Group Title: study of creep in lightweight and conventional concretes
Title: A Study of creep in lightweight and conventional concretes
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Title: A Study of creep in lightweight and conventional concretes
Physical Description: x, 122 leaves. : illus. ; 28 cm.
Language: English
Creator: Cassaro, Michael A., 1931-
Publication Date: 1967
Copyright Date: 1967
 Subjects
Subject: Concrete -- Creep   ( lcsh )
Civil Engineering thesis Ph. D
Dissertations, Academic -- Civil Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 121-122.
General Note: Manuscript copy.
General Note: Vita.
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000568528
oclc - 13676255
notis - ACZ5264

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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

DR-5025 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 .. .. 64-66

11. COMPUTERIZED LOTS OF STRUCTURE STABILITY COEFFICIENT FDR
AGING AGAINST TIME, LIGHTWEIGHT AGGREGATE CONCRETE .. .. 67-69

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) . . ... ... 81-86

19. COMPARISON OF RHEOLOGICAL MODEL AND EXPERIMENTAL DATA
(SPECIMEN, TEST SERIES B) .. .. ... .. .. .. 89-94

20. COMPARISON OF RHBOLOGICAL MODEL AND EXPERIMENTAL DATA
(SPECIMEN, TEST SERIES C) . ... .. .. . .... 95-100







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 pai-days

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 10-16 arg deg-1

h = Planck's constant 6.624 x 10-2 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 non-homageneous, 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 water-cement 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 water-cement ratios for the two mixes could not be so easily

equated. No problem resulted from the selection of a given water-cement

ratio for the limestone concrete. However, a water-cement 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 crose-sectional 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 stone-like 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 colloidal-liquid 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 liquid-colloidal 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 non-linear 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 vice-versa. 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, Hi-Early):

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


Lighewa-igpht (Solite) Concrete

Cement (Type III, Ri-Early):. 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

volume-to-surface area ratios (Table II). The shapes selected, which

had cross, rectangular, and circular cross-sections, 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 cross-section 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- to-crose-section 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 Cross-sectional 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 SR-4 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 (fo-2 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 (10-6) 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*01e-g1E 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 water-cement (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~/, l-la l~ls
Kelvin elements) x10-6dy 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 each-rheological 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

water-cement 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 water-cement 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 area-to-volume 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 for-a 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 so-called plastic zonee contributing to permanent deformations result from






-52-


piling-up 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 piling-up 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 non-recoverable 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 psi-day 207 x 106 pai-day

Rate Theory Constants A* 13.7 x 10-6/day 17.7 x 10-6/day
B .18 x 10-3/psi .18 x 10-3/pai

Initial Stress o 914 psi ~ -914 psi

Calculated Delayed -6 12x1- ni
Elastic Recovery 129 x 106 in/in17x10 na

Actual Delayed '614x1-6 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 physico-chemic~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 non-recoverable 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 10-3

4.774 x 10 3

4.820 x 10-3


-4.851 x 10-6

-5.437 x 10-6

-5.493 x 10-6


-5.270 x 10-6

-5.740 x 10 6

-6.377 x 10-6


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 10-3

2.569 x 10-3

3.530 x 103


3.414 x 10-3

5.224 x 10'3

11.880 x 10-3


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 (A-4)
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 carry-over 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.

c-1.85 (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 (1-K1)1-w (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 twenty-eight 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+ ((0-0m) e(ct~cy (foZ~2 2)]( (1p) ft + Ee1-f ~ 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 is-not 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)




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