Stress-Strain Properties of Concrete
at Elevated Temperatures
April 2009
Adam M. Knaack, Yahya C. Kurama, and David J. Kirkner
0.0
0.000
0.006
Strain, εc
o
1.2
o
70 F (21 C)
392oF (200oC)
752oF (400oC)
1112ooF (600ooC)
1400 F (760 C)
0.012
HSC, Calcareous, Unstressed
Relative Stress, fc / fcmo
Relative Stress, fc / fcmo
1.2 NSC, Calcareous, Unstressed
Report #NDSE-09-01
0.0
0.000
o
o
70 Fo (21 C)o
392oF (200oC)
752 Fo (400 C)
1112 F (600oC)
1400oF (760oC)
0.006
Strain, εc
0.012
Structural Engineering Research Report
Department of Civil Engineering and Geological Sciences
University of Notre Dame
Notre Dame, Indiana
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Stress-Strain Properties of Concrete
at Elevated Temperatures
April 2009
Report #NDSE-09-01
by
Adam M. Knaack
Graduate Research Assistant
Yahya C. Kurama
Associate Professor
David J. Kirkner
Associate Professor
Structural Engineering Research Report
Department of Civil Engineering and Geological Sciences
University of Notre Dame
Notre Dame, Indiana
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ABSTRACT
This report focuses on the compressive stress-strain behavior of unreinforced
North American concrete under elevated temperatures from fire. A database on the
temperature-dependent 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 stress-strain
behavior, including the temperature, aggregate type, test type, and strength at room
temperature as parameters. High-strength and normal-strength, and normal-weight and
light-weight 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 high-strength and
normal-strength 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 Stress-Strain 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 Un-Normalized Regression Coefficients ............................................... 47 4.1.6 Regression Assumptions........................................................................ 48 iii
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 High-Strength versus Normal-Strength 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: STRESS-STRAIN RELATIONSHIP................................................... 99 8.1 Temperature Modified Stress-Strain 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 iv
APPENDIX A: DATABASE ENTRY......................................................................... 109 BIBLIOGRAPHY ......................................................................................................... 129 v
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vi
FIGURES
CHAPTER 2
Figure 2.1: North American temperature-dependent stress-strain models: (a) ASCE; and
(b) Kodur et al....................................................................................................... 10 Figure 2.2: Eurocode temperature-dependent stress-strain models: (a) NSC, siliceous; (b)
NSC, calcareous; (c) NSC, light-weight; (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 temperature-dependent compressive strength models: (a) Eurocode;
(b) CEB Model Code 90; and (c) Rak MK B4. .................................................... 14 Figure 2.6: Other temperature-dependent 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 temperature-dependent modulus of elasticity models: (a)
ACI 216; and (b) ASCE and Kodur et al. ............................................................. 16 Figure 2.8: Eurocode temperature-dependent modulus of elasticity models: (a) NSC; and
(b) HSC. ................................................................................................................ 17 Figure 2.9: ASCE, Kodur et al., and Eurocode temperature-dependent strain at peak stress
models. .................................................................................................................. 18 Figure 2.10: ASCE, Kodur et al., and Eurocode temperature-dependent 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 temperature-dependent compressive strength, fcm relationships.
(Note: 1 ksi = 6.859 MPa) .................................................................................... 29 vii
Figure 3.3: Distribution of the normal-strength 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 high-strength 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 temperature-dependent modulus of elasticity, Ec relationships.
(Note: 1 ksi = 6.859 MPa) .................................................................................... 32 Figure 3.7: Distribution of the normal-strength 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 high-strength 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 temperature-dependent strain, εcm at peak stress relationships.
(Note: 1 ksi = 6.859 MPa) .................................................................................... 36 Figure 3.11: Distribution of the high-strength 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 temperature-dependent ultimate strain, εcu relationships. (Note:
1 ksi = 6.859 MPa)................................................................................................ 38 Figure 3.14: Distribution of the normal-strength 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 high-strength 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,
viii
unstressed; (g) NSC – light-weight, residual; (h) NSC – light-weight, stressed; (i)
NSC – light-weight, 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 – light-weight, residual; (h) NSC – light-weight,
stressed; (i) NSC – light-weight, 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
normal-strength concrete and high-strength 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 – light-weight,
residual; (h) NSC – light-weight, stressed; (i) NSC – light-weight, 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 – light-weight, 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 – light-weight, unstressed; (c) HSC – calcareous, residual;
(d) HSC – calcareous, stressed; and (e) HSC – calcareous, unstressed................ 77 Figure 5.3: Effects on the proposed temperature-dependent 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 – lightix
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 best-fit 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 light-weight aggregate data for the ultimate
strain of normal-strength 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, light-weight 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, light-weight and calcareous, unstressed; and (b)
HSC, calcareous and siliceous, unstressed. .......................................................... 97
CHAPTER 8
Figure 8.1: Proposed stress-strain relationships: (a) NSC – calcareous, unstressed; and (b)
HSC – calcareous, unstressed. ............................................................................ 100 Figure 8.2: Comparison of ASCE, Kodur et al., Eurocode, and T-modified Popovics fc-εc
functions: (a) using ASCE parameters; (b) using Kodur et al. parameters; and (c)
using Eurocode parameters. ................................................................................ 101 x
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 temperature-dependent 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 Kolmogorov-Smirnov 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 Komolgorov-Smirnov test for the
modulus of elasticity regressions .......................................................................... 72 Table 5.4: Proposed modulus of elasticity relationship regression coefficients............... 73
xi
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 Komolgorov-Smirnov 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 Komolgorov-Smirnov 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 xii
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.
xiii
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xiv
LIST OF SYMBOLS
ACI
ASCE
ASTM
b
C
Cm
CEB
CMU
Ec
Eco
fc
fcm
fcmo
HSC
K-S test
L
m
n
NIST
NSC
PCA
r
R
R2
S
s
T
t
v
wc
z
American Concrete Institute
American Society of Civil Engineers
American Society for Testing and Materials
Vector of ones used to constrain the regression equation
Coded variable for the calcareous aggregate type
Constraint matrix
Comites Euro-International Du Beton
Concrete masonry unit
Concrete modulus of elasticity at elevated temperature
Concrete modulus of elasticity at room temperature
Concrete compressive stress
Peak concrete compressive stress (i.e., strength) at elevated temperature
Peak concrete compressive stress (i.e., strength) at room temperature
High-strength concrete
Kolmogorov-Smirnov test
Coded variable for the light-weight aggregate type
Number of regression coefficients
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
National Institute of Standards and Technology
Normal-strength concrete
Portland Cement Association
Number of data points at room temperature
Coded variable for the residual test type
Coefficient of determination
Coded variable for the stressed test type
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
Maximum exposure temperature
Student’s t-Test statistic
Critical test statistic at a specific significance level
Number of statistical degrees of freedom
Concrete mix water to cement ratio
Independent variable matrix
Sample mean of the ith variable in the regression
Response vector containing the regression dataset
Number of possible values for a single variable
xv
α
γc
εc
εcm
εcmo
εcu
εcuo
λ
Significance level for hypothesis and Kolmogorov-Smirnov 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
Concrete unit-weight (lb/ft3)
Concrete compressive strain
Concrete compressive strain at peak stress at elevated temperature
Concrete compressive strain at peak stress at room temperature
Concrete compressive ultimate strain at elevated temperature
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
Penalty function parameter
xvi
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, performance-based 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 stress-strain 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 temperature-dependent mechanical and thermo-physical 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 fire-resistant
design of concrete structures.
