AIR LEAKAGE CHARACTERISTICS IN REINFORCED CONCRETE

By Sami H. Rizkalla/ M. ASeE, Bon L. Lau/ and Sidney H. Simmonds,' M. ASeE

ABSTRACT: Eight reinforced concrete specimens were fabricated and subjected to tensile membrane forces and air pressure to study the air leakage charac­teristics in cracked reinforced concrete members. A mathematical expression for the rate of pressurized air flowing through an idealized crack is presented. The mathematical expression is refined by using the experimental data to describe the air flow rate through any given crack pattern . Graphical charts are also presented for the calculation of the air leakage rate through concrete cracks. The concept,of equivalent crack width for a given crack pattern is introduced. The mathematical expression and graphical charts are modified to include this equivalent crack width concept. The proposed technique is applicable for the prediction of the leakage from concrete containment structures or any similar structures due to high internal pressure sufficient to initiate cracking.

INTRODUCTION

When unlined concrete containment structures are subjected to high internal pressure, the membrane stresses induced may exceed the ten­sile strength of concrete, producing cracks passing through the thickness of the concrete and resulting in leakage from the containment. This pa­per provides a technique to predict the rate of air leakage as related to any given crack pattern, The proposed technique is applicable to the prediction of the leakage from concrete containment structures or any similar structures due to high internal pressure.

The first attempt to examine the leakage of air through concrete cracks was undertaken by Buss (2), He examined the rate of flow of air through a pre-existing crack in a slab under air pressure. Later, researchers at the University of Alberta (8) initiated an experimental program by build­ing prestressed concrete segments loaded uniaxially and biaxially to sim­ulate the loading condition of containment structures under internal pressure. Neither of these studies was in a general form to predict air leakage through any given crack pattern in a reinforced concrete struc­ture, The first dealt with a pre-existing crack pattern, while the later study was based on a single specimen under a limited range of internal pressures. Thus, the data were insufficient for the development of a gen­eralized predictive technique,

The main objective of this paper is to develop a general expression which could predict the rate of flow of pressurized air through cracked

lAssoc. Prof., Civ. Engrg. Dept., Univ. of Manitoba, Winnipeg, Manitoba, Canada.

2Grad. Student, Civ. Engrg. Dept., Univ. of Manitoba, Winnipeg, Manitoba, Canada.

3Prof., Civ. Engrg. Dept., Univ. of Alberta, Edmonton, Alberta, Canada. Note.-Discussion open until October 1, 1984. 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 August 11, 1982. This paper is part of the Jour­nal of Structural Engineering, VoL 110, No.5, May, 1984, <1:>ASCE, ISSN 0733-9445/84/0005-1149/$01.00, Paper No, 18809,

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concrete. The results of the experimental program were used to refine and calibrate the proposed mathematical expression, and test its pre­dictive ability .

MATHEMATICAL FORMULATION OF LEAKAGE RATE

In general, the geometric configuration of a particular crack extending through a cracked concrete section is extremely complex. It is almost impossible to model accurately the crack path. Since each crack is unique, modeling a particular crack would be of limited application. However, for pure membrane loads, the width of any given crack should be rea­sonably uniform through the concrete thickness. Thus, the flow through a concrete crack may be idealized as a flow through a gap between two parallel plates as shown in Fig. 1. Applying the momentum-balance theorem, which states that the summation of forces should equal the change of momentum (3); PA - (P + dP)A - 2,"Bdx = pAV(V + dV) -pAV(V), or

B dP + pVdV + 2'"A:dx= 0 ....... . ... . .. . . . ...... . ............... (1)

In which A = BW ff; B = extent of crack in feet; W = gap between two parallel plates ~crack width) in feet; P = absolute air pressure at any section, in lb-ft i V = velocity of air at any section" in Ib-ft2; To = shear stress due to wall friction, in lb-ff; and p = mass air density, in slug/ ft'.

