A STUDY OF CREEP IN LIGHTWEIGHT AND
CONVENTIONAL CONCRETES
By
MICHAEL A. CASSARO
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE RIEQUIRehtENTS FOR THIE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
April, 1967
TO
Kay
Mike
Kevin
Molly
Katie
Maggie
Matt
ACKNOWLEDGMENiTS
This report vae prepared as a part of a study under Research Project
DR5025 at the University of Florida under contract with the Florida State
Road Department and in cooperation with the United States Bulreau of Public
Roads.
For the University of Florida, the work covered in this report vae
carried out under the general administrative supervision of T. L. Martin,
Jr., Dean of the Gallege of Engineering; M. E. Foreman, Director of the
Engineering and Industrial Experiment Station; R. W. Kluge, Chairman of
the Department of Civil Engineering.
The author wishes to thank James Ganmmage, Engineer of Materials,
Research and Training of the Florida State Road Department, R. W. Kluge,
Chairman, Department: of Civil Engineering, Ujniversity of Florida, K.
Majidzadeh, Associate Professor of Civil Engineering at Ohio State
University, and D. Sawyer, Professor of Civil Engineering, Auburn University,
Alabama, for their valuable contributions to this study. This study was
started under the direction of D. Sawyer while he was with the University
of Plorida and most of the planning is credited to him. Appreciation is
also rendered to J'. L. Holsonback for his efforts in development of
computer program used i~n the study.
The opinions, findings, and conclusions expreased in this publication
are those of the author and not necessarily those of the Florida State Road
Department or the Bureau of Public Roada.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS .. .. .. .. ... .. .. .. . ... iii
LIST OF FIGURES .. .. .. .. . .. .. .. ... .. .. vi
LIST OF TABLES ... .. .. ... .. .. .. .. .. viii
LIST OF SYMBOLS................,, ..... ix
ABSTRACT . .. .. .. . .. . ... .. .. .. x
CHAPTER
I. INTRODUCTION .. .. .. .. .. .. .. .. .. . 1
A. Objective .. .. .. . .. .. .. .. 2
B. Scope .. .... .. .. .. .. .. . .. 3
II. REVIEW OF THE CREEP BEHAVIOR OF CONCRETE .. .. 5
A. Coarse Aggregate Ingredients .. .. .. . 5
B. Mix Proportions ... .. .. .. .. .. 5
0. Shape of Specimen . ... .. .. . .... 6
D. Nature of Creep in Concrete . .. .. .. .. 7
III. THE VISCOELASTIC APPROACH . . .. ... . .. 9
A. Basic Mathematical Models .. ... .. .. .. 9
B. The Nature of the Rheological Model .. .. .. .. 11
IV. THE STATISTICAL MECHANICS APPROACH .. .. .. . 14
A. The Rate Process Theory .. .. .. ... .. .. 14
V. EXPERIMENTAL PROCEDURE ,. .. .. ... .. ... 20
A. Description of the Test .. .. .. .... .. 20
B. Materials .. .. .. .. .. .. .. .. .. . 21
C. Description of Specimens .. .. ... .. .. 24
D. Instrumentation .. .. .. .... . .. 29
VI. DEVELOPMENT OF A MODEL FOR THE CREEP MECHANISM . .. 33
A. Separation of Creep Components . ... .. .. 33
(a) Elastic Creep . .. .. . .. ... 34
(b) Viscous Greep ... .. .. .. .. .. .. 34
Page
Bi. The Generalized Model for the Creep Mechanism . ... 34
(a) Coefficients of Structural Stability .. .. .. 36
(b) The Rheological Parameters in the Greep Equation 39
C. The Analysis of Rheological Assemblies From a Statistical
Mechanics Viewpoint ........r.......... 41
(a) Spacing of Equilibriumn States .. .. .. .. 44
(b) Free Energy of Activation .. .. .. .. .. .. 46
D. Application of the Structural Parameters for the
Rheological Assemblies to the Rheological Model .. 53
(a) Retarded Elastic Recovery .. .. . .. .... 53
(b) Total Change in Fluidity of the Concrete .. .. 54
(c) Inelastic Behavior and t~he Coefficients of
Structural Stability . .. .. . ... .. 57
VII. ANALY'SIS OF RESULTS... .. .. .. . .. 76
A. Application of the Nodel to Concrete Under Sustained
St~ress (Test Series A) ... .. .. .. ... 76
b. Application of the Model t.o Concrete Under Decreasing
Stress (Test Seri.es B and C) ... .. .. .. .. 87
C. Influence of Shape on Creep . ... .. .. .. 101
D. Effect of Aggregate on Creep . ... .. .. ... 102
VIII. CONCLUSIONS..............,.... ..... 105
A. Geraral .. .. .. .. .. .. .. ... .. .. 105
B. Test Results .. .. ... . .. .. ... . 106
C. The Mbdel for Creep Prediction . ... . .. ... 107
D. Procedure For Using the Model . .. ... .. ... 109
APPENDICES
A. CALCULATION OF STRAIN FROM THE ACTIVATION OF A RHEOLOGICAL
ASSEMBLY ................... ...... 114
B. DEVELOPMENT OF THE STRUCTURAL FACTORS FOR COEFFICIENT OF
STABILITY ... ... .. .. .. .. .. . .. .. 116
C. METHOD FOR DETERMINATION OF THE COEFFICIENTS IN THE RATE
PROCESS EQUATION . .. ... . .. . . .. 119
LIST OF REFERENCES .. .. ... . .... .. .. .. .. 121
LIST OF FIGURES
Figure Page
1. LINEARIZED RHEOLOGi:CAL, MODEL FOR CONCRETE .. . .... 10
2. POTENTIAL ENERGY BARRIER OPPOSING MOVEMENT OF TLOW
ASSEMBLIES TO NEW EQUILIBRIUM POSITIONS .. .. ... 16
3. STRENGTH GAIN FOR STANDARD MIST CURED CONCRETE ,. .. 23
4. SHRINKAGE AND CREEP SPECIMEN .. .. .. ... . ... 26
5. DIAGRAM OF CREEP SPECIMEN UNDER STRESS .. .. .. . 28
6. HISTORY OF RELATIVE HTMIDITY IN LABORATORY ... .. .. 30
7. HISTORY OF AMBIENT TEMCER~ATURE IN LABORATORY .. .. .. 31
8. STRESS HISTORY OF CONCRETE SPECIMEN ... .. .. .. 32
9. PROPOSED RHEOLOGICAL, MODEL TO REPRESENT CREEP IN CONCRETE 38
10. COMPUTERIZED PLOTS OF STRUCTURE STABIILITY COEFFICIENT FOR
AGING AGAINST TIME, LIMESTONE AGGREGATE CONCRETE .. .. 6466
11. COMPUTERIZED LOTS OF STRUCTURE STABILITY COEFFICIENT FDR
AGING AGAINST TIME, LIGHTWEIGHT AGGREGATE CONCRETE .. .. 6769
12. COMPARISON OF SHRINKAGE FOR LIMESTONE CONCRETE AND SOLITE
CONCRETE, CIRGULAR SHAPED SPECIMEN . .... .. ... 10
13. COMPARISON OF SHRINLKAGE FOR LIMESTONE CONCRETE AND SOLITE
CONCRETE, RECTANGULAR SHAPED SPECIMEN .. ... . . 71
14. COMPARISON OF SHRINKAGE FOR LIMESTONE CONCRETE AND SOLITE
CONCRETE, CROSS SHAPED SPECIMEN .. .. .. .. .. .. 72
15. COMPARISON OF TOTAL STRAIN FOR LIMESTONE CONCRETE AND SOLITE
CONCRETE (CONTROL SPECIMEN, TEST SERIES A) CIRCULAR SHAPED
SPECIMEN .. .. .. .. .... .. .. .. .. .. 73
16. COMPARISON OF TOTAL STRAIN FOR LIMESTONE CONCRETE AND SOLITE
CONCRETE (CONTROL SPECIMEN, TEST SERIES A) RECTANGULAR
SHAPED SPECIMEN . .... .. ... . ... .. .. 74
17. COMPARISON OF TOTAL STRAIN ~FDR LIEBSTONE CONCRETE AND 80LITE
CONCRETE (CONTROL SPECIMEN, TEST SERIES A) CROSS SHAPED
SPECIMEN ... . .... .. .. .. .. .. .. .. 75
Figure Page
18. COMPARISON OF RHEOLOGICAL MODEL AND EXPERIMENTAL DATA
(AL CONTROL SPECIMEN, TEST SERIES A) . . ... ... 8186
19. COMPARISON OF RHEOLOGICAL MODEL AND EXPERIMENTAL DATA
(SPECIMEN, TEST SERIES B) .. .. ... .. .. .. 8994
20. COMPARISON OF RHBOLOGICAL MODEL AND EXPERIMENTAL DATA
(SPECIMEN, TEST SERIES C) . ... .. .. . .... 95100
LIST OF TABLES
Tr.'l* Page
I. MIX PROPORTIONS PER CUBIC YARD OF CONCRETE . . . 22
II. TEST CONDITIONS ... .. ... .. .. .. .. .. 25
'III. COEFFICIENTS F'OR THE LINEAR VISCOELASTIC MODEL OF FIGURE 1 40
IV, COMPUTED STRUCTURE COEFFICIENTS IN THE RATE FUNCTION . .. 45
V, CHANGE IN RATE FUNCTION STRUCTURE COEFFICIENTS .. . ... 47
VI. ELASTIC RETARDED RECOVERY  COMPARISON OF DATA AND THEORY 55
VII. EQUATIONS FOR THE AGING COEFFICIENTS, ct, OF STRUICTrURAL
STABILITY FROM COMI~1TERIZED LEAST SQUAI FIT .. .. .. 59
VIII. TOTAL OBSERVED TRANSIENT AND STEADY STATE CREEP OF CONCRETE
UNDER VARYING STRESS LEVELS .. .. .. . .. .. 103
LIST OF SYMBOLS
C P concrete strain
& = strain rate, time derivative of strain
E vis E all non recoverable components of total strain
f = concrete stress in pai
fo = initial concrete stress
fD P decaying concrete stress
t I time in days
E = elastic maodulus of concrete in pai
= viscosity in paidays
kno = final viscosity
ko = initial viscosity
0 I fluidity, the reciprocal of viscosity
E, a, Or s pring and dashpot constants respectively of the elastic element
in the theological equation for creep
En> n, s similar constants for any Kelvin element
e base of the natural logarithms
k Boltzmann's constant 1.380 x 1016 arg deg1
h = Planck's constant 6.624 x 102 erg see
T absolute temperature
Eo a free energy of activation
ct time dependent component of the coefficient of structural stability
et coefficient of the stress dependent component of the coefficient of
structural stability
A, B coefficients in the equation governing strain rate based on the
rate process theory
w = weighted flow distance over which a particle of moisture must flow
before reaching the surface of a specimen
Abstract of Dissertation Presented to the Graduate
Council in Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy
A STUDY OF CREEP IN LIGHTWEIGHT
AND CONVENTIONAL, CONCRETE
By
Michael A. Cassaro
April, 1967
Chairman: Professor R. W. Kluge
Major Department: Civil Engineering
Statistical Mechanice and Rheological approaches have been utilized
to establish a mathematical model to represent the long term creep of
concrete.
The generalized model presented predicts behavior of lightweight and
conventional concretes irrespective of size and shape variations. A
coefficient of structural stability has been introduced to incorporate the
effect of aging and of stress variations in the analysis of data. The
use of the age and stress dependent components of the stability coefficient
permits the calculation of creep at any time and for any stress level.
