2 and commentary

1

This draft is not final and is subject to revision. Do not circulate or publish

CODE REQUIREMENTS FOR REINFORCED CONCRETE CHIMNEYS 1

AND COMMENTARY 2

Reported by ACI Committee 307 3

4

F. Alan Wiley

Chair

5

David J. Bird Shu-Jin Fang Joseph Peters John C. Sowizal

Victor A. Bochicchio Jon Galsworthy Denis J. Radecki John Wilson

David C. Mattes Kelly D. Scott

Special acknowledgements to Robert A. Porthouse (deceased) and Scott D. Richart (de-6

ceased) for their contributions to this Code. 7

The committee acknowledges Denis J. Radecki for his contributions in leading the develop-8

ment of this revision of the Code as the Chair from 2014 to 2021. 9

10

This Code provides material, design and detailing requirements for cast-in-place and precast 11

reinforced concrete chimneys. It sets forth minimum loadings for design and contains methods 12

for determining the concrete and reinforcement required to obtain the strength required by the 13

loadings. The methods of analysis apply primarily to circular chimney walls, but guidance is 14

included for applying the general principles to noncircular chimney walls. 15

Keywords: reinforced concrete chimneys; wind load; across-wind load; earthquake load; ther-16

mal load; bending capacity; structural design; reinforced concrete; vortex shedding; load com-17

binations; construction requirements 18

19

2


CONTENTS 1

Preface 2

Chapter 1—General 3

1.1—Scope 4

1.2—General 5

1.3—Purpose 6

1.4—Applicability 7

1.5—Interpretation 8

1.6—Building official 9

1.7—Licensed design professional 10

1.8—Construction documents and design records 11

1.9––Testing and inspection 12

Chapter 2—Notation and terminology 13

2.1—Scope 14

2.2—Notation 15

2.3—Terminology 16

Chapter 3—Referenced standards 17

3.1—Scope 18

3.2—Referenced standards 19

Chapter 4—Structural system requirements 20

4.1—Scope 21

4.2—Materials 22

4.3—Design loads 23

3


4.4—Structural system 1

Chapter 5—Concrete: Materials, design and durability requirements 2

5.1—Scope 3

5.2—General 4

5.3—Materials 5

5.4—Concrete properties for design 6

5.5—Durability requirements 7

Chapter 6—Reinforcing steel: Materials, design and durability requirements 8

6.1—Scope 9

6.2—General 10

6.3—Materials 11

6.4—Steel properties for design 12


Chapter 7—Loads 14

7.1—Scope 15

7.2—General 16

7.3—Dead load 17

7.4—Temperature gradient load 18

7.5—Wind load 19

7.6—Earthquake load 20

7.7—Load combinations 21

Chapter 8—Design strength of horizontal cross-sections 22

8.1—Scope 23

4


8.2—General 1

8.3—Design limits 2

8.4—Required strength 3

8.5—Design strength 4

8.6—Reinforcement limits 5

8.7—Reinforcement detailing 6

Chapter 9—Strength of vertical cross-sections for circumferential ring moments 7

9.1—Scope 8

9.2—General 9


9.4—Required Strength 11




Chapter 10—Opening details 15

10.1—Scope 16

10.2—General 17

10.3—Minimum wall thickness at openings 18

10.4—Vertical reinforcement at openings 19

10.5—Circumferential reinforcement at openings 20

10.6—Corner reinforcement at openings 21

10.7—Seismic detailing 22

Chapter 11—Foundation 23

5


11.1—Scope 1

11.2—Foundation geotechnical capacity 2

11.3—Foundation structural design 3

Chapter 12—Construction requirements 4

12.1—Scope 5

12.2—General 6

12.3—Concrete strength 7

12.4—Concrete strength tests 8

12.5—Formwork 9

12.6—Concrete placement 10

12.7—Concrete curing 11

12.8—Reinforcement placement 12

12.9—Construction tolerances 13

12.10—Precast erection 14

Appendix A––Horizontal cross-section strength for circular chimneys by modified 15

stress block method 16

A.1—Scope 17

A.2—Notation 18

A.3—Procedure for computing combined nominal compression and nominal bending mo-19

ment capacity 20

A.4—Stress block modification factor 21

A.5—Derivation of equations 22

6


Appendix B––Horizontal cross-section strength for circular chimneys by stress-strain 1

relationship integration method 2

B.1—Scope 3

B.2—Notation 4

B.3—Procedure for computing combined nominal compression and nominal bending capac-5

ity 6

B.4—Derivation of equations 7

Appendix C––Temperature gradients for circular chimneys 8

C.1—Scope 9

C.2—Notation 10

C.3—Unlined chimney 11

C.4—Chimney with lining material applied directly to the inside concrete surface 12

C.5—Chimney with insulation completely filling the space between the liner and the chim-13

ney wall (no annular air space) 14

C.6—Chimney with unventilated air space between liner and chimney wall 15

C.7—Chimney with ventilated air space between lining and chimney wall 16

Appendix D––Thermal stresses for circular chimneys 17

D.1—Scope 18

D.2—Notation 19

D.3—Vertical thermal stresses 20

D.2—Horizontal thermal stresses 21

Appendix E––Cracking moment 22

E.1—Scope 23

7


E.2—Notation 1

E.3—Cracking moment calculation 2

Commentary references 3

4

5

8


Preface 1

ACI 307-08 is based on ACI 318-02 and ASCE 7-02. Revisions of these codes as well as 2

recent studies and research have resulted in the revisions of this Code. Changes include: 3

Code completely reorganized to follow the current ACI 318 format as much as pos-4

sible. 5

For the along-wind load, a strength level wind speed is used to determine strength 6

level wind forces. 7

For the along-wind load, the importance factor has been eliminated. Instead, strength 8

level wind speeds are provided for individual risk categories. 9

The directionality factor has been changed from 0.95 to 1.0. 10

The across-wind/along-wind load combination includes a variable load factor de-11

pendent on the wind speed at which the maximum combined load occurs. 12

For the across-wind load, the maximum damping occurring at the strength level wind 13

speed has been reduced from 4% of critical to 2.5% of critical. 14

For the seismic load, the response modification factor has been increased from 1.5 to 15

2.0. 16

For the seismic load, an over-strength factor is required near large openings and for 17

the foundation for chimneys in Seismic Design Categories D, E, and F. 18

Seismic detailing of the jamb area may be required, depending on the Seismic Design 19

Category. 20

All commentary sections have been reviewed. Information deemed not appropriate 21

for the ACI Code format has been removed. Some of the previous commentary in-22

formation has been reformatted as stand-alone reports and has been made available 23

9


on the Committee 307 web page. 1

Revision History 2

As industry expanded in the years immediately following World War I and as a result of the 3

development of large pulverized coal-fired boilers for the electric power-generating utilities 4

in the 1920s, a number of large reinforced concrete chimneys were constructed to accommo-5

date these new facilities. A group of interested engineers who foresaw the potential need for 6

many more such chimneys, and who were members of the American Concrete Institute, em-7

barked on an effort to develop rational design criteria for these structures. The group was or-8

ganized into ACI Committee 505 (predecessor to the present Committee 307) to develop such 9

criteria in the early 1930s. 10

Committee 505 submitted a "Proposed Standard Specification for the Design and Construc-11

tion of Reinforced Concrete Chimneys," an outline of which was published in the ACI JOUR-12

NAL in 1934. This specification was adopted as a tentative standard in February 1936. Alt-13

hough this tentative standard was never accepted by ACI as an official standard, it was used 14

as the basis for the design of many chimneys. As these chimneys aged, inspections revealed 15

considerable cracking. When the industrial expansion began following World War II, other 16

engineers recognized the need for developing an improved design specification for reinforced 17

concrete chimneys. 18

In May 1949, Committee 505 was reactivated to revise the tentative standard specification, 19

embodying modifications that were found desirable during the years it had been in use. The 20

section dealing with the temperature gradient through the chimney liner lining and the chimney 21

wall was completely revised and extended to cover different types and thicknesses of liners or 22

linings and both unventilated and ventilated air spaces between the liner and the chimney wall. 23

10


In 1954, this specification was approved as ACI 505-54. 1

The rapid increase in the size and height of concrete chimneys being built in the mid-1950s 2

raised further questions about the adequacy of the 1954 version of the specification, especially 3

concerning earthquake forces and the effects of wind. 4

In May 1959, the ACI Board of Direction reactivated Committee 505 (renamed Commit-5

tee 307) to review the standard and to update portions with the latest design techniques and 6

the then-current knowledge of the severity of the operating conditions that prevailed in large 7

steam plants. The material in the standard was reorganized, charts were added, and the meth-8

ods for determining loads due to wind and earthquakes were revised. The information on the 9

design and construction of various types of linings was amplified and incorporated in an ap-10

pendix. That version included criteria for working stress design. It was planned to add ultimate 11

strength criteria in a future revision. 12

In preparing the earthquake design recommendations for ACI 307-69, the committee incor-13

porated the results of theoretical studies by adapting them to existing United States codes. The 14

primary problems in this endeavor stemmed from the uncertainties still inherent in the defini-15

tion of earthquake forces and from the difficulty of selecting the proper safety and servicea-16

bility levels that might be desirable for various classes of construction. Committee investiga-17

tions revealed that with some modifications (such as the K factor), the base shear equations 18

developed by the Seismology Committee of the Structural Engineers' Association of Califor-19

nia (SEAOC) could be applied to chimneys. Similarly, the shapes of the force, shear, and 20

moment distributions, as revised in their 1967 report, were also suitable for chimneys. A use 21

factor (U factor) ranging from 1.3 to 2.0 was introduced in the specification, and it was em-22

phasized that the requirements of Section 4.5 of ACI 307-69 that related to seismic design 23

11


could be superseded by a rational analysis based on evaluation of the seismicity of the site and 1

modal response calculations. The modifications were approved in ACI 307-69. In that version, 2

the commentary and derivation of equations were published separately as a supplement to 3

ACI 307-69. 4

In 1970, the document was reissued with corrections of typographical errors. This issue of 5

ACI 307-69 was also designated ANSI A158.1-1970. At the time, as a result of numerous 6

requests, the commentary and derivation of equations were bound together with the specifica-7

tion. 8

ACI 307-79 updated its requirements to agree with the then-accepted standard practices in 9

the design and construction of reinforced concrete chimneys. The major changes included the 10

requirement that two layers of reinforcing steel be used in the walls of all chimneys (previ-11

ously, this only applied to chimney walls thicker than 18 in.) and the requirement that hori-12

zontal sections through the chimney wall be designed for the radial wind pressure distribution 13

around the chimney. Formulas were included to compute the stresses under these conditions. 14

Many revisions of less importance were included to bring the specification up to date. 15

The editions of the specifications before 1979 included appendixes on the subjects of chim-16

ney linings and accessories. In 1971, Committee 307 learned of buckling problems in steel 17

chimney liners. The committee also noted that, in modern power plant and process chimneys, 18

environmental regulations required treatment of the effluent gases that could result in ex-19

tremely variable and aggressively corrosive conditions in the chimneys. These facts led the 20

committee to agree that the task of keeping the chimney liner recommendations current was 21

not a responsibility of an ACI committee and could be misleading to licensed design profes-22

sionals using the chimney specification. By committee consensus, the reference to chimney 23

12


liner construction was dropped from future editions of the specification. Committee 307 then 1

made a recommendation to the Brick Manufacturers' Association and the American Society of 2

Civil Engineers that each appoint a task force or a committee for the development of design 3

criteria for brick and steel liners, respectively. The Power Division of ASCE took up the rec-4

ommendation and appointed a task committee that developed and published a design guide in 5

1975 titled "Design and Construction of Steel Chimney Liners." ASTM established two task 6

forces for chimney liners: one for brick and one for fiberglass-reinforced plastic. 7

The committee had extensive discussion on the question of including strength design in the 8

1979 specification. The decision to exclude it was based on the lack of experimental data on 9

hollow concrete cylinders to substantiate this form of analysis for concrete chimneys. The 10

committee continued, however, to consider strength design, and encouraged experiments in 11

this area. 12

Shortly after ACI 307-79 was issued, the committee decided to incorporate strength design 13

provisions and update the wind and earthquake design requirements. 14

ACI 307-88 incorporated significant changes to ACI 307-79. The major changes incorpo-15

rated were: 16

Modified certain requirements for smaller chimneys; 17

New procedures for along-wind loads were added using equivalent static load distri-18

butions from a simplified dynamic analysis and based on a mean hourly wind speed; 19

Added procedures for calculating across-wind loads corresponding to the peak dis-20

placement induced by vortex shedding using a simplified approach developed by 21

Vickery and Basu (1983) and included provisions for grouped chimneys; 22

Added procedures for combining along-wind and across-wind loads; 23

13


Added procedure for calculating earthquake loads using either dynamic response 1

spectrum analysis or equivalent static lateral force method; 2

Deleted the working stress procedure for design; 3

Added procedure for strength design; 4

Added deflection criteria; 5

The incorporation of the strength design method generally resulted in chimneys with thinner 6

walls and higher deflections. The deflection criteria were added to require that chimney de-7

flections at service loads would be comparable to those of chimneys designed by the previous 8

working stress method and not vary excessively from historical experience. The effects of the 9

changes in ACI 307-88 resulted in designs with relatively thin walls governed mainly by steel 10

area and, in many instances, across-wind forces. 11

The subject of across-wind loads dominated the attention of the committee between 1988 12

and 1995. ACI 307-95 introduced extensively modified procedures for across-wind loads to 13

reflect more recent data, full-scale observations, and theory to address conservatism included 14

in the previous procedures. Precast chimney design and construction techniques were intro-15

duced as this type of design became more prevalent for chimneys as tall as 300 ft. The subject 16

of noncircular shapes was also introduced in ACI 307-95. Due to the infinite array of possible 17

configurations, however, only broadly defined procedures were presented. Because of dissim-18

ilarities between the load factors required by ACI 307 and ACI 318, the committee added 19

guidelines for determining bearing pressures and loads to size and design chimney founda-20

tions. 21

The major changes incorporated into the ACI 307-95 were: 22

Modified procedures for calculating across-wind loads; 23

14


Added requirements for precast concrete chimney columns; 1

Added procedures for calculating loads and for designing noncircular chimney col-2

umns; 3

Deleted exemptions previously granted to smaller chimneys regarding reinforcement 4

and wall thickness; and 5

Deleted static equivalent procedures for calculating earthquake forces. 6

For the ACI 307-98, revisions to the ASCE 7-95 relating to wind and seismic forces required 7

several changes to be made to the ACI 307-95. The changes incorporated into the ACI 307-98 8

were: 9

Site-specific wind loads were calculated using a 3-second gust speed determined 10

from Fig. 6-1 in ASCE 7-95, instead of the previously used fastest-mile speed; 11

Site-specific earthquake forces were calculated using the effective peak velocity-re-12

lated acceleration contours determined from Contour Map 9-2 in ASCE 7-95 instead 13

of previously designated zonal intensity; 14

The vertical load factor for along-wind forces was reduced from 1.7 to 1.3; 15

The vertical load factor for seismic forces was reduced from 1.87 to 1.43; 16

The load factor for across-wind forces was reduced from 1.40 to 1.20: and 17

The vertical strength reduction factor φ was reduced from 0.80 to 0.70. 18

The reduced load factors are used in concert with the revised strength reduction factor and 19

the wind and seismic loads specified in ASCE 7-95. 20

Revisions to ASCE 7 again caused Committee 307 to revisit and revise ACI 307-98. The 21

changes incorporated applicable ASCE 7-02 wind and seismic load factors and methods. The 22

changes in the ACI 307-08 code were: 23

15


Corrected constant in mean along-wind pressure, Eq. (4-5); 1

Added directionality factor 𝐾𝑑 to mean along-wind pressure, Eq. (4-5); 2

Limited the top region where the drag coefficient 𝐶𝑑 for the mean along-wind is 1.0 3

to a maximum of 50 ft, Eq. (4-4); 4

Changed the required reinforcement at the top from twice the amount required for 5

strength to the greater of the amount required for strength and 0.2% in each face; 6

Included procedure in Section 4.3, Earthquake load, compatible with ASCE 7-02 and 7

the ASCE 7 seismic risk maps; 8

Updated the load factors and load combinations to be more in line with ASCE 7-02 9

values and presentation; and 10

Changed the vertical strength reduction factor ɸ back to 0.80. 11

12

13

16


CHAPTER 1—GENERAL 1

1.1—Scope 2

1.1.1 This chapter addresses the following: 3

(a) General requirements of ACI 307 4

(b) Purpose of ACI 307 5

(c) Applicability of ACI 307 6

(d) Interpretation of ACI 307 7

(e) Definition and role of the building official 8

(f) Definition and role of the licensed design professional 9

(g) Construction documents 10

(h) Testing and inspection 11

12

R1.1.1 This Code includes provisions for the design of cast-in-place and pre-cast concrete chim-13

neys utilizing nonprestressed reinforcement. 14

15

1.2—General 16

1.2.1 ACI 307, “Code Requirements for Reinforced Concrete Chimneys,” is hereafter referred 17

to as “this Code.” 18

1.2.2 In this Code, the general building code refers to the building code adopted in a jurisdic-19

tion. When adopted, this Code forms part of the general building code. 20

1.2.3 This Code provides minimum requirements for the materials, design, construction and 21

strength evaluation of cast-in-place or precast reinforced concrete chimneys designed and con-22

17


structed under the requirements of the general building code. A precast reinforced concrete chim-1

ney is defined as a chimney constructed from precast reinforced concrete sections (full cross-sec-2

tions only), assembled one on top of another, to form a self-supporting cantilevered structure. 3

Vertical reinforcement and grout are placed in cores as the precast sections are erected to provide 4

structural continuity and stability during construction and for the completed structure. Chimneys 5

constructed using precast panels as stay-in-place forms are considered cast-in-place chimneys. 6

R1.2.3 This Code provides minimum requirements and exceeding these minimum require-7

ments is not a violation of the Code. The licensed design professional is permitted to specify pro-8

ject requirements that exceed the minimum requirements of this Code. 9

1.2.4 Modifications to this Code that are adopted by a particular jurisdiction are part of the 10

laws of that jurisdiction, but are not a part of this Code. 11

12

1.3—Purpose 13

1.3.1 The purpose of this Code is to provide for public health and safety by establishing mini-14

mum requirements for strength, stability, serviceability and durability of reinforced concrete chim-15

neys. 16

R1.3.1 This Code provides a means of establishing minimum requirements for the design and 17

construction of reinforced concrete chimneys as well as for acceptance of the design and construc-18

tion of reinforced concrete chimneys by the building officials or their designated representatives. 19

1.3.2 This Code does not address all design considerations. 20

R1.3.2 The minimum requirements in this Code do not replace sound professional judgement or 21

the licensed design professional’s knowledge of the specific factors surrounding a project, includ-22

ing its design, the project site, and other specific or unusual circumstances related to the project. 23

18


1.3.3 This Code does not address all construction means and methods. 1

2

1.4—Applicability 3

1.4.1 This Code applies to circular and non-circular cast-in-place or precast reinforced concrete 4

chimneys. 5

R1.4.1 Design equations in this Code have been developed for reinforced concrete chimneys 6

having a circular cross-section. 7

1.4.2 If non-circular shapes are used, their design shall be substantiated in accordance with the 8

principles of this Code and, where applicable, in accordance with ACI 318-19, “Building Code 9

Requirements for Structural Concrete.” 10

R1.4.2 Due to the many possible configurations of non-circular cross-sections, it is not possible 11

to develop specific procedures for every design situation. However, the general principles of this 12

Code and ACI 318-19 may be applied to the design of reinforced concrete chimneys having a non-13

circular cross-section. 14

1.4.3 This Code does not apply to the design and construction of chimney liners. This code does, 15

however, require consideration of the inertia effect, stiffness effect, and insulation properties of 16

liners on the concrete chimney structure. 17

R1.4.3 The presence of a chimney liner (flue) may affect the vertical load, wind load, thermal 18

load and seismic load on the concrete chimney structure. Typically, the chimney is designed con-19

sidering these loads both with and without the liner effects. 20

21

1.5—Interpretation 22

19


1.5.1 This Code consists of chapters and appendices, including text, headings, tables, figures, 1

footnotes to tables and figures, and referenced standards. 2

1.5.2 The Commentary is intended to provide contextual information but is not part of this Code. 3

1.5.3 This Code shall be interpreted in a manner that harmonizes and avoids conflict between or 4

among its provisions. Specific provisions shall govern over general provisions. 5

1.5.4 This Code shall be interpreted and applied in accordance with the plain meaning of the 6

words and terms used. Specific definitions of words and terms in this Code shall be used where 7

provided and applicable, regardless of whether other materials, standards, or resources outside of 8

this Code would provide a different definition. 9

R1.5.4 ACI Concrete Terminology (CT-18) is the primary resource to help determine the mean-10

ing of words or terms that are not defined in this Code. Dictionaries and other reference materials 11

commonly used by licensed design professionals are permitted to be used as secondary resources. 12

1.5.5 The following words or terms in this Code shall be interpreted in accordance with the 13

following: 14

(a) The word “shall” is used to indicate mandatory requirements. 15

(b) Words used in the present tense shall also apply to the future. 16

(c) The word “and” indicates that all of the connected items, conditions, requirements, or events 17

shall apply. 18

(d) The word “or” indicates that the connected items, conditions, requirements, or events are 19

alternatives, at least one of which shall be satisfied. 20

1.5.6 In any case in which one or more provisions of this Code are declared by a court or tribunal 21

to be invalid, that ruling shall not affect the validity of the remaining provisions of this Code, 22

20


which are severable. The ruling of a court or tribunal shall be effective only in that court’s juris-1

diction and shall not affect the content or interpretation of this Code in other jurisdictions. 2

R1.5.6 This Code addresses numerous requirements that can be implemented fully without mod-3

ification if other requirements in this Code are determined to be invalid. This severability require-4

ment is intended to preserve this Code and allow it to be implemented to the extent possible fol-5

lowing legal decisions affecting one or more of its provisions. 6

7

1.6—Building official 8

1.6.1 All references in this Code to the building official shall be understood to mean persons 9

who administer and enforce this Code in the legal authority having jurisdiction. 10

R1.6.1 Building official is defined in 2.3. 11

1.6.2 Actions and decisions by the building official affect only the specific jurisdiction and do 12

not change this code. 13

R1.6.2 Only the American Concrete Institute has the authority to alter or amend this Code. 14

15

1.7––Licensed design professional 16

1.7.1 All references in this Code to the licensed design professional shall be understood to 17

mean the person who is licensed and responsible for, and is in charge of, the structural design or 18

inspection. 19

R1.7.1 Licensed design professional is defined in 2.3. 20

21

1.8––Construction documents and design records 22

21


1.8.1 The licensed design professional shall provide in the construction documents the infor-1

mation required in Chapter 26 of ACI 318-19 and that required by the jurisdiction, as applicable. 2

R1.8.1 The provisions of Chapter 26 of ACI 318-19 for preparing project drawings and specifi-3

cations are, in general, consistent with those of most general building codes. Additional infor-4

mation may be required by the building official. 5

1.8.2 Calculations pertinent to design shall be filed with the construction documents if required 6

by the building official. Analyses and designs using computer programs shall be permitted pro-7

vided design assumptions, user input, and computer-generated output are submitted. Model anal-8

yses shall be permitted to supplement calculations. 9

R1.8.2 If a computer program is used, sufficient information should be provided to allow the 10

building official to perform a detailed review. Documentation of a model analysis should be pro-11

vided with the related calculations. Model analysis should be performed by an individual having 12

experience in this technique. 13

14

1.9––Testing and inspection 15

1.9.1 Concrete shall be tested in accordance with the requirements of Chapter 26 of 16

ACI 318-19, as applicable. 17

1.9.2 Concrete construction shall be inspected in accordance with the general building code 18

and in accordance with Chapter 12 of this code and Chapter 26 of ACI 318-19, as applicable. 19

1.9.3 Inspection records shall include information required in Chapter 12 of this code and 20

Chapter 26 of ACI 318-19, as applicable. 21

22

22


CHAPTER 2—NOTATION AND TERMINOLOGY 1

2.1—Scope 2

2.1.1 This chapter defines notation and terminology used in this Code. 3

4

2.2—Notation 5

𝑨𝒔 = area of reinforcing at top and bottom of the opening, in2 (Ch. 10) 6

𝑩 = band-width parameter (Ch. 7) 7

𝒃𝒐 = width of the opening in chimney wall, in (Ch. 10) 8

�̅� = mean hourly wind speed factor (Ch. 7) 9

𝑪𝒃 = coefficient of thermal conductivity of an uninsulated liner or of the insulation around 10

liner or of the lining material applied directly to concrete chimney wall, Btu∙in/(hr∙ft2∙°F) 11

(Ch. 7, App. C) 12

𝑪𝒄 = coefficient of thermal conductivity of concrete, Btu∙in/(hr∙ft2∙°F) (Ch. 7, App. C) 13

𝑪𝒅 = earthquake deflection amplification factor (Ch. 7) 14

𝑪𝒅𝒓 = drag coefficient for along-wind load (Ch. 7) 15

𝑪𝑬 = end effect factor (Ch. 7) 16

𝑪𝑳 = RMS lift coefficient (Ch. 7) 17

𝑪𝑳𝒐 = RMS lift coefficient modified for local turbulence (Ch. 7) 18

𝑪𝒔 = coefficient of thermal conductivity of insulation filling the space between liner 19

and concrete wall, Btu∙in/(hr∙ft2∙°F) (Ch. 7, App. C) 20

𝒄 = ratio of distance from extreme compression fiber to neutral axis to the total 21

thickness for vertical stresses (Ch. 8, App. D) 22

𝒄′ = ratio of distance from extreme compression fiber to neutral axis to the total 23

23


thickness for horizontal stresses (Ch. 9, App. D) 1

𝑫 = dead load (Ch.7, 11) 2

𝒅𝒃 = mean diameter of uninsulated liner or insulation around steel liner, ft (Ch. 7, App. C) 3

𝒅𝒃𝒊 = inside diameter of uninsulated liner or insulation around steel liner, ft (Ch. 7, App. C) 4

𝒅𝒄 = mean diameter of concrete chimney wall, ft (Ch. 7, App. C) 5

𝒅𝒄𝒊 = inside diameter of concrete chimney wall, ft (Ch. 7, App. C) 6

𝒅𝒄𝒐 = outside diameter of concrete chimney wall, ft (Ch. 7, App. C) 7

𝒅𝒔 = mean diameter of space between liner and concrete chimney wall, ft (Ch. 7, App. C) 8

𝒅(𝒉) = top outside diameter of chimney, ft (Ch. 7, 9) 9

𝒅(𝒖) = mean outside diameter of upper one-third of chimney, ft (Ch. 7) 10

𝒅(𝒛) = outside diameter of chimney at height z, ft (Ch. 7) 11

𝒅(𝒛𝒄𝒓) = outside diameter of chimney at critical height zcr, ft (Ch. 7) 12

𝒆𝒄 = eccentricity of concrete stress resultant, ft (Ch. 8) 13

𝒆𝒔𝒄 = eccentricity of steel compressive stress resultant, ft (Ch. 8) 14

𝒆𝒔𝒕 = eccentricity of steel tensile stress resultant, ft (Ch. 8) 15

𝑬 = effect of earthquake-induced forces (Ch. 7, 11) 16

𝑬𝒄 = modulus of elasticity of concrete, psi (Ch. 5, 8, 9, App. D) 17

𝑬𝒔 = modulus of elasticity of reinforcement, psi (Ch. 6, 8, 9, App. A, B, D) 18

𝑭𝒂 = short-period site coefficient (Ch. 7) 19

𝑭𝒄 = compressive force in concrete, lb (Ch. 8, App. A, B) 20

𝑭𝒄′ = moment due to compressive force in concrete, lb-ft (Ch. 8, App. A, B) 21

𝑭𝒗 = long-period site coefficient (Ch. 7) 22

𝑭𝟏𝑨 = Strouhal number parameter (Ch. 7) 23

24


𝑭𝟏𝑩 = lift coefficient parameter (Ch. 7) 1

𝒇𝒄′ = specified compressive strength of concrete, psi (Ch. 5, 8, 9, 10, App. A, B) 2

