ACI 349.1 R-91 (Reapproved 2000)
Reinforced Concrete Design for Thermal Effects on Nuclear Power Plant Structures Reported by ACI Committee 349 For a list of Committee members, see page 30.
NOTATION
This report presents a design-oriented approach for considering thermal loads on reinforced concrete structures. 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 axisymmetric shells, are addressed. For frame frame structures, structures, a rationale is described described for determining determining the extent of member cracking which can be assumed for purposes of obtaining the cracked structure structure thermal forces and moments. Stiffness coefficients and carryand 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 expressed in terms of these factors for 1) a tem perature perature gradient gradient across the depth of the member, and 2) end displacements displacements due to a uniform temperature change along the axes of adjacent members. For the axisymmetric shells, axisymmetric shells, normalized cracked section thermal moments arepresented in graphical form. These moments are normalized with respect to the cross section dimensions and the temperature gradient across the section. 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 ill ustrated by examples.
General
A s '
d d ’
e
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, pg. 349.1R-1 Chapter l-Introduction, pg. 349.1R-2
n t T m T b
Chapter 2-Frame structures, pg. 349.1R-3 2.l-Scope 2.2-Section cracking 2.3-Member cracking 2.4-Cracked member fixed-end moments, stiffness factors, and carry-over factors 2.5-Frame design example
o( V
Q
Chapter 3-Axisymmetric structures, pg. 349.1R-17 3.l-Scope 3.20.7 for compressive N and tensile M 3.3-General e/d 3.4-Design examples Chapter 4-References, pg. 349.1R-29 4.l-Recommended 4.l-Recommended references 4.2-Cited references are intended for guidance in designing planning, executing, or inspecting shall not be made in the Project Documents. If items found in these documents are desired to be part of the Project Documents, they should be phrased in mandatory language and incorporated into the Project Doc-
area of tension reinforcement within width area of compression reinforcement within width width of rectangular cross section distance from extreme fiber of compression face to centroid of tension reinforcement distance from extreme fiber of compression face to centroid of compression reinforcement eccentricity of internal force N on the rectangular section, measured from the section center line modulus of elasticity of concrete modulus of elasticity of reinforcing steel specified compressive strength of concrete specified yield strength of reinforcing steel ratio of the distance between the centroid of compression and centroid of tension reinforcement to the depth d modular ratio = E s /E c thickness of rectangular section mean temperature, deg F base (stress-free) temperature, temperature, deg F linear temperature gradient, deg F concrete coefficient of thermal expansion, in./in./deg F Poisson’s ratio of concrete ratio of tension reinforcement = ratio of compression reinforcement =
Chapter 2 a
- Frame structures
= the length of the cracked end of member at which the stiffness coefficient and carry-
ACI 349.1R-91 replaces ACI 349.1R-80, Reapproved 1986 effective July 1,199l. 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 1980, American Concrete Institute. Al l 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 any electronic or mechanical device, printed or 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. l
= = = = =
=
= = = = = = = = = = = =
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, beam , (Fig. 2.8 through thro ugh 2.1l), a is the length of the uncracked end of member at which the stiffness coefficient and carryover 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 section uncracked section moment of inertia (excluding 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 2f 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
M
N
= internal moment at section center line due to factored mechanical loads, including factored moment due to T m - T b = internal axial force at section center line due to factored mechanical loads, including factored axial force due to T m - T b = final internal moment at section center line resulting from M and _ ^ _T M , = = thermal moment due to ^ T final cracked section strain at extreme fiber of compression face = + = 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 to 0 = final cracked section curvature change = + -
=
CHAPTER 1- INTRODUCTION AC I 349, Appendix A, provides general considerations 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 “mechanical loads.” One approach is to consider the structure uncracked under the mechanical loads and cracked under the thermal loads. The results of two such analyses 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 temperature distribution, which can generally be nonlinear. Then the linear temperature distribution is separated into a pure gradient _ ^T and into the difference between the mean and base (stress-free) temperatures T m - T b .
