RELATIONSHIPS BETWEEN YOUNG’S MODULUS, COMPRESSIVE STRENGTH,

POISSON’S RATIO, AND TIME FOR EARLY AGE CONCRETE

Ryan P. Carmichael

ENGR 082 Project Final Report

Advisor: Prof. Frederick L. Orthlieb

Swarthmore College Department of Engineering

May 2009

ii

iii

Table of Contents

List of Tables....................................................................................................................... iv

List of Figures ..................................................................................................................... iv

Acknowledgments ................................................................................................................ v

Abstract ............................................................................................................................... vi

1. Introduction ...................................................................................................................... 1

1.1. Technical Introduction ......................................................................................................................1 1.2. Personal Background and Goals.......................................................................................................1 1.3. Planning .............................................................................................................................................2

2. Theory .............................................................................................................................. 2

2.1. Piezoelectric Correlation...................................................................................................................2 2.2. Young’s Modulus and Compressive Strength .................................................................................3

3. Testing Regimen............................................................................................................... 4

4. Experimental Set-up & Procedure..................................................................................... 4

4.1. Casting................................................................................................................................................4 4.2. Experimental Testing ........................................................................................................................7

4.2.1. Young’s Modulus / Poisson’s Ratio Tests................................................................................7 4.2.2. Compressive Strength Tests ......................................................................................................8

5. Results & Analysis ........................................................................................................... 8

5.1. KaleidaGraph Fits..............................................................................................................................8 5.2. Young’s Modulus-Compressive Strength Relation.........................................................................9 5.3. Poisson’s Modulus-Age Relation .....................................................................................................9 5.4. Compressive Strength- and Young’s Modulus-Age Relations.................................................... 10

6. Discussion ...................................................................................................................... 11

6.1. Testing Difficulty ........................................................................................................................... 11 6.2. Young’s Modulus-Compressive Strength Relation...................................................................... 12 6.3. Poisson’s Modulus-Age Relation .................................................................................................. 13 6.4. Compressive Strength- and Young’s Modulus-Age Relations.................................................... 13 6.5. Execution ........................................................................................................................................ 14

7. Conclusions .................................................................................................................... 14

iv

7.1. Academic Conclusion .....................................................................................................................14 7.2. Project Assessment..........................................................................................................................15

References...........................................................................................................................16

Appendix A: Summary of Testing Results...........................................................................17

Appendix B: Miniature Version of the Poster Report...........................................................18

List of Tables Table 1: Concrete mix design [Gu et al, p. 1840] .................................................................. 1 Table 2: Summary of compressive testing results.................................................................17

List of Figures Figure 1: Specimens immediately after casting ..................................................................... 5 Figure 2: Specimens curing in moist tent for first 24 hours ................................................... 6 Figure 3: Set-up for Young’s modulus & Poisson’s ratio tests .............................................. 7 Figure 4: Set-up for compressive strength tests ..................................................................... 8 Figure 5: Young’s modulus vs. compressive strength curve fit.............................................. 9 Figure 6: Poisson’s ration vs. age curve fit............................................................................ 9 Figure 7: Compressive strength vs. age curve fit..................................................................10 Figure 8: Young’s modulus vs. age curve fit........................................................................10

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Acknowledgments

I would like to thank my advisor, Professor Fred Orthleib, as well as Professor Faruq

Siddiqui for their guidance and support with this project. I would also like to thank Thomas

(TK) Kelleher who worked in collaboration with me on a project for ENGR 090: Engineering

Design. Without his guidance and the many hours he spent running tests with me, this

project would not have been possible.

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Abstract

Regular strength concrete specimens were tested for Young’s modulus (E), Poisson’s

ratio (v), and compressive strength (fc’) during 16 of the first 25 days of curing. Results of

this testing include a refined version of the existing ACI relationship between Young’s

modulus and compressive strength that is specified to the batch of concrete used. Likewise, a

time-independent Poisson’s ratio of 0.186--specific to the batch--was found. Further results

include a moderately accurate relationship between compressive strength and time as well as

a loose relationship between Young’s modulus and time.

