ACI 349.1R-91 (Reapproved 2000) supersedes ACI 349.1R-91 (Reapproved 1996)
and became effective July 1, 1991. In 1991, a number of minor editorial revisions
were made to the report. The year designation of the recommended references of the
standards-producing organizations have been removed so that the current editions
become the referenced editions.
*Prime authors of the thermal effects report.
Copyright
 2000, 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 reproduc-
tion or for use in any knowledge or retrieval system or device, unless permission in
writing is obtained from the copyright proprietors.

ACI Committee Reports, Guides, Standard Practices, and Commen-
taries are intended for guidance in planning, designing, executing, and
inspecting construction. This document is intended for the use of indi-
viduals 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 document shall not be made in contract documents.
If items found in this document are desired by the Architect/Engineer to
be a part of the contract documents, they shall be restated in mandatory
language for incorporation by the Architect/Engineer.
349.1R-1
Reinforced Concrete Design for Thermal Effects
on Nuclear Power Plant Structures
ACI 349.1R-91
(Reapproved 2000)

This report presents a design-oriented approach for considering thermal
loads on reinforced concrete structures. A simplified method is provided for
estimating reduced thermal moments resulting from cracking of concrete
sections. The method is not applicable to shear walls or for determining
axial forces resulting from thermal restraints. The global effects of temper-
ature, such as expansion, contraction, and thermal restraints are not spe-
cifically addressed. However, they need to be considered as required by
Appendix A of ACI 349. Although the approach is intended to conform to
the general provisions of Appendix A of ACI 349, it is not restricted to
nuclear power plant structures. Two types of structures, frames, and axi-
symmetric shells, are addressed. For frame structures, a rationale is
described for determining the extent of member cracking, which can be
assumed for purposes of obtaining the cracked structure thermal forces
and moments. Stiffness coefficients and carryover factors are presented in
graphical form as a function of the extent of member cracking along its
length and the reinforcement ratio. Fixed-end thermal moments for cracked
members are expressed in terms of these factors for: 1) a temperature gra-
dient across the depth of the member; and 2) end displacements due to a
uniform temperature change along the axes of adjacent members. For the
axisymmetric shells, normalized cracked section thermal moments are pre-
sented in graphical form. These moments are normalized with respect to
the cross section dimensions and the temperature gradient across the sec-
tion. The normalized moments are presented as a function of the internal
axial forces and moments acting on the section and the reinforcement ratio.
Use of the graphical information is illustrated by examples.
Keywords: cracking (fracturing); frames; nuclear power plants; reinforced
concrete; shells (structural forms); structural analysis; structural design;
temperature; thermal gradient; thermal properties; thermal stresses.
CONTENTS
Notation, p. 349.1R-1

Chapter 1—Introduction, p. 349.1R-2
Chapter 2—Frame structures, p. 349.1R-3
2.1—Scope
2.2—Section cracking
2.3—Member cracking
2.4—Cracked member fixed-end moments, stiffness fac-
tors, and carryover factors
2.5—Frame design example
Chapter 3—Axisymmetric structures, p. 349.1R-17
3.1—Scope
3.2—|e/d|
≥ 0.7 for compressive N and tensile M
3.3—General e/d
3.4—Design examples
Chapter 4—References, p. 349.1R-29
4.1—Recommended references
4.2—Cited references
NOTATION
General
A
s
= area of tension reinforcement within width b
A
s
′ = area of compression reinforcement within width b
b = width of rectangular cross section
d = distance from extreme fiber of compression face to
centroid of compression reinforcement
d
′ = distance from extreme fiber of compression face to

centroid of tension reinforcement
e = eccentricity of internal force N on the rectangular
section, measured from the section centerline
E
c
= modulus of elasticity of concrete
E
s
= modulus of elasticity of reinforcing steel
f
c
′ = specified compressive strength of concrete
f
y
= specified yield strength of reinforcing steel
j = ratio of the distance between the centroid of com-
pression and centroid of tension reinforcement to
the depth d
n = modular ratio = E
s
/E
c
t = thickness of rectangular section
T
m
= mean temperature, F
T
b
= base (stress-free) temperature, F
∆T = linear temperature gradient, F

α = concrete coefficient of thermal expansion, in./in./F
ν = Poisson’s ratio of concrete
= ratio of tension reinforcement = A
s
/bd
= ratio of compression reinforcement = A
s
′/bd
Chapter 2—Frame structures
a = length of the cracked end of member at
which the stiffness coefficient and carry-
Reported by ACI Committee 349
For a list of Committee members, see p. 30.
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=

=
over factor are determined, in the case of
an end-cracked beam (Fig. 2.4 through
2.7). In the case of an interior-cracked
beam, (Fig. 2.8 through 2.1l), a is the
length of the uncracked end of member at
which the stiffness coefficient and carry-
over factor are determined.
cracked member carry-over factor from
End A to End B
cracked member carry-over factor from
End B to End A
cracked member carry-over factor from the
a end of the member to the opposite end
modulus of rupture of concrete
cracked section moment of inertia about
the centroid of the cracked rectangular sec-
tion
uncracked section moment of inertia (ex-
cluding reinforcement) about the center
line of the rectangular section
ratio of depth of the triangular com-
pressive stress block to the depth d
cracked member stiffness at End A
(pinned), with opposite end fixed
cracked member stiffness at End B
(pinned), with opposite end fixed
cracked member stiffness at End
a
(pinned), with opposite end fixed

dimensionless stiffness coefficient
=
KL/E
c
I
g
total length of member
cracked length of member
cracking moment = bt
2
f
r
/6
cracked member fixed-end moment due to
^
_
T or T
m
-
T
b
,
at End a
moment at center line of rectangular cross
section
axial force at center line of rectangular
cross section
transverse displacement difference between
ends of cracked member, due to T
m

- T
b
acting on adjoining members
Chapter 3
-
Axisymmetric structures
f
c
=
f
CL
=
k
=
k
L
=
final cracked section extreme fiber com-
pressive stress resulting from internal sec-
tion forces
M, N, and
^
_
T
cracked section extreme fiber compressive
stress resulting from internal forces
M
and N
ratio of depth of
pressive stress block

sulting from internal
and
^
_
T
ratio of depth of
pressive stress block
sulting from internal
N
the triangular
com-
to the depth d, re-
section forces
M, N,
the triangular
com-
to the depth d, re-
section forces
M and
M =
N
=
iv

=
M,r
=
6,
=
L

CL
=
E
CT
=
4,.

