in-service durability performance of water tanks

10
In-Service Durability Performance of Water Tanks Sudhir Singh Bhadauria 1 and Mahesh Chandra Gupta 2 Abstract: With an alarmingly increased rate of deterioration in reinforced concrete structures due to durability performance, efforts are being made to quantify in situ performance. Performance is a function of time and it is related to degradation and the parameters influencing it. Although state-of-the-art modeling of various deterioration mechanisms is available in the literature, evaluation of the influence of various deterioration mechanisms that decrease performance with time is difficult. However, in situ condition documentation, survey, and assessment of deteriorated structures reflect the resultant deterioration process and also helps in validation of experimental and theoretical methods of performance evaluation. In this research, a systematic in situ condition documentation, survey, and assessment of water tank structures has been done based on an empirical damage scale similar to that suggested in the literature and a bilinear graphical deterioration model for such water retaining structures in a semitropical region like India is presented on the basis of case studies. DOI: 10.1061/ASCE0887-3828200620:2136 CE Database subject headings: Water tanks; Corrosion; Deterioration; Service life; Monitoring; Documentation. Introduction In this age of state-of-the-art technology in civil engineering, it is indeed paradoxical that the profession and public should be in- creasingly concerned with the safety of structures. Modern mate- rials are of better quality and of substantially increased strength. The advances in technology should have produced better and more durable structures; unfortunately, this is not the case. Engi- neers use numerical methods to provide adequate strength, stiff- ness, stability, and serviceability in the final structure. These new skills help in providing this adequately at the lowest cost. The margins and factors of safety of structure in this process are assumed to prevail as soon as the structure is completed and during the entire useful life of the structure, this is not checked in any numerical sense, but tacitly assumed to be so, if material and workmanship specifications are complied with during the con- struction phase. Durability and Design Life Durability is defined as “the ability of the structure to maintain its level of reliability and serviceability during its lifetime.” Durability, as per British Standards BS 5750, is the ability of an item to perform its required function under stated conditions of preventable or corrective maintenance until a limiting state is reached. In the European Commission Standing Committee on Con- struction 2002 durability is defined as follows: “A construction product or work is durable if it is designed and constructed in such a way that under conditions of normal maintenance and use, the essential requirements are satisfied during an economically reasonable working life.” Design life is the minimum period for which the structure can be expected to perform its designated function, without significant loss of utility nor requiring much maintenance. Various aspects of the life of structures have been brought into focus recently such as design life, service life, working life, useful life, predicted life, financial life, etc. The concept of design life is not recent. In 1950, the British Code of Practice CP3: Chapter 9 Durability, London, British Standards Institutions 1950 introduced the concept of durability for both building components and buildings as a whole. Few countries, e.g., United Kingdom, United States, Japan, Norway, Sweden, and Holland so far are in the process of incorporating durability and design life requirements in quantitative terms in their National Standards/Codes of Practices. In United Kingdom, the British Standard Code on Bridge Structures BS:5400 speci- fies a design life of 120 years. The Indian Roads Congress ex- pects a life of 50 years for bridge structures. More recently, the theme was developed by Bratchell 1985, who also proposed values for nominal design lives. The approach is same as for fire resistance, in principle, where nominal design lives are prescribed albeit, in hours, rather than years, and a range of solutions is developed to meet these basic performance criteria. Brief Review of Work Already Done in Field Worldwide concern about unexpectedly low durability is being observed. Low durability is being perceived as one of the poten- tial threats to the future of the concrete industry Mehta and Ger- wick 1996. A report of the U.S. National Materials Advisory Board, for instance, indicates that in 1987 approximately 253,000 concrete bridge decks, some less than 20 years old, were found to be in varying states of deterioration. A survey of automobile park- 1 Professor and Head, Civil Engineering Dept., Univ. Institute of Technology, Rajiv GandhiTechnological Univ., Bhopal-462036, Madhya Pradesh, India corresponding author. E-mail: [email protected] 2 Vice-Chancellor, Rajiv Gandhi Technological Univ., Bhopal 462036, Madhya Pradesh, India. Note. Discussion open until October 1, 2006. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and pos- sible publication on October 5, 2004; approved on May 13, 2005. This paper is part of the Journal of Performance of Constructed Facilities, Vol. 20, No. 2, May 1, 2006. ©ASCE, ISSN 0887-3828/2006/2-136–145/ $25.00. 136 / JOURNAL OF PERFORMANCE OF CONSTRUCTED FACILITIES © ASCE / MAY 2006 J. Perform. Constr. Facil. 2006.20:136-145. Downloaded from ascelibrary.org by University Of British Columbia on 11/25/14. Copyright ASCE. For personal use only; all rights reserved.

