STUDY OF CE
VISCOELASTIC
E'iN T PASTE
by
SB,
JOSEPH NEMC,
Massachusetts Institute
SM,
Massachusetts
JR.
of Technolcgy
(1963)
Institute
of Technolog,
(1964)
Submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
at the
Massachusetts
Institute
of Technology
1967
Signature of Author. .
..
.
. .
.
Department of Civil Engineering , June 24,
Certified by
.
.
.
.
.
.
.
.
.
..
1D67
v
Thesis Super-visor
Accep te(d by
. .
.
. .
Chairman,
Departmen
.. ........
. . .
. .
. .
. . .
.
al Committee
Students
tn c- - on
o Grdu.ate
=
t
St'
ABSTRACT
VISCOELASTTC
"'DY
OF CEMENT PASTE
by
JOSEPH NEMEC, JR.
Submitted to the Department of Civil Engineering on June 24,
1967, in partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
Creep of concrete presents serious problems to the
structural designer. The currently available design techniques are mostly based upon empirical formulae consisting
of additional factors of safety. The major portion of the
creep of concrete is reported to be due to the creep of
cement paste. It is thus the purpose of this thesis to
study the applicability of the theories of linear viscoelasticity to the time dependent mechanical response of cement paste. This is expected to be the first step in the
development of a time dependent composite model of concrete.
Specimens of cement paste of water cement ratio of 1/3
are subjected to creep and relaxation in uniaxial compression tests, to creep in triaxial compression, and to flexural creep tests.
The results of this study indicate that cement paste
specimens tested under conditions of constant moisture, exhibit a linear viscoelastic response within the range of
stresses and strains used in the study.
To verify the applicability of the principle of superposition and the interchangibility of the data within the
context of the laws of viscoelasticity, the results of creep
tests are transformed to the Laplace domain and compared
with the values of the results of the stress relaxation
tests transformed into the Laplace domain.
The characterization is utilized in a simple problem of
viscoelastic deflection analysis using a simple beam made
-2-
The predicted deflections agree with the
of cement paste.
experimental data using the characterization obtained on
beams.
The creep behavior of larger size specimens is investigated.
The results are similar to that of creep of smaller
specimens. A limited number of long range tests are performed in axial creep, relaxation, and flexure creep. The
behavior of cement paste under long range creep is found to
be similar
to that
of shorter
duration tests.
Thesis Supervisor:
Title:
Fred NMoavenzadeh
Associate Professor of Civil Engineering
ACKNOWL EDG
NT S
The author wishes to express his gratitude to Professor
Fred Moavenzadeh, his thesis advisor, for his valuable advice and suggestions during the preparation of this thesis.
The author wishes to thank Professor F. J. McGarry,
Division Head, for his encouragement during the work of
this thesis.
The author wishes to acknowledge the financial support
of the Materials Division of the Department of Civil Engineering, the Ford Foundation, the Province of Quebec, and
the Owens Corning Fiberglas Corporation.
The author is very thankful to Miss Caroline G. Whitney
for the typing of this thesis.
-4-
CONTENTS
. .
Title Page
Abstract
. .
.
Acknowledgment .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
Table of Contents .
Chapter I.
& 0
P0_9
Creep of Concrete . . . . .
.
.
.a
.0
.
.. . . . . . . . .
Concrete as a Composite
Viscoelasticity . . . . . . . . . . . . . . .
15
16
Objective and Scope
Chapter IV.
Chapter V.
Numerical Methods . .
.
0
.
1
.
.
.
Experimental Results . .
. .
. .
.
.
. ...
36
Flexure Cree .........
. . . . . .. . . .
Creep of 4" x 8" Cylinders. . . . . . . .
Properties.
. . . . . .
. . . . . .
..........
Discussions of Experimental
.
Flexure Creep and Its Prediction. . . . .
Creep of 4" x 8" Cylinders. . ..
........
Chapter IX.
.
.
.
.
.
.
.
. . .
Recommendations for Future Work.
-5-
.
.
.
36
36
38
39
41
42
44
Results
Characterization . . . .
............
Interrelation of Functional Representation
of Cement Paste . . . . . . . . . . . . .
Conclusions. .
25
27
27
28
31
E.
F.
Elastic
. . . .
. . . .
. .
22
. . .
.
. . . . . .
. . . . . .
. . ..
Strength and Apparent
Stress Relaxation . .
Axial Creep Tests . .
Triaxial Creep Tests .
Chapter VIII.
.
25
A.
B.
C.
D.
C.
D.
.
.
Functional Representation . . . .
Linear Viscoelastic Behavior.
Laplace Transformation.
. .
..
.....
Ramp Transformation
Correspondence Principle.
..
Inverse Laplace Transformation .
Chapter VII.
9
20
Materials and Procedures
Chapter VI.
A.
B.
5
0
Review of Literature 0 a 0 0 0
Chapter III.
A.
B.
C.
D.
E.
F.
. . .
Introduction......
Chapter II.
A.
B.
C.
.
..
. .
2
45
.
50
50
54
.
55
.
57
CONTENTS
(Continued)
Page
Figures
.
. . . .
Bibliography .
. .
.
. . .
. .
. . .
.
. . .
. . .
. .
. .
. . .
. .
.
59
. .
. . .
. .
.
140
Appendix A.
Chemical Analysis of Cement ......
146
Appendix B.
Computer Programs .
147
.
. . . .
.
B-I.
Ramp Transformation . . . . . . .
B-II. Collocation. . . . . . . . . . .
B-III. Laplace Transformation. . ... .
B-IV. Inverse Laplace Transformation .
Appendix C.
147
151
154
157
.
161
C-I. Creep Tests . . . . . . . . . . .
C-II. Relaxation Tests . . . . . . . .
C-III. Flexure Tests . . . . . . . . .
161
164
167
Long Range Tests. . . . . . . . . . .
-6-
CHAPTER I.
INTRODUCTION
The existence of time dependent deformations of portland
cement presents serious problems to the structural engineer.
The time dependent deformations due to sustained loads are
known as creep.
as:
The manifestations of creep are common such
the sagging of reinforced concrete floor beams and the
loss of prestressing force in the steel cables used in prestressed concrete.
Since the phenomenon is not well under-
stood, the solution to the design of structures for creep is
difficult.
In general, the methods utilized today make use of
a reduced modulus of elasticity to compute deformations due
to creep.
Other widely used techniques simply multiply the
elastic deformations by an arbitrary safety factor.
The cur-
rently available design techniques must be improved since concrete structures are being designed with increasing geometric
and analytic complexity.
The advent of ultimate design em-
phasized the importance of a more accurate technique of time
dependent stress analysis.
There exist several theories that attempt to explain the
mechanisms of creep of concrete.
In general, it may be said
that none of the currently available theories are adequate to
fully describe the
creep behavior of concrete.
Therefore,
more work is also needed to achieve better understanding of
the phenomena.
One approach to the problem consists in
examining the be-
havior of concrete along the lines of a composite.
Concrete
is a composite made of aggregates, matrix (cement paste) and
voids.
The properties of the whole are determined by the
properties of the components.
-7-
In general, concrete creep is
primarily due to the matrix.
In other words, the time de-
pendent behavior of the binder (i.e. cement paste) will govern the time dependent properties of the concrete.
There-
fore, for these reasons, it was decided to limit this to the
study of time dependent properties of the matrix.
This thesis will study the time dependent properties of
cement paste which is a major component of concrete.
The
study utilizes the theory of viscoelasticity in the characterization and stress analysis of cement paste.
-8-
CHAPTER II.
In
this
REVIEW OF LITERATURE
section a brief review of the pertinent litera-
ture is presented.
The areas covered will include creep of
concrete, considerations of concrete as a composite, and
viscoelasticity.
A)
Creep of Concrete
The time dependent behavior of concrete was considered
very early in the development of concrete technology (1).
The increasing deflections of reinforced concrete structures
with time, the loss of prestressing in prestressed structures, and the cracking of plaster on the partitions as the
load carrying frame bears on them due to increased deflections caused by creep are a few manifestations of creep.
Since creep deflections can reach a magnitude of two or
three times greater than the elastic deformations (2, 3),
it
is clear that the structural engineer must consider creep in
his design of structures.
The mechanisms of creep of portland cement and the methods to control the creep have received significant attention
in concrete literature (4, 5).
One of the earliest theories
that was proposed is the plastic slip analogy (6), which explains the creep of concrete in a manner analogous to the
plastic deformations of metals.
The absence of yield stress
and the sensitivity of creep of concrete to moisture are among the discrepancies which could not be explained by the
plastic slip theory.
Freudenthal (7) and others considered hardened cement
paste as a viscous fluid surrounding the loose and rigid
-9-
aggregate components and the creep was considered due to the
flow of
fluid.
viscous
this
with the absence of a
although
This view,
limiting
consistent
or yield stress for creep,
does not explain the observed creep recovery.
The morphology of hydrated
cement
(8,
9)
shows the presThese
ence of a crystalline and a non-crystalline component.
components, of colloidal size, are saturated by adsorbed
moisture.
The gel, under load, is thought to behave as a
phases of varying elas-
composite body consisting of several
tic and
viscous properties.
The externally applied load is
As the viscous
initially borne by the elastic phase.
ponent deforms, the overall body is
creeps.
com-
deformed and the body
This is known as delayed elasticity.
This theory
cannot explain the recovery phase present in concrete creep.
Freyssinet
(10)
suggested that
creep is
due to the ex-
change in the surface tension forces arising in the capillary
pores of the hardened cement paste.
This theory cannot ex-
plain the experimental result which shows that the creep of
a concrete with practically empty capillaries is not significantly lower than that for saturated concrete.
said about this
More will be
later.
The seepage of gel water theory
(11) is based on the as-
sumption that hardened cement paste may be considered as a
limiting swelling gel.
The gel equilibtrium
ton and the externally applied load,
vapor pressure of the gel water.
lates
that
creep is
with its skele-
is determined by the
The seepage theory postu-
caused by the disturbance
of this
equi-
librium due to externally applied loads, and its gradual reestablishment by a moisture exchange with the
environment.
This theory seems to be supported and also contradicated by
-10-
the work of Dolsch and Mullen (12).
Their work consisted of
a creep study of neat cement paste specimens.
obtained under humid environmental
Their results
conditions tend to sup-
port the theory of gel water seepage.
The tests performed
on oven dried specimens (110 0 C), showed no measurable creep.
This would tend to contradict the theory of gel seepage and
also that of delayed elasticity.
Ali and Kesler (13)
hypothesized that creep deformations
could be explained in terms of the modification, by the externally applied loads, of the shrinkage and swelling due to
changes in moisture content and in terms of the viscoelastic
deformation of the structural elements of the gel.
From this brief review of the theories of concrete creep
it becomes apparent that none is really adequate in explaining the entire behavior.
dictions are present.
In several cases outright contra-
It is clear that more work is needed
in order to determine the true mechanism.
One of the aspects
that this additional work may follow would use the general
theories of rheology.
Rheology is the study of flow of materials.
of concrete,
the rheological approach was followe/d by Bingham
and Reiner (14),
(16).
In the creep
Forshind and Torroja
(15),
Paez and Mason
An up-to-date review of the rheology of concrete is
presented by Ali and Kesler (17).
The attempts in applying
the principles of rheology to the creep of concrete are based
on the theory of viscoelasticity.
Since viscoelasticity is
a study of materials exhibiting both elastic and time dependent behavior, and since it has been used with considerable
success with other materials such as polymers (18), it seems
to be feasible to attempt its use in the study of the creep
-11-
of concrete.
Such a first
attempt took the form of model fit--
ting to empirical time dependent data,
(19), Cowan
and was made by Flugge
(20), Hansen (21), and Glucklich and Ishai (22).
The procedure followed by these authors consisted of selecting phenomenological models and determining their parameters
so that the model response closely approximates the empirical data.
The authors did not attempt to use the empirical
formulations to stress analysis.
The structural engineer analyzing a concrete structure
must design it in such a manner that the finished product
will provide adequate service iTmediately after construction
and throughout its useful life.
For this reason, the engi-
neer must know how much the structure will defle-ct and how
long it will take to achieve this deflection, therefore, the
data required by the structural engineer includes the magnitude of creep deformations and the rate of the deformations.
The technique that is used by most engineers is undoubtedly
that of the American Concrete Institute (23).
tion of the ACI Code specifies that:
The 1963 Edi-
"reinforced concrete
structures subject to bending should be designed to have adequate stiffness to prevent deflections or other deformations
which may adversely affect the strength or serviceability
of the structure."
However as far as the design procedure is
concerned, the ACI Code specifies that instantaneous deflections are to be computed by the usual elastic formulae and
that the long time deflections are to be obtained by multiplying the instantaneous deflections by 2.0.
The Code im-
poses certain deflection limits which are expressed in terms
of the ratios of the spans.
This design for creep is clearly
empirical and entirely arbitrary.
-12-
The method does not
consider loading conditions, the history of loading, and the
possibility of changing loads.
The method seems to be that
of an increased safety factor rather than a method of rational design.
Among the other available design methods there are the
rate of creep method (24),
method
the reduced or equivalent modulus
(25) and various design methods developed by Disching-
er (26), Leonhardt
(27), and Bieger (28).
The rate of creep
method uses a creep coefficient derived for a constant load.
This coefficient is then introduced into a differential
equation of equilibrium as a function of time, and the solution yields a time dependent solution for deformation.
solution is
often quite tedious and time consuming.
major drawback of this technique
creep coefficient.
lies
in
The
The
the selection of the
The coefficient is usually computed on
the basis of constant load tests and thus does not reflect
the influence of changing loads.
The method also assumes
that the magnitude of the load does not affect the creep coefficient.
Finally, the assumption of a constant rate of
creep coefficient with time is questionable.
The method of
reduced or equivalent modulus uses elastic methods, with a
reduced modulus of elasticity, to compute creep deformations.
It is similar to the ACI method.
The design methods of
Dischinger et al, use a differential equation relating creep
deformations to the applied load.
The creep shortening is
assumed to be proportional to values of creep obtained for
constant loads with modifications due to the ambient humidity.
In many ways, it is similar to the rate of creep meth-
od with some considerations given to age at loading and environmental conditions.
-13-
Recently there have been a few attempts at the correlation of theoretical and experimental estimates of the deflection of reinforced concrete structures.
were performed by Gorley and Sozen
(30),
and Washa and Fluch
(29),
(31, 32).
Such studies
Bresler and
Selna
Each of these studies
used one type of beam and are thereby of limited application
in a general case in which the
from the beams tested.
structures used are different
A more rational approach to creep has
been used by Finnie and Heller
(33).
The authors attempted
to describe creep properties mathematically and use these
descriptions
in a few cases of structural analysis.
Another
promising approach was followed by Polivka and Best
(34) and
Tielz and Dorn
(35).
The authors tried to analyze the creep
of concrete along the lines of reaction rate theories and
activation energies.
This has theoretical promise but is
still far from practical applications.
The discussion of the currently available design methods
shows that a more rational and universal design method is
needed.
This may take the form of a large scale testing de-
signed to cover all major design cases.
This approach must
be rejected, not only due to the high costs involved, but
much rather due to the basic approach to the problem since
such a large scale testing program does not examine the problem at its source.
Another basic approach could consist in
determining the mechanisms of creep and then in estimating
the time dependent deformations based upon the creep mechan-
isms.
Among the disadvantages inherent in this approach are
that it is not clear structural analysis could be performed
on the basis of microscopic mechanisms and that the creep
mechanisms have not been determined.
-14-
A more rational and practical approach could use the concepts of composite materials.
forms.
Such a study could take two
First, the time dependent behavior could be studied
at the microscopic level.
This is quite difficult since the
morphology of cement is not known accurately (36).
Second,
the study could follow the phenomenological approach.
From
the standpoint of the designer such a study is very appealing
since it does not concern itself with many details that are
unfamiliar to the engineer.
As far as the engineer is con-
cerned, the topics of interest include mathematical characterization of the material and the application of such a
characterization to structural analysis.
B.
Concrete as a Composite
Concrete is a multiphase material consisting of mineral
aggregates, stones and sand, bound by hydrated portland cement.
The study of concrete as a composite began with La Rue
(37) studying the influence of the aggregates on the properties
of the composite as a whole.
A theoretical study of the
subject was performed by Dantu (38).
The author examined the
time independent properties of concrete, such as the distribution of stresses in the material.
tended by Hirsch
(39),
Similar studies were ex-
to include not only the binder but
also the influence of the volume content of aggregates on the
properties of the concrete.
Kaplan
(40) used ultrasonic meth-
ods to perform a study of the strength of concretes made with
different aggregates.
This type of study was extended to
cover the time dependent properties of concrete.
Kordina
(41) was among the first authors to examine the influence of
the mineralogical properties of the aggregates on the creep
-15-
of concrete.
He concluded that creep decreased with in-
creasing stiffness of the aggregate.
followed by Counto
(42).
A
similar approach was
Counto proposed a mathematical
model which he used to describe the elastic and the time dependent properties of concrete.
Similar models, used mainly
for numerical descriptions of the distribution of internal
stresses were proposed by Baker
(45), and Hsu
(46).
(43), Reinius
(44), Pickett
Pickett and Hsu studied the behavior of
concrete as a composite during shrinkage.
The models were
used by Baker to describe and rationalize the mode of failure
of concrete.
Reinius' model examined the failure properties
of concrete by assuming that the hydrated cement gel consisted mainly of crystals in needle shapes.
been supported by electromicrographs
This assumption has
(47).
A very thorough
study of multiphase systems was made by Hansen
(48).
He sum-
marized the various models and developed mathematical formulations to describe their behavior.
C.
Viscoelasticity
The concept of viscoelastic materials evolved from the
studies of the theory of elasticity during the latter part of
the nineteenth century.
The theory was based on Maxwell's
superposition principle and was first proposed by Butcher
(49).
Jeffreys
Further progress was made during the 1920's when
(50) considered creep in solids and attempted to fit
mathematical formulae to experimental data.
During the 1940's,
polymeric materials were first used extensively.
(51)
Leaderman
used the theory of viscoelasticity to describe the time
dependent properties of cellular fibers.
The theory was fur-
ther extended by the work of Alfrey (52),
Tobolsky (53),
-16-
Ferry (54), and Lee (55).
The growth of the field became
explosive in the 1960's with the development of solid fuel
rocket engines (56).
The mathematical representation of viscoelastic materials
may take many forms.
The general theory of viscoelasticity
may be developed in terms of simple spring dash pot models
or in terms of a series of such models.
The early work in
viscoelasticity was concentrated on materials that could be
represented by these simple models.
In general, however,
materials commonly used have viscoelastic characteristics of
such a nature that simple models are not sufficient to represent their behavior accurately.
One way to improve the de-
gree of mathematical representation consists in using integral operator laws based on the hereditary theory of Volterra
(57).
Another such improved representation consists in the
complex modulus representation (Bland (58)].
A good presen-
tation of the various characterization techniques and mathematical representations have been presented by Gross (59).
Each mathematical form has definite advantages and disadvantages as far as the adequacy of representation of a real
material is concerned.
The ease with which material con-
stants or functions may be measured, and the adaptability of
such forms .to stress analysis manipulations, are important
criteria affecting the selection of any given characterization form.
A good discussion of the merits of each form is
presented by Lee (60).
When the characterization process and the material properties have been determined, the stress analysis of viscoelastic materials becomes possible.
-17-
The general method of viscoelastic stress analysis consists in solving the differential equations of equilibrium,
and the time dependent consti-
the compatibility equations,
tutive equations.
In the case of linearly viscoelastic
materials,atechnique known as the principle correspondence,
simplifies this general technique.
The correspondence
principle consists in using an associated elastic solution,
that is an elastic solution with the material properties expressed in the Laplace domain.
The time dependent solution
is obtained by taking the inverse Laplace transform of the
p dependent parameters.
The requirements of this technique
may be summarized as follows.
linearly viscoelastic.
First, the material must be
Second, the boundary conditions and
their type may not vary.
Third, the associated elastic solu-
tion must exist and must be known.
Fourth, the inverse La-
place transform must exist and must be unique.
The limits of
linearity are determined during characterization.
The re-
quirement of the boundary conditions was first advanced by
Alfrey (61).
The technique is best applied in
the case of
quasi-static conditions.
The existence of the inverse Laplace transform presents
some problems.
The structure of the mathematical theory of
this procedure has been presented in detail by Gurtin and
Sternberg (62).
and Onat (63).
A simplified presentation was made by Breuer
The authors have shown that the solution of
the linear viscoelastic problem is unique provided that the
relaxation modulus is a steadily decreasing function of time
and is convex from below and tends towards a non-negative
constant asymptotic value.
-18-
The actual computation of the inverse transform in a
closed form is difficult.
In order to eliminate this diffi-
culty a large number of numerical inversion programs were
developed.
(64).
Another technique was presented by Churchill
This method takes the form of a direct numerical in-
tegration for the evaluation of the inversion integral in
terms of a numerically equivalent elastic solution.
method is rather complex.
This
Schapery (65) presented another
approximate inversion method which is simple but which provides for an evaluation of the error introduced and which
permits the reduction of the error by the introduction of
additional terms.
This technique is known as the collocation
method and will be discussed in detail in Chapter V.
The scope and objective of this thesis, determined on the
basis of this brief literature survey, will be expanded in
Chapter III.
-19-
CHAPTER III.
OBJECTIVE AND SCOPE
This dissertation is a study of the time dependent behavior of cement paste.
The study of the available litera-
ture showed that while a great number of studies of creep of
concrete were performed beginning as early as 1900
(66,
67),
the studies reported on special cases and did not attack the
problem from a rational point of view.
Since concrete is a
composite made of aggregates and portland cement, one obvious
way of studying the properties of the whole is to study the
properties of the components.
As it was pointed out in the
survey of literature, the binder is responsible for the major
portion of the creep in concrete.
The aggregates will modify
the creep properties but will not cause creep.
the
Therefore,
study of cement paste was performed and represented a
first step in a composite time dependent analysis of concrete.
This thesis studied the time dependent properties of cement paste for several testing variables.
clude the age at loading and the stress
study will follow a dual approach.
The variables in-
(strain) levels.
First, the engineering
properties of the material must be determined.
as characterization.
follows:
The characterization
The
This is known
will proceed as
the creep and relaxation properties of the material
will be determined for different stress and strain levels.
The limits of linear viscoelastic behavior are determined by
graphical methods.
These limits are then verified using the
relationship between the transformed values of the creep compliance and the transformed value of the relaxation modulus.
The creep compliance J(t) is defined as the strain divided by
t)
and the relaxation
the con'stant stress (i.e., J(t) , and the relaxation
0-
-20-
modulus
strain
is defined as the stress divided by the constant
(i.e.,
E(t)
=--
.
After transformation
Laplace domain here denoted by
[J(p) and E(p)]
into
the
the functions
are related in the following manner:
E(p)
=
1
(Williams)
The characterization is then utilized in a simple case
of structural analysis, consisting of the creep behavior of
a simply supported point load cement beam.
The midspan de-
flections are measured as a function of time and are compared
to the deflections computed on the basis of the characterization.
The correspondence principle
form the calculations of deflections.
(69) was used to perThe mathematical
operations have been programmed for the Civil Engineering
Systems Laboratory IBM 360-40.
The mathematical methods will
be presented in detail in Chapter V.
In all,
a total of
[176]
specimens were tested.
These
included 54 specimens used for the strength determination for
2"
x 4"
cylinders.
16 specimens.
The 4"
x 8"
cylinder series consisted of
The beam series consisted of 24 specimens.
There were 42 specimens tested in stress relaxation and 40
specimens tested under creep both axial and triaxial.
The
specimens were tested at various stress, strain levels as it
was discussed in the preceding chapters.
-21-
CHAPTER IV.
v.ATEI:ALuS AND PROCEDURES
The material used throughout this thesis was Type I ce-
ment manufactured to ASTM Specification C-150 (70).
The
chemical analysis of this cement is presented in Appendix
(A).
The specimens were prepared in batches of 2800 gm. with
a water/cement ratio of 1/3.
In general, the preparation of
the specimens followed ASTM C-187-58 (71).
The creep and re-
laxation specimens were cast in brass moulds and the flexure
specimens were cast in Plexiglass moulds.
After vibration,
and smoothing of the surface, the specimens were placed in a
moist room for 24 hours and under water after stripping and
capping of the cylinders.
The creep tests were run in a frame designed for this
purpose.
a steel
The
See Figures (1, 2).
The triaxial cell consisted of
cylinder blocked off at both ends by steel plates.
steel plates were clamped together by four one-inch di-
ameter threaded rods.
Through the top plate, a piston guided
by brass bushings applied a load on the specimen.
The load
was applied using a standard railroad car spring.
The stiff-
ness of the spring was such that the load remained constant
during the creep of the specimen.
The hydrostatic pressure
was applied using water as a pressurizing medium.
sure was generated by a pneumatic-hydraulic
system coupled
with a manifold that permitted the filling and
emptying of
the 6 pressure cells and allowed the use of a small
gallon) Greer accumulator.
compressed Nitrogen.
strain gauges.
The pres-
(one
The pressure was generated by
The springs were calibrated using
'Thir
creep deflections were measured using an
-22-
Ames dial gauge #0112 with divisions of 0.00005".
The tests
were carried out in a jacketed condition for about five
days and were performed at several stress levels.
The values of Young's modulus in compression were determined using the Instron Machine.
The internal amplifier
and servo-mechanism drive chart of the Instron were used in
conjunction with the "U-Bars" to measure the strain due to
the load applied.
The sensitivity of the U-Bars and ampli-
fier system was of the order of .00001".
In this fashion a
continuous record of deformation-load was obtained graphically.
The U-Bars were calibrated using a dial gauge.
To
maintain the humidity equilibrium of the specimens, the specimens were coated by paint.
This effectively prevented the
weight losses for the duration of the modulus tests and the
relaxation tests.
loss is
The influence of the paint on the weight
illustrated by Figure 3 which shows that the coated
specimens showed no weight loss up to 20 hours whereas the
uncoated control specimenslost up to 1-5%.
A similar tech-
nique was used in the case of the relaxation tests.
The stress relaxation tests were performed on the Instron
Machine.
After coating the specimens to prevent moisture
losses, the specimens were deformed at a constant rate of
strain (X = .05"/M) until the desired value of initial strain
was reached.
At this point, the cross head was stopped, and
the decay of the load was monitored with time.
The duration
of the test at each strain level was of two hours.
Each
specimen was tested at four strain levels.
The beams were tested in a controlled environment (700 F
and 50%oRH).
The beams were coated to prevent moisture losses
and shrinkage stresses.
The load was applied using a
-23-
mechanical lever with a 2:1 ratio.
The mid-space deflections
were monitored with time and were recorded for various loads.
Strain gauges (A-7) were used to obtain the tensile and compressive extreme fiber strains.
These were used to compute
values of Young's modulus in flexure for the tensile and compressive cases.
The tests were performed for 5 days.
NUMLERICAL METHODS
CHAPTER V.
The experimental results of the tests performed in this
In order to
thesis are presented graphically in Chapter VI.
achieve a form of viscoelastic representation, certain mathematical functions must be fitted to the data.
As it was
mentioned in Chapter II, a large number of different mathematical functions is available.
A.
Functional Representation
One of the most
The methods vary greatly in complexity.
convenient techniques consists in a finite exponential series,
As an illustra-
(68).
known as the Dirichlet or Prony series
tion, the time dependent modulus E(t) could be represented as
E,+ E
D
follows:
where
Ea is the initial value
of the modulus, E. are the constants attached to each expoential term, 6; are the coefficients of the exponential, and
t is time.
This method has several attractive features.
First, the curve fitting is easy; second, the accuracy of the
fit increases with the addition of more terms in the series
and is therefore not limited to any arbitrary value.
this method is
Third,
particularly well suited to the computation of
the Laplace transformation and the inverse transformations.
used thrloughout this
For these reasons this representation is
thesis.
The Dirichlet series collocation may be presented as
follows:
a series of discrete ordinates Do, Dl, ...Dn is
available over a domain D
consists
at the Y)
of collocating
data points.
(i.e.,
the series
n+l points).
D(t)
=
For the creep test
-25-
Et
The technique
-+
E;
-V
the formulation
is modified slightly such that
J~t)
C -t L
i--1
-
where J(t) is the creep compliance, Je is the instantaneous
value of the ccmpliance,T .
*
are constants, and t is time.
The collocation will be exact in that the fitted series will
agree exactly with the data at each abscissa.
For conven-
ience the constants are selected such that they lie
mid span of each of the time decade.
by Brisbane
(72)
at the
This step was proposed
since it effectively triangulizes
the coef-
ficient matrix and thereby simplifies the matrix inversion.
The problem consists in finding the column matrix of the
coefficients in the Dirichlet series.
This column matrix is
obtained by pre-multiplying the column matrix formed with the
experimental results, by the inverse of the matrix formed by
the exponential coefficients.
matrix is
This exponential coefficient
formed from the exponential of the product of the
by the various times corresponding to the experimental
results.
This system of equations may be shown as follows:
E
Se
D,-E
The procedure will yield the coefficients of the Dirichlet
series and since the exponential coefficients have been preassigned the problem is thus fully described.
The matrix
multiplications and inversions programs are shown in
dix B.
-26-1)
JI_
Appen-
)
B.
Linear Viscoelastic Behavior
Linear viscoelastic behavior is defined by Tobolsky (53)
as being the domain within which the strain, in a creep test,
is directly proportional to stress, and in the stress relaxation test, the stress is directly proportional to strain.
Therefore, one way to represent linear viscoelastic behavior
consists in showing graphs of stress and strain at constant
time.
Within the linear viscoelastic region, the points at one
given time will lie on a straight line.
Deviations from the
straight line provide an indication of the nonlinearity of
the system.
Another analogous technique consists in plotting
the creep compliance
T
. E~4/
versus time for various
stress levels; within the linear domain the curves should
superimpose.
Similarly, for a stress relaxation, the re-
laxation modulus
zL)
.,,/Fs
E
should be plotted versus time
for several strain levels and again the curves should superimpose within the linear domain.
