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ACI Structural Journal/May-June 2006 399

ACI Structural Journal, V. 103, No. 3, May-June 2006.MS No. 04-306 received September 27, 2004, and reviewed under Institute publication

policies. Copyright 2006, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including authors closure, if any, will be published in the March-April2007ACI Structural Journal if the discussion is received by November 1, 2006.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Large structures present numerous possible test locations for anondestructive evaluation. Challenges lie in selecting testlocations, managing data collected, and stating testing results.This research evaluated the feasibility of using sampling methodsto assist in these tasks. To assess the methods applicability,sampling was applied to data from actual structures that had

previously been extensively tested. The researchers could thencompare their predictions based on sampling to actual results fromcomprehensive testing. These studies demonstrated that samplingmethods are useful at determining the number of samples and theirlocations. The results can effectively be stated as a confidenceinterval, presenting a range for the prediction based on an acceptable

uncertainty. In Part I, a brief description of some samplingmethods is given and the procedure (including simple random,stratified, and adaptive sampling) is applied to a post-tensionedbridge, which was nondestructively tested to locate air voidswithin grouted tendon ducts.

Keywords: post-tensioned; sampling; test.

INTRODUCTIONIn recent years, the development of a wide variety of

nondestructive testing methods for concrete structures hasprovided engineers with numerous possibilities for evaluatingstructures.1 While greatly expanding an engineers capabilities,this availability of testing techniques has also introduced itsown set of challenges, particularly when evaluating a largestructure. The engineer faces the challenge of dealing withhundreds to many thousands of possible test locations. Timeand cost constraints work to limit the test number to a minimumwhile the desire to accurately assess the state of the structureargues for the maximum number of tests.

This paper and its companion, Sampling Techniques forEvaluating Large Concrete Structures, Part II, which willappear in the July-August issue of the ACI StructuralJournal, present research aimed at examining the use ofsampling techniques to assist the engineer in making choicesconcerning the number and location of tests and in stating theextent of knowledge gained from the testing. Two case studiesare presented. The nondestructive test data for the structureshighlighted in the case studies were initially collected for allpossible test locations in structural investigations; therefore,the authors had the unique opportunity to compare samplingpredictions to the actual state of the structures to evaluate theaccuracy of various sampling approaches.

The structures examined include a post-tensionedbridge,2,3 on which nondestructive testing was performed tolocate air voids within grouted tendon ducts, and a 7.5 mi(12 km) long, reinforced concrete seawall,4 where the aimwas to locate delaminations caused by corrosion of thereinforcing bars. In the first case, sampling methods,including simple random, stratified, and adaptive sampling,were used to determine the number and location of test

points along the bridge. The information collected fromthese tests was used to estimate the level of damage in theentire bridge within a given confidence; these results werethen compared with actual damage statistics. In the secondcase, sampling methods, including simple random, systematic,and adaptive sampling, were employed to make predictionsabout the state of the walls based on tests on only a fractionof the wall panels. Again, the results were compared to theactual results from testing the entire structure. In addition,the seawall data was also used to construct probabilisticmodels to examine patterns in the damage. Subsequently,

repair options were incorporated into these models to determinetheir reliability. The results of these studies were stated interms of the cost of repair versus the predicted cost of failure.This work is summarized in Reference 5.

This paper provides background information on samplingmethods and focuses on the case study of the post-tensionedbridge. In the companion paper, the case study of the reinforcedconcrete seawall is presented and key conclusions are givenbased on the results of both case studies. For a more in depthdiscussion of sampling concepts and their application in thecase studies, the reader is referred to Reference 6.

RESEARCH SIGNIFICANCE

This research has led to the development of a method fordetermining the number and locations of tests in nondestructiveassessment of large concrete structures. The method showshow the information obtained from these tests can then beused to make a prediction about the state of the entire structureusing confidence intervals. This is the first time thatsampling techniques were used to establish the damage statein concrete structures. The results of the studies presentedindicate that sampling techniques are very useful in makingthe collection and analysis of data from nondestructive testsmore efficient and cost effective.

BACKGROUND ON SAMPLINGSampling methods, which allow statements to be made

about an entire group based on data collected for only acertain portion, were applied to the nondestructive testing ofstructures for flaw detection. The inspection schemes thatare the focus of the research are those in which the data takenat each test point is in the form of a Binomial variable (a yes/no answer) such as flaw/no flaw information. For example,tests may be performed at locations on the surface of a

Title no. 103-S42

Sampling Techniques for Evaluating Large Concrete

Structures: Part I

by Tamara Jadik Williams, Linda K. Nozick, Mary J. Sansalone, and Randall W. Poston

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ACI Structural Journal/May-June 2006400

concrete structure to determine whether or not the reinforcingsteel has corroded and caused delaminations in the concreteor beneath the surface or to determine whether there arevoids in grouted tendon ducts. It is assumed that the engineer

performing the tests has only limited knowledge of the currentstate of the specific structure and its state in the recent past. Alsothe main aspect of the evaluation consists of nondestructivetests that provide localized results on a point-by-point basis. Inthese cases, the possible number of test points may be many.Sampling theory is presented as a possible assistant tochoosing the number and location of tests to obtain themaximum information about the entire structure.

Some basic concepts are presented in this section to assistin the understanding of the case studies in this paper and itscompanion. A population is the entire set of a known, finitenumberNof sampling units. A sampling unit is the particularsection of the population for which the data is collected. Aunit may be a single person or institution or it may be a

geographic unit such as a plot of land. The data or valuetaken for each unit is referred to as they-values of the unit.

