical section beyond the 801
shear reinforcement generally has a polygonal shape. Equations for calculating shear stresses on 802
such sections are given in Reference 11.69. <R.11.11.5> 803
9.6.9.1 — Headed shear stud reinforcement shall be permitted to be used as shear 804
reinforcement in slabs and footings if the placement and geometry of the headed shear stud 805
reinforcement satisfies 12.7.7. <11.11.5> 806
9.6.9.2 — For two-way members with headed shear stud reinforcement, sv shall be 807
calculated as: <11.4.7.2> <11.11.5.1> 808
v yts
o
A fv
b s (9.6.9.2) 809
where vA is the sum of the area of all shear studs on one peripheral line that is geometrically 810
similar to the perimeter of the column section, and s is the spacing of the peripheral lines of 811
headed shear stud reinforcement in the direction perpendicular to the column face. 812
Page 8744
ACI 318-14 CR094/LB13-3 2 May 2013
9.6.9.3 — If headed shear stud reinforcement is provided, vA
s shall satisfy Eq. (9.6.9.3). 813
<11.11.5.1> 814
2v oc
yt
A bf
s f (9.6.9.3) 815
9.6.10 — Design provisions for two-way members with shearheads 816
9.6.10.1 — Each shearhead shall consist of steel shapes fabricated with a full penetration 817
weld into identical arms at right angles. Shearhead arms shall not be interrupted within the 818
column section. <11.11.4.1> 819
R9.6.10.1—Based on reported test data,11.70
design procedures are presented for shearhead 820
reinforcement consisting of structural steel shapes. For a column connection transferring 821
moment, the design of shearheads is given in 9.6.10.12. 822
Three basic criteria should be considered in the design of shearhead reinforcement for 823
connections transferring shear due to gravity load. First, a minimum flexural strength should be 824
provided to ensure that the required shear strength of the slab is reached before the flexural 825
strength of the shearhead is exceeded. Second, the shear stress in the slab at the end of the 826
shearhead reinforcement should be limited. Third, after these two requirements are satisfied, the 827
negative moment slab reinforcement can be reduced in proportion to the moment contribution of 828
the shearhead at the design section. <R11.11.4> 829
9.6.10.2 — A shearhead shall not be deeper than 70 times the web thickness of the steel 830
shape. <11.11.4.2> 831
9.6.10.3 — The ends of each shearhead arm shall be permitted to be cut at angles of at least 832
30 degrees with the horizontal if the plastic moment strength, pM , of the remaining tapered 833
section is adequate to resist the shear force attributed to that arm of the shearhead. 834
<11.11.4.3> 835
9.6.10.4 — Compression flanges of steel shapes shall be within 0.3d of the compression 836
surface of the slab. <11.11.4.4> 837
9.6.10.5 — The ratio v between the flexural stiffness of each shearhead arm and that of the 838
surrounding composite cracked slab section of width 2c d shall be at least 0.15. 839
<11.11.4.5> 840
R9.6.10.5 — The assumed idealized shear distribution along an arm of a shearhead at an interior 841
column is shown in Fig. R9.6.10.5. The shear along each of the arms is taken as v cV n , where 842
cV equals c ov b d and cv is defined in 9.6.6.1. <R11.11.4.5 and R11.11.4.6> 843
Page 8745
ACI 318-14 CR094/LB13-3 2 May 2013
844
Fig. R9.6.10.5—Idealized shear acting on shearhead. 845
9.6.10.6 — For each arm of the shearhead, pM shall satisfy Eq. (9.6.10.6). 846
1
2 2
up v v v
V cM h
n
(9.6.10.6) 847
where corresponds to tension-controlled members in 9.4.2.1, n is the number of shearhead 848
arms, and v is the minimum length of each shearhead arm required to comply with 9.6.10.8 849
and 9.6.10.10. <11.11.4.6> 850
R9.6.10.6 —The peak shear at the face of the column is taken as the total shear considered per 851
arm Vu/n minus the shear considered carried to the column by the concrete compression zone of 852
the slab. The latter term is expressed as / 1c vV n , so that it approaches zero for a heavy 853
shearhead and approaches Vu/n when a light shearhead is used. Equation (9.6.10.6) then follows 854
from the assumption that is about one-half the factored shear force . In this equation, pM 855