The viscoelastic functions are interrelated.
pointed out by Gross (59),
Laplace
domain
E )
I'/i
As it was
the following is verified in the
where E(J) and J(r) are the La-
place transforms of the relaxation modulus and creep compliance respectively.
In order to achieve this equality,
the functions must be transformed into the Laplace domain.
C.
LaDlace Transformation
The Laplace transform of a function
follows:
-
formed function and
6
where
I
()
)-
is
defined as
is
the trans-
the Laplace parameter.
Extensive
q
tables of Laplace transforms
are available e.g.
-27-
Hodgman
(73),
and in the case of exponential expressions, solutions are
available in closed form.
pressed in a Dirichlet
For the stress relaxation ex-
series
CLt)= E
EL e
the
following expressions are obtained for the Laplace trans-
form
D
)= E--
-
Williams (68).
Simi-
lar results are obtained for the creep compliance whose
/
A'
transform is as follows
A computer program has been written to generate values of
U(
) and D(y)
as a function of
This program is pre-
sented in Appendix B.
D.
Ramn Transformation
In order to obtain E(t), the data obtained for the stress
relaxation must first be transformed from the ramp loading
used to the step strain input that is required by the definition of stress relaxation.
As it was mentioned in Chapter
IV, the stress relaxation tests were performed using the
Instron testing machine.
The limitations of the machine are
such that, for the specimen size used in this thesis, the
maximum rate of cross head deformation that could be used
without losing accuracy was of the order of .05"/min.
At
this rate, the average length of time required to deform
the specimen was of the order of .01 min.
During the load-
ing time the specimen relaxed so that when the desired constant deformation had been achieved, the load registered by
the Instron was less than it would have been for the step
input case.
(I)
This
is
best
illustrated graphically in Table
which shows the strain
conditions and their stress re-
sponses
for the ramp loading used
experimentally
and the
step strain input required
theoretically.
can be illustrated graphically
The transformation
(see Table I).
of the ramp should be significant
The effects
£
at time
1 , where
,
is the duration of the ramp, and should become negligible
at times
tt)
.
For polymeric materials Hilton
(77) found
I L.
that the effect of the ramp becomes negligible at times
This transformation may take several forms depending on
the form of the mathematical representation used.
The tech-
nique that is best suited to the Dirichlet representation has
He assumed that the strain
been proposed by Brisbane (72).
applied to the specimen had a linear rise to a constant value
and could be expressed as follows:
E
in./min.
where
R = Strain rate in
i (-,)
)=
g
o
'
D
LI
S= Loading time in./min.
= The Heaviside step function such that
H
I
t<L,
Using the above strain input, the stress response is given by
for
L b
the stress
SR+
responses
reduces
to
A
(v-2)
wlhere
fl
-v-DLth
re e
4;
~0
-29-
D
Ac,~
D
D~
4
L1
"
time
Ramp Strain Input
time
Step Strain Input
r)
()
Cl)
time
Stress due to Ramp
I
Stress Corrected for
Table.
Stress due to Step Strain
time
aPRamp Loading
Strain Inpit for Stress Relaxation.
-30-
To simplify the formulation,
may be rewritten as fol-
(V.-2)
lows:
C
where
where
DL :
(V-3)
C LLg)L
CL
e
then the relaxation modulus may be written as
E (t)= E +2jIL E; e
where
E
and
E;
e
=
E, 0
C -,
.ii,
;,
)
:=
Therefore, once the stress is known for
may be obtained by the series representation.
of the relaxation modulus
is
the modulus
The accuracy
highly dependent upon the accu-
racy of the curve fit in the time domain corresponding to
the stress response immediately after
hI
.
Therefore, the
curve fitting technique is improved by taking more data
points in this region and the
rway c
I/2~-l
decades.
.vhere
; is
\1 5 are chosen in the following
centered
in
each of the time
The transformation for ramp effects has been pro-
grammed and is presented in Appendix B.
E.
Corresoondence
Princip
-le
As it was mentioned in Chapter II
analysis may take several forms.
Alfrey
(52)
and extended by Lee
-31-
the viscoelastic stress
The technique proposed by
(69) makes use of the corres-
pondence principle.
formed in
In the case of the stress analysis per-
this thesis,
the associated elastic solution is
obtained by taking the elastic
solution of the deflections
of a simply supported point load beam and replacing
value of E
(modulus of elasticity) by tE(
],
the
Mandel
(74).
The numerical constants required are derived in Table II.
The associated solution is of the following form:
-2
3 Ey )
(V-4)
where the various symbols are defined in Table II.
The time
dependent solution is obtained by taking the inverse Laplace
transform of both sides of (V-4).
Therefore, the problem
consists in determining the inverse Laplace transformation
of
F.
Inverse Laplace Transformation
The inverse Laplace transform is defined as the function
such
(
that the function SOc)
Hildebrand
(75).
satisfies
(-
)
The determination of the inverse
is quite difficult in a closed form.
co
i
function
There are however,
several approximate methods that can be used to obtain the
Laplace transform in
these are by Schapery
a numerical
form.
The best known among
(65) and Cost and Becker (76).
This
technique uses the fact that most general stress or displacement response for viscoelastic problems where the Laplace
technique is
wchere
(t)
applicable,
is
may be represented as follows:
a general stress or displac-ment
-32-
function and
T.ABLE II
-
rZ
The elastic deflection y at the mid
span of a simply supported point
loaded beam is:
SA
PL /
4 EI
where L is the span, P is the load, E the Young Modulus and
I the transverse moment of inertia.
In the viscoelastic case, the load is considered to be
applied in a step fashion so that it can be represented by
a Heaviside function which multiplies Po , the magnitude of
SI
the load.
?%t(C)
There for time dependent
deformation we have
48
In the Laplace domain we have:
45t3
E(-0
E(1)
Ekt)
by replacing the following values for the constants, the desired formula is achieved:
= i0o
.
The moment of inertia
in the transverse direction is computed as follows:
then
K=
-
.73 -,-Y
.
then the time dependent deflections will be achieved by taking the inverse Laplace transform of Loth sides of the equation:
A
')
-
Ps
1
33-33 -
are constants with respect to time and,"i
is
the
transient component.
The constants may vanish depending on
the type of problem.
If - (t) is a stress then
since the stress must remain finite as t--~
if'-(t) is a displacement the i
vanishes
, conversely
is zero when the body does
not manifest steady flow under constant loads.
Schapery (65)
-%- Si
suggests that a Dirichlet seriesa&
where
46%"is the approximation to transient response, can be used
as a good approximation to- the solution .1/-().
uses a series in which the
The technique
,,are prescribed positive con-
stants and the Si are unspecified coefficients to be determined by minimizing the total square error between,7' and
'
This total square error is E2
=
Applying the minimizing criterion, derivation with respect
to Si yields
In this way
relations are obtained between the Laplace
transforms of
evaluated at
.
-
This can be
represented as follows:
(/
By letting p = 1/f
.
-
:.,
',
,,
-)
and multiplying throughout the following
result is obtained:
ris
-
.
V
Explicitly, a more useful expression is achieved
The system ( 1k
for the given '
) of equations is
.
used to calculate the
These operations yield the transient
-34-
'
component of A
the constants
).
For the complete viscoelastic response,
7!and)'.
of tp(i ) and
%A(IO9
must be determined from the behavior
as p tends to zero.
This technique has
been programmed and the determination of the coefficients
proceeds along steps similar to the series collocation described earlier in this chapter.
in Appendix B.
-35-
The program is presented
CHTIP'ER VI.
EXPERIEbTNTAL RESULTS
This section presents the results on the elastic properties of cement paste, stress relaxation, creep in axial,
triaxial, and flexural modes, and creep of larger specimens.
There is no attempt to analyse the results.
The analysis of
the results is presented in Chapter VII.
A.
Strength and Elastic Properties.
The compressive strengths of 2" x 4" cement paste cylin-
ders (w/c = 1/3), as function of age, are shown in Figure 4.
Each point represents the average of nine individual tests.
The stress-strain curves obtained on 2" x 4" cylinders, using
the U Bar extensometers are shown in Figure 5.
The stress-
strain curves are obtained for the four testing ages of 1, 3,
14, and 28 days.
The variation of the modulus of elasticity
with age at testing is shown in Figure 6.
B.
Stress Relaxation.
The stress relaxation tests are performad on 2" x 4" cyl-
inders that are paint coated to prevent the loss of moisture
during testing.
The tests are performed on an Instron testing
machine and the specimens are subjected to several levels of
strains to verify the linear viscoelastic behavior of cement
paste in stress relaxation.
The Instron recorder shows the
load and its decay with time.
Figure 7 is a typical result
obtained from a relaxation test.
This figure shows the varia-
tion of stress with time for one day old specimens, tested at
the two strain levels shown on the graph.
could not be applied
instantaneously,
-36-
it
Since the strain
becomes necessary
to correct for the relaxation of the specimen which takes
place during the loading time.
This correction is performed
using techniques discussed in Chapter V.
The influence of
the ramp loading is shown in Figure 8 which shows the results
of a stress relaxation test corrected for the ramp effect.
In this figure, the original stress relaxation data is comIt is seen that the
pared with the ramp corrected results.
stress relaxation results when corrected for the effects of
the ramp loading have larger initial maximum values than the
original uncorrected data.
The corrected stress relaxation
results coincide with the original data in about 50tl where
tl is the duration of the ramp loading.
Similar results are
obtained for the other relaxation tests.
Figure 9 shows the
stress relaxation tests performed at four strain levels for
1 day old specimens.
The data presented in this figure are
corrected of the influence of the ramp loading.
Figure 1
shows the results for stress relaxation tests performed at
several levels of strain for the same 1 day old specimens.
The results are corrected for the effects of the ramp loading.
Figures 11, 12, and 13 show similar results obtained
for specimens 3, 14, and 28 days old.
average of two separate tests.
Each curve is the
In order to determine whether
the material behaves in a linear viscoelastic manner, the
stresses for constant values of imposed strains are plotted
at constant time.
The values of stress are taken from Figure
10 and are plotted for the 0 minute and for the 120 minutes
isoclines.
fashion,
If the material behaves in a linear viscoelastic
the isoclines must be straight lines.
Figure 14
shows the stress-strain curves obtained for the 0 and the 120
minutes isoclines.
ures 15,
16,
This data is for I day old specimens. Fig-
and 17 show similar results for specimens 3,
-37-
14,
and 28 days old.
The material is seen to behave linearly.
Figure 18 shows a typical relaxation modulus curve for
the 28 days old specimens.
This figure is obtained by taking
the stress versus time curves shown in Figure 14, and dividing it by the corresponding imposed strain.
Similar results
are obtained for the other three ages at testing.
The re-
laxation moduli versus time are shown in Figure 19 for the
four ages at testing.
Data used in Figure 19 are obtained
from Figures 10, 11, 12, and 13.
A limited number of stress relaxation tests were performed
for long time durations.
weekend.
In general, the tests lasted over a
The long range tests are presented in Appendix C.
The stress relaxation results are discussed in detail in
Chapter VII.
C.
Axial Creep Tests.
The creep tests are performed on 2" x 4" cylinders using
the creep frames described in Chapter IV.
filled with unpressurized water.
The cells are
The creep tests are per-
formed at several levels of stress to verify whether the viscoelastic behavior is linear or not.
cal creep test.
Figure 20 shows a typi-
In this figure the creep curves are shown
for various levels of stresses used for 1 day old specimens.
Figure 21 shows creep strains versus time for 1 day old
specimens loaded at several levels of stress.
and 24 show similar results for specimens 3,
old.
Figures 22, 23,
14,
and 28 days
Each curve represents the average of two separate tests.
In order to verify the linearity of viscoelastic behavior in
creep, similar methods to those in the stress relaxation
tests are used.
If
the material behaves linearly, the
-38-
isoclines joining creep st:rains plotted for constant values
of stress must be straight
lines.
Figure 25 shows stress
and strain curves plotted for the 0 and 120 hours isoclines.
The data is obtained from Figure 21.
Figures 26, 27, and 28
show similar results for specimens 3, 14, and 28 days old.
The figures are obtained from the data presented on Figures
22, 23, and 24.
The material is seen to behave linearly un-
der axial creep.
When creep strains are divided by their imposed stresses,
a creep compliance J(t) is obtained i.e.,
~0/60)/
EJO()
Figure 29 shows the creep compliance versus time for 3 days
old specimens.
The data presented in the figure is obtained
by dividing the creep strains of Figure 22 by their constant
imposed values of stress.
the other ages at loading.
Similar curves are obtained for
Figure 30 shows the creep com-
pliance versus time for the four ages at loading.
A limited
number of long range tests are shown in Appendix C.
The
axial creep results are discussed in detail in Chapter VII.
D.
Triaxial Creep Tests.
The triaxial creep tests are performed on 2" x 4" cement
paste cylinders.
The rationale for the tests is two fold.
First, triaxial creep tests bring out the effects of complex
states of stress on the creep behavior of cement paste.
Sec-
ond, terms similar to Poisson's ratios in elasticity can be
evaluated using the results of triaxial and axial creep tests.
This evaluation completes the characterization of cement
paste under the assumption of isotropic and homogeneous material conditions.
The triaxial creep tests are performed using
the creep frames described in Chapter IV and are performed in
-39-
a jacketed condition.
The tests are executed at several levels
of stress and the ratiol/!30
with(
O=
; that is, the
ratio of axial to transverse stresses is kept constant at 3.
Figure 31 shows the curves of axial creep strains versus time
for the several values of stresses and for 1 day old specimens.
Figure 32 shows axial creep strains versus time for several
values of stresses shown and for 1 day old specimens.
Similar
curves are obtained for the other ages at loading and are
shown in Figures 33, 34, and 35 for specimens 3, 14, and 28
days old.
Each curve is the average of two separate tests.
The triaxial creep curves are not corrected for the influence
of the transverse stress.
The isoclines are plotted for the
uncorrected data and the influence of Poisson's ratio will be
discussed in Chapter VII. To verify whether the viscoelastic
behavior of cement paste, under triaxial conditions, is linear
or not, the method of time isoclines is used again.
The con-
dition of linearity requires that the time isoclines joining
creep strains at several levels of stress must be straight
lines.
Figure 36 shows stress and strain for triaxial creep
for 1 day old specimens.
The data is obtained from Figure 32.
Similar results are obtained for the other ages at loading.
Figures 37, 38, and 39 show time isoclines for specimens 3,
14, and 28 days old.
The data are obtained from Figures 33,
34, and 35.
The material is seen to behave linearly.
When creep strains are divided by their corresponding imposed stresses, a triaxial complianceS't)
is obtained i.e.,
S')=
'6
where
formation and
and
0- --
CO.
is
E!A)
is
the time dependent axial de-
the axial stress such that
The creep compliances
uncorrected uniaxial data.
COI /o-i
are calculated for the
The influence of Poisson's ratio
will be discussed in Chapter VII.
-40-
Figure 40 shows creep
compliance versus time for several values of stresses and for
1 day old specimens.
This figure is derived from data pre-
sented in Figure 32.
Similar results are obtained for the
three other ages at loading.
Figure 41 shows the creep com-
pliance versus time for specimens 1, 3, 14, and 28 days old.
The figures are obtained from data available in Figures 32, 33,
34, and 35.
The triaxial creep results are discussed in de-
tail in Chapter VII.
E.
Flexure Creep.
The flexure creep tests are performed on 1" x 1" x 12"
cement paste beams.
for two reasons.
The flexure creep tests are performed
First, the data obtained is used to apply
and to verify the viscoelastic characterization obtained for
the cement paste cylinders.
Second, the flexure creep tests
illustrate the influence of a stress gradient and the influence of a tensile stress on the creep behavior of beams.
The
flexure creep tests are performed using painted beams to prevent moisture losses during the tests.
The creep flexure
tests are performed for three mid span point loads.
This will
permit to verify whether the viscoelastic response of cement
paste beams when tested in flexure, is linear or not.
In
addition to the mid span deflections, readings of extreme
fiber strains are taken both in tension and in compression.
Figure 42 shows the mid span deflection creep versus time for
the three loads levels and for 1 day old specimens.
Similar
results are obtained for the three other ages at loading.
Figures 44, 45, 45, and 47 show mid span creep deflections for
the three loads and for specimens 1, 3, 14, and 28 days old.
Each curve is the average of two separate tests.
-41-
The extreme
fiber strains are obtained in tension and compression and are
plotted in microinches/inches on a common graph.
Figure 43
shows the extreme fiber strains in microinches/inches versus
time, for the three load levels, and for 1 day old beams.
Similar curves are obtained for the other ages at loading.
Figures 48, 49, 50, and 51 show the extreme fiber strains
versus time for the three loads used and for specimens 1, 3,
14, and 28 days old.
The linear viscoelastic determination
makes use of the technique of time isoclines.
Figure 52
shows the load versus mid span deflections for the three loads
and for the four ages at loading.
The material is seen to
behave linearly under flexure.
When the extreme fiber strains in compression are divided
by corresponding stress, a flexure creep compliance
achieved i.e.,
tS')is
Figure 53 shows the flexure
A
creep compliance for 1 day old specimens.
Similar results are
obtained for the three other ages at loading and are shown in
Figure 54.
In this figure flexure creep compliance versus
time is shown for specimens 1, 3, 14, and 28 days old, and the
data is obtained from Figures 44, 45, 46, and 47.
A limited number of long time creep flexure tests are presented in Appendix C.
The flexure creep tests are discussed
in detail in Chapter VII.
F.
CreeD of 4" x 8" Cylinders.
Axial creep tests are performed on 4" x 8" cement paste
cylinders to investigate the effects of specimen size on the
creep properties of cement paste.
The creep tests are per-
formed using the creep frames described in Chapter IV and are
conducted under fully saturated conditions.
-42-
The creep tests
are performed at several levels of stress to determine whether
the viscoelastic behavior of 4" x 8" cement paste cylinders
is linear or not.
The strength properties of 4" x 8" cylinders are shown
in Figure 55 which shows the compressive strength versus age.
Each point represents the average of 6 specimens.
The creep
strains versus time, for 1 day old specimens, are shown in
Figure 56.
Similar curves are obtained for creep strains
versus time for 4"
x 8" cylinders loaded at several stress
levels and for specimens 3, 14, 28 days old.
in Figures 57, 58, and 59.
This is shown
To verify linear viscoelasticity,
the concept of time isoclines is used again.
Figure 60 shows
the stress and strain, for 1 day old specimens, for the 0 and
120 hours isoclines.
Figures 61, 62, and 63 show similar re-
sults for specimens 3, 14, and 28 days old.
The isoclines
are derived from the data presented in Figures 57, 58, 59,
and 60.
The material is seen to behave linearly under axial
creep for 4" x 8"
cylinders.
The axial creep compliance
'"\b]
is
obtained by dividing
the axial creep strain by the imposed constant stresses i.e.,
/<1
.
Figure 64 shows the creep compliance
versus time for I day old specimens.
Similar results are
available for the 3, 14, and 28 days old specimens.
Figure
65 shows the creep compliances versus time for the 1, 3, 14
and 28 days old specimens.
The axial creep strains for 4" x 8" cylinders are dis
cussed in detail in Chapter VII.
-43-
CHAPTER VII.
DISCUSSION OF EXPERIMENTAL
RESULTS
The review of literature and the preliminary tests performed confirmed that cement paste does not behave in an
elastic manner.
The manifestations of deviations from the
elastic behavior are quite common.
These include the creep
of specimens, the relaxation of stresses, and the sensitivity of cement paste to rate of loadings.
This last point
is illustrated by Trost (78) in studies on concrete cylinders subjected to compressive loads applied at different
rates of deformation.
The deviations of cement paste from
elastic behavior, bring about the need for another approach
to the problem.
The approach proposed by several authors
suggests the application of the theories of linear viscoelasticity to the solution of the non-elastic behavior of
cement paste.
It should be pointed out that the study of
cement paste represents the first step in the composite
analysis of the time dependent behavior of concrete.
The
need for a new approach to the solution of the time dependent behavior of concrete is illustrated in the review of
literature presented in Chapter II.
proach is followed in this thesis.
The viscoelastic apIt becomes therefore
necessary to verify whether or not the theory of linear viscoelasticity is applicable to the study of cement paste.
The verification takes the form of creep and relaxation tests.
The tiie dependent tests are performed under the following
conditions and for the following variables.
The creep and
relaxation tests are performed on 2" x 4" cement paste cylinders tested at four different ages:
1, 3, 14, and 28 days.
The specimens are tested in the absence of shrinkage since
all specimens are tested coated or fully saturated so that
the humidity equilibrium is
maintained in
the specimens.
specimens are tested for the water/cement ratio of 1/3.
All
The
casting and curing conditions remain identical throughout
the tests.
The discussions presented in this chapter are
for the tests carried out within the scope of the experimental techniques described above.
A.
Characterization.
The results of the axial creep tests as presented in
Figures 20 through 30,
indicate that the material behaves in
a linearly viscoelastic manner as is illustrated by the use
of time isoclines presented in Figures 25 through 28.
This
linearity is valid within the stress levels used in this
thesis.
age
The magnitude of the creep strains decreases as the
(strength) of the specimens increases.
to the results of Neville et al
This is similar
(79) reporting on the re-
lationships of creep and the gain in strength of concrete.
The decrease is due to the change in the structure of cement
paste as the hydration proceeds.
At first the morphology of
cement paste resembles a loose network of tubes or needles
arranged in three dimensional arrays.
with water.
The pores are filled
As the hydration proceeds, the products of hy-
dration fill up the voids.
water present, both iT the
There is still considerable
gel and capillary pores, but the
density of the structure increases considerably.
The result
is a stiffer structure that is somewhat less sensitive to deformations since more of the load is carried by the skeleton
than was the case for the early cement.
stiffness reduces the deformations
extent of creep.
This increase in
and thereby reduces the
The decrease in creep magnitudes is further
-45-
illustrated
in
Figure 30 which shows the creep compliance
versus time for the four ages at loading.
decreases with increasing age.
to explain this decrease.
The compliance
Similar arguments are used
The functional representations of
the compliances shown in Figure 30 are obtained using the
collocation program shown in Appendix B-I.
A limited number of long range axial creep tests are
presented in Appendix C-I.
The long range axial creep tests
show that creep continues to occur at a decreasing rate. The
long range axial creep tests are allowed to creep for approximately 25 days.
The long range specimens continue to
creep at reduced rates.
This is related to the arguments
on the hydration processes mentioned earlier.
is continuing at reduced rates.
The hydration
The strength increase rate
slows down and the creep decreases.
The results of the stress relaxation tests when corrected
for the influence of the ramp indicate that cement paste behaves in a linear viscoelastic manner; this is illustrated
by the use of time isoclines presented in Figures 14 through
17.
The magnitude of the relaxation decreases with increas-
ing specimen age.
This is again similar to the data presented
by Neville et al (79).
The argument explaining this behavior
is the same as that presented for creep.
The reduction in re-
laxation is further illustrated by the curves shown in Figure 19.
This figure shows the variation of the relaxation
modulus versus time for the four ages at loading.
The modu-
lii increase with age, indicating a reduction in the extent
of relaxation as the specimens increase in
functional representations,
strength.
The
for the relaxation modulii,
shown in Figure 19 are obtained using the ramp transformation
program shown in
Appendix B-II.
range tests are shown in
A limited number of long
Appendix C-II.
The long range re-
laxation tests are usually performed over a weekend.
The
long range tests indicate that stress relaxation continues
to occur at decreasing rate.
The longest time that the
specimens are allowed to relax is of the order of 3500 minutes
(approximately 60 hours).
The relaxation tests carried
out at long times indicate that the behavior of cement paste
under relaxation is similar to the observed long time creep
behavior.
The results of the triaxial creep tests are shown in
Figures 31 through 41.
formed on jacketed 2"
The triaxial creep tests are perx 4"
cylinders with
$ L
,
The
.
ratio of the axial to transverse stresses remains constant
throughout the test series.
This ratio is selected as being
fairly representative of the state of stress found in concrete structures.
Before the linearity of the cement paste under triaxial
loadings is examined, the values of Poisson's ratios must be
determined.
The time variation of Poisson's ratio is obtained
as follows:
Two values of the modulus of elasticity are
picked up from Figure 15, which show the axial creep isoclines.
The selected values are for the 0 and 120 hours
isoclines.
Under the assumption of isotropic and homogeneous
material, this is represented as follows:
-- ~
.
U
Using
the same times, values of axial creep, measured under triaxial conditions, are obtained from Figure 37.
points correspond to the following equation:
.
I~-o
-47-
The triaxial
A e
at
0 Hours
120 Hours
0.282
0.272
3.5
0.199
0.192
3.5
14
0.144
0.141
2.1
28
0.128
0.126
1.6
Loading
TABLE III.
% Difference
Poisson's ratio and the variations with age at
loading and time of tests.
-48-
The triaxial conditions used in the experimental tests are
as follows:
:.~' e- :
l
.
The technique yields values
of Poisson's ratios when values of the axial modulus obtained from the axial creep are introduced
for triaxial creep for the two times.
ratios obtained
into the equations
The values of Poisson's
for the 0 and 120 hours isoclines and for
the four ages at testing are shown in Table III.
Since the
variation of Poisson's ratio does not exceed 4%, it is assumed that for all practical purposes the values of Poisson's
ratio remain constant under creep.
strains measured on 2"
Figure 66 shows the creep
x 4" cylinders aged for 3 days, for
the axial and triaxial creep conditions.
The triaxial creep
strains remain consistently smaller than the corresponding
axial creep strains.
The difference between the strains due
to the modes of loading is constant with time.
This further
illustrates that Poisson's ratio remains constant during
creep.
The differences
increase with increasing stress since
the ratio of axial to transverse
equal to 3.
L'Hermite
stress is
kept constant and
(80)presents similar results on the
time variation of Poisson's ratios.
The consequence of the
constant values of Poisson's ratio during creep is that
cement paste behaves elastically in volumetric strains.
The
stress strain isoclines for the triaxial case are shown in
Figures 36 through 39.
The influence of the t riaxial stress
component consists in reducing the strain by a constant value
at each stress level.
This reduction does not affect the
linearity of the time isoclines since the same constant is
subtracted from the two values of strain,
the two time isoclines.
behave in
corresponding
Therefore cement paste is seen to
a linear viscoelastic manner under triaxial
conditions such that:
to
~~
= ~til
-49-
.
The creep
creep
compliance
is
shown in
Figure 41.
The reduction in
the creep
magnitude with increasing age of the specimen is again apparent in this figure.
Uniaxial and triaxial creep tests
and uniaxial relaxation tests indicate that cement paste behaves in a linearly viscoelastic manner.
B.
Interrelation of Functional Representation of Cement
Paste.
The linearity is further investigated by means of the
interrelations between the viscoelastic functions.
terrelationships are based on the work of Gross
The in-
(59).
Gross
presents the mathematical structures of linear viscoelasticity.
One such interrelationship is established between the creep
compliance and the relaxation modulus.
This technique yields
an algecbraic relation between the Laplace transforms of the
creep compliance and the relaxation modulus.
formation
into the Laplace domain is
The trans-
performed using the com-
puter program presented in Appendix B-III.
Figures 67, 68,
69, and 70 show the curves of E(p) and l/J(p) versus p the
Laplace parameter.
The curves are shown for 1, 3, 14, and
28 days old specimens respectively.
Since the curves of the
creep compliance and the relaxation modulus are superimposable
the relationship obtained by Gross is verified.
This implies
that it is possible to transfer creep to relaxation in the
Laplace domain and that the accuracy of the numerical techniques is good.
C.
Flexure Creep and Its Prediction.
The flexure creap tests are performed on 1" x 1" x 12"
simply supported beams.
The beams are testLed at the four
-50-
ages of loading of 1,
3,
14, and 28 days.
The results of the
flexure creep tests are shown in Figures 42 to 52.
Cement
paste is shown to behave in a linear viscoelastic manner
under flexure.
This is shown by the load-deflection curves
plotted for time isoclines presented in Figure
52.
The
readings of the extreme fiber strains as function of time
are shown in
Figures 48 through 51.
The tensile face ex--
treme fiber strains are consistently larger when compared to
the compressive face extreme fiber strains corresponding to
the same load.
at loading.
This increase is present for the four ages
Similar results are noticed by Glucklich
on experiments with cement paste beams.
(22)
The explanation
proposed by Glucklich attributes this consistent increase in
tensile strains
to the microcracks induced on the tensile
face of the beams.
merits
further
strain
gauges
This explanation seems rational and
comments.
In order
to permit the mounting of
on the cement paste beams,
the specimens
are
removed from the curing tanks and are placed in the constant
temperature and humidity room.
are maintained
at 70
0
The temperature and humidity
F and 60% RH.
During this
time the
strain gauges are mounted on the beams and are allowed to
set for 24 hours.
the
Immediately after the gauges are mounted,
specimens are coated.
It is conceivable that some
shrinkage cracking occurs during the short time limited
drying of the beams.
The shrinkage
on the compressive face, but are
face.
cracks.
This probably is
Their main
in
the strain
compression.
gauges
The
the process
This results
show greater
reduction
-51-
are kept closed
forced open on the tensile
of formation of the
influence consists
geometry of the beam.
and
cracks
in
in
altering
higher actual
readings
in
the
in
stress,
tension than
creep with age
is
evident
in Figure 54 which shows values of creep compliance versus
time.
Again the magnitude of creep diminishes with in-
creasing age of specimens.
The values of flexure creep
complianceare computed using the measured values of extreme
fiber strains and the stresses that are causing them.
This
is shown in Figure 71 where the creep compliances are plotted
for the four ages at loading.
Figure 72 shows the comparison
between the compliance obtained for cylinders and the compliance computed on the basis of the extreme fiber strains
measured on beams.
are large.
The differences between the compliances
This can be explained partly in
terms of the
tensile stress field present at the bottom of the beam which
may have the effect of formation of microcracks as indicated
by the larger tensile strains measured on the bottom face of
the beam.