The general description of how the sample is taken andanalyzed is referred to as the sampling strategy whichconsists of the sampling design together with the inferencemethods. The sampling design is the procedure for selectingthe sampling units. The design must address such concerns assize, selection, and observation method. The samplingdesigns can be placed into one of three general categories,namely conventional, adaptive, and nonstandard. An inferencemethod helps to draw conclusions about the entire populationbased on the data from the samples observed. These infer-ences may take the form of estimates such as the populationmean, tests of hypothesis, and confidence intervals (whichstate the accuracy or confidence of the estimates).7

Of the sampling designs, the conventional design has unitsselected prior to the data collection. Some conventionalsampling methods include simple random sampling, stratifiedsampling, and systematic sampling. Simple randomsampling is a design in which n units are selected from thepopulation in a random order. In this design, each possiblecombination ofn units is equally likely to be selected, and ateach step every unit has an equal probability of selection. Agraphic representation of a random sample of 10 units froma population total of 100 is shown in Fig. 1(a). Stratifiedsampling is a design in which the population is partitioned

into regions (strata) and a sample is selected by some designwithin each stratum. This sampling design is of the most benefitwhen the units within a stratum are as similar as possible. One ofthe types of stratified sampling is stratified random sampling inwhich the units in each stratum are selected by simple randomsampling. When the number of units sampled in each stratum isproportional to the size of the stratum, the sampling is said to bedone with proportional allocation. A graphic illustration of this isshown in Fig. 1(b) where five units are sampled from the stratumof size 50, three from the size 30 stratum, and two from the size20 stratum. Systematic sampling consists of selecting a startingpoint (such as by simple random sampling) and then selecting allthe units spaced in a systematic fashion throughout the popula-tion. In a sample, there may be one starting point or several.

Figure 1(c) shows a systematic sample with two starting points.

7

In the conventional designs, the units for sampling couldbe selected before any observation began. In an adaptivesampling design, the procedure for selecting the units isbased on the values that are observed during the samplingprocess and includes gathering more information in theneighboring area of an observed high value. These units maybe selected in different ways, with the major differencesbetween the designs existing in the initial sample selection.

In adaptive random sampling, an initial set of units isselected by simple random sampling (as was done in Fig. 1(a)).But as the values are observed, the sample can be adaptivelyincreased to include units in the neighborhood of observedunits fitting a certain criteria. For example, if the criterion for

further sampling is an observed nonzero value, then when-ever a unit with a nonzero value is found, the neighboringunits are observed. A graphic representation of such asample is shown in Fig. 2(a). After the initial simple randomsample is taken (shown in dark gray), units are added directlyabove, below, and to each side of the units with nonzerovalues. If any of the added units have nonzero values (shownwith a black dot), then the neighboring units to those are alsoincluded. This process continues until all adjacent units withnonzero values are added to the sample. All of the units thatwould be sampled in addition to the initial random sampleare shown in the light gray checkered blocks in Fig. 2(a).

Tamara Jadik Williams is an adjunct professor in the Department of Civil andEnvironmental Engineering at Lafayette College, Easton, Pa. She received her BSE in

civil engineering from Princeton University, Princeton, N.J., and her MS and PhD in

structural engineering from Cornell University, Ithaca, N.Y. Her research interests

include reliability and sampling methods for the nondestructive testing of large structures.

Linda K. Nozick is a professor in the School of Civil and Environmental Engineering

and the Director of Graduate Studies for the Program in Systems Engineering at Cornell

University. She received her BSE in systems engineering from George Washington

University, Washington, D.C., and her MS and PhD in systems engineering from the

University of Pennsylvania, Philadelphia, Pa. Her research interests include the

development of mathematical models for use in the management of complex systems.

Mary J. Sansalone, FACI, is a professor of civil and environmental engineering atCornell University. She is a member of ACI Committee E 803, Faculty Network

Coordinating Committee. Her research interests include nondestructive evaluation of

materials and structures.

Randall W. Poston, FACI, is a principal of Whitlock Dalrymple Poston and Associates,

Austin, Tex. He is a member and Past Chair of ACI Committee 224, Cracking, and a

member of ACI Committees 222, Corrosion of Metals in Concrete; 228, Nondestructive

Testing of Concrete; 318, Structural Concrete Building Code; 318-F, New Materials,

Products, and Ideas; and 562, Evaluation, Repair, and Rehabilitation of Concrete

Buildings.

Fig. 1Conventional sampling designs: (a) simple random;

(b) stratified; and (c) systematic.

Fig. 2Adaptive cluster sampling: (a) random; (b) stratified;and (c) systematic.

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The stratified and systematic adaptive cluster samples bothbegin in the same manner as their conventional counterparts.Stratified adaptive cluster sampling begins with an initialstratified sample (as was shown previously in Fig. 1(b)), andadditional neighboring units are added if the additionalsampling criteria is met. A graphic representation of such asample is shown in Fig. 2(b). Similarly, systematic adaptivecluster sampling begins with a systematic sample (as inFig. 1(c)), and additional neighboring units are added if thecriteria are met as shown in Fig. 2(c).