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Page 1: In-Service Durability Performance of Water Tanks

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In-Service Durability Performance of Water TanksSudhir Singh Bhadauria1 and Mahesh Chandra Gupta2

Abstract: With an alarmingly increased rate of deterioration in reinforced concrete structures due to durability performance, efforts arebeing made to quantify in situ performance. Performance is a function of time and it is related to degradation and the parametersinfluencing it. Although state-of-the-art modeling of various deterioration mechanisms is available in the literature, evaluation of theinfluence of various deterioration mechanisms that decrease performance with time is difficult. However, in situ condition documentation,survey, and assessment of deteriorated structures reflect the resultant deterioration process and also helps in validation of experimental andtheoretical methods of performance evaluation. In this research, a systematic in situ condition documentation, survey, and assessment ofwater tank structures has been done based on an empirical damage scale similar to that suggested in the literature and a bilinear graphicaldeterioration model for such water retaining structures in a semitropical region like India is presented on the basis of case studies.

DOI: 10.1061/�ASCE�0887-3828�2006�20:2�136�

CE Database subject headings: Water tanks; Corrosion; Deterioration; Service life; Monitoring; Documentation.

Introduction

In this age of state-of-the-art technology in civil engineering, it isindeed paradoxical that the profession and public should be in-creasingly concerned with the safety of structures. Modern mate-rials are of better quality and of substantially increased strength.The advances in technology should have produced better andmore durable structures; unfortunately, this is not the case. Engi-neers use numerical methods to provide adequate strength, stiff-ness, stability, and serviceability in the final structure. These newskills help in providing this adequately at the lowest cost.

The margins and factors of safety of structure in this processare assumed to prevail as soon as the structure is completed andduring the entire useful life of the structure, this is not checked inany numerical sense, but tacitly assumed to be so, if material andworkmanship specifications are complied with during the con-struction phase.

Durability and Design Life

Durability is defined as “the ability of the structure to maintain itslevel of reliability and serviceability during its lifetime.”

Durability, as per British Standards BS 5750, is the ability ofan item to perform its required function under stated conditions ofpreventable or corrective maintenance until a limiting state isreached.

1Professor and Head, Civil Engineering Dept., Univ. Institute ofTechnology, Rajiv Gandhi Technological Univ., Bhopal-462036, MadhyaPradesh, India �corresponding author�. E-mail: [email protected]

2Vice-Chancellor, Rajiv Gandhi Technological Univ., Bhopal 462036,Madhya Pradesh, India.

Note. Discussion open until October 1, 2006. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on October 5, 2004; approved on May 13, 2005. Thispaper is part of the Journal of Performance of Constructed Facilities,Vol. 20, No. 2, May 1, 2006. ©ASCE, ISSN 0887-3828/2006/2-136–145/

$25.00.

136 / JOURNAL OF PERFORMANCE OF CONSTRUCTED FACILITIES © AS

J. Perform. Constr. Facil.

In the European Commission Standing Committee on Con-struction �2002� durability is defined as follows: “A constructionproduct or work is durable if it is designed and constructed insuch a way that under conditions of normal maintenance and use,the essential requirements are satisfied during an economicallyreasonable working life.”

Design life is the minimum period for which the structure canbe expected to perform its designated function, without significantloss of utility nor requiring much maintenance. Various aspects ofthe life of structures have been brought into focus recently such asdesign life, service life, working life, useful life, predicted life,financial life, etc.

The concept of design life is not recent. In 1950, the BritishCode of Practice �CP3: Chapter 9 Durability, London, BritishStandards Institutions 1950� introduced the concept of durabilityfor both building components and buildings as a whole. Fewcountries, e.g., United Kingdom, United States, Japan, Norway,Sweden, and Holland so far are in the process of incorporatingdurability and design life requirements in quantitative terms intheir National Standards/Codes of Practices. In United Kingdom,the British Standard Code on Bridge Structures �BS:5400� speci-fies a design life of 120 years. The Indian Roads Congress ex-pects a life of 50 years for bridge structures.