The cross section under load is reduced and the
beams creep more since the load remains the same.
This
seems to be supported by the observed static fatigue of
several trial beams.
is reduced.
The failures cease when the load level
The difference may also be explained in terms
of a different behavior of cement paste under axial compression and under a stress gradient.
In order to verify the characterization, the deflections
of the beam are predicted using the computer program presented
in Appendix B-IV.
The calculations are performed for each of
the load levels used in the flexure creep and for the four
ages at loading.
The calculations are first performed using
the data obtained on cylinders.
The results of the calcula-
tion are shown in Figures 73, 74, 75, and 76.
The results
of calculations based on the creep compliances obtained on
cylinders are poor.
This could be expected due to the large
deviations in the compliances as shown in Figure 72.
-52-
The
deflection calculations
obtained on beams.
are repeated using the creep data
The results of the calculations are
shown in Figures 77, 78, 79,
and 80.
In this case the
agreement between theoretical values and experimental data
is good.
The results deviate at most by 8%.
The difference in the predicted deflections using the
data obtained on cylinders and the results based on data obtained on beams, point out several things.
It is apparent
that the influence of the tensile stress is larger than first
suspected.
The calculations of creep in reinforced concrete
are based on cylinder creep data.
In the case of reinforced
concrete the influence of the tensile stresses is not very
significant since the steel is assumed to carry all the tensile forces.
However in the case of cement paste beams,
with no reinforcement, the influence of the tensile stress
on the creep behavior of the beam is significant as is shown
by the results.
One of the consequences of the experimental
results is that in order to achieve full characterization,
the tensile creep behavior of cement paste should be determined.
Some data is
crete.
Hallaway (81) presents data on the creep properties
available on the tensile creep of con-
of concrete in tension.
Similar experiments could be carried
out on cement paste.
Another way of explaining the large discrepancies observed
between the compressive axial compliance and the flexure creep
compliance could be as follows.
Since the general shape of
the compliance curves is similar but the initial values differ,
it
is
possible that the values used for the strain on
the cylinders are not exact,
The strain is computed on the
basis of the change of overall length divided by the initial.
-53-
length.
The deflections of the beams are taken directly
using mechanical gauges.
This difference would also ex-
plain the variations obtained in the predicted deflections.
D.
Creep of 4" x 8"
Cylinders.
The creep tests on 4" x 8" cylinders are performed to
investigate the effects of size on the creep of cement paste
cylinders.
The effects of size are also evident in Figure
55 showing the compressive strength of 4" x 8" cement paste
cylinders.
The strength of 4" x 8" cylinders is approxi-
mately 50%/ of the strength of 2"
corresponding ages.
x 4"
cylinders at the
The difference is explained partly on
the basis of the strength of brittle materials.
The creep
behavior of 4" x 8" cylinders is shown in Figures 56 through
65.
The stress-strain curves for time isoclines are shown
in Figures 60 through 63.
in
The material is seen to behave
a linear viscoelastic manner.
the 4" x 8"
The creep compliances for
cylinders are presented in Figure 65.
The comparison between the creep compliances of the
2"
x 4" cylinders and the 4" x 8" cylinders is shown in Fig-
ure 81.
ders
This figure shows that the creep of 2"
x 4"
is greater than the creep of 4" x 8" cylinders.
cylinAl-
though the trend is not entirely consistent as, evidenced by
the behavior of the 3 days old specimens, the difference
points to the tentative conclusion that cement paste creep
decreases with increased specimen size.
reported by Ali (82).
-54-
Similar trends are
CHAPTER VIII.
CONCLUSIONS
Based on the discussions of the experimental data obtained
in this study, the
following are the major conclu-
sions.
1.
Several types of tests show that cement paste does not
behave in an elastic manner.
The tests include
the creep
test, under uniaxial dompression, the relaxation under uniaxial compression, and the creep flexure test performed on
beams.
2.
Under the
following conditions and for the
following
tests, the mechanical response of cement paste is linear
viscoelastic.
The conditions for which linear viscoelastic
behavior holds, are as follows:
i.
ii.
iii.
iv.
one water/cement ratio equal to 1/3.
the specimens are tested in the absence of shrinkage.
This is achieved
by testing under constant humidity
conditions.
the specimens are tested for ages 1,
3,
14, and 28 days only.
the stresses and
the
levels used
strains are within
in this study.
The testing modes for which linear viscoelasticity hold
are:
i.
ii.
iii.
iv.
uniaxial compressive creep on 2"
cylinders.
triaxial
x 4"
compressive creep such that
uniaxial
on 2"
x
creep
flexure of 1 x
compressive stress relaxation
4" cvlinders.
1 x
12"
supported point loaded beams.
simply
3.
The mechanical linear viscoelastic response of cement
paste obtained under the conditions and testing modes
stated above can be represented functionally in such a manner that it is possible to achieve the interrelationship
between the creep tests and relaxation tests in the Laplace
domain.
4.
The mechanical linear viscoelastic response of cement
paste obtained under the conditions stated above can be used
to perform a simple viscoelastic stress analysis of the deflections of a simply supported cement paste beam, loaded
in creep.
The accuracy of the analysis, when compared to
the experimental data, is good.
5.
The mechanical response of creep tests performed on larg-
er cement paste cylinders, is linearly viscoelastic and is
similar to the response obtained on 2" x 4" cylinders.
The
long range response of axial compressive creep, and axial
compressive stress relaxation, and flexural creep is similar
to the response obtained for shorter duration tests.
-56-
CHAPTER iX.
RECO
ETDATIONS
FOR FURTHER WORK
This dissertation slhows that cement paste behaves in a
linearly visccelastic manner when subjected to different
modes of loading.
The work included in this thesis could be
extended further.
There exist several interesting possi-
bilities that this extension could take.
From a purely
materials science point of view, the determination of the
frequency domain viscoelastic response is very interesting.
Such a determination can help define the creep mechanism of
cement paste.
The present work could also be extended by
a more detailed study of the influence of the various variables involved.
These include the water/cement ratio, speci-
men size, environmental conditions, curing conditions, types
of cements, etc.
The influence of the triaxial loadings
could be examined more closely.
The tensile creep behavior
of paste should be investigated.
bilities
The above mentioned possi-
are mostly concerned with the materials character-
istics per se.
The present work could also be extended in
of interest to the structural engineer.
the direction
This type of work
differs from the previously mentioned areas.
could take several of the following paths.
The extensions
The material
could be modified by the inclusion of sand.
The creep proper-
ties could be calculated using the composite models,
compared to the experimental data.
This is
towards the composite study of concrete.
and then
a step forward
This idea could be
examined further by studying the influence of aggregate on
the creep properties of concrete,
proach.
using the composite ap-
This particuar area shows the greatest promise as
far as the ultimate goal of a rational design method for
-57-
creep.
The composite model must then be modified to include
the effects of reinforce ment.
-58-
FIGURE 1. TRIAXIAL CREEP CELL
~i- ssma~ 6 oit-let ta-nifold
dra in
o 1 lgh- prrassu~c.
accjn,!1ator cp
al
Compre ssed lNitrogen
Figj~
2.Pn~Ttic~i~aIic
Pressure System.
,
t....
! ,
'I .
. . .
j:
t
II .I...inted
.
. .:
i
:i
l-:
_,.. -: 1 : : '::' :'I
: 1"
'''|:I :: F T [I .. .7::'
:
:
:1
lit
Lii
. :..
.
: :
::
"
.;.:'
i
' t
"I
"
l'4
'.:".
V
:
I
i
,
... . ..2
..
-L..I
:
- .
tti
I
'
_:j -
1
*i
t "
2&
•~~I-
.... i
.
.t
,
'
i
:
" ..
.....
-
.....:..
...
6:
"
I
'
_
'
i-
.j;
ii
•
....
r
i
-
'
i
:.,:
i
. ...
-i
:
...
I
.
1--i,----
,-!'
I--
t
lr .: ; :
i l:i
! :,
..
,.......
i. ;. ,t '
i
i .....
"
...
. ....
1-I -.
i
""
'
i
t
I. .
I
:
,
.- ...
.
:
i
4'
.....
I-:,
: ',
-: -i :.i
t
i;
..--...
- . .........----
...
: l
~ i ~~~i
,;.
"
-
..
:
.I -----
i
I :-I
FA1
"
.
,i 1 j ! ''-I '.- . '...
1 j,
'.... . '.
I
..
'..
, : i
. .: +
...... ....
i--i ! -!-.--f-....
i ...
....
i....
.'.
.
->
...
.i....---
" 'I', I
:i
aiited saoecixment
! - D:1:
|
i
,.
i - -:
L- i-i
- it
...
-
-- --
i ; I::I:
]
-
: l
'
l, I
l.
:1 ::
I_
I '
r
;
-
:
en..s
~"
:
j.i
,.I.
.
!
JII
.
1 -.
0
.
peci
!
'
.-
j
i_
I _ i
.
.
r
1-71
II
i...
a
t
.
..
i
1
-
'
t.:.e , "-
".. .. ;
'
.I
'
_.i
l
I
t
,
i
i
k
. ..... i. . .. .. ..
Iipoo
i
I
I
.. . .. . .
:
.
.
. .. .. . .. . i i
i .. .. :..
:I
j
l
,.
'I ...."" :
: ""
J
iI
i
lo ooo
1000
•(.
rn
I
.:
i
,,
,
i
"
,
i
.:
!
.I i
i
"
iiiI
t
:1 i . .
,
f
0J
... i..::...
rTO00
;:
I
I
.
V)
-i~ ".
6000
;L-, .. .
; " i. . . . , . .
i.- :I-~-
!
.
:
....
:I
....
i ti
.
.i
:;
;
.....
.
' ". !
1 i, . I . ,-- ~-
,:
I_-
,
, ...
!
"
"..
.....
"
.
.
.! ;
.
. .
i . ..
.
. . .. ...
i ..
ii
I oo
4
S5000
o
t
t.,
4000
I. . ......... .. ......... ....... 1. ..
°-i' . '.
. ..
.
.-
.
i
-t. . ;-t-. ^i( .
. 1. . . . . . ......
-. .
.. l
.
,
'
I_.
i
. i _ . '..-I . ._1;.
!
_.. " "i ' '
: : !
.
. . ... .. .. .
..
... - .. ..
.
.-
-- _
1
i .. . ,.
2000
1000
0-
.
--
1----
--
20
00
Age, days
Figure 4. Compressive Strength of Cement Paste as Fnction of Age
i
. ...
. .. .
.
- --
1
-~1~~~-s1---~--slll~-L-L-------~----~
2
14
3
3400
1
-
I
,,
.,
-
..
ar,
)
1800--
...
.l
4-.
1400
--
-
.
.
-
1000
2600
180
Figure 5.
0
.02- .04 .06
_.
.08
0,
z
St rain
-63-
Stress-Strain Curmes for Cetent
Paste, for fo-ir Ages.
.
.1
.1
20
I
i.60
* ..
I
k
K
*.
-- H
~IL42
*
I
11'
I
j~
~jt
'I
.-II
9,
7~T'
I7T777T"~
ii
-4---
I
**
2.0
K
K
I
' -I
It-
~I
I
If
i-it
I
[7r7CL77Iij
I::.
I
-4-.-
*1 I.
d--4-.4
-I-'
LAO-
~***~
4<
LII
~Ii:7j
174
"<1
.iA2; I.,Ij
7
7F7'T 17FHHFF'
.LHRIIL
t±iiiti+:V+
I
I.
'LLJ I
7171
I-
KI
,.
IliLI~iiK:
l~...fK
7
-
K 777
KLI III I
I.
.t'.
I.
I.;
i7~I~~
I I
:7 rel 6
.ZiL~..
Ij
7K
-'
A
uyRJ
_
Mb'dL tUB3 :C
1V
_
stici
~1
FF*Kj.4.
61D t 2;
V.":
I
i
I
e.
I
~~~:i
;~~!
'
00~
: i
-
-.
:
..
,
,
,4
.
I
:
1 day old secimen
201
i
,
,.
..
...
. -I
.
]..
4
•
[,
~-
.-I
..
•.1.
....
...
.;
•
..
ii
"!
i ...
i'
4
1 ,01
.OZ ,/,
i
i4
,;
,
.
.
.
....
.
'
"
.
I-
.
. .. .
i... . : ... . .. ..
'
;
.
.
.
i
,
I
...
.,.;!i
;
....
'
..
...
'
61
I
60O135Z:a
!
.
i
i . i::
.. . .I
..
,
I
:
|:
..i,
:
Time
.0
-
..
.
100
_
-
...
.
I
61
.minutes
I0
.
61
101
121
minutes
:4 **I1
:
1
..
.
. jj
.:
.____
_1
_
_____~~_I___
_
0
"
:I:I
:i
....
-
,~~~
.~
i-:.I.¢
,.....
ii'7--
~....
....
......
.. .1:-- .:
I,
iI.i. ..
,
~~L~j~
i_ :.
7
I
,"
________
..
-:--.... - I ..
711
"
9-h-
.,-:
;
i:-
.i.LLLF
1
W.ji
Lo
-- ,i,_
I
" .
__L,
t7."
-
;
e I.
m
:-..-::....
•..'..1..
. .L
-~
-
...
T-- -- ......
o
I
::
it
Pr: ,C--.
I :j-
-- -1--
r
-
T i,
I-i--
-
._I.:...I;l.._~i~..r--_:
-^e
-i-----j-----
: 1
t I--- ---- i
t-- i l
A
I
-- r--~---~--r -
I----L. y--
,
:t-
--- ~- ~
i4+
I ~7iL~__ --I-4- -tift--'VA-v- r-t$t>j
KI:
:
i
w.
-
I
-_._~
---I-I-;J-L-;~.
: : .1:-1,
i
j-
------
..
i
I
]
,
........
i --- --i .....
. -.!.--------
i
I
77
I
t
ii
--
Ii-
;7=-:i
-
7---
-
-- i----
--7
----------
-A
a
T ::
0I
:
:
i
.iT
M
.
:11:::
t_:f
..
-
:: I
;
1 4
.:
--
i
i:
.4:
' i--~
I
I
1~1--
-1-. :_~__ i
..
7
.
a
'777*7
T:~-
~
iii::
I
t
i
-t-i----
Tf
:
q
i
I:'00
4AC
::
'
---
71,
S ec:
:i
7;
w
7
:t7-1.
7'"
,
m 4M
:T :
zo
to
60
Time,
-67-
E
minutes
(00
120 . -
140
-
-1~-
-: !:
-.. "
-r_
i..
:::; ~
-:
-
..
..
--
-
-4
i
:I
:i:
r--v
--
-----
-- --i-- --'
.i----
-:. 1
'1 ''
:!. :: :::.1::
_____
I
l
I
-,
.I
-- -. ,--I
.i '- :
|
7-
-
'
t :;.
__
I,:.
i-,,i
--
!
*
---_-: . !---i..--;
:
V.7
--
'
_-
- I--
-
*
'
. . : -. . .
: . . .....
_
_
I1 _ _
i......
. . .1_..
.. ..
.
..--- -T
i
U
--
- -
.....
- : T . -I,; i.
-~
iI
,1
i:
.----
.
. -_
_
-
:t-I-t - T -.£:L ..I- --- ? . 1 -i-t+-? - ¢ -
41
$.4
4,,
0(
-
, '
.-
;
"
i_i
:
_- 2_
_-
_._ .l... _.,_
-t--0
. .
: ::i -!:
_.
.
l
:
-
,
;
"I
""
.
.i
.1•-
60
ii t
~.:
:
: :
i
,
t.
-
-
i
l.
:"
-1
:
t
1_i _-i T- . -_.__
,
. _L
60
Time,minutes
-68-
-
t
.-:Og'
tZ" !--i:
".
: _' :
f.
:
';::
.
1
-i-7)
.. _
_
j i
, ,, .:.
i
l-k.. .L -£-.'
:
: '
_2 _£
.....
i-; : i
-; - :- -
:
! t i:-
n,
_ -- La
'
_i- --
. "" : t
I00
.
- -r
IZ0
-
:! i : :-
:-
.2
:: '
140
TOO
-
--
7
'
,
..4 .. ... . .. .
L----- -- . . ...4 .....
- --
- -4- -:---4
II
_I
~
I
-
----
i
days oid specmes
:.3
. ... . . .. . . ... . -
... .
-
-
: ........
i ..
T- -- 7-
600
-- ---.--...
. - •
-
-
I
...
...
S 0 -....
?
- +i----: -4----.03500
+
- -t
--
'
•-
~
•-----•
---
i
.
,
.0o
-i .
- --
---
:' --
- - '
0%
... i.:
-
-
"
---.
L I---
. . .
t--
-00
--
1
S1
..
.
-
100
I
___
..
"C.L-_
100
T me
11 - Str s Re axat +o. .with
igur, -.
. .. 40I
__--.L
100 ---.:
s .--nute
I- Time,
I
--.
....-+
!
--L-+..i..
..
i60
j:
.. ..+
..+
..-.
-
0Tme
-69-
nat
i0
h:
4
-
L
I
__.. _
~__
_i._i;
:
--;
:r-T-I +• I "~-~---"I._ _ _t
' t i_
------ i--:--- .
-,...7
900
TIJ;
-
-
I
±122412•
.
.... 14
o1t-'
'___
-.
Si
80C
1.
-
-
:1
------ti
,'
:i_
700
t-'
_.;tJ_. -
_
iF
I -:
-----t
-i-
:
. I:
I
.'i-i
...- :-
-
T
F
i-I
4-
--
el'
.ys-
'
i
x1
- --:-- -- --i.. .t----r--c"----- -;T
60C
I--'-
" .--.
-I~--- --:-.....
!-"....
_' i....
. ..:
-j -li:tl~-!-.i
._ __t_.i_
50C
-F
7- ~i
_.. ..
1 --_7
-_----_-:--4-- ; 1.
-.-- :-i -:-
----------II -" ----i
I
I
-4-,-
-:--ii :_L _ ....-__ i ! !
Ii _ .
- r- . -i--r -
'
i1
--:.... f...
......
.-;
; ..
.i
.. ..--. ..-0b17Z07r
ii
30C
i
,
:
4~
..
- --
..
! .--
I
I:--
i
- - .!
1- ! -..
-~_
f
-',- ----t---:
- -- i-..-:~ ,--~.~-i--------1
r
:r
t
i
1_
_~_~
iY
-- -~-i
.iiH
-
-
2 0 ) ~-~-
oc
.--I.f
..
Ti
I
I
:-
I
I
t -I----r--C
I
I-I
I
I
I
r
'
60
Time,
-70-
80
minutes
100oo
120
140
:
-
.
iI--1
-
.....
. ..
•: ,
i
'
i
.
!
:
-i:
... . ..
...
'
:
i
~_-.- . . + .....
~.. : F..
:-- ... -- -
--
-
-
.
I
i
i
;
....
+! •
ll
.!
i
i
i
:1
ii
: _. i
"
. .. . . .
! ._ I.
:_. I
i
. . . . ..-. . . -I--
1.
-I-
.....- .... . ..
.
j
'
+i-' .....-
.. .
.
I
~g_
~___
~__
_
,--r
.. . . I
1 ...
. . . . . .+
- -
,
._: _I - , . ..
l......4
......._--
__
_~_~__1_1__~~~~_
__
_
I
I
I...
."-..... - ....,
. . +---- + i - L..
- . . . . . I.. . . .:--- i -..I.I!
..i .i . . . . . : i ....... !- . . .- . ...
! '
. ! ? :
i.. . i4
' :. .. : .: - ' ' .!i .. . .1. I,
I . .. . .
7O[ .. . .j
i
t i
0
1
'- ;
-
.r4
i
! ,
I
t
i
,
i
f
i ..
.....
...!_.
- tT---i-
______________________________
7•771
i
4)4
0C
-P
'4'
2.
i
I"
0
i
~-. ~:~
_.___.
: i .
..... .
-- - - --;
i ......-
i---...
.........
; .---.........:
....
--- -
- .
-
"1..
•-
i -- i
+
i------
",.....---!. ...---:.....
.
4 ..-
..!
7
F
, --
-
..r-
....
:l'
i
.
i
i---
-;"
i-..
. :.... +
"i..... _...
....
..
I
..-
--....
...
'
. .....
-::-
'" +~ :
20
... r"-
-
..
. .
.. -- ....--:
........
+-
!
(
.. :
_
_
, ..----
. ...
...I . ..i ..
I--.---rI
---
+.
0o 1
:
4
+...
-- i--!
.......
.
-:
--
"
.J..
i
:
...
.+
+-
--
.
...-
Tim,
a
-71-
m
'ate
:
"
-"
.
t! --
- ....
r----i.
..
---
..
i_.l.
. --....
i- - - 1........
!
._-.i
--....•.~i--
i. . . i
;1- - .. :
I0
..
...
;. .. -
..... .....
-_i--
~.!.....
"
-i !
...I
1--T--T! . F
i
!-
!-I--- :r.- ;-:-----
..
Li
!
_:.
.. ..
..
I
........
-
,;, ---
7
+i
100
.
'1
!
--
i
:
I
!--- :+..
..
-i._
i -..
-:..
... ...
' -.
:i.. : 'j
" ... _
l:---
. r .....
.. . .. . .
....
i
i i
-+---- ,-.....
.....
•1-
--
:
"...
.
~
~
1
i
'
i
i i
.....
.........
401
i
.
• :I ii .
.d
:
1£0t4,
I
1.......
.
-
-- I
---
-.-
'--E-
-- 4-1
c
LW]i.
I
-T-
-I-1--
4
* -:-i --
900
I
.
.
-..
4_i -
-
---1~-~'
--- .--.-- (l-I-----
---
I-
-- I-
I
'-I
i:_._If
--i r
~...
C-4-
K- -J..:-~
AA
YN"
4 -
i
-- :--
jJ_-
i
,.....
--
7 --T
-.
4r
-1
i
TOO
i ~~~~i1
I !i :1
i-: :
3-.
i
4
--I'--j
1
• -_ ....
-- i-I -d
.. . --
----.
..
lr4
/
700
ZI
W
I-
h----
(44J
CI
'.4
I .
--.--- 1-.-
•
'__
---
K
1+/Ix
/
'-i---~-----'--'
T-~7
-Ti
----------1------
-_
7
I-- '
i-:
i..:
_
:Ir
--:
-!I
•~~ -l
--: - -
t
:-fi :
2:
i-l-;
;;: i- -t;
i-i:
:-.:!::i:
...
:
:
.(
'
iolAt spinenj
Ildey
---11 Ii
* 400I0 -
,_
IOO
4
•,
] I ._. :
-
_.
:
.
- ,- -1 :- ,. --
_--I
b
r
-
r:i
.Jl---
T
300
II
o
-i
--. .
.dI
.
.4_._
. .
.
i
I
.
..
.
i
I
V
---I--I
i
'
-
- . . . . - ---
Hi
I
f
I-.
:i~~~
~~:
i i
.
-
! ...
H
] "> .I
. - ,-
. ... ,i ! .
:
-
-: - - --. . .
ii
fi-
. -: .;
_:,
. i I . 1 "
-- - - - - l-..... : ---... ... .. . . - -.. .. .. . .....--;- ---..
--";i.~ ~~~~
.~
, !_ "
i
'
.
.
.I
!
"i
i"
.. . _ .. .. . . . : ... . .. . . .. . . .. . I .. _'_-._ . . .I- . . . . . .
- :_4l-_:.
0loo
"
.t--
- "
'--i--
!- i.:.
t::i -:1.
O
•
j
----
iir
-
i
-44-.-
t
--
:
^-1---
-t-
200
)
T
-+-....
i---.. . I :i : -::
r
C
i-C
-77
-'
ii -.
'
i' - :I~~~~
-------
.1: -
---
I
.OZ
.03
,04
Strain,
-72-
.05
;
. .
-- .
~
900
n
W
.
0
0
00
0
•~
0
"
•----
~
/----
---
•
I
,I-
0
(00
too
Figure 15.
.O0
.0Z
.03
St
Stress-Strain for Time Isoclines
Strain,-73- 04
-73-
.05
.06
_
_
000
.1
"
oo
-. t
00-
.0
...
:. . ."
.. .
.
-
:
,
.90C-"
400
.
U,
.
.
...
.
-
...
.05
...
.
...... .
l*
:104
...
+'+.
-
I
"
- .
.
....
.
-I
+
... .-..
.
...
-.-..
-...
...
i..
.
_
:-.
_.. .
- ..
- .. ....
.
Figure 16. Stress-Strain for Time soclines
-
g
-.
- .
......
+- . . .01.x
i
..
.....
tI
100
c
St r
i
/-
...
--
.
•
.
0
---
--------
--
: ..........
. ..
. -...+.
.....
i..02+ '... . . . .03.:. . . :+....... 04....
..
... .:..... ..... ...
.. ..r--. ...:...--.
o
+
"r
+~~~S
T in +......:..
.. ..........:r...r. .
-74-
II __ _
_
100
: '
CTj.
:
i-:---1"
; -__
:: -
::.
" __: I _t
A
I
90(
:::t.
- ,I
8OC
___
_
__
]
i_._
l 4_
7 1 -;
.
700
.
12
,"
...
.
:-
1-.
.
:
.
1--
:. "
.
. -
' "--L._ {1.. .:-
it-_t__
:..
r- ..
i
. -i ).: -:-:-.. .
1.-11
:
} -!-
ir ...
i--] -;- :
I--21:---1 . :1
-1 1
i--li
::I : X . -I'T
I, II~_
K1
1
1
ji
I
-
~-
__
-----r ------- t-.--: -;-......
~ :; 7
--
.
.. .:
... - ..
.:- -: ...
.. -:t...:-=
-- -:.. : ...
ii, - - - ...
..f-i
--
•-
-i
:
__
:
-- -I:
500
~::
- .: :
:!_:_.
_71.._i
_]_. . :.
_. :!.__ii._: _L ::_1 _i:
.£
._
__.
-) -_ tj
---.....
600
" r; -:
: :;
-"--t-,"- '" ' "-!::: ; ...
:
-.-. :
:!
_
,;
"
I
.
.
•
1
I 111 *j ..
..
)..
,f
.
r}~
I
i -: I,..
-:
i
,
11
--- .i_:_.:
I-i:-1
.1[I
I.-~
...
..
--
:
1
.
i
t
400
I
1
1__:-i.l
1li~--
ri-.~_...'i
__: 1' I i
1
300
1
Fi;are 17
-
11 -Stes
.
Sta2r
,_,_,
_cLf---1-
200
100
0
.01
.02
.0otrian,%.04
-75-
.05
for Tun
_-i~--1---
1L-~
Isocline
0
4,
7..
:: :::::::::~
:. .i:
::
. :::
: :
'4
:': " :..
. :7
.. .. .... .
til- .
. . ..t . ..
::: :
::
..
.
---I-
0
r-4
. . ,L
.
If....
..
2.0
-,
i
i7,:::
7.
...
0w
x
r4
I:
:
.
Relaato
..
I
-,
-
:
I-i:
'
Hd.usA
-1
-:
.t:19j
- -
-.
'
ft
...
.....
3 4 ......
I:--~
,nct
o
a,
:
7
.
.-
e d
eI...
w,
C
. ..-
IiI
L!
t7'
.
. ..........
.......
... T0
*ii
,
..
..
o.004%
....
40*
T. e
I.
I
-1------
_I
I
.
!
.
~--J
A-Fncin.otTie
1.0
.
40
I
.I ;
_
60
6O
Time,
100
1
!
mlnutes2i0O
160
'
- ---- ---- -
aaaP~f~---------------
-
i
r-
17-
I
i
,
I
4,
4
l'
.0'
w.;
+.r
'-4 ,0
"-3.1 ,
I-X.' ',
o L __
I !--
0
-
i--.
t i- - L---L
:i
I
.
,
..
4
147
t
~,
.,
.
..
0 e
~o 3:q!~:- i 10 -- "- -..-'. - -....to' -. '-- It'-:+i
.....
!.. :. . . ..
- -(;
.to ~
=:;
_:-,a':
......h . .:1-...............
....,I
.........
......... t;,
-. ..........
(...
- . ,.. . ....
--..
i
1
1
.4
i
+ -ZS.:+t
,+
_'+
!
i .... ... ..........
64
!~i
....
x
t
~
R:C
Ki '0 C1
.'L
'
+ -+
iIi-
,;~.~16
0)
r.J,
7
~C~L~
L~r-RIIU._-C
.- :.......
ii, ...
iI
ti
v
.
,
.
1
*
i
i : ....
20
20
1ir
40
40
i
kelaxation, Moddujl
80 Tie
60
a
Time,
.*
*
I
°
oTmIe
Futil'fTm
z
onnte
lOOmlnutes 11O
fdr the Four Age i at Loaditig
4
140
IbO
T----
---
;1-----~11~
~
0
r -
Jo.
C -rr
""
-e
;rbcuk~L
i;li ~.. i i
l~l L..-'""' --- ~
~~;~~'~i"il""t~rli:iliii.iiiiif
*f
l~t
-77
Iii
...
1
I
.i ! "
:..
, -::.... I__:...-i .l411l-11
..
.. . : .,. :-, " .. _, ..:..:,,.
, , ::':i,'i.:,;:i
. : , ::::: ::::::: ,.' ,i , - ' ... ;. ...i: -::,::.;
i ,.:
. .. .. .:. :.I... .- - - .
. ..
i_..., .-.
fiiiif1ti:Itr
~i~iiliili
.!. ,,l : ....
,- ::;i
- .,- . :
.025
4:4
: i'ZIL JI: UZ: 1;;;
:
:
... ..
.
. .i'.
....
'
~~
i~~~~~
~
:
4.I
I
fil77ii
"''"~'''~''~'''~''
'~ '
":~'~~
.020 :;'!::i
':
'
~
':::t:;i~
!;I"
I1-i
!tj:
;'
II
I
..
, (7,....
''''
V
;I
..LI
.4
.010
2'!.!
I:i,ii
... I
i
-.4
:PT:
.oo
.j.2.
I:
: :IIII'
_:
I:
j: i:-i
t[
-.
