CIVL 513 - Project No. 1 - 13Mar30 - Complete Report [Final]

7/29/2019 CIVL 513 - Project No. 1 - 13Mar30 - Complete Report [Final]

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CIVL 513 TERM PROJECT #1:ESTIMATING DEFLECTIONS OFREINFORCEDCONCRETESLABS

UBC COURSE:CIVL 513 CONCRETE STRUCTURES

SUBMITTED TO:Dr. Perry Adebar, P.Eng.Professor of Structural EngineeringUBC Department of Civil Engineering

SUBMITTED BY:Brandon Paxton (44770089)

SUBMITTED:April 3, 2013


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CIVL 513 Project #1 - Estimating Deflections of Reinforced Concrete Slabs P a g e | 2

Brandon Paxton (44770089) March 18, 2013

Table of Contents

TABLE OF CONTENTS ....................................................................................................................................... 2

1.0 EXECUTIVE SUMMARY ........................................................................................................................... 4

2.0 INTRODUCTION ......................................................................................................................................... 5

2.1 PURPOSE .................................................................................................................................................. 5 2.2 SIGNIFICANCE ......................................................................................................................................... 5

2.3 SCOPE ...................................................................................................................................................... 6

2.3.1 Part 1 Initial Predictions ............................................................................................................... 6

2.3.2 Part 2 Improved Estimates ............................................................................................................ 6

2.4 DESCRIPTION OF EXPERIMENT ................................................................................................................ 6

2.4.1 Description of Specimens ........... .......... ........... .......... ........... .......... ........... ........... .......... ........... ....... 7

2.4.2 Description of Testing ........... .......... ........... .......... ........... .......... ........... .......... ........... .......... ........... .. 9

3.0 DATA & ANALYSIS .................................................................................................................................. 10

3.1 MATERIAL PROPERTIES ......................................................................................................................... 10

3.1.1 Concrete Compressive Strength ..................................................................................................... 10

3.1.2 Concrete Modulus of Rupture ........................................................................................................ 10

3.1.3 Concrete Modulus of Elasticity ...................................................................................................... 11

3.1.4 Creep Parameters .......................................................................................................................... 13

3.1.5 Effects of Shrinkage ......... ........... .......... ........... .......... ........... .......... ........... ........... .......... ........... ..... 14

3.2 SECTION PROPERTIES ............................................................................................................................ 14

3.2.1 Basic Properties ........... .......... ........... .......... ........... .......... ........... .......... ........... ........... .......... ......... 14

3.2.2 Moment-Curvature Models (Detailed Immediate Approach) ......... ........... ........... .......... ........... ..... 15

3.2.3 Effective Stiffnesses (Simplified Immediate Approach) .......... .......... ........... .......... ........... .......... .... 16

3.3 INITIAL PREDICTIONS (PART 1) ............................................................................................................. 17

3.3.1 Immediate Deflections ........... .......... ........... .......... ........... .......... ........... .......... ........... .......... ........... 17

3.3.2 Long Term Deflections .......... .......... ........... .......... ........... .......... ........... .......... ........... .......... ........... 18

3.3.3 Comparison of Detailed and Simplified Predictions ...................................................................... 19

3.4 EXPERIMENTAL DATA ........................................................................................................................... 20

3.4.1 Measured Deflections .......... .......... ........... .......... ........... .......... ........... ........... .......... ........... .......... .. 20

3.4.2 Comparison to Initial Predictions .................................................................................................. 21

3.5 IMPROVED ESTIMATES (PART 2) ............................................................................................................ 22 3.5.1 Immediate Deflections ........... .......... ........... .......... ........... .......... ........... .......... ........... .......... ........... 22


4.0 SUMMARY OF RESULTS & DISCUSSION ........... .......... ........... .......... ........... ........... .......... ........... ..... 34

4.1 INITIAL PREDICTIONS ............................................................................................................................ 34



4.2 IMPROVED ESTIMATES .......................................................................................................................... 35


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5.0 CONCLUSIONS & RECOMMENDATIONS .......................................................................................... 38

5.1 THEORETICAL CONCLUSIONS ................................................................................................................ 38

5.2 PRACTICAL CONCLUSIONS .................................................................................................................... 38

5.3 RECOMMENDATIONS FOR RIGOROUS DEFLECTION ESTIMATES ............................................................. 38

5.4 RECOMMENDATIONS FOR DESIGN PRACTICE ......................................................................................... 39

6.0 CLOSING COMMENTS ............................................................................................................................ 40

6.1 POTENTIAL IMPROVEMENTS TO ANALYSES ........................................................................................... 40

6.2 CLOSURE ............................................................................................................................................... 40

APPENDICES

APPENDIX A: P ART 1 INITIAL PREDICTIONS (ORIGINAL SUBMISSION )APPENDIX B: DESCRIPTION OF TESTING (ORIGINAL DOCUMENT )APPENDIX C: COMMON CONCRETE MATERIAL PROPERTIES APPENDIX D: IMPROVED DETAILED IMMEDIATE ESTIMATE SUPPORTING DATA APPENDIX E: IMPROVED SIMPLIFIED IMMEDIATE ESTIMATE SUPPORTING DATA APPENDIX F: IMPROVED DETAILED LONG TERM ESTIMATE SUPPORTING DATA APPENDIX G: IMPROVED SIMPLIFIED LONG TERM ESTIMATE SUPPORTING DATA


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1.0 EXECUTIVESUMMARY

The Canadian standard for structural design of concrete, CSA A23.3-04 (A23.3) specifies limits on span-to-depth ratios for reinforced concrete slabs, below which the designer need not calculate deflections;for flat plate slabs with Grade 400 reinforcement (the only grade in common use in Canada), Cl. 13.2.4

gives a maximum span-to-depth of 30. However, common practice in Vancouver is to design such slabswith span-to-depth values of 36 and check deflection requirements, as given in Table 9.3. Thus, theability to accurately and efficiently estimate deflections is an essential part of design.

The effects of construction processes on deflection are a critical factor in accurately estimatingdeflection [1]. However, structural designers typically have little knowledge or control of theseprocesses. Moreover, recent studies [2, 3] have shown that common code provisions underestimatedeflections in lightly reinforced members, particularly in the lightly cracked range, due to anoverestimation of the tension stiffening effect; floor slabs are lightly reinforced and often fall into thisload range during construction and in service. The early age at loading typical of short formwork cycles

in high-rise building construction can also lead to increased creep deflections.

This report presents the results of an experimental study conducted at the University of British Columbiaby Adebar [4] on the deflections (both immediate and creep) of reinforced concrete slabs accounting fortypical construction practices, including early-age construction loading. Also presented, is thedevelopment and improvement by the author of detailed and simplified methods of estimatingdeflections. As will be seen herein, an accurate estimate of the immediate deflections is crucial indetermining the short term and long term (creep) deflections.

The report shows that use of the CSA A23.3-04 deflection provisions without the 50% reduction in themodulus of rupture (f r) now specified for both one-way and two-way members or consideration of construction loading leads to a severe underestimate of deflections, both short and long term. Whenthe appropriate A23.3 provisions are followed, the estimates are conservative in terms of immediate anddifferential deflections, and can be reasonable in the long term.

Nonetheless, the aforementioned importance of immediate deflections, construction effects, and theresulting long term deflections may require a more accurate assessment in some situations. Using theobserved data, both rigorous and simplified deflection prediction methodologies are developed andpresented herein, such that they are in substantial agreement with each other and with the measuredresults. The report also highlights the fact that errors in the moment-curvature relationship can lead tounacceptable results when using a mechanics-based approach to computing deflections.

Overall, the report finds that the expression proposed by Bischoff [3] for computing effective momentsof inertia is more appropriate for the specimens investigated; that the apparent ultimate creepcoefficient (C u) for use in rigorous creep calculations for these specimens, subject to early age loading, isquite high (3.5 over the A23.3 average value of 2.35); and that in order to accurate calculatedifferential deflections using simplified methods, a correction term must be added to basic codeprovisions to account for the additional cracking and creep due to construction. These terms aredeveloped and presented herein.


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2.0 INTRODUCTION

2.1 Purpose

This report presents the results of an experimental study conducted at the University of British Columbia

by Adebar [4] on the deflections (both immediate and creep) of reinforced concrete slabs accounting fortypical construction practices, including early age construction loading. The intent was to generate bothsimplified design office and more rigorous, detailed predictions, and to subsequently make use of theexperimental data to improve upon both methods

2.2 Significance

The Canadian standard for structural design of concrete, CSA A23.3-04 (A23.3) specifies limits on span-to-depth ratios for reinforced concrete slabs, below which the designer need not calculate deflections;for flat plate slabs with Grade 400 reinforcement (the only grade in common use in Canada), Cl. 13.2.4gives a maximum span-to-depth of 30. However, common practice in Vancouver is to design such slabswith span-to-depth values of 36 and check deflection requirements, as given in Table 9.3. Thus, theability to accurately and efficiently estimate deflections is an essential part of design.

