Statistics Low Strength

8/3/2019 Statistics Low Strength

http://slidepdf.com/reader/full/statistics-low-strength 1/6

Paper: WatkindMcNicholl

Paper

Statistics applied to the analysis of test data fromlow-strength concrete cores

R.A. M. Watkins, BSc, CEng, MICE, MHKIEHarris & Sutherland (Far East)

D. P.McNicholl, MS c, CEng, FIStructE, FICE, FHKIEHong Kong Housing Authority

SynopsisSince the mid 1950s, the Hong Kong Government has housedsome 2.8 m people in approximately 1500 high-rise reinforcedconcrete buildings. Since the early 1980s the Go vernm ent’sHousing Depa rtment has faced growing maintenance problemsrelated in part to the low concrete strength encountered insome older blocks.

The investigation, appraisal and repair of the blocksrequired in excess of 20000 core tests. To begin with,

conventional approa ches to sampling and analysis were notsuccessful. The paper explains the problems encountered andstatistical methods developed or adopted to assess thevariation in, and reliability of , concrete core testing results.Guidance is given on sample locations and sizes and of meansof assessing in situ characteristic strength f r om several sets of

samples. The inherent dangers in using small numbers of coresto assess large quantities of concrete are evident.

IntroductionModern multistorey buildings are usually complex structures. Decisionsrelating to repair, refurbishment, or redevelopment of such structures carrywith them considerable financial responsibilities. A simple condition surveyof the building may need to be supplemented by a more searching structural

appraisal to arrive at appropriate decisions. It is becoming increasinglycommon for owners to seek specialist appraisals from structural engineers

who can assess the building’s current condition and fitness for service.Besides these conventional reasons for appraisal, the structure’s

compliance with contract documents or building regulations has, sometimesto be examined.

Appraisal of a completed building is now accepted as aprocedure which

is different from survey or design. The Institution’s Appraisal of existingstructures ’ has given some guidance to engineers, but there is a need for

continued research and publication of techniques useful to the practising

engineer.

In appraising reinforced concrete structures, the in situ strength of theconcrete has to be determined.Although non-destructive and semi-destructive strength tests have been developed, it is generally considered

that compression testing, of cores taken from theexisting structure, is the

most reliable method of estimating a building’s in situ strength.A very extensive series of appraisals has recently been carried out by the

Hong Kong Housing Authority. These appraisals and the methodology

developed for them are described in a separate paper2. Considerable usewas made of statistical techniques in the appraisals. These techniques arenot unusual but it is believed that their application is unfamiliar to manypractising engineers, and they may be found useful in the appraisal of

concrete structures elsewhere.The Housing Authority blocks are large, typically of 16loors, but varying

from seven to 23 storey. The blocks are grouped into estates containing,

typically, 10 blocks, but again this can vary from six to over 20 blocks.

Each block contains typically about 5 0 0 0 m3 of concrete and may houseup to 3000 people.A coring and testing programme initiated by the Housing Authority in

1983, involved the extraction and testing of 20 OOO cores.

Results of the survey, showing extensive evidence of very low concretestrengths in occupied buildings, were made public in 19853and 19864. Thestrategy of the initial survey and the planning of the follow-up action wereexplained in a paper’ in 1987. More recent actions re dealt with

elsewhere6.

BackgroundBetween 1946 and 1986, the population of Hong Kong increased from 1.5M

The St ruc tura l Engineer /Volume 8 /No .16 21 A ugus t 1990

to over SM. After a fire in December 1953, which destroyed the homes

of 53 OOO people, the Government set up apublic housing programme usingseveral agencies. These were onsolidated in 1973 under the control of Hong

Kong Housing Authority which now houses bout 2.6M people. The current

annual production is 40 OOO rental homes, and a further 10OOO are builtfor sale. The annual budget is HK$6000M (f500M).

As part of the authority’s stock, there are some older housing estates,

taken over from previous building agencies, which have proved difficultto maintain. The domestic accommodation is in the formof large slab blocks

typically containing between 50 0 and 1OOO flats. The blocks are large,typically of 16 floors, but varying from seven to 23 storeys. The blocks

are grouped into estates containing typically 10 blocks but again this canvary from six to over 20 blocks. The normal arrangement is for the structure

to be constructed as a series of parallel, coupled reinforced concrete shearwalls which form the side walls of flats as well as the vertical loadbearingelements. The shear walls are coupled with beams across the longitudinalcorridors, and the floors are formed with slabs and longitudinal beams

spanning onto the hear walls. Each block contains typically about5000m3 of concrete and may house up t o 3000 people.

