ACI 307-08
Code Requirements for Reinforced
Concrete Chimneys (ACI 307-08)
and Commentary
An ACI Standard
Reported by ACI Committee 307
First Printing
November 2008
®
American Concrete Institute
Advancing concrete knowledge
Code Requirements for Reinforced Concrete Chimneys
and Commentary
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ISBN 978-0-87031-307-3
ACI 307-08
Code Requirements for Reinforced Concrete
Chimneys (ACI 307-08) and Commentary
An ACI Standard
Reported by ACI Committee 307
David J. Bird
Chair
Victor A. Bochicchio
Thomas D. Joseph
Robert A. Porthouse
Randolph W. Snook
John J. Carty
Samuel Dilcer
Jagadish R. Joshi
Faris A. Malhas
Ronald E. Purkey
Denis J. Radecki
John C. Sowizal
Barry J. Vickery
Shu-Jin Fang
David C. Mattes
Scott D. Richart
Edward L. Yordy
Sigmund A. Freeman
The committee acknowledges the late Milton Harstein for his contribution to the development of these code requirements.
CONTENTS
R0—Introduction, p. 307-2
This code gives material, construction, and design requirements for castin-place and precast reinforced concrete chimneys. It sets forth minimum
loadings for design and contains methods for determining the concrete and
reinforcement required as a result of these loadings. The method of analysis
applies primarily to circular chimney shells; however, a general procedure
for analysis of noncircular shapes is included.
Equations are provided for determining the temperature gradient through
the concrete resulting from the difference in temperature of the gases inside
the chimney and the surrounding atmosphere. Methods for combining the
effects of dead and wind (or earthquake) loads with temperature, both
vertically and circumferentially, are included in this code. These methods
permit the licensed design professional to establish minimum concrete and
reinforcement requirements.
The Commentary discusses some of the background and considerations
of Committee 307 in developing the provisions contained in “Code
Requirements for Reinforced Concrete Chimneys (ACI 307-08).” Two
appendixes provide the derivation of the equations for nominal strength
and temperature stresses. Commentary provisions begin with an “R,” such
as “R1.1.1,” and are shown in italics.
Chapter 1—General, p. 307-3
1.1—Scope
1.2—Drawings
1.3—Regulations
1.4—Notation
Chapter 2—Materials, p. 307-7
2.1—General
2.2—Cement
2.3—Aggregates
2.4—Reinforcement
Chapter 3—Construction requirements, p. 307-7
3.1—General
3.2—Concrete strength
3.3—Strength tests
3.4—Forms
3.5—Reinforcement placement
3.6—Concrete placement
3.7—Concrete curing
3.8—Construction tolerances
3.9—Precast erection
Keywords: chimneys; compressive strength; concrete construction; earthquake-resistant structures; formwork (construction); foundations; high
temperature; linings; loads (forces); moments; openings; precast concrete;
quality control; reinforced concrete; reinforcing steels; specifications;
static loads; strength; structural analysis; structural design; temperature;
thermal gradient; wind pressure.
ACI Committee Reports, Guides, Manuals, Standard
Practices, and Commentaries are intended for guidance in
planning, designing, executing, and inspecting construction.
This Commentary is intended for the use of individuals who
are competent to evaluate the significance and limitations of its
content and recommendations and who will accept
responsibility for the application of the material it contains.
The American Concrete Institute disclaims any and all
responsibility for the stated principles. The Institute shall not
be liable for any loss or damage arising therefrom.
Reference to this Commentary shall not be made in contract
documents. If items found in this document are desired by the
licensed design professional to be a part of the contract
documents, they shall be restated in mandatory language.
Chapter 4—Loads and general design criteria,
p. 307-8
4.1—General
4.2—Wind loads
ACI 307-08 supersedes ACI 307-98, was adopted August 19, 2008, and published
November 2008.
Copyright © 2008, American Concrete Institute.
All rights reserved including rights of reproduction and use in any form or by any
means, including the making of copies by any photo process, or by electronic or
mechanical device, printed, written, or oral, or recording for sound or visual reproduction
or for use in any knowledge or retrieval system or device, unless permission in writing
is obtained from the copyright proprietors.
307-1
307-2
ACI STANDARD
4.3—Earthquake loads
4.4—Special design considerations and requirements
4.5—Wind deflection criteria
Chapter 5—Design of chimney shells: strength
method, p. 307-17
5.1—General
5.2—Design loads
5.3—Required strength
5.4—Design strength
5.5—Nominal moment strength: circular shells
5.6—Noncircular shapes
5.7—Design for circumferential bending
Chapter 6—Thermal stresses, p. 307-22
6.1—General
6.2—Vertical temperature stresses
6.3—Circumferential temperature stresses
Chapter 7—References, p. 307-23
7.1/R7.1—Referenced standards/Referenced standards
and reports
R7.2—Cited references
Appendix A—Derivation of equations for nominal
strength, p. 307-25
Appendix B—Derivation of equations for
temperature stresses, p. 307-29
R0—INTRODUCTION
As industry expanded in the years immediately following
World War I and, as a result of the development of large
pulverized coal-fired boilers for the electric power-generating
utilities in the 1920s, a number of large reinforced concrete
chimneys were constructed to accommodate these new
facilities. A group of interested engineers who foresaw the
potential need for many more such chimneys, and who were
members of the American Concrete Institute, embarked on
an effort to develop rational design criteria for these structures.
The group was organized into ACI Committee 505 (predecessor
to the present Committee 307) to develop such criteria in the
early 1930s.
Committee 505 submitted a “Proposed Standard Specification for the Design and Construction of Reinforced Concrete
Chimneys,” an outline of which was published in the ACI
JOURNAL (ACI Committee 505 1934). This specification was
adopted as a tentative standard in February 1936. Although
this tentative standard was never accepted by ACI as an
official standard, it was used as the basis for the design of
many chimneys. As these chimneys aged, inspections
revealed considerable cracking. When the industrial expansion
began following World War II, other engineers recognized
the need for developing an improved design specification for
reinforced concrete chimneys.
In May 1949, Committee 505 was reactivated to revise the
tentative standard specification, embodying modifications that
were found desirable during the years it had been in use. The
section dealing with the temperature gradient through the
chimney lining and the chimney shell was completely revised
and extended to cover different types and thicknesses of linings
and both unventilated and ventilated air spaces between the
lining and the concrete shell. In 1954, this specification was
approved as ACI 505-54 (ACI Committee 505 1954).
The rapid increase in the size and height of concrete chimneys
being built in the mid-1950s raised further questions about the
adequacy of the 1954 version of the specification, especially in
relation to earthquake forces and the effects of wind.
In May 1959, the ACI Board of Direction reactivated
Committee 505 (renamed Committee 307) to review the
standard and to update portions with the latest design
techniques and the then-current knowledge of the severity of
the operating conditions that prevailed in large steam
plants. The material in the standard was reorganized, charts
were added, and the methods for determining loads due to
wind and earthquakes were revised. The information on
design and construction of various types of linings was
amplified and incorporated in an appendix. That version
included criteria for working stress design. It was planned to
add ultimate strength criteria in a future revision.
In preparing the earthquake design recommendations for
ACI 307-69 (ACI Committee 307 1969), the committee
incorporated the results of theoretical studies by adapting
them to existing United States codes. The primary problems
in this endeavor stemmed from the uncertainties still
inherent in the definition of earthquake forces and from the
difficulty of selecting the proper safety and serviceability
levels that might be desirable for various classes of
construction. Committee investigations revealed that with
some modifications (such as the K factor), the base shear
equations developed by the Seismology Committee of the
Structural Engineers’ Association of California (SEAOC)
could be applied to chimneys. Similarly, the shape of the
force, shear, and moment distributions, as revised in their
1967 report, were also suitable for chimneys. A use factor
(U factor) ranging from 1.3 to 2.0 was introduced in the
specification, and it was emphasized that the requirements of
Section 4.5 of ACI 307-69 that related to seismic design
could be superseded by a rational analysis based on evaluation
of the seismicity of the site and modal response calculations.
The modifications were approved in ACI 307-69. In that
version, the commentary and derivation of equations were
published separately as a supplement to ACI 307-69.
In 1970, the document was reissued with corrections of
typographical errors. This issue of ACI 307-69 was also
designated ANSI A158.1-1970. At the time, as a result of
numerous requests, the commentary and derivation of
equations were bound together with the specification.
ACI 307-79 (ACI Committee 307 1979) updated its
requirements to agree with the then-accepted standard
practices in the design and construction of reinforced concrete
chimneys. The major changes included the requirement that
two layers of reinforcing steel be used in the walls of all
chimneys (previously, this only applied to chimney walls
thicker than 18 in.) and the requirement that horizontal
sections through the chimney wall be designed for the radial
wind pressure distribution around the chimney. Formulas
REINFORCED CONCRETE CHIMNEYS
were included to compute the stresses under these conditions.
Many revisions of less importance were included to bring the
specification up to date.
The editions of the specifications before 1979 included
appendixes on the subjects of chimney linings and accessories.
In 1971, Committee 307 learned of buckling problems in
steel chimney liners. The committee also noted that, in modern
power plant and process chimneys, environmental regulations
required treatment of the effluent gases that could result in
extremely variable and aggressively corrosive conditions in
the chimneys. These facts led the committee to agree that the
task of keeping the chimney liner recommendations current was
not a responsibility of an ACI committee and could be
misleading to licensed design professionals using the
chimney specification. By committee consensus, the reference
to chimney liner construction was dropped from future
editions of the specification. Committee 307 then made a
recommendation to the Brick Manufacturers’ Association
and the American Society of Civil Engineers that each
appoint a task force or a committee for the development of
design criteria for brick and steel liners, respectively. The
Power Division of ASCE took up the recommendation and
appointed a task committee that developed and published a
design guide in 1975 titled “Design and Construction of Steel
Chimney Liners” (ASCE Task Committee on Steel Chimney
Liners 1975). ASTM established two task forces for chimney
liners: one for brick and one for fiberglass-reinforced plastic.
The committee had extensive discussion on the question of
including strength design in the 1979 specification. The
decision to exclude it was based on the lack of experimental
data on hollow concrete cylinders to substantiate this form
of analysis for concrete chimneys. The committee continued,
however, to consider strength design, and encouraged
experiments in this area.
Shortly after ACI 307-79 was issued, the committee
decided to incorporate strength design provisions and
update the wind and earthquake design requirements.
ACI 307-88 (ACI Committee 307 1988) incorporated
significant changes in the procedures for calculating wind
forces as well as requiring strength design rather than
working stress. The effects of these and other revisions
resulted in designs with relatively thin walls governed mainly
by steel area and, in many instances, across-wind forces.
The subject of across-wind loads dominated the attention
of the committee between 1988 and 1995, and ACI 307-95
(ACI Committee 307 1995) introduced modified procedures
to reflect more recent information and thinking.
Precast chimney design and construction techniques were
introduced as this type of design became more prevalent for
chimneys as tall as 300 ft.
The subject of noncircular shapes was also introduced
in ACI 307-95. Due to the infinite array of possible
configurations, however, only broadly defined procedures
were presented.
Because of dissimilarities between the load factors
required by ACI 307 and 318, the committee added guidelines
for determining bearing pressures and loads to size and
design chimney foundations.
307-3
The major changes incorporated into the ACI 307-95 were:
Modified procedures for calculating across-wind loads;
Added requirements for precast concrete chimney columns;
Added procedures for calculating loads and for
designing noncircular chimney columns;
• Deleted exemptions previously granted to smaller
chimneys regarding reinforcement and wall thickness; and
• Deleted static equivalent procedures for calculating
earthquake forces.
For the ACI 307-98 (ACI Committee 307 1998), revisions
to the ASCE 7-95 relating to wind and seismic forces
required several changes to be made to the ACI 307-95. The
changes incorporated into the ACI 307-98 were:
• Site-specific wind loads were calculated using a 3-second
gust speed determined from Fig. 6-1 in ASCE 7-95,
instead of the previously used fastest-mile speed;
• Site-specific earthquake forces were calculated using
the effective peak velocity-related acceleration contours
determined from Contour Map 9-2 in ASCE 7-95 instead
of previously designated zonal intensity;
• The vertical load factor for along-wind forces was
reduced from 1.7 to 1.3;
• The vertical load factor for seismic forces was reduced
from 1.87 to 1.43;
• The load factor for across-wind forces was reduced
from 1.40 to 1.20; and
• The vertical strength reduction factor φ was reduced
from 0.80 to 0.70.
The reduced load factors should be used in concert with
the revised strength reduction factor and the wind and
seismic loads specified in ASCE 7-95.
Revisions to ASCE 7 again caused Committee 307 to
revisit and revise ACI 307-98. The changes incorporate
applicable ASCE 7-02 wind and seismic load factors and
methods. The changes to the ACI 307-98 were:
• Included procedure in Section 4.3, Earthquake load,
compatible with ASCE 7-02 and the ASCE 7 seismic
risk maps;
• Updated the load factors and load combinations to be
more in line with ASCE 7-02 values and presentation; and
• Changed the vertical strength reduction factor φ back
to 0.80.
As stated previously, the current methods in this document
can only be used in conjunction with the ASCE 7-02.
•
•
•
CHAPTER 1—GENERAL
1.1—Scope
This code covers the minimum design and construction
requirements of circular cast-in-place or precast reinforced
concrete chimney shells. If other shapes are used, their
design shall be substantiated in accordance with the principles
used herein. This code does not include the design of linings,
but does include the effects of linings on the concrete shell.
A precast chimney shell is defined as a shell constructed
wholly from precast reinforced concrete sections, assembled
one on top of another, to form a freestanding, self-supporting
cantilever. Vertical reinforcement and grout are placed in
cores as the precast sections are erected to provide structural
307-4
ACI STANDARD
continuity and stability. The use of precast panels as stay-inplace forms is considered cast-in-place construction.
R1.1 For this revision, ACI 307-98 was updated to an ACI
code. ACI 307, “Code Requirements for Reinforced
Concrete Chimneys,” is written for new construction. The
committee acknowledges that the general analytical
procedures and requirements contained in ACI 307 are
appropriate for the investigation and retrofitting of existing
chimneys. The committee recognizes, however, that not all
code requirements, such as Section 4.4, will be feasible or
appropriate when retrofitting existing chimneys.
The scope of ACI 307-95 was expanded to include precast
chimney shells. Additional information can be found in PCI
manuals (PCI 1977, 1985). Warnes (1992) provides further
guidelines on connection details for precast structures.
Additional information is given in ACI 550R.
1.2—Drawings
Drawings of the chimney shall be prepared showing strength
of the concrete, the thickness of the concrete chimney shell, the
size and position of reinforcing steel, details and dimensions of
the chimney lining, and information on chimney accessories.
1.3—Regulations
1.3.1 This code supplements local building regulations
and shall govern in all matters pertaining to reinforced
concrete chimney design and construction.
1.3.2 Consideration shall be given to the regulations of the
Federal Aviation Administration with respect to chimney
heights and aviation obstruction lighting and marking (AC707460-1K), and the standards of the Underwriters Laboratories
(UL 96A) regarding lightning protection and grounding.