Adaptive sampling is especially effective for rare or clus-tered populations. It helps to obtain a more precise estimateof population abundance or density than is normally possiblewith conventional designs. It also helps to obtain more infor-mation in the area of any interesting observations. If it ischeaper to observe units in clusters, it can be more cost effi-cient than conventional sampling. One of the advantages ofadaptive sampling over plain sequential sampling (in whichboundaries are established to dictate if further testing isnecessary) is that it not only tells you how many more unitsto sample but also where to sample the additional units.8

CONSIDERATIONS FOR APPLICATION OFSAMPLING TO STRUCTURES

The premise for this research is that engineers conductingstructural investigations can benefit from the aforementionedsampling techniques. Instead of simply choosing samplesbased on their expert knowledge, they could use these toolsto supplement their knowledge by more accurately choosingrepresentative samples and making estimates based on thesesamples. Sampling is most applicable to the evaluation ofstructures when the observation area is divided into gridsections (as may be the case for a building faade or bridgedeck) or ones in which the units are a separate physical entity(as in a beam-by-beam sampling of a bridge).

The choice of a particular sampling design for any givenstructure will depend on the specific physical attributes ofthe structure and the budget for testing. In some cases, asimple random sample may not be cost effective if expensivescaffolding has to be erected for every test point. If the engineeris well aware of similar problems in certain areas of the structure(for example, the southern faade has more deteriorationthan the other faade exposures), stratified sampling mayprovide the best alternative.

Although the total size of an adaptive sample can be moredifficult to estimate in advance, adaptive sampling can beespecially beneficial in cases where:

1. Lower costs and convenience can be achieved fromsampling units in close proximity to one another (as may bethe case when scaffolding has to be erected to collect themeasurements);

2. The extent of clustered flaws may be important forassessing structural integrity. (Larger flaws may indicatelocalized weak areas); and

3. Flaws are likely to be located in close proximity to oneanother due to similar environmental conditions, materialproperties, or same contractor.

The research described in this series made use of actualcase studies where complete information existed to see whatcould be learned about the usefulness of sampling methods.In addition, the research addressed challenges encounteredin evaluation of large structures. For example, there might beareas of a structure that are inaccessible to testing or a client

might prefer certain ways of expressing the results of thepopulation estimate.

CASE STUDYPost-tensioned concrete bridge

The bridge under consideration is a precast concrete,segmental bridge whose piers are precast, post-tensionedcantilever beams. These beams support precast girders,spanning between piers and supporting the roadway. Thearea of the bridge where the pier and girder met was the mainfocus of this study and the detailing of the pier/girder junc-tion is shown in Fig. 3. For brevity, all the information aboutthe structure and the repair are not repeated here, but the

interested reader is directed to References 2 and 3.The corbel region (where the load is transferred from the

girder to the cantilever beam) was the main concern for theengineers due to observed deterioration of the concrete andlack of redundancy in the bridge. To assure that the corbelregion could transfer the load from the girder to the beam,the integrity of the bonded post-tensioned system in thecantilevers had to be assured. To be certain of this integrity,the engineers needed to determine whether the tendon ductswere fully grouted and thus protected from intrusion of waterand possible later corrosion. Thus, one of the main objectivesof the site investigation was to determine whether air voidsexisted in the grouted ducts of the beams. The impact-echomethod was used to detect voids in the grouted tendon ducts.3

In the preliminary testing, seven beams (all of which werelocated over land) were selected for testing. Of the seven thatwere tested, two were found to have voids in at least oneduct. As a result, the engineers decided to test all the beamson the bridge to locate voids. There were a total of 170cantilever beams with each beam having three to five ducts.A layout of the beams is shown in Fig. 4. There were a total of644 ducts, of which 444 were accessible to the test equipment(the uppermost duct in each beam was not accessible).9

Sample parametersOne of the first steps in beginning to sample a structure is to

establish the parameters for the sample. Some of the mainparameters include the structural units to be sampled, thevalues to collect for each of the units, the methodology tochoose which units will be sampled, and the size of the sample.

In the bridge, one possible sampling unit is each individualduct. This would provide a binomial variable that wouldprovide yes/no (void/no void) information which wouldsimplify the analysis to sampling by proportions. The maindrawback of choosing the duct as the sampling unit is thatthis is not consistent with the manner in which testing wouldbe performed. It is not practical to set up the equipment togain access to a certain duct and then reposition the accessequipment under another duct without testing the remainingducts on the first beam. If the time and effort is taken to place

Fig. 3Cantilever beam of pier.

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402 ACI Structural Journal/May-June 2006

the engineer at a certain beam, it makes sense to test theremaining ducts on that beam while they are within easy reach.

Thus, a logical choice for the sampling unit is the beamwhich contains the post-tensioning ducts. Each cantileveredbeam, connecting the pier to the girder, contains three, four,or five post-tensioned ducts. Thus, once the engineer hasgained access to the beam, all accessible ducts on that beamcan be tested and the voided number recorded. The beam isalso a good sampling unit in terms of assessing the structuralstability of the bridge. It is more important when performingthe structural analysis to know if the voided ducts are local-ized by beam. For example, three voided ducts in a singlebeam is of more concern than three voided ducts located inthree different beams because the beam with the three voided

ducts is more likely to fail. If the sampling unit were a singleduct, it would also be necessary to tract a correlation coeffi-cient to determine the likelihood of the voided ducts beinglocated in the same beam. Recording flaws by beam elimi-nates the need for such a coefficient.