1
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
temperature-dependent mechanical and thermo-physical 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 stress-strain behavior at elevated
temperatures; (3) the development of a template for how future fire-related 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 stress-strain properties of unreinforced concrete. The heterogeneous nature
of concrete leads to significant variability in its behavior, making deterministic prediction
of its stress-strain properties difficult. Because of this variability, much of the previous
research on the temperature-dependent 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 stress-strain properties of North American concrete and, as such, only
experimental data obtained using North American materials (e.g., aggregates) is
considered. The data includes normal-strength and high-strength concrete specimens with
normal-weight as well as light-weight aggregates. The constitutive aggregates have a
significant effect on the temperature-dependent 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 stress-strain properties of concrete from
previous research and current design codes is presented.
2
Chapter 3 introduces the North American concrete property database developed
by this project. The format, make-up, 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 temperature-dependent property models are discussed in each chapter.
Chapter 8 combines the proposed temperature-dependent relationships for the
compressive strength, modulus of elasticity, strain at peak stress, and ultimate strain of
concrete with a commonly used concrete stress-strain relationship at room temperature to
develop predictive compressive stress-strain 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.
3
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4
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 stress-strain 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 stress-strain 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
normal-strength concrete. The research involved 3-in x 6-in (7.6-cm x 15.2-cm)
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
0.25-0.55fcmo, 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
5
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
light-weight 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 pre-load 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 normal-strength 4-in x 8-in (10.2-cm x 20.3-cm) 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
light-weight expanded slag concrete with about 30% retained strength and both gravel
and sandstone concretes showed similar results at the high-end 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 6-in x 12-in
(15.2-cm x 30.5-cm) normal-strength 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
normal-strength light-weight 1.9-in x 3.8-in (4.8-cm x 9.7-cm) 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 4-in x 8-in (10.2-cm x
20.3-cm) normal-strength 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
6
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 high-strength 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
high-strength concrete (HSC) can behave significantly differently than normal-strength
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 4-in x 8-in (10.2-cm x 20.3-cm) 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
un-reinforced 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 2-in x 4-in
(5.1-cm x 10.2-cm) 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
7
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 stress-strain curve either as a tangential slope at the origin, or as the secant
slope between the origin and a point on the stress-strain curve at approximately 30-40%
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 light-weight concrete does not see as much decrease in stiffness as
normal-weight 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.
8
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 post-peak range of the stress-strain 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 stress-strain 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, time-dependent 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 temperature-dependent 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 stress-strain curves
and relationships for the concrete modulus of elasticity, strain at peak stress, and ultimate
strain also exist.
9
2.2.1 Stress-Strain Relationships
Figure 2.1 shows the temperature-dependent stress-strain (fc-εc) models for
concrete in compression from ASCE (1992) and Kodur et al. (2008). The fc-εc 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 temperature-dependent stress-strain models: (a) ASCE; and
(b) Kodur et al.
The ASCE fc-εc model is for normal-strength 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 fc-εc models in Figure 2.1. It
can be seen that the Kodur et al. fc-εc 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 temperature-dependent fc-εc
relationships for high-strength concrete (HSC) and normal-strength 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 fc-εc 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 fc-εc 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.
10
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2.2: Eurocode temperature-dependent stress-strain models: (a) NSC, siliceous; (b)
NSC, calcareous; (c) NSC, light-weight; (d) HSC, Class 1; (e) HSC, Class 2; and (f)
HSC, Class 3.
11
Models to estimate the compressive stress-strain behavior of concrete as affected
by fire are also available from European sources. For example, Figure 2.2 shows the
Eurocode (2002, 2004) temperature-dependent fc-εc models for concrete in compression.
The fc-εc 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 fc-εc models, the Eurocode models for normal-strength concrete consider three
different aggregate types (light-weight, 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 fc-εc 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 temperature-dependent 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 temperature-dependent behavior of concrete
subjected to loading over short-durations.
2.2.2 Compressive Strength
Temperature-dependent 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) light-weight. 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
light-weight 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 light-weight aggregate) and three test types (residual, stressed, and unstressed).
12
(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
normal-strength concrete (NSC) whereas the model proposed by Kodur et al. was
developed by modifying the ASCE equations for high-strength concrete (HSC). The
ASCE and Kodur et al. fcm models correspond to the peak stress on the respective stressstrain (fc-εc) 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 normal-strength concrete is
expected to behave according to a bi-linear relationship, where there is no strength loss
until a temperature of 842°F (450°C). In contrast, high-strength 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.
13
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 normal-strength concrete and
high-strength concrete.
(a)
(b)
(c)
Figure 2.5: European temperature-dependent 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 normal-strength 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 normal-strength concrete; however, the test
type or aggregate type with which these models should be used was not reported.
14
(a)
(b)
(c)
Figure 2.6: Other temperature-dependent 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 non-North American constitutive materials. The Phan and
Carino (2003) model is the only North-American model not related to the current U.S.
design codes, and was developed based on a large number of tests on high-strength and
normal-strength concrete specimens. It was shown that the ACI 216 curves in Figure 2.3
result in unconservative estimates of the strength loss for high-strength concrete; and
thus, the model in Figure 2.6(c) was proposed for high-strength calcareous concrete as a
conservative estimate for any test type.
2.2.3 Modulus of Elasticity
Temperature-dependent 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 f cmo = 4,000 psi (27.6 MPa), and a single test type (unstressed). These
relationships are provided without any corresponding stress-strain (fc-εc) model. In
15
comparison, the Ec relationships from ASCE and Kodur et al. were determined by taking
the derivative of the corresponding fc-εc 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 temperature-dependent modulus of elasticity models: (a)
ACI 216; and (b) ASCE and Kodur et al.
As described previously, the ASCE fc-εc 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, temperature-dependent 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 fc-εc 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 normal-strength concrete show smaller differences between the different aggregate
types. The differences between the Ec / Eco ratio of the different class high-strength
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.
16
(a)
(b)
Figure 2.8: Eurocode temperature-dependent modulus of elasticity models: (a) NSC; and
(b) HSC.