Nature of the concrete encounteredA summary of all strength results for this concrete as at April 1986 is setdown in Table 1. The authors andll the engineers concerned had to come

to terms with the truecharacteristics of this unusual material which differedfrom normal concrete. Table 2, prepared with hindsight, illustrates some

of the differences.

TABLE 1 - oncrete strength results

No of Arithmetic mean estimated in situ

Handover strength of walls (MPa)uildingsdate (minimum specified strength = 20.7MPa)nvesti-

gated<l0 >205-200-15

Pre-1960 386il1

1960-64

3il Nil Nilost-l98250ilil1980-82

1161il Nil27975-79

2313421970-74

623208965-69

12520il47

Totals 4174452 1528

TABLE2 - haracteristics of low-strength concrete

Normal strengthconcreteoncreteLow-strength

Mean strength

0.25-0.450.15oefficient of

3-7 MPa-5 MPatandard deviation

non-normalormalistribution

7-22 MPa5-50 MPa

variation

Characteristic 20-40 MPa 2-5-15 MPastrength

Matrix Hard, coherent Powdery, friable

Inter granular Good Aggregate may be

adhesion detached by hand

Hand sample Cannot be broken May be broken by

by hand hand in some cases

3 2 7




It became clear that considerable statistical analysis would be requiredin order to analyse this type of low-strength concrete. There are some

theoretical objections to the use of statistical techniques in this situation.

Statistical theory is based on the assumption tha t a population (in thestatistical sense) is the universe of individual observed values which havecertain common characteristics. In industrial practice, this assumption iscustomarily realised by considering individual items produced by a single

continuous automatic process over a relatively short timespan. Taking the

whole of the blocks sampled by the 20 OOO cores, the concrete in thestructures was cast over a long period of time by a large number of individualcontractors using a wide variety of materials, equipment and labour. There

is, therefore, no a priori reason why the concrete in these blocks shouldbe considered as a population. Indeed, ndividual blocks were constructedover 1or 2 years, and it is known that cement and aggregate supplies were

frequently changed during construction. It is also known that, in many

cases, different subcontractors were engaged on the construction of differentsections of a block. In spite of these reservations, it has been found thatthe application of statistical techniques assists with the interpretation of

concrete core test results and produces useful information which could nothave been obtained in any other way.

The remainder of this paper covers the problems encountered in samplingand analysing compressive test data from low-strength concrete structures.

Some of the statistical terms used are explained in Appendix A.

Description of problems encounteredIn theearliest stages of investigation and appraisal, many difficult problemswere faced. The geographical spread of the deficient buildings, the extent

of the strength deficiencies, social problems of arranging access for coring,the extremely large population affected (several hundred thousand people),

and the ethinking of methods and techniques developed from pastexperience which were inapplicable to this situation, all posed special

difficulties.

Sampling techniquesIn devising a sampling programme for any building, the purpose of thesurvey needs to be considered. Different techniques will be required for

appraisal of strength, for appraisal of durability or for assessment ofcompliance with specification. In his case the appraisals were mainly

concerned with structural adequacy, and thus sampling concentrated onthe main loadbearing elements, i.e. walls and columns. Later in the process,sampling of he more critical elements was necessary.For li tigation purposes,

the preliminary sampling and the results therefrom had to be linked tosmaller corroborative surveys carried out under a close supervision to

establish a chain of evidence. Besides the problems of administering thesedifferent sampling programmes, there arose an inevitable need to combine

several sets of esults. Problems arose when sample means and characteristic

strengths from successive samples were apparently different.

Extent of samplingThe first major problem to be considered was the significance of the sample

and the number of individual results required for a valid sample. Mostprevious structural appraisals known to the authors ave been concerned

with structures which marginally failed to meet the specification and inwhich all major loadbearing elements could be inspected and sampled.Indeed such investigations can usually be limited to a few critical elements.Under those conditions, sampling of structural elements is not usually aproblem, and a small number of cores, if they are consistent, may be

sufficient. The engineer can be confident tha t the normal system of analysisand margins of safety can accommodate minor sampling errors.