CHAPTER I
INTRODUCTION
The mechanical properties of concrete, and perhaps all matter, may
be treated from three distinct viewpoints; each viewpoint attempts to
satisfy an engineering need for predictability concerning the mechanical
behavior of the material in specific structural applications; each
viewpoint used to satisfy this need takes a basically separate route;
each viewpoint has distinct advantages which when applied, are comple
mentary. These viewpoints are the Statistical Mechanics, the Structural,
and the Phenomenological approaches to material behavior.20
The Statietical Mechanice approach considers the material as an
association of discrete particles held together by bonds of high energy
content. At this level behavior of the discrete particles is described
by their relative positions in space, their velocities, and the interaction
forces between them. The Structural approach considers the material to
be continuous but nonhomageneous, being formed of elements of different
properties, distributed randomly throughout the material, and having
finite dimensions. The Phenomenological approach considered the observed
or macroscopic material baehvior. It deals with the observation of the
relation between forces and resulting states of motion of finite bodies
assumed to be homogeneous. Mechanical behavior of the material is
described in terms of the relations between stresses and strained, and
their derivatives. This approach usually involves extensive testing and
bases the material behavior on observed responses obtained from the test
and its agyironment.
A. Objective
In this investigation, an attempt has been made to combine the beat
findings of each approach. The primary concern is with an understanding
of the behavior of a specific structural lightweight aggregate concrete
and normal limestone aggregate concrete as the behavior is influenced by
creep alone. In an attempt to clarify the exteting theories for the
mechanism of creep as they pertain to concrete of the type tested, this
study has endeavored to link molecular~ hypothesis and theological models
with macroscopic observations of the concrete structures under investi
gation. In this endeavor it is hoped that a clear understanding of
creep behavior of concrete as a function of time and stress will result.
The parameters in the model developed eae material constants
related to the rheological structure of the concrete in question. If
it is possible to determine the nature of the changes of these material
constants for the concrete structure with respect to stress independently
of aging it is reasonable t0 Bxpect that it will be possible to predict
creep for the concrete under any condition of stress and time.
In all previous methods developed to explain creap it has not been
possible to make a general prediction of creep behavior for a specific
concrete relative to time and stress based on the results from a single
test. The advantages for being able to do so are obvious. By linking
molecular hypothesis to theological description of the concrete it
appears possible to separate aging effects from stress variation effects.
When the two effects eae separated variation in each may be handled
separately thereby producing the desired model.
rhis investigation has three objectives.
1. To determine the difference in the nature of creep between lightweight
Solite concrete and normal limestone concrete.
2. To determine the influence of shape without size interaction on
concrete creep.
3. To investigate the possibility of establishing a model to represent
the creep of concrete under any stress history from a single test
of the concrete which is to be represented by the model.
In establishing the general creep model with respect to stress and
time based on the observed creep response from a test series under
sustained stress, conventional viscoelastic23 and statistical mechanics 1
approaches are employed to develop the coefficients of the creep equation
for the model. Following the conventional viscoelastic approach the
material behavior is separated into type of response as follove:
1. instantaneous elastic response
2. steady state irrecoverable creep
3. transient delayed elastic response
4. transient viscous response for early stage creep
5. transient irrecoverable creep representing long term responses.
The model is established by converting the last four responses listed
into three parameters in the model:
1. The rate of transition of the ge1 from fluid to solid,
2. The total expected change in the structure of the ge1 as measured by
its fluidity.
3. The, purely elastic transient action of the ge1 and aggregate system.
The first parameter constitutes the rate of change of the concrete
structure under the action of stress and due to normal aging. The second
parameter represents the difference in the nature of the structure from
before load is applied to some final state. These two parameters are
involved in the stability of the concrete structure with respect to
aging and stress.
The third parameter is purely elastic in nature and represents the
delayed elasticity term of thre model. It is shown that this term may be
employed to determine the delayed elastic recovery for the unloaded
specimen.
If the model is to be acceptable it must be capable of predicting
creep behavior of lightweight and normal concretes alike; the model
developed does so. The model developed also attempts to predict creep
behavior irrespective of size and shape variations. The variations in
the material constants of the model due to shape appear to adequately
explain the creep mechanism as it pertains to shape. Size was not a
parameter of the experiment. However the anticipated nature of the
interaction between size and shape is developed and is presented in the
interpretation of the results. Since mix variations and environment were
not parameters in this experiment and in view of a lack of sufficient
correlation between size and creep, the scope of this experiment precludes
the formulation of a more generalized model.
CHAPTER II
REVIEW OF THE CREEP BEHAVIOR OF CONCRETE
Many variables influence the magnitude of creep. This test program
was limited to only a few among which are:
Coarse aggregate ingredients
Mix 'proportions
Shape of the specimen
A. Coarse Aggregate Ingredients
In general it is believed that hard dense aggregates, with low
absorption and a high elastic modulus produce a concrete with low creep
tendencies. TRoxell and Davial indicate that particle shape, surface
texture, pore structure and unit weight may also influence concrete creep.
These influences are all involved in the differences between the two
aggregates used in this study. An individual evaluation of each
influencing factor is difficult if not impossible. However, it may be
possible to speculate on the more prominent. characteristics which are
considered to be the modulus of elasticity and the permeability of the
aggregate. Particle shape and surface texture are not considered to
be major factors in creep behavior according to Best.15
B. Mix Proportions
Cement paste content and watercement ratio appear to be intimately
involved in creep activity of concrete. Many authors have testified to
6
the direct relationship between paste content and creep.18 This study
was not intended to resolve the questions concerning the nature of the
involvement of cement, water or paste. Therefore, in an attempt: to
minimize complications in the behavior of concrete due to variations in
these ingredients the paste content vere made equal for the concrete
mixes.
The watercement ratios for the two mixes could not be so easily
equated. No problem resulted from the selection of a given watercement
ratio for the limestone concrete. However, a watercement ratio for the
lightweight aggregate concrete may not be established since a portion of
the free water contained in the aggregate used must be considered to add
to the mix water for the concrete. By preliminary experimentation the
strength of the lightweight concrete was controlled by selection of an
effective mix water content to establish near equal strengths for the
two structural concretes.
C. Shape of Specimen
Very little work has been done to determine the effect of shape on
creep. Several in vestigations have been reported involving the influence
of size of specimen on creep. These reported indicate a decrease in
creep with increase in aise of specimen. The effect is neually
explained as a result of the increased seepage path to the free surface.
This apparent interaction of stress dependent seepage and shrinkage which
is related to creep bears directly on the shape effect of the concrete
specimen. Thus by varying the free surface area and maintaining a
constant crosesectional area it is possible to evaluate the stress
dependent seepage effect on creep as distinguished from pure shrinkage
for these two concrete types.
D. Nature of Creep in Concrete
The deformation characteristics of concrete are related to its
internal structure.111 The mortar paste constitutes the active part
of the concrete in binding coarse aggregate to form a stonelike material.
It is composed of cement, water, interspersed graded fine aggregate, and
air. The relative proportions of these ingrediente vary from one concrete
mix to another. This variation is the essential factor which results in
all behavioral variations of concrete.
The aggregate particles are bound together by the cement paste
matrix. The aggregate plays a less active role in the creep response
of the concrete than the cement paste.
At any stage of hydration, the structure of the cement paste
consists of unhydrated cement, grains, hydrated cement, water, and air.
The cement paste structure is called cement gel which has a more rigid
structure with aging and a less rigid structure under increasing load.
It is the action of the ge1 which gives concrete its viscoelastic
nature. Initially as the cellular structure of the hydrated cement
is developing, the colloidalliquid phase predominates. If a load is
applied at this early stage the cement ge1 behaves as a liquid in that
it flows under any nonisotropic stress, however small. The degree to
which the liquidcolloidal phase exists is manifested in the amount of
creep which results. As the gel matures, due to continued hydration,
the solid cellular structure becomes dominant over the lIquid phase.
An applied stress at this time results in less creep since the gel would
behave more like a solid, exhibiting elastic tendencies.
The nature of the action of the aggregate dispersed throughout the
cement ge1 has been termed passive.2 If, as stated above, it is the
cement gel which creeps, then by including solid aggregate particles
into the ge1 its resistance to creep should be increased according to
Reiner.3 This condition further complicates the problem of determining
the action of the coarse aggregate. As it was explained in connection
with the coarse aggregate effects discussed in section A of this chapter,
this additional volume concentration effect cannot be evaluated for the
lightweight aggregate because it is not known how the moisture contained
in the lightweight aggregate influences ite volume concentration in the
concrete.
Therefore, like most creep findings reported in the literature,
the findings of this report may only be applied to the specific
concretes being investigated. The model constants are applicable only
to the range of specimen sizes used and under similar ambient: conditions
of humidity and temperature.
CHAPTER III
THE VISCOELASTIC 'APPROACH
A. Basic Mathematical Models
The most succeaeful simple phenomenological form of the data from
a creep study follove Ross's) parametric equation relating strain and
time. Most strain data may first be prepared in this form from which
creep, a may be determined at: any time t from
where a is a dimensionless constant and b is a constant with the unite
of time.
The most general phenomenological form for creep data is obtained
from theological theory. Rheology is the science of flow of materials;
it offers a transition between classical theory of elasticity and classical
hydrodynamicse and has been proposed as a theory of the general behavior
of materials on the assumption that every real material must be supposed
to possess all basic deformational properties in varying proportions.
Strain relations may be developed as viscoelastic strains employing the
conventional theory of rheology.2,1 Using the data from test series A
a creep model with four linear elements was developed which was similar
to Freudenthal's$ model in the simplified form. The detailed model used
is shown in Figure 1. The model consists of the following elements in
series.
10
hasa
LINEARIZED RHEOLOGICAL MODEL FOR CONCRETE
FIGURE .1i
1. A Maxwell element with linear spring and dashpot in series, representinBg
the elastic strain and the long time irrecoverable creep resulting from
inelastic strain.
2. A Kelvin element with linear spring and dashpot in parallel representing
the short time viscoelastic effects. This influence vae attributed to
consolidation due to seepage of pore water.
3. A Kelvin element with linear spring and dashpot in parallel representing
intermediate time viscoelastic effects. This influence was attributed,
after Freudeathal, to retarded elasticity or recoverable creep.~
4. A Kelvin element with linear spring and dashpot in parallel representing
long tone viscoelastic effects. This influence was attributed to
destruction of the gel structure under stress.
The equationl7 representing the simple theological model for constant f is
3 Et
c._ i ft (1 "n n ) .(2)
E As n 1E
The first two terms represent the Mazuell element. The three terms in
the summoation represent the Kelvrin elements. All terms in equation (2)
are defined in the list of symbol.
Most creep curves may be satisfied by simply extending the number
of terms in' the sumat~eion to provide a transition curve which fthe the
data between the elastic response (first term) and the final or steady
state response (second term).
B. The Nature of the Rheolog~ical Model
Rheological equations define ideal bodies which serve as models of
comparison in the analysis of material behavior. The theological
variables are stress, deformation and time; the theological parameters
are viscosity and elasticity. The parameters in the rheological equations
characterize the material behavior; they may be constants from which we
obtain linear viscoelastic equations or they may be functions of time or
stress from which we obtain nonlinear viscoelastic equations.
Many models have been proposed for the creep behavior of concrete.
However, the many factors influencing the gel structure seem to preclude
the development of a single model for all conditions. The nature of
hydration in the gel is believed to contribute the major source of
variability in the creep behavior. It is the gel which supplies the
viscoelastic nature of concrete because it is a varying form of matter
lying between the solid and fluid physical states. The gel in concrete,
which is formed initially from a fluid, is distinguished by a change in
its mechanical properties, by a transition into a solid state having high
viscosity, elasticity, and limiting values of stress related to strain.
Snoe authoreS have experienced completely solid state responses (no
viscous creep) from concretes which have been fully dried and cured.
Concrete undergoes the transition from fluid to solid over a period
of several years. After a few hours, however, the solid phase cellular
structure may have developed sufficient elastic strength to withstand a
test: load. At that time the concrete is put into service and loads are
imparted to it. It is clear that the degree of fluidity of the ge1 at
the time of loading will greatly. influence the elastic and creep properties
of the concrete. Rheological models are unable to cope with the many
variables which influence these phase changes of the gel not to mention
the added variation of aggregate interaction with the ge1. It would
13
appear that a single model, no matter how general it may be, is not the
basic answer to prediction of creep in concrete; rather, a general
procedure for predicting creep behavior is more in order. Such a procedure
will provide for the development of an equation which incorporates the
significant parameters for the concrete being investigated into a specific
theological model.