2.2.4 Strain at Peak Stress
The temperature-dependent strain at peak stress, εcm, of concrete obtained from
the ASCE (1992), Kodur et al. (2008), and Eurocode (2002, 2004) fc-εc 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 ε cmo = 0.0025 and the ASCE and Kodur et al. models
assume a room-temperature strain of approximately ε cmo = 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 high-strength 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 fc-εc relationships (see Figure 2.2).
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17
Figure 2.9: ASCE, Kodur et al., and Eurocode temperature-dependent strain at peak stress
models.
2.2.5 Ultimate Strain
Temperature-dependent 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 post-peak strain corresponding to 85% of the peak stress, fcm from the
corresponding fc-εc relationships in Figure 2.1.
Figure 2.10: ASCE, Kodur et al., and Eurocode temperature-dependent 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 ε cuo = 0.006 and the Kodur et al. model assumes a value
of ε cuo = 0.004. The Kodur et al. model shows that high-strength concrete is expected to
have a smaller ultimate strain than normal-strength concrete at elevated temperatures.
18
According to the Eurocode model, the ultimate strain of concrete at room temperature is
reached at
ε cuo = 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 fc-εc
relationships in Figure 2.2.
19
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20
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
stress-strain properties investigated in this report.
3.1 Database Overview
One of the most important limitations of the previous models focusing on the
compressive stress-strain 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 temperature-dependent 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 temperature-dependent 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 user-specified criteria. For
example, to look at the effects of the water-to-cement 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.
21
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/concrete-fire-database. 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 user-defined polynomial equation form of any order;
and (4) to display the user-defined 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 re-read, 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 Arizona-Software, version 3.0, originally downloaded in
June 2007, http://www.arizona-software.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);
22


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
Mix Aggregate/Cement Ratio
Mix Water/Cement Ratio
Mix Unit Weight (lb/ft3)
Slump (in)
Slump Comment
Cement Type
Cement Comment
Cement Content (lb/yd3)
Aggregate Type
Aggregate Origin
Maximum Aggregate Size (in)
d50 Size (in)
Sand Type
Sand Origin
Silica Fume Amount (lb/yd3)
Fly Ash Amount (lb/yd3)
Water Reducer Amount (oz/yd3)
Water Reducer Type
Retarder Amount (oz/yd3)
Retarder Type
Air Entraining Admixture Amount (oz/yd3)
Air Entraining Admixture Type
Air Content (% of total volume)
Air Content Comment
TABLE 3.2:
CURING PROPERTIES COLLECTED IN THE DATABASE
Initial Curing Humidity (RH)
Initial Curing Temperature (°F)
Initial Curing Duration (days)
Subsequent Curing Humidity (RH)
Subsequent Curing Temperature (°F)
Subsequent Curing Duration (days)
Minimum Subsequent Curing Duration Known (days)
TABLE 3.3:
SPECIMEN PROPERTIES COLLECTED IN THE DATABASE
Shape
Shape Comment
Length or Height (in)
Cross-Section Area (in2)
Volume (in3)
Surface Area/Volume Ratio (1/in)
Oven Dry Mass (g)
23
Air Dry Mass (g)
Saturated Surface Dry Mass (g)
Oven Dry Density (lb/ft3)
Air Dry Density (lb/ft3)
Saturated Surface Dry Density (lb/ft3)
End Conditions
TABLE 3.4:
TEST PROPERTIES COLLECTED IN THE DATABASE
Test Type
Stress Level (% of fcmo)
Test Displacement Rate (in/min)
Test Displacement Rate Comment
Conditioning Chamber Type
Conditioning Chamber Humidity (RH)
Conditioning Chamber Temperature (°F)
Conditioning Chamber Duration (hrs)
Conditioning Chamber Comment
Heating Furnace Type
Heating Furnace Specification
Furnace Volume/Specimen Volume Ratio
Heating Furnace Humidity (RH)
Heating Furnace Temperature (°F)
Heating Rate (°F/min)
Heating Rate Comment
Specimen Age When Placed in Furnace (days)
Minimum Age When Placed in Furnace
(days)
Heating Furnace Duration (mins)
Minimum Heating Furnace Duration
Known (mins)
Heating Furnace Comment
Heating Furnace Duration at Equilibrium
(mins)
Water Quenching Duration (mins)
Residual Chamber Type
Residual Chamber Humidity (RH)
Residual Chamber Temperature (°F)
Residual Chamber Duration (days)
Residual Chamber Comment
Subsequent Residual Chamber Type
Subsequent Residual Chamber Humidity
(RH)
Subsequent Residual Chamber
Temperature (°F)
Subsequent Residual Chamber Duration
(days)
Subsequent Residual Chamber Comment
The dependent properties collected in the database relate to the test results for
each specimen. These temperature-dependent 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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24
TABLE 3.5:
MECHANICAL PROPERTIES COLLECTED IN THE DATABASE
Compressive Strength Before Fire, fcmo
(lb/in2)
fcmo Comment
Compressive Strength After Fire, fcm
(lb/in2)
fcm Comment
fcm / fcmo
fcm / fcmo Comment
Strain at fcmo Before Fire, εcmo
εcmo Comment
Strain at fcm After Fire, εcm
εcm Comment
εcm / εcmo
εcm / εcmo Comment
Ultimate Strain Before Fire, εcuo
εcuo Comment
Ultimate Strain After Fire, εcu
εcu Comment
εcu / εcuo
εcu / εcuo Comment
Young's Modulus Before Fire, Eco (lb/in2)
Eco Comment
Young's Modulus After Fire, Ec (lb/in2)
Ec Comment
Ec / Eco
Ec / Eco Comment
Ultrasonic Pulse Velocity Before Fire, Vco
(ft/sec)
Vco Comment
Ultrasonic Pulse Velocity After Fire, Vc
(ft/sec)
Vc Comment
Resonant Frequency Before Fire, RFo
(Hz)
RFo Comment
Resonant Frequency After Fire, RF (Hz)
RF Comment
RF/RFo
RF/RFo Comment
Dynamic Modulus of Elasticity Before
Fire, Edo (lb/in2)
Edo Comment
Dynamic Modulus of Elasticity After Fire,
Ed (lb/in2)
Ed Comment
Ed / Edo
Ed / Edo Comment
Creep Strain, εcr
εcr Comment
Shear Modulus Before Fire, Gco (lb/in2)
Gco Comment
Shear Modulus After Fire, Gc (lb/in2)
Gc Comment
Gc / Gco
Gc / Gco Comment
Modulus of Rupture Before Fire, fro
(lb/in2)
fro Comment
Modulus of Rupture After Fire, fr (lb/in2)
fr Comment
Vc /Vco
Vc /Vco Comment
25
fr / fro
fr / fro Comment
Linear Expansion, εt (10-3 in/in)
εt Comment
Thermal Coefficient of Expansion, αt
(1/F)
αt Comment
TABLE 3.6:
THERMAL PROPERTIES COLLECTED IN THE DATABASE
Thermal Conductivity, K (Btu-in/ft2-hr-°F)
K Comment
Heat Flux, q (Btu/ft2-hr)
q Comment
Specific Heat, c (Btu/lb/°F)
c Comment
Heat Diffusivity, a (ft2/hr)
a Comment
TABLE 3.7:
PHYSICAL PROPERTIES COLLECTED IN THE DATABASE
Moisture Content (% of initial)
Moisture Content (% of final)
Moisture Content Comment
Mass or Weight Loss from Oven Dry Due
to Heat (%)
M.L. or W.L. from Oven Dry Comment
Mass or Weight Loss from Air Dry Due to
Heat (%)
M.L. or W.L. from Air Dry Comment
Mass or Weight Loss from Saturated
Surface Dry Due to Heat (%)
M.L. or W.L. from Saturated Surface Dry
Comment
Spalling Temperature, Ts (°F)
Ts Comment
Spalling Time, ts (min)
ts Comment
3.3 Data Ranges
The results collected include 3026 data points spanning over a broad range of
temperature-dependent 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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26
TABLE 3.8:
NUMBER OF DATA POINTS COLLECTED FOR EACH TEMPERATUREDEPENDENT CONCRETE PROPERTY
Temperature-Dependent Property Number of Papers Reporting Number of Data Points
Compressive Strength
Static Modulus of Elasticity
Ultimate Strain
Dynamic Modulus of Elasticity
Linear Expansion
Thermal Coefficient of Expansion
Mass Loss from Air Dry
Strain at Peak Stress
Creep Strain
Modulus of Rupture
Thermal Conductivity
Specific Heat
Heat Diffusivity
Mass Loss from Oven Dry
Mass Loss from SSD
Ultrasonic Pulse Velocity
Resonance Frequency
Heat Flux
Spalling Temperature
Spalling Time
9
7
3
3
3
3
3
2
2
2
2
2
2
2
2
1
1
1
1
1
647
275
70
186
132
37
82
30
992
46
25
19
22
308
24
26
57
15
16
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
temperature-dependent property, but also a time-dependent 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
