Sampling Techniques for Evaluating Large Concrete Structures: Part I

Williams, Tamara Jadik

Large structures present numerous possible test locations for a nondestructive evaluation. Challenges lie in selecting test locations, managing data collected, and stating testing results. This research evaluated the feasibility of using sampling methods to 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 then compare their predictions based on sampling to actual results from comprehensive testing. These studies demonstrated that sampling methods are useful at determining the number of samples and their locations. The results can effectively be stated as a confidence interval, presenting a range for the prediction based on an acceptable uncertainty. In Part I, a brief description of some sampling methods is given and the procedure (including simple random, stratified, and adaptive sampling) is applied to a post-tensioned bridge, which was nondestructively tested to locate air voids within grouted tendon ducts.

Keywords: post-tensioned; sampling; test.

(ProQuest-CSA LLC: … denotes formulae omitted.)

INTRODUCTION

In recent years, the development of a wide variety of nondestructive testing methods for concrete structures has provided engineers with numerous possibilities for evaluating structures.1 While greatly expanding an engineer’s capabilities, this availability of testing techniques has also introduced its own set of challenges, particularly when evaluating a large structure. The engineer faces the challenge of dealing with hundreds to many thousands of possible test locations. Time and cost constraints work to limit the test number to a minimum while the desire to accurately assess the state of the structure argues for the maximum number of tests.

This paper and its companion, “Sampling Techniques for Evaluating Large Concrete Structures, Part II,” which will appear in the July-August issue of the ACI Structural Journal, present research aimed at examining the use of sampling techniques to assist the engineer in making choices concerning the number and location of tests and in stating the extent of knowledge gained from the testing. Two case studies are presented. The nondestructive test data for the structures highlighted in the case studies were initially collected for all possible test locations in structural investigations; therefore, the authors had the unique opportunity to compare sampling predictions to the actual state of the structures to evaluate the accuracy of various sampling approaches.

The structures examined include a post-tensioned bridge,2,3 on which nondestructive testing was performed to locate air voids within grouted tendon ducts, and a 7.5 m^sub i^ (12 km) long, reinforced concrete seawall,4 where the aim was to locate delaminations caused by corrosion of the reinforcing 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 from these tests was used to estimate the level of damage in the entire bridge within a given confidence; these results were then compared with actual damage statistics. In the second case, sampling methods, including simple random, systematic, and adaptive sampling, were employed to make predictions about the state of the walls based on tests on only a fraction of the wall panels. Again, the results were compared to the actual results from testing the entire structure. In addition, the seawall data was also used to construct probabilistic models to examine patterns in the damage. Subsequently, repair options were incorporated into these models to determine their reliability. The results of these studies were stated in terms of the cost of repair versus the predicted cost of failure. This work is summarized in Reference 5.

This paper provides background information on sampling methods and focuses on the case study of the post-tensioned bridge. In the companion paper, the case study of the reinforced concrete seawall is presented and key conclusions are given based on the results of both case studies. For a more in depth discussion of sampling concepts and their application in the case studies, the reader is referred to Reference 6.

RESEARCH SIGNIFICANCE

This research has led to the development of a method for determining the number and locations of tests in nondestructive assessment of large concrete structures. The method shows how the information obtained from these tests can then be used to make a prediction about the state of the entire structure using confidence intervals. This is the first time that sampling techniques were used to establish the damage state in concrete structures. The results of the studies presented indicate that sampling techniques are very useful in making the collection and analysis of data from nondestructive tests more efficient and cost effective.