In this case, because ofhe size of he blocks, the unexpected low strengths(2- 10MPa), the obvious low margins of safety and the consequences offailure, a more rigorous approach to sampling had to be taken with noconventional assumptions left unchallenged. It was clear that extensive

sampling spread over several loors of hundreds of blocks would be essential.If all critical elements alone were to be significantly sampled, the coring

would disrupt the lives of several thousand families.Although non-compliance with the specification was relatively obvious,

preparation of evidence for civil and criminal litigation required specialsampling to a very high standard.

Random samplingA truly random sampling programme could not be arranged for most

blocks, as to core the main structural cross-walls between adjacent occupiedflats would have caused considerable disturbance and temporary rehousingof thousands of people. The time required for proper access arrangements,

32 8

security measures fo r belongings, and tenant consultation, would have

extended the survey programme into years rather than months.To begin with, therefore, the best that could be done was to confine coring

to walls in public areas, in stairways, and in the sides of liftshafts. Thisleft nagging doubts about the possibility of weaker areas and whether ornot the sample was truly representative. Later on, more extensive coring

was considered.

Broken coresIn one of the first detailed investigations, cores were taken carefully andsystematically at each floor of the block. It was found that, out of 160

cores taken, some 35 broke on extraction. From visual inspection of thecores it was clear that not all the cores that had broken were of the same‘strength’ and therefore t would not be reasonable to assign a zero strength

to all these cores.

Probability density unctionA consistent problem in handling the test data arose because it became

apparent that the strength results obtained were not normally distributed.

Most engineers had been trained in handling normally distributed strengthdata,and almost all standard references and Codes assume normal

distribution. New techniques had to be researched and developed.

Determination of characteristic strengthWith the use of the limit state approach to structural analysis it is usualto adopt the characteristic strength of the materials in question.The

characteristic strength is usually defined as that strength below which only

5 To of the test results fall. In statistical terms, this is the lower statisticaltolerance limit for 95 070 of the population.

In the case of the non-normal distributions which were encountered,

assessment of standard deviation and characteristic strengh using normalmethods yielded results which were clearly erroneous.

The wide range of strength results from a single block also raised thequestion of how reliable were the core ests as an indicator of the strength

of the block and what confidence limits could be placed on the characteristicstrength obtained from the tests.

Critical zonesAs the analysis of the blocks that were thought to be worst proceeded, itbecame clear that only certain zones of individual blocks were critical in

terms of strength. In most cases, these zones were in the lower floors but,in some of the cases, higher floors were critical because of the geometry

of the blocks and the type of applied loading. Very few of the cores that

had been extracted had been taken directly from the critical zones, anda reliable estimate of strength could be obtained only by extracting further

cores. When this had been done, it was of immediate and critical interestto be able to compare the two amples with a view to either pooling them,

if they could be considered to be derived from the same population, oranalysing them separately, if they were significantly different.

PrioritiesWhen the early investigations and preliminary appraisals had been

completed, it was realised that remedial actions on the worst blocks wouldrequire time to implement. Clearly, the immediate concern was the safetyand living environment of the occupantsof the blocks. It was essential thatsome means be found by which these blocks could be ranked so that those

most at risk could be dealt with first.

Statistical techniques developedSampling techniquesElement references were assigned for critical zones of the block or the whole

block, and these were processed through a random number generationprogram in order to select those elements which should be cored. A reservelist of alternative locations was drawn up andused where access could not

be gained to occupied flats.There is no particular reason why, if randomly chosen, the results of

wall core tests in public areas alone should not be representative of the

whole block. Using sample comparison methods, discussed later, it waspossible to show that, in most cases where fullandom sampling was possiblethroughout the block, there was no significant difference between the overallresults and the results from public areas alone. In the case of one block,

the core strength results from public areas alone were not in agreement

with those from the fully random testing exercise. In this case, however,there were doubts as to the choice of core locations in the public areas.

The comparative study, backed up by an independent check by Lumb’,

The Struc tura l Engineer /Volume 68/No.16/21 A ugus t 1990




was used in several court cases to demonstrate that, on the basis of testsfrom public areas alone, he concrete within the mass of the structure was

not in compliance with the specification.

In another use of sample comparison methods, it was possible to showthat specially supervised tests, taken for the purpose, of 11 cases broughtunder the Prevention of Corruption Ordinance corroborated the earlier

general investigation of several blocks. By combining information from

successive investigations, t was possible to proceed to litigation with a faster

programme than would have been the case had comprehensivereinvestigations been ordered from scratch.