The mass flow rate m and the friction coefficient factor f, are assumed to be constant along the flow path. The air flow is also assumed to be isothermal. Thus" in a steady and uniform flow in a conduit of constant cross section, shear stress can be expressed in terms of velocity V, mass air density p, and dimensionless friction coefficient factor f as follows.

f pV' "=42 ......... ... ........... ....... ...... .... ............. (2)

Using this expression and integrating Eq. 1 along the length of the crack in the direction of flow L gives

FIG. I.-Idealization 01 Crack as Gap Between Parallel Plates

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Pi- pi ~ ;~ :~ B;!' [~~ m (~:) + 1] ....... . . .... ... . . ..... . . (3)

in which subscripts 1 and 2 represent conditions at the beginning and at the end of the crack, respectively. Rand T are the gas constant and the absolute temperature respectively; q2 is the rate of flow of air at the end of crack. 'The first term of Eq. 3 in the bracket is very small com­pared to unity and can be neglected. Hence

, , _ /L pi qi PI - P, - 2W RT B'W' .... . ............. . .......... . ............ (4)

The dimensionless friction coefficient factor / depends on the Rey­nolds number R and the wall roughness, k, and it can be expressed in general for both laminar and turbulent flow as

f~ (~) ' ... ..... ........... . ............... . ............... . . . (5)

in which n is the flow coefficient, which can be determined experimentally. By introducing

p,V,D R ~-- ... . ...................... . ... . . . .. . . ... .... .. . .. .... (6)

1'-,

in which I'- is the dynamic viscosity; and the hydraulic diameter D is defined as four times the cross section divided by the wetted perimeter (3). For flow through a crack, D ~ 4BW /2(B + W) which can be ap­proximated closely as 2W since B is much greater than W. Eq. 6 can be written as

k" ~ n

f~ , ...................................... .. ... .. .. .. (7) (p, V,2W)'

Substitutmg Eq. 7 into Eq. 4 and simplifying yields

P' _ p' (k') ( )" IP 1'-' 1 T ~ '2 ~ (RT), - I ~q, W, .... .... .. ..... .. . ....... (8)

'The total flow rate Q, through a wall having j cracks can be expressed as the sum of the flows through the individual cracks or

p' p' (k')()' IP Q 1'-' 1 T ~ '2 ~ (RT),- l ~ ± Wi" .. . . .. .. . .. . ... (9)

i - l

Eq. 9 can be expressed as

P" ~ c l P~Qf ..... ... .. .. ..... ........................ ....... (10)

in which P" ~ pi - Pl!L, modified pressure gradient; C ~ (k" /2)(1'-/2)" (RT)'-I/};(=I Wi, a constant; m ~ 2 - n; and P,Q,jB ~ rate of flow per unit extent of crack.

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It should be noted that in the aforementioned derivation, Wi is the width of the j th crack which must be determined from measurements. The width of other cracks, i.e., j + 1 crack, may be different.

Test Specimens.-In containments and similar structures subjected to internal pressure, the magnitude of the membrane stresses, and hence the crack width, is a function of the internal pressure. However, in leak­age tests, the tensile forces applied to the specimen are independent of the applied air pressure, which permits an evaluation of the leakage rate for different pressure levels. Moreover, planar specimens were used to simplify the fabrication, testing, and interpretation of the results.

All specimens were rectangular in shape and reinforced in two direc­tions with deformed bars. Transverse reinforcement was provided by MI0 bars spaced at 3 in. (76 mm) center-to-center on both faces. Lon­gitudinal reinforcement was also spaced at 3 in. (76 mm) center-to-center with concrete cover of 1/2 in. (12 mm). The three quantities varied from one specimen to another were longitudinal bar size, concrete thickness and steel reinforcement ratio. The overall dimensions and reinforce­ments of the specimens are shown in Table 1.

All concrete used in the specimens had a nominal design strength of 6,000 psi (41 MPa) and was mixed in the laboratory. The volume of each batch was 3 cu It (0.084 m' ) which is the capacity of the mixer, and contained the following quantities: (1) Water, 35 Ib (15.86 kg); (2) cement (type 1 Portland cement), 70 Ib (31.71 kg); (3) sand, 1411b (63 .87 kg); (4) 3/ 8 in. (10 mm) gravel, 186 Ib (84.26 kg); and (5) water/ cement ratio, 0.5.

