Abstract

The original version of the 4SIGHT computer program was written to facilitate the

performance assessment of buried concrete structures based deterministic calculations

of service life, using data supplied by the user. In an eort to ensure reliability in the

results, a means of validation was required. Since the results from the calculations depend

strongly on the diusive and advective transport of ionic species, validating the transport

portion of 4SIGHT was paramount.

The validation of the transport portion of 4SIGHT used both reference and laboratory

data. The reference data include electrolyte densities, activity coecients, and binary

salt diusion coecients. The laboratory experiments included diusion experiments performed on relatively inert ceramic frits, using a range of ionic species and concentrations.

The laboratory experiments were also used to conrm the fundamental assumption that

multi-species transport in non-reacting systems can be characterized by two numbers,

the porosity and the formation factor, regardless of the number of diusing species.

A default analysis for estimating the spacing and thickness of 
exural and drying

shrinkage cracks was added to 4SIGHT. The 
exural cracks are calculated from a rudimentary structural analysis that assumes the roof of the vault acts as a one-way slab.

The drying shrinkage cracks are estimated, in large part, from the total water content

of the concrete mixture, the exposure environment, and the geometry of the slab. The

subsequent predicted crack widths, crack spacings, and location of the neutral axis, are

used to predict the bulk permeability of the concrete structure as a function of depth;

the portion above the neutral axis being uncracked, the portion below the neutral axis

being cracked.

Another addition to 4SIGHT was the means to perform a Monte Carlo calculation,

based on parameter uncertainty, of the duration of useful service. The service life calculation is repeated multiple times using a random number generator to calculate random

deviates for the physical parameters. The results can then be used to make a probabilistic

statement about the performance of the concrete structure.

The resulting computer program is state-of-the-art attempt to characterize both the

response of a concrete to its environment. However, there can be no assurances that the

model is exact for every scenario possible, regardless of parameter uncertainty. Therefore,

in cases that concern public safety, the only way to conrm the response of the concrete

to its environment is through continuous monitoring and periodic testing. The computer

program, in these circumstances, can also serve as a useful tool in establishing a testing

monitoring protocol. The spatial and temporal distribution of the monitoring and testing

protocol can be established using the computer program as a guide.

Keywords: building technology; concrete; degradation; Monte Carlo; parameter

uncertainty; service life; transport

iii
Contents

Abstract iii

1 Introduction 1

2 Degradation Validation Examples 3

2.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Leaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Chloride Diusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 HETEK Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Transport 10

3.1 Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Electrical Measurements . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.2 Diusive Ionic Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.3 Formation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.4 Conservation Equation . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.5 Complete Transport Equation . . . . . . . . . . . . . . . . . . . . 18

4 Transport Validation 19

4.1 Reference Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.1 Electrolyte Density . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1.2 Activity Coecient . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1.3 Electrolyte Diusion Coecient . . . . . . . . . . . . . . . . . . . 21

4.2 Laboratory Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 Ceramic Frit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.4 Formation Factor Measurement . . . . . . . . . . . . . . . . . . . 25

4.3 Formation Factor vs. Diusion Coecient . . . . . . . . . . . . . . . . . 26

4.3.1 Multi-Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Parameter Uncertainty 36

5.1 Random Deviates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.1 Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.2 Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.3 Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.4 Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Output: Cumulative Distribution . . . . . . . . . . . . . . . . . . . . . . 38

iv
6 Signicance of Cracks 40

6.1 Internal Body Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.2 Sorptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.3 Break-Through Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.4 Transport Through Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.4.1 Residence Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.4.2 Evaporative Mass Transfer . . . . . . . . . . . . . . . . . . . . . . 44

7 Predicting Drying Shrinkage and Flexural Cracks 46

7.1 Cracked Reinforced Concrete . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.1.1 Crack Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7.1.2 Composite Permeability . . . . . . . . . . . . . . . . . . . . . . . 48

7.1.3 Composite Permeability Model . . . . . . . . . . . . . . . . . . . 49

7.2 Flexural Crack Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7.3 Shrinkage Crack Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7.4 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.4.1 Mixture Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7.4.2 Concrete Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7.4.3 Steel Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.4.4 Reinforcement Ratio Requirements . . . . . . . . . . . . . . . . . 55

7.4.5 Roof Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

7.4.6 Positive Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.4.7 Uncracked Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.4.8 Cracked Section Analysis . . . . . . . . . . . . . . . . . . . . . . . 59

7.4.9 Simplied Composite Permeability . . . . . . . . . . . . . . . . . 62

7.5 Probabilistic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.5.1 Initial Material Parameters . . . . . . . . . . . . . . . . . . . . . 63

7.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8 User's Guide 68

8.1 Input File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

9 Summary 73

v
List of Tables

1 Estimated and reported diusion coecients, and estimated paste porosities, for the three concrete mixtures considered. . . . . . . . . . . . . . . 8

2 The measured diusion coecients (DHETEK) and the estimated diusion coecient (D4SIGHT) for dierent values of water-cement ratio (w/c). 9

3 Comparison of diusion coecients D from the computer program (CP)

and handbook (HB) (Ref. [30]) values for various salts in aqueous solution. 22

4 The values for the slope (ADb=L) of the experimental data shown in

Fig. 11. Also shown is the ratio Ds=Db, using Dsfor iodide. The uncertainties shown for the slopes are the estimated standard deviation reported by the statistical software, and also characterize the uncertainty in

the ratio Ds=Db reported. . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Measured and calculated formation factors from impedance spectroscopy

(FIS), computer simulation (Fsim), and apparent diusivity (Ds=Db) using

Dsfor iodide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Absolute value of the slopes calculated from the data for the KCl/KI

and the NaCl/KI systems in Figs. 12, regressed to a linear model. The

uncertainties shown are the estimated from standard deviations reported

by the regression software, with a corresponding coverage factor of k = 1. 31

7 Magnitude of the slope for data shown in Fig. 13(a), regressed to Eqn. 35

for t(v1

1 + v12 ) < 8 h/cm3

. For each system, the KI concentration was

0.10 mol/L. Also shown are the values for the formation factor F used

to produce the solid curves Fig. 13(b). The uncertainties shown are the

estimated from standard deviations reported by the regression software,

with a corresponding coverage factor of k = 1. . . . . . . . . . . . . . . . 31

8 Material parameters a concrete with w/c = 0.45, estimated from equations

in Ref. [1]. (Default degree of hydration  = 0.85) . . . . . . . . . . . . . 40

9 Parameters for the composite permeability calculation for m cracks of

width w. The crack model refers to that shown in either Fig. 19(a) or

Fig. 19(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10 The mean and associated coecient of variation (CoV) for the input parameters that would be used in an uncertainty analysis of 
exural and

shrinkage cracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

vi
List of Figures

1 Sample porosity, based upon pores 1 m to 10 m in size, as a function

of distance from the exposure surface. Graph taken from Tragardh and

Lagerblad [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Chloride concentration (percent by mass fraction of cement) for the three

concrete mixtures (solid symbols). The concretes are made with ordinary

sulfate resistant cements. The curves are estimates by 4SIGHT. Data are

reproduced from Sandberg et al. [6]. . . . . . . . . . . . . . . . . . . . . . 7

3 Schematic of the bulk 
ux and the pore 
ux, characterized by the bulk

velocity vb and pore velocity vp, due to a pressure hydrostatic pressure

drop of P1  P2 across the specimen. . . . . . . . . . . . . . . . . . . . . 15

4 Schematic for ionic 
ux conservation through a volume of unit area and

dierential width dx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Density of 1-1 electrolytes (a) and 1-2, 2-1, and 2-2 electrolytes (b) as

a function of molarity. Data from the chemistry handbook appear as

symbols. The predictions by 4SIGHT appear as solid curves. . . . . . . . . 19

6 Density of strong acids and bases. Data from the chemistry handbook

appear as symbols. The predictions by 4SIGHT appear as solid curves. . . 20

7 Activity coecient of chloride ions 
C l as a function of molality m for

various electrolytes. The estimates by 4SIGHT are shown as solid curves.

Results from the PHRQPITZ computer program [31] are shown as lled

symbols. Data from the CRC Handbook of Chemistry and Physics [30] are

shown as open symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8 The pore solution conductivity p and the corresponding bulk conductivity

b for the resin system of McDu and Ellis [36] and the frit system used

here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 Schematic of the constant gradient hypothesis across a sample with thickness L. Vessels 1 and 2 contain an aqueous electrolyte with target species

concentrations c1 and c2, and have volumes v1 and v2, respectively. . . . 25

10 The conductivity cell apparatus(a) and the diusion divided cell apparatus

(b). The schematics depict the conguration of the two cylindrical glass

vessels on either side of a mounted specimen. The system is sealed using

rubber o-rings; the clamps are not shown. The apparatus dier only in the

vertical platinum electrodes in the conductivity cell. The diameter of the

specimen, the glass vessels, and the platinum electrodes are approximately

50 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11 Concentration dierence across each specimen as a function of time normalized by the vessel volumes. The experimental values are shown as lled

symbols, the estimates by 4SIGHT are shown as solid curves. The measurement uncertainties would appear as the same size as the symbols, so are

omitted for visual clarity. The value of  for the 1.0 mol/L KCl system

is divided by ten in order to appear on the same scale as the other data. 27

vii
12 Concentration dierence  of iodide (lled symbols) for the KCl/KI and

the NaCl/KI systems; the time t is scaled by the vessel volumes v1 and

v2. The quantity 0 is the initial KI concentration. The lines shown

are the result of the linear regression analyses reported in Table 6. The

measurement uncertainties are approximately the size of the symbols used. 30

13 Measured concentration dierence  of iodide (lled symbols) for constant

0.10 mol/L KI concentration and varying counter-diusing species; the

time t is scaled by the vessel volumes v1 and v2. In (a) results from linear

regression (solid lines) for t(v1

1 + v12 ) < 8 h/cm3

; the calculated slopes

are reported in Table 7. In (b) the electro-diusion equation is used to

estimate the formation factor F , the values of which are shown in Table 7.

The error bars represent an estimated standard deviation. . . . . . . . . 32

14 The ratio of the apparent diusion coecient Da to the microstructural

diusion coecient D(as calculated by 4SIGHT using Eqn. 14) of iodide

for the various counter diusing electrolytes, as a function of time t. . . . 33

15 Experimental concentration dierence  of each diusing species in the

(a) CaCl2/KI, and the (b) K2CO3/KI systems, each as a function of time

t. The experimental values are shown as lled symbols, and the curves are

the values calculated by 4SIGHT. The error bars represent an estimated

standard deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

16 The cumulative distribution function G(t). . . . . . . . . . . . . . . . . . 39

17 Schematic of water sorption in concrete. The hashed region represents a

cross section, with pore radius R and liquid-air surface tension 
, with

its bottom surface under water. To the right is a plot of the hydraulic

pressure P as one traverses vertically through the pore. Negative pressure

indicates water in tension. . . . . . . . . . . . . . . . . . . . . . . . . . . 41

18 Schematic of the hydraulic pressures in a porous material with a thin layer

of water on its top surface and unsaturated air exposure on its bottom

surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

19 Composite models for the bulk permeability of the roof incorporating the

uncracked permeability kc, crack permeability k0, and having a number m

cracks over the slab length L: (a) slabs with m cracks, each with width w;

(b) slabs with m1 and m2 cracks of width w1 and w2, respectively. . . . . 48

20 Quantities for estimating shrinkage strain: (a) ultimate drying shrinkage

strain under standard conditions; (b) factor 
cp for duration of moist curing; (c) factor 
 for ambient relative humidity; and (d) factor 
s for the

volume to surface ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

21 Roof geometry used for analysis: (a) cross section showing location of steel

reinforcement (lled circles); (b) transformed section used for uncracked

analysis; (c) transformed section used for cracked analysis. . . . . . . . . 56

22 Depth of neutral axis c as a function of the reinforcement ratio  for the

uncracked analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

23 Compressive stress fc and tensile stress ft as a function of the reinforcement

ratio , based on the uncracked analysis. . . . . . . . . . . . . . . . . . . 60

viii
24 Cracked transformed section analysis: (a) relative depth of neutral axis ;

(b) relative tensile stress ft=fy and compressive stress fc=f 0cfor the cracked

section analysis, assuming linear elastic behavior. . . . . . . . . . . . . . 61

25 Results of the cracking model: (a) The ratio of the shrinkage crack spacing

Ls and the 
exural crack spacing Lf ; (b) Flexural crack width wf and

shrinkage crack width ws. . . . . . . . . . . . . . . . . . . . . . . . . . . 62

26 Ratio of the composite permeability k to the uncracked concrete permeability kc: (a) as a function of reinforcement ratio , and (b) as a function

of the relative depth to the neutral axis . . . . . . . . . . . . . . . . . . 63

27 Probabilistic analysis results: (a) Relative depth of neutral axis ; (b)

Relative tensile stress ft=fy and compressive stress fc=f 0cfor the cracked

section analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

28 Results of the probabilistic analysis of the cracking models: (a) the ratio

of the shrinkage crack spacing Ls to the 
exural crack spacing Lf ; (b)


exural crack width wf and shrinkage crack width ws. . . . . . . . . . . . 66

29 From the probabilistic analysis, the ratio of the composite permeability k to the uncracked concrete permeability kc. . . . . . . . . . . . . . . . . . 67

ix
1 Introduction

The 4SIGHT computer program, rst published in 1995 [1], was developed as a resource

for estimating the service life of new underground concrete structures. The program

was the rst to use combined numerical models for ion transport, chemical reaction, and

subsequent changes to transport coecients to model the response of a concrete structure

to its environment. Although the individual components of 4SIGHT duplicated previous

work, the unique ensemble allowed 4SIGHT to model the synergistic degradation eects

that may occur. While this approach is physically sound, the model remained to be

validated in order to ensure condence in the results.

The initial validation concept was to use samples from eld concrete and compare the

prediction of 4SIGHT to the actual response of the concrete. While conceptually sound,

there were unforeseen pitfalls to such an approach. Since 4SIGHT was to be used to

predict the performance over decades, and possibly centuries, comparable concrete data

were needed. While such concretes exist, the documentation of these structure is either

scarce or non-existent. Further, the environmental conditions that the concretes were

sub jected to were also dicult to quantify. These undetermined parameters were either

estimated using other data within the report or left as a free parameter, and adjusted until

the 4SIGHT prediction agreed with the eld concrete response. Since there were always

a number of input parameters that were undetermined, the eld validation procedure

could only be used to determine if 4SIGHT was unreasonable; the calculation could not

be used to validate the program. Although the results of these trials were favorable, they

were by no means a validation of 4SIGHT. The eld performance validation concept, which has been used elsewhere [2], is a

common approach in service life models for concrete. With few exceptions, these models

are empirical and attempt to predict the response of a concrete under certain environmental conditions; they do not attempt to capture every detail of transport and reaction.

Some of these models are based on a transport equation such as Fick's law. Others are

phenomenological modications to Fick's law that neglect the details of transport and

reaction and seek to simply mimic the macroscopic response of the concrete. Generally, the model parameters are estimated from eld data, and then the model is applied to

other concretes.

By contrast, the approach of 4SIGHT is to capture every detail of ion transport and

chemical reaction. In a sense, the objective is to simulate reality in an attempt to predict

future performance. To achieve this, the rst step is to fully characterize ion transport

and reaction within the concrete. The second is to fully characterize the degradation

process, which includes both the eects of reaction on transport properties and the

extent of reaction at which sucient pressures are produced so as to create mechanical

failure within the concrete. This approach is virtually unique among concrete service life

computer models.

In order to validate the transport portion of the 4SIGHT computer program an inert

ceramic frit was used as the model porous material. By using the ceramic frit, the

eects of transport could be isolated from the eects of reaction. Further, the results of

impedance spectroscopy experiments suggested that the frits contained relatively little

surface charge, which would result in relatively little surface binding of the diusing ionic

1
species, further contributing to nearly ideal diusive transport behavior. A practical

advantage of the ceramic over cement pastes is that the diusivity of the ceramic frits

was relatively large, meaning that data could be collected over a comparatively short

period of time.

The results from the diusion and the impedance spectroscopy measurements yielded

two important facts: the diusive transport of ionic species within a porous material

can be characterized by two numbers, the porosity and the formation factor (dened

subsequently), regardless of the number of dierent diusing species. The apparent

diusion coecient that is often reported is of relatively little use, particularly over long

periods of time. These two results are signicant in that the vast majority of concrete

materials transport studies concentrate on the determination of an apparent chloride

diusion coecient that is calculated from a simplistic transport equation. The fact that

formation factor is far more useful has distinct experimental advantage: the formation

factor can be determined from an electrical measurement that can be performed by

virtually every concrete materials laboratory. Further, the formation factor is a concept

that originated in the eld of geophysical research. Therefore, it is a technique that

should also be familiar to geological testing laboratories.

This report summarizes the experiments that were performed in order to validate

the transport portion of the 4SIGHT computer model. Some of the validation consists of

comparing the output from 4SIGHT calculations to physical and chemical handbook data

for various electrolyte properties. The remaining tests are laboratory experiments performed on the aforementioned ceramic frits. The details of the experiments are published

elsewhere, and the results are summarized here.

In addition to the validation, two enhancements to 4SIGHT were requested: a model

to predict crack formation within the concrete; and a means to perform a Monte Carlo

calculation of the service life based on parameter uncertainty. A model for crack formation was necessary due to the dramatic in
uence of cracks on advective transport (Darcy


ow) through a porous material. The Monte Carlo calculation addresses the nature of

uncertainty for any type of service life calculation. No physical parameter can be determined with absolute certainty, and in some cases small variations in a parameter can

have a dramatic eect on the overall performance of a structure. The Monte Carlo calculation was implemented with a robust random number generator and permits the user

to specify the parameter uncertainty. The cracking model considered the possible causes of cracks within the concrete and

divided the sources of cracking between \engineered" cracks and construction cracks. The

engineered cracks (
exural and drying shrinkage) can be reasonably estimated based on

the structural design, the materials specication, and the exposure. By contrast, construction cracks (plastic shrinkage, settlement, and thermal) can be avoided by applying

proper construction techniques; predicting how well builders will perform their duties is

an open-ended question with an indeterminate amount of uncertainty. Hence, the crack

prediction was limited to engineering cracks, and are estimated from a simple structural

mechanics model (
exural) and the relationships provided in the American Concrete

Institute (ACI) Manual of Concrete Practice [3] (drying shrinkage).

2
2 Degradation Validation Examples

The use of either literature data or eld data for validating 4SIGHT lacks rigor. In every

case, a critical piece of information is missing, which means that some parameter used

in the 4SIGHT model remains as an independent parameter. This free parameter can be

adjusted until the calculated output from 4SIGHT coincides with the results reported in

the literature. If the value chosen for the free parameter is typical for similar concretes,

one can only conclude that the 4SIGHT model is not unreasonable. A similar scenario arises with eld concretes. For a eld concrete that has been in

service for decades, many pieces of data are missing. In the rare cases that the construction was documented, many pieces of information required to make accurate estimates of

transport properties would not be present. Further, for these concrete structures there

are virtually no data of the environmental conditions the concrete had been subjected to

over the decades of service.