In order o examine the structural adequacy of the most critical blocks,it was still considered necessary in some cases to sample critical zonesdirectly, despite the problems of access referred to earlier.

Sample sizesAfter the initial investigations, studies indicated that, in general, at least20but preferably 40 or 50 results are necessary to characterise the concretestrength of an individual block. In some cases, however, decisions could

safely be made on as few as 20-25 results, while in other cases over 200results were needed to confirm the strengths in marginal blocks.

Elements chosen fo r samplingMost of our appraisals of structural adequacy have concentrated on analysis

of strength and stress of the main primary lements-the shear walls. Forthis reason, most coring since 1984 has concentrated on shear wall sampling.For civil and criminal legal cases, the effect of non-compliance on safetyhas been of interest, so the sampling has again concentrated on the shearwalls.

Estimating strengths or uncrushed coresAs mentioned earlier, in initial investigations, a significant number of coreswere not extracted intact. In order to assign strengths to these cores, thefollowing procedure was adopted. All cores were visually examinedseparately by two experienced engineers. The visual examination not onlyestimated the excess voidage in accordance with BS 1881: Part 120 butalso ranked each core for strength. A ranking table was devised, from 1

for very weak, to 10 for very strong. The rankings given to each core by

the two judges was compared using Spearman’s rank correlation method.In thecase in question the agreement between judges was found to e verysignificant. All intact cores were crushed, and the estimated in situstrengthswere obtained using the method in BS 188 1 Part 120.These strengths werethen correlated with the mean rankings given by the two judges. Thescattergram of results of strength against rankings showed reasonablecorrelation and a non-linear regression line was fitted to the results. Thisregression line was thus used to assign strengths to cores that had beenranked but could not be crushed. This technique assigned non-zero strengthsto broken cores and improved the overall estimate of material’s strength.

Choice of probability density function (PDF)The 20 OOO results taken from the primary survey in 1984-86 were plotted

and developed to produce a ‘model’ PDF (Fig 1) which was seen to bemarkedly non- normal. This confirmed that the few blocks which had beenearlier surveyed in detail, and which had been shown to have non-normaldistributions, had not been unique.

Two techniques were adopted to find the best fitting PDF so thatcharacteristic indices could be developed. The first method was to plot the

results on natural: or 1ogarithmic:probability paper. A normaldistributionplots as a straight ine on natura1:probability paper, while a lognormal or

other non-normal distribution plots as a curve. Similarly, a lognormaldistribution plots as a straight ine on 1ogarithmic:probability paper (Figs2 to 4). This technique has other useful applications, some of which arementioned later.

The Kolmogorov-Smirnoff (KS) test was the second method used to

determine whether the sample data approximate to an unknown PDF.Standard computerpackages are used to determine compatibility betweenthe sample and a set of reference PDFs. This is a very rapid method of

determining the appropriate PDF. The software package used by the authorsplots the reference PDF overlaying the sample histogram, and this givesa visual check on the statistical calculations.

Assessment of characteristic strength ((29.5)and other indices

Several techniques can be used to calculate characteristic strength. The firstmethod is to count numbers of cores on the histogram and set the

characteristic strength a t the lower 4 percentile point; this is suitable onlywhen there are a considerable number of results. The second method isto use the shape of the distribution shown in Fig 1 as he underlyingprobability density function and to se the properties of this shape as being

applicable to the population under consideration ither by superimposingthe mode of the model on the mode of the sample or by superimposingthe two means. The shape of the positively skewed unimodal distributionfound is similar to a lognormal distribution, and the third method makesuse of a logarithmic transformation of the normal statistic to find thecharacteristic strength.

As a fourth method, the cumulative distribution, plotted on probabilitypaper, is joined by a smooth curve and the lower 5 percentile limit read

off t o give the characteristic strength. A further advantage of this methodis that the 16 and 84 percentile points can be read off the graph to yieldan interval of twice the standard deviation.

