reinforced concrete design for thermal effects on nuclear power plant structures

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 mome

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.

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349.1R-1

Reinforced Concrete Design for Thermal Effects

on Nuclear Power Plant Structures

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.

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

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 Reported by ACI Committee 349

carry-For a list of Committee members, see p 30.

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

transverse displacement difference between

ends of cracked member, due to T m - T b

acting on adjoining members

Chapter 3 - Axisymmetric structures

fc =

f CL =

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, r e section forces M, N,

-the triangular

com-to the depth d, r e section forces M and

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

final cracked section strain at extreme fiber

of compression face = &cL + &Tcracked section strain at extreme fiber ofcompression face resulting from internalsection forces M and N

cracked section strain at extreme fiber ofcompression face resulting from ^ _ T

cracked section curvature change resultingfrom internal forces M and N

cracked section curvature change required

to return free thermal curvature aAT/t to 0final cracked section curvature change =+1 + 4r

CHAPTER 1 INTRODUCTION

ACI 349, Appendix A, provides general erations in designing reinforced concrete structures fornuclear power plants The Commentary to Appendix

consid-A, Section A.3.3, addresses three approaches thatconsider thermal loads in conjunction with all othernonthermal loads on the structure, termed “mechani-cal loads.” One approach is to consider the structureuncracked under the mechanical loads and crackedunder the thermal loads The results of two such anal-yses are combined

The Commentary to Appendix A also contains amethod of treating temperature distributions across acracked section In this method an equivalent lineartemperature distribution is obtained from the temper-ature distribution, which can generally be nonlinear.Then the linear temperature distribution is separatedinto 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 ering thermal load effects which is consistent with theabove provisions The aim herein is to present a de-signer-oriented approach for determining the reducedthermal moments which result from cracking of theconcrete structure Chapter 2 addresses frame struc-tures, and Chapter 3 deals with axisymmetric struc-tures For frame structures, the general criteria aregiven in Sections 2.2 (Section Cracking) and 2.3(Member Cracking) The criteria are then formulatedfor the moment distribution method of structuralanalysis in Section 2.4 Cracked member fixed-endmoments, stiffness coefficients, and carry-over fac-tors are derived and presented in graphical form Foraxisymmetric structures an approach is described forregions away from discontinuities, and graphs ofcracked section thermal moments are presented

consid-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

2.2 - Section cracking

Simplifying assumptions are made below for thepurpose 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 onthese assumptions:

1 Concrete compression stress is taken to be early proportional to strain over the member crosssection

lin-2 For an uncracked section, the moment of inertia

is I g , where I g is based on the gross concrete sions and the reinforcement is excluded

dimen-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 pression reinforcement is excluded and the tension re-inforcement is taken to be located at the tension face;

com-i.e., d = t is used.

3 The axial force on the section due to mechanicaland thermal loads is assumed to be small relative tothe moment (e/d >, 0.5) Consequently, the extent ofsection cracking is taken as that which occurs for apure moment acting on the section

The first assumption is strictly valid only if the treme fiber concrete compressive stress due to com-bined mechanical and thermal loads does not exceed

ex-0.5f ' c At this stress, the corresponding concrete strain

is in the neighborhood of 0.0005 in./in For extremefiber concrete compressive strains greater than 0.0005in./in but less than 0.001 in./in., the differences areinsignificant between a cracked section thermal mo-ment based on the linear assumption adopted hereinversus a nonlinear concrete stress-strain relationshipsuch 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 crete strains not exceeding 0.001 in./in

con-For concrete strains greater than 0.001 in./in., theequations identified above will result in cracked mem-ber thermal moments which are greater than thosebased on the nonlinear theory In this regard, thethermal moments are conservative However, they arestill reduced from their uncracked values This crack-ing reduction of thermal moments can be substantial,

as seen in Fig 3.2 which also incorporates tion 1

Assump-Formulation of the thermal moments based on alinear concrete stress-strain relationship allows thethermal 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

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)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

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 (6jk2)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

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

specified load factors) forms the basis for the crackedstructure used for the thermal load analysis Crackingwill 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 mechanicalmoments will increase the extent of cracking along themember length Recognizing this, in many cases it isconservative for design to consider the member to becracked wherever tensile stresses are produced by themechanical loads if these stresses would be increased

by the thermal loads Any increase in the crackedlength 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 momentswhich 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 isconsidered to be uncracked for the thermal load anal-ysis wherever along its length the mechanical momentsand thermal moments are of opposite sign

Two types of cracked members will result: (1) cracked, and (2) interior-cracked The first type oc-curs for cases where mechanical and thermal momentsare of like sign at the member ends The second typeoccurs where these moments are of like sign at the in-terior of the member Stiffness coefficients, carry-overfactors, and fixed-end thermal moments are developedfor these two types of members in Section 2.4 Acomprehensive design example is presented in Section2.5

end-The above simplification of considering the member

to be uncracked wherever the mechanical and thermalmoments are of opposite sign is conservative due tothe 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 thistwo-phase aspect makes the problem more complex.The conservative approach adopted herein removesthis 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 tion stiffness, which would include reinforcement and

sec-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 thecracked length a along the member This can be seenfrom a comparison of the results for an end-crackedmember and an interior-cracked member for the same

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

the stiffness of the member at Awith B fixed (4EJ,/L for uncrackedmember)

the stiffness of the member at Bwith 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:

Fig 2.2 - /_\ T fixed-end moments - Member cracked

at ends by mechanical loads

.

Fig 2.3 - T, - T, fixed-end moment - Member

cracked at ends by mechanical loads

where k, is the dimensionless stiffness coefficientwhich 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 mostpractical problems In these figures, k, is given at theend 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 fromthis end to the opposite end

Based on the above discussion, the /_\ T fixed-end

moment at the a end of the member can be expressedas:

of Section 2.5 illustrates this

- - - M E C H A N I C A L L O A D S

- M E C H A N I C A L A N D T H E R M A L

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

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 106 psi; f y = 60,000 psi and E s = 29 x 106 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: -52.3BA: -76.0

B C : +76.0CB: -46.0

C D : +46.0

D C : +7.5These are shown in Fig 2.12

The maximum mechanical load moment of 76 kips is less than the section capacity of 141.1 ft-kips.Therefore, the frame is adequate for mechanicalloads

ft-Thermal loads (^ _ T = 80 F and T m - T b = 20 F)

The ^ _ T = 80 F having hot interior and cold rior is expected to produce thermal stresses which aretensile on the exterior faces of all members Thesestresses will add to the existing exterior face tensile

exte-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

-ABABBCBCCDCD_

EndABBCCD

-

-L T /L a/L T

11.8/20 = 0.59 00.59 1(5.3 + 3.4)/30 = 0.29 5.3/8.7 = 0.61

0.29 3.4/8.7 = 0.3917.2/20 = 0.86 10.86 0All members are the end-cracked type Fig 2.5through 2.7 are used to obtain the coefficients k s and

CO, which are shown in Table 2.5.1