All the influences on creep may be categorized by their influence
on three factors:
1. The rate of transition of the gel from fluid to solid.
2. The purely elastic action of the gel and aggregate system.
3. The total change in the structure of the gel.
If these three factors are used as parameters for the theological
model then it would seem that the creep behavior of every concrete may
be interpreted.
For a more detailed study of the theological approach refer to
items 2, 4, and 17 in the bibliography. A supplementary study to this
report, of theology, and the basis for development and behavior of
the model presented in figure 1 is found in reference 22.
CHAPTER IV
THE STATISTICAL MECHANICS APPROACH
A. The Rate Process Theory
Glasstone, Laidler and Eyring6 extended classical quantum mechanics
theory to include statistical treatment of reaction rates. Since all
matter is composed of molecules or assemblies of molecules which vibrate
about some equilibrium position, when the quantum state (or energy level)
is changed new equilibrium positions are obtained. Eyring, et al. employed
a concept of potential energy surfaces to describe the conversion from
vibrational energy to translational kinetic energy and viceversa. The
essential requirements in the development are that:
1. Energy is conserved, following the first law of thermodynamics.
2. The number of assemblies is constant.
3. There existed definite energy levels (second law of thermodynamics).
4. All possible energy levels (or quantum states) for the entire system
have equal probability.
Since there is a given probability that any molecule or assembly
shall have a free energy in any quantum state which is a function of the
entropy of the assembly, the total probability of the occurrence of an
assembly with given entropy is proportional to the sum of all the energy
terms for the assembly. The sum is called the partition function of the
assembly for a given volume of matter. The partition function may include
energy terms for nuclear, electrical, vibrational, rotational, and
translational energy.
In evaluating the reaction rate of an assembly we are concerned about
the velocity at which an activated assembly travels over the potential
energy barrier (Figure 2)9 thus passing from one equilibrium state to
another. The magnitude of the energy barrier is equal to the work or
thermal energy which must be applied to the assembly in order to activate
it. The assembly is considered to be activated when it is at the level
of energy equal to the energy barrier level. The net rate at which the
reaction occurs is determined by the average velocity of the activated
assemblies passing over the top of the barrier.
Eyring et al. have evaluated the partition functions involved in
the thermodynamics of reaction rates to develop an equation for the
specific reaction rate. The specific reaction rate defines the
frequency, v, with which an activated assembly crosses the barrier and
is displaced a distance 6 (Figure 2) from one equilibrium position to
another.
kT Eo/kT
v = e(3)
in which:
T o absolute temperature
k = Boltsmann's constant
h = Planck's constant.
E o the free energy of activation; for a theological assembly it is
defined by Eo = V T S where V is the total energy of the
assembly and S is the entropy (a measure of the disorder of the
assembly) accompanying the activation process.
In the presence of an applied stress the energy barrier is modified
(Figure 2). The effect. of the stress is to reduce the height of the
f =APPLIED FORCE
ACTIVATION OCCURS
INITIAL
EQUILIBRIUM
POSITION I
LL EQUILIBRIUM'
POSITION
DISPLACEMENT OF ASSEMBLY
POTE NTI AL ENERGY BARRIER OPPOS'TNG MOVEMENT OF FLOW
ASSEMBLIES TO NEW EQUILIBRIUM POSITIONS
FIGURE. 2
energy barrier in the direction of the stress thereby permitting a flow
assembly with given free energy level to pass freely over the barrier.
Since there exists an equal probability for the existence of all free
energy levels, a greater stress will result in activation of a greater
portion of the population of flow assemblies.
If: the potential energy surface is considered symrmetrical, then the
energy level required for flow opposite to the applied stress is
increased an amount equal to the decrease in the energy level in the
direction of the applied stress. That is, activation occurs midway
between equilibrium positions.
The passage of an activated assembly over the potential energy
barrier represents the jump of the assembly from one equilibrium
position to the next. Let A be the distance between two equilibrium
positions. The applied force acting on an assembly is f/D2 where D
is a structure factor which is equal to the number of flow assemblies
per: unit length and f is the applied stress. Hence, the energy that the
assembly acquired in advancing to the activated state is
Af/D2 x 1/2 that is, 1/2 fdb /D2
A creep condition may be analyzed by evaluating the coefficients for
the specific reaction rate of the concrete specimen loaded under sustained
stress. The frequency of activation for movement of the flow assemblies
in the direction of the stress, that is, for flow in the forward direction,
is
vg.,[ ( E epo 2D (3a)
18
and the frequency for movement opposed to the stress, that is, for flow
in the backwards direction, is
vb Ti; exp E D (3b)
kT
The net frequency of movement of the flow assemblies combining motions
in the forward and backward directions becomes
kThBF EoRTDL (3
vf vb = 2 kT sn g9) .(c
The creep rate may now be related to the frequency of movement of flow
assemblies. Let
L P the axial component of displacement due to the movement of
an assembly to a new equilibrium position, and recall that
D =1 a structure factor which is equal to the number of flow
assemblies per unit length.
The creep rate is
6" DL (Vf Vb)
or
E 2 DL exp LT sinh ( )ii (4)
which may be writtenl0
E A sinh B f .(4a)
The coefficients, A and B are evaluated based on the findings of
the theological investigation of the parametric equations obtained from
19
the observed responses. Using the simple viscoelastic model of Figure 1
described in Chapter III with the rate proceed solution, the data may be
further evaluated for a better understanding of the creep behavior.
The rate process equation (4a) related creep rate to stress. The
viscoelastic equation (2) for the simple theological model of Figure 1
related creep to time for a given sustained stress. The two applications
are therefore complementary and may facilitate separation of influences
on creep due to aging and stress variations.
CHAPTER 'V
EXPERIMNTAL~ PROCEDURE
A. Description of the Test.
It is generally considered that creep and shrinkage are intimately
related. Their true interaction, however, is not completely understood.
Shrinkage has been defined as the change in length of concrete members
without the influence of applied stresses. Creep is defined as the
change in length of concrete members under the influence of applied
stresses. These definitions provide only a superficial distinction
between creep and shrinkage. In fact, differential shrinkage in
concrete is known to result: in stress gradient causing compressive
streseos in the interior of the concrete,21 Therefore any contribution
to the shortening of the member by these stressed should be classified
as creep according to the definition. On the other hand, moisture forced
to the vicinity of the surface of a creep specimen by applied stressed
may be removed from the specimen by the same phenomena that cause shrinkage.
In spite of this awareness, it is traditional to separate the two effects,
shrinkage and creep, according to the definitions presented above. This
separation is desirable in this investigation since the basic theories
being employed are contingent upon a knowledge of the applied forces.
Consequently creep strains in this investigation were obtained by subtraction
of shrinkage strains obtained from unstressed specimens, from the total
strains obtained from creep specimens.
The purpose of this investigation, concerning the comparative creep
behavior of a typical Florida limestone aggregate concrete and a Florida
lightweight aggregate concrete (Florida Solite an expanded clay ~product)
wee to determine some structurally important aspects of the mechanism of
creep in concrete. Three test series investigated creep. Series A
investigated creep under sustained stress. Test series B and C investigated
creep under different decreasing stress situations. A fourth test series
investigated shrinkage.
Shrinkage of the test specinans is illustrated in Figurea 12, 13, and
14. Total strained for creep series A are illustrated in Figuree 15, 16,
and 17. Both sets of strain data display seasonal influence beyond the
initial stage of their responses. Shrinkage and creap specimens were
maintained in the laboratory environment where periodic ambient readings
were recorded of humidity (Figure 6) and temperature (Figure 7).
B. Materials
In an attempt to minimize effects from mix variations, strength,
variations and environment variations between pours, a single pour was
made for each concrete type. Table I contained proportions for the two
mixes used. Equal paste contents as nearly as could be determined were
selected to minimize the effects of a ~difference between the concrete
gels. influencing creep. As a result of the desire to obtain near equal
strength for the two concrete types, the water contents were selected to
compensate for the aggregate influence on strength. Figure 3 illustrates
the concrete strength gain for both concretes. Each point represents the
average of four 6 x 12 inch test cylinders.
Conventional Limestone Concrete
Cement (Type III, HiEarly):
Sand (Interlachen, FM =2.42):
Stone (3/4" Brooksville Limestone):
Water 36.6 gallons:
Height per cu~bic foot:
Slump:
W/C Ratio by Weight:
Paste Content:
TABLE I
MIX PROPORTIONS PER CUBIC YARD OF CONCRETE
All Specimens
Lighewaigpht (Solite) Concrete
Cement (Type III, RiEarly):. 8.7 sacks
Sand (Interlachen, FM = 2.42): 1213 #f
Stone (3/&" Florida polite): 1080 #
(23.9% moisture)
Water 27.9 gallons: 232 #
Eight per cubic foot: 123 #/c.f.
Slump:' 2 1/2"
W/C Ratio by HFeight: .284
Paste Content: 83 #/c.f.
6.73 sacks
1220 #
1755 #
303 #
145 #/c.f.
5 "
.477
80 #/c.f.
Is Ip, U, U U
4 ~1 rr I I I 1 SOLITE CONCRETE
9,0
b Ito Po o 40 5s so To so s> 100
AGE~(DAYS)
STRENGTH GAIN FOR STANDARD MYOIST CURED CONCRETE
FIGURE 3
The concrete used in the specimen contained type III, high early
strength cement as would be used in most prestressed concrete bridge
construction. The coarse aggregates had maximum sizes of 3/4". The fine
aggregate need in all specimens was Interlachen sand normally used for
construction in the north Florida area.
C. Description of Specimene
Special shapes were selected to test an hypothesis about the shape
factor effect on the rate of creep. Three shapes with identical crose
sectional areas equal to 28.27 square inches but with unlike perimeters
were cast. The specimens obtained contained equal volumes but varying
volumetosurface area ratios (Table II). The shapes selected, which
had cross, rectangular, and circular crosssections, are shown in Figure 4.
The selection of equal volumes was made in order to evaluate shape effect
by blocking out size effect.
The flow path from any point at the interior of the specimen to the
surface is measured by the average weighted distance over which a particle
of moisture must travel before reaching the surface of the specimen. The
average weighted distances for the specimen may be calculated by summing
the products of the incremental areas of the specimen crosssection and
the least distance from thee incremental areas to the surface of the
specimen, and dividing the sum by the area of the specimen. The average
weighted distances are given in Table II for each shape of specimen.
Although perimater tocrosesection area ratio is frequently used as a
measure of shape effectl6 it isconveient and perhaps more accurate to
use the weighted distance as the significant measure of shape effect.
Coarse Crosssectional Average Initial Concerte Age at
Test Shape Aggregate Area to Length Stress Strength Stressing Stress
Series Type Pathmeter of Flow (psi) at Stressing (days) Character
Ratio Path (psi)
A Cylinder Limestone 1.49 1.00 914 4700 3 Sustained
(control) Solite 1.49 1.00 914 4600 3
Rectangle Limestone 1.23 0.77 914 L700 3
Solite 1.23 0.77 914 4600 3
Cross Limestone 0.99 0.55 914 4?00 3
Solite 0.99 0.55 914 4600 3
B Cylinder Limestone 1.49 1.00 914 4700 3 Decreasing
Solite 1.49 1.00 914 4600 3
Rectangle Limestone 1.23 0.77 914 4700 3
Solite 1.23 0.77 914 4600 3
Cross Limestone 0.99 0.55 914 4700 3
Solite 0.99 0.55 914 4600 3
C Cylinder Limecstons 1.49 1.00 904 5700 7 Decreasing
Solica 1.49 1.00 904 5200 7
Rectangle Limescener 1.23 0.77 904 5700 7
Solitt 1.23 0.77 904 5200 7
Cross Limestone 0.99 0.55 904 5700 7
Solite 0.99 0.55 904 5200 7
TABLE II
TEST CONDITIONS
ELEVATIONtS
SECT IONS
CROSSED SHAPE
CIRCULAR SHAPE
RECTANGULAR SHAPE
SHRINKAGE AND CREEP
FIGURE 4
7.75"
SPECIMIEN
27
When this substitution is made, it becomes immediately evident that shape
and size effects interact since length of flow path is the primary
variable governing their influences on creep. In this regard the specimen
behavior in this investigation may be considered to give some indication
of size effect even though the investigation seemingly blocks out: size
effects.