A fifth method adopted was to use a computer software package to

Estate: Kwai Fong Est. Block: X (+) Total core nos. = 14 4

Estate: Anon Block: Y ( * ) Total core nos. = 143

v)

20

W

Oz

1100

1000

900

800

700

600

500

400

300

2 0 0

100

0

Core results contributed from estate s withmean strength > 2 5 M P a

Core results with assigned strength

5 1 0 1 5 20 2 5 30 3 5 40 45 50 55 6 05 70

Equivalent cube strength of co res (fcu (MPa )

Fig 1. Histogram of 20 000 core test results

The Structura l Engineer /Volume 68/No.16/21 Augus t 1990

99.3

99.3

99.a

%.O(

90.01

80.0

70D

60.0(

5o.M

40.0

30.a

20.M

l o a

5.M

1.00

0.10

M 1

3 -

1 -

1 -

l -D -0-

3 -

1-

1 -

1 -l -1-

1 -

1 -

l -

0 10 20 30 40 500 70 W) 90

Strength (MPa)

Fig 2. Cumulative probabilityplot fo r normal distribution (scales: =

natural; y = probability)

329




Estate: Kwai Fong Est. Block: X ( + ) Total core nos. = 144Estate: Anon Block: V ( * ) Totaloreos. = 143

99.99

99 0

t99m

95.00

90.00

80.00

70.00

6030

50.0040.00

30.00

20.00

10.00

5.00

1.00

0.10

0.01

1 2 4 6 8 10 20 30 40 50 100

Strength MPa)

Fig 3. Cumulativeprob ability plot (scales:x = logarithmic;y = probability)

/

Characteristic Medianstrength Strength (MPa)

Fig 4. Cumulative p robabilityplot (scales: x = natural; y = probability)

calculate the areaunder the smoothed PDF curve to interpolate a strengthlevel exceeded by 95 70of the area under the curve.

In examining critical zones for stability, it was found that not all the

samples from a single block approximate to a ognormal distribution.Generally, it was found that samples with higher mean strengths produceddistributions approximating to a normal distribution while samples withlower mean strengths roduced istributions pproximating tohelognormal distribution. As a result, the sample distribution was always

compared with different PDFs before urther analysis. The sampledistribution was then assumed to be of the type which was closest to it.Thereafter either the normal or the lognormal statistic was used to calculatethe characteristic strength.

330

Confide nce limits or the characteristic strengthFor the first structural appraisals, the number of cores available for each

block was usually about 25 but sometimes less. As a result, the reliancethat could be placed on the characteristic strength was in doubt. When thenumber of observed values in the sample is small, the confidence limitson the mean or on the characteristic strength are wide. For generalised

studies and preliminary assessments,graphical methods can be used to assesssuch limits. For analysis of critical zones a more rigorous approach was

adopted. In the first place the lower confidence limit, at the 5 9'0 significancelevel, of the characteristic strength was calculated using the method in BS

2846'. If this value, when used in analysis, passed the appraisal criteria,

no further action was required. If this value, as sometimes happened, failed

to satisfy the appraisal criteria, this indicated that further cores neededto be taken to increase the sample size and improve the confidence limits.As indicated earlier, between 40 and 50 cores produced characteristicstrengths that would not be much altered by further increasing the numberof observed values.

Comparison of samples

As indicated previously, there were several cases which required separatesamples to be compared. As an example, in cases where further coreextraction and testing was carried out from critical zones of a block, thereis no reason from first principles to suppose that this new sample is from

the same population as the previous sample. Fortunately, methods areavailable to deal with this problem. The simplest procedure involves testingthe two samples for homogeneity using Snedecor's 'F' test and thenStudent's 't' est. If the orm of distribution of the wo samples areapproximately the same, this test is probably sufficient. Where the formof distribution is unknown or where it is uncertain that the two sampleshave the same form of distribution, the conformity of the distributionsof both samples can be tested using the KS test. In many cases in practice,there was no reason to reject the hypothesis of the homogeneity of the twosamples and, as a result, both samples were pooled. This resulted in an

increase in confidence in the strength values found.The KS test was usedalmost exclusively for determining whether samples

taken for itigation purposes corroborated each other and could be pooledwith earlier samples.

Asan alternative, approximate method, sample distributions, plottedon probability paper, can be examined by eye for conformability, and this

is all that is necessary in some cases.

Ran kingFrom 1984 onwards, the evident scale of the problem resulted in the needfor a ranking system so as to decide in what order to deal with the blocks.In the first place, blocks were ranked on the basis of visual assessment,gradually broadened into a strength-based priority system.