Keywords: Young’s modulus, compressive strength, Poisson’s ratio, early age concrete

1

1. Introduction

1.1. Technical Introduction The constant pressure of the construction industry to provide shorter construction

schedules leads to the premature removal of concrete forms before concrete has a chance to

properly cure. When concrete is put into service before it has developed sufficient

compressive strength, disasters ensue.1 One method to protect against such catastrophes is

early age concrete strength monitoring.

There are currently two main methods for the early age strength monitoring of

concrete: the ultrasonic-based monitoring method and the hydration heat-based method. The

first requires large equipment and is expensive. The second is inexpensive but inaccurate and

unreliable. A third option also exists, and is the focus of TK’s 2009 senior design project. His

project uses high frequency harmonic excitation of piezoelectrics to determine Young’s

modulus, and from it, early age strength. He claims, “This non-destructive method has the

potential to be cost-effective, accurate, and automated. If successful, it would represent the

best option for early age concrete strength monitoring.”

In order to successfully correlate the piezoelectric signals to compressive strength, the

relationship between Young’s modulus and compressive strength as well as the relationship

between Poisson’s ratio and time must be well defined. These relationships are dependent on

such things as aggregate properties, richness of the concrete mix, etc. As such, for the former

relation, the general ACI equation is only 20% accurate. Using this ACI equation as a base, a

more accurate relationship for specific batches of concrete will be found using ASTM

standardized tests.

1.2. Personal Background and Goals I initially became interested in this project when TK sent an e-mail out to everyone in

Engineering Materials asking if anyone would be interested in collaborating with his senior

design project. I immediately jumped at the opportunity in order to gain my first hands-on

experience with concrete. As I currently plan to attend graduate school for structural

engineering, this project appeared to provide very useful exposure to a material I will be

1 One such disaster is the Willow Island cooling tower collapse in West Virginia.

2

learning a lot about in the next few years. The most useful background readings I did were

the ASTMs. While the other readings provided some useful tidbits on early age concrete, the

ASTMs gave me a detailed fundamental base for concrete testing. I feel that as I move

forward, having this base will be very helpful for future laboratories and projects. As the

main goal of my project was to refine existing relationships for a specific batch of concrete,

these background readings did not alter the direction of my project, but instead gave me the

tools necessary to go ahead with the testing as planned.

1.3. Planning The needs of TK’s design project greatly influenced the planning of this project. The

frequency of testing was chosen such to get enough data to successfully correlate Young’s

modulus and compressive strength to the piezoelectric data. The number of samples was

chosen to allow multiple samples on each required test day, while still remaining feasible to

cast in one session. Two samples for each test day best met those requirements. One sample

for each test day roughly correlated to one batch of concrete in the Engineering Department’s

one cubic foot maximum capacity mixer. By making two batches, the batches could feasibly

be mixed immediately before casting to produce more uniform results. This would not have

been possible with three batches because of the lack of a large enough mixing container.

Furthermore 3”x6” cylinders were chosen over the industry standard, 6”x12” cylinders for

feasibility purposes. Testing two cylinders a day at this size would have required sixteen

batches of concrete.

2. Theory

2.1. Piezoelectric Correlation Piezoelectric materials can be used as both sensors and actuators. For TK’s project,

the one piezoelectric acts as an actuator and sends high frequency stress waves through a

concrete test cylinder. A second piezoelectric, acting as a sensor, then picks up a signal due

to the propagated stress waves at the opposite end of the cylinder. Based on a relation with

the speed of sound in an elastic solid, an approximation of the Young’s modulus, E, can be

made from the equation shown below.

3

(1 )

(1 )(1 2 )

Ev

!

"

" "

#=

+ # (Pierce, p. 130)

where:

speed of sound

Young's modulus

Poisson's ratio

density

v

E

!

"

=

=

=

=

Solving for Young’s modulus produces:

2 (1 )(1 2 )

(1 )E v

!" "

"

+ #=

#

For concrete, Poisson’s ratio and density should be relatively constant during all

stages of curing, while Young’s modulus and the speed of sound should vary with time.

Poisson’s ratio is generally equal to approximately 0.18 and density of normal weight

concrete is typically equal to about 145 lb/ft3. (Oluokun et al, pp. 3-5) Poisson’s ratio and

Young’s modulus are monitored during curing for this experiment, while density and the

speed of sound are monitored during curing as part of TK’s project.

