code requirements for reinforced concrete chimneys …

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ACI 307-20 (Draft 0.5, 11 Nov 2020) 1

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CODE REQUIREMENTS FOR REINFORCED CONCRETE CHIMNEYS 4

AND COMMENTARY 5

Reported by ACI Committee 307 6

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Denis J. Radecki

Chair

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David J. Bird Shu-Jin Fang Joseph Peters F. Alan Wiley

Victor A. Bochicchio Jon Galsworthy Kelly D. Scott John Wilson

David C. Mattes John C. Sowizal

Special acknowledgements to Robert A. Porthouse (deceased) and Scott D. Richart (de-9

ceased) for their contributions to this Code. 10

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This Code provides material, design and detailing requirements for cast-in-place and precast 12

reinforced concrete chimneys. It sets forth minimum loadings for design and contains methods 13

for determining the concrete and reinforcement required to obtain the strength required by the 14

loadings. The methods of analysis apply primarily to circular chimney walls, but guidance is 15

included for applying the general principles to noncircular chimney walls. 16

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Keywords: reinforced concrete chimneys; wind load; across-wind load; earthquake load; ther-1

mal load; bending capacity; structural design; reinforced concrete; vortex shedding; load com-2

binations 3

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CONTENTS 1

Chapter 1—General 2

1.1—Scope 3

1.2—General 4

1.3—Purpose 5

1.4—Applicability 6

1.5—Interpretation 7

1.6—Building official 8

1.7—Licensed design professional 9

1.8—Construction documents and design records 10

1.9––Testing and inspection 11

Chapter 2—Notation and terminology 12

2.1—Scope 13

2.2—Notation 14

2.3—Terminology 15

Chapter 3—Referenced standards 16

3.1—Scope 17

3.2—Referenced standards 18

Chapter 4—Structural system requirements 19

4.1—Scope 20

4.2—Materials 21

4.3—Design loads 22

4.4—Structural system and load paths 23

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Chapter 5—Concrete and concrete materials 1

5.1—Scope 2

5.2—General 3

5.3—Materials 4

5.4—Concrete properties for design 5

5.5—Durability requirements 6

Chapter 6—Reinforcing steel 7

6.1—Scope 8

6.2—General 9

6.3—Materials 10

6.4—Steel properties for design 11


Chapter 7—Loads 13

7.1—Scope 14

7.2—General 15

7.3—Dead load 16

7.4—Temperature gradient load 17

7.5—Wind load 18

7.6—Earthquake load 19

7.7—Load combinations 20

Chapter 8—Strength of horizontal sections 21

8.1—Scope 22

8.2—General 23

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8.3—Design limits 1

8.4—Required Strength 2

8.5—Design strength 3

8.6—Reinforcement limits 4

8.7—Reinforcement detailing 5

Chapter 9—Strength of vertical sections 6

9.1—Scope 7

9.2—General 8


9.4—Required Strength 10




Chapter 10—Opening details 14

10.1—Scope 15

10.2—General 16

10.3—Minimum wall thickness at openings 17

10.4—Vertical reinforcement at openings 18

10.5—Circumferential reinforcement at openings 19

10.6—Diagonal reinforcement at openings 20

10.7—Special seismic detailing 21

Chapter 11—Foundation 22

11.1—Scope 23

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11.2—Soil bearing pressure 1

11.3—Foundation design 2

11.4—Overturning 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

Commentary references 15

Appendix A––Horizontal section strength by modified stress block method 16

A.1—Procedure for computing bending moment capacity 17

A.2—Stress block modification factor 18

A.3—Derivation of equations 19

Appendix B––Horizontal section strength by stress law integration method 20

B.1—Procedure for computing combined nominal compression and nominal bending capac-21

ity 22

B.2—Derivation of equations 23

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Appendix C––Derivation of temperature gradient equations for circular chimneys 1

C.1—Unlined chimneys 2

C.2—Chimneys with lining material applied directly to the inside surface (no air space) 3

C.3—Lined chimneys with unventilated air space between lining and chimney wall 4

C.4—Lined chimneys with ventilated air space between lining and chimney wall 5

Appendix D––Derivation of equations for thermal stresses 6

D.1—Vertical thermal stresses 7

D.2—Horizontal thermal stresses 8

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Preface 1

Revisions to ACI 318 as well as revisions to ASCE 7 are the basis for revising ACI 307-08. 2

ACI 318-14 was substantially reorganized from previous versions. Other committees were 3

encouraged to follow the new ACI 318 organization and Committee 307 agreed to do so for 4

its next revision. ASCE 7-10 provided wind speed maps directly applicable for determining 5

pressures for strength design approach. Maps were provided for each risk category instead of 6

using one map with an importance factor. 7

The reorganizing effort has been time-consuming. Thus, the bases for the current revision 8

are ACI 318-19 and ASCE 7-16. The changes to the ACI 307-08 are: 9

• Code completely reorganized to follow the ACI 318 format as much as possible. 10

• The along-wind load is based on strength level wind speed and so the load factor is 11

1.0. 12

• The across-wind/along-wind load combination includes a variable load factor de-13

pendent on the wind speed at which the maximum combined load occurs. 14

• For the across-wind load, the maximum damping occurring at the strength level wind 15

speed has been reduced from 4% of critical to 2.5% of critical. 16

• For the seismic load, the response modification factor has been increased from 1.5 to 17

2.0. 18

• For the seismic load, an over-strength factor is required near openings and for the 19

foundation for chimneys in Seismic Design Categories D, E, and F. 20

• For seismic loading, special detailing of the jamb area may be required, depending 21

on the Seismic Design Category. 22

• All commentary sections have been reviewed. Information deemed not appropriate 23

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for the ACI Code format has been removed. Some of the previous commentary in-1

formation has been reformatted as stand-alone reports and have been made availab le 2

on the Committee 307 web page. 3

Revision History 4

As industry expanded in the years immediately following World War I and. as a result of the 5

development of large pulverized coal-fired boilers for the electric power-generating utilit ies 6

in the 1920s, a number of large reinforced concrete chimneys were constructed to accommo-7

date these new facilities. A group of interested engineers who foresaw the potential need for 8

many more such chimneys, and who were members of the American Concrete Institute, em-9

barked on an effort to develop rational design criteria for these structures. The group was or-10

ganized into ACI Committee 505 (predecessor to the present Committee 307) to develop such 11

criteria in the early 1930s. 12

Committee 505 submitted a "Proposed Standard Specification for the Design and Construc-13

tion of Reinforced Concrete Chimneys," an outline of which was published in the ACI JOUR-14

NAL (ACI Committee 5051934). This specification was adopted as a tentative standard in 15

February 1936. Although this tentative standard was never accepted by ACI as an officia l 16

standard, it was used as the basis for the design of many chimneys. As these chimneys aged, 17

inspections revealed considerable cracking. When the industrial expansion began following 18

World War II, other engineers recognized the need for developing an improved design speci-19

fication for reinforced concrete chimneys. 20

In May 1949, Committee 505 was reactivated to revise the tentative standard specificat ion, 21

embodying modifications that were found desirable during the years it had been in use. The 22

section dealing with the temperature gradient through the chimney lining and the chimney 23

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shell was completely revised and extended to cover different types and thicknesses of linings 1

and both unventilated and ventilated air spaces between the lining and the concrete shell. In 2

1954, this specification was approved as ACI 505-54 (ACI Committee 505 1954). 3

The rapid increase in the size and height of concrete chimneys being built in the mid-1950s 4

raised further questions about the adequacy of the 1954 version of the specification, especially 5

in relation to earthquake forces and the effects of wind. 6

In May 1959, the ACI Board of Direction reactivated Committee 505 (renamed Committee 7

307) to review the standard and to update portions with the latest design techniques and the 8

then-current knowledge of the severity of the operating conditions that prevailed in large steam 9

plants. The material in the standard was reorganized, charts were added, and the methods for 10

determining loads due to wind and earthquakes were revised. The information on design and 11

construction of various types of linings was amplified and incorporated in an appendix. That 12

version included criteria for working stress design. It was planned to add ultimate strength 13

criteria in a future revision. 14

In preparing the earthquake design recommendations for ACI 307-69 (ACI Committee 307 15

1969), the committee incorporated the results of theoretical studies by adapting them to exist-16

ing United States codes. The primal), problems in this endeavor stemmed from the uncertain-17

ties still inherent in the definition of earthquake forces and from the difficulty of selecting the 18

proper safety and serviceability levels that might be desirable for various classes of construc-19

tion. Committee investigations revealed that with some modifications (such as the K factor), 20

the base shear equations developed by the Seismology Committee of the Structural Enginee rs' 21

Association of California (SEAOC) could be applied to chimneys. Similarly, the shape of the 22

force, shear, and moment distributions, as revised in their 1967 report, were also suitable for 23

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chimneys. A use factor (U factor) ranging from 1.3 to 2.0 was introduced in the specification, 1

and it was emphasized that the requirements of Section 4.5 of AC1 307-69 that related to 2

seismic design could be superseded by a rational analysis based on evaluation of the seismic ity 3

of the site and modal response calculations. The modifications were approved in ACI 307-69. 4

In that version, the commentary and derivation of equations were published separately as a 5

supplement to ACI 307-69. 6

In 1970, the document was reissued with corrections of typographical errors. This issue of 7

AC1 307-69 was also designated ANSI A158. 1-1970. At the time, as a result of numerous 8

requests, the commentary and derivation of equations were bound together with the specifica-9

tion. 10

ACI 307-79 (ACI Committee 307 1979) updated its requirements to agree with the then-11

accepted standard practices in the design and construction of reinforced concrete chimneys. 12

The major changes included the requirement that two layers of reinforcing steel be used in the 13

walls of all chimneys (previously, this only applied to chimney walls thicker than 18 in.) and 14

the requirement that horizontal sections through the chimney wall be designed for the radial 15

wind pressure distribution around the chimney. Formulas were included to compute the 16

stresses under these conditions. Many revisions of less importance were included to bring the 17

specification up to date. 18

The editions of the specifications before 1979 included appendixes on the subjects of chim-19

ney linings and accessories. In 1971, Committee 307 learned of buckling problems in steel 20

chimney liners. The committee also noted that, in modern power plant and process chimneys, 21

environmental regulations required treatment of the effluent gases that could result in ex-22

tremely variable and aggressively corrosive conditions in the chimneys. These facts led the 23

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committee to agree that the task of keeping the chimney liner recommendations current was 1

not a responsibility of an ACI committee and could be misleading to licensed design profes-2

sionals using the chimney specification. By committee consensus, the reference to chimney 3

liner construction was dropped from future editions of the specification. Committee 307 then 4

made a recommendation to the Brick Manufacturers' Association and the American Society of 5

Civil Engineers that each appoint a task force or a committee for the development of design 6

criteria for brick and steel liners, respectively. The Power Division of ASCE took up the rec-7

ommendation and appointed a task committee that developed and published a design guide in 8

1975 titled "Design and Construction of Steel Chimney Liners" (ASCE Task Committee on 9

Steel Chimney 10

Liners 1975). ASTM established two task forces for chimney liners: one for brick and one for 11

fiberglass-reinforced plastic. 12

The committee had extensive discussion on the question of including strength design in the 13

1979 specification. The decision to exclude it was based on the lack of experimental data on 14

hollow concrete cylinders to substantiate this form of analysis for concrete chimneys. The 15

committee continued, however, to consider strength design, and encouraged experiments in 16

this area. 17

Shortly after ACI 307-79 was issued, the committee decided to incorporate strength design 18

provisions and update the wind and earthquake design requirements. 19

ACI 307-88 (ACI Committee 307 1988) incorporated significant changes in the procedures 20

for calculating wind forces as well as requiring strength design rather than working stress. The 21

effects of these and other revisions resulted in designs with relatively thin walls governed 22

mainly by steel area and, in many instances, across-wind forces. 23

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The subject of across-wind loads dominated the attention of the committee between 1988 1

and 1995, and ACI 307-95 (ACI Committee 307 1995) introduced modified procedures to 2

reflect more recent information and thinking. 3

Precast chimney design and construction techniques were introduced as this type of design 4

became more prevalent for chimneys as tall as 300 ft. 5

The subject of noncircular shapes was also introduced in ACI 307-95. Due to the infinite 6

array of possible configurations, however, only broadly defined procedures were presented. 7

Because of dissimilarities between the load factors required by ACI 307 and 318, the com-8

mittee added guidelines for determining bearing pressures and loads to size and design chim-9

ney foundations. 10

The major changes incorporated into the ACI 307-95 were: 11

• Modified procedures for calculating across-wind loads; 12

• Added requirements for precast concrete chimney columns; 13

• Added procedures for calculating loads and for designing noncircular chimney col-14

umns; 15

• Deleted exemptions previously granted to smaller chimneys regarding reinforcement 16

and wall thickness; and 17

• Deleted static equivalent procedures for calculating earthquake forces. 18

For the ACI 307-98 (ACI Committee 307 1998), revisions to the ASCE 7-95 relating to wind 19

and seismic forces required several changes to be made to the ACI 307-95. The changes in-20

corporated into the ACI 307-98 were: 21

• Site-specific wind loads were calculated using a 3-second gust speed determined 22

from Fig. 6-1 in ASCE 7-95,instead of the previously used fastest-mile speed; 23

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• Site-specific earthquake forces were calculated using the effective peak velocity-re-1

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

of previously designated zonal intensity; 3

• The vertical load factor for along-wind forces was reduced from 1.7 to 1.3; 4

• The vertical load factor for seismic forces was reduced from 1.87 to 1.43; 5

• The load factor for across-wind forces was reduced from 1.40 to 1.20: and 6

• The vertical strength reduction factor φ was reduced from 0.80 to 0.70. 7

The reduced load factors should be used in concert with the revised strength reduction factor 8

and the wind and seismic loads specified in ASCE 7-95. 9

Revisions to ASCE 7 again caused Committee 307 to revisit and revise ACI 307-98. The 10

changes incorporate applicable ASCE 7-02 wind and seismic load factors and methods. The 11

changes to the ACI 307-98 were: 12

• Included procedure in Section 4.3, Earthquake load, compatible with ASCE 7-02 and 13

the ASCE 7 seismic risk maps; 14

• Updated the load factors and load combinations to be more in line with ASCE 7-02 15

values and presentation; and 16

• Changed the vertical strength reduction factor ɸ back to 0.80. 17

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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

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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

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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

structed under the requirements of the general building code. A precast reinforced concrete 23

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chimney is defined as a chimney constructed from precast reinforced concrete sections (full 1

cross-sections only), assembled one on top of another, to form a self-supporting cantilevered 2

structure. Vertical reinforcement and grout are placed in cores as the precast sections are erected 3

to provide structural continuity and stability during construction and for the completed structure. 4

Chimneys constructed using precast panels as stay-in-place forms are considered cast-in-place 5

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 8

project 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

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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 design and construction 18

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

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

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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

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1.5—Interpretation 22

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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 tribuna l 21

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

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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

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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

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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

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1.8––Construction documents and design records 22

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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 speci-3

fications 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 8

analyses 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

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1.9––Testing and inspection 15

1.9.1 Concrete shall be tested in accordance with the requirements of Chapter 26 of ACI 318-16

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

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CHAPTER 2—NOTATION AND TERMINOLOGY 1

2.1—Scope 2

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

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2.2—Notation 5

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

𝑩𝑩 = band-width parameter (Ch. 7) 7

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

𝑪𝑪𝒃𝒃 = coefficient of thermal conductivity of uninsulated liner or of insulation around 9

liner or of lining material applied directly to concrete chimney wall, Btu∙in/(hr∙ft2∙°F) 10

Ch. 7, App. C) 11

𝑪𝑪𝒄𝒄 = coefficient of thermal conductivity of concrete, Btu∙in/(hr∙ft2∙°F) (Ch. 7, App. C) 12

𝑪𝑪𝒅𝒅 = earthquake deflection amplification factor (Ch. 7) 13

𝑪𝑪𝒅𝒅𝒅𝒅 = drag coefficient for along-wind load (Ch. 7) 14

𝑪𝑪𝑬𝑬 = end effect factor (Ch. 7) 15

𝑪𝑪𝑳𝑳 = RMS lift coefficient (Ch. 7) 16

𝑪𝑪𝑳𝑳𝒐𝒐= RMS lift coefficient modified for local turbulence (Ch. 7) 17

𝑪𝑪𝒔𝒔 = coefficient of thermal conductivity of insulation filling the space between liner 18

and concrete wall, Btu∙in/(hr∙ft2∙°F) (Ch. 7, App. C) 19

𝒄𝒄 = ratio of distance from extreme compression fiber to neutral axis to the total 20

thickness for vertical stresses (Ch. 8, App. D) 21

𝒄𝒄′ = ratio of distance from extreme compression fiber to neutral axis to the total 22

thickness for horizontal stresses (Ch. 9, App. D) 23

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𝑫𝑫 = dead load (Ch.7, 11) 1

𝒅𝒅𝒃𝒃 = mean diameter of uninsulated liner or insulation around steel liner, ft (Ch. 7, App. C) 2

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

𝒅𝒅𝒄𝒄 = mean diameter of concrete chimney wall, ft (Ch. 7, App. C) 4

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

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

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

𝒅𝒅(𝒉𝒉) = top outside diameter of chimney, ft (Ch. 7, 9) 8

𝒅𝒅(𝒖𝒖) = mean outside diameter of upper one-third of chimney, ft (Ch. 7) 9

𝒅𝒅(𝒛𝒛) = outside diameter of chimney at height z, ft (Ch. 7) 10

𝒅𝒅(𝒛𝒛𝒄𝒄𝒅𝒅) = outside diameter of chimney at critical height zcr, ft (Ch. 7) 11

𝒆𝒆𝒄𝒄 = eccentricity of concrete stress resultant, ft (Ch. 8) 12

𝒆𝒆𝒔𝒔𝒄𝒄 = eccentricity of steel compressive stress resultant, ft (Ch. 8) 13

𝒆𝒆𝒔𝒔𝒔𝒔 = eccentricity of steel tensile stress resultant, ft (Ch. 8) 14

𝑬𝑬 = earthquake loads or forces (Ch. 7, 11) 15

𝑬𝑬𝒄𝒄 = modulus of elasticity of concrete, psi (Ch. 5, 8, 9, App. D) 16

𝑬𝑬𝒔𝒔 = modulus of elasticity of reinforcement, psi (Ch. 6, 8, 9, App. A, B, D) 17

𝑭𝑭𝒂𝒂 = short-period site coefficient (Ch. 7) 18

𝑭𝑭𝒄𝒄 = compressive force in concrete, lb (Ch. 8, App. A, B) 19

𝑭𝑭𝒄𝒄′ = moment due to compressive force in concrete, lb-ft (Ch. 8, App. A, B) 20

𝑭𝑭𝒗𝒗 = long-period site coefficient (Ch. 7) 21

𝑭𝑭𝟏𝟏𝑨𝑨 = Strouhal number parameter (Ch. 7) 22

𝑭𝑭𝟏𝟏𝑩𝑩 = lift coefficient parameter (Ch. 7) 23

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𝒇𝒇𝒄𝒄′ = specified compressive strength of concrete, psi (Ch. 5, 8, 9, 10, App. A, B) 1

𝒇𝒇𝒄𝒄′′(𝒄𝒄) = fc’ modified for temperature effects, circumferential, psi (Ch. 9) 2

𝒇𝒇𝒄𝒄′′(𝒗𝒗) = fc’ modified for temperature effects, vertical, psi (Ch. 8) 3

𝒇𝒇𝑪𝑪𝑪𝑪𝑪𝑪′′ = maximum circumferential stress in concrete inside chimney wall due to temperature, 4

psi (Ch. 9, App. D) 5

𝒇𝒇𝑪𝑪𝑪𝑪𝑪𝑪′′ = maximum vertical stress in concrete inside chimney wall due to temperature, psi 6

(Ch. 8, App. D) 7

𝒇𝒇𝑺𝑺𝑪𝑪𝑪𝑪 = maximum stress in outside circumferential reinforcement due to temperature, psi 8

(Ch. 9, App. D) 9

𝒇𝒇𝑺𝑺𝑪𝑪𝑪𝑪 = maximum stress in outside vertical reinforcement due to temperature, psi 10

(Ch. 8, App. D) 11

𝒇𝒇𝑺𝑺𝑪𝑪𝑪𝑪′′ = maximum stress in inside vertical reinforcement due to temperature, psi 12

(Ch. 8, App. D) 13

𝒇𝒇𝒚𝒚 = specified yield strength of reinforcing steel, psi (Ch. 6, 8, 9, 10, App. A, B) 14

𝒇𝒇𝒚𝒚′ (𝒄𝒄) = fy modified for temperature effects, circumferential, psi (Ch. 9) 15

𝒇𝒇𝒚𝒚′ (𝒗𝒗) = fy modified for temperature effects, vertical, psi (Ch.8) 16

𝑮𝑮 = across-wind peaking factor (Ch. 7) 17

𝑮𝑮𝒅𝒅(𝒛𝒛) = gust factor for radial wind pressure at height z (Ch. 7) 18

𝑮𝑮𝒘𝒘′ = gust factor for along-wind fluctuating load (Ch. 7) 19

𝒈𝒈 = acceleration of gravity, ft/sec2 (Ch. 7) 20

𝒉𝒉 = chimney height above ground level, ft (Ch. 7) 21

𝑰𝑰𝒆𝒆 = earthquake load importance factor (Ch. 7) 22

𝒃𝒃 = local turbulence factor (Ch. 7) 23

24

𝑲𝑲𝒅𝒅 = mean along-wind load directionality factor (Ch. 7) 1

𝑲𝑲𝒃𝒃 = coefficient of heat transmission from gas to inner surface of liner or to inner surface 2

of lining material applied directly to the chimney wall or to inner surface of chimney 3

wall (when chimney is unlined), BTU/(hr∙ft2∙°F) (Ch. 7, App. C) 4

𝑲𝑲𝒐𝒐 = coefficient of heat transmission from outside surface of chimney wall to surrounding 5

air, BTU/(hr∙ft2∙°F) (Ch. 7, App. C) 6

𝑲𝑲𝒅𝒅 = coefficient of heat transfer by radiation between outside surface of liner and inside 7

surface of chimney wall, BTU/(hr∙ft2∙°F) (Ch. 7, App. C) 8

𝑲𝑲𝒔𝒔 = coefficient of heat transfer between outside surface of liner and inside surface of 9

chimney wall for chimneys with ventilated air spaces, BTU/(hr∙ft2∙°F)(Ch. 7, App. C) 10

𝒌𝒌 = ratio of wind speed 𝑪𝑪�(𝒛𝒛𝒄𝒄𝒅𝒅) to critical wind speed 𝑪𝑪𝒄𝒄𝒅𝒅 (Ch. 7) 11

𝒌𝒌𝒂𝒂 = aerodynamic damping parameter (Ch.7) 12

𝒌𝒌𝒂𝒂𝒐𝒐 = mass damping parameter of small amplitudes (Ch. 7) 13

𝑳𝑳 = correlation length parameter (Ch. 7) 14

𝑳𝑳𝑭𝑭𝒄𝒄𝒘𝒘 = combined along-wind/across-wind load factor (Ch. 7) 15

𝒍𝒍𝒘𝒘 = wind end effect length, ft (Ch. 7) 16

𝑴𝑴𝒂𝒂(𝟎𝟎) = across-wind base moment, lb-ft (Ch. 7) 17

𝑴𝑴𝒂𝒂(𝒛𝒛) = across-wind moment at height z, lb-ft (Ch. 7) 18

𝑴𝑴𝒄𝒄𝒘𝒘(𝒛𝒛) = combined across-wind moment at height z, lb-ft (Ch. 7) 19

𝑴𝑴𝒃𝒃(𝒛𝒛) = maximum circumferential moment due to radial wind pressure at height z, 20

tension on inside, lb-ft/ft (Ch. 7) 21

𝑴𝑴𝒍𝒍(𝒛𝒛) = mean along-wind moment at height z at governing wind speed for across-wind 22

load, lb-ft (Ch. 7) 23

25

𝑴𝑴𝒏𝒏 = nominal moment strength at a horizontal cross-section, lb-ft (Ch. 8, 9, App. A, B) 1

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

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

𝑴𝑴𝒖𝒖 = factored moment at a horizontal cross-section, lb-ft (Ch. 8, 9 App. A, B) 4

𝑴𝑴𝒘𝒘� (𝒃𝒃) = base moment due to mean wind pressure, lb-ft/ft (Ch. 7) 5

𝑴𝑴𝒘𝒘� (𝒛𝒛) = moment at elevation 𝒛𝒛 due to mean wind pressure, lb-ft/ft (Ch. 7) 6

𝒏𝒏 = modular ratio, 𝑬𝑬𝒔𝒔 𝑬𝑬𝒄𝒄⁄ (Ch. 8, 9, App. D) 7

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

𝑷𝑷𝒖𝒖 = factored axial load at a horizontal cross-section, lb (Ch. 8, App. A, B) 9