A second ranking system, described in ref. 5 , was based on the ranking

factor of a block, an assessment of the expected maximum stress underdead and modified live loads as a ratio to the calculated characteristic

strength. This ranking system was used to select blocks for early clearanceand demolition and to choose blocks for detailed structural appraisal.

When the first stage of appraisals had been completed, the number of

blocks requiring further attention was comparatively large. A better rankingmethod was therefore developed to rank the blocks by a rudimentary formof risk assessment. It was assumed that applied loads and resistance to loadsare governed by stochastic processes and theprobability of failure isgoverned by the joint probability of the difference between the resistanceto load and the applied loads. It was also assumed that theprobability ofoccurrence of the worst combination of applied loads is related to the imefor which the structure is exposed to hazard. Combining these factorsenabled a ranking of blocks to be made and this in turn allowed managementaction to concentrate on the blocks seen to be most at risk.

DiscussionThere is insufficient space to publish here the full results obtained fromthe investigations described earlier. However, some general findings canbe reported. These findings may be of interest because structural appraisalselsewhere may encounter poor quality concrete with low core strengths.

One major finding as been mentioned earlier-i.e. that the probability

density function for low-strength concrete approaches the lognormal formof distribution while, as the strength increases, the function approachesthe normal form. This has been suggested previously byother researchers"where control is poor , and these extensive results appear to confirm this.A second finding is that, for low- strength concrete, there appears to bea relationship between mean strength and coefficient of variation. As the

The Structural Engineer/Volume 68/No.16/21 August 1990



Paper: W atkins/McNicholl

Coefficient ofvariation

4. Hong Kong Housing Authority (ii) :Addendum to Report on Structural

Regression ine ~ 2nd orderequation 5 . McNicholl, D. P., and Wong, B.: ‘Investigation, appraisal and repair

of large reinforced concrete buildings in Hong Kong’, Proc. 2ndInternational Conference, Deterioration and Repair f Concrete in theArabian Gulf, 1987

6. McNicholl, D. P. , Ainsworth, P. R., Watkins, R. A. M., Harley,M. V . , and Stubbings, B. J . : ‘Public housing blocks in Hong Kong:the identification, investigation and rectification of structural defects’,

A Legend: Investigation, 1986

0.70-

0.60-

0.50-

.* . :.40- The Structural Engineer, 68, No. 16, 21 August 1990

7. Lumb, P.: private communication, 19870.30- . . * . . 8. BS 1881: Part20:ethod fo r determination of the compressive

.. . .. . . . . .. . .. .

I .

strength of concrete cores,London, British Standards Institution, 1983

2846:Part 2: 1981 Estimation of the mean : onfidence interval; BS

London, British Standards Institution

descriptions of the strength of concrete’, Journal of the StructuralDivision, American Society Civil Engineers, 105, No. ST6, 1979 pp

0.20-

0.10-

. . 9. BS 2846: Part I: 1975 Routinenalysis of quantitativeata; BS

I I l I I I I I 1 10. Ali Merza, S . , Hatzinikolas,., and MacGregor, G.; ‘Statistical

2846 :Part 3: 1975 Determination of a statistical tolerance interval,

0f c u

5 10 15 20 25 30 35 40 L5

( M P a )

Fig 5. Plot of coefficient of variation against strength1021-1037

mean strength decreases, the coefficient of variation appears to increase.

The formof this correlation is shown in Fig 5 . This does not coincide withthe description of Teychenne”. However, in most cases reported, low-strengthoncrete has been produced on purpose by using lowaggregate/cement ratios, a constant, if high, water/cement ratio, and normalworkmanship. Thus, as mean strength is lowered, the coefficient of variationremains relatively constant and the standard deviation thus decreases. In

the present case, aggregate/cement and water/cement ratios varyconsiderably within each element and from element to element, and it might

be expected that the low-strength concrete would have a higher coefficientof variation. Of course, it must be realised hat the ‘coefficient of variation’

referred to in this context has been calculated on the basis of the normal

statistic. The skewed distribution plays a part in the increase n the coefficientof variation with decreasing strength.

Before starting any structural investigation, it is necessary to consider

carefully the choice of sample locations and of sample size using statisticaltheory. Failure to do so might otherwise invalidate the results obtained.Naturally, at the same time it is necessary to consider which zones in anyparticular element are appropriate for coring, but considerations of thistype are outside the scope of this paper.

