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StressStrain Properties of Concrete at Elevated Temperatures
Structural Engineering Research Report
Department of Civil Engineering and Geological SciencesUniversity of Notre Dame
Notre Dame, Indiana
April 2009
Adam M. Knaack, Yahya C. Kurama, and David J. Kirkner Report #NDSE0901
Strain, εc
Rel
ativ
e St
ress
, fc / f
cmo
0.0000.0
0.012
1.2 70oF (21oC)392oF (200oC)752oF (400oC)1112oF (600oC)1400oF (760oC)
0.006
NSC, Calcareous, Unstressed
Strain, εc
Rel
ativ
e St
ress
, fc / f
cmo
0.0000.0
0.012
1.2 70oF (21oC)392oF (200oC)752oF (400oC)1112oF (600oC)1400oF (760oC)
0.006
HSC, Calcareous, Unstressed
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StressStrain Properties of Concrete at Elevated Temperatures
Structural Engineering Research Report
Department of Civil Engineering and Geological SciencesUniversity of Notre Dame
Notre Dame, Indiana
April 2009
by
Adam M. KnaackGraduate Research Assistant
Yahya C. KuramaAssociate Professor
David J. KirknerAssociate Professor
Report #NDSE0901
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ABSTRACT
This report focuses on the compressive stressstrain behavior of unreinforced North American concrete under elevated temperatures from fire. A database on the temperaturedependent properties of concrete is developed from previous experimental research. Predictive multiple least squares regression relationships are proposed for the concrete strength, elastic modulus, strain at peak stress, ultimate strain, and stressstrain behavior, including the temperature, aggregate type, test type, and strength at room temperature as parameters. Highstrength and normalstrength, and normalweight and lightweight materials are considered. It is shown that at elevated temperatures, the concrete strength and elastic modulus are significantly reduced, whereas the strain at peak stress and ultimate strain are increased. Differences between highstrength and normalstrength concrete are quantified. In comparison with previous temperaturedependent relationships, the proposed relationships utilize a larger dataset. Furthermore, the previous models implicitly include creep strains, whereas the proposed relationships provide a baseline to which creep strains could be explicitly added.
This report may be downloaded from http://www.nd.edu/~concrete
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CONTENTS
FIGURES.......................................................................................................................... vi
TABLES............................................................................................................................ xi
ACKNOWLEDGEMENTS .......................................................................................... xiii
LIST OF SYMBOLS ...................................................................................................... xv
CHAPTER 1: INTRODUCTION.................................................................................... 1 1.1 Project Background........................................................................................... 1 1.2 Project Objectives ............................................................................................. 1 1.3 Project Scope .................................................................................................... 2 1.4 Report Layout ................................................................................................... 2
CHAPTER 2: LITERATURE REVIEW ....................................................................... 5 2.1 Previous Experimental Data ............................................................................. 5
2.1.1 Compressive Strength .............................................................................. 5 2.1.2 Modulus of Elasticity............................................................................... 8 2.1.3 Strain at Peak Stress, Ultimate Strain, and Creep Strain ......................... 9
2.2 Previous Proposed Relationships...................................................................... 9 2.2.1 StressStrain Relationships .................................................................... 10 2.2.2 Compressive Strength ............................................................................ 12 2.2.3 Modulus of Elasticity............................................................................. 15 2.2.4 Strain at Peak Stress............................................................................... 17 2.2.5 Ultimate Strain ....................................................................................... 18
CHAPTER 3: CONCRETE PROPERTY DATABASE ............................................. 21 3.1 Database Overview ......................................................................................... 21 3.2 Database Properties......................................................................................... 22 3.3 Data Ranges .................................................................................................... 26
3.3.1 Compressive Strength Data.................................................................... 28 3.3.2 Modulus of Elasticity Data .................................................................... 31 3.3.3 Strain at Peak Stress Data ...................................................................... 35 3.3.4 Ultimate Strain Data .............................................................................. 37
CHAPTER 4: COMPRESSIVE STRENGTH ............................................................. 41 4.1 Statistical Analysis.......................................................................................... 41
4.1.1 Preliminary Regression Forms............................................................... 43 4.1.2 Selected Form of Regression Equations ................................................ 44 4.1.3 Normalized Regression Coefficients ..................................................... 45 4.1.4 Constrained Regression Equations ........................................................ 46 4.1.5 UnNormalized Regression Coefficients ............................................... 47 4.1.6 Regression Assumptions........................................................................ 48
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4.1.7 Full Regression Equations Using Coded Variables ............................... 49 4.2 Proposed Relationships................................................................................... 51 4.3 Results and Evaluations .................................................................................. 54
4.3.1 Comparisons with Test Data and Evaluation of Data Fit ...................... 54 4.3.2 Effect of Aggregate Type....................................................................... 59 4.3.3 Effect of Test Type ................................................................................ 60 4.3.4 HighStrength versus NormalStrength Concrete.................................. 61 4.3.5 Comparisons with Previous North American Models ........................... 63
CHAPTER 5: MODULUS OF ELASTICITY ............................................................. 67 5.1 Statistical Analysis.......................................................................................... 67
5.1.1 Modulus of Elasticity Data .................................................................... 67 5.1.2 Normalization of Modulus of Elasticity ................................................ 70 5.1.3 Preliminary Regression Forms............................................................... 71 5.1.4 Regression Assumptions........................................................................ 72
5.2 Proposed Relationships................................................................................... 72 5.3 Results and Evaluations .................................................................................. 74
5.3.1 Comparisons with Test Data and Evaluation of Data Fit ...................... 74 5.3.2 Effects of Aggregate Type, Test Type, and Room Temperature Strength
................................................................................................................. 78 5.3.3 Comparisons with Previous North American Models ........................... 79
CHAPTER 6: STRAIN AT PEAK STRESS ................................................................ 83 6.1 Statistical Analysis.......................................................................................... 83
6.1.1 Strain at Peak Stress Data ...................................................................... 83 6.1.2 Preliminary Regression Forms............................................................... 85 6.1.3 Regression Assumptions........................................................................ 85
6.2 Proposed Relationships................................................................................... 86 6.3 Results and Evaluations .................................................................................. 87
CHAPTER 7: ULTIMATE STRAIN............................................................................ 91 7.1 Statistical Analysis.......................................................................................... 91
7.1.1 Ultimate Strain Data .............................................................................. 91 7.1.2 Preliminary Regression Forms............................................................... 92 7.1.3 Regression Assumptions........................................................................ 93
7.2 Proposed Relationships................................................................................... 94 7.3 Results and Evaluations .................................................................................. 96
CHAPTER 8: STRESSSTRAIN RELATIONSHIP................................................... 99 8.1 Temperature Modified StressStrain Model ................................................... 99 8.2 Results and Evaluations .................................................................................. 99 8.3 Comparisons with Previous Models ............................................................. 100
CHAPTER 9: SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS .... 103 9.1 Summary and Conclusions ........................................................................... 103 9.2 Recommendations for Future Research ........................................................ 104 9.3 Presenting Future Research........................................................................... 105
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APPENDIX A: DATABASE ENTRY ......................................................................... 109
BIBLIOGRAPHY ......................................................................................................... 129
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FIGURES
CHAPTER 2
Figure 2.1: North American temperaturedependent stressstrain models: (a) ASCE; and (b) Kodur et al. ...................................................................................................... 10
Figure 2.2: Eurocode temperaturedependent stressstrain models: (a) NSC, siliceous; (b) NSC, calcareous; (c) NSC, lightweight; (d) HSC, Class 1; (e) HSC, Class 2; and (f) HSC, Class 3. ................................................................................................... 11
Figure 2.3: ACI 216 compressive strength models at elevated temperatures: (a) residual; (b) stressed; and (c) unstressed. ............................................................................ 13
Figure 2.4: ASCE and Kodur et al. compressive strength models at elevated temperatures................................................................................................................................ 13
Figure 2.5: European temperaturedependent compressive strength models: (a) Eurocode; (b) CEB Model Code 90; and (c) Rak MK B4. .................................................... 14
Figure 2.6: Other temperaturedependent compressive strength models: (a) Hertz unstressed; (b) Hertz residual; and (c) Shi et al., Li and Purkiss, and Phan and Carino.................................................................................................................... 15
Figure 2.7: North American temperaturedependent modulus of elasticity models: (a) ACI 216; and (b) ASCE and Kodur et al. ............................................................. 16
Figure 2.8: Eurocode temperaturedependent modulus of elasticity models: (a) NSC; and (b) HSC. ................................................................................................................ 17
Figure 2.9: ASCE, Kodur et al., and Eurocode temperaturedependent strain at peak stress models. .................................................................................................................. 18
Figure 2.10: ASCE, Kodur et al., and Eurocode temperaturedependent ultimate strain models. .................................................................................................................. 18
CHAPTER 3
Figure 3.1: Full set of compressive strength loss data. ..................................................... 28
Figure 3.2: Distribution of room temperature compressive strength, fcmo of the data used in determining the temperaturedependent compressive strength, fcm relationships. (Note: 1 ksi = 6.859 MPa) .................................................................................... 29
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Figure 3.3: Distribution of the normalstrength concrete data used for the compressive strength relationships: (a) test type; (b) aggregate type; (c) specimen height; (d) specimen shape; and (e) furnace type. .................................................................. 30
Figure 3.4: Distribution of the highstrength concrete data used for the compressive strength relationships: (a) test type; and (b) specimen height............................... 31
Figure 3.5: Full set of modulus of elasticity loss data. ..................................................... 32
Figure 3.6: Distribution of room temperature compressive strength, fcmo of the data used in determining the temperaturedependent modulus of elasticity, Ec relationships. (Note: 1 ksi = 6.859 MPa) .................................................................................... 32
Figure 3.7: Distribution of the normalstrength concrete data used for the modulus of elasticity relationships: (a) test type; (b) aggregate type; (c) specimen height; (d) specimen shape; and (e) furnace type. .................................................................. 33
Figure 3.8: Distribution of the highstrength concrete data used for the modulus of elasticity relationships: (a) test type; (b) specimen height; and (c) specimen shape................................................................................................................................ 35
Figure 3.9: Full set of strain at peak stress data................................................................ 36
Figure 3.10: Distribution of room temperature compressive strength, fcmo of the data used in determining the temperaturedependent strain, εcm at peak stress relationships. (Note: 1 ksi = 6.859 MPa) .................................................................................... 36
Figure 3.11: Distribution of the highstrength concrete data used for the strain at peak stress relationships: (a) aggregate type; and (b) specimen height......................... 37
Figure 3.12: Full set of ultimate strain data. ..................................................................... 38
Figure 3.13: Distribution of room temperature compressive strength, fcmo of the data used in determining the temperaturedependent ultimate strain, εcu relationships. (Note: 1 ksi = 6.859 MPa)................................................................................................ 38
Figure 3.14: Distribution of the normalstrength concrete data used for the ultimate strain relationships: (a) aggregate type; (b) specimen height; and (c) furnace type. ...... 39
Figure 3.15: Distribution of the highstrength concrete data used for the ultimate strain relationships: (a) aggregate type; and (b) specimen height. ................................. 40
CHAPTER 4
Figure 4.1: Proposed compressive strength relationships fit to data: (a) NSC – siliceous, residual; (b) NSC – siliceous, stressed; (c) NSC – siliceous unstressed; (d) NSC – calcareous, residual; (e) NSC – calcareous, stressed; (f) NSC – calcareous,
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unstressed; (g) NSC – lightweight, residual; (h) NSC – lightweight, stressed; (i) NSC – lightweight, unstressed; (j) HSC – calcareous, residual; (k) HSC – calcareous, stressed; and (l) HSC – calcareous, unstressed. ................................. 56
Figure 4.2: Proposed compressive strength relationship prediction bands: (a) NSC – siliceous, residual; (b) NSC – siliceous, stressed; (c) NSC – siliceous, unstressed; (d) NSC – calcareous, residual; (e) NSC – calcareous, stressed; (f) NSC – calcareous, unstressed; (g) NSC – lightweight, residual; (h) NSC – lightweight, stressed; (i) NSC – lightweight, unstressed; (j) HSC – calcareous, residual; (k) HSC – calcareous, stressed; and (l) HSC – calcareous, unstressed. ..................... 58
Figure 4.3: Proposed compressive strength relationships showing the effect of aggregate type on the strength loss: (a) NSC – residual; (b) NSC – stressed; and (c) NSC – unstressed.............................................................................................................. 60
Figure 4.4: Proposed compressive strength relationships showing the effect of test type on the strength loss: (a) NSC – siliceous; (b) NSC – calcareous; (c) NSC – lightweight; and (d) HSC – calcareous. ....................................................................... 61
Figure 4.5: Proposed compressive strength relationships showing the difference between normalstrength concrete and highstrength concrete: (a) calcareous, residual; (b) calcareous, stressed; and (c) calcareous, unstressed. ............................................ 62
Figure 4.6: Proposed compressive strength relationships compared with ACI 216, ASCE, and Kodur et al.: (a) NSC – siliceous, residual; (b) NSC – siliceous, stressed; (c) NSC – siliceous, unstressed; (d) NSC – calcareous, residual; (e) NSC – calcareous, stressed; (f) NSC – calcareous, unstressed; (g) NSC – lightweight, residual; (h) NSC – lightweight, stressed; (i) NSC – lightweight, unstressed; (j) HSC – calcareous, residual; (k) HSC – calcareous, stressed; and (l) HSC – calcareous, unstressed. .......................................................................................... 65
CHAPTER 5
Figure 5.1: Proposed modulus of elasticity relationships fit to data: (a) NSC – calcareous, unstressed; (b) NSC – lightweight, unstressed; (c) HSC – calcareous, residual; (d) HSC – calcareous, stressed; and (e) HSC – calcareous, unstressed................ 75
Figure 5.2: Proposed modulus of elasticity prediction bands: (a) NSC – calcareous, unstressed; (b) NSC – lightweight, unstressed; (c) HSC – calcareous, residual; (d) HSC – calcareous, stressed; and (e) HSC – calcareous, unstressed................ 77
Figure 5.3: Effects on the proposed temperaturedependent modulus of elasticity relationships: (a) aggregate type; (b) test type; and (c) NSC versus HSC............ 78
Figure 5.4: Comparison of proposed modulus of elasticity models with ACI 216, ASCE, and Kodur et al. models: (a) NSC – calcareous, unstressed; (b) NSC – light
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weight, unstressed; (c) HSC – calcareous, residual; (d) HSC – calcareous, stressed; and (e) HSC – calcareous, unstressed. ................................................... 80
CHAPTER 6
Figure 6.1: Comparison of HSC and NSC data with bestfit line for each set. ................ 84
Figure 6.2: Proposed strain at peak stress relationships fit to data: (a) NSC, calcareous, unstressed; and (b) HSC, siliceous and calcareous, unstressed. ........................... 88
Figure 6.3: Comparison of proposed NSC and HSC strain at peak stress relationships. . 88
Figure 6.4: Comparison of the proposed strain at peak stress relationships with ASCE and Kodur et al. models: (a) NSC, calcareous, unstressed; and (b) HSC, siliceous and calcareous, unstressed. .......................................................................................... 89
CHAPTER 7
Figure 7.1: Comparison of the calcareous and lightweight aggregate data for the ultimate strain of normalstrength concrete. ....................................................................... 92
Figure 7.2: Cubic and quadratic functions fit to the NSC ultimate strain data................. 93
Figure 7.3: Proposed ultimate strain relationships fit to data: (a) NSC, lightweight and calcareous, unstressed; and (b) HSC, calcareous and siliceous, unstressed. ........ 96
Figure 7.4: Comparison of proposed NSC and HSC ultimate strain relationships........... 96
Figure 7.5: Comparison of the proposed ultimate strain relationships with ASCE and Kodur et al. models: (a) NSC, lightweight and calcareous, unstressed; and (b) HSC, calcareous and siliceous, unstressed. .......................................................... 97
CHAPTER 8
Figure 8.1: Proposed stressstrain relationships: (a) NSC – calcareous, unstressed; and (b) HSC – calcareous, unstressed. ............................................................................ 100
Figure 8.2: Comparison of ASCE, Kodur et al., Eurocode, and Tmodified Popovics fcεc functions: (a) using ASCE parameters; (b) using Kodur et al. parameters; and (c) using Eurocode parameters. ................................................................................ 101
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TABLES
CHAPTER 3
Table 3.1: Mix properties collected in the database ......................................................... 23
Table 3.2: Curing properties collected in the database ..................................................... 23
Table 3.3: Specimen properties collected in the database ................................................ 23
Table 3.4: Test properties collected in the database ......................................................... 24
Table 3.5: Mechanical properties collected in the database ............................................. 25
Table 3.6: Thermal properties collected in the database................................................... 26
Table 3.7: Physical properties collected in the database................................................... 26
Table 3.8: Number of data points collected for each temperaturedependent concrete property ................................................................................................................. 27
CHAPTER 4
Table 4.1: Preliminary equation forms and test statistics ................................................. 43
Table 4.2: Regression statistics before and after constraining room temperature values. 47
Table 4.3: Required significance level to pass the KolmogorovSmirnov test for the compressive strength regression equations........................................................... 49
Table 4.4: Proposed compressive strength relationship regression coefficients............... 52
CHAPTER 5
Table 5.1: R2 statistics for Ec / Eco_ACI, Ec / Eco, and Ec for each data set.......................... 71
Table 5.2: Regression statistics for modulus of elasticity trial equations for each test type and aggregate type combination ........................................................................... 71
Table 5.3: Required significance level to pass the KomolgorovSmirnov test for the modulus of elasticity regressions .......................................................................... 72
Table 5.4: Proposed modulus of elasticity relationship regression coefficients............... 73
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CHAPTER 6
Table 6.1: R2 values of the strain at peak stress trial regressions ..................................... 85
Table 6.2: Required significance level to pass the KomolgorovSmirnov normality test 86
Table 6.3: Proposed strain at peak stress relationship regression coefficients ................. 86
CHAPTER 7
Table 7.1: R2 values of the ultimate strain trial regressions.............................................. 93
Table 7.2: Required significance level to pass the KomolgorovSmirnov normality test 94
Table 7.3: Proposed ultimate strain relationship regression coefficients ......................... 95
CHAPTER 9
Table 9.1: Independent properties to be reported from future fire tests ......................... 106
Table 9.2: Dependent properties to be reported from future fire tests............................ 107
APPENDIX
Table A.1: Database entry............................................................................................... 109
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ACKNOWLEDGEMENTS
This research was funded by the Portland Cement Association (PCA) through a PCA Education Foundation Fellowship. This support is gratefully acknowledged. In addition, the authors would like to thank David N. Bilow, formerly the Director of Engineered Structures at PCA and Dr. T. D. Lin, President of Lintek International, Inc. for providing their expertise and guidance for the research. The opinions, findings, and conclusions expressed in this report are those of the authors and do not necessarily reflect the views of the individuals or institutions acknowledged above.
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LIST OF SYMBOLS
ACI American Concrete Institute ASCE American Society of Civil Engineers ASTM American Society for Testing and Materials b Vector of ones used to constrain the regression equation C Coded variable for the calcareous aggregate type Cm Constraint matrix CEB Comites EuroInternational Du Beton CMU Concrete masonry unit Ec Concrete modulus of elasticity at elevated temperature Eco Concrete modulus of elasticity at room temperature fc Concrete compressive stress fcm Peak concrete compressive stress (i.e., strength) at elevated temperature fcmo Peak concrete compressive stress (i.e., strength) at room temperature HSC Highstrength concrete KS test KolmogorovSmirnov test L Coded variable for the lightweight aggregate type m Number of regression coefficients n Number of data points in the regression set
Number of data points in the first sample of a two sample hypothesis test Number of data points in the second sample of a two sample hypothesis test
NIST National Institute of Standards and Technology NSC Normalstrength concrete PCA Portland Cement Association r Number of data points at room temperature R Coded variable for the residual test type R2 Coefficient of determination S Coded variable for the stressed test type s Sample standard deviation
Sample standard deviation of the first sample of a two sample hypothesis test Sample standard deviation of the second sample of a two sample hypothesis
test Sample standard deviation of the ith variable in the regression
T Maximum exposure temperature t Student’s tTest statistic
Critical test statistic at a specific significance level v Number of statistical degrees of freedom wc Concrete mix water to cement ratio
Independent variable matrix Sample mean of the ith variable in the regression Response vector containing the regression dataset
z Number of possible values for a single variable
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α Significance level for hypothesis and KolmogorovSmirnov testing Regression coefficient vector Modified regression intercept term Normalized regression intercept term Modified regression coefficient Modified regression coefficient for the first data set in a two sample
hypothesis test Modified regression coefficient for the second data set in a two sample
hypothesis test Normalized regression coefficient
γc Concrete unitweight (lb/ft3) εc Concrete compressive strain εcm Concrete compressive strain at peak stress at elevated temperature εcmo Concrete compressive strain at peak stress at room temperature εcu Concrete compressive ultimate strain at elevated temperature εcuo Concrete compressive ultimate strain at room temperature
Proposed coefficient for modulus of elasticity relationship Proposed coefficient for compressive strength relationship Proposed coefficient for strain at peak stress relationship Proposed coefficient for ultimate strain relationships
λ Penalty function parameter
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CHAPTER 1: INTRODUCTION
1.1 Project Background
Structural design in the U.S. does not consider fire as a load condition even though fire can affect the structural performance of buildings to a degree equal to or greater than other load types (e.g., dead, live, wind, earthquake loads). The current U.S. fire design specifications provide prescriptive requirements on the relative fire performance of different building components using the concept of “fire endurance.” For example, according to ACI 216 (2007), the fire endurance of a concrete bearing wall is governed by its ability to confine a fire over a specified period of time rather than by its structural strength or stability, and the fire endurance for a bearing wall can be determined similar to a concrete slab. As such, currently available design methods and analysis tools cannot be used to evaluate structural performance under a specified fire loading. There is a need for predictive, performancebased structural fire design standards and code provisions as an alternative to the current prescriptive design methodology (NIST 2005).
In recent years, fire hazard mitigation problems have become increasingly difficult, in part, due to considerations of increased fire risk and hazard (NIST 2005). At the same time, the current fire design provisions in the U.S. date back to the early 1900s, suggesting that a major overhaul is needed. The most fundamental step in the rational fire design of structural systems is the development of basic knowledge on temperaturedependent material properties. Many of the currently available material property models for concrete structures in the U.S. are based on sparse sets of experimental data. A much larger experimental research database is available on the properties of North American concrete under elevated temperatures, and since the concrete property models are solely based on empirical evidence, a more complete representation of the existing database is needed. The research presented in this report focuses on this issue.
1.2 Project Objectives
The main objectives of this project are to analyze the current state of knowledge on the behavior of unreinforced North American concrete exposed to elevated temperatures from fire and to develop predictive relationships for the temperaturedependent compressive stressstrain properties of concrete. The project is in response to a solicitation by the Portland Cement Association (PCA), which calls for a “synthesis of properties of concrete used in fire resistance calculations of concrete structures.” The project has the following two additional objectives: (1) to evaluate the existing information on the temperaturedependent mechanical and thermophysical properties of concrete considering current and future trends in concrete technology and structural fireresistant design; and (2) to formulate future research needs for the structural fireresistant design of concrete structures.
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As a means of achieving its goals, the research project consists of four major components: (1) the development of a database for use in the understanding of the temperaturedependent mechanical and thermophysical properties of concrete as well as for future use of this data in the creation of a design/analysis tool; (2) the development of predictive relationships for the concrete compressive strength (i.e., peak stress), modulus of elasticity, strain at peak stress, ultimate strain, and stressstrain behavior at elevated temperatures; (3) the development of a template for how future firerelated research should be conducted and reported in order to provide consistent data for the continued study of the properties of concrete under elevated temperatures; and (4) the identification of areas in the current knowledgebase where little data exists and future research efforts should be focused.
