CRACKING OF PIc NUCLEAR CONTAINMENT STRUCTURES

By Sami H . Rizkalla,l Sidney H. Simmonds': and James G. MacGregoe

ABSTRACT: Rules for determining the spacing and widths of both through·the­wall and surface cracks in post-tensioned concrete containment structures un­der internal pressure are presented. These involve the construction details and the average strain obtained from an analysis that accounts for concrete prop­erties in the post-cracking range. To evaluate these ruJes and the concrete con­stitutive relationship used in the analysis, twelve quarter-scale segments were designed to simulate construction and stress conditions at various locations of a containment structure. Major variables included ratio of prestressing, con­crete cover, reinforcement spacing, lap splices and combined axial load and moment. It was observed that the crack spacing depended on the spadng of the reinforcement and prestressing tendons parallel to the cracks and that the final crack pattern was fully developed at the yield strain of the reinforcing steel. Concrete cover was not found to have a significant influence on crack spacing. The procedures developed to determine crack spacing and widths are used to predict the cracking of a one·fourteenth scale model of a containment structure tested later in the research project.

INTRODUCTION

Containment structures for some Canadian nuclear reactors consist of a heavy concrete base, a cylindrical wall, a ring beam, and a spherical dome. Each element contains a grid of conventional reinforcement and prestressing tendons. In the extremely unlikely event of certain mal­functions~ the pressure may reach several times the design accident pressure. This would result in the walls and dome of the containment being stressed in biaxial tension and cracking could occur. To obtain data on the response of such a structure to these loading conditions, 12 quarter­scale segments representative of construction details of the containment were tested. The main objectives of these tests were to investigate the load-deflection and cracking behavior of the containment components under large biaxial tension forces; formation and propagation of cracks were noted carefully in order to develop a technique to predict the num­ber and width of cracks in the containment structure with increasing internal pressure (7). The test results were also used to predict the stress­strain relationship of reinforced concrete in tension, which was used in the nonlinear analysis of a one-fourteenth scale model structure (9) tested later in the research project, and an actual containment structure (8).

lAssoc. Prof. of Civ. Engrg., Univ. of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2.

2Prof. of Civ. Engrg. , Univ. of Alberta, Edmonton, Alberta, Canada, T6G 2G7. lProf. of Civ. Engrg., Univ. of Alberta, Edmonton, Alberta, Canada, T6G 2G7. Note.-Oiscussion open until February 1, 1985. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for re­view and possible publication on March 23, 1983. This paper is part of the Journal of Structural Engineering, Vol. 110, No.9, September, 1984. ©ASCE, ISSN 0733-9445/84/0009-2148/$01.00. Paper No. 19163.

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DESCRIPTION OF SEGMENT SPECIMENS

The selection of the segment size was dictated by the magnitude of the largest tensile forces that can be applied in the laboratory and the size of the smallest available tendon ducts similar to the type used in actual containments. To accommodate the three layers of tendon ducts and reinforcement with spacing comparable to those in the prototype, a wall thickness of 10.5 in. (267 mm) was chosen. This corresponds to quarter-scale for a typical Canadian Gentilly-2 type containment. To al­low enough length to transfer the load and ensure formation of more than one through-the-wall crack, a lateral dimension of three times the wall thickness or 31.5 in. (800 mm) was chosen. A side view of a typical segment and section through the segment are shown in Fig. 1 and Fig. 2. In the prototype the meridional and circumferential tendons were lo­cated at the middle of the wall and near the outer face, respectively. These are represented by the three tendon and four tendon arrange­ments, respectively. The major variables considered were the ratio of prestressing in the two directions, variations in concrete cover and bar spacing, combined axial tension and moment, scale effects, and lap splices of reinforcement. A summary of these variables and loading for each segment is given in Table 1. In describing the segments in this paper the word "face" refers to the 31.5 in. (800 mm) square surfaces. The word "edge" refers to the four 31.5 in. x 10.5 in. (800 mm x 267 mm) surfaces through which the reinforcement extends.