BACKGROUND ON SAMPLING

Sampling methods, which allow statements to be made about an entire group based on data collected for only a certain portion, were applied to the nondestructive testing of structures for flaw detection. The inspection schemes that are the focus of the research are those in which the data taken at 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 concrete structure to determine whether or not the reinforcing steel has corroded and caused delaminations in the concrete or beneath the surface or to determine whether there are voids in grouted tendon ducts. It is assumed that the engineer performing the tests has only limited knowledge of the current state of the specific structure and its state in the recent past. Also the main aspect of the evaluation consists of nondestructive tests that provide localized results on a point-by-point basis. In these cases, the possible number of test points may be many. Sampling theory is presented as a possible assistant to choosing the number and location of tests to obtain the maximum information about the entire structure.

Some basic concepts are presented in this section to assist in the understanding of the case studies in this paper and its companion. A population is the entire set of a known, finite number N of sampling units. A sampling unit is the particular section of the population for which the data is collected. A unit may be a single person or institution or it may be a geographic unit such as a plot of land. The data or value taken for each unit is referred to as the y-values of the unit.

The general description of how the sample is taken and analyzed is referred to as the sampling strategy which consists of the sampling design together with the inference methods. The sampling design is the procedure for selecting the sampling units. The design must address such concerns as size, selection, and observation method. The sampling designs can be placed into one of three general categories, namely conventional, adaptive, and nonstandard. An inference method helps to draw conclusions about the entire population based on the data from the samples observed. These inferences may take the form of estimates such as the population mean, tests of hypothesis, and confidence intervals (which state the accuracy or confidence of the estimates).7

Of the sampling designs, the conventional design has units selected prior to the data collection. Some conventional sampling methods include simple random sampling, stratified sampling, and systematic sampling. Simple random sampling is a design in which n units are selected from the population in a random order. In this design, each possible combination of n units is equally likely to be selected, and at each step every unit has an equal probability of selection. A graphic representation of a random sample of 10 units from a population total of 100 is shown in Fig. 1(a). Stratified sampling is a design in which the population is partitioned into regions (strata) and a sample is selected by some design within each stratum. This sampling design is of the most benefit when the units within a stratum are as similar as possible. One of the types of stratified sampling is stratified random sampling in which the units in each stratum are selected by simple random sampling. When the number of units sampled in each stratum is proportional to the size of the stratum, the sampling is said to be done with proportional allocation. A graphic illustration of this is shown in Fig. 1(b) where five units are sampled from the stratum of size 50, three from the size 30 stratum, and two from the size 20 stratum. Systematic sampling consists of selecting a starting point (such as by simple random sampling) and then selecting all the units spaced in a systematic fashion throughout the population. 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 could be selected before any observation began. In an adaptive sampling design, the procedure for selecting the units is based on the values that are observed during the sampling process and includes gathering more information in the neighboring area of an observed high value. These units may be selected in different ways, with the major differences between the designs existing in the initial sample selection.

In adaptive random sampling, an initial set of units is selected by simple random sampling (as was done in Fig. 1(a)). But as the values are observed, the sample can be adaptively increased to include units in the neighborhood of observed units fitting a certain criteria. For example, if the criterion for further sampling is an observed nonzero value, then whenever a unit with a nonzero value is found, the neighboring units are observed. A graphic representation of such a sample is shown in Fig. 2(a). After the initial simple random sample is taken (shown in dark gray), units are added directly above, below, and to each side of the units with nonzero values. If any of the added units have nonzero values (shown with a black dot), then the neighboring units to those are also included. This process continues until all adjacent units with nonzero values are added to the sample. All of the units that would be sampled in addition to the initial random sample are shown in the light gray checkered blocks in Fig. 2(a).