=
4T

=
4
=
internal moment at section center line due
to factored mechanical loads, including
factored moment due toT
m
-
T
b
internal axial force at section center line
due to factored mechanical loads, including
factored axial force due toT
m
-
T
b
final internal moment at section center line
resulting from
M and

^
_
T
thermal moment due to
^
_
T, M,= =
A?f

-
M
final cracked section strain at extreme fiber
of compression face =
&cL
+
&T
cracked section strain at extreme fiber of
compression face resulting from internal
section forces
M and N
cracked section strain at extreme fiber of
compression face resulting from
^
_
T
cracked section curvature change resulting
from internal forces
M and N
cracked section curvature change required
to return free thermal curvature

aAT/t
to 0
final cracked section curvature change
=
+1.
+
4r
CHAPTER 1 INTRODUCTION
ACI

349,
Appendix A, provides general consid-
erations in designing reinforced concrete structures for
nuclear power plants. The Commentary to Appendix
A, Section A.3.3, addresses three approaches that
consider thermal loads in conjunction with all other
nonthermal loads on the structure, termed “mechani-
cal loads.” One approach is to consider the structure
uncracked under the mechanical loads and cracked
under the thermal loads. The results of two such anal-
yses are combined.
The Commentary to Appendix A also contains a
method of treating temperature distributions across a
cracked section. In this method an equivalent linear
temperature distribution is obtained from the temper-
ature distribution, which can generally be nonlinear.
Then the linear temperature distribution is separated
into a pure gradient
^
_

T and into the difference be-
tween the mean and base (stress-free) temperatures
T
m
- T
b
.
This report offers a specific approach for consid-
ering thermal load effects which is consistent with the
above provisions. The aim herein is to present a de-
signer-oriented approach for determining the reduced
thermal moments which result from cracking of the
concrete structure. Chapter 2 addresses frame struc-
tures, and Chapter 3 deals with axisymmetric struc-
tures. For frame structures, the general criteria are
given in Sections 2.2 (Section Cracking) and 2.3
(Member Cracking). The criteria are then formulated
for the moment distribution method of structural
analysis in Section 2.4. Cracked member fixed-end
moments, stiffness coefficients, and carry-over fac-
tors are derived and presented in graphical form. For
axisymmetric structures an approach is described for
regions away from discontinuities, and graphs of
cracked section thermal moments are presented.
DESlGN
FOR THERMAL
EFFECTS/NUCLEAR

PLANTS
349.1R-3

This report is not intended to represent a
state-of-
the-art discussion of the methods available to analyze
structures for thermal loads. Rather, the report is in-
tended to propose simplifications that can be made
which will permit a cracking reduction of thermal mo-
ments to be readily achieved for a large class of ther-
mal loads, without resorting to sophisticated and com-
plex solutions.Also, as a result of the report
discussion, the design examples, and graphical presen-
tation of cracked section thermal moments, it is
hoped that a designer will better understand how ther-
mal moments are affected by the presence of other
loads and the resulting concrete cracking.
CHAPTER 2
-
FRAME STRUCTURES
2.1
-
Scope
The thermal load on the frame is assumed to be
represented by temperatures which vary linearly
through the thicknesses of the members. The linear
temperature distribution for a specific member must
be constant along its length. Each such distribution
can be separated into a gradient
^
_
T and into a tem-
perature change with respect to a base (stress-free)

temperature T
m
- T
b
.
Frame structures are characterized by their ability
to undergo significant
flexural
deformation under
these thermal loads. They are distinguished from the
axisymmetric structures discussed in Chapter 3 by the
ability of their structural members to undergo rota-
tion, such that the free thermal curvature change of
aAT/f
is not completely restrained. The thermal mo-
ments in the members are proportional to the degree
of restraint. In addition to frames per se, slabs and
walls may fall into this category.
The rotational feature above is of course automati-
cally considered in a structural analysis using
un-
cracked member properties. However, an additional
reduction of the member thermal moments can occur
if member cracking is taken into account. Sections 2.2
and 2.3 of this chapter describe criteria for the crack-
ing reduction of
membe
r
thermal moments. These cri-
teria can be used as the basis for an analysis of the

structure under
thermal
loads, regardless of the
method of analysis selected. In Section 2.4, these cri-
teria are applied to the moment distribution analysis
method.
There are frame and slab structures which can be
adequately idealized as frames of sufficient geometric
simplicity to lend themselves to moment distribution.
Even if an entire frame or slab structure does not per-
mit a simple idealization, substructures can be isolated
to study the effects of thermal loads. Often with
today’s use of large scale computer programs for the
analysis of complex structures, a “feel” for the rea-
sonableness of the results is attainable only through
less complex analyses applied to substructures. The
moment distribution method for thermal loads is ap-
plicable for this work. This design approach uses
cracked member stiffness coefficients and carry-over
factors. These depend on the extent of member crack-
ing along its length due to mechanical loads, as dis-
cussed in Section 2.3.
2.2
-
Section cracking
Simplifying assumptions are made below for the
purpose of obtaining the cracked section thermal mo-
ments and the section (cracked and uncracked)
stiff-
nesses. The fixed-end moments, stiffness coefficients,

