cr094 commentary to chapter 9 - american concrete …...aci 318-14 cr094/lb13-3 2 may 2013 75 r....

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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ACI 318-14 CR094/LB13-3 2 May 2013

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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ACI 318-14 CR094/LB13-3 2 May 2013

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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ACI 318-14 CR094/LB13-3 2 May 2013

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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ACI 318-14 CR094/LB13-3 2 May 2013

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


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

Page 8721

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

722

ACI 318-14 CR094/LB13-3 2 May 2013

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.

Page 8724

ACI 318-14 CR094/LB13-3 2 May 2013

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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ACI 318-14 CR094/LB13-3 2 May 2013

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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ACI 318-14 CR094/LB13-3 2 May 2013

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


9.5.2.1 — 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

'

Page 8728

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


9.6.2.1 — 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


9.6.4.1 — 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


9.7.2.1 — Strength reduction factor for torsion, , shall be 0.75. <9.3.2.3> 944


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)


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


2

gc

cp

Af

p (a)

Prestressed member

2

14

g pcc

cp c

A ff

p f (b)



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


2

4

cpc

cp

Af

p (a)

Prestressed member

2

4 14

cp pcc

cp c

A ff

p f (b)



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

Page 8755

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

cr094 commentary to chapter 9 - american concrete …...aci 318-14 cr094/lb13-3 2 may 2013 75 r....

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