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

Adaptive sampling is especially effective for rare or clustered populations. It helps to obtain a more precise estimate of population abundance or density than is normally possible with conventional designs. It also helps to obtain more information in the area of any interesting observations. If it is cheaper to observe units in clusters, it can be more cost efficient than conventional sampling. One of the advantages of adaptive sampling over plain sequential sampling (in which boundaries are established to dictate if further testing is necessary) is that it not only tells you how many more units to sample but also where to sample the additional units.8

CONSIDERATIONS FOR APPLICATION OF SAMPLING TO STRUCTURES

The premise for this research is that engineers conducting structural investigations can benefit from the aforementioned sampling techniques. Instead of simply choosing samples based on their expert knowledge, they could use these tools to supplement their knowledge by more accurately choosing representative samples and making estimates based on these samples. Sampling is most applicable to the evaluation of structures when the observation area is divided into grid sections (as may be the case for a building façade or bridge deck) 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 given structure will depend on the specific physical attributes of the structure and the budget for testing. In some cases, a simple random sample may not be cost effective if expensive scaffolding has to be erected for every test point. If the engineer is well aware of similar problems in certain areas of the structure (for example, the southern façade has more deterioration than the other façade exposures), stratified sampling may provide the best alternative.

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

1. Lower costs and convenience can be achieved from sampling units in close proximity to one another (as may be the case when scaffolding has to be erected to collect the measurements);

2. The extent of clustered flaws may be important for assessing structural integrity. (Larger flaws may indicate localized weak areas); and

3. Flaws are likely to be located in close proximity to one another due to similar environmental conditions, material properties, or same contractor.

The research described in this series made use of actual case studies where complete information existed to see what could be learned about the usefulness of sampling methods. In addition, the research addressed challenges encountered in evaluation of large structures. For example, there might be areas of a structure that are inaccessible to testing or a client might prefer certain ways of expressing the results of the population estimate.

CASE STUDY

Post-tensioned concrete bridge

The bridge under consideration is a precast concrete, segmental bridge whose piers are precast, post-tensioned cantilever beams. These beams support precast girders, spanning between piers and supporting the roadway. The area of the bridge where the pier and girder met was the main focus of this study and the detailing of the pier/girder junction is shown in Fig. 3. For brevity, all the information about the 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 the engineers due to observed deterioration of the concrete and lack of redundancy in the bridge. To assure that the corbel region could transfer the load from the girder to the beam, the integrity of the bonded post-tensioned system in the cantilevers had to be assured. To be certain of this integrity, the engineers needed to determine whether the tendon ducts were fully grouted and thus protected from intrusion of water and possible later corrosion. Thus, one of the main objectives of the site investigation was to determine whether air voids existed in the grouted ducts of the beams. The impact-echo method was used to detect voids in the grouted tendon ducts.3

In the preliminary testing, seven beams (all of which were located over land) were selected for testing. Of the seven that were tested, two were found to have voids in at least one duct. As a result, the engineers decided to test all the beams on the bridge to locate voids. There were a total of 170 cantilever beams with each beam having three to five ducts. A layout of the beams is shown in Fig. 4. There were a total of 644 ducts, of which 444 were accessible to the test equipment (the uppermost duct in each beam was not accessible).9

Sample parameters

One of the first steps in beginning to sample a structure is to establish the parameters for the sample. Some of the main parameters include the structural units to be sampled, the values to collect for each of the units, the methodology to choose which units will be sampled, and the size of the sample.

In the bridge, one possible sampling unit is each individual duct. This would provide a binomial variable that would provide yes/no (void/no void) information which would simplify the analysis to sampling by proportions. The main drawback of choosing the duct as the sampling unit is that this is not consistent with the manner in which testing would be performed. It is not practical to set up the equipment to gain access to a certain duct and then reposition the access equipment under another duct without testing the remaining ducts on the first beam. If the time and effort is taken to place the engineer at a certain beam, it makes sense to test the remaining ducts on that beam while they are within easy reach.

Thus, a logical choice for the sampling unit is the beam which contains the post-tensioning ducts. Each cantilevered beam, connecting the pier to the girder, contains three, four, or five post-tensioned ducts. Thus, once the engineer has gained access to the beam, all accessible ducts on that beam can be tested and the voided number recorded. The beam is also a good sampling unit in terms of assessing the structural stability of the bridge. It is more important when performing the structural analysis to know if the voided ducts are localized by beam. For example, three voided ducts in a single beam is of more concern than three voided ducts located in three different beams because the beam with the three voided ducts is more likely to fail. If the sampling unit were a single duct, it would also be necessary to tract a correlation coefficient to determine the likelihood of the voided ducts being located in the same beam. Recording flaws by beam eliminates the need for such a coefficient.