1.3 Project Scope
A major outcome from this report is the development of predictive multiple least squares regression relationships between the maximum exposure temperature and the compressive stressstrain properties of unreinforced concrete. The heterogeneous nature of concrete leads to significant variability in its behavior, making deterministic prediction of its stressstrain properties difficult. Because of this variability, much of the previous research on the temperaturedependent behavior of concrete is experimental. As a result, a theoretical development is not sought in this report and the proposed temperaturedependent relationships are based on a statistical analysis of the existing experimental data from previous research. It should be noted that while the experiments utilized in this research represent the elevated temperatures from fire, most of the experiments were conducted using electric furnaces, and thus, they may not fully simulate the heat transfer associated with an actual fire (e.g., convection instead of radiation).
The regression relationships presented are intended to be used as a predictive guide to the stressstrain properties of North American concrete and, as such, only experimental data obtained using North American materials (e.g., aggregates) is considered. The data includes normalstrength and highstrength concrete specimens with normalweight as well as lightweight aggregates. The constitutive aggregates have a significant effect on the temperaturedependent behavior of concrete. Because of the large percentage of aggregate present in a typical concrete mix, and because of the inherent variability in concrete properties, by limiting the research to only North American materials, the consistency of the resulting regression relationships can be controlled to an extent.
1.4 Report Layout
Chapter 2 focuses on the available literature on the behavior of concrete exposed to elevated temperatures. A background review focusing on the existing experimental data as well as predictive relationships for the stressstrain properties of concrete from previous research and current design codes is presented.
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Chapter 3 introduces the North American concrete property database developed by this project. The format, makeup, and use of the database are discussed in detail.
Chapters 4 through 7 detail the statistical analysis and the resulting predictive relationships for the concrete compressive strength, modulus of elasticity, strain at peak stress, and ultimate strain, respectively, developed by the project. The proposed relationships are evaluated for fit to the available experimental data and comparisons are conducted with previous relationships. Furthermore, the effects of different independent parameters (e.g., concrete compressive strength at room temperature, aggregate type, test type) on the temperaturedependent property models are discussed in each chapter.
Chapter 8 combines the proposed temperaturedependent relationships for the compressive strength, modulus of elasticity, strain at peak stress, and ultimate strain of concrete with a commonly used concrete stressstrain relationship at room temperature to develop predictive compressive stressstrain models at elevated temperatures.
Finally, Chapter 9 provides a brief summary of the work conducted and the conclusions reached. Recommendations for future research and for the development and presentation of future experimental data to increase the current state of knowledge in this field are also presented.
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CHAPTER 2: LITERATURE REVIEW
This chapter provides an overview of previous literature focusing on the stressstrain properties of concrete exposed to elevated temperatures. Section 2.1 investigates the findings of past experimental research and Section 2.2 looks at existing property relationships from previous research and current design codes.
2.1 Previous Experimental Data
A collection of 14 papers (Abrams 1971, Castillo and Durani 1990, Cheng et al. 2004, Cruz 1968, Cruz and Gillen 1981, Gillen 1980, Harmathy and Berndt 1966, Kerr 2007, Lankard et al. 1971, Phan and Carino 2001, Philleo 1958, Saemann and Washa 1960, Van Geem et al. 1997, and Zoldners 1960) on the behavior of unreinforced concrete at elevated temperatures was studied. These papers present previous experimental research utilizing North American materials, focusing on the temperature dependency of the mechanical, thermal, and physical properties of concrete. This report is on the compressive stressstrain properties of concrete, and as such, the previous research on the compressive strength (i.e., peak stress), modulus of elasticity, strain at peak stress, ultimate strain, and creep strain is reviewed below. Note that although some of the past research provides experimental stressstrain curves, the results below are summarized with respect to the compressive strength, modulus of elasticity, and strain relationships associated with each research program.
2.1.1 Compressive Strength
The concrete compressive strength is the most commonly presented property in the papers studied. Several researchers (Abrams 1971, Castillo and Durani 1990, Cheng et al. 2004, Harmathy and Berndt 1966, Kerr 2007, Lankard et al. 1971, Phan and Carino 2001, Saemann and Washa 1960, and Zoldners 1960) discuss the effects of elevated temperatures on the concrete strength, with a total of 647 test results documenting the general trend that the compressive strength decreases as the temperature is increased.
Abrams (1971) is one of the most referenced papers in the literature survey on normalstrength concrete. The research involved 3in x 6in (7.6cm x 15.2cm) cylindrical specimens that were subjected to three test scenarios (stressed during heating, unstressed during heating, and residual tests). In the “residual” test, the concrete specimen is first heated to a specified temperature, then allowed to cool to room temperature, and then loaded to failure under uniaxial compression. This test type is intended to evaluate the remaining strength of a concrete structure following a fire. In the “stressed” test, the specimen is heated while subjected to an axial preload of typically 0.250.55fcmo, where fcmo is the concrete strength at room temperature. The objective of the preload is to represent the axial load that may be present in a concrete member (e.g., a column) prior to the start of a fire. Once the specified temperature is reached, further
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axial compression is applied to the specimen until failure. In the “unstressed” test type, the concrete specimen is heated to a specified temperature (with no preload), and then subjected to uniaxial compression until failure. The unstressed test type acts as a baseline for the residual and stressed test types.
Abrams (1971) concluded that the specimens retaining the highest strength at elevated temperatures were the stressed specimens, followed by the unstressed and finally the residual specimens. Also, it was noted that the unstressed carbonate aggregate and lightweight aggregate specimens retained nearly 75% of their strength at temperatures up to 1,200°F (649°C). Abrams (1971) observed that varying the amount of preload and the concrete compressive strength at room temperature did not significantly affect the results. However, the range of the room temperature compressive strength only varied from 3,900 psi to 6,300 psi (26.9 MPa to 43.4 MPa). Lastly, it was shown in some stressed tests that although the general trend was a decrease in compressive strength with increased temperature, there was initially a slight increase in strength with the addition of relatively small amounts of heat.
Zoldners (1960) studied the effects of aggregate type on the compressive strength of unstressed normalstrength 4in x 8in (10.2cm x 20.3cm) cylinders, as well as beamends, for temperatures up to 1,480°F (804°C). These tests showed that limestone concrete had the most retained strength up to 1,480°F (804°C) at about 40%, followed by lightweight expanded slag concrete with about 30% retained strength and both gravel and sandstone concretes showed similar results at the highend temperature with about 20% retained strength. It can be shown from this data that aggregate type significantly changes the effect of increased temperature on the compressive strength of concrete.
Research conducted by Saemann and Washa (1960) investigated 6in x 12in (15.2cm x 30.5cm) normalstrength concrete cylinders at temperatures up to 450°F (232°C). The results showed that the concrete strength tends to initially decrease as much as 15% up to temperatures of about 250°F (121°C) where it then experiences a slight increase up to about 400°F (204°C) and then begins to decrease again through 450°F (232°C). It was concluded that the general trend was in agreement with Abrams (1971) where as the temperature is increased, the compressive strength tends to decrease.
Harmathy and Berndt (1966) collected data on the compressive strength of normalstrength lightweight 1.9in x 3.8in (4.8cm x 9.7cm) specimens cored from concrete masonry unit (CMU) blocks. The unstressed tests confirmed the general trend from Abrams (1971) that even with identical mix designs and similar environmental conditions, there are notable differences in the compressive strengths of the specimens. The results also showed that until about 400°F (204°C), there is little to no decrease in strength and there is a gradual decrease in strength in the range of 500 to 1400°F (260 to 760°C) for these cored specimens.
Lankard et al. (1971) studied the compressive strength of 4in x 8in (10.2cm x 20.3cm) normalstrength concrete cylinders under unstressed and residual tests with temperatures up to 500°F (260°C). The focus of the research was to study the effects of different moisture and pressure conditions present at the time of heating. This included
7
specimens tested at standard temperature and pressure conditions as well as specimens heated using steam pressure. The steam pressure test results are not included in this report because they do not mimic standard atmospheric fire conditions. Results from the standard atmospheric tests from Lankard et al. (1971) seem to show that for temperatures up to 500°F (260°C), there is no significant decrease in strength under the test conditions. In fact, some of the specimens actually saw an increase in compressive strength initially, after which they experienced strength decreases to 500°F (260°C) ending at only slightly less strengths than the initial room temperature compressive strength.
The effect of elevated temperature on highstrength concrete was also investigated in previous research. As noted earlier, Abrams (1971) showed that the use of higher strength concrete mixes did not change the effects of temperature. However, Abrams only studied strengths up to 6,300 psi (43.4 MPa). Other researchers (Castillo 1988, Cheng et al. 2004, Kerr 2007, Phan 2001) studied specimens with room temperature compressive strengths of up to approximately 13,000 psi (89.6 MPa) and concluded that highstrength concrete (HSC) can behave significantly differently than normalstrength concrete (NSC) because of the different materials used and the different makeup of the concrete mix.
The most comprehensive of all the HSC studies to date is a NIST test series conducted by Phan and Carino (2001). Stressed, unstressed, as well as residual tests were conducted on 4in x 8in (10.2cm x 20.3cm) specimens of both HSC and NSC. It was concluded first that in the stressed tests, the amount of preload has no effect on the strength reduction, which is consistent with the results found by Abrams (1971) for NSC. The residual tests produced less strength loss than the unstressed and stressed tests up to about 570°F (299°C) but the trend reversed above this temperature and the residual tests showed more strength loss than the unstressed and stressed tests. Phan and Carino (2001) concluded that for water to cement ratios of wc = 0.22 and 0.33, wc = 0.22 has less loss in strength. However, for the difference between wc = 0.33 and 0.57, the data proves to be inconclusive as to a general trend for the effect of wc on the compressive strength at high temperatures. It was shown that admixtures such as silica fume could affect the strength loss at temperatures before the chemically bound water is allowed to leave the cement matrix (which occurs at about 302 to 480°F [150 to 249°C]). Lastly, the NIST experiments showed that some HSC mixes have the potential for explosive spalling of unreinforced concrete at high temperatures, with the most likely candidates for explosive spalling being the specimens with smaller wc.
Castillo and Durani (1990) conducted stressed and unstressed tests of 2in x 4in (5.1cm x 10.2cm) HSC cylinders. It was concluded that after initial strength losses of about 20% up to 572°F (300°C), there is strength regain in the range of 572 to 752°F (300 to 400°C). After this range, the concrete strength continues to decrease with increased temperature. It was also noted that the stressed specimens retain more strength at higher temperatures than the unstressed specimens. However, nearly one third of all the stressed specimens failed explosively past about 1300°F (704°C).
Two other researchers (Cheng et al. 2004, Kerr 2007) concluded similar trends of HSC specimens in regard to compressive strength as Phan and Carino (2001) and Castillo
8
and Durani (1990). It was observed that for HSC, there is initially a decrease in compressive strength with increased temperature followed by a temperature range of regained strength and ultimately a decrease in strength as the temperature is increased further.
2.1.2 Modulus of Elasticity
The static modulus of elasticity is defined as the slope of the concrete compressive stressstrain curve either as a tangential slope at the origin, or as the secant slope between the origin and a point on the stressstrain curve at approximately 3040% of the peak stress. The static modulus of elasticity is reported in 7 of the 14 papers studied in this report (Castillo and Durani 1990, Cheng et al. 2004, Harmathy and Berndt 1966, Kerr 2007, Lankard et al. 1971, Phan and Carino 2001, and Saemann and Washa 1960), which is only one less than the number of papers presenting data on the compressive strength of concrete. However, there are significantly fewer test results in total for the static modulus of elasticity (275 data points for the static modulus of elasticity versus 647 data points for compressive strength). Also, it is very typical that the previous research focused more on the compressive strength loss than it did the loss of stiffness, as reflected in the smaller amount of discussion given in the related papers relative to the compressive strength.
Researchers typically looked at residual, unstressed, or stressed specimens considering NSC, HSC, or both NSC and HSC. The researchers who looked at NSC (Harmathy and Berndt 1966, Lankard et al. 1971, and Saemann and Washa 1960) all concluded that the general trend for the modulus of elasticity is that as the temperature increases the stiffness decreases. This is what would be expected of the stiffness given the relationship of the stiffness with the compressive strength. It was shown that there is a slight decrease in the modulus of elasticity at temperatures up to 500°F (260°C) after which the rate of decrease of the modulus is greater. Furthermore, Saemann and Washa (1960) reported that lightweight concrete does not see as much decrease in stiffness as normalweight concrete does. Harmathy and Berndt (1966) found that the elevated temperature exposure time generally has an adverse effect on the stiffness. From the data, it is evident that as the heating duration is increased, the concrete stiffness decreases.
The researchers who studied the static modulus of HSC (Castillo and Durani 1990, Cheng et al. 2004, Kerr 2007, and Phan and Carino 2001) showed that the general trends for HSC are the same as for NSC.
The dynamic modulus of elasticity was also studied as a measure of stiffness although it is only presented in three papers (Kerr 2007, Phan and Carino 2001, and Philleo 1958) and has almost 100 fewer test results than the static modulus. The two methods for determining the dynamic modulus include the ultrasonic pulse velocity calculation and the resonance frequency calculation. The data showed that the trend for dynamic modulus followed that of the static modulus. It was shown that at temperatures up to 1400°F (760°C), the dynamic modulus was reduced to less than half of the concrete stiffness at room temperature.
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2.1.3 Strain at Peak Stress, Ultimate Strain, and Creep Strain
The properties discussed in this section each had three or less than three papers that provided data on them. Also, as described above in the modulus of elasticity review, these properties were not a significant focus of the papers and therefore were not discussed in great detail.
Two researchers (Castillo and Durani 1990, Cheng et al. 2004) studied the effects of temperature on the concrete strain at peak compressive stress. It was shown that for temperatures up to 392°F (200°C), the strain at peak stress does not vary significantly. For the temperature range of 572 to 752°F (300 to 400°C), the strain at peak stress increases slightly, beyond which it increases much more significantly.
Test data on ultimate strain was reported by three researchers (Castillo and Durani 1990, Cheng et al. 2004, and Harmathy and Berndt 1966). Ultimate strain is defined as the maximum strain reached by the concrete specimen before failure. For the purposes of this report, ultimate strain is taken as the strain corresponding to 85% of the peak stress in the postpeak range of the stressstrain relationship. In cases where an abrupt drop in stress occurs (indicating failure) prior to reaching 85% of the peak stress, the strain at the stress drop is taken as the ultimate strain. Castillo and Durani (1990) and Cheng et al. (2004) provided experimental stressstrain curves, however Harmathy and Berndt (1966) did not. Instead, the ultimate strain values were reported as “deformation at fracture,” and knowing the size of the specimen allowed for the strain to be calculated. Similar to the strain at peak stress, the general trend is that the ultimate strain increases as temperature is increased.
Lastly, two researchers (Cruz 1968, Gillen 1980) studied creep strains as a function of temperature and time. It was concluded by Gillen (1980) that at extreme elevated temperatures, timedependent strains of statically loaded concrete can be almost 40 times greater than the strains in room temperature specimens. Furthermore, this timedependent behavior is greatly influenced by the amount of moisture present in the concrete prior to heat exposure, aggregate type, as well as the compressive strength of the concrete at room temperature. As a function of time, it was shown that for a given temperature, approximately half of the creep strains occur within the first hour of a fivehour test. This provides the general trend that the creep strain rate continually decreases with time for a given temperature.
2.2 Previous Proposed Relationships
This chapter reviews existing relationships on the temperaturedependent stressstrain properties of concrete from previous research (Hertz 2005, Kodur et al. 2008, Li and Purkiss 2005, Phan and Carino 2003, and Shi et al. 2002) and code documents (ACI 2007, ASCE 1992, Comites 1991, Concrete 1991, Eurocode 2002, and Eurocode 2004). The primary focus of most of the existing relationships is the compressive strength (i.e., peak stress) of concrete at elevated temperatures, although previous stressstrain curves and relationships for the concrete modulus of elasticity, strain at peak stress, and ultimate strain also exist.
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2.2.1 StressStrain Relationships
Figure 2.1 shows the temperaturedependent stressstrain (fcεc) models for concrete in compression from ASCE (1992) and Kodur et al. (2008). The fcεc curves are normalized with respect to the strength of concrete at room temperature, fcmo, and the last point on each curve represents the ultimate strain, εcu, as defined earlier.
(a) (b)
Figure 2.1: North American temperaturedependent stressstrain models: (a) ASCE; and (b) Kodur et al.
The ASCE fcεc model is for normalstrength concrete (both calcareous and siliceous aggregate) whereas the model proposed by Kodur et al. was developed by modifying the ASCE equations for higher concrete strengths. The ASCE and Kodur et al. concrete strength (i.e., peak stress), modulus of elasticity, strain at peak stress, and ultimate strain relationships described later come from the fcεc models in Figure 2.1. It can be seen that the Kodur et al. fcεc curves show a sharper drop in compressive stress beyond the peak stress point, which is likely because of the more brittle nature of highstrength concrete.
The ASCE and Kodur et al. models together provide temperaturedependent fcεc relationships for highstrength concrete (HSC) and normalstrength concrete (NSC); however, these models do not differentiate between different concrete aggregate types or test types. Furthermore, it is stated in the ASCE Structural Fire Protection Manual (1992) that the ASCE fcεc model implicitly takes into account the creep of concrete at high temperatures. As described later, the Kodur et al. model is also expected to implicitly include creep strains, which pose the following difficulties for the use of these models in practice: (1) the creep effects included in the resulting fcεc curves are based on the work conducted by Ritter (1899) and Hognestad (1951), which predate most of the research on concrete creep at high temperatures (e.g., Cruz 1968, Gillen 1980); and (2) since creep strains are not included explicitly, the amount of time needed to accumulate these predicted strains cannot be determined.
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(a) (b)
(c) (d)
(e) (f)
Figure 2.2: Eurocode temperaturedependent stressstrain models: (a) NSC, siliceous; (b) NSC, calcareous; (c) NSC, lightweight; (d) HSC, Class 1; (e) HSC, Class 2; and (f)
HSC, Class 3.
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Models to estimate the compressive stressstrain behavior of concrete as affected by fire are also available from European sources. For example, Figure 2.2 shows the Eurocode (2002, 2004) temperaturedependent fcεc models for concrete in compression. The fcεc curves are again normalized with respect to fcmo, and the last point on each curve represents the ultimate strain, εcu. Similar to the North American models, the Eurocode models show the same general trends that as temperature is increased, significant decreases in peak stress are observed as well as large increases in strain. Unlike the North American fcεc models, the Eurocode models for normalstrength concrete consider three different aggregate types (lightweight, calcareous, and siliceous aggregates). For highstrength concrete, Eurocode makes no distinction for the type of aggregate, but there are separate relationships for three different strength classifications (Class 1: fcmo = 7,977 to 8,702 psi [55.0 to 60.0 MPa]; Class 2: fcmo = 10,153 to 11,603 psi [70.0 to 80.0 MPa]; and Class 3: fcmo = 13,053 psi [90.0 MPa]).
Note that the Eurocode fcεc models are based on concrete samples made from European constitutive materials. These models may not be suitable for North American concrete because of differences in the temperaturedependent properties of the materials used in the concrete mix (especially the constituent aggregates, which make up approximately 70% of the concrete mix volume depending on the mix design). Furthermore, similar to the ASCE and Kodur et al. models, the Eurocode models implicitly account for creep deformations in the concrete strains. As a result, these models cannot be used to predict the temperaturedependent behavior of concrete subjected to loading over shortdurations.
2.2.2 Compressive Strength
Temperaturedependent concrete compressive strength loss models in the U.S. are provided by ACI 216 (2007), ASCE (1992), and Kodur et al. (2008). As shown in Figure 2.3, the ACI 216 models (given as fcm / fcmo versus temperature, T, where fcm is the concrete strength at maximum exposure temperature, T and fcmo is the strength at room temperature) are grouped based on the aggregate type as: (1) siliceous; (2) calcareous; and (3) lightweight. The models are further divided based on the test type as: (1) residual; (2) stressed; and (3) unstressed.
It can be seen from Figure 2.3 that the ACI 216 models predict significant losses in the concrete strength as the temperature is increased. The largest and smallest losses are observed under the residual and stressed test types, respectively. Furthermore, there are significant differences between the siliceous concrete models and the calcareous and lightweight concrete models, especially under the stressed and unstressed test types. It should be noted that these models were developed using test results from a single research program (Abrams 1971), based on a total of 154 data points covering three aggregate types (calcareous sand and gravel, siliceous sand and gravel, and expanded shale lightweight aggregate) and three test types (residual, stressed, and unstressed).
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(a) (b)
(c)
Figure 2.3: ACI 216 compressive strength models at elevated temperatures: (a) residual; (b) stressed; and (c) unstressed.
Figure 2.4 shows the ASCE (1992) and Kodur et al. (2008) compressive strength models at elevated temperatures. As previously described, the ASCE model is for normalstrength concrete (NSC) whereas the model proposed by Kodur et al. was developed by modifying the ASCE equations for highstrength concrete (HSC). The ASCE and Kodur et al. fcm models correspond to the peak stress on the respective stressstrain (fcεc) relationships in Figure 2.1. Although these two models do not distinguish between the aggregate type, they do show the differences between HSC and NSC on the strength loss at elevated temperatures. It can be seen that normalstrength concrete is expected to behave according to a bilinear relationship, where there is no strength loss until a temperature of 842°F (450°C). In contrast, highstrength concrete experiences an immediate strength reduction followed by a constant strength range from 212°F to 752°F (100°C to 400°C), after which the strength linearly decreases with increased temperature.
Figure 2.4: ASCE and Kodur et al. compressive strength models at elevated temperatures.
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Looking at European sources, Figure 2.5(a) shows the Eurocode (2002, 2004) strength loss models, Figure 2.5(b) shows the models proposed by the Comites EuroInternational Du Beton (CEB) (1991), and Figure 2.5(c) depicts the models provided by the Concrete Association of Finland (1991) code provisions Rak MK B4. Different from the ACI 216 guidelines, the Eurocode and CEB models do not consider the test type as a parameter. Furthermore, the Rak MK B4 provisions do not include the aggregate type as a parameter; however, different models are provided for normalstrength concrete and highstrength concrete.
(a) (b)
(c)
Figure 2.5: European temperaturedependent compressive strength models: (a) Eurocode; (b) CEB Model Code 90; and (c) Rak MK B4.
A few additional models for the concrete strength loss with temperature are available in the literature. The models proposed by Hertz (2005), shown in Figures 2.6(a) and 2.6(b), are for normalstrength concrete and consider the effects of aggregate type and test type. According to Hertz, the unstressed test is more conservative than the stressed test (i.e., it results in larger strength loss), and thus, the stressed test type is not included in the prediction models. The models by Shi et al. (2002) and Li and Purkiss (2005), shown in Figure 2.6(c), are also for normalstrength concrete; however, the test type or aggregate type with which these models should be used was not reported.
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(a) (b)
(c)
Figure 2.6: Other temperaturedependent compressive strength models: (a) Hertz unstressed; (b) Hertz residual; and (c) Shi et al., Li and Purkiss, and Phan and Carino.
Note that similar to the Eurocode models, the CEB and Rak MK B4 models in Figure 2.5, and the Hertz, Shi et al., and Li and Purkiss models in Figure 2.6 are based on concrete samples made from nonNorth American constitutive materials. The Phan and Carino (2003) model is the only NorthAmerican model not related to the current U.S. design codes, and was developed based on a large number of tests on highstrength and normalstrength concrete specimens. It was shown that the ACI 216 curves in Figure 2.3 result in unconservative estimates of the strength loss for highstrength concrete; and thus, the model in Figure 2.6(c) was proposed for highstrength calcareous concrete as a conservative estimate for any test type.