Often, the governing deflections will be differential deflections, occurring after the attachment of structural elements. The reason for this becomes clear when one considers that the aforementionedlong term limits are 75-150% of the immediate limits, while the long term multipliers alone specifiedby Cl. 9.8.2.5 typically result in a 50-100% increase over short term values (assuming non-structuralcomponents are attached between 1 and 3 months after concrete placement) and future live loaddeflections are on the same order of the dead load deflections. Design Example 6.2 from the CACConcrete Design Handbook provides a good example. The overall flatness of the floor slabs has alsobecome increasingly important with the growing popularity of rigid flooring finishes and relatedtightening of construction specifications. As a result, the ability to accurately estimate deflections hasbecome increasingly important.

In estimating the deflections, a key issue is the effect of construction loading. This loading, primarily dueto short formwork/reshoring cycles in multi-story building construction, can be over two times the slabdead load [1]. This large, early-age loading causes increased cracking and creep, sometimes beyond thedesign values. Recent studies found that simplified procedures of A23.3 for computing deflections wereunconservative for lightly reinforced flexural elements, such as floor slabs [2, 3]. In 2009, CSA A23.3

adopted an emergency update for beams and one-way slabs, calling for the modulus of rupture (f r) valueused in computing deflections to be reduced by 50% for beams and one-way slabs (similar to what wasalready present in the Clause 13 for two-way slabs), effectively increasing code estimates for bothimmediate and long term deflections. Overall, a study into the deflection behavior of slabs and theaccuracy of current code provisions is thought to be valuable from both a research and design practiceperspective. The primary benefits are:

An improved understanding of the accuracy of predictions for immediate deflections An improved understanding of the actual deflection behavior of the slabs over time Improvements to and validation of simplified methods for predicting deflections


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2.3 Scope

In accordance with the project requirements, several estimates of the deflections over time were made.As aforementioned, both simplified design office and more rigorous, detailed predictions were madeand these were subsequently revised to reflect the observed data. Ultimately, four distinct estimates arepresented, each of which address immediate and long term deflections:

Detailed Approach Initial Predictions Simplified (Design Office) Approach Initial Predictions Detailed Approach Improved Estimates Simplified (Design Office) Approach Improved Estimates

Throughout the report, the Detailed Approach is presented first, partly because the results of thedetailed estimates provide insight into the improvements made to the simplified methods. In developingthe improved estimates, combinations of several adjustments were considered until all three results(simplified, detailed, and measured) were in substantial agreement.

2.3.1 Part 1 Initial Predictions Part one of this project included initial predictions using both the Simplified and Detailedapproaches and was completed as an earlier course requirement. It was submitted January 28,2013 and is included in this report as Appendix A. It should be noted that, as per the courserequirements, only the effects of the initial dead load and sustained load were considered. Theeffects of construction loading (both cracking and additional creep) were not considered in Part 1.In Part 2, both estimates are revised to include the effects of construction loading. Section 3.3describes the analysis and results of the Initial Predictions.

2.3.2 Part 2 Improved Estimates

In Part 2, the experimental results were used to improve upon both the Simplified and Detailedestimates. The effects of construction loading were considered for both methods and the estimateswere refined until all results were in substantial agreement with the experimental data. Part 2 wasnot submitted earlier and is the focus of this report. Section 3.5 describes the analysis and results of the Improved Estimates.

2.4 Description of Experiment

The testing is aimed at assessing the deflections of typical Canadian two-way building slabs. As such, theidealized prototype structure is a strip of a two-way slab, of length (L) 24ft and height (h) 8in; this yields aspan-to-depth ratio of 36, which is common practice in Vancouver [4]. It should be noted that thisexceeds the prescriptive CSA A23.3 (A23.3) limit of 30, and thus calculations would be required todemonstrate that the resulting deflections are within acceptable limits, as specified by A23.3.

To make efficient use of space and materials, balanced cantilever test specimens representing one-quarter of the full-span member described above were used. The original description by Adebar(Appendix C) provides more detailed discussion on the measures taken to effectively replicate a quarter-span of a slab strip and the related advantages and limitations; this document was provided as part of


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the project background is included as Appendix B. However, two consequences that should behighlighted are that using a cantilever allowed for accurate determination of the bending momentthroughout the specimen (as its a determinate structure), but that the specimens lacked the axialrestraint present in a real structure; this would reduce the cracking due to shrinkage. The design aidssection of the CAC Concrete Design Handbook (Chapter 6) notes that the use of a 50% reduction in the

modulus of rupture (f r) is intended to account for the effect of shrinkage restraint cracking occurring intwo-way slabs [1].

2.4.1 Description of Specimens

2.4.1.1 Material Properties

The reinforcement used was grade 400 deformed bars to CSA standard G30.18-09, as is typical fornew reinforced concrete buildings in Canada. The specified 28-day cylinder strength for theconcrete was 25MPa. Typical specified strengths for building slabs in Vancouver are 30 or 35MPa;the lower value of 25MPa was specified with the intent of achieving an actual 28-day cylinder

strength of 30MPa. Compression tests performed on the samples yielded an average 28-daystrength of 30.5MPa for the standard (moist-cured) cylinders. Tests were also performed at 3 and 7days on field-cured samples, for use in calculating early age properties at loading. Table 1summarizes the results of the cylinder tests. Modulus of rupture tests were also performed; theresults are presented and discussed in the original test description (see Appendix B). In all thedeflection calculations presented herein, the actual strength (i.e. field-cured at the appropriate age)were used. Calculated material properties are presented in Section 3.1.

Table 1: Cylinder Compression Test Results

Age CuringMeasured Strength [MPa]

Test #1 Test #2 Test #3 Average3 Field 18.0 17.4 19.0 18.17 Field 26.2 24.8 24.9 25.3

28 Field 27.1 26.6 26.8 26.928 Moist 30.3 30.7 30.5 30.5

2.4.1.2 Physical Description

In total, four unique specimens were constructed and tested. Each specimen was 20 (508mm)wide by 8 (203mm) deep and 72 (1830mm) long; the tip deflections were measured 2 from theend of the cantilever. As aforementioned, the specimens were intended to represent a quarter-

span of a slab strip. Thus, the specimens were subjected to self-weight, plus additional point loadsto simulate the missing portion of the slab (P 1), intermittent construction loading (P 2, plus increasedP1) and, eventually sustained loading (three smaller point loads). Refer to Appendix B for a moredetailed description. Figure 1 provides an elevation sketch of the specimen and loading.


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Each specimen was unique in terms of the reinforcement area and clear cover. Two specimenscontained 2-15M bars ( =0.39%), while two contained 2-20M bars ( =0.58%).Two different clearcovers were also used: 1 (25mm) and 1 (40mm). As such, the four unique specimens were:

2-15M, with 25mm cover to longitudinal reinforcement (R15-C25) 2-15M with 40mm cover to longitudinal reinforcement (R15-C40) 2-20M with 25mm cover to longitudinal reinforcement (R20-C25) 2-20M with 40mm cover to longitudinal reinforcement (R20-C40)

The test specimens were also provided with reinforcement perpendicular to the principallongitudinal bars, similar to that of a two-way slab; the perpendicular bars were placed either aboveor below the principal reinforcement, accounting for the aforementioned variation in cover. Figure2 below provides a cross section illustrating the various test specimens.

Figure 1: Elevation Sketch of Typical Test SpecimenSource: Adebar (not yet published) Estimating Building Slab Deflections Accounting For Typical Construction Practice: An Experimental Study

Figure 2: Cross Section Sketch of Test SpecimensSource: Adebar (not yet published) Estimating Building Slab Deflections Accounting For Typical Construction Practice: An Experimental Study


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2.4.2 Description of Testing

As noted above, the specimens were subjected to three different loadings representing: self-weight,construction loading, and sustained loading. The formwork was stripped after approximately threedays after casting, subjecting the specimens to self-weight. Three cycles of construction loadingbetween days 5 and 17 were then applied, representing a formwork sequence with formwork at

one level, plus two levels of reshoring. The total duration of the construction loading was 7 days.Figure 3 shows the load history.