FIG. 1.-Side View of Wan Segment Specimen (1 In. ~ 25.4 mm)

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F"",·1 w". !~don. ..r t

~.~,~.~,.-4~~, ~~~~±~d~h~*~~~,~~~,~-~ be.,,,,, III", (® ";!I _...1. ____ ....!. ___ ....!.... ___ '_ __ -.J'.'

- - ~ ---- . . _-- ~ - . "",,.,-f ---~---~~i-I---~--

--!--'>--"'-"--"'i" ..--"'-"T==j'- -.-I

Section A-A

Sei:tion B·B

~ ·o •

s,,· x s··o

_ '"lltll1.

A""ho, h ....

lood ttll 1%0'

11 "

FIG. 2.-Sections Through Wall Segment Specimen (1 In. = 25.4 mm)

TABLE I.-Overview 01 Variables Considered in Wall Segment Tests

load- Ap- Thick-ing plied ness,

Speci- Pre- Nonpres!ressed reinforce- Minimum con- ratio mo- in Lap mena stressed men! per layer, lb crete cover HjV men! inches splices

(1) (2) (3) (4) (5) (6) (7) (8)

1 two 10 No . 3@3in. (76 mm) 0.5 (13 mm) 1,2 10.5 (267 mm) 2 two 10 No. 3@3in. 0.5 1,2 10.5 3 two 10 No.3 @ 3 in. 0.5 U 10.5 4 none 8 No.4@4in. 0.5 U 10.5 5 one" 10 No.3 @ 3 in . 0.5 1,0 10.5 6 one" SNo.4@4in. 0.5 100 10.5 7 none 6 No.6 @ 6 in. 0.75 (19 mm) 1,1 15.75 (400 mm) 8 two 10 No.3 @ 3 in. 1.25 (32 mm) 1:2 10.5 9 two 10 No.3 @ 3 in. 0.5 102 10.5 yes

11 two 10 No.3 @ 3 in. 0.5 1 :2 10.5 yes 12 two 10 No.3 @ 3 in. 0.5 1:2 yes 10.5 13 two 10 No.3 @ 3 in . 0.5 1:2 yes 10.5

'Two additional tests involving air leakage (Spedmens 10 and 14) are presented in Ref. 3. bEach face of the spedmen had one such layer in each direction. "Three tendon direction only, other tendons omitted.

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CONSTRUCTION OF SEGMENT SPEC1MENS

Prestressing wire tendons consisting of either 6 or 7 wires were used to post-tension the segments. The tendons were supplied by the man­ufacturer in a preassembled unit including fittings and ducts conforming to the BBRV system (Birkenmeier, Brandestini, Ros, and Vogt). The wires in each tendon were held in a symmetrical pattern by passing them through threaded end fittings and were button-headed to provide end anchorage. Fabrication began by placing electric resistance strain gages and targets for the mechanical strain gages on the steel reinforcement. The plywood form held the reinforcement and tendons in position as shown in Fig. 3. Normal weight concrete with maximum aggregate size of 3/8 in. (10 mm) was used in all the segments. The No.3 bars were Grade 60. After the curing period, a split ring aluminum load cell was inserted at the fixed end of each tendon to measure the applied pre­stressing forces. The stressing sequence was staged to minimize an­chorage losses and to avoid cracking of concrete due to eccentric pres­tress. The total forces applied to each tendon were 48 kips (214 kN) in the three tendon direction of 64 kips (285 kN) in the four tendon direc­tion . Later, the end fittings were sealed and the ducts were grouted un­der a pressure between 30 and 40 psi (0.2 MPa and 0.28 MPa) to ensure complete grouting of the voids inside the tendon sheath . Based on strain measurements and computations proposed by Libby (6), stress losses of 12% in the four tendon direction and 8% in the three tendon direction were assumed for all segments.

TESTlNG ApPARATUS AND PROCEDURE

Loads were applied to the segments by pulling on the reinforcement

FIG. 3.-Segment 2 Before Casting FIG. 4.-Loadlng Frames for Segment Tests

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and prestressing strands using specially designed loading yokes to en­sure uniform strain over the section as shown in Fig. 4. Loads parallel to the four tendon direction were applied vertically using a 1,400,000 lb (6,230 kN) capacity Material Test System testing machine. These rep­resented the circumferential loads in the prototype. Built around the testing machine and independent of it was a lateral loading frame de­signed to support four 200 kip (890 kN) hydraulic rams to apply the horizontal loads as shown in Fig. 5. These represented the meridional loads in the prototype.

