ACI 349.1R-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 p. 30.

2.5—Frame design example

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 temperature, such as expansion, contraction, and thermal restraints are not specifically 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 axisymmetric 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 gradient 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 presented 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 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 factors, and carryover factors ACI Committee Reports, Guides, Standard Practices, and Commentaries are intended for guidance in planning, designing, executing, and inspecting construction. This document 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 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.

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 As = area of tension reinforcement within width b As′ = 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 Ec = modulus of elasticity of concrete Es = modulus of elasticity of reinforcing steel fc′ = specified compressive strength of concrete fy = specified yield strength of reinforcing steel j = ratio of the distance between the centroid of compression and centroid of tension reinforcement to the depth d n = modular ratio = Es /Ec t = thickness of rectangular section Tm = mean temperature, F Tb = 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 = As /bd = ratio of compression reinforcement = As′ /bd Chapter 2—Frame structures a = length of the cracked end of member at which the stiffness coefficient and carryACI 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 reproduction or for use in any knowledge or retrieval system or device, unless permission in writing is obtained from the copyright proprietors.

349.1R-1

= = = = = = = = = = = = = = = = = =

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 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 compressive 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/EcIg total length of member cracked length of member cracking moment = bt2fr/6 cracked member fixed-end moment due to ^_T or Tm - Tb, 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 Tm - Tb acting on adjoining members

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 to Tm - Tb = internal axial force at section center line due to factored mechanical loads, including factored axial force due to Tm - Tb = final internal moment at section center line _T resulting from M and ^ _ T, M,= = A?f - M = thermal moment due to ^ = 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 _T compression face resulting from ^ = 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 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 _T and into the difference beinto a pure gradient ^ tween the mean and base (stress-free) temperatures Tm - Tb .

Chapter 3 - Axisymmetric structures fc

= final cracked section extreme fiber com-

fCL

=

k

kL

pressive stress resulting from internal sec_T tion forces M, N, and ^ cracked section extreme fiber compressive stress resulting from internal forces M and N = ratio of depth of the triangular compressive stress block to the depth d, r e sulting from internal section forces M, N, _T and ^ = ratio of depth of the triangular compressive stress block to the depth d, r e sulting from internal section forces M and N

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. 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 cracked section thermal moments are presented.

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 complex 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 _T and into a temcan be separated into a gradient ^ perature change with respect to a base (stress-free) temperature Tm - Tb. 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 aAT/f 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 member 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 applicable for this work. This design approach uses cracked member stiffness coefficients and carry-over

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 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 Ig, where Ig is based on the gross concrete dimensions and the reinforcement is excluded. For a cracked section, the moment of inertia is Icr, where Icr is referenced to the centroidal axis of the cracked section. In the formulation of Icr, the compression reinforcement is 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. The first assumption is strictly valid only if the extreme fiber concrete compressive stress due to combined 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 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 member 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 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 Icr, in Assumption 2, the assumptions for the compression and tension reinforcement result in the simple expression of (6jk2)Ig for Icr, if the axial

load is small as specified in Assumption 3. The use of (6jk2)Ig 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 (6jk2)Ig 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 by Icr. Icr depends on the axial force N and moment M. The relationship of Icr/Ig 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 3 1, Icr is practically the same as that corresponding to pure bending. For e/d 2 0.5, the associated Icr is within 10 percent of its pure bending value. Most nonprestressed frame problems 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 (6jk2)Ig for Icr. 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 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 fr . 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 fr. 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 results 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 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 Mcr. 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 Mcr 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, _T, will actually act on a section which may such as ^ be cracked under the mechanical loads. Consequently, _T will be the fixed-end moment due to this part of ^ that due to a member completely cracked along its _T will length. Once the cracks close, the balance of ^ 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 Ig for the uncracked section (Assumption 2) rather than its actual uncracked section stiffness, which would include reinforcement and is substantially greater than Ig for Qn 2 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 member and an interior-cracked member for the same

349.1R-5

DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS

0.8

0.7

0.6 F b a-0‘ t 4 % z E s .-5 t

s: ii c5

lllllllllllilt

0.5

Ill

III I

III III

0.4

0 -r, f

0.3

(5 -

0 0 ._ 6 aL

0.2 For e/d + 00, I

I

1 _____. t 0.1

Icr

,

--=

I

kl

6jk2

0

0

0.02

0.04

0.06

0.08

0.10

Tension Reinforcement, pn

Fig. 2. 1 - Effect of axial force on cracked section moment of inertia (No compression reinforcement)