All epecimene were thirty inches long, This provided space for
two gage lengths of ten inches over which measurements were made plue
six inches on each end. The six specimens comprising a creep test series
were strung together end to end with a transition plate between each specimen.
The ends of all specimens were capped with standard capping compound
consisting of flyash and sulfur. Each specimen was cast with a metal
conduit: along its exia through which the stress rod was placed for
loading of the specimen. A ten inch concrete bearing clock and steel
plate were used at the ends of the strung specimens to accept and
distribute the applied test load.
Shrinkage specimens were made identical to creep specimens. Their
end faces were sealed with a layer of wax and metal fail which resulted
in having their exposed surface identical to the exposed surface of the
creep speelmens.
Shrinkage strains were taken on individual unstressed specimens.
Creep stains were taken on specimens under identical stress conditions
in each test series. One specimen of each shape for each of the two
concrete types was used in each creep test series as shown in Figura 5.
Table II summaries the test conditions for each specimen. Specimens in
test series A were maintained under constant sustained stress of 914 10 pai.
9 CRd$$ SP'ECUIMENS
a REcYANGLE sPecIMENS
2 CTL~INER SPECIMENS
ESOLIS UNEStdN SOLITE LMCIEONE sOoft LIMESTONE
PLAN
mcen Tarrssa srassev sron
PILLER 8TRE88I DISTRIBUTION ILOQON
ELEVATION
DIAGRAM OF CREEP SPECIMEN UNDER STRESS.
SIGUA S
Stresses in specimen for test series B and C were permitted to decrease
as shown in Figure 8. The decrease in stress associated with series C
resulted from normal prestrees loss due to creep and shrinkage. Test
series ~B contained filler blocked of concrete and high strength plaster
between the specimens which permitted a greater initial stress decay.
D. Instrumentation
All specimen were instrumented with standard Nhittemore gage points
spaced at ten inches as indicated in Figures 4 and 5. Shrinkage and
creep specimens were maintained in laboratory environment where periodic
concrete strain reading were recorded along with ambient reading of
humidity (Figure 6) and temperature (Figure 7). Strain adjustments
were made for temperature variation by assuming a thermal coefficient of
0.0000055 inches per inch per degree Fahrenheit. However, no adjustments
were made for humidity variations because there is no way of determining
the integration effects of humidity on concrete behavior.
Load readings on the creep apacimen were obtained from SR4 strain
gages on the stress rods. Loads were applied with hydraulic jacks.
8
A
Ogb
so Owe *
.~~ .
*z ** t o oe a
so **a
4 e ga
I1* *o eG o a oS
aa
o *S *~ e
< so a a
10a
4 6 8 10 12 2 4 6 8 10' 12 2 4 6 8 10 12 2
MONTH OF YEAR LOCATION : GAINESVILLE ,FLORIDA
'TIMWE IN MiONTHCS
HISTORY OF RELATIVE HUMI~DITY IN LABORATORY
FIGURE. 6
100
A MINIMUM TEMPERATURE CURVE
CO
40
20
10
S8
FLORIDA
Oli
IS 1 21 e4 6 1012 24
MONTH OF YEAR LOCATION : GAINESVILLE,
4 6 8
TIME IN MONTHS
HISTORY OF AMBIENT TEMPERATURE IN LABORATORY
FIGURE 7
tSTb SERIES
0.9
0.7
TEST SERIlES B
0.3
0.01
820 300 400 SW 8300 tHO
tIME IN DAYS
STRESS HISTORY OF CONCRIEtE SPECIMEN
FIGURE 8
 140
80 90 CA10OO
CHAPTER VI
DEVELOPMENT OF A MODEL FOR THE CREEP MECHANISM
A. Separation of Creep Cqomonent~s
There are two fundamental types of viscoelastic creep'i
1. Delayed elasticity  a completely recoverable strain phenomenon.
2. Viscous creep  an irrecoverable strain which may contain transtent
and steady state components.
Creep has been associated with rearrangement: of molecules due to
thermal movement. According to quantum theory, the movement must result
in a different and stable equilibrium configuration of molecules. Since
fluid contain many vacant sites in their molecular structure, large.
voided (holes) exist into which the migrating molecules may relocate. Any
motion of one molecular assembly may, of course, result in relocation of
neighboring assemblies. The rearrangement of these assemblies is always
in the general direction of the applied stress but, the motion may have
components in any direction thereby resulting in shear distortion. These
shear distortions are viewed as macroscopic creep. When the applied stress
is removed some of these migrated assemblies return to their initial
equilibrium states. The resulting strain recovery is viewed as delayed
elasticity. It is necessary to separate the two fundamental types of
creep, the irrecoverable from the recoverable creep, in the development
of the model for a creep mechanism because under decreasing stresses the
elastically recoverable component will be active.
(a) Elastic Creep
Elastic creep has been described by Oravan8 as resulting from an
applied stress which alters the statistical equilibrium position for each
assembly. A time dependent strain results which continues as the assemblies
seek new equilibrium position within their stressed configuration, If
the force is suddenly removed, the instantaneous elastic response returned
the molecular structure to its former equilibrium configuration. The
assemblies now seek their original equilibrium positions and recoverable
strains are observed. It is tacitly assumed that no deterioration or
damage occurs to the molecular structure during the load cycle and no
additional thermal energy sources are introduced into the system. The
strain recovery will therefore be complete:.by definition of elasticity.
(b) Viscous Creep
During the transition period, while fluidi.ty characterizes the nature
of the ge1 matrix, the flow assemblies are more mobile. The crystalline
structure is not sufficiently rigid to behave completely elaetically and
therefore, once it has moved to a new equilibrium position the flow unit
is completely stable in this position. Upon removal of the load there is
no tendency for the flow unit to return to its original position.
B. The Generalised Model for the Creep Mechaniem
The development of a model which is to represent the creep behavior
of concrete must contain the two elements, delayed elasticity and viscous
14
creep. The basic strain equation must then be of the form
a = [+ q(f, t)vis, + q(f, t)el. *(5)
where the terms represent the elastic train, the viscous component of
creep, and the elastic component of creep respectively.
Since the last term has an elastic nature, a Kelvin element should be
sufficient to represent this component. In its simplest form the term may
have constant coefficients to represent average effects of delayed elasticity
in the concrete. It represents the basic elastic structure of the ge1.
The second term contains all phenomena which result in permanent changes
in the theological structure. Such phenomena are for example:
1. Hydration effects.
2. Irrecoverable strain effects resulting from structural deterioration
due to stress.
3. Viscous flow.
Each, of the two creep terms in the model must contain coefficients
which relate the structural changes of the concrete due to the various
influences to the two major variables, aging and stress. The objective is
to establish coefficients which include as many influencing factors as
possible so as to make the model applicable in a greater number of situations.
For example, if it would be possible to incorporate such factors as aise
and shape, environment conditions, mix proportion, and physical characteristics
of the mix ingredients into .the model, then the enigma of creep behavior would
truly be considered resolved. It should be emphasized at this point that
this study will not: attempt to develop such a general form of the model
for the creep mechanism. It is hoped, however, that the ground work will
be adequately accomplished toward such developments. Obviously to achieve
such a result will involve a etudy of much greater depth and scope than
this investigation encompasses.
Equation (5) may be revised to include the delayed elastic component
referred to and is therefore written in the form:
f4 ft t+E (1 + e .(5a)
(a) Coefficients of Structural Stability
Reiner3 has introduced a coefficient (1) called "The Coefficient
of Structural Stability" which relates viscosity or more accurately its
reciprocal, fluidity,to stress. The ecope of this coefficient may be
broadened to include structural variations in fluidity due to aging also.
As described in Appendix B let:
x =(0 ) / "_a 2 (6)
where 0 represents the fluidity at any time,
On0 represents the ultimate fluidity,
f is the normal stress.
The partial derivatives are taken with respect: to stress squared (2)
and time (t). Reiner has shown that fluidity is influenced by the square
of the shear stress (f2/4). The negative sign is required beceaue an increase
in time results in a decrease in fluidity.
In order to understand the Structural Stability Coefficient consider
the extreme case of I = 0O and X P O When X = 0 O) = 80 which means
that the viscosity of the material is at its steady state value. The
materiall is fully, aged and,under applied stress, it strains at constant rate.
When I c o the partial derivatfives of fluidity with respect to time and
stress are equal to sero. Thie implies that the viscosity of the material
is not changed due to applied stress or due to aging. This latter case is
obviously not possible with concrete since aging, through hydration, wrill
always produce some finite value for the change in viscoeity with time.
Therefore the smaller values for the structural stability coefficient, X,
indicate greater structural stability of the material. The material is less
subject to creep, since 0 is smaller because of aging or stress influences.
Either factor, stress or aging, may have a greater influence or a
lesser influence on structural changes. It is therefore necessary to
separate the aging and stress components of the coefficient of structural
stability. Accordingly components for aging, et, and stress change, cf,
have been developed in Appendix B.
It can be. shown (Appendix B) that the coefficient of structural'
stability will influence the fluidity in the following way:
0 = On o e a,) exp (Ct + ct (fo2 2 f))] (7)
where
On = steady state or ultimate fluidity
Oo = initial fluidity
to = initial stress
f = stress at any time (t)
The coefficient of structural stability contains two components:
1. Cf represents the extent of influence of stress change on the structural
stability.
2. ct is independent of stress change and represents the natural influences,
such as hydration, on the structural stability.
Substituting equation (7) into equation (5a), the strain may now be
written
C =+ 9, + (Bo0) exp ( (ct + cf (fo2 2)}) ft + 1,e ee(8)
This equation represents a general function for creep following the mechanical
model of Figure 9. It becomes necessary to evaluate the coefficients an
parameters of the second and third elements in Figure 9. The parameters appear
38
E ELASTIC. STRAIN
PERMANENT INFLUENCES
ON STRAIN
RETARDED
Ee, @e ELASTIC ELEMENT
PROPOSED RHEOLOGICAL MODEL TO REPRESENT CREEP
IN CONCRETE
FIGURE. 9
39
in the form of fluidity 0 and elasticity E in equation (8). The coefficients
are eg and cf.
(b) The Rheological Parameters in the Creep Equation
The proposed model contains structural coefficients and theological
parameters. Define creep compliance as creep per unit, stress. Employing
the linear viscoelastic model for creep (Figure 1), the creep compliance
for each specimen under sustained stress is obtained and the rheological
parameters evaluated. Table III presents the parameters for equation (2)
and~ the linear viscoelastic model. The resulting creep compliance (cg )
t3 1 *nEnt
Cg 2 (1 e )(9)
hom En
n=1
The first term in the creep compliance, containing the steady state
(ultimate) viscosity of. the concrete may be used directly to obtain the
ultimate fluidity (On ) required for equation (7).
1
0 n*" "o (10)
The remaining Kelvin elements form the basis for determination of the
remaining parameters in equation (8). Since there will be several (usually
three are sufficient) Kelvin elements in the creep compliance, it is
reasonable to assume that one of these elements may be used as the single
linear element comprising the elastic term of equation (8). It may not be
obvious, however, which element represents the delayed elastic component,
Therefore all Kelvin terms should be investigated by the rate process theory.
Initial fluidity (00) may be determined when it is known which
theological elements contribute to the inelastic portion of the creep.
g(106) psi day 148.54 249.49 447.07 111.27 134.50 121.29
TABLE III
COEFFICIENTS FOR THE LINEAR VISCOELASTIC MDDEL OF FIGURE. 1
Data From the Control Specimen, Series A
'Circle
914
361
Rectangle ~
914
341
Crose
914
329
914
179
Stress i psi
Initial (106) infin
Strain &
Ralte g (10 6) in/in/day
E (10 ) psi
E1 (10 )pi
E2 (106) psi
Ez (106) psi
km(106) pai day
.X1(106) psi day
X2(106 pai day
2.533
5.071
3.809
7.160
32,311.