For each segment approximately 150 measurements were recorded at each load level. These measurements were of such quantities as vertical load, horizontal load, forces transferred to tendons, forces transferred to reinforcement, reinforcement strains, concrete strains, elongation of specimen, crack widths, and slip of tendons. Loadings were applied in increments and terminated at approximately 95% of the rupture strength of the tendons to avoid damage to loading apparatus and instrumen­tation. During testing, an intial preload of 5 to 10 kips (22 kN-45 kN) was applied to facilitate alinement of the system and then further load­ing was applied.

In the test of specimen 12, the vertical load was applied eccentrically and the horizontal load was applied axially. For specimen 13, both loads were applied eccentrically so that the maximum tension from both load­ing directions was on the same face.

All strains were expressed as average strains based on readings taken in the middle portion of the segment. Consequently, only those cracks located in the corresponding region were included in the cracking anal­ysis. Throughout the test, measurements were taken on electrical resis­tance strain gages on the reinforcing steel and the concrete, and on De­rnec points on the concrete. Overall extension was measured with Linear

M.T.S" , Moving

_12WF6~ Hnd

_ 12WFe5

v:"" ,W""~ ~.cticn Beam ~Oooo .:~; t o WF 51 r~"HYd'.YIlc j ~ ,,~

t 2WFe5

~ .l I C1r* .L1 o 0 0 o b,. C<lncf"le

0 o ." ] ;=: Spet::;m,.n ' '. 0 0 0)7

9 "t:~ l 08J ee}.{ 0

0

~"J, ,,fIJ 00 00 ~ ,~ 0)\ 0 End FIlling lot '\10 Y,-TOf\doflIOi,ection 4"" e "

_ (WhiHie Tree)

BaHO 01 I r T"'lt,ng MltChine (lab. FIOO<) ,

14" - O·

FIG. 5.-EAST·WEST Section Through Loading Frame (1 It ~ 0.305 m; 1 In. ~ 25.4 mm)

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width measurements were recorded. A detailed description of the in­strumentation, testing procedure, and test results is contained in Ref. 11. A summary of the applied loads and the load at which cracking was first observed in each direction and on each face are given in Table 2.

AVERAGE STRAIN

All observed strains at any load level were expressed as average strains based on readings taken in the middle portions of the segment. There is a correlation between the average strain and the degree of cracking in this region. Thus the first step in predlcting cracking is to be able to predict the average strain.

The results for all wall segment spedmens are compared with analyt­ical solutions obtained from a modified version of the computer program BOSORS (2,3). BOSORS is a finite-difference displacement model spe­cifically designed for use with complex problems in shells of revolution. The code was developed for the analysis of layered, thin shell, metallic structures and, as such, the nonlinear constitutive relationship was based on a Von-Mises flow theory of metal plasticity. In order to adapt this code to the analysis of prestressed concrete structures it was necessary to develop an elastic-plastic flow theory characteristic of concrete (4). The tensile stress-strain curve proposed for the concrete has a degrading branch after cracking, as .shown in Fig. 6.

By comparing the solutions obtained using BOSORS with the results from the segment tests the validity of the proposed tensile stress-strain in predicting the average strain of the segment at any load stage can be determined.

TABLE 2 -Load and Tensile Stress at First Cracking

Vertical Load at Horizontal Load Tensile Compres- Split First Horizontal at First Vertical stress

Seg- Age sive tensile Cracking, in Cracking, in at first ment at strength, in strength, in Thousands of Thousands of cracking, in num· test pounds per pounds per Pounds Pounds pounds per ber days square inch square inch Face A Face B Face A Face B square inch (1) (2) (3) (4) (5) (6) (7) (8) (9)