0.12

ACI COMMITTEE REPORT

349.1 R-6

value of Lr. The method discussed in Section 2.4 accounts 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 moment of inertia for purposes of member deflection calculation 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 section moment of inertia by the amount described previously. 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 carryover 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 member ends. The second type applies for cases where mechanical and thermal loads produce moments of like sign in the interior of the member. These factors are presented for the case of an endcracked 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 apply to a member cracked in its interior. In the above: aATL/2t = the angle change of the member ends with the rotational restraints removed = the stiffness of the member at A K.4 with B fixed (4EJ,/L for uncracked member) = the stiffness of the member at B KS with A fixed (4EJ,/‘L for the uncracked member) co,, = the carry-over factor from A to B (‘/z for uncracked member) co,, = 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

a

I -1

C-

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/LT 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 uncracked 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 =

_ks(1-CO) 12

2

(2-3)

For the purpose of determining the mean temperature 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

t-

LF? d-i!i i ?-+-

i i 1-t i i i j I d'

0 0.02

0.04

0.06

0.08

Tension Reinforcement, pn

Fig. 2.4 - End-cracked beam, ks and CO for LT = 0.1L

0.10

0.12

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

Tension Reinforcement, pn

Fig. 2.5 - End-cracked beam, ks and CO for LT= 0.2L

0.12

0.8

8

0.7

5 g LL

0.6

5 0 2. b v

0.5

0.4

FEFI CRACKED

ZONES,

0.3

0.2

iiI

0.02

0.04

! II

0.06

0.08

0.10

Tension Reinforcement, pn

Fig.2.4 - End-cracked beam, ks and CO for LT

=

0.1

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

Tension Reinforcement, pn

Fig. 2.7 - End-cracked beam, ks and CO for LT = 0.6L

0.12

0.8

0.7 tttHtt i i i i i tttth

0.6 8 .

p

0.5

rt+i--t

0.02

0.04

I

III.

I

I

1IIIII

Ihl. 1”“’ I I I 1 I

0.06

0.08

0.10

Tension Reinforcement, pn

Fig. 2.8 - Interior-cracked beam, ks and CO for LT = 0.1L

0.12

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

Tension Reinforcement, pn

Fig. 2.9 - Interior-cracked beam, ks and CO for LT = 0.2L

0.12

349.1R--13

0.9

0.8

0.7

s . 5 t 0.6 lE ii

& 0.5 k & ” 0.4 0.3

+-i-i--c

0.2

! !

: f 4 f f : ! ! !

0.1

0 0.02

0.04

0.06

0.08

0.10

Tension Reinforcement, pn

Fig. 2.10 - Inferior-cracked beam, ks and CO for LT = 0.4L

0.12

0.8

s

0.7 :I

L‘ e IE F y L. '0

1.

I

I

u’lll,

I

i

1 II 11 I I ill1 I I! I llllllllllllllll~

0.6

0.5

"0.4

0.3

0.2

0.1

3

Al? . 5 ._U

._ r 0

s

2

s:0 -._c s 1

0

0.02

0.04

0.06

0.08

0.10

Tension Reinforcement, p

Fig. 2.11- Interior-cracked beam, ks and CO for LT = 0.6L

0.12

349.1R-15

DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS 137.4

TOUTS~DE = SOOF W D + L = 1086 LB/FT

---

MECHANICAL

LOADS

MECHANICAL

AND

THERMAL

_ T = &lOoF, T, - Tb = WF, UNCRACKED FRAME l^

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 considered is U = D + L + To + Ess. The mechanical loading consists of: WD = 406 lb/ft WL = 680 lb/ft

on member BC and a lateral load of 3750 lb at Joint C due to Ess. The thermal loading To consists of 130 F interior and 50 F exterior on all members. The base temperature Tb is taken as 70 F. For this condition, Tm - Tb _ T = 80 F (hot interior, cold exterior). = + 20 F and ^ The material properties are f,’ = 3000 psi and Ec = 3.12 x 106 psi; fy = 60,000 psi and Es = 29 x 106 psi; and o( = 5 x 10-6 in./in./deg F. Also, n = Es/Ec = 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 Mu = KuF = (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 positive. These values were obtained by moment distribution, and moments due to Ess include the effect of frame sidesway.

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 Tm - Tb = 20 F) Thermal loads (^

_T = 80 F having hot interior and cold exteThe ^ 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 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 0.59 1 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 D CD _ - All members are the end-cracked type. Fig. 2.5 through 2.7 are used to obtain the coefficients ks and CO, which are shown in Table 2.5.1.