Once the sampling unit is chosen, it is necessary to choosethey-value that will be recorded for each unit. In the case ofthe cantilever beam as sampling unit, a possible choice is thetotal number of voided ducts on that beam. This would prob-ably be the best choice if the total number of ducts was thesame for every beam. In the case of this bridge, however, thetotal number of ducts varies from three to five. If the numberof voided ducts per beam was chosen, it would not indicatethe more serious case, for example, of three voided ducts in

a three-duct beam versus three voided ducts in a five-ductbeam. A better choice fory-value is the percentage of ductsthat are voided in each beam. If the voided percentage ischosen as they-value then it is easy to distinguish betweenthe more serious case of three voided ducts on a three-ductbeam (100% voided) versus three voided ducts on a five-duct beam (60% voided).

For each of the case studies in this work, a variety ofsampling methods were investigated and the various predictionsproduced by each method about the total population werecompared with determine their relative effectiveness. Thebasis for comparison will be the results of random samples.Two general types of sampling methods will be used. Aconventional (nonadaptive) sampling technique will bechosen and performed along with a simple random sample.Then the adaptive version of the same technique will be usedalong with an adaptive random sample. For each method, thesame basic procedure was followed, namely unit selection,mean and variance calculation, and confidence interval plot.

Of the basic sampling methods, stratified samplingseemed most appropriate for this case study. This method iswell suited to this population because the data points separateinto two strata easily, namely the beams over water and thebeams over land. The bridge under consideration spanned ariver, thus the beams at each end were over the shore whilethe beams at the center of the bridge were over water. This

distinction between the beams separated them into two strataof equal size.

In stratified sampling, it is desirable to have the y-valueswithin a single stratum as similar as possible. Although itwas not known before testing if the y-values would be anydifferent in the two strata, the strata distinguished them-selves upon first consideration based solely on the fact thatthe cost for testing over land and water was different.Because the beams over water could not be simply reachedfor testing from below, more time and money were necessary

to set up the equipment to access the beams. Stratifiedsampling would allow for the differing testing costs to beused to optimize distribution of testing locations for aspecific testing budget. Further consideration of the twostrata might also lead an engineer to hypothesize thatbecause the beams over water are more difficult to test, theirducts may have also been more difficult to fill with grout,leading to more voided ducts over the water.

After the matters of which units to sample, what values torecord, and how to select the units is settled, the finalpreliminary step is to select the number of units to test. Theformulas to calculate the sample size are not as straightforwardas may be hoped in that they do require the engineer to makesome assumptions about the population which has yet to be

sampled. These are only approximations, and it may be possiblefor the engineer to base the approximations on data fromprevious testing of similar structures. If no previous test data areavailable, the test number formula is not extremely sensitive tothe approximate values, and thus a rough estimate can still bemade based on experience and making educated guesses.

For the study of the different sampling procedures forthese papers, the number of samples was kept fairly constantso that a comparison could be made between the results withsimilar quantities of data input. The sample size was set bythe prediction from the formula for number of tests using asimple random sample and assuming that the estimator usedan unbiased, normally distributed estimator of the populationvalue. This formula states that the number of samples n is

given by

(1)

where ris the relative error, which equals (estimated value true value)/true value;z is the upper /2 point of the standardnormal distribution; is the estimate of the coefficient ofvariation, which equals standard deviation/mean; and N isthe number of total units in population.7

The engineer can state the relative error r he or she iswilling to incur and the approximate confidence interval forwhich he or she is aiming (determiningz). The total numberof units in the populationNshould be known, but the engineermust estimate the coefficient of variation of the population,which has yet to be sampled. The coefficient of variation isthe quantity that may be approximated using data fromsimilar structures. In the case of the bridge, let us say we areaiming for a relative error of 40% (r= 0.40) and for 90%confidence. Thez value corresponding to 90% confidence is1.645. If we estimate the standard deviation to be equal to themean, then = 1. The total number of beams underconsideration for the sample is 168 (N=168). Using thesevalues in the previous equation yields: n =15 beams, or 9%

n1

r2

z2 z

-------------1

N----+

------------------------=

Fig. 4Layout of bridge cantilever beams.

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of the population. (To get an idea of the effect of the relative

error on the number of samples, consider the following. Ifr= 35%, then n =19.5 20 beams or 12% of the population.)For the following studies, an attempt will be made to keep thenumber of beams used in the sample as close to 15 as possible.

Simple random sampleThe benefits of this type of sample are that it will often

produce good estimates of the mean and variance withoutrequiring any prior information about the sample. It is oftenused in modern sampling theory as the most basic ofsampling designs and the one upon which others are oftenbased. Thus, it will be used as a basis for comparison in thesestudies to determine the relative efficiencies of othersampling techniques.

One of the main drawbacks is the inconvenience ofsampling locations, which may increase the cost. In addition,some engineers may also be resistant to pulling out a randomnumber table (or use a random number generator in acomputer) to select the test locations.

Selecting units for a simple random sample is a straight-forward process. In this study, the beams will be selectedwithout replacement, so once a beam is selected it isremoved from the list of beams available for the remainingtests. Locations are randomly generated for each of the 15beams to be tested and each beam in the bridge has an equalprobability of being selected. A typical simple randomsample of the bridge is shown graphically in Fig. 5 in whichthe darkened beams represent the ones selected for sampling.

Once the beams are selected, they are then tested and they-value for each is recorded. The next step is to select whichstatistic about the entire population one wants to predictfrom the sampled data. For this case, the statistic will be themean of the voided duct percentage. In other structures, thepopulation total may be more important than the populationmean, especially in a structure with many redundancieswhere it can be assumed that the integrity will not becompromised until a crucial number of voids is found in theentire structure.

For a simple random sample, the estimated mean of thepopulation is calculated as

(2)

where n = number of units sampled, andyi =y-value for thei-th unit.