I 84 5,094 490 300.0 325.0 194.0 179.3 307 2 92 4,456 436 287.5 300.0 183.5 200.0 174 3 99 5,690 426 350.0 300 206.8 206.8 229 4 109 5,590 536 80.3 2Q 2Q 2Q 259 5 109 5,690 444 - - 280.2 255.5 389 6 127 4,540 325 - - 185.0 196.0 223 7 104 2,39O(A)' 312(A) 75.1 100.0 82.1 82.1 168

3,720(B)' 365(B) 8 92 4,920 424 350.0 350.0 175.0 175.0 321 9 159 3,920 437 350.0 300.0 181 .8 181.8 221

11 186 4,920 411 284 300 175 175 170 12 131 5,930 486 300 340 170 - 292 13 63 6,130 566 - 175 - 150 300

'SpeClmen 7 was made from two batches one on Face A. and the other on Face 8. Note: 1 psi = 6.89 kPa; 1 Ibf = 4.45 N. The reported tensile stresses correspond to the

underlined loads.

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f" ,

f,

, , , 0.411 ,'; I

! "

0,00012 0.0003

EXpoMnll., Form ',_ !.I02 tj .·~e5.r

FIG. S.-Proposed Tensile Stress-Strain Curve for Concrete

'.0 20 1.0

Strain, t

Expt eoSORS lit ... 0.$',1 BOSORS I'," .. 0.32 ' ,' )

C - Cone.ere Cflekll R " Reba,. Yield

FIG. 7.-Load-Strain Response of Seg­ment 4 (1 kip = 4.45 kN)

Nonprestressed Segment.- The measured and predicted load-average strain responses of segment 4 are compared in Fig. 7. This test was typ­ical for the nonprestressed specimens. Segment 4 was loaded biaxially with a load ratio of 1: 1 and the measured traces are quite similar in the two directions. Directions 1 and 2 in the figure refer to the vertical and horizontal directions, respectively. Cracks were first observed visually at 60 kips (267 kN) as shown in Table 2. This corresponds to a tensile strength of 0.48/; . The measured load strain traces showed a major change in stiffness due to cracking at a load, estimated graphically, of 40.6 kips (181 kN) corresponding to a tensile strength of 0.32 I; .

Previous studies (4) had suggested that large masses of concrete dis­played a reduced tensile strength of roughly 0.6/;. When this value was used in the BOSOR5 analysis of segment 4 the predicted load deflection diagram labeled Run 4a was obtained. A second analysis based on the observed tensile strength of 0.321; (Run 4b in Fig. 7) produced a re­sponse that correlates well with the experimental analysis. This indicates that good behavioral predictions can be obtained if the material prop­erties, especially the tensile strength of concrete, are known.

It can be noted also that the analytical prediction overestimates the load, at the time the rebars yield, by an amount roughly equal to the residual strength of the concrete times the net area of the concrete (~ 100 psi x 327.6/1,000 = 32.8 kips). This is a characteristic of constitutive relationships which model tension stiffening by means of a residual ten­sile stength.

Prestressed Segments.- Segment 3 was loaded biaxially with a load ratio 1: 1 which was maintained only up to a load of 375 kips (1,669 kN) followed by an increase of load in the vertical direction (direction 1) only. The experimental and analytical load-strain curves for this segment are shown in Fig. 8, and are representative for other prestressed segments. It is seen that the BOSORS analysis predicts the response well in the horizontal direction (direction 2) but slightly underestimates the stiffness in the orthogonal direction. Here, an effective tensile strength of 0.6 f; was found to work best.

Recommended Tensile Stress-Strain Relationship.- Based on the segment tests, it was found that the response can be simulated ade­quately by using a stress-strain curve in tension having a degrading branch approximated by a straight line and a degrading branch in exponential

2154

.'02 .---,--~-~---r----'.:-n-~-~--'--'

' .0

' .0

0: ;g. 3.0

~ ...J 1,0

'"

[lo'.CI;ons , , BOSOR5 lo.6r~ -0- ___

Exi>!. •• ""''''1 C - CIId'JnQ 01 Conc.t1. R " R_rsY4Id T r r._ y .. kI ' . l - SUI*",;Pl. 10< Or'Kliofto l ' 2