MANUAL OF CONCRETE PRACTICE

349.1R-16

Table 2.5.1 - Cracked frame coefficients and thermal moments on members

Thermal FEM’s, ft-kips Member

End

AB AB BC :cD CD

L*/L 0.59 0.59 0.29 0.29 0.86 -0.86 -

k

co

kJL

k6, 0.39

:z 2'40 2:65

ii

i:;

0.41 0.70 0.43 0.38 0.57 0.47

0.17 0.10 0.08 0.088 0.095 0.120

a/Lt 0

DF

T- 80 ^_FEM

1.0

+60.1

0.56

it: 0:52 1.0

- 17.97 +41.7 -49.2 +24.5 -37.9

T,-T.-u) 1 I

Total FEM +55.31 -21.36 +41.7 -49.2 +28.2 -33.5

FEM -4.79 -3.39 : +3.72 +4.38

Distributed thermal moments,* ft-kips +36.0 -39.9 +39.9 -37.0 +37.0 -33.2

Distributed thermal moments & mechanical moments, ft-kips - 16.3 -115.9 +115.9 -83.0 +83.0 -25.7

*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. ks _T FEM = E,aA Tbt 2 j(1 - CO) (1) ^ l2

Shear stiffness at C = s [ksC(l + COc) + k,D(l + CWI

_ (3.12 x 10*)(5 x 10-“)(80)(12)(24)2 12

x+ ^_T FEM

(1 - CO)

= 59.9k, (1 - CO)/2 ft-kips

(20 x

(b)(k

12)2

)

s

Tm - Tb FEM = 62.4 (A)(k,)(l + CO) ft-kips

_ = total unrestrained change of length of (3) ^ 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 + COA) + ks,(l + COdI

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

= s(8.19)

=

$+0(1.57) + 2.38(1.47)]

Ag = 0.036 in. (6.48/14.67) = 0.036 x 0.44 = 0.016 in.

x (1 + CO)

^_= ^_ =

$ (6.48)

Sum of shear stiffness at B and C = ++ (14.67)

(2) Tm - Tb FEM = $+ (A)&)(1 + CO) = (3.12 x io6)(24)3

=

AC = 0.036 in. (8.19/14.67) = 0.036 x 0.56 = 0.020 in. To demonstrate the effect of cracking on the thermal moments, the fixed-end thermal moments for the uncracked frame are obtained from the final expressions in (1) and (2) using ks = 4, CO = 1/2, and ^_ = (%)o(T, - Tb)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 _B and frame are obtained using the above values for ^ AC and by referring to Table 2.5.1 for ks 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.

61.6

b \

\ b\ 52.3 16.3 ---M E C H A N I C A L

(UNCRACKED

25.7

F R A M E )

MECHANICAL AND THERMAL* l

o ^_T = 8 0 o F , Tm - Tb, = 20 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 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 compressive 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 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 Tm - Tb. Generally, for most axisymmetric structures, a uniform temperature change (Tm - Tb) 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 Tm - Tb varies fairly rapidly along the structure. 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 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 Tm - Tb 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. _T produces internal section forces The gradient ^ (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 _ T are thermal moments caused by the internal due to ^ restraint provided by the axisymmetric geometry of the structure. The cracking reduction of thermal moments which result from internal restraint is the subject 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 corresponding thermal moment whose magnitude depends on the extent of cracking the section experiences. This in turn _ T, the other section forces N and M, and depends on ^ 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 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 simplifying the determination of the cracked section thermal moment. Although not all the loads which comprise the section forces N and M will necessarily be long-term, the selection of 2n for compression reinforcement 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 section 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 values. 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 tensile 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 tensile N and higher QII values. Actual le/d lower limits for tensile N are given in Fig. 3.3. As long as the actual 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 compression face to the compression reinforcement As'. A modular ratio of 2n is used for the compression reinforcement. 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)}

This substitution for kL is made in Eq. (3-7) and Eq. (3-9). The resulting expression for fc 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)

(3-4)

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 a n d 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 section 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 moment M. 3.3 - General e/d Depending on e/d, the extent of section cracking and the thermal moment may be significantly affected by the actual values of N and M. A theory for the investigation of a doubly reinforced rectangular 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 curvature 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 corresponds to the tension face under M. Therefore,

4 = 4L + 4T

(3-2)