1005.4
218.85
2.682
3.581
3.438
11.612
20,961.
726.86
203.16
2.779
2.683
3.245
20.275
14,431.
557.36
198.98
4.665
9.022
5.321
5.394
61,679.
1763.2
303.04
5.108
4.905
3.609
6.491
31,474.
970.68
206.80
4.787
4.105
3.091
5.843
26,134.
814,09
177.64
Lig~htweight (Solite) Concrete
Lime~stone Concrete
Circle Rectangle Cross
r
.02830 .04362 .06336 .01483 .02905 .03499
I
I
That is, after it has been determined which rheological element will be
used to represent the delayed elastic component, the remaining Kelvin
terms of the creep compliance will be used to determine go. Tesri
rate function for two or more Kelvin elements may be obtained from the
creep compliance by differentiating with respect to time, thus for Kelvin
elements 1 and 3
1 E' 13E3t
eg=gl( *e ) 3(1 ) (9a)
eg*01eg1E t +P 0 e gEtL (9b)
At t = 0, ck, o o, by definition, 'therefore
o, P (1 + 3) (11)
C. The Analysis of Rheological Assemblies from a Statistical Mechanics Viewpoint
In the previous section it was proposed that a rate process study of
each Kelvin element in the creep compliance be made. Such a study will
provide an insight into the stress dependent: nature of each element considered
as a theological assembly. Each assembly contains the characteristics of a
molecular structure having energy levels representing the bonds between
similar assemblies and apacings between equilibrium positions of the assemblies
which regulate the rate at which activation occurs. The rate process equations
presented in Chapter IV are therefore applicable for this theological structure.
For the elastic condition there is expected a uniform statistical pacing
between equilibrium positions (Figure 2). This aeaumption is based on the
premise that for any concrete the molecular structure is randomly oriented
and elasticity implies a certain rigidity of the crystalline structure,
therefore the material would be expected to exhibit uniform elastic
behavior throughout the specimen. By the same argument the free energy
of activation should also be uniform throughout the specimen. In case
where no specimen variation iB introduced to form comparisons, such as
was done in this experiment with regard to~ ehape, it may be difficult to
separate the elastically retarded element from the other elements. In
some cases it may be assumed that the intermediate activation energy is
related to the elastic element. This may not always be true, however,
where severe elastic deterioration is present. In general, however,
elastic deterioration will require the largest energies and viscous flow
will require the smallest energies of activation leaving intermediate
levels of activation energy for assignment to elastic characteristics.
The analysis of the Kelvin elements employing the statistical
mechanics approach is derived generally from Herrin's work and
Majidsadeh's work with asphalta. The application of statistical
mechanics to portland cement concretes is straight forward. The theological
equation for the Kelvin elements is known. For each element the strain is
5 = En (1 e ) .(12)
It is required to obtain an expreeston of the element behavior in terms
of the rate theory. According to the rate theory the strain rate, e
is a function of stress.
to A sinh B fD (13)
where fD is the decaying stress resulting from activation of theological
assemblies into new equilibrium positions. For each Ke~lvin element we have
f I fS + fD
and
E' ES D B (15)
The total stress and strain for an element is I and a respectively. The
subscripts S and D refer to the spring and dashpot components of each
Kelvin element. From equations (14) and (15) ve may obtain
fD = f E En (16)
Differentiating the strain with respect to time for the Kelvin elements
represented by equation (12) yields
g, = fe "nn (17)
Using calculated values from equations (16) and (17), equation (13) may
now be solved by simultaneous application of least squares and successive
approximations for the coefficients A and B. Herrin' has prepared a computer
program for this solution which was employed in this investigation.
In avaluating these data it was considered that: the Kelvin elements
represented four distinct effects of the creep behavior :
1. an elastic response
2.. a viscous response
3. a deterioration or separation of elastic bonds
4. a structural growth of strength and viscosity due to hydration shd aging.
Since these responses represent behavior of the entire concrete mass rather
than responses of the separate phases comprising the concrete, it: may be
assumed that the free energy of activation and the equilibrium spacing are
uniform for each theological assembly (representing each Kelvin element)
which. is taken as the basic assembly or flow unit: in the rate process,
Table IV contains the computed values for the coefficients in the
rate function for each type of theological assembly in the specimen under
sustained stress. The method used to determine these coefficients is
described and illustrated in Appendix C. Some immediate relationships
are evident, and are described in the following.
(a) Spacing of Equilibrium States
Within each concrete type the spacings between states of equilibrium
are relatively independent of shape for each theological assembly.
Furthermore in this investigation, it appears that these spacings are
also independent of type of concrete. Since only the coarse aggregate type
and watercement (w/c) ratio differed between these two concretes it seems
that aggregate and w/c ratio would have a compensating influence on the
spacing of equilibrium states. A small influence is evident for theological
assembly number one, which represents the long term deterioration effects,
wherein the less dense structure of the lightweight aggregate yields a
slightly greater spacing than in the denser limestone concrete. In like
manner, theological assembly number three, representing short term
viscous action, indicates lightly greater spacing for the limestone
concrete which may be attributed to the greater w/c ratio in the limestone
concrete.
The effects of these two factor, w/c ratio and aggregate strength,
also compensated each other from a structural standpoint to yield nearly
equal concrete strength which has been considered by some authors to be
an important factor in creep,12 The major factor in determining the
equilibrium spacing of theological assemblies appears to be related to the
concrete structure as measured physically by strength but it appears that
Computed Structure Coefficients Computed Structure
for = A sinh Bf Coefficients
Rheological at 1000 days at 20 days
Concrete Shape Assembly A BAB
Type (Corresponds to 66 3l~/, lla l~ls
Kelvin elements) x106dy x03pi x0 /day x0/s
6 3 6
TABLE IV
COMPUTED STRUCTURE COEFFICIENTS IN THE RATE FUNCTION
For CDnrEo1 Specimen (Series A)
2.4 x 10
23.0
95.0
4.4
25.6
82.5
5.2
28.0
97.6
.12 x 10
.14
.087
.12
.13
.087
.12
.14
.087
.12
.14
.109
.12
.14
.106
.12
.14
.106
3
4.7 x 10
34.0
100.6
8.5
36.9
85.0
10.1
40.9
94.4
8.2
34.0
61.6
11.4
36.8
37.3
14.8
36.9
21.0
.23 x 10
.18
.090
.23
.18
.090
.23
.18
.084
Limestone
Concrete
Circle
Rectangle
Cross
Circle
Lightweight
(Solite)
Concrete
5.7
.27.3
48.3
7.5
28.0
25.6
.24
.18
.083
.24
.18
.087
Rectangle
Cross
the most important factor governing strength is related to the thermodynamic
bonds between theological assemblies in the mix.
(b) Free Energy of Activation
The most significant influence on the physical behavior of the concrete
is the level of free energy of activation (height of the potential energy
barrier) for each theological assembly. This level determines the extent
of the thermodynamic bonds and the frequency of activation for the
contribution to creep from eachrheological assembly. It can be expected
that deterioration effects .resulting from permanent separation of elastic
bonds will require most energy and that viscous flow would require least:
energy for activation.
1. The second theological assembly.
The second theological assembly representing elastic behavior is only
slightly influenced by shape as it affects hydration during the first twenty
days approximately. After about the first month shape has no effect on the
change in the free energy of activation in the limestone aggregate concrete.
However shape appears to have a slight influence on the change in free
energy of activation in the lightweight concrete (refer to Table V). The
slightly larger average value of the coefficient A from Table IV for the
lightweight concrete is indicative of a slightly greater elastic contribution
than for the normal concrete. Therefore, a greater delayed elastic recovery
may be expected from the lightweight concrete than from the limestone concrete.
In Section D elastic recovery is calculated for the specimens in test series A.
The lightweight specimen display a greater recovery than the limestone concrete
specimen.
Theological Change From
Concrete Assembly Specimen 20 day to 1000 %b Change
rype Number Shape day values
A B A B
Limestone 1 Circle 2,3 +.11 49. +92.
Concrete Rectangle 4.1 +.11 48. +92.
Gross 4.9 +.11 49. +92.
2 Circle 11 +.04 33. +29.
Rectangle 11.3 +.05 31. +38.
Cross 12.9 +.04 32. +29.
3 Circle 5.6 +.003 1. +3.
Rectangle 2.5 +.003 3. +3.
Cross +3.2 .003 +3. 3.
Lightweight 1 Circle 4.0 +.11 49. +92.
(Solite) Rectangle 5.7 +t.12 50. +100.
Concrete Cross 7.3 +.12 49. +100.
2 Circle 9 +t.04 27. +29.
Rectangle 9.5 +.04 26, +29.
Cross 8.9 +.04 24. +29.
3 Circle +17.5 .025 428. 23.
Rectangle +11.0 .023 +30. 22.
Cross t4.6 .02 +22. 18.
TABLE V
CHANGE IN RATE FUNCTION STRUCTURE COEFFICIENTS~
For Control Specimen (Series A)
Between 20 days and 1000 days
For the second theological assembly, representing elastic behavior
of the concrete, the results of the statistical mechanics approach shown
in Tables IV and V indicate the nature of the stability of the elastic
structure. In the limestone concrete the first twenty days produce an
energy level which is influenced by shape. Ebwever, following the first
twenty days shape has no apparent influence on the increase in energy as
seen from Table V. These effects appear to be related to the shrinkage
characteristics of the concrete specimen.
In the lightweight concrete the influence of shape extended beyond
the first twenty days. This effect was probably caused by the moisture
contribution from the coarse aggregate to continued hydration. In spite
of this continuing supply of water in the lightweight aggregate concrete
and a greater quantity of cement, the lightweight concrete was apparently
deficient in moisture to achieve its potential activation energy level.
The limestone concrete gained energy at a greater rate than the lightweight
concrete. The limestone concrete also contained a greater amount of water
in the mix, though not an excessive amount, from which a greater amount
of water was colloidally contained by chemical bond in the gel. It
appears, therefore, that the rate of strength gain may be related to the
quantity of free mix water in the concrete. That is, even though water
may be included in the mix as mechanically contained moisture in the coarse
aggregate, only a portion of this water will become chemically activated
in the hydration process after the water is released from the aggregate.
The remainder, and perhaps the largest portion, will be treated as capillary
moisture, ultimately becoming associated with shrinkage characteristics of
the concrete in moderate to small size specimens. In large specimens a
greater portion of this moisture originating in the coarse aggregate may
become chemically hydrated resulting in strength gain.
If the level of energy for the elastic state of the structure is
considered to be directly related to the strength, a study of the
difference in energy levels in the two concretes for the circular shsaed
specimens would be informative. The circular shaped specimens were
identical in cross section to the strength test cylinders. The limestone
aggregate specimen showed a larger energy by about 10 per cent from Table
IV, The limestone aggregate specimen indicated a greater strength by about
12 per cent from strength tests. These results appear to represeent
reasonably acceptable trends but not enough data are presented to justify
any hypothesis relating strength with activation energy level of the
selected elastic theological assembly.
2. The third theological assembly.
The third theological assembly represents short time deformation and
exhibits a decidedly different: behavior for the normal limestone concrete
than for the lightweight concrete. In the former, the rate coefficient A
are large indicating a small activation energy resulting from the high
watercement ratio leading consequently to greater fluidity. Uniformity
of the activation energy levels for each shape means that the fluidity
of the system is not influenced by the escape of free capillary water
resulting from shrinkage nor is the quantity of water molecules held in
chemical bond influenced by the movement of free water as it evacuates
the concrete.