𝒑𝒑�(𝒛𝒛) = pressure due to mean hourly wind speed at height z, lb/ft2 (Ch. 7) 10

𝒑𝒑𝒅𝒅(𝒛𝒛) = radial wind pressure at height z, lb/ft2 (Ch. 7) 11

𝑹𝑹 = earthquake response modification factor (Ch. 7) 12

𝒅𝒅 = mean radius of a horizontal cross-section, ft (Ch. 8, App. A, B) 13

𝒅𝒅(𝒛𝒛) = mean radius of a horizontal cross-section at height z, ft (Ch. 7) 14

𝒅𝒅𝒒𝒒 = ratio of heat transmission through chimney wall to heat transmission through liner for 15

chimneys with a ventilated air space (Ch. 7, App. C) 16

𝑺𝑺𝟏𝟏 = mapped maximum considered earthquake, 5% damped, spectral response acceleration 17

at a period of 1 second, g (Ch. 7) 18

𝑺𝑺𝒂𝒂 = design spectral response acceleration, g (Ch. 7) 19

𝑺𝑺𝒄𝒄 = compressive force where steel stress is below the yield strength, lb (Ch. 8, App. A, B) 20

𝑺𝑺𝒄𝒄′ = moment about the neutral axis of the compressive force where steel stress is below 21

the yield strength, lb-ft (Ch. 8, App. A, B) 22

𝑺𝑺𝒄𝒄𝒚𝒚 = compressive force where steel stress is at the yield strength, lb (Ch. 8, App. A, B) 23

26

𝑺𝑺𝒄𝒄𝒚𝒚′ = moment about the neutral axis of the compressive force where steel stress is at the 1

yield strength, lb-ft (Ch. 8, App. A, B) 2

𝑺𝑺𝑫𝑫𝟏𝟏 = design spectral response acceleration at short periods, g (Ch. 7) 3

𝑺𝑺𝑫𝑫𝑺𝑺 = design spectral response acceleration at a period of 1 second, g (Ch. 7) 4

𝑺𝑺𝑴𝑴𝟏𝟏 = maximum considered earthquake, 5% damped, spectral response acceleration at a 5

period of 1 second adjusted for site class effects, g (Ch. 7) 6

𝑺𝑺𝑴𝑴𝑺𝑺 = maximum considered earthquake, 5% damped, spectral response acceleration at 7

short periods adjusted for site class effects, g (Ch. 7) 8

𝑺𝑺𝒑𝒑 = across-wind spectral parameter (Ch. 7) 9

𝑺𝑺𝒔𝒔𝒗𝒗 = across-wind mode shape factor (Ch. 7) 10

𝑺𝑺𝑺𝑺 = mapped maximum considered earthquake, 5% damped, spectral response acceleration 11

at short periods, g (Ch. 7) 12

𝑺𝑺𝒔𝒔 = Strouhal number (Ch. 7) 13

𝑺𝑺𝒔𝒔 = tensile force where steel stress is below the yield strength, lb (Ch. 8, App. A, B) 14

𝑺𝑺𝒔𝒔′ = moment about the neutral axis of the tensile force where steel stress is below the yield 15

strength, lb-ft (Ch. 8, App. A, B) 16

𝑺𝑺𝒔𝒔𝒚𝒚 = tensile force where steel stress is at the yield strength, lb (Ch. 8, App. A, B) 17

𝑺𝑺𝒔𝒔𝒚𝒚′ = moment about the neutral axis of the tensile force where steel stress is at the yield 18

strength, lb-ft (Ch. 8, App. A, B) 19

𝒔𝒔 = center-to-center spacing of two identical chimneys, ft (Ch. 7) 20

𝑪𝑪 = temperature effect (Ch. 7) 21

𝑪𝑪𝒏𝒏 = n-th mode period of vibration, sec (Ch. 7) 22

𝑪𝑪𝒃𝒃 = maximum design temperature of flue gas, °F (Ch. 7, App. C) 23

27

𝑪𝑪𝑳𝑳 = long-period transition period, sec (Ch. 7) 1

𝑪𝑪𝑺𝑺 = earthquake response spectrum parameter (Ch. 7) 2

𝑪𝑪𝒐𝒐 = minimum design ambient air temperature, °F (Ch. 7, App. C) 3

𝑪𝑪𝟎𝟎 = earthquake response spectrum parameter (Ch. 7) 4

𝑪𝑪𝒙𝒙 = temperature gradient across the chimney wall, °F (Ch. 7,8,9, App.C, D) 5

𝒔𝒔𝒃𝒃 = thickness of uninsulated liner or thickness of insulation around liner or thickness of 6

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

𝒔𝒔𝒄𝒄 = thickness of concrete wall, in (Ch. 7,10, App. C, D) 8

𝒔𝒔𝒔𝒔 = thickness of insulation filling the space between liner and chimney wall, in 9

(Chapter 7, Appendix C) 10

𝑪𝑪𝒖𝒖 = basic wind speed, mi/hr (Ch. 7) 11

𝑪𝑪𝒔𝒔 = serviceability wind speed, mi/hr (Ch. 7) 12

𝑪𝑪𝒄𝒄𝒅𝒅 = critical wind speed for across-wind load, ft/sec (Ch. 7) 13

𝑪𝑪𝑬𝑬𝑳𝑳𝑭𝑭 = base shear using the equivalent lateral force procedure, lb (Ch. 7) 14

𝑪𝑪𝒔𝒔 = total earthquake design base shear, lb (Ch. 7) 15

𝑪𝑪�𝒖𝒖(𝟑𝟑𝟑𝟑) = mean hourly basic wind speed at height of 33 ft, ft/sec (Ch. 7) 16

𝑪𝑪�(𝒛𝒛𝒄𝒄𝒅𝒅) = variable mean hourly wind speed at 𝟓𝟓𝒉𝒉/𝟔𝟔, ft/sec (Ch. 7) 17

𝑪𝑪�𝒔𝒔(𝒛𝒛) = mean hourly serviceability wind speed at height 𝒛𝒛, ft/sec (Ch. 7) 18

𝑪𝑪�𝒖𝒖(𝒛𝒛) = mean hourly design wind speed at height z, ft/sec (Ch. 7) 19

𝑪𝑪�𝒖𝒖(𝒛𝒛𝒄𝒄𝒅𝒅) = mean hourly design wind speed at 𝟓𝟓𝒉𝒉/𝟔𝟔, ft/sec (Ch. 7) 20

𝑾𝑾𝒆𝒆 = effective seismic weight, lb (Ch. 7) 21

𝑾𝑾 = wind load due to radial wind pressure (Ch. 7) 22

𝑾𝑾𝒂𝒂𝒍𝒍𝒐𝒐𝒏𝒏𝒈𝒈 = wind load due to along-wind loading (Ch. 7) 23

28

𝑾𝑾𝒄𝒄𝒐𝒐𝒄𝒄𝒃𝒃 = wind load due to across-wind loading combined with along-wind loading at the 1

same wind speed (Ch. 7) 2

𝒘𝒘(𝒛𝒛) = total along-wind load per unit length at height z, lb/ft (Ch. 7) 3

𝒘𝒘�(𝒛𝒛) = mean along-wind load per unit length at height z, lb/ft (Ch. 7) 4

𝒘𝒘′(𝒛𝒛) = fluctuating along-wind load per unit length at height z, lb/ft (Ch. 7) 5

𝒘𝒘𝒔𝒔(𝒖𝒖) = average weight per unit height for top one-third of chimney, lb/ft (Ch. 7) 6

𝒁𝒁𝒄𝒄 = exposure length, ft (Ch. 7) 7

𝒛𝒛 = height above ground, ft (Ch. 7) 8

𝒛𝒛𝒄𝒄𝒅𝒅 = critical height for across-wind load 𝟓𝟓𝒉𝒉/𝟔𝟔, ft (Ch. 7) 9

𝜶𝜶 = for a horizontal cross-section, one-half the angle subtended by neutral axis, radians 10

(Ch. 8, App. A, B) 11

𝜶𝜶𝒔𝒔𝒆𝒆 = thermal coefficient of expansion of reinforced concrete, 1/°F (Ch. 5, 8, 9, App. D) 12

𝜶𝜶� = mean hourly wind speed power law exponent (Ch. 7) 13

𝜷𝜷 = one-half of the central angle subtended by an opening on the chimney cross-section, 14

radians (Ch. 8, App. A, B) 15

𝜷𝜷𝟏𝟏 = factor relating depth of equivalent rectangular compressive stress block to neutral axis 16

depth (Ch. 8, App. A) 17

𝜷𝜷𝒂𝒂 = aerodynamic damping factor for across-wind load (Ch. 7) 18

𝜷𝜷𝒔𝒔 = structural damping factor for across-wind load (Ch. 7) 19

𝜸𝜸𝟏𝟏 = ratio of inside face vertical reinforcement area to outside face vertical reinforcement 20

area (Ch. 8, App. D) 21

𝜸𝜸𝟐𝟐𝒐𝒐= ratio of distance between inner surface of chimney wall and outside face vertical 22

reinforcement to total wall thickness (Ch. 8, App. D) 23

29

𝜸𝜸𝟐𝟐𝒃𝒃 = ratio of distance between outer surface of chimney wall and inside face vertical 1

reinforcement to total wall thickness (Ch. 8, App. D) 2

𝜸𝜸𝟏𝟏′ = ratio of inside face circumferential reinforcement area to outside face circumferential 3

reinforcement area (Ch. 9, App. D) 4

𝜸𝜸𝟐𝟐𝒐𝒐′ = ratio of distance between inner surface of chimney wall and outside face 5

circumferential reinforcement to total wall thickness (Ch. 9, App. D) 6

𝜸𝜸𝟐𝟐𝒃𝒃′ = ratio of distance between outer surface of chimney wall and inside face 7

circumferential reinforcement to total wall thickness (Ch. 9, App. D) 8

𝜺𝜺𝒄𝒄 = concrete compressive strain (Ch. 8, App. D) 9

𝝆𝝆𝒐𝒐 = ratio of area of vertical outside face reinforcement to total area of chimney wall 10

(Ch. 8, App. D) 11

𝝆𝝆𝒐𝒐′ = ratio of area of circumferential outside face reinforcement per unit height to 12

total area of concrete wall per unit height (Ch. 9, App. D) 13

𝝆𝝆𝒂𝒂 = specific weight of air, lb/ft3 (Ch. 7) 14

𝝆𝝆𝒔𝒔 = ratio of total area of vertical reinforcement to total area of concrete cross-section 15

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

𝝓𝝓 = strength reduction factor (Ch. 4, 8, 9, App. A,B) 17

𝛀𝛀𝟎𝟎 = overstrength factor (Ch. 7) 18

19

2.3—Terminology 20

building official – term used to identify the Authority having jurisdiction or individual 21

charged with administration and enforcement of provisions of the building code. Such terms 22

30

as building commissioner or building inspector are variations of the title and the term “build-1

ing official” as used in this Code, is intended to include those variations, as well as others 2

that are used in the same sense. 3

buttress – an exterior or interior support projecting from the chimney wall 4

design strength – nominal strength multiplied by a strength reduction factor 𝝓𝝓. 5

jamb area – the concrete wall area on either side of an opening containing additional vertical 6

reinforcing bars 7

licensed design professional – an individual who is licensed to practice structural design as 8

defined by the statutory requirements of the professional licensing laws of the state or juris-9

diction in which the project is to be constructed, and who is in responsible charge of the 10

structural design 11

lintel area – the concrete wall area above an opening containing additional horizontal rein-12

forcing bars 13

MRI – mean recurrence interval. 14

nominal strength – strength of a cross-section calculated in accordance with provisions and 15

assumptions of the strength design method of this Code before application of any strength re-16

duction factors 17

serviceability loads – loads imparted on a structure assumed to be present or to occur during 18

the service life of the structure 19

sill area – the concrete wall area below an opening containing additional horizontal reinforc-20

ing bars 21

31

strength design – a method of proportioning structural members such that the computed 1

forces produced in the member by the factored loads do not exceed the member design 2

strength 3

4

32

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 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 8

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

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

Commentary 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

ASCE/SEI 49-12 Wind Tunnel Testing for Buildings and Other Structures 18

3.2.3 ASTM International 19

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

Concrete Reinforcement 21

ASTM A706/A706M-09b Standard Specification for Low-Alloy Steel Deformed and Plain 22

Bars for Concrete Reinforcement 23

33

ASTM C33/C33M-18 Standard Specification for Concrete Aggregates 1

ASTM C150/C150M-20 Standard Specification for Portland Cement 2

ASTM C260/C260M-10a(2016) Standard Specification for Air-Entraining Admixtures for 3

Concrete 4

ASTM C309-19 Standard Specification for Liquid Membrane-Forming Compounds for 5

Curing Concrete 6

ASTM C494/494M-19 Standard Specification for Chemical Admixtures for Concrete 7

ASTM C595/C595M-20 Standard Specification for Blended Hydraulic Cement 8

ASTM C618-19 Standard Specification for Coal Fly Ash and Raw or Natural Pozzolan for 9

Use in Concrete 10

ASTM C989/989M-18a Standard Specification for Slag Cement for Use in Concrete and 11

Mortars 12

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

Producing Flowing Concrete 14

ASTM C1240-20 Standard Specification for Silica Fume Used in Cementitious Mixtures 15

ASTM C1582/C1582M-11(2017)e Standard Specification for Admixtures to Inhibit Chlo-16

ride-Induced Corrosion of Reinforcing Steel in Concrete 17

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

Production of Hydraulic Cement Concrete 19

3.2.4 Federal Aviation Administration (FAA) 20

AC70-7460-1L, Change 2, Effective 8/17/18, Obstruction Marking and Lighting 21

3.2.5 Underwriters Laboratories (UL) 22

34

UL 96A, Edition 13 (12/18/18), Standard for Installation Requirements for Lightning Pro-1

tection Systems 2

3.2.6 National Fire Protection Association (NFPA) 3

NFPA 780-2020, Standard for the Installation of Lightning Protection Systems, 2020 Edi-4

tion 5

6

35

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 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 Chapter 11

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

36

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 combinat ion 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 Connections to concrete used to transmit loads by means of bearing shall be designed 14

in accordance with Section 22.8 of ACI 318-19 using the strength reduction factor 𝜙𝜙 = 0.65 15

and the factored load combinations of Table 5.3.1 in ACI 318-19. 16

17

37

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

38

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 Section 5.5, concrete mixtures shall 17

conform 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 The concrete properties of this section shall apply when the mean temperature of the 20

concrete wall does not exceed 150°F. When the mean concrete wall temperature exceeds 21

150°F, the concrete design properties shall be adjusted to account for temperature depend-22

ence. 23

39

R5.4.1 Values of concrete design properties at temperatures exceeding 150°F should be ob-1

tained from reliable sources. 2

5.4.2 Compressive strength 3

5.4.2.1 The value of 𝐟𝐟𝐜𝐜′ shall be specified in the construction documents. 4

5.4.2.2 The specified value of 𝐟𝐟𝐜𝐜′ shall be in accordance with: 5

(a) Structural strength requirements 6

(b) Durability requirements of Section 5.5 7

R5.4.2.2 Table 19.3.2.1 of ACI 318-19 lists requirements for concrete by exposure class. 8

5.4.2.3 The specified compressive strength shall be used for proportioning of concrete mix-9

tures in accordance with Section 26.4.3 of ACI 318-19. 10

5.4.2.4 Compliance with 𝐟𝐟𝐜𝐜′ shall be based on cylinder tests in accordance with Section 11

26.12 of ACI 318-19. Test age, if other than 28 days, shall be indicated in the contract docu-12

ments 13

5.4.3 Modulus of elasticity 14

5.4.3.1 The modulus of elasticity for concrete, 𝐄𝐄𝐜𝐜 , shall be in accordance with ACI 318-19, 15

Section 19.2.2. 16

5.4.4 Thermal coefficient of expansion 17

5.4.4.1 The thermal coefficient of expansion of reinforced concrete, 𝛂𝛂𝐭𝐭𝐭𝐭, is permitted to be 18

taken as 5.5 x 10-6 per °F. 19

R5.4.4.1 Per ACI 209R-92, the value specified is a reasonable average value for the coeffi-20

cient of thermal expansion. 21

5.5 Durability Requirements 22

40

5.5.1 Concrete durability requirements shall be in accordance with Section 19.3 of ACI 1

318-19 except that 𝒇𝒇𝒄𝒄′ shall not be less than 3000 psi. The licensed design professional shall 2

assign exposure classes in accordance with the severity of the anticipated exposure of the 3

chimney for each exposure category in Table 19.3.1.1 of ACI 318-19. 4

R5.5.1 Section 19.3.1 of ACI 318-19, Exposure categories and classes, is adopted by refer-5

ence. 6

7

41

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 or 12

A706/A706M. 13

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

ultimate tensile strain equal to or exceeding 0.07. 15

6.4—Steel properties for design 16

6.4.1 Yield strength 17

6.4.1.1 Yield strength for reinforcing bars shall be based on the specified grade of reinforce-18

ment but shall not exceed 75,000 psi. 19

6.4.2 Modulus of elasticity 20

6.4.2.1 Modulus of elasticity, 𝐸𝐸𝑠𝑠 , for reinforcing bars is permitted to be taken as 29,000,000 21

psi. 22


42

6.5.1 The specified concrete cover shall be at least 2 inches for cast-in-place chimneys. 1

6.5.2 The specified concrete cover shall be at least 1.5 inches for precast units manufactured 2

under plant-controlled condition. 3

4

43

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

(a) Dead load 9

(b) Temperature gradient load 10

(c) Wind load 11

(d) Earthquake load 12

7.2.2 A chimney shall be assigned a Risk Category as defined in Section 1.5 of ASCE 7-16. 13

The Risk Category shall be Category III unless Category IV is applicable. 14

15

7.2.3 For wind and earthquake loads, natural frequencies or periods shall be computed by a 16

frequency analysis that takes into account the mass and stiffness distribution of the chimney, 17

the effect of any significant non-structural weights such as access platforms and slabs, and the 18

effect of liner(s) supported and/or braced by the chimney. 19

7.3—Dead load 20

7.3.1 For calculation of self-weight, the unit weight for normal weight reinforced concrete is 21

permitted to be taken as 150 lb/ft3. 22

44

7.3.2 Permanent loads, such as the weight of liners, roofs and/or platforms supported by the 1

chimney wall shall be included in the dead load. 2

R7.3.2 When there is a significant time delay between completion of the chimney wall and 3

installation of any permanent load, the design should be checked with and without that perma-4

nent load. 5

7.4—Temperature gradient load 6

7.4.1 General 7

7.4.1.1 Reinforced concrete chimneys shall be designed to resist vertical and circumferentia l 8

stress due to a temperature gradient across the chimney wall. 9

7.4.1.2 For circular chimneys, the temperature gradient across the chimney wall shall be 10

determined according to Section 7.4.2 or by a rational heat balance study. 11

7.4.1.3 For non-circular chimneys, the temperature gradient across the chimney wall shall be 12

determined by a rational heat-balance study. 13

7.4.2 Temperature gradient for circular chimneys 14

7.4.2.1 Eq. (7-1) to (7-5) are permitted to be used to obtain the temperature gradient across 15

the chimney wall of a circular chimney. For circular chimneys with multiple liners, the equa-16

tions are permitted to be used by setting the liner diameter equal to an equivalent diameter that 17

approximates the thermodynamic conditions. 18

R7.4.2.1 For chimneys with multiple liners a rational heat balance study would be needed to 19

accurately determine the temperature gradient across the chimney wall, however, equations 7-20

3 and 7-4 are permitted to be used with an equivalent liner diameter to represent the mult ip le 21

liners. The normal practice is to use a liner diameter equal to the sum of the individual liner 22

diameters, however in certain cases the sum of the individual diameters may be greater than 23

45

the inside diameter of the chimney wall, in which case a smaller diameter should be used 1

providing a nominal airspace. It may also be necessary to check the gradient using several 2

different equivalent diameters to represent the thermodynamic conditions from variations in 3

gas flow and temperature among the multiple liners. 4

7.4.2.2 For unlined circular chimneys, 5

𝑇𝑇𝑥𝑥 =𝑡𝑡𝑐𝑐𝑑𝑑𝑐𝑐𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

�𝑇𝑇𝑐𝑐 − 𝑇𝑇𝑜𝑜

1𝐾𝐾𝑐𝑐

+ 𝑡𝑡𝑐𝑐𝑑𝑑𝑐𝑐𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

+ 𝑑𝑑𝑐𝑐𝑐𝑐𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜

� (7-1)

7.4.2.3 For circular chimneys with lining material applied directly to the inside surface of 6

the chimney wall, 7

𝑇𝑇𝑥𝑥 =𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐


1𝐾𝐾𝑐𝑐

+ 𝑡𝑡𝑏𝑏𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑏𝑏𝑑𝑑𝑏𝑏

+ 𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

+ 𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜

� (7-2)

R7.4.2.3 This case is added to explicitly address glass block or gunite applied to the chimney 8

wall. 9

7.4.2.4 For circular chimneys with insulation completely filling the space between the liner 10

and the chimney wall (no annular air space), 11



1𝐾𝐾𝑐𝑐


+ 𝑡𝑡𝑠𝑠𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑠𝑠𝑑𝑑𝑠𝑠



� (7-3)

7.4.2.5 For circular chimneys with an unventilated air space between the liner and the 12

chimney wall, 13



1𝐾𝐾𝑐𝑐


+ 𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑟𝑟𝑑𝑑𝑠𝑠



� (7-4)

46

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 insula t io n 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



1𝑟𝑟𝑞𝑞𝐾𝐾𝑐𝑐

+ 𝑡𝑡𝑏𝑏𝑑𝑑𝑏𝑏𝑐𝑐𝑟𝑟𝑞𝑞𝐶𝐶𝑏𝑏𝑑𝑑𝑏𝑏

+ 𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑠𝑠𝑑𝑑𝑠𝑠



� (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, Equations (7-4) and (7-5) can be modified to include a term for the 10

liner 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

𝑟𝑟𝑞𝑞 = 0.5; 14

𝐶𝐶𝑐𝑐 = 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

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

𝐶𝐶𝑏𝑏 = to be obtained from the manufacturer of the materials used; 19

𝐾𝐾𝑐𝑐 = to be determined from curves in Fig. 7.1; 20

𝐾𝐾𝑜𝑜 = 12 Btu/(h-ft2-°F); 21

47

𝐾𝐾𝑟𝑟 = 𝑇𝑇𝑐𝑐/120; 1

𝐾𝐾𝑠𝑠 = 𝑇𝑇𝑐𝑐/150. 2

R7.4.2.8 These constants, when entered into equations for temperature differential through 3

the chimney wall, 𝑇𝑇𝑥𝑥, will give a value having accuracy in keeping with the basic design 4

assumptions for the structural design of the concrete. More refined values and/or a more 5

refined method may be needed for other purposes, such as accurately calculating the 6

temperature in the annular air space between the liner and the chimney wall. 7

7.4.2.9 The value of 𝑟𝑟𝑞𝑞 = 0.5 shall apply only where: 8

1. The distance between the liner and the chimney wall is not less than 4 in. throughout 9

the entire height of the liner and ventilation air inlet and outlet openings are provided 10

at the bottom and top of the chimney wall, respectively. 11

2. The free area of inlet and outlet openings, in square feet, shall each be numerica lly 12

equal to two-thirds of the top inside diameter of the chimney wall, in feet. 13

3. Local obstructions in the air space between the liner and the chimney wall shall not 14

restrict the free area of the air space at any horizontal section to less than the free area 15

specified for the inlet and outlet openings. 16

7.5—Wind load 17

7.5.1 General 18

7.5.1.1 Reinforced concrete chimneys shall be designed to resist wind loads in both the 19

along-wind and across-wind directions. In addition, vertical cross-sections shall be designed 20

to resist bending due to the distribution of wind pressure on the perimeter. 21

48

7.5.1.2 The basic 3-second gust design wind speed, 𝑉𝑉𝑢𝑢, in miles per hour shall be determined 1

in accordance with Section 26.5 of ASCE 7-16 for Risk Category III (1700-year MRI) or Risk 2

Category IV (3000-year MRI) as determined in Section 7.2.2. 3

R7.5.1.2 The subscripted symbol 𝑉𝑉𝑢𝑢 is used in this code rather than the symbol V used in 4

ASCE 7-16 to avoid confusion regarding the meaning of the variable mean hourly wind speed 5

𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟) used in Section 7.5.3. 6

7.5.1.3 The serviceability wind speed, 𝑉𝑉𝑠𝑠, in miles per hour shall be the 100-year MRI 3-7

second gust wind speed determined in accordance with Figure CC.2-4 in Appendix CC of the 8

Commentary for ASCE 7-16. 9

R7.5.1.3 Figure CC.2-4 shows the 100-year MRI 3-second gust wind speed. If the 100-year 10

MRI wind speed is not known, 0.80 times the basic wind speed V is permitted to be used for 11

the serviceability wind speed for Risk Category III and 0.75 times the basic wind speed V is 12

permitted to be used for the serviceability wind speed for Risk Category IV. 13

7.5.1.3 The exposure category as defined in Section 26.7 of ASCE 7-16, shall be C for 14

chimneys unless exposure D is applicable. 15

R7.5.1.3 Exposure D, absent in previous versions, has been explicitly included in the revised 16