and carry-over factors of Section 2.4 are based on
these assumptions:
1. Concrete compression stress is taken to be lin-
early proportional to strain over the member cross
section.
2. For an uncracked section, the moment of inertia
is I
g
, where I
g
is based on the gross concrete dimen-
sions and the reinforcement is excluded.
For a cracked section, the moment of inertia is I
cr
,
where I
cr
is referenced to the centroidal axis of the
cracked section. In the formulation of I
cr
, the com-
pression reinforcement is excluded and the tension re-
inforcement is taken to be located at the tension face;
i.e., d = t is used.
3. The axial force on the section due to mechanical
and thermal loads is assumed to be small relative to
the moment
(e/d
>,
0.5). Consequently, the extent of

section cracking is taken as that which occurs for a
pure moment acting on the section.
The first assumption is strictly valid only if the ex-
treme fiber concrete compressive stress due to com-
bined mechanical and thermal loads does not exceed
0.5f'
c
.
At this stress, the corresponding concrete strain
is in the neighborhood of 0.0005
in./in.
For extreme
fiber concrete compressive strains greater than 0.0005
in./in.
but less than 0.001
in./in.,
the differences are
insignificant between a cracked section thermal mo-
ment based on the linear assumption adopted herein
versus a nonlinear concrete stress-strain relationship
such as that described in References 2 and 3. Con-
sequently, cracked member thermal moments given by
Eq. (2-3) and (2-4) are sufficiently accurate for con-
crete strains not exceeding 0.001
in./in.
For concrete strains greater than 0.001
in./in.,
the
equations identified above will result in cracked mem-
ber thermal moments which are greater than those

based on the nonlinear theory. In this regard, the
thermal moments are conservative. However, they are
still reduced from their uncracked values. This crack-
ing reduction of thermal moments can be substantial,
as seen in Fig. 3.2 which also incorporates Assump-
tion 1.
Formulation of the thermal moments based on a
linear concrete stress-strain relationship allows the
thermal moments to be expressed simply by the equa-
tions in Chapter 2 or by the normalized thermal mo-
ment graphs of Chapter 3. Such simplicity is desirable
in a designer-oriented approach.
Regarding
I
cr
, in Assumption 2, the assumptions for
the compression and tension reinforcement result in
the simple expression of (6jk
2
)I
g
for I
cr
, if the axial
load is small as specified in Assumption 3. The use of
(6jk
2
)I
g
will overestimate the cracked section moment

of inertia of sections, for which e/d
>,
0.5, either with
or without compression reinforcement. For a member
with only tension reinforcement typically located at d
= 0.90t, the actual cracked section moment of inertia
is overestimated by 35 percent, regardless of the
amount of reinforcement. For a member with equal
amounts of compression and tension reinforcement,
located at d’
=
0.1d and d
=

0.9t,
its actual cracked
section moment of inertia is also overestimated. The
overestimation will vary from 35 percent at the lower
reinforcement ratio
(Q
‘n
=

Qn

=
0.02) down to 15
percent at the higher values (
Q ‘n
=


gn

=
0.12).
The use of (6jk
2
)I
g
for cracked sections and the
use of Ig for uncracked sections are further discussed
relative to member cracking in Section 2.3.
Regarding the third assumption, the magnitude of
the thermal moment depends on the extent of section
cracking as reflected byI
cr
. I
cr
depends on the axial
force N and moment M. The relationship of I
cr
/I
g
ver-
sus
e/d,
where e =
M/N, is shown in Fig. 2.1. The
eccentricity e is referred to the section center line. In
Fig. 2.1 it is seen that for e/d 3 1, I

cr
is practically
the same as that corresponding to pure bending. For
e/d
2 0.5, the associated I
cr
is within 10 percent of its
pure bending value. Most nonprestressed frame prob-
lems are in the e/d
Z
0.5 category. Consequently, for
these problems it is accurate within 10 percent to use
the pure bending value of (6jk
2
)I
g
for I
cr
. This is the
basis of Assumption 3.
2.3
-
Member cracking
Ideally, a sophisticated analysis of a frame or slab
structure subjected to both mechanical and thermal
loads might consider concrete cracking and the re-
sulting changes in member properties at many stages
of the load application. Such an analysis would con-
sider the sequential application of the loads, and
cracking would be based on the modulus of rupture

of the concrete f
r

.
The loads would be applied in-
crementally to the structure. After each load in-
crement, the section properties would be revised for
those portions of the members which exhibit
extreme
fiber tensile stresses in excess of f
r
. The properties of
the members for a given load increment would reflect
the member cracking that had occurred under the sum
of all preceding load increments. In such an analysis,
the thermal moments would be a result of member
cracking occurring not only for mechanical loads, but
also for thermal loads.
The type of analysis summarized above is consistent
with the approach in Item 2 of Section A.3.3 of the
Commentary to Appendix A. An approximate analy-
sis, but one which is generally conservative for the
thermal loads, is suggested in Item 3 of Section A.3.3
as an alternative. This alternate analysis considers the
structure to be uncracked under the mechanical loads
and to be cracked under the thermal loads. The re-
sults of an analysis of the uncracked structure under
mechanical loads are combined with the results of an
analysis of the cracked structure under the thermal
loads. A simplified method of analysis is discussed be-

low which will yield cracked member thermal mo-
ments that are conservative for most practical prob-
lems.
The extent of cracking which the members expe-
rience under the total mechanical load (including the
specified load factors) forms the basis for the cracked
structure used for the thermal load analysis. Cracking
will occur wherever the mechanical load moments ex-
ceed the cracking moment M
cr
. The addition of ther-
mal moments which are the same sign as mechanical
moments will increase the extent of cracking along the
member length. Recognizing this, in many cases it is
conservative for design to consider the member to be
cracked wherever tensile stresses are produced by the
mechanical loads if these stresses would be increased
by the thermal loads. Any increase in the cracked
length due to the addition of the thermal loads is con-
servatively ignored, and an iterative solution is not re-
quired. However,
the addition of thermal moments
which are of opposite sign to the mechanical moments
that exceed
M
cr
may result in a final section which is
uncracked. Therefore, for simplicity, the member is
considered to be uncracked for the thermal load anal-
ysis wherever along its length the mechanical moments