Once the sampling unit is chosen, it is necessary to choose the y-value that will be recorded for each unit. In the case of the cantilever beam as sampling unit, a possible choice is the total number of voided ducts on that beam. This would probably be the best choice if the total number of ducts was the same for every beam. In the case of this bridge, however, the total number of ducts varies from three to five. If the number of voided ducts per beam was chosen, it would not indicate the more serious case, for example, of three voided ducts in a three-duct beam versus three voided ducts in a five-duct beam. A better choice for y-value is the percentage of ducts that are voided in each beam. If the voided percentage is chosen as the y-value then it is easy to distinguish between the more serious case of three voided ducts on a three-duct beam (100% voided) versus three voided ducts on a fiveduct beam (60% voided).

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

Of the basic sampling methods, stratified sampling seemed most appropriate for this case study. This method is well suited to this population because the data points separate into two strata easily, namely the beams over water and the beams over land. The bridge under consideration spanned a river, thus the beams at each end were over the shore while the beams at the center of the bridge were over water. This distinction between the beams separated them into two strata of equal size.

In stratified sampling, it is desirable to have the y-values within a single stratum as similar as possible. Although it was not known before testing if the y-values would be any different in the two strata, the strata distinguished themselves upon first consideration based solely on the fact that the cost for testing over land and water was different. Because the beams over water could not be simply reached for testing from below, more time and money were necessary to set up the equipment to access the beams. Stratified sampling would allow for the differing testing costs to be used to optimize distribution of testing locations for a specific testing budget. Further consideration of the two strata might also lead an engineer to hypothesize that because the beams over water are more difficult to test, their ducts 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 to record, and how to select the units is settled, the final preliminary step is to select the number of units to test. The formulas to calculate the sample size are not as straightforward as may be hoped in that they do require the engineer to make some assumptions about the population which has yet to be sampled. These are only approximations, and it may be possible for the engineer to base the approximations on data from previous testing of similar structures. If no previous test data are available, the test number formula is not extremely sensitive to the approximate values, and thus a rough estimate can still be made based on experience and making educated guesses.

For the study of the different sampling procedures for these papers, the number of samples was kept fairly constant so that a comparison could be made between the results with similar quantities of data input. The sample size was set by the prediction from the formula for number of tests using a simple random sample and assuming that the estimator used an unbiased, normally distributed estimator of the population value. This formula states that the number of samples n is given by

… (1)

where r is the relative error, which equals (estimated value – true value)/true value; z is the upper α/2 point of the standard normal distribution; γ is the estimate of the coefficient of variation, which equals standard deviation/mean; and N is the number of total units in population.7

The engineer can state the relative error r he or she is willing to incur and the approximate confidence interval for which he or she is aiming (determining z). The total number of units in the population N should be known, but the engineer must estimate the coefficient of variation of the population, which has yet to be sampled. The coefficient of variation is the quantity that may be approximated using data from similar structures. In the case of the bridge, let us say we are aiming for a relative error of 40% (r = 0.40) and for 90% confidence. The z value corresponding to 90% confidence is 1.645. If we estimate the standard deviation to be equal to the mean, then γ = 1. The total number of beams under consideration for the sample is 168 (N =168). Using these values in the previous equation yields: n =15 beams, or 9% of the population. (To get an idea of the effect of the relative error on the number of samples, consider the following. If r = 35%, then n =19.5 [asymptotically =] 20 beams or 12% of the population.) For the following studies, an attempt will be made to keep the number of beams used in the sample as close to 15 as possible.