1.4—Notation
As
= area of reinforcing steel at top and bottom of
opening, in.2 (Chapter 4)
B
= band-width parameter (Chapter 4)
Cb = coefficient of thermal conductivity of chimney’s
uninsulated lining or insulation around steel liner,
Btu⋅in./(h⋅ft2⋅°F) of thickness/h/°F difference in
temperature (Chapter 6)
Cc = coefficient of thermal conductivity of concrete of
chimney shell, Btu⋅in./(h⋅ft2⋅°F) of thickness/h/°F
difference in temperature (12 for normalweight
concrete) (Chapter 6)
Cdr = drag coefficient for along-wind load (Chapter 4)
CE = end-effect factor (Chapter 4)
CL = rms lift coefficient (Chapter 4)
CLo = rms lift coefficient modified for local turbulence
(Chapter 4)
Cs = coefficient of thermal conductivity of insulation
filling in space between lining and shell, Btu⋅in./
(h⋅ft2⋅°F) of thickness/h/°F difference in
temperature (3 for lightweight concrete) (Chapter 6)
c
= ratio of distance from extreme compression fiber
to neutral axis for vertical stresses to total
thickness t (Chapter 6)
c′
= c for circumferential stresses (Chapter 6)
D
d
db
=
=
=
dbi
=
dc
=
dci
=
dco
=
ds
=
d(b)
d(b)
d(h)
d(h)
d(u)
=
=
=
=
=
d(z) =
d(zcr) =
E
Ec
Eck
Es
=
=
=
=
Fa
=
FV
=
F1A
F1B
f
fc′
=
=
=
=
fc′′(c) =
fc′′(v) =
′′ =
f CTC
f CTV
′′ =
fSTC =
fSTV =
′′ =
f STV
fy
=
fy′ (c) =
dead load (Chapter 5)
diameter of chimney, ft (Chapter 4)
mean diameter of uninsulated lining or insulation
around liner, ft (Chapter 6)
inside diameter of uninsulated lining or insulation
around liner, ft (Chapter 6)
mean diameter of concrete chimney shell, ft
(Chapter 6)
inside diameter of concrete chimney shell, ft
(Chapter 6)
outside diameter of concrete chimney shell, ft
(Chapter 6)
mean diameter of space between lining and shell,
ft (Chapter 6)
bottom outside diameter of chimney, ft (Chapter 4)
mean diameter at bottom of chimney, ft (Chapter 4)
top outside diameter of chimney, ft (Chapter 4)
mean diameter at top of chimney, ft (Chapter 4)
mean outside diameter of upper third of chimney,
ft (Chapter 4)
outside diameter of chimney at height z, ft
(Chapter 4)
outside diameter of chimney at critical height zcr ,
ft (Chapter 4)
earthquake loads or forces (Chapter 5)
modulus of elasticity of concrete, psi (Chapter 6)
modulus of elasticity of concrete, kip/ft2 (Chapter 4)
modulus of elasticity of reinforcement, psi
(Chapters 5 and 6)
acceleration-based site coefficient at 0.2-second
period (Section 4.3)
velocity-based site coefficient at 1.0-second
period (Section 4.3)
strouhal number parameter (Chapter 4)
lift coefficient parameter (Chapter 4)
frequency, Hz (Chapter 4)
specified compressive strength of concrete, psi
(Chapter 4)
fc′ modified for temperature effects, circumferential,
psi (Chapter 5)
fc′ modified for temperature effects, vertical, psi
(Chapter 5)
maximum circumferential stress in concrete due
to temperature inside chimney shell, psi (Chapters 5
and 6)
maximum vertical stress in concrete inside
chimney shell due to temperature, psi (Chapters 5
and 6)
maximum stress in outside circumferential
reinforcement due to temperature, psi (Chapters 5
and 6)
maximum stress in outside vertical reinforcement
due to temperature, psi (Chapters 5 and 6)
maximum stress in inside vertical reinforcement
due to temperature, psi (Chapters 5 and 6)
specified yield strength of reinforcing steel, psi
(Chapters 4 and 5)
fy modified for temperature effects, circumferential,
REINFORCED CONCRETE CHIMNEYS
fy′ (v) =
G
=
Gr(z) =
Gw′ =
g
h
I
IE
=
=
=
=
i
K
K1
K2
K3
Kd
Ke
Ki
=
=
=
=
=
=
=
=
Ko
=
Kr
=
Ks
=
k
=
ka
kao
=
=
ks
=
L
l
=
=
Ma =
Ma(z) =
Mi(z) =
Ml(z) =
Mn
=
Mo(z) =
psi (Chapter 5)
fy modified for temperature effects, vertical, psi
(Chapter 5)
across-wind peaking factor (Chapter 4)
gust factor for radial wind pressure at height z
(Chapter 4)
gust factor for along-wind fluctuating load
(Chapter 4)
acceleration due to gravity, 32.2 ft/s2 (Chapter 4)
chimney height above ground level, ft (Chapter 4)
importance factor for wind design (Chapter 4)
occupancy importance factor from Section 4.3.2
(Chapter 4)
local turbulence parameter (Chapter 4)
parameter for nominal moment strength (Chapter 5)
parameter for nominal moment strength (Chapter 5)
parameter for nominal moment strength (Chapter 5)
parameter for nominal moment strength (Chapter 5)
wind directionality factor (Chapter 4)
Es /fy (Chapter 5 and Appendix A)
coefficient of heat transmission from gas to inner
surface of chimney lining when chimney is lined,
or to inner surface of chimney shell when
chimney is unlined, Btu/ft2/h/°F difference in
temperature (Chapter 6)
coefficient of heat transmission from outside
surface of chimney shell to surrounding air, Btu/
ft2/h/°F difference in temperature (Chapter 6)
coefficient of heat transfer by radiation between
outside surface of lining and inside surface of
concrete chimney shell, Btu/ft2/h/°F difference in
temperature (Chapter 6)
coefficient of heat transfer between outside
surface of lining and inside surface of shell for
chimneys with ventilated air spaces, Btu/ft2/h/°F
difference in temperature (Chapter 6)
ratio of wind speed V to critical wind speed Vcr
(Chapter 4)
aerodynamic damping parameter (Chapter 4)
mass damping parameter of small amplitudes
(Chapter 4)
equivalent sand-grained surface roughness factor
(Chapter 4)
correlation length coefficient (Chapter 4)
width of opening in concrete chimney shell, in.
(Chapter 4)
peak base moment, ft⋅lb (Chapter 4)
moment induced at height z by across-wind loads,
ft⋅lb (Chapter 4)
maximum circumferential bending moment due
to radial wind pressure, at height z, tension on
inside, ft-lb/ft (Chapter 4)
moment induced at height z by mean along-wind
load, ft-lb (Chapter 4)
nominal moment strength at section, ft-lb
(Chapter 5 and Appendix A)
maximum circumferential bending moment due
to radial wind pressure, at height z, tension on
Mu
=
Mw(b)=
Mw(z)=
n
n1
=
=
Pcr
=
Pu
=
p(z) =
pr (z)
Q
Q′
Q1
Q2
Q3
R
=
=
=
=
=
=
=
R
RL
r
rq
=
=
=
=
r(z) =
S1
=
Sa
=
SaM =
SD1 =
SDS =
SM1 =
SMS =
Sp
Ssv
Ss
=
=
=
St
s
T
T
T1
=
=
=
=
=
307-5
outside, ft⋅lb/ft (Chapter 4)
factored moment at section, ft-lb (Chapter 5 and
Appendix A)
bending moment at base due to mean along-wind
load, ft⋅lb (Chapter 4)
combined design moment at height z for acrosswind and along-wind loads, ft-lb (Chapter 4)
modular ratio of elasticity, Es /Ec (Chapter 6)
number of openings entirely in compression zone
(Chapter 5 and Appendix A)
pressure due to wind at critical speed, lb/ft2
(Chapter 4)
factored vertical load, lb (Chapter 5 and
Appendix A)
pressure due to mean hourly design wind speed at
height z, lb/ft2 (Chapter 4)
radial wind pressure at height z, lb/ft2 (Chapter 4)
stress level correction parameter (Chapter 5)
parameter for nominal moment strength (Chapter 5)
parameter for nominal moment strength (Chapter 5)
parameter for nominal moment strength (Chapter 5)
parameter for nominal moment strength (Chapter 5)
response modification factor for concrete
chimney from Section 4.3.2 (Chapter 4)
parameter for nominal moment strength (Chapter 5)
response modification factor for liner (Chapter 4)
average radius of section, ft (Chapter 5)
ratio of heat transmission through chimney shell
to heat transmission through lining for chimneys
with ventilated air spaces (Chapter 6)
mean radius at height z, ft (Chapter 4)
mapped maximum considered earthquake, 5%
damped, spectral response acceleration at a
period of 1 second (Section 4.3)
design spectral response acceleration (Section 4.3)
the maximum spectral response acceleration for
site-specific procedures (Section 4.3)
the design spectral response acceleration at a
period of 1 second (Section 4.3)
the design spectral response acceleration at short
periods (Section 4.3)
the maximum considered earthquake, 5%
damped, spectral response acceleration at a
period of 1 second adjusted for site class effects
(Section 4.3)
the maximum considered earthquake, 5% damped,
spectral response acceleration at short periods
adjusted for site class effects (Section 4.3)
spectral parameter (Chapter 4)
mode shape factor (Section 4.2)
mapped maximum considered earthquake, 5%
damped, spectral response acceleration at short
periods (Section 4.3)
strouhal number (Chapter 4)
center-to-center spacing of chimneys, ft (Chapter 4)
normal temperature effect, °F (Chapter 6)
period of structure (Section 4.3)
fundamental period of vibration for unlined shell,
307-6
ACI STANDARD
T2
=
Ti
=
To
=
To
Tx
=
=
t
t
tb
=
=
=
ts
=
t(b)
t(h)
Uc
Uv
V
Vcr
=
=
=
=
=
=
Vcr2 =
Vr
V
=
=
V(33) =
V(h) =
V(z) =
V(zcr) =
W
=
w(z) =
w(z) =
w′(h) =
w′(z) =
w1(z) =
wa(h) =
wa(z) =
wt(u) =
Ymax =
y
=
yL
=
seconds per cycle (Chapter 4)
second mode period of vibration for unlined
shell, seconds per cycle (Chapter 4)
maximum specified design temperature of gas
inside chimney, °F (Chapter 6)
minimum temperature of outside air surrounding
chimney, °F (Chapter 6)
seismic parameter (Section 4.3)
temperature drop across concrete shell, °F
(Chapter 6)
thickness of concrete shell, in. (Chapters 5 and 6)
concrete thickness at opening, in. (Chapter 4)
thickness of uninsulated lining or insulation
around steel liner, in. (Chapter 6)
thickness of air space or insulation filling the
space between lining and shell, in. (Chapter 6)
thickness of concrete shell at bottom, ft (Chapter 4)
thickness of concrete shell at top, ft (Chapter 4)
required circumferential strength (Chapter 5)
required vertical strength (Chapter 5)
basic wind speed, mph (Chapter 4)
critical wind speed for across-wind loads, corresponding to fundamental mode, ft/s (Chapter 4)
critical wind speed for across-wind loads corresponding to second mode, ft/s (Chapter 4)
V(I 0.5), mph (Chapter 4)
mean hourly wind speed at (5/6)h varying over
range of 0.50 and 1.30V(zcr), ft/s (Chapter 4)
mean hourly wind speed at height of 33 ft, ft/s
(Chapter 4)
mean hourly wind speed at top of chimney, ft/s
(Chapter 4)
mean hourly design wind speed at height z, ft/s
(Chapter 4)
mean hourly design wind speed at (5/6)h, ft/s
(Chapter 4)
wind load (Chapter 5)
total along-wind load per unit length at height z,
lb/ft (Chapter 4)
mean along-wind load per unit length at height z,
lb/ft (Chapter 4)
fluctuating along-wind load per unit length at top
of chimney, lb/ft (Chapter 4)
fluctuating along-wind load per unit length at
height z, lb/ft (Chapter 4)
mean along-wind load per unit length as given by
Eq. (4-27), lb/ft (Chapter 4)
across-wind load per unit length at top of
chimney, lb/ft (Chapter 4)
across-wind load per unit length at height z, lb/ft
(Chapter 4)
average weight per unit length for top third of
chimney, lb/ft (Chapter 4)
maximum lateral deflection of top of chimney, ft
(Chapter 4)
total (SRSS or CQC) lateral displacement of
concrete chimney, ft (Chapter 4)
total (SRSS or CQC) lateral displacement of
Zc
z
zcr
α
=
=
=
=
αte
=
β
=
β1
=
βa
βs
=
=
γ
=
γ1
=
γ2
=
γ1′
=
γ2′
=
γd
δ
=
=
εm
=
λ
λ1
μ
=
=
=
τ
=
ψ
=
π
ρ
=
=
ρ′
=
ρa
ρck
ρt
=
=
=
φ
=
ωt
=
liner, ft (Chapter 4)
exposure length, ft (Chapter 4)
height above ground, ft (Chapter 4)
height corresponding to Vcr , ft (Chapter 4)
on chimney cross section, one-half of the central
angle subtended by neutral axis, radians (Chapter 5)
thermal coefficient of expansion of concrete and
of reinforcing steel, 0.0000065 per °F (Chapter 6)
one-half of the central angle subtended by an
opening on the chimney cross section, radians
(Chapter 5 and Appendix A)
factor in Section 10.2.7.3 of ACI 318 (Chapters 5
and 6)
aerodynamic damping factor (Chapter 4)
fraction of critical damping for across-wind load
(Chapter 4)
one-half of the central angle subtended by the
centerlines of two openings on chimney cross
section, radians (Chapter 5 and Appendix A)
ratio of inside face vertical reinforcement area
(Chapter 6)
ratio of distance between inner surface of
chimney shell and outside face vertical reinforcement to total shell thickness (Chapter 6)
ratio of inside face circumferential reinforcement
area to outside face circumferential reinforcement
area (Chapter 6)
ratio of distance between inner surface of
chimney shell and outside face circumferential
reinforcement to total shell thickness (Chapter 6)
d(h)/d(b) (Chapter 4)
γ – β for two symmetric openings partly in
compression zone, radians (Chapter 5)
maximum concrete compressive strain (Chapter 5
and Appendix A)
τ – n1β, radians (Chapter 5)
μ + ψ – π, radians (Chapter 5)
angle shown on Fig. 5.1(a), radians (Chapter 5
and Appendix A)
angle shown on Fig. 5.1(a), radians (Chapter 5
and Appendix A)
angle shown on Fig. 5.1(a), radians (Chapter 5
and Appendix A)
3.1416 (Chapter 5)
ratio of area of vertical outside face reinforcement to total area of concrete shell (Chapter 6)
ratio of area of circumferential outside face
reinforcement per unit of height to total area of
concrete shell per unit of height (Chapter 6)
specific weight of air, 0.0765 lb/ft3 (Chapter 4)
mass density of concrete, kip-s2/ft4 (Chapter 4)
ratio of total area of vertical reinforcement to total
area of concrete shell cross section (Chapter 5)
strength reduction factor (Chapter 5 and
Appendix A)
ρt fy /fc′ (Chapter 5)
REINFORCED CONCRETE CHIMNEYS
CHAPTER 2—MATERIALS
2.1—General
All materials and material tests shall conform to ACI 318,
except as otherwise specified herein.