is the required plastic moment strength of each shearhead arm necessary to ensure that is 856
attained as the moment strength of the shearhead is reached. The quantity v is the length from 857
the center of the column to the point at which the shearhead is no longer required, and the 858
distance 21 /c is one-half the dimension of the column in the direction considered. <R11.11.4.5 859
and R11.11.4.6> 860
9.6.10.7 — Nominal flexural strength contributed to each slab column strip by a shearhead, 861
vM , shall satisfy Eq. (9.6.10.7). 862
1
2 2
v uv v
V cM
n
(9.6.10.7) 863
cV uV
uV
Page 8746
ACI 318-14 CR094/LB13-3 2 May 2013
where corresponds to tension-controlled members in 9.4.2.1. However, vM shall not be 864
taken greater than the least of (a), (b), and (c). <11.11.4.9> 865
(a) 30 percent of uM in each slab column strip 866
(b) Change in uM in column strip over the length v 867
(c) pM as defined in 9.6.10.6 868
R9.6.10.7 — If the peak shear at the face of the column is neglected, and is again assumed 869
to be about one-half of uV , the moment resistance contribution of the shearhead vM can be 870
conservatively computed from Eq. (9.6.10.7). <R11.11.4.9> 871
9.6.10.8 — The critical section for shear shall be perpendicular to the plane of the slab and 872
shall cross each shearhead arm at a distance 13 4 2v c from the column face. This 873
critical section shall be located so ob is a minimum, but need not be closer than d/2 to the 874
edges of the supporting column. <11.11.4.7> 875
R9.6.10.8 — The test results11.70
indicated that slabs containing under-reinforcing shearheads 876
failed at a shear stress on a critical section at the end of the shearhead reinforcement less than 877
'cf4 . Although the use of over-reinforcing shearheads brought the shear strength back to about 878
the equivalent of 'cf4 , the limited test data suggest that a conservative design is desirable. 879
Therefore, the shear strength is calculated as 'cf4 on an assumed critical section located inside 880
the end of the shearhead reinforcement. 881
The critical section is taken through the shearhead arms three-fourths of the distance 21 /cv 882
from the face of the column to the end of the shearhead. However, this assumed critical section 883
need not be taken closer than 2/d to the column. See Fig. R9.6.10.8. <R11.11.4.7> 884
cV
Page 8747
ACI 318-14 CR094/LB13-3 2 May 2013
885
Fig. R9.6.10.8 – Location of critical section defined in 9.6.10.8. 886
9.6.10.9 — If an opening is located within a column strip or closer than 10h from a column in 887
slabs with shearheads, the ineffective portion of ob shall be one-half of that defined in 888
9.6.5.6. <11.11.6.2> 889
9.6.10.10 — Factored shear stress due to vertical loads shall not be greater than 4 cf on the 890
critical section defined in 9.6.10.8 and shall not be greater than 7 cf on the critical section 891
closest to the column defined in 9.6.5.1(a). <11.11.4.8> 892
9.6.10.11 — Where transfer of moment is considered, the shearhead must have adequate 893
anchorage to transmit pM to the column. <11.11.4.10> 894
R9.6.10.11 — Tests11.73
indicate that the critical sections are defined in 9.6.5.1(a) and 9.6.5.2 and 895
are appropriate for calculations of shear stresses caused by transfer of moments even when 896
shearheads are used. Then, even though the critical sections for direct shear and shear due to 897
moment transfer differ, they coincide or are in close proximity at the column corners where the 898
failures initiate. Because a shearhead attracts most of the shear as it funnels toward the column, it 899
is conservative to take the maximum shear stress as the sum of the two components. 900
Section 9.6.10.11 requires the moment pM to be transferred to the column in shearhead 901
connections transferring unbalanced moments. This may be done by bearing within the column 902
or by mechanical anchorage. <R11.11.7.3> 903
Page 8748
ACI 318-14 CR094/LB13-3 2 May 2013
9.6.10.12 — Where transfer of moment is considered, the sum of factored shear stresses due 904
to vertical load acting on the critical section defined by 9.6.10.8 and the shear stresses 905
resulting from factored moment transferred by eccentricity of shear about the centroid of the 906