Another diculty in the use of eld concretes was nding an \old" concrete that

had been suciently documented, yet was in the appropriate state of degradation. If a

concrete structure that was placed in the 1940s degraded after two decades of service,

that concrete would no longer exist today. If the same concrete was appropriate for its

intended use, it would still be in service today, without any indications of degradation.

Old concrete in a measurable state of signicant degradation is rare. It was determined

that extraction and testing of eld concretes would not be cost eective given the rarity

of these specimens and the number of unresolved parameters.

The following examples serve as a demonstrations of the types of data available in

the literature. Although each example presents all of the relevant data presented in the

report, some data are still missing and must be estimated in some way.

2.1 Literature

The task of validating 4SIGHT using published literature required nding suitable reports

of concrete performance under various exposures. There are a large number of publications characterizing test specimens and environments. Because 4SIGHT will be used to

predict the long term response of concrete, publications dealing with \old" concrete were

needed. This fact alone reduced the number of eligible publications dramatically. An additional complication arises from the state of concrete technology at the time

the concrete was produced. Ideally, concretes at least 40 years old would be desirable for

this study. The ma jority of structures having this longevity are typically public pro jects

that consumed large quantities of concrete, the components of which varied over the life

of the project, and the properties of which were not characterized thoroughly. This lack

of detailed information limits present-day characterization of the concrete. Much of the

same can be said for the cements used at the time, for which very little quantitative

information exists regarding the cement neness, oxide composition, and mineral composition. Historical publications such as the Building Science Series [4] published by the

U.S. Department of Commerce can only serve as general guides.

3
0 5 10 15

Distance (mm)

0

2

4

6

8

10

Porosity (%)

Figure 1: Sample porosity, based upon pores 1 m to 10 m in size, as a function of

distance from the exposure surface. Graph taken from Tragardh and Lagerblad [5]

2.1.1 Leaching

In 1906, a drinking water tank was installed in Uppsala, Sweden. The tank was constructed of steel and lined with a concrete mortar. In 1991, the tank was demolished,

and the concrete was analyzed by the Swedish Cement and Concrete Research Institute.

The results of the analyses were published in an institute report written by Tragardh

and Lagerblad [5].

The water exposure of the concrete provided a nearly constant environment to the

concrete. During its service, the tank was periodically drained and lled. There was

relatively little evidence of carbonation, suggesting that the specimen analyzed had been

submerged for nearly its entire period of service.

Characterization of the concrete is more dicult. No information was given regarding

the mixture design. The only characterization the report oered was that the concentration of calcium hydroxide suggested the water-cement ratio might have been approximately 0.40, which would have been quite advanced for that time.

The porosity measurements for pores in the range 1 m to 10 m have been reproduced in Fig. 1. These measurements show that the depth of leaching is no greater than

7.5 mm, and that the bulk concrete porosity is 3 % to 4 %. The corresponding bulk paste

porosity would be approximately thrice this value, or 9 % to 12 %. Also, the measurements show that porosity near the exposed surface is nearly three times greater than the

bulk porosity, which may be due, in part, to the greater paste content near the surface.

Since the chloride diusion coecient of the sample was not measured, it had to be

estimated. The 4SIGHT prediction is based on a simple model for the diusion coecient

(m2s1) that is only a function of water-cement ratio w=c [1]:

log10 DCl = 6:0 w=c  13:84 (1)

Using this equation, the estimated diusivity for the tank is 3.61012 m2/s. This,

4
along with a water-cement ratio of 0.40, a 0.85 degree of hydration (an assumption in

the original 4SIGHT model), and a relation for the porosity  [1],

 = 1 

1+1:16

1+3:2w=c (2)

yields an estimate for the capillary porosity of 0.13, which is consistent with the measured

values of porosity. The 4SIGHT program was then used to estimate the porosity. The program started

with the following input data le (See Ref. [1] for a description of each parameter):

DIFF = 3.6B-12

WC = .40

THICKNESS = 0.100

EXTERNAL Na = 0.00000

EXTERNAL K = 0.00000

EXTERNAL C1 = 0.00000

INTERNAL K = 0.10000

INTERNAL Na - 0.05000

OUTPUT K

OUTPUT Ca

OUTPUT Ca OH

REBAR = .8000

HEAD = 0.0

DEPTH = .2000

JOINT = 0.00000 AT 1.0 UNTIL 100.

CRACK = 0.00000 AT 2.0 DEPTH 0.25

TIME = 33000

The diusivity DIFF is in units of m2/s; the concrete thickness THICKNESS is in units of

meters; the INTERNAL and EXTERNAL ion concentrations are in units of moles per liter.

This approach assumes that there were no cracks or joints in the concrete mortar.

Also, the REBAR and DEPTH variable were set to large values to eliminate the eects of

corrosion or sulfate attack.

The EXTERNAL parameters yield a neutral pH external environment:

Initial state of system:

" ION " "EXTERNAL" "INTERNAL"

" H:" 0.00000 0.00000

" Ca:" 0.00000 0.00035

" Na:" 0.00000 0.05000

" K:" 0.00000 0.10000

" OH:" 0.00000 0.15070

" pH:" 7.00000 13.17810

The results using the 4SIGHT estimate for the diusivity follow:

5
W/C = 0.40

D = 3.6 E -12

Final System state:

"L(m)" "porosity" "pH" "fc" "Ca" "K" "CaOH"

0.0000 0.1622 7.000 4525 0.0000 0.0000 0.000

0.0050 0.1483 12.404 4747 0.0123 0.0003 39.861

0.0100 0.1345 12.411 4990 0.0119 0.0009 42.963

0.0150 0.1346 12.421 4989 0.0114 0.0017 42.944

0.0200 0.1346 12.431 4988 0.0109 0.0024 42.933

Based upon the values for the porosity and the calcium hydroxide, 4SIGHT predicts that

the depth of leaching is between 5 mm and 10 mm. This agrees with the measured values

of porosity. However, the porosity near the surface is only a few percent greater than the

bulk value of 13.5 %.

A slight modication to the concrete diusivity yields a marked change in the 4SIGHT

estimate:

w/c = 0.40

D = 5.0 E -12

Final System state:

"L(m)" "porosity" "pH" "fc" "Ca" "K" "CaOH"

0.0000 0.4123 7.000 2471 0.0000 0.0000 0.000

0.0050 0.2896 12.400 3456 0.0125 0.0000 9.814

0.0100 0.1670 12.400 4991 0.0125 0.0000 31.971

0.0150 0.1671 12.404 4989 0.0123 0.0003 31.953

0.0200 0.1672 12.408 4988 0.0121 0.0005 31.942

Increasing the concrete diusivity from 3.6 to 5.0 (1012 m2/s) increased the estimated

bulk porosity to 17 %; since the correlation between porosity and diusivity has considerable uncertainty, it is dicult to assert that one of these estimates is unfounded.

The depth of leaching was still between 5 mm and 10 mm. However, the porosity at

the surface is now approximately thrice the bulk porosity, as was seen in the published

measurements. Although a bulk porosity of 17 % is greater than the measured value, the

size limit on the measured capillary pore sizes would suggest that additional capillary

porosity would exist in pores smaller than the 1 m limit used for the measurements. At

best, the values of diusivity appear to be reasonable upper and lower bounds.

2.1.2 Chloride Diusion

The Swedish National Testing and Research Institute initiated a research program in

which concrete specimens were placed on pontoons in sea water at the Traslovslage Marine Field Station. Concrete specimens from more than 50 mixture designs (0.25wc0.75)

6
0 10 20 30 40 50

Distance (mm)

0

1

2

3

4

5

6

[Cl−] (% mass fraction cement)

0.3 − 4sight

0.4 − 4sight

0.5 − 4sight

0.3 − Data

0.4 − Data

0.5 − Data

Figure 2: Chloride concentration (percent by mass fraction of cement) for the three

concrete mixtures (solid symbols). The concretes are made with ordinary sulfate resistant

cements. The curves are estimates by 4SIGHT. Data are reproduced from Sandberg et

al. [6].

were arranged in three tiers that dene their exposure zone: submerged zone, splash zone,

and atmospheric zone. A report of chloride ingress after an exposure of ve years was reported by Sandberg, Tang, and Andersen [6]. Samples were extracted from the exposure

site and milled, starting from the exposure surface, in 1 mm increments. The material

at each depth was analyzed for total chloride content.

The report lists a number of concrete mixtures studied in the program, many of which

contain pozzolanic additions. Here, only the results for the ordinary (sulfate resistant)

portland cement (SRPC) concrete are considered. The three mixtures are denoted SRPC

0.3, SRPC 0.4, and SRPC 0.5, where the number indicates the water-cement ratio. Also,

since 4SIGHT assumes the concrete was continuously saturated, only the data for the

submerged exposure are considered.

In the report of Sandberg et al. [6], the chloride ion diusion coecient was estimated

from the chloride concentration prole within the concrete. These concentration proles

are shown as lled symbols in Fig. 2. It is common that chloride concentrations have a

maximum within the concrete, so least squares regression between the solution to Ficks

law and the recorded values was applied to the data on the \falling" side of the curve.

The results of these regression analyses are shown in Table 1, along with the 4SIGHT

predictions for the diusion coecient and the paste porosity, based solely upon the

water-cement ratio. The curves in Fig. 2 are the 4SIGHT predictions for the chloride

concentration, which are based on the estimated diusion coecient, not the reported

value. The surface chloride concentration for the 4SIGHT estimate was chosen to most

7
Table 1: Estimated and reported diusion coecients, and estimated paste porosities,

for the three concrete mixtures considered.

4SIGHT Report

DCl Porosity DCl Mixture (1012 m2/s) (paste) (1012 m2/s)

SRPC 0.3 0.9 0.04 1.5

SRPC 0.4 3.6 0.14 1.4

SRPC 0.5 14.5 0.22 2.3

reasonably approximate the data; no formal regression was used. Qualitatively, only the

estimation for the SRPC 0.3 mixture is accurate. The results for the SRPC 0.4 mixture

are reasonable, but not remarkable. The diusion coecients from both the 4SIGHT and

the Sandberg reports are shown in Table 1. The variation in the diusion coecients

calculated by regression analysis [6] was less than expected. This is striking given the

range in estimated paste porosities.

The reason for the similarity among diusion coecients calculated by Sandberg et

al. [6] can be explained by the presence of magnesium in the sea water. The magnesium

reacts with the available calcium hydroxide to form magnesium hydroxide. The reaction

products consume a greater volume than the reactants, and the pores become lled [7].

The greater the quantity of calcium hydroxide, which is proportional to the water-tocement ratio, the greater is this eect. Since a 0.3 water-cement ratio concrete would

have relatively little available calcium hydroxide, the 4SIGHT estimation for this mixture

was relatively accurate. This eect also explains why the reported diusion coecients

for the other concretes were much lower than the 4SIGHT estimates.

2.1.3 HETEK Model

The Hjkvalitetsbeton-Entreprenrens TEKnolgi (HETEK) fHigh quality concrete, the

Contractors TEChnologyg program was initiated by the Danish Road Directorate to establish a system for estimating concrete service life for exposure to marine environments.

The entire HETEK program is subdivided into a number of tasks, one of which addresses

chloride penetration into concrete. Although HETEK was designed primarily for concrete in marine environments, it would be useful to simply compare the HETEK and the

4SIGHT models.

Since the HETEK program was developed for a specic purpose, the approach to

concrete degradation has a focused approach. In a marine environment, cements with

reduced sulfate content are specied in order to reduce the reaction of magnesium from

within the sea water with sulfate within the cement. The product of this reaction, magnesium sulfate, leads to expansion. An additional eect due to the low sulfate content is

that the magnesium is available to react with the calcium hydroxide within the cement

paste. The product of this reaction, magnesium hydroxide, has a smaller molar volume

than calcium hydroxide, so it does not lead to expansion. Rather, the magnesium hy8
Table 2: The measured diusion coecients (DHETEK) and the estimated diusion

coecient (D4SIGHT) for dierent values of water-cement ratio (w/c).

w/c DHETEK D4SIGHT

0.32 3.9 1.2

0.40 8.5 3.6

0.70 25.1 229.1

droxide seems to \clog" the pore space, eectively reducing the diusion coecient of

the concrete.

Based upon experimental data, the apparent reduction in the diusion coecient due

to the magnesium hydroxide is given by the following relationship [2]:

D(t)

D(to) =



to tex + t

(3)

The quantity tex is the time at which the concrete was exposed to the marine environment. The quantity to is the time at which the concrete diusion coecient has

reached a steady value, typically 6 months. The parameter  varies from 0.37 to 0.23 for

water-cement ratios ranging from 0.30 to 0.75, respectively. The HETEK model is based upon the solution of Fick's equation with a timedependent diusion coecient D(t):

@c

@t = D(t) @2c@x2

The solution to this equation has been published previously [8] for diusion coecients

having a time dependence as given in Eqn. 3 above. The time dependent behavior

for a specic mixture is based upon measurements performed on similar concretes that

have had similar exposures. The result is a manual [9] that consists, in large part, of

nomographs for the estimation of concrete service life, dened as the time until failure

due to corrosion of the steel reinforcement.

To develop the analytical equations for the time-dependent diusion coecient, the

researchers required large quantities of diusion data. Only a fraction of these data are

shown in the reports. As a basic check, a comparison of the measured diusion coecients

with the estimates by 4SIGHT based upon the water-cement ratio is given in Table 2. The

estimates for the 0.32 and the 0.40 water-cement ratio concretes are within an order of

magnitude of the HETEK values. This is encouraging given the single datum, w/c, used

for the estimate. The 4SIGHT estimate for the 0.70 water-cement ratio concrete diers by

an order of magnitude. Fortunately, 0.70 water-cement ratio concretes are undesirable

for structures designed to inhibit advective and diusive 
ow.

9
3 Transport

Validating the transport model within 4SIGHT is paramount to validating its estimate

of service life. Every degradation mechanism considered by 4SIGHT depends on the

transport of ionic species through the concrete. Therefore, a proper validation requires

that the transport model is suciently comprehensive in order that it can predict the

behavior of electrolytic solutions.

The rst characteristic of electrolytic solutions to be considered is the coulomb interactions among the diusing ionic species. To account for ion-ion interactions one

must determine how the ions would respond to charge imbalances. The solution is the

electro-diusion equation, which has been used extensively in solid state physics [10], the

biological sciences [11, 12], and only recently in the concrete materials research [13{16].

The second characteristic is due to changes in the solution due to changes in species

chemical activity. Since the pore solution of cement paste typically has an ionic strength

of approximately 0.5 mol/kg, one must also account for the nonideal diusive behavior

of each species. This nonideal behavior is related to excluded volume and the second

virial coecient. This eect is quantied through the chemical activity of the species.

The chemical activity a of a species can be related to the ionic concentration c through

the mathematical construct of an activity coecient 
:

a = 
 c (4)

The value of the activity coecient, a dimensionless number between zero and one,

depends on the concentration and units used for the concentration: mol/L, mol/kg,

mol/m3, etc. Although there are a number of approximation schemes for calculating

the activity coecient, the Pitzer equations [17] were used in 4SIGHT since relevant

coecients are readily available and the resulting estimate is accurate over the range of

ionic strengths typically found in cement paste pore solution.

The combination of the electrostatic interactions and the activity coecient calculation yielded a robust transport equation. The electro-diusion equation can be used to

predict the single diusion coecient for a binary salt, which are reported in chemistry

reference handbooks, over a range of concentrations, demonstrating the concentration

dependence on the apparent diusion coecient.

3.1 Transport Equation

The pore solution of hardened cement paste has an ionic strength of approximately 0.5

mol/kg. At this ionic strength, the pore solution cannot be modeled using equations that

characterize idealized transport, which generally treat each ionic species independently of

one another. Further, regardless of the ionic strength, the electrostatic interactions among

the various species are quite strong. Although the self-diusion coecient for sodium

ions is smaller than that of chloride ions, the electrostatic interactions enforce electroneutrality, and the two ions diuse with the same apparent self-diusion coecient, giving

a diusion coecient for the salt; these diusion coecients are reported in chemistry

reference handbooks.

10
Further, the chemistry handbooks report the salt diusion coecient as a function

of concentration. The electrostatic interactions, by themselves, are insucient to characterize this phenomenon. One approach to capture this behavior in a transport model

is to use self-diusion coecients that are a function of concentration. However, the

concentration dependence of sodium chloride is not the same as for magnesium chloride.

Therefore, some experimenters have used empirical relations to predict diusion coe-

cients based on both concentration and speciation [18]. This approach, however, can only

be used to interpolate results; any new species, or concentrations outside the range from

which the coecients were determined, would invalidate the result.

The alternative approach is to characterize the ion-ion interactions at a fundamental

level, thereby \calculating" the eects of concentration and speciation. The electrostatic

interactions give rise to an electrical eld that ensures electro-neutrality. This eld is

unique and can be calculated using principles of electrostatics. The macroscopic eect of

the electrostatic interactions is an electrical potential (the diusion potential) that exists

across the specimen, and is measurable.

The concentration dependence arises from mechanical ion-ion interactions that only

manifest themselves at elevated concentrations. The presence of a concentrated ionic

species acts to change the hydrodynamic properties of the electrolyte, thereby changing

the apparent viscosity of the 
uid. This behavior is related to the species activity, which

is a function of its chemical potential.

3.1.1 Electrical Measurements

Before discussing the transport equation, it will be useful to consider the relationship

between conductivity and diusivity. These principles are fundamental to the nal expression of the transport equation. The nal form exploits the formation factor, which is

a material parameter that can be determined directly from conductivity measurements.

Conductivity measurements hold great promise for estimating the formation factor

of saturated porous materials. The two most commonly used conduction techniques are

the electrical migration test (driven diusion) and the conductivity measurement. Migration tests are used to determine the ionic mobility, which can be related to diusivity. Conduction tests are used to estimate the formation factor, which can also be related to

diusivity. While there have been many reports on the use of the electrical migration

tests to determine the diusion coecient of cementitious systems [19], there have been

fewer reports on the use of the formation factor. This is unfortunate, since the formation

factor is directly related to the diusivity of the material pore structure. The relationship

between formation factor and diusivity, however, has subtleties that must be considered

when making estimates of one from the other.

The formation factor F has its origin in geological research on saturated porous materials. For a nonconducting porous solid saturated with a conducting pore solution, the

formation factor is the ratio of the pore solution conductivity p to the bulk conductivity

b [20]:

F =

p

b

(5)

This quantity characterizes the solid microstructure since the only dierence between the

11
conductivities is due to the restricted pathways through which the current is constrained

in the bulk conductivity measurement.