2.2—Cement
The same brand and type of cement shall be used
throughout the construction of the chimney. The cement
used shall conform to the requirements for Types I, II, III, or
V of ASTM C150, or Type IS or IP of ASTM C595.
2.3—Aggregates
2.3.1 Concrete aggregates shall conform to ASTM C33.
2.3.2 The maximum size of coarse aggregate shall be not
larger than 1/8 of the narrowest dimension between inside
and outside forms nor larger than 1/2 the minimum clear
distance between reinforcing bars.
R2.3.2 This requirement differs from the ACI 318 because
most walls are 8 in. thick, and 3/4 or 1 in. aggregate works
best with 8 ft forms.
2.4—Reinforcement
Reinforcement shall conform to ASTM A615/A615M,
A996/A996M, or A706/A706M. Other deformed reinforcement with a specified minimum yield strength fy exceeding
60,000 psi shall be permitted provided that the ultimate
tensile strain shall equal or exceed 0.07.
R2.4 Refer to R5.1.2 for explanation of ultimate tensile
strain limits.
CHAPTER 3—CONSTRUCTION REQUIREMENTS
3.1—General
Concrete quality, methods of determining strength of
concrete, field tests, concrete proportions and consistency,
mixing and placing, and formwork and details of reinforcement shall be in accordance with ACI 318, except as stated
otherwise.
3.2—Concrete strength
The specified concrete compressive strength shall not be
less than 3000 psi at 28 days.
3.3—Strength tests
The 28-day compressive strength of the concrete shall be
determined from a minimum of two strength tests (consisting
of the average of two cylinders per each test) per 8-hour shift
(slipform) or per lift (jump form). For precast sections, a
minimum of two sets shall be taken from each class of
concrete cast each day and from each 100 yd3 of concrete
placed each day.
R3.3 Requirements for testing precast concrete units were
added in ACI 307-95.
3.4—Forms
R3.4 Shear transfer within precast concrete shells should
be considered in design, especially if the structure has
vertical as well as horizontal construction joints.
3.4.1 Forms for the chimney shell shall be made of metal,
wood, or other suitable materials. If unlined wooden forms
307-7
are used, they shall be of selected material with tongue-andgroove joints and shall be kept continuously wet to prevent
shrinking and warping due to exposure to the elements. Form
oil shall not be used unless it is a nonstaining type and it has
been established that specified protective coatings or paint
can be applied to concrete exposed to form oil.
3.4.2 Forms shall be sufficiently tight to prevent leakage
of mortar.
3.4.3 Load shall not be placed on the concrete structure until
that portion of the structure has attained sufficient strength to
safely support its weight and the loads placed thereon.
3.4.4 Forms shall be removed in such manner as to ensure
the safety of the structure. Forms shall be permitted to be
removed after the concrete has hardened to a sufficient strength
to maintain its shape without damage and to safely support
all loads on it, including temporary construction loads.
3.4.5 Ties between inner and outer chimney shell forms
shall not be permitted.
3.4.6 Construction joints shall be properly prepared to
facilitate bonding. As a minimum requirement, all laitance
and loose material shall be removed.
3.5—Reinforcement placement
R3.5 The size, spacing, and location of vertical cores
within precast concrete chimney shells will be determined by
geometry and steel area requirements. It is important that
the design of precast chimneys complies with the minimum
spacing requirements of ACI 318 when arranging reinforcement
within the cores to permit proper bar splicing and concrete
placement.
3.5.1 Circumferential reinforcement shall be placed
around the exterior of, and secured to, the vertical reinforcement
bars. All reinforcing bars shall be tied at intervals of not
more than 2 ft. Bars shall be secured against displacement
within the tolerances of the ACI 318.
R3.5.1 Particular attention shall be paid to placing and
securing the circumferential reinforcement so that it cannot
bulge or be displaced during the placing and working of the
concrete so as to result in less than the required concrete
cover over this circumferential reinforcement.
3.5.2 Vertical reinforcement projecting above the forms
for the chimney shell or cores of precast sections shall be
temporarily supported so as to prevent the breaking of the
bond with the freshly placed concrete.
R3.5.2 It is important to protect the early bond set.
Vertical bars are subjected to movement due to wind forces.
Vertical bars should be tied together or braced to prevent
damaging the bond.
3.5.3 Not more than 50% of bars shall be spliced along any
horizontal or vertical plane unless specifically permitted and
approved by the licensed design professional.
3.5.4 For reinforcement in cast-in-place chimneys, the
minimum concrete cover shall be 2 in. For reinforcement in
precast units manufactured under plant-controlled conditions,
the minimum concrete cover shall be 1.5 in.
307-8
ACI STANDARD
3.6—Concrete placement
Cast-in-place concrete placement shall conform to ACI
318, and shall be placed in layers no greater than 16 in.
Vertical construction joints for cast-in-place chimney shells
shall not be used. Where used, horizontal construction joints
for cast-in-place and precast concrete shall be approximately
evenly spaced throughout the height of the chimney shell.
Grout for setting precast sections shall have a specified
compressive strength equal to or greater than the specified
compressive strength of the set precast sections.
3.7—Concrete curing
3.7.1 Immediately after the forms have been removed, all
necessary finishing of concrete shall commence.
3.7.2 As soon as finishing has been completed, both faces
of concrete shall be cured by coating with a membranecuring compound or other method approved by the licensed
design professional. The curing compound shall comply
with ASTM C309, and shall be applied in strict accordance
with the manufacturer’s recommendations. If coatings are to
be applied to the concrete, the curing compound shall be of
a type compatible with these coatings.
3.8—Construction tolerances
R3.8 A quality control program should be established to
measure, document, and verify compliance with the
construction tolerance requirements of this code. The
program should identify the type, number, and frequency of
the measurements required to document each of the areas
specified in this code.
3.8.1 The chimney shell shall be constructed within the
tolerance limits set forth herein.
3.8.1.1 Vertical alignment of centerpoint—The actual
centerpoint of the shell shall not vary from its theoretical axis
by more than 0.001 times the height of the shell, or 1 in.,
whichever is greater. Locally, the actual centerpoint of the
shell shall not change horizontally by more than 1 in. for any
10 ft of vertical rise.
3.8.1.2 Diameter—The measured outside shell diameter
at any section shall not vary from the specified diameter by
more than 1 in. plus 0.01 times the specified or theoretical
diameter.
3.8.1.3 Wall thickness—The measured wall thickness
shall not vary from the specified wall thickness by more than
–1/4 in., +1/2 in. for walls 10 in. thick or less, or by more
than –1/2 in., +1 in. for walls greater than 10 in. thick. A
single wall thickness measurement is defined as the average
of at least four measurements taken at a uniform spacing
over a 60-degree arc. A negative tolerance decreases the
overall thickness, and a positive tolerance increases the
overall thickness.
3.8.2 Openings and embedments—Tolerances on the size
and location of openings and embedments in the shell cannot
be uniformly established due to the varying degrees of accuracy
required, depending on the nature of their use. Appropriate
tolerances for opening and embedment sizes and locations
shall be established for each chimney.
3.9—Precast erection
3.9.1 Precast sections shall be erected in a manner and at a
rate that ensures that sufficient strength has been attained in
grout, core concrete, and all connecting components to
safely support construction and applicable design loads.
3.9.2 Precast sections shall be grouted to level, and joints
shall be sealed. Shear keys shall be installed if required by
the licensed design professional.
CHAPTER 4—LOADS AND GENERAL DESIGN
CRITERIA
4.1—General
R4.1 For the 1995 edition, the Committee re-evaluated the
previous exemptions regarding two-face reinforcement and
minimum wall thickness for chimneys 300 ft or less in height
and less than 20 ft in diameter. Recent information has
indicated that two-face circumferential reinforcement is
necessary to minimize vertical cracking due to radial wind
pressures and reverse thermal gradients due to the effects of
solar heating. Reverse thermal gradients due to solar
heating may be more pronounced when the air space
between the column and lining is purged by pressurization
fans and gas temperatures are low. Further, the current
committee believes that two-face reinforcement should be
required in all chimney columns, regardless of size, considering
the aggressive environment surrounding chimneys.
4.1.1 The chimney shell shall be designed for the effects
of gravity, temperature, wind, and earthquake in accordance
with ACI 318, except as stated otherwise.
4.1.2 The chimney shell shall be designed for load
combinations in accordance with the provisions of Chapter 5.
4.1.3 Minimum shell thickness
4.1.3.1 The chimney shell shall not be less than 8 in.
thick when cast in place, or less than 7 in. thick when
composed of precast sections.
R4.1.3.1 A minimum wall thickness of 8 in. (7 in. if
precast) is required to provide for proper concrete placement
within and around two curtains of reinforcement.
4.1.3.2 The chimney shell thickness, through openings,
shall not be less than 1/24 times the height of the opening
over a vertical distance extending from 1/2 the height of the
opening below the sill of the opening to 1/2 the height of the
opening above the top of the opening. Properly designed
buttresses or other means of lateral restraint may be
permitted in place of this requirement. However, the
buttresses or other means of lateral restraint shall not be
included when calculating vertical strength.
R4.1.3.2 The committee expressed concern regarding
edge buckling of relatively thin walls through regions where
tall openings are present. The simplified procedure given in
this section will give approximately the same results as the
procedures of Section 10.10 of ACI 318-02 (ACI Committee
318 2002).
The committee defines a buttress to be rectangular or
square in shape and can project either inside or outside (or
both) the chimney wall. A buttress provides additional
stability to the thin wall design.
REINFORCED CONCRETE CHIMNEYS
If jamb buttresses are used, they should be placed
homogeneously with the section or adequately tied to ensure
composite action.
4.1.3.3 When the inside diameter of the shell exceeds 28 ft,
the minimum thickness shall be increased 1/8 in. for each 1 ft
increase in inside diameter.
4.1.4 Shell and liner interaction—A chimney shell that
supports lining loads shall comply with the requirements of
this standard with the lining in place. The loads on the
concrete shell shall include any lining loads resulting from
dead load, thermal, wind, or seismic loads.
4.1.5 Design for temporary construction loads—When
temporary access openings are used during construction,
they shall be designed as permanent openings.
4.1.6 Foundation considerations
4.1.6.1 The maximum foundation bearing pressure shall
be established using service chimney loads.
R4.1.6.1 Service loads are defined in Section 2.4 of
ASCE 7-02.
4.1.6.2 The foundation shall be designed by the strength
method in accordance with the procedures of ACI 318. The
foundation design shall be based on a pseudo-bearing pressure distribution, or pile loads, using the loading combinations given in Section 5.3.1.
R4.1.6.2 Foundation design—The pseudo-bearing pressure/pile loads should be computed by multiplying the unfactored dead and axial bending loads by their appropriate
load factor from Sections 5.3.1.
4.1.6.3 The minimum factor of safety against overturning
shall be 1.50 using service loads.
4.1.6.4 Design shall include the effects of radiant heat of
gases on any part of the foundation, including the foundation
floor area that is exposed within the liner and concrete floors
supported from the concrete shell.
4.2—Wind loads
4.2.1 General—Reinforced concrete chimneys shall be
designed to resist the wind forces in both the along-wind and
across-wind directions. In addition, the hollow circular cross
section shall be designed to resist the loads caused by the
circumferential pressure distribution.
The reference design wind speed in mph, which shall be
denoted as Vr , shall be the 3-second gust wind speed at 33 ft
over open terrain, where Vr = (I)0.5V. This speed V shall be
as specified by ASCE 7-02. The importance factor I for all
chimneys shall be 1.15. Topographic effects referenced in
Section 6.5.7.1 of ASCE 7-02 are omitted.
At a height z ft above ground, the mean hourly design
speed V(z) in ft/s shall be computed from Eq. (4-1)
z-⎞
V ( z ) = ( 1.47 )V r ⎛ ----⎝ 33⎠
0.154
× ( 0.65 )
(4-1)
The provisions with respect to wind load take into account
dynamic action, but are simplified and result in equivalent
static loads. A properly substantiated dynamic analysis shall
be permitted in place of these provisions.
307-9
R4.2.1 The basic wind speed V in the ACI 307-98 standard
was revised from fastest-mile to a 3-second gust speed to reflect
the changes published in ASCE 7-02. Equation (4-1) was
modified accordingly. In Eq. (4-1), 1.47 converts wind speed
from mph to ft/s, and 0.65 converts the 3-second gust speed
to a mean hourly speed. The revised power law coefficient
0.154 (as an approximation of 1/6.5) came from Table 6.2 of
ASCE 7-02 for Exposure C and for flexible or dynamically
sensitive structures; the increase in the exponent increases
the calculated pressures over the chimney height for the
same speed.
The 3-second gust speed is always higher than the previously
specified fastest-mile speed. A fastest-mile wind speed may
be converted to a 3-second gust speed for normal speeds of
interest in chimney design using
3-second gust V = 1.0546 (fastest mile V + 11.94)
The relationship between a 3-second gust speed and any
other averaging time can be found in texts such as Wind
Effects on Structures (Simiu and Scanlon 1986).
The procedure was determined from simplified dynamic
analyses that resulted in equivalent static load distributions.
This approach requires that a wind speed averaged over a
period approximately 20 minutes to 1 hour be used as a basis
for design. Equation (4-1) permits the mean hourly speed at
height z to be determined from the basic design speed that is
the 3-second gust speed at 33 ft over open country. The
conversion is based on the relationship recommended by
Hollister (1969). The specified wind loads presume that the
chimney is located in open country. In rougher terrains, the
overall loads will be reduced, but for a tall chimney (with a
height of approximately 650 ft), the reduction is not likely to
exceed 20%.
In Eq. (4-1), Vr is the product of the square root of the
importance factor I, and V, the basic wind speed as charted
and defined in ASCE 7-02. I can be used to vary probability as
well as to classify the importance of the structure. All chimneys should be designed to be part of an essential facility
classified as a Category IV structure. The importance factor of
1.15 for Category IV buildings and structures corresponds
to a mean recurrence interval of 100 years. Additional information can be found in ASCE 7-02.
The simplified provisions of this standard do not preclude
the use of more detailed methods, and the results of a full
dynamic analysis employing accepted approaches and
recognizing the flow profile and turbulence levels at a
specific site may be used in place of the standard provisions.
The approximate methods have, however, been tested
against more detailed analyses using probabilistic (Vickery
1969; Vickery and Basu 1985) and deterministic (Rumman
1985) approaches. These methods gave acceptable results.
4.2.2 Along-wind load: circular shapes—The along-wind
load, w(z) per unit length at any height z ft, shall be the sum
of the mean load w(z) and the fluctuating load w′(z).