112.1 ..).. -I
ii-~ri~
i~..
i -i -j -r-:I
i.,: ....
~I)I' '
f
, , :::: i~
~~
.i,.' '
~
:!:,:::
LI .
. .
: i , : i': .
; ;i
i
i
i i
1ii
i
-
L IC
' !Opi
i
.,....
:i --I
'''~i~''
~~' '
:1
I
i - iii!:
I:
II:
''
i' ' ''''
' '
ol22
f
l
t
SIL
2
'
i
_I
~.. ' I
t'
1
i
...
i.....
.
''
H-.
' cc--
Il;i~i
...-tI Ir ..
::
:
-7-7-
IT1
I'''
7
i
: :;;i:: ::
-- i
,rt, r:1:!iTTTKi
I
i/ : ' !I
I-i:lf~
.I ,~t.f.:
:~
i
_1
I:1
'
1:; ::I:~1i,11
'' '
i1.LJffilKIhill...LL;
-:i 11I
--i'
Ii
'
:1
[ilK
____
__iiii
.. .. ,
~ ~ Iii
,.
, ; .- : -t . .
:...
.-...
-4
::':
-i-"----1---~-::
I
i
i .. ~...~KrKI
i'"'1;;1
I :I:
-v: i:''i
.....
:
:i.
iifii iii
i:,
....
..
: ...
t
:Iii 'i.
i1iii1..
Iiii :.,ii
.
Ii iII.. ._..
--I :
v'Iu~U
:2
,'V
~
~
:
:
:2:~~
'12::.:
..I
.015
:i
I
:
:
:; ..... ~. ~
....
:::
......
....
';:!,::
I'
:I: j: :~lj
i :: : :' I:I
I:
-i--
-f
-i
'''''''''~''~~ '~''''~''''~''~'
' ' ' ' ''-
I IiiiifliiVi
,
I
~
,..,.
::t
i
.
.. ~.I
I
i , i,
,, I
i .,
..
.
..
...
i
i
In+
5
... ;i
I,
'
...
lme,
iourp0
j
I.
: fi
-4:
:::
;:;: ::
II: : ''
. ..
..
_77...
;------
!
:.
t~n~
: ::
I
75
ic^-i
j.1.
-I--,...
-i1
:I
;- i
90
lo
/
i
1
C~l~,~ll~o,~^UYII.-~~I*
IIIl CLI
211111
..
....
~.lr
0
''
_
I
i:.'+,.::.
"'
i
i
+
-..
+..~
.......:::+'. .,.
i...
..
..
: " ::
.o03
iitil-i
:
..
. ......:~j+ i :+
:
t . .'.... .'. .. . . . ......1
.030
. .. .. .
.....i ... .... ....
. ..
7i 7
.
.025
.i
Tt.
.: i .
::I
--
t::
:
,
I
t
f
I~:o-djiiL
:
i
II
.. ...
... ...
J. . .
..
i
. +. . .
! .
'
1
: I . ..-
+:,i
::
-r
-
. .
i .
.+
..
-- -:
..
:-.
: ..:-......I
.........i I..i.. :
..1.,-.....:............+....I-
- - + ---
. - :
,
i
.
...
.
. -
..;.....
:..
.020
.oao
4-1I
'.~~~~~
.015
"" ...
'"
-7 :-i-.j. > L
,0 I0
1^ I
.
U
_
.
.
_.
...
.
.
:ii
:r:
it-:_. iT] >F I -
..
'
..
:
I
1 . ' +:'
... ...
.. .....
-:
-
'..
"
:
..
_
' '
..
."
.
:+
"
: .
'T
...
":i +
i
.....
.
i
'--I---.---
•
!
I
. . .
. --- t----tt-+---C'
. .. . . .
--
..
1-I1--1-I
I I ::iKF~::i
'i/ ''
"
.. .
-i
i.
'
II .
1
" 'l.' I . : :.
"v . .: .
, •___
t, .. .' "_
].:
.,. I "....
.
;.
;
- :
T
......
--- -= ..:
- ::
--:I ..+ . .L _L .i _ - L. . . . ... . . .: . . .
,
.005
0
i
: ,
"
.
'
t
.. .
. " .........
L~
-
'
128 :,
-- TT-H
-
I
'
..
'
7:
. . . ........----...
-. . . . .
..
..
.
..
4 TIme, hours60
.
...
..
i.
..
Th-17
+.
.
, ,
+
90
.:-L-il
i
'~1
..--- - - .-- . .
t5
:
i
''
..
, .'
..
_-I....
i.......
.....
+ ........
105
/
.
12o
'"'rrvrC~illrm~-----P~r~rc--r~*~*
rrrr
---r-~- ----
0
I~
.00
,:
, ...
i-- -
- --
!....
--
-.025
. "
" i:
:
• -
-
..
... .
.. ..
: . ..
.
.
. ..
-i -;-;
,
.
--i--- (- -if
!..
l--
-- - ..... ,' -: v ...........
-
:~::-I-,:::i.
::
i-'
: .
-]
r-
....
i- -
i
.
...
..;.....-
.
!
I .
!
; t
-----
-,
:
."
.
-- - - - - .. .. t - ..
- ----
+i
i
....
. ... : .' .....
';
.
- i 1 -r-
-
"I - .....I .. i[ . ..
-
,- i
L
...... , ,+
Ci:
ZO20
t1i !.
.t
....
: -: .
..
i .... +... ...
..
i..
.015
1
.,..-ii:
i i.
':::
i''. i ,' ..' ' :~i.
;'
.010o -~,~li--l-L-C--t---*--t~
I i I
::'::i:
f
I
;:
.005
I
U ... ..
'~
: -I
~L .-i
I
i
.e-- -~-L
t--~t~-CIC-)---tYt -- ~-+lrt
.
I 1. I
*1
.
2.
.
ig rm 2
....
--"...
. .
x
AX.alC
45
~~i--
:1
---
-t----t
•-t
i
0a
.~-..:~.-.,..-.....
:
..
~--i - -
. .
-
1-
l---'-C-I-.----------l------1--.-- .--.
-_
IZ ps'~
~
"
.
ep $traiss 'ai Func ion Qf Tie.
Time,
m~,,rC~
hours60
I1)
~r-
Q/
Af
to
IS
Iv
Ci
-i
c
-
IL--YII
II~
~,-PI~.U~.~,-~n~-Psr~
I
I: --IiiiI
i-;" lI:
.
.. i;;;
1....
I
j
.o30
qi -
i:i
'
'''
..,
.---,-::::I
oas
11'' ;: ' I
I-~-I
.77
" I
i:
rI
.- I . . . I . ..
I
i
i'
(
!:
:i;:::
I
I
-I.
I
!
I
*-- .
L
,-
02o
0.5
t
I
/;1.1-1.. .4.l"
.
I''
-- -~f - - ::1
...
010
.
.
...
i
,'
~
1'
1
j'---'
.
IivL-~!~ I! ::I:'-- LI'II_~t"r
1::::
I I-i
-: i}:
wOli..
.006
C-t-'-----r
'
:
.
. .
T
ir
1
i-lI
4
::
--
i
t
i
1
'
ii;iiiiijilii/.,
r---
-f
--
r-r1
mq L--;I- -r--- T--~ :
i
/
i~'~'''
i.
... Fj~
-!
6
ii-:--
i5,s
. ~.-,~...
: _..~.-_..I.
-I;r
1
I
.
K.-
7
1
I
'
'
of T
'
I
-+-•
...
W
4Jvl~fe~fOUM
45 lime, nours 60
I
L-~.,--~i-.-,--i,-L~I
I i2bjpr
1
A
'
!- i... 1i. ..
l
t
I
1 9t
-
_________~7--r.-
un
i
iia~sptc
~
. .
rdp I:St rafn! a$
1---
I
!
L I.LLJ
.. .i..
l
i.iLi I T
::_:
.Iu ,e: .!A
j
-. -i- ---l -L,:....t :: ' :
-- :i
T 1
l
It
'
.
.. ..
...
',....
....... - ...
. ...
.....
...I
-.-i
!"
:-!-t
.. i-:--.i
--- ' i--'
I,
I
11*. .:.-- -*
I
....i.-.
/
L
I- l :
_
II ' ' '
'
:-f-:
i
I
'I
15
I psi: " w;
.
-.
--- ~I:I../-
i.
I
i:
I
.
I! '''
iiiili '' ''1
_ri::::
1 :i:: :tf:1::!::rlt
: ....
i
,
G-3 P4_
'
:i
2T- VIT
74K.K..
1- i-1
I
L: V-. ftr:t~il
r
1::
i ---t-i -:-~--:-t-i
7V
i:-
I
tro
i
-i-l
~--- 77
- t-T~~
f -
.
:iI
- :" ..
!:. L...-
:
I.
:--t
:
i
I
.
: -i-i-:*1:i
'1KJ
... i iq." ......
i, !'' '
54
I
_ ...
i
-- - --------------t
i ''
'I
it:
7I
.
. I
i__
~- -- p------i
.
-ii------- c---r---+---,
1*1*1
I i:
4,ij
.
:: :j;
~~.,----
lt
i
"
'1-.
... f~' "-
: :I :
- i
~ ~~ ~ ~ ~ ~~~~~~
l
J-,-t
I
-_1-l-L;.
I
i:::: :: : :1 : :: i :: irI
:: l
1
I-I;
I~
I
j
!
9-
;:::1; :: :I :: :
I~T~I/
I
i::?T :
CILCICC(I
ii .....-I
t
!1--lii!
.....
.....
.. 7 ...
1 -: :I-!
'
I
:
|
I, ,
'
i
1
'
,
i
m
-'
I
*~C
vJ
.
.
.
.
' "
•
,-----------,------
---------------- ~--
I
------------
-4
~.. ,...,.1
=
.
i .,.~
..
;;
:I
,..,.
---
t
I
z. jilzjI..~
i..
i~ I : i
++
i -
I
i.
I
i- r
i
-lii
* I.
4.
TTrri
I:; ++
it I:'
'"~"'
~~~~~-~
+ ----t
=
- i I _:"
~~'-'-'---';
t
I
i
:.. t
-11 1
i-C---t-i-l---c-C-c
--e--c-i-s,.
--J
' -;
"
+i+-r.
.,.
•
- I
..
[.2.
i ji11i
lt
iiii
..
.. ......
....
"-"'- 'j;
.2 +
+
4-
+
I
ec+
I
-1:FTI
pOzo
.42
/
[.!
:
'
I . .....
iid .....
iei .i n 'oi| i:
- i
"
+
I
c--c-~L--slil----c--s---e
i:.
i
-
.
...1..
.025
I
. '
i
+
.+ . +.
+..
....
~~~~~~~~~~
i
:..
..
.I
......
.. !...-+........1.
.....:; -.. .+
..
.
'
1.. I
.
A- ", i;
'l'^'~l~--~t
..
..
l:
.
I. .....
I
.
.._i--;----:i :
0 s
.
i
:
:
;
I ---I----: -i
i
+ , +,
I
Ii!
.015
.
.
..
.
.
i, ...
+ +.+.+
,
,,
.
.010
I
-
.
,. ....I_
+,,
...
:+.r
jI I
l ++ .
,~
'+
-:'
""
,+ L
i
'+'
+ ~
i'
!:I
I
, +,
.
! ^,t,:'
+ t_
..',
' , -c-..
:-.
. . ;....
v!:.IIi
ur+ . ... A
iI
i.fpi
.
.
.I
;
F
.. . ..t ..
,
;
in
++
+ '-- I'; -
:
S. + ..
...
t i oI
...
+ 1
_+-_ ,....
_
_
..--
_ ______
b '
ls
.
+_ ._
.._ ...
____,
_
.
'
.005
.I . +
..
0
90
-+
...
+..
t +..
..
..
..
...
~-+
..
...
.
~
1
.i..
.
.
i
..
...
_
-i...;.
.
I
....
_..
" - :t :
..
...
...
. i~ure
;.... Qs Time;,
40: A.ra +re-p iraj
1 .. ..
nsl aS Fundtlonof.
:+ i
.
.
.. :
.
.
i ..
"
W43ime,
.
i •
hours
I . i . ,
60
.
.
i.
..,
90
Io'
[
. ... , •
-. .
I00
_
,
c-_--
-I_~~~~_~_
I~ l~li~~--~
---~-----
1o000
1
--
i
I
i
--I
--- '-- '
;J:
-i :..
.. i-;
- -t1
'
. ",1
*
T
600
-~fl7
"•.--.
•
~~~
..
.
.
t. ...
.1
t
--
..
:
i'---Th-4
4
I
i
1
;
~.
1
_.
I ..
--
1
1
cl---~I
I :i
-!----r
-
..... ;( .
'
.
171:i
.i .. :
_
--
:
~
j
T-1 - i
:: - i:::
::.-1:::
": 17-I--
i
-
----------
f
1
:
:
!-:
''
77
I !
'
-_" __. .. . .-
1
:
i
. ..1. i ...
..
_
_
.. .
:_: ::
. ... .
_
.. . . .: -_- ..
.
.
..
. .-
~-
I--i::
TOO
-6.
- 2''
i . ..
..
d ai
yi.
.
,
-
cc
~
7--,.-
4-
t
c:i
___I
i ,-__::' t- .
,to i.:
300 - --:.'.i ....
- .-,-"
- t~~~
-----.
.
..
.
! 2
_r
-- -i-:
-.
,:
~~ :~ ,7/
~ ~~;
,
8
4-:
-~~---------
-
i
I
::
.1
.
-,
" 77
___,_-_
-. : ..! .
- --
k
-
: i
.
____
Iji
. - --
1
<z : i, .: : ' :-1
--I
ez
' 1
_
_
. L::I
I
i --
.1- :il
-::-t ;::
-
'.i
.
II
--
-;~- -c-Il~-.
.....
.
i-r--
Strain,
.04
a
I :i
__
...]
I
I
I::
'. .. .I... ..:
.:.f:::
igi ag 2-:5- Stress- trai4
.03
__
!
t---~--
... ....,i;-. t
--I $-
-----
-- -l
l
t
-_____
/ .
___
. t -. :
...
...
. ..:.......
......
:L
-
-
:i , I
- i ."I
.
' ..
-...-......
--: .. : r---i..•-T
".];, :
r-- - q -:: : i:
:
i
. t.. .1- 7 t (:t
.
I00
(--i1
i
: r ..
i ::
- . .
0
S-7:
:#-r
I---i
p-i-
---:f
-- --------~
---
_--1
_..,400
_ -_
Itt,1l~it....
71/ Ii-
t~~ij_
or TIe
...
socI
1;_ 1:1:-'
.05
-.
...
_
.- . !. i
..
ies
'-
_
.-l~:i...-L~:-..
-
i
I
--
I
•
i
.
;
04
A-.
i
;
:
...
...
4
I
pcr~
si
..1 , -: ...
-. .
.:ii-:i
:
i ii-
-
i
-T-[--
d
6354s
,
7 ,--]- 7 -
-:
''
+i
P
-1-i
I
T
:
-
_
-1-
:
.i
i
ii
.. .J
- ..
.:.....
..
--....
i-'
----....--
r7
--
-----------I~ti.
i
--. ,--,
"' ... ' ::::lt
::: t
..--: ..
r ...
::
1.: .. ri .
--
to
41J
500
400
-:--- --It'------II
-7 -
--
-7
1- -_+- •
------ t-----t--c+
--- 1-
I
I :::;
--1
-
Z
F--VF
•
1 .-
77-
i:
i).r z::I:::- T=Tf
--
i
_ ,,I::
:F
-:I •T-
I::::!------
a. . .
---
:,
7
-7
1:. 1::
:: i
I
+ -I.+
p$4-:
.1
-'--:i-
i:I:
1'' ~'
...
:: ;1:
"'
11
:i-
i
42
-L~;i-C--L--(----C-C--c-3
a'I T-!
.:
7
i
T
i.
t
itll
~-~--l
1
i
-----
-i
I- c-
:
-I ".--
F:
.4
--I
iiiiF
11
i
iLf
4.:1
.. .
T
a+
II
iie
a_
~
....
.
:
41'~
-2 " !
17
o
S.
O2
03,
Strain, o
1 ..i----
--II ---
-----
I
I
------
:i : :
-7i .
i
1::::
~~~ .t
:::'i~~~~~~
;
I
: ....
: :i::::::
== ==== ============
T :.:t
. ::
::
":--iil:i+
:.: :f:
: i. : +:: '--l:: ::i.ii:.:::
--ii
iii
::-.1
: ::
: ::: -.
: + : : :: : ' :
:::::::
::::...
I
..
--
--I --
I :
7::
:::: ;_:::
:r-l! ....
......:
:I:::: ~--"
:1-: :
:rl:1::: '~ ~~'''
: :::'1:::
-t -- ---- --- "--""
j
m
-..1::: :
wr-
--"'---'--
:."" ~'
~iiilii:itiii'
:=-::::::::
:i::::::::i-f
::::::., ::::
-:
:::I:::.::.:- :
i
i
~'~~
~'''~'"'
''
'' "~~
'~~' ~' ~~' '
::::I::-
:r:: 1:1:~lt .. ' ftl:
'';~''' '' --: ::
: : :: : : ::
I ::: i::- :: .t..
''' ' ' ' '
:::
l:
::
iI:-f:::::: -+::===
===== = ==:: ::
w.
ifSL
:
iii!
-~-- "
---
:-
1.-.
:1:::: 1:r- : : r
'' '
:1:tr::
: i: :
:: ' :
S .........
. .
7-il
::
j --~~~
-- ~
~-~ I: If "~:
-- i~
-r...
- 1j
--
t_..
.-.
w
~- - -''~'~'' ~''
:: : ::...
:: :I::::
i
1:::: --
_:::
:::::+::
-7:
i:L
i-+-l
...
:iti- w.:: : i::
:1::
j
t-i
: " 7:
:
r
'' -7
:4--
7-t
1 i-' :ii irf-iil.:11
.~7-.. i f
llj_
:1::1:::
....t..
:jtiiiitiiii
:t.t.rlt.
---II:
'
~
''''~~''
'~~''
:
-~-::
itirlri -- ii....;I::::. ~~~
iiiiiii iii:I ii
iliiiiii ' ''"' ''
iiii i..i::i i:
ii
:T::~::
-C::::!::.
77
..........
-ii:
i' '
777--.
:: :: :: :: :: :: :: :: :
: : :ia
.:
:
7 7
: ::
-.. ...
..
.....
.. ::::::::::
q, i
iii'1:::
40
'-"--
~ij
-iiiiIi -..
i
i
::
.
iliii
I":
~~~~~
""'
rt-r
~- : i: : :. ::
:i ::::
~'77TT
m
iit :li
"i
I"
1
7-r-.--
1
t:
:::'1:
;:
~1
ii
~
iiiiiliti
l
-_i~iiir~%i l~i liii'i7::T.ii
. ..
....
i~iiiiiiii~d
Ft::r:::;::
I1:::r:
..
~ii~j
::~fl:
il:il
ii
:::: t::
--:: t::I--
----.11.1.
:: :: :: 1: L::
7-:: '7:.1'c-
.02
.04
.03
Strain, %
-85-
.05
-:!:::
7-7-
Is______________LI~__11_1 _
_~~
-V__I_
----i----1 ----
I_
IT-I
1
...., .~...
I
i
-
1:
I
i
.
. I
, +i
__
___
__
__
__
,__
_.___
___
___
.. +
U,
,
I_
....
_______
__.----i_____
_+_
_4
_____
_+__
1
__
..
.
-.
'
I
L
T-T.--+-.+~.+:+:. .+.I:+
--
C/
_-
']r--F-.-.
i+
7+:+ +f _l..
....
i :1:
.
'I:-:---:--..
--.--
- +
1
I
+
"
.I--- --........ +...
....
i.
'-
+
'
--
+
_____,
i
.
1
__.
+_
I,
.
: i
_:.
+1.:+. :
_
._j~
~
____i
500
Ic
28. 4 .
i
:
___;
_______:.:
: ,
i
_
_
_
_i
U,
.
,
j-
€
...
.
.
,
J..-'
t
..~,..
.
t
1
1
__
+_ "+ :
! _._
:
_
++
:
I
:
-7
*
----
I:
'...-_.:_: +.7 .
...
5'...
__,_.
I
.
' ...
i ..
i
_.
....
4
_7---- ----t7'
'----+ : '
:
:
.. .... .4
1..
+ +++t + + +..
j+
_ _•_ +__
_ -_+
;
;
i__ : ]
:-!...
-tI/!--i -j jl ,+ +i
+:-i _ L,_
.
ti-
-t
+
+
: "+
+" "
. :
i' !.
J:-:
: 'i :
:
-:
:
~
4str1Lnf
--
"
.
iV
~
S .0
.02
:
..
_J
.i :i:-. +
7 - -.....
----
10-
__.
:'
.03
Strain,
-36-
04
.
n
-I'j~o
f'T_
_
K
"i_
I 12...
~j
Ii
i--
Ii
't.
LLL
~~
i
,-4
JI
JiUL
I~ji~ii.-..I-....
0
c
C)
/ljy
~
i
CL
I
ii
I t::-
I-4
Z tILL
thF'
. 1
-i6
4C
I
. . ' 4 . . .I
25
i
''' ~- "'' I
i:
+
..--I--
'
:I
t
30-'-'""
30
4.-7no4
I I 1 i
k-T
4,"7ixne~hours ~O
4fime, hours Go
-L I---t--- .-
aa4, F
75
75
f- f
r120
_ _
~i---
- II
I
...
0-
c, ., ...
;.C-)L
[.
li--
++~I
:.,+:,
I+:
i:,;
+i,+'
~-
t-%1t
..
t- ~ 1.1 o
i-
uo - -............
- :I
---t4 ---t.+-r-j-, xL&'
. .S
I
e.
e-i
,
..
....•---/
l A:.F.1+. .
iii
,
a
4
.. ' Ii'i
....
I ...
<
+
.........
..
..
+-f-
....-iI.+I
,t,.;.
Fn
I .
.
i
+T.y
t
60,+,
.
.
'+i i+ '+:
'
t <..i
.
.
r4o r--- --1--i-.-i
I
,t
+
90
I ... .
.
.. .
:. .
- ....i.u
.
t
M,
c.
r-)
..
.LL
.- I.l
i
"g
..
.- i.
-- -
...
..
L.;....
.Ii
.. ..I..
... . . .
(X
.r
,,.,.,
._ ..
:,i-++....
"
t:
r-LO
•
+ r -+....
+"
.. I
3 10
...
i
.
C!
.
. .
,
....
i.
-
,
..
A'
. . . ..
'
+ . • ++
. .
',. ,
......
+...,i. -
-"'
. . ...
-.
+.
-*1 C3
T
.-l--+ .
t: t+- -.-
+ ...
-- --- ..
46rime,hours 60
-
i-i
+
.... aj 'F.
'
•
..
.
. . . .. ] . .:.. .
........
t
..
~-- -..... ...
il~. . ...... l .... ..
-i.. l
..;.... ..
- -...
.+ -+..
I...
. . ...
i ...
.i. : -
:J!4 L'.
....
-"3"
I
+:+++
:+
.
i.
,
00
...
,
.:.
..
.. .
'I
-+
--;.i,
...
I lIQ a S---i----l:.:.t:.
:t
i -- ..
... .
...... . . . .. I
I
TS
90
" . ..
1
ILILIKI
: i:e
17:
i i i.t .
.02s
,; "
.
,
,1
i
'1~;~.I.
ir. ~i ~
Ifi~:
I;
--'....
't :'i1' tI
---- I-- -----ii
iiiii~;
~ii;i ic-i~i
~-----m---rrr
Ii
,,1:iI
.
:r ;i
i ii
ii
,, i .. .I .. ~ I .. I
i: i ii
Ii ji IIIi (
.02C
_
i ,I ~ I ~ I ., f I 1 ~
'' I
i'
' ' ' ' ~~ ' ' ; ;
,
II i,,.
t:
i-- - ii: --~wt---rt~
--rrcr~-~--7
;.~
::
~;i1ii ;:-:)(1i' Iii '' '''
I1 ''
____
~~' ' ' ~' ' ' i
-- ": '
~ r:
----I I -r-T
1' '
'I
i-C1
I I ''
::
I
I; t ~
~
:i:
:
..~~1~' ~I' ~i
-
4J
.015
5:
ii4
I.
!
:i l
t
t
j
:1
!W
0
.00
t!
:Ijj
i iiimep
15
iI
30
ii
i
w
::I
T
4S
' '
:I
I
hour S 60
.
::In,
::':
75 '
''
90
--7t- 7
p
,4
.01C
s
I
i
IEtf-,
i
~I0
.l
105
Ir
12I
,..I
.035
.. .
r...
..
. . .. .. .. ..
.
:.
" ' ' 'I
" ' "
i1
c-
.030
,
-TI
----L
~i
- -:
:1
I I
~ _I_
.025
I
'
,.
. i
*
(:
j -t"C:
-- t--;
'~''
'-~~-'~~~~-''~~'''~ '~''~"''~'-~''-'~''
C).0zo --cylw~--
----e-- i
. ..
r:
i : (R
-t-f:
C.
.015
:
I
-
---
.010
.... -I
i
iii:
i
i
i-
1.1
. . . .:. . .:t:
l-
t
. "
.
:
-j
1.
I
t
^"~'
.
'
.
.
--7',
i
i
::i:
I''
•
'
i
.,
... ..
:
I
I_
.
-
..
:- :
i
. . I.
I_i
-j
I . . . . . ..
'
,.. . , , :_.
- -- i---- -
d
I--
.'--I
--'
"
-
-
s
:f :i
- t
. '
2'
i
.
i
Z56p_:~i-:.
-
i
;
,!
.
|
..;. . ..:. .
It
r
..
.
'
2
I
'
'
.
I
.i
t 7-rP
~
0
'
1
i:f
---t-
i:
1__1
..-.I:,.1
i:
, :: i
IIII:
.
.
..
.
I
--i
-----..
+. :.' .' i.....
• '
1 -I I:
I !
L----+-CLCI-t-e--J----t-c-C---'------C----C--( ---~-----( ---
;
i
.
. : ..
V:
.005
; i
.
.7,
.1
I--~-
c -
r
i
11- 12-i
: :
--
,
VI_.. .si...
-4-~
T---- ::
.I
:
'
--~-:i :: -----+
i.
--- r
I
-7
.)
I. I IL
...
r ...
.-.r-....
I
I~
-!~I - -I----^. -r .i --I ~:-I
i
''-'~'~''""'"~'''~' ~'' ~~~'
1 7
--1
,.I .. t .... I
::I::
,...
_ii__lil__~___
__L_-ill___i~l
i: -. ::"t-:i'i: ---I:_11!
--------- ~
~ -. -~
-II
1
-3
1 --~--7
-I---1----f~
I:--~I
-----
i .
'
I - -t--i.
,-.
iiiii:I:_
II
I ::
i
i
--..t
:.:I:--.. .~..,.. .
.-- t I --~--~-
11/
1 *:l:
I-;
I
-
77
i . I
:
I:
I II
~ -~ -CCCII~----* -- C--~i----_ --::
i
-~
i.
''
. I:
T- ~~-i-l
;I
t::
:
--
.r- . ...
.: '
:!'
:;.-1;.11--
---- I
--
--
i • i- UILLI--LII
, T -• ?
. ..
:. :- .
slL^--c~-c-~---tl.
j
:
-
" -' l
1"
~_~;
': ! "
-t-+---Clf~li-C-+--L~
. L:
I
'
i:!!
,
I "f
--
p
iS
S
s 60
hou
4 r meI
43Time, hours 60
..
1jliLi1-1-4.,..;ii
u,4ton V.
LUht{n
oe
.
iu4< 1
.
,,,
•oii~l ii
....
:
... . .
:i
i
: .
i
:"f-''
;
•
,
l-
" ":
-;!,
,,'
..
.
. . . . . .
.... : ! ,. . . '... . .. !:.. . . .
i 1;: [ ! . '_ _. __ ... . . . __
.0 3 0
i:
"':'":'
': "":
"''t i:i
!
i-t i
,
:t
I..
i
i~i...
i
c~
-tL
.
!-.'
-i--
•-
......-'
,
':
i',-'----
..........
Lo
inp
i
h
.
'
irs&0
......,
.
unct.n
o Tim
s.
. .--. .
!- --
..
i. .
.. . .
t
7590
f
'-'i.-...
.
.
. . i .. .
. -.. -- .
,
--
I
)
~i
I
i--l-
--
--,
e
I......
.ii.
.. ; .
.
, .:
. .I -:. . . :. .
i :
4 - --
i
. .... . ..
i - 1I .
.. .
,
1i :::
....
....
1.
....
: ,. ,.1...
....
..
...
..
-
!
..
-.. ,
A .
.015
1
s t . oc'
. p"
. ..
: ... ..........
.2j 7i....
Ii...
-t. -"-~::
i ... .
:-
!,
i
.i .
| ... --.- ....I ...
i
.
,.
t---
'
..... , . .:.... ...i _ I
15
12
0
Lii.-L 4 ...
:;
:.T :11''
:i: ... .
I
I:
.035
.030
*I
m
K1TfTI
1 ' ' 7ITITKK7V
'
-j
....
I'
"
4
.
*
--
...
.....
T:T~
-7
*7:
....
....
7.,.
:
:7
.025
77
ii
v-
.0?0
: :..
Li- Iw:
I
~trI
1:
psi
I
71- 7,n*--. .........
:;:: :;: ...
'
..
.
.....
V
..
-
... ..
7,i
..
........
i
...........
*
...-
.
C/
t._.-.
AbL:-
.015
-li::-~
--
i--
i--
.010
[
.005
_
30+3Time,houra
60
75
90
105
120
0
.035
.030
.025
.020
ca
4.I
.015
.O00
.005
r
--
1000
-
- __ 1..
I,._ _.
-1--___L___J
-
.. ..
:!
--
900
i:i;
-. - ... . -. . ... -L_
....
__ ...
*.1-
-t:--
4
800
ii
.-a_.