Two air chambers were provided, one on each face of the specimen. The upstream chamber was filled with pressurized air and the down­stream chamber was used to collect air that had leaked through the cracks of the specimen. To provide air-proofing along the edges of the speci­men, a 1/2 in. (12 mm) reinforced rubber liner was cast around the edges

TABLE 1 -Specimen Properties

Longi-tudinal

Concrete Thick· Width, rein- Speci· cover, in Steel ness, in in forcement men inches ratio inches inches bar size mark Remarks

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

O.S S 12 #3 (MIO) L1 I. All specimens 0.0147 7 12 MIO L4 had transverse

10 11 #4 (MIS) L7 reinforcement MlO approxi-mately 3 in. (76 mm).

7 11 #4 (MIS) LS 2. The height of aU 0.0207 10 12 MIS L8 specimens is 30

in. (760 mm). S 11 #4 (MIS) 1.3

0.0294 7 12 MIS L6 10 12 M20 L9

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... , ...... ,. ~-~., ...

. , ••.••. _ ... ~ .... o'" ._ . .." . ..,. ........ '"

lr~~· .1n-... --: ... j _ &IOI ..... [T

... . ..... , ........... .... .. .... ,.wo.,..-.

~. ' " i=i==·· ......... ·_-~'~~ ~ .. ·. ' · ... 1O·1

FIG. 2.-0etalls 01 Air Chambers and Rubber Liners 01 Leakage Specimen

. . -!""'it.+-iI-' ,,: " 0'10"

.•. . . . · I H--il-· '· ·· ·· · ~ "', ·"""~·o,

FIG. a.-Cross Section 01 Fabricated Leakage Specimen

of the specimen. The shape of the rubber liner was specially cast so that a 3/8 in. (10 rom) steel plate could be bolted to the rubber liner at the upstream face to form a chamber. Likewise, the downstream chamber was formed by bolting a 1/2 in. (12 mm) plexiglass plate on to the rubber liner. Details of the air chambers and rubber liner are shown in Fig. 2 and Fig. 3. The rubber was bonded to the concrete surface by using steel wires which embedded into the concrete (7).

To further reinforce the rubber liner during testing and to prevent its separation from the concrete around the edges of the specimen under high internal pressure, four angles were provided along each corner of the specimen in the vertical direction. The angles were supported at one third and two thirds of their height by homemade C-clamps as shown in Fig. 4. The C-clamps were loose during the application of the tensile membrane load and were tightened only when the applied load was constant and the pressure was applied on the upstream chamber. Fig. 4 shows the specimen ready for testing.

Test Procedure,-When a concrete containment structure is subjected to high internal pressure, the membrane stresses induced may exceed

FIG. 4.-Speclmen Ready lor Testing

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the cracking strength of concrete, resulting in cracks passing through the wall of the containment. In the testing program, the situation is sim­ulated by applying the tensile membrane forces to the longitudinal rein­forcing bars which protrude along the ends of the specimen and pro­viding pressurized air to the upstream chamber of the specimen.

Before performing the test, the specimen was alined vertically and the load cells were adjusted to provide equal distribution of load on each longitudinal reinforcement. The load was increased in increments of 5 kips or 10 kips (22 kN or 44 kN) as was deemed appropriate to produce a sufficient quantity of data before yielding of the reinforcement oc­curred. At each load increment, the load was held constant and air pres­sure was applied on the upstream chamber while the downstream cham­ber was kept at atmospheric. The air pressure in the upstream chamber was controlled by a pressure regulator. A pressure tranducer attached to the upstream chamber was connected to the Data Acquisition System to automatically record the pressure in the chamber.

Once leakage was detected, it was obvious that through-the-concrete cracks had been formed. The air pressure was increased in increments of 3 or 5 psi (22 kPa or 35 kPa) . The amount of air flowing through cracks in a 30-sec period was measured by two dry test meters connected in series. Two readings were taken for each increment of air pressure to derive an average value of rate of flow for each increment. The maxi­mum air pressure applied was 30 psi (207 kPa). The induced bending stresses due to air pressure were inSignificant in comparison with the applied tensile stresses to create differential crack widths between downstream and upstream faces of the specimen.