time-dependent 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 temperature-dependent stress-strain 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 stress-strain property relationships to
which creep effects could later be added in the form of an explicit time-temperature
model.
27
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 temperature-dependent
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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28
Figure 3.2: Distribution of room temperature compressive strength, fcmo of the data used
in determining the temperature-dependent 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 normal-strength
concrete used for determining the temperature-dependent relationships in this report. The
statistical reasoning for 6,000 psi (41.4 MPa) being the limit between normal-strength
and high-strength 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
light-weight 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, 3-in (7.6 cm) x 6-in (15.2 cm) cylinders tested using radiation/electric
type furnaces.
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29
(a)
(b)
(c)
(d)
(e)
Figure 3.3: Distribution of the normal-strength 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
The high-strength concrete data with f cmo > 6 ksi (41.1 MPa) has a smaller
property range than the normal-strength data with f cmo ≤ 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 4-in (10.2 cm) x 8-in (20.3 cm) cylinders represent the
most common specimen size used in the high-strength 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 high-strength 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 temperature-dependent 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 room-temperature 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.
31
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 temperature-dependent 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 temperature-dependent modulus of elasticity, Ec relationships. (Note:
1 ksi = 6.859 MPa)
32
(a)
(b)
(c)
(d)
(e)
Figure 3.7: Distribution of the normal-strength 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 3.7 displays further distributions of the normal-strength concrete ( f cmo ≤
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 3.8-in (9.7 cm) to 4-in (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 high-strength concrete data for the
modulus of elasticity has a smaller property range than the normal-strength 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 6-in (15.2 cm) to 8-in (20.3 cm), with the
majority being 8-in (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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34
(a)
(b)
(c)
Figure 3.8: Distribution of the high-strength 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 stress-strain curves or
calculated from experimental load-deformation curves. The entire data set (30 points),
normalized with respect to the room-temperature 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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35
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 temperature-dependent 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 temperature-dependent strain, εcm at peak stress relationships. (Note: 1
ksi = 6.859 MPa)
36
The normal-strength concrete ( f cmo ≤ 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 2-in x 4-in
(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 high-strength 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 light-weight aggregate concrete. Also, a majority
of the data comes from 4-in x 8-in (10.2 cm x 20.4 cm) specimens.
(a)
(b)
Figure 3.11: Distribution of the high-strength 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 temperature-dependent 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 stress-strain 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 stress-strain curves, it was
taken as the strain corresponding to 85% of the peak stress in the post-peak range of the
stress-strain 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.
37
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 temperature-dependent 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 temperature-dependent ultimate strain, εcu relationships. (Note: 1 ksi =
6.859 MPa)
38
From the 67 data points used in the statistical analysis of εcu, a total of 48 points
are for normal-strength 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 normal-strength 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 light-weight 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 3.8-in to 4-in (9.2 cm to 10.2 cm) height range,
with the majority of the data from 3.8-in (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 normal-strength 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
39
and specimen height. It can be seen that the majority of the specimens were made from
calcareous aggregate concrete and 8-in (20.3 cm) tall.
(a)
(b)
Figure 3.15: Distribution of the high-strength concrete data used for the ultimate strain
relationships: (a) aggregate type; and (b) specimen height.
40
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, water-to-cement 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 t-test (Mendenhall 2007) was performed. This test
compares the value of the t-statistic for any individual variable with the critical t-statistic
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 t-statistic 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.
41
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, water-to-cement 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) normal-strength concrete
(NSC) with fcmo ≤ 6 ksi (41.4 MPa); and (2) high-strength 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 25-55% 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 25-55% of fcmo, the amount of preload did
not significantly affect the temperature-dependent 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 non-quenched (i.e., air-cooled) 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 non-quenched
specimens, a total of 26 data points from quenched specimens were excluded from the
42
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 stress-strain 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
43
where:
= normalized regression intercept term (discussed later);
= normalized regression coefficients (discussed later).
Throughout the process of finding the best-fit 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 normal-strength 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 normal-strength and
high-strength 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
44
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 normal-strength 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 so-called “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:
45



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 best-fit 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
the required constraint);
= 107 to achieve
= 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
corresponding to the room temperature tests.
values
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 normal-strength and high-strength 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
46
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
NSC Siliceous
NSC Calcareous
NSC Light-weight
HSC Calcareous
Test Type
Unconstrained R2
Residual
Stressed
Unstressed
Residual
Stressed
Unstressed
Residual
Stressed
Unstressed
Residual
Stressed
Unstressed
0.87
0.92
0.95
0.85
0.66
0.71
0.84
0.82
0.75
0.83
0.31
0.54
Constrained R2
0.87
0.92
0.93
0.85
0.61
0.69
0.84
0.79
0.74
0.79
0.22
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 Un-Normalized 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 un-normalized independent variable data (for example, a designer would use
47
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 Kolmogorov-Smirnov (K-S) 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 K-S 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 K-S test, suggesting that the error from the highstrength concrete data may not come from a normal distribution.