~~.'-~0--70~' -~'~.-~L,~-7'.0~~,.7'-~'~.0--7,.,~.~,,~·,~ Strain

FIG. B.-Load-Straln Response of Segment 3 (1 kip = 4.45 kN)

form (4) as shown in Fig. 6. It was observed also that the maximum tensile strength of concrete r; , has an average value of 0.6 r; . CRACKING BEHAVIOR

Crack Spacing.-A survey of existing procedures for predicting crack spacing and widths was made and reported in Ref. 7. The application of these techniques did not predict the crack spacing or width observed in the wall segments. This was not entirely unexpected since previous investigators had not considered biaxially loaded prestressed sections of such large thicknesses compared to the reinforcemen t diameter and cover. This necessitated the development of a means of predicting the spacing and size of cracks from computed mean strains that was applicable to containment type structures.

It was observed in the segment tests that when a load was applied, the specimens initially cracked at one location. With further loading, more cracks occurred, reducing the crack spacing. These cracks were observed to coincide with reinforcing bar locations. After the formation of a suf­ficient number of cracks, further loading produced no new cracks but certain cracks opened to accommodate the increasing strains.

The average crack spacing in three segments is plotted as a function of average strain in Fig. 9. It is concluded that no new cracks form after a strain equal to the yield strain of the reinforcing bars (in this case, 0.002) and the crack spacing is essentially independent of the concrete cover.

ReinforCing bars perpendicular to the direction of the load create a stress concentration condition which reduces the average tensile stress reqUired to crack the concrete. As a result, if a crack is expected in a given region, it will likely occur at a transverse reinforcing bar. This is particularly true if the transverse bar spacing is similar to the expected crack spacing.

For reinforced concrete members subjected to tension, the average crack spacing is given by Beeby (1) as:

A 5 = 1.33c + O.OBd, - .......................... . ... . .. .. ........ (1)

A,

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"

0001 O(XJ2 0003 o .Q).O OOOS 0005

-" FIG. 9.-Effect of Cover, Bar Spacing and Strain on Crack Spacing (1 in. 25.4 mm)

in which c = concrete cover; db = diameter of reinforcing bar; A b = area of reinforcing bar; and A = area of concrete concentric to the bar = (2c + db)' (bar spacing).

If the spacing of the transverse reinforcing bars is between half and twice the expected crack spacing from Eq. I, the srress concentrations at the bars should be enough to cause cracks to form along each bar and should make the formation of additional cracks between bars unlikely. Such crack spacings were typical of those observed in the segments as shown in Fig. 10.

.~- . . . , '. , •

• • 4, • SPECIMEN NO \ • . H LD= 315 K .

VlD=SSO , ''-

FIG. 1 D.-Cracks In Segment 1 at End of Test: (s) Face A; (b) Face B

2156

To determine how the cracks propagated through the wall a number of specimens were sawn in two. The cracks in Segment 1 in Fig. 11 are representative of the meridional cracks expected in the prototype . From an examination of aU the segments it was concluded that roughly one­half of the cracks extended through the segments and in most cases these cracks divided near the surface to form two surface cracks. In all wall segments which had prestressing tendons parallel to the cracks, the through-the-waU cracks occurred at the prestressing tendons. In these cases, surface cracks which did not penetrate through the wall devel­oped at strains greater than about 80% in the yield strain of the rein­forcing bars.

From further observations of the segments, it was concluded that the effects on cracking of the transverse state of stress in segments loaded in biaxial tension were small and could be ignored in crack width cal-

FIG. !!.-Cracks Through Segments! and 2 at End of Tests: (a) Horizontal Cracks Through Segment! (Face A Upwards); (b) Vertical Cracks Through Segment 2 (Face A Upwards)

2157

culations with relatively little error; the presence of bar splices did not significantly affect crack widths for average strains up to 0.002, although at high strains the existing cracks at the ends of the splice tended to open more. While the presence of a bending moment had a predictable effect on surface strains and hence cracking, the presence of a moment about one axis had little effect on the widths of cracks perpendicular to that axis.

From these observations, a series of rules was developed for use in computing the mean spacing of cracks in the wall segment specimens. It is expected that the same rules would apply to the containment structure.