critical section closest to the column defined in 9.6.5.1(a) shall not exceed 4 cf . 907
<11.11.7.3> 908
9.7 — Torsion 909
R9.7 — Torsion 910
The design for torsion in 9.7.1 through 9.7.8 is based on a thin-walled tube, space truss analogy. 911
A beam subjected to torsion is idealized as a thin-walled tube with the core concrete cross 912
section in a solid beam neglected as shown in Fig. R9.7(a). Once a reinforced concrete beam has 913
cracked in torsion, its torsional resistance is provided primarily by closed stirrups and 914
longitudinal bars located near the surface of the member. In the thin-walled tube analogy, the 915
resistance is assumed to be provided by the outer skin of the cross section roughly centered on 916
the closed stirrups. Both hollow and solid sections are idealized as thin-walled tubes both before 917
and after cracking. 918
919
Fig. R9.7 – (a) Thin-walled tube; (b) area enclosed by shear flow path 920
In a closed thin-walled tube, the product of the shear stress τ and the wall thickness t at any point 921
in the perimeter is known as the shear flow, q t . The shear flow q due to torsion acts as 922
shown in Fig. R9.7(a) and is constant at all points around the perimeter of the tube. The path 923
along which it acts extends around the tube at midthickness of the walls of the tube. At any point 924
along the perimeter of the tube the shear stress due to torsion is 2 oT A t where oA is the 925
gross area enclosed by the shear flow path, shown shaded in Fig. R9.7(b), and t is the thickness 926
of the wall at the point where τ is being computed. The shear flow follows the midthickness of 927
the walls of the tube and oA is the area enclosed by the path of the shear flow. For a hollow 928
member with continuous walls, oA includes the area of the hole. 929
cV remains constant at the value it has when there is no torsion, and the torsion carried by the 930
concrete is always taken as zero. The design procedure is derived and compared with test results 931
in Reference 11.31 and 11.32. <R11.5> 932
Page 8749
ACI 318-14 CR094/LB13-3 2 May 2013
9.7.1 — General 933
9.7.1.1 — Provisions of 9.7 apply to members if u thT T , where is defined in 9.7.2 and 934
threshold torsion, thT , is defined in 9.7.5. If u thT T , it shall be permitted to neglect 935
torsional effects. <11.5.1> <~> 936
R9.7.1.1 — Torques that do not exceed approximately one-quarter of the cracking torque crT , 937
defined as threshold torsion thT , will not cause a structurally significant reduction in either the 938
flexural or shear strength and can be ignored. <R11.5.1> 939
9.7.1.2 — Nominal torsional strength shall be calculated in accordance with 9.7.7. <~> 940
9.7.1.3 — For calculation of thT and crT , shall be determined in accordance with 5.2.4. 941
<~> 942
9.7.2 — Strength reduction factor 943
9.7.2.1 — Strength reduction factor for torsion, , shall be 0.75. <9.3.2.3> 944
9.7.3 — Limiting material strengths 945
9.7.3.1 — The value of used to calculate thT and crT shall not exceed 100 psi. <11.1.2> 946
R9.7.3.1 — Because of a lack of test data and practical experience with concretes having 947
compressive strengths greater than 10,000 psi, the Code imposes a maximum value of 100 psi on 948
cf for use in the calculation of torsion strength. <R11.1.2> 949
9.7.3.2 — The values of yf and ytf for longitudinal and transverse torsion reinforcement 950
shall not exceed the limits in 6.2.2.4. <11.4.2> 951
R9.7.3.2 — Limiting the values of yf and ytf used in design of torsion reinforcement to 952
60,000 psi provides a control on diagonal crack width. <11.4.2> 953
9.7.4 — Factored design torsion 954
R9.7.4 — In designing for torsion in reinforced concrete structures, two conditions may be 955
identified:11.34, 11.35
956
(a) The torsional moment cannot be reduced by redistribution of internal forces (9.7.4.1). This is 957
referred to as equilibrium torsion, since the torsional moment is required for the structure to be in 958
equilibrium. 959
For this condition, illustrated in Fig. R9.7.4(a), torsion reinforcement designed according to 960
9.7.8.1 and Chapter 13 must be provided to resist the total design torsional moments. 961
cf
'