The null hypothesis for a K-S 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 K-S test is α = 0.05; the larger the value of α, the
48
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 K-S test with α = 0.10, then the assumption of normality should not be
rejected.
TABLE 4.3:
REQUIRED SIGNIFICANCE LEVEL TO PASS THE KOLMOGOROV-SMIRNOV
TEST FOR THE COMPRESSIVE STRENGTH REGRESSION EQUATIONS
Aggregate Type
NSC Siliceous
NSC Calcareous
NSC Light-weight
HSC Calcareous
Test Type
Required α
Residual
Stressed
Unstressed
Residual
Stressed
Unstressed
Residual
Stressed
Unstressed
Residual
Stressed
Unstressed
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.004
0.002
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 high-strength 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 temperature-dependent properties of high-strength
concrete, the relationships proposed in this report are recommended for use with caution
until additional test data is developed and the statistical models are re-evaluated.
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
49
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 z-1 coded variables in the regression equation. For
example, in the current database, there are three possible settings for the aggregate type
(siliceous, calcareous, and light-weight), and therefore, two coded variables are needed in
the regression equations. One variable can be for the calcareous aggregate and the other
for the light-weight 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 normal-strength concrete (NSC)
and high-strength 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 light-weight 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 high-strength 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
50
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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51
TABLE 4.4:
PROPOSED COMPRESSIVE STRENGTH RELATIONSHIP REGRESSION
COEFFICIENTS
NSC
SILICEOUS
Residual
Stressed
CALCAREOUS
Unstressed
0.963
0.995
0.953
6.448E-04
8.421E-05
7.887E-04
Residual
Stressed
0.997
1.023
0.981
6.514E-05 -4.001E-04
3.109E-04
-1.643E-06 -9.024E-08 -1.645E-06 -3.125E-07
T
Range
(°F)
fcmo
Range
(psi)
R2
Unstressed
1.106E-06 -5.413E-07
5.459E-10 -1.894E-10
5.280E-10 -6.748E-11 -6.998E-10
1.760E-11
[70, 1472]
[70, 1503]
[70, 1600]
[70, 1472]
[70, 1504]
[70, 1600]
[3900,
5500]
[3900,
5500]
[3900,
5500]
[3542,
6000]
[3900,
5600]
[1149,
6000]
0.90
0.92
NSC
0.93
0.86
0.61
HSC
0.69
LIGHT-WEIGHT
Residual
1.037
Stressed
1.065
CALCAREOUS
Unstressed
1.024
Residual
1.088
Stressed
1.151
Unstressed
1.095
-5.483E-04 -1.059E-03 -3.542E-04 -1.371E-03 -2.454E-03 -1.512E-03
3.686E-07
1.829E-06
2.741E-07
1.763E-06
4.517E-06
2.219E-06
-2.208E-10 -9.203E-10 -2.029E-10 -1.045E-09 -2.411E-09 -1.085E-09
T
Range
(°F)
fcmo
Range
(psi)
R2
[70, 1600]
[70, 1501]
[70, 1599]
[70, 1112]
[70, 1292]
[70, 1472]
[1597,
3900]
[3900,
3900]
[2716,
3900]
[6440,
13982]
[7344,
14229]
[7345,
14230]
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 light-weight stressed
52
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 normal-strength
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 normal-strength calcareous concrete under the unstressed test, the
coefficients
are:
(4.11)
53
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 normal-strength
concrete and three for high-strength 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
high-strength 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
high-strength 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.
54
(a)
(b)
(c)
(d)
(e)
(f)
55
(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 – light-weight, residual; (h) NSC – light-weight, stressed; (i) NSC – light-weight,
unstressed; (j) HSC – calcareous, residual; (k) HSC – calcareous, stressed; and (l) HSC –
calcareous, unstressed.
56
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)
57
(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 – light-weight, residual; (h) NSC – light-weight, stressed; (i) NSC –
light-weight, unstressed; (j) HSC – calcareous, residual; (k) HSC – calcareous, stressed;
and (l) HSC – calcareous, unstressed.
58
4.3.2 Effect of Aggregate Type
Figure 4.3 shows the proposed fcm / fcmo curves for normal-strength 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 light-weight concrete has the least amount of strength loss
for, approximately, T > 1100°F (593 °C). In general, light-weight 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 light-weight concrete
models in Figure 4.3 are smaller than the differences in the ACI 216 models in Figure
2.3.
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59
(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 temperature-dependent compressive
strength of normal-strength and high-strength concrete. For each type of aggregate, the
stressed test results in the smallest amount of strength loss at higher temperatures (with
the exception of high-strength concrete), while the largest strength loss occurs in the
residual test type. It is interesting to note that for calcareous and light-weight aggregate
concrete, the stressed specimens demonstrate some strength regain at moderate
temperatures. It can also be seen that for normal-strength 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.
60
(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 – light-weight;
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 normal-strength 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 high-strength 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 High-Strength versus Normal-Strength Concrete
Figure 4.5 compares the strength loss models for high-strength (i.e., fcmo > 6,000
psi [41.4 MPa]) and normal-strength (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,
high-strength concrete behaves in a significantly different manner under elevated
temperatures as compared with normal-strength concrete. In general, high-strength
61
concrete tends to lose more of its relative strength as temperature is increased than
normal-strength concrete. This has been attributed to the greater density (i.e., smaller
porosity) of high-strength 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
normal-strength concrete and high-strength concrete: (a) calcareous, residual; (b)
calcareous, stressed; and (c) calcareous, unstressed.
Furthermore, it can be seen from Figure 4.5 that the behavior of high-strength
concrete is characterized by strength reduction up to approximately 200 °F (93 °C),
followed by a relatively stable range between 200-400 °F (93-204 °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, normal-strength
concrete tends to experience a more gradual strength loss with temperature.
62
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 high-strength concrete. Using these models to predict the strength loss of
high-strength concrete may, in some cases, grossly over-predict 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
light-weight 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 over-predicts 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 over-predicts 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 high-strength 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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63
(a)
(b)
(c)
(d)
(e)
(f)
64
(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 – light-weight, residual; (h) NSC – light-weight,
stressed; (i) NSC – light-weight, unstressed; (j) HSC – calcareous, residual; (k) HSC –
calcareous, stressed; and (l) HSC – calcareous, unstressed.