Code. 17

7.5.1.4 At a height 𝑧𝑧 in feet above the ground, the mean-hourly design wind speed, 𝑉𝑉�𝑢𝑢(𝑧𝑧), 18

and the mean-hourly serviceability wind speed, 𝑉𝑉�𝑠𝑠(𝑧𝑧), in feet per second, shall be computed 19

from Eq. (7-6a) and Eq. (7-6b), respectively. 20

𝑉𝑉�𝑢𝑢(𝑧𝑧) = (1.47) (0.65) 𝑉𝑉𝑢𝑢 �𝑧𝑧

33�𝛼𝛼�

(7-6a)

𝑉𝑉�𝑠𝑠(𝑧𝑧) = (1.47) (0.65) 𝑉𝑉𝑠𝑠 �𝑧𝑧

33�𝛼𝛼�

(7-6b)

49

The value of 𝛼𝛼� shall be 0.154 for exposure C and 0.111 for exposure D. 1

R7.5.1.4 The constant 1.47 ( = 22/15) converts from miles per hour to feet per second. The 2

constant 0.65 converts from 3-second gust to mean hourly. 3

7.5.1.5 For circular chimneys, wind effects shall be determined in accordance with Sections 4

7.5.2, 7.5.3, and 7.5.4. Alternatively, the wind forces are permitted to be determined by a 5

properly substantiated dynamic analysis or by a wind tunnel study in accordance with ASCE 6

49-12. 7

R7.5.1.5 The provisions with respect to along-wind load take into account dynamic action, 8

but are simplified and result in equivalent static loads. 9

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

ASCE 7-16 and shall consider across-wind load effects (including any interference effects). 11

Alternatively, the wind forces are permitted to be determined by a properly substantiated 12

dynamic analysis or by a wind tunnel study in accordance with ASCE 49-12. 13

7.5.1.7 The maximum lateral deflection of the top of the chimney due to the design wind 14

speed forces shall not exceed ℎ 180� . 15

R7.5.1.7 The deflection is determined using strength-level loads, uncracked sections and a 16

fixed base. Limiting deflections limits the effects of secondary moments. The current limit 17

due to strength-level loads is consistent with the previous version deflection limit due to 18

serviceability- level loads. 19

7.5.2 Along-wind load for circular chimneys 20

7.5.2.1 The directionality factor, 𝐾𝐾𝑑𝑑, shall be 0.95 for circular chimneys. 21

7.5.2.2 The topography factor, 𝐾𝐾𝑧𝑧𝑧𝑧, shall be determined according to Section 26.8 of ASCE 22

7-16. The topography factor is permitted to be taken as 1.0 for all elevations. 23

50

R7.5.2.2 The topography factor affects only elevations near grade and has a small effect on 1

the overturning moments in this area. 2

7.5.2.3 The ground elevation factor, 𝐾𝐾𝑒𝑒, shall be determined according to Section 26.9 of 3

ASCE 7-16. The ground elevation factor is permitted to be taken as 1.0 4

R7.5.2.3 Using a ground elevation factor of 1.0 is conservative. 5

7.5.2.2 The total along-wind load in, lb/ft, at height 𝑧𝑧 in feet shall be the sum of the mean 6

load and the fluctuating load. 7

𝑤𝑤(𝑧𝑧) = 𝑤𝑤�(𝑧𝑧) + 𝑤𝑤′(𝑧𝑧) (7-7)

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

𝑤𝑤�(𝑧𝑧) = 𝐶𝐶𝑑𝑑𝑟𝑟(𝑧𝑧)𝑑𝑑(𝑧𝑧)�̅�𝑝(𝑧𝑧) (7-8)

where 9

𝐶𝐶𝑑𝑑𝑟𝑟 = �1.0 for 𝑧𝑧 ≥ ℎ− 𝑙𝑙𝑤𝑤0.65 for 𝑧𝑧 < ℎ− 𝑙𝑙𝑤𝑤

(7-9)

𝑙𝑙𝑤𝑤 = min(1.5𝑑𝑑(ℎ),50 ft) (7-10)

𝑝𝑝̅(𝑧𝑧) = 0.00119𝐾𝐾𝑑𝑑𝐾𝐾𝑧𝑧𝑧𝑧𝐾𝐾𝑒𝑒 [𝑉𝑉�𝑢𝑢(𝑧𝑧)]2 (7-11)

7.5.2.4 The fluctuating wind load, in lb/ft, at height 𝑧𝑧 in feet shall be computed from Eq. (7-10

12). 11

𝑤𝑤′(𝑧𝑧) = 3.0 𝑧𝑧 𝐺𝐺𝑤𝑤′ 𝑀𝑀𝑤𝑤�(𝑏𝑏)

ℎ3 (7-12)

where 12

𝐺𝐺𝑤𝑤′ = 0.30 +11.0 [𝑇𝑇1𝑉𝑉�𝑢𝑢(33)]0.47

(ℎ + 16)0.86 (7-13)

7.5.3 Across-wind load for circular chimneys 13

7.5.3.1 Procedures in this section apply if the outside chimney diameter at ℎ/3 is less than 14

1.6 times the top outside diameter. Across-wind effect for circular chimneys that do not satisfy 15

51

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

to be analyzed using this section. Across-wind effect 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 Strongly tapered circular chimneys (diameter at h/3 greater than 1.6 times the top 5

outside diameter) develop lower across wind loads than non-tapered or slightly tapered 6

chimneys. Using the method given in 7.5.3 to analyze strongly tapered chimneys gives 7

conservative results. Strongly tapered chimneys may be analyzed using a properly 8

substantiated dynamic analysis to consider the effect of the strong taper. 9

7.5.3.2 The first mode critical wind speed shall be determined as 10

𝑉𝑉𝑐𝑐𝑟𝑟 =𝑑𝑑(𝑢𝑢)

(𝑆𝑆𝑡𝑡)𝑇𝑇1

(7-14)

where 11

𝑆𝑆𝑡𝑡 = 0.25𝐹𝐹1𝐴𝐴 (7-15)

𝐹𝐹1𝐴𝐴 = 0.333 + 0.206log𝑒𝑒ℎ

𝑑𝑑(𝑢𝑢) (7-16)

The second mode critical wind speed shall be determined as 12

𝑉𝑉𝑐𝑐𝑟𝑟 =5𝑑𝑑(𝑢𝑢)𝑇𝑇2

(7-17)

The across-wind load need not be considered when the critical wind speed exceeds 1.5𝑉𝑉�𝑢𝑢(𝑧𝑧𝑐𝑐𝑟𝑟). 13

R7.5.3.2 The maximum critical wind speed for which the across-wind load must be 14

considered is based on the results of a computational study of 25 sample chimneys (Radecki 15

2014). A lower bound for the range of consideration is not specified since group effects can 16

magnify the response. 17

52

7.5.3.3 The combined across-wind/along-wind moment at mean-hourly wind speed 𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟) 1

shall be determined using Eq. (7-18). 2

𝑀𝑀𝑐𝑐𝑤𝑤(𝑧𝑧) = �[𝑀𝑀𝑎𝑎(𝑧𝑧)]2 + [𝑀𝑀𝑙𝑙(𝑧𝑧)]2 (7-18)

𝑀𝑀𝑎𝑎(𝑧𝑧) is the across-wind moment at height 𝑧𝑧 at mean-hourly wind speed 𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟). 3

𝑀𝑀𝑙𝑙(𝑧𝑧) is the mean along-wind moment at height 𝑧𝑧 at mean-hourly wind speed 𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟). 4

R7.5.3.3 The combination formula, Eq. (7-18) does not include the fluctuating component 5

of the along-wind load. Equation (7-18), however, is a good approximation for a more 6

accurate, but also more computationally complex formula that includes the fluctua t ing 7

component. Equation (7-18) always underestimates the combined load, but the error has an 8

upper bound of about 15%. The more accurate formula, based on the across-wind component 9

and the fluctuating along-wind component being uncorrelated random variables having a joint 10

Gaussian distribution, can be found in Vickery and Basu (1984) and Radecki (2014). 11

7.5.3.3.1 The across-wind moment, 𝑀𝑀𝑎𝑎(𝑧𝑧), at height 𝑧𝑧 at mean-hourly wind speed 12

𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟) shall be determined by scaling the corresponding mode shape such that the base 13

moment is 𝑀𝑀𝑎𝑎(0) as determined in Section 7.5.3.5. 14

7.5.3.3.2 The mean along-wind moment, 𝑀𝑀𝑙𝑙(𝑧𝑧), at height 𝑧𝑧 at mean-hourly wind speed 15

𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟) shall be determined using Eq.(7-19). 16

𝑀𝑀𝑙𝑙(𝑧𝑧) = 𝑀𝑀𝑤𝑤�(𝑧𝑧)�𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟)𝑉𝑉�𝑢𝑢(𝑧𝑧𝑐𝑐𝑟𝑟)�

2

(7-19)

𝑀𝑀𝑤𝑤�(𝑧𝑧) is the moment at elevation 𝑧𝑧 due to the mean wind load pressures 𝑤𝑤�(𝑧𝑧) from 17

Section 7.5.2.3. 18

7.5.3.4 The maximum combined across-wind/along-wind load (first mode and/or second 19

mode) considered for design shall be the combined across-wind/along-wind load 20

53

corresponding to the maximum factored combined base moment using the wind speed 1

dependent load factor defined in Section 7.7.1.2, the across-wind base moment, 𝑀𝑀𝑎𝑎(0), 2

calculated using Eq. (7-20) and the mean along-wind base moment, 𝑀𝑀𝑙𝑙(0), calculated using 3

Eq. (7-19). The factored combined across-wind/along-wind load shall be evaluated for mean-4

hourly wind speeds in the range 0.5 𝑉𝑉𝑐𝑐𝑟𝑟 ≤ 𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟)≤ 𝑉𝑉�𝑢𝑢(𝑧𝑧𝑐𝑐𝑟𝑟). 5

R7.5.3.4 Since the load factor is not constant, the governing load for design (the maximum 6

factored combined load) will not necessarily be the maximum combined load as defined in 7

Section 7.5.3.3. The across-wind load first becomes noticeable at approximately one-half the 8

critical wind speed so wind speeds below that need not be considered. 9

7.5.3.5 The across-wind base moment 𝑀𝑀𝑎𝑎(0) at wind speed 𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟) shall be calculated using 10

Eq. (7-20). 11

𝑀𝑀𝑎𝑎(0) =𝐺𝐺𝑔𝑔𝑆𝑆𝑠𝑠𝑠𝑠𝐶𝐶𝐿𝐿

𝜌𝜌𝑎𝑎2

(𝑉𝑉𝑐𝑐𝑟𝑟)2𝑑𝑑(𝑢𝑢)ℎ2 �𝜋𝜋

4(𝛽𝛽𝑠𝑠 + 𝛽𝛽𝑎𝑎)�12�

× 𝑆𝑆𝑝𝑝 �2𝐿𝐿

ℎ𝑑𝑑(𝑢𝑢) + 𝐶𝐶𝐸𝐸

�

12�

(7-20)

where 12

𝐺𝐺 = 4.0

𝐿𝐿 = 1.20

𝐶𝐶𝐸𝐸 = 3

𝑆𝑆𝑠𝑠𝑠𝑠 = � 0.57 for 1st mode 0.18 for 2nd mode

𝐶𝐶𝐿𝐿 = 𝐶𝐶𝐿𝐿𝑜𝑜 𝐹𝐹1𝐵𝐵 (7-21)

𝐶𝐶𝐿𝐿𝑜𝑜 = −0.243 + 5.648𝑖𝑖 − 18.182𝑖𝑖 2 (7-22)

𝑖𝑖 =1

loge𝑧𝑧𝑐𝑐𝑟𝑟𝑍𝑍𝑐𝑐

(7-23)

54

𝑍𝑍𝑐𝑐 = 0.06 ft

𝐹𝐹1𝐵𝐵 = −0.089 + 0.337 logeℎ

𝑑𝑑(𝑢𝑢)

but not > 1.0 nor < 0.2

(7-24)

𝛽𝛽𝑠𝑠 =

⎩⎨

⎧ 0.01 for 𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟)≤ 𝑉𝑉�𝑠𝑠(𝑧𝑧𝑐𝑐𝑟𝑟)

0.01 + 0.015 �𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟)−𝑉𝑉�𝑠𝑠(𝑧𝑧𝑐𝑐𝑟𝑟)𝑉𝑉�𝑢𝑢(𝑧𝑧𝑐𝑐𝑟𝑟)−𝑉𝑉�𝑠𝑠(𝑧𝑧𝑐𝑐𝑟𝑟)�

otherwise

(7-25)

𝛽𝛽𝑎𝑎 =𝑘𝑘𝑎𝑎𝜌𝜌𝑎𝑎𝑑𝑑(𝑢𝑢)2

𝑤𝑤𝑡𝑡(𝑢𝑢) (7-26)

𝜌𝜌𝑎𝑎 = 0.0765

𝑘𝑘𝑎𝑎 = 𝑘𝑘𝑎𝑎𝑜𝑜𝐹𝐹1𝐵𝐵 (7-27)

𝑘𝑘𝑎𝑎𝑜𝑜 =−1.0

(1 + 5𝑖𝑖)�1 + |𝑘𝑘 − 1|𝑖𝑖 + 0.10�

(7-28)

𝑘𝑘 =𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟)𝑉𝑉𝑐𝑐𝑟𝑟

(7-29)

𝑆𝑆𝑝𝑝 =𝑘𝑘1.5

𝐵𝐵0.5𝜋𝜋0.25 exp �−12�

1− 𝑘𝑘−1

𝐵𝐵�2

� (7-30)

𝐵𝐵 = 0.1 + 2𝑖𝑖 (7-31)

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 to 5

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

55

the across-wind load factor, reducing the load so that a constant load factor could be used. 1

That adjustment is now included in a variable load factor for the across-wind combination. 2

7.5.3.6 Group effect – For circular chimneys, when two identical chimneys are in close 3

proximity, the lift coefficient 𝐶𝐶𝐿𝐿 shall be increased to account for potential increase in vortex-4

induced motions. 5

If 𝑠𝑠

𝑑𝑑(𝑧𝑧𝑐𝑐𝑟𝑟)≥ 12.75, 6

𝐶𝐶𝐿𝐿 is unaltered 7

If 3≤𝑠𝑠

𝑑𝑑(𝑧𝑧𝑐𝑐𝑟𝑟) < 12.75, 8

𝐶𝐶𝐿𝐿 shall be multiplied by 2.275− 0.10𝑠𝑠

𝑑𝑑(𝑧𝑧𝑐𝑐𝑟𝑟) 9

where 𝑠𝑠 is the center-to-center distance between the two identical chimneys. 10

For chimneys that are not identical, for chimneys that are identical but the spacing ratio 11

𝑠𝑠 𝑑𝑑(𝑧𝑧𝑐𝑐𝑟𝑟)⁄ is less than 3 and for more than two chimneys in close proximity, the across-wind 12

load for each chimney shall be established by reference to model tests, observations, test 13

reports, or analytical models of similar chimney arrangements. 14

7.5.4 Bending at vertical cross-sections of circular chimneys due to radial wind pressure 15

7.5.4.1 The radial wind pressure, in lb/ft2, at height 𝑧𝑧 in feet shall be computed by Eq. (7-16

32). 17

𝑝𝑝𝑟𝑟 (𝑧𝑧) = �1.0 𝑝𝑝̅(𝑧𝑧) 𝐺𝐺𝑟𝑟(𝑧𝑧), 𝑧𝑧 < ℎ − 𝑙𝑙𝑤𝑤1.5 𝑝𝑝̅(𝑧𝑧) 𝐺𝐺𝑟𝑟(𝑧𝑧), 𝑧𝑧 ≥ ℎ − 𝑙𝑙𝑤𝑤

(7-32)

where

𝐺𝐺𝑟𝑟(𝑧𝑧) = �4.0 – 0.8 log10 𝑧𝑧 , 𝑧𝑧 > 1.0 ft4.0, 𝑧𝑧 ≤ 1.0 ft

(7-33)

56

R7.5.4.1 The mean pressure, 𝑝𝑝̅(𝑧𝑧), and the end effect length, 𝑙𝑙𝑤𝑤, are defined in Section 1

7.5.2.3. The increase in loads near the top is consistent with observations (Okamoto and Tagita 2

1973) that the drag coefficient increases significantly in this region. 3

7.5.4.2 The maximum circumferential ring bending moment, in ft-lb/ft, at height 𝑧𝑧, in feet, 4

due to the radial wind pressure distribution resulting in tension on the inside face, shall be 5

computed by Eq. (7-34). 6

𝑀𝑀𝑐𝑐(𝑧𝑧) = 0.31𝑝𝑝𝑟𝑟(𝑧𝑧)[𝑟𝑟(𝑧𝑧)]2 (7-34)

7.5.4.3 The maximum circumferential ring bending moment, in ft-lb/ft, at height 𝑧𝑧, in feet, 7

due to the radial wind pressure distribution resulting in tension on the outside face, shall be 8

computed by Eq. (7-35). 9

𝑀𝑀𝑜𝑜(𝑧𝑧) = 0.27𝑝𝑝𝑟𝑟(𝑧𝑧)[𝑟𝑟(𝑧𝑧)]2 (7-35)

7.6—Earthquake load 10

7.6.1 General 11

7.6.1.1 Reinforced concrete chimneys shall be designed to resist earthquake loads due to 12

ground motion in accordance with this section. 13

7.6.1.2 Earthquake load on a chimney shall be determined by means of the modal response 14

spectrum analysis of Section 7.6.2, by means of a linear response history procedure in 15

accordance with Section 12.9.2 of ASCE 7-16, or by means of a nonlinear response history 16

procedure in accordance with Chapter 16 of ASCE 7-16. 17

7.6.2 Modal response spectrum analysis 18

7.6.2.1 The required periods, mode shapes, and participation factors of the chimney shall be 19

calculated by established methods of structural analysis. The analytical model used shall be 20

sufficiently refined to represent variations of mass and stiffness. The analysis shall include a 21

57

sufficient number of modes to obtain a combined modal mass participation of at least 90% of 1

the actual mass. 2

R7.6.2.1 Significant weights included in the dead load, such as roof structures, platforms and 3

liners should be included in the analytical model. It is generally sufficient to lump such weights 4

at the elevation where they occur. Liners may be included explicitly as structures tied and/or 5

braced to the chimney structure at discrete elevations or as lumped weights. If liners are 6

included as lumped weights, the total weight should be distributed at discrete elevations to 7

approximate both the vertical and horizontal force transfer. 8

7.6.2.2 Modal shears, modal moments, and modal deflections shall be determined using an 9

elastic response spectrum procedure. The response spectrum shall provide the design spectral 10

response acceleration, 𝑆𝑆𝑎𝑎, at any period and shall be obtained by means of the general 11

procedure of Section 7.6.3 or the site-specific procedure of Section 7.6.4. 12

7.6.2.3 The importance factor 𝐼𝐼𝑒𝑒 shall be determined from Table 1.5-2 of ASCE 7-16. 13

7.6.2.4 The seismic design category shall be determined from Section 11.6 of ASCE 7-16. 14

7.6.2.5 The response modification factor 𝑅𝑅 shall be taken as 2.0. The deflection 15

amplification factor 𝐶𝐶𝑑𝑑 shall be taken as 2.0. The overstrength factor Ω0 shall be taken as 1.5. 16

R7.6.2.5 The response modification factor has been increased from 1.5 to 2.0 to be consistent 17

with Table 15.4-2 of ASCE 7-16 and recognizing that well detailed chimneys have some 18

ductility (Wilson 2003). The displacement amplification factor and overstrength factor are 19

new and are also consistent with Table 15.4-2. 20

7.6.2.6 The modal design shears and moments shall be determined by scaling the modal 21

shears and moments due to the design spectrum by the factor 𝐼𝐼𝑒𝑒/𝑅𝑅. The modal design 22

58

displacements shall be determined by scaling the modal displacements due to the design 1

spectrum by the factor 𝐶𝐶𝑑𝑑/𝑅𝑅. 2

7.6.2.7 The total design shears, moments and displacements shall be calculated from the 3

modal design shears, moments and displacements using either the square root of the sum of 4

the squares (SRSS) method or the complete quadratic combination (CQC) method. 5

7.6.2.8 The loads due to the vertical component of earthquakes are permitted to be neglected. 6

R7.6.2.8 The vertical seismic load effect in ASCE 7-16 is 0.2 SDSD. This effect can be 7

included with the dead load as (1.2 + 0.2 SDS)D in Eq. (7-50d) and (0.9− 0.2 SDS)D in Eq.(7-8

50g). 9

7.6.2.9 For chimneys of circular cross section, the horizontal earthquake force shall be 10

assumed to act alone in any direction. 11

7.6.2.10 For chimneys of non-circular cross sections with an assigned Seismic Design 12

Category of C, D, E, or F, orthogonal effects shall be considered by combining, using the SRSS 13

method, the responses due to the design spectrum applied to any two orthogonal directions. 14

7.6.2.11 For chimneys with an assigned Seismic Design Category of D, E or F, overstrength 15

requirements of Section 7.6.2.12 shall apply. 16

R7.6.2.11 The overstrength factor section is added for consistency with ASCE 7-16, Section 17

15.6.2. The increase of R to 2.0 recognizes that chimneys have some ductility (Wilson 2003). 18

To ensure that inelastic behavior does not initiate at critical sections, such as sections near 19

openings an overstrength factor is applied so that these critical sections remain elastic for the 20

seismic load. 21

7.6.2.12 Where the loss of horizontal cross-section area due to openings is greater than 10%, 22

horizontal cross-sections in the regions of the openings shall be designed for the total design 23

59

shears and moments determined according to Section 7.6.2.8 multiplied by the overstrength 1

factor Ω0. The region where the overstrength factor applies shall extend above and below 2

openings by a distance equal to half the width of the largest opening in the affected region. 3

Appropriate reinforcement development lengths shall be provided beyond the required region 4

of overstrength. 5

7.6.3 Response spectrum - general procedure 6

7.6.3.1 The mapped maximum considered earthquake spectral response acceleration at short 7

periods, 𝑆𝑆𝑆𝑆, and at 1 second, 𝑆𝑆1, shall be obtained from Figs. 22-1 through 22-8 of ASCE 7-16 8

(or electronically via the web sites referenced in ASCE 7-16). 9

7.6.3.2 The site class shall be determined from Table 20.3-1 and Section 20.3 of ASCE 7-10

16. When soil properties are not known in sufficient detail to determine the site class, Class D 11

shall be used. 12

7.6.3.3 The short period site coefficient 𝐹𝐹𝑎𝑎 shall be obtained from Table 11.4-1 of ASCE 7-13

16. The long period site coefficient 𝐹𝐹𝑠𝑠 shall be obtained from Table 11.4-2 of ASCE 7-16. 14

7.6.3.4 The maximum considered earthquake spectral response acceleration parameters for 15

short periods 𝑆𝑆𝑀𝑀𝑆𝑆 and at 1 second, 𝑆𝑆𝑀𝑀1, adjusted for site class effects, shall be calculated by: 16

𝑆𝑆𝑀𝑀𝑆𝑆 = 𝐹𝐹𝑎𝑎 𝑆𝑆𝑆𝑆 (7-36)

𝑆𝑆𝑀𝑀1 = 𝐹𝐹𝑠𝑠 𝑆𝑆1 (7-37)

7.6.3.5 The design earthquake spectral response acceleration parameter at short periods, SDS, 17

and at 1 second, SD1, shall be calculated by: 18

𝑆𝑆𝐷𝐷𝑆𝑆 = �23�𝑆𝑆𝑀𝑀𝑆𝑆 (7-38)

𝑆𝑆𝐷𝐷1 = �23�𝑆𝑆𝑀𝑀1 (7-39)

60

7.6.3.6 The design response spectrum curve shall be developed as follows. 1

(a) For periods less than 𝑇𝑇0, the design spectral response acceleration 𝑆𝑆𝑎𝑎 shall be taken as: 2

𝑆𝑆𝑎𝑎 = 𝑆𝑆𝐷𝐷𝑆𝑆 �0.4 + 0.6 𝑇𝑇𝑛𝑛𝑇𝑇0� (7-40)

where 3

𝑇𝑇0 = 0.2 𝑆𝑆𝐷𝐷1/𝑆𝑆𝐷𝐷𝑆𝑆 (7-41)

(b) For periods greater than or equal to 𝑇𝑇0 and less than or equal to 𝑇𝑇𝑆𝑆, the design spectral 4

response acceleration 𝑆𝑆𝑎𝑎 shall be taken as: 5

𝑆𝑆𝑎𝑎 = 𝑆𝑆𝐷𝐷𝑆𝑆 (7-42)

where 6

𝑇𝑇𝑆𝑆 = 𝑆𝑆𝐷𝐷1/𝑆𝑆𝐷𝐷𝑆𝑆 (7-43)