2.2.3 Modulus of Elasticity
Temperaturedependent relationships for the modulus of elasticity, Ec of concrete in compression are provided in the U.S. by ACI 216 (2007), ASCE (1992), and Kodur et al. (2008). Figure 2.5 shows these models, which are normalized with respect to the modulus of elasticity at room temperature, Eco. The relationships provided in ACI 216 come from a sparse data set from a single test program (Cruz 1966) containing as few as five data points per curve, a single concrete mix with a nominal room temperature strength of
�
fcmo = 4,000 psi (27.6 MPa), and a single test type (unstressed). These relationships are provided without any corresponding stressstrain (fcεc) model. In
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comparison, the Ec relationships from ASCE and Kodur et al. were determined by taking the derivative of the corresponding fcεc relationships (see Figure 2.1) with respect to strain. It can be seen that the general trend is for concrete to lose its initial stiffness immediately after heating.
(a) (b)
Figure 2.7: North American temperaturedependent modulus of elasticity models: (a) ACI 216; and (b) ASCE and Kodur et al.
As described previously, the ASCE fcεc curves implicitly take into account the creep of concrete at high temperatures. As a result, it can be seen in Figure 2.7 that these curves result in a significantly smaller modulus of elasticity as compared to the ACI models. It can also be seen that the Kodur et al. modulus loss model does not vary significantly from the ASCE curve; and therefore, the Kodur et al. model is also expected to implicitly include creep strains. Concrete members subjected to loading over short durations may have significantly smaller strains at elevated temperatures than the strains determined from the current ASCE and Kodur et al. models.
In Europe, temperaturedependent Ec models for concrete in compression are provided by Eurocode (2002, 2004). Figure 2.8 shows these models, which are also normalized with respect to the modulus of elasticity at room temperature, Eco. The Eurocode modulus of elasticity models were determined by taking the derivative of the corresponding fcεc relationships (see Figure 2.2) with respect to strain. The general trend can be seen that as fcmo is increased, the Ec / Eco ratio at elevated temperatures gets reduced. As compared with the ACI models in Figure 2.7, the Eurocode Ec / Eco curves for normalstrength concrete show smaller differences between the different aggregate types. The differences between the Ec / Eco ratio of the different class highstrength concrete types in the Eurocode models are also small. Furthermore, the Eurocode curves result in significantly smaller modulus of elasticity at elevated temperatures than the ACI curves and are similar to the ASCE and Kodur et al. curves; therefore, it is expected that the Eurocode modulus models implicitly include creep strains as well.
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(a) (b)
Figure 2.8: Eurocode temperaturedependent modulus of elasticity models: (a) NSC; and (b) HSC.
2.2.4 Strain at Peak Stress
The temperaturedependent strain at peak stress, εcm, of concrete obtained from the ASCE (1992), Kodur et al. (2008), and Eurocode (2002, 2004) fcεc relationships are shown in Figure 2.9. Unlike the modulus of elasticity models, these strain at peak stress models are either given as explicit equations of temperature (ASCE), or explicit equations of temperature and room temperature compressive strength (Kodur et al.), or tabulated based on temperature (Eurocode). At room temperature, the strain at peak stress models experience relatively similar εcmo values with somewhat different levels of increase in εcm with increased temperatures. The Eurocode model assumes a room temperature strain at peak stress of
�
εcmo = 0.0025 and the ASCE and Kodur et al. models assume a roomtemperature strain of approximately
�
εcmo = 0.0026 depending on the room temperature compressive strength (i.e., peak stress, fcmo) of concrete. It can be seen by looking at the ASCE and Kodur et al. models that at elevated temperatures, normalstrength concrete tends to have larger strains at peak stress than highstrength concrete. The Eurocode εcm model does not differentiate between the three aggregate types (lightweight, calcareous, and siliceous) or strength classifications (Classes 1, 2, and 3) considered in the Eurocode fcεc relationships (see Figure 2.2).
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Figure 2.9: ASCE, Kodur et al., and Eurocode temperaturedependent strain at peak stress models.
2.2.5 Ultimate Strain
Temperaturedependent models for the ultimate strain, εcu, of concrete from ASCE (1992), Kodur et al. (2008), and Eurocode (2002, 2004) are given in Figure 2.10. The Eurocode εcu model is given as an explicit tabulated relationship based on temperature; however, ASCE and Kodur et al. do not provide any explicit relationships for εcu. Therefore, in this report, the ASCE and Kodur et al. εcu models were determined as the postpeak strain corresponding to 85% of the peak stress, fcm from the corresponding fcεc relationships in Figure 2.1.
Figure 2.10: ASCE, Kodur et al., and Eurocode temperaturedependent ultimate strain models.
It can be seen in Figure 2.10 that the three models result in very different relationships for εcu. The ASCE model assumes that the ultimate strain of concrete at room temperature is reached at
�
εcuo = 0.006 and the Kodur et al. model assumes a value of
�
εcuo = 0.004. The Kodur et al. model shows that highstrength concrete is expected to have a smaller ultimate strain than normalstrength concrete at elevated temperatures.
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According to the Eurocode model, the ultimate strain of concrete at room temperature is reached at
�
εcuo = 0.020, which is a much bigger strain than those used in ASCE and Kodur et al., and is also significantly larger than the 0.003 value assumed as the maximum usable strain of concrete in Chapter 10 of ACI 318 (2008). This difference may be due to some implicit inclusion of creep strains in the Eurocode model. It can also be seen that similar to εcm, the Eurocode εcu model does not differentiate between the aggregate types or between the different strength classifications considered in the Eurocode fcεc relationships in Figure 2.2.
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CHAPTER 3: CONCRETE PROPERTY DATABASE
A database of previous experimental data on the mechanical, thermal, and physical properties of concrete under elevated temperatures was collected for this research. This chapter provides a closer look at the makeup and use of this database. Section 3.1 gives an overview of the need for and major uses of the database. Section 3.2 provides a description of the properties collected. Lastly, Section 3.3 shows a breakdown of the range of different material and test properties for the data based on each of the stressstrain properties investigated in this report.
3.1 Database Overview
One of the most important limitations of the previous models focusing on the compressive stressstrain properties of North American concrete under elevated temperatures is that these models are based on sparse sets of data. For example, the ACI 216 fcm and Ec models described in Chapter 2 are each from a single research program [Abrams (1971) for fcm, and Cruz (1966) for Ec], containing as few as five data points from a single concrete mix per curve shown. A considerably larger amount of data exists on the temperaturedependent properties of North American concrete, and since the predictive models are based solely on experimental data, a more complete representation of the existing information is needed. In accordance with this need, a database of previous experimental research on the properties of North American unreinforced concrete under elevated temperatures was developed. The primary objectives of this database are: (1) to collect, sort, and store the test data; (2) to synthesize the existing knowledge on the temperaturedependent properties of concrete; and (3) to make recommendations for future research. In addition, guidelines for presenting future test results are discussed in Chapter 9 based on an evaluation of the collected data.
The database includes a total of 14 papers (Abrams 1971, Castillo and Durani 1990, Cheng et al. 2004, Cruz 1968, Cruz and Gillen 1981, Gillen 1980, Harmathy and Berndt 1966, Kerr 2007, Lankard et al. 1971, Phan and Carino 2001, Philleo 1958, Saemann and Washa 1960, Van Geem et al. 1997, and Zoldners 1960) from North American sources, dating from the 1950s to the present, reporting various mechanical, physical, and thermal properties of unreinforced concrete at temperature. The major use of the database, which was created in Microsoft Access®, comes from the ability to quickly sort and plot the experimental results based on userspecified criteria. For example, to look at the effects of the watertocement ratio, wc and aggregate type on the concrete strength at elevated temperatures, a query can be constructed to limit the data to each specific type of aggregate (calcareous, siliceous, etc.) and to specific ranges of wc. If a specific temperature range or a specific range of room temperature concrete strength is desired, those limits can easily be applied as well, and the data can be transferred into a spreadsheet program, such as Microsoft Excel®, or input to MATLAB®, via a text file, for analysis.
22
This database was used as a major tool in the development of the temperaturedependent relationships described in this report. The database can be downloaded from the research website at http://www.nd.edu/~concrete/concretefiredatabase. Through a graphical user interface (GUI) built in MATLAB®, users of the website can access the full database collected in this research. The GUI gives the user the following capabilities: (1) to plot up to four separate sets of data along with the ability to set the x and y axes for each graph; (2) to investigate data subsets by limiting each independent and dependent property stored in the full database; (3) to apply constrained or unconstrained multiple linear regression to the data with a userdefined polynomial equation form of any order; and (4) to display the userdefined regression equations and statistics.
3.2 Database Properties
The structure of the database is such that it is divided into three separate tables: (1) author information; (2) paper information; and (3) data. Each paper added to the database has some necessary information taken before the data is collected. Items such as the paper title, publication name, date of publication, location of research, author names, etc. are entered into the database. Following the entry of paper and author information, each paper in the database was read thoroughly to develop an understanding of the research conducted. Next, the paper was reread, collecting all of the available information about the tests conducted. In some instances, the test data was given in tabular format. However, the data from many of the source papers was presented in graphs. To collect this data, a scanned copy of each graph was digitized using the GraphClick program (© 2007 ArizonaSoftware, version 3.0, originally downloaded in June 2007, http://www.arizonasoftware.ch/graphclick/). Following this process, the data was transferred into Microsoft Excel® spreadsheets. Each paper had its own individual spreadsheet containing the available test information. After one paper was completely documented, its spreadsheet was used as a template for collecting data from the next paper. Note that each paper was different in the way that the data was presented and in the concrete properties that were reported. For example, one paper would conduct tests to determine how the compressive strength was affected by elevated temperatures, and the next paper would test the effect of temperature on the dynamic modulus of elasticity. This meant that the spreadsheet template for data entry was continuously evolving with each paper. Once all 14 papers were processed, the final spreadsheet was much different and much larger than the original. Then, using the final form of the data spreadsheet, each paper was examined one more time to make sure that the results were input consistently. The final data spreadsheets were then combined into a single spreadsheet and input into the database. By using a specific code for each paper in the data table, the author information and paper information could be linked to each piece of data in the database.
The final form of the data table, available in the database, separates the input data into two separate categories: independent properties and dependent properties. The independent properties contain information about each test and are further broken up as:
Mix properties – the makeup of the concrete materials (see Table 3.1); Curing properties – the manner in which the concrete was cured (see Table
3.2);
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Specimen properties – the physical characteristics of the test specimens (see Table 3.3); and
Test properties – the manner in which the specimens were tested (see Table 3.4).
TABLE 3.1: MIX PROPERTIES COLLECTED IN THE DATABASE
Mix Sand/Cement Ratio Sand Type Mix Aggregate/Cement Ratio Sand Origin
Mix Water/Cement Ratio Silica Fume Amount (lb/yd3) Mix Unit Weight (lb/ft3) Fly Ash Amount (lb/yd3)
Slump (in) Water Reducer Amount (oz/yd3) Slump Comment Water Reducer Type
Cement Type Retarder Amount (oz/yd3) Cement Comment Retarder Type
Cement Content (lb/yd3) Air Entraining Admixture Amount (oz/yd3) Aggregate Type Air Entraining Admixture Type
Aggregate Origin Air Content (% of total volume) Maximum Aggregate Size (in) Air Content Comment
d50 Size (in)
TABLE 3.2: CURING PROPERTIES COLLECTED IN THE DATABASE
Initial Curing Humidity (RH) Subsequent Curing Humidity (RH) Initial Curing Temperature (°F) Subsequent Curing Temperature (°F)
Initial Curing Duration (days) Subsequent Curing Duration (days) Minimum Subsequent Curing Duration Known (days)
TABLE 3.3: SPECIMEN PROPERTIES COLLECTED IN THE DATABASE
Shape Air Dry Mass (g) Shape Comment Saturated Surface Dry Mass (g)
Length or Height (in) Oven Dry Density (lb/ft3) CrossSection Area (in2) Air Dry Density (lb/ft3)
Volume (in3) Saturated Surface Dry Density (lb/ft3) Surface Area/Volume Ratio (1/in) End Conditions
Oven Dry Mass (g)
24
TABLE 3.4: TEST PROPERTIES COLLECTED IN THE DATABASE
Test Type Minimum Age When Placed in Furnace
(days) Stress Level (% of fcmo) Heating Furnace Duration (mins)
Test Displacement Rate (in/min) Minimum Heating Furnace Duration
Known (mins) Test Displacement Rate Comment Heating Furnace Comment
Conditioning Chamber Type Heating Furnace Duration at Equilibrium
(mins) Conditioning Chamber Humidity (RH) Water Quenching Duration (mins)
Conditioning Chamber Temperature (°F) Residual Chamber Type Conditioning Chamber Duration (hrs) Residual Chamber Humidity (RH)
Conditioning Chamber Comment Residual Chamber Temperature (°F) Heating Furnace Type Residual Chamber Duration (days)
Heating Furnace Specification Residual Chamber Comment Furnace Volume/Specimen Volume Ratio Subsequent Residual Chamber Type
Heating Furnace Humidity (RH) Subsequent Residual Chamber Humidity
(RH)
Heating Furnace Temperature (°F) Subsequent Residual Chamber
Temperature (°F)
Heating Rate (°F/min) Subsequent Residual Chamber Duration
(days) Heating Rate Comment Subsequent Residual Chamber Comment
Specimen Age When Placed in Furnace (days)
The dependent properties collected in the database relate to the test results for each specimen. These temperaturedependent properties are broken up as:
Mechanical properties – the structural properties of the specimen (see Table 3.5);
Thermal properties – properties describing heat flow through the specimen (see Table 3.6); and
Physical properties – properties describing physical changes in the specimen (see Table 3.7).
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TABLE 3.5: MECHANICAL PROPERTIES COLLECTED IN THE DATABASE
Compressive Strength Before Fire, fcmo (lb/in2)
Resonant Frequency Before Fire, RFo (Hz)
fcmo Comment RFo Comment Compressive Strength After Fire, fcm
(lb/in2) Resonant Frequency After Fire, RF (Hz) fcm Comment RF Comment
fcm / fcmo RF/RFo fcm / fcmo Comment RF/RFo Comment
Strain at fcmo Before Fire, εcmo Dynamic Modulus of Elasticity Before
Fire, Edo (lb/in2) εcmo Comment Edo Comment
Strain at fcm After Fire, εcm Dynamic Modulus of Elasticity After Fire,
Ed (lb/in2) εcm Comment Ed Comment
εcm / εcmo Ed / Edo εcm / εcmo Comment Ed / Edo Comment
Ultimate Strain Before Fire, εcuo Creep Strain, εcr εcuo Comment εcr Comment
Ultimate Strain After Fire, εcu Shear Modulus Before Fire, Gco (lb/in2) εcu Comment Gco Comment
εcu / εcuo Shear Modulus After Fire, Gc (lb/in2) εcu / εcuo Comment Gc Comment
Young's Modulus Before Fire, Eco (lb/in2) Gc / Gco Eco Comment Gc / Gco Comment
Young's Modulus After Fire, Ec (lb/in2) Modulus of Rupture Before Fire, fro
(lb/in2) Ec Comment fro Comment
Ec / Eco Modulus of Rupture After Fire, fr (lb/in2) Ec / Eco Comment fr Comment
Ultrasonic Pulse Velocity Before Fire, Vco (ft/sec) fr / fro
Vco Comment fr / fro Comment Ultrasonic Pulse Velocity After Fire, Vc
(ft/sec) Linear Expansion, εt (103 in/in) Vc Comment εt Comment
Vc /Vco Thermal Coefficient of Expansion, αt
(1/F) Vc /Vco Comment αt Comment
26
TABLE 3.6: THERMAL PROPERTIES COLLECTED IN THE DATABASE
Thermal Conductivity, K (Btuin/ft2hr°F) Specific Heat, c (Btu/lb/°F) K Comment c Comment
Heat Flux, q (Btu/ft2hr) Heat Diffusivity, a (ft2/hr) q Comment a Comment
TABLE 3.7: PHYSICAL PROPERTIES COLLECTED IN THE DATABASE
Moisture Content (% of initial) Mass or Weight Loss from Saturated Surface Dry Due to Heat (%)
Moisture Content (% of final) M.L. or W.L. from Saturated Surface Dry Comment
Moisture Content Comment Spalling Temperature, Ts (°F) Mass or Weight Loss from Oven Dry Due
to Heat (%) Ts Comment
M.L. or W.L. from Oven Dry Comment Spalling Time, ts (min) Mass or Weight Loss from Air Dry Due to
Heat (%) ts Comment
M.L. or W.L. from Air Dry Comment
3.3 Data Ranges
The results collected include 3026 data points spanning over a broad range of temperaturedependent concrete properties. The breakdown of the number of data points for each concrete property is shown below in Table 3.8. Each data point in the database represents an observed concrete property measured at a maximum exposure temperature. For example, if a single concrete cylinder is heated to a maximum temperature and a compression test is conducted until the specimen experiences failure, the compressive strength obtained from that test is considered a data point. Note that, in some cases, a paper would not report the data from each single test but instead report an average value from a series of tests (e.g., average from 3 cylinder specimens) – that data would also be considered a single point in the database because it is the best information available from the paper. As described in Appendix A, comments are provided in the database to identify single point and average test results.
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TABLE 3.8: NUMBER OF DATA POINTS COLLECTED FOR EACH TEMPERATURE
DEPENDENT CONCRETE PROPERTY
TemperatureDependent Property Number of Papers Reporting Number of Data Points
Compressive Strength 9 647 Static Modulus of Elasticity 7 275 Ultimate Strain 3 70 Dynamic Modulus of Elasticity 3 186 Linear Expansion 3 132 Thermal Coefficient of Expansion 3 37 Mass Loss from Air Dry 3 82 Strain at Peak Stress 2 30 Creep Strain 2 992 Modulus of Rupture 2 46 Thermal Conductivity 2 25 Specific Heat 2 19 Heat Diffusivity 2 22 Mass Loss from Oven Dry 2 308 Mass Loss from SSD 2 24 Ultrasonic Pulse Velocity 1 26 Resonance Frequency 1 57 Heat Flux 1 15 Spalling Temperature 1 16 Spalling Time 1 17
It can be seen that there are a significant number more data points collected for creep strain than any other concrete property. This is because creep strain is not only a temperaturedependent property, but also a timedependent property. Therefore, a single data point for creep strain corresponds to a particular time as well as a particular temperature. For example, consider a stressed specimen heated to a certain maximum exposure temperature, with the temperature then held constant for some time. Data points would be collected at specific time intervals during the exposure period to measure the timedependent creep strains. Then, to study the temperature dependency of the creep strains, another specimen would be subjected to a different temperature, and data collected during the same time interval.
Because this report focuses on the temperaturedependent stressstrain properties of concrete, the following sections look at the distribution of the collected compressive strength, modulus of elasticity, strain at peak stress, and ultimate strain data. Note that creep strain is not included in this investigation since it is outside of the scope of this research. The objective is to develop concrete stressstrain property relationships to which creep effects could later be added in the form of an explicit timetemperature model.
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3.3.1 Compressive Strength Data
A total of 647 data points reported by nine papers (Abrams 1971, Castillo and Durani 1990, Cheng et al. 2004, Harmathy and Berndt 1966, Kerr 2007, Lankard et al. 1971, Phan and Carino 2001, Saemann and Washa 1960, and Zoldners 1960) are used to develop the concrete compressive strength loss relationships in Chapter 4 of this report. Figure 3.1 shows the entire set of strength, fcm data collected in the database, normalized with respect to the room temperature strength, fcmo. The maximum exposure temperature, T ranges from 68°F (20°C) to 1,600°F (871°C). It can be seen that there are data points in the low temperature ranges where the compressive strength ratio is greater than 1.0. While this may have occurred due to the inherent variability in the concrete strength, the general data trends in Figure 3.1 suggest that concrete can experience slight strength increases at low elevated temperatures.
Figure 3.1: Full set of compressive strength loss data.
Figure 3.2 shows the distribution of the concrete room temperature compressive strength [which ranges from approximately fcmo = 1,500 to 14,700 psi (10.3 to 101.4 MPa)] from the nine papers used in the regression analysis of temperaturedependent compressive strength, fcm. Note that although there were a total of 647 data points collected for the compressive strength, Figure 3.2 shows the 635 data points, ranging from temperatures of 70°F (21°C) to 1,600°F (871°C), used in the statistical analysis. The remaining 12 data points, with temperatures below 70°F (21°C), were not included in the regression analysis.
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Figure 3.2: Distribution of room temperature compressive strength, fcmo of the data used in determining the temperaturedependent compressive strength, fcm relationships. (Note:
1 ksi = 6.859 MPa)
Figure 3.3 exhibits further distributions for the data used in the regression analysis (635 data points) with ksi (41.4 MPa), which represents the normalstrength concrete used for determining the temperaturedependent relationships in this report. The statistical reasoning for 6,000 psi (41.4 MPa) being the limit between normalstrength and highstrength concrete is discussed later in Chapter 4. Figure 3.3(a) shows that the unstressed test type is the most common, followed by the residual test type, with the stressed test type having the fewest number of data points (72 points). Similarly, Figure 3.3(b) shows that most of the test specimens include calcareous aggregates, followed by lightweight aggregates, with siliceous aggregates having the fewest number of data points (74 points). Lastly, Figures 3.3(c), 3.3(d), and 3.3(e) show that the majority of the data is from small, 3in (7.6 cm) x 6in (15.2 cm) cylinders tested using radiation/electric type furnaces.
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(a) (b)
(c) (d)
(e)
Figure 3.3: Distribution of the normalstrength concrete data used for the compressive strength relationships: (a) test type; (b) aggregate type; (c) specimen height; (d) specimen
shape; and (e) furnace type.
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The highstrength concrete data with
�
fcmo > 6 ksi (41.1 MPa) has a smaller property range than the normalstrength data with
�
fcmo ≤ 6 ksi (41.1 MPa). The highstrength data comes only from cylindrical specimens with calcareous aggregates tested using radiation/electric furnaces. Figures 3.4(a) and 3.4(b) show that the three test types are almost evenly distributed and 4in (10.2 cm) x 8in (20.3 cm) cylinders represent the most common specimen size used in the highstrength concrete tests. Note that the specimen diameter for the data collected was always equal to half of the specimen height.
(a) (b)
Figure 3.4: Distribution of the highstrength concrete data used for the compressive strength relationships: (a) test type; and (b) specimen height.
3.3.2 Modulus of Elasticity Data
The database contains a total of 461 data points obtained from eight papers (Castillo and Durani 1990, Cheng et al. 2004, Harmathy and Berndt 1966, Kerr 2007, Lankard et al. 1971, Phan and Carino 2001, Philleo 1958, and Saemann and Washa 1960) for use in the development of temperaturedependent modulus of elasticity, Ec relationships for concrete. The modulus of elasticity data was measured in two different ways, either as a static modulus of elasticity or a dynamic modulus of elasticity. There are 275 and 186 data points collected for the static modulus and dynamic modulus, respectively.
Figure 3.5 shows the entire set of data (461 points) collected in the database for the modulus of elasticity, normalized with respect to the roomtemperature modulus, Eco. The maximum exposure temperature, T ranges from 73°F (23°C) to 1,412°F (767°C). The general trend from the data is that the modulus of elasticity tends to immediately decrease as the temperature is increased. However, there are a few data points that suggest that at lower temperatures, the modulus of elasticity remains constant or even experiences a slight increase.
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Figure 3.5: Full set of modulus of elasticity loss data.
Figure 3.6 shows the distribution of the room temperature compressive strength, fcmo for the data used in determining the temperaturedependent modulus of elasticity, Ec relationships in Chapter 5. Note that only 419 data points out of a total of 461 are shown for the modulus of elasticity data in Figure 3.6. The full set of 461 data points was not used in the statistical analysis because some of the necessary information for the remaining test data could not be found from the source papers. Overall, for this set of 419 data points, fcmo ranges from 1,150 psi (7.9 MPa) to 14,707 psi (101.4 MPa).