The use of conduction tests to estimate the diusion coecient is due to the relationship between conductivity and diusivity. In an electrolytic solution, the contribution an

ionic species makes to the overall conductivity can be expressed as a function of its conventional (electrochemical) mobility u, amount-of-substance concentration c, and valence

z [21]:

 = z c F u (6)

The quantity F is the Faraday constant. The mobility can then be related to diusivity

through the Einstein relation [21]:

zFD = RT u (7)

The quantity R is the universal gas constant and T is the absolute temperature. Using

these equations, one can estimate the diusion coecient from either measurements of

ion mobility (driven diusion) or from the conductivity contribution of a particular ion

(conductivity).

Alternatively, if a porous material is saturated with a dilute electrolytic solution, the

ratio of the pore solution conductivity p to the bulk conductivity b would be equal to

the ratio of the diusion coecient of an ion in the pore solution Dp to the bulk diusion

coecient Db of that ion:

F =

p

b = DpDb

(8)

Note that this relationship holds even if the Einstein relation is incorrect at large concentrations; the multiplicative error occurring in both Dp and Db would cancel. Since

the self-diusion coecient of ions can be found in books, this is a provocative approach

to determining the bulk diusion coecient Db [22].

Unfortunately, the pore solution of typical cementitious systems can have a large ionic

strength (0.5 molkg1 to 1.0 molkg1), and so care must be exercised when estimating

the bulk diusion coecient from the formation factor. At these large ionic strengths,

the pore solution diusion coecient Dp of an ion is dierent from the self-diusion

coecient, which is the value reported in tables [23]. Equation 8 can still be used, if one

can correctly determine the diusion coecient Dp in the pore solution. Unfortunately, given the constraints of most experimental diusion apparatus, and that experimenters

typically measure time dependent concentrations, an exact value for Dp can be dicult

to dene.

An alternative approach is to determine the formation factor from the diusion data

by separating the microstructural dependence from the concentration dependence in the

diusion coecient. It will be shown that the formation factor can be alternatively expressed as a function of the microstructural diusion coecient of the porous material.

The microstructural diusion coecient is independent of both changes in the pore solution chemistry and the diusion potential. The estimated formation factor could then

be used to estimate the diusive transport of any ion within the studied system since it

uniquely characterizes the solid pore structure; the chemical eects would be calculated

independently.

12
3.1.2 Diusive Ionic Flux

The proper application of the formation factor to porous systems containing concentrated

electrolytes can be best studied by starting with the appropriate transport equation.

For the case of diusing charged species in concentrated electrolytes, the transport is

governed by coupled electrical and diusive transport, and ion-ion interactions must be

also be considered.

The diusion of ionic species in an electrolyte is governed by the electro-diusion

equation. For the i-th ionic species, the electro-diusion equation relates the bulk 
ux

j to the concentration c, the diusion potential D, the bulk microstructural diusion

coecient D, and the bulk conventional mobility u [10]:

j = D 1 +

@ ln 


@ ln c ! rc  z F u c r D (9)

The subscript i has been omitted here in order to improve visual clarity of the equation,

and will be added later. The quantity 
 is the activity coecient for the species. Although this equation neglects adsorption eects, it is otherwise a complete description

of the nonreacting diusive transport of charged species in concentrated electrolytes.

Equation 9 still bears a resemblance to Fick's law for apparent diusion Da [24]:

j = Darc (10)

The quantity D[1 + @ ln 
=@ ln c] is an agglomerated diusion coecient that includes

the eects of changes in both the microstructure and the concentration.

It should be noted that the agglomerated diusion coecient is not the apparent

bulk diusion coecient Db. Strictly speaking, the apparent diusion coecient also

includes the eects of the diusion potential D. In those cases where the self-diusion

coecients of all the diusing species are nearly identical, the diusion potential will be

nearly zero, and the apparent diusion coecient will be nearly equal to the agglomerated

diusion coecient. However, for cementitious systems, there are many species present

with varying self-diusion coecients.

Ideally, one would like to distinguish between the eects due to microstructural

changes and the eects due to changes in the pore solution electro-chemistry. By observation, one can see that the microstructural diusion coecient D characterizes the

solid microstructure and that the quantity in parenthesis characterizes the pore solution

chemistry. The microstructural diusion coecient is itself independent of the pore solution chemistry. A demonstration of how to extract the value of microstructural diusion

coecient from a diusion experiment is accomplished through a detailed discussion of

the formation factor.

3.1.3 Formation Factor

Consider an electrical measurement of the formation factor F performed on a porous

specimen saturated with a concentrated electrolyte. The conductivity of the pore solution

p is a function of the ionic strength I . The conductivity is proportional to the ionic

strength such that as the ionic strength approaches a value of zero, the conductivity also

13
approaches a value of zero. Similarly, the bulk conductivity has the same dependence

on the pore solution conductivity. If the pore solution conductivity doubles, the bulk

conductivity will also double, ignoring surface conduction contributions.

The diusion coecient of an ion in a concentrated electrolyte has a dierent type

of dependence on the ionic strength. The self-diusion coecients reported in tables are

limiting values under the condition that the limit that both the concentration and the

diusion potential go to zero. In the practical dilute limit, the cations and the anions of

a 1:1 valence electrolyte must diuse together due to charge neutrality. This leads to the

creation of the diusion potential. Therefore, the pore solution diusion coecient Dp

must be an function of the ionic strength, activity coecients, and the diusion potential:

Dp = Ds[1 + g(I; 
; D)] (11)

The function g is an arbitrarily complex function of the ionic strength I , the activity

coecient 
, and the diusion potential D. It characterizes the interaction between

diusion ionic species and the bulk electrolyte. The quantity Dsis the self-diusion

coecient reported in tables [23]. A similar relationship characterizes the bulk diusion

coecient Db:

Db = D [1 + g(I; 
; D)] (12)

The function g is the same since the diusion ionic species will see the same electrolyte.

The microstructural diusivity D is the value corresponding to the dilute limit for

the bulk diusivity. It is independent of both the activity coecient and the diusion

potential, and is the same quantity that appears in Eqn. 9.

Using these relations, the formation factor can be written as a function of quantities

that are themselves functions of the ionic strength I , activity coecient 
, and the

diusion potential D:

F =

p(I )

b(I ) = Ds

[1 + g(I; 
; D)]

D [1 + g(I; 
; D)] = Ds D

(13)

However, the resulting formation factor F is independent of both the ionic strength and

the ionic species; a fact that is exploited by numerical calculations of formation factor

on model microstructures [25].

The advantage of using the ratio Ds=D is not immediately obvious. In Eqn. 8, Db

is observable, but Dp is dicult to determine precisely in typical diusion experiments

containing a variety of species, each with varying concentration. In Eqn. 13, Ds can

be found in tables, but the quantity D cannot be observed in electrolytes because the

eects of the diusion potential cannot be eliminated. Serendipitously, the usefulness of

D in service life prediction through its use in the electro-diusion equation is also the

key to determining this quantity from experiment.

The formation factor is extremely useful in models of ionic transport. Accurate service

life predictions require models that can account for independent changes in either the solid

microstructure or the pore solution chemistry. A suitable computer program could solve

the electro-diusion transport equation by using estimates of the ion activity coecient 


from published empirical relations and using the local electro-neutrality condition to solve

14
for the diusion potential. The microstructural diusion coecient D in the electrodiusion equation would be calculated from the formation factor F and the dilute limit

self diusion coecient Ds:

D = Ds F

(14)

Note that this relation holds for all the species, yet there is only a single value for the

formation factor F . Alternatively, one can determine the formation factor from experimental diusion

data by using the same computer program. Using a computer program that implements

Eqns. 9 and 14, one adjusts the formation factor until the computed output matches the

experimentally observed quantities. The experimental program described subsequently

used this approach to determine the formation factor from divided cell diusion measurements.

3.1.4 Conservation Equation

P

2

P

1

vp

v

b









































































































 

Specimen

Figure 3: Schematic of the bulk 
ux and the pore 
ux, characterized by the bulk velocity

vb and pore velocity vp, due to a pressure hydrostatic pressure drop of P1  P2 across

the specimen.

The 
ux in Eqn. 9 was developed for transport within an electrolyte. Transport

through a porous material must account for the solid framework that comprises the

material. The schematic of the Darcy experiment shown in Fig. 3 demonstrates the

dierence between the pore 
ux (through electrolyte) and bulk 
ux (through porous

material).

The experiment shown in Fig. 3 includes a porous specimen saturated with a brine.

The volumes above and below the specimen are lled with the same brine. Due to

a hydrostatic pressure P1 applied above the sample, there is a mass 
ow of the brine

through the porous specimen. The top free surface of 
uid \falls" at a velocity of vb,

15
representing the bulk velocity of the 
uid through the sample. If the 
uid above the

sample were instantaneously lled with a colored dye, the \front" of the dye would

advance through the specimen at the pore velocity vp. The pore 
ux jp of ionic species

within the brine would be proportional to the species concentration c: jp = cvp. Similarly

for the bulk 
ux: jb = cvb. Employing conservation of mass, and assuming that water is incompressible at these

pressures, the pore velocity vp will greater than the bulk velocity vb. The ratio of the

velocities is the connected porosity : vb = vp. The porosity is also the ratio of the pore

area and the bulk area: Ap = Ab. Likewise, by conservation of ionic species, the rate at

which species are transported through the system is same above, inside, and below the

specimen:

jb Ab = jp Ap (15)

jb =  jp (16)

This is an important relationship when considering diusive and advective transport

through porous media. With a relationship between the pore 
ux jp and the bulk 
ux

jb, one can now develop the appropriate 
ux relationships for each of the three transport

mechanisms (diusion, permeation, and conduction) considered in 4SIGHT. As mentioned previously. the hydrodynamic 
ux will occur due to Darcy 
ow under

a hydraulic pressure gradient rP across the specimen. The bulk velocity vb is equivalent

to the Darcy velocity:

vb = kb



rP (17)

This result, and the similar one for the pore velocity, can be substituted into the appropriate 
ux equation:

jb = cvb = ckb



rP (18)

jp = cvp = ckp



rP (19)

kb = kp (20)

In nearly every application of Darcy's law to concrete, researchers report the bulk permeability kb, so the bulk 
ux jb is most convenient.

Although it has been argued that the electro-diusion equation is a more complete

description of diusive transport, the simplied Fick's law is used here as a demonstration

of the distinction between pore 
ux and bulk 
ux:

jb = Dbrc (21)

jp = Dprc (22)

Db = Dp (23)

Reported results from diusion measurements vary between reporting either the pore


ux jp or the bulk 
ux jb . Unfortunately, this distinction is not always made, and so the

reader must exercise care when interpreting the results from diusion experiments.

16
jx (x,t) jx (x+dx,t)

dx

Figure 4: Schematic for ionic 
ux conservation through a volume of unit area and dierential width dx. The conductive 
ux is that which occurs due to the application of an electrical eld.

Strictly speaking, the 
ux is proportional to the conventional ionic mobility u. The pore


ux and bulk 
ux depend on the corresponding pore and bulk mobilities:

jb = zcFubr (24)

jp = zcFupr (25)

ub = up (26)

In general, the mobilities are approximated using the Einstein relation:

u = D

RT (27)

The diusion coecient D is the self-diusion coecient reported in tables [23]. This

relationship, however, is only valid in the dilute limit.

With the pore and bulk 
ux equations established for both diusion and conduction

transport, a time-dependent equation is needed in order to calculate changes in concentration. This equation can be derived from the conservation of species. Figure 4 shows a

volume element with unit area and dierential width dx. The x-component of the species


ux entering the volume is jx(x; t), and the 
ux exiting is jx(x + dx; t). The rate, per

unit area, at which species are lling the volume is jx(x; t)  jx(x + dx; t). The change

in the concentration is this dierence divided by the width dx:

@c

@t =

jx(x; t)  jx(x + dx; t)

dx = @jx @x

@c

@t = r  j (28)

Now this equation must be reformulated to account for a porous material.

The equation for a porous material can be obtained from the denition of concentration. The experimentally useful quantity is the intrinsic, or pore concentration c, in

units of moles of species per unit pore volume Vp. Let the bulk concentration c represent

17
the number moles per unit bulk volume Vb (volume of pore solution and solid structure

combined). Apply Gauss's divergence theorem to Eqn. 28 over the bulk volume Vb:

Z @c

@t dVb = 

Z r  jb dVb (29)

The bulk concentration c can be related to the intrinsic concentration through the porosity: c = c. The resulting conservation equation relates the bulk 
ux jb to the intrinsic

concentration c:

@c

@t = r  jb (30)

@c

@t = r  jp (31)

These relationships have been published elsewhere [26{29], primarily in the eld of geophysical research.

3.1.5 Complete Transport Equation

The complete transport equation has separate terms for the diusion, conduction, and

permeation components of the 
ux. Substituting for the bulk microstructural diusion

D, the self-diusion coecient Ds

i , the formation factor F , and the denition of the

Darcy velocity vb yields the following time-dependent transport equation.

@ci

@t = r  jb;i = Ds

i

F r  

1 +

@ ln 
i

@ ln ci ! rci +

ziF

RT

Ds

i

F r  cir D + r  kbci

rP (32)

This transport equation is signicant in that only three numbers (, F , kb) are required

in order to characterize the transport of an arbitrary number of ionic species.

18
4 Transport Validation

Validation of the transport equation within 4SIGHT is a two-stage process. First, 4SIGHT

will be used to calculate the properties of bulk 
uids. These properties include density, activity coecient, and salt diusion coecient. These results can then be compared

to values reported in chemistry handbooks. Second, 4SIGHT will be used to calculate

the response of a controlled diusion experiment performed on a well-dened porous

material. The porous material is nonreactive and has relatively little surface charge.

4.1 Reference Data

Comparisons between 4SIGHT and reference data are needed in order to ensure that the

fundamental building blocks of 4SIGHT are constructed in an orderly manner. Practical

eld data are nearly always reported in units of molarity (mol/L), or similar units of

quantity per volume. The calculation of the activity coecient, however, is based on

concentration expressed in units of molality (mol/kg). Converting between molarity

and molality, requires an estimate of the solution density. From a coherent means of

calculating density and activity coecient, one can then use the electro-diusion equation

to calculate the diusion coecient of various binary salts and compare the results to

values reported in chemistry handbooks.

4.1.1 Electrolyte Density

0.0 0.2 0.4 0.6 0.8 1.0

Molarity (mol/L)

1.00

1.02

1.04

1.06

1.08

1.10

1.12

Density (kg/L)

NaNO3

NaCl

NaBr

KNO3

KCl

KBr

KI

0.0 0.2 0.4 0.6 0.8

Molarity (mol/L)

1.00

1.02

1.04

1.06

1.08

1.10

1.12

Density (kg/L)

Na2CO3

Na2SO4

K2CO3

K2SO4

MgCl2

CaCl2

MgSO4

Figure 5: Density of 1-1 electrolytes (a) and 1-2, 2-1, and 2-2 electrolytes (b) as a function

of molarity. Data from the chemistry handbook appear as symbols. The predictions by

4SIGHT appear as solid curves.

A review of approximation schemes for various electrolyte properties revealed no

established means of estimating the electrolyte density. The scheme implemented by

19
4SIGHT was to assume that each ionic species contributes independently to the overall

density. The contribution is a function of the density and the mole fraction of that species

within the electrolyte.

In Figs. 5a and 5b are the results for nominally pH neutral salts. The density is

shown as a function of electrolyte concentration. The estimates of 4SIGHT are denoted

by solid curves. The data from the CRC Handbook [30] are denoted by lled symbols.

The results for the 1-1 electrolytes are shown in Fig. 5(a), and those for the 1-2, 2-1, and

2-2 electrolytes are shown in Fig. 5(b). In both cases, the approximations are relatively

accurate over the full range of concentrations. In general, the accuracy for the 1:1 salts

are better than for the 1-2 and 2-1 salts. For both cases, the accuracy is well within one

percent over the concentration range shown.

0.0 0.1 0.2 0.3 0.4 0.5

Molarity (mol/L)

1.00

1.01

1.02

1.03

Density (kg/L)

HNO3

H2SO4

HCl

NaOH

KOH

Figure 6: Density of strong acids and bases. Data from the chemistry handbook appear

as symbols. The predictions by 4SIGHT appear as solid curves.

The density estimates for strong acids and bases are shown in Fig. 6 as a function

of electrolyte concentration. The estimates by 4SIGHT are denoted by solid curves. The

handbook values for acids are denoted by lled symbols, and values for bases are denoted

by open symbols. Generally, the results for acids and bases are not as accurate as for

the pH neutral salts. Nevertheless, the accuracy is still less than a percent over the

concentration range shown.

4.1.2 Activity Coecient

Accurate estimates of activity coecients are necessary for both calculations of transport


ux at elevated concentrations and for calculations of equilibrium species concentrations

in reacting systems. In 4SIGHT, the activity coecient is calculated using the equations

of Pitzer [17]. To reduce complexity and gain computational speed, a \truncated" implementation was used. All interactions composed of two and three charged species were

implemented; all other interactions were been omitted.

20
0.0 0.2 0.4 0.6 0.8 1.0

m

1/2 (mol/kg)1/2

0.6

0.8

1.0

1.2

γCl

−

NaCl

KCl

CaCl2

MgCl2

Figure 7: Activity coecient of chloride ions 
C l as a function of molality m for various

electrolytes. The estimates by 4SIGHT are shown as solid curves. Results from the

PHRQPITZ computer program [31] are shown as lled symbols. Data from the CRC

Handbook of Chemistry and Physics [30] are shown as open symbols.

As a measure of performance, the results from 4SIGHT are compared to both CRC

Handbook data and the results of the PHRQPITZ [31] computer program, which is a full

implementation. Shown in Fig. 7 are the results for the activity coecient of chloride ions


C l as a function of electrolyte concentration. The estimates by 4SIGHT are denoted

by solid curves. The values from PHRQPITZ are denoted by lled symbols, and the

values from the CRC Handbook are denoted by open symbols. Generally, the agreement

among all three is very good over the entire range of electrolyte concentration shown,

suggesting that the 4SIGHT implementation is sucient. This example of chloride ion

activity is characteristic of the accuracy with which 4SIGHT can estimate the activity of

all the ionic species incorporated into the program.

4.1.3 Electrolyte Diusion Coecient

The electrolyte diusion coecient was calculated by 4SIGHT and compared to values

published in the CRC Handbook. The \sample" had a porosity of one, and a formation

factor of one. The change in concentration across the specimen was only a few percent.

The 
ux of salt passing through the solution was then used to calculate an apparent

diusion coecient, based on Fick's law (Eqn. 10). This apparent diusion coecient

was then compared with the handbook value.