1. If the spacing of transverse bars is between 0.5 and 2 times the crack spacing computed from Eq. 1, cracks will form along each of the transverse bars by the end of the test. The cracking will be limited to these cracks.

2. The spacing of cracks at the surface of the specimen is independent of the radial distance from the longitudinal bars (bars perpendicular to the cracks) to the point on the surface where the cracks were observed.

3. In walls containing prestressing tendons parallel to the direction of cracking, through-the-wall cracks will occur at the same spacing as the tendons. Should the tendon spacing exceed twice the wall thickness an additional through-the-wall crack will occur midway between the tendons.

4. In the walls without prestressing tendons parallel to the direction of cracking, through-the-wall cracks will occur at every second reinforc­ing bar and not further apart than the wall thickness.

5. The number of through-the-wall cracks will stabilize by the time the strain reaches 0.002. At any given strain less than 0.002 the number of through-the-wall cracks can be given as

I ' " - ' "" I N = N"", 0.002 _ . :,." ............. . .. .. ...... . ..... ...... ..... (2)

in which N m, = the final number of through-the-wall cracks according to assumptions 3 or 4; N = the number of such cracks corresponding to strain £52 I the strain in the reinforcing bars perpendicular to the crack (see Eq. 4); and "',cr = the average strain immediately after the onset of cracking (see Eq, 8),

6. At a strain of 0.002, all surface cracks have formed so that the final spacing agrees with rules 1 and 2.

Crack Width,-When computing crack widths it is necessary to dis­tinguish between through-the-wall cracks which result in paths of leak­age and surface cracks which do not. In leakage calculations it is suffi­cient to consider only through-the-wall cracks, while for comparison to cracking tests, the widths of both types must be included. Although the crack widths follow a statistical distribution, only computations for the representative or mean width are considered here, since in a structure as large as a containment vessel the leakage can be expressed as a func­tion of the average crack width,

The procedure outlined below may be used to determine crack widths and spacings for prestressed wall sections containing two layers of re-

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inforcement near each face in percentages and spacings normally asso­ciated with the Canadian Gentilly-2 type of containments (9). The pro­cedure is based partly on a modification of a theory proposed by Leonhardt (5) and partly on the observations listed in the previous section.

The width of through-the-wall cracks is computed first since surface cracks that do not penetrate through the wall are assumed to occur to relieve tension built up in the concrete between the through-the-wall cracks and will generally not form until the reinforcing bars have yielded at the through-the-wall cracks. The expected spacings of these cracks was presented earlier.

At any load level, P, subsequent to initial through-the-wall cracking but prior to yielding of the reiniorcing bars, the steel stress, f." and strain, Esl, are given by

P - Fu f., = .... . .......... . .. . .. • .. . . ..• . .. •. .•.. . ... .. .. •. . . . (3) As + Ap

f., and E., = - .. .. . ..... .. ......... .. ........................... (4)

E. in which F~ = effective prestress force after losses; As = area of rein­forcing bars; Ap = area of prestressing tendons; and Es = modulus of elasticity of steel. When the force P exceeds that required to yield the reinforcing bars the value of f., should be taken as fp' in which

P - Ad, ~= .............. .... ................... .... ~

Ap

and the value of E,l can then be obtained from the stress-strain curve for the tendon.

The mean strain, Em, is computed as

Em = •• ,11- (!7.~a)' I ............ .. ............................ · (6)

The average stress and strain in the reinforcement at the beginning or onset of cracking are defined as

Per - F~ f., a = ...... .. ... .. . .. ......................... . . .... .. (7) . A s + A p

~ .-=~ ... ....................................... ~ •

in which P a = the tensile force required to crack the section. The length of "almost lost" bond, 1" is obtained from the expression (5)

{,l.a 1, = 6,500 db ................ . . ... ............. .. ..... ..... . .. .. (9)

in which 1 I) is in inches and 6,500 has units of psi. For a tendon, d II is taken as the diameter of an equivalent bar having the same cross-sec­tional area as the wires in the tendon.