and thermal moments are of opposite sign.
Two types of cracked members will result: (1)
end-
cracked, and (2) interior-cracked. The first type oc-
curs for cases where mechanical and thermal moments
are of like sign at the member ends. The second type
occurs where these moments are of like sign at the in-
terior of the member. Stiffness coefficients, carry-over
factors, and fixed-end thermal moments are developed
for these two types of members in Section 2.4. A
comprehensive design example is presented in Section
2.5.
The above simplification of considering the member
to be uncracked wherever the mechanical and thermal
moments are of opposite sign is conservative due to
the fact that the initial portion of a thermal load,
such as
^
_
T, will actually act on a section which may
be cracked under the mechanical loads. Consequently,
the fixed-end moment due to this part of
^
_
T will be
that due toa member completely cracked along its
length. Once the cracks close, the balance of
^
_
T will

act on an uncracked section. Consideration of this
two-phase aspect makes the problem more complex.
The conservative approach adopted herein removes
this complexity. However, some of the conservatism is
reduced by the use of I
g
for the uncracked section
(Assumption 2) rather than its actual uncracked sec-
tion stiffness, which would include reinforcement and
is substantially greater than I
g
for
Qn
2 0.06.
The fixed-end moments depend not only on the
cracked length L
T
but also on the location of the
cracked length a along the member. This can be seen
from a comparison of the results for an end-cracked
member and an interior-cracked member for the same
DESIGN
FOR THERMAL EFFECTS/NUCLEAR
PLANTS
349.1R-5
0.8
0.7
0.6
F
b

0‘
a-
t
4
0.5
%
z
E
s
5

t
s:
0.4
ii
c5
0
-r,
f
0.3
(5
-
0
0
._
6
aL
0.2
0.1
lllllllllllilt

Ill III
I
III

III
For e/d
+

00,
I
I
1

_____.
t
Icr
,
=

6jk2
I
kl
0
0
0.02
0.04 0.06 0.08 0.10
0.12
Tension Reinforcement,
pn
Fig. 2. 1- Effect of axial force on cracked section moment of inertia (No compression reinforcement)

349.1 R-6
ACI COMMITTEE REPORT
value of
Lr.
The method discussed in Section 2.4 ac-
counts for this. This approach is more applicable for
the determination of the thermal moments than the
use of an effective moment of inertia for the entire
member length. The concept of a single effective mo-
ment of inertia for purposes of member deflection cal-
culation has resulted in Eq. (9-4) of
ACI
349-76. This
equation is empirically based and, as such, accounts
for (1) partially cracked sections along the member,
and (2) the existence of uncracked sections occurring
between
flexural
cracks. These two characteristics are
indirectly provided for (to an unknown extent) by the
use of
(6jk2)1,,
which overestimates the cracked sec-
tion moment of inertia by the amount described pre-
viously.
2.4
-
Cracked member fixed-end moments,
stiffness coefficients, and carry-over factors
The thermal moments due to the linear temperature

gradient /
_
\ \ T, and those resulting from the expansion
or contraction of the axis of the member
T,,,

-

Tb,
are
considered separately. For each type of thermal load,
fixed-end moments, stiffness coefficients, and carry-
over factors were obtained for two types of cracked
members: (1) end-cracked, and (2) interior-cracked.
The first type applies for cases where mechanical and
thermal loads produce moments of like sign at
mem-
ber
ends.
The second type applies for cases where me-
chanical and thermal loads produce moments of like
sign in the interior of the member.
These factors are presented for the case of an
end-
cracked member in Fig. 2.2.
h4*
=
(2-1)
M,
=

Although shown only for a member cracked at the
ends, the above expressions for
MA
and
MR
also ap-
ply to a member cracked in its interior.
In the above:
aATL/2t
=
K.4
=
KS
=
co,,
=
co,, =
the angle change of the member
ends with the rotational restraints
removed
the stiffness of the member at A
with B fixed
(4EJ,/L
for uncracked
member)
the stiffness of the member at B
with A fixed
(4EJ,/‘L
for the
un-

cracked member)
the carry-over factor from A to B
(‘/z
for uncracked member)
the carry-over factor from B to A
(‘/z
for uncracked member)
The expressions for
K and CO can be derived from
moment-area principles. Also,
K can be expressed as:
K

=

-E+

k,
(2-2)
MA
LT-a

1
I
L

-

LT
I

I;
a
C-
-1
Fig. 2.2
-
/
_
\ T fixed-end moments
-
Member cracked
at ends by mechanical loads
.
I

-
,
Fig. 2.3
-
T,

-

T,
fixed-end moment
-
Member
cracked at ends by mechanical loads
where
k,

is the dimensionless stiffness coefficient
which is a function of
LJL
and a/LT. Likewise, CO
can be expressed as a function of
LJL
and a/L
T
.
Fig. 2.4 through 2.7 show
k,
and CO for selected
values of
LJL
and a/L
T
which should cover most
practical problems. In these figures, k, is given at the
end which is cracked a distance a, and CO is the
carry-over factor from this end to the opposite end.
Intermediate values of
k,
and CO can be determined
by linear interpolation of these curves.
For a member cracked a distance
Lr
in its interior,
k,
and CO are determined from Fig. 2.8 through 2.11.
k, is the stiffness coefficient at the end which is

un-
cracked a distance a. CO is the carry-over factor from
this end to the opposite end.
Based on the above discussion, the /
_
\ T fixed-end
moment at the
a end of the member can be expressed
as:
MFE
=
_
k
s
(1-
CO)
2
(2-3)
12
For the purpose of determining the mean temper-
ature effects, it is necessary to develop the
T,

-

Tb
fixed-end moment, which is shown in Fig. 2.3 for a
member cracked at its ends.
The
T,,,


-
Tb
fixed-end moment at the end cracked a
distance a is:
M,.