As can be seen, loading was greatest during the construction period, about 50% greater than self-weight alone and 10% greater than the sustained load. This is reflective of typical constructionpractices, although in some cases, the construction loading could be even greater. For lightlyreinforced two-way slabs, this construction loading will generally result in a notable increase incracking and, thus, a decrease in the effective moment of inertia (I e) of the slab for all future loads.In lieu of a detailed knowledge of the construction loads, the CAC Handbook (Chapter 6)recommends that a construction load of 2.0 to 2.2 times the slab dead load be used in estimating I e [1]. However, the use of a construction loading of 2.0 (rather than about 1.5) times the self-weightwould only serve to suppress the differences in I e of the various specimens, because tension

stiffening (represented by the I e expressions) is most sensitive to differences in the section when theapplied moment is nearest to the cracking moment [2]; this is the primary benefit of using theconstruction loading presented herein.

Figure 3: Loading History For Slab SpecimensSource: Adebar (not yet published) Estimating Building Slab Deflections Accounting For Typical Construction Practice: An Experimental Study


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3.0 DATA & ANALYSIS

The following section presents the experimental data used in the study, the analyses undertaken, andthe calculated/assumed material and section properties used in the analyses.

3.1 Material PropertiesIn order to produce initial predictions of deflections, various material properties had to be assigned. Thevalues presented herein were determined from the initial data and were consistent across all methods,both initial and improved.

3.1.1 Concrete Compressive Strength

As even the initial predictions attempted to use material properties appropriate to the age of loading, values of the standard cylinder strength were required at various times. The data onlyprovided values at 3, 7, and 28 days. As such, additional values from days 2 to 35 were interpolated.

It was assumed that the concrete reached a strength of 27MPa at day 35 and that there was noappreciable strength gain thereafter. Figure 4 shows the strength gain over time, with selectedlabels. Refer to Appendix D for the complete set of values used.

3.1.2 Concrete Modulus of Rupture

Tests completed by Adebar as part of the experimental study confirmed that the modulus of rupture(f r) of the concrete was directly related to the cylinder compressive strength by the well-knownequation:

= 0.6

Figure 4: Cylinder Compressive Strength Gain with Time


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Where f r* and f c

* are the actual values, as opposed to the 28-day values, which are commonly used

in design; Figure 5 below shows the calculated values for f r*, as well as the f c

* for reference. It isworth noting that the modulus of rupture converges to its ultimate value more quickly than does

the cylinder strength, due to the square root in the equation. For example, the 3-day value of f r* is

82% of the 35-day value, while the 3-day f c* is only 67% of its 35-day value. The most critical values

of f r*, were those at 3 days (2.55MPa) and 5 days (2.91MPa). As will be seen in the followingsections, these are the values that were used to calculate the governing effective moment of inertia(i.e. if the previously cracked moment of inertia is smaller, it is used in the applicable deflectioncalculations).

3.1.3 Concrete Modulus of Elasticity

Various forms of the modulus of elasticity were required throughout the analyses: the secant

modulus (E c) was used in computing the deflections, while the tangent modulus (E c,t ) was used incomputing the cracking curvature ( cr).

3.1.3.1 Secant Modulus

The secant modulus is the form commonly used in design and various expressions have beenspecified by design codes. The following expressions were considered in making the initialpredictions:

= .0.043 (2 A23.3-1984)= 4500 (3 A23.3-04)= 3300 +6900 () . (4 A23.3-04)

Figure 5: Modulus of Rupture Values (with f c* for comparison)


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All three equations represent the secant modulus at a compressive stress 40% of f c (or rather, f c* in

this case); this assumption was reasonably fulfilled in the experimental study, as the self-weight and

construction loading resulted in maximum concrete stresses of about 25%-50% f c*.

Equation (2), from the 1984 edition of A23.3, simplified to about 5000*f c for the measured density

of the concrete used in this study ( c=2396kg/m3

). CSA A23.3 notes that Equation (3) is valid onlyfor fc between 20MPa and 40MPa; since the 3-day strength is 18.1MPa, this equation was deemedinappropriate for the study. Equation (3) was essentially derived from Equation (2) by assuming atypical density and applying a reduction of 10%, and therefore was also deemed inappropriate. Itshould be noted that the 10% reduction was adopted in CSA A23.3-94, for which the commentary toCl 8.6.2 notes that the 10% reduction reflects experience with Canadian concrete . As such,Equation (4) was deemed to be most appropriate and was used throughout the study.

Values of the secant modulus were calculated at the same ages as was f r. Figure 6 shows E c as a

function of time (again with f c* for reference). Note that although it gives a value at 0 days, this is

not of significance because deflections do not occur until day 3. Refer to Appendix C for exactvalues.

3.1.3.2 Tangent Modulus

The tangent modulus (E c,t) is the appropriate modulus to use in tension, up until cracking. E c,t was

taken to be 10% greater than E c [5] . It was only used to compute curvatures at initial cracking (i.e.at 3 days); the value used was 24,500MPa (rounded from 1.1*22257MPa).

Figure 6: Secant Modulus Values (with f c* for comparison)


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3.1.4 Creep Parameters

3.1.4.1 Detailed Approach

For the detailed approach to creep, the method outlined in Chapter 1 of the CAC Handbook(henceforth referred to as the C t Method) was used to reduce the concrete secant modulus to theappropriate, time-dependent effective value (E c,eff ). A number of correction factors are specifiedby A23.3 in order to match the given conditions with the standardized conditions of A23.3. Thefollowing equations were used:

The value C u from Equation (6) is the most important factor in computing the creep deflections(aside perhaps from immediate deflection, which is amplified by the creep calculations); itrepresents the total amount of creep that will occur long term. The CAC handbook notes that thisvalue varies from 1.30 to 4.15 and suggests an average value of 2.35; this average value was usedfor the initial (detailed) predictions, and was a key adjustment factor in the improved predictions.The CAC handbook notes that Equations (6) and (7) were developed for sustained compressivestress not exceeding 50% of the concrete strength. Both these requirements were met in this study.Finally, only limited values of correction factors are provided by the Handbook (Table 1.2) and noinformation was provided about some of the necessary parameters. As such, various interpolationsand assumptions were made. Calculations and the parameters used are provided in the respectiveAppendices for each section; however, the common factors can briefly be summarized as follows:

Q a was taken as the appropriate value for each loading stage (eg. 1.14 @ 3 days) Q h was taken as 0.87 (assumed RH of 60%)

Q f was taken as 1.0 (assumed 50% fine aggregate) Q r was taken as 0.82 (defined for the given volume and surface area of the specimen) Q s was taken as 1.08 (assumed 100mm slump) Q v was taken as 1.0 (assumed air content 6% or less)

3.1.4.2 Simplified Approach

A similar concept is applied to determine creep deflections in the simplified approach: a parameters (similar to C t) is specified by Clause 9.8.2.5 for various ages after loading, eliminating the

= . . (6)Where:t time after loading [days]Cu ultimate creep coefficientQcr composite correction factor (see Eqn 7)

, = /(1+) (5)Where:Ec,eff effective (creep-adjusted) secant modulusEc short term secant modulus (as per section 3.1.3.1)C t creep coefficient (as per Eqn 6)

= (7)Where:Qa age at loading correctionQh relative humidity correctionQ f aggregate correctionQ r surface area correctionQs slump correctionQv air content correction


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consideration of the various correction factors or the variability in the ultimate creep coefficient(Cu). Table 2 (based on Figure N9.8.2.6) shows the specified values:

Table 2: The s-Parameter (Cl. 9.8.2.5)Duration of Sustained Load sFive years or more 2.012 months 1.46 months 1.23 months 1.01 month 0.5

3.1.5 Effects of Shrinkage

In a real structure, the effects of shrinkage would be expected to be a significant factor in thedeflections. This is recognized through the 50% reduction if f r used in calculation the crackingmoment, as specified by A23.3, Clauses 9 and 13. However, shrinkage effects are not expected tobe significant in this study for the following reasons:

The axial restraint present in a real slab is not reflected in the cantilever specimens The curing conditions were presumably not such that shrinkage would be exacerbated (ie.

no wind, extreme heat, low humidity) The concrete mix is design presumably did not contain silica fume (expensive and not

common practice) or unusually high cement content.

As such, the effects of shrinkage were neglected throughout the study. As will be seen in comparingthe experimental data and the deflection estimates, this appears to have been a reasonableassumption. Accordingly, the 50% reduction in f r was not applied to the initial predictions(discussed later), but various reductions in f r were applied in some of the improved estimates;

however the need for this correction was attributed to inaccuracies in the code specified equationsfor immediate deflections for the members in question (discussed later).