𝒇𝒄′′(𝒄) = fc

’ modified for temperature effects, circumferential, psi (Ch. 9) 3

𝒇𝒄′′(𝒗) = fc

’ modified for temperature effects, vertical, psi (Ch. 8) 4

𝒇𝑪𝑻𝑪′′ = maximum circumferential stress in concrete inside chimney wall due to temperature, 5

psi (Ch. 9, App. D) 6

𝒇𝑪𝑻𝑽′′ = maximum vertical stress in concrete inside chimney wall due to temperature, psi 7

(Ch. 8, App. D) 8

𝒇𝒓 = concrete modulus of rupture, psi (Ch. 5, 8, App. E) 9

𝒇𝑺𝑻𝑪 = maximum stress in outside circumferential reinforcement due to temperature, psi 10

(Ch. 9, App. D) 11

𝒇𝑺𝑻𝑽 = maximum stress in outside vertical reinforcement due to temperature, psi 12

Ch. 8, App. D) 13

𝒇𝑺𝑻𝑽′′ = maximum stress in inside vertical reinforcement due to temperature, psi 14

(Ch. 8, App. D) 15

𝒇𝒚 = specified yield strength of reinforcing steel, psi (Ch. 6, 8, 9, 10, App. A, B) 16

𝒇𝒚′ (𝒄) = fy modified for temperature effects, circumferential, psi (Ch. 9) 17

𝒇𝒚′ (𝒗) = fy modified for temperature effects, vertical, psi (Ch.8) 18

𝑮 = across-wind peaking factor (Ch. 7) 19

𝑮𝒓(𝒛) = gust factor for radial wind pressure at height z (Ch. 7) 20

𝑮𝒘′ = gust factor for along-wind fluctuating load (Ch. 7) 21

𝒈 = acceleration of gravity, ft/sec2 (Ch. 7) 22

𝒉 = chimney height above ground level, ft (Ch. 7) 23

25


𝑰𝒆 = seismic importance factor (Ch. 7) 1

𝒊 = local turbulence factor (Ch. 7) 2

𝑲𝒅 = mean along-wind load directionality factor (Ch. 7) 3

𝑲𝒊 = coefficient of heat transmission from gas to inner surface of liner or to inner surface 4

of lining material applied directly to the chimney wall or to inner surface of chimney 5

wall (when chimney is unlined), BTU/(hr∙ft2∙°F) (Ch. 7, App. C) 6

𝑲𝒊𝒄 = component of 𝑲𝒊 due to conduction and convection (Ch. 7) 7

𝑲𝒊𝒓 = component of 𝑲𝒊 due to radiation (Ch. 7) 8

𝑲𝒐 = coefficient of heat transmission from outside surface of chimney wall to surrounding 9

air, BTU/(hr∙ft2∙°F) (Ch. 7, App. C) 10

𝑲𝒓 = coefficient of heat transfer by radiation between outside surface of liner and inside 11

surface of chimney wall, BTU/(hr∙ft2∙°F) (Ch. 7, App. C) 12

𝑲𝒔 = coefficient of heat transfer between outside surface of liner and inside surface of 13

chimney wall for chimneys with ventilated air spaces, BTU/(hr∙ft2∙°F)(Ch. 7, App. C) 14

𝒌 = ratio of wind speed �̅�(𝒛𝒄𝒓) to critical wind speed 𝑽𝒄𝒓 (Ch. 7) 15

𝒌𝒂 = aerodynamic damping parameter (Ch.7) 16

𝒌𝒂𝒐 = mass damping parameter of small amplitudes (Ch. 7) 17

𝑳 = correlation length parameter (Ch. 7) 18

𝑳𝑭𝒄𝒘 = combined along-wind/across-wind load factor (Ch. 7) 19

𝒍𝒘 = wind end effect length, ft (Ch. 7) 20

𝑴𝒂(𝟎) = across-wind base moment, lb-ft (Ch. 7) 21

𝑴𝒂(𝒛) = across-wind moment at height z, lb-ft (Ch. 7) 22

𝑴𝒄𝒘(𝒛) = combined across-wind moment at height z, lb-ft (Ch. 7) 23

26


𝑴𝒄𝒓(𝒛) = cracking moment for a horizontal cross-section at height z, lb-ft (Ch. 8, App. E) 1

𝑴𝒊(𝒛) = maximum circumferential moment due to radial wind pressure at height z, 2

tension on inside, lb-ft/ft (Ch. 7) 3

𝑴𝒍(𝒛) = mean along-wind moment at height z at governing wind speed for across-wind 4

load, lb-ft (Ch. 7) 5

𝑴𝒏 = nominal moment strength at a horizontal cross-section, lb-ft (Ch. 8, 9, App. A, B) 6

𝑴𝒐(𝒛) = maximum circumferential moment due to radial wind pressure at height z, 7

tension on outside, lb-ft/ft (Ch. 7) 8

𝑴𝒖 = factored moment at a horizontal cross-section, lb-ft (Ch. 8, 9 App. A, B) 9

𝑴�̅�(𝒃) = base moment due to mean wind pressure, lb-ft/ft (Ch. 7) 10

𝑴�̅�(𝒛) = moment at elevation 𝒛 due to mean wind pressure, lb-ft/ft (Ch. 7) 11

𝒏 = modular ratio, 𝑬𝒔 𝑬𝒄⁄ (Ch. 8, 9, App. D) 12

𝑷𝒏 = nominal axial strength at a horizontal cross-section, lb (Ch. 8, App. A, B) 13

𝑷𝒖 = factored axial load at a horizontal cross-section, lb (Ch. 8, App. A, B) 14

�̅�(𝒛) = pressure due to mean hourly wind speed at height z, lb/ft2 (Ch. 7) 15

𝒑𝒓(𝒛) = radial wind pressure at height z, lb/ft2 (Ch. 7) 16

𝑹 = earthquake response modification factor (Ch. 7) 17

𝒓 = mean radius of a horizontal cross-section, ft (Ch. 8, App. A, B) 18

𝒓(𝒛) = mean radius of a horizontal cross-section at height z, ft (Ch. 7) 19

𝒓𝒒 = ratio of heat transmission through chimney wall to heat transmission through liner for 20

chimneys with a ventilated air space (Ch. 7, App. C) 21

𝑺𝟏 = mapped maximum considered earthquake, 5% damped, spectral response acceleration 22

at a period of 1 second, g (Ch. 7) 23

27


𝑺𝒂 = design spectral response acceleration, g (Ch. 7) 1

𝑺𝒄 = compressive force where steel stress is below the yield strength, lb (Ch. 8, App. A, B) 2

𝑺𝒄′ = moment about the neutral axis of the compressive force where steel stress is below 3

the yield strength, lb-ft (Ch. 8, App. A, B) 4

𝑺𝒄𝒚 = compressive force where steel stress is at the yield strength, lb (Ch. 8, App. A, B) 5

𝑺𝒄𝒚′ = moment about the neutral axis of the compressive force where steel stress is at the 6

yield strength, lb-ft (Ch. 8, App. A, B) 7

𝑺𝑫𝟏 = design spectral response acceleration at short periods, g (Ch. 7) 8

𝑺𝑫𝑺 = design spectral response acceleration at a period of 1 second, g (Ch. 7) 9

𝑺𝑴𝟏 = maximum considered earthquake, 5% damped, spectral response acceleration at a 10

period of 1 second adjusted for site class effects, g (Ch. 7) 11

𝑺𝑴𝑺 = maximum considered earthquake, 5% damped, spectral response acceleration at 12

short periods adjusted for site class effects, g (Ch. 7) 13

𝑺𝒑 = across-wind spectral parameter (Ch. 7) 14

𝑺𝒔𝒗 = across-wind mode shape factor (Ch. 7) 15

𝑺𝑺 = mapped maximum considered earthquake, 5% damped, spectral response acceleration 16

at short periods, g (Ch. 7) 17

𝑺𝒕 = Strouhal number (Ch. 7) 18

𝑺𝒕 = tensile force where steel stress is below the yield strength, lb (Ch. 8, App. A, B) 19

𝑺𝒕′ = moment about the neutral axis of the tensile force where steel stress is below the yield 20

strength, lb-ft (Ch. 8, App. A, B) 21

𝑺𝒕𝒚 = tensile force where steel stress is at the yield strength, lb (Ch. 8, App. A, B) 22

𝑺𝒕𝒚′ = moment about the neutral axis of the tensile force where steel stress is at the yield 23

28


strength, lb-ft (Ch. 8, App. A, B) 1

𝒔 = center-to-center spacing of two identical chimneys, ft (Ch. 7) 2

𝑻 = temperature effect (Ch. 7) 3

𝑻𝒏 = n-th mode period of vibration, sec (Ch. 7) 4

𝑻𝒊 = maximum design temperature of flue gas, °F (Ch. 7, App. C) 5

𝑻𝑳 = long-period transition period, sec (Ch. 7) 6

𝑻𝑺 = earthquake response spectrum parameter (Ch. 7) 7

𝑻𝒂𝒎𝒃 = minimum design ambient air temperature, °F (Ch. 7, App. C) 8

𝑻𝟎 = earthquake response spectrum parameter (Ch. 7) 9

𝑻𝒙 = temperature gradient across the chimney wall, °F (Ch. 7,8,9, App.C, D) 10

𝒕𝒃 = thickness of uninsulated liner or thickness of insulation around liner or thickness of 11

lining material applied directly to concrete chimney wall, in (Ch. 7, App. C) 12

𝒕𝒄 = thickness of concrete wall, in (Ch. 7,10, App. C, D) 13

𝒕𝒔 = thickness of insulation filling the space between liner and chimney wall, in 14

(Ch. 7, App. C) 15

𝑽𝒖 = basic wind speed, mi/hr (Ch. 7) 16

𝑽𝒔 = serviceability wind speed, mi/hr (Ch. 7) 17

𝑽𝒄𝒓 = critical wind speed for across-wind load, ft/sec (Ch. 7) 18

𝑽𝑬𝑳𝑭 = base shear using the equivalent lateral force procedure, lb (Ch. 7) 19

𝑽𝒕 = total earthquake design base shear, lb (Ch. 7) 20

�̅�𝒖(𝟑𝟑) = mean hourly basic wind speed at height of 33 ft, ft/sec (Ch. 7) 21

�̅�(𝒛𝒄𝒓) = variable mean hourly wind speed at 𝟓𝒉/𝟔, ft/sec (Ch. 7) 22

�̅�𝒔(𝒛) = mean hourly serviceability wind speed at height 𝒛, ft/sec (Ch. 7) 23

29


�̅�𝒖(𝒛) = mean hourly design wind speed at height z, ft/sec (Ch. 7) 1

�̅�𝒖(𝒛𝒄𝒓) = mean hourly design wind speed at 𝟓𝒉/𝟔, ft/sec (Ch. 7) 2

𝑾 = wind load due to radial wind pressure (Ch. 7) 3

𝑾𝒂𝒍𝒐𝒏𝒈 = wind load due to along-wind loading (Ch. 7) 4

𝑾𝒄𝒐𝒎𝒃 = wind load due to across-wind loading combined with along-wind loading at the 5

same wind speed (Ch. 7) 6

𝑾𝒆 = effective seismic weight, lb (Ch. 7) 7

𝒘(𝒛) = total along-wind load per unit length at height z, lb/ft (Ch. 7) 8

�̅�(𝒛) = mean along-wind load per unit length at height z, lb/ft (Ch. 7) 9

𝒘′(𝒛) = fluctuating along-wind load per unit length at height z, lb/ft (Ch. 7) 10

𝒘𝒕(𝒖) = average weight per unit height for top one-third of chimney, lb/ft (Ch. 7) 11

𝒁𝒄 = exposure length, ft (Ch. 7) 12

𝒛 = height above ground, ft (Ch. 7) 13

𝒛𝒄𝒓 = critical height for across-wind load 𝟓𝒉/𝟔, ft (Ch. 7) 14

𝜶 = for a horizontal cross-section, one-half the angle subtended by neutral axis, radians 15

(Ch. 8, App. A, B) 16

𝜶𝒕𝒆 = thermal coefficient of expansion of reinforced concrete, 1/°F (Ch. 5, 8, 9, App. D) 17

�̅� = mean hourly wind speed power law exponent (Ch. 7) 18

𝜷 = one-half of the central angle subtended by an opening on the chimney cross-section, 19

radians (Ch. 8, App. A, B) 20

𝜷𝟏 = factor relating depth of equivalent rectangular compressive stress block to neutral axis 21

depth (Ch. 8, App. A) 22

𝜷𝒂 = aerodynamic damping factor for across-wind load (Ch. 7) 23

30


𝜷𝒔 = structural damping factor for across-wind load (Ch. 7) 1

𝜸𝟏 = ratio of inside face vertical reinforcement area to outside face vertical reinforcement 2

area (Ch. 8, App. D) 3

𝜸𝟐𝒐 = ratio of distance between inner surface of chimney wall and outside face vertical 4

reinforcement to total wall thickness (Ch. 8, App. D) 5

𝜸𝟐𝒊 = ratio of distance between outer surface of chimney wall and inside face vertical 6

reinforcement to total wall thickness (Ch. 8, App. D) 7

𝜸𝟏′ = ratio of inside face circumferential reinforcement area to outside face circumferential 8

reinforcement area (Ch. 9, App. D) 9

𝜸𝟐𝒐′ = ratio of distance between inner surface of chimney wall and outside face 10

circumferential reinforcement to total wall thickness (Ch. 9, App. D) 11

𝜸𝟐𝒊′ = ratio of distance between outer surface of chimney wall and inside face 12

circumferential reinforcement to total wall thickness (Ch. 9, App. D) 13

𝜺𝒄 = concrete compressive strain (Ch. 8, App. D) 14

𝝆𝒐 = ratio of area of vertical outside face reinforcement to total area of chimney wall 15

(Ch. 8, App. D) 16

𝝆𝒐′ = ratio of area of circumferential outside face reinforcement per unit height to 17

total area of concrete wall per unit height (Ch. 9, App. D) 18

𝝆𝒂 = specific weight of air, lb/ft3 (Ch. 7) 19

𝝆𝒕 = ratio of total area of vertical reinforcement to total area of concrete cross-section 20

(Ch. 8, App. A, B) 21

𝝓 = strength reduction factor (Ch. 4, 8, 9, App. A,B) 22

𝛀𝟎 = overstrength factor (Ch. 7) 23

31


1

2.3—Terminology 2

building official – term used to identify the Authority having jurisdiction or individual 3

charged with administration and enforcement of provisions of the building code. Such terms 4

as building commissioner or building inspector are variations of the title and the term “build-5

ing official” as used in this Code, is intended to include those variations, as well as others 6

that are used in the same sense. 7

buttress – an exterior or interior support projecting from the chimney wall. 8

crosstie – a continuous reinforcing bar having a seismic hook at one end and a hook not less 9

than 90 degrees with at least a 6 diameter extension at the other end. 10

design strength – nominal strength multiplied by a strength reduction factor 𝝓. 11

jamb area – the concrete wall area on either side of an opening containing additional vertical 12

reinforcing bars. 13

licensed design professional – an individual who is licensed to practice structural design as 14

defined by the statutory requirements of the professional licensing laws of the state or juris-15

diction in which the project is to be constructed, and who is in responsible charge of the 16

structural design. 17

liner/lining – a material employed to prevent conveyed gasses from direct contact with the 18

structural concrete; the word liner is used for cases where there is a resultant air space be-19

tween the liner and the concrete wall; the word lining is used when lining materials are ap-20

plied directly to the concrete wall. 21

lintel area – the concrete wall area above an opening containing additional horizontal rein-22

forcing bars. 23

32


MRI (mean recurrence interval) – estimate of the average time between a specified sever-1

ity of earthquake motion, a specified wind speed, or a specified value of some other ran-2

domly occurring event. 3

nominal strength – strength of a cross-section calculated in accordance with provisions and 4

assumptions of the strength design method of this Code before application of any strength re-5

duction factors. 6

serviceability loads – loads imparted on a structure assumed to be present or to occur during 7

the service life of the structure. 8

sill area – the concrete wall area below an opening containing additional horizontal reinforc-9

ing bars. 10

strength design – a method of proportioning structural members such that the computed 11

forces produced in the member by the factored loads do not exceed the member design 12

strength. 13

tie – a reinforcing bar enclosing vertical jamb bars when seismic detailing is required. 14

tie bar – a bar at right angles to and tied to reinforcement to secure it in place. 15

16

33


CHAPTER 3—REFERENCED STANDARDS 1

3.1—Scope 2

3.1.1 Standards, or specific sections thereof, cited in this Code, including annexes, appen-3

dices or supplements where prescribed, are referenced without exception in this Code unless 4

specifically noted. Cited standards are listed in the following with their serial designations 5

including the year of adoption or revision. 6

R3.1.1 In this Code, references to standard specifications or other material are to a specific 7

edition of the cited document. All such referenced standards are listed in this chapter with the 8

title and complete serial designation. In other chapters of this Code, these references may be 9

abbreviated, but the abbreviation refers to the specific document listed in this chapter. Com-10

mentary references are listed after Chapter 12. 11

3.2—Referenced standards 12

3.2.1 American Concrete Institute (ACI) 13

318-19 Building Code Requirements for Structural Concrete 14

117-14 Specification for Tolerances for Concrete Construction and Materials 15

3.2.2 American Society of Civil Engineers (ASCE) 16

ASCE/SEI 7-16 Minimum Design Loads for Buildings and Other Structures 17

3.2.3 ASTM International 18

ASTM A615/A615M-20 Standard Specification for Deformed and Plain Carbon-Steel Bars 19

for Concrete Reinforcement 20

ASTM A706/A706M-16 Standard Specification for Low-Alloy Steel Deformed and Plain 21

Bars for Concrete Reinforcement 22

ASTM C33/C33M-18 Standard Specification for Concrete Aggregates 23

34


ASTM C150/C150M-20 Standard Specification for Portland Cement 1

ASTM C260/C260M-10a(2016) Standard Specification for Air-Entraining Admixtures for 2

Concrete 3

ASTM C309-19 Standard Specification for Liquid Membrane-Forming Compounds for 4

Curing Concrete 5

ASTM C494/494M-19 Standard Specification for Chemical Admixtures for Concrete 6

ASTM C595/C595M-20 Standard Specification for Blended Hydraulic Cement 7

ASTM C618-19 Standard Specification for Coal Fly Ash and Raw or Natural Pozzolan for 8

Use in Concrete 9

ASTM C989/989M-18a Standard Specification for Slag Cement for Use in Concrete and 10

Mortars 11

ASTM C1017/C1017M-13e1 Standard Specification for Chemical Admixtures for Use in 12

Producing Flowing Concrete 13

ASTM C1240-20 Standard Specification for Silica Fume Used in Cementitious Mixtures 14

ASTM C1582/C1582M-11(2017)e Standard Specification for Admixtures to Inhibit Chlo-15

ride-Induced Corrosion of Reinforcing Steel in Concrete 16

ASTM C1602/C1602M-18 Standard Specification for Mixing Water Used in the Produc-17

tion of Hydraulic Cement Concrete 18

3.2.4 Federal Aviation Administration (FAA) 19

AC70-7460-1L, Change 2, Effective 8/17/18, Obstruction Marking and Lighting 20

3.2.5 Underwriters Laboratories (UL) 21

UL 96A, Edition 13 (12/18/18), Standard for Installation Requirements for Lightning Pro-22

tection Systems 23

35


3.2.6 National Fire Protection Association (NFPA) 1

NFPA 780-2020, Standard for the Installation of Lightning Protection Systems, 2020 Edi-2

tion 3

4

36


CHAPTER 4—STRUCTURAL SYSTEM REQUIREMENTS 1

4.1—Scope 2

4.1.1 The provisions of this chapter identify important aspects of the design and construction 3

of concrete chimneys and the location of related requirements. 4

R4.1.1 This chapter introduces structural system requirements for reinforced concrete chim-5

neys. 6

4.2—Materials 7

4.2.1 Properties of concrete shall be selected to be in accordance with Chapter 5. 8

4.2.2 Properties of reinforcement shall be selected to be in accordance with Chapter 6. 9

4.3—Design loads 10

4.3.1 Loads and load combinations considered in design shall be in accordance with Chap-11

ter 7. 12

4.4—Structural system 13

4.4.1 The structural system shall include the following, as applicable. 14

1. Chimney wall 15

2. Foundation 16

3. All loads introduced into the chimney wall and the foundation 17

4. Connections and anchors as required to transmit forces to the chimney wall and foun-18

dation 19

4.4.2 Design strength of horizontal chimney cross-sections shall be determined in accordance 20

with Chapter 8. 21

R4.4.2 Chapter 8 presents methods used to determine the vertical reinforcing steel required 22

to meet the bending moment demand at horizontal cross-sections. 23

37


4.4.3 Design strength of vertical chimney cross-sections for circumferential ring moments 1

shall be determined in accordance with Chapter 9. 2

R4.4.3 Chapter 9 presents methods used to determine the horizontal reinforcing steel re-3

quired to meet the ring bending moment demand at vertical cross-sections. 4

4.4.4 Detailing near openings shall be determined in accordance with Chapter 10. 5

R4.4.4 Openings are a significant consideration for the design of reinforced concrete chim-6

neys, so a separate chapter is dedicated to detailing near openings. 7

4.4.5 Foundations shall be designed in accordance with Chapter 11. 8

4.4.6 Anchors in concrete used to transmit loads by means of tension, shear, or a combination 9

of tension and shear shall be designed in accordance with Chapter 17 of ACI 318-19. 10

R4.4.6 Chapter 17 of ACI 318-19 (Anchoring to Concrete) applies to cast-in anchors and to 11

post-installed expansion (torque controlled and displacement controlled), undercut and adhe-12

sive anchors. 13

4.4.7 Compressive forces transferred through bearing shall be designed in accordance with 14

Section 22.8 of ACI 318-19 using the strength reduction factor 𝜙 = 0.65 and the factored load 15

combinations of Table 5.3.1 in ACI 318-19. 16

17

38


CHAPTER 5—CONCRETE: MATERIALS, DESIGN AND DURABILITY 1

REQUIREMENTS 2

5.1—Scope 3

5.1.1 This chapter shall apply to concrete, including: 4

(a) Materials 5

(b) Properties to be used for design 6

(c) Durability requirements 7

5.2—General 8

5.2.1 Concrete materials, design properties and durability requirements not addressed in 9

this chapter shall be in accordance with ACI 318-19. 10

5.3—Materials 11

5.3.1 Cementitious materials 12

5.3.1.1 A single brand and type of cement shall be specified for the construction of the 13

chimney wall. 14

5.3.1.2 Portland cement used shall conform to the requirements for Type I, II, III or V of 15

ASTM C150. 16

5.3.1.3 Blended hydraulic cements used shall conform to the requirements for Type IS or 17

IP of ASTM C595. 18

5.3.1.4 Fly ash and natural pozzolan used shall conform to ASTM C618. 19

5.3.1.5 Slag cement used shall conform to ASTM C989. 20

5.3.1.6 Silica fume used shall conform to ASTM C1240. 21

5.3.1.7 All cementitious materials specified in this chapter and the combinations of these 22

materials shall be included in calculating the water-to-cement ratio of the concrete mixture. 23

39


5.3.2 Aggregates 1

5.3.2.1 Aggregates shall conform to ASTM C33. 2

5.3.2.2 The maximum size of coarse aggregate shall not exceed 1/8 of the narrowest di-3

mension between inside and outside forms. 4

5.3.2.3 The maximum size of coarse aggregate shall not exceed 1/2 the minimum clear dis-5

tance between reinforcing bars. 6

5.3.3 Water 7

5.3.3.1 Mixing water shall conform to ASTM C1602. 8

5.3.4 Admixtures 9

5.3.4.1 Water reducing admixtures and setting time modification admixtures shall conform 10

to ASTM C494. 11

5.3.4.2 Admixtures used to produce flowing concrete shall conform to ASTM C1017. 12

5.3.4.3 Air-entraining admixtures shall conform to ASTM C260. 13

5.3.4.4 Admixtures used to inhibit chloride-induced corrosion of reinforcing steel shall 14

conform to ASTM C1582. 15

5.3.5 Concrete mixture requirements 16

5.3.5.1 Based on the exposure classes assigned from 5.5, concrete mixtures shall conform 17

to the most restrictive requirements of Table 19.3.2.1 of ACI 318-19. 18

5.4—Concrete properties for design 19

5.4.1 General 20

5.4.1.1 The concrete properties of this section shall apply when the mean temperature of 21

the concrete wall does not exceed 150°F. When the mean concrete wall temperature exceeds 22

40


150°F, the concrete design properties shall be adjusted to account for temperature depend-1

ence. 2

R5.4.1.1 Values of concrete design properties at temperatures exceeding 150°F should be 3

obtained from reliable sources. 4

5.4.2 Compressive strength 5

5.4.2.1 The value of 𝒇𝒄′ shall be specified in the construction documents. 6

5.4.2.2 The specified value of 𝒇𝒄′ shall be in accordance with: 7

(a) Structural strength requirements 8

(b) Durability requirements of Section 5.5 9

R5.4.2.2 Table 19.3.2.1 of ACI 318-19 lists requirements for concrete by exposure class. 10

5.4.2.3 The specified compressive strength shall be used for proportioning of concrete mix-11

tures in accordance with Section 26.4.3 of ACI 318-19. 12

5.4.2.4 Compliance with 𝒇𝒄′ shall be based on cylinder tests in accordance with Sec-13

tion 26.12 of ACI 318-19. Test age, if other than 28 days, shall be indicated in the contract 14

documents 15

5.4.3 Modulus of elasticity 16

5.4.3.1 The modulus of elasticity for concrete, 𝑬𝒄, shall be in accordance with ACI 318-19, 17

Section 19.2.2. 18

5.4.4 Modulus of rupture 19

5.4.4.1 The modulus of rupture for concrete, 𝒇𝒓, shall be in accordance with ACI 318-19, 20

Section 19.2.3. 21

5.4.5 Thermal coefficient of expansion 22

41


5.4.5.1 The thermal coefficient of expansion of reinforced concrete, 𝜶𝒕𝒆, is permitted to be 1

taken as 5.5 x 10-6 per °F. 2

R5.4.5.1 Per ACI 209R-92, the value specified is a reasonable average value for the coeffi-3

cient of thermal expansion. 4

5.5 Durability requirements 5

5.5.1 Concrete durability requirements shall be in accordance with Section 19.3 of 6

ACI 318-19 except that 𝒇𝒄′ shall not be less than 3000 psi. The licensed design professional 7

shall assign exposure classes in accordance with the severity of the anticipated exposure of 8

the chimney for each exposure category in Table 19.3.1.1 of ACI 318-19. 9

10

42


CHAPTER 6—REINFORCING STEEL: MATERIALS, DESIGN AND 1

DURABILITY REQUIREMENTS 2

6.1—Scope 3

6.1.1 This chapter shall apply to steel reinforcement and include: 4

(a) Materials 5

(b) Properties to be used for design 6

(c) Durability (concrete cover) requirements 7

6.2—General 8

6.2.1 Reinforcing steel materials, design properties, and durability requirements not ad-9

dressed in this chapter shall be in accordance with ACI 318-19. 10

6.3—Materials 11

6.3.1 Reinforcement shall be deformed bars conforming to ASTM A615/615M, Grades 40, 12

60 and 80 or conforming to ASTM A706/A706M, Grades 60 and 80. 13

6.3.2 Other deformed steel reinforcements are permitted provided that the material has an 14

ultimate tensile strain equal to or exceeding 0.06. 15

R6.3.2 The ultimate tensile strain requirement has been changed from 0.07 to 0.06 so that 16