2.2. Young’s Modulus and Compressive Strength

ACI Committee 318 recommends the following empirical relationship between

Young’s modulus and compressive strength of normal strength concrete:

1.5 '

33 cE w f=

3

'

where:

weight of concrete /

28 day compressive strengthc

w lb ft

f

=

=

4

Although this equation is for the 28 day compressive strength, Oluokun et. al. concluded that

this equation holds within 20% error for regular strength concrete 12 hours or older. For the

w =152 lb/ft3value TK ultimately calculated, the ACI equation becomes '62,025 cE f= .

For TK’s project, 20% accuracy is only moderately acceptable. As such, the

following adaptation of the ACI equation will be used, with the value of the proportionally

constant, k, determined by curve fitting the experimental data.

'

cE k f=

where:

curve fit porportionality constantk =

3. Testing Regimen

The experimental protocol involved sixteen days of testing spanning over twenty-five

days. Forty 3”x6” concrete cylinders were cast: thirty-four cylinders for Young’s modulus,

Poisson’s ratio, and destructive compressive testing and five for back-up. For the first seven

days, two regular cylinders were tested each day to determine the Young’s modulus and

Poisson's ratio (test setup shown in Figure 3 on page 6) as well as to determine the

compressive strength of the concrete (test setup shown in Figure 4 on page 7). For the last 21

days of testing, two cylinders were tested roughly every other day.

4. Experimental Set-up & Procedure

4.1. Casting

As mentioned, forty 3”x6” cylinders

were cast for this experiment. The mix

design used is summarized in Table 1. This

design was taken from Gu et al to allow

potential comparison. Because of the

limitations in size of the available concrete

mixer, the concrete had to be mixed in two batches. To create a larger unified test batch, the

first batch was mixed with a shovel while the second batch was mixing in the Engineering

Table 1: Concrete mix design [Gu et al, p. 1840] Component lb/cubic yard

Type I Portland Cement 580 Sand 1535 ½ inch CA* 1697 Water 355 *During testing it was discovered that several specimens had larger CA, as discussed in section 6.

5

Department’s mixer. When the second batch finished mixing, the two batches were then

combined and mixed thoroughly. Test cylinders were then cast and cured according to

ASTM C192/C192M.

After casting, the specimens were placed in a tent at roughly 100% humidity for 24

hours. After this time period the cylinders were taken out of their molds and placed in room

temperature water to cure for the remainder of the experiment. Casting photos can be seen in

Figures 1 and 2 below.

Figure 1: Specimens immediately after casting

6”x12” Cylinder w/ Embedded Piezoelectric Transducers for TK’s Project

40 3”x6” Cylinders for Compressive Tests

6

Figure 2: Specimens curing in moist tent for first 24 hours

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4.2. Experimental Testing For each test day, two 3”x6” cylinders were tested. Two compressive tests were

performed on each cylinder: the first, to determine Young’s modulus and Poisson’s ratio, and

the second, to determine compressive strength. Rubber caps were used for both tests as per

ASTM C1231 / C1231M – 09

4.2.1. Young’s Modulus / Poisson’s Ratio Tests Following ASTM C 469-02e1, the Young’s modulus/Poisson’s ratio tests were

performed in a screw-driven Tinius Olsen machine as seen in Figure 3. One tester would

operate the Tinius Olsen machine, calling out load readings to a second tester who would

record the vertical and horizontal dial readings for each of the called loads.

Figure 3: Set-up for Young’s modulus & Poisson’s ratio tests

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4.2.2. Compressive Strength Tests Following ASTM C39/C39M-05e2, the compressive strength tests were performed in

a hydraulic compression machine as seen in Figure 4. The specimens were carefully

centered by eye and by ruler before being loaded steadily until failure.

Figure 4: Set-up for compressive strength tests

5. Results & Analysis

5.1. KaleidaGraph Fits The following plots were all curve fit using KaleidaGraph. To fully understand these

plots one must understand the format KaleidaGraph uses to display its curve fit data. A brief

summary is provided below. The error column to the right of the parameter values represents the standard error

values of said parameters. Each row should be read as, "parameter value ± error." "Chisq" represents the Chi Square value, which is the sum of the squared error between the original data and the calculated curve fit. The lower the value, the better the fit. "R" represents the correlation coefficient, which indicates how well the curve fit matches the original data. The coefficient ranges from zero to one. The closer the value to one, the better the fit.