The bond transfer length, It, is taken as

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I, = s - I, ............ . ................... .. ... ............ .. _. (10)

in which s = bar spacing. The mean width of a through-the-wall crack computed at the tendon, w """ is

W twc = E. J2 1o + Emi t • .....•.•.. • .•......•.......•. .. ......... . ... (11)

in which Es2 = steel strain at the crack, Eq. 4i 10 = length of "almost lost" bond at a crack, Eq. 9; Em = average strain measured over a length that includes several cracks, Eq. 6 .. or the mean strain from the analysis described in Ref. 9 or 4; and I, = bond transfer length, Eq. 10.

This width is assumed to be divided evenly between two cracks ex­tending to the surface.

Mean widths of surface cracks are computed in a similar manner from the expression:

W s = Es21 0s + Emits .. ..... ... ... .. . , ... .. .............. . ...... , .. (12)

in which E,2 and Em are the same as for through-the-wall crack compu­tations.

The effective unbonded length at a surface crack, l~, may be com­puted from

l ~ = /~~o db ................. . ... .... .. ......... . ...... .. ..... (13)

in which r,.cr = r: AI Ab (psi); and r: = tensile strength of concrete. The bond transfer length at surface cracks, 1'5 I is taken as

I" = s - l ~ . .. . ......... ..... .... .. .................. .... . ..... (14)

COMPARISON OF COMPUTED AND MEASURED CRACK WIDTHS

While the widths of individual cracks and crack width distributions were determined for both the wall segments and the test structure (9,11), only the mean crack or representative crack width is required to predict overall behavior and leakage. An evaluation of the accuracy of the pro­cedures presented in this paper for determining the spacing and mag­nitude of the mean crack can be made by comparing the sum of the computed crack widths in a given gage length, L, which contains several cracks, with the sum of the measured crack widths in the same length.

Prior to yielding of the reinforcement the computed elongation is as­sumed equal to

LW = N"",'w"", ....... . ..... . ....... . ......................... (15)

in which N"" and w"'" are the number and mean width of the through­the-wall cracks in the assumed gage length. After yielding of the rein­forcement the total crack width is

LW = N,~·w"", + N , ·w, ... .. ............................ .. .... (16)

in which N, and w, = the number and mean widths of the surface cracks, respectively.

Comparisons with measured data were on the basis of the dimen­sionless ratio, LwlL. In the range of loading from a strain of 0.0005-

2160

'"r---------------------,

.. ,

0- -- --0 0_______0 w;,.,~% .. - -- ... _~t

0001 ()OO<> G .. ' 0 ~ O~

Strain

FIG. 12.-Measured and Computed Crack Strain. Vertical Cracks In Model Con­tainment (1 psi ~ 6.89 kPa)

0.002, i.e., for the period that cracks are growing and extending, for segment specimens without splices or moments (segments 1- 6 and 8), the mean ratio of measured to computed 2-wiL was 1.07 with a coeffi­cient of variation of 0.347. This coefficient of variation compares well to values obtained by Beeby (1) and others.

Measured and computed values of 2- w IL for a one-fourteenth scale containment test structure computed by the technigue described above are shown as a function of internal pressure in Fig. 12. The agreement between measured and computed values is sufficiently good indicating that the procedures developed for predicting crack spacing and width have adequate accuracy for use in predicting overall response.

ApPLICATION OF CRACKING ANALYSIS

The rules presented earlier for determining crack widths and spacing can be used to estimate cracking of a prototype containment. The av­erage strains are computed at the required load using an analysis tech­nique such as BOSOR5 that accounts for the concrete properties before and after cracking. The surface of the containment is divided into zones of equal surface strain. The rules given earlier are used to determine the number, extent and width of the cracks in each zone . The resulting fam­ily of crack dimensions can then be used in conjunction with the tech­nique described in Ref. 10 to estimate air leakage.

ACKNOWLEDGMENTS

The work described in this paper was sponsored by the Atomic En­ergy Control Board of Canada, Ottawa, Canada, KIP 5S9 and carried out at the University of Alberta, Edmonton, Canada. The project was supervised by Dr. W. D. Smythe, Dr. F. Campbell, Dr. G. ). K. Asmis, and Mr. D. Whelan, all of AECB.