ASTM 615/615M Grade 80 is permitted. 17

6.4—Steel properties for design 18

6.4.1 Yield strength 19

6.4.1.1 Yield strength for reinforcing bars shall be based on the specified grade of reinforce-20

ment but shall not exceed 80,000 psi. 21

6.4.2 Modulus of elasticity 22

43


6.4.2.1 Modulus of elasticity, 𝐸𝑠, for reinforcing bars is permitted to be taken as 1

29,000,000 psi. 2


6.5.1 The specified concrete cover shall be at least 2 inches for cast-in-place chimneys. 4

6.5.2 The specified concrete cover shall be at least 1.5 inches for precast units manufactured 5

under plant-controlled conditions. 6

7

44


CHAPTER 7—LOADS 1

7.1—Scope 2

7.1.1 This chapter shall apply to the determination of loads for the design of reinforced concrete 3

chimneys including: 4

(a) Calculation of loads 5

(b) Load factors and load combinations 6

7.2—General 7

7.2.1 Chimney loads shall include: 8

(c) Dead load 9

(d) Temperature gradient load 10

(e) Wind load 11

(f) Earthquake load 12

7.2.2 Every chimney shall be assigned a Risk Category as defined in Section 1.5 of 13

ASCE 7-16. The Risk Category shall be either Category III or Category IV as applicable. 14

R7.2.2 Previous versions of the code stated that every chimney should be designated as an 15

essential facility classified as a Risk Category IV structure. The revised code provides for the 16

designation of a chimney as a Risk Category III structure if appropriate. 17

7.2.3 For wind and earthquake loads, natural frequencies or periods shall be computed by a 18

frequency analysis that takes into account the mass and stiffness distribution of the chimney, 19

the effect of any significant non-structural weights such as access platforms and slabs, and the 20

effect of liner(s) supported and/or braced by the chimney. 21

7.3—Dead load 22

45


7.3.1 For calculation of self-weight, the unit weight for normal weight reinforced concrete is 1

permitted to be taken as 150 lb/ft3. 2

7.3.2 Permanent loads, such as the weight of liners, roofs and/or platforms supported by the 3

chimney wall shall be included in the dead load. 4

R7.3.2 When there is a significant time delay between completion of the chimney wall and 5

installation of any permanent load, the design should be checked with and without that perma-6

nent load. 7

7.4—Temperature gradient load 8

7.4.1 General 9

7.4.1.1 Reinforced concrete chimneys shall be designed to resist vertical and circumferential 10

stress due to a temperature gradient across the chimney wall. 11

7.4.1.2 For circular chimneys, the temperature gradient across the chimney wall shall be 12

determined according to 7.4.2 or by a rational heat balance study. 13

7.4.1.3 For non-circular chimneys, the temperature gradient across the chimney wall shall be 14

determined by a rational heat-balance study. 15

7.4.2 Temperature gradient for circular chimneys 16

7.4.2.1 Eq. (7-1) to (7-5) are permitted to be used to obtain the temperature gradient across 17

the chimney wall of a circular chimney. For circular chimneys with multiple liners, the equa-18

tions are permitted to be used by setting the liner diameter equal to an equivalent diameter that 19

approximates the thermodynamic conditions. 20

R7.4.2.1 For chimneys with multiple liners a rational heat balance study would be needed to 21

accurately determine the temperature gradient across the chimney wall, however, Eqs. (7-3) 22

and (7-4) are permitted to be used with an equivalent liner diameter to represent the multiple 23

46


liners. The normal practice is to use a liner diameter equal to the sum of the individual liner 1

diameters, however in certain cases the sum of the individual diameters may be greater than 2

the inside diameter of the chimney wall, in which case a smaller diameter should be used 3

providing a nominal airspace. It may also be necessary to check the gradient using several 4

different equivalent diameters to represent the thermodynamic conditions from variations in 5

gas flow and temperature among the multiple liners. 6

7.4.2.2 For unlined circular chimneys, 7

𝑇𝑥 =𝑡𝑐𝑑𝑐𝑖𝐶𝑐𝑑𝑐

(𝑇𝑖 − 𝑇𝒂𝒎𝒃

1𝐾𝑖+𝑡𝑐𝑑𝑐𝑖𝐶𝑐𝑑𝑐

+𝑑𝑐𝑖𝐾𝑜𝑑𝑐𝑜

) (7-1)

7.4.2.3 For circular chimneys with lining material applied directly to the inside surface of 8

the chimney wall, 9

𝑇𝑥 =𝑡𝑐𝑑𝑏𝑖𝐶𝑐𝑑𝑐


1𝐾𝑖+𝑡𝑏𝑑𝑏𝑖𝐶𝑏𝑑𝑏

+𝑡𝑐𝑑𝑏𝑖𝐶𝑐𝑑𝑐

+𝑑𝑏𝑖𝐾𝑜𝑑𝑐𝑜

) (7-2)

R7.4.2.3 This case is added to explicitly address glass block or gunite applied to the chimney 10

wall. 11

7.4.2.4 For circular chimneys with insulation completely filling the space between the liner 12

and the chimney wall (no annular air space), 13




+𝑡𝑠𝑑𝑏𝑖𝐶𝑠𝑑𝑠



) (7-3)

7.4.2.5 For circular chimneys with an unventilated air space between the liner and the 14

chimney wall, 15

47





+𝑑𝑏𝑖𝐾𝑟𝑑𝑠



) (7-4)

R7.4.2.5 The equation suggests that the liner is not insulated. For an insulated liner, it is 1

permissible to still use this equation, but let the terms with subscript “b” refer to the insulation 2

material instead of the liner material. The assumption is that the thermal resistance of the liner 3

material is negligible compared to the thermal resistance of the insulation material. 4

7.4.2.6 For circular chimneys with a ventilated air space between the liner and the chimney 5

wall, 6



1𝑟𝑞𝐾𝑖

+𝑡𝑏𝑑𝑏𝑖𝑟𝑞𝐶𝑏𝑑𝑏

+𝑑𝑏𝑖𝐾𝑠𝑑𝑠



) (7-5)

R7.4.2.6 See R7.4.2.5 for use of the equation with an insulated liner. 7

7.4.2.7 When multiple layers of material are to be considered, additional terms can be added 8

to the equations. 9

R7.4.2.7 For example, Eqs. (7-4) and (7-5) can be modified to include a term for the liner 10

material and a term for the insulation material. See Appendix C. 11

7.4.2.8 Unless complete heat balance studies are made for the particular chimney, it is 12

permissible to use the following approximate values: 13

𝑟𝑞 = 0.5 (see 7.4.2.9); 14

𝐶𝑐 = to be obtained from the manufacturer of the materials used; for normal weight concrete 15

12 (Btu-in.)/(h-ft2-°F) of thickness/h/°F difference in temperature is permitted to be 16

used; 17

𝐶𝑠 = to be obtained from the manufacturer of the materials used; 18

48


𝐶𝑏 = to be obtained from the manufacturer of the materials used; 1

𝐾𝑖 = 𝐾𝑖𝑐 + 𝐾𝑖𝑟 film coefficient to be determined from curves in Fig. 7.1; 2

𝐾𝑜 = 12 Btu/(h-ft2-°F); 3

𝐾𝑟 = 𝑇𝑖/120; 4

𝐾𝑠 = 𝑇𝑖/150. 5

R7.4.2.8 The research data available to establish the coefficients of heat transfer through the 6

liner(s) and the chimney wall, especially as they concern the heat transfer from gases to the 7

surfaces and through the ventilated air space between the liner(s) and the chimney wall, is 8

somewhat meager. Unless complete heat balance studies are made for the particular chimney, 9

it is permissible to use the coefficients as determined or stated in this code. 10

These coefficients, when entered into equations for temperature differential through the 11

chimney wall, 𝑇𝑥, will give a value having accuracy in keeping with the basic design 12

assumptions for the structural design of the concrete. More refined values and/or a more refined 13

method may be needed for other purposes, such as accurately calculating the temperature in 14

the annular air space between the liner and the chimney wall. 15

16

49


1

Fig. 7.1 2

7.4.2.9 The value of 𝑟𝑞 = 0.5 shall apply only where: 3

1. The distance between the liner and the chimney wall is not less than 4 in. throughout 4

the entire height of the liner and ventilation air inlet and outlet openings are provided 5

at the bottom and top of the chimney wall, respectively. 6

2. The free area of inlet and outlet openings, in square feet, shall each be numerically 7

equal to two-thirds of the top inside diameter of the chimney wall, in feet. 8

3. Local obstructions in the air space between the liner and the chimney wall shall not 9

restrict the free area of the air space at any horizontal section to less than the free area 10

specified for the inlet and outlet openings. 11

7.5—Wind load 12

7.5.1 General 13

50


7.5.1.1 Reinforced concrete chimneys shall be designed to resist wind loads in both the 1

along-wind and across-wind directions. In addition, vertical cross-sections shall be designed 2

to resist bending due to the distribution of wind pressure on the perimeter. 3

7.5.1.2 The basic 3-second gust design wind speed, 𝑉𝑢, in miles per hour shall be determined 4

in accordance with Section 26.5 of ASCE 7-16 for Risk Category III (1700-year MRI) or Risk 5

Category IV (3000-year MRI) as determined in 7.2.2. 6

R7.5.1.2 Beginning with ASCE 7-10, wind speed maps were provided that are applicable for 7

determining pressures for the strength design approach. The importance factor was eliminated 8

by providing a separate wind speed map for each risk category. The subscripted symbol 𝑉𝑢 is 9

used in this code rather than the symbol 𝑉 used in ASCE 7-16. 10

7.5.1.3 The serviceability wind speed, 𝑉𝑠, in miles per hour shall be the 100-year MRI 11

3-second gust wind speed determined in accordance with Figure CC.2-4 in Appendix CC of 12

the Commentary for ASCE 7-16. 13

R 7.5.1.3 The 2008 code was based on factoring loads due to a serviceability-level wind 14

speed. The basic wind speed, defined in ASCE 7-02, corresponded to an MRI of 50 years. An 15

importance factor of 1.15 adjusted this wind speed to correspond to an MRI of 100 years. 16

7.5.1.4 The exposure category as defined in Section 26.7 of ASCE 7-16, shall be C for 17

chimneys unless exposure D is applicable. 18

R7.5.1.4 Exposure D, absent in previous versions, has been explicitly included in the revised 19

Code. 20

7.5.1.5 At a height 𝑧 in feet above the ground, the mean-hourly design wind speed, �̅�𝑢(𝑧), 21

and the mean-hourly serviceability wind speed, �̅�𝑠(𝑧), in feet per second, shall be computed 22

from Eq. (7-6a) and Eq. (7-6b), respectively. 23

51


�̅�𝑢(𝑧) = (1.47) �̅� 𝑉𝑢 (𝑧

33)�̅�

(7-6a)

�̅�𝑠(𝑧) = (1.47) �̅� 𝑉𝑠 (𝑧

33)�̅�

(7-6b)

The value of �̅� shall be 0.65 for exposure C and 0.80 for exposure D. The value of �̅� shall be 1

0.154 (1/6.5) for exposure C and 0.111 (1/9.0) for exposure D. 2

R7.5.1.5 The constant 1.47 ( = 22/15) converts from miles per hour to feet per second. The 3

values of �̅� and �̅� are from ASCE 7-16, Table 25.11-1. 4

The procedure used in this standard requires that a wind speed averaged over a period of 5

1 hour be used as the basis for design. Equations (7-6a) and (7-6b) permit the mean hourly 6

design wind speed and the mean hourly serviceability wind speed (at height z and in ft/sec) to 7

be determined from the 3-second gust design wind speed and the 3-second gust serviceability 8

wind speed (at 33 ft and in mph), respectively. 9

7.5.1.6 For circular chimneys, wind effects shall be determined in accordance with 7.5.2, 10

7.5.3, and 7.5.4. Alternatively, the wind forces are permitted to be determined by a properly 11

substantiated dynamic analysis or by a wind tunnel study in accordance with ASCE 7-16, 12

Chapter 31. 13

R7.5.1.6 The provisions with respect to along-wind load take into account dynamic action, 14

but are simplified and result in equivalent static loads. The simplified provisions of this 15

standard do not preclude the use of more detailed methods and the results of a full dynamic 16

analysis employing accepted approaches and recognizing the flow profile and turbulence levels 17

at a specific site may be used in place of the standard provisions. 18

7.5.1.7 For non-circular chimneys, wind effects shall be determined in accordance with 19

ASCE 7-16 and shall consider across-wind load effects (including any interference effects). 20

52


Alternatively, the wind forces are permitted to be determined by a properly substantiated 1

dynamic analysis or by a wind tunnel study in accordance with ASCE 7-16, Chapter 31. 2

R7.5.1.7 Non-circular shapes may be more sensitive than circular shapes to across-wind 3

forces and require analyses beyond the scope of this code. 4

7.5.2 Along-wind load for circular chimneys 5

7.5.2.1 The directionality factor, 𝐾𝑑, shall be 1.0 for circular chimneys. 6

R7.5.2.1 The directionality factor for round and octagonal structures was changed from 0.95 7

to 1.0 in ASCE 7-16. 8

7.5.2.2 The topography factor, 𝐾𝑧𝑡, shall be determined according to Section 26.8 of 9

ASCE 7-16. The topography factor is permitted to be taken as 1.0 for all elevations. 10

R7.5.2.2 The topography factor affects only elevations near grade and has a small effect on 11

the overturning moments in this area. 12

7.5.2.3 The ground elevation factor, 𝐾𝑒, shall be determined according to Section 26.9 of 13

ASCE 7-16. The ground elevation factor is permitted to be taken as 1.0 14

R7.5.2.3 Using a ground elevation factor of 1.0 is conservative. 15

7.5.2.4 The total along-wind load, in lb/ft, at height 𝑧, in feet, shall be the sum of the mean 16

load and the fluctuating load. 17

𝑤(𝑧) = �̅�(𝑧) + 𝑤′(𝑧) (7-7)

7.5.2.5 The mean wind load, in lb/ft, at height 𝑧 in feet shall be computed from Eq. (7-8). 18

�̅�(𝑧) = 𝐶𝑑𝑟(𝑧)𝑑(𝑧)�̅�(𝑧) (7-8)

where 19

𝐶𝑑𝑟 = {1.0 for 𝑧 ≥ ℎ − 𝑙𝑤0.65 for 𝑧 < ℎ − 𝑙𝑤

(7-9)

53


𝑙𝑤 = min(1.5𝑑(ℎ), 50 ft) (7-10)

�̅�(𝑧) = 0.00119𝐾𝑑𝐾𝑧𝑡𝐾𝑒 [�̅�𝑢(𝑧)]2 (7-11)

R7.5.2.5 The recommended drag coefficients are consistent with slender chimneys 1

[ℎ/𝑑(ℎ) > 20] with a relative surface roughness on the order of 10-4 to 10-5. Some reduction 2

in the drag coefficient 𝐶𝑑𝑟 with decreasing ℎ/𝑑(ℎ) can be expected, but unusually rough (for 3

example, ribbed) chimneys would have higher values of 𝐶𝑑𝑟. The variations of 𝐶𝑑𝑟 with 4

roughness and aspect ratio are discussed in Basu (1982), and Vickery and Basu (1984). 5

7.5.2.6 The fluctuating wind load, in lb/ft, at height 𝑧 in feet shall be computed from 6

Eq. (7-12). 7

𝑤′(𝑧) = 3.0 𝑧 𝐺𝑤′ 𝑀�̅�(𝑏)

ℎ3 (7-12)

where 8

𝐺𝑤′ = 0.30 +11.0 [𝑇1�̅�𝑢(33)]

0.47

(ℎ + 16)0.86 (7-13)

R7.5.2.6 The total load per unit length is computed as the sum of the mean component �̅�(𝑧) 9

and the fluctuating component 𝑤′(𝑧). The fluctuating component is evaluated using a slightly 10

modified form of the gust factor approaches described by Davenport (1967), Vickery (1969), 11

and Simiu et al. (1977). The base moment due to the fluctuating component is evaluated using 12

the gust factor approach, but the loads producing this moment are approximated by a triangular 13

distribution rather than a distribution matching the mean component distribution. Eq. (7-13) is 14

a simple empirical fit to values of the gust factor, 𝐺𝑤′, at a structural damping of 1.5% of 15

critical. 16

7.5.2.7 The along-wind deflection used for serviceability checks shall be 17

54


𝑦𝑤,𝑠(𝑧) = 𝑦𝑤,𝑢(𝑧) (𝑉𝑠𝑉𝑢)2

(7-14)

where 𝑦𝑤,𝑢(𝑧) is the deflection at elevation z due to the wind pressure distribution 𝑤(𝑧) 1

defined by Eq. (7-7). 2

R7.5.2.7 Deflections are determined using a model with a fixed base and uncracked section 3

properties. The design along-wind pressure, 𝑤(𝑧), is the pressure due to the basic design wind 4

speed 𝑉𝑢. Eq. (7-14) estimates the deflection at the serviceability wind speed. In a wind load 5

deflection study, along-wind deflections computed using Eq. (7-14) were shown to 6

overestimate the along-wind deflection using the serviceability wind speed by about 5% (ACI 7

Committee 307 2020b). 8

7.5.2.8 The maximum deflection at the top of the chimney shall be limited as follows 9

𝑦𝑤,𝑠(ℎ) ≤ℎ

300 (7-15)

R7.5.2.8 ACI 307-88 was the first revision to implement strength design. The 1988 10

Committee felt that deflections under serviceability loads should be checked and that the 11

deflections of chimneys designed by the strength method should not vary greatly from the 12

deflections of then-existing chimneys designed by the working stress method. The 1988 13

committee also noted that limiting the top deflection limits the effect of secondary bending 14

moments. In an along-wind load deflection study of 25 sample chimneys, the top deflection 15

limit of Eq. (7-15) affected only very flexible chimneys and only at very high design wind 16

speeds (ACI Committee 307 2020b). 17

7.5.3 Across-wind load for circular chimneys 18

7.5.3.1 Procedures in this section apply if the outside chimney diameter at ℎ/3 is less than 19

1.6 times the top outside diameter. Across-wind effect for circular chimneys that do not satisfy 20

55


this condition shall be obtained from a properly substantiated dynamic analysis or are permitted 1

to be analyzed using this section. Across-wind effects for circular chimneys that have a flare 2

or strong taper (nozzle) for more than one diameter near the top shall be obtained from a 3

properly substantiated dynamic analysis. 4

R7.5.3.1 A general solution for the across-wind response of circular chimneys with any 5

geometry was developed by Vickery (1993). This procedure, based on Vickery’s general 6

solution, was simplified to some extent, which requires that the application be restricted to 7

certain geometries. 8

Strongly tapered circular chimneys (diameter at h/3 greater than 1.6 times the top outside 9

diameter) develop lower across wind loads than non-tapered or slightly tapered chimneys. 10

Using the method given in 7.5.3 to analyze strongly tapered chimneys gives conservative 11

results. Strongly tapered chimneys may be analyzed using a properly substantiated dynamic 12

analysis to consider the effect of the strong taper. The procedure for determining shedding 13

forces, however, is not materially affected by the configuration of the lower third of the 14

chimney, which may range from plumb to any degree of taper. 15

7.5.3.2 The first mode critical wind speed shall be determined as 16

𝑉𝑐𝑟 =𝑑(𝑢)

(𝑆𝑡)𝑇1

(7-16)

where 17

𝑆𝑡 = 0.25𝐹1𝐴 (7-17)

𝐹1𝐴 = 0.333 + 0.206log𝑒

ℎ

𝑑(𝑢)

(7-18)

The second mode critical wind speed shall be determined as 18

56


𝑉𝑐𝑟 =5𝑑(𝑢)

𝑇2

(7-19)

The across-wind load need not be considered when the critical wind speed exceeds 1.5�̅�𝑢(𝑧𝑐𝑟). 1

R7.5.3.2 The maximum critical wind speed for which the across-wind load must be 2

considered is based on the results of a computational study of 25 sample chimneys (Radecki 3

2014). A lower bound for the range of consideration is not specified since group effects can 4

magnify the response. The natural periods of the chimney may include the effect of foundation 5

and soil interaction. 6

7.5.3.3 The combined across-wind/along-wind moment at mean-hourly wind speed �̅�(𝑧𝑐𝑟) 7

shall be determined using Eq. (7-20). 8

𝑀𝑐𝑤(𝑧) = √[𝑀𝑎(𝑧)]2 + [𝑀𝑙(𝑧)]2 (7-20)

𝑀𝑎(𝑧) is the across-wind moment at height 𝑧 at mean-hourly wind speed �̅�(𝑧𝑐𝑟). 9

𝑀𝑙(𝑧) is the mean along-wind moment at height 𝑧 at mean-hourly wind speed �̅�(𝑧𝑐𝑟). 10

R7.5.3.3 The combination formula, Eq. (7-20) does not include the fluctuating component 11

of the along-wind load. Eq. (7-20), however, is a good approximation for a more accurate, but 12

also more computationally complex formula that includes the fluctuating component. 13

Eq. (7-20) always underestimates the combined load, but the error has an upper bound of about 14

15%. The more accurate formula, based on the across-wind component and the fluctuating 15

along-wind component being uncorrelated random variables having a joint Gaussian 16

distribution, can be found in Vickery and Basu (1984) and Radecki (2014). 17

7.5.3.3.1 The across-wind moment, 𝑀𝑎(𝑧), at height 𝑧 at mean-hourly wind speed 18

�̅�(𝑧𝑐𝑟) shall be determined by scaling the corresponding mode shape such that the base 19

moment is 𝑀𝑎(0) as determined in 7.5.3.5. 20

57


7.5.3.3.2 The mean along-wind moment, 𝑀𝑙(𝑧), at height 𝑧 at mean-hourly wind speed 1

�̅�(𝑧𝑐𝑟) shall be determined using Eq.(7-21). 2

𝑀𝑙(𝑧) = 𝑀�̅�(𝑧) [�̅�(𝑧𝑐𝑟)

�̅�𝑢(𝑧𝑐𝑟)]

2

(7-21)

𝑀�̅�(𝑧) is the moment at elevation 𝑧 due to the mean wind load pressures �̅�(𝑧) from 3

7.5.2.3. 4

7.5.3.4 The maximum combined across-wind/along-wind load (first mode and/or second 5

mode) considered for design shall be the combined across-wind/along-wind load 6

corresponding to the maximum factored combined base moment using the wind speed 7

dependent load factor defined in 7.7.1.2, the across-wind base moment, 𝑀𝑎(0), calculated 8

using Eq. (7-22) and the mean along-wind base moment, 𝑀𝑙(0), calculated using Eq. (7-21). 9

The factored combined across-wind/along-wind load shall be evaluated for mean-hourly wind 10

speeds in the range 0.5 𝑉𝑐𝑟 ≤ �̅�(𝑧𝑐𝑟) ≤ �̅�𝑢(𝑧𝑐𝑟). 11

R7.5.3.4 Since the load factor is not constant, the governing load for design (the maximum 12

factored combined load) will not necessarily be at the same wind speed as the maximum 13

unfactored combined load using Eq. (7-20) in 7.5.3.3. The across-wind load first becomes 14

noticeable at approximately one-half the critical wind speed so wind speeds below that need 15

not be considered. 16

7.5.3.5 The across-wind base moment 𝑀𝑎(0) at wind speed �̅�(𝑧𝑐𝑟) shall be calculated using 17

Eq. (7-22). 18

𝑀𝑎(0) =𝐺

𝑔𝑆𝑠𝑣𝐶𝐿

𝜌𝑎2(𝑉𝑐𝑟)

2𝑑(𝑢)ℎ2 [𝜋

4(𝛽𝑠 + 𝛽𝑎)]

12⁄

× 𝑆𝑝 (2𝐿

ℎ𝑑(𝑢)

+ 𝐶𝐸

)

12⁄

(7-22)

where 19

58


𝐺 = 4.0

𝐿 = 1.20

𝐶𝐸 = 3

𝑆𝑠𝑣 = { 0.57 for 1st mode

0.18 for 2nd mode

𝐶𝐿 = 𝐶𝐿𝑜 𝐹1𝐵 (7-23)

𝐶𝐿𝑜 = −0.243 + 5.648𝑖 − 18.182𝑖2 (7-24)

𝑖 =1

loge

𝑧𝑐𝑟𝑍𝑐

(7-25)

𝑍𝑐 = 0.06 ft

𝐹1𝐵 = −0.089 + 0.337 loge

ℎ

𝑑(𝑢)

but not > 1.0 nor < 0.2

(7-26)

𝛽𝑠 =

{

0.01 for �̅�(𝑧𝑐𝑟) ≤ �̅�𝑠(𝑧𝑐𝑟)

0.01 + 0.015 [�̅�(𝑧𝑐𝑟) − �̅�𝑠(𝑧𝑐𝑟)

�̅�𝑢(𝑧𝑐𝑟) − �̅�𝑠(𝑧𝑐𝑟)]

otherwise

(7-27)

𝛽𝑎 =𝑘𝑎𝜌𝑎𝑑(𝑢)

2

𝑤𝑡(𝑢) (7-28)

𝜌𝑎 = 0.0765

𝑘𝑎 = 𝑘𝑎𝑜𝐹1𝐵 (7-29)

𝑘𝑎𝑜 =−1.0

(1 + 5𝑖) (1 +|𝑘 − 1|𝑖 + 0.10)

(7-30)

𝑘 =�̅�(𝑧𝑐𝑟)

𝑉𝑐𝑟 (7-31)

59


𝑆𝑝 =𝑘1.5

𝐵0.5𝜋0.25exp [−

1

2(1 − 𝑘−1

𝐵)

2

] (7-32)

𝐵 = 0.1 + 2𝑖 (7-33)

R7.5.3.5 The maximum damping ratio for the across-wind load when the wind speed reaches 1

design level has been reduced from 4% of critical to 2.5% of critical. This is consistent with 2

ACSE 7-16, Section C26.11 which states that damping for concrete buildings under ultimate 3

strength design conditions is commonly assumed to be 2.5% to 3%. The damping ratio is 1% 4

of critical for wind speeds up to the serviceability wind speed and then increases linearly with 5

wind speed to 2.5% at the design level wind speed. Eq. (4-10) of 307-08 was an adjustment 6

factor for the across-wind load factor, reducing the load so that a constant load factor could be 7

used. That adjustment is now included in a variable load factor for the across-wind 8

combination. 9

7.5.3.6 For circular chimneys, when two identical chimneys are in close proximity, the lift 10

coefficient 𝐶𝐿 shall be increased to account for potential increase in vortex-induced motions. 11

If 𝑠

𝑑(𝑧𝑐𝑟)≥ 12.75, 12

𝐶𝐿 is unaltered 13

If 3 ≤𝑠

𝑑(𝑧𝑐𝑟)< 12.75, 14

𝐶𝐿 shall be multiplied by 2.275 − 0.10𝑠

𝑑(𝑧𝑐𝑟) 15

where 𝑠 is the center-to-center distance between the two identical chimneys. 16

For chimneys that are not identical, for chimneys that are identical but the spacing ratio 17

𝑠 𝑑(𝑧𝑐𝑟)⁄ is less than 3 and for more than two chimneys in close proximity, the across-wind 18

60


load for each chimney shall be established by reference to model tests, observations, test 1

reports, or analytical models of similar chimney arrangements. 2

R7.5.3.6 Interactions between closely spaced cylindrical objects have been studied in 3

considerable detail, but virtually all of the test results are for subcritical values of Reynolds 4

numbers, and their application to chimneys is highly questionable. Even with the scale effects 5

introduced by the inequality of the Reynolds number, however, the wind tunnel is presently 6

the only tool that will provide guidance as to the likely magnitude of interference effects. A 7

review of interference effects was given by Zdravkokvich (1977). Vickery (1993) attributed 8

the amplification of shedding forces to increased turbulence and additional buffeting effects 9

that formed the basis for revisions made to this section. 10

At center-to-center spacings in excess of two to three diameters, the prime interference effect 11

is related to across-wind excitation due to shedding. The recommendations in this section is 12

based on the results of Vickery and Daly (1984), and were obtained at subcritical values of 13