=
E&A
L2

k(l

+

CO)
where
k,
and CO are same as that defined above. The
displacement A is produced by
T,,,

-

Tb
acting on an
adjacent member. The comprehensive design example
of Section 2.5 illustrates this.
0.8
0.7


0.6
$
"0
0.5
IL
P
9
0.4
:
"
0.3
0.2
0.1
0
t-
i

i

i

1-t

i

i

i


j

I
LF?

d-i!
i

?-+-
'
d
0.02
0.04
0.06
0.08
0.10
0.12
Tension Reinforcement,
pn
Fig. 2.4
-
End-cracked beam, k
s
and CO for L
T
= 0.1L
0.8
0.7
0.6
g

i
0.5
IL
E
y
0.4
s
”
0.3
0.2
0.1
0.02 0.04
0.06
0.08
0.10 0.12
Tension Reinforcement,
pn
Fig. 2.5
-
End-cracked beam, k
s

and CO for L
T
= 0.2L
0.8
8

0.7
5

g
0.6
LL
5
0
0.5
2.
b
v
0.4
0.3
0.2
FEF
I
CRACKED ZONES,
i
i
I
! I

I
0.02 0.04
0.06
0.08
0.10
0.1
Tension Reinforcement,
pn
Fig.2.4
-

End-cracked beam, k
s
and CO for L
T
=
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.02
0.04
0.06
0.08
0.10

0.12
Tension Reinforcement,
pn
Fig. 2.7
-
End-cracked beam, k
s
and CO for L

T
= 0.6L
0.8
0.7
8
0.6
.
p

0.5
tttHtt

i
i
i
i
i
tttth
rt+i t
I
III.
II
1IIIII
Ihl.

1”“’
I II
1
I
0.02


0.04
0.06
0.08

0.10

0.12
Tension Reinforcement,
pn
Fig. 2.8
-
Interior-cracked beam, k
s
and CO for L
T
= 0.1L
349.1R-12
1.0
0.9
0.8
0.7
g
g
0.6
IL
E
9
0.5
f

6
v
0.4
0.3
0.2
0.1
0.02

0.04

0.06

0.08

0.10
0.12
Tension Reinforcement,
pn
Fig. 2.9
-
Interior-cracked beam, k
s
and CO for L
T
= 0.2L
349.1R 13
0.9
0.8
s
0.7

.
5
t
0.6
lE
ii
&
0.5
k
&
”
0.4
0.3
0.2
0.1
+-i-i c

!
!
:
f

4

f
f :
!
!

!

0
0.02
0.04 0.06
0.08 0.10
0.12
Tension Reinforcement,
pn
Fig. 2.10
-
Inferior-cracked beam, k
s
and CO for L
T
= 0.4L
0.8
s
0.7
L‘
e
IE
0.6
F
y0.5
L.
'0
"0.4
0.3
0.2
0.1
3

Al?
.
5
._
U
._
r
0
s
2
s:
0
c
-
._
s
1
0
:I

1
.
I
I I
i

1
II
11


I

I

ill1

I
I!
I
u’lll,

llllllllllllllll~
0.02

0.04

0.06

0.08

0.10 0.12
Tension Reinforcement,
p
Fig. 2.11
-
Interior-cracked beam, k
s
and CO for L
T
= 0.6L

DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS
349.1R-15
TOUTS~DE =
SOOF
137.4
W
D

+
L = 1086
LB/FT

MECHANICAL LOADS
-
MECHANICAL AND THERMAL
l
^
_

T =
&lOoF,

T,

-

Tb
=
WF,
UNCRACKED FRAME

Fig. 2.12
-
Uncracked
frame moments (ft-kips)
2.5
-
Frame design example
Given the continuous frame shown in Fig. 2.12 with
all members 1 ft wide x 2 ft thick and 3-in. cover on
the reinforcement. The load combination to be con-
sidered is U =
D + L+T
o
+ E
ss
.
The mechanical loading consists of:
W
D
= 406 lb/ft
W
L
= 680
lb/ft
on member
BC
and a lateral load of 3750 lb at Joint
C due to E
ss
.

The thermal loading T
o
consists of 130 F interior
and 50 F exterior on all members. The base temper-
ature T
b
is taken as 70 F. For this condition, T
m

-
T
b
= + 20 F and
^
_
T = 80 F (hot interior, cold exterior).
The material properties are
f,’
= 3000 psi and E
c

=
3.12
x
10
6
psi; f
y
=
60,000 psi and

E
s
= 29 x 10
6
psi;
and o( = 5 x 10
-6
in./in./deg
F. Also, n = E
s
/E
c

=
9.3.
The reinforcement in the frame consists of
2-#8
bars at each face in all members. This results in
Q

=
1.58/(12

x
21) = 0.0063 and
qn

=
9.3 (0.0063) =
0.059. The section capacity is

M
u
= K
u
F = (320)
x (12)(21)
2
/12,000 = 141.1 ft-kips.
Mechanical loads
An analysis of the uncracked frame results in the
member moments (ft-kips) below. Moments acting
counterclockwise on a member are denoted as posi-
tive. These values were obtained by moment distribu-
tion, and moments due to E
ss
include the effect of
frame sidesway.
AB:
-52.3
BA:
-76.0
BC:
+76.0
CB:
-46.0
CD:
+46.0
DC:
+7.5
These are shown in Fig. 2.12.