65
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66
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 temperature-dependent 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 stress-strain curve) and the secant modulus
(i.e., slope of the straight line connecting the origin to a point at 30-40% 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
67
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=
(β
i,1
− β i, 2 )
2
si,1
si, 2 2
+
n1
n2
(5.3)
with,
,
= i-th modified (see Equations 4.4 and 4.5) regression coefficient for the
first and second data sets, respectively;
,
= variance of the i-th 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.
68
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 normal-strength and high-strength 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 temperature-dependent regression coefficients
were compared (i.e., the intercept term, β 0 , 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 fc-εc 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 high-strength concrete (HSC) and normal-strength
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 temperature-dependent 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 light-weight unstressed, HSC calcareous residual, HSC calcareous
stressed, and HSC calcareous unstressed cases. Note that the modulus of elasticity data
69
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 un-normalize the
ratio by a known or well-established quantity for the room temperature modulus of
elasticity, Eco. A well-established equation for the room temperature modulus of elasticity
is given by ACI 318 (2008) as:
E co _ ACI = γ c1.5 33 f cmo ; in psi
(5.5)
E co _ ACI = 57000 f cmo ; in psi
(5.6)
where, Equation (5.5) was used for light-weight concrete and Equation (5.6) was
used for normal-weight concrete with γc = unit-weight of concrete (lb/ft3). Measured
values of γc = 109.5 lb/ft3 and γc = 111 lb/ft3 were used for the light-weight 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
un-normalized 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 light-weight, unstressed case where the R2 = 0
value for Ec obtained from the Ec / Eco ratio suggests a complete lack of fit with the data.
70
TABLE 5.1:
R STATISTICS FOR Ec / Eco_ACI, Ec / Eco, AND Ec FOR EACH DATA SET
2
NSC
Ec/Eco
Ec/Eco_ACI
Ec from Ec / Eco
Ec from Ec / Eco_ACI
HSC
Calcareous
Light-weight
Calcareous
Unstressed
Unstressed
Residual Stressed Unstressed
0.84
0.65
0.70
0.64
0.86
0.66
0.00
0.72
0.91
0.72
0.64
0.73
0.97
0.97
0.96
0.97
0.87
0.89
0.89
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
Equation
Calcareous
Light-weight
Unstressed
Unstressed
0.63
0.65
0.65
0.66
0.66
0.66
HSC
Equation
Calcareous
Residual
71
Stressed
Unstressed
0.66
0.96
0.79
0.72
0.97
0.89
0.72
0.97
0.89
Along with these R2 values for each trial regression equation, the Student’s t Test
as well as the Analysis of Variance F-Test 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 F-Test.
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 K-S 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 K-S 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 re-evaluated.
TABLE 5.3:
REQUIRED SIGNIFICANCE LEVEL TO PASS THE KOMOLGOROV-SMIRNOV
TEST FOR THE MODULUS OF ELASTICITY REGRESSIONS
Aggregate Type
NSC Calcareous
NSC Light-weight
HSC Calcareous
Test Type
Unstressed
Unstressed
Residual
Stressed
Unstressed
Required α
0.10
0.10
0.00
0.10
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
72
T = maximum exposure temperature (°F).
TABLE 5.4:
PROPOSED MODULUS OF ELASTICITY RELATIONSHIP REGRESSION
COEFFICIENTS
NSC
HSC
CALCAREOUS LIGHT-WEIGHT
CALCAREOUS
Unstressed
1.292
-1.271E-03
4.163E-07
T Range
(°F)
fcmo
Range
(psi)
R2 on Ec
[75, 1400]
[1149, 5443]
0.64
Unstressed
0.738
-5.916E-04
1.308E-07
[74, 1412]
Residual
Stressed
Unstressed
1.351
-2.037E-03
9.671E-07
1.058
-1.168E-03
3.180E-07
1.055
-1.146E-03
4.654E-07
[73, 1112]
[77, 1112]
[75, 1400]
[2716, 3893] [6440, 13982] [7102, 14707] [6316, 14707]
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).
73
3) Determine the final modulus of elasticity relationships [given by Equation
(5.7)], where, κ E i = β i 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 high-strength 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 high-temperature data range very well.
The same is true of the NSC light-weight unstressed case and the HSC calcareous
unstressed case.
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74
(a)
(b)
(c)
(d)
(e)
Figure 5.1: Proposed modulus of elasticity relationships fit to data: (a) NSC – calcareous,
unstressed; (b) NSC – light-weight, unstressed; (c) HSC – calcareous, residual; (d) HSC –
calcareous, stressed; and (e) HSC – calcareous, unstressed.
75
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 light-weight and normal-weight concrete, respectively. For light-weight
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 light-weight 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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76
(a)
(b)
(c)
(d)
(e)
Figure 5.2: Proposed modulus of elasticity prediction bands: (a) NSC – calcareous,
unstressed; (b) NSC – light-weight, unstressed; (c) HSC – calcareous, residual; (d) HSC –
calcareous, stressed; and (e) HSC – calcareous, unstressed.
77
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 high-strength 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 temperature-dependent 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 light-weight and calcareous aggregate concrete. It can be seen that
with respect to the Ec / Eco_ACI value at room temperature, light-weight 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
78
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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79
(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 – light-weight,
unstressed; (c) HSC – calcareous, residual; (d) HSC – calcareous, stressed; and (e) HSC –
calcareous, unstressed.
80
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 light-weight 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 temperature-dependent
relationships would be available. Looking at the high-strength 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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82
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 temperature-dependent 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 stress-strain (fc-εc) curves or
experimental load-deformation curves. Utilizing the regression analysis program written
in MATLAB® for the temperature-dependent 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 normal-strength 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 normal-strength concrete data from the high-strength
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
normal-strength and high-strength 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.
83
Figure 6.1: Comparison of HSC and NSC data with best-fit 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 high-strength and normal-strength 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.
84
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 K-S 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 K-S 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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85
TABLE 6.2:
REQUIRED SIGNIFICANCE LEVEL TO PASS THE KOMOLGOROV-SMIRNOV
NORMALITY TEST
Strength Type
NSC
HSC
Aggregate Type
Required α
Test Type
Calcareous
Calcareous and
Siliceous
Unstressed
Unstressed
0.10
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
T Range (°F)
fcmo Range (psi)
R2
NSC
HSC
CALCAREOUS
SILICEOUS AND CALCAREOUS
Unstressed
Unstressed
0.981
2.181E-04
6.426E-07
[73, 1472]
[4504, 4666]
0.93
0.896
1.431E-03
8.772E-08
[73, 1472]
[9102, 11458]
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
86
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 m coefficients in Table 6.3 are the same as the modified
i
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 fc-εc 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 un-normalized using
Equation (6.2); whereas, the data points were un-normalized 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.