More recently, the theme was developed by Bratchell �1985�,who also proposed values for nominal design lives. The approachis same as for fire resistance, in principle, where nominal designlives are prescribed �albeit, in hours, rather than years�, and arange of solutions is developed to meet these basic performancecriteria.

Brief Review of Work Already Done in Field

Worldwide concern about unexpectedly low durability is beingobserved. Low durability is being perceived as one of the poten-tial threats to the future of the concrete industry �Mehta and Ger-wick 1996�. A report of the U.S. National Materials AdvisoryBoard, for instance, indicates that in 1987 approximately 253,000concrete bridge decks, some less than 20 years old, were found to

be in varying states of deterioration. A survey of automobile park-

CE / MAY 2006

2006.20:136-145.

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ing garages in Canada �Litvan and Bickley 1987� found that sev-eral billion dollars would be needed for the repair of concretestructures which had shown serious deterioration much earlierthan their designated service life. Cases of premature and seriousdeterioration have been reported from around the world with un-dersea tunnels and with marine structures in California and east-ern Canada �Gervick 1989�. Even though no such data are avail-able for India, undoubtedly the amount spent on repair andrehabilitation of concrete structures would be staggering.

Durability and service life of concrete structures is mainlygoverned by microstructural and transport properties of concreteand environmental exposed parameters. Some constitutive andsimulation models for theoretical and mathematical prediction ofdurability and service life performance related microstructural en-gineering properties, e.g., porosity, permeability, surface area,volume of phases, etc., have been proposed and modified. Meth-ods for analytical and experimental determination of physical andchemical characteristics affecting the durability of concrete areproposed by Papadakis et al. �1992a,b�. Some in situ and labora-tory methods are devised to measure the durability parameters.The experience in applying these in situ test methods and inter-preting their results in terms of state of durability of reinforcedconcrete structures is limited. Of the available in situ test meth-ods, the better known are the initial surface absorption test afterFigg’s �1973� water absorption and air permeation tests. Testmethods by Pihlajavaara and Paroll �1975� and Kasai et al. �1984�are reported. In addition, many other in situ test methods based onboth water and gas permeation principles are suggested.

Dhir et al. �1991� and Miyagawa �1991�, suggested durabilitybased design methods for concrete structures in chloride environ-ments. Ahmed et al. �1997� proposed experimental method forservice life prediction of Rebar-corroded reinforced concrete�RC� structures.

A mathematical model for durability of cladding and buildingmaterials was proposed by Hjelmstad et al. �1996� incorporatingenvironmental stimuli, material degradation, and structural perfor-mance. Some efforts have been made for reliability based service-life prediction of aging concrete structures by Mori and Elling-wood �1993� and Enright and Frangopal �1998�.

Physical and chemical characteristics affecting the durabilityand service life of concrete structures are highlighted by the re-search of Papadakis et al. �1991b� and they also proposed thefundamental modeling and experimentation for investigation ofconcrete carbonation �Papadakis et al. 1991a�. Analysis of chlo-ride diffusion into partially saturated concrete was reported bySaetta et al. �1993�.

Experimental efforts for service life prediction of Rebar-corroded RC structures were done by Ahmed et al. �1997�.

Neural network techniques have been applied for predictingthe life of concrete structures by Buenfeld and Hassanein �1998�.

Apart from the above research, attempts have been made forthe experimental and theoretical modeling of concrete deteriora-tion and laboratory measurement of related parameters. However,very limited in-service/field durability performance data are avail-able to supplement laboratory or theoretical modeling, particu-larly in India where concrete durability research is yet to gainmomentum. Idorn �1994� has rightly summarized the modelingresearch in concrete technology by suggestion that “scientificmodeling research must be based on field performance monitor-ing rather than the conventional empirical tartans of specimens as

the source for basic data banks for modeling.”