I..l
L
1
I.j. I...:
li
V
L
0r
-:-
K-i--
700
-/--------ir~BI-till
to
600
6w
i
4i-i
--~
-~
pj
... --- irt.7
V,iHL
iy:
i i
cl--T--.
- I --724
i.
,~-
.i
• :1","
No
i::
-
..
d- t
. Ape
I hyold
ape itten
L-___CIICI-7
~~
-1
4-7- J
i -:
:i:
i - --41.;
--- t----l---!
i:
-C:---
i. -
i:
___
-I
*
Ii
( 400
i
L
U
82
I
ili.!:ill
pipsi
3oc
j
20C
I
.
.
.
..
. .
t
,. i. !
..
.
126
.
2i5 ...
iiii~-k':i- -1
1 uz:
boc
.01
.01
.
:i
.03S tra inf
-94-
1
i
I
1--4
____.._______
1000
• :.
I
I
I
---- r-l
-1
---
} I-
]
/
.! ." 1"
i : :: '
:
' I i:7;--! i - t:
-7
.....
L-~
-_ -.L--L.-~...---- ~-~-r-
t--
i-
_--i-.
.4i :
i
F--
•
! !
:i
-- F---
800
-L
_.1~___ .__! -4
. _I I___
----c---e---l--t---t--+-
f--t:::,...
._.:__.~_I ..I~__
~------L-
C----
S1
C-3-C
-- -L-e-;---C-----t--;--e
7oo
!
]
1--1-4 ------: i;
-~-ii
I
Soo
i
It
_
:1
I
-i
I
i
4i
.------ .....
_:
-I
.i-c--~
- . ...
-...
.
..
-3--
--li-_ ----t----i::i
::i::::
:I-:::
~...
700
4
i j -
:~~~_
-t
1--I
i
?-
4
i
Ls.~as.
r
'
1
'
i
•i
ix
100
:_-___J
..
----
"
i
i
S: !
-r
ii~~sBb
i',
.04
.03
Strain, %
-95-
I -
i-r----:i
i
t
i
t
-
:--i
i "i------
- :i _
:
1
I_
__.
_ .-_ __...
:.Z
i:L:
L
t~mi
-:
i -:
I_!i_-l
i
------ --~--:-i
•-il-
I
-S
~l.'
,,~.-_-___._,__1_
'
4
.02
.. .
i
i
:~, ~ ..
~C-L-l_--- ----l
"
I..
171- IF
.ol01
:
, ::,
S p @-lle.$-P, .... i
t
I
r
._.__-~_.t
.
:. :
i
...
.i_
!_-. ..
. .._i _:..
l---i
--..---
resl
-
"
t.-
--- H-
-----
/
.-2..
__.:
i ._.
. ... .:...
..
!---...
- _I
li_
i---^--~-~
' { _-t~_.
---,.
0
S-Ood.
,
'i'
i i7
i
1.
t
..... - -
-.
....
00
+
I: I/
;
I
-4
200
C40
..
i-
;
::!
- i
.oo
t-i
,
1.
.05
i .
l
:
-.
i
: iod
~ ine~ti...
ii. ,. i:;
_
--
YIIIII~LIIIII~III--c~-_~
1000
iooo
--
900
7
TT
i-
.
i
-c
:
~1
.. ....
.--
--
~
-7H
i'
800
I: -
-7
ix;;....
700
--
--
~
i---i
I
~
I
...
7
--- , T77 .......
7-7-
i-
t
---
-
-77~i
-74 ......
.1j
......
....
J-
::.....i
..
_..
777-'=:
i:'
:
: : :::: __, _-t - 1)01
40
777
t
= r7
_77- 777-7
.i
i:
--- ---- --:' i:
1
~~~- -----
-
;.:1 -:
---------
11-!::
f4-
: ; :i:ii
i _-
-t-;---1---
;7_!
3O
20
-.~.~:~~~
"'"~-"
.04
.03
Strain, %
-96-
--
-- ------ ----.05
~--------
7
I.
~
- ---------
1000
900
i
*
1:
I
{. ----
iI
t
4-
*
600
----f--i: I -I
.-F
700
1-
a
I
-7 - 7-_
71
oo
oo4
4J 600
ri
..,- i
.
.1.:-~
!I
t
iT --
I
. .; I
-i-"I
1
1
500
1:
400
Ii
-I
...
TjI:-T-:£Tl .
I ..
'
II
4
1
200
-I
v
*1 _I- .._,.
i:
+ . _: i _!.:...
S Ii
1tT
IOc
_
2
- t-Isl
I.
urj.4.St
reotStra - f
I-- ----
.02
.04
.04
.03
Strain, %
-97-
-
"
i---.-----.;- .-
.. -.....
:.
:'::i~ l:
:.
60
t
tl.o
ii
6.0
I
j
I
:::1::4
!:'
I::
7
. . .i
, - O ..
.;......
---':ii-
T-1
/-
.:
A
ps,
".
...
.I .
..
. ..
B
.(
. . ._. . . ..
.........
. ...I
. .. ..
•~~L~~ . .
o:: t28!
2
..
:I :
i
. .. .. .. I-..!:.
...
...
..
.i
.4
J.
0t
0
v!
2.5 0
7-t
:
Tiai
30
o
ti
2.50
ICr
f-''i-i
... IT...
~-ii--i~-i
461"ME's n"
1..
4
mviace
~ ~o-i---~
6 60
........ .. .. ..... .. .. .
t i ol.i
..
.
...I
.-..
.....
... .. ...
. ... ..
s
..
uhcim
..
6
90
75
o
s Fi
e
.
7
A
9
105
.
...
. .
.
lzot.----
i L1:
..~,-;-h. ii;-i-..
-~
- iI
6.0
----- I- --
0
1ji
--7
I -
~_r
.__.
7_
i__iiI---l--i--~ ----;
1
... ....i
....
.;
7I
7. 7
ii.
. ::!
t : 1
SI1i!!i: 1l<
*
~ 1*'
*
.
I
.
T
........
I
i-
3.0
I
V-1
f.4
----rI:. i
0U
7A77Vi7:t. 71 I i .1
"
---i--i
i
I
,
I
-
1- --
.
1..
.
.,
tti
. ~---of
.,--- .. o ),,:
r
.n l-i... Time
.- !---- . ..at.
..Loading ..
--.
'
, ,--.. .. , .. i- - -- . ......
i.... .
.
.......
t...
-....--l - - ---. - - ..
.
i! .....• 'I
Ii
h
l
! '
71T
*1~'
i
: :--1
IIi~_i
.;..
........
--- '..
'
Io0
I
i
iI
' - -
I
*-
.
I"
......... ....-. ci
-1__.
I
. ..-i ..
0:
.
..
-- --;b DAY6
i -,
.. !----! ---." " i . ....
1. .
I
I
*
[
'.. i . .
14 DAY
- D
.
I
i
...
..
i....
!....
i--.i....
. ,--....
. .i...
.:......
! ...1.. ..
...
r-4
Ei
0
..
5.4
C.,
I
;
!.i
.
..
.....
. . .. .i T.. ...
... .....
Fun~t
of = Tme
on, . f~r iourl
ge at
r
4
:I
p
3.0
2.5
0
45Time,
hourgo
.:" .
! ~T ~
. .I
i 14
i DAY,5
105
"
i
0
0
I :::
:..
1 "1
.... . .. ..T
77
03
... ...
''~:
... ...
'
:: .
41
. I .... ..
,...
.0
' '
....... ;:
... .... .....
1ii::
::;
::
ii
I:
l.:
.. . .. .. . . . ..j
:::
:~i
r-4
(4
w
I)
do
'-4
'4-
0)
-'7-
I I ...
V C!Lw:: 4
:: ,i +:,::: :::.
-t
-- 7 . - t
I
I
i
.0 ea. f6-#tt
i
02
O
AV
:
t
:
1
..
I
:
fl,
10
60
40
o20
5
,
6
I
s
0
+
I!: _li~iii
:iI
'
'' i'
~-
----
----- --
-
'
ii ' ~ -
1 , i
T.T
i
--
--1.
---
'' 1~'
T.V7''17rT
. jj~j
,4
o
I
,.
1i
T : ::iT
i"- : !
: I- T':T
-.1
'
: I
'
i
I
F K
--
1
!
.
7-
a)0
I '' 2
...: .
7.
:iTT+-
:1: II'
~
0
17
L42 2 2
-_
-t
jt17:7
I
....... . .
er.'S-
. I I ~~I ; tI - i I i''I
Il,il Il i ) iisctcrLi
''
,t... :.1..~. i--~*
' ' ' ''
I
rr t-ko
j::::
j
ii
-LLII
' '' '
f
: :i :
cc~i~r
: ::
~
il
.fI, 8 0
0
111;
i /II~
,
0)
ht ~
i-:t
.
hi~
_.
-_7_-
I
F
t i.,-s
I
I
.. ... i i.
I
: ...
.
:
.
ii-
?--,--;Le s -,
.. .i
......
I
-.
4
'
lri;~
''
S60
X
ii ii . __i L; .. Ii i I i. i
C
i
;:----..'W:ii' i1~~---.-_ _ --- I
ii!i
ic
' .
...
* .
1
ii
.
___i
i
f
.
i..........t...
I
I
:__-_j
"
-I ..
,
i i....I
i
. '
.. ....
. ...
P_ .:
~
!......
_
.:
i
! I i i i
i 1.
.... .,...
--; ..'..
. .-.::..
1 ....
_ ..
.....
......
. .
" !... . i..:...
"
iL
f
1:::F
iI
:~
Ilr.. i ~
'"
Iifi ft I Ii I 1
...
i.-
I~~to
4$* xl ekS-HIi!
I: !_L,,s:L
F i:il
I
60
nours 60
4grlme,
4 5F irne, hours
-
- --
"- '
'--'-
-~ j05
105
..
.
m
] ::
};"
:j
~L 1 t 1
i i1
7F 2
T :::
:12::2:2:V
;:-~--.t---
K~~ ~
It;:
.l
7i
;: :VW PE.i~2: ' 7 7iij
:;2
i;
:i::
H:1 :;;iil
V4~
-.-4.-.-.-.
t
itLVJ'
'
.1
.....
r4t
4)
*j
i-
;I:
ii~f:iI
FI7
'47.
IIt
I-"
::
.
'
: : :: ;
~...
7;
.
,.
ti
I
6
7
........
K
b
j.
I'7i77
...
2
V7.P77t.:
K. .
O
7L.4::J
:
t--1I.::
I
d
I
S
2.0
Ii
F
I
..
i
i:
I
'
.....
...
t
:
:
-4,
p
.. ? ame,
oura 60
i
:
i
105
i.
:77.,
figl
2 1Illl
I1
.-,.i.
0,
0
7
1
'1.1:1
F
..
: i:
i:.7
....
....
....
~:
7t-.7t..---I
__
.-
__
o
S.,
jjF
i
0
Ii
:::
i..I... :1,'
..
..
.
.I:
F!:,:
H
U,
o
o1
U
H
::-*-I:;::tli:i
il :777V7
'
J1.
.::...
.....
:
i4.
1::
:
7~t::
{Ii:
iq::
L05b4
i
(4.J
4'
~~
4. i
Crdag-i D*
e hj
*t
I
i7
*r. 1e
. ..... ... .
-t
'::-7
4Inf 1 T tne
;;
m as
45Time, hours 60
i n
i
~i
e
{
:
.
7
-17
o105
Io
in
7-me
, ~
~
,
I
I
I
7
f
-T-
I
-- :--- -
:1I
I
rr--tIr, ;t
V~:
j1 i
I .
i-cC"rteccL~*cecLcCci~-k--cCu~ ~-c+--~
T-T
j
:I
.:~i
.-
-1-1_
.
.
.
!
i
- t
I.
-I;.--.i
.,
i
1
,:
::iCft': -
!
t-~-
r
' lli:
- k"
'--'
. . .
.
..
,.
l ::
. :.
4
4
•4 daye.
. . .-- - - -- ..
.
tf ....
.
"" t' ii1
e --- i:. • --•
--
4_i--l
~---PII-~----C1F
~
.
i
i
.
--
-- -
-
-
;-,.....
....
. ; :, . ,
, . .:
Pecir
i
i..
,-,.
,;.,,,,.:
4t-_rJt
--
)_*i~
_C_.
--I__~.__~~_
~CI~--.11
,--I
I.;
. :.
LC
i.
~ Ii ~
---,4-'- +
. ... i . . .
[...
- L---)^
.1.r
--
"I '
~~"
m:i; , : i:, : :::
.,,.
__.___ L__ -r
1--7~--r-
I
I:1r::
17--
'
l
: . II.:..
. f - - t-, - - I
-
-
----
''1'
* 7.
, .I
_
~
-I
..
.
..
' .---
.
..
: .:. ;. :. ,: : , . :,. ; ." "...
- i . . '.
.
- ". .
.
I
6o
,
I*
'i"
4C i," , , t: i. '
'
;
..
. .
I
. . . ,. .
.
~
..
.
,
,'~~ , ,...... . .....
tl~
.
• i . . .i,.
.
.
: :
. .
.: ,.
? " . .:. . .f
..
..
.
. .
.
.
:
:
..
f
. ' .
'
.
l..
..
- -.
.
I Iri I.:: I I. I I i
1, IV7..-..:.:-
i-r+--Li~~~
..1. .
IiI:I
'--~---'~-'
6.O ee
" ---i '
1 f (ec
I I
leai
i
,
,I
..
.
,
t orai
I0
0
-60
:
1*
-o
,
Iin
40
-
..
i
I
0
105
i
1
12O
-- I
II1U
I:I 11
11C14h-~- -ICUI*:1
"''
"'
I
+"
"
•'
1 i' i
to......
+
!
.....
...
.. ..
00
U
40-"-t-- -
'!
1
4
:t ---'
......... .,A .
...
. .
...i_....
,
u
. ..
..... ..
.--- '.
.
.,.
.... . .. . . . . . . . . .
-
-
"
"-- ----
-
-
- '---- '
-. . - ---. .T...... .. ...
. -- '-
j......
..... ..-- " .
......
.....
.. ...-
.7:. . t.
"--"
-
. . ..
[
4
,,
-- 21.-
.
__
t...
.. ..
.....
.. .....
Ii
I
60'
j .
o. '+........i' 1.-......
...
I .." .... ;....
o
....
I
i
++.
015
...
,.. p...
:- i
SI
'
t .j
'+ ++'
.I j
" '
I
S15
I
i
,
t
'
..
...
r
.l.l
. ree_.p
i
,
30
'
'
-
. .
..+
'
I
+
..
,,
. ..... ,
+
. .
u
+
-;
.ctio..
i
O.f
+,
as Fulc
,~'
Io
i
:'
,
45ime,hoursO
:
9
t
i
•
:.
'
...
.,
:t
.
ef ed.in. of.!.
- :
F
...
of B"eata as
,Pef3ecition
I
-
"r
-;
.
,-
1
......
,...
.
'
, -- ':
f
+
: ' +:1-"'
- '-i ' .. --T -".....-T
'
....
I --,P
":"',"
f-7-~+
+
'
1
+.j_
. ..
++ " i
-.. ...
ime,
hours
60
+I+
Vig r4 47.
:i . i
..+:
:++...
45
+ ,
+ i, ,'
-~ -~
20+ E
'
.I
....i,. . . ..,..
- - '.;
1..
..
1...
. :.+ ... .
... . .....
t
+~~
I
ti.
.i.-. - - - .-' l::
•
0
. . .. .... + +,
.... ...
....i.
...
..
..-.
....
...... ...
.5l .-+
. 1...
S
.
..
i..
1 (.leam.
io
o
'
+''
"t5
+
'
,-
.90
+
+
i_
,
'
|0.
',
|0
I I
-
i
....
77t=:
~ ~ ~~- i1w
;
ot
%F
~
100i
-r4
I
-A- '''
:;
ilJJl:!
--
...
....
~..
0
a
l::
:'
ii
'''1
:i
' i.z
::::
:;:;::
7':::
:;:;
':::
:I: i::f
I:::::; ;~:
1I:;i:::
::
rit!
''
15
:1::
I::
::::
-- 7i
::;::!i::
. $.A
l:O.4iJ;- trI
~'' ' ''
.30
::
.Tpej
;
tbp
4
!.,..
41
::
....
....
. ....
-:
:.:!
-7=77
;::.
:::--
..
r~j
,
i
7 '
. $-!-F
W:Of
~ita
i-t
t- -n .
'Z'
T.T
r tm
n
r :
90
105'
/--I-1--
-nT ile
) I: i
u) '
""
- . .I
i•
i
.. ....
o....-- ...S..(..
°~
v,
I
' i : . i . '.
:
'
xi Ht
~~~~~
ZO~----
'1
.
__L- :---- '
.
1s
30
45 T me, hours0
1
I
.
...
..
i
i
! ' i.
I
•
....
""
iLgure,,40. Extreme Fiber Striins as "unctioi
,,
i
,
0
:
'
"
of,, Tim
, ,.,
7"'
',e,
, ,
,
105
I20
-c-l--
~- rr~IPY~
UFYL
rUr~r*cU3r~llUra-rr~-Ilr~aulJLysl
t
C)
o
,
L!w
U, 60
Cu
(1
;
i
..........
-- . 1.
11:1::::
. .
.........
I
....
".
:
I
:
:
:I
;
:
,
x
Lu
iral
jiti i ns
i
4. time, hours o
rI
Ct
. -OT
.
oe
. .
...
....
i
.... .
Iw5
..
IK
:
! !T... :7V7
::
-:
•~~ I
.. ... ... , .. ...
.~~~
~~
i
... .. .... ...
: .!
i
:
iT~f7
/ !:
i :i
i:i
!,
~~
i .! :
i..""
.
'"~
~..
i i. I i , I
...
. I.!
.I
...~ ......
Zo
71
U
.. }
C
U
•
..i
. . . . . . .l. . . .'.
.....
... .:
:.....
I
t--f----t- --
...
S.....
U,
. ..
......
...
....
....- . ..
i;
Lcm
i5
.
....
..
.
-.
....
.
.... t....
Cpj
,
...
:
. :.
i
I
i
---'
_.
-
i
I
-
i
JI-c---
.--
._
I ...... _
:
.1...;.~;.
I
-i.
E
''-1-
-1-I
.~.._LiI
I
g
r.
ci
-
Rh-....-
-
v)
i4
'
i--:
C
i
-LOS-
13
-4-4--
'-
~
t f4LB
I
x
....
K......
"W40
:.:1....
cc^
'p
-- ,
L: .. i..I.
: .0
t----i-t
,
l
'
L
:... .
rU
r_ _i_~
7771-77
0
15'
''
i.E
'----'
x5.ri
exm
r
30
--
1
. S trA
4505
inde
.
.... *1
t-L
;
o105
Io
__~~__
___
.-! ::
::
;,
-i
] .
j
1.~ -1
,
0 .
.
-
T
-
"l.:d
-- --- r--~
K
8pe
f44-
Seno
: .;: . - .
: : t?;-i
..i7:-!-
7 17
-1+-
::I,::
.::r::w
w:
__
__
-
7 J7--
7-;
1:
-in
:::-::1
2tcyodsIin
I
.-71:
.....
-- -----
_
-:-
-4 b--*-::
1-
_
_
_
--
--
---------
MIS "
7-T
~
~T~-f~_!
lj~_f
.. .
I:_T-~1:l:I
-:77l::~I
47
.. . .:
. .. .. 1:l:t~:-Tr::f
el et
!
Ai
;-:
-(
~ ~~
~on:~;.:~-
~ lob
I_'frur.'
:
.j
i:Il-_-.
- -1-- -
::f
.~;...
; (_..
~ onicr~
n~i'I
____
--
.: .
-,
:.
:~1.
--
:;
________f:i~
I_::::::_
;:_
____
____::
-------------
-
--
i ~_
~~ ~ ~
:.:
_
-*-
-~~
-
i
--
f~t~
~-
--
---
--
__________t--
-110-
LxyQ:lit:
I I-
-i-
zli4:Vi--
:::r::
4 &--"-t::1-;l1::il::
0
2
!2 IL
I
' i2 '
....
:._
_I22L. I£...£
:,i]
2ii]
i,
]T~i!.I,!£/11
:i
iT
I
i
S;;
] -T7l:.-.7I.7
ITI!i~
iI
' T!2L
P -r-I
"'~"~~~~"" '~-
I'''
'~I...ii ' I-i:iiii'' ''''''
::'::::
:::-- I: ~
±LH
:' 'ii f
f
..
I
-4
o
::I
::::1:::lll
~:
---- t--:-
::: i;
-I-:I
:t,
i
i; ' ! '
r44
'40
0I
'
9.-I
Z4)
I
:: ::
:: i ::
~'
::::
' ~' ' '
:I i I.
:::
::.:
:: IIj1!I:IiI.fiii:.i
i
,
::;:
;i
: :::i:::: :,
1::
:: t:r::
:.i
"-':f :
i,;
..Ii
a
1:
ii:'' ' ~ I'':'' --- i
1
;IL: ..: I
i:
;i ; f
:'::
~'''
:L:li-]:::: ~: 'I
i.Ii
: ; ::
,.,.,
::: ::::'
i :::i
I:.I__-_.!I.:_L
~1Ci-
' ::i:
ii
~:::
i - i - 1- -r 1I -: t:
I :i
Cc~pLianee
457
me,noura 60
:: ~ 11 ~:!::i.
Irlli
--:l;--i-6dn.~f
. 11-.
i' I.
t-I
---i-l-i-
:.:
:~ I
i
"'"'~'
..., 1_1
r
....
,....
::1 :-1"
--' -~~~--f ------;--!
:::_i
i:
1 i It--~-:
' r
'
TiFs
i_
:::i.; :~rl
~--i
--1-
asj Fllne
~
i.
i:
i i
i~III
:r::!r:
i
._i'
---'--:
Ctac
';
30
i1jj
1
'
I
-:i t .I --C
~....,...~
!:.:i:~~:li:!i... i;~~ ~; _Tfi~ti~e'
i1 5 1 Plbxdra
t ~
" '~ ''
1 i:
.... il..ll.l;.i
Ijii
ij
''''
;ii
:i-II -
i: 1i.:.
: .it-: I: ;1i..
TT-
J
8.I
i.
i i
i
:I:::
II
:
:~
:i
I:._1.
j-.
,s!
:)ami
a.
rii:iI~irl;iii :ft iIf-I:/I~-,i t :Iii :I::j
::::
....~,~-..i t -11 - -, 1
::::'
:::
~'
: :1 jjjlli':'''
I
:::It
i:i
:f
I-
I :i;il;;
:t;:r
Ij
':::!:: --- :;::::: ::i:.I::i.T'f ~ r: "
r::::
It:
_.I;..
;:i
; : :i :: : :
.: I...i
li ""
f
C3-.0
i: i.
i~__
:Ii'iiiI
ij i;i iii
I::: ;:::;!: : ;I :: : : LL-lL~!iLtt;1 iIII1
T-t:
I:-f
iill;~
''
II
~-i-:i
.., i...
...,
,...i.
I:i; ; i
f
"" C.~
I:
:~..!
i~': .. ::: :::: :::'::!:':::
::;i:
i.: i
:i-i
.
1i
:-.-.::r-~-.-~
:t:
i'
-* ~ T--T-----l
ii --~i -
:::"n ::.: :;-::::: 'I' ' t,;t:::--ii;: f":::s~ir . ;::,_:I::
i:::c''-'-''
:::::
I!
:;;:
I, irii
;,tI ;Tr;:
:::ii
'
I. :i :
: :: : ; : .:i i ::
jTt '.'
ct0
:::
~I
:.:~
::
;-i-...,i.~-.~I
i--if
r ~
i::1
I ~' ' '
jf'' ' ~
r4
-Li.
;...
I
-i-t---i
---;--'-
i
105
-
,~r
0
[
._
.......
.I ,
I
. . . ....
V
.. . .. . 1 .
.. . .
.
•
1
L
, - r--:
" ' ...
i...
: ...r....
.. :
,
+~~~•
i
-L-
* -
--- --- !.....
A --
ir--i Il-i
t -
r;
-:
";..---.. :....
'l-;- , -
'.:-ii
r-4
i
0
]'
.
. " -.. .
,• ' i
-.
; +
.
.... .l+:.:.
!: ~ • ~
} +
i _.
:
!
'
. :
~. '
f
.
.
...
'
.
..
i
".. .
'I---l
. .
.
t. - - --
'
[
..
.
.
] , :
.ii+
+"
.
'
~.. ,.
+.
I+
i'
..-
! '
1'
-I
+.
S---..: :..
..
.
I
'*
--
,
: ' --.. ~~ i
i
-. t1: ir.
:
.
.
.
[
+
.
,
: :,
.
L
.
j:
i
_L
i-
.1*
.i
"
:
" . . ..
'/
-u...
.i
+
. .
;
'
..
. L ;_
+
"
,,
..
.
.
.
'
"
..
'
":
""
'; : .
. . . .
' '' j
I
Ii
Y
i
Ii ''I -------t---t
--~1
I
ir'fl1
I
I
I
I
.--.--
_
I:
' 1: , :.
_..
t
'
S L , j._;_;
I
'.
-
t
-t
I
+
. . .1.. . "
I
IL
IZ.
I:_i
,"
----, +.. c-- ....
. .
.-fL. 7~IT7 V7
+ ,
C-
k
-i' .- 1 i, ..
f
I-'
i
" :~
'1 i
i
.
: -- ..
I
~~i
.+...
!:+....
I.
'+'
:+
,..._i .+. .: 1
..11'+++1
:j . "".;
:I:
'
t .
~i
.
' -'
.
+
-
++
'
I
,
+ - - u -
- ,'jii~i
+
i
~~~~~~~
.~ .
+i
- - -; +- . - ,-++
- I-
. . ."
I : :'ii
I
i
- --l, '-, I
-
+. +...
II
I!
I
-
'
iii
"
..
.
. . t
i
I
'
DAY
7t14O
..
.
.~~~
..
. P+,
~~ :
<1-17.
30
30 ~
omp. 1 .- ~ne
~------I
-i::
.
..
a,
a!nof
,:I
as
45ile~or
6
45 Time, hours60
...
I
~rf----,t--
Hi
ler
.5...
7.....
, ,
:iK1.-.gu
re
I: Flex ,~
;.,
..... ....
...
I,
P
n
10
-is
75
.
.
.
-
+. .
.
..
:. ' j : ..
Ing
Ianc
i
.
f.
It
I's ,a
d,
t
1o
1
0
-
t : -r ----.-.--.1111__
I1~__~__..
1.: ~.~.._~
--T1.
:
:i -:I'::i
'/iI
.J
K
IClc-I-(I_
i~I
11 i ''It iI
i diI
'"
I:
.,--.
I 1 -1-Llt~--
'
-~
ICC
Ii
L
-1-~1-
__
F
. i" '
~.~.~~~_.W
i
i
j
4
i,i-iiI
~---
~_.___jWICII-~----~--i
;:
I
i
:
:
.i
:
i
:: i 1:
...
,
i
i
I ~--- ~--1---C
i
i i;ii~XLI~-~
I 1 ~e---.~
I..
'
iii'ili
'' -.
....,.-~.
_: -1_ _
J
_I _~
L..
SIT
-: :~....
'_ji:;
i
Tr7::o7
::..1._..
I:II I : j*' - i- --!
-i
_i--?
.Y---'
'
~...~..~
ill.
I7
... ,
-,I--i
_I:
-i-;i -
~
I
Ii.
lk I- Ci~ I LJ _~
: i:
t
.-I -- - I
I
-i~,.--.
::
i
I
I-- t-~-rl---t-
jj
::i
I 1 I
;r j.::.J....I..;
1I-'-:i: :f :i
i
:j i
-- ~ .....;....
i
1
''~-' "'
I
_-
-1--
::!
:: rlii ""-"i 1.
iiiii:i
::t
4
-- I'
I iI
:
L
I
i:;
p
4
1 Ii1 t
LL.iLL.o
1-7jT.4411
....1 ...
.1111:lIiT-11
I 1 i i~
.
~..,
i
--F-
.. 1...
__1111
---C----TC----C
i
.vi
-
,....
... I..-.
______~__~ ___________C1_1
::1 iiiI
....i-I..,>
IL-
.'.
_Jji~i
Yiili---l
Ii
t; i
1. _
r
1~I
.; i .
,----
~_I_
-4
I ILIiI
_
AA Lb
1~~~~~~~t
~_7
- -
-:' t::'1
I.:
I.'''
: 1Lr.h-i
____1~
:1
--i:"""'i'-"'
I
-~ICt-C-C-C-1-----~
- -~
---C-t-(--c----1 IJ--eLI-IIC-c-C---e-e-c--
'''''''jlr:
'
I .I,~~..r ~1~L.1.
t-',-'' '''1 '
i
-
i--i-li
rc;
-~--~I_
;' '' '1
i' :~lli.i.
'''~I
I''
t--~)--LI
.--.,I--
i;
t'''~
*:
i
I;.
r ---Ii i
l._li._-:i::i
I
I
-- +---+-y-i-~-~--^r--L-^(
-V..-
'
i.1..
I....,
-4-'
::1_
:---li_;_1 r~~.
i.
i-..-C-)---~i
....
I
1..1--1-. ;
1 : i : i ;1 ... I ..
: ;, :I:I!
)
:-+
-.
.1.:i
i; : i
!
_
_I
.r....
1i
j;j
I
1
jl
L
f;
r0
-7
iii
i:;i
i,......,,.....
r
.
:
ijfii1
1: :::f
-(----C -
0
i'
~l
I
I~
I
I--1--
-i
""'
I'
I~
..
S::::
.00
i
iil
Ij
ii
ii.
?-
:..j
Si::: I ::
:7
i
i-
T
I
4-
) I. L I
:f:
.1
7 .9::::1
T
I'
rI
ITI
4-
.1~l~
*.I
, ~.-
i: : 1
._i,
~r:-
.
7
i
~l
; _(
- 1 _.
_~ __-~-)
-~--
.
ii 1
--
i----
---
I.
i
i.-i-i:
.-1.:..~....I
li--~
*::I :.:
.