87
(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 temperature-dependent explicit total
strain relationship including creep effects.
88
(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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90
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 temperature-dependent 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 stress-strain
(fc-εc) 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 post-peak range
of the stress-strain 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 temperature-dependent 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 normal-strength and high-strength concrete. The only
available data for NSC is for calcareous and light-weight 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 high-strength 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.
91
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 light-weight 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 light-weight 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 light-weight 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 light-weight aggregate data for the ultimate
strain of normal-strength 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.
92
TABLE 7.1:
R VALUES OF THE ULTIMATE STRAIN TRIAL REGRESSIONS
2
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 K-S test in Table 7.2, the assumption that the error
93
comes from a normal-distribution 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 KOMOLGOROV-SMIRNOV
NORMALITY TEST
Strength Type
NSC
Aggregate Type
Light-weight and
Calcareous
Calcareous and
Siliceous
HSC
Test Type
Required α
Unstressed
0.10
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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94
TABLE 7.3:
PROPOSED ULTIMATE STRAIN RELATIONSHIP REGRESSION COEFFICIENTS
NSC
HSC
LIGHT-WEIGHT AND
CALCAREOUS
SILICEOUS AND
CALCAREOUS
Unstressed
Unstressed
0.979
3.377E-04
-7.561E-07
1.186E-09
[73, 1472]
[2716, 4667]
0.86
T Range (°F)
fcmo Range (psi)
R2
0.991
-7.689E-05
2.803E-06
-1.035E-09
[73, 1472]
[9102, 11458]
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 u values shown in Table 7.3.
i
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.
95
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 un-normalized using
Equation (7.2); whereas, the data points were un-normalized 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, light-weight 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.
96
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, light-weight and calcareous, unstressed; and (b) HSC,
calcareous and siliceous, unstressed.
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98
CHAPTER 8:
STRESS-STRAIN RELATIONSHIP
This chapter focuses on the compressive stress-strain relationship of concrete at
elevated temperatures. Section 8.1 introduces the model used to describe the temperaturedependent concrete stress-strain relationship. Section 8.2 gives the results and evaluation
of the proposed stress-strain relationships. Lastly, Section 8.3 compares the proposed
relationships with previous stress-strain models.
8.1 Temperature Modified Stress-Strain Model
Temperature-dependent compressive stress-strain (fc-εc) relationships for concrete
were developed by combining the temperature-dependent strength (fcm), modulus of
elasticity (Ec), and strain (εcm, εcu) models with the room temperature concrete fc-εc
relationship proposed by Popovics (1973), later adapted by Mander et al. (1988) for use
with confined concrete. The resulting stress-strain model takes the following form:
(8.1)
(8.2)
(8.3)
(8.4)
where,
fc = temperature-dependent 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 stress-strain 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.
99
The fc-εc 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 stress-strain relationships: (a) NSC – calcareous, unstressed; and (b)
HSC – calcareous, unstressed.
By comparing the two sets of stress-strain 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
stress-strain 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
100
be determined. In comparison with these previous models, the proposed stress-strain
models provide a baseline to which creep strains can be added in the future, utilizing an
explicit time-dependent relationship.
To validate that the differences between the proposed and existing stress-strain
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 fc-εc 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 T-modified Popovics fc-εc
functions: (a) using ASCE parameters; (b) using Kodur et al. parameters; and (c) using
Eurocode parameters.
101
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 stress-strain 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 fc-εc test data cannot accurately capture the post peak behavior. For
example, Eurocode states that any relationship can be used for the post-peak descending
branch of the stress-strain 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.
102
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 temperature-dependent
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. Normal-strength and high-strength
concrete using both normal-weight and light-weight North American aggregates are
investigated. Using the proposed compressive strength, modulus of elasticity, strain at
peak stress, and ultimate strain relationships, temperature-dependent 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 temperature-dependent 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 stress-strain model to which
time-dependent 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
temperature-dependent 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 high-strength concrete and
normal-strength concrete. High-strength concrete tends to have a much larger
reduction in strength than normal-strength concrete at elevated temperatures,
103




making the prediction of high-strength concrete behavior using normal-strength
equations un-conservative.
High-strength concrete tends to experience less reduction in modulus of elasticity
at elevated temperatures than normal-strength concrete.
At any given temperature, the strain at peak stress of high-strength concrete tends
to be larger than the strain at peak stress of normal-strength concrete.
Similarly, the ultimate strain of high-strength concrete at elevated temperatures
tends to be greater than that of normal-strength concrete, except at very large
temperatures.
The room temperature concrete stress-strain relationship proposed by Popovics
(1973) provides a good model for the development of temperature-dependent
stress-strain 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 stress-strain 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 light-weight aggregate concrete.
Future research in this area is particularly important since the existing data shows
that high-strength 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 stress-strain 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 light-weight 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
104



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 stress-strain properties of concrete in water-quenched
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 stress-strain behavior. Therefore,
future research is needed on the residual stress-strain behavior of heated concrete
after being cooled by water quenching.
Lastly, the models proposed in this research do not include the time-dependent
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 time-dependent 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 stress-strain curve at a particular
temperature, but also allow for time-dependent 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
105
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
Sand/Cement Ratio
Aggregate/Cement
Ratio
Water/Cement Ratio
Unit Weight
Slump
Cement Type
Cement Content
Aggregate Type
Aggregate Origin
Maximum
Aggregate Size
Curing
Specimen
Properties
Properties
Initial Curing
Specimen
Temperature
Shape
Initial Curing
Length or
Humidity
Height
Initial Curing Cross-Section
Duration
Area
Subsequent Curing
Volume
Temperature
Subsequent Curing
Specimen
Humidity
Mass
Subsequent Curing
Specimen
Duration
Density
End Condition
Test
Properties
Test Type
Preload Level
Axial Displacement Rate
Conditioning Chamber
Conditions
Furnace Type
Furnace Specification
Furnace Size
Furnace Humidity
Furnace Temperature
Heating Rate
Specimen Age When in
Furnace
Furnace Duration
Furnace Duration at
Equilibrium
d50 Size
Sand Type
Sand Origin
Air Entrainment
Amount
Air Content
Water Reducer
Amount
Retarder Amount
Silica Fume Amount
Fly Ash Amount
Water Quenching Duration
Residual Chamber Conditions
Subsequent Residual
Chamber Conditions
106
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
Compressive Strength
Strain at Compressive
Strength
Ultimate Strain
Creep Strain
Young’s Modulus
Dynamic Modulus of
Elasticity
Shear Modulus
Modulus of Rupture
Linear Expansion
Coefficient of Thermal
Expansion
Thermal Properties
Physical Properties
Thermal Conductivity
Heat Flux
Moisture Content
Mass or Weight Loss
Specific Heat
Heat Diffusivity
Spalling Temperature
Spalling Time
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108
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/concrete-fire-database. 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 user-defined polynomial equation form of any order;
and (4) to display the user-defined 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
TitleID
Input
Type
0
Description/
Allowable Entry
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
109
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)
110
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)
111
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)
Cross-Section Area
0.00
Cross-sectional 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.)