The cracking patterns were marked and numbered on the surface of the plexiglass at the downstream chamber. Crack widths were measured by using a movable microscope along the center line of the specimen. Widths of all subsequent cracks were measured through the plexiglass for each load increment.

This procedure was repeated for each load increment until the load

P2 all B - Ib/see.

FIG. 5.-Measured Rate 01 Flow per Unit Extent of Crack lor Given Pressure Gra­dient-Specimen L4

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FIG. 6.-Wall Roughness Coefficient K as Function of Summation of Cube of Mea· sured Crack Widths

applied approached the predicted yield load of the longitudinal rein· forcing bars. Testing was terminated when the reinforcing bars reached yielding point to avoid damage to the load cells which were reused in the other test.

Test Results.- Eq. 10 implies a linear relationship between the mod· ified pressure gradient P " and the rate of flow of air per unit extent of crack using a log-log scale, as is clear in

" IP,Q'I log P = log C + m log Ii"" ............ ... .. .. ... .. ......... .. (11)

Based on the experimental results t a graphical representation of the re­lationship for a typical leakage specimen (L4) is shown in Fig. 5. The test results for the other specimens are given in detail in Ref. 7. The slope of each regression line represents the exponent value of m from which the turbulent coefficient n can be determined.

The relationship between the values of k and ~W! for all the leakage specimens, was plotted and is shown in Fig. 6. The relation between k and IW~ was found to have an exponential form.

k = 2.907 X 10' (~W!)o . .,. . . ... .. . ... ... . ........................ (12)

Likewise, the relationship between the flow coefficient n and ~W! is

10'

r z ~

1;' ~

'0' w '" 0 0

• 0 ~ ~

'(f' 10-12 10-10

FIG. 7.-Flow Coefficient n as Function of Summation of Cube of Measured Crack Widths

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. " .~

T .~ '0' ,: Z w 15 • • ~ w • ~ w w w • ~

~,

pz at IB -lb/SISc.

FIG. a.-Rate of Flow of Air for Different Pressure Gradients and Crack Widths

shown in Fig. 7 and was found to be

0.133 n = (LWlf "'l .. . .. ... . . ..... ...... .... ... . .. . ....... ... . . . . ... (13)

Thus, for a given crack pattern, the wall roughness k and flow coef­ficient n can be found by Eq. 12 and Eq. 13, respectively and the constant C can be calculated. With the known value of the modified pressure gradient P and the constant C, the rate of flow of air per unit extent of crack can be found, from which the rate of flow of air at the end of cracks can subsequently be found.

Based on the derived formulas, the solution is presented in graphical form as shown in the chart in Fig. 8. The reliability of these equations can be examined by comparing the rate of flow calculated using Eq. 10,

·IOZ

~ '" ~ a 10- 3

8 W ; ~ IO ~ o

~-'

a (CALCULATED) cu. FT.lSEC.

FIG. 9.-Comparlson of Measured Flow Rate to Predicted Flow Rate Based on Measured Crack Widths

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0 .t2

M

. 0 .06 '2 . "= , 0 .06 z

i-0 .04

W

0 .02 0

0 .OOO".OOC:--0~.0C:2-0~.0=-':--::OL.0=-6 ---:0:':0"' - 0='.'''0-::'0.12

W:'. - 113 X 10- '

FIG. 10.-Relationshlp between Summation of Cube of Measured Crack Widths Divided by Number 01 Cracks 01 Cube 01 Average Crack Width lor Specimen L4

with the measured values obtained from the leakage tests which are shown in Fig. 9. A linear regression analysis was performed on the test data and the results are represented by a solid line with a linear regression coefficient of 1.03. The scattering of the data results for low range flow rates can be attributed to the insensitivity of the instrument used to mea­sure flow rate and crack width.

Equivalent Crack Width Concept.- To eliminate the necessity of con­sidering the width of each individual crack, the concept of equivalent crack width was introduced. This technique permits replacing };W! by an equivalent crack width for a given crack pattern with number of cracks N as follows.

i LW! = 1.42NW;., .. .. . ... . ........ . . . . . . .. ............. . ..... (14) 1-1

in which 1.42 is the average value of all the linear regression coefficients

f 6' , B 1If'

~ ~

.-. o ICALQ.l.ATEQ cu. fT I stC .