,...,..
4
ir. !.
;
-~~
---~i
-I. I-~-I c_~c_,_c_
..-- L7
l
2
1-----.---...~-
I. day o .dap ¢cr
.. I r :.. I. :. ....
1
.4------
4
--,
! t l l- i i
:
: .::: ...,:.ti
- .:
'---:-
i-i;:
1--
600pJi
.04
1
111114
.3
C.0a
,0
1!
F
:
j.1
.1
I--''
VI
I
4' Tie bor 60
)0
ii
__ _
ft
~5
:I1
K
45 Time, hours 60
1
'ors
...,
I
I -
ii:;
''''''
1
.... 1.-:;
i-~
"""'-'
:I
'---t---- L---ICI* I
F
i...
LI-j-t-.-
t~ii
:1
.I:
I.i .!.
I.
-- ~t~"-
I
.. ,,,
,..
.05
pi..--
"
; . 1i .1..
: :: i
.06
'l: l...
l~i,
iL!
: 1
....
1.
__t' i..;: i....
i -i--...,.. ' ''
i,
'!
;:iI-
-wc~-
-i._ -,
i-....
--t
"-, --- -- --I
--+:...
I' '-~--h.L-r
...,
i '
:i'i
'': i "
' ~-e-~--~-.
..1...
1.~..(....
1- --~--I'
:i ': 1i t -
'dl
i;---CI--L
_i
.04
1
.. -I
I
"j;t
....
i :"i
''
"'
I,
"~' "'
;
"
.
,!
-4
4I
*1iil
-~
l....
,:
,I
-i- -;-
--
e .
I, (im
" 00
i
i.-
..
i
.1.
i..... ...
.I. - I.
l -.
----.....-I
:
j.o...
. ...
.
,
..
.1
_.4oo s i ......
i
--1
-r-i
I...
--(i.. -.
-- -t--
I
.. :-
--
-
I-
i.
I~
-
MiLLL
..4rei.
----
... . .. t
~i
::i
i7
i
i
''
'
.-. -i.'
....
...
-; -----. -----
17--1
-4-i
rJe
'
I
....
C--I.i_
,
7T1
TV
....
,
i
.
i
)(
M~t~-
: :: 1' :: ~
~----L----;
I
!!ii
i
.03
T
i:
.
...L _ •
i'
1-~-I,.
I:
4
i...
..
:._.L ...
IT
.;,
....
t
-
1.i
..TZ~i;
z0P pe ..
1
VFn~tin
tTici
t
I'j
x8Cy8 1 idersI
"
50
4
5T1me, houra 60
75I
s
105
120
P
.
I
f ii :i!
,1
:!:i1
I
.1:.
1
1
I I '
1
"
.
T
si
i,..j
I,
...
,
I: i
. . 22_
... -
..... f.-..
....
. .i.....
:-" ....
: .......
i-7- - i".....
...
r....i', .
'
~,I~-1
i
'
' .. . .
i . .
.. ..
.
.
" 2-;-t
. ..,.__
:
j
2
'
2.1-'
i 1'.';]
:l ,~
.1 .
- ",T
'
b '
.0 1
-I-
i--t-1--t-t-Lt-rtit.I:.,li,...
I I1 i
.I
Si .
.01
I
t -.~i j.:.:l ';
~: !"
'
I
I
..
... I
5
"
1"
"..
.
15
..-I '
I
.. , ,__ ji,.. .
.I
0
o
• ".'ii:
i i"
:-::- -
.i ',
-nt-----.--
al+. C re
.
.......
"
60
+..5
T~.we
our8 6o07
I
. ,_
it
0
. .-...
..
75
.
i,
.
4
I400,bI
: .i
iiie .
:1.
:i:i!:
i7
I.
I......
-i
i I
_
I
:- --(......
" :I:" ":';":C-i
~ C-- iC----
t2 LL<Lt:
Lffi:fifliiffi
*1:
w:
...
•
1"' T2
I-
1
K. 1 v
7 7:t77t
it
i
-.,-i
i
i
:1
<
II 1I7IjIi.
ii
'd aY !t+i
..
. . .
i :.i .t1. : .I .L:I,I :- I'1 ,: i. ;
r T777.-..
-7._
I
17
- 1
i
i.
..
'
.+
:i
",- t-- " . . [ ... 1 ... --.----..
. ........ -"
--"ti-....
I
I .. I .- I
'
t .i
I
Ja
.
I
-- i -e3-LICr-*---+--CC~
*CI-)t~----L-(I
i
I ... - .r -
',
--,......
.......
.....
-- -i - , -'
-t--. .-FT
: :::
.,.
i'.!
mr
I:
i..
::
!~~
4.
' " :
i
I
[.',.2...
, /..._£ ...
__,__;.
...
i....
_ -....
L _ 2 i.:- I-~
-----r--i"? ]:
t V7T7:-
!
I:
,L--I
.:+
t
it
'
I
I'
:
i
~......
T-
.."Ar-,.T-T'-
i
i
i
..........
'___
,.
:
0
I.
2
.06
.........
* *
I.
Fiji.
I.
*
ta yo
extP7?s:
t~I~1TF.
*
I
77
*
~1a
Za (
'7'
7 ,-
if
I i&:I
.04
I ;<K:I>k.~
7
7
71
1
L }77I7
Li
.
I
*j7777j
7
17
F<.
ill
.03
psi
*1
4.
.~~~~
.02
.
II i
{
-
~
II--
0__
I
I7
L
1-7
I
.01
30
4
,ime~hours
60
7590
105
12C
__
____
_~_
t
-------
,F
..
..
7:
_.-
.-..
Ii
ji
ii
1
..
7
. ...... ... ....
..
j
- -- - - - --
I
i
---t-i--.....
--...
i/ i - -....
7007
....
-;i1/
1
.....
:....
-.......
$~
' !-:i
...
Y..
---:SI -: T-:.....---i
I-j
--i
I -i
- -- -
.,
......
m. . . .
.....
I
I
-
*- '
i-
I
;-'
-i
-.
I
.
.
4
I
I 7-'
-7
"
"
-- = :
..
- :-~~~~ ~~
....
..
!.........
I
1I :t~'A
I:-P
-
f-
:
I t
i
ii
: : -
-
j
J
I
t•
.i.../ i 1 ! 1
I..
--tI..-t
-..
I- .. .- ,,,
- - ..
-~-~h
-- ---.:-
Ti
-
" :
i
1-
.
-i
- ]
- - t- - :-
ii
I
-
..
.
Y"2
-- t
in--
I
---- -----:-
"
7 71-,
: : .........
I-......
.....
....
- .....
.......
.i_.l..
! ---.I
I
•
.7,
+..
7
I
:!
-:
.
::.:L .
2 II..
- -
I
':
lpc
I
:
"
irz"i-
-
L
1
i ':
r
i
.ri
a_L-.: 2+..
-_
Z_._L!
7 i 7 :: i:
Y-~
.] - -:..-I
-.;
I.._
_?.
: : -i . : I : . : ' . . . I'"
.I
. .
-.
77
--- - .. .: .
T .i .:..
'--7
-~~~i-:"T -:.-- :--7 _: -- -iL
-,].....
.. .
1. . .. . .. - ; .. . ...
.. .: : .. ...::'
{.:-4
*
i
.::::;:
: 1LI
c3-I
-
-118-
d-]
4:
1Jj I4:
I
-:
77--
ar
ii
1
-::-;::-.
I......
+
!i::l
:-_i~..
...
:i-..i
..
+ :.+:_..
:7__+
:......
__i.
:...ii::j
:i:.:
::i_::
i!
ii+::'::'
.4:
+-+::-._
1 ':. :: :.: +..::-+:
T -+-+ ':
-+....+i:+::
. :+
::
.+.
:+ :::.
::
- . .":..
..
.:
.....
..
..
.++++...
.+++
++:
::1::I:
..+
::
++++
+.+
+++.::
.:
: i 7++++
-:-++:
...++ .
:++
- ..
: i: . :: ;
777- 7
""
" -" :
: ::
..... . .
,.
...
.....
.
a
--T
-:
- .
4-
"
---.
:
----
+i
...-.
.-.
... t-
: .
:....
4I
1:
:::I:: +,.+
:::i+:+-+
-+
++
++.+.
. . ... .. . ...
..
....
. ...
a
4-4
4
1. :
,-:
':77
-:
7-77t,.o7:,+_,
:---+ i:7
i ......
S.
1: -
7= -.
7i+..
....
I ....
,---.
,::+.+,:.... . . ..
.
+:.rt . ........
.._ -....
-.
....
+,-.---.:-+++::
+ i'i
+++++
-++:+i.
+:::
,_:_+
!i~
__: -i:+.
I::.:iiii . :'::
i::++
: :+
iii: -.
.I
+ -::1:-:1:
-:+-':;:
.: +++:-++
.l::;-.
,:
:-:+:+++
:.: ::i-'....
.ii-
. :: ;:t:::: :.:+.:
++
.::
-:
:::j
...
..
:: :.:
i
J77-::i~~ij!iil-:
:-~:
]-.
-
-+.
r+++. :.,
1:+.
+
::!:
I:
:+
:-t:
i_::::++
:++++ ::.:-+:.+. :.i+:
,,
+++
-:i+
::
.:r+:
.:-----
.4 ...
-
...... ...
...
1
- +++++--.++:++i-r7
_---AI::::1: - l-1;: t:::I::i l :-:
7:+
I l V :l+++:
+ ++
+
7.iTt, !++l-+.i++
+:++1.;..++r..+o -t.._~o++l~:_ :.++.
' + .1:-++t
.+
I_
le
4 S,
4
++++:+
j:::i:
+++:;.
:
+:
++++.
I:+
i:L:Lz:
:i
:++:
+:/
:+
:
[
litif
.
.
t
,,I
:.t+
T_+(I+
T
fr
~ iriiiil'~i
~
i:-i'1-- -F+
I+++l++++
:,I:::
p.::
:::
_
:iit~
.i
- +;
:::.
-:
7
i
i
:::::i:
::-::+{
: :
. ... .:i: :+:+.
7iliiijli~;tililri-il
-~
-
-~i
-
-119-
.rt
:.
::f::::i::::
S
:: : ::::
: -::.:
:: .::
:4::'+:: :::.+:i
.
140r
7 trtii
.,I
~
w:rr.
-
-
[7~
7
...
V
T
7
iJ.
.
.
*
4
.
,
4.,
t
1~
,
-
mt
it
j
4
1-
__
.L
1
1
Ii
~IL...
11
_____
~I
~~1
'1
*~
'2
11
'1'
1
4
--
~.
,
*
4
.
~1
.
.1..
_
.4.
:.~.}
j
4
I
4
4
~
L...
I
.
1
500
!,'
I
14.daya olA ep'ci!r~nn
.4
4
f:L
1
4.
0
I,,
~i:4
-
~1
.4
L
.1~
I
~..
_
1
I
1
L
1.
I
~
.1
I
ZOG
2
I
T
4.
-t
I
-1-*
4
100
4
Fx~ure ~2
1-
Stre~,s-1Stra
14
2 4-K..
1
-i.'
~4
I
4
~ot~ Ti~i'e Isoctin~z for 4~8 CyluId~rs
___________________________________________
-______
____________________________________
CI
OZ
03
t5
.04
1.
120-
-.
I
j
41
7.
7
4-$
.4
....-....
-
'~~-1--1_~ ~
1
4
-- -
F
...
7
-Li.
_
.4ffi; ,i
ji jl
____f
*l
II
_
- ---- 7- =
I7
_
7--
I
S4r
7!
- I-
I.
.. ,......,
4
13 L-
S:- . ~
..
-
..
____1:t::t:::::::
-
_
4
L
t~ii
___
___
------
.
cp#mc
e$e
417
44
:::-I:!
___r
r0
r I.
d-iye$ old
-i
40:1
.. ...
____ __
_
- -iv 171
111V1. -~
ii
j
"c~j
---
-121-
_
4.::
I I
:::
~--LI*1.-*
::::,1
7 * :: :
:.I- 77-
1.
15-:i:
b-kl *:j~~
iii-r~:
-; ~uw
-
-k-,*\i~-;ll*u . .rllP-r;-:.- i
:'::::
1
1
--
----
7T---
I
1lii
---(--
...
- - ....
'
----i !....
"- -t
....
!...
-i-
S-2
'1
....,....
'~12
.
H
,....
4 i~1....
,
:
":
I ....
')
j_
i^ I-.-.~
..
.I.
i
1
i
I
J pt-<:
-;
i
--I --I
t
::;:1
: : i
:! : :1: :,~, ..I.
.
I
-.
" '
1.. I""
..
... I
I
!
I
i L _ _i
Ii
i
/.,-
I::
,J
/
!
:
I
i
-iT---r--.L
..
.-----
.!
1-*
--- 7'
--
.
--- C---CI*--- C3
i : , ::t;~ Ii iI
r
:: : t: :
l
.:I , i
:iill::;
'-" ,:1.i 1L~
i
:':: 'I-f: :iT1
.:..
'' '''I I
t,
.. .. .- ..
.
t
:
'(
.I
•.
.. ,.
~- :.
, :i0 , o
1tO5 ~--i.
i/I
ijjiI
ji i -ij: 1: :jiiIi 1.j:
:I
--- no j-_._ 1
II-.
!Ia,4
iI: o.d
I i
I
i
...
,.:.....f
k:I.: .IT:-T:
-
I
!
i:
.
.
i,';
:1
~----.--~-
:: j:::i: :: jj:j
:~::
.
i-- ----- 1--i---:-.
I1
0 --~--,
i I
IL.L!~...-.
i
t
---rr--- r-
i
4
F 5. 0
tv
Q
'I I-~r.i Ii
-,l-Lt/
a
0
...
....
.: i
"'
i... - i--
0
-
.... i ...
. ....
II.
-4
.r4
:'%'r --.nrr ;(rl. P) *U~UWrUIP-rr;ra~
~--UIIY-YIW-~
' iI ...':.'
,
1
:
i
1
.! i
. .t
I;
i ;!.
6r4
...._,_,.1.
~' ''
x
-C
o ~i
~I
~~
CR i
' I:::' ''1'"'
'i :
''''
I
::: t:
.7
h"L
) .... . . .. . ..
-1-/
0
-
~
----1----
'' i:i
-~
'
.- i:-
:f;
::;:
7 7'
t
. .
! ! lI~i1
::lil .
IS
30
.~
:
f 4
'
~I
...,
i,
'
''
; I'
1
r
7
-c-t;.
iii::i:::
~
i::
i"'t
r
-- ..~.,..~.
';
''
!
t-:
i.
"
I
i
:I I
;;
I-ii-I
-77
,
li , an*4,
: .
orqp
j
n tin
i
lo
45 Time, houra60
Tim.
7y
f
ar C 4indri
.9
I
12
-~--
I
..... --
. .. . . 1. .. . .
''''
,lrl,*,.r
'''
1.
I
it"
......
,.
t--t
---~.--.. ~i :i
-,..-h" ..
t-
[
r--1
:
..,ri-~..
:
i'
--- I
I
j
'-:
j :i::j
-
:
---
-li
~ -,----.---,
''
.
.~., ----
;
'L -~ -~~-1 .........
~~-~
..i ~i
I,-1;
.......
.
.
T77..
-- L'
-
.
1:
...
-i
I -1__.~
-
..
I
..
.
.
..
. I.
:
-
....-
i-
I
: L2i:
'
-ii-
.if
::p
_
.
-
-
Z..
.
:
1F 4.
77-:
.
t
a
I
4.
"
×
B ,
,
.
-
I
t-j-i A~- -Kz-
,i
T
.-
....
. i.. .
.
~-------i--.L_.~1
... . . .. ..
S.
un .. t-on
1
i-
41 i-A-e---
i
3.0
0
Is
30
i.7.
qi
.
i
~ .i__~,
T.i
-.-I
1'
-77--
~/ ~j 1
.
-t.l~
A .....
iDAY6
! . .-
'I
1-7- ~
S4.0
.,.
.. ' - ~I -
~--------- ; i--
T
x
--....
... ;
S-4
r-4
'-4.0
.9-
I.
-
--. ;---
.-._i-.:.
...ti..:I....i.. ~r
-~ .. ..r
i
---- 1-- ~ -.i-,-.
... ;
'''''
i--.
~----t--- i.
I
I :-i
T 3
::
Si
I
i- 3--i ~''':''''
-I--,..
" ~'"-'
~
--- t---
U
----- C---~
i---/---
.1
i
I
-----;.....
...
-r----t--~-e~tct- -t)-~-+cJ---.
-1~7
j
1
1
-i
i':
~--I '-"
I.
'
.....
t
.
. i
'''' WE
1.1_..
~t--
I~cG-1
.ICj
i
LY~I
-.- j.-..
-T
-
'''~''''~
*1-~i
._. L
Z .o
I
---
-r4
-r4 5.0
o04V
-I
/-:
..
-
I -:.
own
I ....
....
.... .
7_._._._._._._._._.
6.0
; ;i
i::
45rim,
hours 60
75
S i
90
105
12
rT;r*Lj
PJ~4i~
I
-; :h-riiUI--Ti~rr~
-:;
l~.%.-i
r----- -_----
0
I
F0.11llli:
.1
::::i:.
iiiiliii
+4+:
:,
iiiiiliiii
I:
....- FIT
I :I
o30
I:
i.
4
,,
171
zi
7I
l:
i
. .~ '
'.
i
.
i
I:
.
.
I
I
; iI
:
i"
A...
--T
:I I[.=.+
:i.
I i
.
,
.
E
I.;.. i....!:l:.i...:t.. .-!
.4
.91i-
.015
I ': I
1
TIrm
T
" I
~~
ec;F-t-icclcL - t-t--3--7
Q:L:IL..
. 1
t
'-.7" -
7
1I--
i
I--,,: '
'
-7
;,--.,,,,,+..
.,,
~ i i
I"
1 ;
I i
1 i i
_
<FT
+
;
+_ _
.. + _ _ _ _ + ._
.:; : .
- -
--
-..
--- -
---
. ... ..
-
',
:"
c-
-lr-a
i.
1
-,,,,-- ,=
i
.#'s/
-4-
-
-*
i -£J
1
H~1-.i 7;i
i
-I
--
-i
--~-'--
---------
..
Creep.
re~ep .$
kxiall
i
£i
. .
30
30
.
rfine
h i ,
Fu-c-e"iF
Ctio
n
i
'.a
60
45TimeD hours 60
4STimehours
Ti.. lor .for
Aial.
nd Triaxial
Tri---xial Cree
Creepo£f Time•
Aial
ad
...f
.....
.... ', .A
:--
p
7
I-i-
-j
--
..
....... . ...
4jjr.t-- Iir.re .5
.
--
_ .+
--
- . -7-
-
Ii1.r---i
-------.-
.......
...:
!..i+!
,
"!-t , |i. . . ,. . .i ..t
.• 1 .... /"
---
...........
.,., ..
,,,,.
__..
t--
I
*1
*-'
i
7
-n
•
•l , ~
--il~:
1
T
I
I
--
..
-
.
.
-- i
+ 635',
•'
i .i !
i
i;.- "
--
r
...
.....
~
i,
.o0s
!
--
''1r"i
~' ' ' ' '
.010
't
i- I
. . ....i
'
;'
'
II~Lic
i
i
~.
..
....
!.. -.
..__ _ .._ _.....
,-- ..
i- .......
j
.. . ....
i
..~~~..-.
Ii
II
.
I
I~i :-
( :-
.
.1!:
I
.
,
_...
!
1::**'*
I1~
1.
I
_..~,...
, --.
I-~--~ . .i I-~i __ ;-.I .-..1 I
i
I :-3
:I
ciao
1
C--Cr- . r E:
3LI
1..
.... :i .....,
1 _I
i
Ct
i L:
i
.:. |
. -
i
i
i
I:
'iiir-i
" ~
-:1.. I ..- ~ -t•i.:i
+:: iI
-1 .__~i.-._,._
-I
'f~1-,
I:..t i:i-i.,..
S--4.--,
~
i
~ I
.TJ7,'.:IT-H
.. CI j
:::i;:
7-71 '
..a
.
,
'
-'
103
I05
'.
"
.
.
_~~___
___
:_
!..... ... ...
i-
•, -
4..
...
:,...
! ...
!.
......
: : _.
_
. . .
. , :.
..* --,
-
. . . 7. .
,V ~~~
i----:-I
.
4
- r:
-::
...
I -IE
--------
: : ::
:--1; - -: - -. ---.I :
-.. -. -,
-:
t :- .
:----. ::
.-.-
2
-:
. . ... _-. _._'._
.. .
- --- - -- :
li-- !
. . .......
. T ~-1-..
_
.
:4.
r
... :. :: :.. :- ... :-
-:1
1_61
:
.
. ..
-1>1>9
.
.:4. . ;
1
-
. ..
7..
. . 1::_
_-L
. : :J
1:' .
t
i
t
~
.
7
........
.:
I
.....- -
LL.
..
-1
2F
~ __ .....
_i-:
..
I
--
7.-...
:I-O:
:
1-.1-il/--:
72
I..:-'-.-u:!:7:::::
1.---::
.^~.~.~~~~~~~~~~~
,I- : --;'
- -: : i:!
_l:
.::
...- :.I- ::" .I::.:
._ ..I-.
. .=:.:
.
---.....
i- :,_-:: = _;__'...-_
_ ...
.._ . ....
_
._
:_._= _ ,.. ..
.
I
:1
i:
I~
e
w- -: .: :----: -:-__:::
: -_ - : ._::
..._,
. :_ .... . :: : . . .:. . ::
:_
__
._. - ....
:- .-. . .
,---_.-I2 -. ..
:: ;:- i
-- . ..-- : - -
4:
_-_---
:--
-- : "
__,__.Ji
- .
-::-
..[ i
I::: : : i :
4.
: - -- .:-
-125-
41::::
-
7
--- -' - - " '
:
ill':: :- ::
:
.. :. :.....
.......-..
.
:-
- ~ ---------
------
7
-I-_
L-::i--i I:--I:
1-
i1
--..-
-I7
1:
---,-
t---- 1.. 1I -.
It_
-- 1-----
i
i
L:~.
1
4---
---- tIi-:
r~- 1----~--:
i
I
1
--- t--
I
----r_...~
3'
ys
~tc:
s'
-t-------t----+--------t---- -- + ----------f------t----)--t--t
t
---f71- -C~~1
--
Lr~zL1i
j_
. . . ,_..~
--.t,I
fo
14
ii
I
---t-
,;I
h~(iP)'
i-I---
-:---
v-
"'~''~'~~
---~-------1
:
-Ir;rn~l,-.1.i
: ':i
--- ,---
il:il
I'
--i
ii..
::
r-_--I~ i. ::
:: i
-- t-~--i--)l
-t
-C-------t-----L--~---C-t----~- ^t----~------i -r---t-t-~t--~t-_t__C_
i, t----t~-~C.... r....
1~
i.
7 .
-_.-;-:i_
... ~~..
ce-3--cr--t--*--CI--e--c --C--- -1----C--3--?- -- C-L-i-+---C-t-+- I~
-II....
I:
i..
--------CL
?.
----- i--
w-
1-_-I~-
r
IC--
~
a
i ::1
.__~,__I
C- -+-~----- ;-C
---.
-:1
-----c----_1
-:::
------:. . --*
-
.1.
...
I
t.
1-b-
a--}9
. ---
I~1
___________________
77__
-7 -17:
i~r~~
u r c!
Cu4 -> -
i
Ftm.
o
n
Ifr~ce
..
,..
~~~
"elmhcn
E-p--
zj: n- bf p
::
:
1:.
r.,,.EL
-126-
I
:7--:-
LpI~a44le no,
77-
.........
-(--i I
c~---L-t---l
--
V
-ffi
T
1 -~-?
:r
--- ---
T
i:.
I
I
I1
:4:::',
J'
I _ CI
__
-~
L
I
-
:::
1
;
47,
.~
I/ii
I-,~I-L
- --~I
C--~31
~-~-------.
.1 ,
I
I
I'
ii
w7
.
.
-1
r----7--.tf'*~
I'..
'* i :
I
.
-'I
i---ii-;~ :I.-.if-;.:':
zrsiti
U.H::'
:
-- _1
~-
i
1F tT~7 iis
,-1 c-
1--11
11. /
--i-i
-
----.':
:':: iiI :::I';II
)
I *Vij
t-
j
---- i--"---I
I-'--7I
iI
i'
TI -ijI :I ... jI
----j.... /~..-tj__..ll
.:-i.;-.
-7-
j__
t~-----
I
i
: I' --III --r
~-1i
r-
* :1:
::i : ii
I ; 1~(~
, .--
'' " '''''
' ' '' '''
~ : , ~I ~
Li
*1*1
i
.
:::
.
;;;i
i i i':'
I j
'
' ' I'' ~i
"'
~
. _.
_____i::
____LLI::L
ii
:;
1I . ~
: ::i :-7i ~.-TT~-7 : ~
~ 1II ';
T
Art:
7§fI7
1:,-
1(!,
-
' ''
-
''1'
''''
;i
-I
~' ''
1
:r
iI:Ii1tI
'::
i I
.1
''
I 1
: f:r : :
t1)(
1
,,i~-~
:1, - ~,..,
,
I: :: ii i
'' ' ' ~
I1
r;
'I I 'e,
~
1: i,1
t~ t
3
1
AMh
c~c~ I
: i: : ;: : i
~~ilii I :
t
I~~
;7
0
1----'--
i
---
-K
i
!-...
i-__-.
........
. ._
2.:.....
_:..i
..
l.-._ . _
....
..
....
...
:...-.
-;i...
.....
.
..
*
. .i
I
-.
I
.....
. i . 8.
.
;
"1 i
1 ,..
-Pi
i .......
:j
~-~1.
i -t
i
I
't ;
_
__
:
,i_~~
..
i
--...I-'-- iA
~1..-
I
i
1
-
-.-
. . .
...
i ......
i
,
i-
I_
....
..
:7
--1.
_ _. . ....
,
-''<.
-11i.'IIrl:
'--t
-t
..'t
=",
...---....
,-...
,.
...
..-1i F_-;.:
+1.
I --V
I-cc
i:
-: : :
. . I : ";
--7-7-7.. -7j !L
"w
. ....
!....
:
_£
.' T 77
..
.. _ __.;._
-:
6-
~-::
' -rl----7
-------1- -I~--
t.-..-
~
j_
-----i
C---~
:::: if::--:
t
i
Kr!-
-"---
S
-- -
t
i-fI-.*
-
i. :1
I__
--
'
i-_-L
--:1
*
I
-
-
-
1
-:i
-
::;
*
.
...
-
_
-
* I
--
-----
.--
t
...4-.
-
--
•
i
L
1__~1
H
I
----~
.
.lL
.. .
I'-"-
1 -i
I
-- t--i
::
i :::
1::
i
iii!itii
i
i 1I
::::1:.:
I-: i
If~._i.
:
i:-- 1-~i-T
:::-::
13--;rida Ibt-a-,
r:-::
:: _I: 7--
"
.
. 1
i
----
I
i
.i
.4:
'-
I
l
.
---
:
--I
:
F-
:
~---~--
1
._
i- r-- i--.---- 7 ---- ": -
'..... i-
'
7Yi7-:j4 ui 4
YZ-L
I
*.
L----1
----- C
....._
..
___i_~_~
-------- T-
-;.
V
-1
*. 1
.
-----
i- ....- !
-------t
r" ;
___
:
------ _
----
-7 _
_
i
4~-"--~"-1
(
!
----I----"'
-Mir,
4
PM
-
7
t.o.,
iLi~-~
-128-
;~i
~ia
s
i~bFTn
... ....
O(z!
I I
7 A:
_
-i-
KK
l i:i
iii
ii:
,.
-H---
-
i*r
I;
i''''i
jii..i
: I
I
:
,
I- ---
--
-
rI
:1__:
'
I
7
.
..,1
I
liiiCL1.iliilili
' '
'
c~-ce --
77
--- I ))
.I _ _i----i-l-L--l~
i. _C
--
It
"a
x
I
i---------------~1
Ii
:1.2
i
i.iIf1:':
I
--
'-111----'
i i '~:'i': I
-.'i 1:
j. 1Ii
I:.
f
ii-rtiIi
- ' 'I
,
l i ,!
Ic-.
I
i
i
77ITI.*.
01 ,r"a
C)
-i
--
!
,o
Y
.. -.
77'
iiiii77:
...............
!
:
91 l
. ..
ii:.
ji
.. .. . ..'
,~i.
'!jl
0
+*
i."..."
. ....... ..
.. +..
"....
........
.. ... .. +.. ..+ -+ -+}
....
:ZL
. +I... ..
. .LL
. .L
. .. 'I ..
,,
....
...
'
ii.+:
E..
D
+
i
+
+
I
..
.
.
..
..
....
+.
.
+
I
,.
7-71
+
.
1:
+
:.
,
-,.
:+
•
..
,
....
ji
I
'
..
,.,:
+
+
i
j+ :+
++,
11
ii+++: +LL+ +:+' :+.++KKFVI
+ i:
I
+++
' +I
+++
I'++Fll
I ,:
t
7-
;::: :: :
: ::
. : ..
i I+
- _:
+ :. +:
:
. .
:.+ +++++ L
+....
1.
+ .:.++
j:.+
.
.:.-
:
I
-liz--
I
It
.... ..
: .:+ .. ..
..
..
.
+: + +.+
+++T+ . :+
,.
+
+:,
:+
+++
:
+!
+F
,11
,
:Il
..,.
.......
..
-_,
.+
:I}}:
.:::-- . ...,.. ..t.__+
+,...
.d..
..
..
++
.::::;
.._
+++I...++..,..
,
......... :I. ... ...
I:I
,l4
.
hor :o..e..
.
m.u
+....
.T .
..
..+ .. . ..'4'
,+.
....:
.:
:+++;;+:
.e...
' i '.++
.++++
+-+.
So
1
.7
0
., .
. . ..I .. ..
.