Figure 3.6: Distribution of room temperature compressive strength, fcmo of the data used in determining the temperaturedependent modulus of elasticity, Ec relationships. (Note:
1 ksi = 6.859 MPa)
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(a) (b)
(c) (d)
(e)
Figure 3.7: Distribution of the normalstrength concrete data used for the modulus of elasticity relationships: (a) test type; (b) aggregate type; (c) specimen height; (d) specimen
shape; and (e) furnace type.
34
Figure 3.7 displays further distributions of the normalstrength concrete (
�
fcmo ≤ 6,000 psi [41.4 MPa]) data used in the statistical analysis of Ec from the 419 data points. Figure 3.7(a) shows that the majority of the NSC data comes from unstressed tests, with only 6% of the data from residual tests, and none from stressed tests. Figure 3.7(b) shows that there are essentially an equal number of tests conducted using calcareous and lightweight aggregate specimens, with only two tests conducted on siliceous concrete. Figure 3.7(c) shows that all of the data were obtained by testing small specimens with a large portion being in the 3.8in (9.7 cm) to 4in (10.2 cm) height range. Figure 3.7(d) shows that the majority of the data is for cylindrical specimens, but there are several rectangular specimens as well. Most of the rectangular specimens were used in the determination of the dynamic modulus of elasticity. Lastly, Figure 3.7(e) shows again that the radiation/electric furnace is the most widely used, but the testing furnace type is not reported by the source papers for a large portion of the data.
Similar to the compressive strength, the highstrength concrete data for the modulus of elasticity has a smaller property range than the normalstrength concrete data. All of the HSC data comes from radiation/electric furnace tests using calcareous aggregate concrete specimens. Figure 3.8(a) shows that the majority of the data is from residual tests, followed by the unstressed test, with the stressed test type having the fewest number of tests conducted. Figure 3.8(b) shows that the tests were conducted using small specimens ranging in height from 6in (15.2 cm) to 8in (20.3 cm), with the majority being 8in (20.3 cm) tall. Finally, Figure 3.8(c) shows that most of the specimens were cylinders, although there were some that were rectangular. Again, most of the rectangular specimens were tested for the dynamic modulus of elasticity.
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(a) (b)
(c)
Figure 3.8: Distribution of the highstrength concrete data used for the modulus of elasticity relationships: (a) test type; (b) specimen height; and (c) specimen shape.
3.3.3 Strain at Peak Stress Data
There are a total of 30 data points collected for the strain, εcm at peak stress at elevated temperatures from two papers (Castillo and Durani 1990, and Cheng et al. 2004). All of the εcm data was determined from experimental stressstrain curves or calculated from experimental loaddeformation curves. The entire data set (30 points), normalized with respect to the roomtemperature strain, εcmo, is shown in Figure 3.9. The maximum exposure temperatures range from 73°F to 1,472°F (23°C to 800°C).
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Figure 3.9: Full set of strain at peak stress data.
Figure 3.10 shows the distribution of the room temperature compressive strength, fcmo for the 28 data points used in determining the temperaturedependent strain at peak stress, εcm relationships in Chapter 6. As discussed later in Chapter 6, two data points from the original 30 points were removed from the regression analysis because they were considered outlying data, bringing the number of points used in the regression analysis down to 28. It can be seen in Figure 3.10 that since only a small number of data points for εcm are available from only two papers, the range of fcmo is rather sparse from 4,504 psi (31.1 MPa) to 11,458 psi (79.0 MPa).
Figure 3.10: Distribution of room temperature compressive strength, fcmo of the data used in determining the temperaturedependent strain, εcm at peak stress relationships. (Note: 1
ksi = 6.859 MPa)
37
The normalstrength concrete (
�
fcmo ≤ 6,000 psi [41.4 MPa]) data collected for the strain at peak stress does not have any variability with respect to the test type, specimen height, shape, aggregate type, or furnace type. All of the NSC specimens are 2in x 4in (5.1 cm x 10.2 cm) cylindrical specimens made from calcareous aggregate, tested in a radiation/electric furnace using the stressed test type.
Out of the 28 data points used in the statistical analysis of εcm, a total of 19 data points are highstrength concrete specimens. Again, because of the small data set, the property range for HSC is rather sparse. All of the HSC specimens are cylindrical in shape and tested in a radiation/electric furnace using the unstressed test type. Figure 3.11 shows the distributions of the aggregate type and specimen height used in the 19 HSC data points. Most of the tests were conducted on calcareous aggregate concrete specimens, with no tests conducted on lightweight aggregate concrete. Also, a majority of the data comes from 4in x 8in (10.2 cm x 20.4 cm) specimens.
(a) (b)
Figure 3.11: Distribution of the highstrength concrete data used for the strain at peak stress relationships: (a) aggregate type; and (b) specimen height.
3.3.4 Ultimate Strain Data
The data collected for the temperaturedependent ultimate strain, εcu includes a total of 70 points from three papers (Castillo and Durani 1990, Cheng et al. 2004, and Harmathy and Berndt 1966). The data from two of the papers (Castillo and Durani 1990, and Cheng et al. 2004) was collected from the experimental stressstrain curves while the data from the third paper (Harmathy and Berndt 1966) was calculated from the “deformation at fracture” reported in the paper. As previously described in Chapter 2, when the ultimate strain was collected from the experimental stressstrain curves, it was taken as the strain corresponding to 85% of the peak stress in the postpeak range of the stressstrain relationship. In cases where an abrupt drop in stress occurred (indicating failure) prior to reaching 85% of the peak stress, the strain at the stress drop was taken as the ultimate strain. When the strain was calculated from the “deformation at fracture,” it was not possible to determine from the literature at what stress the deformation occurred.
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Figure 3.12 shows the full data set (70 points) normalized with respect to the room temperature ultimate strain, εcuo. The maximum exposure temperatures in the data set range from 73°F to 1,472°F (23°C to 800°C).
Figure 3.12: Full set of ultimate strain data.
Figure 3.13 shows the distribution of the room temperature compressive strength, fcmo for the 67 data points used in determining the temperaturedependent ultimate strain, εcu relationships in Chapter 7. As discussed later in Chapter 7, three data points from the original 70 points were removed from the regression analysis because they were considered outlying data, bringing the number of points used in the regression analysis down to 67. It can be seen that the fcmo values in Figure 3.13 are sparsely distributed between 4,504 psi (31.1 MPa) and 11,458 psi (79.0 MPa).
Figure 3.13: Distribution of room temperature compressive strength, fcmo of the data used in determining the temperaturedependent ultimate strain, εcu relationships. (Note: 1 ksi =
6.859 MPa)
39
From the 67 data points used in the statistical analysis of εcu, a total of 48 points are for normalstrength concrete. While there is some variation in the data, all of the results are from unstressed tests conducted on cylindrical specimens. Figure 3.14 shows the distribution of the 48 normalstrength concrete data points with respect to the aggregate type, specimen height, and the type of furnace used to heat the specimens. Most of the tests were conducted on lightweight aggregate concrete, with some tests on calcareous aggregate concrete and none on siliceous concrete specimens. It can be seen that the data comes from specimens in the 3.8in to 4in (9.2 cm to 10.2 cm) height range, with the majority of the data from 3.8in (9.2 cm) specimens. Lastly, for most of the tests, the type of furnace is not reported in the literature. Note that for all of the distributions in Figure 3.14, the ratio of the properties being compared is 81% to 19%, suggesting that the NSC data comes from only two papers.
(a) (b)
(c)
Figure 3.14: Distribution of the normalstrength concrete data used for the ultimate strain relationships: (a) aggregate type; (b) specimen height; and (c) furnace type.
A total of 19 data points from the 67 ultimate strain data points are for highstrength concrete. All of these tests were conducted in radiation/electric type furnaces using cylindrical specimens subjected to the unstressed test type. Figure 3.15 shows the breakdown of the 19 HSC ultimate strain data points with respect to the aggregate type
40
and specimen height. It can be seen that the majority of the specimens were made from calcareous aggregate concrete and 8in (20.3 cm) tall.
(a) (b)
Figure 3.15: Distribution of the highstrength concrete data used for the ultimate strain relationships: (a) aggregate type; and (b) specimen height.
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CHAPTER 4: COMPRESSIVE STRENGTH
This chapter presents the statistical analysis conducted on the data for the compressive strength of concrete at elevated temperatures. Section 4.1 describes the process used to conduct the statistical analysis. Section 4.2 proposes temperaturedependent compressive strength relationships resulting from the statistical analysis. Lastly, Section 4.3 provides the results and evaluation of the proposed compressive strength relationships along with comparisons of these relationships with previous compressive strength models.
4.1 Statistical Analysis
Multiple regression analysis was conducted for the concrete compressive strength loss as a result of exposure to elevated temperatures using 647 data points, where each data point represents a measured concrete strength, fcm corresponding to a measured maximum exposure temperature, T, as previously described. To determine the form of the strength loss regression model, a correlation matrix was produced with the independent properties (i.e., mix properties, curing properties, specimen properties, and test properties) reported for each test against the measured concrete strength, fcm. Although there were several variables that showed strong correlations, it was determined that the variables most highly correlated with fcm were the cement content, watertocement ratio (wc), and room temperature compressive strength (fcmo). Using the properties that showed strong correlation with fcm as variables in the regression analysis, it was found that some of these properties, although having high correlation with the compressive strength loss, did not have a large effect on the regression results. For example, the amount of slump in the concrete mix tended to show strong correlation with fcm however, when included as a variable in the regression model, it did not have a significant effect on the overall performance of the equation. Note that some variables could not be included in the regression model because there was either not enough data or not enough variation in the data to conduct a meaningful statistical analysis.
To determine if the model variables contribute significant information to the prediction of strength loss, a Student’s ttest (Mendenhall 2007) was performed. This test compares the value of the tstatistic for any individual variable with the critical tstatistic and determines if the regression coefficient for that variable could statistically have a value of zero, suggesting that the variable does not make a significant contribution to the model. Generally, only those variables that had a significant effect on the regression results as determined by the tstatistic were included in the model. Furthermore, as a measure of the global adequacy of the model, the analysis of variance F test (Mendenhall 2007) was performed. This test is used to determine if any (rather than a specific one) of the regression coefficients used in the model could statistically have a value of zero, suggesting that the model is not a useful representation of the data.
42
Some variables did not need to be included in the strength loss model because their effect was seen through more global predictors. For example, by including fcmo as a parameter in the regression, many of the mix properties that were highly correlated with the strength loss (e.g., cement content, watertocement ratio) did not need to be included in the model because their effect was experienced through fcmo. Through this comprehensive investigation, it was found that the most significant independent properties for the concrete strength loss with temperature are: (1) fcmo; (2) aggregate type; and (3) test type. Note that, with the exception of fcmo, these are the same as the parameters used in the ACI 216 models shown in Figure 2.1.
It was decided to divide the data based on fcmo as: (1) normalstrength concrete (NSC) with fcmo ≤ 6 ksi (41.4 MPa); and (2) highstrength concrete (HSC) with fcmo > 6 ksi (41.4 MPa). The cutoff strength of 6 ksi (41.4 MPa) was used because a distinct difference was observed in the strength loss for the higher strength concrete specimens, and the two strength ranges selected produced the best statistical fit to the data. The aggregate type was grouped into the same three general categories as in ACI 216: (1) siliceous (sandstone and other materials containing significant amounts of quartz); (2) calcareous (carbonate, limestone, dolomitic limestone, and dolomite); and (3) lightweight (expanded shale and expanded slag). The test type was also grouped into the same categories from ACI 216 as: (1) residual; (2) stressed; and (3) unstressed. As described in Chapter 2, during the residual test type, the specimen is heated to a maximum exposure temperature, allowed to cool to room temperature and is then tested in uniaxial compression until failure. The database for the stressed test type includes specimens with an axial compressive preload of 2555% of fcmo. Under this preload, the specimen is heated to a maximum exposure temperature, and is then tested in further uniaxial compression until failure. Within the range of 2555% of fcmo, the amount of preload did not significantly affect the temperaturedependent compressive strength loss, and thus, the preload level was not included in the stressed test models. Lastly, during the unstressed test type, the specimen is heated to a maximum exposure temperature and is tested in uniaxial compression until failure. Because of the limitations of the testing apparatus from each paper, the unstressed test specimens were typically removed from the furnace in order to be subjected to uniaxial compression until failure. Since the papers that these data points were collected from did not specify a loss of temperature or a time that the specimens were taken outside the furnace before testing, it was assumed that the temperature loss in each specimen from the time of the removal of heat to the time of failure was negligible.
It should be noted that the residual test data collected for the compressive strength at elevated temperatures includes data from specimens that were quenched in water for a period of time following heating. By introducing water to a heated specimen, there is inherently a significantly different method of heat transfer occurring than otherwise would be experienced by a nonquenched (i.e., aircooled) specimen. From the available data, it was seen that the quenched specimens tend to have larger strength losses in the low to mid temperature ranges than the specimens that were cooled in air. This suggests that a separate equation should be used in the prediction of strength loss for a quenched specimen. Because of the differences between the quenched and nonquenched specimens, a total of 26 data points from quenched specimens were excluded from the
43
statistical analysis of the residual test data. It is important to note that this only occurred with the compressive strength data, as the modulus of elasticity, strain at peak stress, and ultimate strain data did not contain any quenched specimens. Since only a total of 26 compressive strength data points were collected for quenched concrete, there was not enough data to look at separate equations for each aggregate type. Therefore, it is recommended that future work be conducted to investigate the stressstrain properties of quenched concrete specimens. This is especially important in order to assess the remaining strength of a structure following the primary means of fire suppression using water.
4.1.1 Preliminary Regression Forms
TABLE 4.1: PRELIMINARY EQUATION FORMS AND TEST STATISTICS
Equation R2
0.55
0.55
0.56
0.58
0.58
0.59
0.59
0.61
0.56
0.57
0.58
0.60
0.87
0.87
0.87
0.88
0.86
0.86
0.87
0.88
44
where:
= normalized regression intercept term (discussed later);
= normalized regression coefficients (discussed later).
Throughout the process of finding the bestfit regression equation to the available test data, several equation forms were used. An important decision was to determine which set of data the regression equation should fit: fcm, or fcm / fcmo. In order to determine this, the coefficient of determination, R2, was used as a primary indicator of the adequacy of the fit. In general, R2 is a global predictor of how well a regression equation fits a set of data by comparing the regression equation to the mean of the data. R2 ranges from 0 to 1, where a value of R2 = 1 indicates a perfect fit to the data and a value of R2 = 0 indicates a complete lack of fit to the data. Several preliminary equation formats on the entire set of data are shown above in Table 4.1.
It is clear from the preliminary trials in Table 4.1 that in order to get the highest R2 value over the full data set, it is necessary to use an equation form with fcm as an independent variable. However, the use of fcm as an independent variable has important disadvantages. Firstly, if fcm is used as an independent variable, it would not be possible to show a single regression line through the data since at a single temperature, there could be fcm data ranging from 1,500 psi to 14,700 psi (10.3 MPa to 101.4 MPa), thereby making a visual evaluation of the regression equation difficult. Secondly, there would be no way to constrain the equation so that at room temperature, fcm = fcmo can be achieved (which should be the case by definition). As described later, the constraint of fcm = fcmo at room temperature can be applied to the regression equation using fcm / fcmo as the dependent variable. Thus, it was determined that the equation form should be based on fcm / fcmo as the dependent variable.
4.1.2 Selected Form of Regression Equations
Note that the regression equations in Table 4.1 do not take into account the aggregate type or the test type. Since both of these variables were determined as some of the most highly correlated parameters with the strength loss data, they should be included in the regression equation. To include these parameters, it was decided that a set of regression equations should be produced based on subsets of the full data set. For example, one equation would be produced for all of the data that come from normalstrength calcareous aggregate concrete subjected to the unstressed test type, another equation would be for the normalstrength calcareous aggregate concrete subjected to the stressed test type, and so on.
Using subsets of the full data set, regression equations were developed to determine the best fit to the data. Since there are a total of twelve combinations (described later) of aggregate type and test type when accounting for normalstrength and highstrength concrete, it was necessary to keep the equations simple so that they could easily be used in design practice. Keeping simplicity in mind, as well as creating the best
45
possible fit to the data, it was determined that the general regression form shown by Equation (4.1) would be sufficient for each aggregate type and test type combination.
(4.1)
In order to implement the multiple linear regression analysis, the database was saved in a text file, and was imported into Matlab®. A comprehensive regression program was written using Matlab in such a way that the user could select specific ranges of data to carry out the subset regression analyses. To select certain ranges of data, the database column number for the concrete property in question would have to be known, and then the lower bound and upper bound on each property could be set. For example, if the regression was to be conducted on the fcm / fcmo data for normalstrength siliceous aggregate concrete specimens subjected to the residual test type, the subset data for this condition could easily be selected through the user input file.
4.1.3 Normalized Regression Coefficients
The regression program created for this research allows for any polynomial equation form of independent variables selected by the user. Since the terms included in the regression model range over many orders of magnitude (i.e., , , ), which can affect the conditioning of the equations to be solved, each independent variable was normalized to have a mean of zero and a standard deviation of one by subtracting its mean from each data point of the independent variable and dividing by its standard deviation. After this step, the normalized regression coefficients, , were found by following the standard procedure for minimizing the square of the error produced by the regression equation. This minimization leads to a set of socalled “normal” equations. The solution of these equations is shown in Equation (4.2) and yields the regression coefficients as follows:
(4.2)
where,
= normalized regression coefficient vector (m x 1);
m = number of regression coefficients;
= normalized independent variable matrix (n x m);
n = number of fcm / fcmo data points in the dataset; and
= response vector (n x 1) containing the fcm / fcmo values in the dataset.
The columns in the matrix indicate the different variables used in the model (the first column is a column of ones to act as an intercept term) and the rows represent the different data points. As an example, for Equation (4.1) shown above:
46
The first column of the X matrix contains a column of ones, the second column contains the normalized temperature data, the third column contains the square of the normalized temperature data, and the fourth column contains the cube of the normalized temperature data.
The y column vector contains the fcm / fcmo data points for whatever subset of data being regressed.
The first component of the column vector contains the normalized intercept term, . The second component contains the normalized regression coefficient for the temperature term, . The third component contains the normalized regression coefficient for the squared temperature term, . Finally, the fourth component contains the normalized regression coefficient for the cubed temperature term, .
4.1.4 Constrained Regression Equations
As described previously, by definition, fcm / fcmo = 1 at room temperature (70 °F [21 °C]). Since the regression model obtained through a bestfit solution to the available test data would almost certainly not satisfy this condition exactly, it was decided to enforce this simple constraint by employing the classical penalty function approach (Luenberger 1984), resulting in the following modified normal equations:
(4.3)
where,
= penalty parameter (numerical experimentation resulted in = 107 to achieve the required constraint);
= vector of ones (r x 1);
r = number of data points at room temperature; and
= constraint matrix (r x n) containing ones and zeros to select the values corresponding to the room temperature tests.
The constraint equations can be written as Cmy = b, where each constrained equation is yj = 1 and the subscript j indicates a test conducted at room temperature. Constraining the equations in this manner made only small changes to the regression coefficients and statistics, while resulting in models that identically satisfy fcm / fcmo = 1 at room temperature. Table 4.2 shows the changes in the regression statistics as a result of constraining the equations for each of the twelve combinations of aggregate type and test type when accounting for normalstrength and highstrength concrete. Note that Equation (4.3) is only used so that the regression equations could be constrained to a particular value, otherwise Equation (4.2) would be used to determine the regression coefficients. In Table 4.2, the unconstrained R2 values were determined after calculating the regression
47
coefficients according to Equation (4.2) and the constrained R2 values were determined after calculating the regression coefficients according to Equation (4.3).
TABLE 4.2: REGRESSION STATISTICS BEFORE AND AFTER CONSTRAINING ROOM
TEMPERATURE VALUES
Aggregate Type Test Type Unconstrained R2 Constrained R2
Residual 0.87 0.87 Stressed 0.92 0.92 NSC Siliceous Unstressed 0.95 0.93 Residual 0.85 0.85 Stressed 0.66 0.61 NSC Calcareous Unstressed 0.71 0.69 Residual 0.84 0.84 Stressed 0.82 0.79 NSC Lightweight Unstressed 0.75 0.74 Residual 0.83 0.79 Stressed 0.31 0.22 HSC Calcareous Unstressed 0.54 0.36
It can be seen from Table 4.2 that HSC made from siliceous aggregate and lightweight aggregate were not studied. This is because there is not enough data to conduct meaningful statistical analyses for these cases as mentioned in Chapter 3. The R2 values for the calcareous HSC stressed and unstressed models are very low (as described later in more detail). Furthermore, the regression models for these cases experience the largest changes due to the use of constrained equations. As shown through the literature review, HSC begins to lose strength immediately upon heating, whereas NSC experiences a more gradual loss of strength upon heating. The available data for the HSC stressed and unstressed cases do not include test results between 70°F (21°C) and 200°F (93°C), and therefore, by constraining the equations at 70°F (21°C), the regression curve changes more than the data sets where lower temperature results are available. The reduced R2 values shown in Table 4.2 for the HSC stressed and unstressed cases reflect this change in the low temperature ranges, but the regression equations are not severely affected for the high temperature ranges. Note that though there are not many low temperature results for NSC either, the constrained equations are not affected as much as the HSC equations because of the more gradual loss of strength in NSC.
4.1.5 UnNormalized Regression Coefficients
Since the independent model variables were normalized to have a mean of zero and a standard deviation of one in carrying out the statistical analysis described above, the normalized regression coefficients need to be modified to allow for the equations to be used with unnormalized independent variable data (for example, a designer would use
48
a typical value of temperature and not a normalized value for temperature). Once the regression coefficients are modified, they can be used in place of the normalized coefficients using the same form of the regression equations with any typical value of temperature. In order to achieve this goal, the normalized regression coefficients were modified as:
(4.4)
(4.5)
where,
= modified regression intercept term;
= modified regression coefficient;
= sample standard deviation of the original ith independent variable; and
= sample mean of the original ith independent variable.
4.1.6 Regression Assumptions
In standard multiple regression analysis, there are four assumptions made about the distribution of the error between the data and the prediction as follows (Mendenhall 2007): (1) the mean of the error is zero; (2) the error is normally distributed; (3) the variance of the error is constant for each independent variable; and (4) the error is independent for all values of each independent variable. These four basic regression assumptions were checked for the regression models developed in this research. By examining the error plots, it was determined that Assumptions 1, 3, and 4 were met for all cases. To determine if the error for each data set was normally distributed (Assumption 2), a KolmogorovSmirnov (KS) test (Mendenhall 2007) was conducted. It was found that all nine of the NSC compressive strength loss models (corresponding to combinations of three aggregate types and three test types) passed the KS test within a 90% confidence interval (α = 0.10, as described below) suggesting that it is very likely that the error term from these regression models come from a normal distribution. In contrast, none of the three HSC strength loss models (corresponding to three test types for calcareous concrete only) passed the KS test, suggesting that the error from the highstrength concrete data may not come from a normal distribution.
The null hypothesis for a KS test is that the error for the data in question comes from a normal distribution. Passing the test at a significance level α means that there is not sufficient statistical evidence to reject the null hypothesis. A typically accepted value for the significance level to pass the KS test is α = 0.05; the larger the value of α, the
49
stronger the evidence for the normality of the error. The α required to pass the test for each aggregate type and test type combination in the current database is shown in Table 4.3. The maximum value of α tested was 0.10. It was assumed that if the regression passed the KS test with α = 0.10, then the assumption of normality should not be rejected.