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5.2. Young’s Modulus-Compressive Strength Relation

Figure 5: Young’s modulus vs. compressive strength curve fit

5.3. Poisson’s Modulus-Age Relation

Figure 6: Poisson’s ration vs. age curve fit

E = 75,000 (fc’ ).5

v = 0.186

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5.4. Compressive Strength- and Young’s Modulus-Age Relations

Figure 7: Compressive strength vs. age curve fit

Figure 8: Young’s modulus vs. age curve fit

fc’ = 3500 - 2300 e-0.18t

E = 3.3 (106)e0.018t

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6. Discussion

6.1. Testing Difficulty

By and large, both the Young’s modulus/Poisson’s ratio tests and the compressive

strengths were preformed with little difficulty. However, on several test days around the end

of the first week of testing, a bottom corner cracked off the specimen during the Young’s

modulus/Poisson’s ratio test. When a corner broke, the load was immediately removed,

losing anticipated data points.

Not only did this cracking result in less data points, and consequently, less accurate

values for the Young’s modulus and Poisson’s ratio, but it also caused a reduction of the area

receiving the load from the second testing machine. This caused the actual stress experienced

in the cylinder to be more than the calculated amount. True compressive strength was

subsequently likely to be greater than that recorded. Additionally, the cracked corners

provided an uneven surface, which could have introduced some eccentricity to the second

loading. As concrete is poor in tension, small amounts of eccentricity can cause the concrete

to fail at lower loads than it would have otherwise, again causing the recorded compressive

strength to be less than the actual compressive strength. Furthermore, even if the corners did

not crack during the first test, the stress concentration would still be present for the second

test, causing the concrete to fail at a lower load, once again reducing the result for

compressive strength.

After this cracking occurred several times, TK and I investigated the issue and

discovered that the bottoms of the cylinders were consistently concave, with bottom edges

approximately 1/16 inch below the center of the base. This “dome” was enough such that the

rubber caps were not making good contact with the center of the cylinder. This caused stress

to be concentrated on the edges, which in turn resulted in edges cracking at low loads.

It should also be noted that the tops of the cylinders were not perfectly flat. This was

due to how well the top of the cylinder was scraped off during casting and, as such, the

flatness of the top greatly varied from cylinder to cylinder. These irregularities were,

however, not particularly likely to cause stress concentrations at the edges, but could have

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created some eccentricity during loading, which would have skewed the results, particularly

for compressive strength.

6.2. Young’s Modulus-Compressive Strength Relation

Although the experimental relationship of E=75,000(fc’ ).5 is barely outside of the

20% error range of the ACI predicted value E=62,025(fc’ ).5, it does not fit the experimental

data particularly well as represented by the large Chi Square value and relatively low correlation coefficient. This poor fit is a result of the several sources of error that affected this experiment, including the corner cracking difficulty previously discussed.

Another particularly major source of error is the size of the aggregate used in the

cylinders. The mix design required ½” nominal coarse aggregate. However, over one inch

nominal aggregate was discovered in some specimens during testing. This violated the

ASTM rule that course aggregate must be no larger than 1/3 the diameter of the test cylinder,

creating inconsistent results. This error could have been avoided by sieving the aggregate by

hand or by using 6”x12” cylinders. The larger cylinders—the industry standard— certainly

would have given more consistent results, but would have been unfeasible with the resources

at hand.

The large particles were likely a major reason that the specimens to break at nearly

half the compressive strengths than those seen by Gu et al for a supposedly identical mix

design. Examination of the failed specimens revealed that the larger aggregates did not break,

indicating that the specimens instead failed because of bonding failure. While this is likely a

significant factor for the specimens being much weaker than those made by Gu et al, it is not

likely to be the only factor. It’s quite possible that there was another error in executing the

mix design that has not been identified.

Additionally, TK’s piezoelectric data had a strong correlation to Young’s modulus,

but this correlation broke down when Young’s modulus was converted to compressive

strength. This suggests that more error was present in the compressive strength tests than in

the Young’s modulus test. This is logical when one considers the load rates for the two tests.