3.2 Section Properties

3.2.1 Basic Properties

Section properties such as the gross and cracked moment of inertia were calculated for each of thespecimen; the difference between the gross and net area of concrete was neglected. These basicvalues were consistent for both the simplified and detailed estimates.

Table 3: Basic Section PropertiesSpecimen (%) I g [mm

4] Icr [mm4]

R15-C25 0.39 354*10 6 71.4*10 6 R15-C40 0.39 354*10 6 58.4*10 6 R20-C25 0.58 354*10 6 96.0*10 6 R20-C40 0.58 354*10 6 80.5*10 6


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The 3-day value of E c was used in calculating the cracked moments of inertia, which were then inturn used to calculated effective moments of inertia at all ages (eg. even for construction loadingapplied at day 5). Although the time-appropriate E c should be used, the effective moments of inertia were found to be somewhat insensitive to the possible variation in E c and CSA A23.3 notesthat the expressions for E c typically yield values between 80% and 120% of the true values. The

problem is further complicated by the fact that the specimens have previously been cracked whenconstruction/sustained loading is applied (this is accounted for in the expressions for I e). As such,the error associated with using the 3-day E c for calculating all effective moments of inertia wasdeemed to be acceptable.

3.2.2 Moment-Curvature Models (Detailed Immediate Approach)

Moment curvature models were developed for use in the detailed initial predictions (Section3.3.1.1). A tri-linear model was used to define the M- curves: 1) uncracked 2) cracked with tensionstiffening 3) yielding. The values at yielding were extracted from RESPONSE2000, which accountsfor tension stiffening. Although the calculated tension stiffening from RESPONSE is quite

conservative at the yield moment [5], the applied moments in the study are much closer to crackingthan to yielding, and thus, the conservatism is reduced (although it will be seen later that this stilloverestimated the tension stiffening). Table 4 provides the parameters used in defining the curvesand Figure 7 shows a typical curve. The remaining plots are the same shape, with the yield pointshifted.

Table 4: Moment-Curvature ParametersSpecimen M cr [kN-m] cr [rad/km] M y [kN-m] y [rad/km]R15-C25 8.89 1.02 25.1 15.4R15-C40 8.89 1.02 22.7 16.9

R20-C25 8.89 1.02 36.6 16.8R20-C40 8.89 1.02 33.0 18.6

Figure 7: Typical M- Curve Used in Detailed Prediction


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The values at cracking were determined from simple linear elastic theory using I g and E c,t (not thesecant modulus). This neglects the small differences in transformed moments of inertia, as well asthe effects of the transverse (secondary) bars. Based on the data provided (Appendix B), thisappears to have been a reasonable assumption, as three of the four specimens appear to havecracked at about 9kNm. It is postulated that the one specimen that cracked at a higher moment

(R20-C25 at over 10kNm) is due to variation in the tensile strength at the location of the crack.

3.2.3 Effective Stiffnesses (Simplified Immediate Approach)

In the Simplified approach to immediate deflection, effective moment of inertia (I e) expressions areused, in conjunction with the secant modulus and elastic beam equations, to compute deflections.Cl 9.8.2.6 of A23.3 specifies the use of the Branson (1965) [6] equation:

A23.3 also specifies averaging equations for continuous and one-end-continuous beams. In the caseof a cantilever, however, no averaging is required. The Branson equation aims to account for thetension stiffening phenomena (accounted for in the M- curves in the Detailed approach).

Recent studies have noted that the Branson equation overestimates the tension stiffening effect inlightly reinforced members, as it was developed for beams, with =1-2% [2, 3]. Bischoff (2005)proposed the following:

For the purposes of this study, the gross moment of inertia (Ig) will be used in lieu of the uncracked(transformed) moment of inertia (I uncr ). The two key differences are that this equation averages theflexibilities rather than stiffnesses and that the uncracked contribution (representing tensionstiffening) depends only on the ratio M cr/M a, not on the difference between I g and I cr, as per theBranson equation [2]. The Bischoff equation is used in the Improved Estimates.

It should be noted that various values of M cr have been used in these equations, due to differentspecified values of f r. Until 2009, A23.3 specified the full value of f r for one-way member (Clause 9)

and f r/2 for two-way slabs (Clause 13). A23.3 now specifies f r/2 for both cases. Different values of Ma are also investigated in this study; this is because the Initial Predictions (Section 3.3) neglectconstruction load effects (as per project instructions), while the Improved Estimates (Section 3.5)

= + () (8)Where:Icr fully cracked moment of inertia

Ig gross moment of inertiaM cr section cracking moment (see Eqn 9)M a maximum moment (current or previous)

= / (9)Where:f r modulus of ruptureI

g gross moment of inertia

yt distance from centroid to extreme tension fibre

= ( ) + 1 (10)Where:Icr fully cracked moment of inertiaIuncr uncracked moment of inertiaM cr section cracking moment (see Eqn 9)M a maximum moment (current or previous)


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account for construction and, thus, M a becomes the moment from construction for most of the loadhistory.

3.3 Initial Predictions (Part 1)

3.3.1 Immediate Deflections

3.3.1.1 Initial Detailed Approach

In the detailed approach, immediate deflections were predicted from first principles by applying themoment-area theory and numerically integrating the curvature diagram. The deflection wascalculated for each specimen for self-weight and for the sustained load. As per instructions for Part1, construction loading was neglected. Table 5 summarizes the results and Figure 8 shows typicalplots. Refer to the original submission (Appendix A) for detailed plots for each specimen.

Table 5: Immediate Deflection Predictions Detailed Approach

Specimen Tip Deflection, tip [mm]Self-Weight Self-Weight+Sustained Loading

R15-C25 1.15 2.44R15-C40 1.20 2.81R20-C25 1.11 1.97R20-C40 1.13 2.20

3.3.1.2 Initial Simplified ApproachThe initial prediction for the simplified approach makes use of the effective moment of inertia (I e)expressions in Clause 9.8.2.3, the secant modulus, and elastic deflection equations to computedeflections. The effective moment of inertia expression is from Branson (1965) (Equation 8, Section

3.2.3). In calculating the cracking moment for the initial predictions, no reduction in f r is appliedbecause there is axial restraint (except for the negligible amount due to reinforcement) and becauseuntil 2009 (when A23.3 adopted the emergency update to clause 9), this was common practice for

one-way members (i.e. a typical design office approach). The values of E c used were the 3-day and

Figure 8: Typical Curvature Diagram and Deflected Shape (R15-C25 Shown)


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35-day values, respectively; again, the construction loading was neglected, as per the project

instructions. Table 6 presents the calculated values of I e (as a percentage of I g) and the resultingdeflections. Refer to the Part 1 submission (Appendix A) for further comments.

Table 6: Immediate Tip Deflections Simplified Prediction

SpecimenSelf-Weight Self-Weight+Sustained

Ie/I g [%] tip [mm] I e/I g [%] tip [mm] R15-C25 74% 1.56 43% 3.01R15-C40 73% 1.58 40% 3.18R20-C25 76% 1.52 47% 2.73R20-C40 75% 1.55 44% 2.89

3.3.2 Long Term Deflections


In the Detailed approach, the initial predictions made use of the aforementioned creep parameters

(Section 3.1.4.1) to calculate the creep deflections throughout the entire loading history. This wasaccomplished by amplifying the incremental immediate deflections due to self-weight and self-

weight+sustained loading using the appropriate C t and E c,eff values for each load and summing theresults. C u was taken as the A23.3 recommended average value of 2.35. Again, the constructionload was neglected, as per instructions. Table 7 presents the predicted tip deflections at days 3, 35,151, and 180 and Figure 9 provides a typical plot generated in the Detailed approach. It should benoted that the immediate deflections from Simplified approach (Section 3.3.1.2) were used in orderto make a fair comparison between the Detailed and Simplified creep results. Refer to the Partsubmission (Appendix A) for further comments and all relevant plots.

Table 7: Long Term Tip Deflections Detailed PredictionSpecimen

Deflections [mm]Day 3 Day 34 Day 151 Day 180

R15-C25 1.56 2.99 6.53 6.67R15-C40 1.58 3.03 6.87 7.02R20-C25 1.52 2.91 5.96 6.08R20-C40 1.55 2.96 6.30 6.43


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3.3.2.2 Initial Simplified Approach

In the Simplified approach, the initial predictions applied the code-specified factors (see Section3.1.4.2) to the simplified immediate deflections (Section 3.3.1.2) to determine the deflections atdays 3, 30, 90, and 180 (the values for day 30 were not included in the original submission). Incontrast to the Detailed method, this method provides the deflections at only a few key points,which would be typical of a design office approach. Table 8 presents the results. Refer to theoriginal submission (Appendix A) for further comments.