The mean load w(z) in lb/ft shall be computed from Eq. (4-2)
w(z) = Cdr(z) × d(z) × p(z)
(4-2)
307-10
ACI STANDARD
where
Cdr(z) = 0.65 for z < h – 1.5d(h)
(4-3)
Cdr(z) = 1.0 for z ≥ h – 1.5d(h)
(4-4)
and 1.5d(h) shall not exceed 50 ft
p(z) = 0.00119Kd[V(z)]2
(4-5)
where Kd = 0.95 for circular chimneys.
The fluctuating load w′(z), lb/ft, shall be taken equal to
3.0z × G w′ × M w ( b )
w′ ( z ) = ---------------------------------------------3
h
(4-6)
where Mw(b) is the base bending moment, lb-ft, due to w(z),
and
0.47
G w′
11.0 [ T 1 × V ( 33 ) ]
= 0.30 + ------------------------------------------------0.86
( h + 16 )
(4-7)
where V(33) is determined from Eq. (4-1) for z = 33 ft.
For preliminary design and evaluation of the critical wind
speed Vcr , as described in Section 4.2.3.1, the natural period
of an unlined chimney T1, in seconds per cycle, shall be
permitted to be approximated using Eq. (4-8). For final design,
however, the period shall be computed by dynamic analysis
2
ρ ck ⎛ t ( h )⎞ 0.3
h
T 1 = 5 ------------------ ------d ( b ) E ck ⎝ t ( b )⎠
(4-8)
The mass and structure properties of the chimney lining
shall be included in the calculation of the period.
R4.2.2 The recommended drag coefficients are consistent
with slender chimneys [h/d(h) > 20] with a relative surface
roughness on the order of 10–4 to 10–5. Some reduction in
the drag coefficient Cdr with decreasing h/d(h) can be
expected, but unusually rough (for example, ribbed) chimneys
would have higher values of Cdr . The variations of Cdr with
roughness and aspect ratio are discussed by Basu (1982)
and Vickery and Basu (1984).
The total load per unit length is computed as the sum of the
mean component w(z) and the fluctuating component w′(z).
The dynamic component was evaluated using a slightly
modified form of the gust factor approaches described by
Davenport (1967), Vickery (1969), and Simiu et al. (1977).
The base moment is evaluated using the gust factor
approach, but the loads producing this moment are approximated by a triangular distribution rather than a distribution
matching the mean. Equation (4-7) is a simple empirical fit
to values of Gw′ computed as before for a structural
damping of 1.5% of critical. Except for referencing V as the
3-second gust speed, no revisions have been made to the
procedures for calculating along-wind loads.
For the 2008 revision, the directionality factor Kd was
added to Eq. (4-5). The 1.5 × d(h) is now limited to 50 ft, and
the factor of 2 in 4.4.3 is removed. Also, the numeric coefficient
was revised to correct for a previous error.
The natural period of the chimney may include the effect
of foundation and soil interaction.
4.2.3 Across-wind load: circular shapes
R4.2.3 No revisions were made to the procedures for
calculating across-wind forces in the 2008 version.
In 1995, the committee had numerous user comments and
discussions regarding the procedures included in the 1988
standard for across-wind forces. Virtually all of the
commentators felt that the 1988 procedures were unduly
conservative, especially in the absence of any record of
structural failure. As a result of these discussions, and with
the availability of new data and full-scale observations, the
procedures for calculating across-wind loads were extensively
revised.
A general solution for the across-wind response of
circular chimneys with any geometry was developed by
Vickery (1993). These procedures, based on Vickery’s
general solution, were simplified to some extent, which
requires that their application be restricted to certain
geometries. Similar models have provided the basis for
vortex-induced forces incorporated by the National Building
Code of Canada (Canadian Commission on Building and
Fire Codes 1995) and the ASME/ANSI STS-1.
Circular chimneys outside the bounds of these procedures,
or where a flare or strong taper (nozzle) exists for more than
one diameter near the top, may be conservatively analyzed
using the procedures of Section 4.2.3.3 of ACI 307-88 or by
the general approach put forth by Vickery (1993).
The procedures for determining shedding forces, however,
are not materially affected by the configuration of the lower
third of the shell, which may range from plumb to any degree
of taper.
Noncircular shapes may be more sensitive to acrosswind forces and may require analyses beyond the scope of
this standard.
Equation (4-18) establishes a basis for increasing structural
damping from a minimum of 1.0% to a maximum of 4.0%
when the wind speed V exceeds V(zcr). Structural damping of
1% of critical is consistent with measured values and
moderate stress levels with little cracking. Damping of 4.0%,
which would be permitted when V = 1.30V(zcr), is more
consistent with damping values permitted in seismic design.
Eight sample chimneys were studied using the 1988 and
1995 procedures. The geometry is in Table R4.1, and a more
detailed description is in 307-88. Fatigue damage was also
considered using the procedures put forth by Vickery (1993).
The committee concluded that a case-by-case analysis of
fatigue in circular chimneys that would require a supplemental
working stress analysis was not necessary, as fatigue stresses
in the sample chimneys were within acceptable limits.
Results using the 1988 and 1995 procedures are
compared in Table R4.1. These chimneys were selected
from a group of projects where the aspect ratio h/d is at or
near 10, where peak excitation is normally found. Note that
REINFORCED CONCRETE CHIMNEYS
307-11
Table R4.1—Comparison of results: along-wind plus across-wind moments, 1988 versus 1995 procedures
Description of chimneys
Chimney
6
Height, ft
485
Top outside diameter, ft Bottom outside diameter, ft
47.67
53.50
Tapers
3
VI, mph
85.0
h/d at (5/6)h
10.17
Frequency, hz
0.485
13
500
52.17
52.17
1
76.8
09.58
0.428
7
534
51.09
61.55
1
74.9
10.11
0.591
8
9
545
613
33.00
73.00
55.00
73.00
1
1
85.6
74.9
14.86
08.40
0.432
0.406
12
2
978
275
71.50
28.00
114.58
28.00
3
1
74.9
85.6
13.68
09.82
0.295
0.752
4
375
20.00
32.00
1
Calculated wind speeds
85.6
17.05
0.529
Chimney
6
Vcr, mph
Vcr , mph
78.9
93.9
93.3
V, mph
88.3
77.8
k
1.135
13
7
76.2
106.4
84.0
84.8
83.5
84.3
83.5
84.3
76.3
105.2
1.094
0.802
8
54.0
96.0
95.5
55.2
48.6
1.135
9
101.1
86.4
85.9
85.9
104.9
0.820
12
2
72.0
71.8
92.3
87.2
91.7
86.7
66.0
86.7
66.0
71.5
1.000
1.214
4
39.7
91.1
90.6
45.3
Factored base wind moments in ft-tons
34.6
1.310
Per ACI 307-88
V(zcr), mph
Per ACI 307-95
V(zcr), mph
Chimney
6
Per ACI 307-88, RMS combined along- and
across-wind: Bs = 0.015;
LF = 1.40
270,600
Per ACI 307-95, RMS combined
along- and across-wind: Bs = 0.010;
LF = 1.40
209,200
Per ACI 307-88 and ACI 307-95
along-wind only: LF = 1.70
160,900
13
7
283,500
447,800
224,100
238,100
148,000
165,100
8
9
117,500
971,700
79,400
459,100
161,200
320,700
12
2
1,475,800
39,800
977,400
34,100
865,300
28,600
4
16,500
11,600
43,800
for Chimneys 7 and 9, the critical wind speed exceeds the
design wind speed, permitting modification of both
damping (Eq. (4-18)) and Ma (Eq. (4-10)), which significantly reduces the base moments.
4.2.3.1 General—Across-wind loads due to vortex
shedding in the first and second modes shall be considered
in the design of all chimney shells when the critical wind
speed Vcr is between 0.50 and 1.30 V(zcr). Across-wind loads
need not be considered outside this range.
4.2.3.2 Analysis—When the outside shell diameter at h/3 is
less than 1.6 times the top outside diameter, across-wind
loads shall be calculated using Eq. (4-9), which defines the
peak base moment Ma
ρ 2
2
π
Ma = G
---- S sv C L ----a-V cr d ( u )h × -----------------------g
4 ( βs + βa )
2
1/2
Equation (4-9) defines the peak base moment Ma for
values of V, where V is evaluated between 0.5 and
1.30V(zcr). When V ≥ V(zcr), Ma shall be multiplied by
⎛ V – V ( z cr )⎞
1.0 – 0.95 ⎜ -----------------------⎟
⎝ V ( z cr ) ⎠
(4-10)
where G = 4.0, and Ssv = 0.57 (first mode) and 0.18 (second
mode).
CL = CLoF1B
(4-11)
CLo = –0.243 + 5.648i – 18.182i2
(4-12)
where
(4-9)
where
⎛
⎞ 1/2
⎜
⎟
2L
× S p ⎜ -----------------------------⎟
h
⎜ ⎛ ---------⎞⎟
⎝ ⎝ d ( u -) + C E⎠ ⎠
1
i = ------------------------------( 5/6 )h
log e ⎛ ---------------⎞
⎝ Zc ⎠
(4-13)
307-12
ACI STANDARD
Zc = 0.06 ft.
computed by Eq. (4-25), is between 0.50 and 1.30V(zcr),
where V(zcr) is the mean hourly wind speed at (5/6)h
h
F 1B = – 0.089 + 0.337log e ----------d(u)
(4-14)
but not > 1.0 or < 0.20
ρa = 0.0765 lb/ft3
fd ( u )
V cr = ------------St
(4-15)
St = 0.25F1A
(4-16)
where
h
F1A = 0.333 + 0.206loge ----------d(u)
(4-17)
but not > 1.0 or < 0.60
0.10 [ V – V ( z cr ) ]
β s = 0.01 + --------------------------------------V ( z cr )
(4-18)
but not < 0.01 or > 0.04
2
ka ρa d ( u )
β a = -----------------------wt ( u )
(4-19)
ka = kaoF1B
(4-20)
where
– 1.0
k ao = ----------------------------------------------------k–1
( 1 + 5i ) ⎛ 1 + ------------------⎞
⎝
i + 0.10⎠
(4-21)
where
Vk = -----V cr
1.5
(4-22)
–1 2
1 1–k
k
S p = ------------------- exp – --- ⎛ ----------------⎞
0.5 0.25
2⎝ B ⎠
B π
(4-23)
where
B
= 0.10 + 2i
(4-24)
L
= 1.20
CE = 3
After solving for Ma, across-wind moments at any height,
Ma(z), shall be calculated based on the corresponding mode
shape of the chimney column.
4.2.3.3 Second mode—Across-wind response in the second
mode shall be considered if the critical wind speed Vcr2, as
5d ( u )
V cr2 = -------------T2
(4-25)
The period T2 in seconds per cycle for an unlined shell
may be estimated by Eq. (4-26). For final design, T2 shall be
calculated by dynamic analysis
2
ρ ck t ( h )
h
T 2 = 0.82 ----------- --------------d ( b ) E ck t ( b )
0.09
d(h)
----------d(b)
– 0.22
(4-26)
where t(h) and t(b) are the thicknesses at the top and bottom,
respectively, and d(h) and d(b) are the mean diameters at the
top and bottom, respectively.
The effect of a shell-supported liner on the period of the
second mode shall also be included in the design.
Any method based on the modal characteristics of the
chimney shall be used to estimate the across-wind response
in the second mode.
4.2.3.4 Grouped chimneys—When two identical chimneys
are in close proximity, the across-wind load shall be increased
to account for the potential increase in vortex-induced
motions. In such cases, the lift coefficient CL in Eq. (4-11)
shall be modified as follows
a. If s/d(zcr) > 12.75, CL is unaltered; and
b. If 3 < s/d(zcr) < 12.75, CL shall be multiplied by: [0.26
– 0.015s/d(zcr)] + [2 – s/12d(zcr)].
For chimneys that are not identical and for identical
chimneys where s/d(zcr) < 3, the value of CL shall be
established by reference to model tests or observations or
test reports of similar arrangements.
R4.2.3.4 Interactions between closely spaced cylindrical
objects have been studied in considerable detail, but virtually
all of the test results are for subcritical values of Reynolds
numbers, and their applicability to chimneys is highly
questionable. Even with the scale effects introduced by the
inequality of the Reynolds number, however, the wind tunnel
is presently the only tool that will provide guidance as to the
likely magnitude of interference effects. A review of interference
effects was given by Zdravkokvich (1977). Vickery (1993)
attributed the amplification of shedding forces to increased
turbulence and additional buffeting effects that formed the basis
for revisions made to this section.
At center-to-center spacings s in excess of two to three
diameters, the prime interference effect is related to
across-wind excitation due to shedding. The recommendations
in Section 4.2.3.4 are based on the results of Vickery and
Daly (1984), and were obtained at subcritical values of the
Reynolds number. The first term in the modifier (c) is an
enhancement factor to account for buffeting due to vortexes
shed by the upstream structure; the second term accounts for
small-scale turbulence. The same reference also contains
results for two cylinders of different sizes, with the upstream
structure having a diameter 25% greater than the diameter d of
REINFORCED CONCRETE CHIMNEYS
the other. In this case, the amplification of the response of
the downwind chimney is roughly 3.4 – 0.2s/d for 4 < s/d <
12. The amplification of shedding for grouped cylinders has
also been noted at full scale (Ruscheweyh 1984), but the
available data are not sufficient to quantitatively validate
model test results.
4.2.3.5 Combination of across-wind and along-wind
loads—Across-wind loads shall be combined with the
coexisting along-wind loads. The combined design moment
Mw(z) at any section shall be taken as
Mw(z) = {[Ma(z)]2 + [Ml(z)]2}0.5
(4-27)
where
V
w l ( z ) = w ( z ) --------------V ( z cr )
2
(4-28)
except that wl(z) shall not exceed w(z).
4.2.4 Circumferential bending—The maximum circumferential bending moments due to the radial wind pressure
distribution shall be computed by Eq. (4-29) and (4-30)
Mi(z) = 0.31pr(z)[r(z)]2, ft-lb/ft (tension on inside) (4-29)
Mo(z) = 0.27pr(z)[r(z)]2, ft-lb/ft (tension on outside) (4-30)
pr(z) = p(z) × Gr(z), lb/ft2
(4-31)
Gr(z) = 4.0 – 0.8log10z, except Gr(z) = 4 for z ≤ 1.0 ft (4-32)
The pressure pr(z) shall be increased by 50% for a distance
1.5d(h) from the top. (Note: 1.5d(h) shall not exceed 50 ft.)
R4.2.4 The equation for the prediction of the circumferential
moments is based on measured pressure distributions
(Dryden and Hill 1930; ASCE Task Committee on Wind
Forces 1961). Comparative values for the bending moments
as obtained from different distributions are given by Rumman
(1985). The use of a gust factor Gr in this computation is
based on the assumption that the mean pressure distribution
(when expressed in coefficient form) is also applicable for
short-duration gusts.
The increase in the loads near the tip is consistent with
observations (Okamoto and Yagita 1973) that the drag
coefficient increases significantly in this region.
4.2.5 Wind loads: noncircular shapes—The provisions of
ASCE 7-02 shall be followed including force coefficients,
directionality factors, and gust response factors. Unusual
cross-sectional shapes not covered in ASCE 7-02 shall
require wind tunnel testing or other similar documentation to
verify along-wind loads, across-wind loads, or both. Similarly,
horizontal bending due to wind pressure distributions shall
also require wind tunnel testing or other documentation from
reliable sources.