TABLE 4.3: REQUIRED SIGNIFICANCE LEVEL TO PASS THE KOLMOGOROVSMIRNOV
TEST FOR THE COMPRESSIVE STRENGTH REGRESSION EQUATIONS
Aggregate Type Test Type Required α
Residual 0.100 Stressed 0.100 NSC Siliceous Unstressed 0.100 Residual 0.100 Stressed 0.100 NSC Calcareous Unstressed 0.100 Residual 0.100 Stressed 0.100 NSC Lightweight Unstressed 0.100 Residual 0.004 Stressed 0.002 HSC Calcareous Unstressed 0.007
The inability of the HSC models to satisfy the error normality assumption may indicate that additional parameters not included in the current regression equations may play a significant role in the strength loss of highstrength concrete with temperature. However, given the existing database, the lack of sufficient test results on potentially important parameters (e.g., specimen humidity at the time of heating as described later) prohibits these parameters from being included as statistically relevant predictor variables in the regression models. It should be noted that out of the four regression assumptions discussed above, the assumption that the error is normally distributed (Assumption 2) is the least restrictive because of the overall robustness of the regression analysis with respect to normality. It has been suggested that departures from this assumption generally have little impact on the regression results (Mendenhall 2007). Thus, while further research is certainly needed on the temperaturedependent properties of highstrength concrete, the relationships proposed in this report are recommended for use with caution until additional test data is developed and the statistical models are reevaluated.
4.1.7 Full Regression Equations Using Coded Variables
In order to get the regression form in Equation (4.1) for every aggregate and test type combination, coded variables were used instead of performing the regression analysis twelve separate times. Coded variables are placed into an equation to select
50
particular sets of data. Each coded variable takes a value of one or zero depending on whether a specific criterion for the corresponding variable has been met by the data set in question. For example, consider a coded variable to determine whether the data is for calcareous aggregate concrete or not. If the data set in question is for calcareous aggregate concrete, the coded variable would receive a value of one, and a value of zero if not. A characteristic of coded variables is that, for the number of possible settings of a specific property, z, there must be z1 coded variables in the regression equation. For example, in the current database, there are three possible settings for the aggregate type (siliceous, calcareous, and lightweight), and therefore, two coded variables are needed in the regression equations. One variable can be for the calcareous aggregate and the other for the lightweight aggregate. If the data in question is for siliceous aggregate concrete, then, both coded variables would be equal to zero. Using this logic, the full regression relationships to predict the compressive strength loss for normalstrength concrete (NSC) and highstrength concrete (HSC) are shown in Equations (4.6) and (4.7), respectively.
(4.6)
(4.7)
where,
= regression intercept term;
= regression coefficient;
L = coded variable representing lightweight aggregate type;
C = coded variable representing calcareous aggregate type;
R = coded variable representing residual test type; and
S = coded variable representing stressed test type.
Note that the full regression equation for HSC consists of a fewer number of terms since only calcareous aggregate is included in the highstrength concrete data (i.e., no coded variables are needed for the HSC aggregate type). The use of the full equations to determine the proposed relationships for the strength loss of HSC or NSC for the different aggregate types and test types in the database is described in more detail later.
Note also that in constraining the regression equations, if the only independent variable is temperature, then only one room temperature data point is needed for the constraint. For example, if Equation (4.1) is used as the regression equation, then only
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one room temperature data point would be needed in order to properly constrain the equation. However, when other independent variables (such as coded variables) are introduced, then, a room temperature data point is needed for each possible value of the other independent variables in order to properly constrain the equation at room temperature. For example, to constrain Equation (4.6), there would need to be nine room temperature data points – one point for each of the aggregate type and test type combinations (one data point for siliceous unstressed case, one for siliceous stressed case, etc.).
4.2 Proposed Relationships
As described above (see Equation 4.1), the final form of the concrete strength loss model developed in this research is a cubic relationship in temperature given by:
(4.8)
where,
= aggregate and test type dependent coefficient (see Table 4.4); and
= temperature in degrees Fahrenheit.
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TABLE 4.4: PROPOSED COMPRESSIVE STRENGTH RELATIONSHIP REGRESSION
COEFFICIENTS
NSC
SILICEOUS CALCAREOUS Residual Stressed Unstressed Residual Stressed Unstressed
0.963 0.995 0.953 0.997 1.023 0.981 6.448E04 8.421E05 7.887E04 6.514E05 4.001E04 3.109E04 1.643E06 9.024E08 1.645E06 3.125E07 1.106E06 5.413E07 5.459E10 1.894E10 5.280E10 6.748E11 6.998E10 1.760E11
T Range (°F)
[70, 1472] [70, 1503] [70, 1600] [70, 1472] [70, 1504] [70, 1600]
fcmo Range (psi)
[3900, 5500]
[3900, 5500]
[3900, 5500]
[3542, 6000]
[3900, 5600]
[1149, 6000]
R2 0.90 0.92 0.93 0.86 0.61 0.69 NSC HSC
LIGHTWEIGHT CALCAREOUS Residual Stressed Unstressed Residual Stressed Unstressed
1.037 1.065 1.024 1.088 1.151 1.095 5.483E04 1.059E03 3.542E04 1.371E03 2.454E03 1.512E03 3.686E07 1.829E06 2.741E07 1.763E06 4.517E06 2.219E06 2.208E10 9.203E10 2.029E10 1.045E09 2.411E09 1.085E09
T Range (°F)
[70, 1600] [70, 1501] [70, 1599] [70, 1112] [70, 1292] [70, 1472]
fcmo Range (psi)
[1597, 3900]
[3900, 3900]
[2716, 3900]
[6440, 13982]
[7344, 14229]
[7345, 14230]
R2 0.86 0.79 0.74 0.79 0.22 0.36
Along with the regression coefficients, Table 4.4 also shows the acceptable ranges of the room temperature compressive strength, fcmo and the maximum exposure temperature, T to be used with the proposed equations. These range limits are imposed by the currently available data on which the regression analysis is based. It is important to note that some of the cases represent relatively limiting ranges over which the equations should be used. For example, the available data for the NSC lightweight stressed
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equation only allows the equation to be used for one specific value of fcmo. Extrapolation of the proposed equations outside of the acceptable ranges for fcmo and T is not recommended.
Note that the proposed relationship in Equation (4.8) and the corresponding regression coefficients were determined based on the full relationship given by Equations (4.6) and (4.7). Equations (4.6) and (4.7) include the effects of maximum exposure temperature, room temperature compressive strength, aggregate type, and test type; however, these equations are not suitable for use in design since they contain a large number of terms. By looking at the different test type and aggregate type combinations, the regression coefficients can be combined for the third order relationship in Equation (4.8). For example, the combined equation for the strength loss of normalstrength calcareous concrete from an unstressed test is developed as follows:
1) With Equation (4.6) as the regression form, use Equation (4.3) to constrain (4.6) at room temperature and solve for the regression coefficients (with normalized temperature data).
2) Apply the coded variable values C = 1, L = 0, R = 0, and S = 0 in Equation (4.6) as:
(4.9)
3) Combine the like temperature terms and modify the regression coefficients using Equations (4.4) and (4.5) as:
(4.10)
4) Determine the final strength loss relationship [given by Equation (4.8)], where, for a normalstrength calcareous concrete under the unstressed test, the coefficients are:
(4.11)
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4.3 Results and Evaluations
As given by Equation (4.8) and Table 4.4, the multiple regression analysis for the concrete strength loss results in twelve different relationships (nine for normalstrength concrete and three for highstrength concrete). These relationships are evaluated below.
4.3.1 Comparisons with Test Data and Evaluation of Data Fit
As an overall evaluation of the fit between the proposed equations for the concrete compressive strength and the corresponding data, the coefficient of determination, R2 for each of the 12 relationships is given in Table 4.4. It can be seen that the regression model fits the data very well for most cases, with the exception of the highstrength calcareous aggregate model for the stressed and unstressed test cases. These sets of data are dominated by two papers (Castillo and Durani 1990, and Phan and Carino 2001) with the specimens tested by Phan and Carino having smaller ratios (i.e., larger relative strength losses) at a given temperature than those tested by Castillo and Durani.
A major difference between these two research programs is in the way that the test specimens were prepared. In Castillo and Durani (1990), the specimens were dried to room humidity prior to heating. In comparison, Phan and Carino (2001) tested their specimens shortly after they were cured in water for several days. This may have resulted in more water in the pores of the concrete and in higher pore pressures under increased temperatures leading to larger relative strength losses and more extensive explosive spalling in the specimens tested by Phan and Carino.
As a result of these observations, it is possible that a better fit may be obtained to the test data if the specimen relative humidity at test time were included in the regression model. However, this was not possible for the current data since not enough test results exist to include relative humidity as a statistically relevant predictor variable. Further experimental research is needed to determine if there is a significant relationship between the specimen humidity and the concrete strength loss with temperature. Until this data is collected and the regression models are updated as needed, the proposed equations for highstrength concrete should be used with caution.
Figure 4.1 shows the proposed regression equations against the 12 data sets for the strength loss. Both the regression equations and the data are normalized with respect to fcmo. For all twelve cases, the regression line appears to fit the data very well. For the HSC calcareous relationships that have low R2 (stressed and unstressed test data), the regression appears to fit the data very reasonably for such low values of R2. Again, keep in mind that these low R2 values are due in part to constraining the regression equations while having a small number of data points available for temperatures less than 200°F (93°C). As previously discussed, it is possible that there are other important parameters that affect the compressive strength at elevated temperatures that are not possible to include in the regression model because of the lack of data.
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(a) (b)
(c) (d)
(e) (f)
56
(g) (h)
(i) (j)
(k) (l)
Figure 4.1: Proposed compressive strength relationships fit to data: (a) NSC – siliceous, residual; (b) NSC – siliceous, stressed; (c) NSC – siliceous unstressed; (d) NSC –
calcareous, residual; (e) NSC – calcareous, stressed; (f) NSC – calcareous, unstressed; (g) NSC – lightweight, residual; (h) NSC – lightweight, stressed; (i) NSC – lightweight,
unstressed; (j) HSC – calcareous, residual; (k) HSC – calcareous, stressed; and (l) HSC – calcareous, unstressed.
57
Similarly, Figure 4.2 compares the prediction band and the measured fcm values for each aggregate and test type combination in the database. The prediction bands for fcm were generated by multiplying the fcm / fcmo curve from each regression model with the maximum and minimum fcmo values for the corresponding data range (see Table 4.4). By looking at the prediction bands, it is possible to see the upper and lower limit of fcm that the regression models can predict as the temperature is increased. In general, a reasonable fit is observed between the test data and the prediction bands, with most of the available data falling within the corresponding prediction band.
(a) (b)
(c) (d)
(e) (f)
58
(g) (h)
(i) (j)
(k) (l)
Figure 4.2: Proposed compressive strength relationship prediction bands: (a) NSC – siliceous, residual; (b) NSC – siliceous, stressed; (c) NSC – siliceous, unstressed; (d)
NSC – calcareous, residual; (e) NSC – calcareous, stressed; (f) NSC – calcareous, unstressed; (g) NSC – lightweight, residual; (h) NSC – lightweight, stressed; (i) NSC – lightweight, unstressed; (j) HSC – calcareous, residual; (k) HSC – calcareous, stressed;
and (l) HSC – calcareous, unstressed.
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4.3.2 Effect of Aggregate Type
Figure 4.3 shows the proposed fcm / fcmo curves for normalstrength concrete grouped based on the three aggregate types. The HSC curves are not shown because the only proposed relationship for HSC is for calcareous aggregate. It can be seen that the effect of aggregate type on the concrete strength loss is somewhat mixed. For the residual and unstressed test types, the lightweight concrete has the least amount of strength loss for, approximately, T > 1100°F (593 °C). In general, lightweight concrete tends to perform better than siliceous and calcareous concrete at these high temperatures because in creating the expanded shale and slag aggregates, the material has already undergone thermal processing. In comparison, for all three test types, siliceous concrete experiences the most strength loss for, approximately, T > 800°F (427 °C). This may be because of the high amounts of quartz present in siliceous aggregate, which undergoes a phase transformation at 1063 °F (573 °C) accompanied by a significant volume increase resulting in more strength loss than the other aggregate types. Note that the differences between the siliceous concrete model and the calcareous and lightweight concrete models in Figure 4.3 are smaller than the differences in the ACI 216 models in Figure 2.3.
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(a) (b)
(c)
Figure 4.3: Proposed compressive strength relationships showing the effect of aggregate type on the strength loss: (a) NSC – residual; (b) NSC – stressed; and (c) NSC –
unstressed.
4.3.3 Effect of Test Type
Figure 4.4 shows the effect of test type on the temperaturedependent compressive strength of normalstrength and highstrength concrete. For each type of aggregate, the stressed test results in the smallest amount of strength loss at higher temperatures (with the exception of highstrength concrete), while the largest strength loss occurs in the residual test type. It is interesting to note that for calcareous and lightweight aggregate concrete, the stressed specimens demonstrate some strength regain at moderate temperatures. It can also be seen that for normalstrength concrete, the trends for the strength loss under the unstressed and residual test types are similar, whereas the stressed test type results in a significantly different behavior.
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(a) (b)
(c) (d)
Figure 4.4: Proposed compressive strength relationships showing the effect of test type on the strength loss: (a) NSC – siliceous; (b) NSC – calcareous; (c) NSC – lightweight;
and (d) HSC – calcareous.
Note also that because the unstressed test type tends to result in larger strength losses as compared to the stressed test type, the unstressed test has been used in the past as the basis for design recommendations. However, for normalstrength calcareous concrete shown in Figure 4.4(b), the unstressed case can be viewed as being overly conservative compared to the stressed test. Also, for highstrength calcareous concrete shown in Figure 4.4(d), the stressed test type results in a rapid strength loss causing the unstressed test to give unconservative estimates at high temperatures.
4.3.4 HighStrength versus NormalStrength Concrete
Figure 4.5 compares the strength loss models for highstrength (i.e., fcmo > 6,000 psi [41.4 MPa]) and normalstrength (i.e., fcmo ≤ 6,000 psi [41.4 MPa]) concrete with calcareous aggregates. As shown by the test data (Castillo and Durani 1990, Cheng et al. 2004, Kerr 2007, and Phan and Carino 2001) and represented by the proposed models, highstrength concrete behaves in a significantly different manner under elevated temperatures as compared with normalstrength concrete. In general, highstrength
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concrete tends to lose more of its relative strength as temperature is increased than normalstrength concrete. This has been attributed to the greater density (i.e., smaller porosity) of highstrength concrete (Castillo and Durani 1990), which results in higher internal pore pressures, and thus, greater strength losses as water is driven out of the concrete under increased temperatures.
(a) (b)
(c)
Figure 4.5: Proposed compressive strength relationships showing the difference between normalstrength concrete and highstrength concrete: (a) calcareous, residual; (b)
calcareous, stressed; and (c) calcareous, unstressed.
Furthermore, it can be seen from Figure 4.5 that the behavior of highstrength concrete is characterized by strength reduction up to approximately 200 °F (93 °C), followed by a relatively stable range between 200400 °F (93204 °C), and then a sharp strength loss with increased temperatures. The relatively stable range has been attributed to the stiffening of the cement gel and an increase in the cohesive properties of the gel particles (Castillo and Durani 1990, Cheng et al. 2004). In comparison, normalstrength concrete tends to experience a more gradual strength loss with temperature.
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4.3.5 Comparisons with Previous North American Models
Figure 4.6 shows comparisons between the proposed compressive strength loss models and the available North American strength loss models from ACI 216 (2007), ASCE (1992), and Kodur et al. (2008) as shown in Figures 2.3 and 2.4. Note again that the proposed curves represent a much larger data set than any of these previous models, thus increasing statistical robustness. Furthermore, the current ACI and ASCE models do not consider highstrength concrete. Using these models to predict the strength loss of highstrength concrete may, in some cases, grossly overpredict the available strength at elevated temperatures.
While there tend to be some discrepancies, in general, the proposed curves have similar trends as the ACI models, with calcareous concrete subjected to the residual test being the biggest exception. The currently available ASCE curves do not make a distinction between the siliceous and calcareous aggregate types, provide no model for lightweight concrete, and have no distinction based on the test type. Therefore, the results compared with the proposed models are fairly mixed. For siliceous aggregate concrete, the ASCE curve most resembles the stressed test type [Figure 4.6(b)], whereas for calcareous aggregate concrete, the ASCE curve most resembles the unstressed test type [Figure 4.6(f)]. For siliceous aggregate, the ASCE model overpredicts the concrete strength for both the residual and unstressed test types, but predicts a greater strength loss than the proposed model for the stressed test type at elevated temperatures. For calcareous aggregate, the ASCE model overpredicts the concrete strength for the residual test type as compared to the proposed model, but it predicts greater strength loss than the proposed model for the stressed test type. For highstrength concrete, the Kodur et al. model, which provides no distinction based on the aggregate type or the test type, most closely resembles the proposed calcareous unstressed model [Figure 4.6(l)]. Compared to the proposed models, the Kodur et al. model predicts greater strength losses for the stressed and unstressed test types, but predicts smaller losses for the residual test type.
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(a) (b)
(c) (d)
(e) (f)
65
(g) (h)
(i) (j)
(k) (l)
Figure 4.6: Proposed compressive strength relationships compared with ACI 216, ASCE, and Kodur et al.: (a) NSC – siliceous, residual; (b) NSC – siliceous, stressed; (c) NSC – siliceous, unstressed; (d) NSC – calcareous, residual; (e) NSC – calcareous, stressed; (f) NSC – calcareous, unstressed; (g) NSC – lightweight, residual; (h) NSC – lightweight, stressed; (i) NSC – lightweight, unstressed; (j) HSC – calcareous, residual; (k) HSC –
calcareous, stressed; and (l) HSC – calcareous, unstressed.
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CHAPTER 5: MODULUS OF ELASTICITY
This chapter presents the statistical analysis conducted on the data for the modulus of elasticity of concrete at elevated temperatures. Section 5.1 describes the process used to conduct the statistical analysis. Section 5.2 proposes temperaturedependent modulus of elasticity relationships resulting from the statistical analysis. Lastly, Section 5.3 provides the results and evaluation of the proposed modulus of elasticity relationships along with comparisons of these relationships with previous models.
5.1 Statistical Analysis
Multiple regression analysis was conducted on the 461 data points collected for the concrete modulus of elasticity at elevated temperatures, where each data point represents a measured modulus of elasticity corresponding to a measured maximum exposure temperature. Utilizing the regression analysis program written in MATLAB® for the temperaturedependent compressive strength (see Chapter 4), a similar regression analysis procedure was conducted for the modulus of elasticity, Ec.
5.1.1 Modulus of Elasticity Data
As described in Chapter 3, the current concrete property database consists of 275 and 186 data points, respectively, for the static modulus of elasticity and the dynamic modulus of elasticity. The static modulus of elasticity data is further broken down into the tangent modulus (i.e., initial slope of the stressstrain curve) and the secant modulus (i.e., slope of the straight line connecting the origin to a point at 3040% of the peak stress). Similarly, the dynamic modulus of elasticity data includes results obtained using the ultrasonic pulse velocity test and the resonance frequency test. Because of the limited number of data points, it was necessary to determine if the static and dynamic modulus test results could be combined to increase the size of the data pool. A series of statistical analyses were conducted for this purpose using three different processes: (1) modified R2 analysis; (2) hypothesis testing; and (3) coded variables. In conducting these analyses, Equation (5.1) was used as the regression relationship.
(5.1)
It should be noted that this equation form was used as a preliminary model to determine the validity of combining the modulus of elasticity data pool; it was not used as the final equation for the regression analysis. Also note that in this equation, the data is normalized with respect to the room temperature modulus of elasticity, Eco, and therefore the regression relationships are constrained to a value of 1.0 at room temperature. The modified R2 analysis was used to determine if a regression fit to one set of data could be used to predict a second set of data. For example, first, a regression relationship was
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determined for the dynamic modulus data. Then, the same regression relationship (using the same regression coefficients) was applied to the static modulus data, and the coefficient of determination from this equation, referred to as the modified R2, was compared with the R2 obtained from an independent regression analysis on the static modulus data. It was shown through this approach that the dynamic and static test results could be combined.
The results from the modified R2 analysis were validated using hypothesis testing (Mendenhall 2007). Regression coefficients calculated from multiple linear regression are random variables due to the finite sample size. It can be hypothesized that the coefficients obtained from two different sample data sets come from the same distribution, or at least have the same mean values. Hypothesis testing can be used to determine if the hypothesis that the two sample sets come from the same distribution should be rejected or not based on a specified significance level. In this process, separate regression relationships [using the form in Equation (5.1)] were developed for the static modulus of elasticity and the dynamic modulus of elasticity data sets. It can be shown that the hypothesis that the regression coefficients from these data sets have the same mean is false (implying that the data should not be combined) if:
(5.2)
where, t is the calculated test statistic as:
�
t =βi,1 − βi,2( )si,1
2
n1+si,2
2
n2
(5.3)
with,
, = ith modified (see Equations 4.4 and 4.5) regression coefficient for the first and second data sets, respectively;
, = variance of the ith regression coefficient for the first and second data sets respectively;
=
�
s((XTX)1)1/2;
= variance of the data;
X = data matrix where the first column contains values of 1.0, the next column contains the normalized temperature values, and the third column contains the normalized square of the temperature; and
, = number of data points for the first and second data sets, respectively.
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In Equation (5.2), is the critical test statistic, which is tabulated (Mendenhall 2007) for known values of the significance level (using a typically accepted value of α = 0.05) and the number of degrees of freedom, v, calculated as:
(5.4)
Using this procedure for both normalstrength and highstrength concrete, it was determined that there is not enough evidence to reject the hypothesis that the regression coefficients for the static and dynamic modulus ratio data have the same mean values. This supports the conclusion found earlier using the modified R2 method. Note again that the Ec data used in the hypothesis testing analysis was normalized with respect to the corresponding measured values of the room temperature modulus of elasticity, Eco, and thus, the regression models were constrained to a value of 1.0 at room temperature. Since the equations were constrained, only the temperaturedependent regression coefficients were compared (i.e., the intercept term,
�
β0, was not compared).
The final check to determine if the different data sets could be combined to increase the size of the data pool used coded variables to evaluate the significance of a particular parameter. For example, a coded variable for the modulus of elasticity test type (e.g., static or dynamic) was introduced into the regression model. This variable would take a value of one if the data point in question is obtained from a static modulus test and a value of zero if the data point is from a dynamic modulus test. By conducting a Student’s t Test (Mendenhall 2007) to evaluate if this coded variable is important in the multiple regression analysis of the combined set of dynamic and static modulus data, it was possible to determine if the test type is a significant parameter in the prediction of the modulus of elasticity, Ec, at elevated temperatures.
Upon conducting these statistical tests, it was concluded that the dynamic (both ultrasonic pulse velocity and resonance frequency) and static (both secant and tangent) modulus data could be combined to broaden the data pool for Ec. Note that 25 data points (from the original 461 data points) were excluded from the database because either the modulus data were taken outside of the linear range of the fcεc behavior or the test type was unknown and showed a statistical difference from the rest of the modulus of elasticity data. It was also found that highstrength concrete (HSC) and normalstrength concrete (NSC) have a statistical difference at elevated temperatures. The optimum cutoff strength between NSC and HSC was found as fcmo = 6,000 psi (41.4 MPa), which is the same value as that for the temperaturedependent compressive strength discussed in Chapter 4 (i.e., NSC: 0 ≤ fcmo ≤ 6,000 psi [41.4 MPa] and HSC: fcmo > 6,000 psi [41.4 MPa]). Furthermore, similar to the compressive strength data, the modulus of elasticity data was split according to the aggregate type and heating test type. Based on the limited data pool, five different regression relationships were developed for the NSC calcareous unstressed, NSC lightweight unstressed, HSC calcareous residual, HSC calcareous stressed, and HSC calcareous unstressed cases. Note that the modulus of elasticity data
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for the stressed test case was collected from only one source (Phan and Carino 2001). It is not exactly clear how the stress versus strain measurements from these stressed tests were used to determine the modulus of elasticity.