On the other hand, the lightweight: concrete exhibits a significant
influence of shape on activation energy. For greater flow paths the energy
required for activation decreases implying greater fluidity. Since it was
seen that free water loss from the cement ge1 does not influence fluidity
or the free energy level, it is evident that moisture discharged from the
lightweight aggregate contributes to this variation. Evidently, as
moisture is discharged from the coarse lightweight aggregate as free water
and moves toward the surface of the concrete, portions of this free water
are chemically absorbed alonB the way. With longer flow paths, greater
quantities of water are chemically absorbed, causing greater fluidity to
result thereby producing a lower activation energy. This absorption of
free water would also be expected to occur in the normal concrete,
however, the rate of evacuation of the capillaries is considerably
reduced in the lightweight concrete due to the slow emission of water from
the aggregate. This retarded moisture activity is evident in the shrinkage
curves (Figures 12, 13, and 10. The absorption of water released by the.
aggregate may also be considered to increase the watercement ratio for
the concrete. The significance of this absorption is evident in Table V
from the observed reduction in activation energy by about 30 per cent: in
the lightweight specimen.
The third rheological_assembly is changing during the early life of
the structure. As seen from Table V practically no change occurs in the
limestone concrete after twenty days. This activity means that: a steady
state condition is achieved early in so far as creep associated with flow
is concerned. Viscous activity is extended in the lightweight concrete
due to the reserve moisture supply held in the coarse aggregate.
It appears reasonable to assume that. specimens of larger sizes or
which are maintained in high humidity environments will exhibit this viscous
state through a longer period of time and possibly throughout their life.
51
Under these conditions the simple Kelvin element from which the theological
assembly is developed may be bifurcated according to a period of true
~fluidity and a longer period of quasi fluidity. The former etate represents
the condition between initial pouring of the concrete and setting of the
rigid ge1 structure. The latter state represents fluid action of the
elements still in a colloidal state imposed upon the restrictive nature of
the ge1 structure.
3. The first theological assembly.
The first theological assembly exhibiting long time strain activity
is associated with large values of activation energy which appear to be
equally influenced by shape for both concretes. The more rapid evacuation
of free moisture from capillaries of the gel in the concrete specimen with
greater surface areatovolume ratios results in a reduction of the required
energy of activation due to a reduction in the rate of increase of chemical
bonding forces. This effect is directly related to the amount of hydration
associated with each shape.
The larger values of the coefficient A for the lightweight concrete
represent a lower activation energy which results from weaker bonds and
less hydration. Weaker bonds can be attributed to the ge1 in the lightweight
concrete and to the less dense Solite aggregate. The ge1 may be contributing
to this weakness because of insufficient mix water to hydrate a sufficient
quantity~of cement to fora strong and rigid gel structure. However, the
rate coefficient A also contains a factor L equationn (3c)) related to the
displacement resulting from activation of an assembly, which may be considered
to be a modifying factor for non uniformity of behavior. Stress concentrations
and socalled plastic zonee contributing to permanent deformations result from
52
pilingup of rheological assemblies at internal obstructions that are
present within the ge1 matrix. Such obstructions may be, for example,
dense coarse aggregate particles and reinforcement. The conditions under
which deformations of this type occur require suitably oriented obstructions
in the path of the migrating aeaemblies with sufficiently greater density
and crystalline rigidity to prevent penetration of, or displacement by
the migrating assemblies. This pilingup of assemblies results in
bridging between hard coarse aggregate particles which transfers the stresses
throughout the specimen. In soft aggregate concrete the ge1 matrix is
more uniformly stressed and bridging will not be well developed if it is
developed at all. Consequently, the strains resulting from activation in
the lightweight concrete are greater than in normal concrete where assemblies
which have created fully developed bridging prevent large deformation when
activation occurs in regions between the bridges.
For each of the assemblies discussed it must be remembered that
hydration will influence the level of energy of activation and perhaps
the equilibrium pacing also. Hydration will change the properties of
flow unite and adjust free energy levels causing formation of bonds thereby:
affecting the potential energy barrier with time. Table IV contains
coefficients A and B in the rate function for the first twenty days only.
These values may be compared with corresponding final values at 1000 days
from Table IV for an illustration of the influence of energy changes beyond
the first twenty days on creep rates. When little change occur in the level
of activation energy for a theological element the steady state condition
is said to exist for that element. Achievement of a steady state condition
does not preclude further variations, however, if conditions in the concrete
structure are changed so as to cause a variation in the quantum
energy level. For example, the lightweight concrete experienced changes
in energy due to a moisture transfer from the coarse aggregate to the
gel. In this case the increase in fluidity can be ascribed to the
discharge of free internal moisture from the Solite aggregate. If
the moisture had not. been discharged into the gel, a steady state
condition would have been expected to occur sooner.
Rheological assemblies one and two experienced increases in
activation energy beyond twenty days as a consequence of hydration.
Equilibrium spacings in general change proportionately with changes in the
free energy of activation except in the first rheological assembly. The
large increase in this case probably represents the effect of greater
separation of molecules resulting from activation as the material loses
viscous damping capacity and elastic strain energy is converted directly
to kinetic energy upon rupture of the bonds.
D. Application of the Structure Parameters for the Rheological
Assemblies to the Rheological Model
(a) Retarded Elastic Recovery
Development of equation (8) representing the model for the creep
mechanism may now be completed based on the interpretations of the rate
process theory discussed in the preceding sections. Sinch? the first
and third theological elements contained stress and time dependent vari
ations of nonrecoverable creep they are to be included in the second, non
linear term of equation (8). The second theological element satisfies
elastic conditions, and it will therefore comprise the third term of
equation (8).
A check of the elastic influence of this element may be accomplished by
comparing its response with the complete relaxation data for the actual
specimen.
The strain accumulated in the elastic rcheological assemblies may be
computed at any time
e An 3 (f fo) s3 3f~. 'n3) B5 (f5 f n5 (a
s*Ai, 1 1 3 *3 + ***.(8
This equation, derived from the rate process theory for a rheological
element, is developed in Appendix A.
Table VI, comparing results obtained from equation (18) by pooling all
specimens into lightweight and normal concrete categories against actual
test data supports the use of the second Kelvin element to represent
retarded elastic behavior. The calculated elastic recovery should be
related to the activation energy level at the time of removal of stress.
For a greater period under sustained loading the activation energy increases
thus the factor A decreases, thereby reducing the recoverable elastic train.
The approximated final value of the coefficient A io determined by aeeuming
that the 1000 day value from Table IV is the average of the initial value
of A, assumed equal to the 20 day value from Table IV, and the desired
final value.
(b) Total Change in Flpidier of the Concrete
Earlier, initial and final values of the fluidity terms do and On
were determined from a consideration of the basic theological equations.
Fran a rate theory standpoint the fluidity at any time is
0 df = AB coah Bf
Concrete Type Limestone Lightweight (Solite)
Rheological Consteants E 4.0 x 106 pai 3.5 x 106 psi
X 229 x 106 psiday 207 x 106 paiday
Rate Theory Constants A* 13.7 x 106/day 17.7 x 106/day
B .18 x 103/psi .18 x 103/pai
Initial Stress o 914 psi ~ 914 psi
Calculated Delayed 6 12x1 ni
Elastic Recovery 129 x 106 in/in17x10 na
Actual Delayed '614x16 ni
Elastic Recovery 103 x 10 in/in 14x1 ni
Assume: Final A 1 2 A1000 A20 (approximately)
TABLE VI
ELASTPIC RETARDED RECOVERY  COMPARISON OF DATA AND THEORY
Average of All Shapes for Control Specimens (Series A)
of Each Concrete Type
j=1m i (fL fi
It was shown in the previous section that the free energy of activation
increases with time due to the change in the gel structure. Therefore
the values of A and B in the rate process equation are average values
predicting the behavior to any arbitrarily selected time just as the
theological parameters are average values for any applied stress. If we
set f = 0 in equation (19) then
do =AB, (20)
which gives values equal to the results from equation (11) when the average
A and B values at any age are used. however, any value of f greater than
zero will result in greater fluidity than Oo, when employing equation (19).
Therefore it becomes obvious that the fluidity change must include the
true variation in internal energy and equilibrium spacing for anr accurate
account, of the non recoverable strains at any stage of the creep process.
An understanding of the nature of the change in the quantum states of
the concrete is not within the scope of this study. The true value of 0,
should be approximately equal to the value determined from equation (20).
For the statistical mechanics approach which is represented by equation (20),
the values of Oo = AB do not change. when using the data at: twenty days or
at 1000 days Erom Table IV.
Equation (11) provides a decrease in fluidity with respect to time
whereas equation (20) provides an increase in fluidity with respect to stress.
The true fluidity is probably a non linear function of time and stress
requiring a knowledge of the physicochemic~al thermodynamic variations
affecting the quantum states of the maix.
Assuming the ultimate fluidity i, to be established by equation (10) the
total change in fluidity may be approximated by assuming a variation between
the values established by equations (11 or 20) for Bo and equation (10) for On0
57
The total change in fluridity is influenced primarily by the mix
proportions of the concrete and by the ambient conditions of the
environment in which the concrete will be used. For most structural
concretes in any environment the gel structure becomes sufficiently
rigid after a long time that little or no viscous creep occurs even in
the presence of high humidity. Therefore it may be justified to use
zero as a value for CO If this expedient step is taken it only becomes
necessary to evaluate a O from test data obtained from the concrete under
initial loading.
(c) Inelastic Behavior and the Coefficients of Structural Stability
The coefficients of structural stability for equation (7) may be
determined by evaluating' the responses from the control specimens.
~(e+E~ f2 .21
0 = O + (Qo On) e (7)
From a consideration of the behavior of the control specimens (Series A)
for which~ f a fo, a determination of the time dependent component in
the structural stability coefficient may be made by solving for the aging
component, eg. Substituting f = fo into equation (7) yields the solution
for the time dependent component of creep only,
 = e (21)
ao 0
Taking the log of both sides results in a solution for ct
c = l In~ (22)
in which
a = .. (23)
c vis is simply obtained by subtracting shrinkage, initial elastic strains
and delayed elastic strained from the total creep strain for each epeciman.
Time dependent coefficients (et) for the nonrecoverable creep component
are presented in Figures 10 and 11. The coefficient ct is assumed to be
a function of time only, therefore it is entirely independent of stress
variations. Equation (22) is used to obtain the value of ct for the
concrete from experimental data. However equation (22) may not be used
as a general relation for the prediction of creep behavior. The coefficient
ct must include, in addition to the variable time, all of the influencing
factors which contribute to the aging characteristics of creep of concrete.
Hansen and Mattockl6 have evaluated shape effect on creep and presented
a relationship between creep and shape in an exponential form. It~is not
incongruous to assign a similar relationship to other factors which
influence creep. Accordingly the coefficient: ct is assigned an exponential
form
eg: = tn E F() (24)
where Fn(t) is a polynomial function of time. The equations ct are given
in Table VII for each control specimen. The constants in the ct function
are related to the time dependent factors influencing concrete aging,
hydration for example, which cause variation of the energy states in the
concrete independently of stress. The polynomial exponent has been limited
to a cubic function. The, limitation i~s based partly on an acceptable fit
of the function to the experimental data as represented by the error sum
of squares in Table VII. The closeness of fit is indicated in Figurea 10
and 11. The plots were made from a computerized curve fitting program.
The exponent was limited to a cubic function also because it was quite
Concrete Shape of Equation: ct = te es ( (bl + b2t + b3t2 + bgt) Su o
Type Specimen a blg b2 bg bq Squares
TABLE VII
EQUATIONS FOR THE AGING COEFFICIENTS, q ,
OF STRUCTURAL STABILITY FROM COMPUTERIZED LEAST SQUARE FIT
4.524 x 103
4.774 x 10 3
4.820 x 103
4.851 x 106
5.437 x 106
5.493 x 106
5.270 x 106
5.740 x 10 6
6.377 x 106
2.085 x 10
2.429 x 10
2.439 x 10
9
2.604 x 10
3.014 x 10 9
2.038 x 103
2.569 x 103
3.530 x 103
3.414 x 103
5.224 x 10'3
11.880 x 103
3.699
3.762
3.787
3.753
3.988
4.422
Limestone Circle
Concrete
Rectangle
Cross
.971
.965
.969
.959
.965
.993
Lighftweight
(Solite)
Concrete
Circle
Rectangle
Cross
4.676 x
4.831 x
5.006 x
obvious that higher power terms would only tend to influence the creep
at advanced ages and in not a highly significant manner. A reasonably
successful fit may be obtained i~n the early ages by substitution of a
constant for the polynomial function as was done by Hansen and Mattock.16
The complete form of the aging component ct of the structural
stability coefficient is
ct =ta (bl + blt+ bat2 + bq~t3 (24a)
From Table VII the constant a is relatively independent of shape or
concrete mix. Further, it is reasonably close to unity so that setting
a = 1 does not result in a serious error for the concrete specimens
tested. The polynomial exponent is, however, related to shape. By
employing the circular shape as a standard, since it has a weighted flow
path length equal to unity for a six inch diameter specimen, all other
shapes or saiss may be related to the six inch diameter cylinder as a
standard. Redesignating the coefficients bn in the polynomial function
to cn for the six inch diameter cylinder, the coefficients bn may be
approximately determined from
b,= en r 4Uw) eg + (25)
nP1
where v is the weighted flow distance for the concrete specimen from
Table II and K is a constant associated with the concrete mix. K is
equal to six for the lightweight Solite concrete used in this experiment
and K is equal to three for the limestone concrete.