307-13
4.3—Earthquake loads
R4.3 The earthquake load procedure has been completely
changed for the 2008 edition to be consistent with the
ASCE 7-02 procedures and seismic risk maps. These
procedures and maps are based on the “NEHRP Recommended Provisions for Seismic Regulations for New Buildings
and Other Structures” (FEMA 2003a).
For the 1997 NEHRP provisions, new seismic hazard
maps and procedures were developed. Previous maps had
been based on a uniform likelihood of the ground motion
being exceeded. The new maps are intended to provide a
uniform margin against collapse at the design ground motion
by taking into account both probabilistic and deterministic
data, different ground motion attenuation characteristics,
and different recurrence times. The maps define the
“maximum considered earthquake” ground motion. For
more details, refer to the Commentary of the “NEHRP
Recommended Provisions for Seismic Regulations for New
Buildings and Other Structures” (FEMA 2003b).
During the public discussion phase of the ACI 307 draft,
the question of special seismic detailing in accordance with
Chapter 21 of ACI 318 was raised. Chapter 21 of ACI 318
provides special ductile detailing requirements for design of
reinforced concrete members of building structures and
members of nonbuilding structures similar to buildings,
where the structure is expected to undergo significant
inelastic deformation under design-basis earthquake.
According to ASCE 7-02, nonbuilding concrete structures
that are not similar to buildings, such as chimneys and silos,
are not required to meet the ACI 318 Chapter 21 requirements. They are, however, required to be designed for higher
seismic loads using an R value of 3.0. ACI Committee 307
acknowledges the importance of ACI 318 Chapter 21 and
plans to investigate the need for specific seismic design
enhancements in the next code revision for design of breech
openings. Presently, the committee approves the use of ACI
307 seismic loading with R = 1.5 (versus R = 3.0 in ASCE 702) without the use of special seismic detailing requirements
per ACI 318 Chapter 21, with the exception of Section
21.2.4, which states a minimum concrete compressive
strength, and Section 21.2.5, which establishes a ceiling for
the yield strength of the reinforcement.
4.3.1 General—Reinforced concrete chimneys shall be
designed and constructed to resist the earthquake effects in
accordance with this section.
Earthquake loads on chimneys shall be determined by
means of the dynamic response spectrum analysis method
given in Section 4.3.2. Instead of the dynamic response
spectrum method, the time-history analysis method given in
Section 4.3.3 may be used.
R4.3.1 The dynamic response spectrum analysis method is
used to determine seismic loads. The design spectrum can be
developed from the map values of ASCE 7-02, or a sitespecific design spectrum can be used. Alternatively, a timehistory analysis may be used.
4.3.2 Dynamic response spectrum analysis method—The
shears, moments, and deflections of a chimney due to earthquake shall be determined using a response spectrum and the
307-14
ACI STANDARD
elastic modal method. The response spectrum shall provide
the maximum considered earthquake spectral response
acceleration at any period, Sa, and shall be obtained from
either the general procedure of Section 4.3.2.1 or the sitespecific procedure of Section 4.3.2.2.
The occupancy category shall be determined from Table 1.1
of ASCE 7-02.
The seismic use group shall be determined from Table 9.1.3
of ASCE 7-02.
The occupancy importance factor IE shall be determined
from Table 9.1.4 of ASCE 7-02.
The seismic design category shall be determined from
Table 9.4.2.1(a) or Table 9.4.2.1(b) of ASCE 7-02, whichever results in the most severe category.
The response modification factor R shall be taken as 1.5.
The required periods, mode shapes, and participation
factors of the chimney shall be calculated by established
methods of structural analysis. The analytical model used
shall be sufficiently refined to represent variations of mass
and stiffness. If the liner is vertically supported, laterally
restrained, or both, by the concrete chimney, interaction of
the chimney and liner shall be considered.
The modal design shears and moments shall be determined
by scaling the modal shears and moments due to the design
spectrum (either the general procedure design spectrum of
Section 4.3.2.1 or the site-specific design spectrum of
Section 4.3.2.2) by the factor IE /R.
The total design shears and moments shall be computed
from the modal design shears and moments using either the
square root of the sum of the squares (SRSS) method or the
complete quadratic combination (CQC) method. The analysis
shall include a sufficient number of modes to obtain a
combined modal mass participation of at least 90% of the
actual mass.
The loads due to the vertical component of earthquakes
may be neglected. For chimneys of circular cross section, the
horizontal earthquake force shall be assumed to act alone in
any direction. For chimneys of noncircular cross sections
assigned to Seismic Design Categories C, D, E, or F,
orthogonal effects shall be considered by combining, using
the SRSS method, the responses due to the design spectrum
(either the general procedure design spectrum of Section
4.3.2.1 or the site-specific design spectrum of Section
4.3.2.2) applied to any two orthogonal directions.
R4.3.2 The response modification factor R for reinforced
concrete chimneys is 1.5. This is lower than the 3.0 value
given in Table 9.14.5.1.1 of ASCE 7-02. Ongoing research
indicates that the lower value is more appropriate (Wilson
2002).
Stiffness properties for the chimney shell model should be
based on the uncracked section. The chimney shell model
should include at least three elements per half-wave of the
highest mode of the shell included in the modal summation.
More elements may be required to account for variations in
mass, stiffness, and the liner support conditions. Interaction
of the chimney shell and liner should be considered. For a
liner with a mass that is small compared with that of the
chimney shell (such as steel or fiber-reinforced polymer
liners), the effect of the liner on the chimney shell can be
accounted for by lumping the liner mass at the points of
lateral coupling. Alternatively, the liner can be included in
the model as a beam element appropriately coupled to the
chimney shell at elevations of vertical and lateral support.
ACI 307 does not require that the seismic base shear be
scaled to the minimum values set forth in the ASCE 7-02
using the equivalent lateral force procedure and Eq. 9.5.6.8.
The effect of the vertical component of the earthquake on
the chimney has been determined to be of no design significance.
Concrete chimneys are laterally flexible with a very long
fundamental period (typically in a few seconds), but far more
rigid in the vertical direction. An existing time-history study
by the committee shows that the peak responses due to
horizontal and vertical seismic excitation do not occur
simultaneously. Thus, the vertical stresses due to dead load
and horizontal seismic excitation are increased by, at most,
a few percent by the effects of vertical seismic excitations.
For this reason, the committee considers that the load effects
due to the vertical component of earthquakes can be ignored.
The moment due to the maximum allowable misalignment
in Section 3.8.1.1 and vertical acceleration together with the
static moment due to misalignment is negligible compared
with the moments due to wind a lateral earthquake effects.
For circular chimneys, the earthquake forces can be
assumed to act in any one direction only. For noncircular
chimneys, the design earthquake forces are computed as the
SRSS combination of the responses to earthquake motions
acting in any two orthogonal directions.
4.3.2.1 General procedure—The mapped maximum
considered earthquake spectral response acceleration at
short periods, SS, and S1 at 1 second, shall be obtained from
Fig. 9.4.1.1(a) through 9.4.1.1(j) of ASCE 7-02.
The site class shall be determined from Table 9.4.1.2 of
ASCE 7-02. When soil properties are not known in sufficient
detail to determine the site class, Class D shall be used.
The acceleration-based site coefficient Fa shall be
obtained from Table 9.4.1.2.4(a) of ASCE 7-02. The
velocity-based site coefficient FV shall be obtained from
Table 9.4.1.2.4(b) of ASCE 7-02
The maximum considered earthquake spectral response
acceleration for short periods, SMS , and SM1 at 1 second,
adjusted for site class effects, shall be determined as
SMS = Fa SS
(4-33)
SM1 = FV S1
(4-34)
The design earthquake spectral response acceleration at
short periods, SDS, and SD1 at 1 second, shall be determined as
SDS = (2/3)SMS
(4-35)
SD1 = (2/3)SM1
(4-36)
The design response spectrum curve shall be developed as
follows.
REINFORCED CONCRETE CHIMNEYS
For periods less than To, the design spectral response
acceleration Sa shall be taken as
Sa = SDS(0.4 + 0.6T/To)
(4-37)
For periods greater than or equal to To and less than or
equal to TS, the design spectral response acceleration Sa shall
be taken as
Sa = SDS
(4-38)
For periods greater than TS, the design spectral response
acceleration Sa shall be taken as
Sa = SD1/T
(4-39)
To = 0.2SD1/SDS
(4-40)
TS = SD1/SDS
(4-41)
where
4.3.2.2 Site-specific procedure—A site-specific response
spectrum shall provide the maximum considered earthquake
spectral response acceleration at any period, SaM, according to
the site-specific procedures of Section 9.4.1.3 of ASCE 7-02.
The design earthquake spectral response acceleration at
any period, Sa, shall be determined as
Sa = (2/3)SaM
(4-42)
The value of Sa from Eq. (4-42) shall be greater than or
equal to 80% of the Sa determined by the general procedure
of Section 4.3.2.1 for any period.
R4.3.2.2 When a site-specific spectrum is provided, 80%
of the map-generated values, at any period, are required to
be used as a minimum as permitted by ASCE 7-02.
4.3.3 Time-history analysis—Instead of the dynamic
response spectrum analysis method of Section 4.3.2, a timehistory seismic analysis may be used. The suites of ground
motion acceleration histories shall be selected and scaled
from recorded events and be compatible with the design
response spectrum for the site. A linear time-history analysis
shall conform to Section 9.5.7 of ASCE 7-02. A nonlinear
time-history analysis shall conform to Section 9.5.8 of
ASCE 7-02.
R4.3.3 ACI 307 does not require that the seismic base
shear be scaled to the minimum values set forth in the ASCE
7-02 Eq. 9.5.7.3.
4.3.4 Soil-structure interaction—The effects of seismic
interaction between a chimney and soil can be ignored and a
fixed base condition assumed. When a soil-structure interaction
assessment is desired, the procedure given in Section 9.5.9 of
ASCE 7-02 or other generally acceptable procedure shall be
followed.
307-15
R4.3.4 Soil-structure interaction will, in general, lengthen
the chimney’s natural period and reduce the chimney’s seismic
response; thus, the fixed-base condition is conservative.
4.3.5 Lateral displacements—Service level displacements
shall be determined by scaling the total displacements due to
either the general procedure design spectrum of Section
4.3.2.1 or the site-specific design spectrum of Section 4.3.2.2
by the factor R/IE. Total displacements are the modal
displacements combined by either the SRSS method or the
CQC method. When clearances between the chimney shell
and the liner are critical, the minimum separation shall be
computed by
(Ry + RL yL )/IE
(4-43)
R4.3.5 Clearance should be maintained between the
chimney shell and the lining. Computation of lateral displacements for each structure should take into account the
magnification of elastic deflections due to yielding. Typical
values for RL are 1.25 for brick liners and 3 for steel liners.
4.3.6 P-Δ effect—The P-Δ effect between vertical loads
and seismic lateral deflections shall be considered for chimneys
assigned to Seismic Design Categories D, E, or F.
4.4—Special design considerations and
requirements
4.4.1 Two layers of vertical and circumferential reinforcement
are required. The total vertical reinforcement shall be not
less than 0.25% of the concrete area. The outside vertical
reinforcement shall be not less than 50% of the total vertical
reinforcement. Outside-face vertical bars shall not be smaller
than No. 4, nor shall they be spaced more than 12 in. on
centers. Inside-face vertical bars shall not be smaller than
No. 4, nor shall they be spaced more than 24 in. on centers.
4.4.2 The total circumferential reinforcement shall not be
less than 0.20% of the concrete area. The circumferential
reinforcement in each face shall be not less than 0.1% of the
concrete area at the section.
Spacing of outer face circumferential reinforcement shall
not exceed the wall thickness or 12 in. Spacing of circumferential reinforcement on the inner face shall not exceed 12 in.
The minimum size of circumferential reinforcing bars shall
be No. 3.
4.4.3 The circumferential reinforcement for a distance of
0.2d(h) from the top of the chimney or 7.5 ft, whichever is
greater, shall be at least equal to the amount required by
Section 5.7, but shall not be less than 0.2% of the total
concrete area in each face. The maximum spacing of the
circumferential steel in this area shall be limited to one-half
of the wall thickness, but not to exceed 6 in.
R4.4.3 This section was revised to add a strength and
minimum spacing requirement.
4.4.4 Where a wall segment between openings is critical
either by height or width, this segment shall be investigated
as a beam-column.
R4.4.4 This section was reworded. The analysis procedure
for more than two openings, previously addressed herein, is
now covered in 5.5.8 for emphasis. The committee refers the
307-16
ACI STANDARD
Fig. 4.1—Tie bars at jamb and lintel of openings.
user to classical principals of buckling and to slenderness
effects defined by ACI 318-02, Section 10.12.2, when evaluating
the critical nature of a wall segment. Shear effects and their
secondary moments shall also be included in the evaluation.
4.4.5 In addition to the reinforcement determined by
design, additional reinforcement shall be provided at the
sides, top, bottom, and corners of all openings as hereinafter
specified. This additional reinforcement shall be placed as close
to the opening as proper spacing of bars will permit. Unless
otherwise specified, all additional reinforcement shall extend
past the opening a minimum of the development length.
4.4.6 At each side of the opening, the additional vertical
reinforcement shall have an area at least equal to the design
steel ratio times one-half the area of the opening. The additional reinforcement shall be placed within a distance not
exceeding three times the wall thickness unless otherwise
determined by a detailed analysis. If the additional vertical
reinforcement is not placed in the same layer as the inside
and outside vertical reinforcement, tie bars shall be provided
to brace the additional vertical reinforcement. Maximum
horizontal spacing is 12 in., and vertical spacing is 24 in.
These are in addition to Section 4.4.9 requirements. Refer to
Fig. 4.1 for details.
R4.4.6 Figure 4.1 was added to clarify the tie-bar
requirements.
4.4.7 At both the top and bottom of each opening, additional
reinforcement shall be placed having an area at least equal to
one-half the established design circumferential reinforcement
interrupted by the opening. The area As of this additional
steel at the top and at the bottom, however, shall be not less
than that given by Eq. (4-44) unless otherwise determined by
a detailed analysis
0.06f c′ tl
A s = ------------------fy
(4-44)
One-half of this extra reinforcement shall extend
completely around the circumference of the chimney, and
the other half shall extend beyond the opening at a sufficient
distance to develop the bars in tension. This reinforcement
shall be placed as close to the opening as practicable, but
within a height not to exceed three times the thickness t.
4.4.8 For openings larger than 2 ft wide, diagonal reinforcing
bars with a total cross-sectional area, in square inches, of not
less than 1/10 of the shell thickness, in inches, shall be placed
REINFORCED CONCRETE CHIMNEYS
at each corner of the opening. For openings 2 ft wide or
smaller, a minimum of two No. 5 reinforcing bars shall be
placed diagonally at each corner of the opening.
4.4.9 Tie bars shall be provided between inner- and outerface reinforcement around the perimeter of all openings
where reinforcing steel is interrupted and at the top of
chimney shells. Ties shall be a minimum of No. 3 bars, and
shall not exceed a spacing of 12 in. Refer to Fig. 4.1.