Reynolds number. The magnification factor is the sum of a first term, 0.26 −14

0.015(𝑠 12𝑑(𝑧𝑐𝑟)⁄ ), to account for buffeting due to vortexes shed by the upstream structure 15

and a second term, 2 − 𝑠 12𝑑(𝑧𝑐𝑟)⁄ , to account for small-scale turbulence. The same reference 16

also contains results for two cylinders of different sizes, with the upstream structure having a 17

diameter 25% greater than the diameter d of the other. In this case, the amplification of the 18

response of the downwind chimney is roughly 0. 4 − 0.2(𝑠 𝑑⁄ ) for 4 < 𝑠 𝑑 < 12⁄ . The 19

amplification of shedding for grouped cylinders has also been noted at full scale (Ruscheweyh 20

1984). Available data for interference effects of non-identical chimneys, though, are not yet 21

sufficient to provide general guidance. 22

61


7.5.3.7 The combined across-wind/along-wind deflection used for serviceability checks shall 1

be 2

𝑦𝑐𝑤,𝑠(𝑧) =𝑦𝑐𝑤,𝑢(𝑧)

1.4 (7-34)

where 𝑦𝑐𝑤,𝑠(𝑧) is the deflection at elevation z due to the moment distribution 𝐿𝐹𝑐𝑤𝑀𝑐𝑤(𝑧) 3

defined by Eqs. (7-54) and (7-20) at the wind speed for which the combined across-wind/along-4

wind base moment 𝐿𝐹𝑐𝑤𝑀𝑐𝑤(0) is a maximum. 5

R7.5.3.7 Deflections are determined using a model with a fixed base and uncracked section 6

properties. Eq. (7-34) reduces the deflection due to strength level forces to serviceability level. 7

7.5.3.8 The maximum deflection at the top of the chimney shall be limited as follows 8

𝑦𝑐𝑤,𝑠(ℎ) ≤ℎ

300 (7-35)

R7.5.3.8 The deflection limit also applies to the combined across-wind/along-wind 9

deflection. See R7.5.2.8. 10

7.5.4 Bending at vertical cross-sections of circular chimneys due to radial wind pressure 11

7.5.4.1 The radial wind pressure, in lb/ft2, at height 𝑧 in feet shall be computed by Eq. (7-36). 12

𝑝𝑟(𝑧) = {1.0 �̅�(𝑧) 𝐺𝑟(𝑧), 𝑧 < ℎ − 𝑙𝑤1.5 �̅�(𝑧) 𝐺𝑟(𝑧), 𝑧 ≥ ℎ − 𝑙𝑤

(7-36)

where

𝐺𝑟(𝑧) = {4.0 – 0.8 log10 𝑧 , 𝑧 > 1.0 ft

4.0, 𝑧 ≤ 1.0 ft

(7-37)

R7.5.4.1 The mean pressure, �̅�(𝑧), and the end effect length, 𝑙𝑤, are defined in 7.5.2.5. The 13

use of a gust factor 𝐺𝑟(𝑧) in this computation is based on the assumption that the mean pressure 14

distribution (when expressed in coefficient form) is also applicable for short-duration gusts. 15

62


The increase in loads near the top is consistent with observations (Okamoto and Tagita 1973) 1

that the drag coefficient increases significantly in this region. 2

7.5.4.2 The maximum circumferential ring bending moment, in ft-lb/ft, at height 𝑧, in feet, 3

due to the radial wind pressure distribution resulting in tension on the inside face, shall be 4

computed by Eq. (7-38). 5

𝑀𝑖(𝑧) = 0.31𝑝𝑟(𝑧)[𝑟(𝑧)]2 (7-38)

R7.5.4.2 The equation for the prediction of the circumferential moments is based on 6

measured pressure distributions (Dryden and Hill 1930; ASCE Task Committee on Wind 7

Forces 1961). Comparable values for the bending moments as obtained from different 8

distributions are given by Rumman (1985). 9

7.5.4.3 The maximum circumferential ring bending moment, in ft-lb/ft, at height 𝑧, in feet, 10

due to the radial wind pressure distribution resulting in tension on the outside face, shall be 11

computed by Eq. (7-39). 12

𝑀𝑜(𝑧) = 0.27𝑝𝑟(𝑧)[𝑟(𝑧)]2 (7-39)

R7.5.4.3 See R7.5.4.2. 13

7.6—Earthquake load 14

7.6.1 General 15

7.6.1.1 Reinforced concrete chimneys shall be designed to resist earthquake loads due to 16

ground motion in accordance with this section. 17

R7.6.1.1 The earthquake requirements have been revised to be consistent with ASCE 7-16, 18

Section 15.6.2.2. Revisions have been made to (1) the procedure for determining the design 19

base shear, (2) seismic coefficient values, and (3) detailing of openings. Specific requirements 20

are explained in the commentary of the code sections where each requirement appears. 21

63


7.6.1.2 Earthquake load on a chimney shall be determined by using the modal response 1

spectrum analysis procedure of 7.6.2, by using a linear response history procedure in 2

accordance with Section 12.9.2 of ASCE 7-16, or by using a nonlinear response history 3

procedure in accordance with Chapter 16 of ASCE 7-16. 4

7.6.2 Modal response spectrum analysis 5

7.6.2.1 The required periods, mode shapes, and participation factors of the chimney shall be 6

calculated by established methods of structural analysis. The analytical model used shall be 7

sufficiently refined to represent variations of mass and stiffness. The analysis shall include a 8

sufficient number of modes to obtain a combined modal mass participation of at least 90% of 9

the actual mass. 10

R7.6.2.1 Significant weights included in the dead load, such as roof structures, platforms and 11

liners should be included in the analytical model. It is generally sufficient to lump such weights 12

at the elevation where they occur. Liners may be included explicitly as structures tied and/or 13

braced to the chimney structure at discrete elevations or as lumped weights. If liners are 14

included as lumped weights, the total weight should be distributed at discrete elevations to 15

approximate both the vertical and horizontal force transfer at that elevation. 16

7.6.2.2 Modal shears, modal moments, and modal deflections shall be determined using an 17

elastic response spectrum procedure. The response spectrum shall provide the design spectral 18

response acceleration, 𝑆𝑎, at any period and shall be obtained by using the general procedure 19

of 7.6.3 or the site-specific procedure of 7.6.4. 20

7.6.2.3 The seismic importance factor 𝐼𝑒 shall be determined from Table 1.5-2 of 21

ASCE 7-16. 22

R7.6.2.3 Per 7.2.2, the Risk Category for a chimney will be either III or IV. 23

64


7.6.2.4 The seismic design category shall be determined from Section 11.6 of ASCE 7-16. 1

7.6.2.5 The response modification factor 𝑅 shall be taken as 2.0. The deflection amplification 2

factor 𝐶𝑑 shall be taken as 2.0. The overstrength factor Ω0 shall be taken as 1.5. 3

R7.6.2.5 The response modification factor has been increased from 1.5 to 2.0 to be consistent 4

with Table 15.4-2 of ASCE 7-16 and recognizing that well detailed chimneys have some 5

ductility (Wilson 2003). The displacement amplification factor and overstrength factor are new 6

and are also consistent with Table 15.4-2 of ASCE 7-16. 7

7.6.2.6 The modal design shears and moments shall be determined by scaling the modal 8

shears and moments due to the design spectrum by the factor 𝐼𝑒/𝑅. The modal design 9

displacements shall be determined by scaling the modal displacements due to the design 10

spectrum by the factor 𝐶𝑑/𝑅. 11

7.6.2.7 The total design shears, moments and displacements shall be calculated from the 12

modal design shears, moments and displacements using either the square root of the sum of 13

the squares (SRSS) method or the complete quadratic combination (CQC) method. 14

7.6.2.8 The vertical seismic load effect is permitted to be neglected. 15

R7.6.2.8 The effect of the vertical component of the earthquake on the chimney has been 16

determined to be of no design significance. Concrete chimneys are laterally flexible with very 17

long fundamental period (typically a few seconds), but far more rigid in the vertical direction. 18

Because the peak vertical response and the peak horizontal response do not occur 19

simultaneously, the peak responses due to horizontal seismic excitation are increased by, at 20

most, a few percent by the effects of vertical seismic excitations. For this reason, the vertical 21

seismic load effect can be ignored. 22

65


Conservatively, the vertical seismic load effect may be included in accordance with 1

ASCE 7-16, Section 12.4.2.2. 2

7.6.2.9 For chimneys of circular cross-section, the horizontal earthquake force shall be 3

assumed to act alone in any direction. 4

7.6.2.10 For chimneys of non-circular cross-section with an assigned Seismic Design 5

Category of C, D, E, or F, orthogonal effects shall be considered by combining, using the SRSS 6

method, the responses due to the design spectrum applied to any two orthogonal directions. 7

7.6.2.11 For chimneys with an assigned Seismic Design Category of D, E or F, overstrength 8

requirements of 7.6.2.12 shall apply. 9

R7.6.2.11 To ensure that inelastic behavior does not initiate at critical cross-sections, such 10

as cross-sections near openings, an overstrength factor is applied so that these critical cross-11

sections remain elastic for the seismic load. 12

7.6.2.12 Where the loss of horizontal cross-section area due to openings is greater than 10%, 13

horizontal cross-sections in the regions of the openings shall be designed for the total design 14

shears and moments determined according to 7.6.2.7 multiplied by the overstrength factor Ω0 15

in accordance with 7.6.2.5. The region where the overstrength factor applies shall extend above 16

and below openings by a distance equal to half the width of the largest opening in the affected 17

region. Appropriate reinforcement development lengths shall be provided beyond the required 18

region of overstrength. 19

7.6.3 Response spectrum - general procedure 20

7.6.3.1 The mapped maximum considered earthquake spectral response acceleration at short 21

periods, 𝑆𝑆, and at 1 second, 𝑆1, shall be obtained from Figs. 22-1 through 22-8 of ASCE 7-16 22

(or electronically via the web sites referenced in ASCE 7-16). 23

66


7.6.3.2 The site class shall be determined from Table 20.3-1 and Section 20.3 of ASCE 7-16. 1

When soil properties are not known in sufficient detail to determine the site class, Class D shall 2

be used. 3

7.6.3.3 The short period site coefficient 𝐹𝑎 shall be obtained from Table 11.4-1 of 4

ASCE 7-16. The long period site coefficient 𝐹𝑣 shall be obtained from Table 11.4-2 of 5

ASCE 7-16. 6

7.6.3.4 The maximum considered earthquake spectral response acceleration parameters for 7

short periods 𝑆𝑀𝑆 and at 1 second, 𝑆𝑀1, adjusted for site class effects, shall be calculated by: 8

𝑆𝑀𝑆 = 𝐹𝑎 𝑆𝑆 (7-40)

𝑆𝑀1 = 𝐹𝑣 𝑆1 (7-41)

7.6.3.5 The design earthquake spectral response acceleration parameter at short periods, SDS, 9

and at 1 second, SD1, shall be calculated by: 10

𝑆𝐷𝑆 = (2

3)𝑆𝑀𝑆 (7-42)

𝑆𝐷1 = (2

3)𝑆𝑀1 (7-43)

7.6.3.6 The design response spectrum curve shall be developed as follows. 11

(a) For periods less than 𝑇0, the design spectral response acceleration 𝑆𝑎 shall be taken as: 12

𝑆𝑎 = 𝑆𝐷𝑆 (0.4 + 0.6 𝑇𝑛𝑇0) (7-44)

where 13

𝑇0 = 0.2 𝑆𝐷1/𝑆𝐷𝑆 (7-45)

(b) For periods greater than or equal to 𝑇0 and less than or equal to 𝑇𝑆, the design spectral 14

response acceleration 𝑆𝑎 shall be taken as: 15

67


𝑆𝑎 = 𝑆𝐷𝑆 (7-46)

where 1

𝑇𝑆 = 𝑆𝐷1/𝑆𝐷𝑆 (7-47)

(c) For periods greater than 𝑇𝑆, and less than or equal to 𝑇𝐿, the design spectral response 2

acceleration 𝑆𝑎 shall be taken as: 3

𝑆𝑎 =𝑆𝐷1𝑇𝑛

(7-48)

where 4

𝑇𝐿 = long-period transition period obtained from Figs. 22-14 through 22-17 of ASCE 7-16. 5

(d) For periods greater than 𝑇𝐿, the design spectral response acceleration 𝑆𝑎 shall be taken 6

as: 7

𝑆𝑎 =𝑆𝐷1𝑇𝐿𝑇𝑛2

(7-49)

7.6.3.7 If the total design base shear, 𝑉𝑡, calculated in accordance with 7.6.2.7 is less than 8

85% of the calculated base shear using the equivalent lateral force procedure, 𝑉𝐸𝐿𝐹 , the total 9

design shears, moments and displacements shall be multiplied by 10

0.85𝑉𝐸𝐿𝐹𝑉𝑡

(7-50)

The base shear value 𝑉𝐸𝐿𝐹 shall not be less than the largest of the following: 11

𝑆𝑎1𝐼𝑒𝑅𝑊𝑒

0.044𝑆𝐷𝑆𝐼𝑒𝑊𝑒

0.01𝑊𝑒

(7-51)

68


where 𝑆𝑎1 is the design spectral acceleration, 𝑆𝑎, for the first mode of vibration when 𝑇1 is 1

greater than or equal to 𝑇0. When 𝑇1 is less than 𝑇0, 𝑆𝑎1 = 𝑆𝐷𝑆. In addition, if 𝑆1 ≥ 0.6𝑔, 𝑉𝐸𝐿𝐹 2

shall not be less than 3

0.5𝑆1𝐼𝑒𝑅𝑊𝑒 (7-52)

R7.6.3.7 The requirement that the design base shear be at least equal to 85% of the base 4

shear computed for the equivalent lateral force procedure has been added for consistency with 5

ASCE 7-16. Per Section 15.4.4 of ASCE 7-16, the fundamental period used to determine VELF 6

shall be determined using the structural properties of the resisting elements in a properly 7

substantiated analysis. The requirement of Section 12.8.2 of ASCE 7-16 that the fundamental 8

period not exceed the product CuTa does not apply to chimneys. 9

7.6.4 Response spectrum - site-specific procedure 10

7.6.4.1 A site-specific response spectrum, developed in accordance with the site-specific 11

ground motion procedures of Chapter 21 of ASCE 7-16, shall be permitted to be used in lieu 12

of the general procedure of 7.6.3. 13

7.6.5 Soil-structure interaction 14

7.6.5.1 The effect of seismic interaction between a chimney and soil is permitted to be 15

ignored and a fixed base condition assumed. 16

7.6.5.2 When a soil-structure interaction assessment is desired, the procedure given in 17

Chapter 19 of ASCE 7-16 shall be followed or foundation flexibility shall be modeled in 18

accordance with Section 12.13.3 of ASCE 7-16. 19

7.6.6 P-Δ effect 20

69


7.6.6.1 The P-Δ effect between vertical loads and seismic lateral displacement shall be 1

considered for chimneys with an assigned Seismic Design Category of D, E, or F. The 2

maximum design displacements shall be used to determine the P-Δ effect. 3

7.6.7 Lateral clearance between chimney wall and liner 4

7.6.7.1 The relative displacement between the chimney and liners shall be considered. The 5

total design displacement of the liner shall be determined using values of 𝑅 and 𝐶𝑑 appropriate 6

for the lining material and configuration. 7

R7.6.7.1 Clearances should be maintained between the chimney and liners. Computation 8

of lateral displacements for the chimney and liners should take into account inelastic behavior 9

by using values of 𝑅 and 𝐶𝑑 appropriate for each structure. For brick liners, typical values of 10

𝑅 and 𝐶𝑑 are 1.25 and 1.25, respectively. For steel liners, typical values of 𝑅 and 𝐶𝑑 are 3 11

and 3, respectively. For other materials, values for 𝑅 and 𝐶𝑑 should be obtained from 12

documented studies or from testing. 13

7.7—Load combinations 14

7.7.1 Horizontal cross-sections 15

7.7.1.1 The design strength of horizontal cross-sections of the chimney shall equal or exceed 16

the effects of the factored loads in the following combinations: 17

1.4 D (7-53a)

1.2 D + 1.2 T + 1.0 Walong (7-53b)

1.2 D + 1.2 T + 𝐿𝐹𝑐𝑤 Wcomb (7-53c)

1.2 D + 1.2 T + 1.0 E (7-53d)

0.9 D + 1.2 T + 1.0 Walong (7-53e)

0.9 D + 1.2 T +𝐿𝐹𝑐𝑤 Wcomb (7-53f)

70


0.9 D + 1.2 T + 1.0 E (7-53g)

7.7.1.2 The load factor, 𝐿𝐹𝑐𝑤, for combined along-wind load and across-wind load at mean-1

hourly wind speed, �̅�(𝑧𝑐𝑟), shall be 2

𝐿𝐹𝑐𝑤 =

{

1.4 for �̅�(𝑧𝑐𝑟) ≤ �̅�𝑠(𝑧𝑐𝑟)

1.4 − 0.4 [�̅�(𝑧𝑐𝑟) − �̅�𝑠(𝑧𝑐𝑟)

�̅�𝑢(𝑧𝑐𝑟) − �̅�𝑠(𝑧𝑐𝑟)]

for �̅�𝑠(𝑧𝑐𝑟) < �̅�(𝑧𝑐𝑟) ≤ �̅�𝑢(𝑧𝑐𝑟)

(7-54)

R7.7.1.2 A wind load factor of 1.0 is not appropriate when the governing wind speed for the 3

combined along-wind and across-wind load is less than the strength level design wind speed. 4

Eq. (4-10) of 307-08 implicitly modified the load factor for the combined wind load. This 5

variable load factor for combined wind explicitly provides a load factor appropriate for the 6

governing wind speed. 7

7.7.2 Vertical cross-sections 8

7.7.2.1 The design strength of vertical cross-sections shall equal or exceed the effects of the 9

factored loads in the following combinations: 10

1.2 T + 1.0 W (7-55)

R7.7.2.1 The wind load factor for forces on horizontal and vertical cross-sections in the 11

previous version were 1.6 and 1.4, respectively. In this Code, both wind load factors are 1.0 12

for strength level loads. Eq. (7-55) is therefore about 15% more conservative than the 13

corresponding equation in the previous version. 14

15

71


CHAPTER 8—DESIGN STRENGTH OF HORIZONTAL CROSS-SECTIONS 1

8.1—Scope 2

8.1.1 This chapter shall apply to the calculation of the combined flexural and axial strength 3

of horizontal chimney cross-sections. 4

R8.1.1 Detailed equations for calculating the combined flexural and axial strength have been 5

moved to Appendix A which details the calculation using the modified rectangular stress block 6

approaCh. Appendix B details the calculation of strength using the stress-strain relationship 7

that was used to develop the modified rectangular stress block approach of Appendix A. 8

8.2—General 9

8.2.1 Except as noted in 8.3, the nominal strength at a horizontal cross-section of a chimney 10

having circular cross-sections is permitted to be calculated assuming that the strain in concrete 11

and reinforcement at any point on the section is directly proportional to its distance from the 12

neutral axis. For chimneys having noncircular cross-sections the assumption of linear strain on 13

the cross-section, if used, shall be justified by proper engineering analysis or judgement. 14

R8.2.1 The limits in 8.3 assure that (1) local wall buckling will not precede concrete 15

crushing, that (2) the edges of openings, acting as columns, will not be governed by slenderness 16

effects and that (3) a linear strain distribution on the cross-section can be assumed. To justify 17

the linear strain assumption for chimneys having noncircular cross-sections, similar issues 18

must be addressed. 19

8.2.2 In calculating forces in the reinforcing for a chimney having circular cross-sections, 20

the total reinforcement area at a cross-section is permitted to be considered an annular area, 21

centered on the middle of the concrete wall, having thickness 𝜌𝑡𝑡𝑐. For chimneys having 22

72


noncircular cross-sections, a similar assumption is permitted to be used if justified by proper 1

engineering judgement. 2

R8.2.2 Most chimneys have (relatively) thin walls with reinforcing bars (relatively) closely 3

spaced near the inside and outside faces. The force calculation error in assuming the reinforcing 4

is uniformly distributed along the middle of the concrete wall is negligible. 5

8.2.3 The maximum compressive strain in the concrete shall be assumed equal to 0.003. The 6

maximum tensile strain in the reinforcement shall be assumed equal to 0.06. Whichever value 7

is reached first shall be taken as the limiting value. 8

R8.2.3 The maximum tensile strain in the reinforcing steel is assumed to be a fracture limit 9

of 0.06. The strain limit is consistent with maximum elongation properties in tension of 10

reinforcing steel. The tensile strain limit has been reduced from 0.07 to 0.06 so that 11

ASTM 615/615M Grade 80 reinforcing bars are acceptable. The change in bending moment 12

capacity due to the lower tensile strain limit is negligible. If this fracture limit is reached before 13

the maximum concrete strain of 0.003 is reached, the maximum concrete strain must be 14

reduced. This deviates from the design assumptions of ACI 318. For a given vertical steel ratio, 15

this may occur when the ratio of the vertical load to the moment is below a certain value. A 16

total vertical steel ratio in the chimney cross-section less than that per the minimum 17

requirement of ACI 318 for flexural members is permitted. 18

8.2.4 Tensile strength of concrete shall be neglected. 19

8.2.5 The stress-strain relationship for concrete in compression shall be assumed to be in 20

accordance with Fig. 8.1. For chimneys having circular cross-sections and meeting the 21

requirements of 8.3, the modified equivalent rectangular concrete stress block relationship of 22

Fig. 8.2 is permitted to be assumed. The modification factor, Q, is defined in Appendix A. 23

73


1

2

Curve A: 0.85𝑓𝑐′ [2휀𝑐0.002

− (휀𝑐

0.002)2

] 3

Line B: 0.85𝑓𝑐′(1.30 − 150휀𝑐) 4

Fig. 8.1: Concrete stress-strain relationship 5

6

Fig. 8.2: Modified equivalent rectangular stress block 7

R8.2.5 The concrete stress-strain relationship of Fig. 8.1 is permitted to be used for any 8

cross-section. Fig. 8.2 is the equivalent rectangular concrete stress block relationship of 9

ACI 318-19, Section 22.2.2.4, modified by the factor Q for circular chimney cross-sections. 10

The factor Q has been determined by fitting the results of using the relationship shown in 11

74


Fig. 8.1 to the equivalent stress block method. The equations in Appendix B use the 1

relationship shown in Fig. 8.1. 2

8.2.6 For strain in the reinforcement less than 𝑓𝑦/𝐸𝑠, stress shall be taken as 𝐸𝑠 times the 3

strain. For strain in the reinforcement greater than 𝑓𝑦/𝐸𝑠, stress shall be assumed equal to 𝑓𝑦 . 4

R8.2.6 Reinforcing steel is assumed to follow an elastic-perfectly plastic stress-strain 5

relationship. 6

8.2.7 For chimneys having circular cross-sections and meeting the design limits of 8.3, forces 7

in the reinforcement are permitted to be calculated assuming the reinforcement is uniformly 8

distributed around the entire circumference of the section, including the opening areas. For 9

chimneys having noncircular cross-sections, a similar assumption is permitted to be used if 10

justified by proper engineering judgement. 11

R8.2.7 Reinforcing bars interrupted by openings are required to be replaced at the sides of 12

the openings. For simplicity in calculating forces, it is permitted to assume that these bars are 13

uniformly distributed around the entire circumference. The error in assuming the reinforcement 14

is so located is small. This provision and the requirement of 8.2.4 essentially means that 15

openings in the tension zone are ignored and that in the calculation of forces in the compression 16

reinforcement only, openings in the compression zone are ignored. 17

18


8.3.1 General 20

8.3.1.1 For chimneys having circular cross-sections, the assumption of 8.2.1 is permitted to 21

be used to compute the nominal moment strength of horizontal cross-sections only if the 22

requirements of 8.3.2 through 8.3.4 are satisfied. If any of the requirements are not satisfied, 23

75


the nominal moment strength shall be computed taking into account a non-linear strain 1

distribution, local stability of the wall, and edge stability at openings, as applicable. For 2

chimneys having noncircular cross-sections, the assumption of 8.2.1 is permitted to be used to 3

compute the nominal moment strength of horizontal cross-sections only if the assumption has 4

been justified, local stability of the wall has been taken into account, and edge stability at 5

openings has been taken into account. 6

8.3.2 Minimum wall thickness 7

8.3.2.1 For cast-in-place chimneys having circular cross-sections, the minimum wall 8

thickness at any section with an inside diameter of 28 ft. or less shall be 8 in. For precast 9

chimneys having circular cross-sections, the minimum wall thickness at any section with an 10

inside diameter of 28 ft. or less shall be 7 in. When the inside diameter of a section exceeds 11

28 ft., the minimum wall thickness shall be increased 1/8 in. for every 1 ft. increase in inside 12

diameter. 13

R8.3.2.1 The minimum wall thickness for circular cross-sections ensures local buckling does 14

not occur and that linear strain variation on the cross-section can be assumed. 15

8.3.2.2 For chimneys having noncircular cross-sections and for circular chimneys not 16

calculated using the assumption of 8.2.1, minimum wall thickness shall be determined taking 17

into account strain distribution and local wall stability, but shall not be less than 8 in for cast-18

in-place chimneys or 7 in for precast chimneys. 19

8.3.3 Minimum wall thickness at openings 20

8.3.3.1 The wall thickness at or near openings shall be as specified in 10.3. This requirement 21

applies to all chimneys. 22

76


R8.3.3.1 The regions beside openings (jamb areas) are considered columns when they are in 1

the compression zone. This requirement assures that slenderness effects do not limit the 2

column strength. 3

8.3.4 Maximum opening width 4

8.3.4.1 For chimneys having circular cross-sections, the half-angle of an opening, 𝛽, as 5

shown in Fig. 8.3, shall not exceed 𝜋 6⁄ (30 degrees). 6

7

8

Fig. 8.3: Opening half-angle 9

R8.3.4.1 At circular cross-sections that include openings, the strain is not linear, with strains 10

increasing rapidly near the edge of the opening especially near the corners. However, 11

experience has shown that, for cross-sections having openings within this limitation, the error 12

in computing strength using the linear strain assumption is not significant. 13

8.3.4.2 For chimneys having noncircular cross-sections, the regions near openings of 14

significant width shall be investigated for local strains. A significant width opening shall 15

include, but is not limited to, an opening width greater than 15% of the cross-section perimeter. 16

R8.3.4.2 Judgement must be used as to what constitutes “significant” width. Openings for 17

breeching ducts and construction openings would generally be considered significant. 18

77


8.3.5 Multiple openings at a section 1

8.3.5.1 The width of the concrete wall between two openings shall be sufficient to place all 2

of the reinforcing steel required for both openings by 10.4 and 8.5.3.2 in accordance with the 3

spacing and clearance requirements of this code and ACI 318-19. 4

8.3.5.2 For a section with multiple openings, if a wall segment between openings exceeds 5

the slenderness limits for compression members in ACI 318-19, this segment shall be 6

additionally investigated as a beam-column. The wall segment is permitted to be assumed a 7

column of rectangular section for the additional investigation. The investigation shall consider 8

all loads applied to the segment. The design strength of wall segments between openings shall 9

be calculated in accordance with the applicable sections of ACI 318-19, Chapter 6. 10

R8.3.5.2 Wall segments between openings should be checked as columns that may be 11

governed by slenderness effects. External loads include loads such as support bracket reactions 12

and embedded plate reactions. 13


8.4.1 Required strength shall be calculated in accordance with the load combinations in 7.7.1. 15

The temperature gradient effect 𝑇𝑥 of a load combination shall be included in accordance with 16