Table 8: Long Term Tip Deflections Simplified Prediction

SpecimenDeflections [mm]

Day 3 Day 30 Day 90 Day 180 R15-C25 1.56 2.34 6.91 7.60R15-C40 1.58 2.37 7.31 8.03R20-C25 1.52 2.28 6.26 6.88R20-C40 1.55 2.33 6.64 7.31

3.3.3 Comparison of Detailed and Simplified Predictions

3.3.3.1 Comparison of Immediate Deflections

In comparing the immediate deflections, the detailed approach of integrating curvatures appears toyield values about 30% lower than the Simplified approach. Table 9 provides a comparison of the

Detailed and Simplified initial deflections (from Table 5 & Table 6 ).

Figure 9: Typical Deflection vs. Time Plot from Detailed Approach


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Table 9: Comparison of Immediate Deflections Initial Predictions

SpecimenTip Deflections at Day 3 [mm]

Simplified Detailed % Dif (=dif/avg*100%) R15-C25 1.56 1.15 30.3%R15-C40 1.58 1.20 27.3%R20-C25 1.52 1.11 31.2%R20-C40 1.55 1.13 31.3%

It should be recalled that no reduction in f r was applied in either approach. This difference is at

least partly explained by the fact that the simplified approach used the secant modulus (E c), while asubstantial portion of the M- curve of the detailed procedure is defined by the tangent modulus

(Ec,t). The secant modulus is defined at 40% f c, but at this loading (M=10.11kNm), the maximumconcrete compressive stress is only about 25% of f c. It could also be explained by inaccuracies inthe expression for the effective moment of inertia. Several sources note that the Branson equationis notably less reliable near the cracking moment, as it the response here is significantly sensitive tospatial variations in tensile strength, flaws, etc [7].

3.3.3.2 Comparison of Long Term Deflections

The Detailed and Simplified approaches to creep deflections are much more consistent. The totaldeflections at day 180 are all within about 13%. The simplified results appear to be conservative;however, it should be borne in mind that is it often the differential (long term minus short term)deflections that govern a problem. As aforementioned, the Simplified immediate deflections (Table6) were used as the basis for both the Detailed and Simplified creep calculations in order to make ameaningful comparison. Table 10 provides a comparison (from Table 7 and Table 8).

Table 10: Comparison of Long Term Deflections Initial Predictions

SpecimenTip Deflections at Day 180 [mm]

Simplified Detailed % Dif (=dif/avg*100%) R15-C25 7.60 6.67 13.0%R15-C40 8.03 7.02 13.4%R20-C25 6.88 6.08 12.3%R20-C40 7.31 6.43 12.8%

As will be seen shortly, however, all the predictions thus far (immediate/long term &detailed/simplified) will be greatly exceeded by the experimental results and require improvements.

3.4 Experimental Data

3.4.1 Measured Deflections

The four specimens were subjected to the aforementioned loading, and measurements of thedeflection were taken at the tip (70 from the root of the cantilever, as per Figure 1) at severalintervals. Table 1 (provided for Part 2) shows the measured deflections at selected intervals (themore detailed values referenced in the original document were not provided).


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Table 11: Summary of Measured Deflections

Time[hrs]

Time[days]

LoadingP 1

[kN]P 2

[kN]M max

[kNm]

Measured Deflections [mm]

R15-C25 R15-C40 R20-C25 R20-C40

68 3 None 0 0 0 0 0 0 068 3

Self Wt.4.12 0.00 10.23 3.24 4.37 2.41 3.48

115 5 4.12 0.00 10.23 4.46 5.82 3.49 4.71115 5

Const.6.52 1.23 15.14 8.28 11.39 6.38 8.59

166 7 6.52 1.23 15.14 9.54 12.97 7.48 9.98166 7

Self Wt.4.12 0.00 10.23 8.29 11.29 6.46 8.69

234 10 4.12 0.00 10.23 8.39 11.40 6.57 8.81234 10

Const.6.52 1.23 15.14 9.83 13.36 7.71 10.24

286 12 6.52 1.23 15.14 10.54 14.20 8.36 10.99286 12

Self Wt.4.12 0.00 10.23 9.27 12.50 7.33 9.70

334 14 4.12 0.00 10.23 9.22 12.42 7.30 9.65334 14

Const.6.52 1.23 15.14 10.52 14.22 8.33 10.96

407 17 6.52 1.23 15.14 11.21 15.02 8.99 11.76407 17Self Wt.

4.12 0.00 10.23 9.96 13.36 7.99 10.45838 35 4.12 0.00 10.23 10.54 13.94 8.61 11.10838 35

Sust.4.12 13.59 11.39 15.01 9.28 11.98

3622 151 4.12 13.59 15.35 19.23 13.29 16.21

Additional point loads: 1.24 kN at 0.95 m from support, 1.73 kN at 1.17 m, 0.08 kN at 1.55 m.

It can be seen that the values of M max used in the predictions were slightly lower; this was due toslight differences in precision of the loads and distances. It should also be noted that the daysfield was not provided in the original document; the values of days shown were used in thepredictions and are an approximation to the time in hours.

3.4.2 Comparison to Initial Predictions

3.4.2.1 Comparison with Detailed Approach

Table 12 (next page) provides a comparison between the Detailed Prediction and the measuredvalues. Clearly, the prediction significantly underestimated the deflections and adjustments to theprediction were required. However, it is interesting to note that the relative errors for theimmediate (day 3) and long term (day 151) were somewhat similar; this suggested that a simplecorrect to the immediate deflections could yield reasonable results.

Table 12: Detailed Prediction vs. Observed Deflections

SpecimenTip Deflections at Day 3 [mm] Tip Deflections at Day 151 [mm]

Detailed Measured % Error Detailed Measured % Error R15-C25 1.15 3.24 -64.5% 6.67 15.35 -56.5%R15-C40 1.20 4.37 -72.5% 7.02 19.23 -63.5%R20-C25 1.11 2.41 -53.9% 6.08 13.29 -54.3%R20-C40 1.13 3.48 -67.5% 6.43 16.21 -60.3%


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3.4.2.2 Comparison with Simplified Approach

Table 13 provides a comparison between the Detailed Prediction and the measured values. Clearly,the simplified prediction also significantly underestimated the deflections and adjustments to theprediction were required. The immediate values are slightly closer to the measured values, but stillwell beyond a reasonable number for any purposes.

Table 13: Simplified Prediction vs. Observed Deflections

SpecimenTip Deflections at Day 3 [mm] Tip Deflections at Day 151 [mm]

Simplified Measured % Error Detailed Measured % Error R15-C25 1.56 3.24 -51.9% 7.60 15.35 -50.5%R15-C40 1.58 4.37 -63.8% 8.03 19.23 -58.5%R20-C25 1.52 2.41 -36.9% 6.88 13.29 -48.2%R20-C40 1.55 3.48 -55.5% 7.31 16.21 -54.9%

Based on the results and comparisons above significant adjustments were required in the improvedestimates. The fact that the simplified immediate deflections were out was not surprising, since itwas known that the lack of a reduction in f r (and hence the cracking moment) was recently revisedin A23.3 to a 50% reduction (for one-way construction) via an emergency update. The fact that themore detailed method was even further removed from the measured values was interesting. Inexamining the plots of moment vs. immediate deflection in the original paper (Appendix B), thereappears to be discontinuity in the observed behavior, which was not accounted for in the moment-curvature response. Finally, given that the immediate estimated were significantly lower than themeasured results, it is not surprising that the long term estimates fall short by a similar margin.

3.5 Improved Estimates (Part 2)

Part 2 of the project consisted of making improvements to both the Detailed and Simplified estimates.In both cases, several combinations of adjustments were investigated until the improved estimatessubstantially agreed with the measured values and with each other. This section describes the variouschanges that were considered, the final version of the improved estimates, and the results.


3.5.1.1 Improved Detailed Approach

In the detailed approach, applying the moment area theory greatly underestimated the immediatedeflections. Based on the detailed moment vs. deflection plots provided (Appendix B), this

appeared to be due to a discontinuity at cracking, which was not accounted for in the initial M-response. Figure 10a (left) provides a comparison between the detailed prediction with the resultsgiven by Adebar (from Appendix B) for R15-C25.