For the Young’s modulus test, a machine-operated load was applied at a more or less

constant load rate. The compressive strength test on the other hand was performed on a

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hydraulic machine that was operated by a hand pump. Regardless of how smooth and steady

the operator pumped the load handle, the load rate could be nowhere near as consistent as the

automated machine because the machine only loaded on the down stroke and one stroke

applies much more load near the point of failure than at the beginning of loading.

Ideally all the compressive strength tests would have been performed on an

automated machine, but scheduling problems prevented this for the first several test days and

desire for consistent testing thereafter kept the tests on the hydraulic machine.

A further consideration is the fact that the ACI formula was developed to calculate

Young’s modulus from a given compressive strength. In this report, the assumption was

made that the equation could be solved for compressive strength in terms of Young’s

modulus. However, further investigation is required to validate this assumption.

6.3. Poisson’s Modulus-Age Relation

Exactly as expected, Poisson’s ratio stayed nearly consistent at 0.186 with only one

outlier. Such accuracy speaks to both the preciseness of the machine-driven Poisson’s ratio

test and to the careful curve fitting. Horizontal displacements typically only reached three or

four ten-thousandths on the extensometer over the course of around twenty data points. As

such, several stress values were recorded for each given strain. In most tests, the largest strain

reached would have fewer data points than previous strains. Were the testing to continue, the

“missing” strains would be recorded with the next several readings. However, as this high

strain reading did not have a full set of stress values, it would typically distort the sensitive

curve fit. To correct this, the last strain level was omitted from the calculation of Poisson’s

ratio, resulting in accurate and consistent values.

6.4. Compressive Strength- and Young’s Modulus-Age Relations

These two results separate compressive strength and Young’s modulus, allowing for a

more in-depth understanding of these two material properties. The curve fits for these two

graphs are slightly more accurate than that of the Young’s modulus-compressive strength

relationship. More important though, is that these graphs allow observation based on

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individual test days. From these plots, it can be seen that test days 9, 23, and 25 had

particularly strong concrete, while days 11-19 had particularly weak concrete. Ideally, such

analysis would provide a correlation to a significant event on that test day that would allow

the data to be thrown out or considered with more weight. Unfortunately, however, no such

strong correlations were found with this analysis.

6.5. Execution

As I was collaborating with a senior design project, I did not have to ask for, buy, or

build any setups or materials. I was, however, able to obtain all the information necessary to

perform the tests in a reasonably accurate manor. The only change to the testing protocol was

the aforementioned corner cracking issue, which was resolved through filing. Overall, testing

ran very smoothly.

7. Conclusions

7.1. Academic Conclusion In this experiment, a reliable relationship was established between Poisson’s ratio and

age for the batch of concrete used, while a somewhat less reliable relationship was

established between Young’s modulus and compressive strength for the batch. The consistent

0.186 result for Poisson’s ratio matches the expected value of about 0.18, while the

relationship of E=75,000(fc’ ).5 failed to provide a good fit of experimental data.

Error in this experiment stemmed largely from the fact that the test cylinders

consistently broke below their expected compressive strengths. There are several reasons this

could have occurred. The course aggregate violated ASTM standards and was larger than

what the mix design called for. Several bottom corners cracked during the Young’s modulus

testing before the problem of uneven cylinder ends was solved via filing the concrete. The

loading for the compressive strength tests was inevitably jerky. Additionally, there may have

been a mistake made when weighing out one or more ingredient for the concrete mix. This is

particularly plausible given the quality of the scales in the soils lab.

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7.2. Project Assessment My project was very successful in achieving a relationship between Poisson’s ratio

and time, while less successful at achieving an accurate relationship between Young’s

modulus and compressive strength. Overall I did extensive and effective background

research, got all of the materials I needed for my project in a time-efficient manner, and, with

the help of Thomas Kelleher, planned my project with foresight. We got an early and

developed a reasonable plan that would work around both of our schedules several times a

week. When we had a conflict we had backups (Jing, Prof. Siddiqui, and Prof. Orthleib)

prepared to step in. As such, I rate my background research as good and my planning as

excellent. Furthermore, although the data was not perfect, TK and I stuck to our schedule

week after week, and adjusted quickly to any unforeseen difficulties that arose. As such, I

rate my execution of this project as excellent.