Throughout this document, the term actual (actual mean)will be used as a basis of comparison for the estimated values(estimated mean). Use of this term is not meant to indicatethat the true mean of the population has been calculated(using an infinite number of test points). Actual in the

context here means that the value has been calculated using all

of the sample points from the testing conducted on the entirebridge by the engineering consultants. These tests wereperformed on all of the beams in the population but only at afinite number of locations along each beam, resulting in avalue that is still an approximation to a certain extent.

In addition to calculating the population mean, it is oftendesirable to estimate the variance of the mean prediction. Ifa variance estimate is known, a confidence interval can bemade based on the uncertainty of the calculated mean value.An estimate of the variance of the mean s2 is given by10

(3)

The values for the mean and variance estimates were collectedfor a number of samples. These results are shown in the nextsection with their accompanying confidence intervals.

Because the variance quantity is not necessarily easy tointerpret, one of the more intuitive ways for an engineer toexamine the data collected and present the predictions to aclient is the confidence interval. In this manner, the engineerwho calculated a 90% confidence interval may say There isa 90% chance that the actual mean of the population fallsbetween Value A and Value B. Often this presentation iseasier for the client to understand than references to variances.

The upper and lower bounds on a confidence interval may becalculated as follows

(4)

where = estimated value,z = upper /2 point of the standardnormal distribution (for approximately a (1 )% confidence)and = estimated variance of the value.7

As should be expected, to have a greater confidence in theprediction a wider interval is needed. And conversely, anarrower range can be specified if the desired confidence inthe results is not that high. For an illustration of this point,the confidence intervals for a simple random sample of thebridge data is shown in Fig. 6 for confidence percentagesbetween 80 and 99.9%. The dashed horizontal line showswhere the actual mean of the sample falls at 20.6% voidedducts per beam. The circles in the plot above each confidencepercentage value indicate the estimated population mean,based on the specific sample. The solid vertical lines, endingwith the short horizontal bars and intersecting each meanestimate, show the extent of the confidence interval for thatsample. Again, this is a single sample of the data so the meanand variance for each line shown is the same, with the differencein interval length due solely to the value ofz in Eq. (4). Arange between 14 and 34% can be specified for an 80%

y

y

yii 1=

n

n-------------=

s2

yi y( )2

i 1=

n

n 1----------------------------=

z va r ( )

va r ( )

Fig. 5Typical beam selection for random sample.

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confidence interval while a range as wide as 0 and 54% isneeded to predict the mean with 99.9% accuracy.

In the case of the bridge data, ten samples were taken to geta general idea of a typical set of results for each specificsampling method. Almost all samples will generate adifferent estimated mean, variance, and confidence interval,so a group of results is presented in order that the overalltrend be seen. Figure 7 shows the 90% confidence intervalsfor ten different simple random samples. All ten of the confi-dence intervals generated contain the actual mean withintheir boundaries, and the highest 90% intervals prediction is37% for Sample 1 while the lowest boundary is 5% forSample 4. When examining a structure such as the post-tensioned bridge, the engineer is probably most interested inthe higher end of the interval to get the most conservativeestimate of damage.

Post-stratified random sampleA slight variation of the random sample is the post-stratified,

random sample. In this sampling technique, the units areselected randomly, in the same manner as the simple randomsample. After selection, however, the units are studied to seeif they can be separated into strata (groups with similar proper-ties or values). If strata do exist, the units are then groupedaccordingly before the mean and variance are calculated.This method has the advantage over random sampling in thatcalculation of the mean and variance estimates can beimproved if the data within each stratum are very similar toeach other. It also has the advantage over stratified samplingin that the actual values of the units can be observed beforestrata limits must be defined. It has disadvantages in that the

random selection of the units before strata definition intro-duces an additional error term in the variance calculationwhen compared with traditional stratified sampling. Thisincrease represents the uncertainty introduced by randomsample sizes in each strata.

In post-stratified sampling, after the random sample iscollected from the entire population, the strata limits are

defined and the y-values are separated into their respectivegroups. Using data from these groups, the mean and variancefor the post-stratified, random sample can be estimated usingformulas similar to Eq. (2) and (4) for random sampling, butwith additional terms that include the estimated mean andvariance of each stratum and number of units in each stratum.

The post-stratified samples use the same ten sets of initialtest points as those of the random sample in the previoussection. However, before the mean and variance is calculatedfor the post-stratified sample, the data are separated into twostratanamely, the beams over land and those over water. Atypical result for the averages of voided ducts is that ofSample 1 (whose confidence interval is shown in Fig. 8) inwhich the percentage of voids over water (37%) is much

greater than those over land (10%). This is true for the totalpopulation, which is 28% voided over water and 14% voidedover land, although the difference in the full population is notas great as that sample.

One special consideration that must be made for confidenceintervals of stratified samples is the correction for thedegrees of freedom d. In simple random sampling, thedegrees of freedom is equal to the number of units sampled;and this number is used to determine the value z from thestudents tdistribution of an approximate standard normaldistribution. (Az-value is selected from the table, assumingn-1 (or d-1) degrees of freedom.) In stratified sampling, thedegrees of freedom does not necessarily equal the testnumber because the implied normal distribution may not beentirely accurate and thus a correction to the degrees offreedom will require a largerz-value to be chosen.11

A graph of the confidence intervals for the post-stratifiedsamples, with the same format as the one for simple randomsampling in Fig. 7, is shown in Fig. 8. Comparison to therandom sampling one reveals that the average differencebetween the actual mean and the estimated mean increasesfrom 3.5% for the random sample to 3.7% for the post-stratified sample. Some of the confidence intervals getslightly wider with an average width of 21% for the randomsample and 23% for the post-stratified sample. The onlysample whose interval got much wider was Sample 2. For

Fig. 6Confidence intervals for single random sample withdifferent confidence percentages.