Data for various electrolytic solutions are shown in Table 3 over varying concentrations. Generally, 4SIGHT is accurate to within a few percent up to 0.1 mol/L. The error

increases with concentration and is greater for larger valence species. The results are still

reasonable, considering that a species with a 1.0 mol/L concentration within concrete

pore solution is relatively rare.

21
Table 3: Comparison of diusion coecients D from the computer program (CP) and

handbook (HB) (Ref. [30]) values for various salts in aqueous solution.

Salt conc. DHB DCP

DCP DHB DHB molL1 109 m2

s1 109 m2s1

KCl 0.01 1.917 1.902 -0.0078

0.10 1.844 1.807 -0.0201

1.00 1.892 1.801 -0.0481

NaCl 0.01 1.545 1.539 -0.0039

0.10 1.483 1.476 -0.0047

1.00 1.484 1.571 +0.0586

LiCl 0.01 1.312 1.307 -0.0038

0.10 1.269 1.277 +0.0063

1.00 1.302 1.555 +0.1943

LiBr 0.10 1.279 1.286 +0.0055

1.00 1.404 1.647 +0.1731

LiNO3 0.01 1.276 1.277 +0.0008

0.10 1.240 1.242 +0.0016

1.00 1.293 1.467 +0.1346

KNO3 0.01 1.846 1.831 -0.0081

KI 0.10 1.865 1.829 -0.0193

1.00 2.065 1.911 -0.0746

Na2SO4 0.01 1.123 1.059 -0.0570

MgSO4 0.01 0.710 0.568 -0.2000

The result for MgSO4 is rather anomalous. The value calculated by 4SIGHT is consistent with the approximation based on the Nernst relation:

D = (z+ + z) D+D

z+D+ + zD 

1 +

@ ln 


@ ln c !

(33)

The values calculated by 4SIGHT are consistent with this equation, as are many of the

handbook values. The discrepancy in the MgSO4 result may be due to either experimental errors in the handbook value or the use of the Einstein relation in Eqn. 7 as an

approximation for the ionic mobility u. A more rigorous estimate would require a concerted eort to develop an estimate of the ionic mobility as a function of concentration.

4.2 Laboratory Data

The laboratory experiments were performed in two stages. The rst stage validates the

use of the formation factor in the transport equation. Independent electrical measurements of formation factor are compared with values estimated from diusion data. The

second stage consists of a series of diusion measurements that attempts to demonstrate

22
that the transport equation is valid for a range of ionic species. Further, the varied

behavior of these electrolytes was used to demonstrate that the apparent bulk diusion

coecient depends strongly on the species present, and comparatively weakly on the

ionic strength, over the range of concentrations that are of interest.

Results from the two stages of experiments have been presented [32,33] and submitted

for publication [34, 35]. The experimental details and the results are repeated here for

completeness.

4.2.1 Ceramic Frit

The diusion experiments were performed on a commercial porous frit composed of a

sintered alumina ceramic (99.8 % alumina) measuring (50.760.04) mm in diameter and

(6.480.03) mm thick; the uncertainty is an estimated standard deviation calculated from

7 frits. Mercury intrusion porosimetry analysis revealed a porosity of approximately 0.26,

and a pore size distribution concentrated between 350 nm and 450 nm. This is consistent

with the manufacturer's published specication of a pore size less than 500 nm.

The alumina frit was regarded as a nearly ideal porous medium for studying diusive

transport. The alumina is chemically stable at moderate values of pH. In addition,

conductivity measurements suggest that the frit has a relatively low bound surface charge.

As dened in Eqn. 5, the formation factor F is calculated from the ratio of the pore 
uid

electrical conductivity p to the the bulk electrical conductivity b with that pore 
uid.

The calculation can be repeated for dierent pore solutions with varying values of p.

0 10 20 30 40 50

σ

p (S/m)

0

5

10

15

σb (S/m)

Resin

Frit

Figure 8: The pore solution conductivity p and the corresponding bulk conductivity b

for the resin system of McDu and Ellis [36] and the frit system used here.

Such an experiment was performed in 1979 by McDu and Ellis [36] on a model porous

material composed of a packed column of cation exchange resin. Their data for large

23
pore solution conductivities are denoted in Fig. 8 by lled circles. Linear ordinary least

squares regression applied to the data reveal a positive intercept of (0.680.17) S/m; the

uncertainty is an estimated standard deviation. Similarly, one of the frits was saturated

with standard potassium chloride conductivity solutions [37], and the data are also shown

in Fig. 8 as lled triangles. For the frit, the intercept was only (0.002580.00025) S/m.

Because the intercept is proportional to the surface charge of the material [38]. one

expects that the surface charge of the frit is approximately 250 times smaller than that

of the resin. Unfortunately, comparisons with cementitious systems are not possible due

to lack of similar data. Regardless, the data presented herein for the frit system will

show that an analysis that neglects binding still captures all the salient features of the

eect that speciation has on the diusive transport of ions through these ceramic frits.

4.2.2 Experimental Setup

The divided cell arrangement was used as the experimental apparatus because of its

procedural simplicity. Since the ceramic specimens have a relatively large diusion coecient, compared to to hydrated cement paste, usable data could be collected in a

reasonable period of time. As mentioned previously, the ceramic material also has a

relatively small surface charge and surface adsorption eects could be neglected. The

result is a material that should exhibit nearly ideal diusive transport properties.

The target ion for the experiment was iodide. The iodide concentration was determined using a commercial solid-state ion selective combination electrode. Iodide was

chosen as the target ion because of experimental advantages iodide electrodes have over

chloride electrodes. For each measurement, the electrode was standardized using reference iodide solutions.

Each ceramic frit was rst mounted, using epoxy, into an acrylic annulus. After the

epoxy had cured, the specimen was saturated with a potassium iodide solution under

vacuum. The saturated frit was then clamped between two glass vessels, each having a

capacity of approximately 250 mL, using rubber o-rings for a water tight seal. One vessel

was lled with KI and the other with a test electrolyte. Both vessels were sealed to reduce

evaporation, and then placed inside a walk-in environmental chamber maintained at

25 C. All sampling and concentration measurements were performed within the chamber

in order to ensure stability in both the diusion and the measurements; there is a nominal

2 %/C variation in the self-diusion coecient [23].

4.2.3 Analysis

Historically, ionic transport in cementitious systems has been characterized by Fick's law

of diusive transport:

ji = Da

i rci (34)

The quantity Da

i represents an apparent diusion coecient for the i-th species. The

analysis of the experimental diusion data is based on the \constant gradient" assumption

that underlies Fick's law. Based on this assumption, the concentration gradient across

the specimen is constant and equal to the dierence between cell concentrations, divided

by the specimen thickness L. Under these conditions, Fick's law applies. For vessel

24
c2 v2 v1 c1

x=0 x=L

∆

Figure 9: Schematic of the constant gradient hypothesis across a sample with thickness

L. Vessels 1 and 2 contain an aqueous electrolyte with target species concentrations c1

and c2, and have volumes v1 and v2, respectively.

volumes v1 and v2 and specimen area A, the dierence in concentration  between the

two vessels can be shown to decay exponentially [35, 39]:



0

= exp DaA

L



v1

1 + v12 

t(35)

The quantity 0 is the concentration dierence at the onset of a linear concentration

prole. Based on Eqn. 35, a semi-log plot of the concentration dierence between the

two vessels should appear as a straight line under ideal conditions; the magnitude of

the slope being proportional to the apparent diusion coecient Da. Deviations from a

straight line will indicate behavior that cannot be modeled by Fick's law with a constant

apparent diusion coecient.

4.2.4 Formation Factor Measurement

The formation factor measurements were performed using KCl as the pore solution because the conductivity of a few standard solutions are known to a high precision [37]. A

range of concentrations was used to ensure that the surface conduction component was

properly accounted for. For the conductivity measurement, the specimen was vacuum

saturated with potassium chloride solution and then mounted, using rubber o-rings and

clamps, between two glass vessels, each containing a platinum electrode. The setup is

shown schematically in Fig. 10(a). The apparatus, in the absence of a specimen and

holder, had a conductivity cell constant of 0.3567 cm1. To determine the formation factor of the saturated specimen, the cell was lled with

the same solution as the saturation solution and then allowed to thermally equilibrate in

the environmental chamber. The direct current (dc) resistance of the specimen and cell

was determined using a commercial impedance spectrometer that sampled frequencies

between 10 Hz and 1 MHz [22]. The bulk conductivity of the specimen was calculated

25
(a) (b)

Figure 10: The conductivity cell apparatus(a) and the diusion divided cell apparatus

(b). The schematics depict the conguration of the two cylindrical glass vessels on either

side of a mounted specimen. The system is sealed using rubber o-rings; the clamps are not

shown. The apparatus dier only in the vertical platinum electrodes in the conductivity

cell. The diameter of the specimen, the glass vessels, and the platinum electrodes are

approximately 50 mm.

from the cell constant and the specimen geometry. The formation factor was calculated

from the ratio of the conductivity of the KCl solution to the calculated bulk conductivity. Pore solution concentrations of 0.01 molkg1

, 0.10 molkg1, and 1.00 molkg1 were

used to assess the contribution from surface conduction. Due to a small surface conduction contribution, the formation factor increased with increasing pore solution conductivity, converging to a xed value with increasing concentration. The specimen conductivity

measured using the 0.01 molkg1 solution was approximately 85 % of the value using the

1.0 molkg1 solution, and the specimen conductivity using the 0.10 molkg1 solution

was approximately 98 % of the value using the 1.0 molkg1 solution. Therefore, the

formation factor calculated at 1.0 molkg1 was used as the best estimate. Since the

change in formation factor was only 2 % for a ten fold increase in concentration, the true

value is probably not more than a fraction of percent larger than the values reported.

Since the ceramic specimens are the result of a controlled commercial process, they

have relatively little specimen to specimen variation. The four specimens used in this experiment had formation factors ranging from 10.6 to 10.9. These values were determined

after each specimen was allowed sucient time to reach thermal equilibrium, and then

the values of the dc resistance varied by less than 1 %.

4.3 Formation Factor vs. Diusion Coecient

Upon completion of the conductivity measurement to determine the formation factor,

the specimen was then saturated with KI and placed into a divided cell apparatus shown

schematically in Fig. 10(b). The ob jective was to determine the value for the formation

factor F required in Eqn. 32 in order for the calculated iodide concentrations to agree

with the experimental values.

The measured values of  for the iodide concentration are shown as symbols in Fig. 11.

For each of the four systems, the concentration of the KI is equal to the concentrations

26
0.0 2.0 4.0 6.0

t (v1

−1

+v2

−1) (h/cm3

)

0.04

0.06

0.08

0.10

∆ = c −c

2 1 (mol/L)

KCl 0.1 mol/L

NaCl 0.1 mol/L

KOH 0.1 mol/L

KCl 1.0 mol/L

Figure 11: Concentration dierence across each specimen as a function of time normalized

by the vessel volumes. The experimental values are shown as lled symbols, the estimates

by 4SIGHT are shown as solid curves. The measurement uncertainties would appear as

the same size as the symbols, so are omitted for visual clarity. The value of  for the 1.0

mol/L KCl system is divided by ten in order to appear on the same scale as the other

data.

shown in the gure key. The estimated measurement uncertainties are approximately

the size of the symbols, and are not shown as error bars for reasons of visual clarity. The KCl and the NaCl systems behaved similarly; the values of  for the 1.0 mol/L

KCl system were divided by ten so that they could be included on the same plot. The

1.0 mol/L KCl solution has a similar behavior to the 0.1 mol/L KCl system because

the self diusion coecients Dsfor K+, Cl, and Iare nearly equal to one another.

The estimated diusion potential D (Eqn. 32 across the entire specimen for both of the

KCl systems was less than 1 mV, and was less than 5 mV for the NaCl system. The

KOH system showed a marked dierence in behavior. One reason for this is that the

self diusion coecient Dsfor OHis signicantly greater than that of the other ions

present, resulting in a calculated diusion potential of approximately 16 mV. A diusion

potential of this magnitude could have a noticeable eect on the transport of the iodide

ions.

The calculated slopes of the experimental data from semi-logarithmic plots are shown

in Table 4. The estimated standard deviations shown are typically less than 3 % of the

slope value, suggesting that the constant gradient assumption is valid for these systems.

In fact, the systems reached a constant gradient state at a relatively early age. Analysis

using the 4SIGHT computer program suggests that the constant gradient condition is

achieved in less than 12 h.

Also shown in Table 4 are the values for the ratio Ds=Db, using the iodide value for

Ds (2.045 105 cm2 s1[23]). This ratio represents an incorrect use of the formation

factor to determine the apparent bulk diusion coecient. Since the quantity Dsin this

27
Table 4: The values for the slope (ADb=L) of the experimental data shown in Fig. 11.

Also shown is the ratio Ds=Db, using Dsfor iodide. The uncertainties shown for the

slopes are the estimated standard deviation reported by the statistical software, and also

characterize the uncertainty in the ratio Ds=Db reported.

System ADb=L Ds=Db

(cm3h1)

KCl { 0.1 mol/L 0.20040.0032 11.1

NaCl { 0.1 mol/L 0.21180.0032 10.3

KOH { 0.1 mol/L 0.23810.0047 9.3

KCl { 1.0 mol/L 0.20790.0050 10.7

ratio is a constant, the ratio is inversely proportional to the apparent bulk diusivity Db. This ratio does not re
ect the actual formation factor, however, because the iodide self

diusion coecient within the pore solutions is not equal to Ds. Also, it is clear that

future changes in the pore solution will lead to changes in the apparent bulk diusion

coecient of iodide in these systems, while the formation factor is nearly equal for all

systems.

The values of the formation factor F were determined from the experimental data using the 4SIGHT implementation of Eqn. 32, and are shown in Table 5, labelled Fsim. Also

shown in the table are the values of the formation factor calculated from the impedance

spectroscopy measurements, labelled FIS . The uncertainty in FIS re
ects the variation

in the dc resistance measurement as mentioned previously. To estimate F in Eqn. 32,

its value was adjusted until the calculated values of  agreed with the measured values;

this value is referred to here as Fsim. The values of Fsim shown in Table 5 were used to

calculate values for , and these values of  are plotted in Fig. 11, denoted by the lines.

The experimental data and the calculated values for the KCl and the NaCl systems were

all nearly linear, suggesting that the linear Fick's law for diusive transport is valid for

these systems. The values of  for the KOH system are easily distinguished from the

other systems.

The calculated values of Fsim shown in Table 5 were consistant with the measured

values FIS . The values of Fsim varied by approximately 7 %, compared with the 18 %

variation in the values of Ds=Db. The dierences between the values of Fsim and FIS

were less than 3 % for the KCl and the NaCl system, and less than 8 % for the KOH

system. The consistant bias of calculated values of Fsim being larger than Fis may be

attributed to the method of using a single measurement for the specimen conductivity, rather than using a more rigorous approach.

Sighting along the KOH data in Fig. 11 reveals noticeable curvature. It is interesting

to note that the computed output (solid curve) also exhibits this nonlinear behavior. This

suggests that this nonlinear behavior is due to eects of the pore solution chemistry since

output from 4SIGHT indicates that the iodide concentration prole across the sample is

stable within 12 h. The calculated concentration prole of iodide, however, is not linear

28
due to the diusion potential.

Table 5: Measured and calculated formation factors from impedance spectroscopy (FIS ),

computer simulation (Fsim), and apparent diusivity (Ds=Db) using Dsfor iodide.

System FIS Fsim Ds=Db

KCl { 0.1 mol/L 10.70.2 10.9 11.1

NaCl { 0.1 mol/L 10.90.2 11.2 10.5

KOH { 0.1 mol/L 10.60.2 11.4 9.3

KCl { 1.0 mol/L 10.70.2 10.6 10.7

This electro-chemical eect of the KOH system is revealed in Table 5. As shown in

Table 4, the apparent diusion coecient of iodide in this system is considerably greater

than that for the other systems. The computer calculation reveals that the required

formation factor Fsim is comparable to the values for the other test solutions. This

demonstrates the eect of using the apparent diusion coecient Db to characterize a

microstructure. For KOH solution, the apparent diusion coecient describes how the

iodide ion behaves in the presence of KOH, but does not characterize its behavior in the

presence of other test solutions. Similarly, it does not necessarily characterize how other

ions behave in the same, or similar, microstructure.

Since the pore solution of cementitious systems is typically alkaline, the results for the

KOH system have direct relevance to the prediction of ion transport in portland cement

systems. The pore solution ionic strength in cementitious systems can be nearly ten times

greater than the 0.1 mol/L KOH system studied here. Further, there will be number of

additional ions present, with a corresponding number of self diusion coecients. This

raises the question of the correct method for characterizing the microstructure of these

systems. Either estimating the formation factor from diusion data or estimating the

diusion coecient from a formation factor measurement (conduction test) will require

knowledge of the pore solution chemistry. For a formation factor measurement that uses

pore solution extraction, a chemical analysis of the extracted pore 
uid would be a logical

extension of the measurement procedure. However, diusion measurements typically do

not reveal the chemical makeup of the pore solution, and so further experimentation may

be required for experimental programs based upon diusion measurements to characterize

the microstructure of cementitious systems.

4.3.1 Multi-Species

The results for the multi-species experiments are divided between two series. In the

rst series, the kinds of ionic species present were kept constant but the initial iodide

concentration dierences were varied. In the second, the initial iodide concentration

were held constant, but the kinds of other species present were varied. The rst series

attempts to demonstrate the eect of concentration. The second attempts to demonstrate

the eect of speciation.

29
0 2 4 6 8 10

t (v1

−1

+v2

−1

) (h/cm3)

0.1

1

∆/∆0

0.01 mol/L

0.10 mol/L

1.00 mol/L

KCl/KI

0 2 4 6 8 10

t (v1

−1

+v2

−1

) (h/cm3)

0.1

1

∆/∆0

0.01 mol/L

0.10 mol/L

1.00 mol/L

NaCl/KI

Figure 12: Concentration dierence  of iodide (lled symbols) for the KCl/KI and the

NaCl/KI systems; the time t is scaled by the vessel volumes v1 and v2. The quantity 0

is the initial KI concentration. The lines shown are the result of the linear regression

analyses reported in Table 6. The measurement uncertainties are approximately the size

of the symbols used.

The experiments with varying concentration were performed with either potassium

chloride or sodium chloride as the counter-diusing electrolyte, with respect to potasium

iodide. Plots of the experimental results are shown in Figs. 12. For both the KCl/KI and

the NaCl/KI systems, the data exhibit virtually linear behavior on the semi-logarithmic

plots. The apparent linear behavior suggests that these systems behave in a nearly

\ideal" manner, and can be characterized accurately by Fick's law in Eqn. 10. Due to

the nearly equal self-diusion coecients for potassium, chloride, and iodide, there is

only a relatively small electrical diusion potential across the sample.