Simple random sample

The benefits of this type of sample are that it will often produce good estimates of the mean and variance without requiring any prior information about the sample. It is often used in modern sampling theory as the most basic of sampling designs and the one upon which others are often based. Thus, it will be used as a basis for comparison in these studies to determine the relative efficiencies of other sampling techniques.

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

Selecting units for a simple random sample is a straightforward process. In this study, the beams will be selected without replacement, so once a beam is selected it is removed from the list of beams available for the remaining tests. Locations are randomly generated for each of the 15 beams to be tested and each beam in the bridge has an equal probability of being selected. A typical simple random sample of the bridge is shown graphically in Fig. 5 in which the darkened beams represent the ones selected for sampling.

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

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

… (2)

where n = number of units sampled, and y^sub i^ = y-value for the i-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 indicate that 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 entire bridge by the engineering consultants. These tests were performed on all of the beams in the population but only at a finite number of locations along each beam, resulting in a value that is still an approximation to a certain extent.

In addition to calculating the population mean, it is often desirable to estimate the variance of the mean prediction. If a variance estimate is known, a confidence interval can be made based on the uncertainty of the calculated mean value. An estimate of the variance of the mean s^sup 2^ is given by10

… (3)

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

Because the variance quantity is not necessarily easy to interpret, one of the more intuitive ways for an engineer to examine the data collected and present the predictions to a client is the confidence interval. In this manner, the engineer who calculated a 90% confidence interval may say “There is a 90% chance that the actual mean of the population falls between Value A and Value B.” Often this presentation is easier for the client to understand than references to variances. The upper and lower bounds on a confidence interval may be calculated as follows

… (4)

where θ = estimated value, z = upper α/2 point of the standard normal distribution (for approximately a (1 – α)% confidence) and var(θ)= estimated variance of the value.7

As should be expected, to have a greater confidence in the prediction a wider interval is needed. And conversely, a narrower range can be specified if the desired confidence in the results is not that high. For an illustration of this point, the confidence intervals for a simple random sample of the bridge data is shown in Fig. 6 for confidence percentages between 80 and 99.9%. The dashed horizontal line shows where the actual mean of the sample falls at 20.6% voided ducts per beam. The circles in the plot above each confidence percentage value indicate the estimated population mean, based on the specific sample. The solid vertical lines, ending with the short horizontal bars and intersecting each mean estimate, show the extent of the confidence interval for that sample. Again, this is a single sample of the data so the mean and variance for each line shown is the same, with the difference in interval length due solely to the value of z in Eq. (4). A range between 14 and 34% can be specified for an 80% confidence interval while a range as wide as 0 and 54% is needed to predict the mean with 99.9% accuracy.

In the case of the bridge data, ten samples were taken to get a general idea of a typical set of results for each specific sampling method. Almost all samples will generate a different estimated mean, variance, and confidence interval, so a group of results is presented in order that the overall trend be seen. Figure 7 shows the 90% confidence intervals for ten different simple random samples. All ten of the confidence intervals generated contain the actual mean within their boundaries, and the highest 90% intervals prediction is 37% for Sample 1 while the lowest boundary is 5% for Sample 4. When examining a structure such as the posttensioned bridge, the engineer is probably most interested in the higher end of the interval to get the most conservative estimate of damage.

Post-stratified random sample

A slight variation of the random sample is the post-stratified, random sample. In this sampling technique, the units are selected randomly, in the same manner as the simple random sample. After selection, however, the units are studied to see if they can be separated into strata (groups with similar properties or values). If strata do exist, the units are then grouped accordingly before the mean and variance are calculated. This method has the advantage over random sampling in that calculation of the mean and variance estimates can be improved if the data within each stratum are very similar to each other. It also has the advantage over stratified sampling in that the actual values of the units can be observed before strata limits must be defined. It has disadvantages in that the random selection of the units before strata definition introduces an additional error term in the variance calculation when compared with traditional stratified sampling. This increase represents the uncertainty introduced by random sample sizes in each strata.