The stress dependent component, cf, in the structural stability
coefficient is obtained from the control specimens employing the rate
coefficients from the rate process theory presented in Table IV. In
Appendix A, the creep for a single theological assembly was determined.
The combined strain for two such assemblies representing the Kelvin
elements one and three from the theological model of Figure 1 may be
written in the form of an infinite series,
El +031 j'2 (A~i h /E)1 + (ABl 1 /E)31 i igi
j=1, (26)
where i = 2j 1.
The component cf is related to stress changes in the concrete
beyond the initial applied stress to. If the stress remains unchanged
then only aging influences the creep rate and the cf component is not
effective. However under any change in stress, increase or decrease, the
creep rate will be influenced according to the effect of the stress on the
energy required for activation of a theological assembly. If a creep test
is performed on a standard cylindrical specimen and the macroscopic creep
data evaluated by viscoelastic methods, the resulting theological solution
for the viscous non Newtonian elements, such as Kelvin elements one and
three in this investigation, may be converted to the form represented
by equation (26). This process results in an evaluation of non Newtonian
concrete strain in terms of stress change and the linear viscoelastic
parameters from the theological study. The accuracy of the strains
determined from equation (26) is limited to the accuracy of the theological
parameters On and En in reflectinB the true concrete behavior. This
limitation on accuracy is, however, directly related to the degree of
work which the analyst is willing to expend in obtaining more accurate
viscoelastic parameters. Surely if methods as described by Freudenthal
and Roll are employed, the greater effort will yield better results.
The important aspect of the relationship presented in equation (26)
is in having a function which is sensitive to changes in stress rather
than time. This relationship makes it possible for the structural analyst
to evaluate the influence of stress changes on creep behavior with greater
precision than was possible from just a theological study.
In order to relate these non Newtonian strains to the component cf
of the structural stability coefficient, let
Ni, gi (AB /E)1 + (A' /E)3
Equating the strains represented by equation (26) from the average states
of the theological assemblies one and three to the stress dependent
creep term in equation (8) for the theological model yields
(M s) ex cy(f2 ) ft = =12 i = 2j (27)
Dividing by stress f and time t, substituting from Appendix A
t = In (A4)
Eo
and taking the log of both sides of equation (27), the stress component of
the structural stability coefficient is determined
2~ ~ ~ ~~~r 2)7 (B) +(1) i i
(Qo Om) 1n f/fo
63
The coefficients ..1, B, Oo and On have also been determined. Therefore
knowing the stress variation, t~he right side of equation (28) may be srubstituted
for the coefficien ci (fo2 2) and the total creep may be calcullaed
from equation (8),
gI [, +( (ct + eg (fo2 fZ L .gea
C .E o ) t+be 1 .8
LIMESTONE AGGREGATE CONCRETE CIRCULAR SHAbPE
4.000 +++
1 1 I I I I
I I I I I I
I I I I I I.rr
I 'I I I ,....... I
I I I ......* I
II I I ..... I I
3.000 +++...
I I I ,......I I I
I I I ..... I I I
I I *... I I I
I I ....: 1 I 1
1 I .*.. I I II
I I .... I I I I
1 I.. I I I I
2.C00 +.* +
1 .. I I I I .I
I I I I I I
I .. 1 I I I .I
I .. I I I I I
I I I I I
I *I I I I I
I I I 1 I I
1.000O +*+ ++ +
1 I I I ( I
I I I I I I
I, *I I I I I
I I I I I I
I* I I I I
I* I I I I I
I I I I I I
0. .++++
0. 200 400 600 800 10CC.000
Time in Days
COMPUTERIZED PLOT OF STRUCTURE STABILITY COEFFICIENT FOR AGING AGAINST TIME, LIMESTONE AGGREGATE CONCRETE
FIGURE 10A
L~SOIMBTN ACGEGATE CONOCRETE RECTANGULARA SHAPE
4.000 +++++
1 1 I I * * *
3.;00 +++ ....*..+
1 1 I I ..... 1 I
I I I 1..... I I
I I [ ....*. I I
1 I I .*... I I
I I *...... 1 i I
I I ..... I I 1
I I e.. I I I
2.000 ++....+++
I I... I I I 1
i ..* I I I I
1 *. I I I I I
I .. I. I I I L
I I I I 1 I
1 .* I I I I I
1 I I I I I I
1.000 + .+ ++++
1 ** I I I 1 I
I I I I I I
I I I I I I
I I I I I I
In 1 I I I
I+ I I I i I
I I I 1 1 I
G. .+ +++C+
C;. 200 400 600 800 LCCC.CCC
Time in Days
COMPUTERIZED PLOT OF STRUCTURE~ STABILITY COEFFICIENT POR AGING AGAINST TIME, LIMESTONE AGGREGATE CONCRETE
FIGlas 10B
*ruuu +'+''I+''+''
I I I I I I
I I I I I I
1 1 I I I I
I I I I I I
'I I I I I I
1 I I I I I
I I I I I . . .
3.000 ++ + +..... .+
1 I I I ..... I
I I I I ..... 1 I
1 I I .. .I I
I I I e.... I I I
I I I .... I I I
I I ..*.. 1 I I
I I .... I I I I
...*+++
II I
I I I I
I I I I
I I I I
I L I I
I I I I
I I II
 ++C +++ +*
+   + +
LINEgS19NE AMBGGRPEGT CONCHeET CROSS SBAPE
2.000 ++.
I.*
**I
*I
1
I
I
I
I
1.0i00 +.
I
I *
I
I *
I*
I
ICCCCCCC
Time in Days
STABILITY COEFFICIENT FOR AGING
FIGUMB 10C
COMPUT~ERIZED PLOT OF STRUCTURE
AGAINST TIME, LIMESTONE AGGREGATE CONCRETE
LIGHERBIGIR SC.ITB COMICRBTB CIRCULAR SHAPE
4.000i ++++ 
I I I I I [
I I i I (
I I ~I I I I
I I I I I I
I I I I I I
I I I I I I
I I I I I .. ..
3.000 +++...
I I I 1 ...... I
I I I I **** I I
I I I ..c.. I I
I I I e.... I I
I I I *** I I I
I I ..*. I
I I e.... I I [
2.000 ++ .....+
I I... I I I
I ** I I I
I ... I I I I I
I ** I I I I I
I 1 I I I I
.1 .* I I I I
I I I I I I
1 000 + . +  +   +   4   +
I `I I 1 I I
I *I I. I : I
I I I I I I
I I I I I I
I* I I [ 1 I
I, I I I I I
I I I I I I
G.  +, + +   4.. ...... ,..4
o* 200 400 6 00 800 1000.CCC
Time in Days
COMPUTERIZED PLOT OF STRUCTURE STABILITY COEFFICIENT FOR AGING AGAINST TIME, LIGHTWEIGHT AGGREGATE CONCRETE
FIGURE 11A
LIG~IlliIGHTI SOLIT CONCREBT RECTANGULAR SilAP
4. 000 ++   +  +  + +
I I I [ I I
I I I I I
I I I I I I
I I I I I [
.1 I I II I
I I I I I I
1 I I I I I
3.000 +++ +++
1 I I I
1 I I I I
I I I I I..,...........
S I 1 I .. I
1 I I I .~... 1 I
1 I I e. I
I I I .~....1 I I
2.0i00 +++ .....+++
I I: .*... I I I
I~ I ..... I I [
I I ....* I I I I
I .... I I I I
I ...*r I I I
I ** I I. I I I
1 I I r
1.000 +.* + + +  + +
I I I I I I
I I I I
I I I 1 I I
I I I I I I
I I I I r I
I* I I I1 I I
Is I I 1. I r
a. .+ + + +
0. 20 0 00 0 000.000
Time in Days
COMPUTERIZED PLOT OF STRUCTURE STABILITY COEFFICIENT FOR AGING AGAINST TIME, LIGHTIWEIGHIT AGGREGATE CONCRETE
FIGURE 11B
+ + t  *  *  +  +
r ~ ~    + ~ +~+
I.IGHEMTBICE SOfITE CONCRETB CROSS SIAPE
4.000
3.000 +++
I I I I I I
1 1 1 1 1. I
I I I I I If
1 I I I I I
1 I' I I I I
I I I I I . . .
I I I I ....... I
++++...*+
1 II I ... 1 I
r?2.L000
h
.,... I
.I
*
I
1
**...
.....
...*..I
I
1
I .
I
I .... I I I I:
1.000 +.. .*+++
1 ..* I I I I
I .*. I I I I I
.*I
I:
I
I
I
.1
I **
I **
II
1*
o. 200 400 6001 80(0 1005.2100
Time in Days
COMPUTERIZED PLOT GP STRUCTURE STABILITY COEFFICIENT FOR AGING AGAINST TIMe, LIGHTWEIGHT AGGREGATE CONCRETE
FIGURE 11C
1000
1 00
9 100 2( 00 4 O .6* 700 0 0 C* t
TIEINDY
COPRIO OFSRNAEFRLIETN OCEEAN OIECNRT
FIGURE 12
20 / B SOIT AGRGT a
Y QIMETIME IN DAYSGAT
COMPARISON OF SHRINKAGE FOR LIMESTONE CONCRETE AND SOLITEF CONCRETE
FIGURE 13
U00
 a g
40 a '
3 00
1 00
KO 24 30 40 0 0 70 80 90 00 10
ZE I P I I I ~IM IN DAYS SAPDSPC:IE
COPRSNO HIKG O LMSONE CSONCREE COEANDSLT CNRT
FIGUE 14 AY
1000
UUU g "" "SERIES A
CIRCULhR SHAPED SPECIMEN 
SUBTAINED STRESS 914 P81
STRESSED ACT 5 DAYS
sQ an I I I LIMESTONE AGOOREOATE
W 80sLRE AGGREGATE
900
0 100 200 300 400 800 600 70 800 900r 1000 lo Elo
TIME IN DAYS
COMPARISON OF TOTAL STRAIN FOR LIMESTONE CONCRETE AND SOLITE CONCRETE
FIGURE 15
I
.
~f ~ I
 .
) I I I
1800
14000
400
~ oo
SERIES A I I,
RECTANGULAR SHAPED SPEOIMENS
SUISTAINED STRESS 914 P81
STRESSED AT S DAYS
LIMESTONE AGGRE8ATE
I50LITE AGGREGATE I
300 400
800
700 800 900 000 1100
TIME IN DAYS
COMPARISON OF TOTAL STRAIN FOR LIMESTONE CONCRETE AND SOLITE CONCRETE
FIGURE 16
18000
1800
SERIES A
GROSS SHAPED SPECIMENS
SUSTAINED STRESS 914 PSI
STRESSED AT 5 DALYS
O LIMESTONE AOGREOATE
S 8UTE AGOREGATE
O L..A
o loo
soo soo looo
TIME IN DAYS
COMPARISON OF TOTAL STRAIN FOR LIMESTONE CONCRETE AND SOLITE CONCRETE
FIGURE 17
ANALYSIS OF RESULTS
A. Application of the Model to Concrete Under Sustained Stress (Test Series A)
The model is designed from the response of test series A in which the
concrete specimen were maintained under sustained stress for over 1000 days.