Fig. 7Confidence intervals for random samples.

Fig. 8Confidence intervals for post-stratified, randomsamples.

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this group of sample points, the random sample includedeleven samples over land and only four over the water. (Mostof the other samples were closer to an equal allocation.) Thesmall number of samples over water generated a rather largevariance estimate for that stratum, resulting in a large vari-ance estimate for the entire sample and reducing the degreesof freedom to only five. This combination of factors led to awide confidence interval for that sample.

Stratified sampleAnother conventional sampling method chosen for the

study of this bridge was the stratified sample. The maindifference between this and the post-stratified sample is thatthe strata are chosen before any sampling begins and are thusincorporated into the selection process. This affects thecalculation of the mean and variance for the samples.

For the stratified samples in this section, the first step inthe selection of the sampling units is the calculation of thenumber of units from each stratum that are to be included inthe sample. Proportional allocation, which assigns thenumber of samples per strata to be proportional to thenumber of total units within each strata, was chosen.7 Forexample, if 10 units are to be drawn from strata of sizes 30and 70 units, three will be drawn from the smaller strata andseven from the larger.

In the case of the bridge, there are 84 units in each of theabove-land strata and the above-water strata (N1 =N2 = 84)with a total of 168 beams (N= 168). Allocating the 15 samples(n = 15) between these two strata and using the formulaabove yields 7.5 samples in each stratum. Rounding to wholenumbers gives eight samples from one stratum (for example,the above-water strata) and leaves seven (15 8 = 7) samplesfor the other strata (above-land strata). A typical sample forsuch a stratified sample is shown in Fig. 9.

Other options exist for assigning sample sizes to eachstratum. One such option is optimal allocation which isbased on anticipated standard deviations for each stratum.Another option for assigning a sample number to eachstratum involves the cost of the testing. This methodattempts to minimize the total variance of the populationsample while taking into consideration the cost to test a unitin each stratum, the cost to begin and continue testing, andthe total budgeted cost for the testing. Both methods requirethe engineer to anticipate the standard deviation for eachstratum.7 In the current state of practice, there is little or noinformation available for the standard deviations of damagein structures. Possibly with future research, values such asthese can be collected so that the testing engineer will havemore information on which to base approximations. Untilthat time, the engineer can make an educated guess as to thestandard deviation or may choose stratified sampling using

proportional allocation in which no estimate of these valuesis required.

Once the number of units to be sampled in each stratum isdecided, the units are chosen by random sampling from theirstrata. This process is termed stratified random sampling.After the units are sampled and their y-values recorded, thedata is then analyzed to make predictions about the entirepopulation based on the observations in each stratum. The

estimated population mean and variance can be calculatedfrom a stratified random sample according to formulassimilar to the post-stratified sample.6,7 Once the mean andvariance of the sample have been calculated, it is possible toconstruct the confidence intervals. Just as with the post-stratified sample, a correction of the degrees of freedommust be made. Figure 10 shows the confidence intervals forthis sampling design. All of the 90% confidence intervalshave similar widths and all contain the actual mean. Whencompared with the simple random sample, the intervals areapproximately the same width with similar boundary values.

Adaptive random sampleAdaptive sampling approaches were also applied to the

structure. First, an adaptive random sample was performed,and then it was followed by an adaptive stratified sample. Itwas not clear before beginning the testing of the bridge if itsdata would classify it as a rare, clustered population, butsamples were taken using the technique to determine if itwould be a useful option. The advantage of this type ofsampling is that information gathered while conducting thesample is used to make more informed decisions aboutwhere to continue sampling. Thus, adaptive sampling maybe appealing, because the engineer can add to the sample ina certain area if a high value is detected. The engineer maybe curious to see if neighboring beams, which have the same

Fig. 9Typical beam selection for stratified sample.

Fig. 10Confidence intervals for stratified samples.

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exposure, material properties, or contractor, also sharesimilar high values.

One of the limitations of adaptive sampling is that the finalsample size is not known prior to the survey; therefore, itmay be more difficult for a testing engineer to draw up abudget before testing. The expected sample size of anadaptive random sample can be calculated if the probabilityi that unit i is included in the sample is known. In that case

(5)

where ,

N = number of total units in the population, n1 = initialsample size, mi = number of units in the network containingunit i, and ai = number of units in networks that which unit iis an edge unit.8

However, when approaching a population has yet to besampled and which has no similar sampled population from

which to draw data, the precise values of mi and ai areunknown. Therefore, in this study, we will select samplingsizes similar to those in the nonadaptive study to try andobtain more effective comparisons for a similar amount ofmoney and time spent on the testing. To try and get a finalsample size near the 15 test points of the previous samples, anumber lower than 15 test points, namely six, was chosen asthe initial sample size. Those six initial test points were chosenfrom the entire sample at random, in the same manner in whichthe simple random sample was chosen from the population.

Once a unit was sampled, its y-value (void percentage)was examined to determine if units should be added to thesample in the neighborhood of this test point. If they-valuemet or exceeded the cutoff value C, then the units in the

neighborhood of the original test point were also tested. Theneighborhood of a test point in this case will be defined asthe beams that are directly adjacent to a particular beam, that

E ( ) ii 1=

N

=

i 1N mi aij

n1 N

n1 =

is, the two beams on each side of the original and the two

beams at each end of the original, as shown in Fig. 11.