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Four adjustments were made to the M- relationships in order to better match the observed data.

Figure 10b (right) and Figure 11 (next page) show the corrected M- plot the final M- curve forR15-C25. Plots for the remaining three specimens are provided in Appendix D. Table 14 provides asummary of the immediate deflections from the Improved estimate.

The cracking moment (M cr) was adjusted to the observed values (~8kNm for R15-C25) The yield curvature ( y) was increased by 50% (from 15.4 to 23.1kNm for R15-C25) A discontinuity was added in the moment curvature relationship (the curvature increased

by a factor of 4 at cracking) The uncracked stiffness was reduced by 10%

Table 14: Improved Immediate Estimates Detailed Approach

SpecimenTip Deflection,

tip[mm]

Self-Weight Self-Weight+SustainedDetailed Measured % Error Detailed Measured % Error

R15-C25 3.08 3.24 -4.9% 6.16 5.5 +12%R15-C40 3.20 4.37 -27% 6.77 7.5 -9.7%R20-C25 2.07 2.41 -14% 4.48 4.0 +12%R20-C40 3.19 3.48 -8.3% 6.19 5.5 +13%

The results are indeed substantially improved over the initial estimates (see Table 12). Interestingly,when the adjusted M- curves are compared with RESPONSE2000 (R2K) output, they correspondsomewhat more closely with the curves neglecting tension stiffening (see Figure 11) than those with

tension stiffening (which were the basis for the initial prediction). In comparing the three curves, itcan be seen that the increase in y is mostly an artificial adjustment that was needed to match thedeflections in the range of interest (i.e. to achieve equal moments of areas under the resultingcurvature diagrams). It is postulated that the actual R2K curve without tension stiffening moreclosely represents the true response of the member, but the "Improved" curve has the advantagethat it is easier to implement in the numerical integration: a simple lookup function in MS Excel gavethe curvature corresponding to each bending moment; this same function would not work with theactual R2K output, since it has a descending portion.

Figure 10: Initial (Left) & Improved (Right) M- Comparison for R15-C25


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It should be noted that observed data for R15-C25 was unique in the sense that it fit best with theBranson equation. The remaining three fell between the Branson and Bischoff Equations, consistentwith the corrected M- plots for the Improved Detailed approach. Overall, Improved estimate gavereasonable results, but several M- curves could have satisfied the data and the aforementionedadjustments make the method more phenomenological than rational.

3.5.1.2 Improved Simplified Approach

The Simplified approach for Immediate deflections was improved by investigating various I e expressions: the immediate (day 3) deflections were calculated using various adjustments to the a

parameter and cracking moment (via the modulus of rupture, f r). Both the Branson equation (8) aswell as the Bischoff Equation (10) were used. Table 15 provides a summary of the results. The SRSSvalues represent the absolute accuracy of the estimates (i.e. it does not distinguish betweenoverestimates and under estimates) while the mean provides a sense of the skew (over or under). It

can be seen that the Bischoff equation with f r reduced to 85% of its full value and a=2.0 is, onaverage, the most accurate (lowest SRSS) and has little skew (mean nearest zero).

Table 15: Various Improved Simplified Immediate Estimates ConsideredCase R15-C25 %Dif R15-C40 %Dif R20-C25 %Dif R20-C40 %Dif SRSS Mean

Measured 3.24 -- 4.37 -- 2.41 -- 3.48 -- -- --1 Initial (f r=100%; a=3) 1.56 -52% 1.58 -64% 1.52 -37% 1.55 -55% 105% -52%2 f r=50%; a=3 4.63 43% 5.25 20% 3.76 56% 4.25 22% 77% 35%3 f r=75%; a=3 2.78 -14% 2.94 -33% 2.51 4% 2.67 -23% 43% -17%4 f r=50%; a=2.5 4.05 25% 4.50 3% 3.40 41% 3.77 8% 49% 19%5 f r=50%; a=2.25 3.74 15% 4.10 -6% 3.19 32% 3.51 1% 36% 11%6 f r=50%; a=2.0 3.41 5% 3.70 -15% 2.97 23% 3.23 -7% 29% 1.5%7 f r=75%; a=2.5 2.46 -24% 2.57 -41% 2.26 -6% 2.38 -32% 58% -25%8 f r=75%; a=2.0 2.15 -34% 2.23 -49% 2.02 -16% 2.10 -40% 73% -35%9 Bischoff (f r=85%; a=2) 3.47 7% 4.10 -6% 2.74 14% 3.14 -10% 19% 1.2%

Figure 11: Improved M- Curve Compared with Initial and R2K (R15-C25 shown)


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Although Bischoffs equation is not the most accurate for any single specimen, its error is relativelysmall and consistent, and appears to have little skew. This makes it a useful best estimate forengineering purposes. Because it appears to have little skew, a factor could also be applied toachieve estimates with a relatively uniform degree of desired conservatism; this is useful from adesign perspective. Table 16 presents the Improved Simplified estimates for Immediate deflection.

Calculations for each specimen can be found in Appendix E.

Table 16: Improved Immediate Estimates Simplified Approach

SpecimenTip Deflection, tip [mm]

Self-Weight Self-Weight+SustainedDetailed Measured % Error Detailed Measured % Error

R15-C25 3.47 3.24 +7.1% 5.61 5.5 +2.0%R15-C40 4.10 4.37 -6.2% 6.79 7.5 -9.5%R20-C25 2.74 2.41 +13.7% 4.24 4.0 +6.0%R20-C40 3.14 3.48 -9.8% 4.99 5.5 -9.3%

It is not surprising that the Bischoff equation yielded the best results; as aforementioned, theBranson equation overestimates the tension stiffening in lightly reinforced members. Based onthese results and those of the Detailed approach (Section 3.5.1.1), it is clear that correctlyaccounting for tension stiffening is crucial in estimating deflections of reinforced concrete slabs.

Finally, it should also be noted that Case 2 (i.e. the current code specified estimate as per A23.3)provides conservative estimates of immediate deflection and is acceptable from a designperspective. However, it should be recalled that the differential deflections must also be consideredin the long term this is discussed in the following sections.


The estimates of long term deflections (both Detailed and Simplified) were improved such that theysubstantially agreed with each other and the measured results. Similar to the Initial Predictions(Section 3.3.2), both Improved estimates made use of the Simplified Immediate deflections (fromSection 3.5.1.2) in order to make a fair comparison. This is perhaps the most useful section of theproject, as it makes use of the immediate deflections previously calculated and accurately predictsthe long term deflections: these results could readily be applied to code specified limits, includingthose for differential deflection.


In the Improved Detailed Approach, the incremental immediate deflections were calculated (usingCase 9 from Table 15 - the Improved Simplified Approach for Immediate) for the three load stages:

Stage 1: Self Weight Stage 2: Self Weight + Construction Load Stage 3: Self Weight + Sustained Load

Recall that in the Initial Detailed Approach (Section 3.3.2.1), the construction loading (Stage 2). Thethree incremental immediate deflections were applied over the appropriate periods of time and the


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resulting creep deflection was calculated separately for each load; the deflections were thensummed to find the total deflection over the entire load history. Creep calculations were asdiscussed in Section 3.1.4.1. Figure 12 shows a typical resulting plot of the deflections. Appendix Fcontains all the plots and sample excerpts from the Excel spreadsheet calculations.

The construction loading "spikes" are shown in the immediate deflections. An importantassumption in the modeling was that 50% of the immediate deflection due to construction wasrecovered upon unloading; this represents both the additional and creep recovery (which was notexplicitly modeled). This assumption will be shown to be consistent with the experimental data.

Another important important note is that the effect of (previous) construction loading wasaccounted for in calculating the immediate deflection due to the sustained load.

In developing the Improved Detailed Estimates, several parameters were adjusted until a givencombination was found yielded results in substantial agreement with measured values for all fourspecimens. A few of the combinations considered were as follows and shown in Figure 13 (noteonly selected points are shown and full deflection plots would be nonlinear similar to Figure 12):

Prediction D-1: Cu=3.5 & No Construction Loading (increased over the initial value, 2.35) Prediction D-2: Cu=1.35 with Construction Loading & Increased self-weight from Construction

& 25% Elastic Recovery)

Prediction D-3: Cu=3.1 with Construction Loading & 50% Elastic Recovery of construction Prediction D-4: Cu=3.5 with Construction Loading & 50% Elastic Recovery of construction

Figure 12: Typical Deflection vs. Time for the Improved Detailed Approach (R15-C25 shown)

Full Constr. Load Immed. Deflection50% Elastic Recover Assumed

Full Sustained Load Immed. Deflection

Self-Weight


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Exact values from these plots can be found in Appendix F. The following observations can be madebased on the comparison for R15-C25:

Prediction D-1 achieved good values of deflection from Days 35 to 151, but did not addressthe construction deflections, although it did account for the cracking that would be presentas a result of the loading (which led to the acceptable values in this range).