5.1.2 Normalization of Modulus of Elasticity
The ultimate goal of the regression models is to determine a predicted value for Ec rather than a predicted value for . Looking at the normalized relationship given by Equation (5.1), the only way to get a value for Ec would be to unnormalize the ratio by a known or wellestablished quantity for the room temperature modulus of elasticity, Eco. A wellestablished equation for the room temperature modulus of elasticity is given by ACI 318 (2008) as:
�
Eco_ ACI = γ c1.5 33 fcmo ; in psi (5.5)
�
Eco_ ACI = 57000 fcmo ; in psi (5.6)
where, Equation (5.5) was used for lightweight concrete and Equation (5.6) was used for normalweight concrete with γc = unitweight of concrete (lb/ft3). Measured values of γc = 109.5 lb/ft3 and γc = 111 lb/ft3 were used for the lightweight modulus data.
For the regression analysis, it was shown to be better to normalize the measured modulus of elasticity, Ec data with respect to the ACI modulus, Eco_ACI rather than the measured room temperature modulus, Eco. To validate this process, a multiple linear regression analysis was conducted on the Ec/Eco data as well as the Ec/Eco_ACI data. Then, the Ec/Eco_ACI and Ec/Eco ratios obtained from both of these two regression analyses were unnormalized using Eco_ACI to determine Ec. Note that no attempt was made to develop a new regression relationship for Eco since a replacement model for Eco_ACI could not be justified based on the measured Eco data in the database. Note also that the regression coefficients developed in this report for Ec / Eco_ACI would need to be updated if a revised Eco_ACI equation is specified in a future edition of ACI 318 (2008).
Table 5.1 shows the R2 values calculated for Ec / Eco_ACI, Ec / Eco, and Ec for each available aggregate type and test type combination in the database. It can be seen that if a normalized quantity was sought for the modulus of elasticity at elevated temperatures, then, Ec / Eco would be the better way to normalize the data as compared with Ec / Eco_ACI. However, since a predicted value for Ec is more important for design, the Ec / Eco_ACI ratio provides the best fit to the Ec data for all cases except the NSC calcareous, residual case. The small discrepancy for the NSC calcareous, residual case is overshadowed by the difference in the R2 values for the NSC lightweight, unstressed case where the R2 = 0 value for Ec obtained from the Ec / Eco ratio suggests a complete lack of fit with the data.
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TABLE 5.1: R2 STATISTICS FOR Ec / Eco_ACI, Ec / Eco, AND Ec FOR EACH DATA SET
NSC HSC
Calcareous Lightweight Calcareous
Unstressed Unstressed Residual Stressed Unstressed
Ec/Eco 0.84 0.86 0.91 0.97 0.87 Ec/Eco_ACI 0.65 0.66 0.72 0.97 0.89 Ec from Ec / Eco 0.70 0.00 0.64 0.96 0.89 Ec from Ec / Eco_ACI 0.64 0.72 0.73 0.97 0.91
5.1.3 Preliminary Regression Forms
In determining the final form of the regression equations for the modulus of elasticity, the goal was to remain consistent with the compressive strength relationships. Therefore, polynomial equations with linear, quadratic, and cubic terms in temperature were considered for each aggregate and test type combination. A summary of the R2 values for the different trial regression equations is given below in Table 5.2 for each of the combinations.
TABLE 5.2: REGRESSION STATISTICS FOR MODULUS OF ELASTICITY TRIAL
EQUATIONS FOR EACH TEST TYPE AND AGGREGATE TYPE COMBINATION
NSC
Calcareous Lightweight Equation
Unstressed Unstressed
0.63 0.65
0.65 0.66
0.66 0.66
HSC
Calcareous Equation
Residual Stressed Unstressed
0.66 0.96 0.79
0.72 0.97 0.89
0.72 0.97 0.89
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Along with these R2 values for each trial regression equation, the Student’s t Test as well as the Analysis of Variance FTest were used in determining the best equation form. By looking at the R2 values, it can be seen that the addition of a cubic temperature term does not significantly add to the fit of the data. This result was confirmed with the Student’s t Test and the Analysis of Variance FTest.
5.1.4 Regression Assumptions
As described in Chapter 4, four assumptions are made about the error introduced in predicting the data through a multiple regression analysis. By inspection of the residual plots from each of the regression analyses on the modulus of elasticity data, it was determined that Assumptions 1, 3, and 4 were met for all cases. For Assumption 2, the KS test was implemented to determine if the error for the data could statistically come from a normal distribution. Using the quadratic equation form in temperature, the required significance level to pass the KS test is shown below in Table 5.3. Note again that a typically acceptable value for the significance level is α = 0.05, and any value greater than that is more definitive that the error is normally distributed. It can be seen that with the exception of the HSC calcareous residual case, the available data passes the KS test. It has been suggested that departures from the normality assumption may have little impact on the regression results (Mendenhall 2007). Thus, the relationship proposed in this report for the HSC calcareous, residual case is recommended for use with caution until additional test data is developed and the statistical models are reevaluated.
TABLE 5.3: REQUIRED SIGNIFICANCE LEVEL TO PASS THE KOMOLGOROVSMIRNOV
TEST FOR THE MODULUS OF ELASTICITY REGRESSIONS
Aggregate Type Test Type Required α
NSC Calcareous Unstressed 0.10 NSC Lightweight Unstressed 0.10
Residual 0.00 Stressed 0.10 HSC Calcareous
Unstressed 0.10
5.2 Proposed Relationships
The final form of the modulus of elasticity model developed in this research is a quadratic relationship in temperature given by:
(5.7)
where,
= aggregate and test type dependent coefficient (see Table 5.4); and
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T = maximum exposure temperature (°F).
TABLE 5.4: PROPOSED MODULUS OF ELASTICITY RELATIONSHIP REGRESSION
COEFFICIENTS
NSC HSC
CALCAREOUS LIGHTWEIGHT CALCAREOUS
Unstressed Unstressed Residual Stressed Unstressed 1.292 0.738 1.351 1.058 1.055 1.271E03 5.916E04 2.037E03 1.168E03 1.146E03 4.163E07 1.308E07 9.671E07 3.180E07 4.654E07
T Range (°F) [75, 1400] [74, 1412] [73, 1112] [77, 1112] [75, 1400]
fcmo Range (psi)
[1149, 5443] [2716, 3893] [6440, 13982] [7102, 14707] [6316, 14707]
R2 on Ec 0.64 0.72 0.73 0.97 0.91
Along with the regression coefficients, Table 5.4 also shows the acceptable ranges of the room temperature compressive strength, fcmo and the maximum exposure temperature, T to be used with the proposed equations. These range limits are imposed by the currently available data on which the regression analysis is based. Extrapolation of the proposed equations outside of the acceptable ranges for fcmo and T is not recommended.
Note that as described in Chapter 4, two equations with coded variables were used to develop a total of 12 separate regression relationships for the concrete compressive strength. Since the modulus of elasticity data is not as well populated as the compressive strength data, there are only five different test type and aggregate type combinations for the modulus of elasticity. It was found to be more practical to conduct a regression analysis for each of the five subsets of data, rather than implement coded variables in two equations. Furthermore, since the modulus of elasticity, Ec data was normalized with respect to the ACI modulus, Eco_ACI (rather than the measured room temperature modulus, Eco), the regression relationships were not constrained at room temperature. The final regression coefficients for each aggregate type and test type combination as shown in Table 5.4 were determined as:
1) With Equation (5.1) as the regression form (except for using Ec / Eco_ACI instead of Ec / Eco), use Equation (4.2) to determine the normalized regression coefficients without constraining the equation.
2) Modify the regression coefficients according to Equations (4.4) and (4.5).
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3) Determine the final modulus of elasticity relationships [given by Equation (5.7)], where,
�
κ Ei=
�
βi from step 2.
5.3 Results and Evaluations
As given by Equation (5.7) and Table 5.4, the multiple regression analysis for the concrete modulus of elasticity results in five different relationships (two for normalstrength concrete and three for highstrength concrete). These relationships are evaluated below.
5.3.1 Comparisons with Test Data and Evaluation of Data Fit
As an overall evaluation of the fit between the proposed equations for the concrete modulus of elasticity and the corresponding data, the coefficient of determination, R2 for each of the five relationships is given in Table 5.4 (note that the R2 values were calculated based on the Ec data after multiplying the equations with Eco_ACI). It can be seen that the regression model fits the data very well. The three HSC equations have the best fit with R2 values of 0.74, 0.97, and 0.91 for the residual, stressed, and unstressed test cases, respectively. The lowest R2 value was calculated as 0.64 for the NSC calcareous unstressed case, which is still an acceptable fit to the data.
Figure 5.1 shows the proposed regression curves against the five data sets. It can be seen that while the NSC calcareous unstressed equation has the worst fit among the five cases (based on the R2 values), the majority of the variability in the data comes at lower temperatures and the model seems to fit the hightemperature data range very well. The same is true of the NSC lightweight unstressed case and the HSC calcareous unstressed case.
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(a) (b)
(c) (d)
(e)
Figure 5.1: Proposed modulus of elasticity relationships fit to data: (a) NSC – calcareous, unstressed; (b) NSC – lightweight, unstressed; (c) HSC – calcareous, residual; (d) HSC –
calcareous, stressed; and (e) HSC – calcareous, unstressed.
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Similarly, Figure 5.2 compares the prediction band and the measured Ec values for each aggregate and test type combination in the database. The prediction bands for Ec were generated by multiplying the Ec / Eco_ACI values from the regression models with the minimum and maximum Eco_ACI values for each data set, calculated using Equations (5.5) and (5.6) for lightweight and normalweight concrete, respectively. For lightweight concrete, the minimum and maximum known values of γc were 109.5 lb/ft3 and 111 lb/ft3, respectively. By looking at the prediction bands, it is possible to see the upper and lower limit of Ec that the regression models can predict as the temperature is increased. In general, a good fit is observed between the test data and the prediction bands. Note that the NSC lightweight unstressed and the HSC calcareous residual models have acceptable R2 values; however, the prediction bands do not capture a significant number of the data in the low temperature ranges. Therefore, it is concluded that the proposed Ec models for these two cases are better suited for high temperature ranges.
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(a) (b)
(c) (d)
(e)
Figure 5.2: Proposed modulus of elasticity prediction bands: (a) NSC – calcareous, unstressed; (b) NSC – lightweight, unstressed; (c) HSC – calcareous, residual; (d) HSC –
calcareous, stressed; and (e) HSC – calcareous, unstressed.
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5.3.2 Effects of Aggregate Type, Test Type, and Room Temperature Strength
The effects of aggregate type and test type, and comparisons between normalstrength and highstrength concrete tendencies for the proposed Ec / Eco_ACI models are shown in Figure 5.3. Since the regression models were normalized with respect to Eco_ACI and not Eco, it was not possible to constrain the equations at room temperature, thus making visual comparisons between the different curves difficult. To observe the general trends in the relative amounts of stiffness loss, each model was plotted as a percentage of the Ec / Eco_ACI ratio retained from the room temperature Ec / Eco_ACI ratio. To do this, Equation (5.7) was simply divided by the predicted Ec / Eco_ACI value at room temperature for each aggregate type and test type combination.
(a) (b)
(c)
Figure 5.3: Effects on the proposed temperaturedependent modulus of elasticity relationships: (a) aggregate type; (b) test type; and (c) NSC versus HSC.
The effect of the different aggregate types is shown in Figure 5.3(a). Because of the lack of available data for siliceous aggregate concrete, the only comparison that can be made is between lightweight and calcareous aggregate concrete. It can be seen that with respect to the Ec / Eco_ACI value at room temperature, lightweight concrete tends to result in a smaller reduction in Ec / Eco_ACI than calcareous concrete as the temperature is increased; however, the differences are generally very small. Similarly, by looking at
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Figure 5.3(b), the unstressed test type has the smallest reduction in Ec / Eco_ACI over the temperature range. Lastly, Figure 5.3(c) shows that HSC has a similar amount of loss in Ec/Eco_ACI as NSC with increasing temperature.
5.3.3 Comparisons with Previous North American Models
Comparisons between the proposed modulus of elasticity models and the ACI 216 (2007), ASCE (1992), and Kodur et al. (2008) models (see Figure 2.7) are shown in Figure 5.4. Since the proposed models were normalized using Eco_ACI whereas the previous models were normalized using Eco, direct comparisons between these models are difficult. Therefore, similar to Figure 5.3, each curve in Figure 5.4 was generated by dividing the temperature dependent Ec / Eco_ACI ratio (for the proposed models) or Ec / Eco ratio (for the previous models) with the corresponding ratio at room temperature. By comparing the models in this manner, it is possible to determine the relative amount of stiffness lost from the room temperature modulus.
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(a) (b)
(c) (d)
(e)
Figure 5.4: Comparison of proposed modulus of elasticity models with ACI 216, ASCE, and Kodur et al. models: (a) NSC – calcareous, unstressed; (b) NSC – lightweight,
unstressed; (c) HSC – calcareous, residual; (d) HSC – calcareous, stressed; and (e) HSC – calcareous, unstressed.
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It can be seen that the proposed and ACI 216 models for NSC tend to show similar trends as the temperature is increased. The calcareous NSC model from ASCE gives significantly greater stiffness loss than either the proposed or the ACI 216 models (note that the ASCE model is not valid for lightweight aggregate concrete, and therefore is not shown for this case). This discrepancy is most likely due to the implicity inclusion of creep strains in the ASCE model, which are not included in the ACI 216 and proposed models. The proposed models are intended to provide a baseline to which creep strains could explicitly be included in the future so that time and temperaturedependent relationships would be available. Looking at the highstrength concrete results, the Kodur et al. model also implicitly includes creep strains, and therefore, there is a larger percent stiffness loss as compared to the proposed models, with the unstressed case showing the largest difference between the proposed and the Kodur et al. models. It is important to note that the Kodur et al. model does not distinguish between the test type.
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CHAPTER 6: STRAIN AT PEAK STRESS
This chapter presents the statistical analysis conducted on the data for the strain at peak stress of concrete at elevated temperatures. Section 6.1 describes the process used to conduct the statistical analysis. Section 6.2 proposes temperaturedependent strain at peak stress relationships resulting from the statistical analysis. Lastly, Section 6.3 provides the results and evaluation of the proposed strain at peak stress relationships along with comparisons of these relationships with previous models.
6.1 Statistical Analysis
Multiple regression analysis was conducted on the 30 data points collected for the concrete strain at peak stress, where each data point represents a measured strain corresponding to a maximum exposure temperature. All of the data for the strain at peak stress, εcm, was collected from either experimental stressstrain (fcεc) curves or experimental loaddeformation curves. Utilizing the regression analysis program written in MATLAB® for the temperaturedependent compressive strength (see Chapter 4), a similar regression analysis procedure was conducted for the strain at peak stress.
6.1.1 Strain at Peak Stress Data
To remain consistent with the compressive strength and modulus of elasticity models described in Chapters 4 and 5, the strain at peak stress data was split based on the available aggregate type and test type combinations for both normalstrength and highstrength concrete. The only available data for NSC is for calcareous aggregate concrete subjected to the unstressed test type. For HSC, the available data includes both calcareous and siliceous concrete under the unstressed test. Furthermore, to remain compatible with the compressive strength and modulus of elasticity models, a limit of fcmo = 6,000 psi (41.1 MPa) was used to separate the normalstrength concrete data from the highstrength concrete data.
In determining the form of the regression equation for εcm, it was necessary first to determine if the available data showed enough difference between the aggregate types or between the HSC and NSC results in the database. To complete this analysis, first the normalstrength and highstrength concrete data were graphed using different markers, so that the differences between the two sets could be visually observed. Figure 6.1 shows the results of comparing the NSC and HSC data sets.
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Figure 6.1: Comparison of HSC and NSC data with bestfit line for each set.
For the comparison between the HSC and NSC results, a quadratic regression function in temperature was assumed. Because the data in Figure 6.1 is normalized with respect to the room temperature strain at peak stress, εcmo, by definition, the εcm/εcmo ratio is equal to 1.0 at room temperature. Therefore, the regression equations were constrained at room temperature. For the two regression curves in Figure 6.1, the resulting R2 values are 0.74 and 0.93 for HSC and NSC, respectively. The HSC R2 value is a reasonable fit to the data, and the NSC R2 value demonstrates a very good fit. Looking at the results, it is clear that the highstrength and normalstrength data should be fit using different equations. It can also be seen that the HSC data has a larger relative strain at elevated temperatures than NSC and tends to have larger variability as well.
Note that even though the full data set for the strain at peak stress includes 30 data points, only 28 points are plotted in Figure 6.1 and used in the regression analysis. Two of the HSC data points had εcm/εcmo values greater than six [at 1472°F (800°C)], which is much larger than any of the other available data. These data points were removed from the analysis. If the two extreme points had been included, the regression results would have been skewed towards these larger data points and would have resulted in a worse fit for the majority of the data.
It was also necessary to determine if there was a distinguishable difference between siliceous and calcareous aggregate HSC. First, of the 19 data points for HSC, there were only three data points for siliceous aggregate HSC. Furthermore, these data points came from a paper where calcareous aggregate HSC was also tested. By comparing the results for the two aggregate types, three of the four data points for siliceous concrete had the exact same εcm/εcmo values as the calcareous aggregate data. Therefore, it was concluded that because of the lack of available data, a distinction could not be made between siliceous and calcareous aggregate HSC, and the data for these two cases were combined.
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6.1.2 Preliminary Regression Forms
To determine the final form of the regression relationships for the concrete strain at peak stress, linear, quadratic, and cubic polynomial equations in temperature were tried for both NSC and HSC. The resulting R2 values from this analysis are shown in Table 6.1.
TABLE 6.1: R2 VALUES OF THE STRAIN AT PEAK STRESS TRIAL REGRESSIONS
Equation NSC R2 HSC R2
0.87 0.74
0.93 0.74
0.94 0.76
From the R2 values, it can be seen that the cubic equation form provides the best fit to the data. However, the cubic temperature term results in only a very small increase in R2, suggesting that it does not play a significant role in predicting the εcm/εcmo ratio for both NSC and HSC. This conclusion was also validated using the Student’s t Test. It was also found that using a cubic equation for the HSC data, although having the largest value of R2 and providing a good fit to the NSC data, does not fully demonstrate the increasing trend of the HSC strain data. Keeping these factors in mind, it was determined that the best form for the regression analysis is the second order temperature polynomial.
6.1.3 Regression Assumptions
The four assumptions for the strain at peak stress regression analysis were verified similar to the analyses described in Chapters 4 and 5 for the concrete compressive strength and elastic modulus. By inspection of the residual plots from the strain data, it was determined that Assumptions 1, 3, and 4 were met. For Assumption 2, the KS test was implemented to determine if the error for the data could statistically come from a normal distribution. Using the quadratic equation form in temperature to determine the required significance level to pass the KS test in Table 6.2, it is highly probable that the error comes from a normal distribution. Therefore, the regression equations developed in this chapter satisfy all of the underlying statistical assumptions made.
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TABLE 6.2: REQUIRED SIGNIFICANCE LEVEL TO PASS THE KOMOLGOROVSMIRNOV
NORMALITY TEST
Strength Type Aggregate Type Test Type Required α
NSC Calcareous Unstressed 0.10 HSC Calcareous and
Siliceous Unstressed 0.10
6.2 Proposed Relationships
The final form of the strain at peak stress model developed in this research is a quadratic relationship in temperature given by:
(6.1)
where,
= aggregate and test type dependent coefficient (see Table 6.3); and
T = maximum exposure temperature (°F).
TABLE 6.3: PROPOSED STRAIN AT PEAK STRESS RELATIONSHIP REGRESSION
COEFFICIENTS
NSC HSC
CALCAREOUS SILICEOUS AND CALCAREOUS
Unstressed Unstressed 0.981 0.896 2.181E04 1.431E03 6.426E07 8.772E08
T Range (°F) [73, 1472] [73, 1472] fcmo Range (psi) [4504, 4666] [9102, 11458] R2 0.93 0.74
Along with the regression coefficients, Table 6.3 also shows the acceptable ranges of the room temperature compressive strength, fcmo and the maximum exposure temperature, T to be used with the proposed equations. These range limits are imposed by the currently available data on which the regression analysis is based. Extrapolation of
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the proposed equations outside of the acceptable ranges for fcmo and T is not recommended.
As described previously, only two different regression relationships (one for NSC, and one for HSC) were possible to develop based on the available strain at peak stress data. It was found to be more practical to conduct a regression analysis for each of these two subsets of data, rather than implement coded variables using the full data. The final regression coefficients shown in Table 6.3 were determined using the same process described in Chapter 5 for the modulus of elasticity. The only difference is that the normalized regression coefficients were determined using Equation (4.3) rather than Equation (4.2) because the εcm/εcmo ratio was constrained to a value of one at room temperature. Then, the
�
κε micoefficients in Table 6.3 are the same as the modified
regression coefficients determined from Equations (4.4) and (4.5).
Note that in order to calculate εcm from the proposed regression relationships, it is necessary to have an estimate for εcmo. Using the available measured data on εcmo (Castillo and Durani 1990, Cheng et al. 2004) and the assumption that εcmo = 0.002 at fcmo = 3,000 psi (20.7 MPa), an interpolation equation for εcmo as a function of fcmo (in psi) was found as:
(6.2)
In determining Equation (6.2), of all of the available data for the strain at peak stress, only one paper (Cheng et al. 2004) reported on εcmo. Furthermore, the only εcmo result from this paper was a value of εcmo=0.003 obtained from experimental fcεc curves for HSC. The average of the fcmo values for these data points was calculated as 11,407 psi and a linear interpolation was determined from εcmo = 0.002 to 0.003 for fcmo = 3,000 psi to 11,407 psi, resulting in Equation (6.2).
6.3 Results and Evaluations
As an overall evaluation of the fit between the proposed equations for the concrete strain at peak stress and the corresponding data, the coefficient of determination, R2 for the two relationships is given in Table 6.3. Similarly, Figures 6.2(a) and 6.2(b) compare the proposed εcm regression relationships for NSC and HSC, respectively, with the corresponding data set. Note that the regression curves were unnormalized using Equation (6.2); whereas, the data points were unnormalized using the measured εcmo values, except when the εcmo value was not known. Based on these comparisons and the R2 values in Table 6.3, it is concluded that the regression model fits the data well, especially for NSC.
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(a) (b)
Figure 6.2: Proposed strain at peak stress relationships fit to data: (a) NSC, calcareous, unstressed; and (b) HSC, siliceous and calcareous, unstressed.
By looking at Figure 6.3, it can be seen that the proposed HSC εcm model yields larger strains than the NSC model (unlike the ASCE NSC versus the Kodur et al. HSC models in Figure 2.9), and that the HSC strains experience a greater increase as the temperature is increased.
Figure 6.3: Comparison of proposed NSC and HSC strain at peak stress relationships.
Moreover, as shown in Figure 6.4 comparing the proposed regression relationships with the previous ASCE and Kodur et al. models, the predicted strains from the proposed models are significantly smaller (especially at high temperatures). As described earlier, this difference is possibly due to the implicit inclusion of creep strains in the previous models. By comparing the previous models with the available data on εcm, it is clear that creep introduces large strains at elevated temperatures, and therefore there is a need for the future development of a time and temperaturedependent explicit total strain relationship including creep effects.
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(a) (b)
Figure 6.4: Comparison of the proposed strain at peak stress relationships with ASCE and Kodur et al. models: (a) NSC, calcareous, unstressed; and (b) HSC, siliceous and
calcareous, unstressed.
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CHAPTER 7: ULTIMATE STRAIN
This chapter presents the statistical analysis conducted on the data for the ultimate strain of concrete at elevated temperatures. Section 7.1 describes the process used to conduct the statistical analysis. Section 7.2 proposes temperaturedependent ultimate strain relationships resulting from the statistical analysis. Lastly, Section 7.3 provides the results and evaluation of the proposed ultimate strain relationships along with comparisons of these relationships with previous models.