8.5.2. 17

8.4.2 The factored vertical load 𝑃𝑢 and the factored bending moment 𝑀𝑢 occurring 18

simultaneously at horizontal cross-sections for each applicable load combination shall be 19

considered. 20


8.5.1 General 22

78


8.5.1.1 For each applicable load combination, (𝑃𝑢 , 𝑀𝑢), the design strength at horizontal 1

cross-sections shall satisfy (a) and (b). Interaction between load effects shall be considered. 2

(a) 𝜙𝑃𝑛 ≥ 𝑃𝑢 3

(b) 𝜙𝑀𝑛 ≥ 𝑀𝑢 4

In addition, for chimneys with an assigned Seismic Design Category of D, E or F, the design 5

moment capacity 𝜙𝑀𝑛 shall satisfy the requirement of 8.6.4. 6

8.5.1.2 The strength reduction factor 𝜙 shall be 0.80 for horizontal cross-sections. 7

R8.5.1.2 Horizontal chimney sections are tension controlled unless the stress due to the 8

factored axial load, 𝑃𝑢, exceeds about 0.25𝑓𝑐′ which would be extremely rare for a chimney 9

subject to axial load that is mainly self-weight. Therefore, a tension controlled section is 10

assumed and a constant strength reduction factor is specified. 11

8.5.2 Temperature gradient effect 12

8.5.2.1 The temperature gradient across the wall, computed in 7.4, shall be taken into account 13

by reducing the concrete strength 𝑓𝑐′ and the reinforcement yield strength 𝑓𝑦 by the amount of 14

concrete stress and reinforcement stress required to balance the thermal strain. 15

R8.5.2.1 The temperature gradient through the concrete chimney wall reduces the nominal 16

strength of the chimney section. This effect is accounted for by reducing the specified 17

compressive concrete strength and the specified steel yield strength. The derivation of 18

equations for the temperature gradient is included in Appendix C. 19

8.5.2.2 The maximum vertical stress in the concrete, in psi, occurring at the inside of the 20

chimney wall due to a temperature difference shall be computed by Eq. (8-1). The maximum 21

vertical stress in the steel, in psi, occurring in the inside reinforcing layer of the chimney wall 22

due to temperature gradient shall be computed by Eq. (8-2). The maximum vertical stress in 23

79


the steel, in psi, occurring in the outside reinforcing layer of the chimney wall due to 1

temperature shall be computed by Eq. (8-3). 2

𝑓𝐶𝑇𝑉′′ = 𝛼𝑡𝑒 𝑐 𝑇𝑥 𝐸𝑐 (8-1)

𝑓𝑆𝑇𝑉′′ = 𝛼𝑡𝑒 (𝑐 − 1 + 𝛾2𝑖) 𝑇𝑥 𝐸𝑠 (8-2)

𝑓𝑆𝑇𝑉 = 𝛼𝑡𝑒 (𝛾2𝑜 − 𝑐) 𝑇𝑥 𝐸𝑠 (8-3)

where

𝑐 = −𝜌𝑜𝑛(𝛾1 + 1) + √[𝜌𝑜𝑛(𝛾1 + 1)]2 + 2𝜌𝑜𝑛[𝛾2𝑜 + 𝛾1(1 − 𝛾2𝑖)]

(8-4)

R8.5.2.2 The derivation of equations for vertical stresses due to temperature gradient is 3

included in Appendix D. 4

8.5.2.3 For load combinations with temperature effects, replace 𝑓𝑦 with 𝑓𝑦′(𝑣) using Eq. (8-5) 5

and replace 𝑓𝑐′ with 𝑓𝑐

′′(𝑣) using (8-6). 6

𝑓𝑦′(𝑣) = 𝑓𝑦 −

1.2

1 + 𝛾1(𝑓𝑆𝑇𝑉 − 𝛾1𝑓𝑆𝑇𝑉

′′ ) (8-5)

𝑓𝑐′′(𝑣) = 𝑓𝑐

′ − 1.2𝑓𝐶𝑇𝑉′′ (8-6)

R8.5.2.3 The derivation of equations for modified vertical concrete compressive strength and 7

modified vertical steel yield strength due to temperature gradient is included in Appendix D. 8

8.5.3 Nominal moment strength 9

8.5.3.1 For a cross-section having a factored vertical load 𝑃𝑢 and a factored bending moment 10

𝑀𝑢, the iterative procedure of 8.5.3.3 is permitted to be used to determine the total vertical 11

steel ratio 𝜌𝑡 needed to provide the required design strengths, 𝜙𝑃𝑛 and 𝜙𝑀𝑛. 12

8.5.3.2 For chimneys having circular cross-sections that do not exceed the design limits of 13

8.3, either the procedure in Appendix A using the modified equivalent rectangular stress block 14

relationship of Fig. 8.2 or the procedure in Appendix B using the concrete stress-strain 15

80


relationship of Fig. 8.1 is permitted to be used. Where more than two openings occur at the 1

same elevation, appropriate design methods consistent with the cases shown in Appendix A or 2

Appendix B shall be used. 3

R8.5.3.2 The procedure in Appendix A includes a modifying factor Q for the rectangular 4

stress block that is defined to make the results consistent with the results of the procedure in 5

Appendix B. 6

7

8

Fig. 8.4: Stress Resultants 9

8.5.3.3 Given the factored vertical load 𝑃𝑢 and the factored bending moment 𝑀𝑢, the 10

following procedure is permitted to be used to determine the steel ratio required. 11

1) Assume a value for the total vertical steel ratio 𝜌𝑡; 12

81


2) If temperature effects are to be taken into account, determine the concrete and steel thermal 1

stresses due to 𝑇𝑥 and compute the modified concrete compressive strength 𝑓𝑐′′(𝑣) and the 2

modified steel yield strength 𝑓𝑦′(𝑣); 3

3) Determine the nominal vertical load strength required; 4

𝑃𝑛 =𝑃𝑢𝜙

5

4) By trial and error find the value of 𝛼 (see Fig. 8.4) for which force equilibrium is satisfied; 6

𝑃𝑛 = 𝐹𝑐 + 𝑆𝑐 + 𝑆𝑐𝑦 − 𝑆𝑡 − 𝑆𝑡𝑦 7

5) Substitute this value of 𝛼 into the equation for moment equilibrium to obtain the nominal 8

moment strength 𝑀𝑛 ; 9

𝑀𝑛 = 𝑃𝑛𝑟 cos𝛼 + 𝐹𝑐′ + 𝑆𝑐

′ +𝑆𝑐𝑦′ + 𝑆𝑡

′ + 𝑆𝑡𝑦′ 10

6) If 𝜙𝑀𝑛 < 𝑀𝑢 , increase 𝜌𝑡; 11

If 𝜙𝑀𝑛 > 𝑀𝑢 , decreasing 𝜌𝑡 is permitted 12

7) If 𝜌𝑡 is changed, repeat 2 through 5; 13

R8.5.3.3 The procedure determines a nominal axial and flexural strength pair (𝑃𝑛 ,𝑀𝑛) for a 14

given reinforcing steel ratio. By iterating, the required reinforcing steel ratio can be minimized. 15

The symbols in the equilibrium equations represent the stress resultants (axial forces and 16

bending moments) for the concrete, reinforcing steel in the compression zone, and reinforcing 17

steel in the tension zone. 18


8.6.1 Two layers of vertical reinforcement shall be provided. 20

8.6.2 The total vertical reinforcement shall be not less than 0.25% of the concrete area. 21

8.6.3 The outside vertical reinforcement shall not be less than 50% of the total vertical 22

reinforcement. 23

82


8.6.4 For chimneys with an assigned Seismic Design Category of D, E or F, there shall be 1

sufficient vertical reinforcement such that the design moment strength 𝜙𝑀𝑛 exceeds the 2

cracking moment 𝑀𝑐𝑟 at heights from the base to 80% of the chimney height. The cracking 3

moment at any height shall be the bending moment which, when acting concurrently with the 4

service-level vertical load at the same height, produces a maximum concrete tensile stress at 5

the middle of the wall equal to the stress due to the service-level vertical load plus the modulus 6

of rupture 𝑓𝑟 (see Appendix E). 7

R 8.6.4 This is to ensure that the chimney has sufficient ductility so that multiple hinges can 8

form in the chimney (Wilson 2003). Openings and reinforcing steel are disregarded when 9

computing the cracking moment. The middle of the wall location is used, conservatively, for 10

computational simplicity. 11


8.7.1 Vertical reinforcing bars shall not be smaller than No. 4 bars. 13

8.7.2 Outside face vertical reinforcing bars shall not be spaced more than 12 in. on centers. 14

8.7.3 Inside face vertical reinforcing bars shall not be spaced more than 24 in. on centers. 15

8.7.5 For chimneys with an assigned Seismic Design Category of A, B, or C, not more than 16

50% of the vertical bars shall be spliced along any horizontal plane unless specifically 17

permitted and approved by the licensed design professional. To meet this requirement, the 18

centerline of vertical splices shall be staggered by a distance equal to at least the lap splice 19

length. 20

8.7.6 For chimneys with an assigned Seismic Design Category of D, E, or F, not more than 21

50% of the vertical bars shall be spliced along any horizontal plane. To meet this requirement, 22

83


the centerline of vertical splices shall be staggered by a distance equal to at least the lap splice 1

length plus the development length. 2

8.7.7 Tie bars shall be provided between the inner and outer face reinforcement at the top of 3

the chimney wall as indicated in Fig. 8.5. Tie bars shall be a minimum of No. 3 bars and shall 4

not exceed a spacing of 12 in. The length of the tie bar legs shall be at least the development 5

length of the tie bar. 6

7

Fig. 8.5 8

84


CHAPTER 9—DESIGN STRENGTH OF VERTICAL CROSS-SECTIONS 1

FOR CIRCUMFERENTIAL RING MOMENTS 2

9.1—Scope 3

9.1.1 This chapter shall apply to the calculation of the flexural strength of vertical chimney 4

cross-sections subject to circumferential ring moments. 5

9.2—General 6

9.2.1 Any horizontal strip of the concrete wall is permitted to be considered a horizontal 7

beam (of rectangular cross-section) resisting circumferential ring moments. 8


9.3.1 The minimum wall thickness shall be as specified in 8.3.2 to 8.3.5. 10


9.4.1 Required strength shall be calculated in accordance with the factored load combination 12

in 7.7.2. 13

9.4.2 Moments resulting in compression at the outside face and moments resulting in 14

compression at the inside face shall be considered. 15


9.5.1 General 17

9.5.1.1 For each applicable load combination, design strength at all vertical sections shall 18

satisfy the following 19

𝜙𝑀𝑛 ≥ 𝑀𝑢 20

9.5.1.2 The strength reduction factor 𝜙 shall be 0.9 for vertical sections. 21

9.5.2 Temperature effect 22

85


9.5.2.1 The temperature gradient across the wall, computed in 7.4, shall be taken into account 1

by reducing the concrete strength 𝑓𝑐′ and the reinforcement yield strength 𝑓𝑦 by the amount of 2

concrete stress and reinforcement stress required to balance the thermal strain. 3

R9.5.2.1 The temperature gradient through the concrete chimney wall reduces the nominal 4

strength of the chimney section. This effect is accounted for by reducing the specified 5

compressive concrete strength and the specified steel yield strength. The derivation of 6

equations for the temperature gradient is included in Appendix C. 7

9.5.2.2 The maximum circumferential stress in concrete, in psi, occurring at the inside face 8

of the concrete wall due to a temperature differential shall be computed by Eq. (9-1). The 9

maximum stress in the outside circumferential reinforcement shall be computed by Eq. (9-2). 10

𝑓𝐶𝑇𝐶′′ = 𝛼𝑡𝑒 𝑐

′ 𝑇𝑥 𝐸𝑐 (9-1)

𝑓𝑆𝑇𝐶 = 𝛼𝑡𝑒 (𝛾2𝑜′ − 𝑐′) 𝑇𝑥 𝐸𝑠 (9-2)

where

𝑐′ = −𝜌𝑜′𝑛(𝛾1

′ + 1) + √[𝜌𝑜′𝑛(𝛾1′ + 1)]2 + 2𝜌𝑜′𝑛[𝛾2𝑜

′ + 𝛾1′(1 − 𝛾2𝑖

′ )] (9-3)

R9.5.2.2 The derivation of equations for circumferential stresses due to temperature gradient 11

is included in Appendix D. 12

9.5.2.3 For load combinations with temperature effects, replace 𝑓𝑦 with 𝑓𝑦′(𝑐) using Eq. (9-4) 13

and replace 𝑓𝑐′ with 𝑓𝑐

′′(𝑐) using Eq. (9-5). 14

𝑓𝑦′(𝑐) = 𝑓𝑦 − 1.2𝑓𝑆𝑇𝐶 (9-4)

𝑓𝑐′′(𝑐) = 𝑓𝑐

′ − 1.2𝑓𝐶𝑇𝐶′′ (9-5)

R9.5.2.3 The derivation of equations for modified circumferential concrete compressive 15

strength and modified circumferential steel yield strength due to temperature gradient is 16

included in Appendix D. 17

86


9.5.3 Nominal moment strength 1

9.5.3.1 Moment strength of a horizontal strip shall be calculated using the rectangular stress 2

block method of ACI 318. 3

9.5.3.2 Reinforcing bars near the compression edge of the concrete wall are permitted to be 4

neglected. 5


9.6.1 Two layers of circumferential reinforcement shall be provided. 7

9.6.2 The total circumferential reinforcement shall not be less than 0.20% of the concrete 8

area. 9

9.6.3 The circumferential reinforcement in each face shall not be less than 0.10% of the 10

concrete area at the section. 11

9.6.4 The circumferential reinforcement for a distance of 0.2𝑑(ℎ) from the top of the 12

chimney or 7.5 ft., whichever is greater, shall be at least equal to the amount required by 9.5, 13

but shall not be less than 0.20% of the total concrete area in each face. 14


9.7.1 Circumferential reinforcement shall be placed around the exterior of, and secured to, 16

the vertical reinforcement bars in accordance with 12.8.1. 17

9.7.2 Circumferential reinforcing bars shall not be smaller than No. 3 bars. 18

9.7.3 Spacing of outer face circumferential reinforcement shall not exceed the wall thickness 19

or 12 in., whichever is smaller. 20

9.7.4 Spacing of inner face circumferential reinforcement shall not exceed 12 in. 21

9.7.5 Spacing of the circumferential reinforcement near the top of the chimney, as specified 22

by 9.6.4, shall not exceed one-half the wall thickness or 6 in., whichever is smaller. 23

87


9.7.6 Circumferential lap splice length shall meet the requirements for a Class B lap splice 1

length of deformed bars in tension in accordance with Section 25.5 of ACI 318-19. 2

9.7.7 Not more than 50% of the circumferential bars shall be spliced along any vertical plane 3

unless specifically permitted and approved by the licensed design professional. To meet this 4

requirement, the centerline of alternate circumferential splices shall be staggered by a distance 5

equal to at least the lap splice length. 6

7

88


CHAPTER 10—OPENING DETAILS 1

10.1—Scope 2

10.1.1 This chapter shall apply to the detailing around openings, including: 3

(a) Normal detailing around openings 4

(b) Seismic detailing of jamb areas 5

10.2—General 6

10.2.1 In addition to the minimum wall thickness requirements of 8.3.2, the wall thickness 7

near openings shall meet the minimum requirements as specified in 10.3. 8

10.2.2 In addition to the reinforcement determined by design, additional reinforcement shall 9

be provided at the sides, top, bottom and corners of openings as specified in 10.4 to 10.6. 10

10.2.3 For chimneys with an assigned Seismic Design Category D, E or F, at openings where 11

the loss of cross-sectional area is greater than 10%, detailing of the jamb region shall be 12

provided as specified in 10.7. 13

10.3—Minimum wall thickness at openings 14

10.3.1 The chimney wall thickness, at any opening, shall not be less than 1/24 times the 15

height of the opening. This thickness shall extend over a vertical distance from one-half the 16

height of the opening below the sill of the opening to one-half the height of the opening above 17

the top of the opening. 18

10.3.2 Properly designed buttresses or other means of lateral (radial) edge restraint are 19

permitted in place of the requirements of 10.3.1. However, the buttresses or other means of 20

lateral edge restraint shall not be included when calculating vertical strength. Buttresses shall 21

extend over a vertical distance from one-half the height of the opening below the sill of the 22

89


opening to one-half the height of the opening above the top of the opening. Other means of 1

lateral edge restraint shall be justified by a detailed analysis. 2

10.4—Vertical reinforcement at openings 3

10.4.1 At each side of each opening, the additional vertical reinforcement shall have an area 4

at least equal to one-half the design vertical steel reinforcement interrupted by the opening. 5

R10.4.1 This reinforcement is based on the design reinforcement ratio which is assumed to 6

be uniformly distributed around the entire circumference. In that regard, it is not technically 7

additional reinforcement as it is required to be placed at the sides of the openings. 8

10.4.2 The additional vertical reinforcement shall be placed as close as practical to the edge 9

of the opening, within a distance not exceeding three times the wall thickness unless otherwise 10

determined by a detailed analysis. 11

10.4.3 The additional vertical reinforcement shall extend beyond the top and bottom edges 12

of the opening a sufficient distance to develop the bars in tension. 13

10.4.4 Horizontal tie bars shall be provided at the vertical edges of each opening as indicated 14

in Fig. 10.1. The tie bar size shall be No. 3 or larger and the tie bar vertical spacing shall not 15

exceed 12 in. or the wall thickness, whichever is smaller. The length of the tie bar legs shall be 16

at least the development length of the tie bar. 17

10.4.5 If any additional vertical reinforcement is placed in intermediate layer(s) as indicated 18

in Fig. 10.1, these bars shall be secured in position by horizontal ties bars or other means. 19

20

90


1

Fig. 10.1 2

3

10.5—Circumferential reinforcement at openings 4

10.5.1 At the top and bottom of each opening, the additional reinforcement shall have an area 5

at least equal to the one-half the design circumferential reinforcement interrupted by the 6

opening or the area determined by Eq. (10-1), whichever is greater. 7

𝐴𝑠 =0.06𝑓𝑐

′𝑡𝑐𝑏𝑜𝑓𝑦

(10-1)

The requirement of Eq. (10-1) is permitted to be replaced with results of a proper, detailed 8

analysis. 9

10.5.2 The additional circumferential reinforcement shall be placed as close as practical to 10

the top or bottom of the opening, within a distance not exceeding three times the wall thickness 11

unless otherwise determined by a detailed analysis. 12

10.5.3 One-half the area of the additional circumferential reinforcement shall extend 13

completely around the circumference of the chimney. The other half of the additional 14

reinforcement shall extend beyond the vertical edges of the opening a sufficient distance to 15

develop the bars in tension. 16

91


10.5.4 Tie bars shall be provided at the top and bottom edges of each opening as indicated in 1

Fig. 10.2. The tie bar size shall be No. 3 or larger and the tie bar horizontal spacing shall not 2

exceed 12 in. The length of the tie bar legs shall be at least the development length of the tie 3

bar. 4

5

6

Fig. 10.2 7

8

92


10.6—Corner reinforcement at openings 1

10.6.1 For openings wider than 2 ft., the area of diagonal reinforcement provided at each 2

corner of the opening shall be equal, in square inches, to 0.10 times the wall thickness, in 3

inches. 4

10.6.2 As an alternative to diagonal corner reinforcement, orthogonal (horizontal and 5

vertical) reinforcement is permitted to be provided at each corner. The area of horizontal 6

reinforcement and the area of vertical reinforcement provided shall each be equal, in square 7

inches, to 0.10 times the wall thickness, in inches. 8

R10.6.2 This section has been added for slip-form construction where placing of diagonal 9

bars is restricted by the yoke frames. It may also be beneficial in jump-form construction when 10

the construction joint location relative to the top or bottom of an opening makes it difficult to 11

locate a diagonal bar. The effectiveness of orthogonal corner reinforcement has been studied 12

(Kilic, et al. 2016). 13

10.6.3 For openings 2 ft. wide or less, a minimum of two No. 5 bars shall be placed 14

diagonally at each corner of the opening. Alternatively, two No. 5 horizontal bars and two 15

No. 5 vertical bars are permitted to be placed at each corner of the opening. 16

10.6.4 The length of corner reinforcement bars, diagonal or orthogonal, shall be sufficient to 17

develop each bar in tension on either side of a theoretical crack from the corner of the opening 18

propagating at a 1:1 slope. If orthogonal corner reinforcing bars from adjacent corners overlap, 19

single bars are permitted to be provided. 20

10.6.5 If corner reinforcement is provided by orthogonal bars, the areas required by 10.6.2 21

are permitted to be added to the vertical opening reinforcement of 10.4 and the circumferential 22

93


opening reinforcement of 10.5 provided that the area representing corner reinforcement 1

satisfies the development requirement of 10.6.4. 2

10.7—Seismic detailing 3

10.7.1 For chimneys with an assigned Seismic Design Category D, E or F, at openings where 4

the loss of cross-sectional area is greater than 10%, detailing of the jamb region shall be 5

provided as specified in 10.7.2 to 10.7.6. 6

R10.7.1 Seismic detailing of the jamb regions for Seismic Design Categories D, E and F has 7

been added consistent with ASCE 7-16 Section 15.6.2.2. 8

10.7.2 The reinforcement ratio of the jamb area at each side of each opening shall be not less 9

than 0.01 nor more than 0.08. The width of the jamb area shall be the width of the chimney 10

wall containing the additional vertical reinforcement of 10.4 but not less than twice the wall 11

thickness. 12

R10.7.2 – This section defines the minimum width of the jamb area when seismic detailing 13

is required. Section 10.4.2 defines the maximum width of the jamb area. 14

10.7.3 Vertical bars within a region the jamb area defined in 10.7.1 shall be laterally 15

supported by ties. Ties shall be at least No. 3 in size for jamb bars No. 10 or smaller and at 16

least No. 4 in size for jamb bars No. 11 or larger and bundled bars. 17

10.7.4 Vertical spacing of ties shall not exceed the least of (a) through (c): 18

(a) 16 times the jamb bar diameter 19

(b) 48 times the tie diameter 20

(c) wall thickness at the opening 21

10.7.5 Ties shall consist of closed ties and crossties, as necessary, such that every corner and 22

alternate jamb bar shall have lateral support provided by the corner of a tie with an included 23

94


angle of no more than 135 degrees and no bar shall be farther than 6 in. clear on each side 1

along the tie from a laterally supported jamb bar. Consecutive crossties shall be alternated end 2

for end. 3

R10.7.5 Typical arrangements that satisfy these requirements are illustrated in 4

Fig. R25.7.2.3a of ACI 318-19 for column cross-sections. Similar detailing can be applied to 5

the jamb area of chimney openings. 6

10.7.6 Ties shall extend above and below the opening a distance equal to twice the wall 7

thickness or the development length of the jamb bars, whichever is larger. 8

10.7.7 Ties that satisfy 10.7.3 through 10.7.6 shall be considered to satisfy the tie bar 9

requirements of 10.4.4. 10

11

95


CHAPTER 11—FOUNDATION 1

11.1—Scope 2

11.1.1 This chapter shall apply to the design of foundations for reinforced concrete 3

chimneys. 4

11.2 Foundation geotechnical capacity 5

11.2.1 The following load combinations shall be used to determine the loads transmitted 6

through the chimney foundation to the supporting soil or rock. The weight of the foundation 7

shall be considered dead load. The dead load is permitted to include overlying fill. 8

1. D 9

2. D + 0.6 Walong 10

3. D + 0.7 LFCW Wcomb 11

4. D + 0.7 E 12

5. 0.6 D + 0.6 Walong 13

6. 0.6 D + 0.7 LFCW Wcomb 14

7. 0.6 D + 0.7 E 15

R11.2.1 These load combinations are consistent with those listed in Section 2.4.1 of ASCE 16

7-16. Section 7.6.2.8 of this Code permits the vertical seismic load to be neglected. The 17

factor 0.7 for the along-wind/across-wind combined load is consistent with the corresponding 18

load factor for strength design in 7.7.1.1. Load combinations (5), (6) and (7) are a check on 19

stability and replace the requirement for a factor of safety against overturning in the previous 20

revision. 21

96


11.2.2 The overturning effect at the soil-foundation interface due to seismic load is 1

permitted to be reduced by 10% for foundations of chimneys designed in accordance with the 2

modal analysis requirements of 7.6.2. 3

R11.2.2 This reduction is consistent with Section 12.13.4 of ASCE 7-16. 4

11.2.3 For a shallow foundation, the minimum base area of the foundation shall be 5

calculated from forces and moments transmitted by the foundation to soil or rock and the 6

allowable bearing pressure determined through principles of soil or rock mechanics. 7

11.2.4 For a deep foundation, the number and arrangement of piles shall be determined 8

from forces and moments transmitted to these members and the allowable pile capacity 9

determined through principles of soil or rock mechanics. 10

11.3—Foundation structural design 11

11.3.1 Foundation systems shall be permitted to be designed by any procedure satisfying 12

equilibrium and geometric compatibility. 13

11.3.2 The foundation design shall be in accordance with the applicable sections of 14

Chapter 13 of ACI 318-19. 15

11.3.3 Foundations shall be proportioned to resist loads and induced reactions determined 16

by the following load combinations: 17

1. 1.4 D 18

2. 1.2 D + 1.0 Walong 19

3. 1.2 D + LFCW Wcomb 20

4. 1.2 D + 1.0 E 21

5. 0.9 D + 1.0 Walong 22

6. 0.9 D + LFCW Wcomb 23

97


7. 0.9 D + 1.0 E 1

R11.3.3 These load combinations are the same as those in 7.7.1.1 without temperature 2

effects. 3

11.3.4 Where the structural effects of temperature are expected to adversely affect structural 4

safety or performance, the temperature effect shall be considered in the load combinations of 5

11.3.3. 6

R11.3.4 This section is included consistent with ASCE 7-16. 7

11.3.5 For foundations of chimneys with an assigned Seismic Design Category of D, E, or 8

F, the horizontal seismic effect E shall be replaced by the horizontal seismic effect including 9

overstrength Ω0E. The overstrength factor for both moment and shear, Ω0, shall be 1.33. 10

R11.3.5 Since the 𝑅 factor has been increased from 1.5 to 2.0, the flexural and shear 11

capacities of the foundation system are designed for overstrength so that inelastic flexural 12

behavior will develop in the ductile regions of chimneys assigned to Seismic Design 13

Category D, E, or F (Wilson 2003). 14

11.4 – Reinforcement detailing 15

11.4.1 A minimum area of flexural reinforcement of 0.0018 Ag shall be provided near the 16

top face and near the bottom face of the foundation slab. Such reinforcement shall be 17

provided in each of two orthogonal directions. 18

R11.4.1 Commentary R8.6.1.1 of ACI 318-19 suggests that the licensed design 19

professional consider specifying continuous reinforcement in each direction near both faces 20

of thick two-way slabs including mat foundations. The Committee felt this should be a code 21

requirement for chimney foundations. 22

11.5 – Mass concrete 23

98


11.5.1 This code does not address mass concrete considerations. 1

R11.5.5 Chimney foundations are often very thick. The licensed design professional should 2

determine if mass concrete issues need to be considered. ACI PRC-207.1-05 provides 3

guidance. 4

5

6

99


CHAPTER 12—CONSTRUCTION REQUIREMENTS 1

12.1—Scope 2

12.1.1 This chapter shall apply to construction execution and inspection during construction. 3

R12.1.1 A quality assurance program should be established to measure, document and verify 4

compliance with the construction requirements of this code. The program should identify the 5

type, number and frequency of measurements required to document each of the areas specified 6

in the Code. 7

12.2—General 8

12.2.1 Concrete quality, methods of determining the strength of concrete, field tests, concrete 9

proportions and consistency, concrete mixing and placing, concrete formwork and details of 10

reinforcement shall be in accordance with ACI 318, except as stated otherwise. 11