(c) For periods greater than 𝑇𝑇𝑆𝑆, and less than or equal to 𝑇𝑇𝐿𝐿, the design spectral response 7

acceleration 𝑆𝑆𝑎𝑎 shall be taken as: 8

𝑆𝑆𝑎𝑎 =𝑆𝑆𝐷𝐷1𝑇𝑇𝑛𝑛

(7-44)

where 9

𝑇𝑇𝐿𝐿 = long-period transition period obtained from Figs. 22-14 through 22-17 of ASCE 7-16. 10

(d) For periods greater than 𝑇𝑇𝐿𝐿, the design spectral response acceleration 𝑆𝑆𝑎𝑎 shall be taken 11

as: 12

𝑆𝑆𝑎𝑎 =𝑆𝑆𝐷𝐷1𝑇𝑇𝐿𝐿𝑇𝑇𝑛𝑛2

(7-45)

7.6.3.7 If the total design base shear, 𝑉𝑉𝑧𝑧 , calculated in accordance with 7.6.2.8 is less than 13

85% of the calculated base shear using the equivalent lateral force procedure, 𝑉𝑉𝐸𝐸𝐿𝐿𝐸𝐸 , the total 14

design shears, moments and displacements shall be multiplied by 15

61

0.85𝑉𝑉𝐸𝐸𝐿𝐿𝐸𝐸𝑉𝑉𝑧𝑧

(7-46)

The base shear value 𝑉𝑉𝐸𝐸𝐿𝐿𝐸𝐸 shall not be less than the largest of the following: 1

𝑆𝑆𝑎𝑎1𝐼𝐼𝑒𝑒𝑅𝑅𝑊𝑊𝑒𝑒

0.044𝑆𝑆𝐷𝐷𝑆𝑆𝐼𝐼𝑒𝑒𝑊𝑊𝑒𝑒

0.01𝑊𝑊𝑒𝑒

(7-47)

where 𝑆𝑆𝑎𝑎1 is the design spectral acceleration, 𝑆𝑆𝑎𝑎, for the first mode of vibration when 𝑇𝑇1 is 2

greater than or equal to 𝑇𝑇0. When 𝑇𝑇1 is less than 𝑇𝑇0, 𝑆𝑆𝑎𝑎1 = 𝑆𝑆𝐷𝐷𝑆𝑆 . In addition, if 𝑆𝑆1 ≥ 0.6𝑔𝑔,𝑉𝑉𝐸𝐸𝐿𝐿𝐸𝐸 3

shall not be less than 4

0.5𝑆𝑆1𝐼𝐼𝑒𝑒𝑅𝑅𝑊𝑊𝑒𝑒 (7-48)

R7.6.3.7 Per Section 15.4.4 of ASCE 7-16, the fundamental period used to determine 5

VELFshall be determined using the structural properties of the resisting elements in a properly 6

substantiated analysis. The requirement of Section 12.8.2 of ASCE 7-16 that the fundamenta l 7

period not exceed the product CuTa does not apply to chimneys. 8

7.6.4 Response spectrum - site-specific procedure 9

7.6.4.1 A site-specific response spectrum, developed in accordance with the site-specific 10

ground motion procedures of Chapter 21 of ASCE 7-16, shall be permitted to be used in lieu 11

of the general procedure of Section 7.6.3. 12

7.6.5 Soil-structure interaction 13

7.6.5.1 The effect of seismic interaction between a chimney and soil is permitted to be 14

ignored and a fixed base condition assumed. 15

62

7.6.5.2 When a soil-structure interaction assessment is desired, the procedure given in 1

Chapter 19 of ASCE 7-16 shall be followed or foundation flexibility shall be modeled in 2

accordance with Section 12.13.3 of ASCE 7-16. 3

7.6.6 P-Δ effect 4

7.6.6.1 The P-Δ effect between vertical loads and seismic lateral displacement shall be 5

considered for chimneys with an assigned Seismic Design Category of D, E, or F. The 6

maximum design displacements shall be used to determine the P-Δ effect. 7

7.6.7 Lateral clearance between chimney wall and liner 8

7.6.7.1 [to be determined] 9

7.7—Load combinations 10

7.7.1 Horizontal cross-sections 11

7.7.1.1 The design strength of horizontal cross-sections of the chimney shall equal or exceed 12

the effects of the factored loads in the following combinations: 13

1.4 D (7-50a)

1.2 D + 1.2 T + 1.0 Walong (7-50b)

1.2 D + 1.2 T + 𝐿𝐿𝐹𝐹𝑐𝑐𝑤𝑤 Wcomb (7-50c)

1.2 D + 1.2 T + 1.0 E (7-50d)

0.9 D + 1.2 T + 1.0 Walong (7-50e)

0.9 D + 1.2 T +𝐿𝐿𝐹𝐹𝑐𝑐𝑤𝑤 Wcomb (7-50f)

0.9 D + 1.2 T + 1.0 E (7-50g)

7.7.1.2 The load factor, 𝐿𝐿𝐹𝐹𝑐𝑐𝑤𝑤, for combined along-wind load and across-wind load at mean-14

hourly wind speed, 𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟), shall be 15

63

𝐿𝐿𝐹𝐹𝑐𝑐𝑤𝑤 =

⎩⎪⎪⎨

⎪⎪⎧ 1.4 for 𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟)≤ 𝑉𝑉�𝑠𝑠(𝑧𝑧𝑐𝑐𝑟𝑟)

1.4− 0.4 �𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟)− 𝑉𝑉�𝑠𝑠(𝑧𝑧𝑐𝑐𝑟𝑟)𝑉𝑉�𝑢𝑢(𝑧𝑧𝑐𝑐𝑟𝑟)− 𝑉𝑉�𝑠𝑠(𝑧𝑧𝑐𝑐𝑟𝑟)�

for 𝑉𝑉�𝑠𝑠(𝑧𝑧𝑐𝑐𝑟𝑟) < 𝑉𝑉�(𝑧𝑧𝑐𝑐𝑟𝑟)≤ 𝑉𝑉�𝑢𝑢(𝑧𝑧𝑐𝑐𝑟𝑟)

(7-51)

R7.7.1.2 A wind load factor of 1.0 is not appropriate when the governing wind speed for the 1

combined along-wind and across-wind load is less than the strength level design wind speed. 2

Equation (4-10) of 307-08 implicitly modified the load factor for the combined wind load. 3

This variable load factor for combined wind explicitly provides a load factor appropriate for 4

the governing wind speed. 5

7.7.2 Vertical cross-sections 6

7.7.2.1 The design strength of vertical cross-sections shall equal or exceed the effects of the 7

factored loads in the following combinations: 8

1.2 T + 1.0 W (7-52)

R7.7.2.1 The wind load factor for forces on horizontal and vertical cross-sections in the 9

previous version were 1.6 and 1.4, respectively. In this Code, both wind load factors are 1.0 10

for strength level loads. Eq. (7-52) is therefore about 15% more conservative than the 11

corresponding equation in the previous version. 12

13

64

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

8.2—General 5

8.2.1 Except as noted in Section 8.3, the nominal strength at a horizontal cross-section of a 6

chimney having circular cross-sections is permitted to be calculated assuming that the strain in 7

concrete and reinforcement at any point on the section is directly proportional to its distance 8

from the neutral axis. For chimneys having noncircular cross-sections the assumption of linear 9

strain on the cross-section, if used, shall be justified by proper engineering analysis or 10

judgement. 11

R8.2.1 The limits in Section 8.3 assure that (1) local wall buckling will not precede concrete 12

crushing, that (2) the edges of openings, acting as columns, will not be governed by slenderness 13

effects and that (3) a linear strain distribution on the cross-section can be assumed. To justify 14

the linear strain assumption for chimneys having noncircular cross-sections, similar issues 15

must be addressed. 16

8.2.2 In calculating forces in the reinforcing for a chimney having circular cross-sections, 17

the total reinforcement area at a cross-section is permitted to be considered an annular area, 18

centered on the middle of the concrete wall, having thickness 𝜌𝜌𝑧𝑧𝑡𝑡𝑐𝑐 . For chimneys having 19

noncircular cross-sections, a similar assumption is permitted to be used if justified by proper 20

engineering judgement. 21

65

R8.2.2 Most chimneys have (relatively) thin walls with reinforcing bars (relatively) closely 1

spaced near the inside and outside faces. The force calculation error in assuming the 2

reinforcing is evenly distributed along the middle of the concrete wall is negligible. 3

8.2.3 The maximum compressive strain in the concrete shall be assumed equal to 0.003. The 4

maximum tensile strain in the reinforcement shall be assumed equal to 0.07. Whichever value 5

is reached first shall be taken as the limiting value. 6

R8.2.3 The maximum tensile strain in the reinforcing steel is assume to be the fracture limit 7

of 0.07. The strain limit is consistent with maximum elongation properties in tension of 8

reinforcing steel. If this fracture limit is reached before the maximum concrete strain of 0.003 9

is reached, the maximum concrete strain must be reduced. 10

8.2.5 Tensile strength of concrete shall be neglected. 11

8.2.6 The stress-strain relationship for concrete in compression shall be assumed to be in 12

accordance with Figure 8.1. For chimneys having circular cross-sections and meeting the 13

requirements of Section 8.3, the modified equivalent rectangular concrete stress block 14

relationship of Figure 8.2 is permitted to be assumed. The modification factor, Q, is defined 15

in Appendix A. 16

17

18

66

Curve A: 0.85𝑓𝑓𝑐𝑐′ �2𝜀𝜀𝑐𝑐

0.002− �

𝜀𝜀𝑐𝑐0.002

�2� 1

Line B: 0.85𝑓𝑓𝑐𝑐′(1.30− 150𝜀𝜀𝑐𝑐) 2

Figure 8.1: Concrete stress-strain relationship 3

4

Figure 8.2: Modified equivalent rectangular stress block 5

R8.2.6 The concrete stress-strain relationship of Figure 8.1 is permitted to be used for any 6

cross-section. Figure 8.2 is the equivalent rectangular concrete stress block relationship of 7

ACI 318-19, Section 22.2.2.4, modified by the factor Q for circular chimney cross-sections. 8

The factor Q has been determined by fitting the results of using the relationship shown in 9

Figure 8.1 to the equivalent stress block method. The equations in Appendix B use the 10

relationship shown in Figure 8.1 11

8.2.7 For strain in the reinforcement less than 𝑓𝑓𝑦𝑦/𝐸𝐸𝑠𝑠, stress shall be taken as 𝐸𝐸𝑠𝑠 times the 12

strain. For strain in the reinforcement greater than 𝑓𝑓𝑦𝑦/𝐸𝐸𝑠𝑠, stress shall be assumed equal to 𝑓𝑓𝑦𝑦. 13

R8.2.7 Reinforcing steel is assumed to follow an elastic-perfectly plastic stress-strain 14

relationship. 15

67

8.2.8 For chimneys having circular cross-sections and meeting the design limits of Section 1

8.3, forces in the reinforcement are permitted to be calculated assuming the reinforcement is 2

evenly distributed around the entire circumference of the section. For chimneys having 3

noncircular cross-sections, the calculation of forces in the reinforcement shall consider strain 4

distribution at the opening. 5

R8.2.8 Reinforcing bars interrupted by openings are required to be replaced at the sides of 6

the openings. For simplicity in calculating forces, it is permitted to assume that these bars are 7

equally distributed around the entire circumference. The error in assuming these bars are so 8

located is small. 9


8.3.1 For chimneys having circular cross-sections, the assumption of Section 8.2.1 is 11

permitted to be used to compute the nominal moment strength of horizontal cross-sections only 12

if the requirements of Sections 8.3.2 through 8.3.4 are satisfied. If any of the requirements are 13

not satisfied, the nominal moment strength shall be computed taking into account a non-linear 14

strain distribution, local stability of the wall, and edge stability at openings, as applicable. For 15

chimneys having noncircular cross-sections, the assumption of Section 8.2.1 is permitted to be 16

used to compute the nominal moment strength of horizontal cross-sections only if the 17

assumption has been justified, local stability of the wall has been taken into account, and edge 18

stability at openings has been taken into account. 19

8.3.2 Minimum wall thickness 20

8.3.2.1 For cast-in-place chimneys having circular cross-sections, the minimum wall 21

thickness at any section with inside diameter of 28 ft. or less shall be 8 in. For precast chimneys 22

having circular cross-sections, the minimum wall thickness at any section with inside diameter 23

68

of 28 ft. or less shall be 7 in. When the inside diameter of a section exceeds 28 ft., the minimum 1

wall thickness shall be increased 1/8 in. for every 1 ft. increase in inside diameter. 2

R8.3.2.1 The minimum wall thickness for circular cross-sections ensures local buckling does 3

not occur and that linear strain variation on the cross-section can be assumed. 4

8.3.2.2 For chimneys having noncircular cross-sections and for circular chimneys not 5

calculated using the assumption of Section 8.2.1, minimum wall thickness shall be determined 6

taking into account strain distribution and local wall stability, but shall not be less than 8 inches 7

for cast-in-place chimneys or 7 inches for precast chimneys. 8

8.3.3 Minimum wall thickness at openings 9

8.3.3.1 The wall thickness at or near openings shall be as specified in Section 10.3. This 10

requirement applies to cast-in-place and precast chimneys and to chimneys having circular or 11

noncircular cross-sections. 12

R8.3.3.1 The regions beside openings (jamb areas) are considered columns when they are in 13

the compression zone. This requirement assures that slenderness effects do not limit the 14

column strength. 15

8.3.4 Maximum opening width 16

8.3.4.1 For chimneys having circular cross-sections, the half-angle of an opening, 𝛽𝛽, as 17

shown in Figure 8.3, shall not exceed 𝜋𝜋 6� (30 degrees). 18

19

69

1

Figure 8.3: Opening half-angle 2

R8.3.4.1 At circular cross-sections that include openings, the strain is not linear, with strains 3

increasing rapidly near the edge of the opening especially near the corners. However, 4

experience has shown that, for cross-sections having openings within this limitation, the error 5

in computing strength using the linear strain assumption is not significant. 6

8.3.4.2 For chimneys having noncircular cross-sections, the regions near openings of 7

significant width shall be investigated for local strains. A significant width opening shall 8

include, but is not limited to, an opening width greater than 15% of the cross-section perimeter. 9

R8.3.4.2 Judgement must be used as to what constitutes “significant” width. Openings for 10

breeching ducts and construction openings would generally be considered significant. 11

8.3.5 Multiple openings at a section 12

8.3.5.1 For a section with multiple openings, if a wall segment between openings exceeds 13

the slenderness limits for compression members in ACI 318-19, this segment shall be 14

additionally investigated as a beam-column and/or shear wall as applicable. The wall segment 15

is permitted to be assumed a rectangular section for the additional investigation(s). Design 16

strength of wall segments shall be calculated in accordance with ACI 318-19. 17

70

R8.3.5.1 Wall segments between openings must be checked as columns that may be 1

governed by slenderness effects and/or as shear walls that may be governed by shear strength 2

or buckling. 3

8.4—Required strength 4

8.4.1 Required strength shall be calculated in accordance with the load combinations in 5

Section 7.7.1. The temperature gradient effect 𝑇𝑇𝑥𝑥 of a load combination shall be included in 6

accordance with Section 8.5.2. 7

8.4.2 The factored vertical load 𝑃𝑃𝑢𝑢 and the factored bending moment 𝑀𝑀𝑢𝑢 occurring 8

simultaneously at horizontal cross-sections for each applicable load combination shall be 9

considered. 10


8.5.1 General 12

8.5.1.1 For each applicable load combination, (𝑃𝑃𝑢𝑢,𝑀𝑀𝑢𝑢), the design strength at horizonta l 13

cross-sections shall satisfy (a) and (b). Interaction between load effects shall be considered. 14

(a) 𝜙𝜙𝑃𝑃𝑛𝑛 ≥ 𝑃𝑃𝑢𝑢 15

(b) 𝜙𝜙𝑀𝑀𝑛𝑛 ≥ 𝑀𝑀𝑢𝑢 16

8.5.1.2 The strength reduction factor 𝜙𝜙 shall be 0.80 for horizontal cross-sections. 17

R8.5.1.2 Horizontal chimney sections are tension controlled unless the non-dimensiona l 18

factored axial load, Pu (2πrt⁄ fc′) exceeds about 0.25 which would be extremely rare for a 19

chimney subject to axial load that is mainly self-weight. Therefore, a tension controlled section 20

is assumed and a constant strength reduction factor is specified. 21

8.5.2 Temperature gradient effect 22

71

8.5.2.1 The temperature gradient across the wall, computed in Section 7.4, shall be taken into 1

account by reducing the concrete strength 𝑓𝑓𝑐𝑐′ and the reinforcement yield strength 𝑓𝑓𝑦𝑦 by the 2

amount of concrete stress and reinforcement stress required to balance the thermal strain. 3

R8.5.2.1 The temperature gradient through the concrete chimney wall reduces the nomina l 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

8.5.2.2 The maximum vertical stress in the concrete, in psi, occurring at the inside of the 8

chimney wall due to a temperature difference shall be computed by Eq. (8-1). The maximum 9

vertical stress in the steel, in psi, occurring in the inside reinforcing layer of the chimney wall 10

due to temperature gradient shall be computed by Eq. (8-2). The maximum vertical stress in 11

the steel, in psi, occurring in the outside reinforcing layer of the chimney wall due to 12

temperature shall be computed by Eq. (8-3). 13

𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ = 𝛼𝛼𝑧𝑧𝑒𝑒 𝑐𝑐 𝑇𝑇𝑥𝑥 𝐸𝐸𝑐𝑐 (8-1)

𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶′′ = 𝛼𝛼𝑧𝑧𝑒𝑒 (𝑐𝑐− 1 + 𝛾𝛾2𝑐𝑐 ) 𝑇𝑇𝑥𝑥 𝐸𝐸𝑠𝑠 (8-2)

𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶 = 𝛼𝛼𝑧𝑧𝑒𝑒 (𝛾𝛾2𝑜𝑜 − 𝑐𝑐) 𝑇𝑇𝑥𝑥 𝐸𝐸𝑠𝑠 (8-3)

where

𝑐𝑐 = −𝜌𝜌𝑜𝑜𝑛𝑛(𝛾𝛾1 + 1) + �[𝜌𝜌𝑜𝑜𝑛𝑛(𝛾𝛾1 + 1)]2 + 2𝜌𝜌𝑜𝑜𝑛𝑛[𝛾𝛾2𝑜𝑜 + 𝛾𝛾1(1− 𝛾𝛾2𝑐𝑐 )]

(8-4)

R8.5.2.2 The derivation of equations for vertical stresses due to temperature gradient is 14

included in Appendix D. 15

8.5.2.3 For load combinations with temperature effects, replace 𝑓𝑓𝑦𝑦 with 𝑓𝑓𝑦𝑦′(𝑣𝑣) using Eq. (8-16

5) and replace 𝑓𝑓𝑐𝑐′ with 𝑓𝑓𝑐𝑐′′(𝑣𝑣) using (8-6). 17

72

𝑓𝑓𝑦𝑦′(𝑣𝑣) = 𝑓𝑓𝑦𝑦 −1.2

1 + 𝛾𝛾1(𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶 − 𝛾𝛾1 𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶′′ ) (8-5)

𝑓𝑓𝑐𝑐′′(𝑣𝑣) = 𝑓𝑓𝑐𝑐′ − 1.2𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ (8-6)

R8.5.2.3 The derivation of equations for modified vertical concrete compressive strength and 1

modified vertical steel yield strength due to temperature gradient is included in Appendix D. 2

8.5.3 Nominal moment strength 3

8.5.3.1 For a cross-section having a factored vertical load 𝑃𝑃𝑢𝑢 and a factored bending moment 4

𝑀𝑀𝑢𝑢, the iterative procedure of Section 8.5.3.3 is permitted to be used to determine the total 5

vertical steel ratio 𝜌𝜌𝑧𝑧 needed to provide the required design strengths, 𝜙𝜙𝑃𝑃𝑛𝑛 and 𝜙𝜙𝑀𝑀𝑛𝑛. 6

8.5.3.2 For chimneys having circular cross-sections that do not exceed the design limits of 7

Section 8.3, the procedure in Appendix A using the modified equivalent rectangular stress 8

block relationship of Figure 8.2 is permitted to be used. 9

10

73

1

Figure 8.4: Stress Resultants 2

8.5.3.3 Given the factored vertical load 𝑃𝑃𝑢𝑢 and the factored bending moment 𝑀𝑀𝑢𝑢, the 3

following procedure is permitted to be used to determine the steel ratio required. 4

1) Assume a value for the total vertical steel ratio 𝜌𝜌𝑧𝑧; 5

2) If temperature effects are to be taken into account, determine the concrete and steel thermal 6

stresses due to 𝑇𝑇𝑥𝑥 and compute the modified concrete compressive strength 𝑓𝑓𝑐𝑐′′(𝑣𝑣) and the 7

modified steel yield strength 𝑓𝑓𝑦𝑦′(𝑣𝑣); 8

3) Determine the nominal vertical load strength required; 9

𝑃𝑃𝑛𝑛 =𝑃𝑃𝑢𝑢𝜙𝜙

10

4) By trial and error find the value of 𝛼𝛼 (see Figure 8.4) for which force equilibrium is 11

satisfied; 12

74

𝑃𝑃𝑛𝑛 = 𝐹𝐹𝑐𝑐 + 𝑆𝑆𝑐𝑐 + 𝑆𝑆𝑐𝑐𝑦𝑦 − 𝑆𝑆𝑧𝑧 − 𝑆𝑆𝑧𝑧𝑦𝑦 1

5) Substitute this value of 𝛼𝛼 into the equation for moment equilibrium to obtain the nomina l 2

moment strength 𝑀𝑀𝑛𝑛 ; 3

𝑀𝑀𝑛𝑛 = 𝑃𝑃𝑛𝑛𝑟𝑟 cos𝛼𝛼 + 𝐹𝐹𝑐𝑐′ + 𝑆𝑆𝑐𝑐′ +𝑆𝑆𝑐𝑐𝑦𝑦′ + 𝑆𝑆𝑧𝑧′ + 𝑆𝑆𝑧𝑧𝑦𝑦′ 4

6) If 𝜙𝜙𝑀𝑀𝑛𝑛 < 𝑀𝑀𝑢𝑢 , increase 𝜌𝜌𝑧𝑧; 5

If 𝜙𝜙𝑀𝑀𝑛𝑛 > 𝑀𝑀𝑢𝑢 , decreasing 𝜌𝜌𝑧𝑧 is permitted 6

7) If 𝜌𝜌𝑧𝑧 is changed, repeat 2 through 5; 7

R8.5.3.3 The procedure determines a nominal axial and flexural strength pair (Pn ,Mn) for a 8

given reinforcing steel ratio. By iterating, the required reinforcing steel ratio can be minimized. 9

The symbols in the equilibrium equations represent the stress resultants (forces) for the 10

concrete, reinforcing steel in the compression zone and reinforcing steel in the tension zone. 11


8.6.1 Two layers of vertical reinforcement shall be provided. 13

8.6.2 The total vertical reinforcement shall be not less than 0.25% of the concrete area. 14

8.6.3 The outside vertical reinforcement shall not be less than 50% of the total vertical 15

reinforcement. 16


8.7.1 Vertical reinforcing bars shall not be smaller than No. 4 bars. 18

8.7.2 Outside face vertical reinforcing bars shall not be spaced more than 12 in. on centers. 19

8.7.3 Inside face vertical reinforcing bars shall not be spaced more than 24 in. on centers. 20

8.7.5 For chimneys with an assigned Seismic Design Category of A, B, or C, not more than 21

50% of the vertical bars shall be spliced along any horizontal plane unless specifica lly 22

permitted and approved by the licensed design professional. To meet this requirement, the 23

75

centerline of vertical splices shall be staggered by a distance equal to at least the lap splice 1

length. 2

8.7.6 For chimneys with an assigned Seismic Design Category of D, E, or F, not more than 3

50% of the vertical bars shall be spliced along any horizontal plane. To meet this requirement, 4

the centerline of vertical splices shall be staggered by a distance equal to at least the lap splice 5

length plus the development length. 6

8.7.7 Tie bars shall be provided between the inner and outer face reinforcement at the top of 7

the chimney wall. Ties shall be a minimum of No. 3 bars and shall not exceed a spacing of 12 8

in. 9

10

76

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 horizonta l 7

beam (of rectangular cross-section) resisting circumferential ring moments. 8


9.3.1 The minimum wall thickness shall be as specified in Sections 8.3.2 to 8.3.5. 10

9.4—Required strength 11

9.4.1 Required strength shall be calculated in accordance with the factored load combinat ion 12

in Chapter 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

𝜙𝜙𝑀𝑀𝑛𝑛 ≥ 𝑀𝑀𝑢𝑢 20

9.5.1.2 𝜙𝜙 shall be 0.9 for vertical sections. 21

9.5.2 Temperature effect 22

77

9.5.2.1 The temperature gradient across the wall, computed in Section 7.4, shall be taken into 1

account by reducing the concrete strength 𝑓𝑓𝑐𝑐′ and the reinforcement yield strength 𝑓𝑓𝑦𝑦 by the 2

amount of 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 nomina l 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

𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ = 𝛼𝛼𝑧𝑧𝑒𝑒 𝑐𝑐′ 𝑇𝑇𝑥𝑥 𝐸𝐸𝑐𝑐 (9-1)

𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶 = 𝛼𝛼𝑧𝑧𝑒𝑒 (𝛾𝛾2𝑜𝑜′ − 𝑐𝑐′) 𝑇𝑇𝑥𝑥 𝐸𝐸𝑠𝑠 (9-2)

where

𝑐𝑐′ = −𝜌𝜌𝑜𝑜′𝑛𝑛(𝛾𝛾1′ + 1) + �[𝜌𝜌𝑜𝑜′𝑛𝑛(𝛾𝛾1′ + 1)]2 + 2𝜌𝜌𝑜𝑜′𝑛𝑛[𝛾𝛾2𝑜𝑜′ + 𝛾𝛾1′(1−𝛾𝛾2𝑐𝑐′ )] (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 𝑓𝑓𝑦𝑦 with 𝑓𝑓𝑦𝑦′(𝑐𝑐) using Eq. (9-13

4) and replace 𝑓𝑓𝑐𝑐′ with 𝑓𝑓𝑐𝑐′′(𝑐𝑐) using Eq. (9-5). 14

𝑓𝑓𝑦𝑦′(𝑐𝑐) = 𝑓𝑓𝑦𝑦 − 1.2𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶 (9-4)

𝑓𝑓𝑐𝑐′′(𝑐𝑐) = 𝑓𝑓𝑐𝑐′ − 1.2𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ (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

78

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 0.2𝑑𝑑(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 13

Section 9.5, 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. All reinforceing bars shall be tied at intervals of not more than 17

2 ft. Bars shall be secured against displacement within the tolerances of ACI 318-19. 18

9.7.2 Circumferential reinforcing bars shall not be smaller than No. 3 bars. 19

9.7.3 Spacing of outer face circumferential reinforcement shall not exceed the wall thickness 20

or 12 in. 21

9.7.4 Spacing of inner face circumferential reinforcement shall not exceed 12 in. 22

79

9.7.5 Spacing of the circumferential reinforcement near the top of the chimney, as specified 1

by Section 9.6.4, shall not exceed one-half the wall thickness or 6 in., whichever is smaller. 2

9.7.6 Circumferential lap splice length shall meet the requirements for a Class B lap splice 3

length of deformed bars in tension in accordance with Section 25.5 of ACI 318-19. 4

9.7.7 Not more than 50% of the circumferential bars shall be spliced along any vertical plane 5

unless specifically permitted and approved by the licensed design professional. To meet this 6

requirement, the centerline of alternate circumferential splices shall be staggered by a distance 7

equal to at least the lap splice length. 8

9

80

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) Special seismic detailing of jamb areas 5

10.2—General 6

10.2.1 In addition to the minimum wall thickness requirements of Section 8.3.2, the wall 7

thickness near openings shall meet the minimum requirements as specified in Section 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 Sections 10.4 to 10

10.6. 11

10.2.3 For chimneys with an assigned Seismic Design Category D, E or F, at openings where 12

the loss of cross-sectional area is greater than 10%, detailing of the jamb region shall be 13

provided as specified in Section 10.7. 14

10.3—Minimum wall thickness at openings 15

10.3.1 The chimney wall thickness, at any opening, shall not be less than 1/24 times the 16

height of the opening. This thickness shall extend over a vertical distance from one-half the 17

height of the opening below the sill of the opening to one-half the height of the opening above 18

the top of the opening. 19

10.3.2 Properly designed buttresses or other means of lateral (radial) edge restraint are 20

permitted in place of the requirements of Section 10.3.1. However, the buttresses or other 21

means of lateral edge restraint shall not be included when calculating vertical strength. 22

Buttresses shall extend over a vertical distance from one-half the height of the opening below 23

81

the sill of the opening to one-half the height of the opening above the top of the opening. Other 1

means of 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 evenly distributed around the entire circumference. In that regard it is not technica lly 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 vertical tie spacing shall not 15

exceed 12 in. 16

10.4.5 If any additional vertical reinforcement is placed between the normal innermost and 17

outermost vertical reinforcement layers, horizontal tie bars shall be provided as indicated in 18

Fig. 10.1. Horizontal tie spacing shall not exceed 12 in. and vertical tie spacing shall not 19

exceed 24 in. 20

10.5—Circumferential reinforcement at openings 21

82

10.5.1 At the top and bottom of each opening, the additional reinforcement shall have an area 1

at least equal to the one-half the design circumferential reinforcement interrupted by the 2

opening or the area determined by Eq. (10-1), whichever is greater. 3

𝐴𝐴𝑠𝑠 =0.06𝑓𝑓𝑐𝑐′𝑡𝑡𝑐𝑐𝑏𝑏𝑜𝑜

𝑓𝑓𝑦𝑦 (10-1)

The requirement of Eq. (10-1) is permitted to be replaced with results of a proper, detailed 4

analysis. 5

10.5.2 The additional circumferential reinforcement shall be placed as close as practical to 6

the top or bottom of the opening, within a distance not exceeding three times the wall thickness 7

unless otherwise determined by a detailed analysis. 8

10.5.3 One-half the area of the additional circumferential reinforcement shall extend 9

completely around the circumference of the chimney. The other half of the additiona l 10

reinforcement shall extend beyond the vertical edges of the opening a sufficient distance to 11

develop the bars in tension. 12

10.5.4 Tie bars shall be provided at the top and bottom edges of each opening as indicated in 13

Fig. 10.2. The tie bar size shall be No. 3 or larger and the horizontal tie spacing shall not 14

exceed 12 in. 15

10.6—Corner reinforcement at openings 16

10.6.1 For openings wider than 2 ft., the area of diagonal reinforcement provided at each 17

corner of the opening shall be equal, in square inches, to 0.10 times the wall thickness, in 18

inches. 19

10.6.2 As an alternative to diagonal corner reinforcement, orthogonal (horizontal and 20

vertical) reinforcement is permitted to be provided at each corner. The area of horizonta l 21

83

reinforcement and the area of vertical reinforcement provided shall each be equal, in square 1

inches, to 0.10 times the wall thickness, in inches. 2

R10.6.2 This section has been added for slip-form construction where placement of diagonal 3

bars is restricted by the yoke frames. It may also be beneficial in jump-form construction when 4

the construction joint location relative to the top or bottom of an opening makes it difficult to 5

locate a diagonal bar. The effectiveness of orthogonal corner reinforcement has been studied 6

by Kilic, et al.[2016]. 7

10.6.3 For openings 2 ft. wide or less, a minimum of two No. 5 bars shall be placed 8

diagonally at each corner of the opening. Alternatively, two No. 5 horizontal bars and two No. 9

5 vertical bars are permitted to be placed at each corner of the opening. 10

10.6.4 The length of corner reinforcement bars, diagonal or orthogonal, shall be sufficient to 11

develop each bar in tension on either side of a theoretical crack from the corner of the opening 12

propagating at a 1:1 slope. If orthogonal corner reinforcing bars from adjacent corners overlap, 13

single bars are permitted to be provided. 14

10.6.5 If corner reinforcement is provided by orthogonal bars, the areas required by Section 15

10.6.2 are permitted to be added to the vertical opening reinforcement of Section 10.4 and the 16

circumferential opening reinforcement of Section 10.5 provided that the area representing 17

corner reinforcement satisfies the development requirement of Section 10.6.4. 18

10.7—Special seismic detailing 19

10.7.1 The reinforcement ratio of the jamb area at each side of each opening shall be not less 20

than 0.01 nor more than 0.08. The width of the jamb area shall be the width of the chimney 21

wall containing the additional vertical reinforcement of Section 10.4 but not less than twice 22

the wall thickness. 23

84

10.7.2 Vertical bars within a region the jamb area defined in Section 10.7.1 shall be laterally 1

supported by deformed bar ties. Ties shall be at least No. 3 in size for jamb bars No. 10 or 2

smaller and at least No. 4 in size for jamb bars No. 11 or larger and bundled bars. 3

10.7.3 Vertical spacing of ties shall not exceed the least of (a) through (c): 4

(a) 16 times the jamb bar diameter 5

(b) 48 times the tie bar diameter 6

(c) wall thickness at the opening 7

10.7.4 Ties shall be arranged such that every corner and alternate jamb bar shall have lateral 8

support provided by the corner of a tie with an included angle of no more than 135 degrees and 9

no bar shall be farther than 6 in. clear on each side along the tie from a laterally supported jamb 10

bar. 11

10.7.5 Ties shall extend above and below the opening a distance equal to twice the wall 12

thickness or the development length of the jamb bars, whichever is larger. 13

14

15

16

Fig. 10.1 17

18

19

85

1

Fig. 10.2 2

3

86

CHAPTER 11—FOUNDATION 1

11.1—Scope 2

11.1.1 This chapter shall apply to the design of foundations for reinforced concrete 3

chimney. 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. (1.0 + 0.14 SDS)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 – 0.14 SDS)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. The factor 0.7 for the along-wind/across-wind combined load is consistent with the 17

corresponding load factor for strength design in Section 7.7.1.1. Load combinations (5), (6) 18

and (7) are a check on stability and replace a requirement for a factor of safety against 19

overturning in the previous revision. 20

11.2.2 The overturning effect at the soil-foundation interface due to seismic load is 21

permitted to be reduced by 10% for foundations of chimneys designed in accordance with the 22

modal analysis requirements of Section 7.6.2. 23

87

R11.2.2 This reduction is consistent with Section 12.13.4 of ASCE 7-16. 1

11.2.3 For a shallow foundation, the minimum base area of the foundation shall be 2

calculated from forces and moments transmitted by the foundation to soil or rock and the 3

allowable bearing pressure determined through principles of soil or rock mechanics. 4

11.2.4 For a deep foundation, the number and arrangement of piles shall be determined 5

from forces and moments transmitted to these members and the allowable pile capacity 6

determined through principles of soil or rock mechanics. 7

11.3—Foundation structural design 8

11.3.1 Foundations shall be proportioned to resist loads and induced reactions determined 9

by the following load combinations: 10

1. 1.4 D 11

2. 1.2 D + 1.0 Walong 12

3. 1.2 D + LFCW Wcomb 13

4. (1.2 + 0.2 SDS)D + E 14

5. 0.9 D + 1.0 Walong 15

6. 0.9 D + LFCW Wcomb 16

7. (0.9 – 0.2 SDS)D + E 17

R11.3.1 These load combinations are the same as those in Section 7.7.1.1 and are repeated 18

here for convenience. 19

11.3.2 For foundations of chimneys with an assigned Seismic Design Category of D, E, or 20

F, the horizontal seismic effect E shall be replaced by the horizontal seismic effect including 21

overstrength Ω0E. The overstrength factor Ω0 shall be 1.5. 22

88

R11.3.2 The foundation system should not fail in a brittle manner. The response of the 1

foundation system to earthquake actions is dependent on the type of foundation system and 2

the soil conditions. Thus, overstrength is required for chimney foundations assigned to the 3

Seismic Design Categories indicated [Wilson 2003]. 4

11.3.3 Foundation systems shall be permitted to be designed by any procedure satisfying 5

equilibrium and geometric compatibility. 6

11.3.4 The foundation design shall be in accordance with the applicable sections of Chapter 7

13 of ACI 318-19. 8

9 10

89

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 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 Section 5.4. 15

12.4—Concrete strength tests 16

12.4.1 Unless otherwise specified, 𝑓𝑓𝑐𝑐′ shall be based on 28-day tests. If other than 28 days, 17

test age for 𝑓𝑓𝑐𝑐′ 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

90

be taken from each class of concrete cast each day and from each 100 cubic yards of concrete 1

placed each day. 2

12.4.4 For foundations, the frequency of strength tests shall be at least the minimum required 3

by Section 26.12.2 of ACI 318-19. 4

12.4.5 Acceptance criteria for all strength tests shall be per Section 26.12.3 of ACI 318-19. 5

12.5—Formwork 6

12.5.1 Formwork for the chimney shell shall be made of metal, wood, or other suitable 7

material. 8

12.5.2 Forms shall be sufficiently tight to prevent leakage of mortar. 9

12.5.3 If unlined wooden forms are used, they shall have tongue and groove joints and shall 10

be kept continuously wet to prevent shrinking and warping due to exposure to the elements. 11

12.5.4 Form oil shall not be used unless it is a non-staining type and it has been established 12

that specified protective coatings or paint can be applied to concrete exposed to form oil. 13

12.5.5 Forms shall be removed in such manner as to ensure the safety of the structure. Forms 14

shall be permitted to be removed after the concrete has hardened to a strength sufficient to 15

maintain its shape without damage and to safely support all loads on it, including temporary 16

construction loads. 17

12.5.6 Ties between inner and outer chimney shell forms shall not be permitted. 18

12.6—Concrete placement 19

12.6.1 Cast-in-place concrete shall be placed in layers no greater than 16 in. 20

12.6.2 Vertical construction joints shall not be used for cast-in-place chimney shells. 21

12.6.3 When used, horizontal construction joints for cast-in-place and precast concrete shall 22

be approximately evenly spaced throughout the height of the shell. Grout for setting precast 23

91

sections shall have a specified compressive strength equal to or greater than the specified 1

compressive strength of the precast sections. 2

12.6.4 Construction joints shall be properly prepared to facilitate bonding. As a minimum, 3

all laitance and loose material shall be removed. 4

12.7—Concrete curing 5

12.7.1 All necessary finishing of concrete shall commence immediately after the forms have 6

been removed. 7

12.7.2 As soon as finishing has been completed, both faces of the concrete shell shall be 8

cured by coating with a membrane-type curing compound or other method approved by the 9

licensed design professional. 10

12.7.3 The curing compound shall comply with ASTM C309 and shall be applied in strict 11

accordance with the manufacturer’s recommendations. 12

12.7.4 If coatings are to be applied to the concrete, the curing compound shall be of a type 13

compatible with the coatings. 14

12.8—Reinforcement placement 15

12.8.1 All reinforcing bars shall be tied at intervals of not more than 2 ft. Bars shall be 16

secured against displacement within the tolerances of ACI 117. 17

12.8.2 Vertical reinforcing bars projecting above the forms for the chimney wall or cores of 18

precast sections shall be temporarily braced so as to prevent the breaking of the bond with the 19

freshly placed concrete. 20

12.9—Construction tolerances 21

12.9.1 Vertical alignment of centerpoint—The actual centerpoint of the shell shall not vary 22

from its theoretical axis by more than 0.001 times the height of the shell or 1 in., whichever is 23

92

greater. Locally, the actual centerpoint of the shell shall not change by more than 1 in. for any 1

10 ft. of vertical rise. 2

12.9.2 Diameter – The measured outside shell diameter at any section shall not vary from 3

the specified diameter by more than 1 in. plus the 0.01 times the specified or theoretical 4

diameter. 5

12.9.3 Wall thickness – For walls 10 in. thick or less, the measured wall thickness shall not 6

vary from the specified wall thickness by more than -1/4 in., +1/2 in. For walls thicker than 7

10 in., the measured wall thickness shall not vary from the specified wall thickness by more 8

than -1/2 in., +1 in. A single wall thickness is defined as the average of at least four 9

measurements taken at a uniform spacing over a 60-degree arc. A negative tolerance decreases 10

the overall thickness and a positive tolerance increases the overall thickness. 11

12.9.4 Openings and embedments – Tolerances on the size and location of openings and 12

embedments in the shell cannot be uniformly established due to the varying degree of accuracy 13

required, and the varying nature of their use. Appropriate tolerances for each opening or 14

embedment shall be established and included in the construction drawings. 15

12.10—Precast erection 16

12.10.1 Precast sections shall be erected in a manner and at a rate that ensures that suffic ient 17

strength has been attained in grout, core concrete, and all connecting components to safely 18

support construction and applicable design loads. 19

12.10.2 Precast sections shall be grouted to level and joints shall be sealed. Shear keys shall 20

be installed if required by the licensed design professional. 21

22

93

COMMENTARY REFERENCES 1

American Concrete Institute 2

ACI 307-08 Code Requirements for Reinforced Concrete Chimneys and Commentary 3

ACI 318-19 Building Code Requirements for Structural Concrete 4

ACI 209R-92 Prediction of Creep, Shrinkage, and Temperature Effects in Concrete 5 (Reapproved 2008) Structures 6

CT-18 ACI Concrete Terminology 7

American Society of Civil Engineers 8

ASCE 7-16 Minimum Design Loads for Buildings and Other Structures 9

Authored Documents 10

Kilic, S. A., Altay, S. and Akyniyazov, D. 2016, “Investigation of the Benefits of Additional 11

Rebars for the Cyclic Response of Reinforced Concrete Chimneys,”, CICIND Research Report. 12

Okamoto, T., and Yagita, M. 1973, “The Experimental Investigation on the Flow Past a Circu-13

lar Cylinder of Finite Length Placed Normal to the Plane Surface in a Uniform Stream,” Bulletin, 14

Japanese Society Of Mechanical Engineers, No. 16, 805pp. 15

Radecki, D. J. 2014, “Recommendations for ACI 307-1x Across-wind/Along-wind Load 16

Combination Consistent with ASCE 7-10 Strength Design Wind Provisions,” International 17

Symposium on Industrial Chimneys and Cooling Towers, Prague, Czech Republic, pp. 421-429. 18

Vickery, B. J., and Basu, R. I. 1984, “ Response of Reinforced Concrete Chimneys to Vortex 19

Shedding,” Engineering Structures, V. 6, No. 4, pp. 324-333. 20

Wilson, J. L. 2003, “Earthquake Response of Tall Reinforced Concrete Chimneys,” 21

Engineering Structures, V. 25, No. 1, pp. 11-24. 22

23

94

APPENDIX A—HORIZONTAL SECTION STRENGTH FOR CIRCULAR 1 CHIMNEYS BY MODIFIED STRESS BLOCK METHOD 2

Appendix A consists of three sections: 3

1. Procedure for computing combined nominal compression and nominal bending moment 4

capacity 5

2. Stress block modification factor 6

3. Derivation of equations 7

Notation 8

The following notation is used in Appendix A only. Any notation used in the Code and in 9

Appendix A is defined in Section 2.2 of the Code. 10

𝐴𝐴 parameter in 𝑄𝑄2 expression

𝐾𝐾 parameter in 𝐾𝐾2 expression

𝐾𝐾1 parameter in 𝑀𝑀𝑛𝑛 expression

𝐾𝐾2 parameter in 𝑀𝑀𝑛𝑛 expression

𝐾𝐾𝑒𝑒 𝐸𝐸𝑠𝑠 𝑓𝑓𝑦𝑦⁄

𝑛𝑛1 number of openings in the compression zone

𝑄𝑄 stress block modification factor

𝑄𝑄1 parameter in 𝐾𝐾1 expression

𝑄𝑄2 parameter in 𝐾𝐾2 expression

𝑅𝑅� parameter in 𝐾𝐾2 expression

𝛼𝛼deg one-half the angle subtended by the neutral axis, degrees

𝛽𝛽1 factor relating depth of equivalent rectangular stress block to neutral axis

depth

95

𝛾𝛾 for two openings, one-half the angle between the opening centerlines,

radians

𝜀𝜀𝑚𝑚 maximum compressive strain

𝜃𝜃 variable indicating location on the section, radians

𝜆𝜆 parameter in 𝐾𝐾1 expression

𝜆𝜆1 parameter in 𝐾𝐾1 expression

𝜇𝜇 angular location of the yield strain in the compressive zone, radians

𝜏𝜏 angular location of the depth of the compression zone, radians

𝜓𝜓 angular location of the yield strain in the tension zone, radians

𝜔𝜔𝑧𝑧 𝜌𝜌𝑧𝑧𝑓𝑓𝑦𝑦 𝑓𝑓𝑐𝑐′⁄

1

A.1—Procedure for computing combined nominal compression and nominal bending 2

moment capacity 3

Consider the following demand at a horizontal section 4

𝑃𝑃𝑢𝑢 = factored vertical load at the section 5

𝑀𝑀𝑢𝑢 = factored bending moment at the section 6

The following procedure is permitted to be used to determine the minimum reinforcement to 7

provide the required combined nominal vertical load and nominal bending moment capacity 8

(𝑃𝑃𝑛𝑛,𝑀𝑀𝑛𝑛). 9

In general, a trial reinforcement ratio is chosen. Then, by iteration, the angle corresponding 10

to the neutral axis location, 𝛼𝛼, is determined by considering force equilibrium with 𝑃𝑃𝑛𝑛. Once 11

𝛼𝛼 is determined, 𝑀𝑀𝑛𝑛 is determined from moment equilibrium. If 𝜙𝜙𝑀𝑀𝑛𝑛 is not equal to 𝑀𝑀𝑢𝑢, the 12

reinforcement ratio is adjusted and the procedure is repeated. 13

96

Referring to Figures A.1, A.2 and A.3, the following equations apply to sections with no 1

openings, sections with one opening (centered in the compression zone) and sections with two 2

equal width openings symmetric with respect to the bending direction. For two openings, the 3

openings may be completely within the compression zone, partially within the compression 4

zone, or completely within the tension zone. 5

Step 1: compute the nominal vertical load capacity 6

𝑃𝑃𝑛𝑛 =𝑃𝑃𝑢𝑢𝜙𝜙

7

Step 2: determine opening parameters 8

No openings: 𝑛𝑛1 = 𝛽𝛽 = 𝛾𝛾 = 0 9

One opening: 𝑛𝑛1 = 1, 𝛽𝛽 ≠ 0, 𝛾𝛾 = 0 10

Two openings: 𝑛𝑛1 = 2, 𝛽𝛽 ≠ 0, 𝛾𝛾 ≠ 0 11

Step 3: determine parameters not dependent on 𝛼𝛼 or 𝜌𝜌𝑧𝑧 12

𝛽𝛽1 = 0.85− 0.05 �𝑓𝑓𝑐𝑐′ − 4000

1000� but 0.65 ≤ 𝛽𝛽1 ≤ 0.85 13

𝐾𝐾𝑒𝑒 = 𝐸𝐸𝑠𝑠𝑓𝑓𝑦𝑦

14

Step 4: assume a value for 𝜌𝜌𝑧𝑧 and compute 𝜔𝜔𝑧𝑧 15

𝜔𝜔𝑧𝑧 =𝜌𝜌𝑧𝑧𝑓𝑓𝑦𝑦𝑓𝑓𝑐𝑐′

16

Step 5: assume a value for 𝛼𝛼 and compute the following 17

𝜀𝜀𝑚𝑚 = 0.07 �1− cos𝛼𝛼1 + cos𝛼𝛼

� ≤ 0.003 18

cos𝜓𝜓 = cos 𝛼𝛼 − �1−cos𝛼𝛼𝜀𝜀𝑚𝑚

��𝑓𝑓𝑦𝑦𝐸𝐸𝑠𝑠� ≥ −1.0 19

cos𝜇𝜇 = cos𝛼𝛼 + �1 − cos𝛼𝛼

𝜀𝜀𝑚𝑚� �

𝑓𝑓𝑦𝑦𝐸𝐸𝑠𝑠� ≤ 1.0 20

97

cos𝜏𝜏 = 1− 𝛽𝛽1(1− cos𝛼𝛼) 1

𝑄𝑄 = (See Section A.2) 2

𝜆𝜆 = 𝛾𝛾 − 𝛽𝛽 if 𝑛𝑛1 = 2 and 𝛾𝛾 − 𝛽𝛽 < 𝜏𝜏 < 𝛾𝛾 + 𝛽𝛽 3

= 𝜏𝜏 if 𝑛𝑛1 = 2 and 𝜏𝜏 < 𝛾𝛾 − 𝛽𝛽 4

= 𝜏𝜏 − 𝑛𝑛1𝛽𝛽 otherwise 5

(Note: 𝜆𝜆 in radians) 6

𝑄𝑄1 = sin𝜓𝜓−sin 𝜇𝜇−(𝜓𝜓−𝜇𝜇) cos𝛼𝛼1−cos𝛼𝛼

7

𝜆𝜆1 = 𝜇𝜇 +𝜓𝜓 − 𝜋𝜋 (radians) 8

𝐾𝐾1 = 1.7𝑄𝑄𝜆𝜆 + 2𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑄𝑄1 + 2𝜔𝜔𝑧𝑧𝜆𝜆1 9