The maximum mechanical load moment of 76
ft-
kips is less than the section capacity of 141
.1
ft-kips.
Therefore, the frame is adequate for mechanical
loads.
Thermal loads (
^
_
T = 80 F and T
m
- T
b
= 20 F)
The
^
_
T =
80
F having hot interior and cold exte-
rior is expected to produce thermal stresses which are
tensile on the exterior faces of all members. These
stresses will add to the existing exterior face tensile
stresses due to the mechanical loads. Hence, the
L
T
and a values are arrived at from the mechanical load
moment diagram in Fig. 2.12.
Member


AB
AB
BC
BC
CD
CD
_
End
A
B
B
C
C
D

L
T
/L
a/L
T
11.8/20
= 0.59
0
0.59
1
(5.3 + 3.4)/30 = 0.29
5.3/8.7
=
0.61

0.29
3.4/8.7
=
0.39
17.2/20
= 0.86 1
0.86
0
All members are the end-cracked type. Fig. 2.5
through 2.7 are used to obtain the coefficients k
s
and
CO, which are shown in Table 2.5.1.
349.1R-16
MANUAL OF CONCRETE PRACTICE
Table 2.5.1
-
Cracked frame coefficients and thermal moments on members
Member
AB
AB
BC
:cD
CD
End
L*/L
a/Lt
k
co
kJL

0.59 0
0.41
0.17
0.59
:z
0.70 0.10
0.29
k6,

2'40
0.29
0.39
2:65
0.43
0.08
0.38 0.088
0.86
0.57 0.095
0.86
ii
i:;
0.47 0.120

DF
1.0
0.56
it:
0:52
1.0
*Corrected for sidesway

DF,

=
(k,,E,I,,/L,)
+
Zk,.E,I,,/L,
The expressions from Section 2.4 for fixed-end
moment (FEM) are evaluated below.
(1)
^
_

T FEM =
E,aA
Tbt
2
k
s
l2
j-
(1
-
CO)
Thermal
FEM’s,
ft-kips
^
_

T- 80

T,-T u)
Total
FEM
1
FEM
I
+60.1
-
17.97
+41.7
-49.2
+24.5
-37.9
-4.79
-3.39
:
+3.72
+4.38
FEM
Distributed
thermal
moments,*
ft-kips
Distributed
thermal
moments
&
mechanical
moments,
ft-kips

+55.31 +36.0
-
16.3
-21.36 -39.9 -115.9
+41.7
+39.9
+115.9
-49.2 -37.0
-83.0
+
28.2
+37.0
+83.0
-33.5
-33.2 -25.7
_
(3.12
x

10*)(5

x

10-“)(80)(12)(24)2
12
x+
(1
-
CO)
^

_
T
FEM
=
59.9k,
(1
-
CO)/2 ft-kips
(2) T
m

-
T
b
FEM =
$+
(A)&)(1
+
CO)
=
(3.12

x

io6)(24)3

(b)(k

)
(20

x
12)
2
s
x
(1 + CO)
T
m

-
T
b
FEM
= 62.4
(A)(k,)(l
+ CO) ft-kips
(3)
^
_
=
^
_
=
^
_
=
total unrestrained change of length of
member BC =
a(T,,,


-
T,)L
(5
x

10-6)(20)(30

x
12)
0.036 in.
Distribute 0.036 in. to Ends B and C of Members
AB and CD, respectively, in inverse proportion to
the shear stiffness at these ends.
Shear stiffness at B
= $+

[k&l
+ CO
A
)
+
ks,(l

+

COdI
=-
$
[3.4(1.41) + 2.00(1.70)]
=

s(8.19)
Shear stiffness at C
=
s
[k
sC
(l + CO
c
)
+
k,D(l
+
CWI
=
$
(6.48)
=
$+0(1.57)
+
2.38(1.47)]
Sum of shear stiffness at B and C
=
++
(14.67)
Ag

=
0.036 in.
(6.48/14.67)


=
0.036
x
0.44
= 0.016 in.
AC
=
0.036 in.
(8.19/14.67)

=
0.036
x
0.56
= 0.020 in.
To demonstrate the effect of cracking on the ther-
mal moments, the fixed-end thermal moments for the
uncracked frame are obtained from the final expres-
sions in (1) and (2) using k
s

=
4, CO =
1/2,
and
^
_

=


(%)o(T,

-
T
b
)L,
L
being the length of Member
BC. A moment distribution is performed, and the
resulting distributed moments are added to the
mechanical moments. The combined moments are
shown in Fig. 2.12 for purpose of comparison with
the cracked frame moments.
The fixed-end thermal moments for the cracked
frame are obtained using the above values for
^
_

B
and
AC
and by referring to Table 2.5.1 for k
s
and CO.
These fixed-end moments and the resulting
distributed thermal moments are given in Table
2.5.1. The distributed thermal moments include the
effect of sidesway, which occurs because the frame is
unsymmetrically cracked.
Combined loads

The final frame moments are shown in Table 2.5.1
and Fig. 2.13. These can be compared with Fig. 2.12
to see the effect of the cracking reduction of thermal
moments.
b
\
61.6
\
b\
16.3
52.3
25.7

MECHANICAL
(UNCRACKED
FRAME)
MECHANICAL AND THERMAL*
l
^
_

T
=
80
o
F,
T
m

-

T
b,
=
20
o
F,
CRACKED FRAME
Fig. 2.13
-
Final frame moments (ft-kips)
Although not shown, the member axial forces were
evaluated to confirm that section cracking still cor-
responds to the pure bending condition of Assump-
tion 3. Recall that e/d must be at least 0.5 for this
condition. For Members AB and CD, the axial forces
result primarily from the mechanical loads and are
compressive. For Member BC, the axial force is com-
pressive and includes the compression due to the
20 F increase on the member.
CHAPTER 3
-