. .: I..
,
~-
. ....
.
. •
+ :+
+: :.
+:+: IL..;\
i- .
..+I ..
14
lz
:
::::t
''
10
+: :
:;::
O
i.:t
+ I++
+II
:
.:..
:::.'
~::
i'I
t iI +
'+++:+
. ...
..
l++: : : ii
i :! i . PiI '
I'
.
+
'
..t-'
1
.: ..I
I . . :l.
: I..
. i+
..+.i ....
.i.:
...
...
'I
'
.
'
4 Time, hourso
+
.
l:. :
j+.
"li .. !-
m-i
ili::
+
iJ
r-
:i
I
. t++L!
'
+
+,
.I
L
+ t '.
:;:
:: +: ..
:i:
l++t+!+. +
!:;
+:
+:
:+.
;t ++
. .. :
..'.. i + I+ . ...
;+
:: ...
0
or Tim.+f r ou .
.. ....
Fi
.II..-
ea s.
. .... ..
"A.T
m.
1 L-r
•. ...
1.: " .. +: .. .. .
,-4
o
.+
:.
.. +-+IN
.L ,
. :.+
-. .+
.-
i- t
I 1+.
+ ... i.
J-'
+
.----
-------
-
7-): 1::+4 : ::;-
:
+
..
.
i
I"%ap
.zoh
;-T1..• i.....
-I
- - .r . .. -iW, _-;
, i, i -I
t
.... ~...., :: ,: *1:I:::
j- :*:i#4I
.... 1.. ..,
...~..-i_,
,
t
i~
~ I ~~~
I.
i ( 1
..
.
.
7--1, 4I
I
i1,,
L^:i
._-,I ---i: : :
i..-
------- r:t_-jl-It III i
.
l:
too
- :::: ;: .:"
----1I--1---i
i
U)
0
r-4
I
.
C to
I : ' ':
i.
:
.
.
. .I
;
.
i i
i:
..
I :.: :::iI •iI:
.. .
I:
..
,
....
--I-i
-I----
: S: c|
! aiu--at
i- ..
..
.
-
!
til
:
j Vj
.
.
i "
•
..
.:_ ",--:
. .
L -
*
"
...
... . ..
.
4
4...
,j
;L.7
. K
.!. _ .. _.:_,. .
- ...
. .
-
i
....
. .. .
camsured
.. , II
i
-. .
:.
-
i
I LI
:...
i
.
..
I-
:i
. .
..
I
A
i I i
L
.
,
p
I
i
.
Ii
I"
..
;
+
I
I;
i
i : .":
..
I.
:
I
.;';
.ii
. .. ... . . .
.
:i
1
i
j
...I:_
I---
. ... .. . ":
.
i:;. 1.. i : !' :... .-. '
: ,
41
I44
W
i i--' -L----L~-~I
:ii_--: i:: ~1-I---)-+~-~ ~---~+----c^--I
::
i : ii.-..
..- ,.... Ii-~^~
- - -.
:
.---.----
-, -I --
i.--i:I-ll1
i..., i.!_I--t
: : :1
.,..,~...
I':I j
r
.-i.
.- ' .'
.
, . _-_.-
.
. .
.
9
. 4. .
.
.
.
.
:-,
'-_~:--::~I...
::
--
•
. ...jl. .......-I .+;. . . . .
0
iI~:t i::'
0:;::
:
'::
4:
':
-
:
30
1
i
-j
I
'
'.....,...........
4
_ ., --
'
-
----
---
1
.
49 Time, hour iO ----
I- '
--
-
--..
I
1
± -
.
-..
-
i-
... ..
-
Tm ,husaOi
I:: I
-t;;-;--r..-,
_;
: : :... ':--::
..-....
- --..
-~ l:-l.::!
.- - - - :-.-:- -..-:-'..,.::-.-...-,...... -. . r
t
cf~~pte'I
-V- '
i :
I
tonriiM4sureo Ani
'-
- -. --' " - -
g
---
"--
4epla~tea as
uncttoi of Timn
.
.,
.
.
. .i
, , ,
.
- '
-
0
0
I
..
::. ., .
I:
f
...r~c---. .
7-.+
.. ..
.
. . . .. ..
..
:
.
i
-
. . . .
!.
: ..
..
+~
i
~
.. .....
.
. .
+.
.
.
.
.
*
i
...
..
i
--i4....
..
ilii
...:
... . ..
t _+ . . ..
.
------ ---
..
t: -7 ,
-77 . .1..
::
-
.
. . .t+
,
:...
- '+
:+
+l
- -
-
=+ IT
i
+ I +T
+
.. .. ..
.+ ..I... ..
.
.
.
.
.
.
Loo
:
I ::T,
i I 1:.-.
. .....
.....
.......
o
0
. . I . ...
I
I.
l
+ | . . ..
. ...I
..
-.- r-.
- . ..
1 -
44
.. . . .. .........
...
+
pi
aGO
.
.
.......
...
.+
'~~
.--
. . . . ..
.
. ..
.
.1 .ei ..
+
++_ I
i
++
a --- diLi.
....
......
i-..
.........+
..... '....+
..... ..
.......
....
.. ..
+ ....
....
:
:
:
. + . ...
+i
•-
: . ""*i
1-
!
"-
I
"
:
1
+
-
....
+'
....---..
afed'.:++. +...
...
.......
1+4+
i-: + 1+P,
d'
+
.
__ '
..++
.++
+ ..
-
''~
+
+
[
/r
+
..
~"
:
-i-
J.
...
'
-77
:
~
:
-
:
c
.++++
I
i'-
'
-rt
....
I
.t
-
Ilr
L
4
ue*[UV~'0151.
O0
--
t ..
i
.++++++
.
....
I
45 1menourm
::I
..,.
...t ..i...
+
'"
o90
l
-
"
,i,,
lo5
,
I
,
•
10
0
~
::i ; i ~~
77i .
I
'
I~:
"'
""
:I ::
' ' I:Ilj"'' .]rIiiI:
~ ~ 1~
i
II
-t,..I....
i
i:
1
I': j:~ ~ ~-~L~
1
--
-.-i ~
i '
I
i :i
''''~'''I
--cc~-~
T
"
.+
III
i
,l, 7
i)
::i:; ""iii ii '''' ' '' I::t-- ' '"' ' ~-
+,i
,.
::il:::!
::I:I
i~~lii;
i :i IIi:i IiiIi
JL~ti~-~
40
...i.~:
ij: j 1:;
I;
1:::;-";
'1 '.
:i
"c"-^
'I
.I
!++ Yi!. "+-
:1.
i'"
'
i
!
i
'
- 1:. I.]
'~:
cU
1
1:r:t I..
.......
12
'
i i:
::I
l~eLb~
I
'
iii
"+!
1-: ;!;-:7 71
-I:771,
L '
i
+:
77
;77:7
' '' i ~~'
'
-r
. .
,il
I:
,
'
1i
+. ' .
.." .
I'
ilii-: :'::::
ite
~1~i
:;
:i
t
17.::
i :; I
C
t::s4
I
j
'J{iI':
:
if
tctc1:;
,1Iti~~P
; .:t
o
i
10
os.
:L~ii..!i
i{.i. :ed1d
:;..
,
Ti
e
i II "~'
'''
i
i 1: ~I
r
h
u s
o f
Co
i.l {
ii
3J
' ::0ni ili~ ii;fi
"!
r
I'
i
. ia'
7::71
-:!ii
-i. c,,
"!.
: :: i
i;;(,.~.(....., I:,;
ii 'I
:::
'iii:ii'
77 7 :I:
;
/i Ilj i:
i,
• '
iI
i.;
Ii
1jII ,
,I
.... ''1I
i
i
--*
l :i:
'''
i:
I ::
i:-I:i
Line'iia'-::
:7~-"'
+I
.-ii
I..."
--.~ik--*-c~
ii,-: ir--~"
-:tt--j.l i
I:I'
I I.
i:-Spec ,,,
iI:.
i 1-t-
'"i .. .,..i7
iII; tr;i ir!
fiii iiI
:!I.,!!ii
:+:
14 - :
I_7++7
I" I
-.--L. r,
ii,
' 'iI :i
i
.......
'i
i
t"7.. t--'t~t:-r
,.
'''
I :ii
i ''
"'~''''''
i iI
IL--c,
''
.e
.~~~ ~~
; ~~
..
::::ii
:i
I::I-.L_.
I
1
I
I
'""'""'x~~~I'
i : : i.:
7
'''-'
' I
..
i
I . I:::
:i
...1
C---tiI--
i
::
L-li~( i- ..
j
LL
t" ~ -iI: : : ,(::
i
I
i[
T
.
,
T
1 -- -
It:
21-i
. . i..
-it:
iiii:j;I
l"r
i
o
.:
.
..
I
i
I'I"---i:l!i
. 17
7'-
ed'
c a cul ato
1~
,,c,..,
''''~~'~-~;il' -~---"
I-,
....:.
.....
I
S:
:17
::;:' ~:::
-I---L-l
/
--
*:1!'
;;; j j:i ii
j
I~
i'l
" ....
'. j i
'i:
::
~t~L~L~L~L~L~L~L~L~L~~~
;i i
'''
0
LW.1....
~~1
7~i
'!
I
-7
,
Apecimen5t
I~1128,daysold
'*1
I
0
4
-.
____
r-4
I,
*--1---
*I
-4-
20
-4-
0
0
3
4
*nur
73;0
O5120
------ I------- ---------
--~ccc--~c- ~-cu ~-~-~-U.-----;-----
*----ls~-~
- .rY
IlrI-
it..,: i
:
I-
..
...
I.:
'"
I'
C-)
i-o
"
..
.
.
..
.
.
L~C~-pW-~rl*IUL---3
-*rCx.r
....
,.:
..........
...
:i
1
:
[.
!
- '
I- 2
I
.... ..
ii
I.'
i
!Aj
,,r
.. ....
C111- Ill~-~YL-u-- --I
:I
..:,,
....
:.
:: :..
.....
.. .
I
_t:7-'---i
. . : :. i:
I, :
Ibg
I ---
-II-CI1
--
CO
g
°4
O
co
(a
..
. '.
:
l
i
40
2o
:. .:...:
~.-,...'':T::qgj
~ ;
-
---
::
me, hours
...
..
..
- 1o .......
SoI :
-
i.:II
.
t:
I
0
__
:- t-- i_--
I 'I . T
----~I--------
____
I
__
ILi. I. . .--- = !I::- .i--L--
-*
.
|!i....
Ii:
_:
. ..:i ..:.
!.
...
i:
!.
I
-
- .. ...
-- .. I..
.
"
i
•~~~~
..
....
...
.
C
,.
I
I.
-*--
.'
i
- •;"
" "
"'I!
.
. .
S
q
' 1
. I
.
. . . .
:
_.
..
.:~
.
,
.
.
-
.I
.
.
-
1 q -!
.
..
....
.
.
.
...
.
.
I
'
.. !
.
.
SIC
•
,
,•
i
0
.
-
-- -'
. --!:
. .
..
,
!
i'me
.
1
q;
.1
--.
-~
......
i .
1
..-... -
. . . ..
.. . .
.. .
.
.
..
'-4
C-)s
4)
I
:
-I1-o
1
.;
. ..
, .i
i...
J :I .....
ii.,
..":I.
....... ...
_ .=..: .. -7 7
.
;i
I
I
'
'l .
"
il--i---i--li
....
'-4
4)
1-111
1-*
{
I
1 .7
c-r---,
I
i:.
t
,
C
-j7
..
L
.I_I
^-
--:"
;-
----+--
.I
...
I-
t !7
7 . Ci
.
.... . I.
:.. I.-.. iI ...-:
I.'I
. .
i
i
.--- . 01..- -
-
.-. ,....
iT
I Z
L - jY
C-ll -t----l ~-i-'----i ~-.- ~ ..:.1.. ..-.i.~.~l..-.:.~:;
ij I 3)'I
I i.
ulated -a- Fundtiotofi'ljm
0
...
..
i
- ,.
i
* -".l-,-:*..:,--...-
P:i :tV
i
3-S IS
.: '
.
I:1- :
. ..
,
- i--I - r - -
i: i i
, ~--+---": ' i
t--,--I~
,0
::.r
t:
I
i...-....
....I--i-i-~t'
-+e-
" 'I-; - "
k :!'
: :::
;
I'
I
/1! !
/7
.---:!
.- *
:. . . ,
B aed--8t"
..
.
..
,
0
: iI:: I
-: I--I- i ;I
i:
i
I.I
., -I
ii
i: i
i
''
:i;'i
i
r: 1
n
*r4
'' "?-
-
i
"
"
:, :I'--.-.
• -
i -:--i----
I tI~I~.-j--,I-.,-i--,..
-----r---- )---~)--.-1
i
: II
i)
I
-(
- t---i
i---) --- 1----i---i--
i...-
SII
!
. . ..
I
r' ".
-i
...
. ... . . .....
' -.......... .... ...
.... ..
i....
'.....
... : ...
...I~
.
i-- ;-
I'
i...-...
:..~ . .. j...
:I • '
,
.I~~:~1
....i....
....
i......
. . . ..
. ....:
. ... ....
.
',..
i
I.
!
.
l
•
m..
.... i
i
.
.- --
i -; -
-
'
]
"-l-,,
. .
-- " - -
.
.
:
---I--
!
-".--.
-"
.
--
... t ..
..
" -T. . ..
I
'
.. ''
deyl old 6teetrens
i .,i4
i.i
.I
4
I
i
r ....!
t
i
'
.. ........
-:i-.-:/ -.Ii ../
i
t I
:
i
i::I::.:
i..:..~. :I:i :--1
:: 1
i
1
I
~ I1
..._:....i .-. 1 1-I............
,'''
~
. .
2 U "-
h
.
0
.. ..
. .
.
.
........
.........
.
.
-= -4 = . _
.
...
.
.
.
t•
S
go
' EL . __ . - .= 4 - . . . . ... ... - - -.'- .-- .. . - ' . _._.... . . .. ... . . _,. . .. .
"
"
,
"
i -;_ :... i..
t.i
.-, ... IT
_.. .-. -. - . --.. . .m.
I_ = .a
_.
-"
= "
=
.5.n~~~~
..., .
..
. " 'd" "on ' Beams
'~ ~~~~
.
.
- : i. -:.... . . : . . - t ; tl -..... . -. .. . ..
, :I..
.,.~~~
....
:
,
I
=
i.-
-
.!
7
-
•
i_:
-e
•
:
i
,
.i
-
..
,
-
•
.
:
-i
-
.
-
I^
.
t~
r
l
--
----
-(-
(-
-
s
-
-
~---'- -~-~~
777F77,
I~:
,-i :--.i-ll
.-.-
-~---
[1
:
-~----
I
. 4 4..-..
1
I-i
~~~~
.
f
ivi
" ' .....
.. .... i..
I iI i
.....f
i .I
.
; :: I:
(. :
IVI::T
SI
tl!
~~
i
•
:
1oo
r-4
'
1.
I
i
. .
i :
' '
! '
. .
.
. : |
..- . . . ...
."
.
. .... ....
...I1
i
!
f !-.
.i . . ,
.
2
.
,
.
0
I
.1-i
r4r
... j" _i21
- "i
a
44
60
,
..-
t----:l"7,
/? I i
i
4-
!
, ,
::-i
..
I- 111
I
: :
o
-
----
1
.......-.
I ,.........
f~...~
.-..
-
t--f---r--j
-- i
I
•
CL~-~-3
i
.. .
.. ..
..
,-- h
.
. Z.4
.I~...... I i
--
"
L-
-i:l-i
-----
---
:& .-
-.-
i
I
_..
[
II~-
...-
i
IJ
IS
.
.]i
.
.
I
.
.;.
.
.
.
... i
-
lJ
-
'
'
--
- - -'I
-L
-=-b::.
T
•
S ,i- '
' " -t , -'
I
-I.- ---
-
.
,
*
K--'-- --
--- -
O
6- "
ii
'
Meaurld and
alculated as P ncti
4,5 Timehourso
.T
'L
..
i i-- '
T
C-
.
-- i
Lss
L
....
oS
..
!
.i
"i ~j
I
-!
Dr lac
!
. ... ..
i..
I
s~p
r
i
~.i.,.
n of Time ;Baae
'o
-s
on Beama
o
Z
C
S
: : .!:: i
i~~.:
.o --
-:i
~...
~
!,::.,
:
i
~
-
.. . ;, .
-, . .
~
~
...
i'.
.
I
L
..t-
1/
' I
ii:
.....
'j'i
°
-- - p....--
.. t -3
.. !r:..
1~.l.
ii
.
I
--- -T..--.... -- i
; ; c-7:~....,.i .. i... ~-...
I : j II
-~^c--1
-------
?
I
I
-- i :
i
I
i i -----
:
---------------i
1
'
1
-,--..
.
__
.
'
j1
1
: i -" -, .~
.
W.d.
al--....~-r,:
|i
..- ...!-- .. ..
Comp
:-,
--t - ......
-T .
..
.
.
i"
...
.... ..
..
.
....
i
-- -
j.-
: I-
i- - - -i - --
Ii
' ..
I
" ; ""_
i.r4
_
~I
1
j2x .
L
m:
,
.. -
...... .. ..
... ..
:,......
:I
35
.
I
i ...
4 ......
-- ,-.. .-
.....
...
. . . - --, ..... --.
-
: - *-
4.0
I
I
......
I '. .....
....
:.:...{
I
'
" I
tt
7-"
'H
....
. .... ..
i: i i:
04.5
j3
i
-. ~,.-.I-cc,_-. i..- i._
: -I-: :I:: i LI
i' t rt-I-c- :''i
''
t _ic~rrr~-~1
i: i....
.. 1 . ;..,.
I-
0
-
._I
=-"
i II :
. . . . - ..-, I ... " : ..... i . !
.. ...
r --.. .r-T
-- - tt
...
i
'
:
I
'-4
"I
nd, 4*8 Cylindes as Funiction
4.5
TifOt
..
U~
I
4s Time, hourwo
I.
.
.
75
75
of Ti4
atiFouriAgms
ic-s
las"
i
-/ 20
120
~.__I
II
I
__I__
~II
I
____llllIIIIll
BIBLIOGRAPHY
1.
Hatt, W.K.,
"Notes on Effect of Time Element in Load-
ing Reinforced Concrete Beams," Proceedings, American
Society for Testing Materials, V. 7, 1907.
2.
Ferguson, A.M., "Fundamentals of Reinforced Concrete,"
John Wiley & Sons, New York., 1964.
3.
Large, G.E., "Basic Reinforced Concrete Design-Elastic
and Creep," Ronald Press, New York, 1960.
4.
Neville, A.M., "Theories of Creep of Concrete," Journal,
American Concrete Institute, V. 52, 1955.
5.
"Symposium on Creep of Concrete," SP-9, American Concrete Institute,
1965.
6.
Freudenthal, A.M., "The Inelastic Behavior of Engineering Materials and Structures," John Wiley & Sons, New
York, 1950.
7.
Freudenthal, A.M. and Roll, F., "Creep and Creep Recovery of Concrete under High Compressive Stress,"
Journal, American Concrete Institute, V. 54, 1958.
8.
Grudemo, A., "The Microstructure of Hardened Cement
Paste," Chemistry of Cement, Proceeding of the Fourth
International Symposium, Washington, D.C., 1960.
9.
Taylor, H.F.W., Editor, "The Chemistry of Cements,"
Academic Press, New York, 1964.
10.
Freyssinet, E.,
"The Deformation of Concrete," Magazine
of Concrete Research, London, V. 3, 1951.
11.
Lynam, C.G., "Growth and Movement in Portland Cement
Concrete," Oxford University Press, London, 1934.
12.
Mullen, W.G. and Dolsch, W.L., "Creep of Portland
Cement Paste," Proceedings, American Society for
Testing Materials, V. 64, 1964.
13.
Ali, I. and Kesler, C.E., "Symposium on Concrete Creep,"
American Concrete Institute, SP-9, -1965.
14.
Bingham, E.C. and Reiner, M., "Rheological Properties
of Cement-Mortar-Stone," Physics, No. 4, 1933.
15.
Forshind, E. and Torroja, E., "Elasticity and Inelasticity of Building Materials," North Holland, Amsterdam, 1954.
-140-
__I__ ^_
~___~________
I~----~IIYU*-
IL~-XI~ -~-^~-^-^
~s~--------------- il
BIBLIOGRAPHY (Continued)
16.
Paez, A. and Mason, P., "Elasticity and Inelasticity of
Building Materials," North Holland, Amsterdam, 1954.
17.
Ali, I. and Kesler, C.E., "Rheology of Concrete, Review
of Research," University of Illinois, Engineering Experimental Station, Bulletin 476, 1965.
18.
Tobolsky, A.V., "Stress Relaxation Studies of Viscoelastic
Properties of Polymers," Rheology, M. Reiner,
Editor, Academic Press, New York, 1956.
19.
Flugge, W., "Mechanical Models in Plasticity and Their
Use for Interpretation of Creep in Concrete," Report
No. 8, Office of Naval Research, Stanford University,
1950.
20.
Cowan, H.J., "Inelastic Deformation of Reinforced Concrete in Relation to Ultimate Strength," Engineering,
London, 1952.
21.
Hansen, T.C., "Creep and Stress Relaxation in Concrete,"
Swedish Cement and Concrete Research Institute, Stockholm, Report No. 31, 1960.
22.
Gluckich, J. and Ishai, 0., "Further Study of the Rheological Behavior of Hardened Cement Paste," Journal,
American Concrete Institute, V. 57, 1961.
23.
American Concrete Institute, Building Code 1963.
24.
Yu, W.W. and Winter, G., "Instantaneous and Long Time
Deflections of Reinforced Concrete Beams under Working Loads," Journal, American Concrete Institute, V. 57,
1960.
25.
"Deflections of Prestressed Concrete Members," American
Concrete Institute,
Report 60-72.
26.
Dischinger, F., "Elastische und Plastische Verformung
der Eisenbetontrabwerke," Bauingenieur 20, 1939.
27.
Leonhardt, F., "Prestressed Concrete," W. Ernst & Sohn,
Berline, 1964.
28.
Bieger, K.W., "Prestressed Concrete," Indian Concrete
Association, Sahu Cement, Bombay, 1963.
29.
Corley, W.G. and Sozen, M.A., "Time-Dependent Deflections of Reinforced Concrete Beams," Journal, American Concrete Institute, V. 63, 1966.
-141-
~~~----
--
30.
Bresler, B.
--
,I
and Selna, L.,
"Analysis of Time Dependent
Behavior of Reinforced Concrete Structures,"
American Concrete Institute, 1965.
SP-9,
31.
Washa, G.W. and Fluck, P.G., "Effects of Reinforcement
on the Plastic Flow of Reinforced Concrete Beams,"
Journal, American Concrete Institute, V. 48, 1952.
32.
Washa, G.W. and Fluck, P.G., "Plastic Flow of Reinforced
Concrete Continuous Beams," Journal, American Concrete
Institute, V. 52, 1955.
33.
Finnie, I. and Heller, W., "Creep of Engineering Materials," McGraw Hill, New York, 1959.
34.
Polivka, M. and Best, C.H., "Investigation of the Problems of Creep in Concrete by Dorn's Method," Department of Civil Engineering, University of California,
Berkeley, 1960.
35.
Tietz, T.E. and Dorn, J.E., "Creep of Copper at Intermediate Temperatures," Transactions, American Institute of Mechanical Engineers, V. 206, 1956.
36.
Powers, T.C., "Physical Properties of Cement Paste,"
Chemistry of Cement, Proceedings of the Fourth International Symposium, Washington, D.C., 1960.
37.
La Rue, H.A., "Modulus of Elasticity of Aggregates and
its Effects on Concrete," American Society for Testing Materials, V. 46, 1946.
38.
Dantu, P., "Etude des Contraintes dans les Milieux
Heterogenes, Applications au beton," Annales de
1'Institut Technique du Batiment et des Travaux Publics, Paris, 1958.
39.
Hirsch, T.J., "Modulus of Elasticity of Concrete Affected by Elastic Moduli of Cement Paste Matrix and
Aggregate," Journal, American Concrete Institute,
V. 59, 1962.
40.
Kaplan, M.F., "Ultrasonic Pulse Velocity, Dynamic Modulus of Elasticity, Poisson's Ratio, and the Strength
of Concrete Made with Thirteen Different Coarse Aggregates," RILEM, #1, 1959.
41.
Kordina, K., "Experiments on the Influence of the Minerological Character of Aggregates in the Creep of Concrete," RILEM, #6,
1960.
-142-
_~~r~ss~
I
_I
~__~_
BIBLIOGRAPHY
_1 ___1
I
(Continued)
42.
Counto, U.J.,
"The Effects of Elastic Modulus of the
Aggregate on the Elastic Modulus, Creep and Creep
Recovery of Concrete," Magazine of Concrete Research,
London, 1964.
43.
Baker, A.L.,
"An Analysis of Deformation and Failure
Characteristics of Concrete," Magazine of Concrete
Research, London, 1959.
44.
Reinius, E., "A Theory of the Deformation and Failure
of Concrete," Betong, No. 1, 1955.
45.
Pickett, G., "Effect of Aggregate on Shrinkage of Concrete and a Hypothesis Concerning Shrinkage," Journal,
American Concrete Institute, V. 52, 1956.
46.
Hsu, T.
and Slate, F.O.,
"Tensile Bond Strength Between
Aggregate and Cement Paste or Mortar," Journal, American Concrete Institute, V. 60, 1963.
47.
Brunauer, S. and Copeland, L.E., "The Chemistry of Concrete," Scientific American 1964.
48.
Hansen, T.C., "Theories of Multiphase Materials Applied
to Concrete, Cement Mortar and Cement Paste," Swedish
Cement and Concrete Research Institute Reprint 39,
Stockholm, 1966.
49.
Butcher, J.G., "Introductory Concepts of Combined Elasticity and Viscosity," Proceedings London Mathematical Society, V. 8, 1876.
50.
Jeffreys, H., "On Plasticity and Creep in Solids,"
ceedings Royal Society, 1932.
51.
Leaderman, H., "Elastic and Creep Properties of Filamentous Materials and Other High Polymers," Textile
Foundation, Washington, D.C., 1943.
52.
Alfrey, T., "Mechanical Behavior of High Polymers,"
terscience Publishers, New York, 1948.
53.
Tobolsky, A.V., "Properties and Structure of Polymers,"
Joyn Wiley & Sons, New York, 1960.
54.
Ferry, J.D., "Viscoelastic Properties of Polymers," John
Wiley & Sons, New York, 1961.
55.
Lee, E.H.,
"Stress Analysis in Viscoelastic Bodies,
Quarterly of Applied Mechanics, V.
-143-
13,
1955.
Pro-
In-
I~_-
I
_~
_
_
III
I
I
_
56.
"Solid Rocket Fuel Abstracts,"
cal Society.
57.
Volterra, E., "On Elastic Continua with Hereditary
Characteristics," Journal of Applied Mechanics, V.
Published by the Chemi-
73.
58.
Bland, D.R., "The Theory of Linear Viscoelasticity,"
Pergamon Press, New York, 1960.
59.
Gross, B., "Mathematical Structure of the Theories of
Viscoelasticity, Herman et Companie, Editeurs, Paris,
1953.
60.
Lee, E.H., "Some Recent Developments in Linear Viscoelastic Stress Analysis," Eleventh International Congress of Applied Mechanics, Munich, 1964.
61.
Alfrey, T., "Non Homogeneous Stresses in Viscoelastic
Media," Quarterly of Applied Mechanics, #2, 1944.
62.
Gurtin, M.E. and Sternerg, E., "On the Linear Theory of
Viscoelasticity," Archives for Rational Meclhanics and
Analysis, V. 11, 1962.
63.
Breuer, S. and Onat, E.T.,
"On Uniqueness in Linear Vis-
coelasticity," Quarterly of Applied Mechanics, V. 19,
#4.
64.
Churchill, R.V., "Operational Calculus," McGraw Hill,
New York, 1958.
65.
Schapery, R.A., "Approximate Methods of Transform Inversion for Viscoelastic Stress Analysis," Graduate
Aeronautical Laboratories, California Institute of
Technology, GALCIT 119, 1962.
66.
Smith, E.B., "The Flow of Concrete Under Sustained Load,"
Proceedings, American Concrete Institute, V. 12, 1916.
67.
Fuller, A.H. and More, C.C., "Time Tests of Concrete,"
Proceedings, American Concrete Institute, V. 12, 1916.
68.
Williams, M.L., "Structural Analysis of Viscoelastic
Materials," AIAA, Journal, 1964.
69.
Lee, E.H. and Rogers, T.G., "On the Finite Deflection
of a Viscoelastic Cantilever," Division of Applied
Mechanics, Brown University, 1961.
70.
American Society for Testing Materials, Standard C-150.
71.
American Society for Testing Materials, Standard C-188458.
72.
Brisbane, J.J., "Characterization of Linear Viscoelastic Materials," Rohm and Haas Company, 1963.
-144-
L~ -.
I
I
--
----- ---
BIBLIOGRAPHY (Continued)
73.
Hodgman, M.S., "Mathematical Tables from the Handbook
of Chemistry and Physics," Chemical Publishers,
Chicago, 1954.
74.
Mandel, J., "Les Corps Viscoelastiques Boltzmanniens,"
Rheologica Acta, Band 1, No. 2-3, 1958.
75.
Hildebrand, F.G., "Advanced Calculus for Applications,"
McGraw Hill, New York, 2nd Edition, 1958.
76.
Cost, T.L. and Becker, E.B., "The Multidata Method of
Approximate Laplace Transform Inversion," Rohm and
Haas Company, Special Report S-51, 1965.
77.
Hilton, H.H., "Analytical Determination of Generalized,
Linear Viscoelastic Stress-Strain Relations from Uniaxial, Biaxial, and Triaxial Experimental Creep and
Stress Relaxation Data," Aero-General Solid Rocket
Plant, Sacramento, California, 1961.
78.