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Objectives and Scope of Present Study

The above suggestion along with field investigations of concretestructures in the Arabian gulf region �Suad Al-Bahar et al. 1998�coupled with the recent reporting of relatively higher failure rateof concrete structures has been the major motivation for thisstudy. Repeated requests from the Public Health engineering de-partments of the provincial State of Madhya Pradesh and Rajast-han in India regarding the investigation of the failures of a seriesof water tanks and the anticipated amount of expenditure likely tobe incurred in repair and rehabilitation of such concrete structuresfurther encouraged the research.

Databanks, especially in terms of feedback from the field, haveto be developed and these are required to be used for calibrationlaboratory research and development of design procedures. Loadsare to be quantified, especially in environmental terms, involvingmacro- and microclimates, both natural and manmade. There is acontinuing need to monitor changes in materials, design, and con-struction practices and to assess their significance.

The objectives of the present study have been summarized asfollows:1. To develop a methodology for field surveys, condition as-

sessment, and service life prediction of existing concretestructures;

2. To undertake a case study of field condition survey of dete-riorated water tank structures; and

3. To propose a deterioration model for water tank structuresbased on field condition surveys.

Deterioration Mechanisms of Concrete Structures

The deterioration of concrete in service is the result of a variety ofphysical and chemical processes such as attack by acids, sul-phates or alkalis, alkali–aggregate reactions �AAR�, freeze–thawcycles, etc. In RC, apart from above factors, the most seriousdeterioration mechanisms are those leading to corrosion ofreinforcement, resulting in the reduction in the effectivecross-sectional area of reinforcing bars and, ultimately, in thedeterioration of concrete due to spalling of the concrete cover.Reinforcement corrosion occurs only after depassivation due tocarbonation of the surrounding concrete, penetration of chlorideions, or a combination of both. All processes causing deteriorationof the concrete itself or corrosion of the reinforcement involvetransport phenomena through the pores of the concrete. Thesetransport mechanisms include diffusion of such gases as oxygen�O2�, carbon dioxide �CO2�, or sulphur dioxide �SO2� in the gas-eous phase of the pores from the external environment to theinterior regions. It also includes diffusion of aggressive ionsdissolved in the pore water. These can be ions such as C+, H+, orSO4

++ originating from dissociation of dissolved acids or alkaliions, etc.

Various causes of deterioration are neatly summarized byManning and Somerville �1992� in the mathematical form

Carbonation

d = kt

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Corrosion

d = to + kta

where a=0.5 for diffusion control or 1.0 for reaction control

Sulphate attack

d = to + ktb

Alkali–silica reaction

d = to + ktb

where b�1.0

Frost attack

d = to + kt

where d=deterioration; to=initiation time; t=time; andk=constant for a particular mechanism and concrete.

The effect of above referred various degradation mechanismson the structure is studied and in the case of a combination ofmore than one effect, which is usual, weighing of various mecha-nisms is performed and the equation for resultant rate of deterio-ration is formulated.

In Situ Condition Documentation and AssessmentSurvey of Water Tanks

In an effort to determine the nature, extent, and causes of watertank deterioration, a systematic In situ Condition Documentationand Assessment Survey of water tanks of varying capacities be-tween 50 and 2,500 kL constructed around Gwalior, Bhind, Datia,Guna, Dholpur, etc. districts of India has been conducted in whichfield information/data sources are pursued. The investigated watertanks are situated in rural as well as urban areas. A significantcorrelation among structural type, age, and condition rating, etc.has been established

A case-study data collection questionnaire is planned and

Table 1. Damage Scale for In Situ Condition Survey of Water Tanks �S

Conditionrating

Failureclass

Crack with �mm�and steel cross

section areareduction �% age�

01 Imminentfailure

8.0;30 Structural elements hwater, likely to fall a

02 Critical 5.0;20 Structural elements cto repair.

03 Serious 2.0;10 Structural elements cspecial techniques, e

04 Poor 0.50;0 Visible cracks, minodamaged, reinforcem

05 Fair Hairline cracks Surface cracks patteof minor seepage, le

06 Satisfactory — Reinforcement not e

07 Good — Reinforcement not enot well maintained.

08 Very good — Visible excellent consalt deposits, periodi

09 Excellent — Newly constructed aserviceability, detaili

drafted. It is shown in Appendix I. Efforts have been made to

138 / JOURNAL OF PERFORMANCE OF CONSTRUCTED FACILITIES © AS

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trace the history of water tanks during the stages of construction/maintenance. Condition rating �CR� of water tanks is done on ascale of 0–9 prepared on the basis of similar scales proposed byKaminetzky �1985� and Hollis and Gibson �1991�.