7.1 Statistical Analysis
Multiple regression analysis was conducted on the 70 data points collected for the concrete ultimate compressive strain at elevated temperatures, where each data point represents a measured strain corresponding to a maximum exposure temperature. All of the data for the ultimate strain, εcu, was collected from either experimental stressstrain (fcεc) curves or calculated from the measured “deformation at fracture.” The ultimate strain is taken as the strain corresponding to 85% of the peak stress in the postpeak range of the stressstrain relationship. In cases where an abrupt drop in stress occurs (indicating failure) prior to reaching 85% of the peak stress, the strain at the stress drop is taken as the ultimate strain. Utilizing the regression analysis program written in MATLAB® for the temperaturedependent compressive strength (see Chapter 4), a similar regression analysis procedure was conducted for the ultimate strain.
7.1.1 Ultimate Strain Data
To remain consistent with the fcm, Ec, and εcm models described in the previous chapters, the ultimate strain, εcu data set was split based on the available aggregate type and test type combinations for both normalstrength and highstrength concrete. The only available data for NSC is for calcareous and lightweight aggregate concrete subjected to the unstressed test type. For HSC, the available data includes calcareous and siliceous concrete under the unstressed test. Furthermore, to remain compatible with the fcm, Ec, and εcm models, a limit of fcmo = 6,000 psi (41.1 MPa) was used to separate the normalstrength ultimate strain data from the highstrength data.
Similar to εcm, there were a few εcu data points that did not seem to fit the rest of the data pool. Looking at the normalized results with respect to the room temperature ultimate strain, εcuo, most of the data had εcu/εcuo ratios of less than 4.5, however at the highest temperatures, there were three values with εcu/εcuo well over 7.0. Because of the difficulty in accurately measuring the post peak behavior of concrete, and because these three data points were much larger than the rest of the data, they were removed from the data pool.
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To determine the form of the regression equations using the remaining 67 data points, it was first necessary to determine how many aggregate type combinations could be used. For the HSC data, this was a simple task. Of the 19 data points collected for HSC εcu, there were 14 for calcareous aggregate concrete and only five for siliceous aggregate concrete. Furthermore, all of the five siliceous aggregate data points had exactly the same εcu values as several of the calcareous data points at the same temperatures. Therefore, a distinction could not be made between the two aggregates, and only one regression relationship was developed for HSC using the combined calcareous/siliceous aggregate data pool.
For NSC, the decision was not quite as simple. There were 39 NSC εcu data points for lightweight aggregate concrete and nine points for calcareous aggregate. The NSC ultimate strain data is shown in Figure 7.1. By visually inspecting the data, it was determined that a significant distinction could not be made between the lightweight and calcareous aggregate data. Furthermore, in conducting a modified R2 analysis on these two data sets, it was determined that a line fit to the lightweight aggregate data would still provide a reasonable fit to the calcareous aggregate data (R2 = 0.62). As a result, it was decided to develop a single equation for NSC using the combined lightweight/calcareous aggregate data pool.
Figure 7.1: Comparison of the calcareous and lightweight aggregate data for the ultimate strain of normalstrength concrete.
7.1.2 Preliminary Regression Forms
After it was determined that there would be one regression equation for NSC and one equation for HSC, the best form of these equations was determined. Similar to the process used for εcm, linear, quadratic, and cubic polynomial equations in temperature were tried for both the NSC and HSC εcu data sets. The resulting R2 values from this analysis are shown in Table 7.1.
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TABLE 7.1: R2 VALUES OF THE ULTIMATE STRAIN TRIAL REGRESSIONS
Equation NSC HSC
0.60 0.81
0.84 0.82
0.86 0.83
From the R2 values, it can be seen that the cubic temperature term results in only a very small increase in R2, suggesting that it does not play a significant role in predicting the εcu/εcuo ratio for both NSC and HSC. This conclusion was also validated using the Student’s t Test. However, comparing the quadratic equation to the NSC data (see Figure 7.2), it was found that when the equation was constrained at room temperature, the quadratic model predicted a loss in εcu/εcuo at lower temperatures. The cubic model on the other hand predicted that εcu/εcuo would stay above 1.0 through the entire temperature range, providing a better trend to the available data. As a result, the cubic equation was determined as the best regression form for the NSC εcu/εcuo data. To remain consistent, the cubic equation model was also used for the HSC data.
Figure 7.2: Cubic and quadratic functions fit to the NSC ultimate strain data.
7.1.3 Regression Assumptions
Similar to the fcm, Ec, and εcm models described in the previous chapters, regression Assumptions 1, 3, and 4 were validated by inspection of the residual plots from each of the regression analyses on the ultimate strain data. For Assumption 2, the KS test was implemented to determine if the error for the data could statistically come from a normal distribution. Using the cubic equation form in temperature to determine the required significance level to pass the KS test in Table 7.2, the assumption that the error
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comes from a normaldistribution cannot be rejected. Therefore, the regression equations developed in this chapter satisfy all of the underlying statistical assumptions made.
TABLE 7.2: REQUIRED SIGNIFICANCE LEVEL TO PASS THE KOMOLGOROVSMIRNOV
NORMALITY TEST
Strength Type Aggregate Type Test Type Required α
NSC Lightweight and Calcareous
Unstressed 0.10
HSC Calcareous and Siliceous
Unstressed 0.10
7.2 Proposed Relationships
The final form of the ultimate strain model developed in this research is a cubic relationship in temperature given by:
(7.1)
where,
= aggregate and test type dependent coefficient (see Table 7.3); and
T = maximum exposure temperature (°F).
Along with the regression coefficients, Table 7.3 also shows the acceptable ranges of the room temperature compressive strength, fcmo and the maximum exposure temperature, T to be used with the proposed equations. These range limits are imposed by the currently available data on which the regression analysis is based. Extrapolation of the proposed equations outside of the acceptable ranges for fcmo and T is not recommended.
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TABLE 7.3: PROPOSED ULTIMATE STRAIN RELATIONSHIP REGRESSION COEFFICIENTS
NSC HSC
LIGHTWEIGHT AND CALCAREOUS
SILICEOUS AND CALCAREOUS
Unstressed Unstressed 0.979 0.991 3.377E04 7.689E05 7.561E07 2.803E06 1.186E09 1.035E09
T Range (°F) [73, 1472] [73, 1472] fcmo Range (psi) [2716, 4667] [9102, 11458] R2 0.86 0.83
As described previously, only two different regression relationships (one for NSC, and one for HSC) were possible to develop based on the available ultimate strain data. Similar to the strain at peak stress, it was found to be more practical to conduct a regression analysis for each of these two subsets of data, rather than implement coded variables using the full data. The final regression coefficients in Table 7.3 were determined using the same process described in Chapter 6 for the strain at peak stress. Again, the normalized regression coefficients were determined using Equation (4.3) because the εcu/εcuo ratio was constrained to a value of one at room temperature. These normalized regression coefficients were then modified according to Equations (4.4) and (4.5), which are exactly equal to the
�
κε uivalues shown in Table 7.3.
Note that in order to calculate εcu from the proposed regression relationships, it is necessary to have an estimate for εcuo. By looking at the available test data (Castillo and Durani 1990, Cheng et al. 2004, and Harmathy and Berndt 1966), it was found that εcuo ranges from 0.0027 to 0.0038. Since most of the data falls around a value of 0.003 and since this value is also assumed as the maximum usable strain of concrete in Chapter 10 of ACI 318 (2008),
(7.2)
is recommended for use with Equation (7.1) regardless of the concrete strength or aggregate type.
It should also be noted that a limit imposed on εcu is that it must be greater than or equal to εcm. Thus, εcu is taken as equal to εcm if the value of εcu obtained from Equations (7.1) and (7.2) is less than εcm.
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7.3 Results and Evaluations
As an overall evaluation of the fit between the proposed equations for the concrete ultimate strain and the corresponding data, the coefficient of determination, R2 for the two relationships is given in Table 7.3. Similarly, Figures 7.3(a) and 7.3(b) compare the proposed εcu regression relationships for NSC and HSC, respectively, with the corresponding data set. Note that the regression curves were unnormalized using Equation (7.2); whereas, the data points were unnormalized using the measured εcuo values except when the εcuo value was not known. Based on these comparisons and the R2 values in Table 7.3, it is concluded that both models provide a good fit to the available data.
(a) (b)
Figure 7.3: Proposed ultimate strain relationships fit to data: (a) NSC, lightweight and calcareous, unstressed; and (b) HSC, calcareous and siliceous, unstressed.
By looking at Figure 7.4, it can be seen that the proposed HSC εcu model yields larger strains than the NSC model (unlike the ASCE NSC versus the Kodur et al. HSC models shown in Figure 2.10), except where the two graphs intersect at room temperature and at a high temperature of approximately 1472°F (800°C). For NSC, εcu remains almost constant up to a temperature of about 400 to 600°F (204 to 316°C); whereas εcu for HSC increases with even relatively small increases from the room temperature.
Figure 7.4: Comparison of proposed NSC and HSC ultimate strain relationships.
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Furthermore, as shown in Figure 7.5 comparing the proposed regression relationships with the previous ASCE and Kodur et al. models, the predicted ultimate strains from the proposed models are significantly smaller. As described earlier, this difference is possibly due to the implicit inclusion of creep strains in the previous models.
(a) (b)
Figure 7.5: Comparison of the proposed ultimate strain relationships with ASCE and Kodur et al. models: (a) NSC, lightweight and calcareous, unstressed; and (b) HSC,
calcareous and siliceous, unstressed.
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CHAPTER 8: STRESSSTRAIN RELATIONSHIP
This chapter focuses on the compressive stressstrain relationship of concrete at elevated temperatures. Section 8.1 introduces the model used to describe the temperaturedependent concrete stressstrain relationship. Section 8.2 gives the results and evaluation of the proposed stressstrain relationships. Lastly, Section 8.3 compares the proposed relationships with previous stressstrain models.
8.1 Temperature Modified StressStrain Model
Temperaturedependent compressive stressstrain (fcεc) relationships for concrete were developed by combining the temperaturedependent strength (fcm), modulus of elasticity (Ec), and strain (εcm, εcu) models with the room temperature concrete fcεc relationship proposed by Popovics (1973), later adapted by Mander et al. (1988) for use with confined concrete. The resulting stressstrain model takes the following form:
(8.1)
(8.2)
(8.3)
(8.4)
where,
fc = temperaturedependent concrete stress;
εc = concrete strain;
fcm = peak stress obtained from Equation (4.8);
εcm = strain at peak stress obtained from Equations (6.1) and (6.2); and
Ec = modulus of elasticity obtained from Equations (5.5) and (5.6).
8.2 Results and Evaluations
As shown in Figure 8.1, two different sets of stressstrain curves can be drawn from the collected data and the relationships created as follows: (1) calcareous NSC subjected to the unstressed test; and (2) calcareous HSC subjected to the unstressed test.
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The fcεc curves are normalized with respect to fcmo, and the last point on each curve represents the εcu value obtained from Equation (7.1). By looking at the NSC calcareous curves, it can be seen that a slight increase in strength occurs as temperature is increased to T = 392°F (200°C). It can also be seen that as the temperature is increased to T = 1,400°F (760°C), the calcareous concrete retains only about 33% of its original strength.
(a) (b)
Figure 8.1: Proposed stressstrain relationships: (a) NSC – calcareous, unstressed; and (b) HSC – calcareous, unstressed.
By comparing the two sets of stressstrain curves for calcareous NSC and HSC, it can be seen that whereas NSC experiences a slight strength gain followed by a gradual strength loss after T = 392°F (200°C), HSC experiences a significant strength drop immediately as temperature is increased followed by a period of relatively constant strength for T = 392°F, 752°F, and 1,112°F (200°C, 400°C, and 600°C). For the NSC stressstrain curves, there is a range of stress drop after reaching εcm, whereas for HSC, εcu is the same as εcm (i.e., no stress drop range) for T = 70°F, 392°F, and 752°F (21.1°C, 200°C, and 400°C). Note that for HSC at T = 70°F and 392°F, the predicted value of εcu was less than the value of εcm. As described in Chapter 7, εcu was taken as equal to εcm for these cases. Finally, it can also be seen that at T = 1,400°F (760°C), HSC is expected to lose a larger proportion of its original strength at room temperature as compared with the strength loss of NSC (about 80% loss versus 70% loss, respectively).
8.3 Comparisons with Previous Models
As a major difference between the proposed and existing models, the proposed fcεc curves in Figure 8.1 do not include creep strains whereas the ASCE (1992), Kodur et al. (2008), and Eurocode (2002, 2004) models in Figures 2.1 and 2.2 implicitly take creep into account. This results in significantly smaller strains in the proposed models as compared to the existing models. As described previously, the implicit consideration of creep in the ASCE, Kodur et al., and Eurocode models poses the following difficulties: (1) the creep effects are based on work conducted prior to most of the research on concrete creep at high temperatures; and (2) since the creep deformations are not included explicitly, the amount of time needed to accumulate the predicted strains cannot
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be determined. In comparison with these previous models, the proposed stressstrain models provide a baseline to which creep strains can be added in the future, utilizing an explicit timedependent relationship.
To validate that the differences between the proposed and existing stressstrain curves do not originate from the form (i.e., shape) of the Popovics function, Equations (8.1) through (8.4) were used to represent the ASCE, Kodur et al., and Eurocode stressstrain models. For this purpose, the concrete peak stress, fcm, modulus of elasticity, Ec, and strain at peak stress, εcm, from these previous models were used in Equations (8.1) through (8.4). Calcareous aggregate concrete was used in the study. For example, Figure 8.1(a) compares the Popovics, ASCE, Kodur et al., and Eurocode fcεc curves using the fcm, Ec, εcm, and εcu values from ASCE (1992) at T = 1,400°F (760°C). Similar comparisons using the Kodur et al. (2008) and the Eurocode (2004) fcm, Ec, εcm, and εcu values are depicted in Figures 8.1(b) and 8.1(c), respectively.
(a) (b)
(c)
Figure 8.2: Comparison of ASCE, Kodur et al., Eurocode, and Tmodified Popovics fcεc functions: (a) using ASCE parameters; (b) using Kodur et al. parameters; and (c) using
Eurocode parameters.
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The results from this process show that there are very small differences between the four curves up to the point of peak stress, demonstrating that the differences between these models do not stem from the use of different stressstrain function forms. Note that there are somewhat greater differences between the different curves beyond the peak stress; however, these differences are in general not meaningful since the measurements for most concrete fcεc test data cannot accurately capture the post peak behavior. For example, Eurocode states that any relationship can be used for the postpeak descending branch of the stressstrain curve, without even giving a value for the stress at the prescribed ultimate strain. For the purposes of this research, the stress reached at the ultimate strain for the Eurocode model was taken as 85% of the ultimate stress.
It is also important to emphasize that, as compared to the North American concrete models by ASCE and Kodur et al., the proposed models are based on a more comprehensive study of the previous concrete property test results at elevated temperatures, including a larger set of more diverse data. Comparing the ASCE model with the proposed calcareous NSC model, the ASCE model does not include any strength loss until T = 1,112°F (600°C). Also, at T = 1,400°F (760°C), the proposed model retains more than 13% of the room temperature strength than the ASCE relationship. Similarly, comparing the proposed HSC model with the Kodur et al. model, at T = 1112°F (600°C), the proposed model retains almost 20% more of the room temperature strength.
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CHAPTER 9: SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
This chapter summarizes the findings of the research presented in this report and how those findings could be applied to future research. Section 9.1 provides the conclusions resulting from the collection of the concrete property database and the statistical analysis. Section 9.2 presents areas where future research could be focused based on the existing knowledge of concrete properties at elevated temperatures. Lastly, Section 9.3 suggests how future research should be presented so that the test results could be more easily combined with the current set of data.
9.1 Summary and Conclusions
The focus of this research was the development of temperaturedependent compressive strength (i.e., peak stress), modulus of elasticity, strain at peak stress, and ultimate strain models for use in the fire design of concrete structures. The available experimental data from previous research on North American concrete was collected in a property database. The proposed relationships are based on a comprehensive multiple regression analysis of this existing experimental data. Normalstrength and highstrength concrete using both normalweight and lightweight North American aggregates are investigated. Using the proposed compressive strength, modulus of elasticity, strain at peak stress, and ultimate strain relationships, temperaturedependent compressive stressstrain models are also produced. The primary conclusions resulting from the study are:
In general, the proposed concrete property models provide a good statistical fit to the available experimental data considering different aggregate types, test types, and room temperature strength ranges for the concrete.
Unlike previous models, creep deformations are not included in the proposed models. As a result, the temperaturedependent strain at peak stress and ultimate strain of concrete from the proposed models are significantly smaller than those from the previous models. This results in a baseline stressstrain model to which timedependent creep strains can be explicitly added in the future.
As compared with existing models, the proposed concrete relationships represent a much larger data set for some cases, thus increasing statistical robustness, and depending on the available data, include the effects of aggregate type, heating test type, and room temperature strength, thus giving designers the ability to predict temperaturedependent relationships for a larger range of conditions.
At elevated temperatures, the compressive strength and modulus of elasticity of concrete is significantly reduced, whereas the strain at peak stress and ultimate strain are significantly increased.
Significant differences in behavior are shown between highstrength concrete and normalstrength concrete. Highstrength concrete tends to have a much larger reduction in strength than normalstrength concrete at elevated temperatures,
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making the prediction of highstrength concrete behavior using normalstrength equations unconservative.
Highstrength concrete tends to experience less reduction in modulus of elasticity at elevated temperatures than normalstrength concrete.
At any given temperature, the strain at peak stress of highstrength concrete tends to be larger than the strain at peak stress of normalstrength concrete.
Similarly, the ultimate strain of highstrength concrete at elevated temperatures tends to be greater than that of normalstrength concrete, except at very large temperatures.
The room temperature concrete stressstrain relationship proposed by Popovics (1973) provides a good model for the development of temperaturedependent stressstrain relationships, when combined with appropriate models for the concrete compressive strength, modulus of elasticity, strain at peak stress, and ultimate strain at elevated temperatures.
9.2 Recommendations for Future Research
Based on the currently available experimental data, it is evident that there are significant gaps in the existing knowledge on the stressstrain properties of concrete at elevated temperatures as follows:
One of the major areas where more research is needed is on the behavior of high strength concrete using different aggregate types. The proposed HSC equations in this report use calcareous aggregate because there is not enough data to make significant statistical predictions for siliceous or lightweight aggregate concrete. Future research in this area is particularly important since the existing data shows that highstrength concrete may perform significantly differently than normalstrength concrete at high temperatures.
Aggregate type and test type in general should be studied further with respect to the concrete stressstrain properties. Every property except for compressive strength had lack of data in the possible aggregate type and test type combinations for both NSC and HSC. It was shown for the compressive strength that there are significant differences in the results obtained from unstressed, stressed, and residual test specimens, as well as from siliceous aggregate, calcareous aggregate and lightweight aggregate specimens and it is probable that the same is true for the concrete modulus of elasticity, strain at peak stress, and ultimate strain.
The effect of specimen relative humidity on the concrete properties at elevated temperatures also needs to be studied. While it was inferred that the relative humidity could have had a significant effect on the compressive strength, there was not enough data to be able to determine the relationship between specimen humidity and strength loss as temperature is increased.
Other areas where additional data is needed include the specimen size and furnace type (i.e., electric versus gas furnaces). Since most of the results in the current database are from small specimens tested in electric furnaces, it was not possible to include these variables as parameters in the regression models. It is possible that the type of heat transfer associated with an electric furnace would cause the
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concrete to behave differently than in an actual fire. In addition, a large specimen could be affected more by temperature gradients than small cylindrical specimens.
Next, there is insufficient data on the effect of the amount of preload on a stressed specimen, which may be an important parameter to include in the statistical models. Furthermore, the current residual test database only includes unstressed specimens. Residual strength tests of specimens with various levels of axial preload should be conducted to simulate the remaining strength of an axially loaded structural member following a fire. In a “stressed residual test,” the specimen would first be subjected to an axial preload, mimicking typical service gravity load levels. Then, the specimen would be heated to a specified temperature, allowed to cool to room temperature (still under preload), and finally further loaded to failure under uniaxial compression.
There is currently insufficient data to adequately predict the temperaturedependent residual stressstrain properties of concrete in waterquenched specimens. Because the primary means of fire suppression in building structures use water, it is important to know how the addition of water to heated concrete affects the residual behavior of the material. Not only is this the case for the compressive strength, but also for the entire stressstrain behavior. Therefore, future research is needed on the residual stressstrain behavior of heated concrete after being cooled by water quenching.
Lastly, the models proposed in this research do not include the timedependent effects of creep. While the ASCE (1992), Kodur et al. (2008), and Eurocode (2002, 2004) models consider creep implicitly, they do not explicitly allow for creep as a timedependent parameter. In building upon the proposed models, it is recommended that the existing experimental data on creep strains at elevated temperatures be studied in a similar statistical manner. This would allow the models to predict not only the concrete stressstrain curve at a particular temperature, but also allow for timedependent creep effects to be considered.
9.3 Presenting Future Research
While examining the existing literature, it was apparent that the fire tests conducted by previous research have been reported in very different manners by different researchers. The procedures and results from future tests (e.g., mix design, specimen preparation and properties, test setup) should be reported in a more consistent manner to facilitate better utilization of the data. Table 9.1 shows the recommended independent properties to be reported in four categories: (1) mix properties; (2) curing properties; (3) specimen properties; and (4) test properties.
For mix properties, it is important to provide information on how a measurement was taken (e.g., air content, include ASTM designations if applicable) as well as information about specific products used in the mix (e.g., water reducer type). Similarly for curing properties, the temperature, humidity, and duration for all stages of curing should be included. Under specimen properties, the papers in the current database report the densities and masses in different ways (e.g., oven dry, air dry, or saturated surface dry mass), which makes the assessment of the data difficult. Lastly, information on test
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properties should report the type, temperature, humidity, and duration of the chambers where the specimens were kept before and after heating (i.e., conditioning and residual chambers), as well as information on the heating furnace (e.g., type, size, heating rate, where temperatures were measured). The loading displacement rate used in the uniaxial compression testing of the specimens also needs to be reported, including information on how the test displacements were measured (i.e., specimen deformation or machine cross head movement).
TABLE 9.1: INDEPENDENT PROPERTIES TO BE REPORTED FROM FUTURE FIRE TESTS
Mix Properties
Curing Properties
Specimen Properties
Test Properties
Sand/Cement Ratio Initial Curing Temperature
Specimen Shape Test Type
Aggregate/Cement Ratio
Initial Curing Humidity
Length or Height Preload Level
Water/Cement Ratio Initial Curing Duration
CrossSection Area Axial Displacement Rate
Unit Weight Subsequent Curing Temperature Volume Conditioning Chamber
Conditions
Slump Subsequent Curing Humidity
Specimen Mass Furnace Type
Cement Type Subsequent Curing Duration
Specimen Density Furnace Specification
Cement Content End Condition Furnace Size Aggregate Type Furnace Humidity
Aggregate Origin Furnace Temperature Maximum
Aggregate Size Heating Rate
d50 Size Specimen Age When in Furnace
Sand Type Furnace Duration
Sand Origin Furnace Duration at Equilibrium
Air Entrainment Amount Water Quenching Duration
Air Content Residual Chamber Conditions Water Reducer
Amount Subsequent Residual Chamber Conditions
Retarder Amount Silica Fume Amount
Fly Ash Amount
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The results from future fire tests may include mechanical, thermal, and physical properties of concrete as shown in Table 9.2. In presenting these results, it is important to report not only the measured properties at elevated temperatures but also at room temperature (e.g., fcm and fcmo) so that the changes in the concrete behavior under fire can be assessed. Also, for each property, information on how the test was conducted and how the measurement was taken should be described. For example, if the measurement is for the concrete ultimate strain, a clear definition of this property should be provided (e.g., strain at 85% of peak stress). If there is a standard (e.g., ASTM) that was followed, this information should be reported.