4.5—Wind deflection criteria
The maximum lateral deflection of the top of a chimney
before the application of load factors shall not exceed the
limits set forth by Eq. (4-45)
Y max = 0.04h
------------12
(4-45)
R4.5 The incorporation of the strength design method into
the standard generally results in chimneys with thinner walls
in the lower portion and with higher deflections. The
committee felt that deflections under service loads should be
checked and that the deflections of chimneys designed by the
strength method should not vary greatly from the deflections
of existing chimneys designed by the working stress method.
Limiting deflections also reduces the effects of secondary
bending moments.
The procedures in the ACI 307-88, however, were found to
be too restrictive for shorter chimneys and were modified in
the ACI 307-95. The deflection limit is compared against the
deflection calculated using uncracked concrete sections and
a fixed base.
Operation, access for inspection, lining type, and wind- or
earthquake-induced deflection should be considered when
establishing shell geometry.
CHAPTER 5—DESIGN OF CHIMNEY SHELLS:
STRENGTH METHOD
5.1—General
R5.1 Two significant revisions were made to this section in
this document, most notably the load factors specified in
Section 5.3 and the strength reduction factor φ specified in
Section 5.4.
5.1.1 Except as modified herein, design assumptions shall
be in accordance with ACI 318, Chapter 10. The chimney
shell shall be designed by the strength method.
5.1.2 The equivalent rectangular concrete compressive
stress distribution described in Section 10.2.7 of ACI 318
and as modified herein shall be used. For vertical strength,
the maximum strain on the concrete is assumed to be 0.003,
and the maximum tensile strain in the steel is assumed to be
0.07. Whichever value is reached first shall be taken as the
limiting value.
In place of the equivalent rectangular concrete compressive
stress distribution used in this chapter, any other relationship
between concrete compressive stress and strain may be
assumed that results in prediction of the strength of hollow
circular sections in substantial agreement with results of
comprehensive tests.
307-17
R5.1.2 The maximum compressive strain in the concrete is
assumed to be 0.003, or the maximum tensile strain in the
steel is assumed to be the fracture limit of 0.07, whichever is
reached first. Refer to the strain diagram in Fig. 5.1(a). The
strain limit of 0.07 is consistent with minimum elongation
properties in tension of reinforcing steel. If the steel fracture
limit is reached first, the maximum concrete strain computed
from the linear strain diagram is below 0.003. This deviates
from the design assumptions of ACI 318. For a given total
vertical steel ratio, this may occur when the ratio of the
vertical load to the moment is below a certain value. A total
vertical steel ratio in the chimney cross section less than that
per the minimum requirement of ACI 318 for flexural
members is permitted.
Even when the maximum concrete compressive strain εm is
less than 0.003, the stress block is still considered rectangular.
In these instances, however, the stress level is modified by a
correction factor called the parameter Q. Refer to the
Commentary on Section 5.5.1.
5.1.3 The design and detailing of precast chimney shells
shall parallel the design of cast-in-place chimney shells
unless specifically stated otherwise herein. Particular
attention should be given to the spacing and reinforcement
of cast-in-place cores and closures joining precast units to
ensure that the requirements of this code and other applicable
standards listed in Chapter 7 are met.
5.1.4 Refer to Section 5.7 for design procedures of
noncircular shells.
5.2—Design loads
5.2.1 Dead loads and wind or earthquake forces before the
application of load factors shall be in accordance with
Chapter 4 of this code. Thermal effects at service conditions
shall be in accordance with Chapter 6.
5.3—Required strength
5.3.1 Required vertical strength Uv to resist dead load D,
wind load W or seismic load E, and normal temperature T
shall be the largest of the following
Uv = 1.4D
(5-1)
Uv = 0.9D + 1.2T + 1.6Walong
(5-2)
Uv = 1.2D + 1.2T + 1.6Walong
(5-2a)
Uv = 0.9D + 1.2T + 1.4Wcombined along+across
(5-3)
Uv = 1.2D + 1.2T + 1.4Wcombined along+across
(5-3a)
Uv = 0.9D + 1.2T +1.0E
(5-4)
Uv = 1.2D + 1.2T + 1.0E
(5-4a)
(Note: E to be calculated with 1.0D loads only).
R5.3.1 The committee noted that the fastest-mile provisions
in ACI 307-88 resulted in an increase in wind moments of
between 0 and 50% when compared with ACI 307-79. The
use of a 3-second gust wind speed resulted in further
307-18
ACI STANDARD
increases in axial bending moments that are 10 to 20%
higher than moments calculated using fastest-mile speeds.
Because the committee had no data or information
concerning axial bending failures of chimney shells
designed using previously established procedures, it was
decided that the load factor for along-wind loads could be
safely reduced from 1.7 in ACI 307-88 to 1.3 when 3-second
gust wind speeds are used. (A wind load factor of 1.3 is
consistent with that recommended by ASCE 7-95.)
Similarly, the Committee determined that the wind load
factor for both along- and across-wind loads can be reduced
from 1.4 to 1.2.
The lower vertical load factor reductions incorporated in
the ACI 307-98 should be accompanied by a decrease in the
strength reduction factor φ from 0.80 to 0.70. The net effect
of the ACI 307-98 revision to the vertical load factors,
coupled with the change in the strength factor, is relatively
minor.
For the 2008 version, the ASCE 7-02 revisions caused the
committee to revisit the load factor values and combinations.
The committee agreed to the vertical strength checks
presented in Eq. (5-1) through (5-5). The net effect of the
revised load factors Kd and φ is minor and results in a slight
increase in vertical steel compared with ACI 307-98.
Table R5.1 summarizes the effects of the revisions on 12
sample chimney shells over a range of wind speeds. The
geometry of the chimneys studied is detailed in Table R5.2.
5.3.2 Required circumferential strength Uc to resist wind
load W and normal temperature load T shall be
Uc = 1.2T + 1.4W
Table R5.1—Comparison of along-wind design
moments*(ACI 307-98/ACI 307-95)
Chimney
no.
1
2
3
4
5
6
7
8
9
10
11
12
5.4—Design strength
5.4.1 Design strength of a section in terms of moment shall
be taken as the nominal moment strength calculated in
accordance with the requirements of this standard multiplied
by a strength reduction factor φ equal to 0.80 for vertical
strength and 0.90 for circumferential strength.
R5.4.1 The strength reduction factor for vertical strength
was changed back to 0.80 from 0.70. φ was revised because
the load factors in Section 5.3 were increased. The net effect
is an increased conservatism in the 2008 version.
The formulas are also derived for cross sections with
openings. Additional vertical compression reinforcement on
each side of the opening is provided per Section 4.4.6. No
change in moment capacity is made for the reduced distance
from the neutral axis of the additional jamb reinforcement.
120(3sg)/100(fm)
0.973
0.976
0.980
0.983
0.988
0.991
0.993
0.993
0.998
1.00
1.002
1.008
150(3sg)/130(fm)
0.940
0.944
0.947
0.950
0.955
0.958
0.960
0.960
0.965
0.967
0.969
0.976
*
Values of [1.3 × M(3sg)/0.7]/1.7 × M(fm)/0.8] for sample chimneys, where M is the
wind speed in miles per hour.
†
This abbreviation means 90 mph wind defined by the 3-second gust divided by 70 mph
wind defined by the fastest mile measurement technique.
Table R5.2—Geometry of chimneys studied
1
250
Top outside
diameter, ft
13.50
2
3
275
325
28.00
15.00
28.00
20.00
4
375
20.00
32.00
5
6
425
485
35.00
47.67
39.00
53.50
7
8
534
545
51.09
33.00
61.55
55.00
Chimney no.
(5-5)
R5.3.2 In ACI 307-98, the load factor for determining the
circumferential strength required to resist wind load was not
revised, although the reinforcement necessary to satisfy the
higher moments may increase up to 15% on large-diameter
chimneys. The committee believed, however, that this
additional reinforcement is justified to minimize vertical
cracking of chimney shells.
The 2008 version increased the load factor because kd in
Eq. (4-5) would reduce the loads. For round chimneys, kd is
0.95.
90(3sg)/70(fm)†
1.054
1.058
1.062
1.065
1.069
1.072
1.073
1.074
1.079
1.082
1.084
1.090
Height, ft
Bottom outside
diameter, ft
19.75
9
613
73.00
73.00
10
700
60.00
78.00
11
12
773
978
43.00
73.00
70.00
114.78
5.5—Nominal moment strength: circular shells
R5.5 The formulas for the nominal moment strength of
chimney cross sections are obtained based on the design
assumptions of ACI 318, except as modified under Section
5.1.2 of ACI 307-08. The derivations of the formulas are
given in Appendix A.
The formulas are derived for circular hollow cross
sections with uniform distributions of vertical reinforcing
steel around the circumference.
5.5.1 The following equations apply (refer to Fig. 5.1(a),
(b), and (c))
Pu/rtfc′ = K1 = 1.7Qλ + 2εmKeωtQ1 + 2ωtλ1
(5-6)
where
Ke = Es /fy
(5-7)
ωt = ρt fy/fc′
(5-8)
λ = τ – n1β (radians)
(5-9)
REINFORCED CONCRETE CHIMNEYS
where n1 is the number of openings entirely in the compression
zone. The maximum number of openings is two
sin ψ – sin μ – ( ψ – μ ) cos α
Q 1 = ------------------------------------------------------------------1 – cos α
(5-10)
λ1 = μ + ψ – π (radians)
(5-11)
where the angles β, α, μ, τ, and ψ are shown in Fig. 5.1(a)
307-19
cosτ = 1 – β1(1 – cosα)
(5-12)
1 – cos α f
cos ψ = cos α – ⎛ ---------------------⎞ ⎛ ----y-⎞ ≥ – 1.0
⎝ ε m ⎠ ⎝ Es ⎠
(5-13)
1 – cos α f
cos μ = cos α + ⎛ ---------------------⎞ ⎛ ----y-⎞ < 1.0
⎝ ε m ⎠ ⎝ Es ⎠
(5-14)
307-20
ACI STANDARD
where, for fc′ ≤ 4000 psi,
β1 = 0.85
(5-15)
β1 = 0.85 – 0.05(fc′ – 4000)/1000 ≥ 0.65
(5-16)
εm = 0.07(1 – cosα)/(1 + cosα) ≤ 0.003
(5-17)
Mn/Pur = K3 = cosα + K2/K1, Mn = PurK3
(5-18)
K2 = 1.7QR + εmKeωtQ2 + 2ωtK
(5-19)
for fc′ > 4000 psi
For α ≤ 5 degrees
2
2
Q = ( – 0.523 + 0.181α – 0.0154α ) + ( 41.3 – 13.2α + 1.32α ) ⎛ -t ⎞
⎝ r⎠
(5-20)
For 5 degrees < α ≤ 10 degrees
Q = (–0.154 + 0.01773α + 0.00249α2)
(5-21)
+ (16.42 – 1.980α + 0.0674α2)(t/r)
For 10 degrees < α ≤ 17 degrees
Q = (–0.488 + 0.076α) + (9.758 – 0.640α)(t/r)
(5-22)
For 17 degrees < α ≤ 25 degrees
Q = (–1.345 + 0.2018α + 0.004434α2)
(5-23)
+ (15.83 – 1.676α + 0.03994α2)(t/r)
For 25 degrees < α ≤ 35 degrees
Q = (0.993 – 0.00258α) + (–3.27 + 0.0862α)(t/r) (5-24)
For α > 35 degrees
Q = 0.89
(5-25)
where
2
[ ( ψ – μ ) ( 1 + 2cos α ) + ( 1/2 ) ( 4 sin 2α + sin 2ψ – sin 2μ ) – ς ]
Q2= ------------------------------------------------------------------------------------------------------------------------------------------------------ (5-26)
( 1 – cos α )
where ς = 4cosα(sinα + sinψ – sinμ).
K = sinψ + sinμ + (π – ψ – μ)cosα
(5-27)
R = sinτ – (τ – n1β)cosα – (n1/2)[sin(γ + β) – sin(γ – β)] (5-28)
where γ = 1/2 angle between centerlines of two openings and
for no openings, n1 = γ = β = 0; for one opening in compression
zone, n1 = 1, γ = 0; for two openings in compression zone,
n1 = 2.
R5.5.1 Parameter Q—The use of a rectangular compression
stress block for rectangular and T-shaped reinforced concrete
beams came to be accepted after extensive comparative study
between the analytical results using this stress-strain
relationship and the test data. The acceptability of the
rectangular stress block was based on the good correlation
between the results of the analyses and the tests, comparing:
(a) Concrete compression force; and
(b) Moment of the compression force about the neutral
axis (for a rectangular section, this is equivalent to the
distance of the center of gravity of the compression stress
block from the neutral axis).
The preceding comparative study was based on the limited
test data available on reinforced concrete members of
hollow circular sections subjected to axial and transverse
loads (Mokrin and Rumman 1985).
Another problem in arriving at the compressive stress
block for the analysis of reinforced concrete chimneys was
the fact that the maximum concrete compressive strain is less
than 0.003 when the fracture limit of steel is reached. That
is, the compressive stress block is not fully developed (refer
to Section R5.1.2). Thus, the previous attempts at specifying
the rectangular stress block for chimney cross sections
needed to be modified.
A numerical study was undertaken by the committee in
1988 to find an equivalent rectangular stress block to calculate
the strength of chimney cross sections.
For a given value of α, the results of the rectangular
concrete compression stress block, expressed by dimensionless modifications of (a) and (b) previously stated, were
compared with the corresponding results using a more exact
concrete stress-strain relationship (Rumman and Sun 1977)
given by Hognestad (1951) using a limiting strain of 0.003.
The comparisons were made for hollow circular sections
without openings and with single openings with values of β
of 10, 20, and 30 degrees.
It was concluded that for values of α above 20 degrees, or
when the limiting strain of concrete is reached first, an
equivalence between the two approaches is reached if the
stress level of the rectangular compression block is reduced
by a factor of 0.89. For values of α below approximately 20
degrees, a further correction is required, leading to the
values of the parameter Q defined in Section 5.5.1.
Thus, the correction factor, or the parameter Q, achieves a
close equivalence between the resulting values of (a) and (b)
for the thereby corrected rectangular stress block and the
stress block based on the Hognestad stress-strain relationship, yet retains the simplicity of the rectangular stress block.
5.5.2 Two symmetric openings partly in compression
zone—Refer to Fig. 5.1(c). This condition exists when γ + β
> τ and γ – β < τ. For this case, let δ = γ – β. Then in Eq. (5-6),
λ = δ.
And in Eq. (5-19)
R = sinδ – δcosα
(5-29)
REINFORCED CONCRETE CHIMNEYS
5.5.3 Openings in tension zone—Openings in the tension
zone are ignored because the tensile strength of the concrete
is neglected and the bars cut by the openings are replaced at
the sides of the openings.
5.5.4 Openings in compression zone—In calculations of
the forces in the compression reinforcement only, openings
in the compression zone are ignored because the cut bars are
replaced at the sides of the openings.