AXISYMMETRIC
STRUCTURES
3.1
-
Scope
Axisymmetric structures include shells of revolution
such as shield buildings or, depending on the particu-
lar geometry, primary and secondary shield walls. In

the structural analysis, the structure is considered to
be uncracked for all mechanical loads and for part of
the thermal loads. The thermal load is assumed to be
represented by a temperature which is distributed lin-
early through the wall of the structure. The linear
temperature distribution is separated into a gradient AT
and into a uniform temperature change T
m
- T
b
.
Generally, for most axisymmetric structures, a uni-
form temperature change (T
m
- T
b
) produces signifi-
cant internal section forces (moment included) only at
the externally restrained boundaries of the structure
where free thermal growth is prevented, or in regions
where T
m
- T
b
varies fairly rapidly along the struc-
ture. The magnitude and extent of these discontinuity
forces depend on the specific geometry of the structure
and on the external restraint provided. If cracking oc-
curs in this region, a prediction of the cracking reduc-
tion of the discontinuity forces is attainable through a

re-analysis using cracked section structural properties.
A discussion of such an analysis is not within the
scope
of the present report. Therefore, forces re-
sulting from an analysis for the T
m
- T
b
part of the
thermal load are considered to be included with corre-
sponding factored mechanical forces. These combined
axial forces and moments are denoted as N and
M.
The gradient
^
_
T produces internal section forces
(moment included) at externally restrained boundaries
and, also, away from these discontinuities. At
dis-
continuities, the most significant internal force is usu-
ally the moment, primarily resulting from the internal
restraint rather than the external boundary restraint.
Away from discontinuities, the only significant forces
due to
^
_
T are thermal moments caused by the internal
restraint provided by the axisymmetric geometry of
the structure. The cracking reduction of thermal mo-

ments which result from internal restraint is the sub-
ject of this chapter.
Due to the axisymmetric geometry of the subject
structures, the free thermal curvature change
aAT/t is
fully restrained. This restraint produces a correspond-
ing thermal moment whose magnitude depends on the
extent of cracking the section experiences. This in turn
depends on
^
_
T, the other section forces N and M, and
the section properties. With the ratio M/N denoted as
e, referenced to the section center line, and the dis-
tance from the concrete compression face to the ten-
sion reinforcement denoted as d, two cases of e/d are
identified in Sections 3.2 and 3.3.
349.1R-18
MANUAL OF CONCRETE
PRACTICE
The results in Sections 3.2 and 3.3 include the effect
of compression reinforcement. For this reinforcement,
a modular ratio of 2n is used for purposes of sim-
plifying the determination of the cracked section ther-
mal moment. Although not all the loads which com-
prise the section forces N and
M
will necessarily be
long-term, the selection of 2n for compression re-
inforcement is consistent with design practice.

The results in Sections 3.2 and 3.3 are based on a
linear stress-strain relationship for the compressive
concrete. The basis of this assumption was discussed
in Section 2.2. From this discussion, the cracked sec-
tion thermal moments can be considered to represent
upper-bound values when compared with those which
would result from a nonlinear stress-strain concrete
relationship. Nevertheless,
the thermal moments
herein do offer a reduction from their uncracked val-
ues. The extent of this reduction is shown in Fig. 3.2
and Fig. 3.4 through 3.9.
3.2
-

le/c/l

2
0.7 for compressive N and ten-
sile N
For this range of e/d, a parametric study based on
the results of Section 3.3 indicates that the cracked
thermal moment
MAT
is not strongly influenced by the
axial force as expressed by the ratio
N/(bdE,aAT’).
A
practical range of N/(bdE,aAT) from 0 to
2300

was
used in this study. Therefore, for ranges of e/d and
N/bdE,aAT specified herein,
MbT
can be calculated
from the neutral axis location corresponding to N
=
0.
The
le/dl
lower limit of 0.70 is conservative for ten-
sile N and higher
QII
values. Actual
le/d
lower limits
for tensile N are given in Fig. 3.3. As long as the ac-
tual
le/dl
value for tensile N exceeds this lower limit
curve, the thermal moments given in this section are
applicable.
For doubly reinforced rectangular sections, the
cracked section neutral axis is kd. For N = 0:
k
=

v(2q’n
+
Qn)’

+
2[2Q’n

(d//d)
+
on]
-
(2&n +
Qn)
(3-1)
in which Q'
=

A,il6d,

Q

=

AJbd,
and n =
Es/E,.
Also, d’ is the distance from the concrete compres-
sion face to the compression reinforcement A
s
'. A
modular ratio of 2n is used for the compression rein-
forcement.
The corresponding thermal moment for a section
in which

t

=(
l.l)d is:
aATbd2
MAT

=

E.1
{-0.152k

J
+
1.818Q’n[(d’/d)
-

k](d’/d)
+
0.909Qn(l

-
k)}
(3-2)
The expression for
M,,
given by Eq. (3-2) is ob-
tained from the results of Section 3.2 in the follow-
ing manner. For sections in which
le/dj


2
0.7, the
location of the neutral axis does not change under
the application of
^
_
T, and this results in k
L
. = k.
This substitution for k
L
is made in Eq. (3-7) and Eq.
(3-9). The resulting expression for
f
c
given by Eq.
(3-7) is used in Eq. (3-11). Then
Mbf
is obtained by
subtraction of Eq. (3-9) from Eq. (3-11).
For singly reinforced rectangular sections and
N = 0,
k
=
\/
(on)’ +
2Qn

-


Qn
and the corresponding thermal moment is
MbT
=
E,aA
Tbd
J(jk

‘)
2t(l
-

v)
(3-3)
(3-4)
where
j
= 1
-
k/3.
Eq. (3-2) for
f&r
for a doubly reinforced section
reduces to
Mbf
for a singly reinforced section (Eq.
[3-4]),
with the substitution of (l.l)d for
t

in Eq.
(3-4) and 0 for
Q’n
in Eq. (3-2). In addition, the
substitution of k
2
/2
for
Qn
(1
-
k) in Eq. (3-2) must
be made.
The thermal moments given by Eq. (3-2) and (3-4)
are presented in Fig.3.2 for the special case:
t = (l.l)d for both sections and
g’n
=
gn
and
d’/d
= 0.10 for the doubly reinforced section. For
values of
Q’
less than Q, linear interpolation between
the two curves should yield sufficiently accurate
results. From Fig. 3.2, it is seen that the cracked sec-
tion thermal moment is substantially reduced from its
uncracked value.
The thermal moment