112
TEST PROPERTIES
Test Type
TEXT
Type of test conducted (unstressed pre-fire,
stressed pre-fire, 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)
113
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)
114
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)
115
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
116
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 non-specimen 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 non-specimen 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
117
εcm/εcmo Comment
TEXT
Specify if point was extrapolated from
experimental curve (extrapolated), Specify if
specimen includes non-specimen 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
non-specimen 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
non-specimen 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
118
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
non-specimen 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 non-specimen 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 non-specimen
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
119
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 non-specimen
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
120
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)
121
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
122
εcr Comment
TEXT
Specify if specimen includes non-specimen
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)
123
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 10-3 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)
124
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) (Btu-in/ft2-hr-°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/hr-ft2)
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
125
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
126
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)
127
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128
BIBLIOGRAPHY
Abrams, M., “Compressive Strength of Concrete at Temperatures to 1600 deg F,” ACI
SP-25, 1971.
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 High-Strength
Concrete,” ACI Materials Journal, V.87, No.1, 1990, pp. 47-53.
Cheng, F. P.; Kodur, V. K. R.; and Wang, T. C., “Stress-Strain Curves for High Strength
Concrete at Elevated Temperatures,” Journal of Materials in Civil Engineering,
V.16, No.1, 2004, pp. 84-90.
Concrete Association of Finland, High Strength Concrete Supplementary Rules and Fire
Design, Rak MK B4, 1991.
Comites Euro-International 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. 36-42.
Cruz, C. R., “Elastic Properties of Concrete at High Temperatures,” PCA Journal, V.8,
No.1, 1966, pp. 37-45.
European Committee for Standardization, Eurocode 2: Design of Concrete Structures,
Part 1-2: General Rules – Structural Fire Design, 2004.
European Committee for Standardization, Eurocode 4: Design of Composite Steel and
Concrete Structures, 2002.
Gillen, M. P., “Short-Term Creep at Elevated Temperatures,” PCA R&D Serial No.
1657, 1980.
Harmathy, T. Z. and Berndt, J. E., “Hydrated Portland Cement and Lightweight Concrete
at Elevated Temperatures,” Journal of the American Concrete Institute, V.63,
No.1, 1966, pp. 93-112.
Hertz, K. D., “Concrete Strength for Fire Safety Design,” Magazine of Concrete
Research, V.57, No.8, 2005, pp. 445-452.
Hognestad, E., “A Study of Combined Bending and Axial Load in Reinforced Concrete
Members,” Bulletin No. 399, University of Illinois Engineering Experiment
Station, Urbana, IL, 1951.
Kerr, E. A., “Damage Mechanisms and Repairability of High Strength Concrete Exposed
to Elevated Temperatures,” University of Notre Dame, 2007.
129
Kodur, V.K.; Dwaikat, M.M.; and Dwaikat, M.B., “High-Temperature Properties of
Concrete for Fire-Resistance Modeling of Structures,” ACI Materials Journal,
V.105, No.5, 2008, pp. 517-527.
Lankard, D. R.; Dirkimer, D. L.; Fondriest, F. F.; and Snyder, M. J., “Effects of Moisture
Content on the Structural Properties of Portland Cement Concrete Exposed to
Temperatures up to 500F,” ACI SP-25, Farmington Hills, MI, American Concrete
Institute, 1971.
Li, L. and Purkiss, J., “Stress-Strain Constitutive Equations of Concrete Material at
Elevated Temperatures,” Fire Safety Journal, V.40, No.7, 2005, pp. 669-686.
Luenberger, D., Linear and Nonlinear Programming, Addison-Wesley Inc., 2nd ed, 1984.
Mander, J.B.; Priestley, J.N.; and Park, R., “Theoretical Stress-Strain Model for Confined
Concrete,” Journal of Structural Engineering, V.114, No.8, 1988, pp.1804-1849.
Mendenhall, W. and Sincich, T., Statistics for Engineering and the Sciences, Ed. S.
Yagan, 5th ed, 2007.
NIST, “Final Report on the Collapse of the World Trade Center Towers,” National
Institute of Standards and Technology, 2005.
Phan, L. T. and Carino, N. J., “Code Provisions for High Strength Concrete StrengthTemperature Relationship at Elevated Temperatures,” Materials and Structures,
V.36, No.256, 2003, pp. 91-98.
Phan, L. T. and Carino, N. J., “Mechanical Properties of High-Strength Concrete at
Elevated Temperatures,” NISTIR 6726, Building and Fire Research Laboratory,
National Institute of Standards and Technology, Gaithersburg, MD, 2001.
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American Concrete Institute, V.29, No.10, 1958, pp. 857-864.
Popovics, S. “A Numerical Approach to the Complete Stress-Strain Curves for
Concrete,” Cement and Concrete Research, V.3, No.5, 1973, pp. 583-599.
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1899.
Saemann, J. C. and Washa, G. W., “Variation of Mortar and Concrete Properties with
Temperature,” ACI Journal, V.60, 1960, pp. 1087-1108.
Shi, X.; Tan, T. H.; Tan, K. H.; and Guo, Z., “Concrete Constitutive Relationships Under
Different Stress-Temperature Paths,” Journal of Structural Engineering, V.128,
No.12, 2002, pp. 1511-1518.
Van Geem, M. G.; Gajda, J.; and Dombrowski, K., “Thermal Properties of Commercially
Available High-Strength Concretes,” Cement, Concrete, and Aggregates, V.19,
No.1, 1997, pp. 38-53.
Zoldners, N. G., “Effect of High Temperatures on Concrete Incorporating Different
Aggregates,” Mines Branch Research Report R 64, Department of Mines and
Technical Surveys, Ottawa, Canada, 1960.
130
STRUCTURAL ENGINEERING RESEARCH REPORT SERIES
LIST OF TECHNICAL REPORTS
NDSE-01-01
“Design of Rectangular Openings in Unbonded Post-Tensioned 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/).
NDSE-01-02
“Capacity-Demand Index Relationships for Performance-Based 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/).
NDSE-06-01
“Experimental Evaluation of Unbonded Post-Tensioned 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/).
NDSE-06-02
“Seismic Analysis, Behavior, and Design of Unbonded Post-Tensioned
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/).
NDSE-07-01
“Friction-Damped Unbonded Post-Tensioned 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/).
NDSE-09-01
“Stress-Strain 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/).
NDSE-09-02
“Behavior and Design of Unbonded Post-Tensioning 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/).