5.5.5 Limitation—The one-half opening angle β shall not
exceed 30 degrees.
5.5.6 Calculation procedure—Given r, t, fc′, β, γ, Pu, Mu,
and the number of openings (where Pu and Mu are the
factored vertical load and the factored moment, respectively), use the following procedure:
Step 1—Assume a value for the total vertical steel ratio ρt;
Step 2—By trial and error, find the value of α that satisfies
Eq. (5-6);
Step 3—Substitute this value of α in Eq. (5-18) and
calculate Mn;
Step 4—If φMn < Mu, increase ρt; if φMn > Mu, decrease
ρt; and
Step 5—Repeat Steps 2 through 4 until φMn = Mu.
R5.5.6 Due to thermal exposure of the concrete chimneys,
the temperature drop across the wall reduces the nominal
strength of chimney sections. This effect is accounted for by
reducing the specified yield strength of steel and specified
compressive strength of concrete.
The derivation of equations is included in Appendix A.
5.5.7 For load combinations with temperature effects,
modify fy and fc′ using Eq. (5-30) and (5-31).
Replace fy with
1.2
f y′ ( v ) = f y – -------------- ( f STV – γ 1 f STV
″ )
1 + γ1
(5-30)
Replace fc′ with
′′
fc′′(v) = fc′ – 1.2fCTV
(5-31)
″ , and fCTV
″ are as defined in Chapter 6.
where γ1, fSTV , fSTV
5.5.8 Special considerations—Where more than two openings
occur at the same elevation, appropriate design methods
consistent with the cases shown by Fig. 5.1(a), (b), and (c)
shall be used.
R5.5.8 This document does not limit the number of openings
at any horizontal cross section.
5.6—Noncircular shapes
5.6.1 General—All applicable sections of this code shall
be followed, including horizontal bending and temperature
effects.
5.6.2 Design assumptions—Strain in reinforcement and
concrete shall be assumed directly proportional to the
distance from the neutral axis.
For vertical strength, the maximum strain in the concrete
is assumed to be 0.003, and the maximum strain in the steel
307-21
Fig. 5.2—Stress-strain curve for concrete.
is assumed to be 0.07. Whichever value is reached first shall
be taken as the limiting value.
Stress in reinforcement below the specified yield strength
fy for grade of reinforcement used shall be taken as Es times
steel strain. For strains greater than that corresponding to fy ,
stress in reinforcement shall be assumed equal to fy.
Tensile strength of concrete shall be neglected.
The relationship of concrete compressive stress and
concrete strain shall be assumed in accordance with the
stress-strain curve as shown in Fig. 5.2.
5.6.3 Calculation procedure—For a given geometry and
given Pu and Mu (where Pu is the factored vertical load and
Mu is the factored moment), use the following procedure:
Step 1—Assume a value for the total vertical steel ratio ρt;
Step 2—By trial and error, find the location of the neutral
axis that makes the total vertical force in the section equal
and opposite to Pu;
Step 3—With this location of the neutral axis, calculate
Mn, the nominal moment strength of the section;
Step 4—If φMn < Mu, increase ρt; if φMn > Mu, decrease
ρt; and
Step 5—Repeat Steps 2 through 5 until φMn = Mu.
5.6.4 Horizontal bending—Design for horizontal bending
shall comply with the requirements of Section 5.7.
5.7—Design for circumferential bending
5.7.1 Any horizontal strip of the concrete column shall be
designed as a horizontal beam resisting circumferential
bending moments, as given in Section 4.2.4, and thermal
effects, described in Section 6.3.
5.7.2 For loads combined with temperature effects,
modify fy and fc′ using Eq. (5-32) and (5-33)
307-22
ACI STANDARD
Replace fy with fy′ (c) = fy – 1.2fSTC
(5-32)
″
Replace fc′ with fc″ (c) = fc′ – 1.2fCTC
(5-33)
″ are as defined in Chapter 6.
where fSTC and fCTC
R5.7.2 The commentary on Section 5.5.6 applies equally
to this section.
CHAPTER 6—THERMAL STRESSES
6.1—General
R6.1 Due to a temperature drop only across the chimney
wall, the derivations of the formulas for the vertical and
horizontal stresses in concrete and steel are given in
Appendix B. No revisions were made to this section in the
2008 edition.
6.1.1 The equations for temperature stresses given in this
chapter are based on working stress procedures and shall be
used with the appropriate load factor in the calculation of the
nominal moment strength in Chapter 5.
6.2—Vertical temperature stresses
6.2.1 The maximum vertical stresses in the concrete and
″ and fSTV
″ , in psi, occurring at the inside of the
steel, fCTV
chimney shell due to temperature shall be computed by
Eq. (6-1) and (6-2), respectively
″ = αte × c × Tx × Ec
fCTV
(6-1)
″ = αte(c – 1 + γ2) × Tx nEc
fSTV
(6-2)
where
c = –ρn(γ1 + 1) +
2
[ ρn ( γ 1 + 1 ) ] + 2ρn [ γ 2 + γ 1 ( 1 – γ 2 ) ]
(6-3)
and
n = Es /Ec
(6-4)
The temperature gradient across the concrete shell, Tx ,
shall be computed by Eq. (6-5) through (6-8), or by a
complete heat-balance study for all operating conditions.
a. For unlined chimneys
⎛
⎞
⎟
Ti – To
td ci ⎜
T x = ----------- ⎜ --------------------------------------------⎟
Cc dc ⎜ 1
td ci
d ci ⎟
----- + ----------- + --------------⎠
⎝K
C c d c K o d co
i
(6-5)
b. For lined chimneys with insulation completely filling
the space between the lining and shell
⎛
⎞
⎟
Ti – To
td bi ⎜
T x = ----------- ⎜ ---------------------------------------------------------------------------------⎟
C c d c ⎜ 1 t b d bi t s d bi td bi
d bi ⎟
----- + ------------ + ----------- + ----------- + -------------⎠
⎝K
C
d
C
d
C
d
K
i
b b
s s
c c
o d co
(6-6)
c. For lined chimneys with unventilated air space between
the lining and shell
⎛
⎞
⎟
td bi ⎜
Ti – To
T x = ----------- ⎜ ---------------------------------------------------------------------------------⎟
C c d c ⎜ 1 t b d bi
d bi
td bi
d bi ⎟
----- + ------------ + ---------- + ---------- + ------------⎝K
C b d b K r d b C c d c K o d co⎠
i
(6-7)
d. For lined chimneys with a ventilated air space between
the lining and shell
⎛
⎞
⎟
td bi ⎜
Ti – To
T x = ----------- ⎜ -------------------------------------------------------------------------------------------⎟ (6-8)
t b d bi
d bi
td bi
d bi ⎟
Cc dc ⎜ 1
- + ---------------- + ---------- + ---------- + ------------⎝ --------r q K i r q C b d b K s d s C c d c K o d co⎠
6.2.2 Unless complete heat balance studies are made for
the particular chimney, it is permissible to use the approximate
values given as follows. These constants, when entered into
equations for temperature differential through the chimney
shell, Tx , will give values of accuracy in keeping with the
basic design assumptions:
= 0.5;
rq
Cc = 12(Btu⋅in.)/(h⋅ft2⋅°F) of thickness/h/°F difference
in temperature;
Cs = to be obtained from the manufacturer of the
materials used;
Cb = to be obtained from the manufacturer of the
materials used;
= to be determined from curves in Fig. 6.1;
Ki
Ko = 12 Btu/(ft2⋅h⋅°F);
Kr = Ti /120; and
Ks = Ti /150.
The value of rq = 0.5 shall apply only where the distance
between the lining and the chimney shell is not less than 4 in.
throughout the entire height of the lining and air inlet and outlet
openings are provided at the bottom and top of the chimney
shell. The area of the inlet and outlet openings, in square feet,
shall numerically equal two-thirds of the inside diameter in feet
of the chimney shell at the top of the lining. Local obstructions
in the air space between the lining and the chimney shell shall
not restrict the area of the air space at any horizontal section to
less than that specified for air inlet or outlet.
R6.2.2 The research data available to establish the coefficients of heat transfer through the chimney lining and shell,
especially as they concern the heat transfer from gases to the
surfaces and through ventilated air spaces between lining and
shell, are somewhat meager. Unless complete heat balance
studies are made for the particular chimney, it is permissible to
use constants as determined or stated in this standard.
REINFORCED CONCRETE CHIMNEYS
307-23
Fig. 6.1—Curves for determining Ki.
6.2.3 The maximum stress in the vertical steel, fSTV , in psi,
occurring at the outside face of the chimney shell due to
temperature shall be computed by Eq. (6-9)
fSTV = αte × (γ2 – c) × Tx × Es
(6-9)
6.3—Circumferential temperature stresses
6.3.1 The maximum circumferential stress in concrete,
″ , in psi, occurring at the inside of the chimney shell due
fCTC
to temperature, shall be computed by Eq. (6-10)
″ = αte × c′ × Tx × Ec
fCTC
2
ASCE
ASCE/SEI 7-02 Minimum Design Loads for Buildings and
Other Structures
(6-11)
ASTM International
A615/A615M-05a Standard Specification for Deformed and
Plain Carbon-Steel Bars for Concrete
Reinforcement
A706/A706M-05a Standard Specification for Low-Alloy
Steel Deformed and Plain Bars for
Concrete Reinforcement
A996/A996M-05a Standard Specification for Rail Steel and
Axle-Steel Deformed Bars for Concrete
Reinforcement
and Tx = value determined for vertical temperature stresses.
All other notations are the same as for vertical temperature
stresses.
6.3.2 The maximum stress in psi in the outside circumferential reinforcement fSTC due to temperature shall be
computed by Eq. (6-12)
fSTC = αte × (γ2′ – c′) × Tx × Es
American Concrete Institute
318-02 Building Code Requirements for Structural
Concrete
(6-10)
where
c′ = – ρ′n ( γ 1 ′ + 1 ) + [ ρ′n ( γ 1 ′ + 1 ) ] + 2ρ′n [ γ 2′ + γ 1′ ( 1 – γ 2′ ) ]
CHAPTER 7—REFERENCES
7.1—Referenced standards
Standards referred to in this standard are listed in the
following with their serial designations, including the year of
adoption or revision, and are declared to be a part of this
standard as if fully set forth herein.
(6-12)
307-24
C33-03
C150-04
C309-03
C595-03
ACI STANDARD
Standard Specification for Concrete
Aggregates
Standard Specification for Portland
Cement
Standard Specification for Liquid
Membrane-Forming Compounds for
Curing Concrete
Standard Specification for Blended
Hydraulic Cement
Federal Aviation Administration
AC70-7460-1K Obstruction Marking and Lighting
Underwriters Laboratories
UL 96A
Installation Requirements for Lighting
Protection Systems
R7.1 Referenced standards and reports
American Concrete Institute
318
Building Code Requirements for Structural
Concrete
550R
Design Recommendations for Precast Concrete
Structures
American Society of Civil Engineers
ASCE 7-95
Minimum Design Loads for Buildings and
Other Structures
ASCE/SEI 7-02 Minimum Design Loads for Buildings and
Other Structures
American Society of Mechanical Engineers
ASME/ANSI STS-1 Steel Stacks
The above publications may be obtained from the
following organizations:
American Concrete Institute
P.O. Box 9094
Farmington Hills, MI 48333-9094
www.concrete.org
American Society of Civil Engineers
1801 Alexander Bell Dr.
Reston, Va. 20191
www.asce.org
American Society of Mechanical Engineers
Three Park Ave.
New York, NY 10016-5990
www.asme.org
R7.2 Cited references
ACI Committee 307, 1969, “Specification for the Design
and Construction of Reinforced Concrete Chimneys (ACI
307-69),” ACI JOURNAL, Proceedings V. 66, No. 8, Aug.,
pp. 610-611.
ACI Committee 307, 1979, “Specification for the Design
and Construction of Reinforced Concrete Chimneys (ACI 30779),” American Concrete Institute, Farmington Hills, MI, 63 pp.
ACI Committee 307, 1988, “Standard Practice for the
Design and Construction of Cast-In-Place Reinforced Concrete
Chimneys (ACI 307-88) and Commentary (307R-88),”
American Concrete Institute, Farmington Hills, MI, 16 + 14 pp.
ACI Committee 307, 1995, “Standard Practice for the
Design and Construction of Reinforced Concrete Chimneys
(ACI 307-95) and Commentary (307R-95),” American
Concrete Institute, Farmington Hills, MI, 16 + 14 pp.
ACI Committee 307, 1998, “Design and Construction of
Reinforced Concrete Chimneys (ACI 307-98) and Commentary
(307R-98),” American Concrete Institute, Farmington Hills,
MI, 17 + 14 pp.
ACI Committee 318, 2002, “Building Code Requirements
for Structural Concrete (ACI 318-02) and Commentary
(318R-02),” American Concrete Institute, Farmington Hills,
MI, 443 pp.
ACI Committee 505, 1934, “Proposed Standard for the
Design and Construction of Reinforced Concrete Chimneys,”
ACI JOURNAL , Proceedings V. 30, Mar.-Apr., pp. 367-368.
ACI Committee 505, 1954, “Standard Specification for the
Design and Construction of Reinforced Concrete Chimneys,”
ACI JOURNAL , Proceedings V. 51, No. 9, Sept., pp. 1-48.
ASCE Task Committee on Steel Chimney Liners, 1975,
Design and Construction of Steel Chimney Liners, American
Society of Civil Engineers, New York, 226 pp.
ASCE Task Committee on Wind Forces, 1961, “Wind
Forces on Structures,” Transactions, ASCE, V. 126, Part II,
pp. 1124-1198.
Basu, R. I., 1982, “Across-Wind Responses of Slender
Structures of Circular Cross-Section to Atmospheric
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“NEHRP Recommended Provisions for Seismic Regulations
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REINFORCED CONCRETE CHIMNEYS
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307-25
APPENDIX A—DERIVATION OF EQUATIONS FOR
NOMINAL STRENGTH
Equations for the nominal strength of concrete chimney
sections, with and without openings, are derived in this
Appendix.
The factored vertical load Pu and the corresponding
nominal moment strength Mn are expressed in dimensionless
form, as given in Section 5.5.1 by Eq. (5-6) and (5-18),
respectively. The Q factor shown in Eq. (5-20) through (5-25)
is omitted from the general derivation.
A procedure to account for the temperature effects in the
vertical and horizontal directions is also outlined.
Forces are designated as follows (see also Section 1.4):
= design moment strength of section
MDS
P
= total force in concrete compressive stress
block
P′, S1′,
S2′, S3′, S4′ = moments of P, S1, S2, S3, and S4 about
neutral axis, respectively
S1
= tensile force where steel stress is below
yield strength, from α to ψ
S2
= tensile force where steel stress is at yield
strength, from ψ to π
= compressive force in steel where stress is
S3
below yield strength, from μ to α
= compressive force in steel where stress is at
S4
yield strength, from 0 to μ
Reference Fig. 5.1(a) and (b).