Page 8750
ACI 318-14 CR094/LB13-3 2 May 2013
962
Fig. R9.7.4(a) – Design torque may not be reduced (9.7.4.1) 963
964
Fig. R9.7.4(b) – Design torque may be reduced (9.7.4.2) 965
(b) The torsional moment can be reduced by redistribution of internal forces after cracking 966
(9.7.4.2) if the torsion arises from the member twisting to maintain compatibility of 967
deformations. This type of torsion is referred to as compatibility torsion. 968
For this condition, illustrated in Fig. R9.7.4(b), the torsional stiffness before cracking 969
corresponds to that of the uncracked section according to St. Venant’s theory. At torsional 970
cracking, however, a large twist occurs under an essentially constant torque, resulting in a large 971
redistribution of forces in the structure.11.34, 11.35
The cracking torque under combined shear, 972
flexure, and torsion corresponds to a principal tensile stress somewhat less than the 4 cf 973
quoted in R9.7.5.1. 974
When the torsional moment exceeds the cracking torque, a maximum factored torsional moment 975
equal to the cracking torque may be assumed to occur at the critical sections near the faces of the 976
supports. This limit has been established to control the width of torsional cracks. 977
Section 9.7.4.2 applies to typical and regular framing conditions. With layouts that impose 978
significant torsional rotations within a limited length of the member, such as a heavy torque 979
loading located close to a stiff column, or a column that rotates in the reverse directions because 980
of other loading, a more exact analysis is advisable. 981
When the factored torsional moment from an elastic analysis based on uncracked section 982
properties is between the values in 9.7.5.1 and the values given in this section, torsion 983
reinforcement should be designed to resist the computed torsional moments. <R11.5.2.1 and 984
R11.5.2.2> 985
9.7.4.1 — If u thT T and uT is required to maintain equilibrium, the member shall be 986
designed to resist uT . <11.5.2.1> 987
9.7.4.2 — In a statically indeterminate structure where u thT T and reduction of uT in a 988
member can occur due to redistribution of internal forces after torsional cracking, uT shall be 989
Page 8751
ACI 318-14 CR094/LB13-3 2 May 2013
permitted to be reduced to crT , where the cracking torsion, crT , is defined in 9.7.6. 990
<11.5.2.2> 991
9.7.4.3 — If uT is redistributed in accordance with 9.7.4.2, the factored moments and shears 992
used for design of the adjoining members shall be in equilibrium with the reduced torsion. 993
<11.5.2.2> 994
9.7.5 — Threshold torsion 995
9.7.5.1 — Threshold torsion, thT , shall be calculated in accordance with Table 9.7.5.1(a) for 996
solid cross sections and Table 9.7.5.1(b) for hollow cross sections, where uN is positive for 997
compression and negative for tension. <11.5.1> 998
Table 9.7.5.1(a) — Threshold torsion for solid cross sections 999
Type of member thT
Nonprestressed member
2
cpc
cp
Af
p (a)
Prestressed member
2
14
cp pcc
cp c
A ff
p f (b)
Nonprestressed member
subjected to axial force
2
14
cp uc
cp g c
A Nf
p A f (c)
1000
Table 9.7.5.1(b) — Threshold torsion for hollow cross sections 1001
Type of member thT
Nonprestressed member
2
gc
cp
Af
p (a)
Prestressed member
2
14
g pcc
cp c
A ff
p f (b)
Nonprestressed member
subjected to axial force
2
14
g uc
cp g c
A Nf
p A f (c)
1002
R9.7.5.1 —The threshold torsion is defined as one-quarter of the cracking torque crT . For solid 1003
members, the interaction between the cracking torsion and the inclined cracking shear is 1004
approximately circular or elliptical. For such a relationship, a torque of 0 25. crT , as used in 1005
9.7.5.1, corresponds to a reduction of 3 percent in the inclined cracking shear. This reduction in 1006
Page 8752
ACI 318-14 CR094/LB13-3 2 May 2013
the inclined cracking shear was considered negligible. The stress at cracking 4 cf has 1007
purposely been taken as a lower bound value. 1008
For torsion, a hollow member is defined as having one or more longitudinal voids, such as a 1009
single-cell or multiple-cell box girder. Small longitudinal voids, such as ungrouted post-1010
tensioning ducts that result in g cpA A greater than or equal to 0.95, can be ignored when 1011