12.2.2 Load shall not be placed on the concrete structure until that portion of the structure 12

has attained sufficient strength to safely support its weight and the loads placed thereon. 13

12.3—Concrete strength 14

12.3.1 The specified concrete strength shall be in accordance with 5.4 and 5.5. 15

12.4—Concrete strength tests 16

12.4.1 Unless otherwise specified, 𝑓𝑐′ shall be based on 28-day tests. If other than 28 days, 17

the test age for 𝑓𝑐′ shall be indicated in the construction documents. 18

12.4.2 All strength tests shall be in accordance with Section 26.12.1.1 of ACI 318-19. 19

12.4.3 For chimneys, the compressive strength of the concrete shall be determined from a 20

minimum of two strength tests per 8-hour shift for slip form construction or two strength tests 21

per lift for jump form construction. For precast sections, a minimum of two strength tests shall 22

100


be taken from each class of concrete cast each day and from each 100 cubic yards of concrete 1

placed each day. 2

R12.4.3 Note that each strength test consists of the average of two or three cylinder tests, 3

depending on the cylinder size. 4

12.4.4 For foundations, the frequency of strength tests shall be at least the minimum required 5

by Section 26.12.2 of ACI 318-19. 6

12.4.5 Acceptance criteria for all strength tests shall be per Section 26.12.3 of ACI 318-19. 7

12.5—Formwork 8

12.5.1 Formwork for the chimney wall shall be made of metal, wood, or other suitable 9

material. 10

12.5.2 Forms shall be sufficiently tight to prevent leakage of mortar. 11

12.5.3 If unlined wooden forms are used, they shall have tongue and groove joints and shall 12

be kept continuously wet to prevent shrinking and warping due to exposure to the elements. 13

12.5.4 Form oil shall not be used unless it is a non-staining type and it has been established 14

that specified protective coatings or paint can be applied to concrete exposed to form oil. 15

12.5.5 Forms shall be removed in such a manner as to ensure the safety of the structure. 16

Forms shall be permitted to be removed after the concrete has hardened to a strength sufficient 17

to maintain its shape without damage and to safely support all loads on it, including temporary 18

construction loads. 19

12.5.6 Ties between inner and outer chimney wall forms shall not be permitted. 20

12.6—Concrete placement 21

12.6.1 Cast-in-place concrete shall be placed in layers no greater than 16 in. 22

12.6.2 Vertical construction joints shall not be used for cast-in-place chimneys. 23

101


12.6.3 When used, horizontal construction joints for cast-in-place and precast concrete 1

chimneys shall be approximately evenly spaced throughout the height of the chimney. Grout 2

for setting precast sections shall have a specified compressive strength equal to or greater than 3

the specified compressive strength of the precast sections. 4

12.6.4 Construction joints shall be properly prepared to facilitate bonding. As a minimum, 5

all laitance and loose material shall be removed. 6

12.7—Concrete curing 7

12.7.1 All necessary finishing of concrete shall commence immediately after the forms have 8

been removed. 9

12.7.2 As soon as finishing has been completed, both faces of the concrete wall shall be cured 10

by coating with a membrane-type curing compound or other method approved by the licensed 11

design professional. 12

12.7.3 The curing compound shall comply with ASTM C309 and shall be applied in strict 13

accordance with the manufacturer’s recommendations. 14

12.7.4 If coatings are to be applied to the concrete, the curing compound shall be of a type 15

compatible with the coatings. 16

12.8—Reinforcement placement 17

12.8.1 All reinforcing bars shall be secured at intervals of not more than 2 ft. Bars shall be 18

positioned within the tolerances of ACI 117. 19

12.8.2 Vertical reinforcing bars projecting above the forms for the chimney wall or cores of 20

precast sections shall be temporarily braced to prevent the breaking of the bond with the freshly 21

placed concrete. 22

12.9—Construction tolerances 23

102


12.9.1 Vertical alignment of centerpoint 1

12.9.1.1 The actual centerpoint of the chimney shall not vary from its theoretical axis by 2

more than 0.001 times the height of the chimney or 1 in., whichever is greater. Locally, the 3

actual centerpoint of the chimney shall not change by more than 1 in. for any 10 t. of vertical 4

rise. 5

12.9.2 Diameter 6

12.9.2.1 The measured outside chimney diameter at any section shall not vary from the 7

specified diameter by more than 1 in. plus the 0.01 times the specified or theoretical diameter. 8

12.9.3 Wall thickness 9

12.9.3.1 For walls 10 in. thick or less, the measured wall thickness shall not vary from the 10

specified wall thickness by more than -1/4 in., +1/2 in. For walls thicker than 10 in., the 11

measured wall thickness shall not vary from the specified wall thickness by more than -1/2 in., 12

+1 in. A single wall thickness is defined as the average of at least four measurements taken at 13

a uniform spacing over a 60-degree arc. A negative tolerance decreases the overall thickness 14

and a positive tolerance increases the overall thickness. 15

12.9.4 Openings and embedments 16

12.9.4.1 Tolerances on the size and location of openings and embedments in the chimney 17

cannot be uniformly established due to the varying degree of accuracy required, and the 18

varying nature of their use. Appropriate tolerances for each opening or embedment shall be 19

established and included in the construction drawings. 20

12.10—Precast erection 21

103


12.10.1 Precast sections shall be erected in a manner and at a rate that ensures that sufficient 1

strength has been attained in the grout, core concrete, and all connecting components to safely 2

support construction and applicable design loads. 3

12.10.2 Precast sections shall be grouted to level and joints shall be sealed. Shear keys shall 4

be installed if required by the licensed design professional. 5

104


APPENDIX A—HORIZONTAL CROSS-SECTION STRENGTH FOR CIRCULAR 1

CHIMNEYS BY MODIFIED STRESS BLOCK METHOD 2

A.1—Scope 3

Appendix A consists of three sections: 4

a) Procedure for computing combined nominal compression and nominal bending 5

moment capacity 6

b) Stress block modification factor 7

c) Derivation of equations 8

A.2—Notation 9

The following notation is used in Appendix A only. Any notation used in the Code and in 10

Appendix A is defined in 2.2. All equations in this appendix are to be evaluated using consistent 11

force and length units, so no force or length units are specified for the symbols below. The force 12

and length units listed for a symbol in 2.2 will not necessarily be the units used for the symbol in 13

this appendix. 14

For load combinations with temperature effects, modified values of 𝑓𝑐′ and 𝑓𝑦 are defined in 15

8.5.2.3. 16

𝐴 = parameter in 𝑄2 expression

𝐾 = parameter in 𝐾2 expression

𝐾1 = parameter in 𝑀𝑛 expression

𝐾2 = parameter in 𝑀𝑛 expression

𝐾𝑒 = 𝐸𝑠 𝑓𝑦⁄

𝑛1 = number of openings in the compression zone

𝑄 = stress block modification factor

105


𝑄1 = parameter in 𝐾1 expression

𝑄2 = parameter in 𝐾2 expression

�̅� = parameter in 𝐾2 expression

𝑡 = thickness of concrete wall

𝛼deg one-half the angle subtended by the neutral axis, degrees

𝛽1 = factor relating depth of equivalent rectangular stress block to neutral

axis depth

𝛾 = for two openings, one-half the angle between the opening centerlines,

radians

휀𝑚 = maximum compressive strain

𝜃 = variable indicating location on the cross-section, radians

𝜆 = parameter in 𝐾1 expression

𝜆1 = parameter in 𝐾1 expression

𝜇 = angular location of the yield strain in the compressive zone, radians

𝜏 = angular location of the depth of the compression zone, radians

𝜓 = angular location of the yield strain in the tension zone, radians

𝜔𝑡 = 𝜌𝑡𝑓𝑦 𝑓𝑐′⁄

1

A.3—Procedure for computing combined nominal compression and nominal bending 2

moment capacity 3

Consider the following demand at a horizontal cross-section 4

𝑃𝑢 = factored vertical load at the cross-section 5

𝑀𝑢 = factored bending moment at the cross-section 6

106


The following procedure is permitted to be used to determine the minimum reinforcement to 1

provide the required combined nominal vertical load and nominal bending moment capacity 2

(𝑃𝑛 ,𝑀𝑛). 3

In general, a trial reinforcement ratio is chosen. Then, by iteration, the angle corresponding 4

to the neutral axis location, 𝛼, is determined by considering force equilibrium with 𝑃𝑛. Once 𝛼 5

is determined, 𝑀𝑛 is determined from moment equilibrium. If 𝜙𝑀𝑛 is not equal to 𝑀𝑢, the 6

reinforcement ratio is adjusted and the procedure is repeated. 7

Referring to Figs. A.1, A.2 and A.3, the following equations apply to cross-sections with no 8

openings, cross-sections with one opening (centered in the compression zone) and cross-9

sections with two equal width openings symmetric with respect to the bending direction. For 10

cross-sections with two openings, the openings may be completely within the compression 11

zone, partially within the compression zone, or completely within the tension zone. 12

Step 1: compute the nominal vertical load capacity 13

𝑃𝑛 =𝑃𝑢𝜙

14

Step 2: determine opening parameters 15

No openings: 𝑛1 = 𝛽 = 𝛾 = 0 16

One opening: 𝑛1 = 1, 𝛽 ≠ 0, 𝛾 = 0 17

Two openings: 𝑛1 = 2, 𝛽 ≠ 0, 𝛾 ≠ 0 18

Step 3: determine parameters not dependent on 𝛼 or 𝜌𝑡 19

𝛽1 = 0.85 − 0.05(𝑓𝑐′ − 4000

1000) but 0.65 ≤ 𝛽1 ≤ 0.85 20

𝐾𝑒 =𝐸𝑠

𝑓𝑦 21

Step 4: assume a value for 𝜌𝑡 and compute 𝜔𝑡 22

107


𝜔𝑡 =𝜌𝑡𝑓𝑦𝑓𝑐′

1

Step 5: assume a value for 𝛼 and compute the following 2

휀𝑚 = 0.06(1 − cos 𝛼

1 + cos 𝛼) ≤ 0.003 3

cos𝜓 = cos 𝛼 − (1−cos𝛼

𝑚) (

𝑓𝑦

𝐸𝑠) ≥ −1.0 4

cos 𝜇 = cos 𝛼 + (1 − cos𝛼

휀𝑚) (𝑓𝑦𝐸𝑠) ≤ 1.0 5

cos 𝜏 = 1 − 𝛽1(1 − cos 𝛼) 6

𝑄 = (See A.4) 7

𝜆 = 𝛾 − 𝛽 if 𝑛1 = 2 and 𝛾 − 𝛽 < 𝜏 < 𝛾 + 𝛽 8

= 𝜏 if 𝑛1 = 2 and 𝜏 < 𝛾 − 𝛽 9

= 𝜏 − 𝑛1𝛽 otherwise 10

(Note: 𝜆 in radians) 11

𝑄1 =sin 𝜓−sin 𝜇−(𝜓−𝜇)cos𝛼

1−cos𝛼 12

𝜆1 = 𝜇 + 𝜓 − 𝜋 (radians) 13

𝐾1 = 1.7𝑄𝜆 + 2휀𝑚𝐾𝑒𝜔𝑡𝑄1 + 2𝜔𝑡𝜆1 14

Step 6: check 𝐾1 vs. non-dimensional nominal vertical load capacity 15

𝑃𝑛

𝑟𝑡𝑓𝑐′ = 𝐾1 ? 16

If yes, go to Step 7. If no, return to Step 5. 17

Step 7: compute the following 18

�̅� = sin(𝛾 − 𝛽) − (𝛾 − 𝛽) cos𝛼 19

if 𝑛1 = 2 and 𝛾 − 𝛽 < 𝜏 < 𝛾 + 𝛽 20

= sin 𝜏 − 𝜏 cos𝛼 21

108


if 𝑛1 = 2 and 𝜏 < 𝛾 − 𝛽 1

= sin 𝜏 − (𝜏 − 𝑛1𝛽) cos𝛼 − 𝑛1cos 𝛾 sin 𝛽 2

Otherwise 3

𝐴 = 2(𝜓 − 𝜇)cos2𝛼 + sin 𝜓 cos𝜓 − sin 𝜇 cos 𝜇 − 4 cos 𝛼(sin𝜓 − sin 𝜇) + (𝜓 − 𝜇) 4

𝑄2 =𝐴

1−cos𝛼 5

𝐾 = sin𝜓 + sin 𝜇 + (𝜋 − 𝜓 − 𝜇) cos𝛼 6

𝐾2 = 1.7𝑄�̅� + 휀𝑚𝐾𝑒𝜔𝑡𝑄2 + 2𝜔𝑡𝐾 7

Step 8: compute the nominal bending moment capacity 8

𝑀𝑛 = 𝑃𝑛𝑟 [cos𝛼 +𝐾2

𝐾1] 9

Step 9: compare nominal bending moment capacity to required bending strength 10

𝑀𝑢 = 𝜙𝑀𝑛 ? 11

If yes, done. If no, return to Step 4. 12

13

109


A.4 —Stress block modification factor 1

(Note: 𝛼 in degrees for this calculation only) 2

For 𝛼deg ≤ 5° 3

𝑄 = (−0.523 + 0.181𝛼deg − 0.0154𝛼deg2) + (41.3 − 13.2𝛼deg + 1.32𝛼deg

2) (𝑡

𝑟) 4

For 5° < 𝛼deg ≤ 10° 5

𝑄 = (−0.154 + 0.01773𝛼deg + 0.00249𝛼deg2) + 6

(16.42 − 1.980𝛼deg + 0.0674𝛼deg2) (

𝑡

𝑟) 7

For 10° < 𝛼deg ≤ 17° 8

𝑄 = (−0.488 + 0.076𝛼deg) + (9.758 − 0.640𝛼deg) (𝑡

𝑟) 9

For 17° < 𝛼deg ≤ 25° 10

𝑄 = (−1.345 + 0.2018𝛼deg − 0.004434𝛼deg2) + 11

(15.83 − 1.676𝛼deg + 0.03994𝛼deg2) (

𝑡

𝑟) 12

For 25° < 𝛼deg ≤ 35° 13

𝑄 = (0.993 − 0.00258𝛼deg) + (−3.27 + 0.0862𝛼deg) (𝑡

𝑟) 14

For 𝛼deg > 35° 15

𝑄 = 0.89 16

RA.4 The 1988 committee felt that an equivalent rectangular stress block approach could be 17

used to determine the strength of chimney cross-sections. This approach had been accepted for 18

rectangular and T-shaped beams after extensive comparisons between the analytical results using 19

this stress-strain relationship and results of tests comparing the concrete compression force and the 20

moment of the concrete compression force about the neutral axis (Mattock et al., 1961). 21

110


Two problems confronted the committee’s task. First, there was limited test data available for 1

comparison on reinforced concrete members of hollow circular cross-sections subjected to axial 2

and transverse loads (Mokrin and Rumman 1985). Second, the maximum concrete compressive 3

strain is less than 0.003 if the fracture limit of steel controls. That is, the compressive stress block 4

is not fully developed (see 8.2.3). 5

A numerical study was undertaken by the committee to find an equivalent rectangular stress 6

block to calculate the strength of chimney cross-sections. 7

For a given value of α (neutral axis location), the results using the rectangular stress block were 8

compared with the corresponding results using a more exact concrete stress-strain relationship 9

(Rumman and Sun 1977) given by Hognestad (1951) using a limiting strain of 0.003. The 10

comparisons were made for hollow circular cross-sections without openings and with single 11

openings with values of β of 10, 20 and 30 degrees. 12

It was concluded that for values of α above 20 degrees, or when the limiting strain of concrete 13

was reached before the rupture strain of steel was reached, an equivalence between the two 14

approaches is reached if the stress level of the rectangular compression block is reduced by a factor 15

of 0.89. For values of α below approximately 20 degrees, a further correction is required, leading 16

to the values of the parameter Q defined in this section. 17

Thus, the correction factor, or the parameter Q, achieves a close equivalence between the 18

resulting values of (a) and (b) for the thereby corrected rectangular stress block results and the 19

stress block based on the results using the Hognestad stress-strain relationship, yet retains the 20

simplicity of the rectangular stress block relationship. 21

22

A.5 Derivation of equations 23

111


Referring to Fig. A.1, locations on the cross-section are referenced to the middle of the wall. 1

Strain is assumed to vary linearly across the cross-section. A polar coordinate system (𝑟, 𝜃) is 2

defined with angle 𝜃 = 0 at the point of maximum compressive strain and angle 𝜃 = 𝜋 at the 3

point of maximum tensile strain. Concrete stress is compressive only and is represented by a 4

(modified) stress block. Steel stress is assumed to follow an elastic-perfectly plastic rule. 5

Figures A.1, A.2 and A.3 illustrate the strain and stress distribution on the cross-section. 6

For simplicity, reinforcement is considered to be distributed uniformly around the full 7

circumference, even in the presence of openings, since the reinforcement interrupted by an 8

opening is placed adjacent to the opening. The error introduced by this simplification is 9

negligibly small. 10

Also for simplicity, the maximum concrete strain is considered to occur at 𝜃=0 even if there 11

may be no concrete at that location (single opening case). Again, the error introduced by this 12

simplification is negligibly small. 13

To determine the bending moment capacity, the neutral axis angle 𝛼 is determined, by 14

iteration, such that force equilibrium is satisfied at the cross-section. 15

𝑃𝑛 = 𝐹𝑐 + 𝑆𝑐 + 𝑆𝑐𝑦 − 𝑆𝑡 − 𝑆𝑡𝑦 (A-1)

The reinforcement forces are computed as follows: 16

𝑆𝑡 = 2∫ 𝐸𝑠𝑟(cos𝛼 − cos 𝜃)휀𝑚

𝑟(1 − cos𝛼)𝑟𝜌𝑡𝑡𝑑𝜃

𝜓

𝛼

= 2𝐸𝑠휀𝑚𝑟𝜌𝑡𝑡

(1 − cos𝛼)[𝜃 cos 𝛼 − sin 𝜃]𝛼

𝜓


(1 − cos𝛼)[(𝜓 − 𝛼) cos 𝛼 − sin𝜓 + sin 𝛼]

17

112


𝑆𝑡𝑦 = 2∫ 𝑓𝑦𝑟𝜌𝑡𝑡𝑑𝜃𝜋

𝜓

= 2𝑓𝑦𝑟𝜌𝑡𝑡(𝜋 − 𝜓)

𝑆𝑐 = 2∫ 𝐸𝑠𝑟(cos𝜃 − cos 𝛼)휀𝑚

𝑟(1 − cos𝛼)𝑟𝜌𝑡𝑡𝑑𝜃

𝛼

𝜇


(1 − cos𝛼)[sin 𝜃 − 𝜃 cos 𝛼]𝜇

𝛼


(1 − cos𝛼)[sin 𝛼 − sin 𝜇 − (𝛼 − 𝜇) cos 𝛼]

𝑆𝑐𝑦 = 2∫ 𝑓𝑦𝑟𝜌𝑡𝑡𝑑𝜃𝜇

0

= 2𝑓𝑦𝑟𝜌𝑡𝑡𝜇

1

Defining 2

𝜔𝑡 =𝜌𝑡𝑓𝑦𝑓𝑐′

and 𝐾𝑒 =𝐸𝑠𝑓𝑦 so that 𝐸𝑠𝜌𝑡 = 𝐾𝑒𝜔𝑡𝑓𝑐

′ 3

𝑆𝑡 = 2휀𝑚𝐾𝑒𝜔𝑡𝑓𝑐′𝑟𝑡

[(𝜓−𝛼) cos𝛼−sin 𝜓+sin 𝛼]

(1−cos𝛼) 4

𝑆𝑡𝑦 = 2𝜔𝑡𝑓𝑐′𝑟𝑡(𝜋 − 𝜓) 5

𝑆𝑐 = 2휀𝑚𝐾𝑒𝜔𝑡𝑓𝑐′𝑟𝑡

[sin 𝛼−sin 𝜇−(𝛼−𝜇) cos𝛼]

(1−cos𝛼) 6

𝑆𝑐𝑦 = 2𝜔𝑡𝑓𝑐′𝑟𝑡𝜇 7

The concrete stress resultant for no openings is: 8

𝐹𝑐 = 2∫ 0.85𝑄𝑓𝑐′𝑟𝑡𝑑𝜃

𝜏

0

= 1.7𝑄𝑓𝑐′𝑟𝑡𝜏

9

113


For one opening centered in the compression zone: 1

𝐹𝑐 = 2∫ 0.85𝑄𝑓𝑐′𝑟𝑡𝑑𝜃

𝜏

𝛽

= 1.7𝑄𝑓𝑐′𝑟𝑡(𝜏 − 𝛽)

2

For two symmetric openings completely in the compression zone, that is, when 𝜏 > 𝛾 + 𝛽 3

𝐹𝑐 = 2 [∫ 0.85𝑄𝑓𝑐′𝑟𝑡𝑑𝜃 +∫ 0.85𝑓𝑐

′𝑟𝑡𝑑𝜃𝜏

𝛾+𝛽

𝛾−𝛽

0

]

= 1.7𝑓𝑐′𝑄𝑟𝑡(𝜏 − 2𝛽)

4

The above three cases can be combined. 5

𝐹𝑐 = 1.7𝑓𝑐′𝑄𝑟𝑡(𝜏 − 𝑛1𝛽) 6

For two symmetric openings partially in the compression zone, that is, when 𝛾 − 𝛽 < 𝜏 <7

𝛾 + 𝛽 8

𝐹𝑐 = 1.7𝑓𝑐′𝑄𝑟𝑡(𝛾 − 𝛽) 9

For two symmetric openings completely in the tension zone, that is, when 𝜏 < 𝛾 − 𝛽 10

𝐹𝑐 = 1.7𝑓𝑐′𝑄𝑟𝑡𝜏 11

The reinforcement forces can be combined as follows 12

𝑆𝑐 − 𝑆𝑡 = 2휀𝑚𝐾𝑒𝜔𝑡𝑓𝑐′𝑟𝑡𝑄1 13

where 𝑄1 =[sin𝜓 − sin 𝜇 − (𝜓 − 𝜇) cos𝛼]

(1 − cos 𝛼) 14

𝑆𝑐𝑦 − 𝑆𝑡𝑦 = 2𝜔𝑡𝑓𝑐′𝑟𝑡𝜆1 15

where 𝜆1 = 𝜇 + 𝜓 − 𝜋 16

So Eq. (A-1) becomes, in non-dimensional form 17

114


𝑃𝑛𝑟𝑡𝑓𝑐′⁄ = 1.7𝑄𝜆 + 2휀𝑚𝐾𝑒𝜔𝑡𝑄1 + 2𝜔𝑡𝜆1 1

where 𝜆 = 𝛾 − 𝛽 if 𝑛1 = 2 and 𝛾 − 𝛽 < 𝜏 < 𝛾 + 𝛽 2

𝜆 = 𝜏 if 𝑛1 = 2 and 𝜏 < 𝛾 − 𝛽 3

𝜆 = 𝜏 − 𝑛1𝛽 otherwise 4

Once 𝛼 has been determined for a cross-section, the nominal moment capacity of the cross-5

section, 𝑀𝑛, can be determined by summing moments about the neutral axis. 6

𝑀𝑛 − 𝑃𝑛𝑟 cos𝛼 = 𝐹𝑐′ + 𝑆𝑐

′ + 𝑆𝑐𝑦′ + 𝑆𝑡

′ + 𝑆𝑡𝑦′ (A-2)

The reinforcement moments are computed as follows: 7

𝑆𝑡′ = 2∫ 𝐸𝑠

𝑟2(cos𝛼 − cos𝜃)2휀𝑚𝑟(1 − cos 𝛼)

𝑟𝜌𝑡𝑡𝑑𝜃𝜓

𝛼

= 2𝐸𝑠휀𝑚𝑟

2𝜌𝑡𝑡

(1 − cos 𝛼)[𝜃cos2𝛼 − 2 cos𝛼 sin 𝜃 +

𝜃

2+sin 2𝜃

4]𝛼

𝜓

= 2𝐸𝑠휀𝑚𝑟2𝜌𝑡𝑡

(1 − cos 𝛼)[(𝜓 − 𝛼)cos2𝛼

− 2 cos 𝛼 (sin𝜓 − sin 𝛼)+𝜓 − 𝛼

2+sin 2𝜓 − sin 2𝛼

4]

𝑆𝑡𝑦′ = 2∫ 𝑓𝑦𝑟

2(cos 𝛼 − cos𝜃)𝜌𝑡𝑡𝑑𝜃𝜋

𝜓

= 2𝑓𝑦𝑟2𝜌𝑡𝑡[𝜃 cos 𝛼 − sin 𝜃]𝜓

𝜋

= 2𝑓𝑦𝑟2𝜌𝑡𝑡[(𝜋 − 𝜓) cos 𝛼 + sin𝜓]

𝑆𝑐′ = 2∫ 𝐸𝑠

𝑟2(cos𝜃 − cos 𝛼)2휀𝑚𝑟(1 − cos 𝛼)

𝑟𝜌𝑡𝑡𝑑𝜃𝛼

𝜇

= 2𝐸𝑠휀𝑚𝑟

2𝜌𝑡𝑡

(1 − cos 𝛼)[𝜃

2+sin 2𝜃

4− 2 cos𝛼 sin 𝜃 + 𝜃cos2𝛼]

𝜇

𝛼

115


= 2𝐸𝑠휀𝑚𝑟2𝜌𝑡𝑡

(1 − cos 𝛼)[𝛼 − 𝜇

2+sin 2𝛼 − sin 2𝜇

4− 2 cos𝛼 (sin 𝛼 − sin 𝜇)

+ (𝛼 − 𝜇)cos2𝛼]

𝑆𝑐𝑦′

= 2∫ 𝑓𝑦𝑟

2(cos𝜃 − cos 𝛼)𝜌𝑡𝑡𝑑𝜃𝜇

0

= 2𝑓𝑦𝑟2𝜌𝑡𝑡[sin 𝜃 − 𝜃 cos𝛼]0

𝜇

= 2𝑓𝑦𝑟2𝜌𝑡𝑡[sin 𝜇 − 𝜇 cos 𝛼]

1

Substituting 𝜔𝑡 and 𝐾𝑒defined as above 2

𝑆𝑡′ =

2 𝑚𝐾𝑒𝜔𝑡𝑓𝑐′𝑟2𝑡

(1−cos𝛼)[(𝜓 − 𝛼)cos2𝛼 − 2 cos𝛼 (sin𝜓 − sin 𝛼)+

𝜓−𝛼

2+

sin 2𝜓−sin 2𝛼

4] 3

𝑆𝑡𝑦′ = 2𝜔𝑡𝑓𝑐

′𝑟2𝑡[(𝜋 − 𝜓) cos 𝛼 + sin 𝜓] 4

𝑆𝑐′ =

2 𝑚𝐾𝑒𝜔𝑡𝑓𝑐′𝑟2𝑡

(1−cos 𝛼)[𝛼−𝜇

2+

sin 2𝛼−sin 2𝜇

4− 2 cos𝛼 (sin 𝛼 − sin 𝜇) + (𝛼 − 𝜇)cos2𝛼] 5

𝑆𝑐𝑦′ = 2𝜔𝑡𝑓𝑐

′𝑟2𝑡[sin 𝜇 − 𝜇 cos𝛼] 6

For no openings: 7

𝐹𝑐′ = 2∫ 0.85𝑄𝑓𝑐

′𝑟2𝑡(cos 𝜃 − cos 𝛼)𝑑𝜃𝜏

0

= 1.7𝑄𝑓𝑐′𝑟2𝑡[sin 𝜃 − 𝜃 cos𝛼]0

𝜏

= 1.7𝑄𝑐′𝑟2𝑡(sin 𝜏 − 𝜏 cos𝛼)