θ
= variable function of α
S1 = 2
ψ r ( cos α
– cos θ )
- ⋅ ε m E s ρ t rt dθ
∫α ------------------------------------r ( 1 – cos α )
2ε m E s ρ t rt
ψ
= ------------------------- ( θ cos α – sin θ ) α
( 1 – cos α )
2ε m E s ρ t rt
= ------------------------- [ ( ψ – α ) cos α – sin ψ + sin α ]
( 1 – cos α )
but
Esρt
= Esρt · (ωt fc′/ρt fy)
= Es /fy · ωt fc′
= Keωt fc′
therefore
[ ( ψ – α ) cos α – sin ψ + sin α ]
S 1 = 2ε m K e ω t rtf c′ ⋅ -----------------------------------------------------------------------( 1 – cos α )
or
S1 = 2εm Ke ωtrtfc′ · Q′
307-26
ACI STANDARD
S2 = 2(π – ψ)ρt rtfy
Ke = Es /fy
but
ωt = ρt fy /fc′
ρ t fy = ω t fc ′
S1 ′ = 2
S2 = 2(π – ψ)rtωt fc′
2
= 1.7rtfc′(τ – n1β)
ψ
2
2
∫α ( cos α – 2 cos α cos θ + cos θ ) dθ
2
2ε m K e ω t r tf c′ ⎛
2
θ sin 2θ ψ
= --------------------------------- ⋅ ⎝ θcos α – 2 cos α sin θ + --- + --------------⎞⎠
( 1 – cos α )
2
4 α
= 1.7rtfc′ · λ
2
2ε m K e ω t r tf c′
= --------------------------------- · [(ψ – α)cos2α – 2cosα(sinψ – sinα)
( 1 – cos α )
+ (1/2)(ψ – α) + (1/4)(sin 2ψ – sin 2α)]
where
λ = τ – n1β
α r ( cos θ
– cos θ )
r -------------------------------------- ⋅ ε m E s ρ t rt dθ
r ( 1 – cos α )
α
2ε m E s r ρ t rt
= ----------------------------–
( 1 – cos α )
P = 2(τ – n1β)rt · 0.85fc′
S3 = 2
∫
2
ψ 2 ( cos α
– cos α )
- ⋅ ε m E s ρ t rt dθ
∫μ ------------------------------------r ( 1 – cos α )
2ε m E s ρ t rt
α
= ------------------------- ( sin θ – θ cos α ) μ
( 1 – cos α )
[ sin α – sin μ – ( α – μ ) cos α ]
= 2ε m K e ω t rtf c′ ----------------------------------------------------------------------( 1 – cos α )
Let J = (ψ – α)cos2α + 2 sinα cosα – 2 cosα sinψ + (1/2)sinψ
cosψ – (1/2)sinα cosα + (1/2)(ψ – α)
or
2J = 2(ψ – α)cos2α + 3sinα cosα – 4cosα sinψ + sinψ cosψ
+ (ψ – α)
therefore
= 2εm Ke ωt rtfc′ · Q3
S4 = 2μρtrtfy
= 2ωtrtfc′ · μ
The sum of vertical forces must equal zero, therefore
Pu = P + S3 + S4 – S1 – S2
= 1.70rtfc′ λ + 2εmKeωtrtfc′ Q3 + 2ωtrtfc′ μ
– 2εmKeωtrtfc′Q′ – 2ωt rtfc′ (π – ψ)
Pu/rtfc′ = K1
= 1.70λ + 2εmKeωt(Q3 – Q′) + 2ωt[μ – (π – ψ)]
= 1.70λ + 2εm Ke ωt Q1 + 2ωt λ1
S1′ = εm r2 tfc′ Ke ωt J1
where
J1 = 2J/(1 – cosα)
or
J1 = [2(ψ – α)cos2α + 3sinα cosα – 4cosα sinψ + sinψ
cosψ + (ψ – α)]/(1 – cosα)
S 2′ = 2
α
∫ψ ρt rtfy ⋅ r ( cos α – cos θ ) dθ
2
2
= 2r ρ t tf y [ ( π – ψ ) cos α – sin ψ ]
where
λ = τ – n1β
π
= 2r ρ t tf y ( θ cos α – sin θ ) ψ
but
ρ t fy = ω t fc ′
sin ψ – sin μ – ( ψ – μ ) cos α
Q 1 = ------------------------------------------------------------------( 1 – cos α )
therefore
λ1 = μ + ψ – π
S2′ = 2r 2tfc′ωt J2
REINFORCED CONCRETE CHIMNEYS
where
therefore
J2 = (π – ψ)cosα + sinψ
P′ = 1.70r2tfc′ [sinτ – (τ – β)cosα – sinβ]
S3 ′ = 2
∫
α r 2 ( cos θ
2
For P′ with two openings in compression zone (Fig. 5.1(b))
– cos α )
------------------------------------------- ⋅ ε m E s ρ t rt dθ
r ( 1 – cos α )
μ
2
2ε m K e ω t r tf c′
= --------------------------------( 1 – cos α )
α
2
307-27
sin τ- – r cos α⎞ – γ + βr ( cos θ – cos α ) dθ
P′ = 2rt0.85fc′ · τ ⎛ r-----------⎝ τ
⎠ ∫ γ–β
2
∫μ ( cos θ – 2 cos θ sin α + cos α ) dθ
2
α
2ε m K e ω t r tf c′ ⎛ θ sin 2θ
2
= --------------------------------- ⋅ --- – -------------- – 2 cos α sin θ + θcos α⎞
⎝
⎠
( 1 – cos α )
2
4
μ
= 1.70r2tfc′[sinτ – τcosα – sin(γ + β) + sin(γ – β) + 2βcosα]
therefore
P′ = 1.70r2tfc′[sinτ – (τ – 2β)cosα – sin(γ + β) + sin(γ – β)
2
2ε m K e ω t r tf c′
= --------------------------------- · [(1/2)(α – μ) + (1/4)(sin2α – sin2μ)
( 1 – cos α )
– 2cosα(sinα – sinμ) + (α – μ)cos2α]
Let
Generalizing
P′ = 1.70r2tfc′ · R
where
J3 = 2/(1 – cosα)
R = sinτ – (τ – n1β)cosα – (n1/2)[sin(γ + β) – sin(γ – β)]
or
For no openings
J3 = [α – μ + sinα cosα – sinμ cosμ – 4cosα(sinα – sinμ)
n1 = γ = β = 0
2
+ 2(α – μ)cos α]/(1 – cosα)
For one opening in the compression zone
therefore
n1 = 1
S3′ = εmr2tfc′Keωt J3
γ=0
S 4′ = 2
μ
∫0
For two openings in the compression zone
ρ t rtf y ⋅ r ( cos θ – cos α ) dθ
2
n1 = 2
The sum of moments about the neutral axis must equal
zero, therefore
Mn = Purcosα + P′ + S1′ + S2′ + S3′ + S4′
= Purcosα + 1.70r2tfc′ R + εmr 2tfc′Keωt J1 + 2r2tfc′ωt J2 +
εmr2tfc′KeωtJ3 + 2r2tfc′ωtJ4
= Purcosα + 1.70r2tfc′ R + εmr2tfc′Keωt(J1 + J3) +
2r2tfc′ωt(J2 + J4)
μ
= 2r ρ t tf y ( sin θ – cos α ) 0
= 2r2ρt tfy(sinμ – μcosα)
therefore
S4′ = 2r2tfc′ωt J4
therefore
where
Mn/r 2tfc′ = (Pucosα/rtfc′) + K2
J4 = sinμ – μcosα
For P′ with one opening in the compression zone (Fig. 5.1(a))
r sin τ
P′ = 2rt0.85fc′ · τ ⎛ ------------- – r cos α⎞ –
⎝ τ
⎠
β
∫0 r ( cos θ – cos α ) dθ
= 1.70r2tfc′(sinτ – τcosα – sinβ + βcosα)
where
K2 = 1.70R + εmKeωt(J1 + J3) + 2ωt(J2 + J4)
or
K2 = 1.70R + εmKeωtQ2 + 2ωtK
307-28
ACI STANDARD
Q2 = [(ψ – μ)(1 + 2cos2α) + (1/2)(4sin2α + sin2ψ – sin2μ)
– 4cosα(sinα + sinψ – sinμ)]/(1 – cosα)
′′ =
fSTV
and
ρ
1
---------------------- = -------------- = ratio, outside steel area to total steel area
ρ ( 1 + γ1 )
1 + γ1
K = sinψ + sinμ + (π – ψ – μ)cosα
Multiply both sides of the equation by
1/K1 = rtfc′/Purtfc′/Pu · Mn/r2tfc′ = rtfc′/Pu · Pucosα/rtfc′
+ 1/K1 · K2
therefore
K3 = Mn/Pur = cosα + K2/K1
or
Mn = K3Pur
and require
MDS = φMn ≥ Mu
For two symmetric openings partly in the compression
zone (Fig. 5.1(c))
γ+β>τ
and
γ–β<τ
let
δ=γ–β
The situation is the same as for no openings in the
compression zone with
τ=δ
compressive temperature stress in inside steel,
′′ , at service loads
fSTV and fSTV
γ1 ρ
γ
--------------------- = -------------- = ratio, inside steel area to total steel area
1 + γ1
ρ ( 1 + γ1 )
Ft(v) = load factor for temperature combined with W or E
= 1.2
At ultimate, effect on fy on windward side
Usable yield force = yield force – Ft(v) · tensile force in outside steel + Ft(v) · compressive force in inside steel
Dividing by total steel area As
γ1
1 ⋅A ⋅f
″
⋅ A s ⋅ f STV
F t ( v ) ------------F t ( v ) ------------s
STV
1 + γ1
1 + γ1
f y ′ ( v ) = f y – -------------------------------------------------- + ----------------------------------------------------As
As
therefore
Ft ( v )
f y ′ ( v ) = f y – ------------–γ f″ )
-(f
1 + γ 1 STV 1 STV
It is conservative and convenient to use the same value for
fy′ on the leeward side as well.
Vertical temperature stresses in concrete effect on fc′
′′ = concrete compressive stress due to temperature
fCTV
alone at service loads
At ultimate, effect on fc′ is
′′
fc′′ (v) = fc′ – Ft(v) · fCTV
Nominal strength for circumferential bending
(compression on inside)
λ=δ
R = sinδ – δcosα
f y′ ( c ) = f y – 1.2f STC ⎫
⎬ for combination with temperature
″ ⎭
f c″ ( c ) = f c ′ – 1.2f CTC
and all other values are the same as before.
Openings in the tension zone—Openings in the tension
zone are ignored because the tensile strength of the concrete
is neglected, and the bars cut by the openings are replaced at
the sides of the openings.
Openings in the compression zone—Openings in the
compression zone are ignored in calculations of the forces in
the compression reinforcement only because the cut bars are
replaced at the sides of the openings.
Vertical temperature stresses in reinforcement; effect on fy
fSTV = tensile temperature stress in outside steel
Refer to Fig. A.1 for strain and load diagrams.
ρ′
= ratio of outside steel area to total area
γ1′ = ratio of inside steel area to outside steel area
ρ′t = area outside steel, in.
γ1′ρ′t = area inside steel, in.
Stress in compression steel
[ ( a ⁄ β 1 ) – ( 1 – γ 2′ ) ]
f CS = -----------------------------------------------⋅ 0.003E s
a ⁄ β1
REINFORCED CONCRETE CHIMNEYS
307-29
ΣV = 0, PCB + PCS – PTS = 0
(A-6)
Find the value of a that satisfies this equation
ΣM about PTS , Mn = {PCB[γ2′ – (a/2)] + PCS(2γ2′ – 1)}t
MDS = φMn ≥ Mu
(A-7)
Note: For compression on outside
fy′(c) = fy
fc′′(c) = fc′
Therefore, ignore temperature. Equation (A-3) becomes
PCS = fCS ρ′t
and Eq. (A-4) becomes
PTS = fTS γ1′ρ′t
APPENDIX B—DERIVATION OF EQUATIONS FOR
TEMPERATURE STRESSES
The equations for maximum vertical stresses in concrete
and steel due to a temperature drop only, across the concrete
wall with two layers on reinforcement, are derived as follows.
Unrestrained rotation caused by a temperature differential
of Tx , Fig. B.1(a)
θte = αte Tx /t
Because rotation is prevented, corresponding stresses are
induced, Fig. B.1(b)
In concrete (inside)
Fig. A.1—Circumferential bending.
a – β 1 ( 1 – γ 2′ )
f CS = ---------------------------------- ⋅ 0.003E s ≤ f y′ ( c )
a
(A-1)
εc = θtect = αteTxc
Stress in tensile steel
and
γ 2′ – ( a ⁄ β 1 )
f TS = ----------------------------- ⋅ 0.003E s
a ⁄ β1
f TS
β 1 γ 2′ – a
= -------------------- ⋅ 0.003E s ≤ f y′ ( c )
a
′′
= αtecTxEc
f CTV
In outside reinforcement
(A-2)
Load in compression steel
PCS = fCS γ1′ρ′t
εs = θte(γ2 – c)t
and
(A-3)
fSTV = αte(γ2 – c)TxEs
Load in tensile steel
PTS = fTSρ′t
(A-4)
Load in concrete compression block
PCB = 0.85fc′′ (c)ta
(A-5)
ρ
=
γ1
=
ratio of total area of vertical outside face reinforcement to total area of concrete chimney shell at
section under consideration
ratio of inside face vertical reinforcement area to
outside face vertical reinforcement area
307-30
ACI STANDARD
For c
′′ γ1ρt
′′ (ct/2) + f STV
ΣV = 0, f CTV
= fSTVρt
αtecTxEc(ct/2) + αte(c – 1 + γ2)TxnEcγ1ρt
= αte(γ2 – c)TxnEcρt
c2 + 2nγ1ρc + 2nγ1ρ(γ2 – 1) + 2nρc – 2nργ2 = 0
c2 + 2ρn(γ1 + 1)c + 2ρn[γ1(γ2 – 1) – γ2] = 0
c2 + 2ρn(γ1 + 1)c – 2ρn[γ2 + γ1(1 – γ2)] = 0
c = –ρn(γ1 + 1) +
Fig. B.1—Vertical temperature stresses.
( c – 1 + γ 2 )n
″
f STV
″ ------------------------------ f CTV
c
= αte(c – 1 + γ2)TxnEc
2
[ ρn ( γ 1 + 1 ) ] + 2ρn [ γ 2 + γ 1 ( 1 – γ 2 ) ]
The derivation for the equations for the maximum horizontal
stresses in concrete and steel due to a temperature drop only,
across the concrete wall with two layers of reinforcement, is
similar to that for the vertical temperature stresses.
Replace
ρ
with ρ′;
γ1
with γ1′;
″
″ ;
f CTV
with f CTC
fSTV with fSTC;
c
with c′; and
γ2
with γ2′ .
then
″ = αtec′TxEc
f CTC
fSTC = αte(γ2′ – c′)TxEs
c′
= –ρ′n(γ1′ + 1) +
2
[ ρ′n ( γ 1 ′ + 1 ) ] + 2ρ′n [ γ 2 ′ + γ 1 ′ ( 1 – γ 2 ′ ) ]
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Code Requirements for Reinforced Concrete Chimneys
and Commentary
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