A deterioration scale for condition rating of water tanks hasbeen designed with various classification levels as indicated inTable 1.

Analysis and Inferences of Survey Data

The following inferences have been made on the basis of analysisof data collected during the Condition Documentation and As-sessment Survey.1. The summary of structural componentwise deterioration ob-

served is provided in Table 2 and the same is illustrated inFig. 1. Out of 204 tanks, 65% �132� are found to be badlydeteriorated. Among these badly distressed tanks �132�, 18%�24 of 132� are structurally deficient and functionally obso-lete. Around 17% �22 of 132� of badly deteriorated watertanks are in serious condition. Despite this, these tanks arebeing used for storage of water and 7% �9 of 204� tanks arenot being used since their construction, despite their betterstructural condition because of reasons other than structural/durability failure, e.g., nonavailability of water source or userhabits, particularly in rural areas. Around half of these havedeteriorated without any use.

2. The seepage through the water retaining part of the tanks,e.g, cylindrical/conical wall, bottom dome, bottom ringbeams, etc., has been observed in about one fourth �23%� ofthe deteriorated tanks.

3. Corrosion/rusting of reinforcement of various components isobserved as a visible reason of failure in 85% of the deterio-rated tanks. The corrosion/rusting of water retaining compo-nents of water tanks, e.g., top dome, vertical wall, bottomdome, etc., is observed to the extent of 37%. The corrosion/rusting of nonwater retaining components of water tanks,e.g., columns, bracings, stairs, etc., is observed to be as high

to Scales by Kaminetzky �1985� and Hollis and Gibson �1991��

Description

cracked, wide cracks, highly corroded reinforcement, seepage ofe.

, wide cracks, reinforcement corroded, and seepage of water, not likely

, reinforcement corroded, water seepage, repairable with difficulty byoxy grouting, guniting, epoxy treatment etc. for partial capacity use.

ge of water, initiation of corrosion few structural componentsposed in gallery, stairs flight corroded, repairable.

ible spalling/chipping of plaster, reinforcement not exposed, initiationsalt deposits, not well maintained, no immediate repair required.

, no seepage. No leaching salt deposits, not well maintained.

, no seepage, leaching salt deposits. Visibly fair construction quality,

on quality control, reinforcement not exposed, no seepage No leachingell maintained.

xcellent condition with respect to codal provisions of design,d aesthetic of concrete, workmanship, and maintenance.

imilar

eavilyny tim

racked

racked.g., ep

r seepaent ex

rns, visaching

xposed

xposed

structically w

nd in eng, an

as 47%. Therefore, the corrosion rate of nonwater retaining

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structures is not observed to be slow, rather it is slightlyhigher.

4. The main cause of failure is observed to be corrosion ofreinforcement of both water retaining and staging compo-nents, and spalling and chipping of concrete cover as a con-sequence of increased volume of rusted reinforcement.

5. Photographs in Figs. 2 and 3 of some of the water tankssituated at Rajakheda, Nabab Basai, Patai, etc., in the districtof Dholpur of the provincial state of Rajasthan, whose con-dition rating is evaluated around 01 �i.e., imminent failure� to03 �i.e., serious� show badly deteriorated staging, i.e., col-

Table 2. Summary of Report of Distress Condition of Water Tanks �Tot

S. number Name of element

1. Top dome �i� Crack

�ii� Rust

2. Cylindrical wall/conical wall �i� Crack

�ii� Rust

3. Bottom dome/ring beam �i� Crack

�ii� Rust

4. Columns �i� Crack

�ii� Rust

5. Bracings �i� Spall

�ii� Corr

6. Stairs/flights/gallery flights/gallery �i� Spall

�ii� Corr

7. Shaft �i� Spall

�ii� Corr

8. Foundation �i� Settle

Fig. 1. Componentwise deterioration of distressed water tanks �total o

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umns and bracings and stairs flights. Surprisingly, the tankbody has been found to be relatively less deteriorated inmany cases, although corrosion has been initiated in the bot-tom dome and the bottom ring beam reinforcement. The ageof these tanks is only around 28 years �between 25 and 30�.