FIG. 11.-Flow Coefficient as FUnction 01 Average Crack Width Parameter-All Tests

~ .' w

" ~ ~ .' 0

0 0 t • g , 'O-f •. , c '" .-. .-,

.42. H w:.. -II '

FIG. 12.-Wall Roughness Coefficient k as Function of Average Crack Width Parameter-All Tests

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10·1'':;;~~~~':;;-~~~='::;;;~~~'"'-'-". 10- 12 10-1 1 I(flO I(f9

1.42 N Wa~. - fI~

FIG. f3.-Comparlson of Measured Flow Rate to Predicted Flow Rate Based on Average Crack Width

between LW~ and NW:ve for each specimen as shown in Fig. 10 for a typical leakage specimen (14). Thus Eq. 10 can be modified as follows.

P" = C1 IP~Qf ............................................... (15)

in which C1 = (1('/2)(1'-/2)" (RTr 1/1.42 NW!y,. Following the same arguments as before, the flow coefficient nand

the wall roughness coefficient k can be related to the parameter NW;y, from the test data. Based on the experimental results of all tests, these relationships were found from Figs. 11 and 12 to be

0.195 n = 3 0063 ............ • • ..... .. ... . ........ .. .. ...... . .. • (16)

(NW,y,) .

k = 8.702 X 10' (NW;y,)0367 .. . ....... ... ............ . ........... (17)

Thus, if the value of the average crack width and average crack spac­ing can be predicted, the flow coefficient n and wall roughness coeffi­cient k can be determined by using Eq. 16 and Eq. 17, respectively.

The calculated rate of flow using Eq. 15 is compared with the mea­sured values for all the leakage specimens, as shown in Fig. 13. The perfect relationship between the calculated and the measured rate of flow is presented by the dotted line in this figure. A linear regression analysis was performed on the test data and the results are presented by the solid line with a linear regression coefficient of 0.893. The discrepancy be­tween the mean value and the perfect relationship is greater than the one presented in Fig. 9. This discrepancy can properly be attributed to the fact that the formation of additional cracks would lower the value of the average crack width which would reduce the calculated rate of flow in comparison with the measured values.

AVERAGE CRACK SPACING AND WIDTH

The second procedure described in this paper for obtaining air flow requires information to obtain the average crack width and average crack

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spacing. Many experimental and analytical research projects have been conducted to study the phenomenon of cracking in reinforced concrete members (1,4-6). So far, there is no universally accepted theory or equa­tion for the prediction of crack widths and spacing. Recently, two major experimental programs (5,6) have been conducted to study the cracking behavior of reinforced concrete members subjected to tensile membrane forces. The studies examined the applicability of the existing equations for predicting crack spacing and widths, and the influence of the trans­verse reinforcement on crack behavior. It was concluded (6), that Leon­hardt's expression is valid and provides better accuracy in prediction of average crack spacing. However, Leonhardt's prediction of the average crack widths is highly overestimated, since the ratio between the cal­culated values and those measured suggest an average of 2.38. The pre­diction of average crack widths based on Beeby's work (1), was found to be more accurate since the average ratio between the measured and calculated values is 1.16.

The following is a brief summary of the required expressions for av­erage crack spacing and average crack widths based on the above dis­cussion. The complete derivation is given in Refs. 1 and 4 .

Average Crack Spacing Save.-Based on experimental results, Leonhardt proposed an expression to predict the average crack spacing as follows.

1 S ... = :2 I, + I, ........ .... ...... .. .... . . .. ........ . . ..... . . .... (18)

in which I, is the almost lost bond length and I, is the transfer length.

I, = :'~;o db (f,2.cr in psi units) .............................. . .... (19)

in which db is the bar diameter and t52,a is the stress in reinforcement at a crack immediately after cracking.

db 1,= K1(c,a) + 0.1- ....... ... . . ... ... ........ . . .. ... . .......... (20)

P in which P is the steel reinforcement ratio; Kl is function of the concrete cover c; and a is the spacing between the reinforcement in the direction of the applied load (4) Kl = 1.2c for a '" 2c; and Kl = 1.2(c + a - 2c/4) for a > 2c, with a '" 14d.