Prediction D-2 accounted for the construction deflection (including 25% elastic recovery)and also amplified the deflection attributed to self-weight due to additional cracking underconstruction loading: this method was found to systematically overestimate the 35-daydeflection. It also would likely be unconservative from a design perspective, since the

differential deflection from Day 35 to Day 151 is underestimated. Prediction D-3 was a good match. The 50% elastic recovery appeared to well represent the

combined phenomena of permanent immediate cracking and creep recovery However, D-3was found to overestimate deflections in the remaining three specimen

Prediction D-4 showed the same merits and was found to be the best fit, on average, for thefour specimens. The increased creep rate (Cu=3.5) ensured that the differential deflectionbetween days 35 and 151 was not significantly underestimated for any of the specimens.

Ultimately, Prediction D-4 was selected as the final Improved Detailed estimate of deflections.Figure 14 compares the estimated and measured total deflection at selected times; again, only a

few points are shown and linear plots are only to give a sense of the accuracy. Full detailed plots foreach specimen (broken down into immediate, creep, total similar to Figure 12) are provided inAppendix F.

Figure 13: Various Models Considered For Improved Detailed Approach (R15-C25 shown)


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Table 17 shows the relative errors at selected times, for comparison to Table 12 and Table 13. Thepredicted values are generally less than 1mm, and well under 10%.

Table 17: Measured vs. Predicted Total Deflections - Improved Detailed Estimate

SpecimenDeflections [mm]

Day 3 Day 35 Day 151

Measured Predicted % Error Measured Predicted % Error Measured Predicted % ErrorR15-C25 3.24 3.47 +7.1% 11.39 12.11 +6.3% 15.35 16.69 +8.7%R15-C40 4.37 4.10 -6.2% 15.01 14.56 -3.0% 19.23 20.15 +4.8%R20-C25 2.41 2.74 +14% 9.28 9.25 -0.3% 13.29 12.67 -4.7%R20-C40 3.48 3.43 -1.4% 11.98 10.81 -9.8% 16.21 14.87 -8.3%


In the Improved Simplified Approach to for Long Term deflections were calculated similar to thosein the Initial Simplified Approach (Section 3.3.2.2): values of immediate deflections at selected keypoints were amplified using the code specified S-values. However, the differences were,conceptually, that various expressions for immediate deflections (from Table 15) were investigatedand the effects of construction loading (additional cracking and creep) were accounted for.

Several combinations of adjustments were applied until a combination was found that substantiallyagreed with the measured 3-day, 35-day, and 151-day values and, by extension, the ImprovedDetailed Approach (Section 3.5.2.1) for all four specimens. The combinations are summarized asfollows and as shown Figure 15.

Prediction S-1: Branson Equation, f r=100% (i.e. the Initial Prediction)

Prediction S-2 : Branson Equation, f r=50%

Figure 14: Measured vs. Predicted Total Deflection From The Improved Detailed Estimate


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Prediction S-3: Branson Equation, f r=50% & M a from Construction (i.e. current code) Prediction S-4: Bischoff Equation, f r=85%, M a from Construction (i.e. Improved Simplified) Prediction S-5: Prediction S-4, plus additional multipliers at days 35 and 151 accounting for

creep and residual immediate deflection from construction loading

Exact values from these plots can be found in Appendix G. The following observations can be made

based on the comparison for R15-C25: Prediction S-1 greatly understimates the deflections, although the differential deflection

between days 35 and 151 is reasonable. Prediction S-2 gives a nearly exactly correct value at day 151, but underestimates the

defelction at day 35 and overstimates the deflection at day 3; this would likely beconservative in design

Prediction S-3 gives results similar to S-2, with a slight boost at days 35 and 151 due tocracking from construction. This is the current-code approach and is even moreconservative from a design perspective

Prediction S-4 gives accurate results at day 3, but underestimates the values at days 35 and

151; this would likely be unconservative in design Prediction S-5 gives results almost identical to the Detailed prediction of Section 3.5.2.1 (it

was developed to do this) and is also very similar to the measure values at all days

Ultimately, Prediction S-5 provided the best fit, on average, for all specimens; however, thedifferential deflection between days 34 and 151 was slightly unconservative, but not so much that asimple common multiplier couldn't be used to provide a reliable level of conservatism. Figure 16shows the measured and (S-5) predicted values for all specimens.

Figure 15: Various Models Considered For Improved Simplified Approach (R15-C25 shown)


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Table 18 shows the relative errors at selected times, for comparison to Table 12, Table 13, and Table17. Again, the predicted values are generally within 1mm of the measured values and less than10%.

Table 18: Measured vs. Predicted Total Deflections - Improved Simplified Estimate

Specimen

Deflections [mm]

Day 3 Day 34 Day 151Measured Predicted % Error Measured Predicted % Error Measured Predicted % Error

R15-C25 3.24 3.47 +7.1% 10.54 11.51 +9.1% 15.35 16.38 +6.7%R15-C40 4.37 4.10 -6.2% 13.94 14.01 +0.5% 19.23 19.86 +3.3%R20-C25 2.41 2.74 +14% 8.61 8.80 +2.2% 13.29 12.54 -5.6%R20-C40 3.48 3.43 -1.4% 11.10 10.47 -5.7% 16.21 14.82 -8.6%

3.5.2.2.1 Development of The Construction-Related Multipliers for Prediction S-5

The major improvement of Prediction S-5 was the aforementioned additional emprirical multipliersat days 34 and 151 accounting for the effects of construction. Separate multipliers were used toaccount for:

A. Residual immediate deflection after the construction load was removed; this multiplierapplies to the time between the end of construction loading and the start of the sustainedload application (the only applicable point in the simplified prediction is day 34)

B. Additional creep deflections accumulated during the time the construction load is applied;this applies at any point subsequent to construction and was developed as a function of thetotal days in which construction loading is applied (the only applicable points in thesimplified prediction are days 34 and days 151; in these cases the full 7-day constructionload duration was accounted for

Figure 16: Measured vs. Predicted Total Deflection From The Improved Simplified Estimate


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The multipliers were applied to the differential immediate deflection due to construction, which isthe difference between the self-weight and self-weight+construction values calculated by theBischoff equation (10). This gives a construction correction term to be added to the usualsimplified long term deflection estimate, i.e. the deflection obtained by amplifying the immediatedeflections by the S parameter given in A23.3.

The A multiplier was taken as 0.5 for all specimens. This represents the 50% residual immediatedeflection that was also used in the Improved Detailed estimates (Section 3.5.2.1); in the detailedestimates, this value was found to be consistent with the observed data.

The B parameter was determined based on linear regression of data obtained using the ImprovedDetailed estimates of Long Term deflection (3.5.2.1), as this method was deemed to be sufficientlyaccurate to calculate the accumulated creep deflections for various durations of constructionloading. Using the spreadsheets developed for this method, the construction loading was reducedfrom 7 days (as was the case in the experiment) to 6 days, then 5 days, and so forth, by removingconstruction loading from starting at day 17, then 16, etc; the resulting total deflection at Day 34was recorded for each day removed. Figure 17 shows one such plot: in this case, the last 3 days of construction loading (Days 14-17) have been removed.

Next the difference between each new deflection at Day 34 (34d,detailed

) and the deflection at Day

34 calculated by Prediction S-4 ( 34d,simplified ) was calculated and this difference was plotted. Figure17 provides an example. Various corrections were then applied as follows:

i. Step 1: Baseline Corrected for the to zero at 0 daysii. Step 2: 50% differential immediate deflection applied (to account for multiplier A)

iii. Step 3: Remaining increase normalized by differential immediate deflection (squared for plotting)

Figure 17: Plot Illustrating Removal of Construction Loads to Determine Accumulated Creep

Construction Load Removed

New Defl. @ Day 34 Recorded


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The final results show a strong linear correlation. The linear function represents the additionalsquare of the creep accumulated during construction. Thus, the B multiplier is taken as the squareroot (rounded for simplicity):

A similar procedure was performed for R20-C25, giving B= (0.8*t); the laer was also used for R20-C40, while the former was also used for R15-C40. This was a relatively minor correction, and basedon the required accuracy for code equations a single value could likely be specified. Of course, theregression was based on calculated creep values and thus, B multiplier simply represents acorrection of the simplified method using first principles (i.e. the detailed C t, method). However,it represents a convenient correction that is easy to implement with the existing code predictions.Appendix G contains complete calculations from the development of the B parmeter.