6. The cross sectional area of reinforcement in columns andbracings is found to have reduced to the extent of more than50% and stirrup rings in columns as well as in bracings havealmost vanished.

7. The foundation settlement is also observed in 5% of thewater tanks.

04 Tanks Are Surveyed and 132 Are Found Badly Deteriorated�

ype of distressNumber of tanks

�out of 132�Age of tanks

�%�

d deterioration of concrete 08 6

steel reinforcement 10 8

d deterioration of concrete 14 11

steel reinforcement 17 13

d deterioration of concrete 13 10

steel reinforcement 23 17

d deterioration of concrete 25 19

steel reinforcement 29 22

ipping of concrete 18 14

f reinforcement 21 16

ipping of concrete 11 8

f reinforcement 12 9

ipping of concrete 04 3

f reinforcement 05 4

sinking, etc. 06 5

water tanks are surveyed and 132 are found in deteriorated condition�

al of 2

T

ing an

ing of

ing an

ing of

ing an

ing of

ing an

ing of

ing, ch

osion o

ing, ch

osion o

ing, ch

osion o

ment,

f 204

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Fig. 2. �a� View of deteriorated bracings and columns of water tankat Rajakheda, District–Dholpur, Rajasthan, India and �b� corrosion inreinforcement and spalling of concrete in bracings in water tank atRajakheda, District–Dholpur, Rajasthan, India

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Fig. 3. �a� Vertical cracks and spalling of concrete in columns atNawab Basai, District–Dholpur, Rajasthan, India and �b� bottomdome with leaching salts deposits and corrosion initiation at Patai,District–Dholpur, Rajasthan, India

Fig. 4. Percentage distribution of water tanks in various conditionratings �CR�

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8. The various surveyed water tanks have been regrouped intovarious CR categories and it is found that highest number oftanks �36%� fall under the fair category whereas 14% arepoor and 12% are very good, and 2% are approaching immi-nent failure. A detailed illustration is shown in Fig. 4.

9. On the basis of the record of condition ratings of water tanksand their age, a correlation has been established betweencondition rating and age of the water tank. The study hasbeen made for water retaining tank body, staging component,and then overall tank rating has been considered. As shownin Fig. 5, the pattern of deterioration rate is uniform up to acertain age, then a hinge point is observed beyond which thedeterioration rate suddenly increases. The staging perfor-mance is a little better than the water tank body.This plot presents a practical bilinear graphical deteriorationmodel for water tanks. The data related to various stages ofdeterioration are tabulated in Table 3.

10. The average service life of water tanks, when first visibledeterioration �e.g., cracks, etc.� were observed and mainte-nance was carried out, has been observed as 12 years and theaverage total service life with routine/minor repair and main-tenance has been observed as around 46 years. However, insome cases the water tanks become structurally deficient inservice life of around 25 years.

Table 3. Stages of Graphical Deterioration Model

Tankcomponent

Rating atage zero

Average conditionpoints/year�first slope�

Hin

Water retainingbody �B�

8.5 0.091

Staging �A� 9.3 0.059

For overalltank �C�

9.0 0.071

Fig. 5. Deterioratio

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Conclusion

In the present research, in situ condition assessment of watertanks structures situated in the central part of India has been donebased on a specially designed condition—documentation ques-tionnaire. A survey data of condition documentation has been ana-lyzed and presented. The causes and extent of deterioration havebeen determined. Deterioration of various components of watertank structures along with their condition ratings is evaluatedbased on the condition survey data. A practical bilinear graphicaldeterioration model based on condition rating with time hasevolved. Efforts have been made to evaluate the average servicelife of water tank structures at the first visible cracks and theaverage total service life for staging and water retaining compo-nent and overall tank structure.

Acknowledgment

The writers are grateful to All India Council of Technical Educa-tion, New Delhi for providing financial assistance in the form ofthe R&D Project entitled “Knowledge based expert system forevaluation of residual service life and repair strategies for con-crete structures.”

nt ages�

Hinge pointrating

Average conditionpoint/year

�second slope�

Age atrating zero

�years�

5.5 0.55 43.0

7.0 0.78 48.0

6.5 0.59 46.0

el for water tanks

ge poi�year

33.0

39

36

n mod

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Appendix

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