For one opening centered in the compression zone: 8

𝐹𝑐′ = 2∫ 0.85𝑄𝑓𝑐

′𝑟2𝑡(cos 𝜃 − cos 𝛼)𝑑𝜃𝜏

𝛽

= 1.7𝑄𝑓𝑐′𝑟2𝑡[sin 𝜃 − 𝜃 cos𝛼]𝛽

𝜏

= 1.7𝑄𝑓𝑐′𝑟2𝑡[sin 𝜏 − sin 𝛽 − (𝜏 − 𝛽) cos𝛼]

116


For two symmetric openings completely in the compression zone, that is, when 𝜏 > 𝛾 + 𝛽 1

𝐹𝑐′ = 1.7𝑄𝑓𝑐

′𝑟2𝑡 [∫ (cos 𝜃 − cos𝛼)𝑑𝜃 +∫ (cos 𝜃 − cos𝛼)𝑑𝜃𝜏

𝛾+𝛽

𝛾−𝛽

0

]

= 1.7𝑄𝑓𝑐′𝑟2𝑡[[sin 𝜃 − 𝜃 cos 𝛼]0

𝛾−𝛽+ [sin 𝜃 − 𝜃 cos𝛼]𝛾+𝛽

𝜏 ]

= 1.7𝑄𝑓𝑐′𝑟2𝑡[sin 𝜏 − (𝜏 − 2𝛽) cos 𝛼 − 2 cos 𝛾 sin 𝛽]

The above three cases can be combined. 2

𝐹𝑐′ = 1.7𝑓𝑐

′𝑄𝑟2𝑡[sin 𝜏 − (𝜏 − 𝑛1𝛽) cos 𝛼 − 𝑛1cos 𝛾 sin 𝛽] 3

For two symmetric openings partially in the compression zone, that is, when 𝛾 − 𝛽 < 𝜏 <4

𝛾 + 𝛽 5

𝐹𝑐′ = 2∫ 0.85𝑄𝑓𝑐

′𝑟2𝑡(cos𝜃 − cos𝛼)𝑑𝜃𝛾−𝛽

0

= 1.7𝑄𝑓𝑐′𝑟2𝑡[sin 𝜃 − 𝜃 cos𝛼]0

𝛾−𝛽

= 1.7𝑄𝑓𝑐′𝑟2𝑡[sin(𝛾 − 𝛽) − (𝛾 − 𝛽) cos𝛼]

For two symmetric openings completely in the tension zone, that is, when 𝜏 < 𝛾 − 𝛽 6

𝐹𝑐′ = 1.7𝑄𝑓𝑐

′𝑟2𝑡[sin 𝜏 − 𝜏 cos 𝛼] 7

The reinforcement moments can be combined as follows 8

𝑆𝑐′ + 𝑆𝑡

′ = 휀𝑚𝐾𝑒𝜔𝑡𝑓𝑐′𝑟2𝑡𝑄2 9

𝑆𝑐𝑦′ + 𝑆𝑡𝑦

′ = 2𝜔𝑡𝑓𝑐′𝑟2𝑡𝐾 10

where 𝑄2 =𝐴

(1 − cos 𝛼) 11

𝐴 = [2(𝜓 − 𝜇)cos2𝛼 − 4 cos𝛼(sin 𝜓 − sin 𝜇) + (𝜓 − 𝜇) + sin 𝜓 cos𝜓 − sin 𝜇 cos 𝜇] 12

𝐾 = (𝜋 − 𝜓 − μ) cos 𝛼 + sin 𝜓 + sin 𝜇 13

So Eq. (A-2) becomes, in non-dimensional form 14

117


𝑀𝑛𝑟2𝑡𝑓𝑐′⁄ =

𝑃𝑛 cos 𝛼

𝑟𝑡𝑓𝑐′+ 1.7𝑄�̅� + 휀𝑚𝐾𝑒𝜔𝑡𝑄2 + 2𝜔𝑡𝐾 1

where 2

�̅� = sin(𝛾 − 𝛽) − (𝛾 − 𝛽) cos 𝛼 3

if 𝑛1 = 2 and 𝛾 − 𝛽 < 𝜏 < 𝛾 + 𝛽 4

�̅� = sin 𝜏 − 𝜏 cos𝛼 5

if 𝑛1 = 2 and 𝜏 < 𝛾 − 𝛽 6

�̅� = sin 𝜏 − (𝜏 − 𝑛1𝛽) cos𝛼 − 𝑛1cos 𝛾 sin 𝛽 7

otherwise 8

Letting 9

𝐾1 = 1.7𝑄𝜆 + 2휀𝑚𝐾𝑒𝜔𝑡𝑄1 + 2𝜔𝑡𝜆1 10

𝐾2 = 1.7𝑄�̅� + 휀𝑚𝐾𝑒𝜔𝑡𝑄2 + 2𝜔𝑡𝐾 11

The nominal bending moment strength can be expressed as 12

𝑀𝑛 = 𝑃𝑛𝑟 [cos𝛼 +𝐾2

𝐾1⁄ ] 13

118


1 Fig. A.1: Stress-strain relationships, no opening or one opening 2

3

119


1

2

Fig. A.2: Two openings fully in the compression zone 3

(Dimensions not shown; same as Fig. A.1) 4

5

6

Fig. A.3: Two openings partially in the compression zone 7

(Dimensions not shown; same as Fig. A.1) 8

9

120


APPENDIX B—HORIZONTAL CROSS-SECTION STRENGTH FOR CIRCULAR 1

CHIMNEYS BY STRESS-STRAIN RELATIONSHIP INTEGRATION METHOD 2

B.1–Scope 3

Appendix B consist of two sections 4

a) Procedure for computing combined nominal compression and nominal bending mo-5

ment capacity 6

b) Derivation of equations 7

B.2–Notation 8

The following notation is used in Appendix B only. Any notation used in the Code and in 9

Appendix B is defined in 2.2. All equations in this appendix are to be evaluated using consistent 10


and length units listed for a symbol in 2.2 will not necessarily be the units used for the symbol in 12

this appendix. 13

For load combinations with temperature effects, modified values of 𝑓𝑐′ and 𝑓𝑦 are defined in 14

8.5.2.3. 15

𝐹𝑐1 = force in concrete for the linear-varying stress region

𝐹𝑐1′ = moment of 𝐹𝑐1 about neutral axis

𝐹𝑐2 = force in concrete for the parabolic-varying stress region

𝐹𝑐2′ = moment of 𝐹𝑐2 about neutral axis

𝐹𝑂𝑐1 = force in concrete for the linear-varying stress region to subtract for two open-

ing case

𝐹𝑂𝑐1′ = moment of 𝐹𝑂𝑐1 about neutral axis to subtract for two opening case

121


𝐹𝑂𝑐2 = force in concrete for the parabolic-varying stress region to subtract for two

opening case

𝐹𝑂𝑐2′ = moment of 𝐹𝑂𝑐2 about neutral axis to subtract for two opening case

𝑡 thickness of concrete wall

𝑥0 = depth to 0.002 compressive strain (see Fig. B.1)

𝑥𝑛𝑎 = depth to neutral axis (see Fig. B.1)

𝑥𝑦𝑐 = depth to compressive steel yield strain (see Fig. B.1)

𝑥𝑦𝑡 = depth to tensile steel yield strain (see Fig. B.1)

𝛾 = for two openings, one-half the angle between the opening centerlines, radians

𝛿1 = min[max(𝜏, 𝛾 − 𝛽), 𝛼] , radians

𝛿2 = max[min(𝛼, 𝛾 + 𝛽), 𝜏] , radians

휀𝑐𝑢 = concrete strain limit ( + = compression)

휀𝑚𝑎𝑥 = maximum strain ( + = compression)

휀𝑚𝑖𝑛 = minimum strain ( + = compression)

휀𝑠𝑢 = steel strain limit ( + = compression)

휀𝑦𝑐 = compressive steel yield strain ( + = compression)

휀𝑦𝑡 = tensile steel yield strain ( + = compression)

𝜃 = variable indicating location on the cross-section, radians

𝜆1 = min(𝜏, 𝛾 − 𝛽) , radians

𝜆2 = min(𝜏, 𝛾 + 𝛽) , radians

𝜇 = angular location of the yield strain in the compressive zone, radians

𝜏 = angular location where the compressive strain is 0.002, radians

122


𝜓 = angular location of the yield strain in the tension zone, radians

1

B.3–Procedure for computing combined nominal compression and nominal bending 2

moment capacity 3

Consider the following demand at a horizontal cross-section 4

𝑃𝑢 = factored vertical load at the cross-section 5

𝑀𝑢 = factored bending moment at the cross-section 6

The following procedure is permitted to be used to determine the minimum reinforcement 7

to provide the required combined nominal vertical load and nominal bending moment capac-8

ity (𝑃𝑛 , 𝑀𝑛). 9

The radius to the middle of the wall is 𝑟. The neutral axis is located by its distance, 𝑥𝑛𝑎, 10

from middle of the wall at 0 degrees as shown in Fig. B.1. 11

12

Procedure (see B.2 for notation and B.4 for derivation) 13

1. Set reinforcement ratio; the minimum reinforcement ratio required is the initial value 14

2. By iteration, determine the location of the neutral axis, 𝑥𝑛𝑎, so that axial force equilibrium is 15

achieved 16

𝑃𝑢𝜙= 𝑃𝑛 = 𝐹𝑐1 + 𝐹𝑐2 + 𝑆𝑐 + 𝑆𝑐𝑦 + 𝑆𝑡 + 𝑆𝑡𝑦 17

3. Determine the moment capacity corresponding to the axial load capacity 𝜙𝑃𝑛 18

𝑀𝑛 = 𝑃𝑛(𝑟 − 𝑥𝑛𝑎) + 𝐹𝑐1′ + 𝐹𝑐2

′ + 𝑆𝑐′ + 𝑆𝑐𝑦

′ + 𝑆𝑡′ + 𝑆𝑡𝑦

′ 19

4. If 𝑀𝑢 > 𝜙𝑀𝑛 , increase the reinforcement ratio and go to Step 2 20

5. If 𝑀𝑢 ≈ 𝜙𝑀𝑛 or reinforcement ratio is equal to the minimum required, Stop 21

6. If 𝑀𝑢 < 𝜙𝑀𝑛 , decrease reinforcement ratio and go to Step 2 22

123


1

Referring to Figs. B.1, B.2 and B.3, the following equations apply to cross-sections with no 2

openings, cross-sections with one opening (centered in the compression zone) and cross-sec-3

tions with two equal width openings symmetric with respect to the bending direction. For 4

cross-sections with two openings, the openings may be completely within the compression 5

zone, partially within the compression zone, or completely within the tension zone. 6

B.4–Derivation of equations 7

Referring to Fig. B.1, locations on the cross-section are referenced to the middle of the wall. 8

Strain is assumed to vary linearly across the cross-section. A polar coordinate system (𝑟, 𝜃) is 9

defined with angle 𝜃 = 0 at the point of maximum compressive strain and angle 𝜃 = 𝜋 at the 10

point of maximum tensile strain. Concrete stress is compressive only and is represented by a 11

parabolic-varying stress region and a linearly-varying stress region. Steel stress is assumed to 12

follow an elastic-perfectly plastic rule. 13

Figs. B.1, B.2 and B.3 illustrate the strain and stress distribution on the cross-section. 14

For simplicity, reinforcement is considered to be distributed uniformly around the full cir-15

cumference, even in the presence of openings, since the reinforcement interrupted by an open-16

ing is placed adjacent to the opening. The error introduced by this simplification is negligibly 17

small. 18

Also for simplicity, the maximum concrete strain is considered to occur at 𝜃 = 0 even if 19

there may be no concrete at that location (single opening case). Again, the error introduced by 20

this simplification is negligibly small. 21

22

Cross-sections with no openings or one opening in the compression zone 23

124


The neutral axis is located a distance 𝑥𝑛𝑎 from the point of maximum concrete strain. The 1

following derivation is valid for any positive value of 𝑥𝑛𝑎. When 𝑥𝑛𝑎 ≥ 2𝑟 the entire cross-2

section is in compression. An angle 𝛼 also locates the neutral axis and is defined as follows 3

𝛼 = cos−1 (𝑟 − 𝑥𝑛𝑎𝑟

) if 𝑥𝑛𝑎 < 2𝑟 4

= 𝜋 otherwise 5

The maximum strain limits for concrete and steel, with compression strains positive, are 6

휀𝑐𝑢 = 0.003 7

휀𝑠𝑢 = −0.06 8

For a given neutral axis location, either the maximum concrete or maximum steel strain will 9

govern. The minimum and maximum strains on the cross-section are 10

휀min = max [ 휀𝑠𝑢 , 휀𝑐𝑢𝑥𝑛𝑎 − 2𝑟

𝑥𝑛𝑎] 11

휀max =1

max [ 1휀𝑐𝑢

,𝑥𝑛𝑎 − 2𝑟𝑥𝑛𝑎

휀min] 12

The strain at any angle 𝜃 between 0 and 𝜋 is 13

휀 = 휀max

𝑟 cos𝜃 − (𝑟 − 𝑥𝑛𝑎)

𝑥𝑛𝑎 14

The concrete force is divided into two regions; a region of parabolic-varying stress from the 15

neutral axis to a strain of 0.002 and a region of linearly-varying stress from a strain of 0.002 to 16

a strain of 0.003. The strain is 0.002 at a distance 𝑥0 from the point of maximum concrete 17

compressive strain (see Fig. A.4). This distance and the corresponding angle 𝜏 are defined as 18

follows 19

𝑥0 = 𝑥𝑛𝑎 (1 −0.002

휀max

) if ε < 휀max ; = 0 otherwise 20

125


𝜏 = 𝑐𝑜𝑠−1 (𝑟 − 𝑥0𝑟

) if x0 < 2r ; = π otherwise 1

The concrete compressive force in the linearly-varying stress region is 2

𝐹𝑐1 = 2∫ 0.85𝑓𝑐′𝑟𝑡(1 − 𝜌𝑡) [1.3 − 150 [(1 −

𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos 𝜃] 휀𝑚𝑎𝑥] 𝑑𝜃

𝜏

𝛽 3

= 𝐴1[𝑎1(𝜏 − 𝛽) + 𝑏1(sin 𝜏 − sin 𝛽)] 4

where 5

𝐴1 = 1.7𝑓𝑐′𝑟𝑡(1 − 𝜌𝑡) 6

𝑎1 = 1.3 − 150(1 −𝑟

𝑥𝑛𝑎) 휀𝑚𝑎𝑥 7

𝑏1 = −150𝑟

𝑥𝑛𝑎휀𝑚𝑎𝑥 8

The moment about the neutral axis due to force 𝐹𝑐1 is 9

𝐹𝑐1′ = 2∫ 0.85𝑓𝑐

′𝑟𝑡(1 − 𝜌𝑡) [1.3 − 150 [(1 −𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos 𝜃] 휀𝑚𝑎𝑥] [(𝑥𝑛𝑎 − 𝑟)

𝜏

𝛽

10

+ 𝑟 cos𝜃]𝑑𝜃 11

= 𝐴2 [𝑎2(𝜏 − 𝛽) + 𝑏2(sin 𝜏 − sin 𝛽) +𝑐22(sin 𝜏 cos 𝜏 − sin 𝛽 cos𝛽 + 𝜏 − 𝛽)] 12

where 13

𝐴2 = 1.7𝑓𝑐′𝑟𝑡(1 − 𝜌𝑡) 14

𝑎2 = [1.3 − 150(1 −𝑟

𝑥𝑛𝑎) 휀𝑚𝑎𝑥] (𝑥𝑛𝑎 − 𝑟) 15

𝑏2 = 𝑟 [1.3 − 150(1 −𝑟

𝑥𝑛𝑎) 휀𝑚𝑎𝑥] − 150

𝑟

𝑥𝑛𝑎휀𝑚𝑎𝑥(𝑥𝑛𝑎 − 𝑟) 16

𝑐2 = −150𝑟2

𝑥𝑛𝑎휀𝑚𝑎𝑥 17

The concrete stress is parabolic-varying from 0 strain to 0.002 strain, according to the fol-18

lowing stress-strain relationship 19

126


𝑓𝑐 = 0.85𝑓𝑐′ [

2휀

0.002− (

휀

0.002)2

] 1

Substituting the expression for strain at angle 𝜃 2

𝑓𝑐 = 0.85𝑓𝑐′ [2휀𝑚𝑎𝑥0.002

((1 −𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos𝜃)3

− (휀𝑚𝑎𝑥0.002

)2

((1 −𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos𝜃)

2

] 4

The concrete compressive force in the parabolic region is 5

6

𝐹𝑐2 = 2∫ 0.85𝑓𝑐′𝑟𝑡(1 − 𝜌𝑡) [

2휀𝑚𝑎𝑥0.002

((1 −𝑟

𝑥𝑛𝑎) +

𝑟


𝛼

𝜏

7

− (휀𝑚𝑎𝑥0.002

)2

((1 −𝑟

𝑥𝑛𝑎) +

𝑟


2

] 𝑑𝜃 8

= 𝐴3[𝑎3(𝛼 − 𝜏) + 𝑏3(sin 𝛼 − sin 𝜏) +𝑐32(sin 𝛼 cos𝛼 − sin 𝜏 cos 𝜏 + 𝛼 − 𝜏)] 9

where 10

𝐴3 = 1.7𝑓𝑐′𝑟𝑡(1 − 𝜌𝑡)

𝑚𝑎𝑥

0.002 11

𝑎3 = 2(1 −𝑟

𝑥𝑛𝑎) − 𝑚𝑎𝑥

0.002(1 −

𝑟

𝑥𝑛𝑎)2

12

𝑏3 = 2𝑟

𝑥𝑛𝑎[1 − 𝑚𝑎𝑥

0.002(1 −

𝑟

𝑥𝑛𝑎)] 13

𝑐3 = −𝑚𝑎𝑥

0.002(𝑟

𝑥𝑛𝑎)2

14

The moment about the neutral axis due to force 𝐹𝑐2 is 15

127


𝐹𝑐2′ = 2∫ 0.85𝑓𝑐

′𝑟𝑡(1 − 𝜌𝑡) [2휀𝑚𝑎𝑥0.002

((1 −𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos 𝜃)

𝛼

𝜏

1

− (휀𝑚𝑎𝑥0.002

)2

((1 −𝑟

𝑥𝑛𝑎) +

𝑟


2

] [(𝑥𝑛𝑎 − 𝑟) + 𝑟 cos𝜃]𝑑𝜃 2

= 𝐴4[𝑎4(𝛼 − 𝜏) + 𝑏4(sin 𝛼 − sin 𝜏) +𝑐42(sin 𝛼 cos𝛼 − sin 𝜏 cos 𝜏 + 𝛼 − 𝜏) 3

+𝑑43(sin 𝛼 cos2𝛼 − sin 𝜏 cos2𝜏 + 2(sin 𝛼 − sin 𝜏))] 4

where 5

𝐴4 = 1.7𝑓𝑐′𝑟𝑡(1 − 𝜌𝑡)

𝑚𝑎𝑥

0.002 6

𝑎4 = [2 (1 −𝑟


0.002(1 −

𝑟

𝑥𝑛𝑎)2

] (𝑥𝑛𝑎 − 𝑟) 7

𝑏4 = [2 (1 −𝑟


0.002(1 −

𝑟

𝑥𝑛𝑎)2

] 𝑟 + 2𝑟


0.002(1 −

𝑟

𝑥𝑛𝑎)] (𝑥𝑛𝑎 − 𝑟) 8

𝑐4 = 2𝑟


0.002(1 −

𝑟

𝑥𝑛𝑎)] 𝑟 − [ 𝑚𝑎𝑥

0.002(𝑟

𝑥𝑛𝑎)2

] (𝑥𝑛𝑎 − 𝑟) 9

𝑑4 = −[ 𝑚𝑎𝑥

0.002(𝑟

𝑥𝑛𝑎)2

] 𝑟 10

The steel compressive force is divided into two regions; a region of constant stress and a 11

region of linear-varying stress. The compressive steel stress is constant 𝑓𝑦 when the compres-12

sive strain exceeds the compressive yield strain 휀𝑦𝑐. The compressive strain equals the com-13

pressive yield strain at a distance 𝑥𝑦𝑐 from the point of maximum concrete compressive strain 14

(see Fig. B.1). The yield strain, the distance 𝑥𝑦𝑐 and the corresponding angle 𝜇 are defined as 15

follows 16

휀𝑦𝑐 =𝑓𝑦

𝐸𝑠 17

128


𝑥𝑦𝑐 = [1 −𝑦𝑐

max

] 𝑥𝑛𝑎 if 휀max > 휀𝑦𝑐 ; = 0 otherwise 1

𝜇 = cos−1 [𝑟−𝑥𝑦𝑐

𝑟] if 𝑥𝑦𝑐 < 2𝑟 ; = π otherwise 2

The steel compressive force where steel stress is at the yield strength 3

𝑆𝑐𝑦 = 2∫ 𝑓𝑦𝑟𝜌𝑡𝑡 𝑑𝜃𝜇

0 4

= 2𝑓𝑦𝑟𝜌𝑡𝑡𝜇 5

The moment about the neutral axis due to force 𝑆𝑐𝑦 is 6

𝑆𝑐𝑦′ = 2∫ 𝑓𝑦𝑟𝜌𝑡𝑡 (𝑥𝑛𝑎 − 𝑟 + 𝑟 cos 𝜃)𝑑𝜃

𝜇

0

7

= 2𝑓𝑦𝑟𝜌𝑡𝑡[(𝑥𝑛𝑎 − 𝑟)𝜇 + 𝑟 sin 𝜇] 8

The steel compressive force where steel stress is below the yield strength 9

𝑆𝑐 = 2∫ 𝐸𝑠𝛼

𝜇휀max [(1 −

𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos𝜃] 𝑟𝜌𝑡𝑡𝑑𝜃 10

= 2𝐸𝑠휀max𝑟𝜌𝑡𝑡 [(1 −𝑟

𝑥𝑛𝑎) (𝛼 − 𝜇) +

𝑟

𝑥𝑛𝑎(sin 𝛼 − sin 𝜇)] 11

The moment about the neutral axis due to force 𝑆𝑐 is 12

𝑆𝑐′ = 2∫ 𝐸𝑠

𝛼

𝜇휀max [(1 −

𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos𝜃] [(𝑥𝑛𝑎 − 𝑟) + 𝑟 cos𝜃]𝑟𝜌𝑡𝑡𝑑𝜃 13

= 𝐴5 [𝑎5(𝛼 − 𝜇) + 𝑏5(sin 𝛼 − sin 𝜇) +𝑐5

2 (sin 𝛼 cos𝛼 − sin 𝜇 cos𝜇 + 𝛼 − 𝜇)] 14

where 15

𝐴5 = 2𝐸𝑠𝑟𝜌𝑡𝑡휀max 16

𝑎5 = (1 −𝑟

𝑥𝑛𝑎) (𝑥𝑛𝑎 − 𝑟) 17

𝑏5 = (1 −𝑟

𝑥𝑛𝑎) 𝑟 +

𝑟

𝑥𝑛𝑎(𝑥𝑛𝑎 − 𝑟) 18

𝑐5 =𝑟2

𝑥𝑛𝑎 19

129


The steel tensile force is divided into two regions; a region of constant stress and a region of 1

linear-varying stress. The tensile steel stress is constant −𝑓𝑦 when the tensile strain exceeds 2

the tensile yield strain 휀𝑦𝑡. The tensile strain equals the tensile yield strain at a distance 𝑥𝑦𝑡 3

from the point of maximum concrete compressive strain (see Fig. B.1). The tensile yield strain, 4

the distance 𝑥𝑦𝑡 and the corresponding angle 𝜓 are defined as follows 5

휀𝑦𝑡 = −𝑓𝑦

𝐸𝑠 6

𝑥𝑦𝑡 = (1 −𝑦𝑡

𝑚𝑎𝑥) 𝑥𝑛𝑎 7

𝜓 = cos−1 [𝑟−𝑥𝑦𝑡

𝑟] if 𝑥𝑦𝑡 < 2𝑟 ; = 𝜋 otherwise 8

The steel tensile force where the steel stress is at the yield strength is 9

𝑆𝑡𝑦 = −2∫ 𝑓𝑦𝑟𝜌𝑡𝑡 𝑑𝜃𝜋

𝜓 10

= −2𝑓𝑦𝑟𝜌𝑡𝑡(𝜋 − 𝜓) 11

The moment about the neutral axis due to force 𝑆𝑡𝑦 is 12

𝑆𝑡𝑦′ = −2∫ 𝑓𝑦𝑟𝜌𝑡𝑡 (𝑥𝑛𝑎 − 𝑟 + 𝑟 cos𝜃)𝑑𝜃

𝜋

𝜓 13

= −2𝑓𝑦𝑟𝜌𝑡𝑡[(𝑥𝑛𝑎 − 𝑟)(𝜋 − 𝜓) − 𝑟 sin𝜓] 14

The steel tensile force where the steel stress is below the yield strength is 15

𝑆𝑡 = 2∫ 𝐸𝑠𝜓

𝛼휀max [(1 −

𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos 𝜃] 𝑟𝜌𝑡𝑡𝑑𝜃 16

= 2𝐸𝑠휀max𝑟𝜌𝑡𝑡 [(1 −𝑟

𝑥𝑛𝑎) (𝜓 − 𝛼) +

𝑟

𝑥𝑛𝑎(sin 𝜓 − sin 𝛼)] 17

The moment about the neutral axis due to force 𝑆𝑡 is 18

𝑆𝑡′ = 2∫ 𝐸𝑠

𝜓

𝛼휀max [(1 −

𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos𝜃] [(𝑥𝑛𝑎 − 𝑟) + 𝑟 cos𝜃]𝑟𝜌𝑡𝑡𝑑𝜃 19

= 𝐴6 [𝑎6(𝜓 − 𝛼) + 𝑏6(sin 𝜓 − sin 𝛼) +𝑐6

2(sin𝜓 cos𝜓 − sin 𝛼 cos𝛼 + 𝜓 − 𝛼)] 20

130


where 1

𝐴6 = 2𝐸𝑠𝑟𝜌𝑡𝑡휀max 2

𝑎6 = (1 −𝑟

𝑥𝑛𝑎) (𝑥𝑛𝑎 − 𝑟) 3

𝑏6 = (1 −𝑟

𝑥𝑛𝑎) 𝑟 +

𝑟

𝑥𝑛𝑎(𝑥𝑛𝑎 − 𝑟) 4

𝑐6 =𝑟2

𝑥𝑛𝑎 5

6

Cross-sections with two equal openings in the compression zone 7

The case of two equal openings completely in or partially in the compression zone is shown 8

in Figs. B.2 and B.3, respectively. For this case, the force equilibrium is modified by subtract-9

ing the concrete compressive force that was included in 𝐶1 + 𝐶2 at the openings. 10