computing the threshold torque in 9.7.5.1. The interaction between torsional cracking and shear 1012
cracking for hollow sections is assumed to vary from the elliptical relationship for members with 1013
small voids, to a straight-line relationship for thin-walled sections with large voids. For a 1014
straight-line interaction, a torque of 0 25. crT would cause a reduction in the inclined cracking 1015
shear of about 25 percent. This reduction was judged to be excessive; therefore, the expressions 1016
for thT are multiplied by the factor in 2
g cpA A . 1017
Tests of solid and hollow beams11.33
indicate that the cracking torque of a hollow section is 1018
approximately g cpA A times the cracking torque of a solid section with the same outside 1019
dimensions. The additional multiplier of g cpA A reflects the transition from the circular 1020
interaction between the inclined cracking loads in shear and torsion for solid members, to the 1021
approximately linear interaction for thin-walled hollow sections. <R11.5.1> 1022
9.7.6 — Cracking torsion 1023
9.7.6.1 — Cracking torsion, crT , shall be calculated in accordance with Table 9.7.6.1 for 1024
solid and hollow cross sections, where uN is positive for compression and negative for 1025
tension. <11.5.2> 1026
Table 9.7.6.1 — Cracking torsion 1027
Type of member crT
Nonprestressed member
2
4
cpc
cp
Af
p (a)
Prestressed member
2
4 14
cp pcc
cp c
A ff
p f (b)
Nonprestressed member
subjected to axial force
2
4 14
cp uc
cp g c
A Nf
p A f
(c)
1028
R9.7.6.1 — The cracking torsion under pure torsion crT is derived by replacing the actual 1029
section with an equivalent thin-walled tube with a wall thickness t prior to cracking of 1030
0 75. cp cpA p and an area enclosed by the wall centerline oA equal to 2 3cpA . Cracking is 1031
Page 8753
ACI 318-14 CR094/LB13-3 2 May 2013
assumed to occur when the principal tensile stress reaches 4 cf . In a nonprestressed beam 1032
loaded with torsion alone, the principal tensile stress is equal to the torsional shear stress, 1033
2 oT A t . Thus, cracking occurs when reaches 4 cf , giving the cracking torque crT 1034
as 1035
2
'4
cp
cr c
cp
AT f
p 1036
For prestressed members, the torsional cracking load is increased by the prestress. A Mohr’s 1037
Circle analysis based on average stresses indicates the torque required to cause a principal tensile 1038
stress equal to 4 cf is 1 4pc cf f times the corresponding torque in a 1039
nonprestressed beam. <R11.5.1> 1040
If the cracking torsion of hollow section is calculated for uT of a member in a statically 1041
indeterminate structure, the replacement of cpA with gA , as in the calculation of the threshold 1042
torque for hollow sections in 9.7.5.1, is not applied here. Thus, the torque after redistribution is 1043
larger, and hence, more conservative. <R11.5.2.1 and R11.5.2.2> 1044
9.7.7 — Torsional strength 1045
R9.7.7 - The factored torsional resistance nT must equal or exceed the torsion uT due to the 1046
factored loads. In the calculation of nT , all the torque is assumed to be resisted by stirrups and 1047
longitudinal steel with the torsional resistance provided by the concrete equal to zero. At the 1048
same time, the nominal shear strength provided by concrete, cV , is assumed to be unchanged by 1049
the presence of torsion. <R11.5.3.5> 1050
9.7.7.1 — For prestressed and nonprestressed members, nT shall be calculated as the lesser of 1051
(a) and (b): 1052
2cot
o t ytn
A A fT
s (9.7.7.1a) 1053
2tan
o yn
h
A A fT
p (9.7.7.1b) 1054
where oA shall be determined by analysis, shall not be taken less than 30 degrees nor 1055
greater than 60 degrees, tA is the area of one leg of a closed stirrup resisting torsion, A is 1056
the area of longitudinal torsion reinforcement, and hp is the perimeter of the centerline of the 1057
outermost closed stirrup. <11.5.3.6> <11.5.3.7> 1058
R9.7.7.1 — Equation (9.7.7.1a) is based on the space truss analogy shown in Fig. R9.7.7.1(a) 1059
with compression diagonals at an angle , assuming the concrete carries no tension and the 1060
reinforcement yields. After torsional cracking develops, the torsional resistance is provided 1061