Finally, we should emphasise the need for care in the use of the statisticaltechniquesescribed. Fornstance,hemethod of logarithmic

transformations produces a ’mean’ strength which is actually the geometricmean rather than the arithmetic mean. Similarly, the use of statistical tablesrequires care and understanding if misleading results are not to be obtained.

ConclusionsThe use of statistics in the analysis of concrete cores from this investigationhas undoubtedly led to a greater understanding of variation in concretestrengths. It has led to greater confidence in the strengths to be used in

structural appraisals. It has also highlighted the dangers of using smallnumbers of cores to assess large quantities of concrete. It is hoped thatthe methods outlined will find more general applicability to the appraisalof existing concrete structures.

AcknowledgementsThis paper is published withhe kind permission of he Director of Housing,

Mr Y. L. Pang, ISO, JP.Thanks are due to he individual members of staff in the Hong Kong

Housing Department and the three groups of consultants - Harris &

Sutherland (Far East), Ove Arup ( Partners (HK) Ltd., and the MitchellMacFarlane Brentnall & Partners and L. G. Mouchel & Partners jointventure - ho collaborated on this project.

References1. Appraisal of existing structures, London, Institution of Structural

Engineers, 1980.

2. Stubbings, B. J . , Ainsworth, P. R., Crane, R., and Watkins, R. A.M.: ‘Appraisal of thestructuraladequacy of high-rise reinforcedconcrete domestic buildings in Hong Kong’, The Structural Engineer,68 , , No. 16, 21 August 1990.

3. Hong Kong Housing Authority (i) : Summary Report on Structural

Investigation, 1986

11. Teychenne, D. C., : ‘Recommendations for he reatment of thevariations of concrete strength in Codes of Practice’, CP 6/74,

Watford, Building Research Establishment, 1974

Appendix A. Explanation of som e statistical termsPopulation. It is assumed in statistical work that a sample (q.v.) is drawnfrom an underlying population composed of the setof all possibleobservations.

Sample. A sample is assumed to be a subset of a population. It s normallyhoped that, providing samples re properly gathered, the sample will indicatethe character of the population. Samples are composed of a number of

observed values (q.v).The use of theword ‘sample’ in the statistical contextshould be differentiated from its common use as closely synonymous witha specimen.

Observed values.These are the individual readings or observations obtainedin the testing or sampling process.

Average or arithmetic mean. The sum of the observed values divided by

their number. It is the most commonly used, and generally the mostsatisfactory, measure of the centre of a distribution.

Range. The difference between the values of the largest and smallest

observed values.

Median. That value of the variable which divides the total frequency intotwo halves.When a set ofobserved values isarranged in order of magnitudethe median is the middle observation if the number of observed values isodd and by convention is the average of the two middle observations ifthe number is even.

Standard deviation.The commonest and most useful measure of the spreador dispersion of a distribution. When the true mean of the population isknown, the standard deviation is derived by taking the square rootof the

mean of the squared deviations from that true mean. In the absence ofsuch information, the mean of the observed values must be used and thesum of squares of the deviations is then divided by (n-l) instead of n before

extracting the square root. n s the number of observed values in the sample.

Variance. The variance is the square of the standard deviation.

Coefficientof variation. This is another measure of dispersion and is defined

as the standard deviation divided by the arithmetic mean. This is sometimesused as a substitute for standard deviation as a measure of dispersion.

Frequency distribu tion. The number of observations in a set of data lyingbetween specified imits is called he frequency, and the table of frequenciesfor a series of intervals covering the whole range of data is called thefrequency distribution of the set of data.

Histogram. When continuous data have been grouped, their frequencydistribution may be represented by a histogram. On a horizontal scale

representing values of the variable, segments are taken representing the

T he 68/No.16/21 Augus t 1990 3 31



Paper:Watkins/McNicholl. nformal study groups

>C

2ELL

2E

20

15

10

5

Frequency

MPa

0 - 0.9

1 - 1.9

2 - 2.9

3 - 3.9

4 - 4.9

5 - 5.96 - 6.9

7 - 7.9

8 - 8.9

9 - 9.9

10 - 10.9

1 1 - 11.9

12 - 12.9

13 - 13.9

14 - 14.9

15 - 15.9

16 - 16.9

. . . . . . . .

25 - 25.9

Distribution

No.