If I were to do this project again, I would sieve the course aggregate by hand, sand the

ends of every cylinder, and find a way to do all of the compressive strength testing on an

automated testing machine. From this project I gained a good basic knowledge of concrete

and concrete testing. On a more personal level, the most valuable part of this project was

spending lab time with TK, picking his brain about graduate school, summer internships,

career goals, engineering courses, and the like. While exposure to E90-level research was a

monumental benefit, exposure to Thomas Keller’s insights have been invaluable.

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References

American Society for Testing Materials. “Standard Test Method for Compressive Strength of Cylindrical Concrete Specimens (ASTM C39 / C39M - 05e2).” V. 4 Pt. 2. ---. “Standard Test Method for Slump of Hydraulic-Cement Concrete (ASTM C143/C143M).” V. 4 Pt. 2. ---. “Standard Practice for Making and Curing Concrete Test Specimens in the Laboratory ASTM C192 / C192M – 07).” V. 4 Pt. 2. ---. “Standard Test Method for Static Modulus of Elasticity and Poisson’s Ratio of Concrete in Compression (ASTM C 469-02e1).” V. 4 Pt. 2. ---. “Standard Practice for Use of Unbonded Caps in Determination of Compressive Strength of Hardened Concrete Cylinders (ASTM C1231 / C1231M – 09).” V. 4 Pt. 2. The ASTMs were all used to ensure that the tests performed for this project were to standard. As such, they were vital to the success of the project. Gu, H, G Song, H Dhonde, YL Mo, and S Yan. “Concrete early-age strength monitoring using embedded piezoelectric transducers.” Smart Materials and Structures, Vol. 15, Nov. 2006. This article was used for the mix design and to compare compressive strength data. Without this article our error would have been a lot less apparent. Khan, Arshad A., William D. Cook, and Denis Mitchell. “Early Age Compressive Stress-Strain Properties of Low, Medium, and High-Strength Concretes.” ACI Materials Journal, V. 92, No. 6, November-December 1995, pp. 617-624. This article served as a cross check on the Oluokun article. It was a nice find, but the least useful source of those cited here. Pierce, Allan D. Acoustics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America, Woodbury, NY. 1989. This text was used for the relation between Young’s modulus, Poisson’s ratio and the speed of sound through a solid. This source was useful to me only in the sense that it was useful to TK’s project. Oluokun, Francis A., Edwin G. Burdette, and J. Harold Deatherage. “Young’s modulus, Poisson’s Ratio, and Compressive Strength Relationships at Early Ages.” ACI Materials Journal, V. 88, No. 1, January-February 1991., pp. 3-9. This article was invaluable in my attempt to establish the relationships sought after in this project. This article acted as my main source for the theory of the experiment.

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Appendix A: Summary of Testing Results Table 2: Summary of compressive testing results SPECIMEN1 SPECIMEN2 AVERAGE

Day fc UnitWt. fc UnitWt. fc E ν lb psi lb/ft3 lb psi lb/ft3 lb psi psi in/in

1 10740 1519 153.1 11390 1611 155.1 11065 1565 3079082 NA

2 12250 1733 155.3 13780 1949 154.3 13015 1841 3490590 NA3 14360 2032 154.4 17800 2518 153.6 16080 2275 3526864 0.186

4 14490 2050 154.1 19960 2824 155.3 17225 2437 3641757 0.184

5 19190 2715 153.0 19380 2742 155.0 19285 2728 3880445 0.1836 22510 3185 154.5 18090 2559 155.2 20300 2872 3898696 0.178

7 19500 2759 153.8 22920 3243 153.6 21210 3001 4012590 0.2039 23980 3392 153.5 23610 3340 154.6 23795 3366 3935557 0.174

11 22900 3240 154.5 19670 2783 153.9 21285 3011 3891686 0.24813 22930 3244 154.0 21360 3022 154.9 22145 3133 3980031 0.184

14 22030 3117 154.1 23740 3359 154.6 22885 3238 4122259 0.194

15 22170 3136 154.1 22350 3162 154.8 22260 3149 4198048 0.17917 26330 3725 154.2 19830 2805 155.6 23080 3265 4314568 0.184

19 22700 3211 152.4 24250 3431 151.9 23475 3321 4590814 0.18823 25580 3619 154.3 28300 4004 153.4 26940 3811 5264952 0.188

25 27880 3944 154.2 28770 4070 153.4 28325 4007 5359575 0.179

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Appendix B: Miniature Version of the Poster Report