𝑃𝑢𝜙= 𝑃𝑛 = 𝐹𝑐1 + 𝐹𝑐2 − 𝐹𝑂𝑐1 − 𝐹𝑂𝑐2 + 𝑆𝑐 + 𝑆𝑐𝑦 + 𝑆𝑡 + 𝑆𝑡𝑦 11

The required extra terms are: 12

𝐹𝑂𝑐1 = 2∫ 0.85𝑓𝑐′𝑟𝑡(1 − 𝜌𝑡) [1.3 − 150 [(1 −

𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos 𝜃] 휀𝑚𝑎𝑥] 𝑑𝜃

𝜆2

𝜆1

13

= 𝐴1[𝑎1(𝜆2 − 𝜆1) + 𝑏1(sin 𝜆2 − sin 𝜆1)] 14

where 15

𝜆1 = min(𝜏, 𝛾 − 𝛽) 16

𝜆2 = min(𝜏, 𝛾 + 𝛽) 17

𝐹𝑂𝑐2 = 2∫ 0.85𝑓𝑐′𝑟𝑡(1 − 𝜌𝑡) [

2휀𝑚𝑎𝑥0.002

((1 −𝑟

𝑥𝑛𝑎) +

𝑟


𝛿2

𝛿1

18

− (휀𝑚𝑎𝑥0.002

)2

((1 −𝑟

𝑥𝑛𝑎) +

𝑟


2

] 𝑑𝜃 19

131


= 𝐴3 [𝑎3(𝛿2 − 𝛿1) + 𝑏3(sin 𝛿2 − sin 𝛿1) +𝑐32(sin 𝛿2 cos 𝛿2 − sin 𝛿1 cos 𝛿1 + 𝛿2 − 𝛿1)] 1

where 2

𝛿1 = min[max(𝜏, 𝛾 − 𝛽), 𝛼] 3

𝛿2 = max[min(𝛼, 𝛾 + 𝛽), 𝜏] 4

The moment equilibrium equation is modified in a similar way. 5

𝑀𝑛 = 𝑃𝑛(𝑟 − 𝑥𝑛𝑎) + 𝐹𝑐1′ + 𝐹𝑐2

′ − 𝐹𝑂𝑐1′ − 𝐹𝑂𝑐2

′ + 𝑆𝑐′ + 𝑆𝑐𝑦

′ + 𝑆𝑡′ + 𝑆𝑡𝑦

′ 6

The required extra terms are: 7

𝐹𝑂𝑐1′ = 2∫ 0.85𝑓𝑐

′𝑟𝑡(1 − 𝜌𝑡) [1.3 − 150 [(1 −𝑟

𝑥𝑛𝑎) +

𝑟

𝑥𝑛𝑎cos 𝜃] 휀𝑚𝑎𝑥] [(𝑥𝑛𝑎 − 𝑟)

𝜆2

𝜆1

8

+ 𝑟 cos𝜃]𝑑𝜃 9

= 𝐴2 [𝑎2(𝜆2 − 𝜆1) + 𝑏2(sin 𝜆2 − sin 𝜆2) +𝑐22(sin 𝜆2 cos 𝜆2 − sin 𝜆1 cos 𝜆1 + 𝜆2 − 𝜆1)] 10

11

𝐹𝑂𝑐2′ = 2∫ 0.85𝑓𝑐

′𝑟𝑡(1 − 𝜌𝑡) [2휀𝑚𝑎𝑥0.002

((1 −𝑟

𝑥𝑛𝑎) +

𝑟


𝛿2

𝛿1

12

− (휀𝑚𝑎𝑥0.002

)2

((1 −𝑟

𝑥𝑛𝑎) +

𝑟


2

] [(𝑥𝑛𝑎 − 𝑟) + 𝑟 cos𝜃]𝑑𝜃 13

= 𝐴4 [𝑎4(𝛿2 − 𝛿1) + 𝑏4(sin 𝛿2 − sin 𝛿1) +𝑐4

2(sin 𝛿2 cos 𝛿2 − sin 𝛿1 cos 𝛿1 + 𝛿2 − 𝛿1) 14

+𝑑4

3(sin 𝛿2 cos2𝛿2 − sin 𝛿1 cos2𝛿1 + 2 sin 𝛿2 − 2 sin 𝛿1) 15

132


1

Fig. B.1: Stress-strain relationships, no opening or one opening 2

3

133


1 Fig. B.2: Two openings fully in the compression zone 2

(Dimensions not shown; same as Fig. B.1) 3

4

5

Fig. B.3: Two openings partially in the compression zone 6

(Dimensions not shown; same as Fig. B.1) 7

8

134


APPENDIX C—TEMPERATURE GRADIENTS FOR CIRCULAR CHIMNEYS 1

C.1–Scope 2

Appendix C consists of derivations of the equations of 7.4 to determine the temperature gradi-3

ents through the concrete wall of circular chimneys for the following scenarios: 4

a) Unlined chimney 5

b) Chimney with lining material applied directly to the inside concrete surface 6

c) Chimney with insulation completely filling the space between the liner and the chimney 7

wall (no air space) 8

d) Chimney with a liner, optional insulation, and an unventilated air space 9

e) Chimney with a liner, optional insulation, and a ventilated air space 10

C.2–Notation 11

The following notation is used in Appendix C only. Any notation used in the Code and in Ap-12

pendix C is defined in 2.2. Units listed for symbols in 2.2 are consistent with units for symbols 13

listed below. Note in 2.2, diameter units are feet and thickness units are inches. This is consistent 14

with the typical published units for thermal conductivity in U.S. customary units where the con-15

ductivity is expressed per square feet of conducting area per inch thickness of the conducting ma-16

terial (BTU-in/hr-ft2-°F). 17

𝑇1,⋯ , 𝑇4 = various surface temperatures, as defined, °F

135


𝑄 = heat transfer from flue gas to the inside surface of liner or to the inside

surface of liner material applied directly to the concrete wall or to the

concrete wall (when the chimney is unlined), Btu/(hr-ft2)

1

C.3–Unlined chimney 2

Refer to Fig. C.1. In this case, all of the heat leaving the flue gas is transmitted through the 3

concrete wall to the ambient air. The temperature gradient through the concrete wall is 𝑇𝑥 = 𝑇1 −4

𝑇2. Let 𝑄 be the amount of heat transmitted through a unit area on the inside surface of the concrete 5

wall. 6

𝑄 = 𝐾𝑖(𝑇𝑖 − 𝑇1) =𝐶𝑐𝑑𝑐𝑡𝑐𝑑𝑐𝑖

(𝑇1 − 𝑇2) =𝐾𝑜𝑑𝑐𝑜𝑑𝑐𝑖

(𝑇2 − 𝑇𝒂𝒎𝒃) 7

From above, 8

𝑇𝑖 − 𝑇1 =𝑄

𝐾𝑖 𝑇1 − 𝑇2 =

𝑄𝑡𝑐𝑑𝑐𝑖𝐶𝑐𝑑𝑐

= 𝑇𝑥 𝑇2 − 𝑇𝒂𝒎𝒃 =𝑄𝑑𝑐𝑖𝐾𝑜𝑑𝑐𝑜

The difference between the known temperatures 𝑇𝑖 and 𝑇𝒂𝒎𝒃 can be expressed as 9

𝑇𝑖 − 𝑇𝑜 = (𝑇𝑖 − 𝑇1) + (𝑇1 − 𝑇2) + (𝑇2 − 𝑇𝒂𝒎𝒃) = 𝑄 [1

𝐾𝑖+𝑡𝑐𝑑𝑐𝑖𝐶𝑐𝑑𝑐


] 10

Equating two expressions for 𝑄 11

𝑄 =𝑇𝑖 − 𝑇𝒂𝒎𝒃



=𝐶𝑐𝑑𝑐𝑡𝑐𝑑𝑐𝑖

𝑇𝑥 12

The temperature gradient can then be expressed as 13

𝑇𝑥 =𝑡𝑐𝑑𝑐𝑖𝐶𝑐𝑑𝑐

[𝑇𝑖 − 𝑇𝒂𝒎𝒃



] (C-1)

14

136


1

Fig. C.1 2

C.4–Chimney with lining material applied directly to the inside concrete surface 3

Refer to Fig. C.2. In this case, all of the heat leaving the flue gas is transmitted through the lining 4

material and the concrete wall to the ambient air. The temperature gradient through the concrete 5

wall is 𝑇𝑥 = 𝑇2 − 𝑇3. Let 𝑄 be the amount of heat transmitted through a unit area on the inside 6

surface of the lining material. 7

8

𝑄 = 𝐾𝑖(𝑇𝑖 − 𝑇1) =𝐶𝑏𝑑𝑏𝑡𝑏𝑑𝑏𝑖

(𝑇1 − 𝑇2) =𝐶𝑐𝑑𝑐𝑡𝑐𝑑𝑏𝑖

(𝑇2 − 𝑇3) =𝐾𝑜𝑑𝑐𝑜𝑑𝑏𝑖

(𝑇3 − 𝑇𝒂𝒎𝒃) 9

From above, 10


𝐾𝑖 𝑇1 − 𝑇2 =

𝑄𝑡𝑏𝑑𝑏𝑖𝐶𝑏𝑑𝑏

𝑇2 − 𝑇3 =𝑄𝑡𝑐𝑑𝑏𝑖𝐶𝑐𝑑𝑐

= 𝑇𝑥 𝑇3 − 𝑇𝒂𝒎𝒃 =𝑄𝑑𝑏𝑖𝐾𝑜𝑑𝑐𝑜

11


𝑇𝑖 − 𝑇𝑜 = (𝑇𝑖 − 𝑇1) + (𝑇1 − 𝑇2) + (𝑇2 − 𝑇3) + (𝑇3 − 𝑇𝒂𝒎𝒃)13

= 𝑄 [1

𝐾𝑖+𝑡𝑏𝑑𝑏𝑖𝐶𝑏𝑑𝑏



] 14


137






=𝐶𝑐𝑑𝑐𝑡𝑐𝑑𝑏𝑖

𝑇𝑥 1







] (C-2)

3

4

Fig. C.2 5

6

C.5–Chimney with insulation completely filling the space between the liner and the chimney 7

wall (no annular airspace) 8

Refer to Fig. C.3. In this case, all of the heat leaving the flue gas is transmitted through the liner, 9

the insulation, and the concrete wall to the ambient air. The temperature gradient through the con-10

crete wall is 𝑇𝑥 = 𝑇3 − 𝑇4. Let 𝑄 be the amount of heat transmitted through a unit area on the 11

inside surface of the liner. 12

𝑄 = 𝐾𝑖(𝑇𝑖 − 𝑇1) =𝐶𝑏𝑑𝑏

𝑡𝑏𝑑𝑏𝑖(𝑇1 − 𝑇2) =

𝐶𝑠𝑑𝑠

𝑡𝑠𝑑𝑏𝑖(𝑇2 − 𝑇3) =

𝐶𝑐𝑑𝑐

𝑡𝑐𝑑𝑏𝑖(𝑇3 − 𝑇4) =

𝐾𝑜𝑑𝑐𝑜

𝑑𝑏𝑖(𝑇4 − 𝑇𝒂𝒎𝒃) 13

From above 14

138



𝐾𝑖 𝑇1 − 𝑇2 =


𝑇2 − 𝑇3 =𝑄𝑡𝑠𝑑𝑏𝑖𝐶𝑠𝑑𝑠







+𝑡𝑠𝑑𝑏𝑖𝐶𝑠𝑑𝑠



] (C-3)

2

3

Fig. C.3 4

5

C.6–Chimney with unventilated air space between liner and chimney wall 6

Refer to Fig. C.4. In this case, all of the heat leaving the flue gas is transmitted through the liner, 7

the air space and the chimney wall. The temperature gradient through the concrete wall is 𝑇𝑥 =8

𝑇3 − 𝑇4. Let 𝑄 be the amount of heat transmitted through a unit area on the inside surface of the 9

liner. 10

𝑄 = 𝐾𝑖(𝑇𝑖 − 𝑇1) =𝐶𝑏𝑑𝑏𝑡𝑏𝑑𝑏𝑖

(𝑇1 − 𝑇2) =𝐾𝑟𝑑𝑠𝑑𝑏𝑖


(𝑇3 − 𝑇4) =𝐾𝑜𝑑𝑐𝑜𝑑𝑏𝑖

(𝑇4 − 𝑇𝒂𝒎𝒃) 11

From above 12

139



𝐾𝑖 𝑇1 − 𝑇2 =


𝑇2 − 𝑇3 =𝑄𝑑𝑏𝑖𝐾𝑟𝑑𝑠




𝑇𝑖 − 𝑇𝑜 = (𝑇𝑖 − 𝑇1) + (𝑇1 − 𝑇2) + (𝑇2 − 𝑇3) + (𝑇3 − 𝑇4) + (𝑇4 − 𝑇𝒂𝒎𝒃)2

= 𝑄 [1





] 3







=𝐶𝑐𝑑𝑐𝑡𝑐𝑑𝑏𝑖

𝑇𝑥 5








] (C-4)

7

8

Fig. C.4 9

10

C.7–Chimney with ventilated air space between liner and chimney wall 11

140


Refer to Fig. C.4. In this case, all of the heat leaving the flue gas is transmitted through the liner. 1

Some of this heat is carried away by the ventilated air space between the liner and the chimney 2

wall. The remainder of the heat is transmitted through the concrete wall. The temperature gradient 3

through the concrete wall is 𝑇𝑥 = 𝑇3 − 𝑇4. Let 𝑄 be the amount of heat transmitted through a unit 4

area on the inside surface of the liner and let 𝑟𝑞𝑄 be the amount of heat transmitted through the 5

chimney wall. 6

𝑟𝑞𝑄 = 𝑟𝑞𝐾𝑖(𝑇𝑖 − 𝑇1) = 𝑟𝑞𝐶𝑏𝑑𝑏𝑡𝑏𝑑𝑏𝑖

(𝑇1 − 𝑇2) =𝐾𝑠𝑑𝑠𝑑𝑏𝑖


(𝑇3 − 𝑇4)7

=𝐾𝑜𝑑𝑐𝑜𝑑𝑏𝑖

(𝑇4 − 𝑇𝒂𝒎𝒃) 8

From above 9


𝐾𝑖 𝑇1 − 𝑇2 =


𝑇2 − 𝑇3 =𝑟𝑞𝑄𝑑𝑏𝑖𝐾𝑠𝑑𝑠

𝑇3 − 𝑇4 =𝑟𝑞𝑄𝑡𝑐𝑑𝑏𝑖𝐶𝑐𝑑𝑐

= 𝑇𝑥 𝑇4 − 𝑇𝒂𝒎𝒃 =𝑟𝑞𝑄𝑑𝑏𝑖𝐾𝑜𝑑𝑐𝑜


𝑇𝑖 − 𝑇𝑜 = (𝑇𝑖 − 𝑇1) + (𝑇1 − 𝑇2) + (𝑇2 − 𝑇3) + (𝑇3 − 𝑇4) + (𝑇4 − 𝑇𝒂𝒎𝒃)11

= 𝑄 [1


+𝑟𝑞𝑑𝑏𝑖𝐾𝑠𝑑𝑠

+𝑟𝑞𝑡𝑐𝑑𝑏𝑖𝐶𝑐𝑑𝑐

+𝑟𝑞𝑑𝑏𝑖𝐾𝑜𝑑𝑐𝑜

] 12




+𝑟𝑞𝑑𝑏𝑖𝐾𝑠𝑑𝑠

+𝑟𝑞𝑡𝑐𝑑𝑏𝑖𝐶𝑐𝑑𝑐

+𝑟𝑞𝑑𝑏𝑖𝐾𝑜𝑑𝑐𝑜

=𝐶𝑐𝑑𝑐𝑟𝑞𝑡𝑐𝑑𝑏𝑖

𝑇𝑥 14


141



[(𝑇𝑖 − 𝑇𝒂𝒎𝒃)

1𝑟𝑞𝐾𝑖

+𝑡𝑏𝑑𝑏𝑖𝑟𝑞𝐶𝑏𝑑𝑏

+𝑑𝑏𝑖𝐾𝑠𝑑𝑠



] (C-5)

1

142


APPENDIX D—THERMAL STRESSES FOR CIRCULAR CHIMNEYS 1 2

D.1–Scope 3

Appendix D consists of derivations of equations to determine the stresses due to a thermal gra-4

dient across the chimney wall. 5

D.2–Notation 6

The following notation is used in Appendix D only. Any notation used in the Code and in Ap-7

pendix D is defined in 2.2. 8

휀𝑐 = concrete thermal strain

휀𝑠 = steel thermal strain

𝜃𝑡𝑒 = unrestrained rotation caused by temperature gradient

9

D.3–Vertical thermal stresses 10

The equations for maximum vertical stresses in concrete and steel due to a temperature drop 11

only, across the concrete wall with two layers of reinforcement, are derived as follows. 12

The unrestrained rotation caused by a temperature gradient of 𝑇𝑥 is shown in Fig. D.1(a): 13

𝜃𝑡𝑒 =𝛼𝑡𝑒𝑇𝑥𝑡𝑐

14

Since rotation is prevented, stresses are induced as shown in Fig. D.1(b). The concrete strain and 15

stress at the inside surface is 16

휀𝑐 = 𝜃𝑡𝑒𝑐𝑡𝑐 = 𝛼𝑡𝑒𝑇𝑥𝑐 17

𝑓𝐶𝑇𝑉′′ = 𝐸𝑐휀𝑐 = 𝛼𝑡𝑒𝑐𝑇𝑥𝐸𝑐 18

The steel strain and stress in the outside reinforcement is 19

휀𝑠 = 𝜃𝑡𝑒(𝛾2𝑜 − 𝑐)𝑡𝑐 20

143


𝑓𝑆𝑇𝑉 = 𝐸𝑠휀𝑠 = 𝛼𝑡𝑒(𝛾2𝑜 − 𝑐)𝑇𝑥𝐸𝑠 1

The inside steel stress is 2

𝑓𝑆𝑇𝑉′′ =

(𝑐 − 1 + 𝛾2𝑖)

𝑐𝑛𝑓𝐶𝑇𝑉

′′ 3

𝑓𝑆𝑇𝑉′′ = 𝛼𝑡𝑒(𝑐 − 1 + 𝛾2𝑖)𝑇𝑥𝑛𝐸𝑐 4

The sum of vertical forces must equal zero, so 5

𝑓𝐶𝑇𝑉′′ (

𝑐𝑡𝑐2) + 𝑓𝑆𝑇𝑉

′′ 𝛾1𝜌𝑜𝑡𝑐 − 𝑓𝑆𝑇𝑉𝜌𝑜𝑡𝑐 = 0 6

𝛼𝑡𝑒𝑐𝑇𝑥𝐸𝑐 (𝑐𝑡𝑐2) + 𝛼𝑡𝑒(𝑐 − 1 + 𝛾2𝑖)𝑇𝑥𝑛𝐸𝑐𝛾1𝜌𝑜𝑡𝑐 7

−𝛼𝑡𝑒(𝛾2𝑜 − 𝑐)𝑇𝑥𝑛𝐸𝑐𝜌𝑜𝑡𝑐 = 0 8

𝑐2 + 2𝑛𝛾1𝜌𝑜𝑐 + 2𝑛𝛾1𝜌𝑜(𝛾2 − 1) + 2𝑛𝜌𝑜𝑐 − 2𝑛𝜌𝑜𝛾2 = 0 9

𝑐2 + 2𝜌𝑜𝑛(𝛾1 + 1)𝑐 − 2𝜌𝑜𝑛[𝛾2𝑜 + 𝛾1(1 − 𝛾2𝑖)] = 0 10

Solving for 𝑐, 11

𝑐 = −𝜌𝑜𝑛(𝛾1 + 1) + √[𝜌𝑜𝑛(𝛾1 + 1)]2 + 2𝜌𝑜𝑛[𝛾2𝑜 + 𝛾1(1 − 𝛾2𝑖)] 12

144


1

Fig. D.1: Vertical temperature stresses 2

3

4

145


D.5–Horizontal thermal stresses 1

The derivation of equations for the maximum horizontal stresses in concrete and steel due to a 2

temperature drop only, across the concrete wall with two layers of reinforcement, is similar to 3

that for the vertical temperature stresses. 4

Replace 𝜌𝑜 with 𝜌𝑜′ 5

𝛾1 with 𝛾1′ 6

𝑓𝐶𝑇𝑉′′ with 𝑓𝐶𝑇𝐶

′′ 7

𝑓𝑆𝑇𝑉 with 𝑓𝑆𝑇𝐶 8

𝑐 with 𝑐′ 9

𝛾2𝑜 with 𝛾2𝑜′ 10

𝛾2𝑖 with 𝛾2𝑖′ 11

then 12

𝑓𝐶𝑇𝐶′′ = 𝛼𝑡𝑒𝑐′𝑇𝑥𝐸𝑐 13

𝑓𝑆𝑇𝐶 = 𝛼𝑡𝑒(𝛾2𝑜′ − 𝑐′)𝑇𝑥𝐸𝑠 14

𝑐 = −𝜌𝑜′𝑛(𝛾1

′ + 1) + √[𝜌𝑜′𝑛(𝛾1′ + 1)]2 + 2𝜌𝑜′𝑛[𝛾2𝑜

′ + 𝛾1′(1 − 𝛾2𝑖

′ )] 15

16

146


APPENDIX E—CRACKING MOMENT 1 2

E.1–Scope 3

Appendix E defines the cracking moment. 4

E.2–Notation 5

The following notation is used in Appendix E only. Any notation used in the Code and in Ap-6

pendix E is defined in 2.2. All equations in this appendix are to be evaluated using consistent 7


and length units listed in 2.2 will not necessarily be the units used for the symbol in this appen-9

dix. 10

𝐴𝑔(𝑧) = uncracked (gross) area of concrete cross-section at height 𝑧

(openings may be disregarded)

𝑓𝑎(𝑧) = compressive stress at height z due to vertical load 𝑃(𝑧)

𝑓𝑏(𝑧) = bending stress at height z due to cracking moment 𝑀𝑐𝑟(𝑧)

𝐼𝑔(𝑧) = moment of inertia of uncracked (gross) concrete cross-section at height 𝑧

(openings may be disregarded)

𝑃(𝑧) = vertical load at height 𝑧

11

E.3–Cracking moment calculation 12

Cracking initiates when the concrete tension (computed at the middle of the wall for conven-13

ience) is equal to the modulus of rupture 𝑓𝑟 . From Fig. E.1, this occurs when the bending stress is 14

𝑓𝑏(𝑧). 15

𝑓𝑏(𝑧) = 𝑓𝑟 + 𝑓𝑎(𝑧) = 𝑓𝑟 +𝑃(𝑧)

𝐴𝑔(𝑧) 16

The cracking moment 𝑀𝑐𝑟(𝑧) is the moment that yields the bending stress 𝑓𝑏(𝑧). 17

147


𝑀𝑐𝑟(𝑧) =𝑓𝑏(𝑧)𝐼𝑔(𝑧)

𝑟(𝑧)= [𝑓𝑟 +

𝑃(𝑧)

𝐴𝑔(𝑧)] [𝐼𝑔(𝑧)

𝑟(𝑧)] 1

2

Fig. E.1 3

4

148


COMMENTARY REFERENCES 1

American Concrete Institute 2

ACI 207.1-05 Guide to Mass Concrete (Reapproved 2012) 3

ACI 307-08 Code Requirements for Reinforced Concrete Chimneys and Commentary 4

ACI 318-19 Building Code Requirements for Structural Concrete 5

ACI 209R-92 Prediction of Creep, Shrinkage, and Temperature Effects in Concrete 6

(Reapproved 2008) Structures 7

CT-18 ACI Concrete Terminology 8

American Society of Civil Engineers 9

ASCE 7-16 Minimum Design Loads for Buildings and Other Structures 10

ASCE Task Committee on Wind Forces, 1961, “Wind Forces on Structures,” Transactions, 11

ASCE, V. 126, Part II, pp. 1124-1198. 12

13

Authored Documents 14

Basu, R. I., 1982, “Across-wind Responses of Slender Structures of Circular Cross-Section to 15

Atmospheric Turbulence,” PhD thesis, Faculty of Engineering Science, University of Western 16

Ontario, London, ON, Canada. 17

Davenport, A. G., 1967, “Gust Loading Factors,” Proceedings, ASCE, V. 93, No. ST3, June, 18

pp. 11-34. 19

Dryden, H. H., and Hill, G. C., 1930, “Wind Pressure on Circular Cylinders and Chimneys,” 20

Research Paper No. 221, National Bureau of Standards, Washington, DC. Also, NBS Journal of 21

Research, V. 5, Sept. 22

149


Hognestad, E., 1951, “Study of Combined Bending and Axial Load in Reinforced Concrete 1

Members,” Bulletin No. 399, Engineering Experiment Station, University of Illinois, Urbana, IL, 2

128 pp. 3

Kilic, S. A., Altay, S. and Akyniyazov, D. 2016, “Investigation of the Benefits of Additional 4

Rebars for the Cyclic Response of Reinforced Concrete Chimneys,”, CICIND Research Report. 5

Mattock, A. H., Kriz, L. B., and Hognestad, E., “Rectangular Concrete Stress Distribution in 6

Ultimate Strength Design,” ACI Journal Proceedings, V. 57, No. 8, Feb., pp. 875-928. 7

Mokrin, Z.A.R., and Rumman, W.S., 1985, “Ultimate Capacity of Reinforced Concrete 8

Members of Hollow Circular Sections Subjected to Monotonic and Cyclic Bending,” ACI 9

Journal Proceedings V. 82, No. 5, Sept.-Oct., pp. 653-656. 10

Okamoto, T., and Yagita, M. 1973, “The Experimental Investigation on the Flow Past a Circu-11

lar Cylinder of Finite Length Placed Normal to the Plane Surface in a Uniform Stream,” Bulletin, 12

Japanese Society Of Mechanical Engineers, No. 16, 805 pp. 13

Radecki, D. J. 2014, “Recommendations for ACI 307-1x Across-wind/Along-wind Load 14

Combination Consistent with ASCE 7-10 Strength Design Wind Provisions,” International 15

Symposium on Industrial Chimneys and Cooling Towers, Prague, Czech Republic, pp. 421-429. 16

Rumman, W. S., and Sun, R.T., 1977, “Ultimate Strength Design of Reinforced Concrete 17

Chimneys,” ACI Journal, Proceedings V. 74, No. 4, Apr., pp. 179-184. 18

Ruscheweyh, H., 1984, “Problems with In-Line Stacks: Experience with Full-Scale Objects,” 19

Engineering Structures, V. 6, No. 4, Guilford, Oct., pp. 340-343. 20

Simiu, E.; Marshall, R. D.; and Haber, S., 1977, “Estimation of Along-Wind Building 21

Response,” Proceedings, ASCE, V. 103, No. ST7, July, pp. 1325-1338. 22

150


Vickery, B. J., 1969, “On the Reliability of Gust Loading Factors,” Wind Loads on Buildings 1

and Structures, Building Science Series No. 30, National Bureau of Standards, Washington, DC, 2

pp. 93-104. 3

Vickery, B. J., and Basu, R. I. 1984, “ Response of Reinforced Concrete Chimneys to Vortex 4

Shedding,” Engineering Structures, V. 6, No. 4, pp. 324-333. 5

Vickery, B. J., and Basu, R. I. 1983, “Simplified Approaches to the Evaluation of the Across-6

Wind Response of Chimneys,” Journal of Wind Engineering and Industrial Aerodynamics, 7

V. 14, Amsterdam, pp. 153-166. 8

Vickery, B. J., and Daly, A., 1984, “Wind Tunnel Modelling as a Means of Predicting the 9

Response of Chimneys to Vortex Shedding,” Engineering Structures, V. 6, No. 4, Guildford, 10

Oct., pp. 363-368. 11

Vickery, B. J., 1993, “Across-Wind Loading on Reinforced Concrete Chimneys of Circular 12

Cross Section,” Boundary Layer Wind Tunnel Report, BLWT-3-1993, University of Western 13

Ontario, Dec. 14

Wilson, J. L. 2003, “Earthquake Response of Tall Reinforced Concrete Chimneys,” 15

Engineering Structures, V. 25, No. 1, pp. 11-24. 16

Zdravkokvich, M. M., 1977, “Review of Flow Interference Effects between Two Cylinders in 17

Various Arrangements,” Journal of Fluids Engineering,V. 99, p. 618. 18

COMMITTEE STUDIES 19

ACI Committee 307, 1995, “ACI 307-95 vs. ACI 307-88 Across-wind Load Comparison” 20

ACI Committee 307, 2020a, “ACI 307-20 vs. ACI 307-08 Wind Load Comparison” 21

ACI Committee 307, 2020b, “ACI 307-20 Wind Deflection Study” 22