mainly by closed stirrups, longitudinal bars, and compression diagonals. The concrete outside 1062
Page 8754
ACI 318-14 CR094/LB13-3 2 May 2013
these stirrups is relatively ineffective. For this reason oA , the gross area enclosed by the shear 1063
flow path around the perimeter of the tube, is defined after cracking in terms of ohA , the area 1064
enclosed by the centerline of the outermost closed transverse torsional reinforcement. 1065
The shear flow q in the walls of the tube, discussed in R9.7, can be resolved into the shear 1066
forces 1V to 4V acting in the individual sides of the tube or space truss, as shown in Fig. 1067
R9.7.7.1(a). 1068
1069
Fig. R9.7.7.1(a) – Space truss analogy 1070
Figure R9.7.7.1(a) shows the shear forces 1V to 4V resulting from the shear flow around the 1071
walls of the tube. On a given wall of the tube, the shear flow iV is resisted by a diagonal 1072
compression component, sini iD V , in the concrete. An axial tension force, coti iN V , is 1073
needed in the longitudinal steel to complete the resolution of iV . 1074
Figure R9.7.7.1(b) shows the diagonal compressive stresses and the axial tension force, iN , 1075
acting on a short segment along one wall of the tube. Because the shear flow due to torsion is 1076
constant at all points around the perimeter of the tube, the resultants of iD and iN act through 1077
the midheight of side i. As a result, half of iN can be assumed to be resisted by each of the top 1078
and bottom chords as shown. Longitudinal reinforcement with a strength yA f should be 1079
provided to resist the sum of the iN forces, iN , acting in all of the walls of the tube. 1080
In the derivation of Eq. (9.7.7.1b), axial tension forces are summed along the sides of the area 1081
oA . These sides form a perimeter length, op approximately equal to the length of the line joining 1082
the centers of the bars in the corners of the tube. For ease in computation, this has been replaced 1083
with the perimeter of the closed stirrups, hp . <R11.5.3.6> 1084
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ACI 318-14 CR094/LB13-3 2 May 2013
1085
Fig. R9.7.7.1(b) – Resolution of shear force iV into diagonal compression force iD and axial 1086
tension force iN in one wall of tube. 1087
— In Eq. Eq. (9.7.7.1a) and (9.7.7.1b), it shall be permitted to take oA equal to 9.7.7.1.11088
0.85 ohA . <11.5.3.6> 1089
R9.7.7.1.1 - The area ohA is shown in Fig. R9.7.7.1.1 for various cross sections. In an I-, T-, or 1090
L-shaped section, ohA is taken as that area enclosed by the outermost legs of interlocking 1091
stirrups as shown in Fig. R9.7.7.1.1. The expression for oA given in Hsu11.36
may be used if 1092
greater accuracy is desired. <R11.5.3.6> 1093
1094
Fig. R9.7.7.1.1 - Definition of Aoh 1095
— In Eq. Eq. (9.7.7.1a) and (9.7.7.1b), it shall be permitted to take equal to 9.7.7.1.21096
(a) or (b): <11.5.3.6> 1097
(a) 45 degrees for nonprestressed members or members with effective prestress force 1098
less than 40 percent of the tensile strength of the longitudinal reinforcement 1099
(b) 37.5 degrees for prestressed members with an effective prestress force of at least 1100
40 percent of the tensile strength of the longitudinal reinforcement 1101
R9.7.7.1.2 —The angle can be obtained by analysis11.36
or may be taken to be equal to the 1102
values given in 9.7.7.1.2(a) or (b). The same value of should be used in both Eq. (9.7.7.1a) 1103
and (9.7.7.1b). As gets smaller, the amount of stirrups required by Eq. (9.7.7.1a) decreases. At 1104
the same time, the amount of longitudinal reinforcement required by Eq. (9.7.7.1b) increases. 1105
<R11.5.3.6> 1106
Page 8756
ACI 318-14 CR094/LB13-3 2 May 2013
9.7.8 — Cross-sectional limits 1107
9.7.8.1 — Cross-sectional dimensions shall be such that (a) or (b) is satisfied. <11.5.3.1> 1108
(a) For solid sections 1109
22
28
1.7
u u h cc
w woh
V T p Vf
b d b dA (9.7.8.1a) 1110
(b) For hollow sections 1111
2
81.7
u u h cc
w woh
V T p Vf
b d b dA (9.7.8.1b) 1112
R9.7.8.1 — The size of a cross section is limited for two reasons: first, to reduce unsightly 1113
cracking, and second, to prevent crushing of the surface concrete due to inclined compressive 1114