The magnitude of the slope calculated from the data at each concentration is reported

in Table 6. The uncertainties reported are the estimated standard deviations reported

by the regression software, and have the corresponding coverage factor [40] k = 1. For

the three concentrations, the coecient of variation of the estimated slopes was 5 % for

the KCl/KI systems, and 10 % for the NaCl/KI systems. While the dierences among

slopes, being greater than the estimated uncertainties, suggest that these variations are

statistically signicant, for practical use, these variations are minor. Any chemical reactions that might occur within the cementitious matrix can alter the diusion coecient,

easily overwhelming a 20 % uncertainty in the apparent diusion coecient.

The experiments with constant KI concentration and varying speciation were performed with KCl, NaCl, KOH, K2CO3, CaCl2, H2SO4, and MgSO4 as the counterdiusing electrolyte. In all cases, the initial concentration of KI was 0.10 mol/L. The

concentration of the counter-diusion electrolyte was chosen to minimize an osmotic pressure gradient across the sample; it is not asserted here that the values used eliminated

an osmotic pressure gradient.

30
Table 6: Absolute value of the slopes calculated from the data for the KCl/KI and

the NaCl/KI systems in Figs. 12, regressed to a linear model. The uncertainties shown

are the estimated from standard deviations reported by the regression software, with a

corresponding coverage factor of k = 1.

Species concentration jSlopej

(mol/L) (cm3/h)

KCl/KI 0.010/0.010 0.19230.0039

0.100/0.100 0.19780.0029

1.000/1.000 0.21010.0024

NaCl/KI 0.010/0.010 0.23650.0051

0.100/0.100 0.21200.0024

1.024/1.000 0.19310.0059

The experimental data are shown in Figs. 13, and the analyses of the data are divided

between the \short" time and the \long" time results. In Fig. 13(a), a linear model was

regressed to the data for which t(v1

1 + v12 ) < 8 h/cm3

. The results from the linear

regression analyses are shown in Table 7. The uncertainties reported are the estimated

standard deviations reported by the regression software, and have the corresponding coverage factor [40] k = 1. Based on these results, the CaCl2/KI system and the K2CO3/KI

system represent the extrema, and are shown in Fig. 13(b) for all time, along with the

KCl/KI system, which is an example of nearly ideal behavior, as a means to constrast

the results.

Table 7: Magnitude of the slope for data shown in Fig. 13(a), regressed to Eqn. 35

for t(v1

1 + v12 ) < 8 h/cm3

. For each system, the KI concentration was 0.10 mol/L.

Also shown are the values for the formation factor F used to produce the solid curves

Fig. 13(b). The uncertainties shown are the estimated from standard deviations reported

by the regression software, with a corresponding coverage factor of k = 1.

Counter Solution concentration jSlopej F

(mol/L) (cm3/h)

K2CO3 0.0769 0.15810.0014 11.7

H2SO4 0.0859 0.16880.0075 9.6

MgSO4 0.1660 0.18670.0022 13.5

KCl 0.1000 0.19780.0029 11.4

NaCl 0.1000 0.21200.0024 11.8

KOH 0.0879 0.23330.0034 11.2

CaCl2 0.0731 0.28010.0089 12.8

31
02468

t (v1

−1

+v2

−1

) (h/cm3)

0.1

1

∆/∆0

KCl

NaCl

KOH

H2SO4

K2CO3

CaCl2

MgSO4

(a)

0 5 10 15 20

t (v1

−1

+v2

−1

) (h/cm3)

0.01

0.1

1

∆/∆0

KCl

K2CO3

CaCl2

(b)

Figure 13: Measured concentration dierence  of iodide (lled symbols) for constant

0.10 mol/L KI concentration and varying counter-diusing species; the time t is scaled

by the vessel volumes v1 and v2. In (a) results from linear regression (solid lines) for

t(v1

1 + v12 ) < 8 h/cm3

; the calculated slopes are reported in Table 7. In (b) the electrodiusion equation is used to estimate the formation factor F , the values of which are

shown in Table 7. The error bars represent an estimated standard deviation.

The computed slopes given in Table 7 indicate the eect of speciation on the observed

diusion coecient. The magnitude of the slopes shown in the table varies by nearly a

factor two. This contrasts with the previous 20 % variation in slopes due to concentration.

The CaCl2 and the K2CO3 data are shown in Fig. 13(b) to delineate the envelop over

which the concentrations could vary over long times. Note that for long times the slopes

of the curves are not constant. While the change in slope for the K2CO3/KI system is

gradual, the change in slope for the CaCl2/KI system is dramatic by comparison. Therefore, it is inappropriate to characterizing iodide transport in the CaCl2/KI system using

Fick's law and a constant apparent diusion coecient. Further, the apparent iodide diffusion coecient decreases slightly in the K2CO3/KI system and increases greatly in the

CaCl2/KI system. Any \adjustments" to an apparent diusion coecient to account for

one system would certainly not apply to another. This is analogous to determining the

apparent diusion coecient under one set of conditions and using that to extrapolate

to behavior in the eld with varying species exposure.

By contrast, the data for each system were analyzed using the 4SIGHT implementation

of Eqn. 32. The porosity remained xed at 0.26 and the formation factor remained the

only free parameter. The value of the formation factor F required to best approximate

the measured data, for all times, is shown in Table 7 for each system. The coecient of

variation for these values was approximately 10 %. The corresponding results are shown

as solid curves in Fig. 13(b). The curves for the other systems are not shown for clarity. Further, remaining systems had only minor curvature by comparison.

The important feature of the electro-diusion equation is that it can capture the

32
0 20 40 60 80

t (d)

0

1

2

3

4

5

Da/D

μ

KOH CaCl2

MgSO4

NaCl

K2CO3

H2SO4

KCl

Figure 14: The ratio of the apparent diusion coecient Da to the microstructural

diusion coecient D(as calculated by 4SIGHT using Eqn. 14) of iodide for the various

counter diusing electrolytes, as a function of time t.

nonideal behavior of these systems. The apparent diusion coecient can be calculated

at any time from the slope of the curve based on the electro-diusion equation. This

calculated apparent diusion coecient can then be compared with the microstructural

diusion coecient. The results from this comparison are shown in Fig. 14 for all the

systems considered in Fig. 14.

The curves shown in Fig. 14 express quantitatively the deciency in characterizing

multi-species diusion with a single apparent diusion coecient. Accurately characterizing the diusive transport of iodide in these nonreactive systems using Fick's law

requires that the apparent diusion coecient vary in time. For the CaCl2/KI system,

the apparent diusion coecient has increased by a factor of 5 at 60 d.

The surprising result was the behavior of the CaCl2/KI system at very long time.

The curve for the CaCl2/KI system in Fig. 13(b) appears to be diverging at long times.

Analysis of the experimental setup using the electro-diusion equation predicts that the

concentration dierence between the vessels will become negative; this is equivalent to

a negative apparent diusion coecient as the 
ux of iodide ions is in the direction of

the low concentration to high concentration. The concentration dierences  for the

CaCl2/KI system are shown in Fig. 15(a) on a linear scale. After approximately 70 d,

the experimentally measured concentration dierence was, in fact, negative. As a means

of comparison, the corresponding data for the K2CO3/KI system are shown in Fig. 15(b).

Also shown in Figs. 15(a) and 15(b) are the relative concentration dierences for

all the species present, as calculated by Eqn. 35. For the CaCl2/KI system, the \slow"

species is the Ca2+ ion. For the K2CO3/KI system, it is the CO2

3 ion. The eect of these

33
0 50 100 150

t (d)

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

∆/∆0

Ca2+

Cl−

K+

(a)

I

−

0 50 100 150

t (d)

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

∆/∆0

CO3

2−

(b)

K+

I

−

Figure 15: Experimental concentration dierence  of each diusing species in the (a)

CaCl2/KI, and the (b) K2CO3/KI systems, each as a function of time t. The experimental

values are shown as lled symbols, and the curves are the values calculated by 4SIGHT. The error bars represent an estimated standard deviation.

two ions is to greatly control the macroscopic diusion potential across the specimen. The

presence of the Ca2+ generates a diusion potential that is positive with respect to the

KI side. Correspondingly, the CO2

3 ion generates a negative potential with respect to

the KI side. The result is a macroscopic diusion potential that accelerates the iodide

ion in the CaCl2 system, and retards it in the K2CO3 system. In the CaCl2 system,

the macroscopic diusion potential is sucient to drive iodide from the KI vessel to the

CaCl2 vessel, even though the iodide concentration in the KI vessel is less than that of the

CaCl2 vessel. In time, the diusion potential decreases and the concentration dierence

 approaches zero.

4.4 Summary

Since each of the degradation mechanisms considered by 4SIGHT are controlled by the

transport of ionic species throughout the concrete, it is imparitive that the transport

portion of the 4SIGHT computer program be as accurate as possible. The rst step was to

validate the transport equation Eqn. 32 used by 4SIGHT. It was shown that the formation

factor used in Eqn. 32 is equivalent to the formation factor that can be calculated from

conductivity measurements. The second step was to demonstrate that the transport

equation characterizes the transport of all the ions present in the electrolyte, and that

the only two parameters required are the porosity and formation factor. By characterizing

the transport of all the species present, and the resulting macroscopic diusion potential,

the 4SIGHT computer program successfully predicted the occurance of an experimental

setup that could exhibit a negative apparent diusion coecient (based on Fick's law).

Although only iodide concentration measurements were performed, the wide variation

34
of behavior of the dierent experimental setups suggests that the complete system was

modeled accurately. Future experiments where all diusion species concentrations are

measured are needed to conrm this.

35
5 Parameter Uncertainty

As the complexity of service life estimation increases, it is clear that deterministic calculations have diminishing utility. One cannot say with absolute certainty, what the

performance of a facility will be over a period of decades or centuries. Rather, one can

only oer probabilistic statements of possible outcomes.

To facilitate a probabilistic risk assessment, 4SIGHT allows the user to specify input

parameters with both a mean and a level of uncertainty. These uncertainty parameters

are then used to generate random deviates based on a random number generator. Multiple service life calculations are performed, each using a new random deviate. The failure

times for each calculation is recorded. The results are then reported as a cumulative

distribution of failure times.

5.1 Random Deviates

The basis for the random deviate calculation lies in the ability to generate a random

deviate from the uniform distribution over the interval (0,1). This is accomplished using

a random number generator (see Ch. 7 of Ref. [41]). The resulting uniform random

deviate i is then used to calculate a random deviate Xi from an arbitrary probability

density function f (x):

Z Xi 1

f (x)dx = i (36)

The random number generator used by 4SIGHT is from Knuth [42], and referred to as

ran3 in Numerical Recipes [41]. It has been shown that this random number generator

is suciently robust for calculations involving 107 deviates.

5.2 Distributions

Four probability density functions f (x) are provided to the user by 4SIGHT. These four

distributions can be used to characterize random deviates over a broad range of uncertainty.

5.2.1 Delta Function

The delta function (see Refs. [43, 44]) is equivalent to having a deterministic value. The

delta function is, in a practical sense, a Gaussian distribution with zero variance. The

symbol typically used for the delta function is (x  xo), and a plot of the function would

consist of a vertical line at x = xo. A delta function has the following mathematical

denitions [44]:

Z +1

1

(x) dx = 1

Z

b

a g(x)(x  x) dx =

8

><

>:

g(x) a<x < b

0 otherwise

(37)

36
The probability density function f (x) is written mathematically as follows:

f (x) = (x  x) 1 x  +1 (38)

Every random deviate xi will have the same value x:

xi = x

5.2.2 Uniform Distribution

The uniform distribution allows one to specify a range over which values can vary; values

cannot be used that lie outside the specied width of the distribution. As its name

implies, random deviates are chosen uniformly over the specied range, characterized

by a center at x and a width w. Values are chosen uniformly between x  w=2 and

x + w=2:

f (x) = 1

w

x 

w

2  x  x +

w

2

(39)

Using the uniform random deviate i over the interval (0,1), the random deviate xi

is calculated from the center x and width w of the distribution:

xi = x + w



i 

1

2

5.2.3 Gaussian Distribution

The Gaussian distribution is familiar. For a mean  = x and variance 2, the probability

density function can be stated mathematically as follows:

f (x) =

exp 1

2 

xxo  2

p

2

(40)

Using standard software routines [41], a random deviate i is calculated from the Gaussian

distribution with zero mean and unit standard deviation. The normal random deviate i

can then be used to calculate a random deviate from a Gaussian distribution with mean

xo and variance 2:

xi = x + i

5.2.4 Lognormal Distribution

An advantage of the lognormal distribution is that it can characterize uncertainty over

orders of magnitude. Although an entire family of n-th order logarithmic distributions

exist [45], the basic feature of a lognormal distribution is that the logarithms of the deviates are normally distributed. This fact can be exploited to simplify deviate generation.

Consider experimentally determined quantities yi that are the result of a measurement

process, and can be characterized by a lognormal distribution (e.g., hydraulic conductivity). Let the variable zi represent the logarithms (base 10) of these values: zi = log10(yi).

Let xo denote the average of the values of zi. Let  represent the standard deviation of

37
the values zi . Given an normal random deviate i with zero mean and unit variance, a

logarithmically distributed random deviate xi can be generated in the following way:

xi = 10xo+i (41)

5.3 Output: Cumulative Distribution

For the Monte Carlo calculations, the user will specify a number N of iterations. For each

iteration, new random deviates will be calculated and the time at which failure occurs

will be calculated; in the absence of failure, the program returns the time limit set by the

user. Let each failure time be denoted by ti , where 1 i  N. The probability density

function g(t), giving the probability that the service life of a computational iteration

terminated between t and t + dt, can be expressed as a sum of delta functions:

g(t) = 1

N X

N

i=1

(t  ti) (42)

The cumulative probability function G(t) is the integral of the probability density

function g(t):

G(t) =

Z

t

0

g(t

0) dt0

=

1

N X

N

i=1 Z

t

0

(t

0  ti) dt0 (43)

Therefore, under the parameter uncertainty conditions specied by the user, the probability that failure will occur before time  is simply G( ).

This is a mathematically rigorous means of expressing a straightforward concept. An

example will clarify the idea further: Consider a service life calculation performed with

N = 10 iterations, with each iteration using dierent input parameters, resulting in the

following service life termination times:

i ti

1 120

2 324

3 80

4 260

5 250

6 210

7 500

8 170

9 500

10 280

The corresponding cumulative distribution function G(t) is given in Fig. 16. These data

can then be used within a larger service life calculation composed of a number of elements,

38
0 100 200 300 400 500 600

t

0.0

0.2

0.4

0.6

0.8

1.0

G(t)

Figure 16: The cumulative distribution function G(t).

of which this is only one. A random failure deviate can be generated from these data

by picking a uniform random deviate i over the interval (0,1) and solving for ti from

G(ti) = i .

39
6 Signicance of Cracks

Most degradation processes are due to the presence of some specic ion within the cement

paste microstructure. These deleterious ions typically penetrate the microstructure by

diusion and/or advection. To obtain an understanding of the relative magnitude of

these mechanisms, consider their contribution to the ionic 
ux. For diusivity D, ion

concentration c within the capillary pores, and pore solution velocity up, the 
ux j is a

sum of contributions from diusion and advection:

j = Drc + cup = Drc + c

vD

 = Drc + ck

 rP (44)

Here, the permeability k is the bulk permeability that is often reported. The second line

in Eqn. 44 represents the pore velocity as a function of the Darcy (bulk) velocity vD

and capillary porosity . The third line contains the substitution for the Darcy velocity

as a function of the permeability k, pore 
uid viscosity , and hydraulic pressure P . For a 0.45 water:cement ratio (w/c) concrete, the relevant material parameters can be

estimated from the previous 4SIGHT report [1] and are shown in Table 8.

Table 8: Material parameters a concrete with w/c = 0.45, estimated from equations in

Ref. [1]. (Default degree of hydration  = 0.85)

Parameter Symbol Value

diusivity D 71012 m2s1

permeability k 11019 m2

viscosity  103 Pa s

porosity  0.11

The gradients in Eqn. 44 can be estimated from \order of magnitude" values. The

length scale is approximately 1 m, the concentrations on the order of 1 mol, and the

hydraulic pressure a tenth of an atmosphere (104 Pa). The contributions to the 
ux are

71012 for the diusive transport and 91012 for the advective transport. Therefore,

this scenario represents an approximate equality between the relative contributions to

the overall 
ux. Changes in either the concentration or the hydraulic gradients would

change the contributions, respectively. Since cracks in concrete have permeabilities that

are orders of magnitude greater than the permeability of uncracked concrete, their impact

on the ionic 
ux can be dramatic.

6.1 Internal Body Force

The previous report [1] contained a model for advective transport through uncracked

concrete that included a term for the body force of g ( is the 
uid density and g is

40
atm P

2

R

γ

∆P

∆P











 

















 







2R

= L

Figure 17: Schematic of water sorption in concrete. The hashed region represents a cross

section, with pore radius R and liquid-air surface tension 
, with its bottom surface

under water. To the right is a plot of the hydraulic pressure P as one traverses vertically

through the pore. Negative pressure indicates water in tension.

the gravitational acceleration), adding a constant to the pressure gradient. This term

contributed to the overall advective transport through the concrete. While this addition

is accurate for a large saturated specimen with water at both ends, it is not accurate

when either end is exposed to an unsaturated environment. Exposure to an unsaturated

environment will lead to the formation of menisci in the cement paste pore system.

Under many possible conditions, the pressures developed by these menisci are sucient

to prevent the water from \draining" out of the concrete due to gravity.

6.2 Sorptivity

Typical capillary sorptivity tests demonstrate that concrete is not a self-draining material.

A dry concrete specimen placed onto a free surface of water will draw the water into the

concrete due to capillary sorption. If allowed to contact the water for a suciently

long period of time, and assuming the height of the concrete specimen is less than 10

m (hydraulic head of one atmosphere), the concrete will, given sucient time, become

saturated with the water.

Concrete is able to retain water because of the menisci formed by a wetting liquid

with surface tension 
 in pores of radius R. For a concrete of thickness L resting upon a

surface of water, the menisci within the pores at the top surface of the concrete specimen

generate a pressure drop of 2
=R as one passes from the air above into the pore 
uid just

below the menisci surface. The pressure is sucient to overcome the head pressure of gL, thereby drawing water in from the bottom surface. This condition is shown schematically

in Fig. 17. The pores in concrete are suciently small that, for concrete in contact with

water on its bottom surface and in contact with air at the top surface, the concrete

will remain saturated, and there will be no downward 
ow of pore 
uid. This gives an

eective hydraulic head pressure of zero. In fact, for an unsaturated environment at the

41
2

R

γ ∆P =

atm P

 ∆P









 

















 







2R

L

Figure 18: Schematic of the hydraulic pressures in a porous material with a thin layer of

water on its top surface and unsaturated air exposure on its bottom surface.

top surface, the water will wick away due to evaporation, creating an upward movement

of 
uid.