3.5.2.2.2 Example Application of Improved Simplified Method with Multipliers

R15-C25 will be used as an example. First, the immediate deflections due to self-weight and self-weight+construction are computed as 3.47mm and 6.86mm, respectively; thus the differentialimmediate deflection would be 3.39mm:

, = , , = 6.863.47= 3.39 At Day 34, both multipliers A and B would be applied to the value 3.39mm, since the constructionload has been removed, but the sustained load has not been applied. The A and B parametersare 0.5 and (0.06*7)=0.65 respecvely, as previously discussed. Thus, the correction term is:

= 0.6 (11)Where:

t total duration of construction load [days]

Figure 18: Plot Illustrating Removal of Construction Loads to Determine Accumulated Creep


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= , ( + )= 3.39(0.5+.65)= 3.90 Note that in the case of Day 151, the A parameter would not be applied, since the effects of theresidual deflection are mitigated by the application of the sustained load. This value is then added

to the long term deflection given by Prediction S- 4 (i.e. the typical code-type estimate, which

amplifies the immediate deflection using the S parameter from A23.3); this is the long term valueneglecting construction effects (i.e. from self-weight). For R15-C25 it was calculated to be 7.61mm.Therefore the resulting predicted deflection is:

=, +, = 7.613.90= . The measured value of deflection from the experiment was 10.54mm (+9.1% Error). This was thehighest error of all specimens for all points using the correction. The average errors were +1.5% and-1.1% for Days 34 and 151, respectively. Refer to Appendix G for complete calculations for allspecimens.


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4.0 SUMMARY OF RESULTS & DISCUSSION

The following section provides a brief, qualitative summary of the results, and a discussion of the keyshortcomings and improvements of the various predictions.

4.1 Initial Predictions

Initial predictions of both the long term and the immediate deflections were made as Part 1 of thisproject. The results are summarized and discussed herein.



In the Initial Detailed approach, the immediate deflections were estimated from first principles, byapplying the moment-area theorem and numerically integrating moments of areas under thecurvature diagram. The M- relationship used was a trilinear model that accounted for a significantdegree of tension stiffening in the range of interest (see Section 3.3.1.1). This approach predictedthe smallest immediate deflections of any approach and significantly underestimated the immediatedeflections: the predicted deflections were about 33% of the measured results (see Table 12)and

about 50% of the initial approach. Based on the results of the Improved Detailed Immediateestimates and comparison to RESPONSE2000 output, it was determined that the tension stiffeningwas overestimated in the Initial Prediction, leading to very small deflections.


In the Initial Simplified approach, the immediate deflections were predicted using the applicable

equations of CSA A23.3-04 (see Section 3.3.1.2). No reduction to the modulus of rupture (f r) wasapplied (despite the fact that it is now specified for one-way members) because the cantilevers didnot have significant axial restraint. This method also significantly underestimated the measuredimmediate deflections: the predicted deflections were about 50% of the measured values (see Table13). Based on on experimentation with various parameters in the Improved approach, it wasdetermined that it was necessary to reduce f r (i.e. the cracking moment) to achieve satisfactorresults with this method. The 50% reduction specified by A23.3 gave conservative values, about30% greater than than measured values (see Table 15), on average, which would likely beacceptable from a design perspective (differential deflection considerations notwithstanding).



In the Initial Detailed approach to long term deflections, the incremental immediate deflectionsfrom the various loadings were separated and the more rigorous creep calculations provided inChapter 6 of the CAC Handbook were used to calculate the accumulated creep over the entireperiod. The immediate deflections from the simplified approach were used, so as to provided a faircomparison. The sum of the immediate plus accumulated creep for each separate loading gave the


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total deflection over time (Section 3.3.2.1); however, effects of construction loading were neglected(per the project instructions). The ultimate creep coefficient was taken as the A23.3 recommendedaverage value of 2.35. This method significantly underestimated the long term deflections,compared to the measured results: the predicted 151-day total deflections were about 40% of themeasured deflection (see Table 12); however, the differential deflection between days 35 and 151

were about the same. Thus, the error was largely due to the poor estimate of immediate deflection.


In the Initial Simplified approach to long term deflections, the incremental deflections due to self-weight and to the sustained load were amplified by the A23.3-specified S parameter values toobtain long term deflections at selected points (see Section 3.3.2.2); again the construction loadeffects were neglected. The immediate deflections from the simplified approach were used, so asto provided a fair comparison. This method also significantly underestimated the long termdeflections, compared to the measured results: the predicted 151-day total deflection was about50% of the measured deflection (see Table 13); however, the differential deflection between days

35 and 151 were again about the same. Thus, the error was largely due to the poor estimate of immediate deflection.

4.2 Improved Estimates

After obtaining the experimental data (see Section 3.4), significant revisions were made to both theDetailed and Simplified methods of estimating deflection. The changes and results are summarized anddiscussed herein.



In the Improved Detailed approach, the immediate deflections were again estimated from firstprinciples, by applying the moment-area theorem and numerically integrating moments of areasunder the curvature diagram. However, the M- relationship was revised to reduce the tensionstiffening effect; the final M- curve was more similar to the RESPONSE200 curve without tensionstiffening. However, the curve was still partially calibrated to achieve results in the specific range of interest (see Section 3.5.1.1); more adjustments would be required at higher loads. This approachyield reasonable estimates of the immediate deflections: the predicted immediate deflections wereabout 90-110% of the measured results (see Table 14) in that specific load range. A key lesson was

that the tension stiffening effect is very critical in lightly rienforced members just after cracking; italso serves as a reminder that more rigorous methods require accurate input and that they canproduce worse results than more simple methods, if the input is not incorrect.


In the Improved Simplified approach, the immediate deflections were again calculated usingeffective stiffnesses, but expressions other than the equation from A23.3 were investigated, namelythe Bischoff (2005) equation; various reductions to the modulus of rupture (f r) were tried (see


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Section 3.5.1.2). Ultimately the Bischoff equation, with f r reduced to 85% of its full value was foundgive the most accurate results: the predicted deflections were about 90-110% of the measuredvalues (see Table 16), similar to the detailed method. It should be noted that this 10% erroramounts to less than 1mm for the given testing. It was not surprising that the Bischoff equationyielded more accurate results, as the Branson equation specified by A23.3 was developed for beams

with =1-2% and has been found to overestimate the tension stiffening effect in lightly reinforcedmembers [2]. The Bischoff equation, on the other hand gives reliable results for all practicalreinforcement ratios [2, 3].



The Improved Detailed approach to long term deflections was similar to the Initial detailedapproach, but with a few key revisions (see Section 3.5.2.1):

Immediate deflections from the Improved Simplified approach were used (recall that thepoor estimate of immediate deflections was a key issue with the initial predictions)

The construction loading was accounted for (in terms of both cracking and additionalcreep); the effect of cracking was accounted for by using M a values from constructionloading for loads that occurred thereafter

A residual immediate deflection (i.e. only 50% elastic recovery) was accounted for whenconstruction loads were removed

Various values of C u were investigated and the value of 3.5 was found to give the best fit.This significantly higher than the average value of 2.35 per CSA A23.3; this is perhaps dueto the early age of loading (although correction factors per A23.3 were applied). Therewere also a number of other correction factors that could have been underestimated,

leading to the apparent high value of C u

This method produced good results, with errors consistently less than 10% and in many case lessthan 0.5mm (see Figure 14 and Measured vs. Predicted Total Deflections - Improved DetailedEstimate). This is also the only method that gives an accurate and complete prediction throughoutthe load history, although the Improved Simplified Approach (next section) can be used to accuratepredict deflections at any key points.


The Improved Simplified approach to long term deflections was similar to the initial approach inthat the incremental deflections due to self-weight and to the sustained load were amplified by theA23.3-specified S parameter values to obtain long term deflections at selected points. However,some key revisions were made (see Section 3.5.2.2):

Immediate deflections from the Improved Simplified approach were used, similar to theImproved Detailed approach (i.e. the Bischoff Equation with f r set to 85%)

Construction effects were accounted for in two ways:o M a values from construction were used for immediate deflections, as appropriate


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o The residual immediate deflection and additional creep were accounted for bydeveloping two new parameters,

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CIVL 513 - Project No. 1 - 13Mar30 - Complete Report [Final]

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