Hence it is expected that the model will represent the behavior well.
Figure 18 confirms the comparison between creep data and model for each
condition of specimen shape and concrete mix.
The creep data for Figure 18 were obtained by subtracting the
corresponding shrinkage from Figures 12, 13, or 14 from the total strain
for the specimen shown in Figures 15, 16, or 17, with allowance made for
shrinkage occurring up to the time of stressing. There is a prominent
seasonal effect on shrinkage and total strain which becomes inconspicuous
in the carryover to the creep data. The extent of the seasonal effects
in the creep data when compared with the effects in shrinkage is a measure
of the interaction between creep and shrinkage of concrete.
The model which has been developed is in the general form represented
by equation (8).
f +[r i(, (c + Cc (fo2 2 B, e~
The elastic components are not considered to be influenced by shape.
Therefore E, Os, E, are the theological parameters determined directly
from the viscoelastic analysis of the data from the control specimen.
16
CHAPTER VII
The inelastic component is a function of shape. The four paramete~rs
involved are On o,, ct and cf. Each parameter appears to be influenced
by shape or flow path. The value for On may be obtained from equation (10)
employing the results in Table III.
c1.85 (We w)
tO, = n e(29)
where On~ is the ultimate fluidity for the standard control specimen, the
six inch diameter cylinder, and w is the weighted flow distance from Table II.
for the specimen under consideration. Equation (29) is applicable to either
concrete. When the standard eix inch diameter cylinder is used, No = 1.0
and equation (29) becomes
c 1.85 (1 v)
Onm = 0 e (29a)
The value for Oo is obtained from equations (11) or (20). For the
specimens tested, employing values from Table III in equation (11) results
in no noticeable influence of shape on Oo for the limestone concrete.
However, the lightweight Solite concrete exhibited a variation in Oo
influenced by shape. The reason for this difference in behavior may be
attributed to the proximity of the' moisture laden Solite aggregate to the
~concrete surface. The apparent difference in viscosity is caused by
differences in moisture loss from the aggregate. employing equation (11)
from page 41,
Oo 3)c 1.45 M (1 v) ,(0
where the subscript c refers to the values for a standard control specimen,
a six inch diameter cylinder in this case.
78
M = 0 for standard limestone concrete
Mi = 1 for Solite lightweight concrete as used in this experiment.
A determination of the variation of M is not: within the scope of this
experiment .
The structural stability coefficients ct and cf have been fully
described in the previous chapter. The aging component ct was defined
by equations (24a) and (25). For the concrete in this investigation the
value of ct was obtained by omitting the power of t in equation (24a).
et=t e(b1+ bZt + b~t2+ bqt3) (31)
w here bn = Cn~,)",~1 .n + 1 .(25 )
The values of Cn are determined by fitting equation (31), with Cn replacing
bn, to the creep data 'from a standard control specimen made from the
concrete under investigation. The values for ct are determined for any
time t by use of equation (22). At least ten time intervals should be used
in obtaining t~hF values of C Valule~s for bn for any other shape or size
of specimen, are obtained from equation (25) where K = 3 (1 + M)
K = 3 for limestone aggregate concrete
K i 6 for Solite aggregate. concrete.
The value for the structural stability component eg is determined by
substitution defined by equation (28). Values for 0, and On are obtained
as described earlier. Only the rate parameter A is dependent upon flow path
or shape (see Table IV). Determination of the values for the parameter A
to be used in equation (28) may be approximated from values obtained for a
standard control specimen of six inch diameter for each inelastic theological
assembly unit. For the first theological assembly
Al = A1C (1K1)1w (32)
where Alc is the calculated value of A for the first theological assembly
of the cylindrical control specimen. K and v have been previously defined.
The value for the third theological assembly is
A3 = A5c 2.5 M (1 w) ,(33)
where M has been previously defined.
All parameters may be obtained from a single sustained creep test on
a standard control specimen. However, in each of the relations above it is
not possible to cross from one concrete mix to another. Therefore the test
must be made on the type mix to be used in the prototype structure. The
limited ecope of this experiment precludes development of a model which will
include mix variations and environmental variations in its prediction of
creep behavior.
An analysis of the model (Figure 9) reveals certain characteristics
of its behavior. Considering the first and third terms of equation (8) it
is evident that these components of the model reflect no variations in the
molecular structure. The variations due to irrecoverable viscous flow,
destruction of bonds, and growth of new bonds are therefore all contained
in the second term of equation (8). In the case of concrete under sustained
stress these factors are all time dependent and the magnitudes of their
influence on inelastic creep is directly proportional to the sustained
stress level.
Proportionality of creep and stress is limited to low stress ranges.
When the applied stress exceeds about 30 per cent of the twentyeight day
strength, fe, the creep rate increases and proportionality between creep
and stress ceases. Frendenthal and Roll have determined that only the
inelastic components of creep are not proportional to stress. In the
model represented by equation (8) therefore only the second term representing
inelastic creep components would be non linear with respect to stress for
high stress levels. This investigation did not include a study of stress
level. Freudenthal and Roll4 investigated stress level on different mizes
producing concretes varying from very viscous to very fluid in nature.
Employing their results equation (8) may be modified to include consid
eration of non proportional creep response under high stress conditions.
Since the second term of equation (8) is linear with respect to stress
level any creep component obtained from it at any time would be the linear
creep component only. In order to obtain the true creep the linear creep
component must be multiplied by a factor which produces the non linear
component of creep. The non linear component is related to the viscosity
of the mix, A more viscous mix will yield a smaller non linear creep
component. Since Freudeathal and Roll's results represented the extreme
conditions of viscosity it is not known how the non linear creep component
varies with viscosity. Therefore it is assumed that the non linear creep
component is inversely proportional to the initial viscosity of the mix at
the time the stress is applied.
In corporating non proportional relations between creep and stress into
the model, equation (8) becomes
S4001 / SERIES A
U ~ i CIRCULAR SHAPED SPECIMEN
Too LIMESTONE AGGREGATE
SUSTAINED STRESS 914 PSI
100 STRESSED AT 3.DAVS
S1 0 2a 0 as 0 4* 0 500 600 700 800 900 1t 00
TIME IN DAYS
COMPARISON OF RHEOLOGICAL MODEL AND EXPERIMENTAL DATA
FIGURE 18 sA
Equatiorl (8)
SERIES A
RECTANGULAR SHAPED SPECIMN
Ten UME5TONlE AGGREGATE
SUSAINIED STRESS 914 PSI
STRESSED. AT 3 DAYS
1 (0 2< 0 39 0 40 0 54 0 600 7100 8100 9<40 10 00
TIME IN DAYS
OF RHEOLOGICAL MODEL AND EXPERIMENTAL DATA
FIGURE 18 B
COMPARISON
cn
I
c
Z
O
I
v
ur
ul
ar
V
IEquation (8)
ann SERIES A
CROSS SHAPED SPECIMEN
300  UMESTONE AGGREGATE
SUSTAINED STRESS 914 PSI
cTRoSEDs AT 3 DAYS
O
07 0 800 900
EXPERIMENTAL DATA
3uv
0
100 2 0 3 0 **09 5=0 600
TIME IN DAYS
COMPARISON OF RHEOLOGICAL MODEL AND
FIGURE 18 C
Y
O
j
U
4 9 Equationl (8)
ann SERIES A
CIRfCULAR.SHAPED SPECIMEN
Too SOLITE AGGREGATE
SUSTAINED SRE~SS 914 PSI .
100~STRESSED AT 3 DAYS
1 0 > 0 3* 0 4 0 54 0 600
700
800
S1 30
TIME IN DAYS
COMPARISON OF RHEOLOGICAL MODEL AND EXPERIMENTAL DATA
FIGURE 18 D
2
O
z
**
U
ann 4Equationl j8)
RECTANGULAR. SHAPED SPECIMEN
100 SOLITE AGGREGATE
SUSTAINED STRESS 914 PSI
101 1 n STRESSED AT 3.DAYS
07 0 800
30V
e S 0 600
TIME IN DAYS
COMPARISON OF RHEOLOGICAL MODEL AND EXPERIMENTAL DATA
FIGURE 18 E
Z
m
m
g Equation (8)
400 SERIES A
CROSS SHAPED SPECIMEN
900 ~SOLITE AGGREGATE
SUSTAINED STRESS 914 P51
TO@ STRESSED AT 3 DAYS
06 0 700 800 900 9 3
TIME IN DAYS
COMPARISON OF RHEOLOGICAL MODEL AND EXPERIMENTAL DATA
PIGUREB 18 F
87
LP f + m+ ((00m) e(ct~cy (foZ~2 2)]( (1p) ft + Ee1f ~ e (a
where p = (2.5 + lo x 10 ) (Ofl .30)
The ratio f/fe represents the ratio of stress level to strength for concrete
being evaluated. .It is tacitly assumed here that the stress to strength
ratio at which proportionality ceases is about 0.30. The' value for p may
not be negative. Therefore, the value for p is set equal to zero for all
stress levels resulting in values for f/f' less than 0.30.
While grouping all inelastic effects in the second term is mathematically
convenient there are certain disadvantages with respect to the activity of
the third term. When evaluating the delayed elastic recovery upon unloading
the concrete the elastic state, or more precisely the quantum etate of the
molecular structure at the time of unloading determines the amount of
recovery. The constant coefficients of the third term are unable to
satisfactorily predict this variation. This deficiency may also affect ::
recovery values for concrete under decreasing stress. Further investigation
is required to fully understand the manner in which the elastic response
coefficients vary.
B. Application of the Model to Concrete Under Decreasing Stress (Test Series
The model's coefficients were derived from the response of the specimen
held under sustained stress for convenience. The application of the model
to the specimens of test series B (stress decrease of 65 per cent) and C
(stress decrease of 50 per cent) is therefore independent of the test results
for these specimens. Figure 19 compares the model with creep test: data for
specimens of test series B. The model agrees with the data.
88
Figure 20 compared the model with creep test data for specimens of
test series C. The agreement is not as good as for test series B. The
apparent cause of the discrepency seems to be related to the delayed
elastic element (third term) which recovers a greater amount of strain than
it should. This effect was described earlier in connection with delayed
elastic recovery of the specimen of series A. Not only was the average
value of the structure coefficient A used in connection with elastic
recovery, but the coefficient A was also declared independent of time in
the derivation of the stress dependent factor for the stability coefficient.
These two assumptions, based on an insufficient knowledge of the
thermodynamic properties of the material with time, lead to low predicted
values of concrete strain. If it were known how the coefficient A
decreases with time and mix then a smaller recovery could be predicted
and accuracy improved. This effect only influences creep predictions at
advanced ages. The model appears to adequately predict, creep in the early
ages under stress. The early influence of fluidity on creep is controlled
to a great extent by the third Kelvin element of the elementary model of
Figure 1. This element isnot greatly influenced by time beyond 20 days
(Table V). Therefore the assumption of constancy of thermodynamic
coefficients A for this theological assembly does not introduce serious error.
The response of the model for test: series B agrees better than the
response for test series C because the error caused by the discrepency in
the third term of equation (8) is proportional to stress. Since series C
had a larger average stress the deviation of the model from the data will
be greater.
900 SERIES B
CIRCULAR SPECIMEN
800 .LIMESTONE AGGREGATE
INITIAL STRESS 914 PSI
STRESSED AT 3 DriS
700
01 0
> 0 3 0
O1 0
TIME IN DAYS
COMPARISON OF RHEOLOGICAL MODEL AND EXPERIMENTAL DATA
FIGURE 19 AL
EXPERIMENlTAL DATA
900 SERIES B
RECTANGULAR SHAPED SPECIMEN
444 LIMESTONE AGGREGATE
INITIAL.STRESS 914 PSI
STRESSED AT 3 DAYS
Eqain(8)
01 00
t oW 10
TIME IN DAYS
COMPARISON OF RHEOLOGICAL MODEL AND
PICOBB 19 B)