6.3 Break-Through Pressure

The eect of the menisci can also be applied to a concrete with water on its top surface

and exposed to air at the bottom surface. If the bottom surface of a concrete is exposed

to an unsaturated environment, the bottom surface of the concrete will have menisci in

the pores, analogous to the previous discussion. Imagine a very shallow pool of water

exists on top of a concrete specimen with thickness L. The hydraulic pressure within the

concrete increases as gz, with z represented the distance below the surface of the water

above the concrete. At the menisci on the bottom surface, there is a pressure increase

of approximately 2
=R as one passes from the pore 
uid to the air below the specimen.

This is shown schematically in Fig. 18.

The minimum specimen thickness L required for the 
ow of water through the specimen can be calculated from the balance of these pressures:

gL =

2


R

(45)

In SI units,  = 1000 kgm3, g = 9.8 ms2, and 
 = 0.0735 Nml (liquid-vapor surface at

15C). The approximate value for the menisci radii corresponding to a given permeability

can be approximated from the Katz-Thompson relationship:

k =

F

200

d

2

c (46)

The quantities k, F , and dc are the specimen permeability, formation factor, and critical

pore diameter, respectively. For typical concretes of good quality, k = 1017 m2 and

F = 500. Using the critical diameter dc as an estimate of the meniscus diameter, this

42
approximation gives R = 1.0 nm. The corresponding hydraulic head is 15 000 m. This is

also an estimate of the hydraulic head required to overcome the eect of surface tension

or order for 
ow to initiate. And since the hydraulic pressure increases as gz both inside

concrete and in free water, for a thin lm of water over the concrete, the specimen would

have to be 15 000 m thick in order for 
ow to initiate. Although this calculation was a

rough approximation, this minimum thickness is nearly four orders of magnitude greater

than one would expect for a vault roof. Therefore, one would not expect water to 
ow

through a vault roof that had a thin lm of water on its top surface, and an exposure of

unsaturated air on the bottom surface.

This simple calculation demonstrates the signicant eect that an unsaturated environment at the lower concrete surface can have an advective transport through undamaged concrete. By keeping the interior environment of the vault below saturation, the

advective 
ow through the concrete will be virtually zero.

6.4 Transport Through Cracks

The preceding section demonstrated the signicance of having undamaged concrete in

contact with a unsaturated interior vault. In practice, the concrete will likely have

some distress that will cause cracking, and may have a far dierent performance than

undamaged concrete.

One can characterize the eects of cracks by calculating the minimum crack width

before water will 
ow through the concrete. Consider a horizontal concrete slab that is l

m thick. Assume that the slab is saturated, that there is a shallow lm of water on the

top surface, and that the bottom surface is exposed to an unsaturated environment like

that shown in Fig. 18. The gh hydraulic pressure is just able to \push" water through

a 15 m crack (assuming plane, parallel walls) which would cause dripping from, or

pooling on, the bottom surface of the concrete. A 15 m crack is generally invisible

to the naked eye, yet its eect upon the advective transport through the concrete is

dramatic. Clearly, to accurately quantify advective transport through concrete one has

to accurately quantify the cracking within the concrete.

6.4.1 Residence Time

Using this more clearly constructed description of the eects of cracks, it should be much

easier to characterize 
ow through the cracks, and to estimate a residence time for the

time required for water to traverse the concrete member. If there exist cracks either at

the bottom of the slab or through the entire slab, there should be mass 
ow, even if

the atmosphere of the vault is unsaturated with respect to water vapor. Knowledge of

the residence time is needed in order to characterize the chemistry of the euent into

the interior of the vault. If the residence time is suciently short, the 
uid will not

have sucient time to reach equilibrium before passing through the concrete. This eect

directly eects any attempt to predict the chemistry of the euent from the vault.

The residence time for a crack can be calculated using Darcy's law. Within a vertical

crack, there is a body force of g downward on the water. This corresponds to a constant

43
pressure gradient: rP = g. Darcy's law gives the bulk 
uid velocity vb:

vb = 

k



rP (47)

The averaged pore velocity vp, is greater than the bulk velocity by a factor of the porosity:

vp =

vb



(48)

Therefore, the residence time tr for a 
uid to pass through a sample of thickness L is

tr = L=vp. For a quality concrete (k = 1018 m2) 1 m in thickness, with 1 m of head,

the Darcy velocity is less than 1 m/d. The corresponding residence time is 2 700 years.

For cracked concrete, the Darcy velocity is much greater. As an example, consider

a 1 m thick slab with a 100 m crack extending through the entire slab. Since the

permeability of plane, parallel surfaces separated by a distance w is w2=12 [47], the

permeability of the 100 m crack is 8.31010 m2. Assuming the pore 
uid density and

viscosity are approximately equal to those of water, for an equivalent water head of 1 m,

the bulk velocity vb through the crack is 8.3 mm/s. For the 100 m crack, vp = vb, and

the residence time tr is approximately 120 s. Therefore, the issue of pore 
uid residence

time in cracked concrete is important.

6.4.2 Evaporative Mass Transfer

In cases where a concrete is kept saturated and the there are cracks at on the interior

surface, water will probably advance to the surface of the concrete where these cracks

exist. Given the size of typical cracks, the equilibrium relative humidity of an atmosphere exposed will be nearly 100 % RH. Therefore, a vault interior environment below

saturation will cause evaporation of the concrete pore 
uid that is in the cracks.

Assuming that the concrete permeability of the uncracked portion is suciently great

so as to supply these cracks with additional pore solution, there should be a thin lm of

water on the interior surface of the concrete. The rate of evaporation from the bottom

surface can be approximated from formulas for evaporation from a free surface of water.

The saturation pressure PS (Pa) of water can be expressed as a function of its temperature

T (in Celsius) [46]:

PS (T ) = 610 exp  17:3 T

237:3 + T



(49)

Let P cS (Tc) and P aS (Ta) represent the saturation pressure of water at the temperature

of the concrete (Tc) and of the air (Ta), respectively. Exposed to an atmosphere with

relative humidity  and with wind speed (forced convection, if applicable) vw, (m s1)

across the concrete surface, the water evaporation rate E (kgm2s1) is [46]

E =



8:69 105(P cS  P aS )(0:253 + 0:017 vw) (50)

For an underground concrete structure, the temperature of the concrete would be equal

to the vault temperature. Therefore, the evaporation rate equation for a concrete vault

at temperature T simplies slightly:

E(T ) = 8:69 105 PS(T ) (1  ) (0:253 + 0:017 vw) (51)

44
This is the most general result. If the interior atmosphere of the vault is maintained

constant through convection, the wind speed will be non-zero. If the vault interior is

sealed, the evaporation rate will reach zero over time.

Once the evaporation rate is determined, it can be compared with the advective 
ux.

If the evaporation rate is less than the advective 
ux, there will exist accumulations of

pore solution on the bottom surface of the roof, and this accumulation will drip onto the

contents of the vault.

45
7 Predicting Drying Shrinkage and Flexural Cracks

A recent review of the 4SIGHT computer model [47] revealed that the most important

factor aecting ionic transport through concrete is the presence of cracks. The presence

of cracks not only controls the quantity of ions transported, but can also control whether

there will be convective transport in the absence of externally applied hydraulic pressure.

That recent review is reproduced here nearly in its entirety for completeness.

Cracking can originate from within the concrete due to a number of mechanisms.

During the placement of the concrete, if the evaporation rate is great enough, the concrete

surface can develop tensile stresses sucient to crack the plastic concrete. These plastic

shrinkage cracks typically extend through the section of the concrete. If during placement

the concrete member experiences settlement, and the concrete has insucient strength

to withstand the resulting stresses, the concrete will undergo settlement cracking . If the

concrete member is a slab that spans a distance, the concrete surface in tension (typically

the bottom surface) will possibly develop 
exural cracks after removal of the concrete

form work. During the cement paste hydration, a large portion of the initial mix water

will be consumed, with the generation of heat. Dierences between concrete temperature

and ambient temperature can lead to thermal cracking, depending on the mechanical state

of the concrete. The continued removal of water by the hydration process will generate

a chemical shrinkage stress that can initiate autogeneous shrinkage cracks . Subsequent

drying due to ambient conditions will also generate a shrinkage strain that can initiate

drying shrinkage cracks when the concrete is restrained.

Some cracks are to be expected, while others are the result of improper construction

practices. Plastic shrinkage cracking, thermal cracking, and settlement cracking can be

either avoided or minimized by following industry guidelines (ACI Manual of Concrete

Practice) for the placement and curing of concrete. Both 
exural cracks and shrinkage

cracks are to be expected (under the assumptions given above), and the resulting crack

widths and spacings can be estimated. Therefore, the incorporation of crack modeling

by 4SIGHT will be limited to the prediction of 
exural and shrinkage cracking and their

eect on transport.

The models for the crack width and spacing described here will be implemented

into a structural analysis of a vault that is meant to represent a typical design. Under

ideal conditions, a structural engineer would perform an analysis of the specic vault in

question, and then input the calculated parameters into 4SIGHT. The structural analysis

presented is not meant to represent an exhaustive analysis. It is the default analysis,

however, that will be performed in the absence of any analysis by the engineer who is

using 4SIGHT. To proceed, one must assume an approximate geometry for a representative vault

structure. The principles used here for estimating the composite permeability due to

cracking are general enough to apply to many dierent vault designs. A general geometry

and construction plan for the vault will be assumed here to illustrate the analysis.

To partially simplify the structural analysis, the vault will be approximated by a

container with horizontal dimensions L1 and L2 such that their ratio is greater than

two: L1=L2 > 2. The corresponding edge loads f1 and f2 have a quartic dependence on

46
length [48]:

f1

f2 =

L4

2

L4

1

(52)

Since the load carried in the direction of the shorter span is 16 times greater, the vault

analysis will be performed using one-way slab behavior.

The spacing and width of both shrinkage and 
exural cracks depends strongly on the

amount and the location of the steel reinforcement. In addition, the positive moment

due to the load will depend on how the roof is attached to the vault walls. The roof

could either rest on the walls (simply supported) or it could be rigidly attached to the

walls (continuous reinforcement). For a given geometry and load, the simply supported

roof will sub jected to approximately twice the positive moment of a xed end roof.

Although an analysis that assumes a simply supported roof would result in a greater

estimated positive moment, it is quite unlikely that simply supported construction would

occur in practice. By denition, a simply supported roof must be able to rotate and

translate freely over the vault walls. While this type of construction can eliminate drying

shrinkage cracks, the joint provides a likely path for the ingress of 
uids and ionic species.

In addition to the analysis of the 
exural and shrinkage crack formation at the center

of the roof, there will exist negative moments at the connection between the roof and

the walls. Although an analysis of negative moment cracks is not performed here, the

method for incorporating these cracked sections into the overall composite properties

follows from the positive moment analysis. In addition, no analysis of the shear stress at

the joint between the roof and the wall is presented.

The scenario adopted here is that the vault will rst be lled with waste materials,

compacted, and the roof cast in place on top of the vault contents. The expected thickness

of the vault roof is assumed to be 1 m, and the contents of the vault will probably settle

under the dead load of the roof and the soil above the vault. The roof will then 
ex,

possibly forming 
exural cracks. In fact, a small amount of 
exure will be useful in that

it will help close shrinkage cracks at the top surface.

7.1 Cracked Reinforced Concrete

Given the construction plan outlined above, the concrete roof will likely experience shrinkage stresses during the rst months, and later experience 
exural stresses as the vault

contents settle over time. While the 
exural stresses will widen the shrinkage cracks on

the bottom of the roof, they will contract the shrinkage cracks on the top surface of the

vault. The reinforced concrete containing both shrinkage and 
exural cracks will have a

permeability that is greater than the uncracked concrete.

7.1.1 Crack Model

For simplicity, the cracks are approximated by plane parallel walls separated by the

observed crack width. The open cracks on the tension face of the roof propagate upwards

to the neutral axis. While the depth of crack penetration is accurate, the approximation

of the cracks by plane parallel walls is conservative. This assumption does not account

47
for either the \V" shape of the crack or the crack wall surface roughness, both of which

would act to reduce the crack permeability with respect to one with plane parallel sides.

The permeability k0 of a crack with plane parallel sides is solely a function of the

crack width w [47]:

k

0 =

w2

12

(53)

7.1.2 Composite Permeability

αh kc

kc k/

kc

kc k/ k/

m w1 1 m w2 2

m w

L L

h

1 2

(a) (b)

L - m w m w1 1 m w2 2 L - -

Figure 19: Composite models for the bulk permeability of the roof incorporating the

uncracked permeability kc, crack permeability k0 , and having a number m cracks over

the slab length L: (a) slabs with m cracks, each with width w; (b) slabs with m1 and m2

cracks of width w1 and w2, respectively. Composite models for the roof bulk permeability are shown in Fig. 19. The cracks

propagate from the bottom of the slab up to the neutral axis. The thickness of the slab

is h, and the depth to the neutral axis is expressed as the product h. The permeability

of the uncracked concrete is kc. The composite has an additional component with permeability k0 due to m cracks, each with width w, over the entire span L. For Fig. 19(a),

the permeability kb of the two bottom sections kc and k0 in parallel is a weighted average:

kb =

1

L

[(L  mw) kc + mwk0] (54)

The composite permeability k of kb and kc (top section in Fig. 19(a) ) in series is treated

like the corresponding problem of conductors in series:

1

k =



kc +

1  

kb

(55)

k

 =

kckb

(1  )kc + kb

48
Given that a crack with a width of one micrometer has a permeability of 81014 m2

, and that concrete permeability is on the order of 1018 m2

, the second term in Eqn. 56

can be neglected:

k

(kb  kc) = kc

(56)

Therefore, for a given concrete mixture, the control of roof permeability is the control

of the neutral axis depth, given that there is sucient 
exural strain to close shrinkage

cracks at the top surface of the roof.

The relationship between composite permeability and the neutral axis depth should

not be used as a means to reduce vault permeability. Within practical limits, the depth

of the neutral axis can only aect the composite permeability by a factor of two or

three. However, variations in concrete mixture proportions and cementitious materials

can change the concrete permeability by a factor of 10 to 100. Therefore, eorts to reduce

overall permeability should concentrate on controlling the concrete mixture.

7.1.3 Composite Permeability Model

The way in which the eects of shrinkage and 
exural cracks are combined depends upon

the formation and the geometry of the cracks. The shrinkage cracks will likely form

rst, and there will be a number ms of these cracks, each with width ws, separated by

a distance Ls. Similarly for the 
exural cracks, formed independently of the shrinkage

cracks, there will be mf 
exural cracks, each with width wf , separated by a distance

Lf . The ob jective is to consider the formation of the cracks independently, and then

determine the properties of the cracked material.

The resulting width and separation of cracks depends on the cracking scenario. Here,

the assumption is that the shrinkage cracks form rst and the 
exural cracks form after

the vault contents settle. For the case when the crack spacings are approximately equal,

so that Ls  Lf , the later 
exural cracks should simply widen the existing shrinkage

cracks. If Ls  Lf , the 
exure should once again widen the existing shrinkage cracks.

For the third case when Ls  Lf , new 
exural cracks should form in between the

existing shrinkage cracks, each with width wf and separated by Lf . If one assumes that

the existing shrinkage cracks contribute to the 
exural cracking, the shrinkage cracks

should also widen by an amount equal to wf . The particular crack model to choose from in Fig. 19 depends upon the ratio Ls=Lf . For the rst two scenarios presented (Ls=Lf  1; Ls=Lf  1), the 
exural stress widens

the existing shrinkage cracks. Each crack will then have the same width and spacing,

corresponding to the model shown in Fig. 19(a). For the third case (Ls=Lf  1), there

will exist two crack widths and spacings. The existing shrinkage cracks will widen by an

amount wf , and the mf  ms 
exural cracks will each have a width wf , corresponding to

the model shown in Fig. 19(b). The analysis of this ternary composite bulk permeability

kfollows from the binary example given previously. The parameters for each model are

shown in Table 9.

49
Table 9: Parameters for the composite permeability calculation for m cracks of width w. The crack model refers to that shown in either Fig. 19(a) or Fig. 19(b).

Ls=Lf Model w1 m1 w2 m2

< 1 (a) ws + mf ms wf ms  1 (a) ws + wf ms > 1 (b) ws + wf ms wf mf  ms

7.2 Flexural Crack Model

The model for estimating 
exural crack widths and spacings is based upon the recommendations given in ACI 224.2R-90 [3]. The recommendation for estimating 
exural

crack width wf is the Gergely-Lutz equation:

wf = 2:2  r (dcAn)

1=3 (57)

The denitions of the parameters are as follows:

d1 : distance from the neutral axis to the tension face

d2 : distance from the neutral axis to the centroid of the tension steel reinforcement

 : the ratio d1=d2

r : strain in the tension steel reinforcement

dc : minimum concrete cover thickness, measured to center of tension reinforcement

An : the largest area of concrete, perpendicular to the reinforcement, symmetric about the centroid of the tension reinforcement, divided by the

number of bars

The estimate of the spacing between 
exural cracks is based upon the CEB/FIP

model. The maximum distance ls over which slip can occur can be approximated from

the following equation:

ls =

dr 3:6 ps (58)

The quantity dr is the steel reinforcement diameter, and ps is the area of steel divided

by the eective area of concrete in tension. The average spacing between 
exural cracks

Lf is taken to be 2/3 this value:

Lf =

dr 5:4 ps (59)

7.3 Shrinkage Crack Model

The drying shrinkage model is based upon the calculations of Base and Murray [49] for

calculating the width ws and spacing Ls of shrinkage cracks. The model assumes the

member is restrained at its ends and requires the following design parameters:

50
L : length of slab

Es : Young's modulus of the steel reinforcement

Ec : Young's modulus of the concrete

n : modulus ratio = Es=Ec As : total area of steel in cross section

Ac : total area of concrete in cross section

b : slab width

d : depth of reinforcement

 : reinforcement ratio = As=bd

dr : steel reinforcement bar diameter

sh : shrinkage strain

t : concrete tensile strain at crack initiation

The \no-bond" or slip length a of reinforcement near a shrinkage crack can be approximated from the reinforcement diameter dr and the reinforcement ratio p:

a = 0:08

drp

(60)

The number of shrinkage cracks ms that form along a restrained slab of length L can be

estimated from the following equation:

ms =1+

Lnp

2a



sh  t

3 t 

(61)

The spacing between shrinkage cracks Ls can be estimated from the number of shrinkage

cracks ms over the slab span L:

Ls =

L

ms + 1

(62)

The estimate for the number of shrinkage cracks ms can be used to estimate the steel

stress fs:

fs = Es 

sh + 2t

3  

L  2ma

npL + 2ma!