Chapter 3 f c
f CL
k
k L
-
Axisymmetric structures
= final cracked section extreme fiber com-
pressive stress resulting from internal sec , N , and _ tion forces M ^T cracked section extreme fiber compressive stress resulting from internal forces M and N = ratio of depth of the triangular com , r e pressive stress block to the depth d M , N , section forces sulting from internal and _ ^T = ratio of depth of the triangular com , r e pressive stress block to the depth d an d sulting from internal section forces M and =
This report offers a specific approach for considering thermal load effects which is consistent with the above provisions. The aim herein is to present a designer-oriented approach for determining the reduced thermal moments which result from cracking of the concrete structure. Chapter 2 addresses frame structures, and Chapter 3 deals with axisymmetric structures. 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. 2.4. Cracked member fixed-end moments, stiffness coefficients, and carry-over factors are derived and presented in graphical form. For axisymmetric structures an approach is described for regions away from discontinuities, and graphs of
= = = = =
=
= = = = = = = = = = = =
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, beam , (Fig. 2.8 through thro ugh 2.1l), a is the length of the uncracked end of member at which the stiffness coefficient and carryover 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 section uncracked section moment of inertia (excluding 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 2f 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
M
N
= internal moment at section center line due to factored mechanical loads, including factored moment due to T m - T b = internal axial force at section center line due to factored mechanical loads, including factored axial force due to T m - T b = final internal moment at section center line resulting from M and _ ^ _T M , = = thermal moment due to ^ T final cracked section strain at extreme fiber of compression face = + = 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 to 0 = final cracked section curvature change = + -
=
CHAPTER 1- INTRODUCTION AC I 349, Appendix A, provides general considerations 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 “mechanical loads.” One approach is to consider the structure uncracked under the mechanical loads and cracked under the thermal loads. The results of two such analyses 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 temperature distribution, which can generally be nonlinear. Then the linear temperature distribution is separated into a pure gradient _ ^T and into the difference between the mean and base (stress-free) temperatures T m - T b .
Chapter 3 f c
f CL
k
k L
-
Axisymmetric structures
= final cracked section extreme fiber com-
pressive stress resulting from internal sec , N , and _ tion forces M ^T cracked section extreme fiber compressive stress resulting from internal forces M and N = ratio of depth of the triangular com , r e pressive stress block to the depth d M , N , section forces sulting from internal and _ ^T = ratio of depth of the triangular com , r e pressive stress block to the depth d an d sulting from internal section forces M and =
This report offers a specific approach for considering thermal load effects which is consistent with the above provisions. The aim herein is to present a designer-oriented approach for determining the reduced thermal moments which result from cracking of the concrete structure. Chapter 2 addresses frame structures, and Chapter 3 deals with axisymmetric structures. 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. 2.4. Cracked member fixed-end moments, stiffness coefficients, and carry-over factors are derived and presented in graphical form. For axisymmetric structures an approach is described for regions away from discontinuities, and graphs of
DESlGN FOR THERMAL EFFECTS/NUCLEAR PLANTS
This report is not intended to represent a state-ofthe-art discussion of the methods available to analyze structures for thermal loads. Rather, the report is intended to propose simplifications that can be made which will permit a cracking reduction of thermal moments to be readily achieved for a large class of thermal loads, without resorting to sophisticated and com plex solutions. Also, as a result of the report discussion, the design examples, and graphical presentation of cracked section thermal moments, it is hoped that a designer will better understand how thermal 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 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 rotation, such that the free thermal curvature change of is not completely restrained. The thermal moments 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 automatically considered in a structural analysis using uncracked 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 cracking reduction of membe r thermal moments. These criteria 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 criteria 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 permit 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 reasonableness 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
-
349.1R-3
factors. These depend on the extent of member cracking along its length due to mechanical loads, as discussed in Section 2.3. 2.2 Section cracking Simplifying assumptions are made below for the purpose of obtaining obtaining the cracked section thermal moments and the section (cracked and uncracked) stiffnesses. 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 linearly proportional to strain over the member cross section. 2. For an uncracked section, the moment of inertia is I is I g , where I g is based on the gross concrete dimensions 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 of I cr , the com pression reinforcement is excluded excluded and the tension reinforcement 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. section. The first assumption is strictly valid only if the extreme fiber concrete compressive stress due to com bined mechanical mechanical and thermal loads does not exceed ' 0.5f ' At this stress, the corresponding concrete strain c. 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 moment based on the linear assumption adopted herein versus a nonlinear concrete stress-strain relationship such as that described in References 2 and 3. Consequently, cracked member thermal moments given by Eq. (2-3) and (2-4) are sufficiently accurate for concrete 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 moments which are greater than those based on the nonlinear nonlinear theory. In this regard, the thermal moments are conservative. However, they are still reduced from their uncracked values. This cracking reduction of thermal moments can be substantial, as seen in Fig. 3.2 which also incorporates Assumption 1. Formulation of the thermal moments based on a linear concrete stress-strain relationship allows the thermal moments to be expressed simply by the equations in Chapter 2 or by the normalized thermal moment 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
-
load is small as specified in Assumption 3. The use of (6 jk 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 ‘n = = 0.02) down to 15 percent at the higher values ( Q ‘n = = 0.12). The use of (6 jk 2) I g for cracked sections and the use of I g 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 by I cr . I cr depends on the axial force N and moment M . The relationship of I cr /I g versus 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 1, I cr is practically the same as that corresponding to pure bending. For e/d 0.5, the associated I cr is within 10 percent of its pure bending value. Most nonprestressed frame problems are in the e/d 0.5 category. Consequently, for these problems it is accurate within 10 percent to use the pure bending value of (6 jk 2) I g for I cr . This is the basis of Assumption 3.