If any of these newly sampled beams have ay-value which

meets or exceeds C, they are added to the sample data and the

beams in their neighborhood are also tested. If any of the

additional beams have ay-value below C, then these units are

considered edge units and their values are not included in

the calculations described in the next section. The only units

with values less than Cthat are used in the calculations are

those that are already included in the initial sample of the

population. References to a specific sample size will refer tothe size of a sample that includes all the units whose values

are used to calculate the mean and variance. Thus, there will

be additional edge units that may be tested (and the cost of

testing such units will be incurred), but their values are basically

discarded from the sample. Thus, to have a sample size of 15

to compare with previous sampling techniques, more than

15 samples will probably be taken but only about 15 of these

values will be used in the calculations. A typical adaptive

random sample of the bridge is shown in Fig. 12. The six initial

samples are shown in black and the beams that were adaptively

added are shaded gray or patterned. The patterned units are

edge units not included in the original sample and are thus

not used in the calculations. (Here, the sample size will beconsidered to be 18 for our purposes.)

After collecting all the information from the test points, the

next step is to calculate the mean and variance. To calculate

these values, a few terms associated with adaptive sampling

must be defined. One of the main concepts that must be

understood for the calculations associated with adaptive

sampling is the term network. A network greater than size

one is a group of test points that are located adjacent to one

another and whosey-values are greater than or equal to C. A

graphic representation of a network is shown in Fig. 13 and is

taken from the right-center portion of Fig. 12. If one of the units

in the network is tested and its y-value observed, then by

adaptive sampling rules, all of the units in the network willbe tested. The only network that would contain ay-value less

than Cis a network of size one that is formed by a test point

in the initial sample that does not meet C. If this unit were

sampled in an adaptive addition, it would have been

discarded as an edge unit; but if it is in the initial sample, it

remains in the calculations.

To calculate the mean estimate , based on the number of

initial intersections, the following formula is used

(6)

1n1----- wi

i 1=

n1

=Fig. 11Neighborhood definition.

Fig. 12Typical beam selection for adaptive random samples.

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where n1 = the number of units in the initial sample; wi =

; and mi = number of units in networkAi.

Its variance can be estimated as

(7)

whereN= number of total units in the population.12

Some initial studies were conducted to determine a suitablecutoff value Cfor adaptive sampling of the bridge. In termsof structural considerations, one would want to set C lowenough that beams with void percentages that are structurallycompromising are included in the sample, but not so low thatbeams with any damage warrant further testing (unless, ofcourse, there are funds for this rigorous testing or anydamage is considered crucial). In addition, ifCis set too lowso that most beams qualify for further testing, the populationwould not be considered to consist of rare, clustered groupsfor which adaptive sampling is best suited.

The general effect of a change in the cutoff value in termsof the width of the confidence interval can be seen in Fig. 14(a).As the C value is increased, the width of the confidence

interval increases. The main reason for this trend is the factthat a higher Cvalue will include fewer points in the sample,increasing the variance of the sample. The effect of thechange in cutoff value on the number of units tested can beseen by examining Fig. 14(b). For the remainder of adaptivesamples of the bridge, a cutoff value of 40% will be used.This yields about 15 sample units for a typical sample,locates beams at a higher structural risk, and avoids a verywide confidence interval.

The confidence intervals for the adaptive random sample,shown in Fig. 15, are much wider than those of any of thetraditional sampling methods presented previously. Theconfidence intervals yield estimates as high as 53% wherethe highest estimate in the traditional random sample was37%. One of the main reasons for the increased intervalwidth is the fact that the degree of freedom used for theconfidence interval calculations was the number of beams inthe initial sample. Thez value from the distribution table fora 90% confidence interval with 6 samples is 1.94 while thezvalue for 15 samples is 1.75.13 Thus when calculating theinterval bounds using Eq. (4), the term is multipliedby a factor which is greater by 0.19 (1.94 1.75 = 0.19),including an additional 19% of the term.

Because the adaptive random sample requires more testing(edge units must be tested although their values arent used)and is more computationally intensive but still yields wider

1

mi----- yj

j Ai

var [ ]N n1

Nn1 n1 1( )----------------------------- wi ( )

2

i 1=

n1

=

var ( )

var ( )

confidence intervals than its conventional counterpart, itdoes not seem to be a good technique for analyzing thebridge data. The adaptive techniques are best suited forpopulations in which units with y-values exceeding thecutoff limit are rare and spaced in clusters. This does notseem to be true for the void percentages of this bridge.

Adaptive stratified sampleOne final check of the adaptive technique was done for the

adaptive stratified sample to determine if it appears that mostadaptive techniques will yield wide intervals for the data.The selection of the sampling units for the adaptive stratifiedsample is similar to the selection for the adaptive randomsample in that an initial sample is taken and units are addedadaptively in the neighborhood of observedy-values equal to

Fig. 15Confidence intervals for adaptive random samples.

Fig. 13Graphic network representation.

Fig. 14Results for adaptive random sample of bridge datawith different cut-off (C) values: (a) confidence intervals;and (b) number of units sampled.

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or above the cutoff value. The only difference is that theinitial sample in this case is a stratified sample. The numberof units from each stratum (three) is determined by propor-tional allocation and units are added adaptively regardless ofstratum boundaries.