(63)

The average shrinkage crack width ws can be estimated from the following equation:

ws = 2a 

fs Es +sh

3 !

(64)

7.4 Design Example

The following is an example of how one can perform a calculation to estimate the spacing,

width, and depth of shrinkage and 
exural cracks. The calculation begins with the

concrete mixture design, estimates the extent of cracking, and then estimates of the

composite permeability.

51
7.4.1 Mixture Design

The mixture design chosen for this example represents a balance between quality of

concrete and ease of formulation and placement. The design compressive strength f 0cis

35 MPa, which is within the normal strength guidelines of ACI 211.

The constituent material properties are as follows:

Cement Type Type I

Cement Density 3.2 g/cm3

CA Density 2.8 g/cm3

FA Density 2.6 g/cm3

Max. Agg. Diameter 25 mm

For a reinforced foundation with a design strength of 35 MPa, the ACI 211 procedure

results in the following trial mixture proportion:

Constituent Mass (kg/m3)

Water 190

Cement 399

Coarse Agg. 1104

Fine Agg. 757

A high range water reducer is not specied, simplifying the formulation.

Although this mixture is only a trial proportion, it will be used as the nal mixture

proportion for the purposes of demonstration.

7.4.2 Concrete Properties

Based on the ACI 211 procedure, the concrete is assumed to have the following properties:

w/c 0.43

Slump 75 mm

Air Content 1.5 %

Concrete Density 2450 kg/m3

In practice, these properties would be determined from tests performed on trial batches

and not based on the ACI 211 procedure. As a demonstration, however, the ACI values

will be used here.

The equation for estimating the elastic modulus of concrete is described in Nilson [48]:

Ec =



3320 [MPa1=2]q

f 0c + 6890 [MPa] 

c 2330 [kg/m3

]!1:5

(65)

This equations can be applied to normal density concretes over a wide range of design

strengths: 21 MPa  f 0c  83 MPa. For the mixture given above, the estimated elastic

modulus is 28.6 GPa.

52
A concrete member in tension will experience a tensile stress ft. In 
exure, the

member will remain uncracked until the tensile stresses exceed the modulus of rupture.

The estimation for the modulus of rupture fr is taken from Nilson [48]:

0:67 [MPa1=2]q

f 0c  fr  1:0 [MPa1=2]qf 0c For the mixture used in this example, it is expected that the modulus of rupture fr can

be characterized by the following inequality:

4:0 MPa  fr  5:9 MPa

120 140 160 180 200

Water Content (kg/m3)

200

300

400

500

600

700

εsh,u (

μm/m)

(a)

1 10 100

t (d)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

γcp

ACI 209

γcp = 1 − 0.103 ln t/τ

(b)

τ = 7 d

0.40 0.60 0.80 1.00

φ

0.0

0.2

0.4

0.6

0.8

1.0

γφ

(c)

0.0 0.1 0.2 0.3 0.4 0.5

ν (m)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

γvs

γvs = 1.2 exp (−4.72 ν)

(d)

Figure 20: Quantities for estimating shrinkage strain: (a) ultimate drying shrinkage

strain under standard conditions; (b) factor 
cp for duration of moist curing; (c) factor


 for ambient relative humidity; and (d) factor 
s for the volume to surface ratio.

The ACI guideline for estimating the ultimate drying shrinkage strain sh;u is shown

in Fig. 20(a) as a function of the mixture water content. For this mixture, the expected

ultimate drying shrinkage is approximately 600 m/m.

53
The expected shrinkage strain sh will be less than the ultimate value due to mitigating

factors. ACI 209R-82 gives the following mathematical expression for the combined

eects of these factors on the expected ultimate shrinkage strain:

sh = sh;u Y

i


i (66)

The quantity sh;u is the ultimate shrinkage strain under standard conditions: curing

duration of 7 d, exposure to 40 % relative humidity, and a volume-to-surface ratio of

0.038 m.

There are three factors 
i considered here: duration of curing 
cp; ambient relative

humidity 
; and volume-to-surface ratio 
vs. There are other factors outlined in ACI

209, but the factors relating to concrete mixture design do not apply here since we are

using the ultimate drying shrinkage prediction shown in Fig. 20(a).

The factor 
cp for the duration of moist curing is shown in Fig. 20(b) as a function of

curing time t. The lled circles are the recommendations from ACI 209, and the curve

is an approximation:


cp = 1  0:103 ln 

t

 

(67)

The value of  is 7 d, corresponding to the standard curing condition for the duration of

moist curing.

The factor 
 for the ambient environment is shown in Fig. 20(c) as a function relative

humidity . Over the range of relative humidity shown in Fig. 20(c), the factor has two

linear regions:


 =

8

><

>:

1:4  1:0  0:40    0:80

3:0  3:0  0:80    1:00

(68)

It is dicult to estimate the ambient relative humidity for a buried vault. However, most

certainly the relative humidity will be greater than 40 % and less than 100 %. As a

conservative estimate, the factor 
 will be assigned a value of one here. However, in

practice, an assessment should use a value commensurate with the specic soil conditions

being considered.

The factor 
vs for the geometry of the structural element is shown in Fig. 20(d) as a

function of the volume-to-surface ratio . ACI 209 expresses this factor in analytic form:


vs = 1:2 exp (=o) (69)

The quantity o equals 0.212 m, and simply renormalizes . For the present design, the

length and width of the slab are L and 2L, respectively. For a slab thickness h, the

volume-to-surface ratio  is the following:

 =

h

2+3 h

L

(70)

For a slab thickness h equal to 1 m, and a slab length L equal to 10 m, the volume-tosurface ratio  equals 0.435 m, corresponding to a shrinkage factor 
vs of 0.15.

54
The expected ultimate shrinkage strain sh is the product of the ultimate shrinkage

strain under standard conditions sh;u and the factors 
i:

sh = sh;u 
cp 
 
vs = 90 m/m (71)

This is signicantly less than the 600 m/m that was based on standard conditions.

7.4.3 Steel Properties

The steel considered for this example is ASTM A 615 Grade 60 steel; a steel with a

60 000 psi yield strength. The corresponding properties in SI units are as follows:

Es = 200 GPa (72)

fy = 414 MPa

Therefore, the modulus ratio n, dened as Es=Ec, is approximately 7.

7.4.4 Reinforcement Ratio Requirements

One of the most important parameters in estimating the bulk permeability of the reinforced concrete element is the depth of the neutral axis. This quantity is estimated

from a structural analysis of the complete concrete structure. Ideally, an engineer using

4SIGHT will have access to a structural analysis that includes an estimate of the depth

of the neutral axis. In the absence of this information, 4SIGHT must make a reasonable

approximation.

The following section is not meant as a primer on structural analysis. Rather, it

outlines the procedure 4SIGHT will use to estimate the neutral axis in the absence of

such data. If the engineer using 4SIGHT determines that the analysis invoked by 4SIGHT

is insucient, the engineer must then perform a more thorough analysis independently, and report the neutral axis as input to the 4SIGHT computer program.

There is a limit on both the minimum and maximum amount of steel reinforcement

allowed in the design of a 
exural member. The minimum limit is based on the ability

of the element to carry the intended load. The maximum limit is determined by the

ultimate failure mode within the reinforced concrete element.

The two regions of stress within the slab are the tension zone in the bottom and the

compression zone in the top. At the ultimate load, either the steel will fail in tension, or

the concrete will fail in compression. Steel failure in tension is a gradual process, giving

\warning" of impending failure. Compressive failure of concrete, by contrast, is a sudden

mode of failure. Therefore, there is a maximum limit on the amount of steel reinforcement

so as to ensure that the steel will yield before the concrete fails in compression.

The amount of steel at which the two modes of failure coincide at ultimate load is

the balanced reinforcement ratio b :

b =

f

0c fy

0:85

1 + fy 600 [MPa]

= 0:033 (73)

55
For a 35 MPa concrete mixture, the recommended value of  is approximately 0.775.

The maximum allowable reinforcement ratio max is 3/4 of the balance reinforcement

ratio:

max = 0:75 b (74) = 0:025

The minimum reinforcement ratio min can be calculated based on the assumption

that a steel stress fs that is 2/3 of the yield stress is an acceptable limit of steel stress

immediately after cracking. The corresponding reinforcement is the following:

min =

0:220 [MPa1=2] qf 0c fy

(75) = 0:0031

This is in agreement with the ACI 318 recommendation:

min =

1:38 [MPa]

fy

(76) = 0:0033

7.4.5 Roof Geometry

(n-1)A s nAs

λd

























































 

















d

b

h

(a) (b) (c)

Compression Tension

c

Figure 21: Roof geometry used for analysis: (a) cross section showing location of steel

reinforcement (lled circles); (b) transformed section used for uncracked analysis; (c)

transformed section used for cracked analysis.

The dimensions of the roof slab are dened in Fig. 21 and are given the following

values:

56
h : 1.0 m

d : 0.9 m

b : 1.0 m

dr : 35.8 mm

The slab width b is set to 1.0 m to simplify the analysis by normalizing values on a \per

meter" basis. The reinforcement diameter dr of 35.8 mm corresponds to a No. 36 metric

bar size or a No. 11 bar in customary inch-pound units.

7.4.6 Positive Moment

The positive moment at the center of the roof slab is determined from the dead load

due to the concrete mass and the mass of soil above the vault. The thickness h of the

concrete roof is 1 m, and the assumed thickness of soil hs over the roof is 10 m. The

density of the concrete c is 2450 kg/m3, and the density of the soil s is assumed to be

1500 kg/m3 (this soil density value is used for demonstration purposes and is not meant

to be an accurate estimate of any particular type of soil). The nominal uniform load

(pressure) wn on the roof depends upon the thickness of each element:

wn = cgh + sghs (77) = 171; 000 N/m2

The quantity g is the gravitational acceleration (9.8 m/s2). The design load wu:

wu = 1:4 wn (78) = 239; 000 N/m2

The quantity 1.4 is the load factor for dead loads.

The positive moment M+ at mid-span due to the ultimate load wu is the integral of

the load over half the span:

M+ =  Z

b

0 Z L=20

lwu dl db (79) = bwuL2

8

The quantity  is a factor that depends on the end conditions of the slab. For a simply

supported one-way slab,  = 1. Here, it will be assumed that the slab edge condition

corresponds to the end slab of a multi-span roof; one end is restrained and the other is

simply supported. Under these conditions,  = 4/7:

M+ = 1; 707; 000 Nm

7.4.7 Uncracked Analysis

The starting point for the structural analysis is an analysis of the roof assuming that there

are neither 
exural nor shrinkage cracks. The maximum 
exural stress from the design

57
loads can then be compared to the modulus of rupture. If the 
exural stress exceeds the

modulus of rupture, the analysis must be repeated assuming 
exural cracking.

As the slab 
exes, the top portion of the roof will be in compression, and the bottom

portion in tension. There will exist a 2-D surface through the midsection of the slab

that denes where the concrete is neither in tension nor in compression. This surface is

referred to as the neutral axis.

For a member in pure bending (no shearing force) the strains are linearly proportional

to the distance from the neutral axis. Since the coecient of proportionality is same for

both the tensile and the compressive strains (within the elastic limit), the location of the

neutral axis can be determined by balancing the area moments of the slab cross section.

To simplify the calculation of the tensile and compressive area moments, the reinforced

concrete slab cross section is rst transformed into an equivalent concrete cross section.

Specically, the steel is replaced mathematically by an area of concrete equal to (n1)As, and located at the depth of the reinforcement. This is shown schematically in Fig. 21(b)

for this analysis.

The equation for calculating the area moment is the integral of the moment arm r

(absolute distance from the neutral axis) over the slab cross section (da = dr db):

M =

Z

r da (80)

The moment of the area above the neutral axis, which is located at depth c from the top

of the slab, is set equal to the moment of the area below the neutral axis:

bc2

2 =

b (h  c)

2

2 + (d  c) (n  1) As (81)

The quadratic terms in c cancel, yielding an algebraic solution:

c =

bh2 + 2(n  1)Asd

2bh + 2(n  1)As (82)

The steel area can be expressed as a function of the steel reinforcement ratio :

 =

As bd (83)

This can be used to simplify the equation for the neutral axis depth c:

c =

d

2

2

6

4



h

d 2 + 2(n  1)

h=d + (n  1)

3

7

5 (84)

The neutral axis depth c, as a function of the reinforcement ratio , is shown in Fig. 22.

For the uncracked analysis, the depth of the neutral axis is insensitive to the reinforcement

ratio.

The stresses within the transformed concrete section will depend upon the moment

of inertia I of the transformed slab. The moment of inertia about the neutral axis is the

integral of the square of the moment arm over the slab cross section:

I =

Z

r

2 da (85)

58
0.00 0.01 0.02 0.03

ρ

0.45

0.50

0.55

0.60

0.65

c

Figure 22: Depth of neutral axis c as a function of the reinforcement ratio  for the

uncracked analysis.

For the transformed section, the moment of inertia is

I =

by3

3 +

b(h  y)

3

3 + (d  y)

2 (n  1)As (86)

For linear elastic behavior, the compressive stress fc and the tensile stress ft at the

top and bottom surfaces are proportional to the distances from the neutral axis:

fc = M+

y

I ft = M+

h  y

I

(87)

These quantities are shown in Fig. 23 as a function of the reinforcement ratio p. The

compressive stress fc is approximately one-quarter the design strength f 0cfor all values

of the reinforcement ratio. By contrast, the tensile stress ft is considerably greater than

the estimated modulus of rupture range of 4 MPa to 6 MPa. Therefore, one expects that

the roof slab would develop cracks in the middle portion of the slab.

7.4.8 Cracked Section Analysis

Based upon the previous analysis, it is likely that the concrete roof slab will crack due to


exural loading. These cracks will initiate at the tension surface and continue upward to

approximately the neutral axis. Analysis based upon this idea requires a dierent transformed cross section. This transformed cross section is similar to that of the uncracked

transformed cross section except that the concrete below the neutral axis is disregarded.

Determination of the neutral axis for the cracked cross section is similar to the uncracked case. The dierence being that one assumes that the concrete in tension does

59
0.00 0.01 0.02 0.03

p

7

8

9

10

11

f (MPa)

f

c

f

t

Figure 23: Compressive stress fc and tensile stress ft as a function of the reinforcement

ratio , based on the uncracked analysis.

not contribute to the moment calculation. This cracked transformed section is shown

schematically in Fig. 21(c). The portion of the concrete below the neutral axis is disregarded, and the steel is transformed into an equivalent area nAs of concrete.

Let the depth of the neutral axis be a fraction  of the reinforcement depth d. The

balance of the area moments about the neutral axis leads to the following algebraic

equation:

b

(d)

2

2 = nAs(d  d) (88)

The solution for  is found from the quadratic equation:

 = q

(np)

2 + 2np  np (89)

The neutral axis depth factor  is shown in Fig. 24(a) as a function of the reinforcement ratio . In contrast to the uncracked section analysis, the depth of the neutral axis

for the cracked cross section has a strong dependence on the reinforcement ratio.

The stresses at the compression surface and the steel reinforcement are calculated in

much the same way as for the uncracked analysis. The calculation of the cracked section

moment of inertia Icr omits the second term on the right hand side of Eqn. 86 because

this concrete is in tension. Assuming linear elastic behavior, the compressive stress fc in

concrete at the top surface and the tensile stress ft in the steel are proportional to their

distance from the neutral axis:

fc = M+

d

Icr ft = M+

d(1  )

Icr (90)

60
0.00 0.01 0.02 0.03

ρ

0.1

0.2

0.3

0.4

0.5

λ

(a)

0.00 0.01 0.02 0.03

ρ

0.0

0.2

0.4

0.6

0.8

Relative Stress

f

t

 / fy

f

c

 / fc

’

(b)

Figure 24: Cracked transformed section analysis: (a) relative depth of neutral axis ; (b)

relative tensile stress ft=fy and compressive stress fc=f 0cfor the cracked section analysis,

assuming linear elastic behavior.

The compressive stress fc and the tensile stress ft are shown in Fig. 24(b) as a

function of the reinforcement ratio . Further, both stresses are normalized by their

limiting values. The ultimate compressive stress is f 0c, and the limiting tensile stress is

the steel yield stress fy . Based on the assumptions that the vault contents will subside and that the roof slab

will be a span in 
exure, the top surface of the slab should remain in 
exure, regardless

of the shrinkage strain. The eect of the shrinkage strain, if any, will be to increase the

bending moment of the slab. Regardless, the top surface of a linear elastic member in


exure should be in compression, with no open cracks. Therefore, the importance of the

structural analysis is to determine the depth of the neutral axis and the width of the

cracks within the bottom surface of the slab.

Proceeding further with the analysis requires employing a specic model for combined

cracking under 
exural and shrinkage stresses. First, one considers the development of

he cracks to be independent of one another. Based on this estimation, an estimate of the

distances between the nal cracks will be formulated based on physical intuition.

An estimate for the nal crack width and spacing within the bottom surface of the

slab depends on the relative size of the shrinkage and 
exural cracks. The estimate for the

ratio of shrinkage crack spacing Ls to the 
exural crack spacing Lf is shown in Fig. 25(a)

as a function of the reinforcement ratio p. Over most of the allowable range for p, the

ratio Ls=Lf is greater than ve. Therefore, the 
exural cracks should develop between

shrinkage cracks.

An independent analysis of the shrinkage crack width ws and 
exural crack width wf

are shown in Fig. 25(b) as a function of the reinforcement ratio . From an assessment

of the estimate of crack widths, the combined crack widths will certainly be tens, if not

hundreds, of micrometers wide. The cracks will occur at the tension face and extend at

61
0.000 0.010 0.020 0.030

ρ

0

20

40

60

80

100

Ls / Lf

(a)

0.00 0.01 0.02 0.03

ρ

1

10

100

1000

10000

crack width (

μm)

ws

wf

(b)

Figure 25: Results of the cracking model: (a) The ratio of the shrinkage crack spacing

Ls and the 
exural crack spacing Lf ; (b) Flexural crack width wf and shrinkage crack

width ws.

least to the neutral axis.

7.4.9 Simplied Composite Permeability

Since the permeability of the cracked portion of the slab will be orders of magnitude

greater than the uncracked portion of the concrete, further analysis can be simplied by

approximating the composite permeability k using Eqn. 56:

k kc  1 =

h

d

This result is plotted in Fig. 26(a) as a function of the reinforcement ratio . The ratio of

the permeabilities increases by less than a factor of ve over most of the allowable range

of .Since the exact crack width at the bottom surface of the concrete may become inconsequential, the signicance of the structural analysis will be to formulate an accurate

estimate for the depth of the neutral axis. This fact is emphasized in Fig. 26(b) that

shows a plot of the ratio k=kc as a function of the relative depth to the neutral axis , for

three values of the ratio kb=kc. Note that the ratio k=kc converges rapidly with respect

to the ratio kb=kc. In practice, the ratio kb=kc could be orders of magnitude greater than

those shown in the graph.