The expression for M,, given by Eq. (3-2) is obtained from the results of Section 3.2 in the following manner. For sections in which le/dj 2 0.7, the location of the neutral axis does not change under _ T, and this results in kL.= k. the application of ^

(3-3)

(3-5)

The curvatures before and after the application of

4T are shown in Fig. 3.1. In Fig. 3.1: E cL

= concrete strain at compression face due to N and M

DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS

c

_T Fig. 3.1 - Section under M, N, ^ L CT

=

concrete strain is compression face due to

^_ T

= total concrete strain kLd = neutral axis location on section due to N and M kd = neutral axis location on section due to N, _T M, and ^

cc

The thermal curvature change +T is

_T is always taken as positive. where ^ Using this and cc = +kd and c,,. = +,.kLd in Eq. (3-5) gives

f,/(E,kd) = f,,./(E,k,.d) + aAT/t

or (3-7)

To maintain equilibrium of the section both before _ T, the following condiand after the application of ^ tions occur: Before

^_T

1. The internal axial force N is equal to the resultant 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 line) produced by N and M. M = ‘/zfCLbk,.d(t/2 - k,.d/3) + 2Q’nbdfCL x [(kL - d’/d)/kJ [(t/2) - d’] + Qnbdf,,_ x [(1 - k,.)/k,,l[d - (t/2)]

1. The internal axial force N is equal to the resul_ T. tant of the stresses produced by N, M, and ^ N = %f,bkd + 2Q’nbdf,[(k - d’/d)/k] + Qnbdf,[(k - 1)/k]

(3-10)

IW = %f,bkd[(t/2) - (kd/3)] + 2q’nbdf, x [(k - d’/d)/k] [(t/2) - d’] + x [(l - k)/k] [d - (t/2)]

Qnbdf,

(3-l1)

(3-6)

For the case where the concrete stress is a linear function of strain, Eq. (3-6) becomes

f, = (f,,./(E,k,.d) + aAT/t] E,kd

^_ T

2. There exists an internal center line moment m _ T. of the stresses produced by N, M, and ^

97 = aA T/t

E,/kd = &k,.d + aAT/t

After

(3-9)

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 compression reinforcement have been expressed as Q = A,/bd and Q’ = A,Ybd, respectively. A modular ratio of 2n has been used for the compression reinforcement. Also, the reinforcement stresses have been expressed in terms of the concrete stress. From Eq. (3-8), fcL can be expressed in terms of N, kL, and the section properties. Use of this in Eq. (3-7) allows fc to be written in terms of N, kL, 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, kL, 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 properties and e, where e = M/N. Thus, k is determined 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 kL = 0.10 and kL = 1.0. The range 1.0 2 k,. 2 0.10 should cover many practical situations not involving prestressed sections. For kL

0.06 e = M/N n = Es/Ec

f

c’E*

0.04

s 5 E er f 0.03

Uncracked: 0.02

btr) MAP = 0.101 bd’E,cyAT

l

(Based on gross concrete section)

0.08

0.10

Tension Reinforcement, pn

Fig. 3.2 - Cracked section thermal moment for

le/d 1 3 0.70

0.12

0.80 N (positive as shown)

_

- .

.A_.

.

._L--- . . -

CUseFigures 3.4 thru 3.9

Iiiiiiiiiiiiiiiiilr

e = M/N Ic

I. : : 1 I ! : i 1 I I I I III I I II 11’ I I

i-iiIiTii I II il;.:iili”’ P-

0.06 Reinforcement, pn

Fig. 3.3 - e/d limits

Pbd

“““jii:‘iiiiijijiii . .I ! I I

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

349.1R-26

MANUAL OF CONCRETE PRACTICE

IVX)

‘3 CPQ

? (n-1)

‘INMOW lVVVL13Hl

349.1R-27

DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS

s 0

‘h

&I

‘INMOW lVWL13Hl

349.1R-28

ACI COMMITTEE REPORT

Since t/d = 1.1 and d ’ /d = 0.10, use Fig. 3 . 2 . However, since Q’ f Q, interpolate between the two curves.

outside this range, i.e., the entire section being under tension ( kL 2 0.10) or compression (kL 2 1.0), a similar set of equilibrium equations based on a rectangular stress distribution would be required. Special Case:

From Fig. 3.2 for Qn = 0.050, read (1 - Y)&/ (bd ‘E,aA T) as:

Q’ = Q, t/d = 1.1, and d’/d = 0.10

0.0303 for Q’ = 0

For this case, cracked section thermal moments were calculated for a gn range of 0.02 to 0.12 and N/6dE,aAT ranging within 2300. For the case of compressive N, Fig. 3.4 through 3.9 apply. Alternatively, for compressive N, and e/d 2 0.7, Fig. 3.2 may be used with reasonable accuracy. For the case of tensile N, only Fig. 3.2 applies. As discussed above, the thermal moments are valid only for 1.0 2 k, 2 0.10. Associated with these limits are e/d values which are indicated in Fig. 3.3. Also presented in Fig. 3.4 through 3.9 are the uncracked thermal moments based on both gross section (neglecting reinforcement) and actual section (including concrete and reinforcement). It is seen that the cracked section thermal moments are always less than the uncracked thermal moments obtained for the actual section. However, for the combination of higher gn values and lower e/d values, the cracked section thermal moments exceed the uncracked thermal moments based on a gross concrete section. This is due to the fact that the increase in section flexural stiffness (EI) due to inclusion of the reinforcement is greater than the loss of section flexural stiffness that results from the relatively minor cracking associated with the low e/d value. The net effect is to give a larger actual cracked section stiffness than that obtained for the gross uncracked concrete section alone. If the designer finds these cracked section thermal moments to be unacceptably high, a potential reduction may be available through the use of an approach incorporating a nonlinear representation of the concrete stress-strain behavior. Such an approach is described in References 1 and 2.

0.0320 for Q’ = Q

3.4 - Design examples Example 1 - Compressive N and le/dl > 0.70, Q’ =o b = 12 in., t = 36 in., d = 32.7 in., d ’ = 3.3 in., As’ = 2 in.2, As - 3 in.2, Es = 29 x 106 psi, Ec = 4 x 106 psi, v = 0.2, N = 50 kips compression, M = _ T = 80 F 100 ft-kips, ^ n = Es/Ec = 29/4 = 7.25, e = M/N = 100/50 = 2 ft = 24 in., Q ’ = 2/12 x 36 = 0.0046, Q’n = 0.0046 x 7.25 = 0.033, Q = 3/12 x 36 = 0.0069, Qfl = 0.0069 x 7.25 = 0.050 Cracked thermal moment M,, e/d ratio: e/d = 24/32.7 = 0.733 > 0.70; therefore, use Section 3.2 results.

Interpolate for Q’ = 0.0046 (1 - u)MdT/(bd ‘E,aAT) = 0.0303 + (0.0320 - 0.0303)(0.0045/0.0069) = 0.0314 (1- v)M*, = 0.0314(12)(32.7)2(4 x 106) x (5.5 x 10e6)(80) = 709,118 in.-lb

MbT = 73.9 ft-kips Check using equations k N=O

=

f(2Q’fl +

Qn)’ + 2[2Q’n(d’/d) + Qn]

- (2Q'n + QIJ)

= [(2 x 0.033 + 0.050)2 + 2(2 + 0.033 x 0.10 + 0.050)]1/2 - (2 x 0.033 + 0.050) = 0.239 (1 - u)MdT/(bd ‘EaAT) = -0. 152k3 + 1.818 Q’n(d’/d - k) x (d’/d) + 0.909Qn(l - k ) = -0.1 52(0.239)3 + 1.818(0.034) x (0.1 - 0.239)(0.10) + 0.909(0.050) x (1- 0.239) = -0.0317 M,, = 73.9(0.0317/0.0314) = 74.6 ft-kips kips, OK

= 73.9 ft-

Concrete and rebar stresses are calculated from a cracked section investigation with N = 50 kips compression and M = 173.9 ft-kips at section center line Example 2 - Tensile N and le/dl > 0.70, Q’ = Q 6 = 12 in., t = 36 in., d = 32.7 in., d ’ = 3.3 in., A s ’ = As = 3.0 in.2, Es - 2 9 x 106 psi, Ec = 4 x 106 psi, v = 0.2, N = 50 kips tension, M = 100 ft_T = 80 F kips, ^

n = 7.25, e = 100/-50 = -2 ft = -24 in., = 0.0069, Q ‘n = Qn = 0.050

Q’

=Q

DESIGN FOR THERMAL EFFECTS/NUCLEAR PLANTS

MQT e/d = -24/32.7 = - 0 . 7 3 3 , le/dl = 0.733 > 0.7; therefore, use Fig. 3.2.