Calculation of the mean and variance estimators for theadaptive stratified sample follow formulas similar to that forthe adaptive random sample but with additional terms

including the number of total units in each stratum, thenumber of units initially sampled in each stratum, and thenumber of units in each network within each stratum. Theformulas are not as straightforward as their nonadaptivecounterparts. Although the calculations are mainly sums andratios, there is still a fair amount of bookkeeping to be doneto be sure that the correct values are assigned during themany calculations necessary.6,14

The confidence intervals shown in Fig. 16 for the adaptivestratified sample were the widest of any sampling techniquestudied. The intervals are almost meaningless in that manyspan from 0% to more than 70% (while the estimated meanwas only approximately 20%). These wide intervals are dueto the high variances estimated by the equations and by the

fact that there are only a few points in the initial sample,providing only a few degrees of freedom and dictating alargez value.

CONCLUSIONSThe results indicate that there are many possible techniques

that may be used to predict the mean void percentage of thebeams in a post-tensioned bridge from a sample consisting ofapproximately 9% of the total population. The key conclusionsfrom the various techniques studied include:

1. The simple random sample yields good results with someof the narrowest confidence intervals of any of the samples;

2. Post-stratification is a viable option for studying the dataif the strata are only recognized after the sample is taken;

3. The stratified random sample produces results similar tothe simple random sample for this case but does have theadvantage of allowing cost considerations to be introduced intothe sampling process. This is especially useful if there is a largedisparity between the testing costs for different strata; and

4. The adaptive techniques are the least effective of thesampling techniques studied herein. They have the disadvantagethat the final sample size is not known before testing, moneymust be spent on some tests which will be disregarded (edge

units), and the resulting confidence intervals are wide whencompared with the traditional techniques. The computationsare also more difficult, requiring more time and introducingmore chances for error.

The above conclusions apply specifically to this particularcase study of a post-tensioned bridge. While it is early to drawgeneral conclusions, it does seem that simple random samplingworks well if little prior information is available about thestructure and if the engineer is not able to make any predictionsabout the possible results before testing begins. An additional

study of this type is shown in the companion paper.

NOTATIONai = number of units in network of which unit i is edge unitC = cutoff valued = degrees of freedommi = number of units in network containing unit i (or networkAi)

N = number of total units in populationn = number of samplesn1 = initial sample sizer = relative errors2 = estimate of variance of meanwi = average of values of network that includes i-th unit

y = estimated mean of population for simple random sampleyi = y-value for i-th unitz = upper /2 point of standard normal distribution = allowable probability of error = estimate of coefficient of variation

= mean estimate of adaptive sample = expected sample size of adaptive random samplei = probability that unit i is included in sample

= estimated value for confidence interval= estimated variance of value for confidence interval

REFERENCES1. Malhotra, V. M., and Carino, N. J., Handbook on Nondestruct ive

Testing of Concrete, 2nd Edition, CRC Press, 2004, 384 pp.2. Jaeger, B.; Sansalone, M.; and Poston, R., Detecting Voids in the

Grouted Tendon Ducts of Post-Tensioned Concrete Structures Using theImpact-Echo Method, ACI Structural Journal, V. 93, No. 4, July-Aug.1996, pp. 462-473.

3. Jaeger, B. J.; Sansalone, M.; and Poston, R.W., Using Impact-Echo toAssess Tendon Ducts, Concrete International, V. 19, No. 2, Feb. 1997,

pp. 42-46.4. Kesner, K.; Poston, R.; Salmassian, K.; and Fulton, G. R., Repair of

Marina del Rey Seawall, Concrete International, V. 21, No. 12, Dec.1999, pp. 43-50.

5. Williams, T.; Sansalone, M.; Grigoriu, M.; and Poston, R. W., Reliability-Based Nondestructive Testing and Repair of Concrete Seawall, ACIStructural Journal, V. 97, No. 1, Jan.-Feb. 2000, pp. 166-174.

6. Williams, T., Use of Sampling Techniques and Reliability Methodsto Assist in Evaluation and Repair of Large Scale Structures, PhD thesis,Cornell University, Ithaca, N.Y., 1999, 114 pp.

7. Thompson, S. K., Sampling, John Wiley & Sons, Inc., New York,1992, 360 pp.

8. Thompson, S. K., and Seber, G. A. F.,Adaptive Sampling, John Wiley& Sons, Inc., New York, 1996, 288 pp.

9. KCI Technologies, Inc., Report On Evaluation of Condition of Post-Tensioned Cantilever BeamsWashington Bridge No. 700, KCI ProjectNo. 06-94022, June 1, 1994, pp. 19-25.

10. Cochran, W. G., Sampling Techniques, 3rd Edition, John Wiley &Sons, Inc., New York, 1977, 448 pp.

11. Satterthwaite, F. E., An Approximate Distribution of Estimates ofVariance Components,Biometrics Bulletin, V. 2, 1946, pp. 110-114.

12. Thompson, S. K., Adaptive Cluster Sampling, Journal of theAmerican Statistical Association, V. 85, No. 412, Dec. 1990, pp. 1050-1059.

13. Devore, J. L., Probability and Statistics for Engineering and theSciences, 3rd Edition, Brooks/Cole Publishing Co., Pacific Grove, Calif.,1991, 716 pp.

14. Thompson, S. K., Stratified Adaptive Cluster Sampling,Biometrika, V. 78, No. 2, 1991, pp. 389-397.

va r

( )

Fig. 16Confidence intervals for adaptive stratified samples.

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