In post-stratified sampling, after the random sample is collected from the entire population, the strata limits are defined and the y-values are separated into their respective groups. Using data from these groups, the mean and variance for the post-stratified, random sample can be estimated using formulas similar to Eq. (2) and (4) for random sampling, but with additional terms that include the estimated mean and variance of each stratum and number of units in each stratum.

The post-stratified samples use the same ten sets of initial test points as those of the random sample in the previous section. However, before the mean and variance is calculated for the post-stratified sample, the data are separated into two strata-namely, the beams over land and those over water. A typical result for the averages of voided ducts is that of Sample 1 (whose confidence interval is shown in Fig. 8) in which the percentage of voids over water (37%) is much greater than those over land (10%). This is true for the total population, which is 28% voided over water and 14% voided over land, although the difference in the full population is not as great as that sample.

One special consideration that must be made for confidence intervals of stratified samples is the correction for the degrees of freedom d. In simple random sampling, the degrees of freedom is equal to the number of units sampled; and this number is used to determine the value z from the student’s t distribution of an approximate standard normal distribution. (A z-value is selected from the table, assuming n-1 (or d-1) degrees of freedom.) In stratified sampling, the degrees of freedom does not necessarily equal the test number because the implied normal distribution may not be entirely accurate and thus a correction to the degrees of freedom will require a larger z-value to be chosen.11

A graph of the confidence intervals for the post-stratified samples, with the same format as the one for simple random sampling in Fig. 7, is shown in Fig. 8. Comparison to the random sampling one reveals that the average difference between the actual mean and the estimated mean increases from 3.5% for the random sample to 3.7% for the poststratified sample. Some of the confidence intervals get slightly wider with an average width of 21% for the random sample and 23% for the post-stratified sample. The only sample whose interval got much wider was Sample 2. For this group of sample points, the random sample included eleven samples over land and only four over the water. (Most of the other samples were closer to an equal allocation.) The small number of samples over water generated a rather large variance estimate for that stratum, resulting in a large variance estimate for the entire sample and reducing the degrees of freedom to only five. This combination of factors led to a wide confidence interval for that sample.

Stratified sample

Another conventional sampling method chosen for the study of this bridge was the stratified sample. The main difference between this and the post-stratified sample is that the strata are chosen before any sampling begins and are thus incorporated into the selection process. This affects the calculation of the mean and variance for the samples.

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

In the case of the bridge, there are 84 units in each of the above-land strata and the above-water strata (N^sub 1^ = N^sub 2^ = 84) with a total of 168 beams (N = 168). Allocating the 15 samples (n = 15) between these two strata and using the formula above yields 7.5 samples in each stratum. Rounding to whole numbers gives eight samples from one stratum (for example, the above-water strata) and leaves seven (15 – 8 = 7) samples for the other strata (above-land strata). A typical sample for such a stratified sample is shown in Fig. 9.

Other options exist for assigning sample sizes to each stratum. One such option is optimal allocation which is based on anticipated standard deviations for each stratum. Another option for assigning a sample number to each stratum involves the cost of the testing. This method attempts to minimize the total variance of the population sample while taking into consideration the cost to test a unit in each stratum, the cost to begin and continue testing, and the total budgeted cost for the testing. Both methods require the engineer to anticipate the standard deviation for each stratum.7 In the current state of practice, there is little or no information available for the standard deviations of damage in structures. Possibly with future research, values such as these can be collected so that the testing engineer will have more information on which to base approximations. Until that time, the engineer can make an educated guess as to the standard deviation or may choose stratified sampling using proportional allocation in which no estimate of these values is required.