2161

ApPENDIX I.-REFERENCES

1. Beeby, A. W., "A Study of Cracking in Reinforced Concrete Members Sub­jected to Pure Tension," Technical Report 42.468, Cement and Concrete As­sociation, London, England, June, 1972.

2. Bushnell, D., "BOSOR5-A Computer Program for Buckling of Elastic-Plas­tic Complex Shells of Revolution Including Large Deflections and Creep," Vol. 1: User's Manual, Input Data, Lockheed Missiles and Space Company, Inc., Sunnyvale, Calif., Dec., 1974.

3. Bushnell, D., "BOSORS-A Computer Program for Buckling of Elastic-Plas­tic Complex Shells of Revolution Including Large Deflections and Creep/' Vol. 3: Theory and Comparison with Tests, Lockheed Missiles and Space Com­pany, Inc., Sunnyvale, CaliL, Dec., 1974.

4. Chitnuyanondh, L., Rizkalla, S. H., Murray, D. W., and MacGregor, J. G" "An Effective Uniaxial Tensile Stress-Strain Relationship for Prestressed Con­crete," Structural Engineering Report No. 74, Department of Civil Engineering, University of Alberta, Edmonton, Canada, Feb., 1979.

5. Leonhardt, F., "Crack Control in Concrete Structures," IABSE Surveys, 5-4/ 77, IABSE Periodical 3/1977, International Association for Bridge and Struc­tural Engineering, Zurich, Switzerland, Aug., 1977.

6. Libby, J. R , Modern Prestressed Concrete, Van Nostrand-Reinhold, New York, N.Y., 1977.

7. MacGregor, J. G., Rizkalla, S. H., and Simmonds, S. H., "Cracking of Rein­forced and Prestressed Concrete Wall Segments," Structural Engineering Re­port No. 82, Department of Civil Engineering, University of Alberta, Edmon­ton, Canada, Mar., 1980.

8. Murray, D. W., Wong, c., Simmonds, S. H., and MacGregor, J. G., "An Inelastic AnalYSis of the Gentilly-2 Secondary Containment Structure," Struc­tural Engineering Report No. 86, Department of Civil Engineering, University of Alberta, Edmonton, Canada, Apr., 1980.

9. Rizkalla, S. H., Simmonds, S. H., and MacGregor, J. G., "A Test of a Model of a Thin-Walled Prestressed Concrete Secondary Containment Structure/' Paper J4/ 2, SMiRTS, Berlin, Aug. , 1979.

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ApPENDIX II.-NoTATION

The following symbols are used in this paper:

A

Ab A p A ,

C

db E, = E,

effective area of concrete around each bar, equal to area of concrete concentric with one bar and bounded by edges of member or points halfway between two bars; area of one bar; area of prestressing tendons; cross-sectional area of reinforcing bars; minimum cover to surface of bar (measured perpendicular to surface); diameter of bar; modulus of elasticity of steel; modulus of elasticity of concrete;

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FK t: t ;'

f"a f"

f"

/ S2,c:r fy I,

l ~ It L

N N "",

N , P

Pcr s

W "",

W , LW

<m

<" E 5 2,0"

=

effective prestress force after losses; tensile strength of concrete by Brazilian splitting test; maximum tensile strength of concrete; stress in reinforcing bar at initiation of first crack; stress in reinforcing bar at crack prior to yielding of reinforc­ing bars (also equal to change in stress in tendon); equivalent stress in prestressing tendons at crack after yield­ing of reinforcing bars; stress in reinforcement at crack immediately after cracking; yield stress of reinforcing bars; unbonded length of bar at crack; effective unbonded length at surface crack; transfer length; total length over which strain and crack width measurements were made; number of through-the-wall cracks at given load; number of through-the-wall cracks at end of test; number of surface cracks; axial load; cracking load; spacing of cracks; width of through-the-wall crack computed at tendon, Eq. 11; width of surface crack; total crack width after yielding of reinforcement; average strain measured over gage length which includes sev­eral cracks; steel strain at crack; and steel strain at crack immediately after cracking.

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