Step 6: check 𝐾𝐾1 vs. non-dimensional nominal vertical load capacity 10

𝑃𝑃𝑛𝑛𝑟𝑟𝑧𝑧𝑓𝑓𝑐𝑐

′ = 𝐾𝐾1 ? 11

If yes, go to Step 7. If no, return to Step 5. 12

Step 7: compute the following 13

𝑅𝑅� = sin(𝛾𝛾 − 𝛽𝛽) − (𝛾𝛾 − 𝛽𝛽) cos𝛼𝛼 14

if 𝑛𝑛1 = 2 and 𝛾𝛾 − 𝛽𝛽 < 𝜏𝜏 < 𝛾𝛾 + 𝛽𝛽 15

= sin 𝜏𝜏 − 𝜏𝜏 cos 𝛼𝛼 16

if 𝑛𝑛1 = 2 and 𝜏𝜏 < 𝛾𝛾 − 𝛽𝛽 17

= sin 𝜏𝜏 − (𝜏𝜏 − 𝑛𝑛1𝛽𝛽) cos𝛼𝛼 − 𝑛𝑛1cos𝛾𝛾 sin 𝛽𝛽 18

Otherwise 19

𝐴𝐴 = 2(𝜓𝜓 −𝜇𝜇)cos2𝛼𝛼 + sin𝜓𝜓 cos𝜓𝜓 − sin 𝜇𝜇 cos𝜇𝜇 − 4 cos𝛼𝛼(sin𝜓𝜓 − sin 𝜇𝜇) + (𝜓𝜓 −𝜇𝜇) 20

𝑄𝑄2 = 𝐴𝐴1−cos𝛼𝛼

21

𝐾𝐾 = sin𝜓𝜓 + sin 𝜇𝜇 + (𝜋𝜋 − 𝜓𝜓 − 𝜇𝜇) cos𝛼𝛼 22

98

𝐾𝐾2 = 1.7𝑄𝑄𝑅𝑅� + 𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑄𝑄2 + 2𝜔𝜔𝑧𝑧𝐾𝐾 1

Step 8: compute the nominal bending moment capacity 2

𝑀𝑀𝑛𝑛 = 𝑃𝑃𝑛𝑛𝑟𝑟 �cos𝛼𝛼 + 𝐾𝐾2𝐾𝐾1� 3

Step 9: compare nominal bending moment capacity to required bending strength 4

𝑀𝑀𝑢𝑢 = 𝜙𝜙𝑀𝑀𝑛𝑛 ? 5

If yes, done. If no, return to Step 4. 6

7

99

A.2 —Stress block modification factor 1

(Note: 𝛼𝛼 in degrees for this calculation only) 2

For 𝛼𝛼deg ≤ 5° 3

𝑄𝑄 = �−0.523 + 0.181𝛼𝛼deg − 0.0154𝛼𝛼deg2� + �41.3 − 13.2𝛼𝛼deg + 1.32𝛼𝛼deg

2� �𝑧𝑧𝑟𝑟� 4

For 5° < 𝛼𝛼deg ≤ 10° 5

𝑄𝑄 = �−0.154 + 0.01773𝛼𝛼deg + 0.00249𝛼𝛼deg2�+ 6

�16.42− 1.980𝛼𝛼deg + 0.0674𝛼𝛼deg2��

𝑡𝑡𝑟𝑟� 7

For 10° < 𝛼𝛼deg ≤ 17° 8

𝑄𝑄 = �−0.488 + 0.076𝛼𝛼deg� + �9.758− 0.640𝛼𝛼deg� �𝑧𝑧𝑟𝑟� 9

For 17° < 𝛼𝛼deg ≤ 25° 10

𝑄𝑄 = �−1.345 + 0.2018𝛼𝛼deg − 0.004434𝛼𝛼deg2�+ 11

�15.83 − 1.676𝛼𝛼deg + 0.03994𝛼𝛼deg2� �

𝑡𝑡𝑟𝑟� 12

For 25° < 𝛼𝛼deg ≤ 35° 13

𝑄𝑄 = �0.993 − 0.00258𝛼𝛼deg� + �−3.27 + 0.0862𝛼𝛼deg� �𝑧𝑧𝑟𝑟� 14

For 𝛼𝛼deg > 35° 15

𝑄𝑄 = 0.89 16

17

100

A.3 Derivation of equations 1

Referring to Fig. A1, locations on the section are referenced to the mid-surface of the wall. 2

Strain is assumed to vary linearly across the section. A polar coordinate system (𝑟𝑟, 𝜃𝜃) is defined 3

with angle 𝜃𝜃 = 0 at the point of maximum compressive strain and angle 𝜃𝜃 = 𝜋𝜋 at the point of 4

maximum tensile strain. Concrete stress is compressive only and is represented by a (modified) 5

stress block. Steel stress is assumed to follow an elastic-perfectly plastic rule. 6

Figures A.1, A.2 and A.3 illustrate the strain and stress distribution on the section. 7

For simplicity reinforcement is considered to be distributed evenly around the full 8

circumference, even in the presence of openings, since the reinforcement interrupted by an 9

opening is placed adjacent to the opening. The error introduced by this simplification is 10

negligibly small. 11

Also for simplicity, the maximum concrete strain is considered to occur at 𝜃𝜃=0 even if there 12

may be no concrete at that location (single opening case). Again, the error introduced by this 13

simplification is negligibly small. 14

To determine the bending moment capacity, the neutral axis angle 𝛼𝛼 is determined, by 15

iteration, such that force equilibrium is satisfied at the section. 16

𝑃𝑃𝑛𝑛 = 𝐹𝐹𝑐𝑐 + 𝑆𝑆𝑐𝑐 + 𝑆𝑆𝑐𝑐𝑦𝑦 − 𝑆𝑆𝑧𝑧 − 𝑆𝑆𝑧𝑧𝑦𝑦 (A-1)

The reinforcement forces are computed as follows: 17

𝑆𝑆𝑧𝑧 = 2� 𝐸𝐸𝑠𝑠𝑟𝑟(cos𝛼𝛼 − cos𝜃𝜃)𝜀𝜀𝑚𝑚

𝑟𝑟(1 − cos𝛼𝛼) 𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝑑𝑑𝜃𝜃𝜓𝜓

𝛼𝛼

= 2𝐸𝐸𝑠𝑠𝜀𝜀𝑚𝑚𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡(1− cos𝛼𝛼)

[𝜃𝜃 cos𝛼𝛼 − sin 𝜃𝜃]𝛼𝛼𝜓𝜓


[(𝜓𝜓− 𝛼𝛼) cos𝛼𝛼 − sin𝜓𝜓 + sin 𝛼𝛼]

18

101

𝑆𝑆𝑧𝑧𝑦𝑦 = 2� 𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝑑𝑑𝜃𝜃𝜋𝜋

𝜓𝜓

= 2𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡(𝜋𝜋 −𝜓𝜓)

𝑆𝑆𝑐𝑐 = 2� 𝐸𝐸𝑠𝑠𝑟𝑟(cos𝜃𝜃 − cos𝛼𝛼)𝜀𝜀𝑚𝑚

𝑟𝑟(1− cos𝛼𝛼) 𝑟𝑟𝜌𝜌𝑧𝑧 𝑡𝑡𝑑𝑑𝜃𝜃𝛼𝛼

𝜇𝜇


[sin 𝜃𝜃 − 𝜃𝜃 cos𝛼𝛼]𝜇𝜇𝛼𝛼

= 2𝐸𝐸𝑠𝑠𝜀𝜀𝑚𝑚𝑟𝑟𝜌𝜌𝑧𝑧 𝑡𝑡(1− cos𝛼𝛼)

[sin 𝛼𝛼 − sin 𝜇𝜇 − (𝛼𝛼 − 𝜇𝜇) cos𝛼𝛼]

𝑆𝑆𝑐𝑐𝑦𝑦 = 2� 𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝑑𝑑𝜃𝜃𝜇𝜇

0

= 2𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝜇𝜇

1

Defining 2

𝜔𝜔𝑧𝑧 =𝜌𝜌𝑧𝑧𝑓𝑓𝑦𝑦𝑓𝑓𝑐𝑐′

and 𝐾𝐾𝑒𝑒 =𝐸𝐸𝑠𝑠𝑓𝑓𝑦𝑦

so that 𝐸𝐸𝑠𝑠𝜌𝜌𝑧𝑧 = 𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′ 3

𝑆𝑆𝑧𝑧 = 2𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡[(𝜓𝜓−𝛼𝛼) cos𝛼𝛼−sin𝜓𝜓+sin 𝛼𝛼]

(1−cos𝛼𝛼) 4

𝑆𝑆𝑧𝑧𝑦𝑦 = 2𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(𝜋𝜋 −𝜓𝜓) 5

𝑆𝑆𝑐𝑐 = 2𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡[sin 𝛼𝛼−sin 𝜇𝜇−(𝛼𝛼−𝜇𝜇) cos𝛼𝛼]

(1−cos𝛼𝛼) 6

𝑆𝑆𝑐𝑐𝑦𝑦 = 2𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡𝜇𝜇 7

The concrete stress resultant for no openings is: 8

𝐹𝐹𝑐𝑐 = 2� 0.85𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡𝑑𝑑𝜃𝜃𝜏𝜏

0

= 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡𝜏𝜏

9

102

For one opening centered in the compression zone: 1

𝐹𝐹𝑐𝑐 = 2� 0.85𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡𝑑𝑑𝜃𝜃𝜏𝜏

𝛽𝛽

= 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(𝜏𝜏 − 𝛽𝛽)

2

For two symmetric openings completely in the compression zone, that is, when 𝜏𝜏 > 𝛾𝛾 + 𝛽𝛽 3

𝐹𝐹𝑐𝑐 = 2 �� 0.85𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡𝑑𝑑𝜃𝜃 + � 0.85𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡𝑑𝑑𝜃𝜃𝜏𝜏

𝛾𝛾+𝛽𝛽

𝛾𝛾−𝛽𝛽

0�

= 1.7𝑓𝑓𝑐𝑐′𝑄𝑄𝑟𝑟𝑡𝑡(𝜏𝜏− 2𝛽𝛽)

4

The above three cases can be combined. 5

𝐹𝐹𝑐𝑐 = 1.7𝑓𝑓𝑐𝑐′𝑄𝑄𝑟𝑟𝑡𝑡(𝜏𝜏 − 𝑛𝑛1𝛽𝛽) 6

For two symmetric openings partially in the compression zone, that is, when 𝛾𝛾 − 𝛽𝛽 < 𝜏𝜏 <7

𝛾𝛾 + 𝛽𝛽 8

𝐹𝐹𝑐𝑐 = 1.7𝑓𝑓𝑐𝑐′𝑄𝑄𝑟𝑟𝑡𝑡(𝛾𝛾 − 𝛽𝛽) 9

For two symmetric openings completely in the tension zone, that is, when 𝜏𝜏 < 𝛾𝛾 − 𝛽𝛽 10

𝐹𝐹𝑐𝑐 = 1.7𝑓𝑓𝑐𝑐′𝑄𝑄𝑟𝑟𝑡𝑡𝜏𝜏 11

The reinforcement forces can be combined as follows 12

𝑆𝑆𝑐𝑐 − 𝑆𝑆𝑧𝑧 = 2𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡𝑄𝑄1 13

where 𝑄𝑄1 =[sin𝜓𝜓 − sin 𝜇𝜇 − (𝜓𝜓 −𝜇𝜇) cos𝛼𝛼]

(1− cos𝛼𝛼) 14

𝑆𝑆𝑐𝑐𝑦𝑦 − 𝑆𝑆𝑧𝑧𝑦𝑦 = 2𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡𝜆𝜆1 15

where 𝜆𝜆1 = 𝜇𝜇+ 𝜓𝜓 −𝜋𝜋 16

So Eq. (A-1) becomes, in non-dimensional form 17

103

𝑃𝑃𝑛𝑛𝑟𝑟𝑡𝑡𝑓𝑓𝑐𝑐′� = 1.7𝑄𝑄𝜆𝜆 + 2𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑄𝑄1 + 2𝜔𝜔𝑧𝑧𝜆𝜆1 1

where 𝜆𝜆 = 𝛾𝛾 − 𝛽𝛽 if 𝑛𝑛1 = 2 and 𝛾𝛾 − 𝛽𝛽 < 𝜏𝜏 < 𝛾𝛾 + 𝛽𝛽 2

𝜆𝜆 = 𝜏𝜏 if 𝑛𝑛1 = 2 and 𝜏𝜏 < 𝛾𝛾 − 𝛽𝛽 3

𝜆𝜆 = 𝜏𝜏 − 𝑛𝑛1𝛽𝛽 otherwise 4

Once 𝛼𝛼 has been determined for a section, the nominal moment capacity of the section, 𝑀𝑀𝑛𝑛, 5

can be determined by summing moments about the neutral axis. 6

𝑀𝑀𝑛𝑛 − 𝑃𝑃𝑛𝑛𝑟𝑟 cos 𝛼𝛼 = 𝐹𝐹𝑐𝑐′ + 𝑆𝑆𝑐𝑐′ + 𝑆𝑆𝑐𝑐𝑦𝑦′ + 𝑆𝑆𝑧𝑧′ + 𝑆𝑆𝑧𝑧𝑦𝑦′ (A-2)

The reinforcement moments are computed as follows: 7

𝑆𝑆𝑧𝑧′ = 2� 𝐸𝐸𝑠𝑠𝑟𝑟2(cos𝛼𝛼 − cos𝜃𝜃)2𝜀𝜀𝑚𝑚

𝑟𝑟(1− cos 𝛼𝛼) 𝑟𝑟𝜌𝜌𝑧𝑧 𝑡𝑡𝑑𝑑𝜃𝜃𝜓𝜓

𝛼𝛼

= 2𝐸𝐸𝑠𝑠𝜀𝜀𝑚𝑚𝑟𝑟2𝜌𝜌𝑧𝑧 𝑡𝑡(1− cos𝛼𝛼) �𝜃𝜃cos2𝛼𝛼 − 2 cos𝛼𝛼 sin 𝜃𝜃 +

𝜃𝜃2

+sin 2𝜃𝜃

4�𝛼𝛼

𝜓𝜓

= 2𝐸𝐸𝑠𝑠𝜀𝜀𝑚𝑚𝑟𝑟2𝜌𝜌𝑧𝑧 𝑡𝑡(1− cos𝛼𝛼) �

(𝜓𝜓 − 𝛼𝛼)cos2𝛼𝛼

− 2 cos𝛼𝛼 (sin𝜓𝜓 − sin 𝛼𝛼) +𝜓𝜓 −𝛼𝛼

2+

sin 2𝜓𝜓− sin 2𝛼𝛼4

�

𝑆𝑆𝑧𝑧𝑦𝑦′ = 2� 𝑓𝑓𝑦𝑦𝑟𝑟2(cos𝛼𝛼 − cos 𝜃𝜃)𝜌𝜌𝑧𝑧𝑡𝑡𝑑𝑑𝜃𝜃𝜋𝜋

𝜓𝜓

= 2𝑓𝑓𝑦𝑦𝑟𝑟2𝜌𝜌𝑧𝑧 𝑡𝑡[𝜃𝜃 cos𝛼𝛼 − sin 𝜃𝜃]𝜓𝜓𝜋𝜋

= 2𝑓𝑓𝑦𝑦𝑟𝑟2𝜌𝜌𝑧𝑧 𝑡𝑡[(𝜋𝜋 − 𝜓𝜓) cos𝛼𝛼 + sin𝜓𝜓]

𝑆𝑆𝑐𝑐′ = 2� 𝐸𝐸𝑠𝑠𝑟𝑟2(cos𝜃𝜃 − cos𝛼𝛼)2𝜀𝜀𝑚𝑚

𝑟𝑟(1− cos𝛼𝛼) 𝑟𝑟𝜌𝜌𝑧𝑧 𝑡𝑡𝑑𝑑𝜃𝜃𝛼𝛼

𝜇𝜇


𝜃𝜃2

+sin 2𝜃𝜃

4− 2 cos𝛼𝛼 sin𝜃𝜃 + 𝜃𝜃cos2𝛼𝛼�

𝜇𝜇

𝛼𝛼

104


𝛼𝛼 − 𝜇𝜇2

+sin 2𝛼𝛼 − sin 2𝜇𝜇

4− 2 cos𝛼𝛼 (sin 𝛼𝛼 − sin 𝜇𝜇)

+ (𝛼𝛼 − 𝜇𝜇)cos2𝛼𝛼�

𝑆𝑆𝑐𝑐𝑦𝑦′ = 2� 𝑓𝑓𝑦𝑦𝑟𝑟2(cos𝜃𝜃 − cos𝛼𝛼)𝜌𝜌𝑧𝑧𝑡𝑡𝑑𝑑𝜃𝜃

𝜇𝜇

0

= 2𝑓𝑓𝑦𝑦𝑟𝑟2𝜌𝜌𝑧𝑧 𝑡𝑡[sin 𝜃𝜃 − 𝜃𝜃 cos𝛼𝛼]0𝜇𝜇

= 2𝑓𝑓𝑦𝑦𝑟𝑟2𝜌𝜌𝑧𝑧 𝑡𝑡[sin 𝜇𝜇 − 𝜇𝜇 cos𝛼𝛼]

1

Substituting 𝜔𝜔𝑧𝑧 and 𝐾𝐾𝑒𝑒defined as above 2

𝑆𝑆𝑧𝑧′ = 2𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑡𝑡𝑓𝑓𝑐𝑐′𝑟𝑟2𝑧𝑧

(1−cos𝛼𝛼) �(𝜓𝜓 −𝛼𝛼)cos2𝛼𝛼 − 2 cos𝛼𝛼 (sin𝜓𝜓 − sin 𝛼𝛼) + 𝜓𝜓−𝛼𝛼2

+ sin 2𝜓𝜓−sin 2𝛼𝛼4

� 3

𝑆𝑆𝑧𝑧𝑦𝑦′ = 2𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡[(𝜋𝜋 −𝜓𝜓) cos𝛼𝛼 + sin𝜓𝜓] 4

𝑆𝑆𝑐𝑐′ = 2𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑡𝑡𝑓𝑓𝑐𝑐′𝑟𝑟2 𝑧𝑧

(1−cos𝛼𝛼) �𝛼𝛼−𝜇𝜇2

+ sin 2𝛼𝛼−sin 2𝜇𝜇4

− 2 cos 𝛼𝛼 (sin 𝛼𝛼 − sin 𝜇𝜇) + (𝛼𝛼 − 𝜇𝜇)cos2𝛼𝛼� 5

𝑆𝑆𝑐𝑐𝑦𝑦′ = 2𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡[sin𝜇𝜇 − 𝜇𝜇 cos𝛼𝛼] 6

For no openings: 7

𝐹𝐹𝑐𝑐′ = 2� 0.85𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡(cos𝜃𝜃 − cos𝛼𝛼)𝑑𝑑𝜃𝜃𝜏𝜏

0

= 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡[sin 𝜃𝜃 − 𝜃𝜃 cos𝛼𝛼]0𝜏𝜏

= 1.7𝑄𝑄𝑐𝑐′ 𝑟𝑟2𝑡𝑡(sin 𝜏𝜏 − 𝜏𝜏 cos𝛼𝛼)

For one opening centered in the compression zone: 8

𝐹𝐹𝑐𝑐′ = 2� 0.85𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡(cos𝜃𝜃 − cos𝛼𝛼)𝑑𝑑𝜃𝜃𝜏𝜏

𝛽𝛽

= 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡[sin𝜃𝜃 − 𝜃𝜃 cos𝛼𝛼]𝛽𝛽𝜏𝜏

= 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡[sin 𝜏𝜏 − sin 𝛽𝛽 − (𝜏𝜏− 𝛽𝛽) cos𝛼𝛼]

105

For two symmetric openings completely in the compression zone, that is, when 𝜏𝜏 > 𝛾𝛾 + 𝛽𝛽 1

𝐹𝐹𝑐𝑐′ = 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡 �� (cos𝜃𝜃 − cos𝛼𝛼)𝑑𝑑𝜃𝜃 + � (cos𝜃𝜃 − cos𝛼𝛼)𝑑𝑑𝜃𝜃𝜏𝜏

𝛾𝛾+𝛽𝛽

𝛾𝛾−𝛽𝛽

0�

= 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡�[sin𝜃𝜃 − 𝜃𝜃 cos𝛼𝛼]0𝛾𝛾−𝛽𝛽 + [sin 𝜃𝜃 − 𝜃𝜃 cos𝛼𝛼]𝛾𝛾+𝛽𝛽𝜏𝜏 �

= 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡[sin 𝜏𝜏 − (𝜏𝜏 − 2𝛽𝛽) cos𝛼𝛼 − 2 cos𝛾𝛾 sin 𝛽𝛽]

The above three cases can be combined. 2

𝐹𝐹𝑐𝑐′ = 1.7𝑓𝑓𝑐𝑐′𝑄𝑄𝑟𝑟2𝑡𝑡[sin 𝜏𝜏 − (𝜏𝜏 − 𝑛𝑛1𝛽𝛽) cos𝛼𝛼 − 𝑛𝑛1cos 𝛾𝛾 sin 𝛽𝛽] 3

For two symmetric openings partially in the compression zone, that is, when 𝛾𝛾 − 𝛽𝛽 < 𝜏𝜏 <4

𝛾𝛾 + 𝛽𝛽 5

𝐹𝐹𝑐𝑐′ = 2� 0.85𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡(cos𝜃𝜃 − cos𝛼𝛼)𝑑𝑑𝜃𝜃𝛾𝛾−𝛽𝛽

0

= 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡[sin 𝜃𝜃 − 𝜃𝜃 cos𝛼𝛼]0𝛾𝛾−𝛽𝛽

= 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡[sin(𝛾𝛾 − 𝛽𝛽) − (𝛾𝛾 − 𝛽𝛽) cos𝛼𝛼]

For two symmetric openings completely in the tension zone, that is, when 𝜏𝜏 < 𝛾𝛾 − 𝛽𝛽 6

𝐹𝐹𝑐𝑐′ = 1.7𝑄𝑄𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡[sin 𝜏𝜏 − 𝜏𝜏 cos𝛼𝛼] 7

The reinforcement moments can be combined as follows 8

𝑆𝑆𝑐𝑐′ + 𝑆𝑆𝑧𝑧′ = 𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡𝑄𝑄2 9

𝑆𝑆𝑐𝑐𝑦𝑦′ + 𝑆𝑆𝑧𝑧𝑦𝑦′ = 2𝜔𝜔𝑧𝑧𝑓𝑓𝑐𝑐′𝑟𝑟2𝑡𝑡𝐾𝐾 10

where 𝑄𝑄2 =𝐴𝐴

(1− cos𝛼𝛼) 11

𝐴𝐴 = [2(𝜓𝜓− 𝜇𝜇)cos2𝛼𝛼− 4 cos 𝛼𝛼(sin𝜓𝜓 − sin 𝜇𝜇) + (𝜓𝜓− 𝜇𝜇) + sin𝜓𝜓 cos𝜓𝜓 − sin 𝜇𝜇 cos𝜇𝜇] 12

𝐾𝐾 = (𝜋𝜋 −𝜓𝜓 − µ) cos𝛼𝛼 + sin𝜓𝜓 + sin 𝜇𝜇 13

So Eq. (A-2) becomes, in non-dimensional form 14

106

𝑀𝑀𝑛𝑛𝑟𝑟2𝑡𝑡𝑓𝑓𝑐𝑐′� =

𝑃𝑃𝑛𝑛 cos 𝛼𝛼𝑟𝑟𝑡𝑡𝑓𝑓𝑐𝑐′

+ 1.7𝑄𝑄𝑅𝑅� + 𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑄𝑄2 + 2𝜔𝜔𝑧𝑧𝐾𝐾 1

where 2

𝑅𝑅� = sin(𝛾𝛾 − 𝛽𝛽) − (𝛾𝛾 − 𝛽𝛽) cos𝛼𝛼 3

if 𝑛𝑛1 = 2 and 𝛾𝛾 − 𝛽𝛽 < 𝜏𝜏 < 𝛾𝛾 + 𝛽𝛽 4

𝑅𝑅� = sin𝜏𝜏 − 𝜏𝜏 cos𝛼𝛼 5

if 𝑛𝑛1 = 2 and 𝜏𝜏 < 𝛾𝛾 − 𝛽𝛽 6

𝑅𝑅� = sin𝜏𝜏 − (𝜏𝜏 − 𝑛𝑛1𝛽𝛽) cos𝛼𝛼 − 𝑛𝑛1cos𝛾𝛾 sin 𝛽𝛽 7

otherwise 8

Letting 9

𝐾𝐾1 = 1.7𝑄𝑄𝜆𝜆 + 2𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑄𝑄1 + 2𝜔𝜔𝑧𝑧𝜆𝜆1 10

𝐾𝐾2 = 1.7𝑄𝑄𝑅𝑅� + 𝜀𝜀𝑚𝑚𝐾𝐾𝑒𝑒𝜔𝜔𝑧𝑧𝑄𝑄2 + 2𝜔𝜔𝑧𝑧𝐾𝐾 11

The nominal bending moment strength can be expressed as 12

𝑀𝑀𝑛𝑛 = 𝑃𝑃𝑛𝑛𝑟𝑟 �cos𝛼𝛼 +𝐾𝐾2𝐾𝐾1� � 13

107

1 Figure A.1: Stress-strain relationships, no opening or one opening 2

3

108

1

2

Figure A.2: Two openings fully in compression zone 3

(Dimensions not shown; same as Fig. A.1) 4

5

6

Figure A.3: Two openings partially in compression zone 7

(Dimensions not shown; same as Fig. A.1) 8

9

109

APPENDIX B—HORIZONTAL SECTION STRENGTH FOR CIRCULAR 1 CHIMNEYS BY STRESS-STRAIN RELATIONSHIP INTEGRATION METHOD 2

Appendix B consists of two sections: 3

1. Procedure for computing combined nominal compression and nominal bending moment ca-4

pacity 5

2. Derivation of equations 6

Notation 7

The following notation is used in Appendix B only. Any notation used in the Code and in 8