From Fig. 3.2 with in = 0.050 and

Q’

= Q curve,

(1 - v)M,,/(bd *EaAT) = 0.032

349.1R-29

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. Example 4 - Tensile N and /e/d1 < 0.70

Same as Example 3 except N = 60 kips tension

MAr = (1.25)( 12)(32.7)‘(4)(5.5)(80)(0.032) = 75.3 ft-kips

e = 100 ft-kips/(-60 kips) = -1.67 ft = -20 in., e/d = -20/32.7 = -0.612 or le/dl = 0.612

MdT = 75.3 ft-kips

From Fig. 3.3 for en = 0.05, the lower limit on le/dl is 0.575 for the tensile N case. Since 0.612 > 0.575, use Fig. 3.2.

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 Example 3 - Compressive N and je/dl < 0.70

Same section as Example 2 N ==100 kips compression, ^_T = 80 F,

e=

M =-100 ft-kips,

100 ft-kips = 1 ft, = 12 in. 100 kips

From Fig. 3.2, with Qn = 0.05, read (1 - v)MJ (bd *E,aAT) = 0.032 KT = 0.032(1.25)( 12)(32.7)‘(4)(5.5)(80) = 75.3 ft-kips MbT = 75.3 ft-kips

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.

e/d = 12/32.7 = 0.367 < 0.70; therefore, don’t use Fig. 3.2.

CHAPTER 4-REFERENCES

From Fig. 3.3 for en = 0.05, read lower limit on le/d as 0.25. Since 0.367 > 0.25, use Fig. 3.5 and 3.6.

4.1-Recommended references The document of the standards-producing organi-

N/(bdE,aA T) = 100,000/(12 x 32.7 x 4 x lo6

x 5.5 x 1O-6 x 80) = 0.145 For N/(bdE,aAT) = 0.145 and e/d = 0.367, find Mhz. By interpolation between e/d’s: w = 0.04 (Fig. 3.5) @ e/d = 0.367; 0.035

by interpolation

Qn = 0.06 (Fig. 3.6) @ e/d = 0.367; 0.046 by interpolation For en = 0.05, (1 - v)M*J bd *E,aA T = ti(O.035 + 0.046) = 0.0405

J%T = 0.0405(1.25)(12)(32.7)‘(4)(5.5)(80) = 95.3 ft-kips

Mdf = 95.3 ft-kips

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.

ACI COMMITTEE REPORT

CONVERSION FACTORS - U.S. CUSTOMARY TO SI (METRIC) To convert from

in. ft sq in. kip psi in.-lb Ib/ft ft-kip deg Fahrenheit (F)

to

multiply by

mm m sq cm newton (N) MPa m-N kg/m m-N deg Celsius (C)

25.4 0.3048 6.451 4448 0.0069 0.1129 1.4882 1355.8 tc = (tp - 32)1.8

This report was submitted to letter ballot of the committee, which consists of 33 members; ballot results were 28 affirmative, 2 negative, and 3 not returned.

ACI COMMITTEE 349 Concrete Nuclear Structures Richard S. Orr Chairman Omesh B. Abhat Hans Ashar Ted M. Brown Edwin G. Burdette Oral Buyukozturk R. W. Cannon* King-Yuen Chu James F. Costello Vijay K. Datta Thomas J. Duffy Consulting Members: M. Bender Harold S. Davis

James R. Crane Secretary Alan M. Ebner James F. Fulton* Dwaine A. Godfrey Gunnar A. Harstead Robert P. Kennedy John C. King Ronald A. Lang Kenneth Y. Lee Romuald E. Lipinski Thou-Han Liu Frederick L. Moreadith

Richard J. Netzel Dragos A. Nuta Duke Oakes Julius V. Rotz Subir K. Sen Hemant H. Shah Robert E. Shewmaker Eric C. Smith Lee Stern Richard H. Toland

Myle J. Holley, Jr. Morris Schupack

Chester P. Siess

--*Prime authors of the thermal effects

Committee members voting on the 1992 revisions: John K. McCall Chairman Omesh B. Abhat Hans G. Ashar Ted M. Brown Robert W. Cannon Ronald A. Cook Thomas J. Duffy

W. Bryant Frye Dwaine A. Godfrey Herman L. Graves III Gunnar A. Harstead Charles J. Hookham Richard E. Klingner

Marvin A. Cones Secretary T.H. Liu Timothy L. Moore Frederick L. Moreadith Dragos A. Nuta Richard S. Orr Julius V. Rotz

Robert E. Shewmaker Chen P. Tam Richard M. Toland Donald T. Ward Albert Y.C. Wong Charles A. Zalesiak