7.5 Probabilistic Approach

A deterministic calculation, like that performed in the previous section, is useful for estimating the probable behavior of the reinforced concrete structure. In practice, however,

62
0.00 0.01 0.02 0.03

ρ

2

3

4

5

6

k

*

 / kc

(a)

0.0 0.2 0.4 0.6 0.8 1.0

α

1

10

100

k

*

/ kc

10

100

1000

kb / kc

(b)

Figure 26: Ratio of the composite permeability k to the uncracked concrete permeability

kc: (a) as a function of reinforcement ratio , and (b) as a function of the relative depth

to the neutral axis .

the exact size of the cracks that form will vary, as will the depth of the neutral axis. The

resulting variations in these quantities lead to variations in the expected permeability of

the structure. In order to capture this variation a probabilistic approach is used.

The probabilistic method employed here is based on a Monte Carlo calculation using

parameter uncertainty as a means of characterizing random deviates. Each parameter

required for a deterministic calculation has associated with it an estimated uncertainty. With this uncertainty, one can use a computer to repeat the deterministic calculation numerous times, with each iteration using a new random deviate for each parameter. From

the multiple calculations, the estimated composite permeability from the each iteration

form a population of permeability values, from which a population mean and standard

deviation can be calculated. These values can then be incorporated into the service life

calculation, which will, in turn, yield an estimated mean and standard deviation for the

service life of the structure.

7.5.1 Initial Material Parameters

The following probabilistic analysis follows identically from the previous section with the

only exception being that some of the input parameters are normally distributed about

some mean value with a coecient of variation (CoV) (the ratio of the mean to the

standard deviation). The parameters allowed to vary are shown in Table 10. Note that

the mean values are identical to those used in the previous deterministic analysis. The

values for the (CoV) are not meant to be accurate for any particular structure, and are

only given for demonstration of the technique. The reader is referred to the ACI 117

report on construction tolerances for general guidance.

The values for the coecient of variation given in Table 10 are not based on any

63
Table 10: The mean and associated coecient of variation (CoV) for the input parameters

that would be used in an uncertainty analysis of 
exural and shrinkage cracking.

Variable Property Mean CoV

L Slab length (m) 10.0 0.05

Ec Concrete Modulus (GPa) 28.6 0.05

f 0c Concrete compressive strength (MPa) 35.0 0.05

h Slab thickness (m) 1.00 0.10

d Depth of steel (m) 0.90 0.05

s Shrinkage strain (m/m) 90 0.25

particular data. An analysis of particular vault should be done with careful consideration

of the construction plans and the pro ject description. For example, the CoV for the slab

thickness given above is large compared with normal slab on grade construction. The

reason for this is the assumption that the contents of the vault would be compacted

remotely, and so the variability of the slab thickness would be greater than for normal

practices. Construction plans for specic vaults may vary and warrant a smaller CoV.

For the probabilistic analysis, the randomization is done as a function of the designed

reinforcement ratio . For each value of , random values are drawn for the parameters

shown above. Using these, the analysis proceeds as for the deterministic analysis. The

approach used by 4SIGHT would be to establish a xed reinforcement ratio  and then

repeat the analysis at a constant value of , randomizing the input parameters listed

above. 4SIGHT would then report the mean and standard deviation of the results at a

given reinforcement ratio.

7.5.2 Results

The material and structural parameters given in Table 10 are used to recalculate the

same quantities as were calculated for the deterministic analysis. The only dierence is

that the calculation is repeated many times. Each time, a value for the reinforcement

ratio p is chosen and then the computer generates a random number for each parameter

listed in Table 10. These \randomized" values are then used in the deterministic analysis.

The variation in the result indicates the output sensitivity to that parameter, at that

value of CoV. Each gure of the results to be shown includes a solid curve indicating the

values from the previous deterministic calculations.

The results for the depth of the neutral axis and the relative stresses are shown in

Fig. 27 and can be compared directly to Fig. 24. In general, the probabilistic data follow

the same trends as in the deterministic case. The calculated depths of the neutral axis

shown in Fig. 27(a) are a function of the reinforcement ratio . The results indicate that

the uncertainty in the depth of the neutral axis is relatively insensitive to the uncertainty

in the input parameters.

By comparison, the relative compressive and tensile stresses shown in Fig. 27(b) are

much more sensitive. The variability in the compressive stress ratio fc=f 0cis dramatic,

64
0.00 0.01 0.02 0.03

ρ

0.1

0.2

0.3

0.4

0.5

λ

0.00 0.01 0.02 0.03

ρ

0.0

0.2

0.4

0.6

0.8

1.0

Relative Stress

f

c

 / fc

’

f

t

 / fy

Figure 27: Probabilistic analysis results: (a) Relative depth of neutral axis ; (b) Relative

tensile stress ft=fy and compressive stress fc=f 0cfor the cracked section analysis.

with some of the values approaching one, even for relatively large reinforcement ratios.

Since the analysis has been done assuming a linear elastic material, a relative stress

greater than 0.5 is probably unreliable. Regardless, the tensile stresses in the bottom of

the slab will control the crack width.

The ratio of the shrinkage crack spacing Ls to the 
exural crack spacing Lf is shown

in Fig. 28(a) as a function of the reinforcement ratio . Although the ratio has large

variation, the value is typically far greater than one. Therefore, regardless of the variation

in input parameters, one expects that 
exural cracks will be initiated between the existing

shrinkage cracks.

The shrinkage crack width ws and 
exural crack width wf are shown in Fig. 28(b). The

shrinkage crack widths seem to be insensitive to the variations in the input parameters.

Although the 
exural cracks appear to be more sensitive, the data are plotted on a

logarithmic ordinate which enhances the appearance of variability in smaller quantities.

Since the expected crack widths are still on the order of tens or hundreds of micrometers, the estimate of the bulk concrete permeability k can again be simplied with the

following approximation:

k kc h

d

This result is shown in Fig. 29. Since the depth of the neutral axis was fairly insensitive

to the variability of the input parameters, the relative bulk permeability is also relatively

insensitive to parameter variation. The added variation over  is due to the variability

in the roof thickness h and the depth of steel reinforcement d. The result shown in Fig. 29 is signicant because it reveals that the details of the

drying shrinkage model are relatively unimportant. Rather, the most signicant factors

in predicting the permeability of a reinforced concrete structure is the permeability of the

sound concrete, and the depth of the neutral axis. Therefore, the two most important

65
0.00 0.01 0.02 0.03

ρ

0

20

40

60

80

100

Ls/Lf

0.000 0.010 0.020

ρ

1

10

100

1000

10000

Crack Width (m)

ws

wf

Figure 28: Results of the probabilistic analysis of the cracking models: (a) the ratio of

the shrinkage crack spacing Ls to the 
exural crack spacing Lf ; (b) 
exural crack width

wf and shrinkage crack width ws.

requirements of the engineering using 4SIGHT to assess a concrete vault is an accurate

estimation of the sound concrete permeability and an accurate estimate of the depth of

the neutral axis.

66
0.00 0.01 0.02 0.03

ρ

2

3

4

5

6

k

*

/kc

Figure 29: From the probabilistic analysis, the ratio of the composite permeability k to

the uncracked concrete permeability kc.

67
8 User's Guide

An evaluation of the previous user interface to 4SIGHT revealed unnecessary complexity. In an eort to simplify the use of 4SIGHT, a number of assumptions have been

incorporated. The input le is now more restrictive, but the requirements are fewer. The

ob jective was to simplify the user's burden and facilitate the independent development of

a graphical front end for user input and computed output. A graphical front end could

be tailored to a specic software platforms, while the computational portion remains

relatively platform independent.

8.1 Input File

In general, most of the parameters from 4SIGHT remain. In some cases, however, the

expression has been compressed to facilitate the addition of parameter uncertainty. A

number of parameters require a single value, others require the specication of a probability distribution (for parameter uncertainty Monte Carlo calculation), and still others

allow for default values with a specied coecient of variation. A useful means of describing the input le is to give an example:

21 N_cells

4 num_species

8 0.000 0.300 0.000

19 0.000 0.200 0.000

11 0.200 0.100 0.000

17 0.200 0.000 0.000

0 1 free_surface_bool

0.02 0.0 pressure_MPa

-4 -9.00 0.500 permeability_cm/s

-3 1000. 0.25 formation_factor

-3 0.150 0.05 porosity

+3 0.500 0.05 wc

+3 20.0 1.00 fc_MPa

-3 20.0 0.10 Elastic_modulus_GPa

3 10.0 0.10 length_m

3 1.000 0.020 thickness_m

3 0.900 0.010 steel_depth_m

3 10.00 1.0 soil_depth_m

-3 0.010 0.050 As/Ac:reinforcement_ratio

100000. time_limit_days

23204 seed

10 simulations

The parameters shown will be discussed on a line by line basis. The optional text that

follows each numerical parameter(s) is included as an aid to the user, but is ignored by

the computer program.

68
Line 1: N_cells

The number of nite dierence computational elements used to represent the slab for the

1-D calculation. A value of 21 is typically sucient.

Line 2: num_species

The number of ionic species available for transport. The available ionic species are the

following:

Key Species

1 H+

3 Li+

11 Na+

12 Mg+2

19 K+

20 Ca+2

6 CO32

7 NO3

8 OH

16 SO42

17 Cl

35 Br

53 I

In this example there are four species: OH, Cl, K+, Na+. Line 2a,2b,. . . :

The quantity of each of the num_species present. The four values given (AtNo cA cInt

cB)are as follows:

AtNo : atomic number of species. In the case of OHand SO2

4 , it is the atomic number

of the rst species. Ex.: OH = 8, SO2

4 = 16, etc.

cA : molarity of species external to Side A. cInt : molarity of species within the concrete.

cB : molarity of species external to Side B. Side A and Side B are arbitrary. Line 3: free_surface_bool

A boolean value, either 0 or 1, denoting whether Side A or Side B is a free_surface. For a free_surface, the diusive 
ux is zero and ions within the pore solution accumulate in the presence of a Darcy 
ow.

Line 4: pressure_MPa

The hydrostatic pressure at side 0 and side 1, in units of MPa.

69
n.b.: For some of the following parameters, the user is to supply 4SIGHT with information

regarding the type of distribution to use and the parameters of that distribution: xo and

. The meaning of the distribution parameters have been discussed previously. The

particular distribution is specied using the following numbering scheme:

n = 1 : delta function

n = 2 : uniform distribution

n = 3 : Gaussian distribution

n = 4 : Lognormal distribution

If the user is specifying a denite value (and not relying upon 4SIGHT to create and

estimated value for them), the distribution type n is made positive using a leading plus

sign. If the distribution type n is made negative with a leading minus sign, 4SIGHT

will generate a default value based on the remaining specied values. If a default value

is requested, the value for xo appearing in the input le will be ignored and and the

user species the coecient of variation of this default value through the parameter

. The exception to this is the hydraulic conductivity: if its distribution type n is

negative,  represents the standard deviation of the logarithm, which is identical to its

meaning for positive values of the distribution type. The relationships used by 4SIGHT

to calculate default values either appear in this report or can be found in the previous

4SIGHT report [1].

Line 5: permeability_cm/s

The logarithm (base 10) of the hydraulic conductivity of uncracked concrete, when expressed in units of cm/s. The three values shown are the distribution type, xo, and . In the example, the user is requesting that the computer estimate the hydraulic conductivity, and that the standard deviation of the logarithms (base 10) of these values be

0.500. This corresponds to hydraulic conductivity values varying over one half and order

of magnitude.

Line 6: formation_factor

The formation factor of the intact concrete. The three numbers are the distribution type,

xo, and . In the example above, the user is requesting 4SIGHT to estimate the formation

factor from the other parameters, and that these predicted values will have a coecient

of variation of 25 %.

Line 7: porosity

The capillary porosity of the intact concrete. The three numbers are the distribution

type, xo, and . In the example above, the user is requesting 4SIGHT to estimate the

porosity from the other parameters, and that these estimated values will have a coecient

of variation of 5 %.

70
Line 8: wc

The water cement ratio at the time of mixing. The three numbers are the distribution

type, xo, and . In the example above, the user is specifying that the water cement ratio

shall be chosen from a Gaussian distribution with mean 0.50 and standard deviation 0.05.

Line 9: fc_MPa

The design compressive strength of the concrete. The three numbers are the distribution

type, xo, and . In the example above, the user is specifying that the compressive

strength shall be chosen from a Gaussian distribution with mean 20 MPa. and standard

deviation 1.0 MPa.

n.b.: At least one of the above six parameters must be specied by the user. Therefore,

at least one of these values must have a positive distribution type. Otherwise, 4SIGHT

has no information from which to characterize the concrete. Of course, the user may

specify all of these parameters. In the above input le, the + and - are used to remind

the user of this requirement.

Line 10: Elastic_modulus_GPa

The elastic modulus of the concrete, in units of GPa. The three numbers are the distribution type, xo, and . In the example above, the user is specifying that the elastic modulus

be estimated from the other material parameters, and that these values be chosen to have

a coecient of variation of 10 %.

Line 11: length_m

The length of the slab, in units of m. The three numbers are the distribution type, xo, and . In the example above, the user is specifying that the length be estimated from a

Gaussian distribution with mean 10.00 and standard deviation 0.10.

Line 12: thickness_m

The vertical thickness of the slab, in units of m. The three numbers are the distribution

type, xo, and . In the example above, the user is specifying that the thickness be

estimated from a Gaussian distribution with mean 1.00 and standard deviation 0.02.

Line 12: steel_depth_m

The depth of reinforcement steel, in units of m. The three numbers are the distribution

type, xo, and . In the example above, the user is specifying that the depth of steel

reinforcement be estimated from a Gaussian distribution with mean 0.90 and standard

deviation 0.01.

Line 12: soil_depth_m

The depth of soil over the vault, in units of m. The three numbers are the distribution

type, xo, and . In the example above, the user is specifying that the depth of steel

reinforcement be estimated from a Gaussian distribution with mean 10.00 and standard

71
deviation 1.00.

Assigning a negative value to the depth of soil informs 4SIGHT that the user does not

want a crack analysis, and that the concrete should remain entirely intact, without any

cracks.

Line 12: As/Ac:reinforcement_ratio

The reinforcement ratio, dened as the ratio of the area of steel (As) to the area of concrete

(Ac). The three numbers are the distribution type, xo, and . In the example above,

the user is specifying that 4SIGHT should estimate a \best" guess from the available data

and that deviates should have a coecient of variation of 5 %.

Line 12: time_limit_days

The time limit for the service life calculation, in units of days.

Line 12: seed

A seed for the random number generator. Using the same seed will result in identical

values being generated.

Line 12: simulations

The number of times to perform a parameter uncertainty calculation.

72
9 Summary

These eorts directed towards 4SIGHT have moved it towards a rigorous model of advectivediusive transport within concrete. This is a fundamental part of establishing a service

life model based on the simulation of reactions within the concrete. Every degradation

mechanism relevant to concrete performance depends, in some way, on the transport

characteristics of the material. Although applying transport models to both reference

data and carefully controlled experiments appear to be logical endeavors, they are rare

procedures in the concrete materials research literature.

The work done for this pro ject has shown that a carefully constructed model that

accounts for ion-ion interactions, due to both electrostatic interactions and excluded

volume interactions, can then be used to model \real world" behavior. Further, the

resulting transport equation reveals the meaningful physical quantities that are required

from experiments performed on the material of interest. It was serendipitous that the

more rigorous equation actually requires fewer physical parameters, regardless of the

number of dierent species being transported.

An unforeseen result of this work has been a critical evaluation of the meaningfulness of the apparent diusion coecient. Experiments veried theoretical expectations

that the apparent diusion coecient can vary over time in nonreacting porous systems.

Data from experiments done here have shown that the apparent diusion coecient of a

particular ion can vary by a factor of ten, depending on the other ionic species present.

Further, a particular collection of species can lead to a negative apparent diusion coef-

cient. In contrast, a rigorous transport equation, using a xed porosity and formation

factor, can capture this behavior by incorporating the ion-ion interactions into the transport equation.

The addition of a simple structural analysis into 4SIGHT will allow users to make, in

the absence of a more thorough analysis, reasonable approximations of the width, depth,

and spacing of drying shrinkage and 
exural cracks. An analytical investigation of these

relationships revealed that the most important factors in
uencing the bulk permeability

of a reinforced concrete slab in 
exure is the permeability of the uncracked concrete

and the depth of the neutral axis. The permeability is insensitive to the details of the

particular crack model employed.

The addition of a Monte Carlo calculation based on parameter uncertainty addresses

the inherent issue of variability. A deterministic service life calculation is meaningless

since it would be impossible to guarantee the predicted result. Rather, by at least

incorporating the parameter uncertainty into the result, one can make a statement about

the result uncertainty as a function of the parameter uncertainty. Further, a probabilistic

result is a better representation of a physical problem since one can then ask for prediction

intervals of the computed results.

73
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74
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75
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In Brines," Water-Resources Investigations Report 88-4153, U.S. Geological Survey, Reston, VA, 310 pp., 1988.

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Theory and Experiment," (Presented) Advances in Cement and Concrete (United

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Non-Reactive Systems," (Presented) Symposium FF: Materials Science of HighPerformance Concrete (Materials Research Society), Boston, MA, November 2000.

[34] K.A. Snyder, \The relationship between the formation factor and the diusion coe-

cient of porous materials saturated with concentrated electrolytes: Theoretical and

experimental considerations," (Submitted to) Concrete Science and Engineering , 2000.

[35] K.A. Snyder and J. Marchand, Eect of speciation on the apparent diusion coef-

cient in nonreactive porous systems," (Submitted to) Cement and Concrete Research, 2000.

[36] R.E. McDu and R.A. Ellis, \Determining diusion-coecients in marine-sediments

{ Laboratory study of the validity of resistivity techniques," Am. J. Sci. 279 (1979)

666{675.

[37] F.A. Settle, ed., Handbook of Instrumental Techniques for Analytical Chemistry, Prentice Hall, 1997.

[38] L.M. Schwartz, P,N. Sen, and D.L. Johnson, \In
uence of rough surfaces on electrolytic conduction in porous media," Phys. Rev. B 40 (1989) 2450{2458.

[39] H.J.V. Tyrrell and K.R. Harris, Diusion in Liquids , Butterworths, 1984.

[40] B.N. Taylor and C.E. Kuyatt, \Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results," National Institute of Standards and Technology, NIST Technical Note 1297, Gaithersburg, MD, 1993.

[41] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, Numerical Recipes , Cambridge University Press, New York, 1988.

[42] D.E. Knuth, Seminumerical Algorithms , 2nd ed., Vol. 2 of The Art of Computer

Programming , Addison-Wesley, 1981.

76
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