Appendix B is defined in Section 2.2 of the Code. 9

𝐹𝐹𝑐𝑐1 force in concrete for the linear-varying stress region, lb

𝐹𝐹𝑐𝑐1′ moment of 𝐹𝐹𝑐𝑐1 about neutral axis, lb-ft

𝐹𝐹𝑐𝑐2 force in concrete for the parabolic-varying stress region, lb

𝐹𝐹𝑐𝑐2′ moment of 𝐹𝐹𝑐𝑐2 about neutral axis, lb-ft

𝐹𝐹𝐹𝐹𝑐𝑐1 force in concrete for the linear-varying stress region to subtract for two opening

case, lb

𝐹𝐹𝐹𝐹𝑐𝑐1′ moment of 𝐹𝐹𝐹𝐹𝑐𝑐1 about neutral axis to subtract for two opening case, lb-ft

𝐹𝐹𝐹𝐹𝑐𝑐2 force in concrete for the parabolic-varying stress region to subtract for two open-

ing case, lb

𝐹𝐹𝐹𝐹𝑐𝑐2′ moment of 𝐹𝐹𝐹𝐹𝑐𝑐2 about neutral axisto subtract for two opening case, lb-ft

𝑡𝑡 thickness of concrete wall, ft

𝑥𝑥0 depth to 0.002 compressive strain , ft (see Figure B.1)

𝑥𝑥𝑛𝑛𝑎𝑎 depth to neutral axis, ft (see Figure B.1)

𝑥𝑥𝑦𝑦𝑐𝑐 depth to compressive steel yield strain, ft (see Figure B.1)

110

𝑥𝑥𝑦𝑦𝑧𝑧 depth to tensile steel yield strain, ft (see Figure B.1)

𝛾𝛾 for two openings, one-half the angle between the opening centerlines, radians

𝛿𝛿1 min[max(𝜏𝜏 , 𝛾𝛾 − 𝛽𝛽),𝛼𝛼] , radians

𝛿𝛿2 max[min(𝛼𝛼 , 𝛾𝛾 + 𝛽𝛽), 𝜏𝜏] , radians

𝜀𝜀𝑐𝑐𝑢𝑢 concrete strain limit ( + = compression)

𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥 maximum strain ( + = compression)

𝜀𝜀𝑚𝑚𝑐𝑐𝑛𝑛 minimum strain ( + = compression)

𝜀𝜀𝑠𝑠𝑢𝑢 steel strain limit ( + = compression)

𝜀𝜀𝑦𝑦𝑐𝑐 compressive steel yield strain ( + = compression)

𝜀𝜀𝑦𝑦𝑧𝑧 tensile steel yield strain ( + = compression)

𝜃𝜃 variable indicating location on the section, radians

𝜆𝜆1 min(𝜏𝜏 , 𝛾𝛾 − 𝛽𝛽) , radians

𝜆𝜆2 min(𝜏𝜏 , 𝛾𝛾 + 𝛽𝛽) , radians

𝜇𝜇 angular location of the yield strain in the compressive zone, radians

𝜏𝜏 angular location where the compressive strain is 0.002, radians

𝜓𝜓 angular location of the yield strain in the tension zone, radians

1

B.1 Procedure for computing combined nominal compression and nominal bending mo-2

ment capacity 3

Consider the following demand at a horizontal section 4

𝑃𝑃𝑢𝑢 = factored vertical load at the section 5

𝑀𝑀𝑢𝑢 = factored bending moment at the section 6

111

The following procedure is permitted to be used to determine the minimum reinforcement 1

to provide the required combined nominal vertical load and nominal bending moment capac-2

ity (𝑃𝑃𝑛𝑛,𝑀𝑀𝑛𝑛). 3

The radius to the middle surface of the wall is 𝑟𝑟. The neutral axis is located by its distance, 4

𝑥𝑥𝑛𝑛𝑎𝑎, from the maximum compression location at the middle surface of the section (ignoring 5

any opening). In Fig. B.1, this is the middle of the wall at 0 degrees. 6

Procedure (see Section B.2 for notation) 7

1. Set reinforcement ratio; the minimum reinforcement ratio required is the initial value 8

2. By iteration, determine the location of the neutral axis, 𝑥𝑥𝑛𝑛𝑎𝑎, so that axial force equilibrium is 9

achieved 10

𝑃𝑃𝑢𝑢𝜙𝜙

= 𝑃𝑃𝑛𝑛 = 𝐹𝐹𝑐𝑐1 + 𝐹𝐹𝑐𝑐2 + 𝑆𝑆𝑐𝑐 + 𝑆𝑆𝑐𝑐𝑦𝑦 + 𝑆𝑆𝑧𝑧 + 𝑆𝑆𝑧𝑧𝑦𝑦 11

3. Determine the moment capacity corresponding to the axial load capacity 𝜙𝜙𝑃𝑃𝑛𝑛 12

𝑀𝑀𝑛𝑛 = 𝑃𝑃𝑛𝑛(𝑟𝑟 − 𝑥𝑥𝑛𝑛𝑎𝑎) + 𝐹𝐹𝑐𝑐1′ + 𝐹𝐹𝑐𝑐2′ + 𝑆𝑆𝑐𝑐′ + 𝑆𝑆𝑐𝑐𝑦𝑦′ + 𝑆𝑆𝑧𝑧′ + 𝑆𝑆𝑧𝑧𝑦𝑦′ 13

4. If 𝑀𝑀𝑢𝑢 > 𝜙𝜙𝑀𝑀𝑛𝑛 , increase the reinforcement ratio and go to Step 2 14

5. If 𝑀𝑀𝑢𝑢 ≈ 𝜙𝜙𝑀𝑀𝑛𝑛 or reinforcement ratio is equal to minimum required, Stop 15

6. If 𝑀𝑀𝑢𝑢 < 𝜙𝜙𝑀𝑀𝑛𝑛 , decrease reinforcement ratio and go to Step 2 16

17 Referring to Figures B.1, B.2 and B.3, the following equations apply to sections with no 18

openings, sections with one opening (centered in the compression zone) and sections with two 19

equal width openings symmetric with respect to the bending direction. For two openings, the 20

openings may be completely within the compression zone, partially within the compression 21

zone, or completely within the tension zone. 22

B.2 Derivation of equations 23

112

Referring to Fig. B.1, locations on the section are referenced to the mid-surface of the wall. 1

Strain is assumed to vary linearly across the section. A polar coordinate system (𝑟𝑟, 𝜃𝜃) is defined 2

with angle 𝜃𝜃 = 0 at the point of maximum compressive strain and angle 𝜃𝜃 = 𝜋𝜋 at the point of 3

maximum tensile strain. Concrete stress is compressive only and is represented by a parabolic-4

varying stress region and a linearly-varying stress region. Steel stress is assumed to follow an 5

elastic-perfectly plastic rule. 6

Figures B.1, B.2 and B.3 illustrate the strain and stress distribution on the section. 7

For simplicity reinforcement is considered to be distributed evenly around the full circum-8

ference, even in the presence of openings, since the reinforcement interrupted by an opening 9

is placed adjacent to the opening. The error introduced by this simplification is negligib ly 10

small. 11

Also for simplicity, the maximum concrete strain is considered to occur at 𝜃𝜃 = 0 even if 12

there may be no concrete at that location (single opening case). Again, the error introduced by 13

this simplification is negligibly small. 14

15

Sections with no openings or one opening in the compression zone 16

The neutral axis is located a distance 𝑥𝑥𝑛𝑛𝑎𝑎 from the point of maximum concrete strain. The 17

following derivation is valid for any positive value of 𝑥𝑥𝑛𝑛𝑎𝑎. When 𝑥𝑥𝑛𝑛𝑎𝑎 ≥ 2𝑟𝑟 the entire section 18

is in compression. An angle 𝛼𝛼 also locates the neutral axis and is defined as follows 19

𝛼𝛼 = cos−1 �𝑟𝑟 − 𝑥𝑥𝑛𝑛𝑎𝑎

𝑟𝑟� if 𝑥𝑥𝑛𝑛𝑎𝑎 < 2𝑟𝑟 20

= 𝜋𝜋 otherwise 21

The maximum strain limits for concrete and steel, with compression strains positive, are 22

𝜀𝜀𝑐𝑐𝑢𝑢 = 0.003 23

113

𝜀𝜀𝑠𝑠𝑢𝑢 = −0.07 1

For a given neutral axis location, either the maximum concrete or maximum steel strain will 2

govern. The minimum and maximum strains on the section are 3

𝜀𝜀min = max � 𝜀𝜀𝑠𝑠𝑢𝑢 , 𝜀𝜀𝑐𝑐𝑢𝑢𝑥𝑥𝑛𝑛𝑎𝑎 − 2𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎

� 4

𝜀𝜀max =1

max � 1𝜀𝜀𝑐𝑐𝑢𝑢

, 𝑥𝑥𝑛𝑛𝑎𝑎 − 2𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎

𝜀𝜀min� 5

The strain at any angle 𝜃𝜃 between 0 and 𝜋𝜋 is 6

𝜀𝜀 = 𝜀𝜀max𝑟𝑟 cos𝜃𝜃 − (𝑟𝑟 − 𝑥𝑥𝑛𝑛𝑎𝑎)

𝑥𝑥𝑛𝑛𝑎𝑎 7

The concrete force is divided into two regions; a region of parabolic-varying stress from the 8

neutral axis to a strain of 0.002 and a region of linearly-varying stress from a strain of 0.002 to 9

a strain of 0.003. The strain is 0.002 at a distance 𝑥𝑥0 from the point of maximum concrete 10

compressive strain (see Fig. A.4). This distance and the corresponding angle 𝜏𝜏 are defined as 11

follows 12

𝑥𝑥0 = 𝑥𝑥𝑛𝑛𝑎𝑎 �1−0.002𝜀𝜀max

� if ε < 𝜀𝜀max ; = 0 otherwise 13

𝜏𝜏 = 𝑐𝑐𝑐𝑐𝑠𝑠−1 �𝑟𝑟 − 𝑥𝑥0𝑟𝑟

� if x0 < 2r ; = π otherwise 14

The concrete compressive force in the linearly-varying stress region is 15

𝐹𝐹𝑐𝑐1 = 2∫ 0.85𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧) �1.3 − 150 ��1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

� + 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

cos𝜃𝜃� 𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥� 𝑑𝑑𝜃𝜃𝜏𝜏𝛽𝛽 16

= 𝐴𝐴1[𝑎𝑎1(𝜏𝜏− 𝛽𝛽) + 𝑏𝑏1(sin 𝜏𝜏 − sin 𝛽𝛽)] 17

where 18

𝐴𝐴1 = 1.7𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1−𝜌𝜌𝑧𝑧 ) 19

𝑎𝑎1 = 1.3− 150�1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

�𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥 20

114

𝑏𝑏1 = −150 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥 1

The moment about the neutral axis due to force 𝐹𝐹𝑐𝑐1 is 2

𝐹𝐹𝑐𝑐1′ = 2� 0.85𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1−𝜌𝜌𝑧𝑧 ) �1.3 − 150 ��1 −𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎

�+𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎

cos𝜃𝜃� 𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥� [(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟)𝜏𝜏

𝛽𝛽3

+ 𝑟𝑟 cos𝜃𝜃]𝑑𝑑𝜃𝜃 4

= 𝐴𝐴2 �𝑎𝑎2(𝜏𝜏− 𝛽𝛽) + 𝑏𝑏2(sin 𝜏𝜏 − sin 𝛽𝛽) +𝑐𝑐22

(sin 𝜏𝜏 cos𝜏𝜏 − sin 𝛽𝛽 cos𝛽𝛽+ 𝜏𝜏 − 𝛽𝛽)� 5

where 6

𝐴𝐴2 = 1.7𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧) 7

𝑎𝑎2 = �1.3− 150�1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

�𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥 � (𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) 8

𝑏𝑏2 = 𝑟𝑟 �1.3− 150�1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

�𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥 � − 150 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) 9

𝑐𝑐2 = −150 𝑟𝑟2

𝑥𝑥𝑛𝑛𝑛𝑛𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥 10

The concrete stress is parabolic-varying from 0 strain to 0.002 strain, according to the fol-11

lowing stress-strain relationship 12

𝑓𝑓𝑐𝑐 = 0.85𝑓𝑓𝑐𝑐′ �2𝜀𝜀

0.002− �

𝜀𝜀0.002

�2� 13

Substituting the expression for strain at angle 𝜃𝜃 14

𝑓𝑓𝑐𝑐 = 0.85𝑓𝑓𝑐𝑐′ �2𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥0.002

��1−𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎


cos𝜃𝜃�15

− �𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥0.002

�2��1−

𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎

� +𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎

cos 𝜃𝜃�2

� 16

The concrete compressive force in the parabolic region is 17

18

115

𝐹𝐹𝑐𝑐2 = 2� 0.85𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧) �2𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥0.002



cos𝜃𝜃�𝛼𝛼

𝜏𝜏1


�2��1−



cos 𝜃𝜃�2

� 𝑑𝑑𝜃𝜃 2

= 𝐴𝐴3[𝑎𝑎3(𝛼𝛼 − 𝜏𝜏) + 𝑏𝑏3(sin 𝛼𝛼 − sin 𝜏𝜏) +𝑐𝑐32

(sin𝛼𝛼 cos𝛼𝛼 − sin 𝜏𝜏 cos𝜏𝜏 + 𝛼𝛼 − 𝜏𝜏)] 3

where 4

𝐴𝐴3 = 1.7𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧)𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚0.002

5

𝑎𝑎3 = 2 �1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

� − 𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚0.002

�1 − 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

�2 6

𝑏𝑏3 = 2 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

�1 − 𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚0.002


�� 7

𝑐𝑐3 = − 𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚0.002

� 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

�2 8

The moment about the neutral axis due to force 𝐹𝐹𝑐𝑐2 is 9

𝐹𝐹𝑐𝑐2′ = 2� 0.85𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧) �2𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥0.002



cos𝜃𝜃�𝛼𝛼

𝜏𝜏10


�2��1−



cos 𝜃𝜃�2

� [(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) + 𝑟𝑟 cos𝜃𝜃]𝑑𝑑𝜃𝜃 11

= 𝐴𝐴4[𝑎𝑎4(𝛼𝛼 − 𝜏𝜏) + 𝑏𝑏4(sin 𝛼𝛼 − sin 𝜏𝜏) +𝑐𝑐42

(sin𝛼𝛼 cos𝛼𝛼 − sin 𝜏𝜏 cos𝜏𝜏 + 𝛼𝛼 − 𝜏𝜏) 12

+𝑑𝑑43

(sin𝛼𝛼 cos2𝛼𝛼 − sin 𝜏𝜏 cos2𝜏𝜏 + 2(sin 𝛼𝛼 − sin 𝜏𝜏))] 13

where 14

𝐴𝐴4 = 1.7𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧)𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚0.002

15

116

𝑎𝑎4 = �2 �1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛


�1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

�2� (𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) 1

𝑏𝑏4 = �2 �1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛


�1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

�2�𝑟𝑟 + 2 𝑟𝑟

𝑥𝑥𝑛𝑛𝑛𝑛�1− 𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚

0.002�1− 𝑟𝑟

𝑥𝑥𝑛𝑛𝑛𝑛�� (𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) 2

𝑐𝑐4 = 2 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

�1 − 𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚0.002


�� 𝑟𝑟 − �𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚0.002


�2�(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) 3

𝑑𝑑4 = − �𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚0.002


�2� 𝑟𝑟 4

The steel compressive force is divided into two regions; a region of constant stress and a 5

region of linear-varying stress. The compressive steel stress is constant 𝑓𝑓𝑦𝑦 when the compres-6

sive strain exceeds the compressive yield strain 𝜀𝜀𝑦𝑦𝑐𝑐 . The compressive strain equals the com-7

pressive yield strain at a distance 𝑥𝑥𝑦𝑦𝑐𝑐 from the point of maximum concrete compressive strain 8

(see Fig. B.1). The yield strain, the distance 𝑥𝑥𝑦𝑦𝑐𝑐 and the corresponding angle 𝜇𝜇 are defined as 9

follows 10

𝜀𝜀𝑦𝑦𝑐𝑐 = 𝑓𝑓𝑦𝑦𝐸𝐸𝑠𝑠

11

𝑥𝑥𝑦𝑦𝑐𝑐 = �1− 𝜀𝜀𝑦𝑦𝑐𝑐𝜀𝜀max

� 𝑥𝑥𝑛𝑛𝑎𝑎 if 𝜀𝜀max > 𝜀𝜀𝑦𝑦𝑐𝑐 ; = 0 otherwise 12

𝜇𝜇 = cos−1 �𝑟𝑟−𝑥𝑥𝑦𝑦𝑐𝑐𝑟𝑟

� if 𝑥𝑥𝑦𝑦𝑐𝑐 < 2𝑟𝑟 ; = π otherwise 13

The steel compressive force where steel stress is at the yield strength 14

𝑆𝑆𝑐𝑐𝑦𝑦 = 2 ∫ 𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡 𝑑𝑑𝜃𝜃𝜇𝜇0 15

= 2𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝜇𝜇 16

The moment about the neutral axis due to force 𝑆𝑆𝑐𝑐𝑦𝑦 is 17

𝑆𝑆𝑐𝑐𝑦𝑦′ = 2� 𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡 (𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟 + 𝑟𝑟 cos𝜃𝜃)𝑑𝑑𝜃𝜃𝜇𝜇

0 18

= 2𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡[(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟)𝜇𝜇+ 𝑟𝑟 sin 𝜇𝜇] 19

117

The steel compressive force where steel stress is below the yield strength 1

𝑆𝑆𝑐𝑐 = 2 ∫ 𝐸𝐸𝑠𝑠𝛼𝛼𝜇𝜇 𝜀𝜀max ��1− 𝑟𝑟

𝑥𝑥𝑛𝑛𝑛𝑛�+ 𝑟𝑟

𝑥𝑥𝑛𝑛𝑛𝑛cos𝜃𝜃�𝑟𝑟𝜌𝜌𝑧𝑧 𝑡𝑡𝑑𝑑𝜃𝜃 2

= 2𝐸𝐸𝑠𝑠𝜀𝜀max𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡 ��1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

� (𝛼𝛼 − 𝜇𝜇) + 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

(sin𝛼𝛼 − sin 𝜇𝜇)� 3

The moment about the neutral axis due to force 𝑆𝑆𝑐𝑐 is 4

𝑆𝑆𝑐𝑐′ = 2 ∫ 𝐸𝐸𝑠𝑠𝛼𝛼𝜇𝜇 𝜀𝜀max ��1− 𝑟𝑟


𝑥𝑥𝑛𝑛𝑛𝑛cos𝜃𝜃� [(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) + 𝑟𝑟 cos𝜃𝜃]𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝑑𝑑𝜃𝜃 5

= 𝐴𝐴5 �𝑎𝑎5(𝛼𝛼 − 𝜇𝜇) + 𝑏𝑏5(sin 𝛼𝛼 − sin 𝜇𝜇) + 𝑐𝑐52

(sin𝛼𝛼 cos𝛼𝛼 − sin 𝜇𝜇 cos𝜇𝜇 + 𝛼𝛼 − 𝜇𝜇)� 6

where 7

𝐴𝐴5 = 2𝐸𝐸𝑠𝑠𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝜀𝜀max 8

𝑎𝑎5 = �1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

� (𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) 9

𝑏𝑏5 = �1 −𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎

�𝑟𝑟 +𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎

(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) 10

𝑐𝑐5 = 𝑟𝑟2

𝑥𝑥𝑛𝑛𝑛𝑛 11

The steel tensile force is divided into two regions; a region of constant stress and a region of 12

linear-varying stress. The tensile steel stress is constant −𝑓𝑓𝑦𝑦 when the tensile strain exceeds 13

the tensile yield strain 𝜀𝜀𝑦𝑦𝑧𝑧. The tensile strain equals the tensile yield strain at a distance 𝑥𝑥𝑦𝑦𝑧𝑧 14

from the point of maximum concrete compressive strain (see Fig. B.1). The tensile yield strain, 15

the distance 𝑥𝑥𝑦𝑦𝑧𝑧 and the corresponding angle 𝜓𝜓 are defined as follows 16

𝜀𝜀𝑦𝑦𝑧𝑧 = − 𝑓𝑓𝑦𝑦𝐸𝐸𝑠𝑠

17

𝑥𝑥𝑦𝑦𝑧𝑧 = �1 − 𝜀𝜀𝑦𝑦𝑡𝑡𝜀𝜀𝑚𝑚𝑛𝑛𝑚𝑚

�𝑥𝑥𝑛𝑛𝑎𝑎 18

𝜓𝜓 = cos−1 �𝑟𝑟−𝑥𝑥𝑦𝑦𝑡𝑡𝑟𝑟

� if 𝑥𝑥𝑦𝑦𝑧𝑧 < 2𝑟𝑟 ; = 𝜋𝜋 otherwise 19

The steel tensile force where the steel stress is at the yield strength is 20

118

𝑆𝑆𝑧𝑧𝑦𝑦 = −2∫ 𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡 𝑑𝑑𝜃𝜃𝜋𝜋𝜓𝜓 1

= −2𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡(𝜋𝜋 −𝜓𝜓) 2

The moment about the neutral axis due to force 𝑆𝑆𝑧𝑧𝑦𝑦 is 3

𝑆𝑆𝑧𝑧𝑦𝑦′ = −2∫ 𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡 (𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟 + 𝑟𝑟 cos𝜃𝜃)𝑑𝑑𝜃𝜃𝜋𝜋𝜓𝜓 4

= −2𝑓𝑓𝑦𝑦𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡[(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟)(𝜋𝜋 − 𝜓𝜓)− 𝑟𝑟 sin𝜓𝜓] 5

The steel tensile force where the steel stress is below the yield strength is 6

𝑆𝑆𝑧𝑧 = 2∫ 𝐸𝐸𝑠𝑠𝜓𝜓𝛼𝛼 𝜀𝜀max ��1− 𝑟𝑟


𝑥𝑥𝑛𝑛𝑛𝑛cos𝜃𝜃� 𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝑑𝑑𝜃𝜃 7

= 2𝐸𝐸𝑠𝑠𝜀𝜀max𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡 ��1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

� (𝜓𝜓− 𝛼𝛼) + 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

(sin𝜓𝜓 − sin 𝛼𝛼)� 8

The moment about the neutral axis due to force 𝑆𝑆𝑧𝑧 is 9

𝑆𝑆𝑧𝑧′ = 2∫ 𝐸𝐸𝑠𝑠𝜓𝜓𝛼𝛼 𝜀𝜀max ��1− 𝑟𝑟

𝑥𝑥𝑛𝑛𝑛𝑛� + 𝑟𝑟

𝑥𝑥𝑛𝑛𝑛𝑛cos𝜃𝜃� [(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) + 𝑟𝑟 cos𝜃𝜃]𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝑑𝑑𝜃𝜃 10

= 𝐴𝐴6 �𝑎𝑎6(𝜓𝜓 − 𝛼𝛼) + 𝑏𝑏6(sin𝜓𝜓 − sin 𝛼𝛼) + 𝑐𝑐62

(sin𝜓𝜓 cos𝜓𝜓 − sin 𝛼𝛼 cos𝛼𝛼 + 𝜓𝜓 −𝛼𝛼)� 11

where 12

𝐴𝐴6 = 2𝐸𝐸𝑠𝑠𝑟𝑟𝜌𝜌𝑧𝑧𝑡𝑡𝜀𝜀max 13

𝑎𝑎6 = �1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

� (𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) 14

𝑏𝑏6 = �1− 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

� 𝑟𝑟 + 𝑟𝑟𝑥𝑥𝑛𝑛𝑛𝑛

(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) 15

𝑐𝑐6 = 𝑟𝑟2

𝑥𝑥𝑛𝑛𝑛𝑛 16

17

Sections two equal openings in the compression zone 18

119

The case of two equal openings completely in or partially in the compression zone is shown 1

in Figures B.2 and B.3, respectively. For this case, the force equilibrium is modified by sub-2

tracting the concrete compressive force that was included in 𝐶𝐶1 + 𝐶𝐶2 at the openings. 3