Member cracking 2.3 Ideally, a sophisticated analysis of a frame or slab structure subjected to both mechanical and thermal loads might consider concrete cracking and the resulting changes in member properties at many stages of the load application. Such an analysis would consider 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 incrementally to the structure. After each load increment, 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 analysis, 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-
-
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 below which will yield cracked member thermal moments that are conservative for most practical problems. The extent of cracking which the members experience 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 exceed the cracking moment M cr . The addition of thermal 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 conservatively ignored, and an iterative solution is not required. 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 analysis wherever along its length the mechanical moments and thermal moments are of opposite sign. Two types of cracked members will result: (1) endcracked, and (2) interior-cracked. The first type occurs 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 interior 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 to a 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 section stiffness, which would include reinforcement and is substantially greater than I g for 0.06. The fixed-end moments depend not only on the cracked length LT 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
DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS
349.1R-5
0.8
0.7
0.6
a-
t
0. 5
0.4
0
0. 3 0 0 6
0.2 For e/d Icr 0.1
0
0
0.02
0.04
0.06
0.08
0.10
Tension Reinforcement,
Fig. 2. 1 - Effect of axial force on cracked section moment of inertia (No compression reinforcement)
0.12
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
Tension Reinforcement,
Fig. 2.4
- End-cracked beam, k s and CO for LT = 0.1 L
0.10
0.12
0. 8
0. 7
0.6
0. 5
0. 4
0. 3
0. 2
0.1
0.02
0.04
0.06
0.08
0.10
Tension Reinforcement, pn
Fig. 2.5
- End-cracked beam, k s and CO for LT = 0.2 L
0.12
0.8
0.7
0.6
0.5
0.4 I 0.3
0.2
CRACKED ZONES,
1.0 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.8
0.7 i
0.6
i
i
0.5
I
I
I I I
0.02
0.04
I
I
0.08
0.06
0.10
Tensi on Rei nforcement ,
Fig. 2.8
- Interior-cracked beam, k and CO for L s
T =
0.1L
0.12
349.1R-12
1.0
0. 9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.02
0.04
0.06
0.08
0.10
0.12
349.1R--13
0.9
0.8
0.7
. 0.6
0.5
0.4
0.3
0.2
0.1
0.02
0.04
0.06
0.08
0.10
Tension Reinforcement,
Fig. 2.10
Inferior-cracked beam, k and CO for L = 0.4L
0.12
0.8
0.7 II
I!
I
0.6
0.5
0.4
0.3
0.2
0.1
3
.
U
2
1
0
0.02
0.04
0.06
0.08
Tension Reinforcement,
0.10
0.12
349.1R-15
DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS 137.4
= WD
+ L = 1086 L B / F T
---
-
MECHANICAL LOADS MECHANICAL l _ ^T
=
2.5
=
-
Fig. 2.12
AND
-
THERMAL UNCRACKED FRAME
Uncracked frame moments (ft-kips)
- 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 considered is U = D + L + T o + E ss. The mechanical loading consists of: W D = 406 lb/ft WL = 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 temperature 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 = 3000 psi and E c = 6 6 3.12 x 10 psi; f y = 60,000 psi and E s = 29 x 10 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 = 1.58/(12 x 21) = 0.0063 and = 9.3 (0.0063) = 0.059. The section capacity is M u = K u F = (320) 2 x (12)(21) /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 positive. These values were obtained by moment distribution, and moments due to E ss include the effect of
AB: BA: BC: CB: CD: DC:
-52.3 -76.0 +76.0 -46.0 +46.0 +7.5
These are shown in Fig. 2.12. The maximum mechanical load moment of 76 ftkips is less than the section capacity of 141 .1 ft-kips. Therefore, the frame is adequate for mechanical loads.