TABLE 9.2: DEPENDENT PROPERTIES TO BE REPORTED FROM FUTURE FIRE TESTS
Mechanical Properties Thermal Properties Physical Properties
Compressive Strength Thermal Conductivity Moisture Content Strain at Compressive
Strength Heat Flux Mass or Weight Loss
Ultimate Strain Specific Heat Spalling Temperature Creep Strain Heat Diffusivity Spalling Time
Young’s Modulus Dynamic Modulus of
Elasticity Shear Modulus
Modulus of Rupture Linear Expansion
Coefficient of Thermal Expansion
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APPENDIX A: DATABASE ENTRY
Each of the papers examined in this research reported the test data in significantly different manners, and it was imperative that the data be input into a database in a consistent manner. This appendix provides a guide for the entry of the existing data in the database, which can be downloaded from the URL at http://www.nd.edu/~concrete/concretefiredatabase. As discussed previously, through a graphical user interface (GUI) built in MATLAB®, users of the website can access the full database collected in this research. The GUI gives the user the following capabilities: (1) to plot up to four separate sets of data along with the ability to set the x and y axes for each graph; (2) to investigate data subsets by limiting each independent and dependent property stored in the full database; (3) to apply constrained or unconstrained multiple linear regression to the data with a userdefined polynomial equation form of any order; and (4) to display the userdefined regression equations and statistics.
The type of data input, and a description of the data entered in each column of the concrete property database are given in Table A1, in the order in which the columns appear in the database. First, the database column heading is listed. Next, the input type for the column is given. For example, if text is allowed, then the input type shows “TEXT.” If a given column of the database requires numerical data type, then the number format is listed (e.g., two decimals = 0.00, scientific notation = 0.00E+0). Lastly, a description of the data in each database column is provided. If only a specific entry is allowed, it is displayed along with a description. In general, if there are multiple allowable entries for a particular database column, they are separated by a comma.
TABLE A.1: DATABASE ENTRY
Database Column Heading
Input Type
Description/ Allowable Entry
TitleID 0 ID number of paper that data comes from DataID 0 ID number of specific data point Figure # TEXT Figure number in paper where data was found, if
multiple separate by a commas Table # TEXT Table number in paper where data was found, if
multiple separate by a commas
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MIX PROPERTIES
Mix Sand/Cement Ratio
0.00 Ratio of fine aggregate or sand to amount of cement in concrete mix
Mix Aggregate/Cement Ratio
0.00 Ratio of course aggregate to amount of cement in concrete mix
Mix Water/Cement Ratio
0.00 Ratio of amount of water to amount of cement in concrete mix
Mix Unit Weight 0.0 Unit weight of concrete mix (lb/ft3) Slump 0.0 Slump of concrete mix (in) Slump Comment TEXT ASTM designation followed for slump test Cement Type 0 Type I, II, III etc., listed as a number Cement Type Comment
TEXT ASTM designation of cement used, or name of cement (e.g., Portland)
Cement Content 0 Amount of cement in concrete mix (lb/yd3) Aggregate Type TEXT Specific course aggregate type in mix (i.e.,
carbonate, dolomite, limestone, etc.) Aggregate Origin TEXT City and state, or country, where course
aggregate was mined Maximum Aggregate Size
0.00 Maximum size of course aggregate (in)
d50 Size 0.00 Diameter of 50th percentile of course aggregate
(in) Sand Type TEXT Specific type of fine aggregate or sand in mix
(i.e., natural river sand, siliceous, etc.) Sand Origin TEXT City and state, or country, where fine aggregate
or sand originated Silica Fume Amount 0.0 Amount of silica fume in mix (lb/yd3) Fly Ash Amount 0.0 Amount of fly ash in mix (lb/yd3)
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Water Reducer Amount
0.00 Amount of water reducing admixture in mix (oz/yd3)
Water Reducer Type TEXT HRWR type or product name Retarder Amount 0.00 Amount of retarding agent in mix (oz/yd3) Retarder Type TEXT Product name of retarding agent Air Entraining Admixture Amount
0.00 Amount of air entraining admixture in mix (oz/yd3)
Air Entraining Admixture Type
TEXT Product name of air entraining admixture
Air Content 0.0 Amount of air in concrete mix (as a percent of
total volume) Air Content Comment
TEXT ASTM designation of test used to measure air content
CURING PROPERTIES Initial Curing Humidity
0.0 Relative humidity of initial stage of curing
Initial Curing Temperature
0.0 Temperature of initial stage of curing (°F)
Initial Curing Duration
0.0 Duration of initial stage of curing (days)
Subsequent Curing Humidity
0.0 Relative humidity of any curing that took place after initial curing stage
Subsequent Curing Temperature
0.0 Temperature of any curing that took place after initial curing stage (°F)
Subsequent Curing Duration
0.0 Duration of any curing that took place after initial curing stage (days)
Minimum Subsequent Curing Duration Known
0.0 If subsequent curing duration is not known exactly, this can be used as a way to determine minimum time that a specimen was cured (days)
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SPECIMEN PROPERTIES Shape TEXT Shape of specimen tested (e.g., cylindrical,
rectangular) Shape Comment TEXT If specimen was cored from an existing concrete
member, correct input would specify that tested specimen was cored and dimensions of original member would be listed (in)
Length or Height 0.00 Length or height of tested specimen (in) CrossSection Area 0.00 Crosssectional area of face of specimen over
which concrete property is measured (e.g., for a compression test on a cylinder, this would be area of circle outlining cylinder) (in2)
Volume 0.00 Total volume of tested specimen (in3) Surface Area/Volume
0.00 Ratio of surface area of specimen to volume of specimen (1/in)
Oven Dry Mass 0.00 Total mass of specimen after oven drying, prior
to testing (g) Air Dry Mass 0.00 Total mass of specimen after air drying, prior to
testing (g) SSD Mass 0.00 Total saturated surface dry mass of specimen
prior to testing (g) Oven Dry Density 0.0 Density of oven dried specimen prior to testing
(lb/ft3) Air Dry Density 0.0 Density of air dried specimen prior to testing
(lb/ft3) SSD Density 0.0 Density of saturated surface dried specimen
prior to testing (lb/ft3) End Conditions TEXT Specify how specimen ends were finished (i.e.,
if they were ground plane, grouted, etc.), how specimen ends were restrained (i.e., simply supported, fixed, etc.)
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TEST PROPERTIES Test Type TEXT Type of test conducted (unstressed prefire,
stressed prefire, residual, or thermal) Stress Level (% of fcmo)
0.0 Amount of preload on specimen prior to heating (as a percent of fcmo)
Test Displacement Rate
0.000 Rate of compression during test (in/min)
Test Displacement Rate Comment
TEXT Specify whether displacement rate given is machine movement rate (machine) or rate of deformation in specimen (specimen), and direction of movement (e.g., compression)
Conditioning Chamber Type
TEXT Specify chamber in which specimen was placed after curing and before heating (e.g., room, convection furnace, radiation furnace)
Conditioning Chamber Humidity
0.0 Relative humidity of conditioning chamber
Conditioning Chamber Temperature
0.0 Temperature of conditioning chamber (°F)
Conditioning Chamber Duration
0.0 Amount of time specimen spent in conditioning chamber (hrs)
Conditioning Chamber Comment
TEXT Specify whether specimen reached equilibrium with conditioning chamber (equilibrium reached, equilibrium not reached)
Heating Furnace Type
TEXT Type of heat transfer used by furnace to heat specimen (convection, radiation)
Heating Furnace Spec.
TEXT ASTM designation used to heat specimen (e.g., ASTM E119), or designation of furnace itself
Furnace Vol. / Specimen Vol.
TEXT Relative ratio of volume of furnace to volume of specimen (i.e., small, medium, or large) (small furnace had heat source closely surrounding specimen, large furnace meant specimen was placed in a large heating chamber, medium furnace was in between these two limits)
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Heating Furnace Humidity
0.0 Relative humidity of heating furnace
Heating Furnace Temperature
0.0 Maximum exposure temperature reached by furnace (°F)
Heating Rate 0.0 Rate of increase in temperature from room
temperature to maximum exposure temperature (°F/min)
Heating Rate Comment
TEXT Specify location where heating rate was measured (e.g., specimen core)
Specimen Age When Placed in Furnace
0.0 Exact age of specimen when placed in heating furnace (days) (this should be equal to sum of durations of curing and conditioning, if available)
Minimum Age When Placed in Furnace
0.0 If exact specimen age is unknown, minimum known age of specimen is given here (days)
Heating Furnace Duration
0.0 Total time specimen spent in heating furnace while being heated (mins) (If specimen was tested at temperature, this duration would be total time from placing specimen in furnace to time when specimen was tested; if this was a residual test, duration would be total time spent heating specimen)
Minimum Heating Furnace Duration Known
0.0 If exact duration of heating is not known, minimum known total duration specimen spent in heating furnace while it being heated would be placed here (mins)
Heating Furnace Comment
TEXT Specify whether specimen reached equilibrium with heating furnace (i.e., equilibrium reached, equilibrium not reached)
115
Heating Furnace Duration at Equilibrium
0.0 Total time specimen spent in furnace after equilibrium conditions were met (mins) (if specimen was tested at temperature, this would be duration from specimen reaching equilibrium to time that specimen was tested; if this was a residual test, duration would be time from specimen reaching equilibrium to time that either specimen was removed or heat was turned off)
Water Quenching Duration
0.0 Amount of time that specimen was drenched in water after heating (mins)
Residual Chamber Type
TEXT Type of chamber that specimen was placed in after being removed from heating furnace (e.g., room, convection furnace, radiation furnace, water bath)
Residual Chamber Humidity
0.0 Relative humidity of residual chamber
Residual Chamber Temperature
0.0 Temperature of residual chamber (°F)
Residual Chamber Duration
0.0 Total amount of time spent in residual chamber (days)
Residual Chamber Comment
TEXT Specify whether specimen reached equilibrium with residual chamber (i.e., equilibrium reached, equilibrium not reached)
Subsequent Residual Chamber Type
TEXT If specimen was placed in a second residual chamber, this is type of chamber it was placed into (e.g., room, convection furnace, radiation furnace, water bath)
Subsequent Residual Chamber Humidity
0.0 Relative humidity of subsequent residual chamber
Subsequent Residual Chamber Temperature (F)
0.0 Temperature of subsequent residual chamber (°F)
Subsequent Residual Chamber Duration
0.0 Total amount of time spent in subsequent residual chamber (days)
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Subsequent Residual Chamber Comment
TEXT Specify whether specimen reached equilibrium with subsequent residual chamber (i.e., equilibrium reached, equilibrium not reached)
MECHANICAL PROPERTIES Compressive Strength Before Fire, fcmo
0.0 Room temperature compressive strength (lb/in2)
fcmo Comment TEXT Specify whether data was a single test or an
average of multiple tests (i.e., single test, average), Specify if specimen had different properties than specimen tested under fire conditions (e.g., fcmo is determined from a beamend specimen, but fire test was conducted on a standard cylindrical specimen), Specify age when specimen was tested if it was different than age of specimen tested under fire conditions (e.g., if a cylinder was tested at 28 days for fcmo, but then cylinder for fcm was tested at 60 days), Specify ASTM designation used to test specimen
Compressive Strength After Fire, fcm
0.0 Specimen compressive strength at elevated temperature (lb/in2)
fcm Comment TEXT Specify whether data was a single test or an
average of multiple tests (i.e., single test, average), Specify age when specimen was tested if it was different than age of specimen tested at room temperature (e.g., if a cylinder was tested at 28 days for fcmo, but then cylinder for fcm was tested at 60 days), Specify ASTM designation used to test specimen
fcm/fcmo 0.00 Ratio of compressive strength at elevated
temperature to compressive strength at room temperature
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fcm/fcmo Comment TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify age of heated specimen/age of room temperature specimen if different
Strain at fcmo Before Fire, εcmo
0.00E+0 Room temperature strain at peak stress
εcmo Comment TEXT Specify if point was extrapolated from
experimental curve (extrapolated), Specify if specimen includes nonspecimen deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Strain at fcm After Fire, εcm
0.00E+0 Strain at peak stress at elevated temperature
εcm Comment TEXT Specify if point was extrapolated from
experimental curve (extrapolated), Specify if specimen includes nonspecimen deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
εcm/εcmo 0.00 Ratio of elevated temperature strain at peak
stress to room temperature strain at peak stress
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εcm/εcmo Comment TEXT Specify if point was extrapolated from experimental curve (extrapolated), Specify if specimen includes nonspecimen deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify age of heated specimen/age of room temperature specimen if different
Ultimate Strain Before Fire, εcuo
0.00E+0 Ultimate strain at room temperature
εcuo Comment TEXT Specify what stress value point was taken as if
different from 0.85fcmo, Specify if point was extrapolated from experimental curve (extrapolated), Specify if specimen includes nonspecimen deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Ultimate Strain After Fire, εcu
0.00E+0 Ultimate strain at elevated temperature
εcu Comment TEXT Specify what stress value point was taken as if
different from 0.85fcm, Specify if point was extrapolated from experimental curve (extrapolated), Specify if specimen includes nonspecimen deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
εcu/εcuo 0.00 Ratio of ultimate strain at elevate temperature to
ultimate strain at room temperature
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εcu/εcuo Comment TEXT Specify what stress value point was taken as if different from 0.85fcm, Specify if point was extrapolated from experimental curve (extrapolated), Specify if specimen includes nonspecimen deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify age of heated specimen/age of room temperature specimen if different
Young's Modulus Before Fire, Eco
0.00E+0 Static modulus of elasticity at room temperature (lb/in2)
Eco Comment TEXT Specify if point was extrapolated from
experimental curve (extrapolated), Specify if specimen includes nonspecimen deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated, Specify if this data point is a secant modulus or a tangent modulus and at what point is it measured
Young's Modulus After Fire, Ec
0.00E+0 Static modulus of elasticity at elevated temperature (lb/in2)
Ec Comment TEXT Specify if specimen includes nonspecimen
deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify if this data point is a secant modulus or a tangent modulus and at what point is it measured, Specify specimen age when measured if different than specimen age when heated, Specify if this data point is a secant modulus or a tangent modulus and at what point is it measured
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Ec/Eco 0.00 Ratio of static modulus of elasticity at elevated temperature to static modulus of elasticity at room temperature
Ec/Eco Comment TEXT Specify if specimen includes nonspecimen
deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify if this data point is a secant modulus or a tangent modulus and at what point is it measured, Specify age of heated specimen/age of room temperature specimen if different
Ultrasonic Pulse Velocity Before Fire, Vco
0.0 Ultrasonic pulse velocity at room temperature (ft/s)
Vco Comment TEXT Specify whether data was a single test or an
average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Ultrasonic Pulse Velocity After Fire, Vc
0.0 Ultrasonic pulse velocity at elevated temperatures (ft/s)
Vc Comment TEXT Specify whether data was a single test or an
average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Vc/Vco 0.00 Ratio of ultrasonic pulse velocity at elevated
temperature to ultrasonic pulse velocity at room temperature
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Vc/Vco Comment TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify age of heated specimen/age of room temperature specimen if different
Resonant Frequency Before Fire, RFo
0 Resonance frequency at room temperature (Hz)
RFo Comment TEXT Specify whether data was a single test or an
average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Resonant Frequency After Fire, RF
0 Resonance frequency at elevated temperatures (Hz)
RF Comment TEXT Specify whether data was a single test or an
average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
RF/RFo 0.00 Ratio of resonance frequency at elevated
temperature to resonance frequency at room temperature
RF/RFo Comment TEXT Specify whether data was a single test or an
average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify age of heated specimen/age of room temperature specimen if different
Dynamic Modulus of Elasticity Before Fire, Edo
0.00E+0 Dynamic modulus of elasticity at room temperature (lb/in2)
122
Edo Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated, Specify if this point was calculated from ultrasonic pulse velocity or resonance frequency (i.e., V or RF)
Dynamic Modulus of Elasticity After Fire, Ed
0.00E+0 Dynamic modulus of elasticity at elevated temperatures (lb/in2)
Ed Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated, Specify if this point was calculated from ultrasonic pulse velocity or resonance frequency (i.e., V or RF)
Ed/Edo
0.00 Ratio of dynamic modulus of elasticity at elevated temperature to dynamic modulus of elasticity at room temperatures
Ed/Edo Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated, Specify if this point was calculated from ultrasonic pulse velocity or resonance frequency (i.e., V or RF), Specify age of heated specimen/age of room temperature specimen if different
Creep Strain, εcr
0.00E+0 Creep strain given at a specific heating furnace
temperature and heating furnace duration
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εcr Comment
TEXT Specify if specimen includes nonspecimen deformations, Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Shear Modulus Before Fire, Gco
0.00E+0 Shear modulus at room temperature (lb/in2)
Gco Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Shear Modulus After Fire, Gc
0.00E+0 Shear modulus at elevated temperatures (lb/in2)
Gc Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Gc/Gco
0.00 Ratio of shear modulus at elevated temperature
to shear modulus at room temperature Gc/Gco Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify age of heated specimen/age of room temperature specimen if different
Modulus of Rupture Before Fire, fro
0.0 Modulus of rupture at room temperature (lb/in2)
124
fro Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Modulus of Rupture After Fire, fr
0.0 Modulus of rupture at elevated temperatures (lb/in2)
fr Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
fr/fro
0.00 Ratio of modulus of rupture at elevated temperature to modulus of rupture at room temperature
fr/fro Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify age of heated specimen/age of room temperature specimen if different
Linear Expansion, εt
0.00 Linear expansion of concrete at room or elevated temperature (determined by heating furnace temperature) given as 103 in/in
εt Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Thermal Coefficient of Expansion, αt
0.0E+0 Thermal coefficient of expansion of concrete at room or elevated temperature (determined by heating furnace temperature) (°F)
125
αt Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
THERMAL PROPERTIES Thermal Conductivity, K
0.0 Thermal conductivity at elevated or room temperature (determined by heating furnace temperature) (Btuin/ft2hr°F)
K Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Heat Flux, q
0.0 Heat flux at elevated or room temperature (determined by heating furnace temperature) (Btu/hrft2)
q Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Specific Heat, c 0.00 Specific heat of concrete (Btu/lb/°F) c Comment
TEST Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
126
Heat Diffusivity, a
0.00 Heat diffusivity measured at elevated or room temperature (determined from heating furnace temperature) (ft2/hr)
a Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
PHYSICAL PROPERTIES Moisture Content (% of initial)
0.00 Moisture content as a percentage of initial volume
Moisture Content (% of final)
0.00 Moisture content as a percentage of final volume
Moisture Content Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated, Specify depth of measurement in specimen
Mass or Weight Loss from Oven Dry Due to Heat (%)
0.00 Amount of mass or weight loss from a oven dried specimen due to heating furnace as a percentage of total mass before heating
M.L. or W.L. from Oven Dry Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Mass or Weight Loss from Air Dry Due to Heat (%)
0.00 Amount of mass or weight loss from a air dried specimen due to heating furnace as a percentage of total mass before heating
127
M.L. or W.L. from Air Dry Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Mass or Weight Loss from SSD Due to Heat (%)
0.00 Amount of mass or weight loss from a SSD specimen due to heating furnace as a percentage of total mass before heating
M.L. or W.L. from SSD Comment
TEXT Specify whether data was a single test or an average of multiple tests (i.e., single test, average), Specify if room temperature specimen had different properties than specimen tested under fire condition, Specify specimen age when measured if different than specimen age when heated
Spalling Temperature, Ts
0.0 Temperature at which spalling occurred (°F)
Ts Comment
TEXT Specify type of spalling that occurred (e.g., explosive), Specify location of temperature reading if different from heating furnace temperature
Spalling Time, ts (min)
0.0 Time during heating at which spalling occurred (mins)
ts Comment
TEXT Specify type of spalling that occurred (e.g.,
explosive)
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BIBLIOGRAPHY
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ACI 216R, Guide for Determining the Fire Endurance of Concrete Elements, Farmington Hills, MI, American Concrete Institute, 2007.
ACI 318, Building Code Requirements for Structural Concrete and Commentary, Farmington Hills, MI, American Concrete Institute, 2008.
ASCE, Structural Fire Protection, ASCE Manuals and Reports on Engineering Practice No. 78, New York, American Society of Civil Engineers, 1992.
Castillo, C. and Durani, A. J., “Effect of Transient High Temperatures on HighStrength Concrete,” ACI Materials Journal, V.87, No.1, 1990, pp. 4753.
Cheng, F. P.; Kodur, V. K. R.; and Wang, T. C., “StressStrain Curves for High Strength Concrete at Elevated Temperatures,” Journal of Materials in Civil Engineering, V.16, No.1, 2004, pp. 8490.
Concrete Association of Finland, High Strength Concrete Supplementary Rules and Fire Design, Rak MK B4, 1991.
Comites EuroInternational Du Beton, Fire Design of Concrete Structures in Accordance with CEB/FIP Model Code 90 (Final Draft), CEB Bulletin D'Information No. 208, Lausanne, Switzerland, 1991.
Cruz, C. R. and Gillen, M. P., “Thermal Expansion of Portland Cement Paste, Mortar, and Concrete at High Temperatures,” PCA RD074.01T, 1981.
Cruz, C. R., “Apparatus for Measuring Creep of Concrete at High Temperatures,” Journal of the PCA Research and Development Laboratories, Bulletin 225, V.10, No.3, 1968, pp. 3642.
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European Committee for Standardization, Eurocode 4: Design of Composite Steel and Concrete Structures, 2002.
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Hertz, K. D., “Concrete Strength for Fire Safety Design,” Magazine of Concrete Research, V.57, No.8, 2005, pp. 445452.
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STRUCTURAL ENGINEERING RESEARCH REPORT SERIES
LIST OF TECHNICAL REPORTS
NDSE0101 “Design of Rectangular Openings in Unbonded PostTensioned Precast Concrete Walls,” by M. Allen and Y.C. Kurama, April 2001, 142 pp. (this report may be downloaded from http://www.nd.edu/~concrete/).
NDSE0102 “CapacityDemand Index Relationships for PerformanceBased Seismic Design,” by K.T. Farrow and Y.C. Kurama, November 2001, 260 pp. (this report may be downloaded from http://www.nd.edu/~concrete/).
NDSE0601 “Experimental Evaluation of Unbonded PostTensioned Hybrid Coupled Wall Subassemblages,” by M.A. May, Y.C. Kurama, and Q. Shen, April 2006, 212 pp. (this report may be downloaded from http://www.nd.edu/~concrete/).
NDSE0602 “Seismic Analysis, Behavior, and Design of Unbonded PostTensioned Hybrid Coupled Wall Structures,” by Q.Shen, Y.C. Kurama, and B.D. Weldon, December 2006, 596 pp. (this report may be downloaded from http://www.nd.edu/~concrete/).
NDSE0701 “FrictionDamped Unbonded PostTensioned Precast Concrete Moment Frame Structures for Seismic Regions,” by B.G. Morgen and Y.C. Kurama, March 2007, 610 pp. (this report may be downloaded from http://www.nd.edu/~concrete/).
NDSE0901 “StressStrain Properties of Concrete Under Elevated Temperatures,” by A.M. Knaack, Y.C. Kurama, and D.J. Kirkner, April 2009, 130 pp. (this report may be downloaded from http://www.nd.edu/~concrete/).
NDSE0902 “Behavior and Design of Unbonded PostTensioning Strand/Anchorage Systems for Seismic Applications,” by K.Q. Walsh and Y.C. Kurama, April 2009, 130 pp. (this report may be downloaded from http://www.nd.edu/~concrete/).