𝑃𝑃𝑢𝑢𝜙𝜙

= 𝑃𝑃𝑛𝑛 = 𝐹𝐹𝑐𝑐1 + 𝐹𝐹𝑐𝑐2 − 𝐹𝐹𝐹𝐹𝑐𝑐1 − 𝐹𝐹𝐹𝐹𝑐𝑐2 + 𝑆𝑆𝑐𝑐 + 𝑆𝑆𝑐𝑐𝑦𝑦 + 𝑆𝑆𝑧𝑧 + 𝑆𝑆𝑧𝑧𝑦𝑦 4

The required extra terms are: 5

𝐹𝐹𝐹𝐹𝑐𝑐1 = 2� 0.85𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧) �1.3− 150 ��1−𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎


cos𝜃𝜃� 𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥� 𝑑𝑑𝜃𝜃𝜆𝜆2

𝜆𝜆1 6

= 𝐴𝐴1[𝑎𝑎1(𝜆𝜆2 − 𝜆𝜆1) + 𝑏𝑏1(sin 𝜆𝜆2 − sin 𝜆𝜆1)] 7

where 8

𝜆𝜆1 = min(𝜏𝜏 , 𝛾𝛾 − 𝛽𝛽) 9

𝜆𝜆2 = min(𝜏𝜏, 𝛾𝛾 + 𝛽𝛽) 10

𝐹𝐹𝐹𝐹𝑐𝑐2 = 2� 0.85𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧) �2𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥0.002



cos𝜃𝜃�𝛿𝛿2

𝛿𝛿111


�2��1−



cos 𝜃𝜃�2

� 𝑑𝑑𝜃𝜃 12

= 𝐴𝐴3 �𝑎𝑎3(𝛿𝛿2 − 𝛿𝛿1) + 𝑏𝑏3(sin 𝛿𝛿2 − sin 𝛿𝛿1) +𝑐𝑐32

(sin𝛿𝛿2 cos𝛿𝛿2 − sin 𝛿𝛿1 cos𝛿𝛿1 + 𝛿𝛿2 − 𝛿𝛿1)� 13

where 14

𝛿𝛿1 = min[max(𝜏𝜏 , 𝛾𝛾 − 𝛽𝛽),𝛼𝛼] 15

𝛿𝛿2 = max[min(𝛼𝛼 ,𝛾𝛾 + 𝛽𝛽),𝜏𝜏] 16

The moment equilibrium equation is modified in a similar way. 17

𝑀𝑀𝑛𝑛 = 𝑃𝑃𝑛𝑛(𝑟𝑟 − 𝑥𝑥𝑛𝑛𝑎𝑎) + 𝐹𝐹𝑐𝑐1′ + 𝐹𝐹𝑐𝑐2′ − 𝐹𝐹𝐹𝐹𝑐𝑐1′ − 𝐹𝐹𝐹𝐹𝑐𝑐2′ + 𝑆𝑆𝑐𝑐′ + 𝑆𝑆𝑐𝑐𝑦𝑦′ + 𝑆𝑆𝑧𝑧′ + 𝑆𝑆𝑧𝑧𝑦𝑦′ 18

The required extra terms are: 19

120

𝐹𝐹𝐹𝐹𝑐𝑐1′ = 2� 0.85𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧) �1.3− 150 ��1−𝑟𝑟𝑥𝑥𝑛𝑛𝑎𝑎


cos𝜃𝜃� 𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥� [(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟)𝜆𝜆2

𝜆𝜆11

+ 𝑟𝑟 cos𝜃𝜃]𝑑𝑑𝜃𝜃 2

= 𝐴𝐴2 �𝑎𝑎2(𝜆𝜆2 − 𝜆𝜆1) + 𝑏𝑏2(sin𝜆𝜆2 − sin 𝜆𝜆2) +𝑐𝑐22

(sin𝜆𝜆2 cos𝜆𝜆2 − sin 𝜆𝜆1 cos𝜆𝜆1 + 𝜆𝜆2 − 𝜆𝜆1)� 3

4

𝐹𝐹𝐹𝐹𝑐𝑐2′ = 2� 0.85𝑓𝑓𝑐𝑐′𝑟𝑟𝑡𝑡(1− 𝜌𝜌𝑧𝑧) �2𝜀𝜀𝑚𝑚𝑎𝑎𝑥𝑥0.002



cos𝜃𝜃�𝛿𝛿2

𝛿𝛿15


�2��1−



cos 𝜃𝜃�2

� [(𝑥𝑥𝑛𝑛𝑎𝑎 − 𝑟𝑟) + 𝑟𝑟 cos𝜃𝜃]𝑑𝑑𝜃𝜃 6

= 𝐴𝐴4 �𝑎𝑎4(𝛿𝛿2 − 𝛿𝛿1) + 𝑏𝑏4(sin𝛿𝛿2 − sin 𝛿𝛿1) + 𝑐𝑐42

(sin 𝛿𝛿2 cos𝛿𝛿2 − sin 𝛿𝛿1 cos𝛿𝛿1 + 𝛿𝛿2 − 𝛿𝛿1) 7

+ 𝑑𝑑43

(sin 𝛿𝛿2 cos2𝛿𝛿2 − sin 𝛿𝛿1 cos2𝛿𝛿1 + 2 sin 𝛿𝛿2 − 2 sin 𝛿𝛿1) 8

121

1

Figure B.1: Stress-strain relationships, no opening or one opening 2

3

122

1 Figure B.2: Two openings fully in compression zone 2

(Dimensions not shown; same as Fig. B.1) 3

4

5

Figure B.3: Two openings partially in compression zone 6

(Dimensions not shown; same as Fig. B.1) 7

8

123

APPENDIX C— TEMPERATURE GRADIENT FOR CIRCULAR CHIMNEYS 1

2

Appendix C consists of derivations of the equations of Section 7.4 to determine the tempera-3

ture gradient through the concrete wall of circular chimneys for the following scenarios: 4

1. Unlined chimneys 5

2. Chimneys with lining material applied directly to the inside concrete surface (no air 6

space) 7

3. Chimneys with a liner, optional insulation, and an unventilated air space 8

4. Chimneys with a liner, optional insulation, and a ventilated air space 9

10

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 Section 2.2 of the Code. 13

𝑇𝑇1,⋯ ,𝑇𝑇4 = various surface temperatures, as defined, °F

𝑄𝑄 = heat transfer from flue gas to inside surface of liner or inside surface

of liner material applied directly to the concrete wall or to the concrete

wall (when chimney is unlined), Btu/(hr ft2)

14

C.1 Unlined chimneys 15

Refer to Figure C.1. In this case, all of the heat transmitted from leaving the flue gas is trans-16

mitted carried through the concrete wall to the ambient air. The temperature gradient through the 17

124

concrete wall is 𝑇𝑇𝑥𝑥 = 𝑇𝑇1 − 𝑇𝑇2. Let 𝑄𝑄 be the amount of heat transmitted through a unit area on the 1

inside surface of the concrete wall. 2

𝑄𝑄 = 𝐾𝐾𝑐𝑐(𝑇𝑇𝑐𝑐 − 𝑇𝑇1) =𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐𝑡𝑡𝑐𝑐𝑑𝑑𝑐𝑐𝑐𝑐

(𝑇𝑇1 − 𝑇𝑇2) =𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜𝑑𝑑𝑐𝑐𝑐𝑐

(𝑇𝑇2 − 𝑇𝑇𝑜𝑜) 3

From above, 4

𝑇𝑇𝑐𝑐 − 𝑇𝑇1 =𝑄𝑄𝐾𝐾𝑐𝑐

𝑇𝑇1 − 𝑇𝑇2 =𝑄𝑄𝑡𝑡𝑐𝑐𝑑𝑑𝑐𝑐𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

= 𝑇𝑇𝑥𝑥 𝑇𝑇2 − 𝑇𝑇𝑜𝑜 =𝑄𝑄𝑑𝑑𝑐𝑐𝑐𝑐𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜

The difference between the known temperatures 𝑇𝑇𝑐𝑐 and 𝑇𝑇𝑜𝑜 can be expressed as 5

𝑇𝑇𝑐𝑐 − 𝑇𝑇𝑜𝑜 = (𝑇𝑇𝑐𝑐 − 𝑇𝑇1) + (𝑇𝑇1− 𝑇𝑇2) + (𝑇𝑇2 − 𝑇𝑇𝑜𝑜) = 𝑄𝑄 �1𝐾𝐾𝑐𝑐

+𝑡𝑡𝑐𝑐𝑑𝑑𝑐𝑐𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

+𝑑𝑑𝑐𝑐𝑐𝑐

𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜� 6

Equating two expressions for 𝑄𝑄 7

𝑄𝑄 =𝑇𝑇𝑐𝑐 − 𝑇𝑇𝑜𝑜

1𝐾𝐾𝑐𝑐



=𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐𝑡𝑡𝑐𝑐𝑑𝑑𝑐𝑐𝑐𝑐

𝑇𝑇𝑥𝑥 8

The temperature gradient can then be expressed as 9

𝑇𝑇𝑥𝑥 =𝑡𝑡𝑐𝑐𝑑𝑑𝑐𝑐𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐


1𝐾𝐾𝑐𝑐



� (C-1)

10

C.2 Chimneys with lining material applied directly to the inside concrete surface (no an-11

nular air space) 12

Refer to Figure C.2. In this case, all of the heat transmitted from leaving the flue gas is trans-13

mitted carried through the lining material and the concrete wall to the ambient air. The temperature 14

gradient through the concrete wall is 𝑇𝑇𝑥𝑥 = 𝑇𝑇2 − 𝑇𝑇3. Let 𝑄𝑄 be the amount of heat transmit ted 15

through a unit area on the inside surface of the lining material. 16

17

125

𝑄𝑄 = 𝐾𝐾𝑐𝑐(𝑇𝑇𝑐𝑐 − 𝑇𝑇1) =𝐶𝐶𝑏𝑏𝑑𝑑𝑏𝑏𝑡𝑡𝑏𝑏𝑑𝑑𝑏𝑏𝑐𝑐

(𝑇𝑇1 − 𝑇𝑇2) =𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐

(𝑇𝑇2 − 𝑇𝑇3) =𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜𝑑𝑑𝑏𝑏𝑐𝑐


From above, 2


𝑇𝑇1 − 𝑇𝑇2 =𝑄𝑄𝑡𝑡𝑏𝑏𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑏𝑏𝑑𝑑𝑏𝑏

𝑇𝑇2 − 𝑇𝑇3 =𝑄𝑄𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

= 𝑇𝑇𝑥𝑥 𝑇𝑇3 − 𝑇𝑇𝑜𝑜 =𝑄𝑄𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜

3


𝑇𝑇𝑐𝑐 − 𝑇𝑇𝑜𝑜 = (𝑇𝑇𝑐𝑐 − 𝑇𝑇1) + (𝑇𝑇1− 𝑇𝑇2) + (𝑇𝑇2 − 𝑇𝑇3) + (𝑇𝑇3 − 𝑇𝑇𝑜𝑜) = 𝑄𝑄 �1𝐾𝐾𝑐𝑐

+𝑡𝑡𝑏𝑏𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑏𝑏𝑑𝑑𝑏𝑏

+𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

+𝑑𝑑𝑏𝑏𝑐𝑐




1𝐾𝐾𝑐𝑐




=𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐

𝑇𝑇𝑥𝑥 7




1𝐾𝐾𝑐𝑐




� (C-2)

9

C.3 Chimneys with insulation completely filling the space between the liner and the chimney 10

wall (no annular airspace) 11

12

Refer to Figure C.3. In this case, all of the heat leaving the flue gas is transmitted through the 13

liner, the insulation, and the concrete wall to the ambient air. The temperature gradient through 14

the concrete wall is 𝑇𝑇𝑥𝑥 = 𝑇𝑇3 − 𝑇𝑇4. Let 𝑄𝑄 be the amount of heat transmitted through a unit area on 15

the inside surface of the liner. 16

𝑄𝑄 = 𝐾𝐾𝑐𝑐(𝑇𝑇𝑐𝑐 − 𝑇𝑇1) = 𝐶𝐶𝑏𝑏𝑑𝑑𝑏𝑏𝑧𝑧𝑏𝑏𝑑𝑑𝑏𝑏𝑏𝑏

(𝑇𝑇1 − 𝑇𝑇2) = 𝐶𝐶𝑠𝑠𝑑𝑑𝑠𝑠𝑧𝑧𝑠𝑠𝑑𝑑𝑏𝑏𝑏𝑏

(𝑇𝑇2 − 𝑇𝑇3) = 𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑏𝑏

(𝑇𝑇3 − 𝑇𝑇4) = 𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜𝑑𝑑𝑏𝑏𝑏𝑏


126

From above 1



𝑇𝑇2 − 𝑇𝑇3 =𝑄𝑄𝑡𝑡𝑠𝑠𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑠𝑠𝑑𝑑𝑠𝑠






1𝐾𝐾𝑐𝑐


+ 𝑡𝑡𝑠𝑠𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑠𝑠𝑑𝑑𝑠𝑠



� (C-3)

3

C.4 Chimneys with unventilated air space between liner and chimney wall 4

Refer to Figure C.4. In this case, all of the heat transmitted through the liner leaving the flue gas 5

is transmitted through the liner, the air space and the chimney wall. The temperature gradient 6

through the concrete wall is 𝑇𝑇𝑥𝑥 = 𝑇𝑇3 − 𝑇𝑇4. Let 𝑄𝑄 be the amount of heat transmitted through a unit 7

area on the inside surface of the liner. 8

𝑄𝑄 = 𝐾𝐾𝑐𝑐(𝑇𝑇𝑐𝑐 − 𝑇𝑇1) =𝐶𝐶𝑏𝑏𝑑𝑑𝑏𝑏𝑡𝑡𝑏𝑏𝑑𝑑𝑏𝑏𝑐𝑐

(𝑇𝑇1 − 𝑇𝑇2) =𝐾𝐾𝑟𝑟𝑑𝑑𝑠𝑠𝑑𝑑𝑏𝑏𝑐𝑐


(𝑇𝑇3 − 𝑇𝑇4) =𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜𝑑𝑑𝑏𝑏𝑐𝑐

(𝑇𝑇4− 𝑇𝑇𝑜𝑜) 9

From above 10



𝑇𝑇2 − 𝑇𝑇3 =𝑄𝑄𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑟𝑟𝑑𝑑𝑠𝑠




𝑇𝑇𝑐𝑐 − 𝑇𝑇𝑜𝑜 = (𝑇𝑇𝑐𝑐 − 𝑇𝑇1) + (𝑇𝑇1− 𝑇𝑇2) + (𝑇𝑇2 − 𝑇𝑇3) + (𝑇𝑇3 − 𝑇𝑇4) + (𝑇𝑇4− 𝑇𝑇𝑜𝑜)12

= 𝑄𝑄 �1𝐾𝐾𝑐𝑐


+𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑟𝑟𝑑𝑑𝑠𝑠

+𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

+𝑑𝑑𝑏𝑏𝑐𝑐


127



1𝐾𝐾𝑐𝑐





=𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐

𝑇𝑇𝑥𝑥 2




1𝐾𝐾𝑐𝑐





� (C-4)

4

C.5 Chimneys with ventilated air space between liner and chimney wall 5

Refer to Figure C.4. In this case, a portion of the heat leaving the flue gas is transmitted through 6

the liner. Some of this heat is carried away by the ventilated air space between the liner and the 7

chimney wall. The remainder of the heat is transmitted through the concrete wall. The temperature 8

gradient through the concrete wall is 𝑇𝑇𝑥𝑥 = 𝑇𝑇3 − 𝑇𝑇4. Let 𝑄𝑄 be the amount of heat transmitted 9

through a unit area on the inside surface of the liner and let 𝑟𝑟𝑞𝑞𝑄𝑄 be the amount of heat transmitted 10

through the chimney wall. 11

𝑟𝑟𝑞𝑞𝑄𝑄= 𝑟𝑟𝑞𝑞𝐾𝐾𝑐𝑐(𝑇𝑇𝑐𝑐 − 𝑇𝑇1) = 𝑟𝑟𝑞𝑞𝐶𝐶𝑏𝑏𝑑𝑑𝑏𝑏𝑡𝑡𝑏𝑏𝑑𝑑𝑏𝑏𝑐𝑐

(𝑇𝑇1 − 𝑇𝑇2) =𝐾𝐾𝑠𝑠𝑑𝑑𝑠𝑠𝑑𝑑𝑏𝑏𝑐𝑐


(𝑇𝑇3 − 𝑇𝑇4)12

=𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜𝑑𝑑𝑏𝑏𝑐𝑐


From above 14



𝑇𝑇2 − 𝑇𝑇3 =𝑟𝑟𝑞𝑞𝑄𝑄𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑠𝑠𝑑𝑑𝑠𝑠

𝑇𝑇3 − 𝑇𝑇4 =𝑟𝑟𝑞𝑞𝑄𝑄𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

= 𝑇𝑇𝑥𝑥 𝑇𝑇4 − 𝑇𝑇𝑜𝑜 =𝑟𝑟𝑞𝑞𝑄𝑄𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜


128

𝑇𝑇𝑐𝑐 − 𝑇𝑇𝑜𝑜 = (𝑇𝑇𝑐𝑐 − 𝑇𝑇1) + (𝑇𝑇1 − 𝑇𝑇2) + (𝑇𝑇2 − 𝑇𝑇3) + (𝑇𝑇3 − 𝑇𝑇4) + (𝑇𝑇4 − 𝑇𝑇𝑜𝑜)1

= 𝑄𝑄 �1𝐾𝐾𝑐𝑐


+𝑟𝑟𝑞𝑞𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑠𝑠𝑑𝑑𝑠𝑠

+𝑟𝑟𝑞𝑞𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

+𝑟𝑟𝑞𝑞𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜

� 2



1𝐾𝐾𝑐𝑐


+𝑟𝑟𝑞𝑞𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑠𝑠𝑑𝑑𝑠𝑠

+𝑟𝑟𝑞𝑞 𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐

+𝑟𝑟𝑞𝑞𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑜𝑜𝑑𝑑𝑐𝑐𝑜𝑜

=𝐶𝐶𝑐𝑐𝑑𝑑𝑐𝑐𝑟𝑟𝑞𝑞 𝑡𝑡𝑐𝑐𝑑𝑑𝑏𝑏𝑐𝑐

𝑇𝑇𝑥𝑥 4



�(𝑇𝑇𝑐𝑐 − 𝑇𝑇𝑜𝑜)

1𝑟𝑟𝑞𝑞𝐾𝐾𝑐𝑐

+ 𝑡𝑡𝑏𝑏𝑑𝑑𝑏𝑏𝑐𝑐𝑟𝑟𝑞𝑞𝐶𝐶𝑏𝑏𝑑𝑑𝑏𝑏

+ 𝑑𝑑𝑏𝑏𝑐𝑐𝐾𝐾𝑠𝑠𝑑𝑑𝑠𝑠



� (C-5)

6

7

129

Figure C.1

Figure C.2

Figure C.3

Figure C.4

1

2

130

APPENDIX D—THERMAL STRESSES FOR CIRCULAR CHIMNEYS 1 2

Appendix D consists of derivations of equations to determine the stresses due to a thermal gra-3

dient across the chimney wall. 4

5

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 Section 2.2 of the Code. 8

𝜀𝜀𝑐𝑐 = concrete thermal strain

𝜀𝜀𝑠𝑠 = steel thermal strain

𝜃𝜃𝑧𝑧𝑒𝑒 = unrestrained rotation cause by temperature gradient

9

D.1 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 𝑇𝑇𝑥𝑥 is shown in Fig. D.1(a): 13

𝜃𝜃𝑧𝑧𝑒𝑒 =𝛼𝛼𝑧𝑧𝑒𝑒𝑇𝑇𝑥𝑥𝑡𝑡𝑐𝑐

14

Since rotation is prevented, stresses are induced as shown in Fig. D.1(b). The concrete strain 15

and stress at the inside surface is 16

𝜀𝜀𝑐𝑐 = 𝜃𝜃𝑧𝑧𝑒𝑒𝑐𝑐𝑡𝑡𝑐𝑐 = 𝛼𝛼𝑧𝑧𝑒𝑒𝑇𝑇𝑥𝑥𝑐𝑐 17

𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ = 𝐸𝐸𝑐𝑐𝜀𝜀𝑐𝑐 = 𝛼𝛼𝑧𝑧𝑒𝑒𝑐𝑐𝑇𝑇𝑥𝑥𝐸𝐸𝑐𝑐 18

The steel strain and stress in the outside reinforcement is 19

𝜀𝜀𝑠𝑠 = 𝜃𝜃𝑧𝑧𝑒𝑒(𝛾𝛾2𝑜𝑜 − 𝑐𝑐)𝑡𝑡𝑐𝑐 20

131

𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶 = 𝐸𝐸𝑠𝑠𝜀𝜀𝑠𝑠 = 𝛼𝛼𝑧𝑧𝑒𝑒(𝛾𝛾2𝑜𝑜 − 𝑐𝑐)𝑇𝑇𝑥𝑥𝐸𝐸𝑠𝑠 1

The inside steel stess is 2

𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶′′ =(𝑐𝑐 − 1 + 𝛾𝛾2𝑐𝑐 )

𝑐𝑐𝑛𝑛𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ 3

𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶′′ = 𝛼𝛼𝑧𝑧𝑒𝑒(𝑐𝑐 − 1 + 𝛾𝛾2𝑐𝑐 )𝑇𝑇𝑥𝑥𝑛𝑛𝐸𝐸𝑐𝑐 4

The sum of vertical forces must equal zero, so 5

𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ �𝑐𝑐𝑡𝑡𝑐𝑐2�+ 𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶′′ 𝛾𝛾1𝜌𝜌𝑜𝑜𝑡𝑡𝑐𝑐 − 𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶𝜌𝜌𝑜𝑜𝑡𝑡𝑐𝑐 = 0 6

𝛼𝛼𝑧𝑧𝑒𝑒𝑐𝑐𝑇𝑇𝑥𝑥𝐸𝐸𝑐𝑐 �𝑐𝑐𝑡𝑡𝑐𝑐2�+ 𝛼𝛼𝑧𝑧𝑒𝑒(𝑐𝑐− 1 + 𝛾𝛾2𝑐𝑐 )𝑇𝑇𝑥𝑥𝑛𝑛𝐸𝐸𝑐𝑐𝛾𝛾1𝜌𝜌𝑜𝑜𝑡𝑡𝑐𝑐 7

−𝛼𝛼𝑧𝑧𝑒𝑒(𝛾𝛾2𝑜𝑜 − 𝑐𝑐)𝑇𝑇𝑥𝑥𝑛𝑛𝐸𝐸𝑐𝑐𝜌𝜌𝑜𝑜𝑡𝑡𝑐𝑐 = 0 8

𝑐𝑐2 + 2𝑛𝑛𝛾𝛾1𝜌𝜌𝑜𝑜𝑐𝑐 + 2𝑛𝑛𝛾𝛾1𝜌𝜌𝑜𝑜(𝛾𝛾2 − 1) + 2𝑛𝑛𝜌𝜌𝑜𝑜𝑐𝑐 − 2𝑛𝑛𝜌𝜌𝑜𝑜𝛾𝛾2 = 0 9

𝑐𝑐2 + 2𝜌𝜌𝑜𝑜𝑛𝑛(𝛾𝛾1 + 1)𝑐𝑐− 2𝜌𝜌𝑜𝑜𝑛𝑛[𝛾𝛾2𝑜𝑜 + 𝛾𝛾1 (1− 𝛾𝛾2𝑐𝑐 )] = 0 10

Solving for 𝑐𝑐, 11

𝑐𝑐 = −𝜌𝜌𝑜𝑜𝑛𝑛(𝛾𝛾1 + 1) +�[𝜌𝜌𝑜𝑜𝑛𝑛(𝛾𝛾1 + 1)]2 + 2𝜌𝜌𝑜𝑜𝑛𝑛[𝛾𝛾2𝑜𝑜 + 𝛾𝛾1 (1− 𝛾𝛾2𝑐𝑐 )] 12

132

1

Figure 2: Vertical temperature stresses 2

3

4

133

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

D.2 Horizontal thermal stresses 1

2

The derivation of equations for the maximum horizontal stresses in concrete and steel due to a 3

temperature drop only, across the concrete wall with two layers of reinforcement, is similar to 4

that for the vertical temperature stresses. 5

Replace 𝜌𝜌𝑜𝑜 with 𝜌𝜌𝑜𝑜′ 6

𝛾𝛾1 with 𝛾𝛾1′ 7

𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ with 𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ 8

𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶 with 𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶 9

𝑐𝑐 with 𝑐𝑐′ 10

𝛾𝛾2𝑜𝑜 with 𝛾𝛾2𝑜𝑜′ 11

𝛾𝛾2𝑐𝑐 with 𝛾𝛾2𝑐𝑐′ 12

then 13

𝑓𝑓𝐶𝐶𝐶𝐶𝐶𝐶′′ = 𝛼𝛼𝑧𝑧𝑒𝑒𝑐𝑐′𝑇𝑇𝑥𝑥𝐸𝐸𝑐𝑐 14

𝑓𝑓𝑆𝑆𝐶𝐶𝐶𝐶 = 𝛼𝛼𝑧𝑧𝑒𝑒(𝛾𝛾2𝑜𝑜′ − 𝑐𝑐′)𝑇𝑇𝑥𝑥𝐸𝐸𝑠𝑠 15

𝑐𝑐 = −𝜌𝜌𝑜𝑜′𝑛𝑛(𝛾𝛾1′ + 1) +�[𝜌𝜌𝑜𝑜′𝑛𝑛(𝛾𝛾1′ + 1)]2 + 2𝜌𝜌𝑜𝑜′𝑛𝑛[𝛾𝛾2𝑜𝑜′ + 𝛾𝛾1′(1− 𝛾𝛾2𝑐𝑐′ )] 16

17