Mhr
occurs at the center line
of the section.
MdT
should be multiplied by its code
specified load factor before it is added to the mo-
ment M.
3.3
-
General e/d
Depending on
e/d, the extent of section cracking
and the thermal moment may be significantly af-
fected by the actual values of N and M. A theory
for the investigation of a doubly reinforced rec-
tangular section is presented below. The axisymmetric
effect increases the section thermal moment due to
^
_
T by an amount l/(1
-

v).
Although this effect is
not shown in the derivations below, it is included in
the final results, Fig. 3.2 and 3.4 through 3.9.
It is assumed for the section that the final cur-
vature change
0
is equal to the curvature change due

to N and M,
4,.,
plus the curvature change required
to return the free thermal curvature
4T
to 0.
oL
and
Or
are additive when the cold face of the section cor-
responds to the tension face under M. Therefore,
4

=

4L

+

4T
(3-5)
The curvatures before and after the application
4T
are shown in Fig. 3.1.
In Fig. 3.1:
E
cL
=
concrete strain at compression face due
N and

M
of
to
DESIGN
FOR
THERMAL EFFECTS/NUCLEAR
PLANTS
L
CT
=
cc

=
k
L
d
=
kd
=
c
Fig. 3.1
-
Section under M, N,
^
_
T
concrete strain is compression face due to
^
_
T

total concrete strain
neutral axis location on section due to N
and M
neutral axis location on section due to N,
M, and
^
_
T
The thermal curvature change
+T
is
97
=
aA
T/t
where
^
_
T is always taken as positive.
Using this and
cc
=
+kd
and
c,,.
=
+,.kLd
in Eq.
(3-5) gives
E,/kd

=
&k,.d

+

aAT/t
(3-6)
For the case where the concrete stress is a linear
function of strain, Eq. (3-6) becomes
f,/(E,kd) =
f,,./(E,k,.d)
+
aAT/t
or
f, =
(f,,./(E,k,.d)
+
aAT/t]

E,kd
(3-7)
To maintain equilibrium of the section both before
and after the application of
^
_
T, the following condi-
tions occur:
Before
^
_

T
1. The internal axial force N is equal to the resul-
tant of the stresses produced by N and
M.
N =
1//2f,,.bk,.d
+ 2~‘nbdf,,[(k,.
-

d’/d)/k,.]
+
gnbdf,,.[(kL

-

I)/kLl
(3-8)
2. The internal moment M is equal to the internal
moment of the stresses (about the section center
produced by N and
M.
M=
‘/zfCLbk,.d(t/2

-

k,.d/3)
+
2Q’nbdfCL
x

[(k
L

-

d’/d)/kJ
[(t/2)
-
d’] +
Qnbdf,,_
x

[(1

-
k,.)/k,,l[d
-

(t/2)]
line)
(3-9)
After
^
_
T
1. The internal axial force N is equal to the resul-
tant of the stresses produced by N,
M, and
^
_

T.
N=
%f,bkd + 2Q’nbdf,[(k
-
d’/d)/k]
+
Qnbdf,[(k

-

1)/k]
(3-10)
2. There exists an internal center line moment
m
of the stresses produced by N, M, and
^
_
T.
IW
= %f,bkd[(t/2)
-

(kd/3)]
+ 2q’nbdf,
x
[(k
-
d’/d)/k] [(t/2)
-
d’] +

Qnbdf,
x
[(l
-

k)/k]

[d

-
(t/2)] (3-l1)
3. The internal thermal moment
M,,
at the section
center line is equal to
M

-
M.
In
Eq. (3-8) through (3-11) the tension and com-
pression reinforcement have been expressed as
Q
=

A,/bd
and Q’ =
A,Ybd,
respectively. A modular
ratio of 2n has been used for the compression rein-

forcement. Also, the reinforcement stresses have been
expressed in terms of the concrete stress.
From Eq.
(3-8),
f
cL
can be expressed in terms of
N,
k
L
, and the section properties. Use of this in
Eq. (3-7) allows f
c
to be written in terms of N, k
L
,
k,
E,aAT,
and the section properties. Substitution of
this expression for fc into Eq. (3-10) results in a
quadratic equation in k which is solved in terms of
the section properties,
k
L
, and the quantity
N/bdE,aAT.
However, by dividing Eq. (3-9) by Eq.
(3-8),

kL

can be written in terms of the section prop-
erties and e, where e = M/N. Thus, k is deter-
mined for a specified section e and
N/bdE,aAT.
The
above results also allow
Mdf
to be determined from
these specified quantities.
The equilibrium equations, appearing as Eq. (3-8)
through (3-ll), are based on a triangular concrete
stress distribution. The two extremes of the stress
distribution are at k
L
= 0.10 and k
L
= 1.0. The
range 1.0
2
k,.
2 0.10 should cover many practical
situations not involving prestressed sections. For k
L
0.06
f
c’
E*
0.04
s
5

E
er
f
0.03
0.02
e

=
M/N
n
=

E
s
/E
c
Uncracked:
btr)

MAP

=
0.101
bd’E,cyAT
l
(Based on gross concrete section)
0.08
0.10

0.12

Tension Reinforcement,
pn
Fig. 3.2 -
Cracked section thermal moment for
le/d

1

3
0.70
0.80
Pbd
e

=
M/N
_

Ic
N (positive
as
shown)
-

.
CUseFigures
.A_.

.
._L

. .
-
3.4 thru 3.9
I.

:

:

1

I
!
:
i
1

I

I

I

I

III

I
I II
11’

I I
Iiiiiiiiiiiiiiiiilr

i-iiIiTii

il;.:iili”’
I

II
P-
“““jii:‘iiiiijijiii
.
.I

! I

I
0.06
Reinforcement,
pn
Fig.
3.3
-
e/d
limits
349.1R-22
MANUAL OF CONCRETE PRACTICE
DESIGN FOR THERMAL EFFECTS/NUCLEAR
PLANTS
f

d
c
C
d
‘h

M-1)
‘lN3WOW

lVWkl3Hl
249.1R-24
MANUAL OF
CONCRETE
PRACTICE
Y
3
E
8
DESIGN
FOR THERMAL EFFECTS/NUCLEAR PLANTS
0
ci