0

3

2

7

13

1724

23

17

8

9

7

2

4

3

2

2

-1

885 l 0 15 20 25

Strength (Mpal

Number of observed values = 144

Average = 6.42MPaStandarddeviation = 2.87MPa

Median = 5.9MPa Mode = 6.5MPaVariance = 8.25

Minimum = 1 .l MPa Maximum = 25.4MPa Range = 24.3MPa

Ceoff. of variation = 0.45

Fig A . . Histogram illustrating statistical terms

groupntervals. On each segment is erected a rectangle with area

proportional to the number of observations in the interval. (See Fig A.1which illustrates this and previous terms.)

Normal distribution. The term refers to a bell-shaped symmetrical form

of frequency distribution with a defined mathematical equation to whichmany practical distributions approximate. The theoretical curve is definedcompletely by the mean and the standard deviation.

Lognormal distribution.This form of probability density function is relatedto the normal distribution but the natural logarithm of the variable issubstituted in the normal function. The resultant distribution is positivelyskewed with no negative values.

Confidence limits. In experimental work, it is usually possible to obtainonly a limited number of observed values. It is recognised that any quantity

(e.g. the mean) calculated from them may differ from the true value ofthe population. Using the estimated variation of the observed values andmaking certain assumptions regarding their distribution, it is possible toform an estimate of the uncertainty of any statistical measure derived. Theuncertainty is conveniently given in the form of confidence limits which

can be obtained from the standard deviation of the observations, the numberof degrees of reedom, and the degree of confidence which has been decidedusually 95 Yo. (Partextracts from BS 2896 for thisexplanation areacknowledged with thanks.)

Informal study groups

The purpose of the Study Group scheme is to createopportunit ies fo r m embers of the Institution to exchange ideas

and work on deepening and developing their knowledge ofstructural engineering, thus stimulating a greater interest inand promoting the art and sc ienceof structural engineering.

Members wishing to take part in the work of a StudyGroup or wh o require further information about a StudyGroup should w rite to the appropriate Convener.

History of Structural EngineeringConvener: R. J. M. Sutherland, BA, FEng, FIStructE, FICE

Harris & Sutherland, 82-83 Blackfriars Road, London SEI8H AThe Structural Engineer, Ma rch 1973, p1 10

Model Analysis as a Design ToolConvener: F. K. Garas, PhD, CEng, FIStructE, MICE

Taylor Woodrow Construction Ltd., Taywood House, 345Ruislip Road, South hall, Middlesex UBI 2QXThe Structural Engineer, Febru ary 1977, p63

Qualitative Analysis of Structural

BehaviourConvener: D. Johnson, BSc(Eng), PhD, CEng, MIStructE, MICE

Departme nt of Civil & Structural Engineering, TrentPolytechnic, Burton Street, Nottingham N G l 4BUThe Structural Engineer, Novem ber 1978, p309

The Design of Steel Portal FramesConvener: L . J . M orris , BSc(Eng), PhD, ACGI, DIC, CEng, FIStructE

Simon Engineering L aborato ries, U niversity of Man chester,Manchester M13 9PLThe Structural Engineer, Par t A, June 1983, p170

Vibration Problems in StructuresConvener: J . W. Smith , BSc(Hons), PhD, ACGI, CEng, MIStructE

Department of Civil Engi neerin g, University of Bristo l, BristolBS8 1TRThe Structural Engineer, Pa rt A, June 1983, p170

Advanced Computing TechniquesConvener: A. T. Humphrey, CEng, MIStructE, MIMechE

Analysis & Test Division, GE C Research, Marcon i ResearchCentre, West Haningfield Road, Gt. Baddow, EssexThe Structural Engineer, Ma rch 1987, p83

Advanced Composite Materials and

StructuresConvener: P. R. Head, BSc(Eng), ACGI, CEng, MIStructE, MICE

Maunsell Structural Plastics Ltd., Yeoman House, 63 CroydonRoad, London SE20 7TPThe Structural Engineer, Pa rt A, June 1987, p221

~~ ~~ ~ ~ ~~~~ ~~ ~~~

Management and Maintenance of

BridgesConvener: G. Davison, BSc, CE ng, MIS tructE

c/o The Institution of Structural Engineers, 11 Upper BelgraveStreet, London SW lX 8BHStructural news, 23 January 1990, p4

332 The Structura l Engineer /Volum e 68/No.16/21 August 1990

Documents

Statistics Low Strength