_ T = 80 F and T m - T b = 20 F) Thermal loads ( ^
The _ ^T = 80 F having hot interior and cold exterior 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 LT and a values are arrived at from the mechanical load moment diagram in Fig. 2.12. a/ LT LT /L -Member - End 0 A 11.8/20 = 0.59 AB 1 0.59 AB B B (5.3 + 3.4)/30 = 0.29 5.3/8.7 0.61 BC 3.4/8.7 = 0.39 BC C 0.29 17.2/20 = 0.86 1 C CD 0 0.86 CD -D=
All members are the end-cracked type. Fig. 2.5 through 2.7 are used to obtain the coefficients k s and
61.6
52.3
16.3 M E C H A N I C A L
--l
(UNCRACKED
F R A M E )
MECHANICAL AND THERMAL*
_ T= 8 0 F , Tm - Tb = 20 o F, ^ o
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 corresponds to the pure bending condition of Assumption 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 3.1
25.7
- AXISYMMETRIC
STRUCTURES
- Scope
Axisymmetric structures include shells of revolution such as shield buildings or, depending on the particular 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 linearly 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 uniform temperature change (T m - T b) produces significant 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 structure. The magnitude and extent of these discontinuity forces depend on the specific geometry of the structure
curs in this region, a prediction of the cracking reduction 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 resulting from an analysis for the T m - T b part of the thermal load are considered to be included with corresponding 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 discontinuities, the most significant internal force is usually 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 moments 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 is fully restrained. This restraint produces a corresponding thermal moment whose magnitude depends on the extent of cracking the section experiences. This in turn , the other section forces N and M , and depends on _ ^ T the section properties. With the ratio M/N denoted as e, referenced to the section center line, and the distance from the concrete compression face to the tension reinforcement denoted as d, two cases of e/d are
0.06 e = M/N n = Es /Ec
0.04
er 0.03
=
Uncracked:
0.02
0.101 l
(Based on gross concrete section)
0.08
0.10
Tension Reinforcement,
Fig. 3.2 - Cracked section thermal moment for
0.70
0.12
0.80 N (positive as shown)
M/N
3.4 thru 3.9
0.06 Reinforcement,
F ig.
3.3
- e/d limits
. .
!
i
I II
I I
349.1R-22
MANUAL OF CONCRETE PRACTICE
DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS
249.1R-24
MANUAL OF CONCRETE PRACTICE
DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS
0
349.1R-26
MANUAL OF CONCRETE PRACTICE
349.1R-27
DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS
0
349.1R-29
DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS
e/d = -24/32.7 = - 0 . 7 3 3 ,
= 0.733 > 0.7;
therefore, use Fig. 3.2. From Fig. 3.2 with
= 0.050 and
(1 -
Q ’
=
curve,
= 0.032
Concrete and rebar stress are calculated from a cracked section investigation with N = 100 kips compression and M = 195.3 ft-kips at section center line. Exampl e 4
- Tensil e N
e = 100 ft-kips/(-60 kips) = -1.67 ft = -20 in., = 0.612 e/d = -20/32.7 = -0.612 or
= 75.3 ft-kips = 75.3 ft-kips Concrete and rebar stresses are calculated from a cracked section investigation with N = 50 kips tension and M = 175.3 ft-kips at section center line Exampl e 3
- Compressive
From Fig. 3.3 for = 0.05, the lower limit on is 0.575 for the tensile N case. Since 0.612 > 0.575, use Fig. 3.2. From Fig. 3.2, with (bd = 0.032
< 0.70
N an d
75.3 ft-kips
N = 100 kips compression,
= 75.3 ft-kips
M = 100 ft-kips,
_ T = 80 F, ^
Concrete and rebar stresses are calculated from a cracked section investigation with N = 60 kips tension and M = 175.3 ft-kips at section center line.
1 ft, = 12 in.
e/d = 12/32.7 = 0.367 < 0.70; therefore, don’t use
CHAPTER 4-REFERENCES
Fig. 3.2. = 0.05, read lower limit on From Fig. 3.3 for as 0.25. Since 0.367 > 0.25, use Fig. 3.5 and 3.6. T) = 100,000/(12 x
For
5.5
x
x
x
32.7
x
4
x
80) = 0.145
= 0.145 and e/d = 0.367, find By interpolation between e/d ’ s: = 0.04 (Fig. 3.5)
e/d = 0.367; 0.035
by interpolation = 0.06 (Fig. 3.6) by interpolation For
e/d = 0.367; 0.046
= 0.05, ( 1 bd T = + 0.046) = 0.0405 = = 95.3 ft-kips
= 95.3 ft-kips
= 0.05, read (1 -
= 0.032(1.25)(
Same section as Example 2
100 ft-kips 100 kips
< 0.70
Same as Example 3 except N = 60 kips tension
= (1.25)(
e=
and
4.1-Recommended references The document of the standards-producing organi-
zation referred to in this document is listed below with its serial designation. American Concrete Institute 349 Code Requirements for Nuclear Safety Related
Concrete Structures and Commentary
4.2-Cited
references 1. Gurfinkel, G., "Thermal Effects in Walls of Nuclear Containments-Elastic and Inelastic Behavior,” Proceedings, First International Conference on Structural Mechanics in Reactor Technology (Berlin, 1971), Commission of the European Communities, Brussels, 1972, V. 5-J, pp. 277-297. 2. Kohli, T., and Gurbuz, O., “Optimum Design of Reinforced Concrete for Nuclear Containments, Including Thermal Effects,” Proceedings, Second ASCE Specialty Conference on Structural Design of Nuclear Plant Facilities (New Orleans, 1975), American Society of Civil Engineers, New York, 1976, V. 1-B. pp. 1292-1319.