stresses due to shear and torsion. In Eq. (9.7.8.1a) and (9.7.8.1b), the two terms on the left-hand 1115
side are the shear stresses due to shear and torsion. The sum of these stresses may not exceed the 1116
stress causing shear cracking plus8 cf , similar to the limiting strength given in 9.5.1.2 for 1117
shear without torsion. The limit is expressed in terms of cV to allow its use for nonprestressed or 1118
prestressed concrete. It was originally derived on the basis of crack control. It is not necessary to 1119
check against crushing of the web because this happens at higher shear stresses. 1120
In a hollow section, the shear stresses due to shear and torsion both occur in the walls of the box 1121
as shown in Fig. R9.7.8.1(a) and hence are directly additive at point A as given in Eq. (9.7.8.1b). 1122
In a solid section, the shear stresses due to torsion act in the “tubular” outside section while the 1123
shear stresses due to uV are spread across the width of the section as shown in Fig. R9.7.8.1(b). 1124
For this reason, stresses are combined in Eq. (9.7.8.1a) using the square root of the sum of the 1125
squares rather than by direct addition. <R11.5.3.1> 1126
1127
1128
Fig. R9.7.8.1 —Addition of torsional and shear stresses. 1129
Page 8757
ACI 318-14 CR094/LB13-3 2 May 2013
— For prestressed members, the value of d used in 9.7.8.1 need not be taken 9.7.8.1.11130
less than 0.8h. <11.5.3.1> <11.4.3> 1131
R9.7.8.1.1 — Although the value of d may vary along the span of a prestressed beam, 1132
studies11.2
have shown that, for prestressed concrete members, d need not be taken less than 1133
0.80h . The beams considered had some straight tendons or reinforcing bars at the bottom of the 1134
section and had stirrups that enclosed the steel. <R11.4.3> 1135
— For hollow sections where the wall thickness varies around the perimeter, 9.7.8.1.21136
Eq. (9.7.8.1b) shall be evaluated at the location where the term 21.7
u u h
w oh
V T p
b d A
is a 1137
maximum. <11.5.3.2> 1138
R9.7.8.1.2 — Generally, the maximum torsional stress will be on the wall where the torsional 1139
and shearing stresses are additive [Point A in Fig. R9.7.8.1(a)]. If the top or bottom flanges are 1140
thinner than the vertical webs, it may be necessary to evaluate Eq. (9.7.8.1b) at points B and C in 1141
Fig. R9.7.8.1(a). At these points, the stresses due to the shear force are usually negligible. 1142
<R11.5.3.2> 1143
9.7.8.2 — For hollow sections where the wall thickness is less thanoh
h
A
p, the term
21.7
u h
oh
T p
A
1144
in Eq. (9.7.8.1b) shall be taken as 1.7
u
oh
T
A t
, where t is the thickness of the wall of the 1145
hollow section at the location where the stresses are being checked. <11.5.3.3> 1146
Page 8758
ACI 318-14 CR094/LB13-3 2 May 2013
Approved changes to Chapter 2 during balloting of Chapter 9 1147
1148
NOTATION: 1149
pdA = total area occupied by duct, sheathing, and prestressing reinforcement, in.2 1150
ptA = total area of prestressing reinforcement, in.2
1151
tr = transfer length of prestressed reinforcement, in. 1152
db = debonded length of prestressed reinforcement at end of member, in. 1153
thT = threshold torsional moment, in.-lb 1154
crT = cracking torsional moment, in.-lb 1155
ty = value of net tensile strain in the extreme layer of longitudinal tension reinforcement used to 1156
define a compression-controlled section, see 9.4.2.2. 1157
cv = stress corresponding to nominal two-way shear strength provided by concrete, psi 1158
sv = equivalent concrete stress corresponding to nominal two-way shear strength provided by 1159
reinforcement, psi 1160
nv = nominal shear stress equivalent concrete stress corresponding to nominal two-way shear 1161
strength of slab or footing, psi 1162
uv = maximum factored two-way shear stress calculated around the perimeter of a given critical 1163
section, psi 1164
1165
DEFINITIONS: 1166
Compression-controlled section – A cross section in which the net tensile strain in the extreme 1167
layer of longitudinal tension reinforcement at nominal strength does not exceed ty . 1168
Compression-controlled strain limit – the net tensile strain at balanced strain conditions. See 1169
10.3.3. 1170
Page 8759