Once the number of units to be sampled in each stratum is decided, the units are chosen by random sampling from their strata. This process is termed stratified random sampling. After the units are sampled and their y-values recorded, the data is then analyzed to make predictions about the entire population based on the observations in each stratum. The estimated population mean and variance can be calculated from a stratified random sample according to formulas similar to the post-stratified sample.6,7 Once the mean and variance of the sample have been calculated, it is possible to construct the confidence intervals. Just as with the poststratified sample, a correction of the degrees of freedom must be made. Figure 10 shows the confidence intervals for this sampling design. All of the 90% confidence intervals have similar widths and all contain the actual mean. When compared with the simple random sample, the intervals are approximately the same width with similar boundary values.

Adaptive random sample

Adaptive 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. It was not clear before beginning the testing of the bridge if its data would classify it as a rare, clustered population, but samples were taken using the technique to determine if it would be a useful option. The advantage of this type of sampling is that information gathered while conducting the sample is used to make more informed decisions about where to continue sampling. Thus, adaptive sampling may be appealing, because the engineer can add to the sample in a certain area if a high value is detected. The engineer may be curious to see if neighboring beams, which have the same exposure, material properties, or contractor, also share similar high values.

One of the limitations of adaptive sampling is that the final sample size is not known prior to the survey; therefore, it may be more difficult for a testing engineer to draw up a budget before testing. The expected sample size ? of an adaptive random sample can be calculated if the probability π^sub i^ that unit i is included in the sample is known. In that case

… (5)

…

N = number of total units in the population, n^sub 1^ = initial sample size, m^sub i^ = number of units in the network containing unit i, and a^sub i^ = number of units in networks that which unit i is an edge unit.8

However, when approaching a population has yet to be sampled and which has no similar sampled population from which to draw data, the precise values of m^sub i^ and a^sub i^ are unknown. Therefore, in this study, we will select sampling sizes similar to those in the nonadaptive study to try and obtain more effective comparisons for a similar amount of money and time spent on the testing. To try and get a final sample size near the 15 test points of the previous samples, a number lower than 15 test points, namely six, was chosen as the initial sample size. Those six initial test points were chosen from the entire sample at random, in the same manner in which the 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 the sample in the neighborhood of this test point. If the y-value met or exceeded the cutoff value C, then the units in the neighborhood of the original test point were also tested. The neighborhood of a test point in this case will be defined as the beams that are directly adjacent to a particular beam, that 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 a y-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 a y-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 C that 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 to the 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 be considered 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 whose y-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 will be tested. The only network that would contain a y-value less than C is 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 µ, based on the number of initial intersections, the following formula is used

… (6)

where n^sub 1^ = the number of units in the initial sample; … ; and m^sub i^ = number of units in network A^sub i^.

Its variance can be estimated as

… (7)

where N = number of total units in the population.12

Some initial studies were conducted to determine a suitable cutoff value C for adaptive sampling of the bridge. In terms of structural considerations, one would want to set C low enough that beams with void percentages that are structurally compromising are included in the sample, but not so low that beams with any damage warrant further testing (unless, of course, there are funds for this rigorous testing or any damage is considered crucial). In addition, if C is set too low so that most beams qualify for further testing, the population would not be considered to consist of rare, clustered groups for which adaptive sampling is best suited.

The general effect of a change in the cutoff value in terms of 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 fact that a higher C value will include fewer points in the sample, increasing the variance of the sample. The effect of the change in cutoff value on the number of units tested can be seen by examining Fig. 14(b). For the remainder of adaptive samples 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 very wide confidence interval.

The confidence intervals for the adaptive random sample, shown in Fig. 15, are much wider than those of any of the traditional sampling methods presented previously. The confidence intervals yield estimates as high as 53% where the highest estimate in the traditional random sample was 37%. One of the main reasons for the increased interval width is the fact that the degree of freedom used for the confidence interval calculations was the number of beams in the initial sample. The z value from the distribution table for a 90% confidence interval with 6 samples is 1.94 while the z value for 15 samples is 1.75.13 Thus when calculating the interval bounds using Eq. (4), the term is … multiplied by 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 aren’t used) and is more computationally intensive but still yields wider confidence intervals than its conventional counterpart, it does not seem to be a good technique for analyzing the bridge data. The adaptive techniques are best suited for populations in which units with y-values exceeding the cutoff limit are rare and spaced in clusters. This does not seem to be true for the void percentages of this bridge.