Trost, H., "Spannungs-Dehnungs-Gesetz eines Viskoelastischen Festkorpers wie Beton und Folgerungen fur
Stabtgwerke aus Stahlbeton und Spannbeton," Sonderdruch aus Beton, H. 6, 1966.
79.
Neville, A.M. and Staunton, M.M. and Bonn, G.M., "A
Study of the Relation between Creep and the Gain of
Strength of Concrete," Civil Engineering Report,
University of Alberta, Calgary, 1965.
80.
L'Hermite, R.G., "Volume Changes of Concrete," Proceedings Fourth International Symposium on the Chemistry of Cement, Washington, 1960.
81.
Hallaway, L.C.,
"The Creep and Elastic Properties of
Concrete in Tension," M.S. Thesis, University of
London,
82.
1965.
Ali, J.,
"Creep in Concrete With and Without Exchange
of Moisture With the Environment," Ph.D. Thesis,
University of Illinois, 1964.
-145-
-
~
I
__
APPENDIX A
The chemical analysis of the cement used throughout this
thesis is presented in Appendix A.
The chemical analysis is
provided through the courtesy of the Hercules Cement Company
of Stockertown, Pennsylvania.
A.
1.
Chemical
Silica (SiO ) . . . .
Alumina (Al 0). . .
Ferric Oxide(e2 03).
Lime (CaO)
3..
. . . .
....
. . . . . . .
.
........
.
.
a 5.81%
a 2.53%
. 62.96%
. 3.80%
2.48%
.
0.99%
a
Magnesia (MgO) ....
.
Sulfur Trioxide (SO3).
.
Ignition Loss . . . . . .
. .
.
.
.
.
. .
..
. .
.
.
.
.
Potential Compounds
Tricalcium Silicate (3CaO-SiO2).
.
Tricalcium Aluminate (3CaO-A12 03). .
Dicalcium Silicate .. .
........
Tetracalcium Alumino Ferrite . . .
CaSO 4 .
...............
.
4
2.
20.12%
.
.
.
54.00%
.
.
.
.
11.20%
16.80%
.
.
7.60%
4.30%
.
Physical
Fineness, Specific Surface, Wagner
Air Permeability, Blaine
Soundness, Autoclave Expansion
Time of Setting, Gillmore
initial (Hr.: Min.) .
......
Final (Hr.:Min.) . . . . . . . .
Compressive Strength, psi
1-Day. . . . . . . . . . . . . .
3-Day .
.
.
.
.
.
.
.
.
.
.
.
.
7-Day.
..............
Air Entrainment, % by Vol ......
-146-
.
1793
3209
0.32%
.
.
.
.
.
.
.
.
a
a
.
.
.
.
.
.
.
.
.
3:20
5:30
1633
3158
4158
9. 800/0
_
I
I
~L_
APPENDIX B.
B-I.
COMPUTER PROGRAMS
Ramp transformation program for stress relaxation.
-147-
~--
_
/JOB
/FTC
C
C
C
C
C
C
C
C
C
999
1
100
2
101
3
102
C
4
103
5
98
6
99
104
7
106
~~
__
NEIEC
14610.
TRANFORMATION DF STRESS RELAXATION FROM I<AMP 1C STEP S TRAIN
COLLOCATION METHOD
CNT=ASY,',PTTIC VALUE OF STRESS
N=NUpbbR OF COLLOCATION POINTS
SIlGrtiA(I)=STRESS FkiOM INSTRON
ALPHA(I)=COEFFICIENTS OF EXPONENTIALS
TNUT=LOADING TIME
RATE=RATE CF CROSS HEAD DISPLACEMlENT
J=IDENTIFICATION CODE IN TERMS OF AGE AT LOADING
uIMENSIUN SIGM4A(50O)T(50),ALPHA(50),HCCOF(50),D(50,50)
DIENSIGN A(50,50),FRS(50),F(50,50),FR(50)
DI;MENSION COEF(50),DC0EF{50)
READ(5,
)J,N, M,CNTI, TNCT, RATE
FORMAT(315,3F10.5)
WRITE(6,1O0)J,N, M,CNOT,TNOT,RATE
FORMAT (315,3FI0.5)
READ(5,2)(SIGMkA(I),I=,N)
FORMAT (7F10.5)
WRITE(6,101)(SIGMA(I),I=1,N)
FORMAT (7F10.5)
RFAO (5,3) (T( I ), I=1,N)
FORMAT (7F10.5)
WvRITE(6,102)
(TlI),I=1,N)
FORMAT (7F10.5)
CO2MPUTE CUEFFICIENTS
DO 4 I=1,N
ALPHA(I)=-1./(2.:T(I)-TNOT)
WRITF(6,103)
(ALPiA( I) ,I=1,N)
FORMAT'COEFFICIENTS OF EXPONENTS.=',7FlO.4,//)
DO 5 I=1,N
D(I, 1)=SIGMA(I)-CNOT
wRITE (6,98)D(
,1)
FOR Al (IF.10.4)
DV 6 I=1,M
DU) 6 J=1,M
A(I,J)=O.0
FRS(I)=-T(I)''ALPHA(J)
CONTINUE
DO 99 I=1,M
wv ITEI(6,104) (A(I,J),JI1,M)
FOR.MAT(7F10.5)
CALL MATINV(A,M)
0DO 7 I=I,M
WRITE(6,106)(A(I,J),J=1,M)
FORCAT(4X,7F10.5)
CALL NULTP(Ai),F,M ,M, M1,ISSUC,0)
WRIT E(6,105) (F(i, 1), I=1,)
105
C
8
FO MAT(4X,'RAiP COEFFICIENTS.=*,iOE10.3,//)
COQPUTE MODULUS COEFFICIENrS
DC 8 I=1,N
HCuEF()=F(I,1)
00 9 I=1,N
-148-
I
INPUT
I __
9
10
107
C
108
/FTC
C
1160
1170
10
11
1180
1190
1100
1110
/FTC
C
C
C
C
C
1
2
COEF(I)=1iCOEF (I)E(XP(ALPHA(I)*TNOT))
DO 10 I=I,N
DCOEF(I)=(COEF(! ) ALPHA(I))/(RATE (1.-_XP(ALPH(tI)'TNFT)))
RIT E(6, 1E-F
iO7) (0v;
I I ),I=iN)
F ORMkAT(4X, 'MODiULJS COEFFICIENTS.='t 7E.
3,//)
CO!PUTE CCNSTANT VALUE OF MOCULUS
D NUT=CNUITI/( TNOT''R ATE)
FORMAT (4X,'CUN!STANT VALUE OF MODULUS.=', 1E10.3,//)
GO TO 999
END
MATRIX INVERSION BY PARTITIONING
SUBROUTINE MATINV(A, N)
DIMjENSION A(20,20),B (30),C(30)
NN=N-1
A(I,i1)=1./A(1,1)
DO 1110 M=1tNN
K=M+I
I)0 1160 T=,M
83(I)=0.0
00 1160 J=1,M
8(I)=B (I)+A ( I ,J)A (J ,K)
R=0.0
00DO 1170 I=I,M
R=R+A(K,I)*8(I)
R=-R+A (K,K)
IF (R-.O0001) 0, 10,11
R=.000001
CONTINUE
A(K,K)=I./R
DO 1180 I=1,M
A (I,K)=-B(I)*A(K,K)
DO 1190 J=1,M
C(J)=O.O
C(J)=C(J)+A(K,I )*A( I ,J)
DO 1100 J=1,M
A(K,J)=-C(J)*A(K,K)
DO 1110 I=1,M
00 1110 J=1,M
A (IJ )=A( I,J)--B(I)*A (K,J)
R ETURN
END
A=MA'NA MATRIX
= M BNB MATRIX
C=PRODUCT MATRIX
ISSUC=TEST FOR CONFORMABILITY
IB=TEST FOR HOMOGEI4EOUS SYSTEM
SUBROUT INE fMULTP(AB, C,MA,NA,M8B, NB, ISStU , I 8)
DIM'ENSIO)N A(20,20) ,B(20,20),C(20,20 ), TEBAP (4030)
IF(NA-M3) 1,2,1
ISSUC=2
WRITE (6,40)
GO TO 35
ISSUC=1
-149-
_
_~
__
_ _
IF( I B-0) 15, 5 15
00 10 I=i, MA
DO 10 J=1, NB
C( I, J )=0.
0D 10 K=I,NA
10
C(1,J)=C(l,J)+A(I ,K)*
(K,J)
GO TO 35
DO 30 J=INB
DO 20 I=1,'-A
TE!P(I )=0.0
DO 20 K=1,NA
20
TEMP(I)=T EP {)I +A(.I, K)* (KJ
DO 30 I=1,MA
B( I,J)=TEMP( I
30
CONTINUE
40
FORMAT ('MATRIC ES ARE NUT COMFORMABLE
35
RETURN
END
/ENO OF FILE
5
-150-
' )
~
~I
I
B-II.
Collocation program for curve fitting.
-151-
II__
_~ ~
/JOB
/FTC
C
C
6iC
C
C
C
C
C
999
L
5
2
6
C
9
3
20
21
60
70
15
/FTC
C
____
__________
1- e610.
NEMEC
SERIES FITTING BY COLLOCATION
,==;UH ERFF
CO iLL_iCA ION PO I N TS
-(1 )=TI-iE AT CLLCATION P(.IINTS
GA''A(I)=VISCJELASTIC RESPCNSE
MAT INV=:MARI X iNVFrSION SUBRLUTINF
AULTP=-"ATRIX 4ILTIPLICATIf'.N SUBROUTINE
L=l
RELAXATION DATA
L=2
CREEP DATA
IN TH F 1 NTFRVAL
kETA(1)=COEFFICIENTS OF EXPONENTS CENTERE)
SUN ( 1 ),RSU ( I L ) ,b t TA1 20)
DI
1NSION T( 1I ) ,G(10),DELTA(10)
OIMENSION FRS( 10),FR(10),GAMMA( 10),A(20,20,20),F(2 ,20) ,t)(20,20)
RFAD(5,1)J,L,M,
1N, GAMNOT
F]ORAAT(415, IF10. 5)
WRITE( 6,5 )J ,L ,M ,N,GAMNOT
FOkMAT(4I5,1FIO.5)
REAC(5,2)(T(I), 1=1,N)
FOi<MAT (7F10.5)
R EAD(5,2 ) (GAMA ( I) ,I=1,N)
WRITE(6,6)(GAMM4A(I),I=1,N)
FORMAT (7F10.5)
COMPUTE BETA
D-i 9 I=1,N
TA( I )=1./(2. T(I))
WR I T (6,
) (,F FA (I) , I=1, N)
DO 13 I=1,M
' A ( I) --GAMNOT
, 1) =G
3 ( iA
FORXATi(LF1O.5)
00 20 J=,
A(I,J)-= .O
F .S(I) =) BETA(J)
A(I,J) A( ,J)
r-EXP(FRS I))
OiI INUE
DO 21 I= I,M
RITE(6,A6)(A(I,J) ,Jl,M)
CALL MT.iNV(A,M)
DO 60 I=1,M
RITE(6 ,7J)(A(i 1,J) J= 1M)
FU'MAT ( 7F 10.3 )
CAILL_ '"JITiP 1,D, F M,M M 1 ISSUC 0)
(F(I, 1), I=1iiM) t,GAMNOT
,RITE ( ,15)
FO[RJ-AT (4X,'CREEP FUNCTION COEFFS.=',1OF10.5,//)
GO TO 999
END
.IATRIX INVERSION BY PARTITIONING
SUROUTINE IATINV(AN)
DI-JCENSION A(20,20) ,B(30),C(30)
NN=N-1
A(1,I1)= I./ A ( 1 , I)
)Dj 1110 M=1,NN
K= +1
.- 152-
__
_
~__
116
1170
1180
1190
1100
1110
I
DO
B (I
DO
B( I
1160 I=1,M
)=0.0
1160 J=1,M
)=3( I )+A( I ,J)
J,K)
DO 1170 I=1,M
R=R+A(K,I )* (I)
A(K,K)=l./R
O0 1180 I=1,M
A(I ,K)=-B( I) *Af K,K)
DO 1190 J=1,M
C (J)=O.0
C(J)=C(J)+A(K,I )*A(I,J)
no 1100 J=i,M
A(K,J)=-C(J)*A(K,K)
DO 1110 I=1,M
DO 1110 J=1,M
A( I,J)=A( I,J)(I )*A(K,J)
RETURN
END
/ FTC
C
C
C
C
C
A=MA*NA MATRIX
B=MB'NB MATRIX
C=PRODUCT MATRIX
ISSUC=TEST FOR CUNFORMABILITY
IB=TEST FOR HOl;i;GENEOUS SYSIEM
SUBROUTINE MULT P(A,B,C,MA,NA, MB, NB, I SSUC, [13)
DIMENSION A(20,20) , B (20,20) ,C(20,20 ),TEMP (400)
IF(NA-R8)1,2,1
1
ISSUC=2
WRITE (6,40)
GO TO 35
2
ISSUC=1
-0)15, 5,15
IF(
5
DO 10 I=1,4A
DO 10 J=1,N3
CII,J)=O.O
00 10 K=1,NA
10
C(I,J)=C(I,J)+A(I,K)*B(K,J)
GU TO 35
15
CONTINUE
DO 30 J=1,NB
DO 20 I=1,MA
TEiP (I )=0.O
DO 20 K=1, 1 NA
20
TEMP( I)=TE
I) +A I, K)'1 (K,J)
DO 30 I=1,MA
B(I,J)=TEMP(I)
30
CONTINUE
40
FORcMAT ('MATRICES ARE NOT COMFORMAB LE')
35
RETURN
END
/ END OF FILE
-153-
_r
__
__
B-III.
_ _______ _~__ _
Laplace transformation program for
relaxation modulus,
-154-
creep compliance.
________
_ _
_
/ JO B
/ FTC
C
C
C
C
C
C
C
C
C
C
999
100
104
101
105
102
106
103
107
7
8
2
108
109
98
3
9
20
11I
_~_______ _
__
14610.
LAPLACE TRANSFJORMATIGN OF D.IRICHLET SERIES
ANUT=CflNSTANT TE1[r' IN SERIES
ALPHA(I)=COEFFICIENTS UF EXPF(NETIALS
GAM<A( I)=CCFFICIENTS
DIRICHLET
'
SERIES
P(I)=LAPLACE PARAiETERS
NN=NULBER
PF P PARAMEIERS
CLAPL(I)=LPLACE TRANSFORA OF CREEP
'1=1 CREEP
M=2 RELAXATION
ALAPL(I)=LAPLACE TRANSFORM OF MODULUS
oDIMENSION ALPHA( 5,) ,GAMMA(50),P( I00),LAPL( 50),SL APL(50)
0I'bENSI-N PSU4'i (50),SUP(50)
DI ENSION CLAPL)(50),ILAPL(50),OLAPL(53),EL APL (50)
READ 5,1 C ) i, ,NN,ANOT
FORMAT (315,1E10.5)
,RITE(6,104)N,M,NN,ANOT
FORMAT (315,1E10.5)
READ(5,i01)(ALPHA( ),T=l,N)
FORMAT (7F10.5)
W;ITE(6,105) (ALPHA(I),I=1,N)
FORMAT (7F10.5)
R EAo ( 5, 102 )(GA 'IMA ( 1) , =1 ,N)
FOPMAT(7E10.5)
WRITE (6,106) (GAMMA(I),I=1,N)
FOR IMAT(7F10. 5)
P(J ),J=INN)
R EAD (5 ,103)
FOr MiAT ( 7F 10.5)
WRITE(6, 107) P( J ,J=1, NN)
FOPfMAT (7F10.5)
IF(M-2)3,7,7
J=I
I=1
SUM(I)=GAMPA(I)/(P(J)+ALPHA(I))
DO 2 I=2,N
PSU'k ( I )=GAMAA (I) / P( J )+ALPHA( I))
SJ (i )=SU ,( I-1)+PSUM( I)
WRITF(6, I8)M,I, J , SUM(I )
PCtAT(3I5,4X,1-10.5)
SIA PL ( J )= ANOT/P (J)
MiLAPL( J )=DI APL( J )+SUM(N)
Ir
,109) - IJ r LAPLI(J) DLAPL(J)
,(6
X, IE1O.5)
FOR.MAT(3I15,4X,IE10.5,
J=J+1
IF (J-NN)8,8,98
CONTINUE
J=1
I=1
SU(I)=GAMA(I)/IP(J)+ALPHA(I))
00 20 I=2,N
PS
I )=G
A (I)/ I P( J )+ALPHA( I) )
S Ui (I )=SUM (I- 1) +PSUM( I)
WRITE( 6, 1ll), I , J,SUNII)
FORMAT(315,4X,1EI0.5)
-155-
____
__
__
___~_~__1~1__
110
99
/END
DLAPL(J)=ANOT/P (J)
CLAPLIJ)=1./(,)LAPL(J)-SUM(N))
WKITE(6,11)MI,J,CLAPL(J) 0LAPL(J)
E10.5)
FORMAT (3I 5,4X,.lEO.5,4X,
J =J +-1
IF (J-NN) 9,9,99
C ONT I NUE
ENO
CF FILE
-156-
I_
~~_
B-IV.
Inverse Laplace transformation program
using Schapery's collocation.
-157-
_
~
_
/JOB
/FTC
C
C
C
C
C
C
C
C
C
C
999
i
99
2
101
100
102
3
109
C
C
17
16
110
15
103
C
19
104
C
11
10
_~ _
__
NEMEC
14610.
INVERSCELAPLACE TRANSFCRMATICN BY COLLOCATION
FERMS IN SERIES
iF
4=N=NU,
-- -I'U 4R
BETA(I)=COEFFICIENIS OF EXPLiNENT.S
LLAST=ASSOCIATED ELASTIC SOLUTION
SNCT=CONSTANT VALUE OF FIRST TERM
GAiMMA (I)=OIRIPCHLET SERIES COEFFICIENTS
P(I)=P PARAMETERS
R
ATIX OF COEFFICIENTS EVALUATED AT P-PA-RA'ET ~S
KAPPA( I )=COLUtN
F(UI,)=TIMF DEPENDENT STRESS RESPONSE
iREAL TIIE EXPONENTS AS INPUT
ETA(I)=l./P(J),
KAPPA(23),BKAPPA(20)
DIMENSION BETA(20),P(1C) ,GAMMA(20),
DIMENS ION A(20 ,20) ,f(2C,20),KAPPA( IC ),FRS(50),F (20,20)
DIi;ENSIGN PSUi(50), PKAPPA(50),ETA(2J)
READ(5,1 )N,, SNOT,ELASTLCAD
5,3F10.5)
F ,ORMAT(21
WRITE (6,99) N,M,SNOT, ELAST, LOAD
FORMAT(215,3FI0.5)
READ(5,2) (BETA(I),=1,N)
FlRMAAT(7FIO.5)
WRITE(6,101)(BETA(I),I=1,N)
FO.YAT(4X,'EXPONENTS OF SERIES.=',7F10.5,//)
A( ) ,I=l ,N)
READ (5, 100)(GAG
FO
fr AT (7F 10.5)
WRITE(6, 102) (GAMMA(I),I= ,N)
FORMAT(4X,'COEFFICIENTS OF SERIES.=',7FIO.5,//)
(ETA(I),I=1,N)
READ(5,3)
FORMAT ( 7E10.5)
WRITE(6,109)(ETA(I),I=I,N)
EX PNENTS.=',7El0.5,//)
F tRAT(i r IE
GENERATE P PARAMETERS
GETNERATE PGAMMAIP) VECTOR
DO 15 J=I,N
I=l
AKAPPA( I-1)=0.0
PSU; (I )=GAi>MM-A(I)/(ETA(J) +BETA(I))
I)+AKAPPA(I-1)
AKAPPA(I)=PSU
I=1+1
IF(I-N)17,17,16
;R I T ( 6,11 ) ,AKAPPA (I)
FOR:AT (H 5,4XiE 10.5)
KAPPA (J )=(ETA (J)."AKAPPA ( I ))+SNT ) ELAST LOAD
WRITE(6,103)I,J,BKAPPA(J)
AT ( 15,4X ,I 10.5)
FU
WR ITE VECTOR IN COLUMN FORM
DO 19 I=1,N
(1, I)BKAPPA(I )
;,R ITE{6, 104) (O( 1,1), I=1, N)
FORAT ( 'P VECTOR.=' ,IEIO.5,//
G. ,NERATE COEFFICIENT MATRIX
I=1
DO 10 J=1,N
A(I,J)=ETA(I)/tETA(I )+ETAJ))
I=I+L
-158-
TIMES
I
_II
_I
~._~_.__1.___.
_ __._.~___
____
12
13
105
C
14
106
C
C
107
108
/FTC
C
1160
1170
1180
1190
1100
1110
/FTC
C
C
C
C
C
.
IF(I-N)11,t11,12
CONT INUE
DO 13 I=1,M
00 13 J=l,M
WR ITE(6, 105)( A( I, J ) ,J=l1 M)
FOR AT(7F10. 5)
CU IPUTE IfNVERSE OF COEFFICIENT MATRIX
DC 14 I=IM
WRITE(6,106)(A(I,J),J=1,M)
FORViAT (4X, 7F10.5)
COUiPUTE MATRIX PRODUCT
CALL tULTP(A,D, F, M,M,M,
ISSUC,O )
COniPUTE TIME UEPENDENT SOLUTION
WRITE(6,107) (FI I, ),I= ,M)
FORPAT(4X, 'REAL TINE COEFFICIENTS.=',I OFI0.3,//)
WR IT E (6, 108) SNOT
FURMAT(1F10.5)
GO TO 999
END
MATRIX INVERSION BY PARTITIONING
SUBROUTINE iATINV(A ,N)
DIMENSION A(20,20), B(30),C(30)
NN=N-1
)=1./A(1,)
A(,1
DO 1110 M=1,NN
K =M +
DO 1160 I=1,M
B(I )=0.0
DO) 1160 J=I,M
A( J,K)
B(I)=B(I)+A(I,J)
R=0.0
DO 1170 I=l,MA
) 3(I)
R=R+AK,I
R=-R+A(K,K)
A(K,K)=I./R
DO 1180 I=1,M
A(I,K)=-8( I )A(KK)
0D 1190 J=1,M
C(J)=O.0
C(J)=C(J)+A(K,I)-A( I ,J)
DO 11CO J=1,M
A(K,J)=-C(J)*A(K,K)
DO 1110 I=1,M
DO 1110 J=1,M
A (KJ)
A(I,J)=A( I,J)-B3 (I)
RETURN
END
A=MA*NA MATRIX
8=1Bi*NB MATRIX
C=PRODUCT MATRIX
ISSUC=TEST FOR CONFORMABILITY
IB=TEST FOR H&OMCGENEOUS SYSTEM
SUBROUTINE MULTP(A,B,C,MA,NA,M3,NB,
-159-
ISSUC ,3)
-
~1~11_
~~~ _~_^~
I
2
10
15
20
30
40
35
/END
DIMtENSION A(20, 20),(2C,20) ,C(20,20), TEMiP('40)
IF (NA-MB ) 1,2, 1
ISSUjC=2
WRITE (6,40)
GO TO 35
ISSUC=l
IF( IB- )15,5,15
DO 10 J=1,NB
C ( I,J)=O.0
DO 10 K=I,NA
C ( I,J)=C( I ,J)+A( I ,K)*BI(K, J)
GO TO 35
C ONT INUE
DO 30 J=1,NB
DO 20 I=1,MA
TEMP(I)=0.0
DO 20 K=1,NA
)+A( IK)B (K, J)
TEMP( I)=TEPI(
DO 30 I=1,MA
B ( I, J )=TEMP(I)
CONT INUE
FORMAT ('MATRICES ARE NOT COMFORMABLE')
RETURN
END
CF FILE
-160-
_
I~_~
~
APPENDIX C
Long Range Tests
C-I.
Long range creep tests.
-161-
~-
-~-----r---rs_~--
_I ~
_
_-_-_r~ ni -;r ~--P9D~ix
-s-
----' I-
----~ - --r~-~---T 1 ' :
I-dw
,.
.. ..
..
..
,
...
....
:
-,.-.-..
I; . + I
+
....
,.,
. ... ...
......
I
1
I"
I
i-.. ..
:
,i
-.
-1*
:'i,
I
I
n5-TT 7.1,*E T.____+
.
..
.~...~
.:
"4 ' ++
- 4 - .
I-__
- __
____
-- ;r--"
---;..
:
.-:...
"
.
:".-t
I
,
+
:..
....!
.,
.....
....
i-i- L .. . .
:I'
+-- -+ .'.
."'... - ...... t-.-:'
.- . ..
.-- t-..
-i '
i;
71:171
. ...
"............
.....
'
+--
I
.
'
-<- ---
......
.
...
..
. -"
...
.
I
-- *L-V-,.+-ti
,...
-++-+
_.
+,__,...
..
....
...
++__
....
:+
_+:+
i+
~.
~~~~
iii .. ... +....
..
...
,:,+++...
''+
., .:.:
t --
•
"
:
. . ..+-
-'+'
-,+t
" -:.
::i
:
,
- •-
-+
!-
I '
i
-_i: !
. .. :
..
::
.
"j
"
-
': ;I:
.. .
I£
.-~-i-r
7
"1-
.i.~~ i K..
-I--
. -..
..
1*:
*
WTI To : motaoung-SIV s
I
17V
iIp
(
Ij
r
Fl
1
--
'
-- i
....
........
i-- ..._-
I
1f77Ji
iI
_
1
-t
. . 1
i
I::
i
:-i
_
_
t7;:7171:'
_
_
_
_
_
_
___
!::
i ii
_
_
_
_
_
i
r
-T-
IrP
I
.i
-
II
o
_
1
44_rt
r
,
It
j-h
It......i'fI:I..
_
_
_
l/:11i..
4 ...
..
i
_
. ;i-, ------ P
____CII^_~~___I__II___
_~I~ ~~1_1__~__ _
C-II.
___
Long range stress relaxation tests.
-164-
2
3
4
5
67891
2
3
4
5
67891
....
3 4
q
5
4
5.A..
6 7- 8 9
t-M
.-.-- i
.,
. 1:.-:.471 T
f ii
::
.
t:
H
i4:44~A
.I--
i
I
T1_:
IAi
.44. 1?
-------c----~---i---e-r-)
:
I 7
i: i! !i: i• 'il
.
f:---~-tfi~f:;t
.
67
A a
I
7"
I -I----l-i-i--l-----+--C
:
5
q
A
i
-- "~"--"
IZIITVV
.r
-~~rI!
Io
~iu.l-r--
-i
I:r]
:;;--i
171
2
I.
too
A
A
3
...
A
l r-i-i
I
9
V
7
't' ,
I
,, " ! '
.
i i
..
_.
i
:
..
.
. .
::
I '
C
ar4
5
4
T T h
3
2]
j
,
L1L
..
II
1
___TI1~
1
,t
41
to
00
- 7-
7--.I
_
it
_
9
68
7
5
!S ...... r .:3i..
'
'
-
4
3
...
*.
.
,.
I
.
.
ljr .
.
.
...
.
L
:t q''*:O
"''
....
K m~" H
' ''''
too100.
Time, minutes
loo.
.
sv.Qol"ai'
10041
ft
0
_
__~____.
II
_~ .__
C-III.
....
Long range flexure creep tests.
-167-
-
--
Ii
1:1~~~.:
I.
I
.......
1....
I
T.~~t
:4
~~
i:...i.--.
'I
1
~.
4
I.-
irl:i
i-.i ~L.jI
i
J~P61
ii
t---l-- ---- ~
:: : i ' ~ i
i
i.
j
!.lfl-
.1.1,.
r
ii ' ;
i
I
Ii:i
II
t
1:1
i:
4...
4
I
I
LILL'
'' '~l I
IL
+----j
j; r' ' )''''~;
ii j i
I:
-1
I
~----------ii
.I
<-1
~----
1 : t : Cln
'.1'
~~~ ~~
( ''W
--
I
.--.-
*1*
4...-..-----.-----
7I
.I
--
---4-
-
It
j;:::/
-
I:
2
7A-
I
; :-+I.-:.I:+ :
1
:
~---f I?
A
I;
tf
7 .!
I
-- T7
:
I
I. . .
.1
1
11--tii
.
C 1,6a
1
61.....lixu e rr.4b
4-K
II: .
III t ,
1,
tt-On -0 Tim
t
PUn4
__
4
-____
__
_
_
__
_
--
~-
l---------
t
.
.
.
.
.....
..-
.
----
:---;-.
I
.....
i
I'li..l..;.
.---.
-- - ---- ,....
. . .....
-
i-
. I .... . .i . .
----...- t
,
r
-I..-
•
i
r ,
i.
.. I _:.
,
I
-T,
.. '~
..
j
......
,
......
I
...
.
,
4----~I~.. -
---- ' i
,
;-_',-
. .........
I
.i
.
T-:
.
•
.
!-~
;-;
-
. .
I-
t
.
I
f'--t-
"
i:
. - . . ..
-
:
• .
--,
•:----
-- I
•
,
.
.
.'
:
'
--
. . - - - -T,
..
.,.
,
--.
-
1
;.
,
'to
'
i_I
- I -T . -- .:__. .!-
:'
It;
'*:
I':--:
i
..
I
.
-
. . .
.L
- -
!I1 N~
.
.. .
; jC4
. ..
. .
;.
.
.
I
. :I
II
.
.
,. ., l _:,_ _ l , , : .j
'
" ....
, " 'I
; '- . .
.- _::_ . . ......
:
.
.
1
- -F "--T-
. ., i , ,
:,
- - :
:I ::i ...i
Ii....
'
,,,es
r
..
P
:
t
..
'
.~.....
",
. .
.. . . .
'
bo|
.
!""' i TT -T ..
- - - --- ---. ... .. . "T
...- -. --. i-... . ..-- ...... . . . ..
. : . ... . •
.
-;.
-
Ci
..
.. .
1
,1 1
.
-_
1
.
..
..
.
..
1 I I
:.j
.....
.
.
.
.
.
. .
II.-
-' :