Average Crack Width W •••. -Beeby proposed an expression to predict the average crack width in terms of the average crack spacing Save and the average gross strain of the member Em as follows.

Wave = SaveEm ............ . ....... .. ... . . . . .. .. .. .............. (21)

Kd: tS2,a Em = E,2 ••• .•••••••••• • •••••• •••.• ... . ..•. •• . . . . •.••••• (22)

pE,f'2 in which Es2 is the steel strain in the cracked section corresponding to the steel stress fa; K2 is a constant depending on the type of reinforce­ment; t: is the tensile strength of concrete; tS2,a is the stress in the re­inforcement at a crack immediately after cracking; and t$2 is the stress in

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the reinforcment at a crack at any given load after cracking and before yielding of the reinforcement.

Numerical Example.-Based on the two proposed methods for pre­dicting air leakage through cracked reinforced concrete, specimen L4 will be used as an example to illustrate the steps for calculating air leakage through cracked concrete due to applied membrane tensile loads. The air flow rate will be predicted, initially, using the first approach, which requires information about each individual crack for a given crack pat­tern. Subsequently, the equivalent crack width approach will be used. The predictions based on both methods are compared to the measured values to reflect the reliability of the two proposed approaches.

Using Measured Crack Spacing and Widths.-The air leakage rate will be predicted using the measured crack width and spacing for specimen L4 as an example of an applied membrane load of 45 kips (200 kN) and an upstream pressure of 30 psi (207 kPa). The measured crack widths, sec­tion properties, and material properties of specimen L4 are given in Ta­ble 2, from which values of k and n were found to be 1,359 and 0.878, using Eqs. 12 and 13, respectively. At the specified upstream pressure of 30 psi (207 kPa), the air flow rate can be predicted using Eq. 10, and was found to be 0.021 cu ft'/sec (5.88 x 10- ' mJ/s).

Using Predicted Equivalent Crack Width Approach.-This method first re­quires prediction of crack width and spacing under the applied tensile load. Based on Leonhardt's expression, Eq. 18, the almost-lost bond length I, and the transfer length It were found to be 1.97 in. (50 mm) and 4.23 in. (110 mm) using Eqs . 19 and 20, respectively. The values were based on an estimation of the cracking load, using 60% of the split strength of concrete t: (5). Using Eq. 21 the average crack spacing obtained is 5.22 in. (130 mm), from which 6 cracks were estimated for the specimen length of 30 in. (760 mm).

TABLE 2.-Materlal Properties and Experimental Results of Specimen L4

Number (1 )

1 2 3 4 5 6

Crack width x 103 inches (2)

Cracks Data

3.15 0.79 1.57 3.15 1.57 3.93

Note: Loading Condition: Applied membrane tensile load = 45 kips (200 kN); Upstream pressure = 30 psi (207 kPa)i Downstream pressure = 14.7 psi (101 kPa).

Sectional and Material Properties: Cross-sectional area of concrete (7 in. x 12 in.) = 84 sq in. (54,200 mml); Cross-sectional area of steel (8-MlO) = 1.24 sq in. (800 mm'); Modulus of elasticity of steel, E, = 29 x 10' ksi (200 x 10' MPa); Concrete split strength, t: = 710 psi (4.89 MFa); Laboratory temperature, T =

70' F (530' R); Gas constant, R = 1,716 cu It/sec" R (287 J/kg k); Dynamic vis­cosity, ~ = 0.38 x 10-' lb sec/sq It (1.8 x 10-' N· S· /m').

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At an applied membrane force of 45 kips (200 kN), the calculated av­erage strain Em was 0.0046, using Eq. 22. Based on the average crack spacing and the average gross strain, the average crack width WOlve of 1.98 X 10-' ft (0.06 mm) was obtained using Eq. 21, which is in strong agreement with the average measured values given in Table 2. In this case, the values of k and n were obtained using Eq. 17 and Eq. 16 and were found to be 1,405 and 0.873 respectively, which are again close to the values obtained in the first method.