Adaptive stratified sample

One final check of the adaptive technique was done for the adaptive stratified sample to determine if it appears that most adaptive techniques will yield wide intervals for the data. The selection of the sampling units for the adaptive stratified sample is similar to the selection for the adaptive random sample in that an initial sample is taken and units are added adaptively in the neighborhood of observed y-values equal to or above the cutoff value. The only difference is that the initial sample in this case is a stratified sample. The number of units from each stratum (three) is determined by proportional allocation and units are added adaptively regardless of stratum boundaries.

Calculation of the mean and variance estimators for the adaptive stratified sample follow formulas similar to that for the adaptive random sample but with additional terms including the number of total units in each stratum, the number of units initially sampled in each stratum, and the number of units in each network within each stratum. The formulas are not as straightforward as their nonadaptive counterparts. Although the calculations are mainly sums and ratios, there is still a fair amount of bookkeeping to be done to be sure that the correct values are assigned during the many calculations necessary.6,14

The confidence intervals shown in Fig. 16 for the adaptive stratified sample were the widest of any sampling technique studied. The intervals are almost meaningless in that many span from 0% to more than 70% (while the estimated mean was only approximately 20%). These wide intervals are due to 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 a large z value.

CONCLUSIONS

The results indicate that there are many possible techniques that may be used to predict the mean void percentage of the beams in a post-tensioned bridge from a sample consisting of approximately 9% of the total population. The key conclusions from the various techniques studied include:

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

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

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

4. The adaptive techniques are the least effective of the sampling techniques studied herein. They have the disadvantage that the final sample size is not known before testing, money must be spent on some tests which will be disregarded (edge units), and the resulting confidence intervals are wide when compared with the traditional techniques. The computations are also more difficult, requiring more time and introducing more chances for error.

The above conclusions apply specifically to this particular case study of a post-tensioned bridge. While it is early to draw general conclusions, it does seem that simple random sampling works well if little prior information is available about the structure and if the engineer is not able to make any predictions about the possible results before testing begins. An additional study of this type is shown in the companion paper.

NOTATION

a^sub i^ = number of units in network of which unit i is edge unit

C = cutoff value

d = degrees of freedom

m^sub i^ = number of units in network containing unit i (or network A^sub i^)

N = number of total units in population

n = number of samples

n^sub 1^ = initial sample size

r = relative error

s^sup 2^ = estimate of variance of mean

w^sub i^ = average of values of network that includes i-th unit

y = estimated mean of population for simple random sample

y^sub i^ = y-value for i-th unit

z = upper α/2 point of standard normal distribution

α = allowable probability of error

γ = estimate of coefficient of variation

µ= mean estimate of adaptive sample

v = expected sample size of adaptive random sample

π^sub i^ = probability that unit i is included in sample

φ= estimated value for confidence interval

var(φ)= estimated variance of value for confidence interval

REFERENCES

1. Malhotra, V. M., and Carino, N. J., Handbook on Nondestructive 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 the Impact-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 to Assess 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,” ACI Structural Journal, V. 97, No. 1, Jan.-Feb. 2000, pp. 166-174.

6. Williams, T., “Use of Sampling Techniques and Reliability Methods to 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 Beams-Washington Bridge No. 700,” KCI Project No. 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 of Variance Components,” Biometrics Bulletin, V. 2, 1946, pp. 110-114.

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

13. Devore, J. L., Probability and Statistics for Engineering and the Sciences, 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.

Tamara Jadik Williams is an adjunct professor in the Department of Civil and Environmental 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 at Cornell 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.

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