For the same loading conditions, and upstream pressure of 30 psi (207 kPa), the air flow rate can be predicted using Eq . 15 and was found to be 0.0187 cu ft/sec (5.24 x 10- 4 m' Is) from a surface area of 2.5 sq ft (0.23 m'). The measured rate of flow under the same described loading conditions was found to be 0.0185 cu ft/sec (5.18 x 10-' m' Is) which is in excellent agreement with the predicted values based on the two pro­posed methods.

Practical Application.-The procedures developed can be used to es­timate the rate of leakage of unlined concrete containment structures subjected to internal pressure high enough to produce cracks through the concrete thickness of the wall [typically 60 psi in a prototype vessel (9)J. The methods as illustrated in this paper are for leakage of air. How­ever, they should be applicable to other fluids if appropriate fluid con­stants are used . The procedure presented also assumes that it is possible to maintain constant internal pressure during the evaluation of leakage, in spite of the fact that the maximum pressure that could be developed within the containment is related to the rate of leakage from the building and the rate of delivery of the pressurizing medium to the building.

CONCLUSIONS

Two dependent methods were introduced to predict the air leakage through cracked concrete. The mathematical formulation of the first method is designed to predict the air leakage through cracked concrete with known width of each individual crack in any given crack pattern. The concept of equivalent crack width for a given crack pattern is intro­duced in the second method. This concept removes the requirement of knowing every individual crack width of a given crack pattern, by in­troducing an equivalent crack width, provided that the number of cracks is known. Both methods were compared with the measured values and were found to be in good agreement.

To verify the reliability of the proposed techniques and to take into account the effect of prestressing and biaxial loading conditions, further independent testing should be undertaken.

ACKNOWLEDGMENTS

This study was carried out in the Department of Civil Engineering at the University of Manitoba with financial assistance from the Natural Sciences and Engineering Research Council of Canada and the Univer­sity of Manitoba Research Board.

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ApPENDIX.-REFERENCES

1. Beeby, A. W., "The Prediction of Crack Widths in Hardened Concrete," The Structural Engineer, Vol. 57A, No.1, Jan., 1979, pp. 9-17.

2. Buss, W., "Proof of Leakage Rate of a Concrete Reactor Building," Paper 34-61, Concrete For Nuclear Reactors, AO Special Publication Sp-34, Vol. ill, 1972, pp. 1291-1320.

3. John, J. E. A., Gas Dynamics, Allyn and Bacon, Boston, 1969. 4. Leonhardt, F., "Crack Control in Concrete Structures," IABSE Surveys, 5-4/

77, IABSE Periodica 3/1977, International Association for Bridge and Struc­tural Engineering, Zurich, Aug., 1977.

5. MacGregor, J. G., Rizkalla, S. H., and Simmonds, S. H. , "Cracking of Rein­forced and Prestressed Concrete Wall Segments," Structural Engineering Re­port, No. 82, Dept. of Civ. Engrg., Univ. of Alberta, Edmonton, Canada, Mar., 1980.

6. Rizkalla, S. H., EI-Shahawai, M., and Kwok, C. K., "Cracking Behavior of Reinforced Concrete Members," Proceedings of the Annual Conference of Ca­nadian Society for Civil Engineering, May 27-28, 1982, Edmonton, Alberta, Canada.

7. Rizkalla, S. H., and Lau, B. L., "Leakage of Pressurized Gases through Cracks in Reinforced Concrete Structures," Structural Engineering Report, Dept. of Civ. Engrg., Univ. of Manitoba, Winnipeg, Manitoba, Apr., 1982.

8. Rizkalla, S. H., Simmonds, S. H., and MacGregor, J. G., "Leakage Tests of Wall Segments of Reactor Containments." Structural Engineering Report No. 80, Dept. of Civ. Engrg., Univ. of Alberta, Edmonton, Canada.

9. 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, Dept. of Civ. Engrg., Univ. of Alberta, Ed­monton, Alberta, Canada, Apr., 1980.

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